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i
Fourth

International

Congress for

Stereology

NATIONAL BUREAU

OF STANDAKDS

]5,IBRARY

JUL 7 1976 762516

ISS

PROCEEDINGS OF THE

Fourth International

Congress for Stereology

HELD AT THE

National Bureau of Standards

Gaithersburg, Maryland, U.S.A.

September 4-9, 1975

EDITED BY:

Ervin E. Underwood

Georgia Institute of Technology

Atlanta, Georgia 30332

Roland de Wit and George A. Moore

Institute for Materials Research

National Bureau of Standards

Washington, D.C. 20234

SPONSORED BY

International Society for Stereology

Societe internationale pour la Stereologie Internationale Gesellschaft fur Stereologie

The National Bureau of Standards and the National Science Foundation

U.S. DEPARTMENT OF COMMERCE, Rogers C. B. Morton, Secretory

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson, Assisfarif Secreiary for Science and Technology

, NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acfing Director

Issued January 1976
Library of Congress Catalog Card Number: 76-8668

National Bureau of Standards Special Publication 431

Nat. Bur. Stand. (U.S.), Spec. Publ. 431, 547 pages (Jan. 1976) CODEN: XNBSAV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1976

For sale by the Superinlendenl of Documents, U.S. Government Printing Office, Washington, D.C. 20402

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ABSTRACT

These proceedings contain most of the papers presented at the technical sessions

and workshops of the Fourth International Congress for Stereology, held at the National Bureau of Standards, Gaithersburg, Maryland, from 4-9 September 1975. Of the 113 papers recorded here, 10 are invited lectures, 65 are submitted

papers, and 37 come from the six workshops. Three of these workshops containing 17 papers were integrated into the technical sections for these proceedings.

Topics covered in the Technical Sections are Principles and Mathematical

Developments, Pattern Recognition, Instrumentation, Three-dimensional Reconstruction, and Stereological Applications in Materials and Biology. Additional

topics covered in the Workshops are Mathematical Foundations of Stereology,

Particle Science, and Size Distributions.

Key Words: Anatomy, automated microscopy, biology, convex particles, curvature,

geometrical probability, image analysis, materials science, mathematics, microscopy, microstructure , morphometry, oriented structures, particles, pattern

recognition, probability theory, quantitative microscopy, reconstructions,

sections, serial sectioning, shape parameters, size distributions, spacing

distributions, statistical analysis, statistics, stereology, stereometry,

stereoscopy, stochastic models, structures, ' three-dimensional reconstruction.
PREFACE

These Proceedings represent, in compact form, a record of all talks delivered

at the Fourth International Congress for Stereology, held at NBS near Washington, D. C, from 4 to 9 September 1975. Previous proceedings of the congresses

of the International Society for Stereology (ISS) have appeared every foiir years, starting with the First International Congress for Stereology in Vienna

in 1963, followed by Chicago in I967 and Berne in 1971.

The ISS is a truly international and interdisciplinary society. It was founded in I96I by a small group of anatomists and mathematicians who found

they were working on the same problems of expressing microstructures in quantitative terms, Wow-a-days, the ISS represents equally experimentalists,

theoreticians and instrumentalists, and its growth is due largely to the interchange of information among workers in many different fields. Thus, one finds

anatomists and ceramicists, metallurgists and biologists, petrographers and

mathematicians, using the basic equations of stereology and striving to solve problems that have applicability to all.

Stereology may be defined as a body of methods for relating the micro structure

as seen in the random section or projection plane to the true spatial quantities. It means, in a broad sense, the study of structure in three-dimensions.

Thus it includes the narrower and more restrictive terms, such as stereometry

(measurement in three-dimensions) and morphology (the study of shape). The basic equations of stereology are well-known by nowo Consequently, the main

efforts of the ISS are devoted to advancing stereological theory, solving new problems, improving the efficiency of measurement methods, determining accuracy and error, and contributing to the development of automatic image

analysis. The usefulness and the success of the Society depends on its ability to attack these problems on a broad front, whereby the different

talents of many people contribute to the solution of the problem.

A continuing objective of the ISS is educational in nature, whereby the organization of meetings, symposia, lectures and the publication of survey

and specialized papers on stereology are constantly encouraged and supported.

Major contributions to the literature of stereology are the quadrennial

congress proceedings. In order to keep the size of the 1975 Proceedings down

to manageable proportions, it has been necessary to severely limit the length

of the papers. Even so, to collect and publish all these papers in one volume

is a tribute to the Managing Editor of the Proceedings, Dr. Roland deWit, and

to the support provided by NBS. The Congress Program Committee, headed by the President of the ISS, has also contributed to this volume through the initial

selection and continuing editorial supervision of the submitted papers.

I am confident that the readers of these Proceedings will find something of interest, whether basic or advanced, old or new, routine or stimulating, and

that all will profit from a more extensive use of the powerful, yet elegantly

simple, relationships of stereology.

Ervin E. Underwood, Sc. D. President, 1971-1975

International Society for Stereology

iv
INFORMATION ON THE INTERNATIONAL SOCIETY FOR STEREOLOGY

The International Society for Stereology (ISS) was founded in I961 as an interdisciplinary society, gathering among its members scientists of disciplines seemingly as disparate as biology, medicine, mineralogy, metallurgy

and mathematics. What brought them together was the common problem of quantitatively interpreting the micro struct\are in three dimensions from two- dLmensional sections which reveal only part of the true spatial geometry of

the material. Measurements performed on these sections are not necessarily

representative of the dimensions of the spatial structures. But the relationships between measurements on sections and the spatial microstructural

features can be established by means of appropriate mathematical treatment in the form of stereological principles.

What is "Stereology"?

This term was coined at the Foundation Meeting of the ISS on the Feldberg in 1961. The founder and first president of the ISS, Hans Elias, Chicago, has proposed the following definition

:

"Stereology, sensu stricto, deals with a body of methods for the exploration

of three-dimensional space, when only two-dimensional sections through solid

bodies or their projections on a surface are available,, Thus, stereology

coTild also be called extrapolation from two- to three-dimensional space".

The general problem is evident: the interior of opaque solids can only be

investigated if the aggregate is somehow "opened". A common way of "opening"

is to produce sections or thin slices, thus sacrificing the bulk of the material, but preserving an "image" of the relationship between the components

at least in the plane of the section.

This image is distorted: solids become reduced to plane profiles; contact

surfaces between component phases appear as linear contours or boundaries of the profiles; linear features of the material (filaments, intersection lines

between surfaces) are seen on the section as points. In short, on the section any particular feature -of the solid is represented by another, related feature

whose dimension is reduced by 1. Furthermore, the diameter of a given profile

is not representative of the particle from which it was derived: sectioning

of spheres of equal diameters gives rise to a population of circular profiles

of varying diameter.

We expect stereology to provide answers to the following f\mdamental questions:

What is the quantitative relationship between the area of the profile and the volume of the solid from which it was derived, or of the profile boundary to the solid's surface, etc.? What is the true shape and size distribution of particles if only profiles can be studied? These are problems for mathematics

to solve, particularly for integral geometry and geometric probability theory,

as the sections will, most of the times, cut through the solids at random,

f4any of these problems await solution; many have been solved and it remains to put them to fruitfvil use. Improvements in optical and electron microscopes

have appreciably improved the resolving power of the most important instruments

used to penetrate into the depth of inorganic materials and living organisms;

stereological methods supplement these tools by indicating the proper measurements. The development of adequate technical means and the introduction of electronic data analysis help to render these methods efficient in application.

V
The use of Stereology

Metallurgists are applying stereological principles to study the composition

of metal alloys, the size distribution of grains and particles, the contact surface area between the different phases, and the anisotropy of the grains,

i.e., microstructural properties which are related to the physical and mechanical properties of materials. The problems investigated by these methods in mineralogy, petrography and geology are quite similar; it is in these fields

that many of the foundations of basic stereologic principles have been laid,

starting in 18U7 with the fundamental principle of Delesse who demonstrated

that the volume fraction of rocks can be directly estimated from the areal

profile fraction of sections. In biology and medicine the internal structure

of cells, tissues and organs is now being quantitatively investigated in many

fields of stereological methods. Besides providing new insights into the internal geometrical properties of living organisms, such data can be success- fully applied to quantitative correlations between structure and function - - applications that have previously been virtually impossible.

Periodicals

The ISS publishes an annual Newsletter (around 200 pages), under the Editorship of Dr. Gerhard Ondracek, which contains outstanding stereological papers, discussions, meeting notices and literatiire surveys. In between, there are messages of interest to the membership from the President or other officers.

During the past year, special arrangements were made to provide free to interested members a copy of the "Seattle" conference, edited by Dr. W. L. Nicholson (260 pages), where statisticians and materials experts met and

discussed stereological topics of quite general interest, and the Handbook

on Powder Science and Technology, by Dr. Brian Kaye (about I50 pages).

The ISS has also arranged with the Royal Microscopical. Society for ISS members

to subscribe at a reduced rate to the Journal of Microscopy, and to publish

their papers on stereology therein, if acceptable. The current Editor for the

ISS is Dr. E. R„ Weibel, with Drs. E. E. Underwood and G. C. Arastutz serving on the Editorial Advisory Board.

Membership in the ISS

Membership is open to anyone who wishes to help to promote stereology, make use

of its methods, and exchange ideas with colleagues of related interests in various disciplines. Membership dues are presently US $5.00 per year, or US $15.00 for h consecutive years. Information on student rates or industrial

affiliates can be obtained from the Secretary-Treasurer:

Dr. Anna-Mary Carpenter

University of Minnesota

Department of Anatomy

Minneapolis, Minn. 55^55, USA

Activities of the ISS

One of the major activities of the ISS is the organization of scientific

meetings. The ISS International Congresses have been held every foxuc years, starting in I963 at Vienna, and followed by Chicago in I967 and Berne in 1971. The Fourth International Congress for Stereology was held from '+-9 Sept, 1975,

at the National Bureau of Standards near Washington, D. C. Another major ob- jective of the Society is educational in nature. The ISS has organized numerous local seminars and has helped to introduce stereology into many larger

meetings. For example, a Stereology Seminar was organized and presented at the 8th International Congress on Electron Microscopy, at Canberra, Australia, in August 197^, and at the kth International Metallography Congress held at Leoben,

Austria in October, 197^. Also, smaller local meetings and symposia are held.

vi
HOW STEREOLOGY CAME ABOUT

by Hans Eli as, Honorary President

International Society for Stereology

In the process of teaching and producing teaching films for veterinary

and human micro-anatomy, I drew stereograms, i.e., generalized pictures having

a three-dimensional appearance, based on verbal textbook descriptions. However, it was impossible to take photomicrographs that would confirm those stereograms. I noticed (19A5) that alleged tubular glands in the chicken's

stomach were instead concentric trenches, and that the so-called liver cords,

assumed to be connected cylinders, were in reality interconnected walls forming a continuum (1948). These corrections of traditional textbook descriptions

were arrived at by common sense reasoning without the aid of reconstruction and

of mathematics.

Three years after the discovery of the liver wall continuum it dawned on me

that all this was a matter of geometrical probability and that such common sense determination of structure could be verified mathematically, if one measured and classified sectional profiles. I wrote a number of papers (1951, 1954) which

dealt with the geometry of sectioning. August Hennig showed me a few minor

mistakes I had made in two of those papers, and we began corresponding. From

him I learned that others before me had used geometrical probability for the spatial interpretation of flat sections. The method of determining three-dimensional structure from single random sections by mere logical thinking is described in Stereology 3 (1972).

It should be noted that I used quantitative methods chiefly for shape

determination and problems of continuity versus discreteness. My paper (1954) on size distribution of spheres was written just for fun, but I used the results for determining the size distribution of fat drops in fatty liver (1956).

By a chance meeting with Dr. Herbert Haug on a pleasure boat sailing around

Manhattan Island while we were attending the International Congress of Anatomists

in New York (1960) , we discovered that we were both using statistico-geometrical

methods to obtain information on three-dimensional structure from sections. We

decided to call a gathering of persons interested in three-dimensional interpretation of flat images.

An announcement in a few journals brought 11 scientists together on the Feldberg Mountain in the Black Forest (11-12 May, 1961). The participants of this meeting and hence charter members were: G. Bach (first secretary), A. Bohle, H. Elias (first president), H. Haselmann, H. Haug (first vice president),

A. Hennig, A. Hossmann (first treasurer), M. Gihr, R. Lorenz, H. Sitte, and

W. Treff (first assistant secretary) . The word stereology was then coined and

the International Society for Stereology was founded and incorporated as a nonprofit organization in Neustadt, Schwarzwald. Dr. Ervin E. Underwood had read

the announcement and helped organize the Feldberg meeting, although he could not

attend. Later when Dr. Hossmann resigned as treasurer. Dr. Underwood took his office and also became editor of our first periodical publication STEREOLOGIA.

The Society was subsequently incorporated as a non-profit organization in Stuttgart (1962); Vienna (1963); and Deerfield, Illinois (1964).

On my way home from the Feldberg, I stopped in New York and, while walking

along Central Park, met my friend Dr. Ewald R. Weibel. We entered a little

cafe to talk and he told me he had just written on the morphometry of the lung. He suggested we organize a symposium on the use of geometry in the evaluation

of sections. I then related to him what had transpired at the Black Forest and

that the official word now was stereology.

vii
ELECTED OFFICERS OF THE INTERNATIONAL SOCIETY FOR STEREOLOGY

1972-1975 1976-1979

President

:

Ervin E. Underwood

Atlanta, Georgia

U.S.A.

Herbert Haug

Liibeck, Germany

Vice-Presidents , Regional : for America: John E. Hilliard

Evanston, Illinois

U.S.A.

Robert T. DeHoff

Gainesville, Florida

U.S.A.

for Australasia;

for Europe

:

Vice-President , Corporate

:

Herbert Haug

Kiel, Germany

Robert S. Anderssen

Canberra, Australia

H. Eckart Exner

Stuttgart, Germany

John E. Hilliard

Evanston, Illinois

U.S.A.

Secretary- Treasurer

;

Anna-Mary Carpenter

Minneapolis, Minnesota

U.S.A.

Anna-Mary Carpenter

Minneapolis, Minnesota

U.S.A.

Secretary for

Publications : G. Christian AmstutzHeidelberg, Germany

Gerhard Ondracek

Karlsruhe, Germany

HONORARY MEMBERS OF ISS

Prof. Dr. Hans Elias

(Honorary President)

San Francisco, U.S.A.

Dr. R. Buckminster Fuller

Philadelphia, Pennsylvania

U.S.A.

Dr . Otto Rohm

Lausanne, Switzerland

Prof. Dr. S. A. Saltikov

Erevan, Armenia

U.S.S.R.

Dr. med. August Hennig

Miinchen, Germany

Dr. Harold W. Chalkley

(Deceased)

viii
APPOINTED OFFICERS OF THE INTERMATIONAL SOCIETY FOR STEREOLOGY

1972-1975 1976-1979

Conmittee Chairmen ; (and Assistant Chairmen for 1976 - 1979)

Mathematics

:

Biological

Sciences

:

Robert T. DeHoff

Gainesville, Florida

U.S .A.

Alden V. Loud

Valhalla, New York

U.S.A.

Roger E. Wiles

Canberra, Australia

Wesley L. Nicholson

Richland, Washington

U.S.A.

Ewald R. Weibel

Bern, Switzerland

J. L. Binet

Paris, France

Materials

Science

:

H. Eckart Exner

Stuttgart, Germany

James H. Steele, Jr,

Middletown, Ohio

U.S.A.

Instrumentation

:

George A. Moore

Washington, D.C.

U.S.A.

Jean-Louis Chermant

Caen, France

Hans P. Hougardy

Dusseldorf, Germany

Gustav Bernroider

Salzburg, Austria

Publications G. Christian Amstutz

Heidelberg, Germany

Regional Representative for Eastern Europe;

Geza M. Timc^k

Kosice, Czechoslovakia

ISS Representatives on the Journal of Microscopy;

Editor

;

Editorial

Advisory Board;

Ewald R. Weibel

Bern, Switzerland

Ervin E. Underwood

Atlanta, Georgia

U.S.A.

Ewald R. Weibel

Bern, Switzerland

Herbert Haug

Lubeck, Germany

G. Christian Amstutz

Heidelberg, Germany

Gerhard Ondracek

Karlsruhe, Germany
CONGRESS ORGANIZING COMMITTEE

Chairman

:

Deputy

:

Arrangements

:

Support Staff:

Treasurer

:

International Relations:

Public Relations:

George A. Moore

Roland de Wit

Sara R. Torrence

Robert F. Martin

Ronald B. Johnson

Michael B. McNeil

Madeleine Jacobs

CONGRESS PROGRAM COMMITTEE

Ervin E. Underwood (Chairman)

Atlanta, Georgia, U.S.A.

G. Christian Amstutz

Heidelberg, Germany

Anna-Mary Carpenter

Minneapolis, Minnesota, U.S.A.

Robert T. DeHoff

Gainesville, Florida, U.S.A.

H. Eckart Exner

Stuttgart, Germany

Herbert Haug

Liibeck, Germany

John E. Hilliard

Evanston, Illinois, U.S.A.

George A. Moore

Washington, D.C., U.S.A.
WELCOMING REMARKS

by John D. Hoffman, Director

Institute for Materials Research, National Bureau of Standards

Good evening ladies and gentlemen. Welcome to the National Bureau of Standards and to the 4th International Congress for Stereology cosponsored by

the International Society for Stereology and the National Bureau of Standards.

I would like to add a special welcome to our guests from abroad and wish you a pleasant visit in the United States.

Because this is the first congress of this society to be held at the Bureau of Standards, some of you are visiting here for the first time and may

not be familiar with the range of activities at the Bureau. I hope that the tour which many of you took today gave you a first-hand view of the breadth

and depth of the work performed here. I would like to take a few minutes now

to describe some things about the Bureau that you might not know.

The Bureau of Standards was created in 1901 to develop and disseminate

the national standards of measurement, to determine physical constants and properties of important materials, to develop test methods, to aid in the establishment of standard practices, and to provide technical services to other

government agencies. More recently, legislation has added new responsibilities

in the areas of computer technology, standard reference data, and fire research. Measurements, standard practices and data are the common themes in all our work,

and we are perhaps best described as the Nation's measurement laboratory.

The work of the Bureau is carried out in four institutes: The Institute

for Basic Standards, The Institute for Applied Technology, The Institute for Computer Sciences and Technology, and The Institute for Materials Research.

The Institute for Basic Standards is responsible for the custody,

maintenance and development of the national standards, for methods of physical measurement and for the provision of means for making measurements consistent

with those standards. This work includes the basic units of measurement such as mass, length, and time; and NBS is perhaps best known to some of you for its basic standards work. Many of you, I am sure», are also familiar with the NBS work in pattern recognition and computer analysis of images begun about 15 years

ago. This work is carried out in the applied mathematics group of The Institute for Basic Standards.

The Institute for Applied Technology is concerned with engineering

standards, codes, and test methods. It works to promote the development and use of technology in important areas such as building construction, fire prevention,

electronics, and consumer product safety.

The Institute for Computer Sciences and Technology has the responsibility

to improve federal management and utilization of computer technology, and to lead federal efforts to stimulate the use of computer technology to meet national needs. With the growing use of remote terminals and computer-computer

communications, problems of access, privacy, and rapid turn-around inevitably

arise. The computer networking laboratory you visited today is one in which

experiments are conducted to develop guidelines for the efficient use of computer

networks

.

The Institute for Materials Research is concerned with the development of methods for measuring the key properties of materials, the development of methods for relating these properties to the performance of materials, and with

the development of standard reference materials. Three of the facilities you

visited today - the image analysis laboratory, the reactor, and the dental

research laboratory - are concerned with different facets of materials

xi
research and illustrate the breadth of our activities. Incidentally, the image analysis laboratory is one of several institute central facilities that we have established during the last few years. The work being carried out in this laboratory includes metallographic work in support of our metallurgy

program, particulate analysis in support of our environmental program and analysis of blood cells in support of our clinical standards program. The materials research program at NBS also includes work in nondestructive

evaluation of materials, corrosion, test methods development for materials

to be used in advanced energy generation systems, and polymer research.

I note that many of you, perhaps a majority of you, are interested in the stereology of biological systems. While NBS is not an agency concerned with

biological problems per se, understanding the behavior of biological systems

and delivering health care increasingly require accurate measurement and characterization. Beginning with calibrating clinical thermometers almost

75 years ago, NBS has cooperated with the medical community to develop better

methods of measurement for medical applications. The dental research program

described to you earlier today has been contributing to improved dental care

for more than 50 years. This program, jointly supported by the American

Dental Association, the National Institute for Dental Research and NBS has resulted in new dental restoratives, the high speed drill and the panoramic

x-ray. The dental program has recently been expanded to include the development of techniques for characterizing polymeric implant materials.

Earlier, I alluded to our clinical standards program. For the last six

years, NBS has cooperated with a number of groups including the College of American Pathologists and the American Association for Clinical Chemists to develop standard reference materials and analytical reference methods needed

to improve the accuracy of measurements in the clinical laboratory. This

year, about 13000 private and hospital affiliated clinical laboratories will

perform about four billion tests. The general trend toward early detection

of diseases indicates that measurement and standards will play an ever greater

role in the total health care system of the future.

Another area of medical diagnosis, closer perhaps to the interests of this congress, involves the use of ultrasound to characterize human tissue and to detect abnormalities such as malignant tumors. .NBS has recently undertaken

work in this area in collaboration with the National Institutes of Health. Work on image reconstruction in connection with ultrasonic testing is under consideration. Interestingly, the NBS ultrasonic diagnostic work is an outgrowth of our program in nondestructive evaluation of materials. The NBS work is intended to improve the methods for characterizing flaws in materials

or abnormalities in tissue. This work is one of the many examples where

developments in one area stimulate developments in another and for which the same techniques are adopted for diverse purposes.

The fact that this conference brings together scientists from many disciplines - metallurgy, medicine, earth resources, mathematics - for the common purpose of utilizing the science of stereology indicates to me that your society is well aware of the benefits that may be derived from such sharing of common methodology and mathematical analysis.

I would like to take this opportunity to thank Dr. George Moore of NBS who

is serving as chairman of the organizing committee for this congress. For many years, George has worked to advance the science of stereology, most

recently, in connection with the analysis of inclusions in metals. To facilitate his metallographic work, George designed, built and applied a system

for automated image analysis. He has also been active in ASTM affairs where he has devoted much time to the development of standard practices for metallographic

analysis.

Once again, welcome to the National Bureau of Standards and have a good meeting.

xii
ACKNOWLEDGMENTS

Many people have contributed to the technical, organizational, and social success of this Congress and its Proceedings, and we are grateful to all of

them. We wish to single out a few for special mention.

The highlight of the Thursday Night Opening Session was the light-hearted

talk "Toronto Daze" by J. H. L. Watson from the Edsel B. Ford Institute for Medical Research in Detroit. He covered the gamut and liberally sprinkled

his talk with song and dance. For this he was ably accompanied by pianist

William R. Ott from the Optical Physics Division at NBS, who did a superlative

job at the keyboard on one day notice, when the previously engaged pianist fell

ill. He also provided background piano music through the rest of the Opening

Session. Another fascinating feature of that evening was E. E. Underwood's

talk on "Art in Microscopy."

The organizational and social program would not have been possible

without the unstinting efforts of Mrs. Sara K. Torrence from the Office of Information Activities at NBS. She kept us moving in the right direction at all times and also arranged the Saturday Night Bar-B-Que and Old Fashioned Hoe-Down and the Sunday Tours with the help of Gretchen Poston. We are also indebted to the firms of Bausch and Lomb, IMANCO, and E. Leitz for their con- tribution to the success of the Thursday Reception.

For help with the editorial work on the Proceedings we particularly want

to thank Mrs. Betty L. Hurdle and Mrs. Mildred C. Reid from the Office of Technical Publications at NBS. Their meticulous efforts, in close cooperation

with the Visual Arts Section at NBS, led to the attractive physical appearance

of this volume.

xiii
ARNOLD LAZAROW

1916 - 1975

Arnold Lazarow's contributions to science are documented in some

250 publications which include original articles, reviews, abstracts and lectures both to students and at scientific meetings. His influence on colleagues, medical students, graduate students and post-doctoral fellows was so personal that each responded with loyalty and increased productivity.

Dr. Lazarow was born in Detroit, Michigan in 1916. He studied at the University of Chicago where he was granted the Bachelor of Science (1937) as well as both Ph.D. and M.D. (1941). He was on the staff of the Department

of Anatomy at Western Reserve University for a decade when he was invited

to the University of Minnesota where he served as professor and head of the department of Anatomy for two decades. Honors included: Phi Beta Kappa,

Sigma Xi, Alpha Omega Alpha, the Capp Prize and the Banting medal of the American Diabetes Association. A member of 23 scientific societies, Arnold

was elected to office in many of them; he was secretary of the International

Society for Stereology from 1968 to 1971. Dr. Lazarow died June 25, 1975 after a brief illness.

Many of his contributions to science were in advance of their time: he pioneered in establishing the concept that intracellular glycogen is in par- ticulate form, that mitochondria contain respiratory enzymes and that sul- phhydryl groups protect against alloxan. His interest in mathematical ex- pression of morphologic data and his attraction to instrumentation resulted

in the development of the first quantitators for the application of stereologic methods to light and electron microscopy (1958) . These tools were

used in establishing the percentage of islet tissue in the total pancreas,

the volume per cent of the cellular components of the islets, the cytotoxic

effects of alloxan on the beta cells and the precise increment in beta cell mass during embryogenesis . Using these parameters as a basis, sufficient

numbers of beta cells grown in organ culture were transplanted to alloxan

diabetic rats; the animals became normoglycemic and remained normal for as long as two years. Challenged by the computer, Arnold applied information storage and retrieval methods to the development of the Diabetes Literature Index , a monthly publication, which NIH distributes to members of the American

Diabetes Association.

Dr. Lazarow had clarity of thought, drive, and an innate inquisitiveness

coupled with a perfectionism that enabled him to achieve his goals and to inspire others. Smilingly he interrupted his own activities to help those who queried, "Do you have a minute?"; these minutes are precious to all who knew him.

A-M Carpenter

xiv
HAROLD W. CHALKLEY

1 886 - 1 975

Harold W. Chalkley's principal research efforts were in the field of protozoan cell division. He was the author of approximately 100 papers con- cerned with the mechanics of cell division in Amoeba proteus , its chemistry

and other aspects of protozoan physiology. His contributions to stereology

came late in his research career.

Dr. Chalkley was bom in London, England, in 1886. He moved to Lawrence,

Kansas, in 1905 and entered Kansas University. Two years later, finding

university life less fascinating than the still expanding western frontier,

he left school and went to work on a Rock Island railway crew surveying across

Indian country in the Texas panhandle. He worked later on canal surveys in Louisiana and as a railroad hand, enginewright , and stationary engineer.

After completing World War I service, he entered the Mississippi Agricultural and Mechanical College. He graduated with honors in 1925 and entered

the Johns Hopkins University as a Johnston Fellow. He received his Ph.D. in zoology in 1929 and was elected a member of both Phi Beta Kappa and Sigma Xi. He entered government service the same year as a physiologist of the old Hygenic Laboratory, now the National Institutes of Health of the United States Health Service.

His early experience in engineering was frequently reflected in his

research. This included the design of a split-field microscope that permitted

simultaneous visualization of an object from both the top and side, the adaptation of planimetric methods for the determination of amoeboid volume

and a variety of other devices to assist, not only in his own work, but in the work of colleagues. One of these efforts led to his adaptation (1943) of the "flying spot" method to the quantitative analysis of sectioned tissues. He often remarked that this technique, which he developed in less than an hour,

would probably be the only finding for which he would be remembered. Later contributions to the armamentarium of stereology included a method for determining numbers of cellular nuclei (1945) and for determining surface/volume

ratios (1949) . He collaborated in several studies involving the application

of these techniques to biological investigation. He retired from government

service in 1952.

Dr. Chalkley retained a vigorous interest in the world about him, an all- encompassing sense of humor, and a willingness to respond to challenges that lasted until the very minute of his death on the evening of September 25, 1975^ sixteen days after his election as an Honorary Member of the I.S.S.

Donald T. Chalkley

XV

CONTENTS

Page

ABSTRACT iii PREFACE iv INFORMATION ON THE INTERNATIONAL SOCIETY FOR STEREOLOGY v HOW STEREOLOGY CAME ABOUT vii ELECTED OFFICERS AND HONORARY MEMBERS OF ISS viii APPOINTED OFFICERS OF ISS ix

CONGRESS ORGANIZING AND PROGRAM COMMITTEES x WELCOMING REMARKS xi ACKNOWLEDGMENTS xlii ARNOLD LAZAROW xiv HAROLD W. CHALKLEY xv

PART I. TECHNICAL SECTIONS

1. PRINCIPLES AND MATHEMATICAL DEVELOPMENTS

ON ESTIMATING AGGREGATE AND OVERALL CHARACTERISTICS FROM THICK

SECTIONS BY TRANSMISSION MICROSCOPY 3 R. E. Miles, Invited Speaker

COMPUTATIONAL METHODS IN STEREOLOGY 13

R. S. Anders sen and A. J. Jakeman

ESTIMATION OF LINEAR FUNCTIONALS BY MAXIMUM LIKELIHOOD 19

W. L. Nicholson

THE GEOMETRY AND STEREOLOGY OF ' CUBO-SPHERICAL ' PARTICLES 25

R. Warren

QUANTITATIVE MICROSCOPY OF LINEAL FEATURES IN THREE DIMENSIONS 29

R. I. DeHoff and S. M. Gehl, Invited Speakers

CONNECTIVITY OF 'DISPERSED' PARTICLES: A PROBABILISTIC COMPUTATION Al

G. A. Moore

ANALYTIC RESULTS FOR THE CONNECTIVITY OF DISPERSED PARTICLES 45

R. de Wit

TOPOLOGICAL ANALYSIS OF DENDRITIC TREES 49

M. Berry

THEORETICAL AND EXPERIMENTAL STUDIES OF THE STEREOLOGICAL PROPERTIES

OF POROSITY, SPECIFIC SURFACE, AND CONNECTIVITY 55

R. Bruce Martin

ASSESSMENT OF SAMPLING ERRORS IN STEREOLOGICAL ANALYSIS 59

J. E. Hilliard, Invited Speaker

ON OPTIMAL FORMS FOR STEREOLOGICAL DATA 69

A. J. Jakeman and R. S. Anderssen

MULTIVARIATE DATA ANALYSIS TO DESCRIBE INTRA- AND INTERGRANULAR

RELATIONS IN THIN SECTIONS 75

W. Good

CORRECTION OF STEREOLOGICAL PARAMETERS FROM BIASED SAMPLES ON NUCLEATED PARTICLE PHASES 79

L. -M. Cruz-Orive

xvii
Contents

STOCHASTIC MODELS IN STEREOLOGY: STRENGTH AND WEAKNESSES 83

J. P. Serra, Invited Speaker

A STEREOLOGICAL VIEW OF DATA ANALYSIS 87

G. A. Moore and G. T. Eden

THREE-DIMENSIONAL SHAPE PARAMETERS FROM PLANAR SECTIONS 91

E. E. Underwood

INTERPRETATION OF SOME OF THE BASIC FEATURES OF FIELD-ION IMAGE PROJECTIONS FROM A HEMISPHERICAL TO A PLANAR SURFACE USING MOIRE

PATTERNS 93

P. D. Ownby, R. M. Doerr and W. Bollmnn

NETWORK SIMULATION OF CONNECTIVITY AND AGGREGATE SIZE IN TWO-PHASE MIXTURES 99

W. D. Leahy and J. H. Steele, Jr.

2. PATTERN RECOGNITION

WHAT CAN PATTERN RECOGNITION DO FOR STEREOLOGY? 107

G. C. Cheng, Invited Speaker

THEORY AND APPLICATIONS OF IMAGERY PATTERN RECOGNITION 113

C. H. Chen

A NEW, FAST AND STORAGE-SAVING IMAGE ANALYSIS PROCEDURE FOR INVESTIGATING INDIVIDUALS BY A DIGITAL COMPUTER 117

M. Rink

SYSTEM FOR COMPUTER INPUT AND PROCESSING OF TWO-DIMENSIONAL

PICTURES 121

I. Krekule and M. Indra

PATTERN RECOGNITION ON THE QUANTIMET 720 IMAGE ANALYZING COMPUTER 123

K. E. Lawson

RECOGNITION AND CLASSIFICATION OF STRUCTURE BY MEANS OF STEREOLOGICAL METHODS IN NEUROBIOLOGY 127

G. Bernroider

PROJECTORS IN PATTERN RECOGNITION CATEGORIES 133

M. Pavel

3. INSTRUMENTATION

AUTOMATIC IMAGE ANALYSING INSTRUMENTS TODAY 141

H. P. Hougardy, Invited Speaker

TEXTURE ANALYSIS OF STEREOLOGICAL SPECIMENS 149

W. Hunn

A COMPUTERIZED IMAGE PROCESS FOR ISOLATING INDIVIDUALS IN AN ORIGINALLY NETTED PATTERN 155

M. Rink

AN IMAGE ANALYSIS SYSTEM BASED ON FEATURE DESCRIPTORS 159

R. R. A. Morton

xviii
Contents

IMPLICATIONS OF RECENT ADVANCES IN IMAGE ANALYSIS FOR STEREOLOGICAL

THEORY 163

H. Schmeisser

EXPERIENCES WITH OPTOMANUAL AUTOMATED EVALUATION-SYSTEMS IN BIOLOGICAL RESEARCH, ESPECIALLY IN NEUROMORPHOLOGY 167

H. Haug, Invited Speaker

A USER-ORIENTED AUTOMATIC SCANNING OPTICAL MICROSCOPE 173

J. E. Hilliard

THE DETERMINATION OF PARTICLE SHAPE AND SIZE DISTRIBUTIONS USING AUTOMATIC IMAGE ANALYSIS TECHNIQUES 177

J. Slater and B. Ralph

SOME SHAPE FUNCTIONS FOR USE IN AUTOMATIC IMAGE ANALYSERS FOR

THE CLASSIFICATION OF PARTICULATE MATERIALS 181

A. C. Terrell and R. J. Willes

DETERMINATION OF PARTICLE SIZE AND SHAPE DISTRIBUTION BY AUTOMATIC

FEATURE ANALYSIS 185

J. R. Schopper

STEREOLOGY AND THE AUTOMATIC LINEAR ANALYSIS OF MINERALOGICAL

MATERIALS 189

M. P. Jones and G. Barbery

A NEW OPTO-MANUAL SEMI-AUTOMATIC EVALUATION SYSTEM 19 3 H. P. Rohr

AUTOMATED MEASUREMENTS FOR DETERMINATION OF THE STATISTICAL

NATURE OF THE DISTRIBUTION OF INCLUSIONS IN STEEL 19 7 G. A. Moore

4. THREE-DIMENSIONAL RECONSTRUCTION

RECONSTRUCTION FROM PROJECTIONS: A SURVEY OF APPLICATIONS 201

R. Gordon

MORPHOMETRIC ANALYSIS OF NEURONS IN DIFFERENT DEPTHS OF THE

CAT'S BRAIN CORTEX AFTER HYPOXIA 203

0. Hunziker, U. Schulz, Ch . Walliser, and J. Serra

THREE-DIMENSIONAL RECONSTRUCTION FROM SERIAL HIGH VOLTAGE

ELECTRON MICROGRAPHS 207

L. D. Peachey, C. H. Damsky, and A. Veen

DETERMINING THREE-DIMENSIONAL INTRACELLULAR STRUCTURE DATA

FROM SCANNING PROTON MICROSCOPY LUMINESCENCE DATA 211

D. G. Oldfield

ARTIFACTS IN THREE DIMENSIONAL RECONSTRUCTION FROM MEDICAL RADIOGRAPHIC DATA 215

E. L. Hall, G. C. Huth, R. A. Cans, and I. S. Reed

ARTIFICIAL STEREO: A GENERALIZED COMPUTER ALGORITHM FOR COMBINING MULTI-CHANNEL IMAGE DATA 219

W. G. Pichel, R. L. Brower, D. R. Brandman, and R. J. Moy

xix
Contents

STEREOSCOPIC PHOTOMACROGRAPHY AND STEREOSCOPIC PROJECTION 221

Y. Fujimoto and Y. Ohta

HOLOGRAPHIC METHODS FOR IN-DEPTH VIEWING OF MICROSCOPIC SUBJECTS 223

R. J. Schaefer and J. A. Blodgett

EVALUATION OF A NEW TECHNIQUE FOR DETERMINING THE VOLUME OF THE HUMAN LEFT VENTRICLE FROM ONE-PLANE RADIOGRAPHIC MEASUREMENTS 225

R. Moore, I. Beranek, S. Kim, and K. Amplatz

MOVIE: PRIM-9. VIEWING HIGH DIMENSIONAL POINT CLOUDS 229

J. W. Tukey, Narrator

5. APPLICATIONS IN MATERIALS

APPLICATIONS OF QUANTITATIVE MICROSCOPY IN METAL SCIENCE 233

F. N. Rhines, Invited Speaker

APPLICATION OF STEREOLOGICAL TECHNIQUES TO THE QUANTITATIVE CHARACTERIZATION OF WOOD MICRO STRUCTURE , 245

J. H. Steele, Jr., G. Ifju, and J. A. Johnson

TEXTURE ANALYZER MEASUREMENTS AND PHYSICAL ANISOTROPY OF TROPICAL

WOODS 257

A. Mariaux, 0. Peray, and J. Serra

DECISION CRITERIA IN QUALITY CONTROL OF STEEL WITH RESPECT

TO NONMETALLIC INCLUSIONS 261

S. Johansson

AUTOMATIC RECOGNITION OF NON-METALLIC STRINGERS IN STEEL 265

D. Jeulin and J. Serra

METALLOGRAPHIC CHARACTERIZATION OF FRACTiJRE SURFACE PROFILES

ON SECTIONING PLANES 269

J. R. Pickens and J. Gurland

THE EQUIVALENT FOIL THICKNESS PRINCIPLE 273

S. M. El-Soudani

STEREOLOGICAL METHOD FOR ESTIMATING RELATIVE MEMBRANE

SURFACES IN FREEZE-FRACTURE PREPARATIONS 277

E. R. Weibel, G. A. Losa and R. P. Bolender

APPLICATION OF VARIOUS STEREOLOGICAL METHODS TO THE STUDY OF THE GRAIN AND THE CRACK STRUCTURE OF CONCRETE 281

P. Stroeven

THE USE OF QUANTITATIVE METALLOGRAPHY AND STEREOLOGICAL

METHODS TO DESCRIBE COMPOSITE MATERIALS 287

J. -L. Chertnant and M. Coster

A DEFINITION OF THE TWO-DIMENSIONAL SIZE OF IRREGULAR BODIES 291

G. M. Timcak

STEREOMETRIC MICROSTRUCTURE AND PROPERTIES OF TWO-PHASE MATERIALS 295

G. Ondracek

USE OF MATHEMATICAL MORPHOLOGY TO ESTIMATE COMMINUTION EFFICIENCY 299

Ph. Cauwe

XX
Contents

MICROSCOPY, RADIAL DISTRIBUTION ANALYSIS, AND SORTABILITY 305

R. M. Doerr

SIZE DISTRIBUTION ANALYSIS IN SITU ON INDIVIDUAL OR INTERCONNECTED

PHASES BY IMAGE ANALYSES 309

C. Gateau and J. M. Prevosteau

STEREOLOGICAL DETERMINATION OF THE PROPORTIONAL VOLUME OF DENDRITES IN CAST IRON BY MEASURING PLAN MICROGRAPHS 315

R. Wlodawer

6. APPLICATIONS IN BIOLOGY

MORPHOMETRY, STEREOMETRY AND STEREOLOGY APPLIED TO CANCER DEVELOPMENT IN THE LARGE INTESTINE OF MAN 321

H. Elias, Invited Speaker, D. Bokelmann, and R. VOgtle

ANALYSIS OF THE HETEROGENEOUS COMPOSITION OF CENTRAL NERVOUS TISSUE 327

S. Eins and J. R. Wolff

STEREOLOGICAL ANALYSIS OF THE HETEROGENEOUS COMPOSITION OF CENTRAL

NERVOUS TISSUE: SYNAPSES OF THE CEPvEBRAL CORTEX 331

J. R. Wolff

THE DEVELOPMENT OF MYELINATED FIBERS IN THE VISUAL CORTEX OF CATS 335

H. Haug

THE VOLUMES OF ELASTIC TISSUE AND SMOOTH MUSCLE CELLS IN THE ARTERIAL WALL 337

V. Levick^

PROGRESS, SUCCESS AND PROBLEMS IN APPLYING STEREOLOGY IN BIOLOGICAL RESEARCH 341

E. R. Weibel, Invited Speaker

HEXACHLOROBENZENE-INDUCED SMOOTH ENDOPLASTIC RETICULUM IN RAT LIVER: A CORRELATED STEREOLOGIC AND BIOCHEMICAL STUDY 351

T. Kuiper-Goodman, D. Krewski, H. Combley, M. Doran, and D. L. Grant

AUTOMATED MEASUREMENT OF AIRSPACES IN LUNGS FROM DOGS CHRONICALLY

EXPOSED TO AIR POLLUTANTS 355

D. Hyde, A. Wiggins, D. Dungworth, W. Tyler and J. Orthoefer

VOLUMETRIC DETERMINATIONS OF CELLS AND CELL ORGANELLES FROM TWO-DIMENSIONAL TRANSSECTIONS 359

L. G. Lindberg

MORPHOMETRIC ANALYSIS OF THYMIC CELLS DURING MURINE RADIOLEUKEMOGENESIS 363

J. Boniver, R. Courtoy, and L. J. Simar

STUDIES ON THE PARTIALLY-ORIENTED SURFACES OF SKELETAL

MUSCLE MITOCHONDRIA 367

N. T. James and G. A. Meek

DETERMINATION OF THE NUMBER OF CELLS WITH MULTIPLE NUCLEOLI

IN HISTOLOGICAL SECTIONS 371

A. Schleicher, H. -J. Kretschmann, F. Wingert, and K. Zilles

COMBINED LIOiT AND ELECTRON MICROSCOPY OF SERIALLY SECTIONED

CELLS AND TISSUES 375

Y. Collan

XXI
Contents

MORPHOMETRIC "ORGANELLE PROFILES" AS METABOLIC INDICATORS 379

S. Malamed and L. C. Zoller

A QUANTITATIVE ANALYSIS OF SOME ULTRASTRUCTURAL ASPECTS OF

SEED DEVELOPMENT 385

C. E. Hughes and L. G. Briar ty

THE BRAIN AS A "STEREOLOGICAL DEVICE" 389

T. Radii-Weiss and J. RadilovS

STEREOLOGICAL ANALYSIS OF NEURAL ORGANOGENESIS IN THE CHICK EMBRYO 393

0. Mathieu and P. -E. Messier

APPLICATION OF STEREOLOGICAL METHODS TO THE STUDY OF PRE- IMPLANTATION EMBRYOGENESIS IN MICE 397

R. L. Deter

THE DETERMINATION OF SIZE DISTRIBUTION ON LYMPHOBLASTS IN ACUTE LEUKEMIA AOl

E. Feinermann and G. A. Langlet

A MORPHOMETRIC STUDY OF HUMAN PLACENTAE OF DIFFERENT

GESTATIONAL AGES FROM NORMAL AND TRIPLET PREGNANCIES 405

1. Bhargava, K. Kamashki, and Y. Dodge

THE EFFECTS OF OPTICAL RESOLUTION ON THE ESTIMATION OF STEREOLOGICAL PARAMETERS 409

H. J. Keller, H. P. Friedli, P. Gehr,

M. Bachofen, and E. R. Weibel

QUANTIFICATION OF RARELY OCCURRING STRUCTURES IN ELECTRON MICROSCOPY 411

U. Pfeifer

THE MORPHOMETRY AND STEREOLOGY OF CEREBRAL ARTERIAL BIFURCATIONS 415

P. B. Canham, J. G. Walmsley and J. F. H. Smith

KARYO-INTERKARYOMETRY AS A PERIODIC CONTROL FOR CLINICAL PROGNOSIS 419

E. C. Craciun and C. Tasca

NUCLEIC ACIDS CONTENT-NUCLEAR VOLUME CORRELATION AS A

QUANTITATIVE PARAMETER IN CELLULAR MALIGNANCY 423

C. Tasca and E. C. Craciun

IS THERE AN UNRECOGNIZED SYSTEMATIC ERROR IN THE ESTIMATION OF SURFACE DENSITY OF BIOMEMBRANES? 427

A. Reith and T. Barnard

QUANTITATIVE STUDIES WITH THE OPTICAL AND ELECTRONIC MICROSCOPE OF THE PATHOLOGICAL CELLS OF THE ACUTE LEUKEMIA AND CHRONIC LYMPHOCYTIC LEUKEMIA 429

J. L. Binet, P. Debre, P. d'Athis, D. Dighiero and F. de Montaut

SIMPLE DEVICES FOR STEREOLOGY AND MORPHOMETRY 431

H. Elias and E. Botz

THE EDEMA SYNDROME IN MOTION PICTURES 435

0. C. Jaffee

xxii
Contents

PART II. WORKSHOP SECTIONS

7. MATHEMATICAL FOUNDATIONS OF STEREOLOGY

OVERVIEW A 39

R. T. DeHoff

PROBABILISTIC FOUNDATIONS OF STEREOLOGY 443

R. E. Miles and P. Davy

DERIVATION OF THE WORKING RELATIONSHIPS OF STEREOLOGY

A. VOLUME FRACTION ANALYSIS 446

H. E. Exner

B. SURFACE AREA 448

E. R. Weibel

C. LENGTH OF LINEAL FEATURES IN SPACE 455

R. T. DeHoff

D. THE AREA TANGENT COUNT AND THE TOTAL CURVATURE 45 7 R. T. DeHoff

GRANULOMETRY FOR ISOLATED PARTICLES OR CONNECTED MEDIA 460

J. Serra

STATISTICS OF MEASUREMENT 461

W. L. Nicholson

8. PARTICLE SCIENCE

INTRODUCTION REMARKS 467

B. H. Kaye

STEREOLOGY AND PARTICLE TECHNOLOGY 469

H. E. Exner

REPRESENTATION OF PARTICLE SIZE AND SHAPE 471

B. Scarlett

THE APPLICATION OF STEREOLOGY TO PARTICLE TECHNOLOGY 47 7 B. Scarlett and P. J. Lloyd

TECHNIQUES FOR PARTICLE MEASUREMENT USING IMAGE ANALYSIS 483

R. R. A. Morton

MICROSCOPIC CHARACTERIZATION OF FINE POWDERS 487

P . Ramakrishnan

STEREOLOGY OF PAINT FILMS 491

B. H. Kaye and I. Robb

CONCLUDING REMARKS 493

B. H. Kaye

xxiii
Contents

9. SIZE DISTRIBUTIONS

ESTIMATION OF LINEAR PROPERTIES OF SPHERICAL BODIES IN THIN FOILS FROM THEIR PROJECTIONS 49 7 F. Piefke

ON SIZE DISTRIBUTION METHODS 499

F. A. L. Diillien and K. S. Chang

SIZE DISTRIBUTION OF RETAINED AUSTENITE PHASE IN A . ' ,\ . . r QUENCHED STAINLESS STEEL 503

A. V. Samudra and Om Johari

ANALYSIS OF A SET OF SPHERICAL CELLS RELATIVE TO THEIR VOLUME 507 Ph. d'Athis

APPENDIX - BASIC STEREOLOGY 509

E . E . Underwood

ABBREVIATED GLOSSARY OF EQUIVALENT SYMBOLS 513 DIRECTORY OF MEMBERS OF THE ISS 515 CONTRIBUTOR INDEX 531 SUBJECT INDEX 533

xxiv
PART I. TECHNICAL SECTIONS

PRINCIPLES AND MATHEMATICAL DEVEL0PMJ:NTS

Chairpersons

:

R. E. Miles

W. L. Nicholson

R. T. DeHoff

R. de Wit

J. E. Hilliard

R. S. Anderssen

J. C. Serra
NOTE

In order to assist the reader with the symbolism

in the more mathematical papers of Miles (p. 3),

DeHoff and Gehl (p. 29), and Rhines (p. 233), an abbreviated glossary of equivalent symbols has been assembled on p. 513.
National Bureau of Standards Special Publication 437

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

ON ESTIMATING AGGREGATE AND OVERALL CHARACTERISTICS FROM THICK SECTIONS BY TRANSMISSION MICROSCOPY

by R, E. Miles Department of Statistics. Institute of Advanced Studies, Australian National University, P.O. Box 4. Canberra, A.C.T. 2600, Australia

ABSTRACT

The random spatial structure considered is the union X of an aggregate

of random opaque convex particles in transparent space, the particle centroids

being sited at the points of a homogeneous Poisson process. The model is thus the union of an aggregate of not—necessarily-convex disjoint regions.

Expressions are derived for the fundamental stereological quantities Vy, S^,

(integrated mean curvature density) and (integrated gaussian curvature

density) for X ; and for Aj^, LJ^ and (integrated curvature density) for

the projection of a slice of thickness t of this model onto a plane parallel

to the slice, where t may take any non-negative value. The estimates

relevant to transmission microscopy suggested by this theory are presented.

The stochastic model considered is an aggregate of random 'opaque' convex

particles (with V, S and M denoting their mean volume, mean surface area and

mean mean {sio) caliper diameter, respectively) distributed randomly throughout

a 'transparent' medium as follows. Their centroids (or any other well-defined

particle point) are sited at the points of a Poisson process with constant

density D , and their orientations are (independently) isotropic about these

points. Thus the particles may interpenetrate, a prerequisite for any

effective mathematical theory; and so the model is essentially one of disjoint

opaque regions, each of which is the union of one or more overlapping particles.

General properties of this rather ' natural stochastic model have been given by

Giger and Hadwiger [3] . The sets of points contained in at least one particle,

and not contained in any particle, are denoted by X and Y , respectively.

As usual in transmission microscopy, a foil (or section) bounded by two

parallel planar faces a distance t apart is taken, and the content of this

section is projected perpendicularly onto an observation plane parallel to the
R. E. Miles

faces. At each point of the observation plane it may only be discerned whether

it is covered by at least one particle, or is uncovered. With a dash relating

to the observation plane, we write X' and Y' for the covered and uncovered

areas, respectively.

The stereological problem is to estimate V, V, S and M from

measurements solely in the observation plane; these estimates may in turn be

used to furnish estimates for the following fimdamental 'overall' quantities:

the fraction

(la) = 1 - exp(-W)

of the volume which is occupied by X ; the interface area

(lb) = ^'^ exp(-DV)

between X and Y per unit volume; the total (or integrated) mean curvature

(Ic) Ky = \2-nDM - n'^n'^S^]^ exp{-DV)

of this interface per unit volume (at each point of surfaces under surface

tension, the pressure difference between the two sides divided by the surface

tension equals twice the mean curvature) ; and the integrated gaussian

curvature of this interface per unit volume

(Id) = |4TrD - IW'^MS + ^

ir^O^^j exp(-OI0

(for aggregates of non-overlapping particles, not necessarily convex, Gy is

Afr times their number per unit volume). It is of interest that (la) - (Id)

remain true when the particles are no longer convex - in that case M is no

longer the mean mean caliper diameter, but represents (27t) ' times the mean

value of the integral of mean curvature of the particles.

In the observation plane, define A'. to be the covered area fraction, L' A A

to be the length of interface curve between X' and Y' per unit area, and

to be the total curvature of this interface curve per unit area. Then

(2a)

'^i = ' " exp[-D{?+(t5/4)}] ,

(2b) = TT£'{(5/A)+tM} exp[-D{7+(t5/4)}] ,

and

4
On estimating characteristics frort) tliicl(^ sections by transmission microscopy

(2c) = [2TTO(A/+t)-2TrV{(5/8) + (tW/2)} ] exp[-D{7+(t5/4)}] .

The derivations of formulae (1) and (2), by the methods of geometrical

probability, are given in the Appendix. With D small, (1) and (2) reduce to

the relations (20)-(22) in DeHoff [1]. Standard stereological relations for a

planar section result from (I) and (2) upon setting t = 0 .

Using (2), we may estimate 0{7+(t5/4)}, D{(S/it)+tM} and D(M+t) . Thus,

if t (see [6; pp. 188-190]) and any one of D, V, S and M is known, the

other three may be estimated. Alternatively, if sections of differing but

unknown thicknesses t^, are available, we have the estimates

(3a) ^7 = 1I - e

•

(3b) = Uva e

21

(3c) =

1

(3d) =

1

(4) D : V : S 1: M :

and

(5) : : 1

where ^21 = ^2"^1 ' etc .

,

1 ^2 ,2 2 -u

-«^"S.^1 ^l^'"'4l

2 21 2 21 12 2 1

LI , C\

(6a) a = -ln(l-A;), h = - j=p , o-^^^—.

A A

2 (6b) u = {a2j (e|a2~C2aj)+fc2l ^'^1^2~^2^1-' ^^(^21~'^21^2I^ '

and

2 (6c) V = (e,fc2-e2Z',)/(fe2r'^21^2l) '

If at least one of tj, ^2 known, or sections of three or more thicknesses

(known or unknown) are at hand, estimates of a more statistical nature may be

obtained from (1) and (2).

A feature of our theory is its generality: t is not necessarily small,

and the convex particles may have any shape distribution (surprisingly, even

with this generality, no approximations are made). As observed above, since

the particles may interpenetrate, our model is really one of disjoint opaque

5
R. E. Miles

regions, each of which is the union of one or more overlapping convex

particles

.

When 0(27+^ S) is small, most of these are isolated and therefore

convex. For larger values of D{2V+M 5), estimates of D, V, S and M lose

their practical interpretation, unlike those of Vy, S^, Ky and Gy . In

fact, it seems likely that (3) (and (5)) will often give useful estimates for a

thick section from a general homogeneous and isotropic two-phase (one opaque,

one transparent) structure, e.g. a porous medium.

The only work similar to this appears to be some of Giger [2], but the

stereological implications of his paper seem a little obscure.

In this appendix the derivations of the basic relations (la)-(ld) and

(2a)-(2c) are presented. The remaining formulae given above follow from these,

essentially by simple algebra.

The model may be thought of as being constructed in two stages:

(i) The centroids of the particles are located at the points of a Poisson

point process in space having constant density D . Thus the number of process

points in disjoint volumes V^, are mutually independent and Poisson

distributed, with mean values DV^, DVj^ , respectively.

(ii) 7or each process point x , a random convex set K is sampled from

some isotropic distribution of convex sets having the origin as centroid.

Then the particle associated with x is x + K , i.e., K translated so as to

have X as centroid. This procedure is carried out independently for each

process point, using the same convex set distribution. The random set X of

interest is the union of the particles so formed.

First we have the key

Theorem (Hitting Numbers). The number of particles hitting any fixed

convex domain K. is Poisson distributed with mean value

APPENDIX

0

(7)

In particular the probability that K„ is not hit is g -6

6
On estimating characteristics from thick sections by transmission microscopy

Proof of Theorem. Write dP for the proportion of particles with

dimensions in the range {M, M+dM; S, S+dS; V, V+dV) . Then by the three

dimensional form of the principal formula of integral geometry [5; equation

(66)] the number of particles in this range hitting is Poisson distributed

with mean value \i = D dP * ^

S^M + jM^S + • follows from the

'complete independence' of Poisson processes and the model that the number

hitting is Poisson (| y) distributed. Q.E.D.

Although not mentioned, the following theory depends heavily upon this

Poisson independence, and would scarcely be possible without it.

Taking Kq to be a point, (la) follows. The particle surface area per

unit volume is DS . For each such surface point, the probability it is

uncovered by other particles is e . (!b) follows at once from these two

observations. (We shall repeatedly use 'uncovered' in this sense below.)

Since the mean caliper diameter of a convex set is (27r) ' times its

integral of mean curvature, the integral of mean curvature of the particle

surfaces per unit volume is 1t\DM . Hence the contribution of these surfaces

to is

(2) — -DV

(8) K!j^ ' = 2-nDM e .

To simplify the theory, we suppose the particles have smooth surfaces, so

that pairs of surfaces intersect in smooth curves. These we call \-curves\

likewise, points of intersection of triples of particle surfaces we call

l-points . In fact, our results extend without change to general convex

particles. It only remains to find the contribution ^y'^ to Ky from the

uncovered portions of I-curves, since it is readily verified that the

contribution from uncovered I-points is zero.

But first we determine the length of I-curve and the number Ny of

I-points per unit volume. Consider a small fixed sphere Q of radius

e « 1 . The following statements are true in the limit e 4- 0 . Intersections

of particle surfaces with Q are planar, and their number n is Poisson

(2e05) distributed. Further, given n = Hq , these Hq hitting planes are

7
R. E. Miles

independent isotropic uniform random (lUR) planes in § . We require the

following specializations of the results (2.13), (1.35P) and (2.31T) of [4]:

(9) F (2 independent lUR planes in Q intersect in a line

2 hitting = 7T /16 ;

(10) P (3 independent lUR planes in Q intersect in a point of Q)

= ;

and given that 2 independent lUR planes in 5 intersect in a line hitting

Q , this line is lUR in Q , so that the mean length of its intersection with

Q is 4e/3 . It follows from the above that

(M) Ny = eQ (77^/48) /(4TTe-^/3) = TTr)%^/48

and

(12) = ^(5 (4e/3)/[4TC^/3) =W^S^/8 .

Now, since the integral of mean curvature for unit length of a wedge of

dihedral angle (j) is (•n"-<J))/2 , we have

(,3, 4" . ^-.7

.

Here (}> has a distribution symmetric about E((^) = 3it/2 , and (Ic) follows

from Ky = X^'^ + K^^'^ .

We now derive (Id). Since the integral of gaussian curvature for any

convex domain is 471 , the contribution to Gy from particle surfaces is

(2) -DV

(14) g!^ ' = h-n D e .

The contribution from each uncovered I-point is the same as the integral of

gaussian curvature for the complementary cone C at the I-point, which itself

equals the exterior solid angle o) [4; p. 216] of the cone dual to C .

Hence, since £(0)) = 7r/2 , we have

(15) Gf^ = Ny (77/2) e~^^ .

It remains to find the contribution G^'^ from the uncovered portions of

I-curves. We suppose the particles are polyhedral. This is permissible, since

a convex set may be .approximated arbitrarily closely within and without by

convex polyhedra. Then the I-curves are equivalently (i) line segments of

intersection of pairs of polyhedral faces; and (ii) points of intersection of

8
On estimating characteristics from thick sections by transmission microscopy

polyhedral faces and edges. We need only consider (ii), since the contribution

of (i) to '^y'^ is zero. Consider in particular the contribution from a

polyhedral edge of length , dihedral angle (|)^ [O < (})^ < tt) (and hence

exterior angle ir - (j)^ [4; p. 216]). By considering the intersections of a

half-plane with a 'ripple surface' in which the dihedral angles are alternately

(j)^ and 2tt - (f)^

, and noting that the sum contribution of the vertices to the

integral of gaussian curvature is zero, we see that the contribution from the

intersection of a polyhedral face with our edge of dihedral angle (j)^ is minus

that from the intersection of the face with an edge with dihedral angle

2Tr - (j)^ . Averaging over mutual orientations, the average value of the latter

contribution is seen to be it - (f)^ . Now, by the classical stereological

formula "P_ = \r A " , the mean number of intersections of the (Z. . , (j).) edge

with polyhedral faces is ^l^DS . Hence the average contribution of a

polyhedron is

(16) DS g-^^

.

That is, since M = (4tt) ^

\ 'i * -] for polyhedra,

(17) =-2.1^2^7 5.-^^.

(Id) follows from G = . V ^0 V Now we turn to the transmission microscopy theory for our model, and

derive (2a)-(2c). Taking in the Theorem to be a line segment of length

t , for which M = t/2 , we obtain

(2a) ' " = exp [-£>{?+ (tS/4)}] = 6 , say.

Next we derive (2b) . We find by application of the classical

stereological formula "L' = (77/2) N' ", where N' relates to a line section A Lf Lj of X' . To determine N' , we consider the number m of particles hitting a

rectangle R of width t and length T » 1 , representing an orthogonal

section of the foil of length T . Since T » 1 , we may assume each hitting

particle projects onto each rectangle side of length T as a segment. Since

each segment end point contributes towards ^17' with probability 9 , we have

L

9
R. E. Miles

(18) 29 E{m)lT N' as T -> oo . L/ For R , V = 0 , S = 2tr and M = (t+T)/2 [A; (2.29T)] and so, by the

Theorem,

(19) E(.m) D{(S/U)+tM}T as T ^ «>

,

and (2b) follows.

Finally we derive (2c), adopting the following terminology: portion for

that part of a particle within the foil, image for the projection of a portion

onto the observation plane, and vertex for the intersection point of the

boundaries of a pair of images when this point is uncovered by any other image.

Now

(20) = Cj°> . C]') ,

where the addends are the contributions to from the vertices and image

boundaries, respectively. Clearly

(21) = - (Ti/2) ff; ,

where NJ^ is the number of vertices per unit area, and

(22) c]'^ = E{K) , where k is the curvature at a uniform random point of the boundary of X' .

First we find . In the following, the symbols ± and o refer to

the two foil faces and the foil interior, respectively. We describe the

intersection of a particle boundary with the ± faces as a ± arc (which it

should be noted is closed). Again, for a particle intersecting the foil we

describe a point on the boundary of the corresponding portion at which the

tangent plane is perpendicular to the foil as a tangent point, and describe the

arc of projection of the set of tangent points as the associated O arc (which

is not necessarily closed) . The aggregates of ± and o arcs are (dependent)

planar Poisson aggregates of essentially the same type as the basic spatial

particle aggregate, and we now investigate their densities p (corresponding

to D } and mean lengths L . We have

(23) = p_ = DM E .

The mean perimeter of the section of a convex set by an lUR plane is vS/hM

10
On estimating characteristics from thicl< sections by transmission microscopy

[4; <2.31T)]. Since the 'probability' each foil face hits a particle is M ,

we have

(24) ^+ = ^_ = • M^/E(.M) = Tr5/4W = ,

The determination of and is not so simple; however, fortunately all

we need is their product - the arc length per unit area. Since the mean

perimeter of the projection of a convex set onto an isotropic random plane is

ttW , we have

(25) pl^ = tn TT M .

Now each vertex is of one of the six types ++, —, oo; -o , o+ and +- ; corresponding to the two associated particles. It is easily shown that the

corresponding densities of arc-arc intersections per unit area are -12—2 -1

IT L^, 2ir p p^ L L^, ... . Whether such an arc-arc

intersection is a vertex depends on

(i) whether, for the relevant point on a ± arc, it is within or

without the projection of the associated portion (each case has

probability %) ; and

(ii) whether the relevant segment of length t is hit by any other

particle

.

Taking all this into account, we have

(26) i7j = (p/^+ P^rj2 g ^

Finally we determine E{k) . For a single fixed closed arc,

(27) E{k) = I

k ds/| ds = 2Tr/L .

Hence for an aggregate of such arcs

(28) EM =

I (2TT/LJ LJ[1 L^] = 2^/1 .

As t increases, the o arcs become closed and so (28) applies, yielding

£(<) = 2lM for the o arcs. It follows, taking ± and o arcs with

weights % and I , respectively, that

11
R. E. Miles

P+ + p L {IIM)

(29) EM = -^—^

= 8(t+M)/ (S+^tM) .

(2c) now follows from (20)-(29).

REFERENCES

[l] DeHoff, R.T., 1968. Quantitative microstructural analysis. ASTM STP.

430, Am. Soa. Testing Mats.: 63-95.

[2] Giger, H., 1972. Grundgleichungen der Stereologie II. Metrika 18: 84-93.

[3] Giger, H. and Hadwiger, H. , 1968. Uber Tref f zahlwahrscheinlichkeiten in

Eikorperf eld . Z. Wahrsaheinliahkeitstheorie verw. Geb. 10: 329-334.

[4] Miles, R.E., 1969. Poisson flats in Euclidean spaces. Part I: A finite

number of random uniform flats. Adv. Appl. Prob. 1: 211-237.

[5] Santalo, L.A., 1936. Integralgeometrie 5: Ifber das kinematisohe Mass im

Raum. Hermann, Paris (Act. Sci. Indust. No. 35 7).

[6] Underwood, E.E., 1970. Quantitative Stereology . Addi son-Wesley (Reading,

Mass . )

.

12
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

COMPUTATIONAL METHODS IN STEREOLOGY

by R. S. Anderssen and A. J. Jakeman

Computer Centre, Australian National University, Canberra, ACT, 2600, Australia

mrnypucnw

A classical problem in stereology is the determination of three-dimensional

(structural) properties of an a-phase in a 3-matrix from two-dimensional observations of the a-phase. Although numerous numerical methods have been proposed

for the approximate solution of the resulting mathematical formulations (such as the random spheres and thin section models), when they exist and are known, none

appear to be completely satisfactory. Little attention appears to have been

paid to the actual classification of the mathematical formulations involved, and

the use of appropriate results from the numerical analysis literature for the numerical solution of the resulting classes. A reason for this was the lack,

until recently, of suitable results in the literature.

This situation has now changed since many of the questions concerning the construction of basic methods in computational mathematics have been resolved.

For this reason, a systematic and comparative study of the different classes of methods available for the different types of stereological estimations (inversions) is now possible. For the estimation of the size distribution of spheres

from planar or thin section observations of their circular contours, the following classes of methods can be identified: (1) Degradation of Problem Formulation; (2) Non-parametric Methods; (3) Parametric Methods; (4) Evaluation of Inversion Formulas; (5) Regularization Methods; and (6) Filtering Methods.

There is little justification for the degradation of the problem formulation (when available) such as the use of the Random Spheres Model {see, for example, Baudhuin and Berthet (1967)} to determine the size distribution of

spheres from thin sectional observations of circles. The use of degraded

problem formulation leads to the necessity to adjust for its effect which can be

avoided, if the correct formulation is used. Thus, the necessity to correct for

the Holmes effect is avoided, if the Thin Section Model is used for thin section

data {see, for example, Jakeman and Anderssen (1975a)}.

It follows from the systematic study of methods by Anderssen and Jakeman

(1974) and Jakeman and Anderssen (1974) that, except under special circumstances, the use of non-parametric and parametric methods as well as the evaluation of inversion formulas can behave unsatisfactorily as a consequence of the following factors: stereological data is observational and therefore non-exact,

and the mathematical formulations involved are improperly posed (small perturbations of the data can correspond to arbitrarily large perturbations of the

solution) . The non-parametric methods do not take account of the improperly

posed nature of the formulation and were developed for exact rather than nonexact data. They can be used to yield results, but at the expense of accuracy

in order to control error growth. For example, when using finite difference

methods, the choice of the class width (grid spacing) must be a trade-off of accuracy against error growth by ensuring that it is not taken to be too small. As indicated by Anderssen (1975), the same comment applies to the use of direct

numerical methods to evaluate inversion formulas.

The difficulty posed by the parametric methods, and the use of pseudoanalytic methods to evaluate an inversion formula, is the choice of functions

(distributions) which model correctly (not approximately) the structure in the

data. In general, this can only be done approximately, since the level of information about the underlying structure is virtually non-existent. Even in the exceptional situation where it is possible, this does not necessarily cir- cumvent the difficulty associated with the improperly posed nature of the

13
R. S. Anderssen and A. J. Jakeman

formulation.

Although regularization methods {see, for example, Lavrentiev (1967)}

stabilize the solution of an improperly posed problem formulation, they were not designed to filter the error from the signal in non-exact data, whereas the filtering methods were designed to do the latter and not the former. It is clear then that satisfactory numerical methods must combine the advantages of both the regularization and filtering methods at the expense of their disadvantages. One possibility is the application of a filtering method to the data

followed by the use of a regularization method to invert the appropriate problem

formulation for the filtered data. It is not necessary however to limit atten- tion to such a simplistic implementation of these two methods. Other forms for the problem formulations, such as inversion formulas, could be used when known. Methods which implicitly combine the advantages of both regularization and filtering should be sought.

The methods proposed recently by Anderssen and Jakeman (1974) and Jakeman

and Anderssen (1974) aim to achieve this through the evaluation of known inversion formulas using product integration and spectral differentiation. In order

to illustrate, consider the thin section model for spherical particles

c(y) = 2a \ t ^ sW[x' - y']""' dx v t s(y)[ , (1)

where s(x) denotes the probability density for the radii of the spheres, c(y)

= dC(y)/dy the probability density for the radii of the circles in thin sections

oo of thickness t, and a = / x s(x) dx is the expectation of the radii of the

0 spheres. For the model, explicit inversion formulas are known, including

s(x) = - [2/^"' ^2^-^-^ ^] f{[27T(y2 - x^)]^} C(dy) , (2)

where

1 f(w) = /2? exp(wV2) {1 - / exp(-uV2)du}

/2-n- -00 Using product integration and spectral differentiation, the method proposed

in Anderssen and Jakeman (1974) was:

(a) Evaluate, using product integration

V(x^) = / f{[2^(y2 - x^)]Vt} (dCj^/dy)dy (3) ^k where {yj^^ (k=0,l,2, . .

. ,n) denotes the non-uniform grid on which the data

is given, {x^.} (k=0,l,2, . .

. ,N) the uniform grid onto which the data is integrated, and '^i^iv) a- localized Lagrange interpolation smoothing of the data

CC(yj^)} {see, Anderssen and Jakeman (1975), for details}.

(b) Use spectral differentiation to determine

S(x^) = - [2/.]''^ X2a^^ HV(x,)}]^^^^ . (4)

CXMPARISON OF METHODS

Using the exact (effect of sampling not included) synthetic data of

Anderssen and Jakeman (1974) , Jakeman and Anderssen (1974) have compared the above method with the non-parametric method of Bach (1967) and the parametric

method of Keiding et al. (1972). Their graphs clearly illustrate the superiority of the above method over these two and exemplify the above general comments

about parametric and non-parametric methods. In Figure 1, we compare the above method with a pseudo-analytic method {of the type discussed by Piessens and Verbaeten (1973) for an Abel equation in a non-stereological context} for (2) based on an exact inversion result of Bach (1966), §§ VI, VII, : viz., if

14
Computational methods in stereology

(2a + t) c(y) = p (y) J a 9^ y'^ , 6 > 0 , (5) i=0 then m . . s(x) = pg(x) I A. 6^ x^^ . (6) i=0

9y^) with the p£ likelihood argument) by

Here Pa (y) = 29y exp(-9y^) with the parameter 9 estimated (using a maximum

9 = n / 5 y^ . (7) i=0

The constants a. and A. are related by

1 m

A. = [9^^ a. - (1 - 6^) I [j] r(j - i + |) A ]/(t9''^ + , (8) j=i+l ^ ' i=m,m-l ,...,1,0 , with r(x) denoting the gamma function with argument x , and Sl" equal to one when m = i and zero otherwise.

The actual computational implementation was: (a') For given m , fit the representation (5) to the data {(2a + t) c(y]^)} (k=0, 1

, 2 , . . . ,n) using linear

least squares. (b') Substitute in (8) the {a^} (i=0, 1 , 2 , . . . ,m) obtained to

find the corresponding {A.} , and hence the representation (6) for s(x) .

-» 1 1 1 1 1 • 1 1 r

1

1 0.00 0.40 0 80 1.20 1 60 2.00 0 00 0.40 0.80 1.20 1 60 2.00 Radius Radius

* 1 1 1 1

1 * 1 1 1 1

1 aOO O40 0.80 1.20 1.60 2.00 0.00 0.40 0 80 1 20 1 60 2.00 Radius ' Radius Figure 1

15
R. S. Anclerssen and A. J. Jakeman

It is clear from Figure 1 that there exist data for which this pseudoanalytic method is unable to yield a satisfactory solution. Potentially, it will

only work for data which have the structure of (5) with m finite. The solution

obtained from the proposed method (using the same exact data) was quite close to the exact solution {see Figure 1(b) of Jakeman and Anderssen (1975a)}.

In the comparisons mentioned above, exact synthetic data (with sampling

effects neglected) were used. When synthetic sample cumulative data are used,

methods will not perform as well as when exact synthetic data are used. This is a consequence of the properties of the sample cumulative rather than the methods.

Thus, methods which fail for exact data will also fail for the sample cumulative

data, while methods which yield satisfactory results for the exact will exhibit

degraded performance on the sample cumulative. For the last m.entioned class of methods, however, the degraded performance can be improved by increasing the number of data points. This is again a consequence of the properties of the sample cumulative data (when the method under consideration is stable) . It is best explained by observing that the sample cumulative data H (y) = {y. ^ i/n ; i=l ,2 , 3, . . . ,n} sampled from H(y) have variance satisfying

var[H^(y)] = - H(y) (1 - H(y)) (9)

Derivative of sample

from C(v) = (3-2y)y2 800 points

Derivative of sample

from C(y) = (3-2y)y2

1 600 points

Figure 2(a)

0 00 0 20 0.40 y 0 60 0 80 100 Figure 2(b)

Derivative of sample

from C(v) = (3-2y)y2 3200 points

0.20 0.40 y 0.60 0.80 Figure 2(c)

1.00

16
Computational methods in stereology

Consequently, the error between H^(y) and H(y) behaves like l//n .

The nature of the problem can be fully illustrated by considering the numerical differentiation of sample cumulative data sequences collected from the cumulative distribution function C(y) = (3 - 2y)y^ on the interval [0,1] . The spectral derivative of one such sample cumulative is shown in Figures 2(a), (b) and (c) . For the same amounts of exact synthetic data, the spectral derivatives

were graphically indistinguishable from the exact solution shown. The improvement resulting in the accuracy of the spectral derivative through increasing the number of points in the sample cumulative can be seen from a comparison of Figure

2(a) with 2(b) and 2(c).

In implementing the spectral methods, other points which require future

attention are the use of the even ordinate grid of size distribution data, and

the grouping of size distribution data onto an even abscissae grid.

TRUNCATED MCMENTS FOR THIN SECTICN DATA

Some of the numerical difficulties posed by the estimation of three dimensional particle size distribution functions can often be avoided when only linear

properties of the size distributions are required. A general procedure has been

developed by Nicholson (1970) for estimating such linear properties from given

truncated data. It requires that the particles be completely specifiable by one parameter. It has been applied to spheres as well as to a certain class of cylinders with both sampled by planar sections. The theory also extends to observations in a thin section, though its specific application to thin section

data has yet to be published.

When an inversion formula is known, which is the case for the thin section

model for spherical planar data, the approach used by Jakeman and Anderssen

(197Sb) for deriving linear estimators of the Nicholson type from planar sections

of random spheres can be applied. Using the inversion formula, they first transform the linear property into a linear functional defined on the size distribution function of the observations. Next, they stabilize the actual evaluation of

the functional on the observational (cumulative) data {C(y^)} by evaluating it,

using product integration, on a localized Lagrange interpolation smoothing Cj^(y) of data. For example, when using thin section data for spherical particles, the truncated zeroth moment ^

= / s(y)dy (10)

transfoms using the inversion formula (2) to the functional

m; = [2A]'^I^^-^ / f{[2^(y2 - yl)]^Vt} c(y)dy . (11)

Thus, using an appropriate form for Cj^(y) {see, for example, Jakeman and Anderssen (1975b)}, the estimation of M* reduces to the evaluation of

M* = [2/^^-^^^-^-^ / f{[2Tr(y^ - y^)]^} C[(y)dy (12) Xo using product integration, where C^(y) = d Cj^(y)/dy .

These stabilized estimators have been shown to be better than those previously proposed {see Anderssen and Jakeman (1975)}.

REFERENCES

R.S. Anderssen and A.J. Jakeman (1974) On computational stereology. Proa. 6th. Aust. Comp. Conf._, Sydney, Vol. 2, 353-362.

17
/?. S. Anderssen and A. J. Jakeman

R.S. Anderssen and A.J. Jakeman (1975) Product integration for functionals of

particle size distributions, Vtilitas Mathematica (in press).

R.S. Anderssen (1975) A stable procedure for the inversion of Abel's equation, JIMA (to appear)

.

G. Bach (1966) Zuffallssahnitte duroh Haufwerke von KoTpen, (Habilitationsschrift) , Braunschweig.

G. Bach (1967) Size distribution of particles derived from the size distribution

of their sections. In Stereology , Elias, H. (ed.). Springer Inc., New York, 174-

186.

P. Baudhuin and J. Berthet (1967) Electron microscopic examination of sub- cellular fractions II: Quantitative analysis of the mitochondrial population

isolated from rat liver, J. Cell. Biol. 35, 631-648.

A.J. Jakeman and R.S. Anderssen (1974) A note on numerical methods for the thin

section model. Electron Microscopy 1974, Vol. 2, 4-5.

A.J. Jakeman and R.S. Anderssen (1975a) On optimal forms for stereological data,

Fourth International Congress for Stereology, September 1975, National Bureau of Standards, Gaithersburg , Maryland.

A.J. Jakeman and R.S. Anderssen (1975b) Abel type integral equations in stereology I: General Discussion, J. Mioroso. (to appear).

N. Keiding, S. Tolver Jensen, and L. Ranek (1972) Maximum likelihood estimation

of the size distribution of liver cell nuclei from the observed distribution in

a plane section. Biometrics 28, 813-829.

M.M. Lavrentiev (1967) Save Improperly Posed Problems of Mathematical Physics,

Springer Tracts in Natural Philosophy, Vol. 11, Springer, Berlin.

W.L. Nicholson (1970) Estimation of linear properties of particle size distributions, Biometrika 57, 273-298.

R. Piessens and P. Verbaeten (1973) Numerical solution of the Abel integral

equation, BIT 13, 451-457.

18
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

ESTIMATION OF LINEAR FUNCTIONALS BY MAXIMUM LIKELIHOOD

by W. L. Nicholson

Pacific Northwest Laboratories, A Division of Battelle Memorial Institute, Richland, Washington 99352,

U.S.A.

1. INTRODUCTION

A population of spherical particles, with cumulative size distribution

GCx) , is randomly distributed in an infinite three-dimensional matrix. The position of particle centers is defined by a Poisson field. is the volume

number density of sphere centers (centers per unit volume). Thus, N^GCx) is the volume number density contribution from spheres of diameter ^ x. y is the mean sphere diameter, assumed to be finite. This model is an idealization for

physical situations where the sphere density is low (so there is small chance

for crowding) and spatially homogeneous. A classical problem of stereology is the estimation of linear functionals

of the form

oo /

e = Ny / Jl(x)G(dx) (1)

from linear or planar probe data. Nicholson [5] describes an estimation theory

for a more general situation of geometrically similar particles distributed in

a finite matrix. The theory is a mathematical statement of an entire class of stereology problems, treated more or less independently in the literature. Re- sults are extended to the Poisson field infinite matrix in [6]. For simplicity

we consider here spheres and planar probes. More general treatment is possible,

but mathematically tedious.

A planar probe of finite area A is inserted into the matrix to obtain a sample of n circular intersections. The measured diameters of these intersections y-j < y2 ••• y 3re the event points for a realization of a generalized

Poisson process {n(y)IO^y}. The process is defined by: i) n(0) = 0 with probability one; ii) non-overlapping increments are independent; iii) and for

0 ^ y' < y" the increment n(y") - n(y') is Poisson distributed with mean

E{n(y") - n(y')} = N^[H(y") - H(y')]. (2)

In (2) N^ = yNy is the area number density (circle centers per unit area) on the probe and

H(y) = 1 - / (xV)^/2G(dx). (3)

The contribution to area number density from circles of diameter = y is N^H(y). The measured diameters are an ordered random sample with cumulative distribution function H(y). The sample size is n = n{+^) . Differentiation of (3) gives the familiar Abel's integral equation relationship between the circle

frequency function h(y) = H'(y) and the sphere frequency function g(x) = G'(x)

first derived by Wicksell [9].

Research sponsored by U.S. Energy Research and Development Administration

under contract E(45-l ) :1830.

19
W. L, Nicholson

From [5] and [6] an unbiased estimate of the functional (1) is

e = A"^ / mCy)nCdy), (4) i

where m(y) is a solution to the integral equation

X[(x^-y^)"^/^ym(y)ydy = ^Cx). C5)

•o Uniqueness (non-existence of the solution) implies that (4) is the only (there

is no) linear estimate which is unbiased over the class of all discrete and absolutely continuous G(x). Watson [8] points out that in many applications

the variance of (4) is infinite, even though the estimator is consistent. The estimator of p(xq) = Ny[l-G(xg)] is such a case. Here the solution to (1) is unique. The estimator is p(xj = (2/A7T) z (y?-^o)"^^^- (^6^ ^i^^O

With Xg = 0, (6) is Pullman's formula [4] for estimating N^.

Anderssen and Jakeman [1] consider a family of product integration for- mulas which are smoothed versions of (4). Their simplest formula, a trapezoidal

rule, is

^i+l m(y)dy. (7)

^i A factor 1/n is replaced by 1/A in (7) since Anderssen and Jakeman estimate

functional s of G(x) not NyG(x). With appropriate regularity conditions, An- derssen and Jakeman show that their estimators have finite variance and are con- sistent. Tall is [7] employs a special case of product integration formulas to grouped data to get finite variance estimates of and l-G(x).

An alternative form of smoothing the estimate (4) is to construct a smooth

estimate N^H(y) from the sample function n(y). The estimate of (1) is (4) with

AN^H(dy) substituted for n(dy). One method of construction is to use the probability structure of the data and the theory of maximum likelihood estimation.

Section 2 is an outline of the method. Section 3 is a discussion of relative

merits of the several approaches. Section 4 is an example of maximum likelihood applied to estimation of Ny[l-G(x)]. Data are collected at several magnifications, a complicating feature for other approaches, but not for maximum

likelihood.

2. MAXIMUM LIKELIHOOD ESTIMATION Consider k fixed cells with boundaries 0 = Zq < z^ < ... < Z|^. The number of circle diameters in the j^*^ cell is An^ = "(zj) - "(Zj,-])- Because of the generalized Poisson process character of the data, An-j , An2, ... , An|^ are mutually independent Poisson random variables with means, respectively,

Zj

EAnj = AN^ JH(dy) = AN^h(Zj)AZj. (8)

20
Estimation of linear functionals by maximum likelihood

In (8) z- is a specific point satisfying z. < z. < z- and Az- = z. - z.

J J ' J J J J J The observed frequencies in the form An^/AAZj (number of circles per unit area

per diameter unit) are estimates of the area! density function, say, f^.) = N^h(^.). (9)

Suppose that f(z) = f{z\<\>) is known up to a p-dimensional parameter (|) = ((jj-j

,

<i>2> ••• ' 'J'p)- Then maximum likelihood theory provides a method for estimating

(J). The likelihood function for the observed Poisson frequencies is

k* -B.($) An.

L(n|((,) = n e

J [B. (<{.)] ^/An.!- (10)

j=l ^

^

In (10) Bj(({)) = AAZj f(Zj|(})). Because z^ is unknown, we approximate by the

cell midpoint zj = (z^ + z^_^)/2. The maximum likelihood estimate of (J) is that value if, which maximizes (10). In practice ^ is taken as a solution to the likelihood equations

3log L(n|())) N An.-B.((f>) 9B.((t)) = —i— = ^ = ° ^''^

1 = 1 1"^'

with a = 1 , 2, ... , p. A maximum likelihood smoothed estimate of the linear functional (1) is

O^U) = J

ni(y)f(y|$)dy. (12)

0 With appropriate regularity conditions on f(z|(j)) (see e.g., [10]) ^ is asymp- totically distributed as multivariate normal with mean <!p^, the true value of (f), and covariance matrix {AN.C (6^,)}'^ where

A pq'^0

Cpq(<t) = /•••/ Sp(n|<l,)Sq(n|<f)L(nU)dAn. (13)

0 0 Further,

X^*o) = -2 logg[L(n|*Q)/L(n|^)] (14)

is asymptotically distributed as chi -square with p degrees of freedom. In these limiting results AN^ plays the role of the large sample size. That is, the distributional results are approximately true for fixed and A sufficiently large. Let T((i)|p,a) be the set of ({> values which satisfy the inequality,

X (*) X (P>ct). Here, x (P>a) is the 100 a perc^entile for the chi-square

distribution with p degrees of freedom. Let 9^ and be the minimum and maximum value of (14) over the set T((t)|p,a). Then (e^, is a 1006% confidence

interval for the linear functional 9 of (1) where a = 6 = 1 whenever A is sufficiently large.

3. DISCUSSION

The present status of linear functional estimation in stereology is that

many classical procedures exist (that is, those that reappear periodically in the many application oriented journals of the diverse specific fields where

stereological methods are used) which are in the linear category of (4). In

21
W. L, Nicholson

actual practice the properties of these estimators are not as bad as mathematical theory predicts, though admittedly poor in some instances. Natural truncation of small features by observational thresholds removes infinite variance

stigma from reciprocal type estimators, like Pullman's density formula. Vari- ances are known specifically and can be estimated from the data with the Poisson assumption. But, now there is a negative bias because the contribution

from small features is omitted.

Grouping of data into cells by observational instruments and use of a single cell point as representative gives finite known variance, again at the expense of bias. With many observations so cells can be narrow, the bias is probably tolerable but, without investigation, specifically not known. The Anderssen and Jakeman formulas are an attractive alternative which in many situations give more stable estimates. These formulas are non-linear in the y's and hence, probabilistically speaking, complicated. Exact variances,

and- hence, variance estimates are not yet available. Maximum likelihood

includes a theory for asymptotic variance estimation, which is a plus. However,

the dependence on a family of smooth frequency functions f(z|(j)) to represent

the reality of area! density is a minus. In practical situations the classical

distributions don't seem to apply. Even lognormal, when appropriate for actual particle size, is not correct for observed intersections. Often competing

evolutionary processes give rise to compound multi-modal distributions. Work is in progress on low order spline functions which may offer a general solution.

A tremendous plus for maximum likelihood is its proper weighting of information in the cell frequencies. Other methods lack this sophistication.

Weighting is critical when frequencies range from zero to several powers of

ten, for example, when data are a composite of observotions from several microscopes using distinct magnifications. Composite observation is necessary

when feature sizes range over several orders of magnitude. In an overlap

region there are a few large features on the right tail of higher magnification

data and many small features on the left tail of lower magnification data. With

insufficient experimental planning there may be voids in the middle of the

data. Large right tail features completely disappear while left tail features

are still too small to be observed. Another complicating aspect of composite

observation is that the observation area for high magnification is often a sub area for that of low magnification. Maximum likelihood can be made to handle

all of these situations if a family f{z|(|)) is available.

4. AN EXAMPLE

In connection with an experimental characterization of fuel [3],

planar probe data were collected on near-spherical pores distributed in as- fabricated fuel pellets using four distinct magnifications. Higher magnification (21,000X and 6440X) fields were observed with a scanning electron microscope over a size range of 0.05 to 5.0 micrometers. Lower magnification {850X

and 61X) fields were observed with an optical microscope over a size range of

1.0 to 100.0 micrometers. The features contained on 76 micrographs were measured using a Zeiss Particle Size Analyzer. For simplicity of presentation,

data from the Zeiss 48 exponential scale cells were combined into 24 cells. Pore frequency vs. size data are given in Table 1 as a function of magnification. Arrows in the first three columns indicate the beginning points for

the next lower magnification data. Figure 1 is a plot of the area density

estimates (An/AAz) of Section 2 versus cell midpoint z'. These data are used to illustrate maximum likelihood estimation of Ny[l-G(3.0)], the contribution to the density from pores of diameter > 3.0 micrometers. To obtain this estimate, it is sufficient to model the right

tail of the size distribution, say starting at 1.0 micrometer. For that re- gion, a quadratic relationship between log density and log diameter seems appropriate from Figure 1. Thus, the parametric family of Section 2 was taken as

f(z|(t.) = az^^"^^"^ 1.0<z< 100.0 (15)

22
Estimation of linear functionals by maximum likelihood

where ^ = (a,B,Y) is the unknown parameter and z is intersection diameter.

TABLE 1. PORE FREQUENCY VS SIZE DATA

MICROSCOPE

SEM OM OM

MAGNIFICATION

21000X 6440X 850X 6DC CELL MICROGRAPH AREA MEASURED mmxmm) mMl nDDI 1^ 1 u ru 1 iiM (mm) 1008640 5W320 501320 378240 L18 2 360 1415 232

1.37 4 384 701 50

1.57 11 499 481 41

1.81 23 599 334 24 Z.08 38 648 194 17

2.41 64 652 151 6 2.77 58 633 78 5 3.Z0 81 510 63 0

3.63 403 38 2 4.18 90 320 26 0 4.77 122 209 13 0

5.43 131 132 7 0

6l19 138 69 6 0

7.05 151 37 4 1 &(B 151 18 1 0

9.15 125 15 1 0 ia42 100 13 I0

11.87 92 4^- 0

13.53 56 0 0

15.41 39 3 0 0

17.56 22 1 0 0

20.00 9 0 0 0

22.39 4 1 0 0

25.96 2 0 0 0 * AS MEASURED ON MICROGRAPH USING

ZEISS PARTICLE SIZE ANALYZER

The products in the likelihood

function (10) and the likelihood equations (11) involve a double indexing

to handle the composite nature of the

data. Let An,, (i=l ,2,3, . . . ,24; j=l

,

2,3,4) be the frequency in the i cell for the j^^ magnification. The products are taken over all index

pairs (i,j) with cell midpoints z' satisfying 1.0 = z^. ^ = 100.0.

,th

Also,

1.0 10

INTERSECTION DIAMETER - z' (ul

FIGURE L AREAL DENSITY OF PORES ON PLANAR SECTIONS OF

B^.j(*) = Aj.Az.j f(zlj|<i)) where = d./M.,

Here, A. is the total area observed for the j**^ magnification. M. is the j^*^ th magnification factor, and d^j is the upper boundary of the i Zeiss cell.

The maximum likelihood equations were solved with LEARN program [2]. LEARN

is a general purpose non-linear estimation program which uses second degree

Taylor expansion logic. Initial parameter guesstimates came from least squares regression of the logarithm of non-zero areal density data on the quadratic

polynomial in log-diameter. The specific maximum likelihood estimate of (}) is

23
W. L. Nicholson

a = 27930.0 ± 3060.0

e = -3.43 ± 0.14

Y

= -0.147 ± 0.037. The precisions are standard deviation estimates from asymptotic theory. The smoothed function f(z|(j)) is plotted as the curve in Figure 1. Substitution of f(z|(t)) into (12) gives the maximum likelihood smoothed es- timate of the density contribution from pores of diameter > 3.0 micrometers as

with Xg = 3.0. Since the data end at x-j = 100.0 micrometers, the estimate is truncated at that point. With the contribution near x^ down a factor of 10^ from that near Xq, and since the observed areal density has a negative slope near x^ the density estimate is clearly not dependent on any detail at or be- yond x^

.

The final estimate is

The approximate 95% confidence interval from likelihood contours and asymptotic theory is given in parenthesis.

5. ACKNOWLEDGMENTS

The author is indebted to J. L. Daniel and H. A. Treibs for discussion

which helped formulate the mathematical model for microscopy data. B. H. Duane obtained the solution to the maximum likelihood equations.

[1] Anderssen, R. S. and A. J. Jakeman, "Product Integration for Functional

s

of Particle Size Distributions", to be published in Utilitas Mathematica . [2] Duane, B. H., Maximum Likelihood Non-linear Correlated Fields (BNW Pro- gram LIKELY ), September, 1967, BNWL-390, 46-49.

[3] Freshley, M. C, P. E. Hart, J. L. Daniel, D. W. Brite, T. D. Chikalla,

Proceedings of Joint Topical Meeting on Commercial Nuclear Fuel Technology

Today, Toronto, Canada, April 28-30, 1975, CNS ISNN 0068-8517, 2-106—2-126.

[4] Fullman, R. L., (1953), J. Metals , 5, 447-52.

[5] Nicholson, W. L., (1970), Biometrika , 57, 273-297.

[6] Nicholson, W. L. , (1970), "Sampling and Estimation Theory for Thinly Distributed Particulate Materials", presented at the Symposium on Sampling

Theory in Characterization of Bulk Materials Under Japan-U.S. Cooperative

Science Program, Honolulu, Hawaii, October, 1970.

[7] Tallis, G. M., (1970), Biometrics , 26, 87-103.

[8] Watson, G. S., (1971), Biometrika . 58, 483-490.

[9] Wicksell, S. D., (1925), Biometrika , 17, 84-99.

[10] Wilks, S. S., (1962), Mathematical Statistics , John Wiley and Sons, Inc., New York.

(16)

Ny[l-G(3.0)] = 1.46x10° (1.35x10° - 1.58x10°) pores/cc.

REFERENCES

24
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

/Je/rf at NBS, Gaithersburg, Md.. September 4-9. 1975 (Issued January 1976)

THE GEOMETRY AND STEREOLOGY OF 'CUBO-SPHERICAL' PARTICLES

by R. Warren Department of Engineering Metals, Chalmers University of Technology, Gothenburg, Sweden

SUMMARYA class of three-dimensional shapes is defined, consisting of rounded cubes

and orthogonally-faceted spheres. Two linear parameters, namely the radius of the spherical surfaces and the distance between parallel faces, are sufficient

to define the shape and size of the bodies in this class, while the shape alone

is defined by their ratio. The surface areas and volumes, and consequently the mean intercept lengths, of the bodies have been derived as a function of the parameters. Conditions and possible methods of measurement are presented for the determination of the shape, size and nxjmber per unit volume of particles in the shape class when observed on two-dimensional sections. Some implications and applications of the analysis are discussed briefly.

1. INTRODUCTION A particle shape often observed in metallic and other crystalline systems

is the rounded cube. In fact, there exists a continuous class of shapes, ranging

from the sharp-edged cube to the sphere. These can be described as orthogonally

faceted spheres and as rounded cubes, formed by the intersection of a cube and

a sphere having a common centre. Two representative members of this shape class

are shown in Figures 1 and 2. The shape and size of any body in the class is defined uniquely by two parameters, e.g. the half distance between two parallel

flat faces, a, and the radius of the spherical surfaces, r. The shape alone can

be defined by a single parameter, e.g. 3, the ratio a/r.

It is convenient to divide the class into two distinct groups, namely (i) faceted spheres, for which 1 > B > l//2~; and (ii) rounded cubes for which

l/VZ > 3 > 1/ 3. The boundary between the two groups occurs at B = 1//2, i.e. when the circular flat surfaces on adjacent sides just touch.

Fig. 1. Orthogonally faceted Fig. 2. Rounded cube sphere (group 1) (group 2)

2. GEOMETRIC ANALYSIS OF THE SHAPES Facetted spheres (1 > B > 1//2) For shapes in this group, the surface area of the flat surfaces is given by

Sp = 6Tr (r^ - ah (2.1) The total surface area of the body is found simply by replacing the spherical

surface area of the six spherical caps by S„ giving:

r S = 27r(6cr - - 3c?^) (2.2) A similar approach allows derivation of the volume of the body:

25
/?. Warren

3 3 4r - 3e )/3 (2.3) A general equation for convex bodies (Underwood, 1970) gives the mean intercept

length of the body:

= kV/S = 4(9cr kr^ - 3e^)/3(6cp - - 3c^) (2.4)

Since for a given shape.

4r(96 - 33^ - 4)/3(63 - 33^ 1) (2.5)

Values of 5^ 7 and L3 for different shapes, i.e. different values of S, are included in Table 1.

Rounded Cubes (l//2"> 6 > ll/T)

The geometry of this group is somewhat more complicated. The area of the flat surfaces is:

6TT(r^ - o^) - 2MA{r^ 2, , 2 c ) - a{,r

where

:

A = sm L (2^ „ 2-1/2,. 2 2a/2i

(2.6)

Derivation of the total surface area involves double integration (Warren, 1972a) and is given by: S + 4lT2' + llvov

r 24S

where:

B = tan '^[o'^ lr{v^ - a^)^^^] + 2cr tan ^[(r^ - 2o^)/a^]^^^ (2.7)

Derivation of 7 and for this group is omitted here. It will be seen that the omission does not seriously affect the subsequent analysis.

Table 1. Geometric Properties of Cubo-Spherical Particles.

Particle

shape

B = a/r

(sphere) 0.9

0.8

Surface area

S

(for ^=1)

4lT

3.94ir

3.76TT 0.707 3.4841T (i.e. l/ZT) 9.577 2.5467r (i.e. l/ZT) (i.e. 8) (cube)

Volume V

(for r=l)

1.3331T

1.275Tr 1.109TT

0.8687T

0.490Tr

Mean intercept length

£3

(general)

1.333r

1.295r

l.lSOr

0.997P

0.770r

Surface area

of unit

volume

particle

4.835

4.908

5.140

5.604

for unit

volume

particle

0.827

0.815 0.778 0.713

0.667

(i.e. 2/3)

0

0.29

0.59 0.86

3. MEASUREMENTS ON SECTIONS Combination of equations 2.1 with 2.2 and 2.6 with 2.7 shows that, for a given shape in the class, the fraction S^/S is a function solely of 3 i.e. it

is independent of size. The fraction is plotted as a function of 3 for the whole

shape range in Figure 3. Thus, for an assembly of randomly-oriented particles

of a single shape, 3 can be determined from a two-dimensional section by measurement of S-p/S. This involves counting the niimber of intersections made

by test lines with the curved and flat surfaces of the particle sections

(Warren, 1972b).

If the particles are of uniform size and shape, measurement of the mean

intercept length will, together with 3, allow determination of the partic»le

size using, for example, equation 2.5. Similarly, application of general

stereological equations for convex particles allows the determination of other

26
The geometry and stereology of cubo-spherical particles

properties of the dispersed system. For example, the number of particles per

unit volume, N-^ = U Nj^/S (Underwood, 1970) and so for the present class of shapes N can be determined by measurement of N , 3 and L .

T I

1

> 1 1 1 1——I r

FRACTION OF FLAT SURFACE, Sp/S

Fig. 3. Relationship between S„/S and a/r for cubo-spherical particles

4. SOME IMPLICATIONS AND APPLICATIONS

The dependence of mean intercept length on shape

Figure 4 shows the dependence of mean intercept length on shape for particles of fixed volume. This clearly has implications for studies of particle

size in systems in which a shape change occurs. For example, in particle growth

stud^ies, account must be taken of shape changes if growth is measured in terms

of L., particularly at low growth rates. Similarly, in studies of the pressing

or sintering of spherical powders, the development of interparticular contact

areas will lead to an apparent reduction in particle size if measurement is made in terms of L- and shape is not accounted for.

O

z

LU

Q.

LUO

q:LU

<

UJ

li = c/r

Fig. 4. The dependence of mean intercept length on the shape of cubo-spherical particles of unit volume.

27
R. Warren

Anisotropy of swcfaoe energy

In certain circumstances crystalline particles develop an equilibrium

cubic or cubo-spherical shape because the surface or interfacial energy of their (100) planes is lower than that on the other crystallographic planes. The equilibrium shape is given by the Wulff construction, in which the distance

from the centre of the particle to a given plane on the surface is proportional

to the energy of that plane (Winterbottom, 1967). For 'cubo-spherical' particles

therefore, the surface energy ratio, ^loo/^hkl' given by civ. Thus in a system of particles, all having the equilibrium shape, the energy anisotropy

can be determined by determination of 3 (e.g. by measurement of S_/S).

r REFERENCES

Underwood, E.E. (1970) Quantitative Stereology , Chapter 4. Addison-Wesley

,

Reading, Mass. Warren, R. (1972 a) The Misrostruoture of Cemented Carbides Produced by LiquidPhase Sintering (Ph.D. Thesis) University of Surrey, Guildford.

Warren, R. (1972 b) Carbide Grain Shape in Cemented Carbide Alloys of Cubic

Refractory Carbides. J. Inst. Metals, 100 , 176. Winterbottom, W.L. (1967) Crystallographic Anisotropy in the Surface Energy

of Solids. In: Surfaces and Interfaces (Ed. by J.J. Burke, N.L. Reed, and V. Weiss), p. 33. Syracuse University Press, New York.

28
National Bureau of Standards Special Publication 437

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaittiersburg, Md., September 4-9, 1975 (Issued January 1976)

QUANTITATIVE MICROSCOPY OF LINEAL FEATURES IN THREE DIMENSIONS

by R. T. DeHoff and S. M. Gehl, Department of Materials Science and Engineering. University of Florida, Gainesville, Florida 32611, U.S.A.

SUMMARY

Mi crostructural features that may be characterized as lines

in three dimensional space possess, at every point along their

length, two properties: curvature ^ which is the rate of change of

direction of the tangent to the curve, and tors i on , which is the

rate of change of direction of the binormal to the curve. If, in

addition, the lineal feature is an edge, i.e., is defined by the

meeting of tv/o surfaces, the feature also possess a third

characteristic; the d i hedral angle , defined to be the angle

between the surface normals at the edge.

A set of three new fundamental relations In stereology are presented which permit the estimation of the total and average

curvature and torsion of a set of lineal features, and the average

dihedral angle of a set of edges in a microstructure.

INTRODUCTION

Lineal features are common in the three dimensional

structures that are explored in microscopy. In materials science,

the archtype of lineal features is the dislocation; a multitude of

these one dimensional, tortuously curved crystal imperfections

pervade the three dimensional lattice of any crystalline solid.

More generally across the spectrum of mi crostructural analysis,

any structure composed of contiguous cells contains cell edges of

a variety of classes; these triple lines form partially or wholly

connected networks that frame interfaces in the structure.

Finally, there is the class of features that are actually three

dimensional, but have one dimension that is very much larger than

the other two, so that the approximation that the feature is one dimensional is valid for some purposes. Capillaries, nerve

fibres, or slag stringers fall into this class of essentially one dimensional features in three dimensional space.

Methods for characterizing the extent of lineal features were

first proposed by Saltykov (1958) and Smith and Guttman (1953).

The description of the anisotropy of such an array was elegantly

developed by Milliard (1962). These results are now part of

standard texts in stereology (DeHoff and Rhines, 1958; Underwood,

1970; and Weibel and Elias, 1967). In addition to their length,

lineal features also possess other unambiguously definable

geometric properties, associated with their local curvature and

torsion. Further, If the lineal feature is an edge, it has an additional property that derives from the implicit fact that an edge Is defined by two surfaces that meet in space. This property

is the dihedral angle, defined as the angle between the normals to the mating surfaces at the edge. This paper explores the

relations between these additional properties of space curves and

observations that may be made In two dimensions. The result is three new s tereol og i ca 1 relations, presented here for the first

t i me

.

THE GEOMETRY OF SPACE CURVES

The concept of curvature at a point P on a plane curve is usually defined by establishing two other points on the curve. Figure la, and constructing a circle through the three points

29
R. J. DeHoff and S. M. Gehl

(a) (b)

Figure 1. Alternate constructions defining the

local curvature on a plane curve.

(Schwartz, 1967). The outlying points are allowed to approach P, and in the limit, the circle approaches the osculating circle.

Its center is the center of curvature at P; its radius is the

radius of curvature of the curve at P, and the reciprocal of the

radius is called the curvature of the curve at the point P. Alternatively, the curvature may be formulated by considering

an element of arc, dX, Figure lb. The ci rcular i mage of that

element is defined as the angle, d0, swept by the normals to dX.

The angle swept out by the tangent as it moves along dX is also

de. Simple geometric considerations yield dX = r d0, or (Schwartz, 1967)

k = 1/r = d0/dX (1)

This construction of the concept of curvature may be extended

to an element of arc of a space curve. Figure 2a (Struik, 1950).

As the points A and B approach P, succeeding constructions of the

circle through A, P, and B do not generally lie in the same plane.

Nonetheless, there exists a limiting oscu 1 a t i ng c i rc 1

e

that

defines the center and radius of curvature of the space curve at

P. If dQ is considered to be the rotation of the tangent vector

to the space curve as an element, dX, is traversed, then equation

(1) holds for space curves.

The trajectory of a space curve has an additional degree of

freedom, not associated with plane curves, that is described as the rotation of the oscu 1 at i ng p 1 ane as P moves along the curve. Figure 2b. Locally, this rotation may be described as the angle

dij/ traced out by the normal to the osculating plane as the element

dX is traversed. A local property, analogous to the curvature at

P may be defined:

T E dii)/dX (2)

where t is called the tors i on of the curve at P (Struik, 1950 ). The torsion may be thought of as a measure of the out-of -pi aneness

of the curve.

It is convenient to define a local reference system at P, Figure 2c: t is the tangent vector; n, the vector pointing to the

30
Lineal features in three dimensions

center of curvature is called the norma 1 vector at P; and E, the

vector that is perpendicular to both t and n, and hence to the

osculating plane, is called the b i norma 1 . The two vectors, n and

B, taken together define a plane that is perpendicular to the

tangent vector, called the norma 1 plane . Rotations of this

coordinate system as the curve is traversed define its local

curvature and torsion.

THE GEOMETRY OF EDGES ON SURFACES

The intersection of two curved surfaces. Figure 3, defines a space curve, usually called an edge on the figure defined by the

surfaces. (In cell structures, such edges are usually defined by

the incidence of three cells and the intersection of three

surfaces; in such cases, one must consider that three separate,

though not independent, edges are thus defined.) In addition to curvature and torsion, such edges possess, at each point, a di hedral ang 1 e, X' Figure 5a. Note that the dihedral angle is defined as the angle between the surface normals at the edge, and

not the angle between the tangent planes, which is ambiguously

defined. Edges thus have some of the aspects of surfaces, and

some of space curves. For example, an edge may be convex,

concave, or have a saddle configuration. At the same time, an edge possesses local curvature and torsion.

THE STEREOLOG I CAL RELATIONS

The Length of L i nea 1 Features . When a three dimensional

structure containing lineal features is sectioned with a test plane, the features appear as points on the microsection. If the

number of points is counted, and the result divided by the total area scanned, the a rea po i nt count

,

Py^, is obtained. This

Figure 2. Construction illustrating local curvature (a)

torsion (b) and the reference trahedron (c) for a segment of a space curve.

31
/?. T. DeHoff and S. M. Gehl

quantity has been shown to be an unbiased estimator of the total

length of the lineal features thus measured, per unit volume of

mi crostructure (DeHoff and Rhines, 1968).

Pa = (3)

2 This established counting measurement will be used to normalize

the new s tereol og i ca 1 measurements presented below in order to obtain average values of curvature, torsion, and dihedral angle.

Curva ture of L i nea 1 Features . Focus upon an element of arc,

dA, of a collection of space curves. Figure k. Figure 2 shows the

normal plane at a point P on such an element. The spherical image

of this normal plane is obtained by centering P in a unit sphere

(the orientation sphere) and noting the intersection of the plane

with the sphere. The spherical image of the normal plane is thus

a great circle whose pole is the spherical image of the tangent

direction. As the point P moves along the arc, dX, in Figure ka,

the tangent rotates, as does the normal plane. This rotation

traces out a spherical image that (in the first order of

differentials) has the shape corresponding to the region between

two longitudinal lines on the globe. Figure ^a. The area of this

spherical image is simply related to the angle d9 of rotation

of the tangent, t:

dfip, = I* do (i»)

Let this structure be sampled by sweeping planes through its

volume, Vq. The event to be counted in this statistical

experiment is the number of tangents formed by these sweeping

planes and the element dX. This event occurs when the sweeping

32
Lineal features In three dimensions

(a) (b)

Figure k. Spherical images of (a) the normal plane, and

(b) the osculating plane for an element of a space

curve, dX.

plane contains the tangent direction, somewhere in the interval

dX. An equivalent condition is that the normal to the sweeping

plane must lie within the spherical image dfip.

If a large number, M, of orientations of sweeping test planes

is chosen at random from the uniform distribution of directions on the unit sphere, the fraction that form tangents with dX Is

df^p/itir. The number of tangents formed Is The total

volume sampled is MVq. The number of tangents formed with dX, per

unit volume of structure sampled is:

dTy = (Md:Jn/i»7T) • (I/MVq) = d^^/h-n^^ (5)

Apply equations (1) and (i*) to convert the spherical image to the

local curvature:

dTy = (1/i»ttVo) (itdQ) = de/7rVo = kdX/irVo (6)

This result holds for each element of arc in the set of space

curves. The number of tangents formed with all elements is the

integral over the length of these features:

Jy = (I/ttVq) /kdX (7)

The quantity /l<dX may be usefully defined to be the total curva ture of 1 i neal features in the structure. The average

curvature may be defined by dividing by the length of arc:

Ic 5 /kdX//dX = ir Ty/Lv (8)

This volume tangent count may be obtained by sampling the

structure with a series of closely spaced mi c rosec t i ons, and

noting appearances and disappearances of the lineal features;

these events correspond to tangencies of the feature with a plane

swept in a direction normal to the sectioning plane.

Alternatively, If the structure can be viewed in transmission, the

operation of sweeping a line across the transmitted Image

corresponds to sweeping a plane through the volume of the

structure. Tangencies with this sweeping plane will be observed

as tangencies with the sweeping line on the projected image. If

33
R. T. DeHoff and S. M. Cehl

^Aproj number of tangents formed per unit area of projected

image traversed, then

""Aproj = Tyt

and, from equation (7),

TAproj = (t/irVo) /kdX (9)

where t is the thickness of the thin section being viewed in transmission. Again applying equation (3), and noting that, on a transmitted image observed in projection, the P/^ count may be made

by a line intercept count, i.e., by counting the number of

intersections a test line makes with lineal features, and dividing

by the length of test line, one obtains (Underwood, 1972)

^Lproj = PAt = (J) Lyt (10)

Substituting these results into equation (8) gives an expression

for the average curvature of lineal features in terms of counting

measurements that may be made upon a projected image:

^ = (^^T^proj/t'/^^NLp.^j/t) = f Vroj/\proJ ^^^^ Tors i on of L i nea 1 Features . Just as the curvature may be

expressed in terms of the rotation of the tangent vector or the

normal plane, the torsion may be visualized as the rotation of the

binormal vector, or the osculating plane. The spherical image of

the osculating plane is a great circle whose pole is the binormal.

As the element dX is traversed. Figure itb, the binormal rotates,

and the spherical image of the osculating plane traces out a double-lune shape similar to that obtained for the normal plane in Figure ha. The area of this spherical image dfig is simply related

to the angle of rotation of the binormal, dtj;, and thence to the

local torsion at dA:

d^Q = h6^ (12)

Imagine that the array of lineal features is viewed in projection. Let the projection direction, i.e., the orientation

from which the structure is viewed, be randomly chosen from the

uniform distribution of orientations on the unit sphere.* Focus

upon the element dX as the structure is viewed. The event to be

counted is the number of times that the projected image of

dX contains an i nf 1 ec t i on po i nt . This event will occur for a given projection direction if the orientation of the point of view

of the observer 1 i es 1 n the oscu 1 a t i ng plane . Thus, those

projection directions that lie within the spherical image of the

osculating plane, d^Q/ will produce inflection points at dX on the

projection plane.

Suppose the structure is viewed from a large number, M, of

orientations. Each view samples the entire volume of the

structure. The fraction of these observations of dX that show an inflection point is dS2_/inT. The number of inflection points thus

*ln practice it will not be practical to make such a sequence of

observations. Usually only one direction of projection is available. Nonetheless, the thought experiment required to obtain

the fundamental relationship requires averaging the observations

over the sphere of orientation. This procedure is inherent in essentially all of the basic s tereolog i ca 1 relationships; the

present case is not an exception.

34
Lineal features in three dimensions

observed is MdnQ/i*TT. The total volume sampled is MVq. The number

of inflection points counted per unit volume of structure sampled

i s :

div = (Md:2o/'»Tr)<l/MVo) = d'^Jkti'^Q (13)

Evaluate the spherical image in terms of the local torsion of

dA by applying equations (2) and (12)

dly = Udii)/i»TrVQ = TdX/TrVg (li*)

This result is valid for each element of arc in the structure.

Thus, for the structure as a whole, the number of inflection

points observed in a projected image, per unit volume of structure

sampled in the projected image is:

ly = (l/TrVg) /TdX (15)

where the integration is carried out over all of the length of

space curves in the structure. The average torsion of the set of curves may be defined by dividing by the total length of lineal

features in the structure:

T = /TdX//dX = TT ly/Ly (16)

In practice, the inflection point count is performed over an area. A, of a projected image of the structure. The structure

itself is a thin film or section of thickness t. Let lAproj be

the number of inflection points counted per unit area of projected

image. Then

'Aproj = 'v t

and, from equation (15),

'Aproj = (t/irVo) /xdX (17)

The average torsion of the set of space curves becomes

^ " ' Aproj/'^ ^/(2NLproj /t) = 7 'Aproj/Nlproj ^18)

Thus, the average torsion of a collection of space curves in a transparent structure may be obtained from two counting

measurements made upon a representative projected image. If the

structure is opaque, estimation of the torsion requires serial

sectioning, with the counting of projected inflection points as the sectioning planes traverse the three dimensional structure.

Tota 1 Curvature and the D i hedra 1 Ang 1 e at Edges . The total

curvature of a collection of smoothly curves surfaces is defined

by (Cahn, 1967; DeHoff, 1967)

M = //Hds (19)

where H Is the local mean curvature of the surface,

H = 1/2(ki + (20)

<1 and K2 being the local principal normal curvatures (Struik,

1950). One of the established relationships of stereology permits

estimation of total curvature per unit volume of surfaces. My, by means of the area tangent count, T/^, applied to a representative

35
/?. 7". DeHoff and S. M. Gehl

section through the structure (DeHoff, 1967).*

^Anet = '^V/^ <21)

In its original formulation, this rel a t i onsh i p was limited in application to smooth surfaces, i.e., to surfaces possessing no edges or vertices. However, in a note added in proof to his

paper, Cahn (1967) suggested that the notion of total curvature

could be extended to include convex polyhedral surfaces.** The

present development accomplishes this extension, and in addition

removes the limitation that the surfaces must be convex. Thus,

the result derived in this section applies to surfaces with edges

of arbitrary configuration, excluding as usual the collection of

mathematical curiosities classed as topol og i ca 1 1 y "wild" surfaces.

The derivation mal<es use of the concept of density of

features developed in integral geometry, and presented by Santalo

( 1967) and others (Blaschl<e, 1955 ). Figure 3a shows an edge

defined by two surfaces, together with the dihedral angle, x» The

structure containing this element of edge is Imagined to be

sectioned with a random sample of test planes drawn from the

uniform distribution of planes in space. Within each plane, the

sectioned structure is sampled by selecting randomly from the set

of sweeping test lines that traverse the structure. The event of

interest is the formation of a tangent between such sweeping test

lines and the vertex which is the emergence of the edge on the

sectioning plane. This event requires the simultaneous occurrence

of two independent events: namely that the plane intersects the

element of edge length, dX, and that the direction of sweep of the

test line in the plane is contained between the normals to the

meeting traces. Figure 3b.

The density of planes in space is given by dp s i n(j) d0 6^,

(Santalo, 1967) where dp is an element of length along a direction

specified by the usual spherical coordinates of an orientation (0,

<() ) . The density of sweeping test lines in a plane is equal to the

density of orientations in two dimensions, which is an angle, da.

Thus, the density of sweeping test lines in three dimensional

space is

dT = da dp s I n(|) d0 d(f) The measure of the set of sweeping test lines that form tangents

at dX is obtained by integrating this density over the set of

values of (a, p, 0 and 0) that are contained within the event.

The measure of the total area swept out by the set of sweeping

lines as the Volume of the structure, Vq, is sampled is //// Adp

da sin<f> d0 dcfi where A is the area of the local sectioning plane.

This tangent count is similar to that applied in equations (9)

and (11); it differs in that it is determined upon a section

through the structure observed in reflection, as opposed to a projected image obtained in transmission. The tangents counted

are those observed to form between a sweeping test line and the

traces of the surface whose total curvature is being measured.

**lt is not clear whether Cahn limited his result to convex polyhedra because he was interested primarily in the relation of

total curvature to the caliper diameter of the particle

(Minkowski's formula (Minkowski, 1903)) or because he had not

explored nonconvex cases. He merely states the equation, without

proof or reference.

36
Lineal features in three dimensions

Thus the number of tangents formed with all edge elements in the

i s

:

volume Vq, normalized to unit area of structure swept over,

Ta = //// dadpsin(!)d0d<f)/////Adpdasin({>d0d(f> (22)

(Lines form tangents)

Since for any orientation / Adp is the volume of the sample, Vq,

the denominator integrates to Stt^Vq. For any given orientation,

(0, _ 0), the domain over which the plane intersects the edge dX is

dX'N where N is the unit vector normal to the plane. For the

convention shown in Figure 3a,

dp = dX COS0 sin(J) (23)

For a fixed plane that intersects dX, the domain of a over which

tangents are formed is the dihedral angle in the plane at the

edge, a(9, (j), x^/ Figure 3b. Thus, the net number of tangents per

unit area of test plane traversed is

Ta = ///a COS0 sin2(j) d0 d<^ dX/Sir^M. (2k)

The value of the dihedral angle observed on the sectioning plane,

a, depends upon the dihedral angle x three dimensions at the

edge segment and the orientation of the sectioning plane specified

by (0, 4>). Numerical integration of the function //a(0, <)), X)cos0 sin^^ d0 d<() for a sequence of values of x ^rom 0 to tt gave

the resul t

:

//a(0, (f, x)cos0 sin2({i d0 d(j) = kirx Inserting equation (25) into (2h) yields:

T^ = UTr/xdX/87r2Vo = ( 1/tt ) ( 1/2 ) /xdX

(25

(26)

Comparison with equation (21) suggests that the quantity

(l/2)/xdX is analogous to the concept of total curvature of

Test

L i ne lOum

Figure 5, Projected image of a structure containing lineal

features. Tangents formed with sweeping test lines

are indicated by ^ ; inflection points by

37
R. T. DeHoff and S. M. Gehl

surfaces in the volume and may reasonably be defined as the total

curvature of edges as suggested by Cahn (1967):

f'^edges = 2^XdX (27)

where the integration is carried out over the length of the edges

in the structure. Then relation (26) becomes:

TAedges = Mg^ges/Vo = Mvedges/^^ (28)

where Mvedges the total curvature of edges per unit volume of

structure. Thus, the application of the area tangent count

separately to the edges that exist in a structure permits

estimation of the total curvature of edges in unit volume of

structure, where the contribution of edges to the total curvature

is defined in terms of the dihedral angle at the edge according to equation (27).

The average dihedral angle at edges may be defined as:

X

H /xdX//dX

Substituting from equations (26) and (3) yields:

X

= (2t:Vo T^edges )/(2Vo PAedges^ = TAedges/PAedges ^29)

An area point count and a tangent count at edges together permit

the estimation of the mean dihedral angle at edges in a three

dimensional structure.*^ 5 Oum

SOpm

Figure 6. The minor phase (3) has edges (aaB); the dihedral

angle may be estimated by counting aaB triple

points and edge tangents with the test line

(A).

Sections through edges may be convex or concave; the

corresponding dihedral angle on the section may thus be positive

or negative. Tangents to convex corners are counted as positive,

and those at concave corners as negative. In the preceding

equations T^ must be interpreted to mean Ta(CONVEX) - Ta(CONCAVE),

just as is the case for smooth surfaces (DeHoff, 1967).

38
Lineal features in three dimensions

MECHANICS OF THE MEASUREMENTS

While it is possible to estimate torsion and curvature of

lineal features from serial sections through opaque structures,

these measurements are more likely to be made upon projected

images through transparent structures. Accordingly, the

illustrations presented in this section dealing with torsion and

curvature are based on analysis of projected images. Estimates of

the dihedral angle at edges requires that the observations be made

upon a section, as opposed to a projection.

Curvature . Figure 5 is a containing curved lineal features,

sampled are shown; the thickness,

number of tangents formed

structure is found to be

The area tangent count is area sv;ept out: T^j^p^oj = unit vol ume i s

:

projected image of a structure

The dimensions of the image

t, is known to be 10"^ cm. The

when a vertical line is swept across the

13, with each tangent indicated by a A • obtained by dividing this number by the

1.3xl07 (cm"2). The total curvature per

1/Vq /kdX = (TT/t)TAproj = '*.lxlol2 (l/cm^)

A line intercept (Nl) count applied to the structure yields 3200

counts per centimeter. From this, the total length of features

per unit volume is estimated, using equation (10):

Lv = 2NLproj/t = B.ijxlO'' cm/cm^

The average curvature of these lineal features

applying equation (11)

Is obtained by

(c = I*. lxl0^2/5_ i^xIqS = 61*00 (1/cm)

Tors i on . The lineal features in Figure 5 also possess

torsion. The total or integral torsion may be estimated by

applying the Inflection count to th& projected image. Each

inflection point is designated by a ; the number in the area

shown is 6. The number of inflections per unit area of Image Is:

1;^ = 6x10^ (cm"2). The total torsion per unit volume is obtained

by applying equation (17), and is found to be:

(l/Vo)/TdX = t: lAproj/t 1.88x10^2 (l/cm^)

The average torsion of these space curves may be determined from

equation (18):

T = 1. 88x10^2/6. UxlO^ = 2900 (1/cm)

Edge Curvature . Figure

in which the minor constituent

between cells of the major

structure contains triple lines

A or A • A vertical test producing tangents at aaB tripl

number of tangents formed

2.5x10"^ cm^. The area tangent

total curvature of B edges

from equation (28) to be:

shows a two phase structure (a+3)

(B) appears at the boundaries

constituent. As a result, the

of the type aaB, Indicated with a line Is swept across the structure

e points, indicated by A • The

Is 20. The area swept out Is count Is thus 8.0xl05 (cm~2). The

along aaB triple lines Is estimated

ges

= tt{8.0x105) = 2.5x10^ (cm/cm^)

The total length of triple lines of the type aaB per un may be estimated by counting aaB triple points (A+A) and

equation (3). This length Is estimated to be:

t vol ume

appl y I ng

39
/?. T. DeHoff and S. M. Gehl

Lv"^ = 2P^ = 2(30/2.5x10-5) = 2.itxl06 cm/cm^

The average dihedral angle along 3 edges of aa3 triple lines in

the three dimensional structure is thus estimated to be:

X

= 2Mygjgg3/Lv = 2.09rad = 120°

The interior angle is (180°-x) or 60°.

SUMMARY

The average curvature of a set of lineal features in a three

dimensional structure may be estimated by combining a tangent

count on a projected image of the structure (which estimates the

integral of the curvature over all arc length in the sample) with

the line intercept count on the projection (which estimates the

length of features):

1^ = f Vroj/^proj (11'

The average torsion of a set of lineal features may be

estimated from an inflection point count on a projected image:

^ = 1 'Aproj/f^Lproj (18>

The contribution of the edges bounding particles in a structure to the total curvature of the particles may be usefully

defined in terms of the dihedral angle x 3t the edge: W i^xdX (27)

The average dihedral angle at an edge may be estimated by counting

tangents formed by a sweeping test line with the emergence of

edges on the section, and combining the result with the

established area point count:

X = ^Aedges/f'Aedges ^29)

ACKNOWLEDGEMENTS

This paper was the result of interaction between

investigators from two projects; one was sponsored by the National

Science Foundation, the other by the United States Atomic Energy

Commission. Their support is gratefully acknowledged. The

authors also thank Mr. A. Gokhale and Mr. C. Iswaran for their

participation in the discussions that led to these results.

References

Blaschke, W. (1955), Vorlesungen uber Integral- Eeometrlc. 3rd Ed., Berlin.

Cahn, J. W. (1967) Trans. Met. Soc. AIME. 22X,

610. DeHoff, R. T. (1967) Trans. Met. Soc. AIHE. iii,

617. Hilliard, J. E. (1962 ) Trans. Met. Soc. AIME. Uk.

1201. Minkowski, H. ( 1903) Hath. Ann. . 52, I|ii7.

Saltykov, S. A. (1958) Stereometric Metal lograohv.

2nd. Ed., Me tal 1 urgi zdat, Moscow. Santalo, L. A. (1967) Studies In Global Geometry

and Analysis. S. S. Chern, Ed., Prent Ice-Hal 1

,

p. Iii7.

Schwartz, A. (1967) Calculus and Anal ytic Geometry.

2nd, Ed., Holt, Relnhart and Winston, New York,

p. Ii7l4.

Smith, C. S. & Guttman, L. ( 1953 ) Trans. AIME. iH,

81. Strulk, 0. J. (1950) Lectures on Classical Differential Geometry . Add i son-Wes ley, Cambridge,

(lass., p. 15. Underwood, E. E. (1970) Quantitat ive Stereoloty.

Addi son-Wes ley, Reading, Mass.

Underwood, E. E. ( 1972 ) Jnl. Mic 95 - 1. 25.

Uelbel, E. R. « Ellas, H. (1967) Quantitative

Methods In Morphometry. Springer Verlag, Inc., Berl i n.

40
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg. Md., September 4-9, 1975 (Issued January 19761

CONNECTIVITY OF "DISPERSED" PARTICLES: A PROBABILISTIC COMPUTATION

by George A. Moore

National Bureau of Standards, Washington. D.C. ,20234. U.S. A ,

The false appearance of total dispersion of particles usually seen on plane sections through an aggregate is a persistent and serious stereological

error. This is inherent in the fact that it is topologically impossible to represent more than one continuous phase on an extended plane, although an un- limited number of independent and totally interconnected systems or phases can co-exist in volume. Diff icvilties in tracing actual connections require a method of estimating contact frequency from simple information.

In ball models of either of the regular close packed structures, each ball

has a "hard contact" with 12 neighbors in space. The number of such "hard contacts" per ball will here be used as the measure of connectivity for any aggregate. It is easily seen that only a few specifically oriented and positioned section planes through the close-packed model will show contactconnected circles, while most sections falsely show a dispersion of small

circles. Other regular arrangements of balls, as used as models of crystal

structures, have 4 to 8 hard contacts. For these models the number of contacts increases quite regularly with the fraction of total volume occupied by the balls, as noted in Figure 3. This suggests that random arrangements of particles in an aggregate should, on the average, make increasing numbers of contacts as the volume fraction of particles increases.

The first key to modeling a random aggregate lies in determining the probability that one added particle comes to rest in a position at which it generates a contact with some other particle. The probability of finding the center of any specified particle within any defined small volume is equal to the volume fraction occupied by the defined small volume. To make the model

independent of absolute particle size the unit of measurement of volume is taken eqxial to the average volimie of 1 particle. For spheres of uniform size any one sphere will acquire a contact whenever the center of some other sphere

falls within a shell of doubled radius. The sphere in which contact is possible thus has a volume of 8 units. It is, however, more realistic to consider

that placing the new center within the original particle is either impossible

or forms a single larger particle. Thus a shell volume of 7 units gives a more realistic model. The intrinsic probability of contact, P, is thus taken as 7 for this particular uniform sphere model.

We may consider two forms of the uniform sphere model. In the "soft" form spheres in motion deform on contact with final center to center distance of any value between r and 2r. Growth of spheres upon small nuclei by precipitation

from solution yields the same configuration. In this soft case each added

particle shields against future additions by a factor, s = 7, by occupying one- seventh of the contact shell.

In the "hard" form spheres are assumed to be subject to a mixing motion

in a viscous fluidized matrix which will subsequently solidify. Elastic

rebound between particles is prevented. Thus whenever the relative motion of

two particles would result in a center to center distance less than 2r the particles will roll about each other, while remaining in contact, until they

come to rest at 2r spacing. In this condition a shielding factor, s = 12,

applies as each contact uses one-twelfth of the possible contact positions.

There is no assumption that the new sphere rolls into a pocket or ends in any other specific configuration.

It is also assumed that there are no significant attractive or repulsive

forces between the particles, or surface forces from the matrix, and that the

41
G. A. Moore

densities of the two phases are sufficiently similar to eliminate gravitational

effects. If these assumptions do not hold, the intrinsic probability, P, will

be different from 7. Irregular particle shapes generally will call for P>7. Nonuniform particle sizes could be computed in several classes of configurations, using different values of P for each class. , .

.

We now proceed toward the construction of an iterative computer program which models the series of microprocesses which occur each time an incremental

addition of new particles is made to an existing, but still fluidized, mixture.

The quantities Fq, F]^, F2, . . . F]^2 represent the fractions of all particles

currently present which exist in states having 0, 1, 2, . . . 12 contacts. Starting with an originally negligible particle concentration Vy = 0, all particles are in the zero contact state where Fq = 1 and F]^, . . . F]^2 ~ 0. A small incremental particle addition, AVy, is now made. The part of the present zero state particles which are involved in contacts with this addition

P • F^ • AV^ Each such contact-forming action forms 2 contacts, one on each particle. While

the 2 particles may in fact end in different configurations in later steps, the new particles are not in fact distinguishable from old particles. Therefore,

the part of the zero state portion promoted to the one contact state is

''^ AF^ = -AFq = 2P • Fq • AV^ [1] At each small addition of particles, this first action is followed by up

to 11 more actions in each of which some portion of the particles in one contact state are promoted to the next higher state by contact with part of the remaining portion of the addition. For calculation, these actions are treated as successive, with each new state formed only from the next lower state. The fraction in the lower state is corrected before each new action. The probability V is multiplied by a shielding factor for all states above zero. The contacting ability of AVy is decreased after each usage. Each action thus

is represented by 4 computations as follows.

^^n+l) = 2P F'(^) AV^(^) [2]

^(n+1) = ^n+1) ^ ^^n+1) f^]

^\(n^-l) = %(n) -

^n = - ^^n+1) f5]

The index "n" represents the number of contacts in the starting state undergoing reaction with the addition. F' represents the temporary fraction

in the state after the new contacts are formed but before this state is again

decreased by reaction to the next higher state [eq. 5], Rather than attempt

integration of these operational equations, they have been incorporated as an iterative cycle in a programmable calculator operation which accepts chosen

values of P, s, and AVy and prints values of the fractions, Fq, F^ . . . '^12^ and the mean number of contacts per particle, M, after each AVy addition.

The state fractions first rise to individual maxima, then fall as most are con- verted to higher states (Figures 1 and 2) . The mean number of contacts per particle rises continuously with increasing volume fraction, with moderate

downward concavity (Figure 3) . The mean contact number reaches a maximum

of 6.6 for hard spheres at the highest concentration obtainable in practice

(about = 0.64) and of 5.1 for soft spheres at the same concentration.

The computed mean contact, or coordination, numbers for the hard sphere

random aggregate may be compared with experimentally known combinations of coordination and occupied volume fraction (Figure 3) . This curve passes slightly above the regular diamond cubic array (M = 4) and slightly below the simple cubic (M = 6) , then cuts through the lower corner of the region covered

by reports for "dense random packing" (Ref . 1&2) obtained either from "heaps" of

42
Connectivity of "dispersed" panicles: a probabilistic calculation

real balls or computer construction of such heaps. All "dense random" determinations inclvide either an effect of gravity or a condition of fitting new

spheres into 3-sphere "pockets," hence it is expected that the true coordination of spheres in a matrix should be slightly lower than by these models.

Selected distributions from Figures 1 and 2 have been checked for variance and

their standard deviations found to agree well with Poisson statistics. Poorly

mixed aggregates would have an excess number of particles of high coordination

number in the more concentrated regions and therefore a higher average coordination number.

Average contact numbers of 2, 3, and 4 correspond to structures dominated

by lines (filaments), sheets, and 3-d networks of particles respectively.

The compositions at which these structures would be dominant are noted in Figure 3.

The probabilistic model of an aggregate thus appears to give valid estimates for contact connectivity as a direct function of volume fraction for the assumed uniform random mixture of noninteracting uniform hard spheres. Departures from this ideal model can be treated by the same program by inserting

different constants, P and s. Computation for Bemal's (Ref. 2) model of a liquid with contacts to r = 1.05 gave 3% lower coordination number but 2.4 times

greater state variance than Bernal reported. The present computation will

apparently give a realistic model of a liquid by using P based on an enlarged

sphere of attraction and the appropriately larger value of s.

REFERENCES

1. C. H. Bennett, "Serially Deposited Amorphous Aggregates of Hard Spheres,"

J Appl. Phys. 43, 2727-34 (1972).

2. J. D. Bernal, "The Geometry of the Structure of Liquids" (pp. 25-50) in

Liq\u.ds, Structure Properties and Solid Infractions , ed. T. J. Hugel,

Elsevier (1965).

Sph«re Concentrolion, V„ Figure 1 - Distribution of Contact States for "Hard Sphere" Model.

43
G. /4. Moore

Sphere Concenlrolion, Vy Figure 2 - Distribution of Contact States for "Soft Sphere" Model.

Sphere Concentration,

Figure 3 - Mean Contact Coordination Number as a Function of Concentration.

44
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

ANALYTIC RESULTS FOR THE CONNECTIVITY OF DISPERSED PARTICLES

by Roland de Wit National Bureau of Standards, Washington, D.C. 20234, U.S,.A.

In the previous paper of these Proceedings G. A. Moore has presented a probabilistic computation of the connectivity of dispersed spheres. He constructed an iterative computer program, based on a set of four operational

equations, which he used to generate the curves for the state fractions. We wish to examine whether it is possible to integrate his operational equations

and thus find an analytical solution to his problem. The present paper shows

that such a solution always exists for the no-contact state and that analytic

solutions can be foun,d for the other states if a certain approximation is made.

We use the same notation as Moore and derive the equations ab initio.

Hence, Fq, F] , F2,..., Fp,..., Fs are defined as the fractions of all spheres

in states having 0, 1, 2,..., n,..., s contacts with neighbouring spheres. Here s is the maximum number of contacts a sphere can have, i.e., the maximum

coordination number. Suppose now that we have a certain distribution of contact states at a particular volume fraction of spheres. Then the effect

on this distribution of a small increment aVv is that a portion of spheres in each contact state is promoted to the next higher state as shown in the following scheme:

2P FoAVy

2P^F^ AV, (1

2^^^-i (i4)---n-%)

F3 J 2p1f3.t AV, (1 . . .(1

Here the factor 2 occurs because each contacting action changes the state of 2 spheres. The factor P is the intrinsic local probability of contacting action

for a sphere with no contacts. As the sphere acquires n contacts, this probability is decreased by the shielding factor (s-n)/s. The portion promoted

to a higher contact state Fp+i is naturally proportional to the fraction in the contact state Fp and the concentration increment AVy. Finally, the product of

the expressions in parentheses represents the decrease in contact forming

ability of the concentration change AVv 3t each step. This scheme leads to the following set of s differential equations

^=Q,.i - Q,, {n=0, 1, . . s), (1)

Qn

= 2P^F^ n^^ (1 -^). Q.^ = 0, (2)

which are completely equivalent to Moore's 4 operational equations. Here Q represents the fraction of spheres promoted from the contact state Fp to the contact state Fp+i as the composition Vy is increased. We note a point of consistency. By definition the complete set of state fractions Fq, F],..., Fj must add up to unity:

45
/?. de Wit

S

n=0

1

Now it follows from (1) and (2) that

s - I V n=0

0,

so that the condition (3) can be maintained at all concentrations.

The no-contact state Fq satisfies the simple equation

dF, -2P F, 0

dV, 0-

C3)

(4)

(5)

With the initial condition that Fq = 1 when Vy = 0 the solution is

Fo = exp (-2P Vv). (6) Hence the no-contact state undergoes a simple exponential decay with increase

in composition, as is also borne out by floore's Curves. For n > 1 the differential equations are nonlinear due to the product factor in (2)t As a consequence, an analytic solution does not appear feasible. However, if we linearize the equations by deleting the product factor, i.e., if we take

Q = 2P — F (7)

instead of (2), then there exist simple solutions, which can be obtained by the standard methods of solving differential equations. With the initial conditions that Fn = 0 when = 0 for n >^ 1 , they are

s: n:(s-n)

n

1i=0

-1)^" n: rHviT: exp (-2P V, s-n+i

), (n=0, 1,..., s] (8)

It is straightforward to show that these solutions satisfy the consistency

condition (3) by a double application of the binomial theorem. Note that the result (6) is included in (8). The result (8) can also be applied to two dimensions, if Vy is replaced by

Aa, and to one dimension, if V\/ is replaced by Ll. The following table summarizes the best values of the parameters:

Dimension 1 2

3

Mara spheres Soft spheres

s 2 6 12 7 P = 2D - 1 1 3 7 7 Maximum of 1 tt/2/3 tt/3/2

L|_, A^, or Vv 0.907 0.7405

The maximum values of Ll, Aa, and Vv are those for the close-packed arrangement.

In three dimensions Moore has made a distinction between hard spheres (s=12) and soft spheres, for which he has chosen s=7. Our results for hard spheres are plotted in Figure 1 and for soft spheres in Figure 2. These curves differ only

by an insignificant amount from Moore's results. Another comparison is afforded by the mean coordination number, which is defined as

s M = z n n=0

(9)

The differential equation for M is found as follows

46
Analytic results for the connectivity of dispersed particles

VOLUME FRACTION OF SPHERES

Figure 1. State fractions versus concentration for hard spheres

0 0,1 0.2 0.3 0.4 0.5 0.6 0.7 VOLUME FRACTION OF SPHERES Vy

Figure 2. State fractions versus concentration for soft spheres

47
R. de Wit m

dV

sz n

V n=0

dF

r dV„

s

In=0n (Q n-1

s = 1: 2P

n=0

s-n

s 2P CI - M/s), (10)

from (9), (1), (7), and (3). the solution is With the initial condition that M=0 when Vv=0

M = s [1 - exp (-2P Vy/s)]. (11) The same result could also have been obtained from (8) and a double application

of the binomial theorem and its derivative. Equation (11) is plotted in Figure 3 for hard spheres (s=12) and soft spheres (s=7), using P=7. The curves

are lower than Moore's result by 5% for hard spheres and 1% for soft spheres.

The comparison with Moore's work would seem to indicate that the approximation (7) is not too severe, and that the product factor in (2) supplies only

a small correction. We believe that the analytic solutions may sometimes have certain advantages over the numerical ones, even though we had to approximate.

One advantage is that equation (8) directly gives the concentration F^ for any value of Vy any choice of P and s, thus showing the qualitative dependence of

the solution on those parameters.

Some computer calculations and experimental results for the packing of

spheres have been summarized by Norman and others. Their findings for the mean coordination number agree qualitatively with ours, but differ in the details.

REFERENCES

"Computer simulation of particulate systems" by Lindsay D. Norman and others,

U.S. Bureau of Mines Bulletin 658 (1971).

"Connectivity of dispersed particles: A probabilistic computation" by

George A. Moore. This volume.

1 1 1 1 1

1 1

Hard Spheres

s = 12

" Soft Spheres s= 7

/ 1 \ 1 1 1 1 1

UJm

o

iro

o

o

<

UJ

0.1 0.2 0.3 0.4 0.5 0.6 VOLUME FRACTION OF SPHERES

Figure 3. Mean coordination number versus concentration

0.7

48
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

TOPOLOGICAL ANALYSIS OF DENDRITIC TREES

by Martin Berry Anatomy Department, Medical School, University of Birmingham, England

The dendxitic trees of neurons offer a large surface area for contact with

presynaptic elements over wide areas of neuropil. At the same time, the biophysics of the membrane and the impedance properties of the core of dendrites

impose constraints on the flow of current such that the characteristics of the branching patterns of dendrites are thought to have profound effects on both

the logical and information processing capacity of neurons (Rail, 1970) . In the light of this widely held view, that the branching patterns of dendrites

are in some way related to the integrative capabilities of neurons, it is surprising that a technique has not been developed for the analysis of dendritic topology. In fact, since the 1940' s geographers have been studying the networks created by rivers, roadways and other lines of communication, using

the method of network analysis (Haggett and Chorley, 1969) and it has recently

been shown that the topological and ordering methods they use are directly

applicable to the investigation of both the branching patterns and the growth

of the dendritic fields of neurons (Berry et al, 1975; Hollingworth and Berry,

1975; Berry and Bradley, 1975a, b)

.

The method of network analysis requires the tree to be viewed as a plane

graph, made up of points, called vertices , interconnected by lines called arcs . The outermost tips of the tree are termed pendant vertices and the respective

arcs, pendant arcs . The point of origin of the tree is called the root vertex . Translated into dendritic terminology arcs are the segments and pendant arcs

the terminal segments of the tree. All vertices are nodes, the root vertex is taken to be the axon hillock and the pendant vertices are the tips of the terminal segments. Different forms of branching can be defined by the number

of arcs draining into each vertex and the order of magnitude of branching at a vertex may be described as dichotomous if it drains three arcs, trichotomous if

it drains four arcs, and so on. Branching patterns can be established by a variety of processes of growth. Terminal growth occurs when arcs are added

to pre-existing pendant vertices and segmental growth when arcs are added to pre-existing arcs including pendant arcs . Monochotomous , dichotomous or trichotomous growth patterns are generated by the addition of one, two or three arcs, respectively, at the newly established vertices. The connectivity

of dendritic trees is thus described by defining both the mode of interconnection

of arcs in the networks and the frequencies of vertices of differing orders of magnitude of branching.

Topological analysis can be used to study branching in both small and large

dendritic trees, e.g., the small arrays of the basal dendrites of neocortical

pyrcimids and the massive trees of Purkinje cells in the cerebellum (Hollingworth

and Berry, 1975) . In the former case the analysis is performed on the entire

tree whilst, in the latter case, only the peripheral parts of the network, s\ibtending a small number of pendant arcs, are analysed. The latter strategy is adopted because the numbers of topologically distinct patterns which are possible

in networks with only small numbers of pendant arcs is so large that the number

of examples needed for analysis becomes unmanageable (Berry et al , 1975).

Growth of dendrites

The growth of dendrites occurs at specialised sites called growth cones which can be located along segments (Merest, 1969) or at the tips of dendrites

(Bray, 1973) . This latter location probably predominates in most growing trees

and growth of any new branches from segments is only likely to be transitory

since growth cones will be carried to the tips of all newly formed dendrites

(Berry and Bradley, 1975) . Thus terminal growth should predominate in dendritic

systems. Vaughn et al (1974) have suggested that synaptic engagement of filopodia may determine branching and Berry and Bradley (1975a) have elaborated this idea

predicting that the order of magnitude of branching at nodes, the direction of

growth and the length of segments will be dependant on an interplay between the growth cone and synaptic potential in the surrounding axon field.

49
M. Berry

(^^X-X X X

(ii) '

4:1 Figure 1. Topological branching patterns formed by (a) segmental and (b) terminal growth. The Roman numerals refer to the consecutive stages of branching and the patterns to the right of each arrow depict where branching can occur

on the pattern to the left of the arrow. The branching patterns seen under

each bracket represent the topological types drawn in a standard format, used

throughout this paper, into which the patterns above the bracket may be resolved. The rationale for resolving mirror images and rotational differences are given by Berry et al (1975) . The ratios against the products of the third

stage of branching refer to the frequency distribution of each topological

branching pattern.

There are, of course, an infinite number of growth hypotheses but perhaps

the most useful formulations are stochastic models which highlight any deviation of growth from a purely random process. Random models also seem relevant

in terms of growth cone/axon field interaction since filopodial synaptogenesis is likely to be a chance event as may be the number, length and stability of

filopodia emanating from a growth cone at any one point in time. To test the prediction of Berry and Bradley (1975a) , that terminal growth predominates in dendritic systems, two models of growth were fomulated (Fig. 1). In the

first, called the terminal growth hypothesis , two arcs were added randomly to existing pendant vertices and, in the second, called the segmental growth

hypothesis , a single arc was added randomly to existing arcs including pendant

arcs. Starting from the root vertex the method of generating specific topological branching patterns according to each hypothesis is illustrated in fig.l. It can be seen that by the third stage of branching the frequency distribution

of topological branching patterns discriminates between the two growth models.

If the calculation for each hypothesis is continued, the distribution of topologies at subsequent stages of branching can be worked out. The complete

series of topological types possible at each stage of branching is defined as an absolute pendant arc series and the frequencies of topologies in the 1st - 9th absolute pendant arc series for random terminal and random segmental growth

models are given in figures 2a and b, respectively.

Hollingworth and Berry (1975) have compared their observed data with those

in figs . 2a and b and shown that the mature tree appears to grow in a manner indistinguishable from random terminal growth. In fact, certain constraints

during development do tend to deviate growth from a purely random process

(Berry and Bradley, 1975a) , as do the effects of irradiation (Bradley and

Berry, 1975; Berry and Bradley in preparation) and starvation (McConnell and Berry in preparation) although growth at pendant vertices always predominates.

The lengths of segments can easily be incorporated into the analysis and the mean probability of branching in the network can be calculated from the frequency distribution of arcs in the mature tree. Thus, if there are arcs of

length then the frequency of arcs, of length will be given by the expression:

Y^ = Y^ (1 - p) ^^2

"

where p = mean probability of branching in the tree. Y^ and X^ , Y^ and X^ can be obtained from the frequency distribution of segment lengths of the tree and statistical adjustments between observed and expected distribution verify and

50
Topological analysis of dendritic trees

PIMOAJT

ARCS POSSIBLE TOPOLOGICAL TYPES

I

2

1 X

1- A

(a) 1 J<

5

8

1_ 1 1 1 l_ 1 1 II II

a If rllllffrllrl<FrFFiy M

9

UKS POSSIBLE TOPOLOGICAL TYPES

1

2

3 A

X J?

5 "" ^ f B

•

• III

aaaaaaaaaaa aaaaaaaaa

Figure 2. Topological types generated by random segmental (a) and random

terminal growth (b) cire listed together with the frequency of occurrence (%) of types in networks with from 1 to 9 pendant arcs. (see Berry et al. (1975)

and Berry and Bradley (1975a) for sources of data)

.

51
M. Berry

refine the value of p (Berry and Bradley, 1975a) . This estimate of p is based

on the assumption that segment lengths remain fixed once a vertex, defining the distal end of a segment, has been formed. Evidence supporting the validity of

such an assumption has come from cinematographic studies of growing 'neurites'

in tissue culture (Bray, 1970, 1973) and from studies on the microtubular

skeleton within neural processes which appears to stabilise established segments

(Bunge, 1973; Yamada, Spooner and Wessells, 1971; Yamada and Wessells, 1971).

Estimates of the frequency of different orders of magnitude of branching

are important because they also provide an estimate of the intensity of interaction between axons and dendritic growth cones and tl.us it is to be expected

that the magnitude of this parameter should be inverseley correlated with the mean frequency of segment lengths (Berry and Bradley, 1975a; Hollingworth and

Berry, 1975)

.

Figure 3. Illustration of the Strahler

method of ordering. The tree can be ordered

from the most peripheral branches inwards by

assigning all terminal branches (pendant arcs) as order 1. Where two order 1 branches join, an order 2 branch is formed. Where two order

2 branches join, an order 3 branch is foirmed, etc. Daughter branches of 'n - 1' Strahler

order form parent branches of Strahler order

'n'. Collateral branches of 'n - 1' Strahler

order, or less, divide the parent branch into

segments but do not change the order of the Parent branch. To reverse, the highest

Strahler order branch, draining into the root

Vertex, is assigned order 1 and all other orders

are consecutively assigned order 2,3... etc. In this illustration Strahler order branches

are given in parenthesis and the reversed order

numbers occur to the side of each of these bracketed numbers

.

The topological analysis of small trees defines the mode of interconnection

of arcs in a given network pictorially. The difficulties of quantifying

connectivity are enormous and centre on choosing a method of assigning a ranking

order to all arcs. Such methodologies are fraught with problems (Berry et al, 1975; Uylings et al, 1975) but the method we have chosen, called the Strahler

technique can be used to order the tree both centripedally and centrifugally

(Fig. 3) which does give this technique some advantages (Uylings, 1975; Hollingworth and Berry, 1975; Berry and Bradley, 1975b)

.

The Strahler technique defines a branch as a series of arcs of identical

Strahler order. Daughter branches of 'n - 1' Strahler order create parent

branches of Strahler order 'n' and collaterals draining into the parent branch

are always 'n - 1' Strahler order, or less. By estimating the relative

frequencies of branches of each order the relationship between orders can be expressed in terms of the bifurcation ratio i.e. the ratio between adjacent orders. The number of branches of different Strahler order in a given network tends to approximate to an inverse geometric series defined by the mean, or overall, bifxir- cation ratio. Some examples of mean bifurcation ratios of different branching

systems are given in figure 4. The application of ordering methods to the study of dendritic connectivity

is complicated by changes in the order of magnitude of branching at different vertices in the same tree and also by conceptual difficulties in the interpretation

of the meaning of the bifurcation ratio parameter in the context of connectivity.

In this latter respect, the greatest difficulties arise when different networks

give the same overall bifurcation ratio (Berry et al, 1975). This lack of

52
STRAHLER ORDERS Figure 4 . Graphic illustration of

some of the possible bifurcation ratios

(numbers in parentheses) of networks

grown by the addition of one arc with

the formation of each new vertex (n = total number of pendant arcs)

.

Topological analysis of dendritic trees

discrimination by the method comes about for two reasons; firstly, because the overall bifurcation ratio

represents the slope of an inverse

geometric series and thus is a measure of the size of an arborescence in terms of the maximal Strahler order

attained by a given tree and secondly,

Strahler ordering defines branches and

not segments in the tree and thus gives only general information about

connectivity (Berry et al, 1975). These objections can be overcome

either by choosing to quantify relationships between all arcs in terms of

daughter and parent Strahler orders or of combining the topological analysis

with the method of Strahler ordering

and calculating absolute bifurcation

ratios . This latter measure is obtained by computing the mean bifurcation ratios between successive

Strahler orders for each topological

branching pattern contained in an absolute pendant arc series for a given hypothesis of growth. Thus,

absolute bifurcation ratios give in- formation about both growth and connectivity in absolute and observed

networks.

Co-nclusioris Topological analysis of dendritic networks together with the application

of ordering systems defines the connectivity of trees so that their anatomy

and growth can be comprehended in precise quantitative terms. The application

of the method is thus likely to yield a fuller understanding of important

problems in neurobiology like the degree of interaction between axons and dendrites during the establishment of connections, the plasticity of axonal

and dendritic fields and the relative contribution of nurture and nature in determining form and function in the central nervous system. The precise

definition of the morphology of dendritic trees afforded by network analysis

means that the effects of treatments in experimental studies on growing and adult networks can be assessed quantitatively and that accurate models of

dendritic networks can now be built from which their logic can be deduced for

a given set of assumptions about the physical properties of their membrane.

References

Berry, M. , Hollingworth , T. Anderson, E.M. and Flinn, R.M. (1975) Application

of network analysis to the study of the branching patterns of dendritic

fields. In Advances in Neurology Vol . 12 Ed. G.W. Kreutzberg. Raven Press, New York pp. 217-245. Berry, M.and Bradley, P. (1975a) . The application of network analysis to the study of branching patterns of large dendritic fields. Brain Res . (s\ib- mitted for publication)

.

Berry, M. and Bradley, P. (1975b) . The growth of the dendritic trees of Purkinje cells in the cerebellum of the rat. Brain Res , (submitted for publication)

.

Bradley, P. and Berry, M. (1975). The effects of reduced climbing and parallel

fibre input on Purkinje cell dendritic growth. Brain Res . (Submitted for

p\iblication) . Bray, D. (1970) Surface movements during growth of single explanted neurons.

Proc. Natn. Acad. Sci . U.S.A. 65, 905-910.

53
M. Berry

Bray, D. (1973) Branching patterns of sympathetic neurons in culture. J. Cell . Biol . 56, 702-712.

Bunge, M.B.^Tl973) Fine structure of nerve fibres and growth cones of isolated

sympathetic neurons in culture. J . Cell . Biol . , 56 , 713-735. Haggett, P and Chorley, R.J. (1969) Network analysis Tn geography , Arnold,

London

.

Hollingworth, T and Berry, M. (1975) . Network analysis of dendritic fields of

pyramidal cells in neocortex and Purkinje cells in the cerebellum of the

rat. Phil. Trans. Roy. Soc. B . 270 227-264. Horton, R.W. (1945) Erosicnal development of streams aid their drainage basins,

hydro-physical approach to quantitative morphology. Bull, geol . Soc. Am .

,

56 275-370.

Mores"t7 D.K. (1969) . The growth of dendrites in the mammalian brain. Z. Anat . Entwgesch. 128 290-317. Rail, W. (1970) Cable properties of dendrites and effects of synaptic location.

In Excitatory synaptic mechanisms . Proceedings of the Fifth International

Meeting of Neurobiologists . Ed. P. Anderson and J.F.S. Jansen. Oslo,

Bergen, TronsO, Universitetsforlaget . pp. 175-188. Strahler, A.N. (1953) Revisions of Morton's quantitative factors in errosional

terrain. Trans. Am. geophys. Un . , 34 . 345. Uylings, H.B.M. Smit, G.J. and Veltman ,"~W. A. B. (1975) Ordering methods in quantitative analysis of branching structure of dendritic trees. In Advances in Neurology Vol. 12. Ed. G.W. Kreutzberg Raven Press. New York

pp. 247-254. Vaughn, J.E. Henrikson, C.K. and Grieshaber, J. A. (1974). A quantitative study

of synapses in motor neuron dendritic growth cones in developing mouse

spinal cord. J. cell. Biol . 60 , 664-672. Yamada, K.M. Spooner, B.S. and Wes¥ells, N.K. (1971) Ultrastructure and

function of growth cones and axons of cultured nerve cells. J. Cell Biol . 49, 614-635. Yamada, K.M. and Wessells, N.K. (1971) Axon elongation. The effect of nerve growth factor on microtubular proteins. Exptl. Cell. Res. 66, 346-352.

54
National Bureau of Standards Special Publication 437

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

THEORETICAL AND EXPERIMENTAL STUDIES OF THE STEREOLOGICAL PROPERTIES OF POROSITY, SPECIFIC SURFACE, AND CONNECTIVITY

by R. Bruce Martin West Virginia University, Morgantown. W. Va. 26506. U.S.A.

Theoretical proofs of the equivalence of linear, areal, and volumetric

measures of porosity, specific surface, and connectivity are discussed. Most

such equivalencies are found to be independent of the orientation of a cross- section or transect line in an anisotropic material. The application of these results to materials of various kinds and degrees of porosity is discussed, and experimental data is presented illustrating the utility of the theory in regard to dimensionally-reduced modeling and connectivity.

INTRODUCTION

As an engineer who is interested in analyzing bone as a structural- material

and, furthermore, in studying the relationships between the physiology of this

material and its mechanical properties, this author has been compelled to enter

the realm of stereology. Specifically, in attempting to analyze the mechanical

properties of bone in terms of its histology, and in trying to formulate mathematical representations of physiologic processes, the following questions arise.

(1) Is the areal porosity (p^, void area/total area) of a cross-section

numerically equal to the volumetric porosity (pv, void volume/total volume) of

the original specimen?

(2) Is the areal specific surface (c^, void perimeter/total area) of a cross-section numerically equal to the volumetric specific surface (sy, void

surface/total volume) of the original specimen?

(3) How may one relate the connectivity of various features in a cross- section to three-dimensional connectivity patterns?

In surveying the history of these problems the author found that while

geologists and others had for some time accepted the equivalence of areal and volumetric measures of relative composition or porosity, no rigorous mathematical proofs of this equality were available. The writer has therefore attempted

to formulate such proofs and to analyze the other questions raised above. This

work has been reported elsewhere (1) ; in this paper some additional results are derived and an application of these results in the problems just outlined is discussed.

55
/?. B. Martin

The writer has previously shown that areal measures of porosity can be expected to equal volumetric porosity. Furthermore, it was shown that linear measures of porosity (pl, the fraction of a transect line which falls in inclusions) can be expected to equal p^ and py, and the specific surface measured in two dimensions can be expected to equal the actual value in the whole

specimen if the cross-section is representative. In all these cases the ortho- gonality of certain geometric relationships requires that the above results are insensitive to the orientation of the transecting plane or line (1) . The next result stands in contradistinction to this.

y

THEORY

Theorem: The perimeter, c, of any simply connected plane figure is equal

to the projected length of the figure in

a certain direction, _Ay, multiplied by a geometric constant, gy, for that direction.

Using this theorem, it can be shown that

the frequency, f^j^, with which a transect

line hits inclusions on a cross-section

is related to the two-dimensional specific surface measure, c^., as follows:

"A/g (1) Fig. 1

where g^ is a geometric property of the inclusions associated with the i direction. The linear measure f^ is proportional to but can not be expected

to equal the areal measure of internal surface area, c^. We have then,

Pt = P. = P,. . (2)

V

(3)

It is seen that linear and areal measurements of porosity on a representative

cross-section or transect line of a solid body can be expected to equal the volumetric value, and that this equivalency is entirely independent of the orientation of the section or transect line. A similar situation connects areal

and volumetric measures of specific surface area, but the linear measure of this parameter is neither equal to its higher order analogs nor independent of direction. In all cases the inclusions need only be simply connected.

DISCUSSION

The requirement that the cross-section (or transect) be "representative"

means that it must reflect the homogeneity of the original specimen; if the specimen contains regions of greater inclusion density, the cross-section which

misses such regions will not be representative. If one has specimens which are reasonably homogeneous with regard to inclusion density, only a few cross- sections may be required to give an accurate result.

In this paper the symbol = is used to mean "can be expected to equal" in the sense that if enough measurements are made on representative sections, the mean value of the equation's left side will approach that of the right side.

56
On porosity, specific surface, and connectivity

Although the equivalence of specific surface and porosity measures in two and three dimensions does not depend on the geometry of the pores or inclusions, the relationship between specific surface and porosity does. Bone of low porosity has cylindrical, tunnel-like voids that run parallel

to one another several void-diameters apart. High porosity hone has large,

polyhedral voids. Fig. 2 shows the relationship between specific surface

and porosity as measured on microradiographs of human bone specimens. The dimensional equivalence of these variables allows one to make two dimensional

models of porous bodies . Fig. 2 also shows the <^p^V^ relationship for two such models of bone in which the voids are allowed to grow and break into one another in a natural way. Model 2 is better than Model 1 because its features were more nearly scaled to those of real bone. The usefulness of such models in studying dynamic porous materials is enhanced by the insensitivity to anisotropy of the material. If the patterns of change are realistic,

one orientation with respect to the actual body is as good as another, but changes in anisotropy e-xperienced during the remodeling will not be apparent

unless simultaneous models on orthogonal planes are studied.

Consider now the relationship between f^ and "connectivity." Others have

used the frequency with which a transect line intercepts an inclusion or other

feature as an indication of the degree to which these features are connected

across the path of the transect (2) (3). For instance if" f^ = 25 cm" ^ and

fx = 50 cm"-*^, there would be twice as many cross connections in the z-direction.

Equation 3 shows that such a measure of connectivity is directly related to internal surface area. In fact, one might well use specific surface (s^ or c^)

as a non-directional measure of connectivity.

Referring again to Fig. 2, it is clear in the light of equation 3 that the maximum value of c^ is associated with the greatest connectivity in the bone,

both of the voids and the solid matrix. The connections between voids increase

as one moves from left to right and the growing voids break into one another.

As even greater porosity values are reached, however, the voids tend to engulf

the inter-connections themselves, and the connectivity decreases. The reverse

scheme is true for the connectivity of the solid matrix.
R. B. Martin

REFERENCES

(1) Martin, R. Bruce, Equivalence of porosity and specific surface measures

one, two, and three dimensions. Pore Structure and Properties of Materials , Prague, Academia, 1973.

(2) Rose, R.M., Dynamic properties of subchondral bone. Proceedings , 5th Annual Biomaterials Symposium, Clemson, S. C, 1973.

(3) DeHoff, R. T. and Rhines, F. N. , Quantitative Microscopy . New York,

McGraw-Hill, 1968.

58
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

ASSESSMENT OF SAMPLING ERRORS IN STEREOLOGICAL ANALYSES

by J. E. Hilliard Department of Materials Science and Engineering, Northwestern University. Evanston, Illinois 60201, U.S.A.

Theoretical and experimental results cire presented for the sampling

errors in (1) Volume fraction analysis by point counting; (2) The estimation

of boundary area per unit volume; and (3) The estimation of the length of

lineal features per unit volume.

In all three cases the coefficient of variation of the estimate has the form:

in vAich is the total number of points counted and k is a parameter which may depend on the form of the structure and the test probe. For volume

fraction analysis by point counting k = (1 - Vy) provided the point density

is not too high. In the estimation of boundary area of contiguous grains

k 1.0 or 1.4 for straight or circular probes respectively as the probe

length tends to zero. For long probes k as 0.6 for both types. If the boundaries are surfaces of discrete bodies, k = 1.4 for long, straight probes. For an analysis of line length k = 1.0 if the lineal features are randomly

dispersed. A procedure is described for determining k experimentally for structures that do not conform to the ones treated here.

INTRODUCTION

Stereological analyses are subject to various types of errors. These

include: (1) Statistical sampling errors which, provided that certain

sampling conditions are satisfied, are inversely proportional to the square- root of the number of observations; (2) Errors due to the limited resolution

of the microscope and scanning instrument (if the latter is used); (3) Biases

introduced by incorrect sampling procedures; (4) Observational errors (for

example, miscounts if measurements are made visually or lapses in discrimination with automatic instruments). We shall be considering only sampling errors

and we will further limit the treatment to three types of analysis: Volume

fraction estimation by point counting; the estimation of boundary area per unit volume; and, the estimation of the length per unit volume of lineal features. These three properties can be determined from measurements on a section

without any assumption about the form of the structure. In addition, they

require only a counting measurement (as distinct from an analog one). Before

considering the individual analyses, we will first derive some statistical

relationships that are common to all three.

STATISTICS OF COUNTING MEASUREMENTS

Consider an analysis i.n which a point, linear, or planar probe is applied

n times to a section. If P is the average number of point intersections of

the probe with structural features then, by definition, the total number of

intersections P„ is given by:

ABSTRACT

a(x)/x = k/yp^

nP. (1)

We next utilize a well known result in statistics that if: X = CjX^ + C2X2 + (2)

where the x's are random variables and the c's are constants, then:
J. E. Hilliard

X = c^x^ + C2X2 + . . . . (3)

If, in addition, the x's are independent of one another (we will later examine

the implications of this condition in stereological measurements) then:

2 2 2 2 2 a (x) = c^a (x^) + {^^) + . , . (4)

2 in which ct (•) is the variance [i.e., the square of tLa standard deviation

Applying Eq. (4) to Eq. (1) we obtain:

a2(P^) . n^a^CP). (5)

But since: _

P = (l/n)(P^ + P2 + . . . + P^),

it also follows from Eq. (4) that:

a^CP) = (l/n)a2(P). (6)

[The distinction between a (P) and a (P) is that the latter is the variance for

a single application of the test probe whereas the former is the variance for the average of n applications.] Substituting into Eq. (5) we find:

a2(P^) = na^CP).

2 —

Finally, a division by P^ (= nPP^) yields:

a2(P^)/p2 = kV^, (7)

where 9 9 _

k = a (P)/P. (8)

As we will later see, k depends on the form of the structure and on the shape

and size of the test probe. However, once k is known it is possible to esti- mate a_ priori the standard deviation of an analysis in terms of the number of

observations. Also, since an analysis usually comprises a 100 or more individual observations, it can be assumed from the central limit theorem that the results will be normally distributed. It is therefore possible to attribute

a probabilistic interpretation to the standard deviation; for example, that

there is approximately a 2/3 probability that the true value of a result x

lies in the range x ± a(x).

Another result that we will be needing is the probability, p(P) that

there will be exactly P intersections on a single application of the probe. If the intersections are "randomly" distributed on the probe and if they are independent of one another, then p(P) follows a Poisson distribution; i.e.:

p(P) = (l/P!)P^exp( -P). (9) The variance of this distribution is given by:

CT^(P) = P. (10)

VOLUME FRACTION ANALYSIS

There are three different procedures for estimating the volume fraction

of a constituent: (1) An areal analysis in which the areal fraction of the constituent intercepted by a planar probe is determined. (2) The measurement of the fractional length of a linear probe (applied to a section) that intercepts the constituent. (3) The application of a grid of points to a section

and a measurement of the fractional number of points falling within the con- stituent. In each case the fractional occupancy gives an unbiased estimate of

the volume fraction. (This is true even if the measurements are made on a

60
Assessment of sampling errors in stereological analyses

non-randomly oriented section through an anisotropic structure.) A theoretical analysis made by Hilliard and Cahn (1961) revealed that the standard deviation for all three analyses was determined almost entirely by the number of observations and that the type of observation (estimation of the area of a profile, of the length of an intercept, or the identification of the constituent occupied by a point) had little effect. They therefore concluded

that point counting would be the most efficient method of estimation. They

also found that if the grid spacing was such that not more than one point can fall on a profile of the constituent being estimated (if the profiles are not discrete the corresponding condition is that there should be little or no correlation between the occupancy of adjacent points) then, in the limit as the volume fraction V^.— 0,

a2(Vy)/v2 = 1/P^, (11)

where is the total number of points falling on the constituent being estimated. It is apparent that Eq. (11) cannot be valid for large volume fractions

because it predicts that as Vy — 1, cr2(Vy)/Vy — 1/P (where P is the total num- ber of points applied) whereas the variance has to go to zero at this limit. An expression for the variance must satisfy the following two conditions if

it is to be valid over the whole range of volume fractions. (1) It should be symmetric in Vy and (1 - Vy) [this follows from the requirement that a2(Vv)

be independent of the constituent on which the count is made] and that Eq. (11)

should be satisfied in the limit Vy = 0. Hilliard (1968) has proposed that

the simplest expression* meeting these requirements is:

a2(Vy) = Vy(l - Vy)/P, (12)

^^(Vy)/Vj = (1 - Vy)/P^. (13)

Recently, Andersen, James and Hilliard (to be published) have made a study

of point-counting sampling errors on a series of two-phase p-brasses. Six

different samples were used which were of the same composition but which had

been rolled by various amounts to produce elongations varying between 20 and

70 pet. There was thus an appreciable variation in grain shape. Altogether,

approximately 8,500 applications of a 9-point grid were made, by two observers, on the specimens. The average volume fraction was 28.1 pet and the average value of c7(Vy)/Vy for one application of the grid was 0.532. Setting

Vy = 0.281 and P(y = 9 X 0.281 = 2.53 in Eq. (13) yields a(Vy)/Vy = 0.533. [The agreement between the theoretical and observed values of (J(Vv)/Vy is actually better than would be expected on the average.] From this investigation it can be concluded that Eq. (12) or (13) can be used with some confidence for the prediction of the standard deviation of a volume fraction analysis by point counting. However, it is to be emphasized

that they are valid only for low point densities. If, on the average, there

are several points per profile these expressions will underestimate the standcird deviation.

BOUNDARY AREA ANALYSIS

The boundary (or surface) area per unit volume, S^, can be estimated by

applying a linear probe to a section and counting the number of intersections,

Pl, per unit length of the probe with boundary profiles. Then, by a well known

relationship: _

The probe need not necessarily be straight but can be of any shape. However,

* The same expression has also been given by Gladman and Woodhead (1960) and

others. However, the derivations were based on a random point count performed

on a single sub-area and they fail to reveal that Eq. (12) is completely

invalid if a high point density is used.

61
J. E. Hilliard

if the structure is anisotropic, Eq. (14) is only valid if the probe and the section to which it is applied are randomized with respect to orientation.

It follows from Eqs. (4) and (14) that: a^Sy) = 402(P^) = 402(P^) /L^, .; (15)

where Pt is the total number of intersections on a total probe length (i.e., the probe length times the number of applications). Dividing through by

we obtain: a2(Sy)/s2 = a2(p^)/p2, (15)

or from Eq. (8): a2(Sy)/s2 = k^/P^. (17)

The value of k depends on the form of the structure. We will consider two

cases: (1) Contiguous grains of the same constituent. (2) Discrete particles

embedded in a matrix.

Contiguous Grains

We will assume that three (and only three) grains meet along an edge.

(This is usually so if surface tensions are a significant factor.) We first

consider the case when the probe is in the form of a circle and will derive

a theoretical estimate for kg for the limit when the expected number of intersections, P, on the probe approaches zero.

Fig.. 1. In an analysis for the boundary area of contiguous

grains it is only necessary

to consider the outcomes: 0,

2, and 3 intersections for a circular probe and 0, 1, and 2 intersections for a straight

probe as the length of the probe tends to zero.

The condition P — 0 is equivalent to the diameter of the probe tending

to zero. We therefore need only consider the three possible outcomes: 0, 2 and 3 intersections illustrated in Fig. 1. Let p(0), p(2) and p(3) be the corresponding probabilities of these outcomes, then:

P = Op(0) + 2p(2) + 3p(3). (18)

Applying the usual expression for the variance:

a2(P) = -p2 + SP^p(P.),

we obtain: „ _„

a'^(P) = Op(0) + 4p(2) + 9p(3) - P . ; (19)

62
Assessment of sampling errors in stereological analysis

Multiplying Eq. (18) through by 2 and subtracting from Eq. (19) we find after

rearrangement:

kj = a2(P)/P = 2 + {[3p(3)/p2] - l} P , [P ^ 0], (20)

The next step is to estimate p(3). This probability is equal to the area of

the circulcir probe (= -nd^/A, where d is the diameter) times the number, P^,

of triple junctions per unit area of the section. Thus:

p(3) = (TTdV4)P^. (21)

If Ly is the edge length per unit volume, then:

Pa = V^- (22)

It also follows from Eq. (14) that:

P = (TTd)P^ = (TTd/2)Sy. (23)

Substituting from the last three equations into Eq. (20) we obtain:

= 2 + [(3y2.Ts5) - 1]P, [P - 0]. (24)

2 The ratio Ly/S„ can be determined for a particular specimen. In order to get

a rough estimate of k^ that is generally applicable, we will assume that the specimen is composed of egual size tetrakaidecahedral grains. In this case it

is easily shown that Ly/Sy = 0.7570 which yields:

kg = 2 - 0.639P, [P - 0]. (25)

A plot of this theoretical k is shown by the upper dashed curve in Fig. 2.

Fig. 2. The dependence of

kg on the average_number

of intersections P for boundary traces from con- tiguous grains on a probe

for (1) circular probes

(upper curve) and (2)

straight probes (lower

curve). The dashed curves

are the theoretical values

calculated from Eqs. (25) and (34).

p

The points are from measurements made by Milliard (1962) on a sample of silicon

iron. It will be seen that both the theoretical, and experimental curves display an initial decrease in kg with increasing P (or, equivalently , with in- creasing size of the probe). However, at P «s 6, there is a leveling off and thereafter kg remains constant at a value of ~ 0.63. We next consider the case of a straight test probe. At the limit P -• 0 we can neglect any possibilities other than 0, 1 and 2 intersections (Fig. 1). If p(0), p(l) and p(2) are the corresponding probabilities, then:

63
J. f. Hilliard

and

P = Op(0) + lp(l) + 2p(2),

a^(P) = Op(0) + lp(l) + 4p(2) - P^.

(26)

(27)

Taking the difference between the two equations aud dividing by P we obtain:

= a2(P)/P = 1 + {[2p(2)/P^] - l}p, [P - 0]. (28)

In order to estimate p(2) we will assume that the angle between the boundary

traces at a triple junction is 120° . Consider a probe of length i oriented

at an angle 9 to one of the boundary traces. Then, as shown in Fig. 3, in order to obtain two intersections, the center of the probe has to fall within

Fig. 3. The center of a straight probe of length

I (= ab) oriented at an angle 6 to a boundary must

fall within the triangle

abc in order to generate

two intersections

.

the triangle abc, where the construction is such that ap = pb = qc = 1/2, and

ab is parallel to qs. Thus the probability that a probe randomly oriented

over the range 0^9^ n/3 will yield two intersections is:

p(2) = AP^ (29)

where A is the average area of the triangle and P^ is the number of triple

junctions per unit area. Noting that:

pq = jlsin9/sin(2n/3) .= {2i/J^)siT&,

and that the height of the triangle is given by:

h = pqsin[(n/3) - 9]

,

we obtain for the area of the triangle abc:

A(9) = (jeV'y3)sinesin[(TT/3) - e]. (30)

We next have to determine the average area of the triangle for a randomly

oriented probe. For 0^9^ n/3 the distribution function, f(9), for 9 is

3/tt. Hence:

jT/3 jt/3

A= A(9)f(9)de = (3/n) A(e)d9. ^0 ^0 Substituting for A(9) from Eq. (29) and performing the integration we obtain:

A = (l/4ny3)(3y3'- tt);^^ =i 0.0944^^.

64
Assessment of sampling errors in stereological analyses

Hence from Eq. (29):

p(2) = 0.0944^^P^. (31)

From Eq. (14):

I = 2P/Sy.

Substitution of i into Eq. (31) together with the value of given by Eq. (22)

yields: „ _„

p(2) = 0.1888(LySpP\ (32)

We thus obtain from Eq. (28):

kj = 1 + [0.3776(ysJ) - 1]P, [P - 0]. (33)

If we again assume that the grains are equal-size tetrakaidecahedra, then:

kg = 1 - 0.714P, [P - O] . (34)

A plot of this theoretical k is shown by the bottom dashed curve in Fig. 2. The experimental points were detemined by Philofs'ky (1966) on the same specimen of silicon iron as was used for the circular probe measurements. The grains in this specimen were somewhat elongated and Philofsky found that k^ for probes parallel to the direction of elongation were about 15 pet lower

than those in the transverse direction. (The values plotted in Fig. 3_are

the averages for the two directions.) It will be seen that for large P, kg for the straight probe is slightly less than that for a circular probe.

However, this is probably due to the anisotropy in the grain shape since, in general, one would expect the properties of a circular probe to approach those

of a straight probe as the diameter of the former is increased.

Non-Contiguous Particles

For the case of non- contiguous particles we will restrict the treatment

to linear probes that are long compared with the mean free path between par- ticle profiles. With this condition it is permissible to neglect instances

when the probe starts or terminates_within2a profile and assume that all intersections occur in pairs. Thus, if N and a (N) are the average and variance

in the number of intercepts on the probe, we have from Eqs. (3) and (4) with

respect to the number of point intersections on profile boundaries:

P = 2N, (35) and „ „

CT (P) = 4j^(N). (36)

We cannot proceed further without some assumption about the dispersion of the

profiles on the section. A reasonable hypothesis is that the particles are randomly dispersed. (Some measurements we are currently making indicate that

this is a valid approximation for nodular and flake graphite particles in cast

irons.) The assumption of a random dispersion requires that the volume frac- tion of particles be small (say, < 15 pet) as, otherwise, there is an excluded area effect. For a random dispersion, the number, N, of intercepts on the probe will follow a Poisson distribution. Hence, from Eqs. (10), (35), and (36):

kg = a2(P)/P = 2. (37)

Thus, for non-contiguous particles, k^ is about twice as large as it is for a long probe on contiguous grains.

Discussion

The theoretical and experimental results indicate that for contiguous

grains kg decreases with increasing probe length until a limit is reached

65
J. E. Milliard

corresponding to P s» 2 for a straight probe and P 6 for a circular probe.

Thereafter kg rs 0.6 for both types of probe. It thus follows from Eq. (17)

that the relative standard deviation of the analysis can be estimated from:

a(Sy)/Sy 0.6/yp^,

• (38)

in which Prj. is the total number of intersections counted. For non-contiguous

particles occupying a small volume fraction we have:

a(Sy)/Sy ^ I.Wp^. (39)

Two qualifications are necessary in the use of these expressions. The first is that they should not be applied when the structure does not approximate the two cases considered [i.e., a space filling cellular type for Eq. (38) and a low density of discrete bodies for Eq, (39)]. For other types of struc- ture kg should be estimated experimentally as described in a later section. The second qualification is that Eq. (7) [and therefore Eqs. (38) and

(39)] are valid only if the individual measurements are uncorrelated. As an extreme example consider an estimation of V-^ in which a series of traverses

are made that exactly overlap one another. It is obvious that, no matter how

many traverses are made, the result can never be any more reliable than that

obtained on the first traverse. Thus, if parallel traverses (or probes) are used, the spacing between them should be commensurate with the scale of the structure. This condition is not satisfied by automatic image scanners using

closely spaced scans. However, an approximate estimate of the sampling error

can be obtained in such cases by setting P^p in Eqs, (38) or (39) equal to the actual number of intersections divided by the average number of scanlines

intercepting a grain or particle profile,

LINEAL FEATURE ANALYSIS

Lineal features within a specimen yield point intersections on a planar

probe. If Ly is the length per unit volume and P^ the number of intersections per unit area, then:

Lv = 2Pa. (40)

By analogy with Eq. (39):

CT(Ly)/Ly = k^/VP^ (41)

where Pip is the total number of intersections counted. If the features are randomly dispersed then it follows from Eqs. (8) and (10) that: \ = 1. (42)

An experimental study by Hilliard (1966) showed that the assumption of a random dispersion was valid for grain edges in a polycrystalline metal specimen

and that k-|^ was independent of the size of the probe. However, these results

will not necessarily be true in general and k-|^ should therefore be determined

experimentally.

EXPERIMENTAL DETERMINATION OF k

As an illustration of the experimental determination of k we will consider

some of the data collected in the study of grain edge length referred to in the previous section. A circle was applied 200 times to non-overlapping regions

on a set of micrographs and the number, P, of triple junctions falling within

the circle was counted. The observed frequency of counts is given in the Table,

66
Assessment of sampling errors in stereological analyses

TABLE

Distribution of Counts in Test Circle

P.

1 Frequency, n^

0

1234

114

> 5

6118

6

10

The average number of counts is calculated from:

P = En.P./n, (43)

where n = En.._ For the data given in the Table, n = 200 and Sn.P^ = 119,

which yield P = 0.595. The variance is given by:

which yields ct (P) = 0.664. Thus:

= CT(P)/yp = 1.057,

which differs slightly from the theoretical value of 1. In order to test

whether the difference is statistically significant we need to calculate the standard deviation of k. It can be shown that:

The data in the Table yield (^^(P) = 2.106 which on substitution into Eq. (45)

gives o'(k) =0.09. We can therefore conclude that observed deviation of 0,06

from unity is not significant.

REFERENCES

Gladman, T. & Woodhead, J. H. (1960) The accuracy of point counting in metallographic investigations. J. Iron Steel Inst., 194 , 189. Milliard, J. E. (1962) Grain-size estimation. General Electric Research

Laboratory Report 62-RL-3133M.

Milliard, J. E. (1966) Sampling error in the estimation of grain-edge length.

Trans. Met. Soc. AIME 236, 589. Milliard, J. E. (1968) Measurement of volume in volume. In: Quantitative

Microscopy, p. 45. (Ed. by R. T. DeHoff and F. N. Rhines) McGraw-Hill, New York. Milliard, J. E. & Cahn, J. W. (1961) An evaluation of procedures in quantitative metallography for volume- fraction analysis. Trans. Met. Soc. AIME _221, 344. Philofsky, E. M. (1966) MS Thesis, Northwestern University.

(45)

in which ti^(P) is the fourth moment of P about the average; i.e

^^(P) = (l/n)Ln.(P. - P)^. (46)

67

I

National Bureau of Standards Special Publication 437

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithereburg, NW.. September 4-9, 1975 (Issued January 1976)

ON OPTIMAL FORMS FOR STEREOLOGICAL DATA

by A. J. Jakeman and R. S. Anderssen Computer Centre, Australian National University, Canberra, ACT, 2600, Australia

INTRCDCCITCW

A basic aim in stereology is the detennination of three-dimensional Cstruc- tural) properties of some given object from lower-dimensional data (Z-d data) of

it. Recent research has concentrated on the construction and inversion of the formulas which define some particular three-dimensional (structural) property in terms of the various forms of l-d observational data which can be used. See, for example, the work of Santalo (1955) on the size distribution of similarly

shaped convex particles, and that of Matheron (1974) on the application of

random set theory to the construction of stereological formulas, as well as the examinations of Jakeman and Anderssen (1974) and Anderssen and Jakeman (1974) of

the different computational methods proposed for the inversion of the formulas

for the thin section and random spheres models.

For any three-dimensional structural property, it seems that a classification of the various forms of £-d data would be useful, so that optimal forms

for it can be delineated for the determination of the given property. Other

advantages of such a classification are obvious. Among other things, it can be used by experimenters as the basis for the design of stereological experiments

to determine specific three-dimensional properties. For each three-dimensional

(structural) property, the aim is to highlight, and thereby avoid the use of,

sxiboptimal forms of i-d data.

It is clear that the actual classification must be based on the accuracy

and efficiency with which the required three-dimensional property can be

obtained. The implementation of a classification along such lines is not straightforward. The first complication is the fact that the types of lower- dimensional observational data which can be used to estimate the given property

depend heavily on underlying assimrptions . For example. Milliard's (1968) con- clusion that systematic point counting data is optimal for volume-in-volume

estimation rests largely on his definition of efficiency and on his choice of

minimi.nil grid spacing.

In trying to produce a classification, the problem of the underlying assumptions has even deeper ramifications than this. For example, let us consider a distribution of ovoids in a field. If no assimiption is made about these ovoids

other than that they are convex, then it follows from the general Minkowski

functional formulation for Crofton's theorem {see, for example, Hadwiger (1957),

p. 209} that it is not possible to estimate the number density of ovoids from i.-d observations - i.e., no formula relating the number density of the ovoids to i.-d observations exists.

Thus, the essential steps in such a classification must be: Step 1 . For a particular three-dimensional (3-d) property and given assumptions, the determination of the types of £-d data for which the existence of

formulas (which relate this data to the required 3-d property) is guaranteed.

Step 2 . The specification of the conditions under which the classification

is made. Step 3 . The classification.

Since space does not allow a full classification of all stereological data

at this stage, we limit attention to the following points:

(i) The determination of the size distribution and number density of

69
A. J. Jakeman and R. S. Anderssen

spheres in some given matrix.

(ii) Optimal properties of linear versus plarar data. For brevity, we do not pursue the question of the exclusion of t-d data on the basis of Crofton's theorem or other grounds such as smallness of sample size. In fact, with respect to a given 3-d property, we will only work with Ji-d data for which formulas relating the two are known, and will assume that the amount of available data is adequate for the computational methods to be considered.

In this note, the classification will be based on the numerical stability

and reliability of the computational methods available for the evaluation of the 3-d property from the known formula when using observational (noisy) data. It is also assumed that the wrong formula is not used, as has been the practice in the past when, for example, the random spheres model has been used to invert thin

section data. Use of the thin section model avoids the Holmes effect {see, for example, Tallis (1970)} and the necessity to correct for it. In fact, computing

expertise has reached the point where it is no longer necessary to use expedients

such as model degradation and subsequent correction of the abovementioned type. The correct formula should be solved on its own merits using available techniques. If such techniques do not yield satisfactory results, it is preferably to seek new ones which do, rather than appeal to some model degradation strategy.

THE SIZE DISTRIBUTICM AND NUMBER CENSITY OF SPHERES

Under the assumption that the population of spherical particles, with a size

distribution (probability density) function g(x) for the diameters, is randomly

dispersed throughout space and that their centers are defined by a Poisson process of low intensity such that overlapping of spheres does not occur, it is known that the following formulas exist for defining g(x), and the number den- sity (number per unit volume), in terms of £-d data:

1. Plane-of-Polish Data (The Random Spheres Model)

h(y) = J

/ (x' - y')"^ g(x)dx, 0 $ y < oo

, (la) y

= N^/m , (lb)

where m is the expectation of the spherical diameters, h(y) is the size distribution of the circular diameters observed in the random plane-of-polish

sections, and is the expected number density of circles (number per unit

area)

.

2. Thin Section Planar Data (The Thin Section Model)

c(y) = (y / (x^ - y^)'"' g(x)dx + t g(y)| , (2a)

Ny = N^/(m + t) , (2b) where c(y) is the size distribution of circular diameters observed in random

planar thin sections of thickness t .

3. Linear Probe Data In Plane-of-Polish

--h {^}/f'CO) ,

• , C3a)

= i f(0) ,

' (3b)

where f(x) is the observed size distribution of the length of line intercepts

between the linear probes and circles in the plane-of-polish, and N^^ is the expected number density of intercepts.

Note. The above formulas for number density exist because the assumption

70
On optimal forms for stereological data

that the ovoids are simply convex has been greatly strengthened by assuming that they are in fact spherical particles with centers defined by a Poisson process.

This is much stronger than the minimum assumptions required to yield number

density of ovoids from l-d data; viz., similarity of the convex particles distributed homogeneously and isotropically throughout space.

Now, Moran (1972) states in §6 in his review of "the probabilistic basis of

stereology" that

:

"linear probes should not be used, if the density of centers and diameter distribution are required because

the use of plane sections is more effective".

The basis for his conclusions is Nicholson's (1970) work on the use of linear

probe data to estimate diameter distribution and the density of centers.

Classifying this 2.-d data on the basis of the numerical stability and reliability of the computational methods available for the evaluation of the above

formulas, we aim to confirm and substantiate Moran's conclusion.

For the numerical solution of (la) and (2a) , we use the procedures proposed

by Anderssen and Jakeman (1974) . For synthetic planar data, which yield a bimodal size distribution of spheres, the results are shown in Figures 1(a) and

1(b). We compare the results with those obtained for the solution of 3(a) by the following two methods.

Ml. Group the given cumulative data {F(X)^); k=0,l,2, . . . ,n} onto an even grid {tj^} in the abscissae direction to obtain {F(tj^); k=0,l,2, . . . ,N,N < n/2} . Then estimate g(t, ) of (3a) as

g(t,)

1 d'F(t)

^k _ dt^ _ t=t.

dF(t)

dt

t=t.

d"F(t)

dt^

t=o

for k=0,l,2,, ,N . Here [d F(t)/dt ]^_^ denotes the FFT spectral derivative

of order m of Anderssen and Bloomfield (1974) evaluated at the point t=t, for the data {F(tj^)}.

M2. Using the even grid of cumulative size distribution data in the ordinate direction, estimate g(x^) of (3a) as

g(x,)

* * ^ d^x dx

_dF^_ x=x^/

dF

. X=Xj^.

1/

* 1 * 1

^

2 dx dx d^x

^k dF

x=x,

k^ li

dF x=o| / LdF^J

For synthetic linear probe data, which yield the abovementioned biomodal size

distribution of spheres, the results obtained using these two method are shown

in Figures 2(a) and 2(b).

It is clear that a comparison of Figures 1 and 2 justifies Moran's conclusion. In addition, we see that the inversion of planar and thin section data

71
A. J. Jakeman and R S. Anderssen

Sphere size distribution 400 points using

exact planar data

Sphere size distribution 400 points using

exact thin section data

1 60 2 00

Residuals Residuals

CE

'0.00 0.40 0.80 120

diameter

1.60 2.00 '0.00 0.40 0.80 1.20 diameter

1.60 2.00

Figure 1(a) Figure ICb)

Sphere size distribution 1000 points using

exact linear probe data

8n Sphere size distribution 1000 points using

exact linear probe data

0.00 0.40 0.80 1.20 diameter

1.60 2.00

Figure 2(a) Figure 2(b)

72
On optima/ forms for stereological data

does not suffer from such severe numerical problems when methods of the proposed

type are used. As shovm in the Figures, even the use of exact data does not alleviate the problem posed by linear probe data. Thus, we conclude that, for

size distribution estimation, linear probe data are suboptimal. The two factors

which contribute most to yield this conclusion are the effect of the 1/x factor

in the neighbourhood of the origin and the near impossibility of being able to accurately estimate the constant of proportionality f ' (0) = F"(0) from truncated data. Because of the latter problem it follows that the best which can be achieved with linear probe data is the determination of the shape of g(x) but

not its true-scaled value.

VOIZJ>E IN VOLUME

Even though we have concluded above that there exists a context in which

planar data are superior to linear probe, this does not establish a general rule

of thumb. In fact. Milliard (1968^ has shown that there do exist situations

where the linear probe are superior to the planar.

When estimating volume in volume, the computational problem (viz., evaluation of a fraction) is trivial, and so, the classification of data on the basis

of the numerical stability and reliability of the resulting solutions is inappropriate, and sane more suitable criterion is required. Using efficiency, defined

in terms of the least effort required to yield a given accuracy, as the basis

for classification, and excluding automatic measuring processes fron consideration, Hilliard (1968) concludes that point counting data are optimal for the measurement of volume in volume. The conclusion hinges on the assimption that it

is easier and simpler to decide whether a point falls on a region than it is to specify the boundaries of the region.

This corroborates the above point that the optimality of Jl-d data for the determination of a given 3-dimensional structural property depends heavily on underlying assumptions and the criterion of optimality.

CAVEAT

A conclusion that, with respect to a given optimality criterion, a particular

type of i-d data is suboptimal for the determination of some 3-d structural pro- perty does not necessarily exclude its use. However, it does indicate that there

exist other l-d data for which the required structural property can be obtained

with greater reliability and/or efficiency. For example, in concluding above

that, compared with planar data, linear probe data are suboptimal when determining

a size distribution for spheres, it is not implied that linear probe data cannot

be used; but rather than they should be avoided unless there exists for their use

a strong independent argument which is outside the context of the present considerations.

REFEFENCES

R.S. Anderssen and P. Bloomfield (1974) A time series approach to numerical

differentiation, Teohnanetrics 16, 69-75.

R.S. Anderssen and A.J. Jakeman (1974) On computational stereology. Proa. 6th. Aust. Comp. Conf. J Sydney, Vol. 2, 353-362.

H. Hadwiger (1957) VorlesKngen uber Inhatt, OberfVaehe vend Isoperimetrie

,

Springer, Berlin.

J.E. Hilliard (1968) Measurement of volume in volume, in Quantitative Metallograpky (Eds. R.T. DeHoff and F.N. Rhines) , McGraw Hill, New York.

A.J. Jakeman and R.S. Anderssen (1974) A note on numerical methods for the thin

section model. Electron Microscopy 1974, Vol. 2, 4-5.

73
A. J. Jakeman and R. S. Anderssen

G. Matheron (1974) Random Sets and Integral Geometry, J. Wiley Interscience

,

New York.

P. A. P. Moran (1972) The probabilistic basis of stereology, Suppl. Adv. Appl.

Prob. (1972), 69-86.

W.L. Nicholson (1970) Estimation of linear properties of particle size distributions, Biometrika 57, 273-298.

L.A. Santalo (1955) Sobre la distribucion de los tamanos de corpusculos

contenidos en un cuerpo a partir de la distribucion en sus secciones o proyecciones, Trabajos Estadist 6, 181-196.

G.M. Tallis (1970) Estimating the distribution of spherical and elliptical

bodies in conglomerates from plane sections. Biometrics 26, 87-103.

74
National Bureau of Standards Special Publication 431 Proceedings of tfie FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, IVW., September 4-9, 1975 (Issued January 1976)

MULTIVARIATC DATA ANALYSIS TO DESCRIBE INTRA-AIMD IIMTERGRAIMULAR RELATIONS IN THIN SECTIONS

by W. Good

Sw/'ss Federal Institute for Snow and Avalanche Research, CH-726Q Weissfluhjoch/Davos, Switzerland

Introduction

Structure and texture are only part of the physical reality.

The physical and mechanical properties of matter are other important qualities. The description of structural properties therefore

should aim at a physical understanding of the material being investigated , The scope of this paper is to present a semi-quantitative

correlation between structural elements and mechanical properties

.

Experimental part

Samples of homogeneous snow were altered under controlled

conditions. The resulting samples were tested according to ram

hardness R(N) and tensile strength ax(N/m2)

.

In parallel experiments, thin sections were prepared and

structural elements investigated.

Numeriaal treatment of thin seation information

The primary information for each thin section is a point

grid of light intensities. Each observation point is implicitely

characterized by the orientation of the crystallographic c-axis

.

The computer program MSTER.FTN either generates coherent

areas joining points with the same crystallographic information

(Fig 1) , or links contiguous areas even if crystallographically

inhomogeneous (Fig 2)

,

A parametrization of these simple or complex areas yields 28 structural parameters (1) divided into:

- Parameters of point information (Point density e.g.).

- Stereological parameters (Grain surface per unit volume, mean

grain diameter, mean void diameter etc.).

- Geometrical parameters (Grain area and circumference, mean radius of convex and concave borders, mass distribution

(Principal moments of inertia tensor) etc.).

- Crystallographic information (Azimuthal and zenithal angle of

c-axis)

.

Out of this redundant set of parameters, 21 are kept for a subsequent treatment.

Paotoriat analysis

Each thin section, being a sample of the corresponding snow,

is therefore characterized by its p parameters (or 2p parameters

for inter- and intra- grain conditions)

.

N samples with p parameters form a nxp matrix. Substituting

each value in this matrix by its centered and reduced one, a pxp

dimensional correlation matrix may be obtained. It can be shown

(2) that the eigenvalue problem for this quadratic and symmetric

matrix yields an optimum, .orthogonal coordinate frame. The axes are linear combinations of the parameters . Parameters and samples

may both be located in this coordinate system and the euclidian

75
VJ. Good

distance between them is a measure of their differences.

Discussion of results

For each sample two sets of p parameters have been generated . Fig 1 corresponds to the crystallographically homogeneous

areas generated ( intragrain representation) whereas Fig 2 is an intergrain representation of the same sample with connected

structures

.

The second one is better fitted for the discussion of

mechanical properties. Fig. 3 and 4 are projections of samples

and parameters for Fig 3 in the first two principal planes. It

is seen from Fig. 3 that Fi is mainly a linear combination of

parameter groups 111 and IV (see further discussion)

.

In addition, sample groups with similar tensile strength

have been delimited (from left following the arrows O'p = 12000 , 4-07 50 , 55500 and 217000 N/m2) . This evolution in physical phase

space corresponds more or less to the path from parameters groups

I, II, ... to VI. Briefly they are characterized as follows:

- I Important variances of domain area and circumference as well as cry stallographic index.

- II Large voids and few grains.

- Ill Well defined border conditions i.e. lengths and radius of convex and concave circumferences.

- IV High density and large grain areas.

- V Well defined mass distribution and orientation of the grains,

-VI Large Ellipticity

(a^/iiirV I1I2' > a = area, Ii, I2 principal moments of

inertia) .

Conal usion

In additionto this description, it is possible for a given

sample to compute by discriminant analysis its probability to belong to one of the different subgroups.

This kind of semi-quantitative comparison is a first step

in the finding of models without assumptions, connecting structural and physical parameters.

Aaknowledgment

I am indebted to Dr. H.U. Gubler (Swiss Federal Institute

for Snow and Avalanche Research, Weissfluhjoch/Davos) for preparing

the snow samples and performing the mechanical experiments as well

as for valuable discussions.

Bibliography

1 Good, W. 197 5. Numerical Parameters to Identify Snow Structure.

To be published in:

lAHS Publ. No. 114 (International Association of Hydrological

Sciences (Proceedings of the Grindelwald meeting 1974).

2 Lebart, L. and Fenelon, J. P. 1973. Statistique et Informatique

appliquees, Paris, Dunod 2nd Edition.

76
eneous Areas

Multivariate data analysis in thin sections

Fig. 2 Contiguous Areas
W. Good

-0,4^

Fig. 4. Intergrain relations (samples 1-22, Gu, Kr, Ho) and tensile strength

in plane of factors 3 and 4 (84,4% of total information).

78
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

field at NBS, Gaithersburg, Md., September 4-9. 1975 (Issued January 1976)

CORRECTION OF STEREOLOGICAL PARAMETERS FROM BIASED SAMPLES ON NUCLEATED PARTICLE PHASES

by Luis-M. Cruz Drive, Department of Human Biology & Anatomy, The University. Sheffield S10 2LA, and Department of Probability & Statistics. The University. Sheffield S3 7fiH. England

INTRODUCnON

A system of 'free' cells in a medium can be modelled by a 3-d particulate

phase a dilutely embedded in a matrix 6. The particles, however, are not homogeneous; in general, each of them consists of a body contained in a

'bigger' one K-j^ (i.e., K-j^^K^). K-j^ may be regarded as a cell, and its nucleus. The matrix may consist of many distinct phases. It is required to estimate, (i) the true ratio Vj/V-j^ of the total

nuclear volume to the total cellular volume, and (ii) the specific surface area

(S-| /V-j^)rj, of the cells, from measurements made on random plane sections through

the aggregate. In normal circumstances, the corresponding frequency-free consistent estimates of the above parameters are well known, namely,

Vy^ = (V^/V-^)^ (E^ h^/fl \) ; (S-^/V-^)^ J a" B^/e" A^), (1)

viiere A. , A^ denote cellular and nuclear profile areas, respectively, whereas

B-|^ is the perimeter length of a cell profile. The symbol becomes = as the

number n of profiles monitored becomes infinity.

Suppose, ho^/ever, that the cell profiles cannot be identified unless a nuclear fraction is present in them. In other words, only transnuclear profiles

are analyzed, whereas the paranuclear ones - arising when the plane of section T

hits K-|^ but not - are discarded. Under this sampling regime, the observed

ratios

A^/Zj A^ I T+K^) ; (S^/V^)o ^ J a\ B^/eJ A^ | T+K^). (2)

are' no longer consistent estimates of the required parameters (1), and correc- tion functions vdiich relate VTM to V on the one hand, and (S /V )q to (S-j^/V-|^)rp on the other, are necessary. This is, in essence, the statement of the problem.

(The bars in (2) denote 'conditional to', and 'T+Kj' means 'T hits Kj').

THE MATHEMATICAL MODEL: GEMERAL CORRECTION FUNCTIONS

Let us assume that the cells K-j^ have a fixed shape and a variable size

determined by a single parameter X; likewise, Y.^ has a fixed shape (different

fron that of K-j^, in general) and a variable size, described by another parameter

Y. In general, (X,Y) will be a 2-d random variable with a distribution function

G(x,y). We then speak of a 'polydispersed phase of nucleated particles', and

G(x,y) represents the size distribution function of the nucleated particles in

the system. When X and Y are constant, we speak of a 'monodispersed phase of

nucleated particles '

.

In the general case, the stereological parameters sought are,

= k2.E(Y^)/ k-^.E(X^) ; (S^/V^)^ k2.E(X^)/ E(X^) , (la)

respectively, where k-j^, k„ and k^ are constant shape coefficients, and E(.)

denotes 'expectation' with respect to G. The correction functions are obtained by evaluating the right hand sides of

(2) when n ->- <», and combining the results with the respective equations (la). For simplicity, it can be assumed that the relative orientation of K2 with

respect to the containing K-j^ does not change, so that their relative position is fully determined by a randan vector h - 0^0, joining their respective centroids.

With the above requirements and notation, the equations (2) become

2Trk„.E(Y^) „ E E/„Bi(T|x,h) dT

%0 - E g;^i(f x,h) dT ' ^V^l^O " ¥ E g/^i(T|x,h) dT

^^^^

79
L.-M. Cruz Drive

which ccmbined with (la) give rise to the correction functions of biased nuclear

volume fraction and specific surface area estimates, respectively, on a polydispersed phase of nucleated particles. E and E denote expectations with respect

to the distribution of (X,Y) and of h,Srespefetively , whereas dT = sin9 dp d()) de

is the element of invariant measure for random planes in 3-d (see, e.g., Santalo

1953, p. 121; Kendall&Moran, 1963). fiz denotes the set {T : T+K„}.

The corresponding correction formulae for monodispersed phases of nucleated

particles are obtained by deleting E in the above expressions.

CORRECTION FORMULAE FOR MONODISPERSED PHASES CONSISTING OF A CERTAIN CLASS OF NUCLEATED PARTICLES. Suppose that the bodies K-^(X) Z) K2(Y), (X, Y, constant), constituting each

particle are convex and satisfy the additional conditions,

(i) is a contraction of K-j^ with ratio 0 < X < 1, and the centrxDid 0 of K-, as invariant point.

(ii) K-j^ and K2 belong to a class of bodies in which the area and boundary

length of a plane section of coordinates (p, (j), 9) factorize as

A(p,(t),e) = f{p/H((i),e)}.g((i),e) ; B(p,c|),e) = u{p/H((|),9)}-v((t),e), O)

where H((t>,9) is the function of support of the boundary of the body, and

represents the distance of the origin 0 from the tangent plane perpendicular to the direction ((1),9) (see, e.g., Bonnesen&Fenchel, 1934, p. 23). If and satisfy the above conditions, the equations (la) and (2a)

yield the following correction formulae,

1

= F(i).-J^ ; (s^/v^)^ = gi}. —f--(s,/v^),, W

z f^^vt) z ^

where F(Z) = / f(z) dz, and U(Z) = / u(z) dz. 0 0 EXAMPLE. Let the particles consist of two triaxial ellipsoids K, Z) K„ with the same eccentricities, a constant size and the three principal axes in common. It can be seen that the conditions set out above are fulfilled; the relevant

functions are found to be f(z) =_l-z , u(z) = (1-z )i, and the corrections (4) become (after appropriate inversion).

V = ( V 5 cs /V ) = ^—- ;-i_i_^2_,.(s /V )q.(5)

The above formulae coincide with that of a 'concentric spheres' model; the

result is logical, since this cell results after applying a linear transformation

to the ellipsoid-ellipsoid cell described above, and conversely. The first (5) was first given by Konwinski&Kozlowski(1972 )

, and rederived by Mayhew&Chruz (1973). CORRECTION FORMULAE FOR MONODISPERSED PHASES WHERE NUCLEUS AND CELL ARE

DISSIMILAR SPHEROIDS. When K^, K2 are either dissimilar or eccentric with respect to each other,

the corrections (4), (5) do not hold, and the general formulations (la) and (2a)

have to be recalled. This we have done for monodispersed phases where K-. and K2 are ellipsoids of revolution (spheroids) of the same kind - either prolate or

oblate - since the nucleus usually becomes prolate as the cell elongates, and

oblate as the cell flattens. The correction formulae are collected in Tables 1 and 2, together with the description of the models.

In Table 1, formula (6a) should be applied when the nucleus can be

tangential at the poles of K-j^ (i.e., when K2 is rather unequiaxed, or small),

whereas (6b) is more appropriate when these circumstances are not met. Formula

(6a) generalizes (17) of Mayhew&Cruz (1973) for eccentric spheres.

General (S /V, ) corrections for eccentric spheroids can readily be derived,

but unfortunately, they are far too complicated. By assuming that = X2 = 0 or that h = 0, (eccentric spheres and concentric spheroids, respectively) , the corrections become more tractable (Table 2).

3 3 i7 .0 «, ^ TT ^^VT " \t

80
Correction of biased samples

o

KO me

[i-. o s

+-> O

o>

>

\ [to

CN O

C-J 1— -IIX

I

C5

iH o

CM 1—

1

+O

CM I—

1

cr

+

CM

a'

iH |co

CM

CD

1

.

(6b)

1 rH

H |CM

<^

1

+

rH

1

CO CD

ir II II II II CM o

CO Ln

p a

X

I

<SJ.-t X

I

iH |m

'iH ft

CM r- ^ LX

CM rH -H|X

rH CU

CM r- rH| X

CM o

rnTo'

I CO

CNi Iro cj' CD

I

rH + 1

rH CO 'cr

II II 11 CO

CD

I

I

•!3

o

8

4h

CD

CO I

.

HO -P •H W

Mh <U Mh 3 8 S

W -P rH to

Q) &5

81
L.-M. Cruz Drive

TABLE 2. THE SPHEROID-SPHEROID CORRECTIONS OF BIASED SPECIFIC SURFACE AREA.

(Si/V-l)o

(Si/Vi)x

V

VT

Estimate obtained from nuclear-biased profiles.

'True' (corrected) estimate. True nuclear volume fraction (see Table 1).

MODEL (S) : Nucleus and cell are eccentric spheres, the distance hetQeen their

centers being uniformly distributed.

Correction formula:

i , (s^/Vt = ^^V^-^V^^o' where, putting = k,

1 C(V ) = \ k(l-k) (4+k-2k^){k / ^sin"-'"(k+(l-k)t) - sin~-'-(k-(l-k) t) }dt +

Q

t + (6-3k-2k"^) -

"I (30-llk+20k^)/k(l-k) -

-I (6-9k+2k'^) sin""'"(l-2k) +

+ i(2+k^)/mr^ log(l+2k+2/k(l+k) }""" (7a)

MODEL (P): Nucleus and cell are concentric prolate spheroids. MODEL (0) : Nucleus and cell are concentric oblate spheroids.

MODEL (P): K^(a^,b^,b^)Z> K^(a^,b^,b^). MODEL (0) : K^(a^,a^,b K^(a^,a^,b ^)

:

Correction formula: ,„ ,,, s 2 2. , . , ^ l' l^T

" C(V^^,x^,X2)

•

(S^/V^)q, (8)

, , , kl - -k^{(x /x )^I + (l-(x /x )^)I } where C(V^^,x^,xp = |^ Q(x^)--*^ ^ ± ^ (8a)

n=l

(2n-l) !

!

(2n) ! !

2n

''l ~:r M

2n-l n

Q(x^) = (1/x^) sin'-'-x^ + (1-x^)^; Q(x^)=l+{ ( 1-x^) /2x^}log{ (1+x^) / (1-x^) } ;

M = / (l-t^)'^{sin"-'-(ku)+ku/l-(ku)^}dt;M = / —'^K^ J , {sin"-'-(ku)+ku/l-(ku)^}dt; " 0 " 0 (l-x^t^)"-^

u = {(l-X2(l-t^))/(l-x^(l-t^))}2; u = {(l-X2t^)/(l-x^t^)}^;

1 1

I = / u dt ; I = / u dt ; 0 0

I = / u/{l-x^(l-t^)} dt ; '^7 = ^ u/(l-x^t^) dt.

0 0

2 2 (Curves for a direct obtention of (7a) and of (8a) when either x^^ = 0 or X2 = 0, are available from the author, on request.)

REFERENCES

Bonnesen, T. and Fenchel, W. (1934) Theorie dev konvexen Korper. Springer, Berlin

Kendall, M.G. & Moran, P. A. P. (1963) Geometrical Probability. Griffin, London. Konwinski, M. & Kozlowski, T. (1972) Zellforsch. mikrosk. Anat. , 129, 500. Mayhew, T.M. & Cruz, L. M. (1973) J. Microsc. , 99, 287. Santalo, L.A. (1953) Introduction to Integral Geometry. Hermann, Paris.

82
National Bureau of Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

field at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

STOCHASTIC MODELS IIM STEREOLOGY: STRENGTH AND WEAKNESSES

by Jean P. Serra Ecole Nationale Superieure des Mines de Paris, Centre de Morpfiologie Mathematique, 77305-Fontainebleau, France

SUMMAJIYIn stereology, the necessity of a model appears as soon as one wants to extrapolate, predict or interpret the geometrical properties of a structure.

One can imagine questions being asked at several levels when modelling. For statistical extrapolation (estimation variance of a volume percentage, for instance) it is sufficient to fit a punctual covariance to a mathematical function.

To predict the mill degradation of ore, one needs axioms for the granulometries

"in situ". To define and characterize clusters of features, one has to compare

the experimental results with moments of random functions. At the present time, the main handicap in modelling is the lack of statistical theory when using the notion of a random function.

INTRODUCTION.

In stereology, we have two main aims, firstly the aim of describing the geometric structure structural analysis, secondly the linking of this description to physical properties. On one hand, we want to extract several significant parameters, or functions, from complicated geometrical sets. On the other

hand, we hope to emerge from pure geometrical considerations and explain, or simply correlate, physical phenomena with morphological data. For example, we may want to relate mechanical strengths to sizes of metallic grains, or, in the case of a rock, petrographic arrangements to ease of milling (1), or, in biology, morphological changes in neurons to hypoxia (2), etc., etc.. From a practical point of view, the second of these objectives is the most

important. The people who use stereology place little importance in the manner

in which the correlation was found, as long as they are satisfied that it truly

exists. Nevertheless, the first objective is fundamental, for only it gives an overall view of the features of interest. In fact, in practical studies it becomes clear that these two objectives are two faces of the same reality and complement each other.

LEVEL OF SOPHISTICATION.

Let us for the moment focus our attention on the structural analysis approach. Classically, we define several basic parameters, such as the phase areas, perimeters, connectivity numbers, size distributions, etc.. By doing this, we implicitely assume that the features studied are the realisations of a random

function. We use random functions here as a means to an end. We do not want to say whether the features observed are indeed random or deterministic phenomena,

but only that at the scale of observation, they appear chaotic. Our interest

however, is not in local description, but in mean values, which carry 'physical

meanings. For example, let us consider the porosity of a rock. We are not interested in the fact that a given point x belongs to the grains or to the pores,

but only in the proportion of such points : this is exactly a changing of scale,

and the theoretical tool for approaching it is the concept of a random function.

Along this line of thought, the classical stereological parameters appear

as the various moments of the random function. The proportion of a phase (classically denoted by A^ or V^) is an estimate of the first order moment. In the same way, one can show that the other basic stereological parameters (surface, mean and total curvature) are the moments associated with groups of two, three

or four points. In many applications we need no more than this first set of moment s

.

But one can easily imagine some problems for which we need to go to higher

order moments. Let us give two examples. First the estimation variance of a votume percentage : it is known (3) that this variance estimation depends only

83
J. p. Serra

on the punctual covariance, which is the moment of order two (probability of finding two points simultaneously in the chosen phase). Second example : the size measurement of one particular medium (chord distribution for example).

G. Matheron proved that this kind of measurement depends only on the probability that on an arbitrary point x, we can put a given convex set that is contained totally within the medium (infinite moment)

.

The two steps above (first moments, and more complicated ones) are not sufficient for solving all the problems. Let us consider for example the following question : "Does the structure that we see, exhibit clustering?". The concept of a cluster is not defined on the same level as the perimeter, and needs, for its description, the reference to a random function with clusters

(or without)

.

Another reason for fitting a model comes from the very large number of data provided by the moment measurements. Each size and shape of the structuring element corresponds to a different probability. It seems often that the key parameters are marked by too much data. The model allows us to predict the theoretical value of every moment, and, as soon as it is correctly fitted,

substitutes its own two or three parameters into the moments.

^ 3 The last reason is that the stereological inference from R to R becomes

easier when using models.

Today, the library of models in stereology is not very large, and we do not pretend to give here an exhaustive list of them. The most developed are the point processes (4). Starting from the Poisson Point process, one can sophisticate it by varying the density, and introducing clusters around the Poisson

Points

.

A second group of models consists of the tessellations of the space R (5). The basic one is the Poisson flat process, studied by R.E. Miles and later by

G. Matheron. In the same group of models, one can quote the doublet of Poissonnian planes of Cauwe, and the Voronoi polyhedra (Meijering, Gilbert, Miles). A third group consists of the boolean schemes. We will present later the basic prototype.

The three groups probably represent the building blocks in model construction, since the Poisson point processes, the Poisson flats and the boolean

scheme are indefinitely divisible.

THE BOOLEAN SCHEME.

In order to illustrate the previous considerations, let us present now, with more details, the boolean scheme. The definition of the Poisson point process in R is well known. It consists of a random set of points which satisfies the two following properties :

a) if B and B' are two sets such as B fl B' =0 , the numbers N(B) and N(B')

of points falling in B and B' are two independent random variables.

b) the elementary volume dv contains one point with the probability 9(dv) and none with the probability I-e(dv). The measure 6 is called density of the process. Here we will take 6 = constant, because it leads to more geometrical results.

By integration, the two properties imply that for any bounded B the random

integer N(B) satisfies the Poisson law with parameter

e(dv)

P{N(B)=n} = ^ e' (I)

Let us take a realization of a Poisson process of constant density

This realization has points which may be indexed by the set I. We consider each point as the germ of a crystalline growth. If two crystals

84
Stochastic models in stereology

meet each other, they are not disturbed in their growth, which stop independently for each component. Let us transpose this description in terms of random

sets. The points of the Poisson realization are in x. (i € I). The elementary

grain is a nonstationary random set A'. Successively, we pick out various reaI ! lizations A. of A' from its space fi of definition and implant each A. at the corresponding point x. . The different A. are thus independent of each other. We

will call the realization A of a Boolean scheme the union of the A. associated

with a given realization of the points x..

A' = UA^

i € I

The Boolean scheme is extremely fertile. It represents one of the first steps,

in modelling, when one admits negligible interactions between the particles A.. We had the opportunity to use it in sedimentary petrography, in dendritic

crystalline growth, in studies of forests, and all its fields of application

are probably not yet discovered.

The basic study of the Boolean scheme has been made by Matheron.

We center the primary grain A' at the origin. This random set is known by

the datum of the two families of functionals oj(B) and x(^)'

5r(B) = P i B C A'} and x (B) = P 1 B fl A ' ?^ 0! (2)

If A' is translated from the origin to the point z, it admits the new

functionals w^(B) and X.^(B) easily deducible from (2)

u^CB) = P {b C A^i = P jB_^ C A' } = a!(B_^) (3)

X^(B) = x(B_^)

Let us consider now the Boolean scheme itself, namely the union of all

the A|, and compute the probability of B being included in the pores ^A

of the scheme. According to the property (a) of the Poisson process, each element of volume dz of the space makes its contribution independently of the others. In

dz, centered in z, two incompatible favourable events may happen :

1 - no germ in dz : probability l-Gdz.

t 2 - one germ in dz, but the grain A does not reach B : probability

edz. x(B_^).

^

By composition of these tyo probabilities and summation in z extended to the space r"^, we find

-eX t'- x(B_^) ]dz

Q(B) = P jBC^A} = e ^ (4)

To interpret the geometrical meaning of the integral in (4) , let k(z) be

the indicator function of the random set A' ® B, that is

k(z) = 1 when A' f] B^ / 0

k(z) = 0 when A' f| B =0

85
J. p. Serra

and mes A' @ h = f k(z)dz.

By taking the mathematical expectation, and inverting the operations E and

J

, we find

E[mesA®B]= [ 1- x(B_ ) ] dz R

Thus formula (4) may be written

- 6E [ mes (A' © B)] Q(B) = e (5)

which is the fundamental formula for the Boolean scheme. It links the functionals

after randomisation to those of the primary grains. In other words, the proportion l-Q(B) of the dilated A © B (final state) is related to the measure

E [mes A' © B

J of the dilated primary grain by an exponentiation.

Unfortunately, one cannot have such a beautiful result for the erosions.

The principle of the construction of the scheme gives priority to the logic of intersection of the grains.

Formula (5) leads directly to the computation of the usual functionals

(a) The covariance :

rrM 2 r 6K(h) ,T - BE [Mes A'] C(h) = q [e -IJ q = porosity = e

K(h) = E[Mes(A'n A^)

]

(b) Intercepts distribution function ( dir. a) :

-eD I

I - F (1) = e D = diametral variation of A

(dir. a)

The intercepts law is a negative exponential. This important property ap- pears as a particular case of the general notion of semi Markov processes

(Matheron (3)). In practice, it may be used for testing the Boolean hypothesis.

REFERENCES.

1. Ph. Cauwe - See paper in the same Congress.

2. 0. Hunziker et Al.- See paper in the same Congress.

3. G, Matheron - Random Sets and Integral Geometry.- John Wiley and Sons, New

York, 1975.

4. Stochastic Point Processes.- Edited by Peter A. Lewis, John Wiley and Sons, New York, 1972.

5. R.E. Miles - in Proceedings of the Symposium on Statistical and Probabilistic Problems in Metal.- Adv. in Appl. Prob.- 1972.

6. J. Serra - Mathematical Morphology and Image Analysis.- SIAM, Washington, D.C.- To be published.

86
National Bureau of Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

A STEREOLOGICAL VIEW OF DATA ANALYSIS

by George A. Moore and George T. Eden

National Bureau of Standards, Washington, D.C. 20234, U.S.A.

The basic stereological reasoning by which structure in a two-dimensional

image is interpreted as structure of a solid can be usefully applied to rational

deduction of meaning in collections of numerical data. Data collections from

the real world often involve many variables and fail to conform to theoretical

ideals of independence of variables or rectangularization of design. Customary

statistical tests often fail to support logical or apparently obvious relationships while at the same time developing "statistically significant" associations which have no logical basis for acceptance as functional relations.

Visualization of simultaneous numerical observations as points plotted in data

space or hyperspace is a substantial aid in rational sorting of functional

relationships from accidental or indirect associations.

In a stereological investigation of a material problem, parameters of a three-dimensional substance are determined with the expectation that they will

predict, explain, or control some subsequent behavior. A large number of re- dundant parameters may be invented; hence functional meaning may be expected

only for a limited subset of parameters which properly define a Gestalt (1) controlling the behavior considered. A proper description of this Gestalt, to- gether with a tracing of the effect of treatment on the descriptive parameters,

is usually necessary before these treatments can be effectively modified to improve behavior.

In the geometrical visualization each specimen supplies the position of one point in a data hyperspace having behavior as the dependent dimension. Some geometrical figure descriptive of the functional relationships may be discovered

either by computational processes or by visual study of 2-d or 3-d abstracts.

Projections into 2-d are subject to the same types of stereological illusions

encountered when real 3-d structures are examined from a single viewpoint. Com- plex illusions in interpreting higher dimensionality must also be expected.

Among the usual analytical operations simple correlation and ordinary regression

computations treat only 2-d projections of the data. Multivariate analysis

methods and in particular simultaneous regression computations attempt to interpret the n-dimensional form of the data structure.

Total regression is in bad repute for producing false relations when

redundant dimensions are present in the data. We tested the actual computer program by deliberately introducing a column of pseudo-data derived by a linear

relationship from a column of real data. High slope coefficients resulted but even higher standard deviations of these coefficients clearly showed them to have zero significance. This implementation thus may produce no solution when

an excess number of dimensions are used. The original representation of all available data as a single hyperspace normally offers excess dimensions. Analysis is facilitated by logically dividing this hyperspace into lower dimensional

spaces with each lower space limited to a set of parameters which might be

functionally related. A rule that cause must be present immediately before

effect can occur enables data on a material problem to be divided into one space representing possible effects of structural parameters on behavior and other

spaces representing effects of preparation and treatment on each structural

parameter separately.

Limited data can support determination of only a limited number of equation

terms. Thus it is usually necessary to forego the luxury of estimating curvature and synergistic relationships and confine attention to the approximation

given by the best fit plane or hyperplane. For this the "statistical significance" of each parameter can be conveniently indicated by the ratio t = slope

87
G. 4. Moore and G. T. Eden

coefficient i standard deviation of this coefficient. A high statistical asso- ciation may be found when a parameter has a small overall effect on the dependent variable. Hence, "functional significance" also requires a slope coefficient of sufficient magnitude that the total range of the "causative" parameter

accounts for a substantial fraction of the range of the dependent variable.

The product of t with this fraction appears to be useful in judging relative

functional significance.

An illustration of the highly variable and often contradictory results

obtainable by various methods of analyzing the same experimental data will be

presented using data previously obtained by one of us (GTE) on brittle fracture

of dental amalgams (2) . Eight screen separated fractions of spray-formed

"spherical" silver-tin alloy powder were used, along with commercial alloy in

the form of turnings as a control. Following customary dental practices, 30 cylindrical amalgam restorations were made from each powder. Six cylinders

from each particle size were compacted at each of five specific pressures. After setting, five specimens from each of the 40 sets were broken in diametrical compression and the nominal (tensile) fracture strength determined. A full cross section micrograph of one specimen of each set was analyzed by scanner to determine parameters of the void structure. Mercury analysis was obtained from

the broken pieces. One determination of unreacted silver-tin core size was made for each particle size, with the finding that the unreacted core size was directly related to the original nominal particle size. Very early scanner

data have been discarded as unreliable. Data Set B uses void measurements made

at an arbitrary threshold setting and includes amalgams made from commercial

alloy turnings. For data Set C the scanner data were recomputed at objectively

determined thresholds and additional parameters computed. Only the spherical

alloys are included in this set.

Simple (2-d) correlation coefficients for all data pairings are shown in

the lower triangle of Table 1. Partial correlation coefficients for the overall

n-dimensional systems are shown in the upper triangle of this table. Of the 45 correlation coefficients for which full observations are available in Data C, 25 are statistically significant at the 95% confidence level (*) . The matching

partial correlation coefficients show reversed sign in 13 cases, of which 6 are above the 95% confidence level for 1 or both coefficients. Thus it is apparent

that no conclusions regarding functional relationships can be drawn from these correlations. It is, however, useful to note that the cross-correlation between void content and void width is slightly lower than the correlation of fracture strength to void content for Data B, but higher than the correlation of fracture strength to either void content or void width for Data C.

Two-dimensional linear regressions. Figure lA & B, were computed for fracture strength against each of- the six potential causative parameters in Data B and against the nine parameters in Data C. Ranking of the potential significance of these parameters by the ratio t was essentially the same as by simple

correlation and unrelated to the partial correlation coefficients. The fraction

of the total variance of fracture strength ascribed to the "causative" parameter

was determined for each regression and summed for each set. Vci^anee was found

to be overaccounted for by a factor of 5.3 for Data B and of 7.9 for Data C. Such excessive explanation of the variance may be taken to indicate that each

two- dimensional view is more dependent on the unstated hidden factors than on

the parameter supposedly represented. In subsequent multidimensional regressions, it was usvially found that most of the variance was assigned to whichever

parameter had been listed first in the programmer's order, although the coefficients produced and their t values did not change with this order. Thus for these data at least, assignment of variance is of no value in determining the proper dimensions of the functional relationships.

Being unable to discover the functional causes of lowered fracture strength

by two-dimensional methods, simultaneous regressions were tried in various com- binations of dimensions. A method of sorting dimensions is illustrated graphically in Figure IC & D for the ambiguity between void content and void size as

88
A stereological view of data analysis

Correlation Coefficients Between AnHlgan Data

: Triangle = Simple CorreUtion Coefficients; Data C/Data northeast Triangle = Partial Correlation Coefficients; Data C/Dat; •Indicates significance aiwve 95S confidence level. Fig. IB

Figure 1: Fracture Strength vs. Void Parameters in 2, 3, and 7 Dimensions.

Figure 2 Influences of Preparation on Micros true tural Parameters and on Fracture Strength of Soherlcal Powder Aiwlnams (Data C)

89
6. A. Moore and G. T. Eden

the major cause of decreased fracture strength within the choices of Data B. Figure IC represents a three-dimensional box in which an observed fracture

strength is plotted simultaneously against void content and void width. Data

for each amalgam are represented by a ball mounted on a stalk whose height indi- cates the fracture strength and whose base is planted at the combination of void content and width measured for this specimen. The substantial cross-correlation

of void parameters can be seen in the pattern of the base points. The two- dimensional regression lines (lA & IB) of the projections of the cloud of balls against the two vertical planes are shown as dashed lines in these planes. If another plane is visualized as defined by these two regression lines, it may easily be seen that this plane lies well below most of the balls. Thus one or both regression lines must be an illusion of the projection into two dimensions. The best fit three-dimensional regression plane is represented by the lower of the two shaded planes in Figure ID. This plane has nearly the same slope In the void content direction as the linear regression. However, the slope of the 3-d plane in the width direction is much less than that of the linear regression, indicating a lowered functional significance for width. In a final step, the best fit hyperplane regression was computed in seven-dimensional hyperspace.

The three-dimensional abstract of this hjrperplane is shown as the upper shaded plane in Figure ID. The slope of this plane in the void content direction is little changed from that of the 3-d plane, but the slope in the void width

direction has been reduced nearly to zero. It is thus demonstrated that void content is the functionally significant parameter included in this data set and that void width, while showing apparently high significance in correlation and linear regression has no actual functional significance in its own right.

Attempts to determine the best functional parameters of the structural

Gestalt from the larger choice of parameters offered in Data C have encountered

difficulties which apparently trace to the high cross-correlations in this data

set and which may result from omission of the commercial alloys as strangers to the spherical particle system. Generally no more than five dimensions could be

included in one regression equation and only one parameter of the voids could be evaluated at one time. Regression equations in several 3 to 5-diinensional selections have resulted in the general map of apparent relationships shown in Figure 2. Relative weights are shown as the product of t times the fraction of range accounted for.

Failure of any of the simple void measurements to assume a dominant status led to the development of "relative surface coverage," the ratio of the measured

Sy of the amalgam, as generated by voids, to the Sy^ value characteristic of dry alloy particles of the nominal particle size. This constructed parameter

is of moderate functional significance when used in combination with total void content (which contributes to Sy) but becomes the dominant functional parameter

in a three-dimensional analysis including mercury content. Mercury content has

an adverse effect as logically expected but in contrast to the small positive

influence indicated by the correlation and linear regression computations.

In this example of analysis of less than adequate data, it is found

that representation of alternate Gestalt models by hyperplanes in data spaces leads to logically acceptable representation of functional effects. In contrast, two-dimensional analyses were usually ambiguous and frequently directly

contrary to the conclusions from controlled multidimensional analysis. Use of single viewpoints when analyzing complex data arrays thus is shown to involve hazards in interpretation which are analogous to those encountered when

viewing real three-dimensional systems from a single viewpoint.

REFERENCES NOTE - Discussions o£ multiple regression analysis

will be found in:

1. Moore, G. A., "Gestalt Properties of Aggregate Materials" - Practical Metallography, Vol. 9, 3. Draper, N. and Smith, H. , "Applied Regression

No. 2 (2/72) p. 76-97. Analysis," New York, Wiley Interscience, 1966.

2. Eden, G. T. and Waterstrat, R. M. , "Effects of 4. Daniel, C. and Wood, F. , "Fitting Equations to Packing Pressure on the Properties of Data: Computer Analysis of Multlfactor Data

Spherical Alloy Amalgams" - JADA - Vol. 74 for Scientists and Engineers," New York, Wiley

(4/67) p. 1024-9. Interscience, 1971.

90
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

field at NBS. Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

THREE-DIMENSIONAL SHAPE PARAMETERS FROM PUANAR SECTIONS

by Ervin E, Underwood, President of the International Society for Stereology

Visiting Scientist, Max-Planck-lnstitut fur Metallforscf)ung, Stuttgart, Germany On leave from Georgia Institute of Technology, Atlanta, Georgia 30328, U.S.A.

Shape parameters for features of the microstructixre may be expressed

in many ways, depending on the type of system being studied and the purpose

in mind (l) . The preferred shape parameters are those based on the quantitative, assumption-free, statistically exact equations of stereology (2)

,

obtained from simple micro structural measurements.

In this paper we utilize general relationships that equate measurements made on the random 2-d microsection to the spatial quantities

belonging to the 3-d features of the microstructure. Five 2-d and five 3-d

quantities have been selected for study.

2-d Q;uantities 3-d Quantities

A, intercept area V, volume

L , perimeter length S, surface area tangent diameter D, tangent diameter

Lg, intercept length L_, intercept length

k, curvature of lines K , curvature of surfaces

' m' These quantities may be expressed simply for convex bodies, or aggregates

of bodies, in space or in the section plane, (Note that primes denote

projected quantities.)

A = V^A V = V = V /N

v' v L

P

=7rs/ij.D = V\ s = hk' = S /N V v

-5. = S/llD = V^A

= V^L

D = A'/d

A' V \

= Uv/s ^3 = A/d = k = 8D/S = 2^a/\ K.m

= 7r/2d

For systems of convex particles of the same shape, we can express the 3-d

quantities in terms of an arbitrary linear dimension or "size", X (such as the mean intercept length L-,, or mean tangent diameter, 5), and a dimensionless Form Factor, k .

' n

n

jfonodispersed Polydispersed Log-normal Size Distribution

V = k^X^ V = k^M^ V = k2exp(3inXg+i+.5inV)

S = kgX^ S = kgMg S = k2exp(2inXg+2£nV)

D = k^X D = k^I.L D = k exp(inX +0.5in^r) K = k ^X" K = k ^ K = k ,exp(-inX +0.5jtn <r ) m -1 m -1 -1 m -1 g g The moments of the size distributions are defined by M (X) =Tf.X^/^t. = X

n 1 i' 1 For log-normal distributions, X is the geometric mean, O" is the geometric

S ! S . standard deviation, and the moments about the origin^. (X) = exp(i/nX + {l/2)±Hn^(r^) . (2) Various dimensionless groups of the above four Form Factors were

examined in order to determine their usefulness as Shape Parameters,

Several basic classes are distinguishable. Class I Shape Parameters are fimctions of «f/K , where e/, the specific surface, equals S/V. Class II
E. f. Underwood

Shape Parameters are functions of J'D, while Class III parameters are functions of D K . Class IV parameters consist of miscellaneous categories,

including ratios that are not dimensionless (3, 6).

Examples of the four classes of Shape Parameters are given below for monodispersed systems of convex particles, for polydispersed systems, and

for log-normal size distributions.

Class I Shape Parameters (functions of iq/k ) Monodispersed

-to— Polydispersed

DV

l6d ^p: N PA P K_

^A^P

Class II Shape Parameters (functions of;^D) 2 ^-1^2 2TrDS V

, A' ^A^L

Vp

Class III Shape Parameters (functions of D K ) m^

kg D 2P^N

' L V 2P^N

L V

Class IV Shape Parameters (miscellaneous)

Log-normal

hV^ \ ,ff 2 > L_] exp(i'n <r).

Vp

Vp

2P^N

L V

exp (/nV),

kgk^ S^ 16A'^

^^P

V P / ^P / It can be seen that the Class I Shape Parameters need only counting measurements for their evaluation, as noted by DeHoff (h) and Fischmeister (5). Consequently, several Class I parameters were formed from the five 2-d quantities

for possible application to different types of microstructures.

Examples of Shape Parameters deemed suitable for expressing particle

shape are L /Lg and L /Ak; for interlocking microstructures, L /A and kL /Lgj

and for single-phase materials, d/Lg, d/Ak or L^/Lg. Of course, other parameters (6) may perform as well or better, depending on the particular application desired. In any case, the above treatment provides a generalized

framework for devising new or modified parameters that relate featiares on the microsection to the 3-d spatial quantities of the microstruct\rre.

References ; Til Underwood, E. E. , "Stereology in Automatic Image Analysis", The Microscope, 22, First Qtr. (197M P. 69.

(2) Ifodei-wood, E. E. , Quantitative Stereology , Addison-Wesley Publ. Co.,

Reading, Mass, (197O) p. 2k; IhO.

(3) Ifederwood, E. E. , in Quantitative Analysis of Microstructures in Medicine , Biology and Materials Development , Edited by H. E. Exner, Dr. RiedererVerlag GmbH, Stuttgart (1975) P. 23h.

(k) Quantitative Microscopy , Edited by R. T. DeHoff and F. R. Rhines,

McGraw-Hill Book Co. (I968) p. 133.

(5) H. F. Fischmeister, Zt. Metallk. 65 (197^) p. 558.

(6) Iftiderwood, E. E. "Quantitative Shape Parameters for Microstructural

Features" Inter/Micro-75, Cambridge, England, June 23-26, 1975. To be

published in The Microscope, 2U, First Qtr. (1976).

92
National Bureau of Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9. 1975 (Issued January 1976)

INTERPRETATION OF SOME OF THE BASIC FEATURES OF FIELD-ION IMAGE PROJECTIONS FROM A HEMISPHERICAL TO A PLANAR SURFACE USING MOIRE PATTERNS

by P. Darrell Qwnby*, Robert M. Doerr,*' Walter Bollmann,*** 'Ceramic Engineering Dept., University of Missouri, Roila, Missouri 65401 , U.S.A. "U.S. Bureau of Mines, Roila, Missouri 65401, U.S.A.

***Battelle Institute, Geneva, Switzerland

A common problem In projection geometry is that of analyzing the pattern

formed in a planar observation surface by the projection of point, lineal, or areal features on a hemispherical surface. In this study, the most prominent

ellipses representing successive ledges on off-axis oriented planes, are simulated by the Moire patterns produced by the intersection of a simple grid

representing the crystal lattice parallel to the projection plane and a spherical projection of concentric circles representing successive ledges of atomic planes proceeding normal to the projection plane. The Moire patterns

are analyzed mathematically by considering the coincidence of the equivalent

points in both the circle set and the parallel line (grid) set. Computer

plotted patterns using this analysis are shown to coincide with the Moire

patterns. Examples of orthographic and stereographic projections are shown

for comparison with the actual field-ion image.

The surface geometry of a single crystal which produces a field-ion microscope image is to a first approximation, a hemispherical surface which is intersected by sets of lattice planes. For cubic crystals, three sets of or- thogonally oriented planes intersect the surface. These intersections determine

the ledge-step geometry of the surface which is seen projected onto a plane in atomic detail in field-ion microscopy.

In the present work the Intersection of each plane of the three sets with

the surface of the sphere is projected to a plane, forming three families

of lines. Each of these sets of curves interacts with the others to form a Moire pattern which faithfully represents many of the main features of the spherical surface geometry and therefore resembles, to a limited degree, the field-ion micrograph.

Curves representing the Moire interference bands can be produced by an analytical calculation without actually drawing the families of lines. The

guiding principle for calculating the Moire pattern between two parametrized

sets of curves is the "coincidence of equivalent positions" (1). For simplicity consider first the orthographic projection. The (001)

oriented, face-centered cubic crystal lattice geometry will be used for illustration. Orthographically projected Moires of other orientations and Bravais

lattices are shown elsewhere (2). The first set of projected curves becomes a spherical projection of concentric circles. The circles tend to overlap as the great circle^' is approached, making a full hemisphere projection impractical.

The second and third sets of curves are simply two orthogonal sets of parallel

straight lines. Note that in the FCC lattice those lines are A5° to the cube

axes. The resulting indexed Moire pattern is shown in figure 1. Each set of parallel lines produces Moire ellipses along a circle diameter perpendicular to the lines. The three sets of lines combine to produce the ellipses which lie

along the cube axes (at 45° to the straight line sets). For the analytical calculation of the Moire ellipses, consider the interaction of a set of parallel equidistant straight lines numbered from - oo to <» and a set of concentric circles numbered from 1 to N, from the smallest to the

The radius of the reference sphere and the maximum radius of the projected

circles in both projections shown herein was 4 inches (101.6 mm) before

reduction for publication.
p. D. Ownby, R. M. Doerr and W. Bollmann

largest. The "Coincidence of equivalent positions" are the intersections of circle No. 1 with line No. 1, of circle No. 2 with line No. 2, etc. Then we consider a continuous distribution of lines and circles so that, i.e., line No. 1.376 intersects circle 1.376, etc. This gives a continuous curve of intersections which represents a curve of the Moire pattern. A second set of curves

(ellipses in the present example) is obtained by the intersection of circle No.

1 with line No. 2, circle No. 2 with line No. 3, circle No. 1.376 with line

2.376, etc. For the set of straight lines parallel to the y-axis, x = na where a is the spacing of the lines (for FCC (001), a is the unit cell dimension/ /2 ) and n

is an integer. The other set of straight lines are given by y = na. The set of circles is given by N equidistant cuts of thickness d through a hemisphere,

(for FCC (001) this thickness is h of the unit cell dimension) so that

N • d = R, the radius of the sphere. The radius, p, of each circle is given by

= d^ (2Nn - n^) (1)

Now we change the integer n into a continuous z and introduce in x the integer

b, which is the difference in the numbering of the two sets.

X =a(z+b)

2 2 2 2 2 p = X + y = d (2Nz - z )

(2)

(3)

Now we calculate z from these equations as the solution of a quadratic

equation. ^ i b a + y

2 ^ j2 / 2 _^ ,2 a + d / a + d

2 2 Nd - ba

2

J. ^2

a + d

2 2 Nd - ba

(4)

Next we eliminate z by introducing equation (4) into (2); this yields the equation of an ellipse.

(x - x ) y ^-^ + = 1 (5)

with ad (N + b)

2 X ^2

a + d (6)

a [N d - (2Nb - b )a d ] 2 2 2 (a^ + d^r (7)

a (8)

where x is the center and a and b are the two semi-axes of the ellipse. By

varying°b, the whole set of ellipses is obtained. However, analytically, the values of b cannot be chosen fully arbitrarily. There is an upper and lower

limit of b which depends upon the relative placement of the origin of the intersecting sets of curves, and is determined by the condition a^ > 0. For

a^ = 0, we obtain the equation

[N^d^ - (2Nb + b^) a^d^] = 0 (9)"

which is a quadratic equation for b with the solutions:

94
Study of field-ion image projections by Moire patterns

max min

- N (10)

The next Integer smaller than b corresponds to the smallest ellipse of the set and the smallest integer larger than b to the largest ellipse.

Other sets of Moire ellipses are obtained if e.g. the first circle inter- sects the second line, the second circle the fourth line, etc. This means

that the Moire appears as though the spacing, a, were doubled or tripled, etc.,

i.e. a is replaced by 2a, by 3a, and so on. The smaller the ratio d/a, the more circle-like become the ellipses which is seen from equation (8). As can be seen on Figure 1, ellipses perpendicular to^ the line sets, i.e. those along the cube diagonals, have the spacing a = nd/ / 2. For ellipses inclined by 45° to the straight line sets, i.e. those along the cube axes, the spacing a becomes

nd.

If a Moire pattern such as on figure 1 is already given, the data can be

analyzed in the following way. From the ratio of the square of the radius of the inner circle with number n, to that of the limiting circla.

2 £_

r2

* = 2Nn

(11)

we obtain

1 + (1 - <I>) (12)

Since $ depends on n, N should prove to be independent of n. In the case of Figure 1, for n > 10, (where the errors are small) N = 100, and since Nd = R,

d = 1.016 mm and the unit cell dimension in the projection = 2.032 mm (before

reduction)

.

In the second quadrant of Fig. 1 some analytical results are shown. Only a few of the ellipses from each set were drawn and they show very good agreement

with the Moire. For example, for the set where a = d, 83 ellipses are possible

according to equation (10), whereas only 5 ellipses were drawn. Also, not all poles predicted by the analysis were drawn - only those corresponding to the most obvious ones in the Moire pattern. The "b" value for each ellipse is shown

as well as the"a"value for each set. The poles are iftdexed in the fourth

quadrant.

Having given the explanation of the approach, we will proceed to the

stereographic projection, which has the following advantages:

(a) The complete hemisphere can be visibly projected.

(b) It more closely resembles the projection obtained in the field-ion

microscope.

(c) The angular relationships between poles is preserved so that indexing

with a Wulff net is facilitated. This characteristic of stereographic

projections renders the pole steps circular Instead of elliptical,

preserving their spherical surface character.

The computer program by which the Moire is plotted is a general one which

considers three (or four in the hexagonal case) sets of planes intersecting the surface of a sphere. The interplanar spacings for each set and the angles

between sets are variables. The point of projection can also be varied.

In stereographic projection, all three sets of circles on the reference

sphere produced by the planar-sphere intersections are projected as circles

95
p. D. Ownby. R. M. Doerr and W. Bollmann

(as opposed i.e. to the orthographic, where two sets project as straight lines)

as shown on Figure 2. The Moire pattern is seen to Include many more poles

and more closely resembles the FIM. The analytical Moire circles are shown in the second quadrant. They were produced by projecting the analytical ellipses

from the orthographic projection, back to the reference sphere, and then repro- jecting them stereographically . As in the orthographic case only a few of the possible circles are drawn for each pole (not necessarily the same ones),

however, in this case, all possible poles from the orthographic analysis are represented by at least one circle. The general position of these circles

coincides very closely with the Moire pattern as before. The exact size of the small circles in the Moires varies considerably and is not Important here. Small changes in the relative position of each of the original sets of planes

make large differences in the size of the smallest Moire fringes. This should

not surprise anyone who has observed the field evaporation process in a fieldion microscope where all of the circles representing the atomic plane edges are continually collapsing to zero radius as the last atom from each plane evaporates. Consequently, in field-ion images the size of the smallest ledge circle

from the same type of plane will vary from quadrant to quadrant in the same image as is also true and can be seen in the Moires.

The most notable discrepancy between the stereographic Moire pattern and

the analytical circles produced by reprojection from the orthographic is that many prominent outer poles are not predicted. This is because they require that

the circles from the 2nd and 3rd sets be projected as non-straight lines. This

requires a modification of the details of the described procedure, but the same approach is applicable and should yield equally good agreement with all of the Moire poles. Differences between the FIM image and the Moires arise with

deviations from sphericity, changes in local radii of curvature, and physical

properties beyond the scope of purely geometrical considerations such as finite

sizes of atoms, ionization potentials, field sublimation energies, directional

bonding, distribution of surface charges, etc. The use of Moire models for the geometrical description of crystalline

Interfaces has been well established (1). The present work has demonstrated

that the same general geometry which produces the successive rings of planar

edge atoms around low index poles in field-ion images can be used to produce

Moire patterns of rings around "low index poles" which greatly resemble those

of the basic field-ion image.

The assistance of Dr. H.L. Lukas with the electronic calculations and graphics

is gratefully acknowledged.

REFERENCES

1. Bollmann, W. , (1970), Crystal Defects and Crystalline Interfaces, SpringerVerlag, Berlin.

2. Doerr, R.M., and Ownby, P.D. (1975), Moire Simulation of Field-Ion

Micrographs, Pract. Metallog. 12, 78.

96
Study of field-ion image projections by Moire patterns

a: 1.437mm

FCC (001) ORTHOGRAPHIC

Figure 1: Indexed Orthographic Moire with corresponding analytical ellipses

labeled.

97
p. D. Ownbv, R. M. Doerr and W. Bollmann
National Bureau of Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

NETWORK SIMULATION OF CONNECTIVITY AND AGGREGATE SIZE IN TWO PHASE MIXTURES

by W. D. Leahy* and J. H. Steele, Jr.** Division of Minerals Engineering, Virginia Polytechnic Institute and State University. Blacksburg.

Virginia 24061, U.S.A. INTRODUCTION

The theory of random clumping (1) or clustering encompasses some of the most difficult unsolved problems in stereology. One such problem involves

predicting the distribution of aggregate sizes and connectivity in a random

mixture of polyhedral grains of two phases. Cahn(2) has described a model

which predicts aggregate connectivity in an average sense, that is connectivity

per grain. His model does not, however, provide statistical estimates of aggregate size. Bishop et al. (3) have also used Cahn's model for describing

connectivity in a graded mixture of two phases. In this note initial results

of a Monte Carlo computer simulation using a network model for a polycrystalline structure will be described.

DESCRIPTION OF COMPUTER SIMULATION

A network model (4) for a polycrystalline microstructure can be defined if each vertex or node represents a polyhedral-like grain and each edge represents

a face shared by an adjoining grain. This type of model allows aggregate

properties to be calculated after the nodes are randomly associated either

with one or the other two phases forming the mixtiire. For illustrative purposes the phases will be called or and /S . A random process such as this is similar to a coloring(5) where two colors, e.g. black and white, are associated

with the nodes of the network. The network information was stored within a digital computer using its vertex matrix, (Vj_j) where V^^j = 1 if the ith and jth grains form a face, and

Vj^j = 0 otherwise. In order to store the required information for a 1000 node

network, involving (1000)^ matrix values, a sparse matrix technique was

utilized. Briefly this involved two lists, the first indicating the niimber of edges emanating from each node and the second indicating the number labels for each of the connecting nodes.

In order to define a network which would be a representative model for a polycrystalline aggregate, the following procedure was used. First, 1000 points

were positioned randomly within a unit cube using a random number generator to select {x,y,z) coordinates. Then all of the points within a fixed radial

distance of each point were found and used as neighbors to define the faces for

the grain at each point. This procedure is similar to that used by Meijering(6)

cind Gilbert (7) to define random tesselation models for cellular aggregates

such as the polycrystal. Varying the radial distance changes the distribution

and average number of neighbors. A radial distance which produced an average

of 14 faces (or edges) per grain (or node) was used to define the network. Periodic boundary conditions were set up so that external surface effects on the unit cube would be minimized.

Random labeling of the grains or nodes as either o< or /8 was carried out

to produce a mixture within the network structure. The initial results

presented are based upon a single network model with different nxanber fractions

of grains applied randomly upon it. Aggregate size and distribution are determined from the vertex matrix by

a sorting algorithm which partitions the matrix according to the separate

parts. The procedure involves a search and a relabeling of the nodes in a connected part so that they occur sequentially within the vertex matrix. In this manner the number of separate parts and their size can be obtained from

the partitioned vertex matrix. An example network with its partitioned

vertex matrix is shown in Figure 1. Presently with Corning Glass, Corning, New York

•Presently with Armco Steel Corporation, Middletown, Ohio

99
W. p. Leahy and J. H. Steele, Jr.

Figure 2. Variation in connectivity per separate part (data points shown

as squares) and separate parts per grain (shown as circles) with

the fraction of grains associated with a phase. Note that the plot is symmetric about 0.5. The dashed curve is a plot of the function exp. (-8V„) for the minor phase.

100
Network simulation of connectivity and aggregate size in two phase mixtures

RESULTS AND DISCUSSION

The effect of increasing nuniier fraction (and also volume fraction) upon

the number of aggregates per grain and upon the connectivity per aggregate is shown in Figure 2. An aggregate is defined as a connected clump or cluster of

grains of -che same phase. Connectivity in this model represents the number of

independent circuits in a network sense in each aggregate. Number of separate

parts per grain indicates by its reciprocal the number of grains per separate

part. An extremely good fit for this function (separate parts per grain) is given by e~^^V where Vy is the volume fraction of the phase. This is in- teresting since this exponential represents the probability for an isolated

sphere among randomly positioned spheres in three dimensions. The dashed line

in Figure 1 shows how well this random isolation probability estimates the aggregate size function for volume fractions up to 15%. Distribution data for the various aggregates are presented in Table I for the initial simulation run. Although the single simulation does not provide

quantitative statistical estimates of the probabilities of grain pairs,

triplets, etc., it does give a qualitative indication of the tendency for these configurations. It should be noted from Figure 2 and the tabulated data

that the random labeling process is symmetric with regard to the minor con- stituent. Hence, one simulation run provides two statistical data points.

Approximate probabilities for a randomly selected grain belonging to a given

size configuration are presented in Figure 3. These probabilities are somewhat

below the functions given by Roach (1) as approximations for the pair, triplet,

and higher order configurations.

Connectivity (denoted by /? ]_) , which represents the niomber of linearly

independent circuits, in a connected network, is given by,

>9l = Ni - Nq + 1 (1)

where Ng = number of nodes, and N^^ = number of edges. If this equation is summed over the separate parts, then it becomes,

1 = Nl - No + Np^rts <2) TOT TOT TOT

Hence, when the vertex matrix is partitioned to allow the number of nodes and edges in each separate part to be counted, the connectivity of each separate

part and the total connectivity may be calculated rather easily.

Cahn's statistical equation (2) for connectivity in a similar type ag- gregate involves the Euler characteristic, Q', of the phase boundary. This is not the same as the connectivity plotted in Figure 1 since internal circuits

among o( -grains within a single aggregate are involved in the network model,

but not in Cahn's model. Effectively, Cahn's model treats each aggregate

according to the properties of its phase boundaries. A comparison between the network simulation and Cahn's Q'/G' (Euler characteristic per grain) can be made by noting that for each ith separate part.

2-2/? Q| = li C3) gT

1

Hence, summing over the separate parts yields.

sfNparts -fil I Q' = _|-L TOTJ (4) G' Vains

101
W. D, Leahy and J. H. Steele. Jr.

Amount

TABLE I - SIMULATION DATA

Total No. Number of Separate Parts Separate With N Grains

Separate

Parts With

Of Phases Parts N=l 2 3 4 5 6 7 8 9 N > 10

%ol %/3 /} /3 o/ 5% 95% A c X 24 6 3 0 1 0 10 - 90 *4X ot J_ 19 7 7 2 2 3 10 15 - 35 34 & 18 1 2 5 2 2 1 10-10-55

20 - 80 24 & 12 2 2 1 2 2 0 15-23-114

25 - 75 13 & 1 5 2 0 1 1 1 1 1 211

30 - 70 4 & 1 2 0 0 0 1 293

35 - 65 Z9 c6f i. 1 349 40 - 60 9 r X 1 399

45 - 55 1 6f X

x 50 - 50 1 & 1 55 - 45 1 & 1 ~/3 /3 60 - 40 1 & 2 399

65 - 35 1 S 3 1 0 1 346

70 - 30 1 S 7 6 294

75 - 25 1 & 9 8 242

80 - 20 1 s 27 10 4 6 2 2 0 1 0 1 130

85 - 15 1 & 47 24 5 6 4 4 1 1 24 & 25 90 - 10 1 s 50 30 10 5 2 2 0 0 17 95 - 5 1 & 38 31 5 0 1 1

0)

IB

U

4 —

Simulated Data

Singlets: —^—

Doublets:

Triplets: •

SOLID CURVES - ROACH (1) Singlets: P^ Doublets: P2 = e~^^V (l-e°^V)

Triplets: P-, = e"^^V (l-e'^^V) ^

Figure 3. Distribution of aggregates which consist of separate grains,

(singlets) pairs of grains (doublets) , and triplets. Simulated

data points are shown with dashed curves drawn through them. Solid curves are plots of approximations given by Roach (1)

.

102
Network simulation of connectivity and aggregate size in two phase mixtures

A plot of this characteristic per grain fiinction using the computer simulation

data is presented in Figure 4. Negative values are produced by larger con- nectivity, and the effect of internal circuits within Q- -aggregates causes the network analysis to be significantly lower than Cahn's equation between V^. = .2 and Vy S .8. toother property which is of interest in clustering problems is the critical percolation probability (8) (denoted P(-) • This value represents the fraction of nodes Cor grains) required in a phase to produce infinite size

clusters and thus provide continuous paths through the network. Applying the techniques used by Dean and Bird (9) who plot the fraction of nodes in the largest part within the network, and a modified second moment of the cluster

distribution versus fraction of nodes in the phase as shown in Figure 5. The modified second moment used is given by,

/< =Z:n{i) i2/(Z; n(i)i)2

i i

where*^ represents the sum over all size clusters, and n(i) is the number of

nodes in the ith size cluster. The critical percolation probability is estimated as the fraction of grains where the maxim\im slope occurs for

Thus, the estimated value is approximately, P^. = .20. This value is likely to be an over estimate because of the small size of the network utilized in this

simulation. However, it is quite close to the value of for the face- centered cubic lattice arrangement {Pj, = .195) which has twelve edges per

node.

CONCLUSIONS

Polycrystalline aggregates may be simulated by network models of various

types to study clustering, connectivity, and percolation phenomena. This

study has indicated qualitatively the type of results that can be obtained

using a network model and Monte Carlo simulation techniques.

Qualitatively, the results show that in a two phase polycrystalline

aggregate where both phases have similar size and shape distributions , that

between volume fractions of 0.2 and 0.8, both phases would be expected to have

large highly interconnected clusters permeating the structure.

REFERENCES

1. Roach, S. A., The Theory of Random Clumping , Methuen S Co. Ltd., London,

1968.

2. Cahn, J. W. , "A Model for Connectivity in Multiphase Structures," Acta

Met., 14 (1966), 477.

3. Bishop, G. H., Quinn, G. D. , and Katz, R. N. , "Connectivity in Graded

Two-Phase Structures," Scripta Met., 5_ (1971), 623.

4. Steele, J. H., Jr., "Application of Topological Concepts in Stereology,"

Quantitative Metallography and Stereology , ASTM STP 504, Philadelphia,

Pennsylvania, 1972, Page 39.

5. Busacher, R. G. , and Saaty, T. L. , Finite Graphs and Networks , McGrawHill Company, New York, 1965.

6. Meijering, J. L. , "Interface Area, Edge Length, and Number of Vertices in Crystal Aggregates with Random Nucleation, " Philips Research Reports, 8^ (1953), 270.

7. Gilbert, E. N., "Random Subdivisions of Space into Crystals," Ann. Math. Stat., 32_ (1962), 958.

8. Shante, V. K. S. and Kirkpatrick, S., "An Introduction to Percolation

Theory," Adv. Phys., 20 (1971), 325.

9. Deem, P. and Bird, N.F., "Monte Carlo Studies of the Percolation Properties

of Two and Three Dimensional Lattices," National Physical Laboratory

Report Ma 61, Mathematics Division, November 1966.

I

103
IV. p. Leahy and J. H. Steele, Jr.

Cahn ' s 2(1-Vv) -^7= Vv(l-V^)

FRACTION OF GRAINS = V,

8 1.0

G

' 2(N - Connectivity ) /N surfaces grains

Figure 4. Connectivity per grain comparing Cahn's equation (2) to computer

simulation results (dashed curve)

.

W

<

u

z

o

H

EhU

<

Fraction of Nodes

in Largest Part

j Modified Second

1 Moment of Aggregate

/ Size Distribution

Figure 5.

0 .1 .2Vy.3 .4 .5 Results showing the variation in the fraction of nodes associated

with the longest part aind the modified second moment with fraction

of grains associated with minor phase. The percolation probability is estimated from the fraction of grains at which the slope in the modified second moment is a maximum.

104
PATTERN RECOGNITION

Chairpersons

:

G. Bernroider

G. C. Cheng

C.-H. Chen

National Bureau of Standards Special Publication 437

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg. Md., September 4-9, 1975 (Issued January 1976)

WHAT CAN PATTERN RECOGNITION DO FOR STEREOLOGY

by George C. Cheng

University of Georgia, Athens, Georgia 30601, U.S.A.

ABSTRACT*

What is Pattern Recognition? Hundreds of scientists have devoted research

to Pattern Recognition for more than twenty years. More than a hundred

books have been written on this subject. Thousands of articles and reports

have been published. The Pattern Recognition Society and its official journal

were organized eight years ago. However, no one has seriously attempted to define the term 'Pattern Recognition'. Because of the concepts, methodologies,

and applications of Pattern Recognition are very diverse in research, no two researchers would agree upon a precise definition. There are also diverse

opinions about whether Pattern Recognition should be considered as an art or

a science. Among the research in the past twenty years, the main interest

has been in the development of automatic systems and machines to perform

pattern recognition.

From the methodological viewpoint . Pattern recognition research has been

generally approached from one of the following directions:

1. Heuristic: cognitive process, Gestalt, artificial intelligence, machine

intelligence, theorem proving, problem solving, robot, automata, and learning machine are frequently associated terms.

2. Structural or linguistic: syntax-directed methods, picture gramma, state

diagram, sequential decision, and tree searching are related terms.

3. Mathematical/statistical: multivariate analysis, factor, clustering,

classification, and discriminant analyses are the terms associated with

the statistical approach. Set theory, category theory, and formulism are

those related with other mathematical approaches.

In addition, Perceptron, Neuron-models, and Cybernetics are also related to Pattern Recognition.

The heuristic and structural approaches are discussed in this talk through

several simple illustrative examples. The statistical approach is discussed

by Dr. C. H. Chen in this session. The Category Theory oriented approach is discussed by Dr. M. Pavel in the session on Mathematics and Pattern Recognition

of this Congress.

*Due to copy rights, the pictures and data presented at the Congress are not included in this printed version.

107
G. C. Cheng

A major research area in Pattern Recognition has been the pictorial (or visual) Pattern Recognition. Stereologists may find that this class of Pattern Recognition is most useful for their work.

From the operational viewpoint . Pattern Recognition, particularly pictorial Pattern Recognition can be considered in the following four phases:

1. Data acquisition: instrumentation is a major concern. Representative

works include that of Dr. G.A. Moore (NBS), Dr. L.E. Lipkins (NIH)

,

Dr. R.S. Ledley (Georgetown University), Quantimet, Leitz Texture Analysing System, Illiac III, etc.

2. Data preprocessing: picture or image digitization, edge/line/boundary

detection, feature extraction, scale transformation, spatial transformations, etc. are associated terms.

3. Pattern Recognition (of preprocessed data): factoring, clustering,

classifying, discriminant-analysing, syntax-directed recognition, tree- searching, etc. are the employed operations.

4. Theory forming: hypothese forming and testing, speculating/forecasting,

decision making, etc. are the consequences of Pattern Recognition.

What has Pattern Recognition helped to accomplish in solving practical

problems ? The most successful application of Pattern Recognition research

has probably been Optical Character Recognition (OCR). For examples, started

in 1967, the Japanese automatic mail sorting machine could recognize the handwritten postal code (zip code) and sort 600 pieces of mail in a minute.

The OCR machines made by IBM for the U.S. Social Security Office could

recognize typed multifont characters at a speed of more than a thousand

characters per second. The state of the art in OCR is about 2000 characters

per second with a less than 0.02% error rate.

How does a Pattern Recognition machine recognize a handwritten character ? A simplified example of OCR is presented for illustration of the structural

approach of Pattern Recognition. One of the purposes of showing this example

is to illustrate the picture processing operations, including the formation

of the picture matrix, feature extraction from the picture matrix, and

syntax-directed recognition. Another purpose is to point out that in OCR,

the quality and/or form of the raw data can be controlled. For examples,

the Japanese postal code is written in red colored boxes on the envelope,

and for the typed and printed OCR, two new fonts of English letters were

designed specially for OCR machines, namely the OCR-A font in America and

the OCR-B font in Europe.

Examples of raw data that have been subjected to Pattern Recognition research . In stereology as well as in many other disciplines, the raw data cannot be

controlled as well as does in OCR. Some thirty selected examples of this

108
What can pattern recognition do for stereoiogy?

kind raw data that have been subjected to Pattern Recognition research are shown herewith. (Many more examples are shown by other colleagues in this

Congress.) A purpose of showing these pictures is to provide comparison

facility to stereologists . If there is a close similarity between the stereologist's own data and some of the tried examples, the degree of assistance that Pattern Recognition can offer to his problem and the cost

of using this technique can be estimated from the previous experience.

Otherwise, more effort would be needed to study his problem from the viewpoint of Pattern Recognition.

The purpose of showing the last set of three pictures is to point out that in the computer processing of pictorial data, the data is usually preprocessed

and converted into line drawings before Pattern Recognition is applied.

For this reason, the following examples used to illustrate the heuristic

approach of Pattern Recognition are all line drawings.

A purpose of showing the example of a 'bear behind a tree' is to point out that

in the real world, we use information from the environment, the context

of the situation, the purpose of our investigation, and our build-in cognitive

system as aids in doing pattern recognition. When we see four paws on the tree trunk, it is not difficult for us to recognize that there is a bear

behind it. However, in an automatic Pattern Recognition system or machine,

the raw data is converted into a line drawing like that shown in the slide. The information from the environment or context is lost. In most Pattern

Recognition research, the contextual information is neither presented to the researchers nor requested by the researchers to be provided. In other

cases, the mathematical models are so rigidly designed, there is no room

for the exogenous information to be inserted.

The purpose of showing the 'man or girl' example is to point out the existence

of the overlapping situation. Statistics provides many sophisticated

techniques for handling the population-overlapping situations. However,

a situation like the 'man or girl' would create a gap between the raw data

and the kinds of measurement required by the statistical models.

What can Pattern Recognition do for Stereoiogy ? There is no Pattern Recognition cook-book available for stereologists to use. The immediately useful

contribution of automatic Pattern Recognition to Stereoiogy would be in the instrumentation of data acquisition and data preprocessing. The transfer

of Pattern Recognition technology from one discipline to another is not straightforward. This is mainly because the effectiveness of a particular

Pattern Recognition technique depends on the quality and quantity of the

data. Either a stereologist needs to devote a lot of effort to learn Pattern Recognition or the Pattern Recognition researchers need to devote

effort to learn about the data and problems of the stereologist. The cost

109
6. C. Cheng

of these trainings is high, but it may be worthwhile if the case is like

that of OCR. From the viewpoint of the Pattern Recognition discipline, any research effort in this area by stereologists would be beneficial to its advance. We hope that it would also be beneficial to Stereology. In conclusion,

- If your data is or can be controlled in quality and in quantity as in OCR, the structural and/or statistical

approaches of Pattern Recognition can be of great

assistance to Stereology

- If you fancy the modern algebraic way of thinking, the mathematical Pattern Recognition concept may provide

you with mental enjoyment and may lead to some break

through

.

- In one degree or another, we all use heuristic approaches

in solving problems. The concept of heuristic Pattern Recognition may reinforce your intuitive attitude in solving Stereological problems.

Selected Pattern Recognition bibliography for Stereologists

I. Review articles:

Cormack, R.M. , "A Review of Classification," the Journal of Royal Statistical

Society , V.134, (1971) pp. 321-367. (227 references were included.)

Duran, B.S. & Odell, P.L., Cluster Analysis - A Survey , Springer-Verlag, 197A.

(409 references were included.)

Das Gupta, S., "Theories & Methods in Classification - A Review," in Discriminant Analysis and Applications , edited by T. Cacoullos, Academic

Press, 1973. (260 references were included.)

Nagy, G. , "State of the Art in Pattern Recognition," IEEE Proceedings , V.56,

No. 5, (May, 1968) pp. 836-862. (148 references were included.)

Toussaint, G.T., "Subjective Clustering and Bibliography of Books on Pattern

Recognition," Information Sciences , V.8, (1975) pp. 251-257. (117

references were included.)

II. Pictorial and 3-dimensional objects Pattern Recognition:

Barnhill, R.E. & Reisenfeld, R.F., (ed.) Computer Aided Geometric Design , Academic Press, 1974.

Cheng, G.C., Ledley, R.S., Pollock, D.K. , Rosenfeld, A., (ed.) Pictorial Pattern Recognition , Tompson Book Co., 1968.

Gips, J., "A syntax-directed program that performs a three-dimensional

perceptual task," Pattern Recognition , V.6., (1974) pp. 189-199.

Grasselli, A. (ed.) Automatic Interpretation & Classification of Images , Academic Press, 1969.

110
What can pattern recognition do for stereology?

Ivancevic, N.S., "Stereometric Pattern Recognition by artificial touch," Pattern Recognition , V.6. (1974) pp. 77-83.

Maruyama, K., A study of visual shape perception . University of Illinois,

University of Computer Science. UIUCDS-R-72-533, Oct., 1972.

McKee, J.W., & Aggarwal, J.K., "Finding the edges of 3-D objects," Pattern Recognition , V.7, (1975) pp. 25-52.

Muller, W., "The Leitz Texture Analysing System," Leitz Scientific & Technical Information , (April, 1974) pp. 101-116.

Rosenfeld, A., Picture Processing by Computer , Academic Press, 1969.

Serra, J., "Theoretical Bases of the Leitz Texture Analysing System,"

Leitz Scientific & Technical Information , (April, 1974) pp. 125-136.

Sobel, I., "Towards Automated 3D Micro-Anatomy Via Reconstruction from

Serial section micrographs," 1975 Engineering Foundation Conference

on Automatic Cytology.

Swann, P.R., Humphreys, & Goringe, (ed.) High Voltage Electron Microscopy , Academic Press, 1974.

Tenenbaum, J.M., Garvey, T.D., Weyl, S., Wolf, H.C., An Interactive Facility

For Scene Analysis Research , Stanford Research Institute, Artificial

Intelligence Center, Tech. Note 87, Jan., 1974.

III. Books:

Andrews, H.C., Introduction to Mathematical Techniques in Pattern Recognition , John Wiley & Sons, 1972.

Arbib, M.A., The Metaphorical Brain , John Wiley & Sons, 1972.

Banerji, R.B., Theory of Problem Solving - An approach to artificial intelligence , American Elsevier Publishing Co., 1969.

Bongard, M. (ed. by Hawkins, J.K) Pattern Recognition , Spartan Books, 1970.

Braffort, P., L' intelligence artificielle . Press Universitaires De France,

1968.

Carne, E.B., Artificial Intelligence Techniques , Spartan Books, 1965.

Chen, C.H., Statistical Pattern Recognition , Spartan Books, 1973.

Duda, R.O., Hart, P.E., Pattern Classification and Scene Analysis , John

Wiley & Sons, 1973.

Findler, N.V. , Meltzer, B., (ed.) Artificial Intelligence & Heuristic

Programming , American Elsevier Pub., 1971.

Fukunaga, K., Introduction to Statistical Pattern Recognition , Academic Press,

1972.

Grenander, U. , A Unified Approach to Pattern Analysis , Brown University, Center for Computer and Information Sciences, 1969.

Kanal, L.N., (ed.) Pattern Recognition , Thompson Book, 1968.

Ill
G. C. Cheng

Kolers, P. A., Eden, M. (ed.) Recognizing Patterns, Studies in Living & Automatic Systems , the MIT Press, 1968.

Meltzer, B. & Michie, D. (ed.) Machine Intelligence , Am. Elsevier, 1971

Mendel, J.M., Fu, K.S. (ed.) Adaptive, Learning, & Pattern Recognition

Systems , Academic Press, 1970.

Minsky, M. , Paper t, S., Perceptrons, An introduction to computational

Geometry , the MIT Press, 1969.

Moore, G.A. , Application of Computers to Quantitative Analysis of Microstructures , National Bureau of Standards, Report No. 9428, Oct., 1966.

Nilsson, N.J., Learning Machines , McGraw-Hill, 1965.

Nilsson, N.J., Problem Solving Methods in Artificial Intelligence , McGrawHill, 1971.

Patrick, E.A., Fundamentals of Pattern Recognition , Prentice-Hall, 1972.

Pavel, M. , Fondements mathematlgues de la Reconnaissance des Structures , Actualistes Scientif iques. No. 1342, Hermann, Paris, 1969.

Sklansky, J. (ed. ) Pattern Recognition - Introduction & Foundation , Dowden,

Hutchinson & Ross, 1973.

Tsypkin, Y.Z., Foundations of the Theory of Learning Systems , Academic Press,

1973.

Watanabe, S., Methodologies of Pattern Recognition , Academic Press, 1969.

Watanabe, S., Frontiers of Pattern Recognition , Academic Press, 1972.

Young, T.Y., Calvert, T.W. , Classification, Estimation, and Pattern Recognition , American Elsevier Publishing Co. , 1974.

IV. Additions:

Bellman, R. , Kalaba, R. , Zadeh, L. , "Abstraction and Pattern Classification,"

J. of Mathematical Analysis and Applications, V.13, (1966) pp. 1-7.

Kirsch, R. , Cahn, L. , Ray, L.C., Urban, G.H. , "Experiments in Processing Pictorial Information with a Digital Computer," Proceedings of Eastern Joint

Computer Conference , (1957) pp. 221-229.

Moore, G.A. , "Gestalt Properties of Aggregate Materials," Practical Metallography , V.IX, No. 2, pp. 76-97.

Zobrist, A., & Thompson, W.B., "Building a Distance Function for Gestalt

Group," IEEE Trans, on Comp., C-4. #7, (July, 1975) pp. 718-728.

112
National Bureau of Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

THEORY AND APPLICATIONS OF IMAGERY PATTERN RECOGNITION

by C. H. Chen

Southeastern Massachusetts University, North Dartmouth, Massachusetts 02747, U.S.A.

ABSTRACT

In recent years, there has been axi increasingly great demand for auto- matic recognition of imagery patterns which arise in biomedical, space and

stereology and a number of other applications. As the computer hardware cost decreases rapidly, automatic imagery recognition will soon become a reality

in many of these applications. In this paper, a \inified theory of imagery

pattern recognition is developed which includes: (l) image representation and

compression, (2) preprocessing, (3) feature extraction, and (h) pattern classificationo Status of image recognition applications is reported. Practical

considerations such as the computational complexity and operational expenses

are discussed.

I. Introduction

Since the development of pattern recognition area in early fifties, there

have been much efforts made to develop methods of processing pictorial information by computer, optical devices or hybrid systems. Most patterns in real

life are imagery in natiare. The demand for automation is great due to the rapidly increasingly volume of pictorial patterns. Extensive research on various image recognition application has been reported in the literature. For example, Refs. 1-3 provide excellent discussions of automatic image analysis in stereology or vice versa. Extensive list of references on this subject

is also available {h) (5). However the automatic recognition has not been

cost effective for desired performance and the gap between theory and practice

remains wide open. Much effort is needed to make effective use of pattern

recognition theory, which offers a number of mathematical tools, on applications. With this objective in mind, we discuss in this paper both the theoretical and practical considerations of imagery pattern recognition.

II. Image Representation and Compressions

To start with, the image data are in analog forms such as continuous gray

levels or intensity values. For computer processing it is necessary to digitize the images and quantize the discrete set of data points. The result may

be called a picture function which represents the original images completely

with the exception of quantization errors. The picture function consisting

of two-dimensional array of data points usually contains too much information

to be handled effectively by the computer. To reduce the amount of data without any significant loss of useful information, image representation and data

compression must be considered. If the images are considered as stochastic processes, the Karhunen-Loeve transform is optimum in the sense of minimizing

the mean square en-or between the original and transformed pictures. Although

the symmetric property of covariance matrices for imagery data can be used to simplify this optimum transform, the required computation is still excessive.

Two-dimensional orthogonal transforms such as Fourier, Walsh-Hadamard, Haar,

and discrete cosine transforms can be computed much more efficiently because

fast algorithms are available. The area of data compression for picture bandwidth reduction has been well studied in recent years. Its objective is to reconstruct the transmitted image at the receiving site as accurately as possible. For the purpose of pattern recognition, data compression is to

remove the irrelevant details and to retain only the discrimination information.

Thus the data compression objective is to reduce the amoiint of computation for

classification with minimal loss of recognition accuracy. A simple data com- pression technique is to reduce the original picture to a binary (two-level)

picture which can further be compressed with almost no loss of discrimination

113
C. H. Chen

information (6). Picture encoding methods may also be used for image repre- sentation and compression.

IIIo Image Preprocessing Techniques

An important class of preprocessing techniques is the position-invariant

operations. This includes linear operations such as convolution and correlation and point operations such as requantization and gray level normalization

which may be linear or nonlinear operations. Image enhancement for smoothing to reduce the noise effect or sharpening to improve the boundaries or edges

are often necessary before any features can be derived. The enhanced image

which is better for human visualization is also good for computer recognition.

Statistical infonnation of a picture or a subpicture may be derived from gray

level histograms. An optimum threshold can be determined from the histogram.

The threshold is useful to generate a binary picture for picture segmentation,

picture data compression, or even boundary detection (6), (T).

Spatial smoothing of a region can be performed by convolution or fre- quency filtering. If noise spectrum is considerably different from the signal,

then bandpass filtering can suppress noise by deleting a selected band of spatial frequencies while leaving enough at high frequencies to keep edges

unblurredo Optimum (Wiener) filtering may be performed but the boundaries

will be blurred. There are many ways to shairpen or deblur a picture. For

example, this can be done by high-pass spatial filtering or high-frequency

emphasis operation. We know that integration (i.e., averaging) blxirs a picture, it is natural to use some sort of differentiation operation to deblur

a picture. Gradient operation serves this objective; but the resulting isolated bursts have to be removed by further processing. Gray level normalization and distribution equalization also can sharpen the picture. A combination

of sharpening and smoothing is usually needed to arrive at the best result. Gray level slicing is particularly effective in improving certain characteristics of the picture (7). Other preprocessing problems include image matching,

restoration, etc. Digital signal processing techniques such as the use of hormomorphic systems (8) are also very useful for image preprocessing.

IV o Feature Extraction

Feature extraction, consisting of deriving certain characteristic measurements from the input picture function, is considered by far the most difficult step in image recognition. Our approach to the feature extraction

problem, applicable to a restricted, although large class of picture functions,

is to derive measures of a statistical nature. Such measures are most appro- priate with textural patterns. The second-order gray level distribution can

be used to derive a set of texture features or average local property measures. Features may also be derived from two-dimensional transforms which exploit

global property. At present features are extracted fairly heuristically and

the effectiveness of such features is evaluated by the probability of error

as well as the information and distance measures (9). It is usually more

desirable having a small set of effective features followed by a simple classifier than using a large number of features in conjunction with a complicated

classifier. To select good features really requires a good understanding of

the basic (physical) properties of the pattern. Strictly relying on mathematical features usually cannot lead to the best possible performance. Thus in automatic image analysis, the stereologists can be very helpfvil to provide

good features.

V. Pattern Classification

As a final portion of the recognition system design, classification is a decision making process which assigns the input pattern to one of several possible classes (categories). The design objective inmost recognition

114
Theory and applications of imagery pattern recognition

problems is the minimum number of errors incurred in decision making. Statistical decision theory provides an excellent formal approach to the decision

making. Typical classification rule are (lO): l) Bayes decision rule,

2) maximum likelihood decision rule, 3) nearest neighbor decision rule,

h) sequential decision procedure. To reduce the computational effort, a table look-up approach has been proposed.

VI, Practical Considerations

The following are some important practical problems:

1= The sample size, i.e., the number of samples available for learning

and classification is always limited. Pattern recognition theoiy

is based on infinite sample size„ This may cause discrepancy between

practical result and theoretical prediction. Large number of samples,

however, may require excessive computation.

2. Computational complexity including the niimber of computation steps

and the memory requirement is directly related to the cost for the recognition task. There is almost no theoretical work on the prediction of computation costo Programming complexity is another

probleffio At present it all depends on experience and experimentation.

A few simple rules may be useful. If the probability density cannot be conveniently defined, the nearest neighbor decision rule is always

useful though the computation of Euclidean distance is time consuming

for large sample size. If Gaussian assumption is reasonable, then

the maximum likelihood decision rule is useful.

3. The cost difference between human recognition and machine recognition

is still considerable. For example human interpretation of a photo

may cost only a few cents but it may require over a hundred dollars

for machine interpretation. Although the basic problems here are not just computer hardware and software, good computational algorithm is always important in reducing the cost. Larger and faster computers

alone cannot solve the problem.

To resolve some of the practical problems, it is important to note that

the effective preprocessing and feature extraction can reduce considerably

the overall computation cost. And the finite sample size effect must always

be taken into account.

VII. Applications

The great demands for automatic imagery recognition in various application areas have prompted much theoretical studies on imagery pattern recognition. New ima^e processing techniques, on the other hand, have also been

used in various applications. An excellent discussion on various applications

is in the book by Cheng, et.al. (ll). There is more success in image processing than in feature extraction and classification. At present the imagery

recognition has much more success in space, biomedical areas than in military

applications. It appears, however, to be a long way to a fully automated

recognition system. Thus much more work is needed in both theory and applications.

115
C. H. Chen

References

1. G. A, Moore, "Recent progress in automatic image analysis". Journal of Microscopy, vol. 95, Pt. 1, pp. 105-118, Feb. 1972.

2. E. E. Underwood, "Stereology in Automatic image analysis". Microscope,

vol. 22, pp. 69-80, 19TU.

3. I. A. Cruttwell, "Pattern recognition by automatic image analysis".

Microscope, vol. 22, pp. 27-37, 197^+.

h. A. Jesse, "Information sources in automatic image analysis". Microscope,

vol. 22, pp. 81-88, I97I+.

5. A. Jesse, "Bibliography on automatic image analysis". Microscope, vol. 22, pp. 89-109, 197ito

6. C. He Chen, "Theory and applications of image pattern recognition",

TR EE-75-3, April 1975.

7. C. Ho Chen,et.al., "Image analysis, enhancement, display, and classification study with the aerial photographic data", TR EE-75-8, August 1975.

8. A. V. Oppenheim and R. W. Schafer, "Digital Signal Processing", PrenticeHall 1975.

9o C. H. Chen, "On information and distance measures, error bounds, and

feature selection". Information Sciences Journal, to appear.

10. C. H. Chen, "Statistical Pattern Recognition", Hayden Book Co., 1973.

llo G. C. Cheng, et.al., "Pictorial Pattern Recognition", Thompson Book Co. 1968.

116
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

A NEW,FAST AND STORAGE-SAVING IMAGE ANALYSIS PROCEDURE FOR INVESTIGATING INDIVIDUALS BY A DIGITAL COMPUTER

by M. Rink

Institut far Geophysik der TU C/austhal, D-3392 Clausthal-Zellerfeld. Germany

INTRODUCTION

The problem of automatic counting and sizing of particle projections or sections, or any other maybe complex-shaped, but individual parts of a distinct image component - here simply called

figures - is a very old one, ajid many different principles have

been applied to its solution. At present the extension of the problem towards recognizing special individuals of a given collective

by a morphometric characterization by means of a versatile detailed

analysis of individuals has become of interest. The analysing ca- pabilities of modem automatic image analysers are not yet satisfying in this respect. Some of the limitations of these instruments arise from the restrictions of an economically reasonable

perfection of the logical units and of their storage capacities.

The rapid development in computer technology suggests the use

of a general purpose computer for the solution of this problem. It

is the merit of G.A.Moore and co-workers of having given for the

first time, about in 1957» a computer method resulting in an exact

analysis of such individual figures (Kirsch et al., 1957; Moore

and Wyman, I963). But his way of solution, based on the so-called

arrested-scan-principle , is somewhat tedious. For getting information on the interesting figures, the whole digitalized image

must be stored. During the automatic analysis always just one single figure is copied out in a second blank image field, re- served in the computer, and is quantitatively investigated there.

Then this auxiliary image field is cleared again, the next figure

in the original image is searched for, copied out, numerically

analysed, a.s.o. An advantage of this method of time serial figure

by figure analysis is the easy understandability and straight forwardness of the program concept, whereas it is relatively storage—

ajid time consiuning in operation. The latter fact, among others,

has apparently led Moore (l972) himself to a somewhat pessimistic

view on the future chajices of the analysis of individuals.

The author has developed a universal image analysis procedure

(lAP) for digital computers pennitting the speedy and comprehensive analysis of individual figures without any restrictions to shape ajid connectivity (Rink, 1970).

THE IMAGE ANALYSIS PROCEDURE

The digitalized - ajnd with respect to the light intensity dualized - image values serve as input into the computer. The image

digitalizing points define the nodes of a square mesh net. They

are arrajiged in lines and columns of distance x, and can be ex- clusively either a so-called figure point or a complementary point.

The main feature of the lAP is the continuous line scan principle, marked by proceeding step by step in line direction and

continuing line by line. In this characteristic point the lAP

differs from earlier image analysis methods for computers. During

operation, information of two adjacent lines only is necessary at one particular time. In the following these two lines will be named

comparison- and investigation line.

The basic problem of such a two-line comparison consists in

recognizing which individual figure intercept belongs to which

particular figure. The principle of solution involves the quasi

parallel resp, simultaneous recognition, unravelling, and proper

assemblage of all figure parts hit by the scan. This is achieved

117
M. Rink

by using the well-known criterion of overlapping figure intercepts

supplemented by a continuous monitoring and reordering of the

intermediate figure parameters belonging to the figure parts already under treatment.

By shifting the bit pattern of an image line within the computer

this information is converted into the equivalent form of the so- called intercept data, which represent the series of pairs of col- umn coordinates of the left and right figure intercept ends of this

line. The intercept data of the actual comparison- and investigation

line are stored in separately reserved arrays. The association by

overlapping of intercepts then is found by arithmetic relations

between these intercept data.

Directly gainable figure parameters are built up successively

with each increase in the next line. Some of them are figure integral values, as e.g. figure area, and other ones are figure-specific comparison values, as e.g. the column coordinate of the leftest figure point. To each figure just being investigated, a set of intermediate parameter values belongs, with strictly observed

arrangement. All these sets of intermediate parajneter values are stored until the figures are recognized to be complete. The as- signment of such a set of intermediate parameter values to the cor- responding figure is realized by an address calculation resulting

from a systematic figure numbering. Reasons for changing these

numbers are the beginning of a new figure, the merging of several

figure parts, and the completeness of a figure.

If an intercept of the investigation line is overlapped by intercepts of the comparison line more than once, the remaining

number of the so far recognized figure part will be the smallest

number of the overlapping figure parts. In this case, the updating

of the intermediate figure parajneter values consists - besides

considering the usual intercept increase - in taking over the intermediate parameter values of the multiply overlapping figure

parts into the now valid intermediate parameter set of that so far

recognized figure part.

During the scanning of the investigation line, newly recovered

figure parts are characterized as such and their so-called starting

parameter values are stored intermediately as a reduced set of intermediate parameter values.

A criterion for the completeness of a figure is derived from the

increment of a monotonously increasing intermediate figure parameter (e.g. figure area) being checked at the end of each line.

The inserting of the starting parameter values of new figures,

and the sorting-out of the intermediate parameter values of com- plete figures as well as those no longer needed ones of multiply

overlapping figure parts, at the end of a line, in conjunction with

the two-line principle, keeps the working storage at a minimiim.

Parallel to this rearrangement of the intermediate parameter values

also the numbers assigned to the intercepts of the investigation

line have to be actualized. Because of figure parts possibly branching in downward direction those numbers then represent a se- quence beginning with one, but being not necessarily a monotonous

sequence

,

In the computing process the two-line cycle is reflected by

overwriting the content of the storage region of the comparison

line by that of the storage region belonging to the present investigation line. This can be done at the end of each line since then

the former intercept data of the comparison line are not needed

any longer. At the same time the storage region of the investigation line is available again. After this redefinition of the for- mer investigation line as new comparison line, the next line can

be investigated in the same manner. When a figure has proven to be complete, all their directly

118
On fast storage-saving image analysis by computer

gainable parameters are available in the computer. By mathematical operations a series of" meaningful derived individual figure

parameters can be calculated from the direct ones. So after just

one analysing computer nan, the wanted direct amd derived parameters of all figures of the image are determined. Of course, the

recognition of individual figures also permits their exact counting.

To demonstrate the capability of the lAP a compilation of

figure parameters will be given.

Direct figure parameters are: the area a, proportional to the number of figure points;

the smallest (le resp. lo) and largest (ri resp. hi) column resp.

line coordinate, which characterize the figure-circumscribed rectangle;

the multiple projection,p resp. q on the column resp. line direction;

the maximum chord length s^^^ ^ in line direction, also known as so-called Krumbein diameter. ' It should be noticed that the

line-by-line analysis permits determining this pareimeter - often

used as an orientation dependent and therefore statistical one- dimensional size parameter - with reference to the line direction

only, whereas it is possible to measure the multiple and single

projection in the perpendicular direction, too. the number v of rectangular inward corners at the boundary of the

digitalized figure. This information is very important for the

determination of the length of the true figure circumference,

the connectivity, informing about the niimber of holes within the

outer figure limit;

the image frame contact of a figure, which is of interest for

excluding errors caused by freime-cut figures.

Derived figure parameters are: the height h = (hi-lo+1 )x resp. the width w = (ri-le+l)x as the

single projection on the column resp. line direction or so-called

statistical Feret diameters;

the mesin chord length = a/p resp. s = a/q in line resp.

column direction. These ' linear figure * measures are still

orientation dependent, but they are weighted by the figure area.

the nominal diameter after Wadell d = 2 i a./7l ' as diaimeter of the w circle being area-equivalent with the figure. an approximation to the longest figure dimension 1 of convex figures as a non-statistical, unique measure for figure length

1 = (s /w)W(h,w) + {^-{s /^rfh'^^)', , , max , . . max , / . , . ^ ^ ^ the mean dimension d = a/1 perpendicular to 1; the minor axis e = ^a/lTC of the ellipse of area a and major

axis 1; aspect ratio factors, derived from the ratio l/d resp. l/e ; the compactness factor k = a/hw as the area ratio of figure

and circtimscribed rectangle (Moore, I968);

the so-called concavity factor k = (p+q)/(h+w), exactly giving

information about the re—entrance as well in line as in column

direction;

the length c = 2(p+q) of the rectemgular polygon limiting the

digitalized ^ figure;

a good approximat ion to the true figure circumference

c = c - v(2- )x;

The cSmbination of the two orientation independent parameters

area and circumference yields further essential information

about the figure, e.g.:

the so— called hydraulic radius m = a/c as a size and shape dependent parameter, ajid mere shape factors expressing the circularity or roughness of the

119
M. Rink

circumference as the so-called Heywood factor ^ = c/2^ aff ' or its

squared inverse value — ^Tia./ c^ ; further possibilities of combining area and circumference are

the transformations of the figure into the equivalent ellipse and

rectangle after Moore (1968).

At present the prograjn will be extended by the evaluation for

the areal center of mass and the inertial moment - pareiraeters

surely of interest to morphological and stereological problems.

With the great number of those morphometric parameters a characterization of individual figures is possible, and geometrical

filters can be set up to perform image cleaning and shape selection.

The data storage in the computer also permits the derivation

of image related integral values as well as the immediate statistical analysis of the figure parameters. So the prograjn is provided for computing the mean, the variance, and the standard deviation, and higher statistical moments if desired, of all geometrically meaningful figure parajneters , further their frequency

distributions in logarithmic or linear scale. This advanced statistical treatment of the figure morphometry is the necessary prerequisite for any attempt of a three-dimensional particle number-,

size-, and shape analysis.

The essential program part responsible for the figure recognition and evaluation of the direct figure parameters is written in

assembler language, all other parts are written in FORTRAN. The

assembler part including working storage requires about 4200 com- puter words, the FORTRAN part needs about 3500 words.

With the described lAP, the computing time for analysing an image of IO7 200 image points containing about 800 - 900 figures,

and for listing their statistical results, runs up to about I50 sec, with the TR4 computer having a word length of 48 bits and a cycle

time of 9 /Usee.

The difficile task of developing such a program, once done,

pays by the speed and enormous saving in storage space. Therefore

this software method seems well suited for the combination of a computer with a fast television input device.

References

Kirsch,R.A. , Cahn,L., Ray,C. & Urban, G. H. ( 1 957 ) Experiments in

processing pictoral information with a digital computer. Proc.

Eastern Joint Computer Conf . , 221-229, Washington D.C.

Moore,G, A. ( 1 968) Automatic scanning and computer processes for

the quantitative analysis of micrographs and equivalent subjects. In: Pictorial pattern recognition, 275-325 » Thompson

Book Co, Washington

Moore , G, A, ( 1 972 ) Recent progress in automatic image analysis. In: Stereology 3 (ed,Weibel et al,) IO5-II8, Blackwell Sci,Publ,

Oxford London Edinburgh Melbourne

Moore, G, A, & Wyman, L, L, ( 1 963 ) Quantitative metallography with a digital computer: Application to a Nb-Sn superconducting wire,

J,Res.NBS 67A, 127-147

Rink , M, ( 1 970 ) Automatische morphometrische Bildanalyse mit Hilfe

eines elektronischen Digitalrechners , Diss. Techn, Univ. Clausthal, BRD

120
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

he/d at NBS. Gaithersburg, Md.. September 4-9. 1975 (Issued January 19761

SYSTEM FOR COMPUTER INPUT AND PROCESSING OF TWO-DIMENSIONAL PICTURES

by Ivan Krekule and Miroslav Indra

Institi/te of Physiology, Czechoslovak Academy of Sciences, Prague. Czechoslovakia

A hard and software system, based on the laboratory

computer LINC 8 /DEC/ is described, which makes it possible

i/ to input contours of two-dimensional structures from a micrograph into the computer with the aid of an operatormanipulated x-y plotter /resolution better than 0.5 mm/,

ii/ to display the plot, respectively correct and store these contours on magnetic tape, iii/ to evaluate their oarameters /as area, perimeter, length of axes/ and statistics

of these parameters.

Application of the system is illustrated by examples

taken from a morphological study on the develonnent of the

rat brain.

An important prereauisite for practical application

of stereological methods to research in morphologv of the nervous system /2/ is an input of two-dimensional data /micrographs prints or slides/ into a digital computer. Vihen

comparing the information content of a micrograph and the

memory capacity of the computer, it is of use to carrv out some kind of preprocessing of the picture /i.e. recognition

of elements of structures and their classification/ concurrently with its input into the computer. However, the com- plexity of the preprocessing task mav increase owing to so- me imperfections of the picture processed. Therefore , if a digital laboratory computer is used for this purpose,

a human operator providing the classification and interpolation of the structure during the input of the picture is

very often employed /I/.

This communication deals with a system for a/ semiautomatic input of pictures into a laboratorv computer LINC 8 /DEC/; its b/ editing and c/ elementary processina features.

a/ The input is based on application of a x-y plotter,

the pen holder of which is manually controlled to follow

contours of the chosen element /structure/ of the picture.

Two types of plotters v/ere used: the analog /BAK 4T Aritma

Czechoslovakia/ and the incremental /CALCOMP 565 U.S.A./.

The former approach is simpler as far as the control is

concerned, since the holder is moved directly by the operator and the computer samples analog signals fed from precision potentiometers of the plotter. Sampling cvcles are running continuously and values obtained are compared with

those in a previous cycle. Whenever a difference is detected, the sample is stored as a new element of the contour.

121
/. Krekule and M. Indra

The application of the incremental plotter is more convenient as far as speed and processing of the input signal are concerned, since the operator controls the movement of the

plotter carriage via the computer by usina set of push button switches, in both cases the resolution is better than

0.5 mm and the matrix is 512 x 512 points in the former

or 1024 X 1024 in the latter case. The sampled contour

/structure/ of the picture is displayed concurrently on

a CRT. Each structure is stored in a separate file composed

of blocks of 0.25 K words 12 bits each, on DECTAPE.

Individual structures can be divided into 26 classes

labelled by letters. The library of all files /i.e. table

of labels, starting blocks on the DECTAPE and coordinates

of initial points/ is stored in two initial blocks of the

DECTAPE as soon as the input of the whole picture is terminated.

b/ Editing consists of i/ checking and manual correcting of erroneous segments of the selected individual structure and its renaming by using CRT display /featuring cursor

and scaling/ and a teleptype console, ii/ Simultaneous display of initial points /read from the librarv/ of all structures belonging to the chosen classes or sequential display

of full contours of these structures. Instead of the display

the x-y plotter can be used as an output device producing

a hard copy of sampled structures, which can thus be compared with the original picture, iii/ Generation of a new set

of files by selecting files of the same structure sampled

from different micrographs can be done. This will be used

for three-dimensional reconstruction of that structure.

c/ Processing of each structure enables evaluation of

its area, perimeter, length and orientation of axes. Arithmetic mean and standard deviation of these values for different classes of structures can also be computed.

Programming was accomplished in the LAP 6 assembler,

the processing benefits from double-precission arithmetic.

The system has modular structure and some modules are also

used for input and processing of time functions or data in

autoradiography.

The described system is now applied to the quantitative study of development of nervous tissue in the hippocampus

of rats. It may, however, be used also for linear problems

like the study of the lengths of different kinds of neuronal

axons

.

REFERENCES

1/ Cowan W.M. and Wann D.F.: A computer system for the

measurement of cell and nuclear sizes. J. of

Microscopy _99:331-348, 1973

2/ Haug H,: Stereological methods in the analvsis of

neuronal parameters in the central nervous

system. J. of Microscopy 95^:165-180, 1971

122
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

PATTERN RECOGNITION ON THE QUANTIMET 720 IMAGE ANALYZING COMPUTER

by Katheryn E. Lawson

Sandia Laboratories, Albuquerque, New Mexico 871 15 U.S.A.

The Sandia Laboratories Image Analyzing Facility consists of an IMAMCO Quantimet 720 Image Analyzing Computer with an assortment of modules selected

to tackle a wide diversity of problems in materials characterization. The meas- urements most frequently required are particle count, particle area, and particle size distribution. The term "particle" refers to any feature of a material

which contributes to or can be related to its physical or mechanical behavior.

Thus it may be, for example, a void in a material or an inclusion in a matrix.

On the 720 an image is produced from a microscope or epidiascope, then pro- jected onto a high resolution scanner. The video signal corresponding to the image is then detected; i.e., interesting parts of the field of view are isolated on the basis of grey level criteria. The detector output signal is then

fed into a computer module which makes the required measurement, which may range

in complexity from a simple area measurement to an elaborate pattern recognition. Numerical values for the measured parameter are shown on a display moni- tor screen which can also show exactly the features measured. The 720 is interfaced to a Texas Instrument Silent Data Terminal which tab\ilates all measure- ments made on each field and if desired, records on tapes data for input to a EDP-10 time- sharing computer to which the system is acoustically coupled.

The modules available for operating in the pattern recognition mode include

(l) the 1^3 standard computer which accepts the detected or amended video, per- forms chord sizing and controls the remaining modules; (2) two function computers

which measure area, perimeter, horizontal and vertical Feret diameters, and horizontal and vertical projections; (3) a form separator which computes a shape

function involving measured parameters combined in a way suitable for the sepa- ration of mixed populations of features; and ih) a classifier-collector from

which is released only those data meeting the pre- selected criteria. The data

flow is shown in Figure 1.

A number of metal and metal oxide powders, including Fe, W, and oxides of U

and Pu have been subjected to classification and particle sizing. Most often,

the classification has been based on area sizing primarily because the particles

have been near spherical and varies faster than the linear parameters.

As an example of these classifications. Figure 2 shows the monitor display

for the analysis of a Pu oxide aerosol. The aerosol was produced and dispersed

with a Lovelace aerosol particle separator (LAPS) designed at the Lovelace foun- dation for Medical Education and Research in New Mexico. LAPS is essentially a spiral centrifuge which continuously separates micron and submicron particles

according to their aerodynamic equivalent diameters, i.e. "the diameter of a spherical particle of unit density having the same settling speed in air as the particle in question."-'- The particles are collected on electron micrograph grids

strategically placed along the spinning spiral duct of the separator. Individual

area and perimeter measurements from the 5000X electron micrographs were made as the particles were moved successively into the live frame. The variable frame

and the translational-rotational capability of the epidiascope stage were par- ticularly useful for this analysis. As shown, the particles are generally non- spherical and non- uniform. Even so, area was considered to be a suitable criterion for classification and was used as the basis for the distribution shown

by the histogram in Figure 3- The mean area of the particles was 0.68 ± .07m-^. Manual measurements of some of the larger particles agreed with those obtained

by the Quantimet within ^^5%. The software for this histogram was developed at Sandia. It is a family of subroutines directed by a driver program to input sta123
K. E. l.awson

tistical data from the keyboard or to process data already in a file on the PDP-10 by means of a Tektronics kOlO terminal. The program is interactive and has many options for output presentation.

The measurements made on the aerosol utilized the MS3 standard computer,

one function computer and the collector of the pattern recognition system. They

could very well have been made using only the MS3, which can be programmed to provide area-perimeter data. Ideally, this analysis would have utilized the MS3,

one or two function computers, a sizing distributor, and the classifier-collector.

The inability to fully utilize the pattern recognition mode is related to detection problems and the quality of the micrographs . This situation will again be referred to under limitations of the system.

A more extensive application of the pattern recognition mode is illustrated

for an airborne dust sample used in the abrasion of solar reflector materials.

The sample which had passed a Tyler standard screen sieve #35 (^17|-i opening) was dispersed dry onto microscope slides. Some UOOO particles were available for the size distribution. As shown in Figure U, the particles of interest cover a wide range of sizes. Pattern recognition was used to accept and provide data

for particles falling within a certain area range and rejecting all others, as shown in Figure 5- A size distribution into eight classes using automatic

staging and focusing resulting in a curve shown in Figure 6. An alternate method

of sizing would use the form separator set to calculate the dimensionless area/

perimeter^ factor for each feature. The smaller particles have a value closer

to that of a sphere for this function and can be sized separately.

Finally, the suitability of this or any pattern recognition system for stereological analysis is not without its drawbacks. Inherent sources of error

should be fully appreciated when attempting quantification of a microstructure

through the image analysis technique. Few specimens have the high surface per- fection and contrast needed for full utilization of all the functions of an automated image analyzing computer. The choice of modules for use in any one analysis is dictated not only by the information needed, but also by the quality

of the input image. Even when elaborate specialized preparation techniques are resorted to, as was the case for the Pu oxide aerosol previously discussed, the quality of the input to the system is often inadequate. In this particular case

it was virtually impossible to focus on a large area of the electron micrograph.

This resulted in an increase in analysis time, since it was necessary to employ

semi-automatic and even manual operation.

The most frustrating source of error originates in the detection. Even with the most sophisticated automatic detectors, such as the 2D of our system, the wanted feature is often improperly detected. The operator must often man- ually adjust the threshold of detection to insure that all feature boundaries are correctly recorded. The recently available image editor module which permits the operator t.o modify the image for more accurate detection of wanted features will

be added to our system shortly and hopefully will alleviate this problem.

Reference

1. P. Kotrappa and M. E. Light, Rev. Sei. Inst. , 1+3, 1106-1112, 1972.

124
Pattern recognition on the Quantimet 720 Image Analyzing Computer

MS3

FUNCTION

1

FUNCTION

2

FORM

SEPARATOR

CLASSIFIER

FORM SEPARATOR

COLLECTOR CLASS IFIER^

CHORD SIZER RELEASE

SELfCTED DETECTED OR AMENDED V I DEO IN DATA RELEASED ONLY FOR FEATURES WH ICH MEET

MS3, FORM SEPARATOR AND CLASS IFIER CRITERIA

Fig. 1: Modules for Pattern Recognition

Fig. 2: Framing of a Pu Oxide Aerosol Particle for Image Analysis (5000X Electron Micrograph)

0 5 10 15 20 25 30

PARTICLE COUNT

Fig. 3: Distribution of Particles in Pu Oxide Aerosol

125
K. E. l.awson 3

Fig. h: Full field count of airborne dust particles

Fig. 5: Field count of airborne

dust particles having an area > 500pp2

DUNE DUST DISTRIBUTION

100-1

80-

60-

cr 40-

°

« [I I—I

ft I

1 I I I

I—I—I I I—

p

0 1000 2000 3000 4000 5000

AREA, pm

Fig. 6: Size distribution of airborne dust particles

126
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, ^W., September 4-9, 1975 (Issued January 1976)

RECOGNITION AND CLASSIFICATION OF STRUCTURE BY MEANS OF STEREOLOGICAL METHODS

IN NEUROBIOLOGY

by Gustav Bemroider

Zoologisches Institut der Universitat, A-5020 Salzburg, Austria

SUMMARY

The paper presents methods which connect pattern recognition procedures

with the probabilistic formalism of stereology. By taking individual measurements and transforming them to a d-dimensional feature space, the most informative datas can be extracted. Now the question: what are the most informative

variables describing the particular structure?, can be answered. By this supervised data reduction much more information can be obtained than by using mean

values as estimates. An important result of these methods is the possible com- bination of position invariant and dependent characteristics which is strongly

desirable for many neurobiological questions. The realisation of these problems

is based on two software packages named ACOFS and SWIFA (Automatic Comparative

Feature Selection) and (Structure Within Function Analysis) . Both are demonstrated by flow charts. Some results, which are obtained by analysing hypothalamic nuclei of the developing postnatal ratbrain, are shown. The general concept

itself is not versus stereological principles but can be a well integrated ex- tension of them.

INTRODUCTION Among others, there is still one great problem in Stereology which is worth

while to be considered most carefully and the solution of this may lead to a new

concept in today's Stereology. Especially in biology many questions deal with

arrangements of cells into certain groups, where the properties of a single

cell are of as much interest as the specific position of this cell and possible

relation to near or far neighbours. The best example for this is the brain. In neurobiology we are concerned with two main 'neuronal specificities', the first

being the position invariant properties of a single neuron, the second position

dependent locus specificities (Jacobson M. 1973). Now the question arises: how

can both types of information be obtained by using stereological principles , e.g. taking random sections and extrapolate from 2-dim to 3-dim structural information. In this paper an approximate solution is discussed, which is based on 'supervised data reduction' after having transformed single particle variables

from 3-dim observation space to a d-dim feature space.

It should be noted that a rigorous mathematical notation of this problem

is not subject to this paper and after completion will be discussed at another

place. The result of such data extraction methods is numerous. Firstly we can combine position coordinates and their properties. Secondly we can select those variables which contribute most information to our problem. This 'information' content of each variable is tested by its 'discrimination power' e.g. by its ability to discriminate between a class of neurons A" and a second class B". The concept of SWIFA (Structure Within Function Analytical System) mainly

lies in the selection of 'high density boxes' by prior knowledge. This means

that domains (D(l), D(2), D(k)) are selected where the probability of occurrence of single cell characteristics is considerably high.

Rather than program details, the concept of a possible connection between

individual and mean valued measurements will be discussed, since this concept in general enables various practically most useful procedures. Some problems like

comparative feature selection can only be mentioned briefly and the reader who

is not very familiar with pattern recognition terminology may find some topics

rather difficult to understand. He may refer to one of the many excellent books

written about this subject. We shall first consider some information -theoretical

relations and apply them afterwards to the principles of feature selection methods. Classification algorithms are not the decisive part in pattern recognition

and their efficiency is strongly dependent on proper feature selection.

127
G. Bernroider

INFORMATION CONTENT AND CONSISTENT MEAN VALUES Stereological analysis of real structure is based on expected values which

result from various statistical experiments, as random division of space, random

interaction of an element with its structural component etc. As a result we obtain estimates for the fundamental stereological relations

( Pp. Ll' Aa. Nv) (•>

If such an experiment (e.g. the result of an interaction between a structuring

element and a component a on the section plane) is denoted by Z, writing

Z = ( Zi,Z2,--,Zn

)

. (2) Pl>P2»-->Pn

where (zj ,Z2, . . ,Zn) are its possible results and (pi ,P2 > •

• >Pn) the corresponding

probabilities, a measure of Z is H(Z)

H(Z) = E(pi,P2,..,Pn) = - E p(j)lg P(j), (3)

j H(Z) is the well known information content of experiment Z. Using 'mean values' on the test plane , possible results of the experiment (Z) are [ 1 /m . E C(i) ]j where j = l,2,3,....,n and where C stands for any measur - able characteristic (e.g. Area).

Of course, the probability that a certain characteristic C(j) appears witlin

the mask of a test area depends on the relative size of this area (e.g. the random variable 'occurrence of C(J) within a sufficient small test plane will follow

a Poisson distribution with considerably small p(j).) On the other hand the abo\e mean value [ I /n . E C(i) ]j is supposed to be consistent, symbolically,

lim P [ll/n. EC(i) - E(C)|<e]— 1, (4)

n-«o ' j[ While, concerning the total experiment Z(zi,Z2, . . ,Zr,) , the average information

content H(Z) merely depends on the n-tupel P (pj ,P2 , • . ,Pn) with 0^p(J)^l and E P(j) = 1.

j We also know that H(Z) has its maximum where P is equally distributed with

p(j) = l/n. In other words, if there are messages (sections and test planes) with

large p(j), the average information contents per message .is poor. Dealing with ojnsistent mean values only brings a considerable large loss of information per message about our domain D, since by replication for every message, it is most

probable that the case [1/mE C(i)]j = E(C) (5)

i occurs. The information concerning each single C(i) is almost lost. This fact might seem to be obvious now, but nevertheless is hardly ever accounted for in theory nor in practice. It can be simply demonstrated by comparing the computer

storage used before and after calculation of consistent statistics. This loss of information becomes a decisive problem, if 'individual properties' of particles

are important and even more is their position within the domain D is also of interest.

SINGLE CELL CHARACTERISTICS

As a single cell characteristic we simply define an r-dim coliimn vector

^1

C2

(6)

Such an r-tuple is presumed to be the result of the feature extraction operation

Since feature extraction is normally a reduction of a large amount of data to a

128
Recognition and classification of structure in neurobiology

smaller amount of data with the same useful information content, r^k if the number k denotes the dimension of the column vector in 'observation space'. The

amount of objects in observation space (e.g. cells, nuclei etc) leads to a sequence

of t such vectors, denoted by C(")-t = [Cj, ,C^]. FIG I demonstrates a single cell characteristic of a neuron (a typical neuron of Nucleus Supraopticus) . In this exan5)le r=5 and the variables measured are Area (normal), Area(paralysed) , Vertical Projection,Horizontal Projection and Density of the cell nucleus.

A Problem -in Measurement Spaae

Each individual cell characteristic C is obtained from measurements on a section with finite thickness. Because we deal with individuals and not with

'aggregate mean values', each characteristic C(j) of the jth objects must be a good representative. In other words, a consistent estimate for each vector

component should be optained.

Now, if c(l)", c(2)" ,c(r)" are the true parameters, we have to find an estimate c(l), c(2), ,c(r) for which

lim P [| c(i)n - c(i)"| < e] - 1 , as n - « (7)

for i = l,2,...,r and e being a real number however small. How to obtain such estimations within the slice sections has been discussed previously.

(Bemroider, 1974)

Transformations to Feature Space

The above written sequence of vectors are characterized by their components

and their dimensions. The selection of those is most decisive for the following

processes. The underlying philosophy of a selection mechanism is the retention

of class discriminatory information and reduction of class common information.

There are too many different methods of feature selection that they can be discussed at this place. As we have seen above, the average amount of information can be maximized if p(j) = 1 /n for all n. This implies that we learn most from every message ( = object ) if the probability of occurrence of a single cell characteristic is uniformly distributed. We also know, that, if there are two experiments Z and Q, the amount of information which is contained in Z about Q is just

I(Z/Q) = H(Q) - H(Q/Z), (8)

where H(Q/Z)

H(Q/Z) = - E p(i) (Z P(k/i).lg P(k/i), (9)

i k

H(Q/Z) is the mean conditional uncertainty of experiment Q under the assumption

that the result of Z is known. If we now apply this to a conditional probability

P(W(k)/c(i)£) which says that P is the probability that an object is allotted to the kth class W(k) under condition that the ith component c(i) takes a value of t, the amount of information for a given dimension becomes

I(W/c(i)) = H(W) - H(W/c(i)), (10)

Note: If this amount of information is high, c(i)£ also carries high information,

thus ordering the coordinate dimensions (c^, Cj,) such that

I(W/ci) ^ I(W/c2):^> >I(W/Cr), (11)

we can reduce our r - dim observation space by retaining the largest such

dimensions only. If we select the first m components we have reduced this space by r/m. (FIG II represents a selected 2 - dim feature space for n vectors)

Note: Datas can now be reduced with only small loss of information!

129
G. Bernroider

FIG I

A single neuron and its image

display .demonstrating a 5 - dim

single cell characteristic C(5).

Also from the detected image

itself characteristics can be extracted. If in such a case the most informative m features

are extracted, the result of it will be the well known

'skeletonization' of picture

processing.

FIG II

2 - dim feature space for

a sequence of n vectors.

The s)raibol 'X' stands for a density box where it is highly

probable that a neuron of 'Nucleus Suprachiasmaticus

appears. '0' is the box of Nucleus Paraventricularis

The components are Area of the cell nuclei and their

optical density. The cluster of 'X' is clearly

visible.

FIG III

The computed area of box '0' show that the neurons tend to form a physical cluster.

Do they also form a cluster

on the basis of their single

cell characteristic ?

130
Recognition and classification of structure in neurobiology

131
G. Bernroider

WHAT IS WHERE? RANDOM POSITION DEPENDENCE When we are interested in the position of a single cell characteristic we have to select the 3 physical dimensions and transform them to feature space, obtaining

[di,d2,d3,Ci , . . . .Cn] (12)

with r = n + 3 dimensions.

The problem itself lies in the fact that each 'location' should have the same chance to be selected. This can be obtained by a 'uniform random number generator'

and a selection of 'high density boxes' where the probability of occurrence of a single cell characteristic can be adjusted with the number of locations necessary.

FIG IV

Demonstration of 'high density

boxes' and the origin of

a random coordinate frame. The origin of the coordinates

lay within a certain tolerance

region which is due to the normal spread of biological

individuals

.

AUTOMATIC COMPARATIVE FEATURE SELECTION (ACOFS) If we can set up two extremely different feature vectors X and Y and select clearly different categories W(I) and W(2), under replication feature selection

for both vectors will lead to a point where both vectors dicriminate both

categories W(l), W(2) best. As a criterion of "goodness" serves the reclassification rate, i.e. the percentage of objects of category W(J) and W(2) respectively

which have been allotted to the corresponding class W(l)", W(2)''. Flow chart 2 shows a simplified diagram of ACOFS. The procedure stops where both

kind of information have the same effect and allow 'correlative Interpretation',

(e.g. a feature vector obtained from electron microscopical investigations and another one from light microscopy)

CONCLUSIONS

Individual data extraction allows selection of the most informative variables and measurements describing a physical structure. Further , using Pattern Recognition and Cluster techniques a number of operations can be performed which

enable the linkage of position dependent and invariant cell characteristics and comparative feature selection for "interpretation problems". Compared with

conventional estimates the amount of information gained can be decisive for many

problems, especially those of Neurobiology.

ACKNOWLEDGMENT

I would like to thank Prof. Dr. Adam for his encouragement and support,

further Dr. A. Goldschmied for his anatomical advice and Mrs. H. Langer for excellent assistance in preparing the biological material. The work was kindly

supported by the Fond zur Forderung der wissenschaf tlichen Forschung, Austria,

(Progr. 1838/n 39)

REFERENCES

Andrews, H.C. (1972) Introd.to Mathem. Techniques in Pattern Recognition, Wiley

Bernroider, G. (1974) In: Spec . Issues of Pract.Metallogr . ,5,pp. 177-184,Dr.Riederer V. Haug,H. (1972) J.Microsc. ,^,pp. 165 - 180. Miles,R.E. , (1972)J.Microsc. ,95,pp. 181 - 195. Miles, R. E. , (1969)Adv.appl.Prob.J^, pp.21 1 . >,

132
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

PROJECTORS IN PAHEFUM RECOGNITION CATEGORIES

by Monique Pavel Universite Rene Descartes, Paris, France

ABSTRACT

In Pattern Recognition, as well as in particular in Stereology, one aims

to identify patterns starting from incomplete information; we are interested

in this paper in mathematical methods of Pattern Recognition which can, in particular, be useful in Stereology. We proceed in two steps.

(1) We present for the first time a general formalism of recognition problems, using a categorical setting. If the objects IeI are classes of identified configurations into a same image, and if the family A of deformations 6 is a semigroupoid, then we prove that (I, A) is a category. If A contains the group

r of similarities, then (I,r) is proved to be the associated recognition category. We characterize probes and recognition functions as being invariant

functors of categories; probes may in particular be projectors.

(2) Patterns can be rebuilt and studied starting from their projections

and the lifting properties of the latter. We present for the first time a characterization of projectors by means of retracts of images and of categories

of images. In the category (I, A), every image lel is a retract of its similar

images. If T is a normal subgroup of A, and if 6^^ e Morjp(Iy,I^) is a retraction in (I, A), then its class of equivalence is a retraction in the quotient category I^/r. I being a recognition category associated with lA, if

the category P of patterns is P = I/F, then P is a retract of I.

INTRODUCTION

We present in this paper the first general mathematical setting for recognition problems and for projections. Objects or images can be described by

their primitive components and their composition, and (or) be defined by their

global and (or) local topologically invariant properties; recognizing a pattern

then means detecting an equivalence, or isomorphism, with regard to a specified

set of transformations, between the given object and an element of a set of

"canonical patterns," or templates. This identification process has to be accomplished starting from incomplete information about objects and about

possible transformations between them.

In order to systematize the concept of recognition problem, it will be helpful to introduce and adapt some elementary concepts from category theory,

and then to define recognition categories and invariants: the interest of recognition categories, in which all transformations are isomorphisms, lies

in the fact that they provide the adequate setting for the identification

problem associated with the original category, in which transformations are deformations

.

The equivalence mentioned above is generally established by computing and comparing a "sufficient" number of probes, i.e., functions which are invariant

with regard to certain transformations. Probes often used are projectors. The interest of retracts is that they provide the general mathematical setting for projectors in recognition categories and that they focus our attention on their

particularly useful invariance lifting propetties.

Only a slight acquaintance with the elementary definitions of category

theory (see B. Mitchell [1] ) is needed to understand the discussion in this paper

.

133
M. Pavel

1. I^4AGES, PATTERNS, RECOGNITION

We shall refer in this section to M. Pavel [2]

.

1.1 Let S be a set of elements s^, called signs , possibly belonging to different types aj, Op. Configurations Ci = (s^j^, s^j,,) e C are built

by defining on S a composition +; a group G of transformations g, called

similarities , is also defined on S, G: S ^ S. An identification relation R, or

a simplification operator p, both of which are equivalence relations and G- invariant, define I = C/R, respectively 5 = C/p ; the elements lel are classes,

called images: I = {c; cRcj}, respectively, I = {c'; p(c') = p(cj)}. When

defined, the mappings (Ii,l2) I1+I2 ^"^"^ (gjl) ^ gl give rise on I to an additive semigroup structure such that g(Ii+l2) = gIi+gX2 and gi{g2l) = i9\92^^'

I is called a "grammar" of images.

The patterns Pj , . . . ,Pj^ retain only the same "meaning" of possibly different images (e.g. behaviour with regard to transformations, or types which have

been used, etc.): the patterns are defined by means of a family P of disjoint and G-invariant sets Pj , Pj^. 1.2 The recognition problem for I consists in finding a mapping, or recognition function, R: I P, which assigns an image lel to a pattern PjeP

(if it exists) . If in particular P is the family of equivalence classes induced

by G, we shall write P = I/G and then R(I ) = R(I ) iff I = gl . y V V y 1.3 If, more generally, a set T of transformations t, T:l -> 1 , containing

the group G of similarities but not necessarily being a group is given on I, and if there exists a set of templates (i.e., images I^eI such that R(TI,,) = P>,) we shall say that 1^ and I^ are synonymous iff R(Ijj) = R(I^) , i.e., iff

I^ = tl|j, teT. Synonymy is generally established by means of probes P, which

map I on a set V (generally different from I) and which are constant on synonymy

classes: I^ E I^ (T) =>P(I|j) = P(Iy); probes can in particular be projectors.

^ 1.4 Let D be a set of transformations d, called deformations , defined on

I ; I will stand for the set of elements dl , with lel and deD. We shall suppose

that D I, and that similarity transformations as well as the composition + on coincide (on I) with those originally introduced on I . A recognition function

R:I° P the n assigns a deformed image dlel'^ to a pattern P-j-eP (if it exists) : we shall write R(dl) = Pj. If ^efojrmations d are homomorphisms (d(Ii+l2) = dli+dl2, d(gl) = g(dl)), then I = {dl; lel, deD, I a grammar and d homomorphic}

is a "grammar" of deformed images.

2. RECOGNITION CATEGORIES

2 . 1 Defini tions .

Categories will be designated by capital script letters, objects of categories by capital Roman letters A through I, and morphisms of categories by lower case Greek letters. If A and B are objects of the category A, the ex- pression Mor^(A,B) denotes the set of morphisms with domain A and range B. The composition of the morphism a with the morphism B is written ag, i.e., 6 is followed by a. A morphism a e Mor^(A,B) is called an isomorphism if it has a two-sided inverse, i.e., there exists g e Mor^(A,B) such that ag =

I3, 6a = Ift. We write A B if there is an isomorphism a e Mor^(A,B). It is easy to see that = is an equivalence relation. The capital Roman letters

J, P, R, S, and T represent functors from one category to another; only covariant

functors will be encountered in this paper. If S and T are naturally equivalent

functors from a category A to a category 8, we shall write S = T.

134
Projectors in pattern recognition categories

Definition 2.1. A recognition category A is a category in which all mor- phisms are isomorphisms.

In the rest of this section, A will denote a recognition category.

Definition 2.2 . If A,B e Ob A, the recognition problem associated to A consists in deciding whether or not A is isomorphic to B, i.e., whether or not

Mor^(A,B) is non-empty.

Definition 2.3 . An invariant J of A is a functor J: A -* E, where E is a discrete category (i.e., the only morphisms of E are the identity morphisms

1e, E e Ob E) . The last definition means that J is an invariant of A iff A-= B in A

iTnplies J (A) = J(B) in E; J is full (or complete ) if the converse is also

true.

2 . 2 Recognition categories .

Let the objects of Ob be the sets of classes I^el of n-j^-uples of R-

^ identified or p-simplified configurations defined in section 1; taking for Mori

the family of mutually disjoint sets {Mor (ly, I^) } for all objects ly,

e Ob I , whose elements 6^^ e Morj^(Iy,I^) are defined by the triples

6yy = (Iy,dpy,I^) , where I^^ dy^Iy, dy^ e D, and "forgetting" for the moment

the algebraic structure of I*^, we prove:

Theorem 2.1 . If A is a semigroupoid, then the family (I, A) is a category.

We have thus organized images and deformations in a category, and shall

designate this category by I^. Corollary 2.1 . The category in which the objects are the sets I and

the morphisms are the deformations 6yy, is the image by a forgetful functor of

the category in which the objects are grammars and the morphisms are homo- morphisms . Similarities being elements of a group G, and defining V =

{y^v' ^yv ~

<Iji' gyv ' 9yv e G} we prove:

Theorem 2.2 . If A is a semigroupoid of deformations containing the group

r of similarities, then (I,r) is a recognition category associated to the category I^. The only morphisms of (1 ,V) are similarities, therefore isomorphisms,

and hence Mor^j r) (^y^v^ ^ ^ means that ly and 1^ are similar. We shall

designate this category by I. Corollary 2.2 . T tA

(i) i is a (non-full) sxibcategory of L .

(ii) P is a discrete category.

(iii) If A is a group containing F as a normal svibgroup, then the category

P = I/r is mapped by the deformations belonging to the quotient group

A/r on the category 1^^^

.

2.3 Recognition problem .

Given the category and an equivalence or synonymy law T, the recognition

problem for reads as follows: find a way to decide if ly H 1^ (T) . By

associating to its recognition category I , the recognition problem for the latter reads: given Iy,I-y e Ob I, decide whether ly = I^(r), i.e., whether

Morjdy, ly) ^ 0. The solution of the problem can be expressed by:

135
M. Pavel

Theorem 2.3 .

(i) Probes P of the category I are (forgetful) invariant functors from I to a discrete category E:

I = I (T) in I=>P(I ) = P(I ) in E. U V y V

(ii) Recognition functions R defined on I are invariant functors from I to the discrete category P of patterns:

I = gl <S^I cs I in I ^R(I ) = R(I ) in P.

y V y V y v

(iii) If P = I/r then R is the canonical mapping of I into its equivalence

class [1]

.

(iv) If P is a full probe, then

P(I ) = P(I )<=>I = I (T)<?>R(I ) = R(I ).

y V y V y v See for the proofs of the theorems in this section, M. Pavel [3]

.

Let us note that: (1) the equation in (i) is the reason for which probes, and in particular projectors, are used even if they do not solve the problem

entirely: if P(Ip) 7^ P (.1^) , then certainly Ijj 9^ I^. (2) G-invariance is an elementary component of synonymy T; it shows off one of the numerous characteristics of the images in I , G-invariant probes forgetting all but invariance

with regard to the similarity geG. (3) Total discrimination requires a full

(or complete) invariant, which is often built on a basis whose components are "simpler" but each of them being incomplete: one actually operates on "ele- mentary" characteristics of images. (4) The recognition problem for cannot

be reduced to that of constructing a full invariant of I; in order to obtain

significant results, one has to take Py = R(TIjj) ^ 1^: Py has to contain also

other images than those obtained from I by similarity.

3. PROJECTORS

Probes that are often used are projectors. Let P be a projector, i.e., a mapping from a space 8 of objects to the subspace A of B, P:8 A, such that

P(8) = A and that p2(b)=P(b) (which means that P(a) = a for all aeA) . In other

words, if P is a projector to A, then its restriction on A is the identity

mapping of A:P/A = Id/\; this comes to the existence of a mapping P from A to 8 such that PP = Id^. If instead of taking for A a subspace of 8, we more generally take for A an arbitrary space, if we consider the mapping S:8 ^ A, and replace P by an arbitrary mapping T:A 8, then, in order to preserve the projector's property we need to ask that ST be, if not equal to the identity of

A, at least "equivalent," in some sense to be precised, to the identity of

the space A: this can be expressed by the general concept of retract.

The utility of retracts in recognition problems proceeds from their

lifting properties: suppose, more generally, we are given a functor T:A 8, A and 8 categories. Then T can be used to lift every invariant of 8 to an in- variant of A; in fact, it is obvious that if J is an invariant of 8, then

the composite functor JT is an invariant of A. Many familiar invariants

are obtained in this way.

3.1 Definitions .

Let us define first retracts of objects in a category and then retracts

of categories.

136
Projectors in pattern recognition categories

Definition 3.1 . A morphism a:A->B, A,B e Ob A, is a coretraction

(or a section ) if there exists a morphism g:B A such that ga = 1;^. A is called a retract of B.

Definition 3.2 . A morphism a:A B is called a retraction if there

exists a morphism yiB ->- A such that ay = Ib- If a is a retraction and a coretraction, then it is an isomorphism (see Section 2.1).

Definition 3.3 . Let A and B be recognition categories; A is a retract

of B if there exist functors S:B -> A and T:A -»• B such that the composition

ST be naturally equivalent to the identity functor Id^. We shall in this case write K-*^ ,pB or simply A -< B if there is no possible confusion. The condition A -* 6 agrees very well with the intuitive idea that the recognition

problem for A is no more difficult than the recognition problem for 8.

Definition 3.4 . Two categories A and B are equivalent if there exist

functors T:A B and S:B A such that ST = Id^ and TS = Idg. Obviously, if A is equivalent to B, then A is a retract of B and B is a retract of A, but

the converse fails.

Definition 3.5 . Let A be a recognition category. For leObA, define

Aut(I) = Mor^(I,l). For a e Mor^(I^,I^) and ifi s Aut(Ijj), define (Aut(a))

(i(/) = aijja"^.

3 . 2 Projectors in pattern recognition categories .

Theorem 3.1 . In the category I , every image I is a retract of all similar

images gl, geG.

Theorem 3.2 . If 5^^ e Mor-^d ,1^) is a retraction in I^, and if A is a group containing F as a normal subgroup, then the equivalence class [5^^] of

5j.^ is a retraction of the quotient category

^ Theorem 3.3 . If I is a recognition category associated with the category

I , and if the category P of patterns is P = I/G, then the category P is a retract of the category I

.

Probes P:I -> V, P(I) = V ,y a discrete category, and in particular pro-^

jectors, can then be used to lift every invariant of to an invariant of I^; and every invariant of I to an invariant of I^; and every invariant of P to an invariant of I

.

Theorem 3.4 . A recognition category I is equivalent to the discrete

category P of patterns iff Aut(I) is a one element group for every image lel.

See for proofs of the theorems in this section, M. Pavel [4]

.

Ordering recognition categories following the concept of projector given

in Definition 3.3, the last results can be interpreted as follows: (1) discrete

image categories, such as the category P of patterns, are minimal; (2) every recognition category majorizes some discrete category; and (3) a recognition

category which is equivalent to a discrete category need not be discrete but

is very particular: each set of similarities Morj(I ,1^) is either empty,

or else contains exactly one isomorphism; in particular, Mor^d,!) is a one element group for every image lel.

137
M. Pavel

BIBLIOGRAPHY

[1] B. Mitchell. Theory of categories. Academic Press, 1965.

[2] M. Pavel. Fondements mathematiques de la reconnaissance des structures. Actualites Scientifiques no. 1342, Hermann, Paris, 1969.

[3] M. Pavel. Reconnaissance des structures dans les categories. Comptes

Rendus de I'Academie des Sciences de Paris, serie A, v. 280, 1975,

pp. 295-298.

[4] M. Pavel. Retractes dans les categories de reconnaissance. Comptes

Rendus de I'Academie des Sciences de Paris, serie A, v. 280, 1975,

pp. 285-287.

138
INSTRUMENTATION

Chairpersons

:

H. P. Hougardy

R. Ralph

H. Haug

R. Kirsch

G. A. Moore

National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

AUTOMATIC IMAGE ANALYSING INSTRUMENTS TODAY

by Hans P. Hougardy

Max-Planck-lnstitut fur Eisenforschung GmbH. DOsseldorf, Germany

SUMMARY

At this time, there are 12 different instruments for automatic image analysis on the market. Most of them are modular

giving an infinite number of sub-systems. Advices are given for

a technical description of the performance of such instruments.

A survey on the modern pattern recognition systems is given with

some remarks, how to choose the 'best' suited instrument.

Since the last ISS congress in 1971 there has been an explosion in the development of instruments for automatic image

analysis. The variety of only one instrument makes it impossible

to describe a single system in detail. Therefore the basic rules

for the design of such Instruments are given as well as some advice to the user how to form an opinion of such a system and to

test them. The available commercial instruments are listed with a short characterisation.

PRINCIPALS OF AUTOMATIC IMAGE ANALYSIS

The reason for using automatic image analysis is to make the

measurement faster and more accurate. The im.age to be analysed may

be a print, a slide or immediately the output of a light- or electron microscope. In general, special features must be selected

for measurement from all the other parts of the image. That means

that all the information of the image must be transferred to the

analyser, but only a small part is really needed for the measurement. Therefore there must be a rapid decision about each information bit, if it is needed or not, to get a sufficient working

speed

.

At this time the fastest processing and data handling systems

are analog or digital computers. Consequently the first step in an automatic image analysing instrument is a conversion of the

optical information (the image) picture point by picture point

into an electric signal suited for the particular computer used.

The converter - see figure 1 - may be a TV camera, a photo diode,

or a photomultiplier , as well as an x-ray tube. If the image

already has the form of an electric signal - as in scanning

electron microscopy - only an adaption of the voltages and a synchronisation is necessary.

All instruments 'detect' the features to be measured first

according to the signal modulation - the grey level in an image - using a signal discriminator (1), From these detected features, in

a second recognition process, those having the desired shape,

orientation or other criteria are next separated and analysed to

get parameters such as area or perimeter. The accuracy of such a measurement of course depends mainly on the correct image con- version and detection (1,2,9). Therefore the main efforts of the

manufacturers in the last few years were to Improve the converter

accuracy and to develop automatic detectors, in order to be

independent from subjective threshold setting by the operator

(1,2). This was the basis for the realisation of extended pattern

recognition systems (3-7).

141
H. P. Hougardy

DESIGN OF MODERN IMAGE ANALYSERS

Low cost instruments are built according to figure 1 without

a 'feature recognition' unit. In this case the 'feature analyser' measures only 'field specific data'. That means, for example, that

the value of 'projected length' is the mean value of all the

features in a field of view. It is not possible to measure area or perimeter of each feature in a field of view separately, except

setting the measuring frame to each feature separately by hand.

This is a limitation, but for many applications these instruments

give all the desired , results with sufficient accuracy.

If there are several parts of the image having the same grey

level as the features to be measured, the grey level detection

must be extended by a feature recognition according to their

shape, which is realised in modern Instruments using three

different principles: The parameter method, the feature modification and the interactive method. According to figure 1 the first

step is in every case a grey level detection to reduce the amount

of information to be handled. To give a survey, all principal

systems available at the present time are combined in figure 2. Any particular system can be redrawn from figure 2 by deleting one or several units or connections. Of course, depending on the

instruments, the 'units' of figure 2 consist of one or several

hardware or software modules with different capabilities. In some cases, the arrangement of the units as given by figure 2 can be

changed to get another sequence of operations.

The parameter system . Using this method, the shape of

features is defined by the parameters determined by the 'feature

analyser' (3»'J)» These parameters are interconnected in a 'shape

recognition' unit to get such values as 'area divided by the

perimeter squared' which is used as description of a circular

shape, for example (6-8,16). Only the parameters of features

satisfying such a preset shape condition are fed into the

analysing and output computer by gate h. This method is quite

powerful, provided the features have sufficient contrast and a sufficient size on the monitor screen (2,9). Otherwise the scatter

in values of the shape descriptor is too great, giving an over- lapping in identification of the desired shape with other shapes.

If, for example, the coefficient of variation for repeated

measurements of one field of view for feature parameters is 3 ^ - due to insufficient contrast of the features - it is not

possible to distinguish exactly between circles and squares using

the shape descriptor mentioned above (^).

The feature modification . The two-dimensional analysis (1,4,

IO-12I allows a modification of features which can be used for

identification. By 'erosion' (12), for example, features which

appear as two overlapping circles can be separated. Additionally

small particles or scratches can be eliminated Cj). On the other

hand, particles can be identified according to their distance by

the opposite operation, the 'dilatation' (4,12). Therefore this

method is well suited to define features by their neighbourhood,

in addition to their shape, especially if a sequence of erosion

and dilatation is used. The separated features may be analysed to get the standard parameters or, for example, they may be sized by

a subsequent erosion (12-14).

In those machines that provide manual interaction the

features are recognized by marking them with a light pen (4,7,5).

This, of course, cannot be done with the speed of a TV system, for

example. Therefore at first glance manual interaction does not

142
Automatic image analysing instruments today

seem to be part of an automatic but of a semiautomatic system. But

in fact it is much faster than common semiautomatic instruments.

If a feature cannot be recognized automatically, it is only

necessary to redraw its outlines with a light pen. Also the most

complex parameters are then measured automatically by the

instrument. According to figure 2 this interaction is stored in

the appropriate unit and gated into the measuring line, in general

by gate 2. If an instrument allows a connection to gate 3, the

interactive system can be used to correct the automatic feature

recognition only, if necessary, which of course speeds up the

operation. Additionally, interactive systems can be used to draw

any form of measuring frame or to compare a measured feature with

another one which may be a standard shape (4).

According to figure 2 these systems can be combined to any

degree of complexity, which of course will be very expensive. This

is only recommended if complex measurements are to be done

frequently.

PROGRAMMING OF INSTRUMENTS

To program a system as given by figure 2 takes a lot of time

if it must be done manually by interconnection of each module or by the use of patchboards. To simplify this, for most the

instruments a computer connection is available which enables the use of software systems which ask just for the kind of detection,

the shape of the features to be recognized and the parameters to

be measured. All other processing is done automatically (4). This

is very useful, if the parameters to be measured and the detection

criteria must be changed often. If only one kind of measurement is

necessary, specialized instruments are available with a hard wired

program, operated just by pushing the button 'start'. They can be

used for control purposes by unskilled operators (1,4),

METHODS FOR EVALUATION OF INSTRUMENTS

In general, it is not possible to decide - even after an extended test - that this is the 'best' instrument. The best

system is that which provides only the measurements that are necessary with the desired accuracy. This instrument has, in

general, the lowest price compared with the capability used. The

first step to find the most suitable instrument is therefore to

define the parameters to be measured and the method for feature

detection. If the features can be separated by their grey level

only, no feature recognition units are needed. Otherwise the shape

descriptor which is valid only for the desired features must be

defined, whereupon, on the other hand, this specifies the feature

recognition system.

The second step is to decide if field specific data are sufficient, or whether feature specific data must be measured.

The third step is to decide if hardware programming is

sufficient or if software programming is to be preferred. The

latter requirement frequently occurs in a complex system where the

kind of measurement changes often. With these steps the type of

instrument, and its analysing capability is determined according

to figure 2. The fourth step is to define the accuracy necessary and to

select an adequate instrument. The accuracy is determined mainly

by the first two units in figure 1, the image converter, and the

detector. All the other modules handle electrical information by

arithmetic or logic rules using, in general, digital, or in some instruments, analog methods. The latter give a well known accuracy

143
H. P. Hougardy

Which is generally better than that of the converter and detector.

Therefore it is important, to get values for the accuracy of these

modules. The capability of the converter and the detector can be

described by four values, which can be measured by each user (2):

the resolution, the separation of grey levels, the reproducibility

of threshold settings, and the statistical accuracy of a system.

There are values not derived from the common technical description

of such systems but from the needs of a user of automatic image

analysers

.

From this point of view, following a proposal (2), resolution

is defined as the diameter of the smallest circle whose area can

be measured with an error less than + 6 %. Of course other values

can be taken. For 6 a diameter of 1.5 % of the frame width is a reasonable value for this resolution. This value is defined for a contrast of 1 ! At lower contrast the error increases for a circle

of the same diameter. The resolution limit is very important, for

the error arising from the measurement of particles smaller than

the given limit is not detectable in the result by any statistic

calculation of accuracy !

The separation of grey levels . Two phases which should be

separated by the detector must have a minimum difference in their

grey level. This difference can be measured quantitatively using

additional equipment: The camera is Illuminated without any lens

system by parallel monochromatic light, starting with the maximum

light intensity which can be taken for the camera system used. For

this value, the threshold is set to get 99 % area of the live

frame detected. Then the light source is dimmed until the area reading is only 1 %, This is one grey step. After changing the

threshold until again 99 % area is measured, the light is dimmed

again and so on. 30 grey level steps determined by this method

show acceptable performance (2). Using neutral density filters the

absolute sensitivity of the system can be measured. Comparing

these results with photometric measurements of specimens shows

whether a specific feature is detectable due to its grey level or not. The number of grey levels which can be separated depends on the wave length of the light used for illumination, the sensitivity of the camera system, and the detector used. It is only valid

for features greater than the resolution limit defined above and

depends on the supposition that the image has no shading. In general, this is not given if a microscope is used. Otherwise the

values measured as described can be realised only with an image

shading corrector (2).

The reproducibility of threshold setting is given by the area

of a feature as a function of the threshold setting. Standard

detectors give no advice about the correct setting. Therefore

special methods must be used to avoid over- or under-detection

(9). V/ith automatic detection systems there is a plateau at the

correct setting (1) which means that the value of the area does

not change with a change of the threshold. The difference in the area measurement for a + 1 % change of the threshold setting gives

a value for the reproducibility of the threshold setting. It changes with size and contrast of the features (2).

The statistical accuracy of a system is given by the standard

deviation of a set of measurements on the same field of view. For

small particles or low contrast the coefficient of variation can exceed 1 % and go up to 20 % (2). This statistical error must be

added to the error, calculated from different fields of view.

144
Automatic image analysing instruments today

These four values, if known for an instrument, enable the user to decide if his problem can be solved by a particular system

and he can calculate the accuracy of the parameter measurements.

From this, the scatter of the shape descriptors - as far as used - can be estimated which gives a decision if two shapes can be

separated or not, according to step 1 mentioned above.

INSTRUMENTS AVAILABLE AT THIS TIME

There are 12 instruments manufactured by 8 different com- panies on the market at this time. Most of then are modular, giving an infinite number of possible combinations and special

systems. Therefore, here we characterize only briefly the basic

instruments. If not mentioned, the input is a TV system. Details

are given elsewhere (1,4).

The Bausch and Lomb (20) instrument, called 'Omnicon', is a modular system which can be extended to include automatic

detection and complex parameter feature recognition, supplemented

by an interactive system, which i,s then called 'P.A.S.' (Pattern

Analysis System). The sequence of -operations , including feature

recognition, is programmed just by pushing buttons on a keyboard.

Ealing-Beck (21) has developed recently an instrument called

'Histotrak' (19,22) which is combined with a microscope to provide

an automatic change of magnification. It is designed for easy

handling, measuring the most important parameters by means of an interactive system.

The Hamamatsu Corporation (23) manufactures a variety of

systems for special applications such as the 'Iriscorder' for

ophtalmology . General purpose instruments are the 'Multianalyser '

,

an analog instrument for simple measurements, and the digital

•Polyprocessor ' which is hardware programmed using read only

memories. The main part of the processing and data evaluation are done by software.

The 'Quantimet 720', manufactured by Imanco (24), is a system

which can be extended from a simple instrument to a computer

processed and programmable system containing all the capabilities

described in figure 2. Special systems are available for special

purpose as the 'Quantimet 360', a push button instrument for the

measurement of nonmetallic inclusions in steel.

The Kontron (26) instrument, built in cooperation with V/ild

(27), is called 'Digiscan' and uses a photo diode as image con- verter which also gives a high grey-level resolution. The specimen

is moved for scanning by a mechanical stage. The ' Epi quant ' and

the Kontron instruments are powerful systems for measuring

features with low contrast.

Leitz (28) has developed a system which measures the linear

parameters of features, the 'Classimat'. They also build the

'Texture Analysing System' (T.A.S.). This instrument uses the

principles of the random set theory for a two-dimensional feature

analysis. Feature modification, as well as two-dimensional sizing,

can be realised (1,12,13).

VEB Carl Zeiss Jena (25) has developed a system called

'Eplquant ' specially designed for investigations of opaque specimens. The image converter is a photomultiplier which has in

principle a higher grey level resolution than TV systems. All

measurements of a lineal analysis can be done.

The 'Micro Videomat', manufactured by Zeiss (29), is a compact system for making the main important measurements. The

application is aided by a set of software for processing and data

handling, and for statistical calculations to get the accuracy of

a measurement

.

145
H. P. Hougardy

FURTHER DEVELOPMENTS

Looking at the instruments today, they represent different

compromises between the optimal system and the costs. For example,

to increase the speed of a measurement it is necessary to have a simultaneous data processing which can be realised by an increasing amount of hardware and - Increasing costs. On the other hand,

for many applications a digital storage of the image is desired to

have all the advantages of a software analysis. Until now this is

realised only for special applications (17, l8) due to the high

costs of core memories necessary for image storage and the

relatively long time necessary for analysis. But the costs of com- puter memories decrease, the speed of modern computers Increases.

Therefore a digital image storage and a software analysis may be

used more and more in the future. But in every case, to store the

total information of an image needs too much memory. Therefore it

may be useful to reduce the information as far as possible. In principle, figure 2 includes highly sophisticated instruments

which scan the image in a first step very quickly with low

resolution to get a 'survey'. In a second step the image converter

is positioned - directed by the computer (see figure 2) - two

points of special Interest to get more detailed information. This

is the most economical way of image analysis which is used by

human operators if they have to describe an image (15) • It depends

on the development of computer costs and speed, if such a system

will be a good solution for some applications.

New developments of instruments will go in three directions.

First, a series of systems for special purposes will be designed

as push button instruments for quality control and routine

measurements. Only such an instrument guaranties reproducible

measurements working with unskilled operators which again is a question of the costs of quality control. Of course, if a special

system is not yet available, a research instrument may be used

making all switches and knobs for adjustment and calibration

unavailable and starting the measurement by a remote button.

Second, very complex instruments, combined with large computers

according to figure 2, will be built for very difficult, especially research problems. In between, instruments of medium capacity

will be available which are self contained or which utilize only

some modules of a large system. They are - due to their low

costs - very useful, if an image analyser is not used full time

but it is too time consuming to work manually or to use semiautomatic instruments. For all systems the accuracy of the image

converter and the detector will be improved further, at least as an option, using new conversion principles or additional modules

like the shading corrector. Therefore the number of instruments,

especially of different versions of one system, available at the

present time will Increase further. But if the ability to do the

measurement is determined as described above, defining the desired

parameters and their accuracy and comparing these values with that

of an instrument, it should not be difficult to select from an increasing number of instruments the most suitable one for a given

application.

14 6
Automatic image analysing instruments today

Formatting Recognition Analysis

Input (image)

Detector Feature Analyser

Image

Converter

Feature Recognition

Output

( Result

)

Fig. 1 Schematic drawing of automatic image analysing systems.

The input information must be formatted for the

recognition system which extracts all the features to be

measured

.

Formatting Recognition Analysis

Fig. 2 Schematic drawing of a complex image analysing system

with different pattern recognition modes, automatic

processing and digital image storage (connection d)

.

Heavy lines: Information flow.

Thin lines: Processing connections.

147
H. P. Hougardy

The author thanks Mr. E.E. Underwood for his help In writing this

paper

.

REFERENCES

1) Hougardy, H.P. (197'J) Instrumentation

in automatic image analysis.

The Microscope 22, 5/26.

2) Hougardy, H.P. Determination of automatic image analyzing Instruments

accuracy. Praktische Metallographie

( in press )

.

3) Cruttwell, I. A. (197')) Pattern recognition by automatic image

analysis. The Microscope 22, 27/37.

>i) Hougardy, H.P. (1976) Recent progress

in automatic Image analysis instrumentation. The Microscope 2k . January

.

5) Fraser, P.V. (1975) New operator

Interactive light pen system. INTER/ MICRO-75 (Cambridge, GB, 23-26 June

1975)

.

6) Lawson, K.E. (1975) Pattern recognition on the Quantimet 720 image

analyzing computer, kth International

Congress for Stereology (Gaithersburg, MD, 1-9 Sept. 1975).

7) Korton, R.R.A. (1975) An image

analysis system based on feature descriptors. Ith International Congress for Stereology ( Galthersburg,

MD, 1-9 Sept. 1975).

8) Kelly, T. & Heil, R.H. (1975) Automatic data collection for the analysis of porous materials. 1th International Congress for Stereology

(Galthersburg, MD, 1-9 Sept. 1975).

9) Hougardy, H.P. (1975) Fehler bei Messungen mit autoraatischen Bildanalysatoren. In: Quantitative Analyse

von Mlkrostrukturen in Medizin, Biologie und Materlalentwlcklung.

Praktische Metallographie, Sonderband ^, 73/80.

10) Griggs, R. (1975) Application of two- dimensional image amendment in automatic image analysis. INTER/MICRO-75

(Cambridge, GB, 23-26 June 1975).

11) Hunn, W. (1975) Texture analysis of stereology specimens. 1th International Congress for Stereology

(Galthersburg, MD, 1-9 Sept. 1975).

12) Muller, W. (1971) The Leltz-Texture- Analyzing-System, Scientific and Technical Information, .Suppl. I, 1971, p. 101/116.

13) Delflner, P. (1972) A generalisation

of the concept of size. J. Microscopy

ii. 203/16.

11) Serra, I. (1971) Theoretical bases of

the Leit z-Texture -Analyzing-System

.

Scientific and Technical Information,

Suppl. I, _1, 1971, p. 125/136.

15) a) HougardyT H.P. (1972) Gefugebildanalyse im Prinzlp und ihre tech- nische Verwirklichung. Atti della giornata di studio sui metodi quantitative di analisi delle imaginl in metallographia . Nel quadro di met' 72-3^ mostra europea della metallurgia. Torino, 23 settembre - 2 ottobre

1972.

b) Hougardy, H.P. (1971) II principle

dell' analisi strutturale e sua realizzazione tecnica. La metallurgia

italiana. Attl notlzie. Vol. XXIX,

n. 3-1971, p. 183/85.

16) Terrell, A.C. (1975) Characterization

of particles by size, shape, etc. INTER/MICRO-75 (Cambridge, GB, 23-26 June 1975).

17) Rink, M. (1975) A New Fst, storagesaving image analysis procedure for investigation individual features.

1th International Congress for Stereology (Galthersburg, MD, 1-9 Sept. 1975).

18) Rink, M. (1975) A computerized image process of isolating Individual

features in a netted pattern.

1th International Congress for Stereology (Galthersburg, MD, 1-9 Sept. 1975).

19) Hopkins, B. & Healy, P. (1975) The use

of a commercially available image

analyser in obtaining stereological

information on corpora lutea in the mouse ovary. 1th International Congress for Stereology (Galthersburg,

MD, 1-9 Sept. 1975) . 20) Bausch i Lomb, I183I Bausch St., Rochester, NY 11602, USA. W.A. Howe Co. Ltd., 88 Petersborough

Road, London SV/6, England.

Flea Route de Levis St. nom, F-78

Le Mesnil St. Denis, France. D-8013 Unterfohrlng, Postfach 11 28, Germany

.

21) Ealing-Beck Ltd. Graycaine Road, Watford WD2 1PW, England.

22) Healy, P. (1975) The design and applications of a new multi-disciplinary automatic image analyser

(Hlstotrak). INTER/MICRO-75 (Cambridge, GB, 23-26 June 1975).

23) Hamamatsu TV Co., Ltd., 1126, Ichlnocho, Hamamatsu, Japan.

3000 Marcus Ave., Lake Success, N.Y. 11010, USA. D-8031 Hechendorf /Pilsensee, Hauptstr. 2, Germany.

21) Image Analysing Computers Ltd. Melbourn, Royston, Herts SG8 6EJ, England

.

21, bd de Port Royal, F-75 Paris I3, France

.

D-605 Offenbach, Schreberstr. I8, Germany

.

10 Robert Pitt Drive, Monsey, NY

10952, USA.

25) Jenoptik Jena GmbH, DDR-69 Jena, Postfach 190.

26) Kontron, D-8 Munchen 50, Lerchenstr. 8-10, Germany.

27) Wild Heerbrugg Ltd., CH-9135 Heerbrugg, Switzerland.

28) Ernst Leitz, D-633 Wetzlar, Postfach

210, Germany.

30 Mortimer Street, London WIN 8BB, England.

17-19 Rue Danton, F-9I Le KremlinBlcetre, France. Rockleigh, NJ O76I7, USA.

29) Carl Zeiss, D-7O82 Oberkochen, Postfach 35, Germany.

31-36 Foley Street, London V/l, England

.

Les Bureaux de la Collin de Saint

Cloud, F-92 213 St. Cloud, Prance.

Ill Fifth Avenue, New York, NY IOOI8,

USA.

148
National Bureau of Standards Special Publication 437

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

he/d at NBS. Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

TEXTURE ANALYSIS OF STEREOLOGICAL SPECIMENS

by Werner Hunn

E. Leitz. Inc.. Rockleigh. New Jersey 07647. U.S.A.

The major development effort for the commercially available

image analyzers has been directed toward automation of the

measurement process. The classical measuring procedures are essentially applied and implemented in television speed logics.

Even the individual feature analysis is no exception. What

was lacking were new concepts for the quantization of a measuring problem so that data can be extracted from the image

which correlate to the physical characterization of a sample.

New possibilities in the framework of mathematical morphology

were developed by Matheron and Serra (1,2) and found their technical realization in the Leitz Texture Analyzing System (3)

.

Its comprehensive concept of automatic image analysis with

structuring elements contains the classical procedures as specialized cases where it generally reaches deep into the

analysis of textures.

This is achieved by a generalization of the concept of size (4).

the measurement of image correlation functions to probe for

spacial interrelationships, the determination of the rose of

direction for structural anisotropy description and the two

dimensional image transformations. Some of these novel concepts should be briefly discussed.

Covarioqram (5.6) ; This distribution measures image correlation.

It is defined as the frequency of occurrence of point pairs in

direction within the image whereby the distance between the

two points increases incrementally and is not "cared" for.

Only the two end points are tested to be a "hit" in any one structure of the interesting component.

If we consider the covariogram more closely, it becomes obvious that this function reveals very powerful information

about the morphology of a complex body.

1. Periodicities of sample structures in the micro

or macro range which can even be disturbed or hidden by regional phenomena.

2. Characterization of object groups of various

sizes or spacing.

3. The interrelationship of different sample phases

which are probed in pairs in the so-called crosscovariogram.

4. The statistical estimation (7). How many sample

fields and polished section have to be analyzed,

for example, to extract a volume fraction of inclusions in steel with a certain accuracy (8)

.

The morphological answer is uniquely given by the

covariogram.

149
W. Hunn

The rose of direction is a powerful tool to describe orientations or anisotropies within a plane sample. Schmidt (9) proposed in 1917 a polar plot to show the frequency of a certain

parameter in any direction (the number of intersections with

surfaces per unit length in function of the variable direction

a) . . The result allows a quantitative evaluation of the degree of orientation. Serra found from mathematical morphologic consideration . . . p(a) = ^-^io.) + [PlC")]'^

. . . which is extremely powerful because it yields the perimeter length degree by degree. It basically adds one term

(the second derivative in a) to Schmidt's definition which

increases the sensitivity of this function to directional

changes considerably, and anisotropy directions of various

orders of magnitude become obvious (see Figure). The machine

is capable of measuring this polar plot by means of a programmed

optical image rotation in one degree increments in a matter of

minutes

.

Two Dimensional Transformation : If a structuring element of

two dimensional extension and isotropic shape, such as a circle, is used then the original image is transformed by various

yes-no operations:

Erosion: All points which can be occupied by the center

point of the circle so that the circle itself is fully

contained within the original image structure define a new image which is outlined by a minimum distance from

the original structure periphery. This image transform

yields some interesting results. Surface extremities are smoothened, structural bridges smaller than the circle

are broken open and detail smaller than the circle com- pletely suppressed. When this operation is repeatedly

applied with increasing size of the circles, the original

image structures can, for example, be size classified on

the basis of maximum circle to be inscribed. Whereby, we

do not request a priori that the image consist of individual features but rather can be a complex pattern of,

for example, porosity.

Opening: A different transform is obtained if we consider all the points which are covered by the circle as long as the circle stays within the original structure.

Here no marginal zone is deleted and testing with increasing size circles leads to a structural sieveanalysis or porosity analysis by mercury pressure

method implemented in morphometric tv operations. This

distribution lends itself particularly to the analysis

of complex pattern in form of a size-area fraction

distribution as is often required in an intersection

plane of a body. For example, to analyze a lung tissue

network or a continuous, interspersed phase in a metal

section

.

Dilation: If we ask for the image transform which is

given by all the points covered by the circle center

point so that the original structure and the circle

have at least one point in common, then we see that the

image components are expanded in all directions by the same amount and we probe for neighborhood relationships,

such as, orientation independent distance distributions.

150
Texture analysis of -stereological specimens

Closing: If we take this latest transform and erode it

again by the same size circle, then we reconstitute the

original image very closely except for bridges which re- main between structures closer spaced then the circle

diameter and holes are filled in. This transform finds

application in, for example, the spacial analysis of

stringer inclusions in steels.

Serra shows in a beautiful example the power of these new

techniques. Mineral sections of volcanic rock of various

geographical locations over ten square kilometers have been

studied as to their genesis history. The porous medium has

been measured in an opening-closing distribution, whereby

the size-area fractions are plotted starting from openings

at large circle sizes decreasing to zero circle dimension to

closings with step-wise increasing circles. The results

shown are interesting. We can see (Fig. 3) that the curves

of the porous samples are grouped. The bottom group of

nearly parallel lines with each other and the x-axis re- flects a progressive invasion of the space by the porous

medium. Two other groups of curves have various amounts of

incline and come to an intersection to the left of the yaxis. The first fact reveals that the pores and the distance between them increase simultaneously; the second one that a small size porosity seems superimposed on the larger

one as if it were coming from different genesis.

1. G. Matheron: Les variables regionalisees et leurs estimation, Masson, Paris, 1963

2. J. Serra: Introduction a la Morphologie

Mathematique . Les cahiers du Centre de

Morphologie Mathematique, Fasc. 3, 1969 - Ecole des Mines de Paris

3. W. Mueller: The Leitz - Texture - Analyzing - System, Leitz - Mitt. Wiss. u. Techn. Suppl.

1, Nr. 4, June 19 7 3

4. P. Delfiner: A generalization of the concept

of size. Journal of Microscopy, Vol. 95,

Pt. 2, April 1972, pp. 203-216

5. J. Serra: Un critere nouveau de decouvertes

des structures, le variogramme, Sci. Terre

(sous presse)

6. R. Alberny, J. Serra and M. Turpin: Use of

Covariograms for Dendrite Arm Spacing.

Measurements, Trans, of the Met. Soc. of

AIME, Volume 245, Jan. 1969-55

7. J. Serra: Echantillonnage et estimation locale des phenomenes de transition minieres.

These, Nancy, 660 pp

8. J. Serra: L ' echantillonnage des inclusions

dans les aciers, Journies d' etudes de la

metallurgie, Turin, 1972

9. W. Schmidt: "Statistische Methoden beim

Gefugestudium kristalliner Schiefer," Sitz.

Kaiserl. Akad. Wiss. Wien, Math.-nat, Kl.,

Abt. 1, 126 (1918) 515

151
W. Hunn

152
Texture analysis of stereological specimens

153
W. Hunn

37,3,9,

64 ym 4 0 +4 ym + 64 ym

40%

30

16

-64 ym •4 0 +4 ym + 64 ym

Figure 3: Opening-closing percentage area distribution of

various samples (see text)

154
National Bureau of Standards Special Publication 437 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STERE0L06Y

held at NBS. Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

A COMPUTERIZED IMAGE PROCESS FOR ISOLATING INDIVIDUALS IN AN ORIGINALLY NETTED PAHERN

by M. Rink

Institut fUr Geophysik der TU Clausthal. D-3392 Clausthal-Zellerfeld, Germany

INTRODUCTION

In nature, objects mostly are much more complicated than is ac- cessible to theoretical model treatments. In the theories describing microstnactures , the structure-forming elements are approximated by relatively simple geometric bodies. Such an exajnple in the

field of sediment petrophysics is given by the attempt of estimating hydraulic permeability of a porous rock, as an important threedimensional material constajit, from the microstructure of the

pores, measurable in plane sections only. To accomplish this, most

mathematical considerations start from much simplified capillary

bundle models. However, the sections through a porous rock show

very clearly the netting of the pore channels. Especially for a

weaJcly cemented matrix, larger compact pore cross sections are multiple—connected by narrow bridges, axid often extend over wide parts

of a section. In such cin original section image, it is not possible

without restriction to talk of single pore cross sections as individual figures. Therefore, the section image of the pore pattern

must be simplified so that individual pore cross sections only remain, which contribute considerably to the permeability, emd which

are simultaneously simplified in shape so far that their so-called

hydraulic radii ( = area/circximference ) are good enough for the description not only of hydrostatic but also of hydrodynamic effects.

In this case, the problem of figure individualization can be solved

in principle by artificially separating the narrow interconnecting

figure bridges.

Since the author had developed a computerized image analysis

procedure (Rink, 1 970, 1 975) » which yields statistical values of a lot of morphometric figure parajneters of all figures of the investigated image in just one computer run, it became desirable to apply this image analysis procedure even to images, not containing

classical individuals a priori. So the above mentioned pattern simplification had to be realized, (it may be noticed that this demand

does not result from insufficiencies in the image analysing program, but from the necessity of gaining meaningful statistical values of the figure parameters,

)

KNOWN IMAGE PROCESSES

Useful auxiliary image processes for figure individualization

are the areal erosion and dilatation of figures or combinations

thereof, known by the notions opening and closing (Matheron, 1 96 7» Serra, 1 969)

,

The author has written computer programs for perfonning the

said image processes. Of course, the theoretical backgroxind is the same for a hardware or software solution. For practical realization

it is an important fact that the opening figures may be constructed

from the original figures eroded at first and dilatated afterwards,

both with the same structuring element. Furthennore just the programming of an erosion and dilatation subroutine for the smallest

striicturing element only is necessary. By repeated subroutine calling, the proceeding of the image processes is achieved, corresponding to the use of increasing structuring elements.

The effect of the opening process consists in eliminating all

figures resp. figure parts which are smaller thaxi the pre-set

structuring element, being a circle area in the ideal theoretical

case. This leads to a separation of larger more compact figure

155
M. Rink

parts, originally interconnected by narrow bridges. In most cases

the bridges have different width, however. Xo complete the figure

separation, the opening process therefore must be continued with

increasing structuring elements. But thereby also small isolated

figures are lost. Finally the largest and most compact figures only are left. Also, during the opening process a certain shape

variation towards the shape of the structuring element occurs with

all figures involved.

The usual evaluation of such opening series consists in acquiring so-called granulome tries. But this shall not be discussed here.

It is intended rather - starting from a complicated netted original

pattern - to create one image containing all those - eventually

artificially individualized - figures that shall be statistically

evaluated in a common manner. THE CUT PROCESS

For this problem, the mere opening process does not lead to the

wanted aim. Indeed, that process removes narrow bridges between

compact figure parts, and thus separates originally complicated

figures into smaller and simpler ones, as mentioned above. But its

disadvantage is that the already isolated figures, which are smaller than the pre-set structuring element, disappear.

For avoiding this loss, a first idea could be collecting in an

auxiliaary image in advajice all those figures that would be eliminated by the next opening step. Thus, before the n-th opening step,

this auxiliary image contains figures - shape simplified by opening - of all size classes smaller thaji the remaining opening figures. Then the superposition of these last opening figures and

the figures of the auxiliary image yields an individualized Image

pattern.

This method requires a criterion for recognizing in advance the

figures smaller than the pre-set structuring element and a procedure for transferring them to the auxiliary image field. Such a criterion may be derived from the information, whether a digitalized figure consists of boundary points only or not. A continuous

three-line association is necessary for achieving this information.

This way of solution was tried, but it turned out to be unsatisfactory for its time-consuming complexity of the copying process.

Furthermore, the figure separation was of limited success only

because of overlapping and contacting of figures after the necessary image superpositions.

A better effect of individualization is obtained in the following even simpler an.d faster way. At first, in the original pattern,

all figure points on the outline of the circumference of the figures, resulting from the first opening step, will be erased. By

that means the narrow interconnecting bridges and small figure

parts will not be eliminated totally - as by mere opening - but

only separated at the contact points to the larger figures. That

is why this procedure may shortly be called 'cutting'. For advancing the process of figure individualization, successive opening is

continued as an auxiliary process, always erasing the outline

points of the corresponding opening figures in the just preceding

cut image. This process can be continued until no further figure

separation occurs. By the continued cutting, many very small relics of the original bridges result that must not be taken into account. To remove

them, a final opening has to be applied to the so far derived individualized image. However, in this so-called clear cut image the

smallest original figures are missing now. Often this clear cut

image is already welcome for the quantitative analysis. But when

the loss of those smallest original figures cannot be tolerated,

they have to be filtered out of the original, and added to the

clear cut image, after the above mentioned method. Such a so156
On computerized image process for isolating individuals

Fig.1 Fig.

2

Parts of computer outputs of a cross section of the porous sandstone specimen B50. Fig, 1 : Original image; Fig,2:Clear cut 5+ image

Fig. 3 Fig.

4

Number-normalized frequency distributions of the mean chords in

line direction (Fig. 3) resp. hydraulic radii, times k (Fig. 4) of

pore cross sections of specimen B4 5 . 777- Original ; Clear cut 4+

157
M. Rink

called clear cut 5+ image Is partially shown in Fig. 2, 5 being the

number of cut steps.

The elementary processes erosion and dilatation in this soft- ware solution are based on a successive processing of three adjacent image lines. For an advanced image processing however, it is

favourable to reserve storage space for four image fields. The

cutting then is only a logical word by word combination of the

three images involved,

EXAMPLE OF APPLICATION

The pore space of specially prepared sandstone specimen was investigated by the described image analytic computer methods (Rink,

1-973a» b , 1 97^ ) , The cutting effect can be seen by a comparison of

Fig, 1 and 2. A comparison of data of original smd clear cut images clearly shows the size reduction, shape simplification and

number increase of the individualized figures. The frequency distributions of figure parameters reflect details of the sajne facts

(Fig. 3 and 4),

For the stereological determination of usual parameters as porosity or specific surface, integral values of the original images

must be used. But for estimating the hydraulic permeability after

a refined Kozeny-Carman theory, three-dimensional statistical pore

values have to be incorporated. A first calculation has shown that

the original images are not suited for this purpose, whereas images, individualized by the cut process, furnish much better

results.

The presented cut process - relying on the individualizing property

of the opening process, but granting at the same time the coexistence of isolated figures of various sizes in one image - offers

extended possibilities for the analysis of complicated image

patterns

.

REFERENCES

Matheron, G. ( 1 967 ) Elements pour \ine thfeorie des milieux poreux.

Masson, Paris

Rink, M. ( 1 970) Automatische morphometrische Bildanalyse mit Hilfe

eines elektronischen Digitalrechners . Diss. Techn. Univ. Clausthal, BRD

Rink, M. ( 1 973a ) Porengeometrische Untersuchungen an Sedimentgesteinen mit einem Bildeuialyseverfahren fiir Digitalrechner . Z.Geophys. ,39,989-1005

Rink, M. ( 1 973b) An image analysis procedure for digital computers

and its application to investigations on pore stnactures. Proc.

RILEM/IUPAC Int.Symp. 'Pore Structures and Properties of Materials', Prague, Final Report II, C497-C506

RinkjM. ( 1 97^ ) Untersuchungen am Porenraum von Sedimentgestein rait bildanalytischen Methoden im Digitalrechner. Z , Prakt . Metallogr

.

Sonderband 5,308-322

Rink, M. ( 1 975 ) A new fast and storage saving image analysis procedure for investigating individuals by a digital computer. Proc.

4th Int . Congr . Stereology , Stereology 4, Washington D.C.

Serra, J. ( 1 969) Introduction a la morphologie raathfematique. Cahier

du Centre de Morphologie mathfematique , Fasc.III, Ecole des

Mines de Paris

158
National Bureau of Standards Special Publication 437

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

AN IMAGE ANALYSIS SYSTEM BASED ON FEATURE DESCRIPTORS

by Roger R. A. Morton Bausch & Lomb, Rochester, New York 14625. U.S.A.

Image analysis is finding increased acceptance in a diversity of

fields. As a direct result of this acceptance, new applications

are continually being found which require image analysis equipment with more flexible and powerful capabilities.

To meet these requirements, an image analysis system must be able

to quickly solve advanced analysis problems. Furthermore, the

control of the instrument by the operator as well as the instrument's communication to the operator, must be simple and rapid.

Clearly, there is little point in having an instrument which can analyze an image rapidly, but which requires a much greater time

for the operator to tell it what to do.

TM These considerations were foremost in designing the Omnicon Pattern

Analysis System. As a result, the two modes of operation were

chosen. One employs a special purpose keyboard which operates in

conjunction with an alphanumeric presentation on the display. The

other uses the BASIC language to sequence and control the system

operation. When using the BASIC language, a second general purpose keyboard is employed which operates interactively with both

the alphanumeric display and a printer. There are many advantages

to this new type of image analysis control. For example, expandability of the system is not limited by the control panel; incorrect or ambiguous commands may be flagged directly and the

control status of the system is continuously summarized on the

display.

In the keyboard mode, five function phases are defined. They

are: Selection, Measurement, Process, Output, and Mode.

The Selection phase is used to select those features within the

sample which it is desired to measure. The operator selects,

using the keyboard, the ranges of specific measurements, shape

factors, orientation factors and so on, that a feature must have

if it is to be selected. The instrument then performs the desired

measurements and indicates with a tag on the display the features

which meet these criteria. The parameters upon which a selection

may be based are termed "descriptors", and they may be classified

as follows

:

Gray Level

The gray level of a feature may correspond to its density, its

absorbance, reflectance or transmittance, depending on how the

object is imaged. Selection may be based on maximum gray value

per feature, the average value or the requirement that the entire

feature exceed a preset value.

Size

Size of a feature may be defined in terms of large number of different measurements. Examples include area, area of holes, longest

dimension, breadth, perimeter, convex perimeter, or derived measurements. (Derived measurements are simply expressions involving a number of measurements of the type just mentioned.)

159
/?. R. A. Morton

Shape

Shape factors may be constructed from measurements of the type

already discussed, or from additional measurements such as the

niimber of vertices of a feature or the number of holes within a feature. Shape factors are dimensionless expressions and are invarient with both size and orientation of the feature.

Orientation Factors

Orientation factors define the orientation of a noncircular feature. A variety of orientation factors can be constructed using

measurements like the ones discussed or measurements such as the

direction of the longest dimension or direction of breadth which

may be used directly for such factors.

Position

Features may be selected based on their position within the field

of view or position within the sample. The position of the feature may be defined in terms of either the position of the topologically defined point corresponding, for example, to one of the

extremities of the feature or the position of the centroid of a feature.

Adjacency

Adjacency is a position measure which is based on the distance to

other features, or, for example, deteirmines whether a feature

is surrounded by another feature.

Focus

The measurement of average sharpness of the edge of a feature or

some other focus measure is a useful descriptor when the features

of interest do not lie in the same plane.

Notice that these descriptors are in a sense independent of each

other. One may even say orthogonal in the mathematical sense. The value of any descriptor may be changed for a particular feature

without affecting the value of other descriptors of the feature.

Alternatively, they may be considered as dimensions in a decision

space which is used to define the properties of desired features.

Measurement

Selected features may be measured using any of the measurements or descriptor types already described. Measurement expressions are scanned to determine their dimensionality and calibration factors

are appropriately applied.

The Measurement operation produces a single result for every feature in the field of view in a fraction of a second. This large

amount of data is usually of little use to the operator. Consequently, the Process phase occurs concurrently with the measure- ment operation to reduce this data into the format desired. Typical formats include statistical summary indicating sum of measurements, mean, standard deviation, number of features measured,

maximum or minimum values, or size distribution based on linear

or logarithmic intervals. Alternatively, a list of the feature

measurements may be presented on the display.

160
An image analysis system based on feature descriptors

Also, occurring during Measurement and Processing operations is

the Output operation, which involves passing the processed data

to a printer, onto a display or into the BASIC language section

of the software for further processing using operator selected

programs.

BASIC Language

The BASIC compiler implements a full repertoire of BASIC commands.

Additional commands have the same effects as keyboard operations

while others allow the transfer of data from the processing sec- tion of the software into BASIC variables. In addition, BASIC

commands permit control of the microscope stage and sense the

nimiber of the microscope objective currently in use. Consequently, a BASIC programs may be written to perform complex

analysis and recognition operations. Examples include: agglomerating detected areas into single feature for sample where a single

feature may appear as a number of "white blobs", to measure Feret's

diameter at right angles to the longest dimension, and a program

to measure agglomerates, and separate the agglomerates into individual features based on prior knowledge of the shape of the individual features.

Block Diagram

Figure 1 illustrates the major functional blocks of the Pattern

Analysis System. The scanner receives the image from either microscope, macro unit, or transmission electron microscope and generates a corresponding video signal. This signal is passed to the

detection or thresholding electronics where the boundaries of the

features to be measured are defined using one of a number of detection criteria. The detection circuits identify those transitions which are to form the boundaries of features to be analyzed

using either the fixed threshold level or using a threshold level

defined with respect to the background of the image. Once a transition has been identified as forming part of a boundary of a feature to be analyzed, the position of the boundary with respect to

the transition may be defined by determining the maximum and mini- mum video amplitudes in the vicinity of the transition and computing the boundary as occurring at a weighted average, 50% of the

maximum and minimxim. The exact weighting of this average is determined by the gamma of the scanner tube.

One option which may be placed between the scanner and the detection electronics is the "Shading Corrector". This module corrects

shading which may arise in both scanner and image formation optics.

This shading is classified into white or multiplitive shading,

and black or offset shading. The white shading corresponds to

variations in illumination of the sample vignetting in the optics

variations of sensitivity across the faceplate of the scanner,

while the black shading will correspond to dark current offsets

in the scanner and nonuniform glare in the optics, especially when

using incident light illumination. The Shading Corrector independently assesses these two sources of shading by means of two

reference images. Firstly, a black reference image which is

formed from a uniform sample is viewed through the microscope or other image source to produce the darkest gray levels or density

levels anticipated in the specimen. The video profiles corresponding to this resulting black image is stored in the Shading

Corrector. Secondly, a white reference image is produced from a uniform sample which produces an image corresponding to the lightest

or least dense areas of the specimen. By storing the white reference and comparing it with the black, the Shading Corrector is able

161
/?. R. A. Morton

to determine the multiplicative or white shading components and

treat them separately from the offset or black level components of

shading determined on the black reference.

Using the Shading Corrector, the number of shades of gray which

can be separated (measured using the 10%/90% method) is typically

50 and can be as high as 66. In other words, the shading plus

noise taken between the ten percentile extremes is typically

1/46 or 2.2% to 1/66 or 1.5% of the black to white video range.

The detected signal is passed to the Pattern Analysis Module (PAM)

comprising the Pattern Analysis System electronics and associated

processes

.

In this module, the video signal is processed on an intercept

basis in accordance with the commands entered through the keyboard

of the system. A keyboard compiler in the processor processes

the keyboard instructions, accesses the appropriate subroutines,

and generates the appropriate code to cause the system to execute

the required operations.

The processor has a number of peripheral devices connected to it

through an input/output buss. These include the display processor,

which generates graphics, characters and other numerics on the

display, a printer to print the results of the operation; the

special purpose Pattern Analysis System keyboard and its associated

status matrix; a general purpose keyboard for communication to the

BASIC compiler; and the stage control X,Y and Z motions of the

microscope stage. Other peripherals may be added including: automatic focus, and system sequencer to control the functions of other

hardware modules.

SCANNER - -

DETECTIONPAS

ELECIRONIQS_

PROCESSOR

>PAM

Figure 1: Block diagram of Omnicon Pattern Analysis System.

162
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

IMPLICATIONS OF RECENT ADVANCES IN IMAGE ANALYSIS FOR STEREOLOGICAL THEORY

by Hoiger Schmeisser IMANCO Bild-Analysen-Computer GmbH, D-605 Offenbach. Schreberstrasse 18, Germany

INTRODUCTION

One of the most recent technical advances in electronic image analysis is the facility to modify image features in two dimensions in a well defined manner. Unfortunately this innovation has caused considerable confusion. While having been introduced to the system QUANTIMET 720 as an additional option which

increases the structure descriptive power of the system considerably (Schmeisser

1974), the same technique has independently been featured as the heart of a com- pletely new concept of describing structures and interpreting morphometrical

properties. From discussions in the recent past, it is felt that there is a con- siderable lack of understanding of the relations between "traditional" stereology and the new approach. This paper attempts to clarify some of the aspects in- volved. Because of their relevance to practical problems, the concept of two dimensional amendment and the measurement of spatial distributions have been chosen as examples.

THE CONCEPT OF TWO DIMENSIONAL AMENDMENT

While structures displaying well defined features in a background-matrix

allow very detailed characterization by means of feature specific data, one finds

oneself drawn back to fairly general descriptors as soon as penetration structures are under study. Overall measurements e.g. area fraction or mean intercept

length are very useful in many instances, particularly because they are part of

a well evaluated framework of stereologic relations which allow direct interpretation of the data (De Hoff & Rhines 1968, Underwood 1970). There are, however,

a number of situations in which more detailed information is required e.g. where

distributional measurements are to be performed. Due to the technical layout of most image analyzers, a chord size distribution is measured.

Before discussing its problems, the actual method of measurement should be

recalled to fig. 1 : In order to determine whether or not a given chord exceeds a preset length 1 a proportion of the chord corresponding to 1 is suppressed electronically as shown in fig. la. Only chords which "survive" this procedure are

then taken into consideration. So effectively, the first step involves a modification of the image and the result of this amendment is then subject to the actual measurement. Detailed study shows that the results are independent of the particular way

in which the amendment is performed. If one suppresses half the chord length at

the leading edge of a feature and half at the trailing edge (fig. lb), this is exactly the same as suppressing the full chord length

at either the leading or trailing edges. The system

illustrated in fig. II3, however, would be referred to as "erosion of the feature

with respect to 1". The re- verse operation, amendment

by extending the chords, results in the corresponding

"dilation".

The obvious problem

with chord sizing - or in general with linear amendment - is the directional

dependence of the results.

The importance of this fact

is demonstrated in fig. 3.

Fig. 1 Equivalence of chord sizing and linear

erosion

163
H. Schmeisser

As this difficulty originates from the presence of a preferred direction in the evaluation process, the natural solution is to treat all directions equivalently.

This results in a concept of two dimensional amendment as shown in fig. 2: In- stead of shifting the boundaries in one direction only,

"elementary shifts" are ap- plied which result in a new

boundary. This concept is similar to the well known

principle of wave optics.

Fig. 2 shows that also in this case there are two ob- vious derived structures - an eroded and a dilated one. Again it turns out that this

simple generalization of the concept of chord sizing

gives exactly the same results as a strict mathematical treatment. The practical usefulness is demonstrated in

Fig. 2; Relationship between original structure

(bold) and the eroded (dashed) and dilated structures respectively

DDDQQ

(b)

CHORD LENGTH

OCTOGON SIZE

Comparison of directional dependence of chord size distributions (b) and distributions obtained by combined application

of erosion and- dilation (c) - both diagrams show two distributions obtained in perpendicular orientations of the sample

shown in (a)

fig. 3c, which shows the result of successively performed erosions/dilations in

two perpendicular orientations of the sample. The elimination of the ambiguity of

the chord size distributions of fig. 3b is clearly noticeable.

So in the same sense in which the step from linear analysis to true two dimensional image analysis enabled the description of features by real two dimensional feature specific parameters (Gibbard, et al. 1972) rather than individual

chord information, the concept of two dimensional amendment allows a more compre- hensive and less ambiguous description of complex structures than simple chord

sizing.

It is noteworthy that two dimensional amendment in combination with feature specific measurements is particularly useful for some applications (Schmeisser

1974). A direct application is e.g. feature sizing with respect to the "biggest

164
Implications of recent advances in image analysis

inscribable octagon" as explained in fig. 4. This is a valuable supplement to the normally available maximum chord length distribution, although it must be emphazised that it is completely different from a true area size distribution

based on feature specific data (fig. 4). The results obtained will, however, only

be meaningful if 2D amendment is applied in combination with a fiill feature count

logic as "keying" signal - otherwise the well known effect of feature break up

(Cole 1971) will destroy the measuring accuracy (fig. 4).

Fig. 4: Sizing with respect to the biggest inscribable octagon

problem of feature break up

(a) and comparison of obtained size distribution (b) with maximum chord length

and true area distributions

(c).

Fig. 5: Two dimensional dilation clusters

particles with certain distances

(a) , (b) . Appropriate logical set up of the instrument enables particle group analysis (c)

.

Two dimensional dilation is useful for determining neighbourhood relations be- tween features as shown in fig. 5. Besides the measurement of shortest distancedistributions a more sophisticated combination of 2D amendment and feature specific measurements enables particle group analysis (fig. 5c).

165
H. Schmeisser

MEASUREMENTS OF SPATIAL DISTRIBUTIONS

A well known way of determining the spatial relationship of components in a structure is the application of a stepping measuring frame. Fig. 6a shows a slightly different arrangement: the total frame is chopped into many little sub- frames which are assessed individually. Fig. 6b shows the result of an area fraction measurement in each of the individual frames plotted against the respective

position - the banded structure is obvious. As the wave length of such a banding

pattern is an important parameter of the materials properties the distribution

Fig. 6; Spatial distribution measurement

and wave length spectrum of pe-

^ ^ riodicities of the structure

10 GO SHIFT curve in fig. 6b is converted into a wave length spectrum by means of a simple

algorithm (fig. 6c). Applying the algorithm to the distribution curve of only one phase gives the respective wave length spectrum, cross correlation of two phases

shows the spatial relationship of those. Being comparable to a "covariance"-measurement this algorithm is also easily applicable to e.g. concentric patterns.

CONCLUSION

Considering the results presented here the new concepts represent a very natural extension of the existing approach to morphological measurements. Although

there is certainly no necessity for a complete reconsideration of the existing

formulation, the theory which accommodates the new measuring feasibilities to the framework of stereology still has to be evaluated. Looking on the structure descriptive power of these measuring aids, however, it should be a real challenge

to the theoretically working stereologists to fill this gap very soon.

REFERENCES

Cole, M. ( 1 97 1 ) "Instrument Error s in Quantitative Image Analysis", Microscope

11' ' De Hoff, R.T. & Rhines, F.N. (1968) "Quantitative Microscopy", Mc. Graw-Hill

Gibbard, D.W. , D.J. Smith, A. Wells (1972) "Area Sizing and Pattern Recognition

on the Quantimet 720" Microscope 20, 37 Schmeisser, H. (1974) "Zweidimensionale Bildveranderung - eine Erweiterung bildanalytischer MelEmoglichkeiten" in Special Issues of Practical

Metallography No. 5, 60 Underwood, E.E. (1970) "Quantitative Stereology", Addison Wesley

166
National Bureau of Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at N8S, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

EXPERIENCES WITH OPTOMANUAL AUTOMATED EVALUATION-SYSTEMS IN BIOLOGICAL RESEARCH,

ESPECIALLY IN NEUROMORPHOLOGY.

by H. Haug

Medical School Lubeck - Department of Anatomy, D-2400 Liibeck, Ratzeburger Allee 160, Germany

The principles of stereology have a mathematical basis and this science is independent of a practical use (15, 17, 21) . However, stereology is only

significant in the case of its application in morphological sciences (2, 5,

6, 9, 21). The practice needs on the other hand some technical helps for

an expedient evaluation. In the last years this can be observed by the development of more and more sophisticated instruments (3, 12, 17) . Many

papers of this congress deal with the construction and use of such

instruments . The most modern instr\iments have a full-automated device where

the scientist operates mainly with electronic switches of the computer

equipment (4, 11, 13, 14, 15).

Scan-instruments are guided by their computer and in this way the instrument

can distinguish between distinct gray- intensities of different structural

details in a tissue and can calculate the results from this evaluation by

means of stereological formulas. However in many cases this distinction is difficult, due to:

1) the gray-steps are too low, 2l_ more than one structure has an equal grayintensity, 3)_ one structure has different kinds of gray-steps dependent on some factors in the section (7, lo, 12, 22). In such cases the scan-computer

system is unable to differentiate the structures and we are forced to evaluate

with older and simpler but time-consuming procedures. Another possibility

is the extension of the scan-instruments for operating with the human eye- brain system. This eye-brain system is very well suited for the discrimination

of the above mentioned difficulties

.

This paper deals with the advanced practice for operating by the eye-brain

system. The output of the eye-brain system is the hand and therefore such devices are called opto-manual instruments. All of the newer developments

of these instruments work with computers. These enables one to restrict the function of the eye-brain system to the important point of the diagnosis.

It is also possible to improve the full-automated scan-devices for optomanual

treatments, but, such devices need besides the relatively expensive scan- instruments further expensive hard and soft-ware. Therefore it is important

that in the last years some instruments have been developed which can operate directly with the eye-brain system and are on this basis less expensive than the full-automated scan-devices.

Before a description of the possibilities of these optomanual systems will

be given, it is necessary to describe two biological examples which will make

clear that a scan-instrument is unable to work in tissue with minute

dif ferencies. The first one deals with macroscopy of the brain and the second

one with electron-microscopy of the nervous tissue.

A macroscopic section through the human brain shows white and gray matter.

The white matter consists of connections between the different parts of the gray matter by nervous fibres. The gray matter contains many nerve cells

and the complicated network of their processes. However, every gray matter

represents another functional part of the brain. Therefore it must be

differentiated by an evaluation between the distinct gray matters which have,

in many cases, but not all, nearly the same gray value. Such a differentiation

can only be done by the eye-brain system.

167
H. Haug

Fig.l: Neuropil of the cortex cerebri with entangled cell processes. The arrows indicate the possible locations of oblique sectioned cytomembranes (4o,ooo : 1)

.

The second example shows an electronmicrograph of the, so called, neuropil

(Fig. 1) .

This neuropil is lying between the perikarya of the nerve cells in the gray matter and contains the interconnections between the network of nerve cell processes. The network is composed of three kinds of processes with different

functions: one is for input, one is for output and one is for different

auxiliary functions. The three kinds of processes have a similar image and

the human eye needs a certain period of experience to recognize these subtile

differencies. The functional interconnections between the in- and output processes are the synapses with a specific, but, not too different structure.

Furthermore the diagnosis of tliese processes will be implicated by the fact,

that the surface of it, which is represented by cytomembranes, have different

angles to the surface of the section resulting in different gray-values. In this case the structural composition can only be differentiated by the eye- brain system and not by the scaning-instruments.

The optomanual systems. The main problem in the optomanual system is the manual transfer of the optical diagnosis into an advanced system of registration. Today various

solutions of this problem exist.

1) A simple instrument was and is a counter with a keyboard for different

structures. Such instruments have in the meantime been developed as a receiver for a computersystem. The first construction was introduced by

WEIBEL. Such instruments are able to register values for more than one structure (2o)

.

2) An older instrument, the TGZ 3 of Zeiss, has recently been automated and can now be used for registration of more than one diameter of a structure. At the same time it registers the number of measured structures (6, 7)

.

168
Optomanual automated evaluation-systems in biological research

The following instruments are based on the automatic registration of a manually tracing of structural details with an instrument. The technical

solution of such devices are very different. However, three of these devices

need a flat plane for evaluation. The extension of this plane should not be

too small, because oux brain-hand-instrument should have a certain space for

an exact manipulation of the specific instriiment to diminish inexact tracing

errors. The following three devices have solved this problem in a similar

way. It is possible to examine slides within a microscope with the help of

a drawing mirror. All other structures may be evaluated by their photomicrographs or any other kind of projection. In this way these systems are independent of the instrument of observation or of the magnification of the image

.

3) One instrument has been developed by COWAN and WANN. It registers the topographical coordinates of the manual traced instrument by an ultrasonic

device in a fixed board of evaluation. This registration is transfered to

a computer (1, 19)

.

4) The MOP II from Contron comp. registers with the help of a light stylus

the crossing of a line in a lattice as a point. Dr. Rohr suggested the development of such instrument. The optical registered points are counted in different registers of the instrument and can, but must not be transfered on

or off line to a computer-system. This instrument can also be used as a pointcounter. There are different kinds of lattices (e.g. after WEIBEL, I»1ERTZ etc.) and these can be used on every flat sxirface (7)

.

5) The tracing-instruments of some computer-systems which have been developed

for the automated cartography can also be used in stereology. Such a device

has been developed by BLACKSTADT in Aarhous (Denmark) for investigations in the nervous system. The different kinds of technical solutions make it impossible to gather experience with all the mentioned devices. Therefore

the following part of the report is mainly restricted on the application of

the MOP II in point counting and tracing, and of the TGZ 3 Zeiss in measuring

diameters. But it seems that the other both devices can be similarly handled

like the MOP Il-instrument.

In this past year the MOP II and the TGZ 3 of Zeiss have been installed in our institute and they are combined with a smaller computer (Wang 22oo B) . Now

it is possible to report on the first experience with these instruments. The

dimension of this equipment has been directed by financial possibilities. The

supporting instruments of the computer, especially, the Teleprint, are relatively slow. But, I think it is not worthwhile to discuss on a two, three

or four times faster system because the speed of evaluation is surely slower

than one of the computer equipment.

The following report shows that an advanced investigation needs as well

sophisticated optomanual instriaments as a good computer program. One important

problem in biological research especially in brain-research is the inhomogeneous structural distribution of details in the nervous tissue. For

example the cortex cerebri, in which the consciousness is located, is composed of different layers with different functions and structures.

The nerve cells and all other structures of the cortex cerebri are arranged

in layers. For example, the nerve cells place themselves in different

densities and sizes. Thus it is ne cessary to evaluate the different densities

because each layer has another functional significance. Our way to solve this

problem is the following: The measuring of all desired stereological

parameters will be done from one evaluated microscopical field to the next

field (8) . The fields are arranged in rows perpendicular to the surface of

the cortex. However, the distance of every field to the surface as well as the measured values are stored automatically by a tape-p\inch of a teletyp.

169
H. Haug

Twelve different values for each field are possible.

The evaluation of one cortical area needs about 15 to 3o rows , each 2o to 5o fields long. The number of fields in one row depends on the thickness of the cortex. The content of the punched tape will be transferred to our computer

system and for all other calculations stored on a magnetic tape. With the help of a program developed in our institute we can compute our results under

different stereological assumptions, as numbers per volume, or surface areas or length of a structure or the volume-part of one or more details. An

important further advantage of our program is the possibility to divide our samples in the perpendicular direction as well as in the horizontal one. The calculation in horizontal direction can be done by a rotation of the matrix

of values. Such a new calculation needs the stored original values. In this way it is possible to study distinct details in our material without a new measurement step. The fig. 2 shows a result of such an investigation, in which the average values and the confidence limits are plotted by the computer. To avoid irregularities in our samples the computer program makes

it possible to smooth the values by averaging over one^ two or three fields.

SEHRINDE MENSCH 7B J.

DICHTE 1000/CMM

Fig. 2: Computer diagram: Ordinate: Celldensity looo/mm (p = o.o5)

.

Abscisse: Distance from surface in counting fields.

This short example of the evaluation in the cortex cerebri shows the importance that every single value of one field should be stored. The reason

is that in the beginning of a scientific evaluation it is not always known in which way the individual results should be arranged. The storing of the original values makes it possible to compute results based on other and newer arrangements of the values without a new measuring. This is, in my opinion, a further advantage of our system. In this way it compensates its slower

evaluation speed with respect to the scanning devices.

170
Optomanual automated evaluation-systems in biological research

Summary

Some factors of the image, especially low degrees of gray-intensities,

lead to problems for full-automated scan-devices. It is therefore

important to evaluate with the aid of an opto-manual intervention . Newly

developed instruments do not work with scan-procedures , but are based on

the visual diagnosis of structure and a manual transfer of the information

to a full-automated system. By this way the human eye-brain-system can work

like a highly-programmed scan-computer. Three of five optomanual systems

are briefly described. Some more details are given for the TGZ 3 and the MOP II. Furthermore it will be pointed out that it is necessary to store

the original values in order to calculate the different structural

arrangements. Therefore it is possible to calculate certain topographical

details without a new reevaluation. A computer-program has been developed

for such investigations in tissues which are arranged in layers. As an example, investigations in the cortex cerebri of the man are reported

upon.

References

1) Cowan, W. M. and Wann, D. F., J. Microsc. 99 (1973) 331 - 348.

2) Elias, H., Hennig, A. and Schwartz, D. E., Physiol. Rev. 51 (1971) 158 - 2oo.

3) Exner, H. E., in Newsletter 73 in Stereology, Kemforschungszentrum Karlsruhe KFK Ext. 6/37-2 (1973) 6 - 18.

4) Gahm, J., in Quantitative Analyse von Gefugen in Medizin, Biologie

und Materialentwicklung, Ed. H. E. Exner, Dr. Riederer GmbH,

Stuttgart (1975) 29 - 46.

5) Haug, H. , Z. Anat. Entwickl.-Gesch. 118 (1955) 3o2 - 312.

6) Haug, H., J. Microsc. 95 (1972) 165 - 18o.

7) Haug, H., in Quantitative Analyse von Gefugen in Medizin, Biologie

und Materialentwicklung , Ed. H. E. Exner, Dr. Riederer GmbH,

Stuttgart (1975) 91 - lo2.

8) Haug, H., Kebbel, J. und Wiedemeyer, G.-L., Microsc. Acta 71

(1971) 121 - 128.

9) Hennig, A., Bestimmung der Oberflache beliebig geformter Korper mit

besonderer Anwendung auf Korperhaufen im mikroskopischen Bereich.

Mikroskopie Wien 11 (1956) 1 - 2o.

10) Hougardy, H. P., in Quantitative Analyse von Gefugen in Medizin,

Biologie und Materialentwicklung, Ed. H. E. Exner, Dr. Riederer GmbH,

Stuttgart (1975) 73 - 8o.

11) Kluge, N., in Newsletter 73 in Stereology, Kernforschimgszentrum

Karlsruhe KFK Ext. 6/73-2 (1973) 29 - 39.

12) Moore, G.A., J. Microsc. 95 (1972) lo5 - 118.

13) Muller, W. , in Newsletter 73 in Stereology, Kemfor schxingszentrum

Karlsruhe KFK Ext. 6/37t2 (1973) 19 - 28.

14) Ondracek, G., in Newsletter 71 in Stereology, Kernforschungszentrum

Karlsriihe (1971) , 1 - lo9-.

15) SchmeiBer, H., in Quantitative Analyse von Gefugen in Medizin, Biologie

und Materialentwicklung, Ed. H. E. Exner, Dr. Riederer GmbH.,

Stuttgart (1975) 6o - 72.

16) Underwood, E.E., Stereology and quantitative metallography, ASTM STP

5o4, American Society for Testing and Materials (1972) 3 - 38.

171
H. Haug

17) Underwood, E. E., Microscope 22 (1974) 69 - lo9.

18) Underwood, E. E., in Quantitative Analyse von Gefugen in Medizin,

Biologie und Materialentwicklung, Ed. H. E. Exner, Dr. Riederer GmbH.,

Stuttgart (1975) 8-14.

19) Wann, D. F., Woolsey, T. A., Dierker, M. L. and Cowan, W. M.

,

IEEE Trans. Biomed. Eng. BME 2o (1973) 233.

20) Weibel, E. R. , Stereology Proceedings of the Second International

Congress for Stereology Chicago, Springer, New York (1967) 275 - 275.

21) Weibel, E. R. and Ellas, H., in Quantitative Methods in Morphology,

Ed. E. R. Weibel and H. Elias, Springer, Berlin, Heidelberg, New York

(1967) a 3 - 19.

22) Weibel, E. R. and Elias, H. , in Quantitative Methods in Morphology,

Ed. E. R. Weibel and H. Elias, Springer, Berlin, Heidelberg, New York

(1967) b 89 - 98.

172
National Bureau of Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

A USER-ORIENTED AUTOMATIC SCANNING OPTICAL MICROSCOPE

by J. E. Hilliard Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60201, U.S.A.

ABSTRACT

A "Digiscan" optical stage scanning microscope has been interfaced to a DEC PDP8/e computer. .Two important factors in the design of the instrument

were flexibility and ease of operation. The latter has been achieved by making

the instrument self- instructional and by the use of dialog with the user in order to eliminate the large number of controls that are normally necessary

with an instrument of this complexity.

INTRODUCTION

The following were the design objectives for an automatic optical scanning

microscope that has been assembled for use in the metallography facility at Northwestern University: (1) Maximum flexibility - it must be capable of

being easily updated as new types of analyses and computational procedures are developed. (2) Ease of operation - since there will be several different and changing users it is important to avoid the need for lengthy instructions in the use of the instrument. (3) Must allow for visual discrimination in those

cases where there is insufficient difference in contrast between constituents

to permit automatic discrimination. (4) State of the art performance with re- spect to resolution and linearity.

It was decided that these requirements would best be met by a computercontrolled stage scanning microscope, SSM, rather than a TV scanning instrument. The use of a computer provides the required flexibility, since modifications

in the functioning of the instrument require only changes in the programming

and not in the hardware. Also, a computer permits the employment of certain

self- instructional features that will be discussed later.

In an SSM, scanning is accomplished, as the name implies, by a motor

driven stage; a photodetector is used to determine the light intensity at a small spot at the center of the field and, provided there is sufficient difference in contrast between the constituents, the output from the detector can be used for automatic discrimination. Alternatively, the user can signal an event

(the crossing of a boundary, for example) by pressing one of five keys on the base of the microscope. Although it is theoretically possible to make closely

spaced scans, this would be very time consuming. Thus, in practice the SSM

is limited, \ihen automatic discrimination is used, to measurements that can be made with a linear probe. This limitation is not too serious if, as in our

case, one is primarily concerned with measurements on a section through a three-dimensional structure. Most of the properties of interest (volume fractions, boundary area per unit volume, size distribution, shape factors, etc.) can be obtained from linear probe data. [The situation is different when anal- yses are being performed on a structure that is essentially two-dimensional

(dust particles on a slide, for example). In this case there are certain properties, such as the shape of individual particles, which may be of importance

and which cannot be determined with a linear probe.] In other respects, an SSM

compares favorably with image or TV scanning instruments: (1) All measurements

are made on the optical axis of the microscope and therefore maximum resolution

is achieved. Also, it is possible to operate with the field diaphragm almost

completely closed, thereby reducing glcire and enhancing the contrast. (2) There

is no "shading" problem. For this reason, and because of the slower scanning

speed, the gray level resolution is limited only by the stability of the photodetector and light source. (3) The scanning is inherently linear. (4) Because

long traverses can be made, end effects are negligible or, at most, require a minor correction. (5) An SSM is easily operated in a visual discrimination

mode when automatic discrimination is not possible.

173
J. E. Milliard

DETAILS OF THE INSTRUMENT

The instrument comprises a Wild-Kontron "Digiscan" optical microscope

which has been interfaced to a DEC PDP8/e computer. The stage is driven in the X and y directions by pulsed DC motors; the stage velocity can be varied over a range of 150: 1 with a maximum of 1 mm per second. Even at maximum

velocity under an oil immersion objective there is no stage vibration; thus measurements can be (and are) made with the stage in motion. (This is in con- trast with some other SSM's in which the stage is stepped through a finite

interval in between measurements.) A variable joystick is available for manual control of the stage. The response to this joystick is exponential so that precise control can be obtained at low velocities while still allowing

for high- velocity slewing of the stage. Motion of the stage is monitored by

optical encoders having a resolution of 0.5 p,m. A solid-state photodetector

is used and the stability of the system is good enough to resolve over 500 gray levels. The spatial resolution varies inversely as the power of the ob- jective; for a lOOX objective it is 0.9 (j,m for a 5 to 95 pet change in intensity across a sharp boundary.

Components of the computer include an 8-channel A/D converter (used for interfacing the joystick and photodetector), a programmable time clock, and a 12-channel digital input and output interface (the output is used for pulsing

the stage motors and other purposes; the input side is interfaced to the optical encoders and the keys on the microscope). A dual magnetic "DECtape"

unit permits rapid loading of programs and the storage of data. A video

terminal is used for high-speed text display (900 characters per second) and

a teletype is available when a hard copy of the output is required. The com- puter also has a hard-wired, high-speed (150,000 band) connection to a network

maintained by the Department of Computer Sciences. This allows use of the network's magnetic disc and line printers and also permits direct transmission

of data to the University's CDC 6400 computer whose processing is beyond the capabilities of the PDP8/e.

The DEC OS/8 operating system is used for the loading of programs and also

for their editing and assembly. Software developments have included the de- velopment of a high level language FOCUS. This is an extensively modified

version of FOCL/F written by D. E. Wrege which, in turn, is a variant of

DEC FOCAL-8. (The latter is an interpretive language similar in form to BASIC. Unlike FORTRAN, the user's program does not have to be assembled; as a result,

debugging is very much faster.) FOCUS contains the machine- language routines

required to drive the stage, to monitor the optical encoders and photodetector,

and to handle interrupts from the five microscope keys. Since FOCUS eliminates most of the need for machine-language programming, new programs for specific

analyses can be prepared very rapidly even by persons who have had little pre- vious experience in programming. It should also be added that the interfacing

to the microscope does not interfere with the regular use of the computer.

Thus, when the scanning microscope is not being used, a reasonably powerful

computing system is available.

OPERATION OF THE INSTRUMENT

As noted in the introduction, one of the design objectives was simplicity

of operation. This has been accomplished in two ways. First, insofar as possible the unit has been made self-instructional. Secondly, most of the information required by the instrument is elicited by dialog rather than having

the user operate switches. (In fact, excluding the microscope keys and the switches on the video terminal, there are only two switches and one knob on the whole instrument that have to be set by the user. The teletype, for example,

is turned on and off under program control.)

In order to start operation, the user enters a two word command at the keyboard. From then on all programs are chained in automatically as required.

After the initial program has been loaded, it checks that the right magnetic

tapes have been loaded and then runs a 3-second diagnostic which tests all the principal components and reports any malfunctions detected. The user is then

174
A user-oriented automatic scanning optical microscope

asked "Do you need instructions for aligning the microscope (Y/N)". If the response is "Y" the user is queried with respect to the mode in which the microscope will be used (reflected or transmitted light, bright- or dark field).

Step-by-step instructions are then given on the video terminal. The user ad- vances to the next frame by giving a carriage return; the option is also available for backing up to any previous frame. At the completion of the alignment

instructions (which occupy about 20 frames) a directory is displayed of the analyses currently available. The user enters his choice at the keyboard and the program is automatically loaded. As an illustration of these programs we will consider one that involves visual discrimination and another using auto- matic discrimination.

Point Counting

This is for a volume fraction determination using visual discrimination.

The user is first given the option of receiving instructions for performing

the analysis. A request is then made for (a) Specimen identification (this

is printed out with the results); (b) Power of the objective (this determines

the spacing between counting positions); (c) Approximate area of specimen (used

for the estimation of the minimum coefficient of variation, CV, that can be achieved; where CV is the ratio of the standard deviation to the average). All responses are checked and any which are apparently in error are queried (for

example, an objective power outside the range of 10 - lOOX). The user is then

instructed to proceed. A 9-point grid inscribed on a reticule in a focusing

eyepiece is used for counting. One of the microscope keys is used to signal

an occupied point, other keys are used for advancing the stage to the next counting position, reversing direction in the x-direction and incrementing the stage in the y-direction; recording of 1/2 points and the deletion of erroneous

entries. After 20 fields have been counted a buzzer sounds and the user is informed on the video terminal of: (a) The minimum CV that can be attained in the estimate of the volume fraction, assuming that the whole area available for analysis is sampled, and; (b) The counting time required to achieve this CV. These estimates cire based on a theoretical estimate of the CV [Milliard (1975)] and the time taken by the user to count the first 20 fields. The user can then enter any higher value for the CV and will be given the corresponding

counting time. (The time estimates turn out to be surprisingly reliable;

usually they are within 5 pet) If a higher CV than the minimum is selected,

the option is provided of increasing the distance between counting positions

so as to provide a uniform sampling of the specimen.

During the analysis, a current estimate of the CV is displayed on the terminal and this is updated after every 10 frames have been counted. At any time the user can obtain a summary of the results on the video terminal and, if desired, a full print-out on the teletype. The user can then continue the analysis or return to the directory of programs, one of which is the shutdown

procedure.

Intercept-Length Distribution

This program permits the determination of the distribution of intercept

lengths in two different constituents. As before, the user has the option of

receiving instructions for the analysis; these also include the steps required

to adjust the balance control on the photodetector . The next step is to position the stage at various positions within the two constituents , the identity

of which is signaled by one of the microscope keys. As each entry is made, the intensity is plotted with an identifying letter on the video terminal. After

a sufficient number of entries, the user decides whether automatic discrimination is possible and, if so, enters lower and upper discrimination levels.

(If, because of overlap on the gray scale, automatic discrimination is not

possible, the option is provided of branching to a version of the program that

allows visual discrimination.)

During the analysis the stage can be driven with the joystick or at a preset velocity by pressing one of the microscope keys. The intercept lengths

in the two constituents are classified into 128 class intervals. (If necessary

this number can easily be increased.) Initially, the width of the intervals is

175
J. E. Milliard

set at 0.5 p,m giving an upper limit of 64 \im for an intercept length. If an intercept length longer than this is encountered, permission is requested to regroup the data. If this is given, adjacent class intervals are combined so as to produce 1 |im intervals with an upper limit of 128 |j,m. This regrouping

procedure is repeated each time an intercept is encountered that exceeds the current upper limit. (The grouping of the intercepts is treated separately

for the two constituents.) At any time, a histogram of the results can be displayed either on the video terminal or on the teletype. For display purposes the distribution can

be grouped into any number of intervals that can be expressed as a power of

two. Provision is also made for deleting any entry that appears erroneous.

In addition to the distribution of intercept lengths, the volume fractions and the boundary area between the two constituents are reported, together with the experimentally estimated standard deviations of these quantities. After the measurements have been completed, various programs can be called in for the analysis of the data. These include routines for unfolding the intercept distribution to determine sphere-size distributions, testing for a random spatial

dispersion of particles, and the determination of shape factors. Also, as previously mentioned, the data can be transmitted directly to the CDC 6400

computer.

ACKNOWLEDGEMENTS

This instrument would not have been possible without the programming aid

of several undergraduate students; I am particularly indebted to L. E. Suk, who developed FOCUS and was responsible for several of the operating programs.

REFERENCES

Milliard, J. E. (1975) Assessment of sampling errors in stereological analyses.

[This volume, Eq. (13).]

176
National Bureau of Standards Special Publication 437

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

THE DETERMINATION OF PARTICLE SHAPE AND SIZE DISTRIBUTIONS USING AUTOMATIC IMAGE ANALYSIS TECHNIQUES

by Jeff Slater and Brian Ralph Department of Metallurgy and Materials Science, University of Cambridge, England

1) INTKODOCTION

One of the commonly occurring structures in materials science comprises a population of particles dispersed in a matrix. In general the particles

within the population are of the same origin and chemical composition,

consequently having similar three dimensional shape throughout, though there is

a variation of particle size due to the kinetics controlling their formation.

Various workers (e.g. Hennig and Elias, 1963) have developed methods for the determination of particle shape and size distributions from manual measurements where particles belong to a simple analytical shape class. The

development of this approach has been retarded by the amount of data required

for the analysis.

The larger image analysing computers, currently available on the market,

provide the means for the acquisition of the large amounts of feature specific

data (i.e. parameters that may be related to each feature in a field of view)

quickly and efficiently. The aim of this paper is to outline a general method

for the classification of a population of particles, by size and shape, using

data from automatic instruments.

2) THE MATHEMATICAL APPROACH

Before a solution to the particle sizing problem can be achieved, the following assuii5)tions must be made:

i) The volume size distribution of the par-ticles contains particles of

discrete sizes;

ii) The observed distribution of the particles on the section may be

classified as the number of particles in fixed size intervals. The relationship between true and observed size distributions then becomes:

N„.. (P. .) = N . (1) V] 13 Ai

where the probability matrix P. . relates the number of particles of class j to

the observed number of features"' in class i. For three dimensional shapes that

may be expressed analytically, the probability matrix can be calculated

exactly and the equation may then be inverted to give the relationship between

observed and true sizes. This is the method used by Saltykov (1958) , Hyam and Nutting (1956) and others. This method has been extended to cover the case of particles of non- analytical shapes (Schwartz and Ralph (1969), Schwartz (1972)). The particles

in space are described by a faceted model which is then sectioned randomly in a computer to examine the probability of attaining a section of size i from a particle of size j and an equation of type (1) is assembled. However, due to

the fact that it is only possible to make a finite number of random sections,

there are inherent errors in the calculation of the probability matrix. In this

case the final errors are smaller if the equation is not inverted but an iterative procedure is used. Traditionally the model shape has been determined by serial sectioning or

by making an intuitive estimate from observations on the section; or has been

assumed to be spherical so that the simplest sizing procedures could be

followed. Elias (1963) and Myers (1962) independently developed a particle

shape determination method in parallel with the sizing procedures where by a three dimensional particle of known shape was related to a distribution of

observed shapes on a section by a probability matrix. Both of these methods

were developed before the advent of large image analysing computers and were

therefore tailored to manual measuring methods for specific model particle

shapes; both excluded the relationship between feature size and the measured

shape.

177
J. Slater and B. Ralph

When a non-simple particle shape is being evaluated the form factor

(e.g. Fischmeister , 1974) being used must define the shape of the feature on the section as well as possible. With an automatic image analysing computer, like

the Quantimet 720, the feature specific parameters of interest are: Area,

Perimeter, Projections and Feret Diameters at angles of 0°, 45°, 90° and 135°

to the television scan direction. To involve all of these parameters in a single form factor would result in an unwieldy relationship. It is instructive

to use the available parameters to develop two independent form factors for each feature such that the meaning of each form factor can be interpreted in terms of an equation :-

S^. (Q ) = (S^ ) (2) nm nm

2 The measured distribution (S ) is then represented not as a curve bu^ a surface

defined by the number of features having form factors in the limits S and S (this is simply the three dimensional analogue of a histogram) ; for ease of"^ presentation the surface is best shown as a contour map. In principle the shape

of the particles in the population can then be calculated by setting up a m^del

of S in a computer program then random sectioning to compute a value for S for comparison with the measured S . The best fit value of S"^ can then be obtained by following an iterative procedure.

3) EXPERIMENTAL AND COMPUTING PROCEDURES

a) Experimental

The required feature specific experimental data was collected using a Quantimet 720 image analysing computer with pattern recognition, a Calculator

Field/Feature Interface and a 2D Automatic Detector. A test specimen was constructed by accurately cutting several thousand pieces of copper wire such that they all had the same length to diameter ratio. These were then embedded

in cold mounting medium as randomly as possible. The amount containing the rods was then cut into arbitrary pieces which were then remounted as randomly as conditions would permit.

The parameters measured for each feature were Area, Perimeter and the

four Feret Diameters (at 0°, 45°, 90° and 135°) to the T.V. scan direction).

This data was accumulated in a Hewlett Packard 9830 programmable calculator and

the values of S were calculated directly, nm

b) Computing

2 The computed values of the S array were determined using an IBM 370/165

ComputerLand the IBM PL/1 compilers. A particle of a given three dimensional

shape (S ) was set up in the program using a many faceted polyhedral

approximation, the particle being described by the Cartesian co-ordinates of its

vertices and a look-up table to define which vertices were connected by edges.

The particle was randomly sectioned a large number of times and the co-ordinates

of the vertices of each generated feature were stored. The computed data at this point represents the expected distribution of features produced by a section through a dispersion of particles all of the same size and shape. The

parameters require^ for the generation of the form factors were then computed

and the computed array assembled.

4) RESULTS AND DISCUSSION

In taking the faceted computer model approach to particle shape and size

determination, one of the first points to be considered is the number of facets

on the model required to represent the model shape adequately. It was foiind that in the general case a 48 sided model was sufficient; though in special cases much simpler models could be used. In simple situations where the actual particle shape is faceted and contains a low number of facets the actual

shape determination procedure reverts to a simpler situation. In such cases

178
The determination of particle shape and size

COUNTS300

200

CUIE TETtAHEDtON

1000 SKTtONS

F0RM2=flREfl/ 1 FMflX«PERPFMHX)

COMPy T Ep. . . FO««^

OlSTRIByTJON.

60

F0RM2=flREfl/(FnflX« PERPFMflX J 3

0.9 .

0.8 .

0.7 .

0 o

0.6 .

0.5 .

0.4 .

0.3 .

<H>

0.2 .

0.1 ,

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

F0RMUflREfi/FMfiX«»2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

F0RMl=flREfl/FMflX»»2

FIGURE 1. Frequency of occurrence of form factor (Area/Perimeter ) for random

sections of cubes and tetrahedra. FIGURE 2. Computed form factor map for random sections of 5:1 aspect ratio

rods. FIGURE 3. Measured form factor map for random sections of 6:1 aspect ratio

rods.

179
J. Slater and B. Ralph

the distribution of only a single fo^ factor is required. Figure 1 shows

unique computed distributions of A/P (Area/Perimeter ) for cubes and tetrahedra

where the principle of identification of a particle shape from a form factor distribution curve can be seen readily.

Figure 2 shows the computed form factor map for a distribution of rods of aspect ratio 6:1. As is expected the majority of the generated features have

form factors close to the limiting case of a circle, (the point marked 'O' on

the figure corresponds to a circular section) . In the corresponding

experimental data, presented in figure 3, the expected circular sections have

been displaced towards the origin. This is thought to be the compound effect

of resolution errors in the input peripheral and errors in the measurement of

the constituent parameters of the form factors. Any particle of finite size when sectioned will produce sections varying

in size from an upper limit to zero. As the size of the feature defined on the section approaches zero it becomes much harder to define the shape of the feature as the relative errors in the measuring process increase. This effect

is of particular importance when automatic instrumentation is used to make the measurements where analogue to digital conversion of the image takes place. The

computer program that generates S must therefore contain the same digital

logic that is used by the image analysing instrument for the detection and measurement of the computed 'image' of the features. Once this digital logic

has been included comparisons can be made between the measured and computed

values of S It mus^be emphasised that once the instrument digital logic has been

included in the shape determination procedure the apparent shape of the features

on the section is inter-related with the size as well as the shape of the particles in the population. It is not possible therefore, to measure the size

distribution without knowledge of the shape distribution; the converse is also true but for different reasons:

i) Particle shape influences the values of the elements in the probability

matrix P. . in equation (1) , therefore the particle shape must be known

before paiticle sizing can be accomplished,

ii) The particle sizes present determine the image magnification used in a Quantimet type experiment thus influencing the relative spacing of the

'picture point grid' and hence the shape measurements with respect

to the particle size. It is not currently feasible to write a computer program to solve the interrelated problems of size and shape distributions simultaneously, largely

because of the enormous amount of computer time that would be required by such

a program. The problem of particle shape determination must therefore be, solved

by scaling the 'picture point grid' in the computer model so that it has the

same correspondence with the mean feature size (computed) as does the actual

picture point grid with the mean feature size (measured) , the mean feature size

being a readily measurable parameter. The iterative sizing procedures may then

be utilised.

REFERENCES Fischmeister , H.F. (1974) Shape Factors in Quantitative Microscopy; Z. Metal Ikde, 65, 558. Hennig, A. and Elias, H. (1963) Theoretical and Experimental Investigations on Sections of Rotary Ellipsoids; Z. wiss Mikroskopie, 64, 133. Hyam, E.D. and Nutting, J. (1956) The tempering of Plain Carbon Steel; J. Iron

and Steel Inst. , 184 , 148. Myers, E.J. (1962) Quantitative Metallography of Cylinders, Cubes and Other

Polyhedrons. Ph.D. Thesis, University of Michigan.

Saltykov, S.A. (1958) Stereometric Metallography, Moscow. Schwartz, D.M. (1972) A General Method for Calculating Particle Size

Distributions; J. Microsc. , 95, 25. Schwartz, D.M. and Ralph, B. (1969) The Analysis of Particle Size Distributions

from Field Ion Microscope Data, Phil. Mag. , 19 , 1061.

180
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

SOME SHAPE FUNCTIONS FOR USE IN AUTOMATIC IMAGE ANALYSERS FOR THE CLASSIFICATION OF

PARTICULATE MATERIALS

by Adrian C. Terrell' and R. J. Willes** *lmage Analysing Computers Ltd., Me/bourn. England

""Image Analysing Computers Inc., New Yorl<, U.S.A.

INTRODUCTION

Television based Image Analysers are now able to measure 2 Dimensional image features sepczrateZy at very 'high speed. The various available measurements can be combined to form the most appropriate size parameter for a particular

application. Dimensionless form functions may also be derived from the measurements so that a shape distribution can be used to characterise a material or, alternatively, a single shape class can be chosen to pick out a particular

population of features so that its number /density, location and/or size

distribution can be obtained separately. The choice of form function depends

not only on its theoretical ability to recognise the geometric, densitometric or texture characteristics of the required features but also on the precision with

which it can be measured; given the available algorithms of the hardware and software employed and the signal to noise ratio of the optics and electronics

used to form, scan and digitise the image.

AVAILABLE MEASUREMENTS

Some typical feature measurements are shown in fig 1. Many other feature measurements are available from specially equipped machines; for example the features may be classified by integrated brightness or by optical density. The number of sub features within the main feature may be used as a classification

parameter. The sub features as well as the main feature, may be subject to size and shape tests. 2 Dimensional Amendment can be used to generate new

features representing groups of image features by agglomeration in 2 dimensions.

The lobes of interconnected features may be separated so that they can be treated individually.

P^QJECTIONS.

HO/S.I-Z.C>/^TP>L.)

P£/DIMETER.

rER.ET

Df/^METE/^S.

('^ DlfS.ECTIO/sls)

£>-CL.UOeS

TPiNCiENT CO'

O/ZblNRTES.

Figure 1. Quantimet 720, some feature measurements.

SPEED

The Authors are applying two distinct image analysis systems to a wide

variety of applications problems. The first system uses a Quantimet 720 interfaced to a Hewlett Packard 9830 calculator so that either the results of size/shape distributions made in the Quantimet can be transferred to the calculator or the individual feature measurements can be transferred for more

181
A. p. Terrell and R. J. yjilles

sophisticated size and shape tests. Using the former operating mode, particles

might be classified into 14 size classes with selection by a simple shape test

at a speed of about 4 seconds per microscope field. In a typical powder

problem this would represent some 40,000 particles per hour. The speed is slightly slower at the highest optical microscope magnifications because of the time required for the autofocus mechanism to operate. In the feature data

mode the system transfers all the required measurements from one feature at a time. Much more sophisticated shape tests may be used and the size

distribution can be in any number of classes but the speed is reduced to some 5,000 particles per hour. The second system uses a fast Direct Memory Access method to transfer the data to the core of a PDF 11 computer. Four separate measurements plus

co-ordinates of all the features in a field are transferred in one scan

(about 1/lOth second). Overall speeds of some 400,000 features per hour are attainable.

The hardwired classification, by size and/or shape, is frequently used

together with the calculator or computer methods. In the case of the calculator this is to save time by skipping the 'out of range' features from

the feature sequencing. In the case of the computer method it is to save memory by allowing smaller arrays to be dimensioned for the feature data.

CLASSIFICATION PARAMETERS

For specialised applications the Quantimet can be programmed to measure

only the required parameters. For general feature size and shape distributions

it is useful to derive a number of non directional classification parameters

from the raw feature data. It is convenient to express all of these parameters in the same dimensions and with a constant of proportionality for each one chosen to make them mutually comparable; for example if the parameters are calculated as diameters then we might use the following selection.

1. area diameter - diameter of circle having same area as feature

2. perimeter diameter - diameter of circle having same perimeter as feature

.

3. longest dimension - max feret diameter

4. mean feret diameter - a good approximation to the diameter of the circle having the same convex perimeter as the feature

5. convex area diameter - the diameter of the circle having area equal

to the convex area of the feature, derivable from the 8 tangent

co-ordinates

6. mean width - equal to twice the area divided by the perimeter, not

an independent function as it can be derived from 1 and 2 but useful for many classifications

Minimum feret diameter would be a useful indication of the width of the envelope for elongated features but the feret diameter would have to be sampled at, say, 16 directions to give a reliable value.

FORM FUNCTIONS The advantage of arranging the basic size measurements as a set of comparable diameters is that any pair of them can easily be ratioed to give a dimensionless form function which has a value of 1 for a circular feature. The

six chosen parameters give rise to 13 separate form functions (excluding those which are equivalent to, or reciprocals of, each other). It is not easy to visualise the geometric meaning of each one of them in a general sense but their meanings are quite clear in respect of some limited ranges of allowable

shapes. The selection of the best form functions for any particular image

analysis problem depends not only on the geometric interpretation of the function in terms of the expected feature shapes but also on the precision

with which the function can be determined for the range of shapes, sizes

and orientations of features which will be found. For our general purposes feature data programs, used to characterise a wide range of particulate materials; we have found it useful to allow for

two classifications, each of which applies a series of limits to either one

182
Shape functions for use in automatic image analysers

of the classification parameters, as a sise test, or one of the form functions

as a shape test. If we choose a single bin (two limits) for one of the classifications then we have a selection, usually by shape, and a classification, usually by size but the same program will give a matrix

classification when both classifications have more than one bin.

SHAPE DISCRIMINATION

We can predict the variations in the measured values of form functions,

for any particular size and shape of feature from a knowledge of the relevant

system properties; Detection and Digitisation errors have greatest effect on

small features. Large features are affected by detection errors if they have narrow elements as occur in images of thin fibres. Detection errors are largely eliminated by the latest auto delineating detectors which establish

local feature boundary positions by using 2 Dimensional look around logic

to establish the half amplitude point of the video signal across the feature

boundary, independently of the angle which the scan direction makes with the local boundary. (Wadlow et al , 1972) Algorithm errors can give rise to small variations in some parameters with feature orientation. These errors occur with perimeter and diagonal feret diameter measurements for a few feature

shapes which happen to lie in unfavourable orientations.

As 'the proof of the pudding is the eating thereof we decided to compare

some of our predictions with measurements made on a few feature shapes. A

single feature was measured many times, typically 100 times, and the mean

and standard deviation of values for the 6 basic classification parameters and some Form Functions considered useful for the shapes concerned were printed

out. The measurements were repeated with the features being moved small distances to check digitisation errors, and larger distances to check

distortion errors and finally with randomly changed orientation to check

algorithm errors. We also repeated the measurements for variously sized features having the same shape. (Note that the variations in a form function

value cannot be predicted for a knowledge of the standard deviations of the parameters from which it is derived because there is often an unknown

correlation between them) A lot of information is still to be gleaned from

the data but some conclusions can be drawn at once.

PRELIMINARY CONCLUSIONS

In one experiment we picked a family of elliptical features having the smooth sharp edges and covering the eccentricity range from one (circle) to

10. From the results we estimated the number of shapes which could be discriminated within the eccentricity range; a) with constant orientation b) with

random orientation. (The discrimination criterion used was that the widths of

the shape classes should accept the full range of form function values found in

a set of 100 measurements.) The rapid change in discrimination with feature

size reinforced our view that any image analysis involving shape as well as size tests on features should be confined to features larger than a lower

limit which is considerably higher than that usable for ordinary size

distributions. The printout can be made to show the overlapping parts of the distribution separately or to merge the curves by taking the high magnification

figure only when a statistically significant count was achieved.

The second tentative conclusion is that the best form function for separating the family of shapes concerned will often not be the same one that is best

for eliminating other features not belonging to the same family of shapes. For

this reason we now sometimes prefer to store the basic parameters for all the features on cassette or disc data files so that different shape tests can be applied to the original data in subsequent experiments without having to repeat the microscopy. With the high speed computer interfaced Quantimet we

can apply a series of different tests to the features from one field,

displaying the results of the tests on the Quantimet screen against each feature.

The sophisticated display facilities with this instrumentation greatly aid

the choice of form function and the selection of separation limits to be applied to it. Once a classification regime has been established it might be preferable to set up a working system using the less expensive calculator

system which does not have the computer generated display facilities. In

183
A. C. Jerrell and R. J. Wil/es

other cases the computer system might be required for use with the developed

application because of its inherently faster operating speed.

STEREOLOGICAL IMPLICATIONS

The shape tests we have described are limited to 2 Dimensions. We have

reached the stage that we can now generate fast reasonably accurate feature

specific data on 2 dimensional image features. It remains to see if this data

can be used to derive reliable 3 dimensional information in each of a variety

of applications.

REFERENCE

WADLOW^ D.E., HOPKINS, B.M. , GARDNER, G.M., and FISHER, C, "New Detection

Systems for the Quantimet 720", The Microscope, 20, 183-202, (1972)

184
National Bureau oi Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

DETERMINATION OF PARTICLE SIZE AND SHAPE DISTRIBUTION BY AUTOMATIC FEATURE ANALYSIS

by J. R. Schopper

Institut fur Geophysik der TU Clausthal, D-3392 Clausthal-Zellerfeld, Germany

SUMMARY

Particle type, shape and size analysis of mixtures of multishape multi-size particles of various geometrical types, from sec- tion images is possible in principle, since a sufficiently fast

computer method is available for a thorough evaluation of size and

shape of section features. Such features can be classified according to their origin from certain type particles, and a statistical

analysis of distinct geometrical feature parameters can be used

for a stereological transformation into space, resulting in the

number per volume of the different type particles and the mean and

variance of their size and shape parameters. Even some topological

information on cell stnactures or networks might be gained from

such a section image analysis. - As an example, ways for a morphometric and granulometric analysis of a mixture of approximate

balls, prolate spheroids, cylinders, cubes and tetrahedrons of

non-uniform shape, size and orientation are indicated, final solutions yet pending.

1 . INTRODUCTION

There is more information about spatial structures contained in

a section image than can be extracted by point and lineal analysis.

There has been considerable theoretical work already in the past,

about shape and size relations between structures in space and in

sections, but the first one to recognize the practical need for an extensive individual section feature analysis, to recommend computer processing, and to develop himself a working - though yet

slow - computer program, has been George Moore (Moore & Wyman,1963;

Moore, 1 964, 1 966)

.

The development of a sufficiently fast computer procedure for a most complete statistical analysis of the geometry of individual

section features (Rink, 1 970 ), enables stereology now to draw more efficiently on the complex statistical signal from space, a section

image represents. Thus this paper deals with the problem of particle size and shape analysis, of agglomerates of different type particles, of different sizes, and of different shapes (consistent

with each type), using geometrical information from section images

that can be automatically obtained by such an image analysis program.

The here considered collection of model particles of rather regular geometry is very well suited for approximating practically oc- curring real particles, so far as they are singly connected and do

not have extreme concavities. However, in the present preliminary

stage of work, it really is the intention of this paper to discuss

ways and means rather than to present ready solutions,

2. QUALITATIVE EVALUATION OF FEATURE SHAPES IN SECTIONS

Already a common sense approach can clearly show the vast amount

of morphometric information available from a section image. Each

particle shape in space results in some characteristic feature

shapes in sections, and by the selection of most significant feature shapes, much can be said about the presence or absence of particles of various shapes. Such 'geometrical filtering' of the sec- tion image can also be done automatically by computer.

Of course, as the section features become more complex and diversified, more uncertainty is involved in their interpretation, How185
J. R. Schopper

ever, little outside information can greatly reduce this uncertainty in many practical cases.

3. QUANTITATIVE EVALUATION OF FEATURE SHAP.ES IN SECTIONS

With the aid of computer processing, the previously discussed

approach can also be quantified. The frequency of appearence of va- rious characteristic shapes or shape groups of intersects tells

about the frequency of certain type particles in space. Eventually

this information can be used for a complete particle type, shape

and size analysis, as will be shown below.

Myers (l964) lists the probabilities for the intersects of various polyhedrons being polygons of certain numbers of corners. Thus

an analysis of an agglomerate of mixed one-size polyhedrons, resulting in the frequencies N^ of the different polyhedrons, is possible by counting the frequencies N of the different polygons in the

section plane. Note that this method is size independent in principle, and can be adapted to mixtures of multi-type multi— size

polyhedrons. Even certain type-consistent shape variations can be

permitted.

In a similar way other types of particles can be treated. Quite

a number of authors investigated the geometrical probabilities of

sections through ellipsoids and cylinders, and there is no fundamental problem in expanding those findings to cones etc.

4. PARTICLE TYPE, SHAPE AND SIZE ANALYSIS

From the well-known fact that a particle size determination from

sections requires the knowledge of the particle shape, it follows

that shape analysis must always precede the size analysis. But

while the measure of size is a single number, shape requires a - generally large - set of numbers, and a really general mathematical

expression for shape can be very complex and impractical. To simplify the shape description, the occurring particles should be preclassified - exactly or approximately - into particle 'types', re- presented by basic geometrical bodies as cylinders, spheroids, etc. Shape - within each class - then can be expressed by one or a few

numbers, e,g. an axial ratio.

A rigorous definition of size should be independent of type and

shape and thus based on the particle volume. The cubic root of the

volume or the radius of a volume-equivalent sphere would be such a proper measure of size. However, it is often handier to use other

size parameters, differently for each type; always some characteristic length like e.g. a main axis. This latter way is preferred

here for practical reasons. The determination of the complete size distribution, with many

particle types of variable shape present, is a very complex problem, and a mathematically rigorous attack - as that of Wicksell

(1925) - becomes quite involved and requires the solution of integral equations. However, knowledge of the complete distribution is

seldom desired. By means of a binomial or Taylor development, any

distribution can be fully expressed by its arithmetic mean and an infinite series of moments about the mean. But there is hardly ever

a need in practice for extending it further than to the fourth order. In most cases the mean and the variance will suffice, and this

is presumed here in this paper.

The problem of particle type, shape and size analysis thus is

reduced to the determination of the types, and the mean and variance of proper shape and size parameters. This results in a much

easier solution. But there still remains the problem of untangling

the intertwined effects of size and shape variation. That is why

most publications dealing with particle size analysis assume con- stant-shape particles, or at least such of the same type. Here,

ways for an analysis without such limitations will be shown by an example of a mixture of balls, prolate spheroids, prolate cylinders

186
Particle size and shape by automatic feature analysis

of varying size and shape and of" regular tetrahedrons and cubes of

vai^ring size; with type, shape and size assumed statistically independent .

5. EXAMPLE FOR QUANTITATIVE ANALYSIS OF AN AGGLOMERATE

The aforesaid model mixture results in triangles, quadrangles,

pentagons, hexagons, circles, ellipses and tninkated ellipses in

the section image. For its analysis, Fullman (l953) could be followed. However, instead of intersect and intercept, other section

feature parameters are often used with more advantage, and from

their mean and mean square, mean and variance of particle size and

shape quantities can be derived,

5.1, AJIALYSIS FOR CYLINDERS

The minor axes of elliptical sections of prolate cylinders always equal their diameter. Symmetrically trunkated ellipses, necessarily belonging to cylinders, provide those minor axes too, and

also tell about the cylinder length. A proper mean and mean square

analysis then can yield mean and variance of cylinder radius and

length. Finally, the total number of cylinders derives from a count

of some particular or all of the trunkated ellipses,

5.2. ANALYSIS FOR POLYHEDRONS

Hexagons and pentagons in the section result from cubes only.

Quadrangles might result from cubes or tetrahedrons or - less probable, and valid for rectangles only - from cylinders. Triangles

might result from cubes or tetrahedrons. If necessary, the rectangles resulting from cylinders can be calculated from the information of the previous subsection (5.1.) and excluded prior to, or during analyzing the set of polygons.

With Myers's {^^6k) probabilities for an i-sided polygon resulting from a k-faced polyhedron, the relative numbers of cubes and

tetrahedrons can be obtained. Subsequently the mean and variance of

the size of either type polyhedron and their niimber per volume can

be determined in various ways,

5.3, ANALYSIS FOR SPHEROIDS, INCLUDING BALLS

Ellipses in the section result from cylinders and spheroids. The

part from cylinders can be calculated according to the data of subsection 5.'1. and eliminated from the analysis for spheroids.

For prolate spheroids, the minor axes e^ of section ellipses re- present intercepts in planes normal to the rotational axis, while

the major axes e^ are always intercepts of the generating ellipse.

The circular intersects are oriented normal to the rotational axis.

Thus the mean of the elliptical axes and the circular areas yield mean and variance of the radial axis and the mean of the ro- tational axis of the spheroids, rendering the over-all mean intercept and intersect redundant. A variance analysis of e^ could provide an estimate of the variance of the rotational axis too. With mean shape known, the number per volume can be derived,- If desired,

balls can be treated separately after elimination of the circular

intersects pertaining to the non-spherical objects,

5.4. GENERAL BALANCE AND CHECK FOR MODEL FIDELITY

There are more criteria for double-checking the results obtained

so far, and possibly improving them iteratively. Namely the volume

fraction, specific surface and number per volume must balance.

Such a check will also inform about the applicability of the

assumed model agglomerate. On the other hand, larger deviations

from the model will already show up in the section image by particularly shaped features and might be semi -quantitatively assessed

therefrom.

Nevertheless, often not so great an accuracy at all is required

187
J. p. Schopper

in practice, so that the application of the theoretical concept discussed here to a complete morphometric image analysis promises con- siderable merits for practical particle size and shape analysis.

6. AIDS FOR DETERMINING TOPOLOGICAL PROPERTIES

It has been repeatedly stressed, especially in various papers of

DeHoff and Rhines (cf.1968) that complete topological information

about completely irregular spatial structures cannot be gained from

single sections, but by serial sectioning only. However, a complete

knowledge of the topology of a structure is not always required.

Into the hydraulic permeability of a porous material e.g., just a single topological constant enters, namely the ratio of the number

of nodes to the number of branches of the pore channel network

(Schopper, I972).

PXirthermore , often simple shapes can be assigned (truly or approximately) to the structural elements, thus permitting the application of a morphometric principle to the problem of finding topological numbers per unit volume from similar topological numbers

per unit area. Consider a network of channels, topologically to be viewed as a network of branches; or, consider a granular pack, that can be described topologically as a system of space-filling polyhedral cells.

Both those structures can be thought topologically identical by

thinking of the branches as being edges of polyhedrons.

In the section plane, a plane network of polygonal meshes results, of which the corners (nodes) represent the piercing points

of the spatial branches and the sides (branches) are the traces of

the polyhedron faces.

Thus Myers's (l964) probability coefficients can be applied for

calculating the frequency of different type polyhedrons occurring

in the cellular structure and for finally arriving at a 'mean cell

polyhedron', the number of cells per unit volume, and the mean

cell size.

If the common assumption is made, that two cells meet in a face,

three cells in an edge and four cells in a corner, immediately the

number of faces, edges and corners per unit volume is known too,

and e.g. the above hydraulically interesting topological constant

can be calculated.

However, such an assumption reduces generality unnecessarily,

since just the meeting of two cells in a face is obviously a general truth for the considered structure. Even without such an assumption, the number N^ of branches (edges) per unit volume can be determined independently by counting the number of piercing points

N^ per unit area on the section according to = 2N^/ £ and deriving the mean edge length I from the mean polyhedron already calculated. Then the numbers per unit volume of cells, faces and edges

(branches) are known and that of corners (nodes) follows from

Euler's polyhedron theorem, as does the wanted constant.

References

DeHoff, R.T. & Rhines, F. N. ( 1 968) Quantitative Microscopy. McGraw-Hill, New York, Fullman.R.L. ( 1 953) Measurement of Particle

Size in Opaque Bodies. Trans. AIME 197,'*'*7 Moore, G, A. & Wyman, L. L. ( 1 96 3 ) Quantitative

Metallography with a Digital Computer. J.Res.NBS 67A, 127 Moore , G. A. ( 1 964 ) Direct Quantitative Analysis of Photomicrographs by a Digital

Computer. Pho t ogr . Sc i . Engng . 8,152 Moore,G. A. ( 1 966) Application of Computers

to Quantitative Analysis of Microstructures. In: Ceramic Microstructures

,

Wiley & Son, New York. Myers , E. J . ( 1 964 ) Sectioning of Polyhedrons.

1st Congr. Stereology, Vienna, 15 Rink , M. { 1 970 ) Automatische morphometrische

Bildanalyse mit Hilfe eines elektronischen

Digitalrechners. Doct. Diss., Clausthal.

Schopper, J. R. ( 1 972 ) Theoretische Untersuchung elektrischer, hydraulischer und an- derer physikalischer Eigenschaften porbser Gesteine mit Hilfe statistischer Netzwerkmodelle. Habili tationsschrift , Clausthal,

Wicksell, S,D. ( 1 925) The Corpuscle Problem, Eiometrica 17,81*

188
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

STEREOLOGY AND THE AUTOMATIC LINEAR ANALYSIS OF MINERALOGICAL MATERIALS

by M. P. Jones and G. Barbery

Mineral Technology Department, Imperial College, London, England

Abstract

.

The authors have designed an automatic method of linear analysis based on an electron probe X-ray microanalyser. The linear data that are produced are stereologically transformed by a method of moments and can be used to determine

volumetric grain size, grain, shape and other parameters.

The numerical equivalence of point-, length-, area-, and volume-proportions

in modal analyses have been known and used by mineralogists for a long time. The application of these and other stereological relationships to mineralogical

measurements has been limited by the tedium of collecting the necessary data and

by the inconsistencies of the results that are obtained by man\aal measuring

methods. Consequently, a number of attempts have been made to develop automatic

methods of measurement and these include a linear-measuring system that has

been devised by the authors. The measuring equipment is based on a Geoscan

electron probe X-ray microanalyser and this device is controlled by a small dedicated computer. The Geoscan was selected because of its excellent ability

to discriminate between mineral phases; an ability that is achieved by

simultaneously using some of the many characteristic X-ray signals that are produced when a specimen is bombarded by a narrow beam of hi^ energy electrons.

This phase discriminating ability of the Geoscan (Jones and Shaw, 1973) is at least as good (and is often much better) than that achieved by the opticallybased, automatic, linear, image analysers' that are commercially available.

In the Geoscan a siiitably prepared specimen is moved from point to point

under the electron beam by computer-controlled stepper motors. The 'points' are about 2 ymi in diameter and overlap and thu^ provide a means of carrying

out a linear analysis. The X-rays (and, if necessary, other signals) from the specimen are fed into the control computer. The X-ray signals are characteristic

of the elements and of the proportions of those elements that occur in the small volume of specimen (about 10-18 m?) being irradiated by the electron beam. These signals are used to identify the mineral phases that pass under the

stationary electron beam. During such a traverse the instrument measures

only a single mineral phase (althou^ programmes are being prepared that may

enable 5 or 6 phases to be measured during a single traverse) and the following details are collected:

T = total traverse len^h; N = number of grains of the selected

mineral; ^ (l) = total intercept length on the selected mineral; = the sum of the n^th power of each individual intercept,

where n is an integer between 1 and 4; A/B = the nmber of contacts between minerals A and B.

In addition, each intercept is categorised, either in arithmetic or geometric progression, and listed in the form of histogram data. The results

form the basis of a series of stereological transformations that provide

infounation on grain size distributions, grain shape coefficients, specific

surface areas, etc.

Grain Size. The size of a simple geometrical form such as a sphere can be clearly

defined by a single value 'd', its diameter. However, since it is difficult

189
M, P. Jones and G. Barbery

to define the shape of an irregular particle it also becomes difficult to define its size . The most commonly used method for determining the size

distribution of 'free' mineral grains is to pass the grains throTogh a series

of ^ertures of defined shape and size (i.e. by sieving). This procedure

involves an undetermined grain shape factor and, therefore, the 'size' is only

poorly defined. The size of 'locked' grains cannot, in any case, be determined

by this means.

The method of random intercept length measurement allows the shapes of all convex grains to be mathematically defined. It is then possible to transform

the linear intercepts derived from grains of unknown size (whether 'free' or

'locked') into the true three-dimensional size distribution (or equivalent

screen size) of those grains. The basic eqmtion for the transformation was derived by Barbery (1974) and the values required can be established by linearmeasuring methods.

K = K— -^rjtj—

n+1 '^n ^

^

where D is the three-dimensional, 'convenient' size (usually the screen size);

|Ij^(l) is the n*'^ moment (from the origin) of the random chord length distribution; and Kn is the n''''^ order shape factor determined by linear measizrement of

uniformly-sized grains of statistically tuiiform shape.

The first 3 moments of (D^) can be calculated from the first 4 moments of the intercept lengths and a variety of methods can be used to establish estimates

of the actual values of the distribution of grain size. For example, the in- complete beta function can be used to determine estimates of the ciimulative fraction that forms size (Dq). (Jones and Barbery, 1975).

Other Measurements

Linear measurements provide sufficient data to calculate voltmie proportions,

specific surface areas, particle densities, mean free distance between particles,

etc. Thus,

Z L

a) Volumetric proportion is equivalent to — (see Table l).

b) Specific surface area. =

yi<^ (i) ~ T.^) '^^/'^^

f 1 — V )t c) Interparticle distance. X = li- . X is a shape-independent N

parameter and is the uninterrupted inter-particle distance averaged over all possible pairs of particles in a specim^n^

d) Particle density. =

' rr K

d5

cmax

1 . V

V

where p and q = are parameters of the incomplete beta fimction

B(p,q) = beta function

= n*'^ order shape factors

Dj, = maximum particle size - as defined by screening max V = proportion by voliime of the particles.

190
Stereology and the automatic linear analysis of mineralogical materials

TABLE 1.

Linear Measiirement of Shape Coefficients etc. for Monodisperse Beach Sand

Zircon Grains

(after Simovic, 1973) "

T = total traverse length across specimen 684,000 ym

ZL = total length of intercept on zircon grains 33,914 VTM

E(L^)= 3,027,656 pm^

Z(L^)= 312,817,616 pm^

Z(l4)= 37,345,460,660 pm^ N = total number of zircon grains 478

Shape factors = K = ^ n^'^^ where D =97.5 pm, i.e. the zircon grains all

c passed a 100 yim precision screen and were held on a 95 pm screen. = 0.7277 Kg = 0.6663 = O.706I E^ = O.8646

The above data can also be used to determine:

Z L

a. volumetric proportion of zircon = ^7- = 4-96%

A 2 ^ b. specific surface area of zircon = —|

i ^^^^j = 564 cin /cm

1 -z c. particle density = 60.7/mm

d. mean free inter-particle distance (X) = 1350 ym

e. average nearest neighbour distance ( A^) = I4I pm

Grain Shape.

The shape of a convex grain can be described uniquely by the length

distribution of random intercepts through that grain. The length distribution

of random intercepts through a simple geometric form can be determined by calculation or by Monte Carlo simulation. Unfortunately, the length distribution

of random intercepts throu^ an irregular-shaped grain can only be established

by experimental measurements. One method of doing this is to prepare a large

number of grains of uniform size and statistically uniform shape and then dis- perse these grains at random in space (Simovic, 1973). The moments of the length

distribution of random intercepts through these monodisperse grains provide

mathematical expressions of the average shape of those grains, i.e.

E =

c where E^ = dimensionless n*^ order shape factor; Fj!^(l') = 't'^® moment from

the origin of the intercept length distribution; Dq = 'convenient' size - usually the screen size.

Shape coefficients determined by this method ajre given in Table 1

.

It is also possible to derive shape coefficients by comparing the intercept length distribution obtained from polydisperse grains of a mineral embedded

in a rock matrix with the three-dimensional size distribution obtained by sieving

the unbroken grains after they have been removed from the rock by dissolving the

matrix (Foo,1974). The parameters that relate the two distributions provide the average shape coefficients of that mineral over its complete size range.

191
M. P. Jones and G. Barbery

e) By a small modification of the control programme it is possible to calciilate a locking index, ^p^/-Q ~ ^^/^ where P^^-g

contact with mineral B

=A/B

References

Barbeiy, G. (1974)' Determination of particle size distribution from measurements on sections. Powder Technology, 9? 23I

.

Foo, E. (1974) • Process design study for the beneficiation of a tin ore from Queen Hill, Tasmania. M. Sc. Thesis, London University.

Jones, M.P. and Barbery, G. (1975)- The size distributions and shapes of

minerals in multi-phase materials: practical determination and use

in mineral process design and control. Paper 36, Xlth Inter. Miner.

Proc. Cong. Cagliari.

Jones, M.P. and Shaw, J.L. (1973)« Automatic measurement and stereological assessment of mineral data, for use in mineral technology. Proc. Xth

Inter. Miner. Proc. Cong. London. Ed. M.J. Jones, p.757.

Simovic, M. (1973) • Stereological determination of the size distributions

of minerals. Min. Tech. Res. Prog. Rept., Imperial College, London.

192
National Bureau of Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

field at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

A NEW OPTO-MANUAL SEMI-AUTOMATIC EVALUATION SYSTEM

by H. P. Rohr Department of Patfiology, University of Basel, Basel, Switzerland

The iiitroduction of stereological principles to determine volumes,

surfaces or th.e number of particles offers new aspects and possibilities to assign interdisciplinary importance to morphological results. As mostly a greater number of such test samples must be eva- luated for statistic securitj'' of the results, the broadest possible

rationalisation and economy of image analysis have become extremely

desirable to-day.

3asic3.11y there are two approaches for image analysis: 1. The image

analj-'sis through the so-called scanning methods which are usu3.11y

performed mostly without the human eye and 2. the opto-manual seraiautomatic evalus-tion systems . In spite of the rapid and very promising development in the field of scanning, some limitations of

this method have become more and more obvious: As with this method

image analysis is achieved mostly by a differentiation of various

shades of grey, a clear identification of biological structures is

o^uite often not possible. Therefore, we believe that the human eye

will for long remain a determinating factor in image analysis, at

least in biolog;,''. This assumption is finally supported bj' the fact

that at present opto-manual systems develop rather rapidly.

To-day the determinations of volume can be realized rationally ac- cording to the principles of Delesse and G-lagoleff. Surface determinations are based mainly on the Rosival-principle . In volume and

surface determinations the measuring procedures are reduced to simple counting procedures. Iri both cases test plates are used.

The simplest vja;/ to transm.it the counting results is a transmission of the data by hand to a key board which is joined with an adequate memory unit.

At the present time, no suitable appliance which would render the

human eye superflous for automatic picture analysis exists. Aware

of this fact, an instrument (MOP, Manual Optical Picture Analyser)

was developed, which allows a simpler and more efficient way for

conventional picture analyses (Fig.l). The principle of this semiautomatic picture analysing instrument consists therein, that elec193
H. P. Pohr

trie current is carried through the test lines which are placed

upon a plexiaiass plate. By every contact of a special pencil vjith

these lines, an impulse is released which is registered by a storage unit. How are vol'ume and surface densities of certain structures determined vrith this opto-manual system ?

To determine a distinct point density for th^ calculation of the

correspojiding volume density, the intercepts lying within the structure are scanned with the contact pen. At each crossing with a testline (= test point in conventional couiiting) an impulse is re- leased (Pig. 2 )

.

For the calculation of the surface density the nujnber of intersections (I^) per test plate can be determined by tracing the contours

of the structure profiles to be analysed with the contact pen (Fig.

3

The number of structure profiles per test area in order to calculate the numerical density is realized by another pen. The setting

on of this mechanical pen to each structure profile releases an impulse

.

Finally this MOP-system can be used as simple manual coimter.

The counting data, or basic values, of up to 24 different structujres can be accumulated and stored for each sample. An outprint

of the data is possible. With adequate small computers these miprocessed data can be converted into the according volume, surface or numerical densities with regard to the magnification and

the characteristics of the test plate. Furthermore, particle size

distributions can also be performed v;ith this system. If the optonaiiual system is used in combination with a calculator, it is e.g.

possible to accumulate the results of a preliminary evaluation on magnetic cards or another data carrier.

jivery structure definable by the human eye is accessible by aai image aiialysis with consideration of the premises given bj^ the

stereological principles. In image analyses with optomanual sjstems depending on the principles of stereology, the contact test

plates can be adjusted to the structures to be analysed. The lines

of the contact test plates can be arranged very closely. Therefore,

the average va>,lues of the samples offer a greater statistical se- curity. Marking of the analysed structures is possible.

194
A new opto-manual semi-automatic evaluation system

Sunoarizing we have with this instrument a very simple picture

analj'sing system at hand, which allows a very fast morphometric

evaluation of structures at light and electron microscopic level.

Fig. 1 : I-Ianual optical sample analyser (Kontron Messgerate GmbH,

Munich, Germany)

1: Electronic count, memory and display unit

2: Desk computer

5: Illumination box with incorporated test plate

4: Light and mechanical pen

195

National Bureau oi Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

AUTOMATED MEASUREMENTS FOR DETERMINATION OF THE STATISTICAL NATURE OF THE

DISTRIBUTION OF INCLUSIONS IN STEEL

by George A. Moore

National Bureau of Standards. Washington, D. C. 20234. U.S. A

An ASTM committee [1] is composing a recommended practice for quality

control inspection of metals with respect to inclusion content. This must include minimum inspection requirements together with methods of determining

when further inspection is necessary. Knowledge of the statistical nature of

the distribution of field measurements is therefore necessary. Data were published by Allmand and Coleman [2] for concentration Vy and size classified

island counts for 1800 fields on each of three steel specimens. These data

exhibit an unexpectedly high coefficient of variation and extreme abnormality

as compared with either a Gaussian distribution or a positive half Gaussian.

High readings are several times as abundant as expected in a normal distribution. In the presence of this abnormality, it cannot be stated that their

^ three specimens are statistically different. It was thus estimated that 10

to 10^ fields must be measured on specimens of one steel to determine the true nature of the distribution. High level statistical analysis is required to estimate the inflation ratio to be applied to the confidence limits.

A task of this magnitude requires automation at the computer-assisted-by- man level. Measurement acquisition and data processing are required to proceed

automatically without continuous supervision. We are operating a quantitative

television microscope assembly under full control of a mini-computer. A software program of over 25,000 computer words in BASIC operates in several

sections. The system first instructs the operator in validating magnification,

setting up the specimen, and setting necessary machine controls. A test exercise determines that all parts of the system are operational and that the area threshold has in fact been set at that level where addition or removal

of edge delineation does not significantly alter the area measurement. Necessary identification and control information is then demanded and supplied

by the operator. The system then proceeds to automatic measurements of 10-^ fields on the specimen. Protection from off-specimen positioning and damaged

or dirty fields is provided by an area magnitude check, with a decision required

from the operator if a new area maximum is higher than reasonably anticipated.

Analysis is made in sets of 30 to 100 fields forming a coherent band along

the longitudinal direction of the specimen. The first four powers of each

measured parameter are accumulated in both set and specimen stores. The class

size of each measurement is tallied. Two statistical analyses are made at the end of each set. The first examines the data for agreement with a mode zero distribution of positive half Gaussian shape. Normal, RMS, and 4th root of 4th moment averages are shown. The 2nd, 3rd, and 4th moments are listed and the kurtosis computed. The ratio of this kurtosis to the value, 3, for a half Gaussian is a measure of the lack of conformity to this form of distribution.

For the second analysis the four moments are recomputed about the mean.

The statistical parameters usually determined at this level of analysis are

computed. The confidence ranges about the mean are known to require inflation

whenever the coefficient of variation (CV) is excessive or the distribution

abnormal. The confidence limit (CL) value is arbitrarily flagged as unreliable

if the CV exceeds 0.22 and a suggested maximum CL computed from the kurtosis.

At the end of the final set similar statistical analyses are made for the observations over the whole specimen. The standard deviations of the RMS

average and of the standard error of the mean are determined, permitting a new estimate of the inflation factor for the confidence limit. An analysis is also

made for the distribution of set means about the grand mean. The distribution

of set means can be expected to be more nearly normal than the distribution of

obseirvations, and therefore to yield a more reliable estimate of the true CL.

197
G. A. Moore

The final data for each specimen are systematically stored on magnetic

tape files. These can be recalled and merged to yield a combined analysis for several surfaces of the same specimen or for all the specimens from one heat.

A logical decision has been made that total inclusion concentration,

~ transverse projection value of surface to volume ratio, Sy', should be functionally related to the mechanical behavior of steel. Sy' is the stereo logical equivalent of totaled feature length in a standard field area but is computed directly from the intercept count. This count is made at a threshold higher than the proper setting for area determination, but below the snow level. This threshold is normally set at a level where addition of edge

delineation does not significantly increase the intercept count.

As no functional relationships are presently known between measurements

of individual inclusion islands and mechanical behavior of the heat, the additional machine time to mteasure individual features does not appear justified

for quality control applications. The research system does, however, include

total feature counts at each threshold level and a count of large inclusions

more than 20 ym long. Computations to reveal possible dependence of measurements on a Poissonian distribution of inclusion particles are included in the final specimen analysis. Maximum values of each field measurement within each

set are retained and an attempt is being made to find a relation of these to the probable maxima for the material.

The material being measured, a heat of steel intended for wire drawing

material, contains small dark isotropic particles presumably of indigenous

origin, together with lighter stringers presumed exogenous. These stringers

are of duplex structure, frequently incorporating the small particles at their surface. As the two types presumably cooperate in determining mechanical properties, no attempt is currently being made to measure them separately. No evidence of a bimodal distribution has been seen in the histogram tables.

The present research system is memory bound in the data acquisition program

section and computation bound in the rate of processing incoming data. Hence,

advantage is taken of the averaged scan mode for higher precision. About 225 fields are measured per hour, requiring over five hours per specimen. While

acceptable for research purposes, this rate is not acceptable in quality control

applications. Scanners exist which operate at about 500 fields per minute.

These could be fitted with parallel high-speed computation systems to accomplish

whatever portion of the present computations may prove necessary or useful.

Presently available data are only preliminary. Analysis of the first

12,000 fields (120 sets) of the test steel confirms a high coefficient of variation between fields for all five measurements. The distribution apparently is much less uniform than Poissonian. High positive skew values are found. The kurtosis is an order of magnitude higher than normal. Thus confidence limits

computed on the assumption of normality could be even an order of magnitude too small! Fortunately, the distribution of averages for sets of 100 fields does appear to approach normality. Analysis by sets apparently will be acceptable

in control practice. Confidence ranges computed from set averages are 2.5 to 6 times wider than those computed from the observations as a whole. Thus results

based on traditional examination of 100 to 300 fields do not characterize the material. Data on ten sets taken from different specimens may be sufficient to determine if a material is close to an imposed control limit. Measurement of

30 to 100 sets may be necessary for critical decisions.

REFERENCES

1. Am. Soc. for Testing and Materials, Committee E4 on Metallography, Task

Group 14.04 (W. D. Forgeng, Jr., Chm.).

2. T. R. Allmand and D. S. Coleman, The Microscope 20, 1 (Jan. 1972) 57-81.

198
4. THREE-DIMENSIONAL RECONSTRUCTION

Chairpers ons :

R. Gordon

L. D. Peachy

R. Moore

199

National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

RECONSTRUCTION FROM PROJECTIONS: A SURVEY OF APPLICATIONS

by Richard Gordon

I Image Processing Unit, National Cancer Institute, National Institutes of Health, Building 36, Room 4D28

Bethesda, Maryland 20014, U.S.A.

i

. .

^

The problem of reconstructing a function in n dimensions

from its projections to a lower dimensional space arises

in many fields, from astronomy to electron microscopy. The

most active application is in medicine, in which the new

x-ray scanners are revolutionizing the practice of radiology

by providing clear cross sections of the brain and body.

Nuclear medicine is also making advances in the reconstruction

of the distribution of emitting radioisotopes. In astronomy

there is a need to obtain good two dimensional images from

the whole spectrum of waves and particles using instruments

which record strip integrals. Within our own solar system

three dimensional features may be reconstructed, such as the solar corona, the planetary atmospheres, and the earth's

ionosphere. Reconstruction from projections in electron

microscopy is providing low resolution structures of biological

macromolecules , promising to become as important as x-ray

crystallography in molecular biology. There is a whole unex-- plored range of applications to light microscopy.

REFERENCES

A. The following is a good current sampling of the range

of interdisciplinary activity in reconstruction from projections

and also includes a bibliography of the whole field:

1. Digest and Supplement of Technical Papers, Topical

Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections: Theory and Practice in

Medicine and the Physical Sciences, 4-7 August,

Stanford University, chairman R. Gordon. Optical

Society of America, Washington, D.C.

B. The following collections of papers are also valuable:

1. Tomographic Imaging in Nuclear Medicine, Soc. Nuclear

Medicine, New York, 1973.

2. IEEE Trans. Nucl. Sci . , volume NS-21, Number 3 , 1974 .

3. Techniques of Three-Dimensional Reconstruction,

Brookhaven National Laboratory, July 15-19, 1974,

BNL 20425.

4. Workshop on Reconstruction Tomograph in Diagnostic

Radiology and Nuclear Medicine, San Juan, Puerto Rico,

17-19 April, 1974, New York University Park Press,

in press.

201
/?. pardon

C. The following are reviews of the algorithms and the whole

field:

1. R. Gordon & G. T. Herman (1974) . Three dimensional

reconstruction from projections: A review of algorithms.

Int. Rev. Cytol. 38, 111-151.

2. R. Gordon (1976) A Treatise on Reconstruction

from Projections. Plenum Press, New York, in

preparation.

202
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

MORPHOMETRIC ANALYSIS OF NEURONS IN DIFFERENT DEPTHS OF THE CAT'S BRAIN CORTEX AFTER HYPOXIA

by 0. Hunziker,* U. Schuiz,* Ch. Walliser* and J. Serra** *Basic Medical Research Department, Sandoz Ltd.. CH-4002 Basel, Switzerland **Centre de morphologie mathematique, ecole des Mines de Paris, F-77303 Fontainebleau, France

A morphometrical method is described, which allows to determine size and

form of neurons (opening-procedure) in the cat's sensori-motor cortex with the Texture Analyzer of Leitz. The data obtained from control animals are compared

with those of cats having resided a 3 weeks period at high altitude. For this reason the opening values were statistically treated with the multivariate

analysis of Benzecri, which yields the classification of neurons in different

depths of the cerebral cortex.

INTRODUCTION

The aim of this study is to investigate quantitatively neuronal cells of

the cat's cerebral cortex using light microscopy. During morphometric investigations of this kind many problems arise from the great variety of shape ( Gihr , 1963) and size (Colon & Smit , 1970; Haug , 1967, 1972; Ramon - Moliner , 1961; Sholl , 1967) of cortical neurons. Thus, the Texture Analyzer (TAS) of Leitz

( Serra & >fuller , 1973) as an instrument of optical-electronic image-analysis

facilitates considerably quantitative histological studies on the neuronal

system. In order to demonstrate another object of the study and the technique

of the TAS, neurons under normal conditions were compared with experimentally

altered cells.

EXPERIMENTAL AND MORPHOMETRIC PROCEDURE

The experimental part of our investigation was performed as follows: 3 male cats were kept during 3 weeks in a conditioned low-pressure chamber and exposed to a simulated high altitude of 5025 m as previously described ( Hunziker

et al., 1975). During the same time 3 other cats remained as controls at 250 m

altitude in a standard laboratory. At the end of the experiment all animals were deeply anaesthetised with Pentobarbital and the brains perfused with 2.5% glutaraldehyde in phosphate buffer (pH 7.4). The neurons of the temporo-parietal Suprasylvian Gyrus were demonstrated by a Nissl-staining with GallocyaninChromalum (Fig. 1) and subsequently measured with the TAS (magnif. factor of

the objective: 40x) . For the investigation 9 serial frozen sections (thickness

= lAy) per brain region and animal were randomly selected. In each section 3 measuring directions were localised, the directions being identical for all sections. For each direction the scanning stage of the TAS was preprogrammed

to start from the pial surface and to reach the white matter by steps of 50iJ. The total number of fields per measuring direction was 30, covering a total

thickness of ISOOy, I.e., approximately the effective thickness of the temporoparietal cortex. Using an electronically set measuring field (mask) of the TAS,

1 neuron, of which the nucleolus was visible in the section ( Haug , 1972) , was selected from each measuring field. The optical input system of the TAS is a microscope combined with a black-and-white television camera. The theoretical

background involved in the TAS is mathematical morphology and the hit or miss

transformation ( Matheron , 1975). In the following investigation we particularly

used the concept of isotropic opening, defined as follows. Let us consider a circle with a given radius r (structuring element) moving everywhere inside the neuron and satisfying the condition, not to cut the pericaryon border. The set covered by the circle when sweeping over the neuron is called its morphological

opening. The smaller the radius r is, the clearer are the details of the

shape, which the circle can reach. When r increases, the opening area decreases

to zero for r > z^^^ ("^max ~ radius of the circle within the neuron) . For

i^max the neuron shrinks progressively (continuous part of the opening) , and

the remaining part vanishes at one for r = r^^^^ (discontinuous part). Experimentally, a hexagon is sufficient for our purpose. A digital hexagonal frame

can produce very small regular hexagons. Here the elementary step (side of the

203
0. Hunziker, U. Schuiz, Ch. Wa/liser and J. Serra

Fig. 1 : Neurons in the upper layers Fig. 2 : Opening curves of a triangular

of the cat's cerebral cortex. and a circular neuron. Controls. Gallocyanin-Chro- ^2 /J malum (magnif. 230x) . = Pp

hexagon) values a = 0.33p. It was enlarged, step by step, to 6a and by increasments of 2 to 32 a, which represent 18 classes of opening increments. Thus the data are a matrix of 72 x 30 elements (18 opening increments

X 4 cats = lines, 30 depths = columns). As an example we plotted 2 curves of opening increment area (Fig. 2) for 2 different neurons. They differ: i) by

their size or integral of the curve, ii) by the ratio of the discontinuous part

over the total area, or circulatory factor , which is for the triangular neuron

= 0.51 and for the circular one = 0.77, iii) by the distribution of the continuous part (opening steps 16-32 resp. 16-24). All these properties can be quantified more precisely ( Serra , 1975). Here they are evaluated by a multivariate analysis treatment. The key point is, that a shape is characterized not by a finite number of parameters, but by a function.

STATISTICAL TREATMENT

There are several methods of multivariate analysis. The method of choice

was the correspondence analysis theory of J. P. Benzecri, of which the advantage

is to provide results independent from the selected opening classes: If 2 points representing 2 steps of openings are close together in the main factor

subspaces, the 2 corresponding classes may be regrouped in a single one '(they are proportional) without disturbing the remaining constellation of points. Fig.

3, 4 and 5 show the first factorial plane (axis Nr. 1 and 2). Both depth-points

(stars) and openings (circled points) are simultaneously represented. The first

2 axes explain 45% of the total inertia of the point cloud (resp. 28 and 17%).

Their meaning is provided by the projection of the points upon them. The closer

they are to the origin, the less significant they are. Considering the opening

points up to Nr. 9, they may be assumed non-significant on the first 2 axes. After Nr. 9 their first axis coordinate increases systematically, showing that this axis classifies the neurons according to their size. The second axis opposes 2 phenomena: the medium sized openings (10, 11, 12) from the larger ones

(14, 15), and the large openings (17) from the largest one (18). The first

opposition can be explained by the form factor. This interpretation of both axes

204
Analysis of neurons in the cat's brain cortex

a

O' A triangular T form

Oi~control

I>-,^hypOJiiC ®

MULTIVARIATE

ANALYSIS o

9 o. o ® targe neurons —

Fig . 3 : Neurons

of layer 1-4. (cortex thickness

50-200y).

-^t— small jj. neurons '

*

^

ci rcular ®

®

®

A triangular ^ form @

©

small „ (T) neurons [})

(14) p i

control "tfr ^hypoxic

large neurons

circular

1 form (15) o

opening C18)

Fig. 4: rJeurons of layer 5-10. (cortex thickness: 250-500y),

^^hypoxic

(10

control

small neurons —O ^ ® _ ® ^ X)

triangular form

®

®

large neurons

(14 (15)-o

circular form

Fig. 5: Neurons of layer 25-30. (cortex thickness: 1250-1500)i)

.

205
0. Hunziker, U. Schuiz, Ch. Walliser and J. Serra

allows the differentiation between certain groups of measuring fields, depending

on their depths 1 to 30 from the pial surface. Hence it helps to distinguish

certain layers of the cerebral cortex. With reference to the depth of the mea- suring fields, we decomposed the diagram into 4 parts. Fig. 3 contains 2 parts:

In the superficial layers 1 and 2 the neurons of the hypoxic cats have a more spherical shape, but do not seem to be larger. The layers 3 and 4 beyond exhibit

a predominance of larger neurons, all having a similar shape. In the depth of 250-500y (Fig. 4) the neurons of the hypoxic animals are enlarged, without changing their shape. It is interesting to notice that the stressed neurons seem to resemble each other more than the normal neurons do. In the layers of 550-1200y

no apparent differences are observed. From 1250-1500y (Fig. 5) the neurons of all animals are smaller than in the layers of 250-500vi. Between the normal and

the hypoxic animals there is no difference in size, whereas there is one in their shape. These neurons appear to be more circular. The presented morphometrical method allows to determine quantitatively neuronal changes after longer

residence of animals at high altitude.

REFERENCES

Colon, E. J. &Sinit,G.T. (1970) The organization of the cerebral cortex. Quantitative aspects. Acta Morph.Neerl.Scand.2, 309-319. Gihr.M. (1963) Rekonstruktion von Nervenzellen . Int

.

Congr.F.Stereologie 18, 1-37. Haug,H. (1967) Probleme und Methoden der Strukturzahlung im Schnit tpraparat , in:Quantitative

Methods in Morphology (Ed. by E. Weibel) , pp

.

57-78, Springer-Verlap, Berlin-HeidelbergNew York. Haug.H. (1972) Stereological methods in the analysis of neuronal parameters in the central nervous system. J . Microsc . 95 , 165-180

.

Hunziker, 0. , Schuiz , U. , Vet eau, M. J . ,Weihe,W.H. and Meier-Ruge,W. (1975) Morphometric capillary parameters in the cat's brain cortex after

chronic hypoxia. In:Quantitative Analysis of Microstructures in Medicine, Biology and Materials Development (Ed. by H.E.Exner),

pp. 168-176, Dr. Riederer-Verlag, Stuttgart

.

Matheron,G. (1975) Random sets and integral geometry .Wiley- In terse iences , New York. Ramon-Moliner,E. (1961) The histology of Che post- cruciate gyrus in the cat. I. Quantitative

studies. J . Comp . Neurol

.

117 ,4 3-62. Serra, J. (1975) Mathematical morphology and image analysis. SIAM, Washington D.C. Serra, J. &Muller,W. (1973) The Leitz TAS . LeiczMitt.Wiss.Tech.Suppl.J.,4. Sholl.D.A. (1967) The organization of the cerebral

cortex. Hafner Publ. Company, New YorkLondon.

206
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

THREE-DIMENSIONAL RECONSTRUCTION FROM SERIAL HIGH VOLTAGE ELECTRON MICROGRAPHS

by Lee D. Peachey, Caroline H. Damsky and Arthur Veen

Department of Biology, University of Pennsylvania, Philadelphia, Pennsylvania 19174 and Department of

Molecular, Cellular and Developmental Biology, University of Colorado, Boulder, Colorado 80302, U.S.A.

nraRODucTicw

The process of three-dimensional reconstruction from a series of two-dimensional profiles, as in the case of serial slices of biological tissue exam- ined microscopically, is simple and straightforward in principle. In practice,

however, it can be tedious and, if not done accurately, the results can be misleading. Because of the great potential usefulness of three-dimensional recon- struction of fine details of cellular structures from relatively thick serial

slices of biological tissue using high voltage electron microscopy (HVEM) with

accelerating voltages of about 1 MV, we decided to reconsider the methods that

could be used in such a reconstruction process. Our hope was to improve the reliability sind accuracy of the process, to increase its versatility in terms

of the kinds of structures that could be reconstructed, and to improve the rate

of reconstruction while reducing its tediousness. Our approach has been to con- sider each step in the process as separate from the others and subject to its own improvements, and to consider the use of digital computers to replace

hi iman operations whenever the replacement really made sense. Our goal was not

a fully automated system. We felt that such a system, even if it could be produced, probably would not have reasonable accessibility by many workers because

of its complexity and cost. We also felt that a highly automated system would

be likely to have less versatility than a system with a greater degree of human

involvement at certain critical steps in the process of generating the reconstruction. Therefore, the system we have evolved can be considered as a result

of a series of compromises between human and machine operations.

CHARACTERISTICS OF HVEM IMAGES HVEM provides images that have a high degree of lateral resolution com- pared to the thickness of the specimens. Lateral resolution of the order of

10 na. or less can be obtained with specimens of the order of 1 jam. in thickness, with the entire thickness of the specimen in focus in the imeige. Thus

the positions of structures or boundaries of structures, when oriented favorably, can be determined very accurately in two dimensions, but information on

the third dimension is lost in single images. Pairs of images, taken with

the specimen tilted an appropriate aaount between exposures, can be viewed

stereoscopioally, and this often provides considerably ia^jroved visualization

of the third dimension. However, the greatest extent of three-dimensional

information can be obtained with multiple images of serial slices. IHiese can be used even when the structure of interest is too large to be contained within a single slice. In this case, reconstruction of the original struciiaa'e from a series of micTOgraphs becomes a necessity.

RECONSTRUCTION

Reconstruction from serial micrographs is, in essence, a form of re- assembly of a microscopic structure that previously was disassembled when it

was cut into a series of slices. If the assembly is to be valid, then

information of how the structure original 1 y was arranged in three dimensions must be retained or regained as the reconstruction is done. A common form

for such reconstructions is the real physical model. These can be built from

a series of slabs of plastic or other material, with each slab cut out to represent the profiles of tJie structures in one slice or micrograph. If the

slab thickness is chosen correctly in relation to both the slice thickness

and the magnification of the micrograph, and if the slabs are aligned properly

one upon the next, then the model can be a rather accurate and useful representation of the original structure.

207
L. p. Peachey, C. H. Damsky and A. Veen

STEPS IN RECONSTRUCTION

Assuming that one has an adequate set of micrographs of knovn magnification from a set of consecutive serial slices of known thickness, the proc- ess of reconstruction can be divided into a sequence of steps, as follows:

1. Recognizing and selecting in each micrograph profiles of the structures to be included in the reconstruction.

2. Obtaining data on the relative locations of points along the boundaries of the chosen profiles.

3. Aligning the profile data for each slice in relation to adja- cent slices to restore the original spatial relationships, which

were distorted when the slices were cut and separated.

k. Generating a representation of the complete set of profiles

in three dimensions as a physical model, artist's drawing, or

coii5)uter graphic display.

RECOGNITION, SELECTION, AND OBTAINING PROFILE DATA

We have chosen to do steps 1 and 2 visually and manually, rather than

to attempt to develop an automated optical scanning and pattern recognition

system. Using a tracing stylus linked thro\igh a backlash-free cable system

to a pair of high-resolution potentiometers, the operator traces around the selected profiles. Voltages from the potentiometers, proportional to polar

position of the stylus, are converted automatically to digital values and

fed to a PDP-8/l computer. Significant movements of the stylus, usually

greater than about 0.3 mm., are coded and punched on paper tape. Information on magnification, locations of calibration points, number of slices, a title for the reconstruction, etc. axe input to the computer from a tele- type and transferred to the paper tape along with the profile data. Step 2 is completed when this paper tape is read into a PDP-6 computer and the data

are converted into a series of points in rectangular coordinates and aire packed efficiently into 36-bit words on a magnetic storage device. The data

also are scaled to fit within a square of unit size, so that all X and Y

coordinates are in the range 0.0 to 1.0.

ALIGNMENT OF DATA FRCM SUCCESSIVE SLICES An important part of the reconstruction process, and one apparently not often considered fully, is editing of the data to achieve valid alignment of each slice with respect to the next slice. This must be done objectively and

with information not biased incorrectly by any assuii5)tion of what the struc- ture originally was like. For this pu2Tpose, we select certain features of the microscopic images that are external to the features being reconstructed and

that can be expected to maintain positional and/or rotational alignment from

one micrograph to the next. These include spherical objects cut near their

centers and thin filamentous or tubular structures cut exactly transversely,

lines drawn between such structures lying at some distance from and on oppo- site sides of the area being reconstructed are traced, and the data on their

positions and orientations are carried along with information on the recon- struction profiles and fiduciary profiles in the data set.

A program called EDIT is used for visual presentation on an oscilloscope

terminal of the data for individual slices or pairs of slices before, during,

and after editing for alignment. The operator uses a set of teletype com- mands for specifying translation and/or rotation of the data set of one slice

with respect to that of the next slice. Alternately, the criteria for align- ment caji be specified, e.g. two lines should be parallel, two closed profiles

should be superimposed, etc., and the cong)uter calculates and carries out the rotation and translation required to meet the specified criteria as well as possible. Sizes of individual profiles, or of the whole slice, are changed

either isotropically or anisotropically, when such adjustment is specified

and can be justified. The number of data points can be reduced, for efficiency of storage, and profiles used for alignment but not wanted in the final

reconstruction can be eliminated from the data set. When the same operations

208
Three-dimensional reconstruction from serial high voltage electron micrographs

are to be done on a series of more than two slices, global commands can be

used. The restilt of the EDIT operations is a new data set in which the data

for successive slices have been recomputed for best alignment from each slice

to the next, and step 3 above has been con^jleted.

THREE-DIMENSIOML DISPIAY

Step k starts with a conversion of the data for each slice into a tabu- lation of intersections of profiles with a set of about 200 evenly spaced,

parallel lines in the plane of the slice, by a program called DIGIT. The output of this program then is displayed in three dimensions by a program

called FIG3D« The display, on an oscilloscope terminal or hard copy device,

is presented in proper perspective, from any viewing angle in the forward

hemisphere, and from any viewing distance.

The three-dimensional appearance of the final display is enhanced by

a number of features, some optional. Hidden lines are not displayed, so the reconstruction appears solid rather than transparent. Several lines can be

drawn for each slice, giving it a slab-like appearance. Each slab has an apparent thickness in proportion to the individual slice thickness and to the magnification and perspective of the display. A calibrated box can be displayed enclosing the reconstruction. Two displays, presented at slightly different viewing angles, can be viewed stereoscopically in three

dimensions.

SUMMARYWe have described a method for generating three-dimensional representations of structures reassembled from a set of micrographs of serial slices

of the original structures. A combination of human and machine operations

has been selected for each of the sequential steps in making the reconstruction, and particular care has been taken to achieve valid and accurate re- alignment of profiles in successive slices. This computer based procedure

is more rapid than construction of a real model, and is likely to be more

precise in the key step of aligning successive slices in the reconstruction

to produce a valid representation of the original structure.

ACKNOWLEDOffiNT

This work was supported by grants from the National Institutes of Health,

Division of Research Resources (No. RR-I5 and RR-592), the National Heart and

lung Institute (Pennsylvania Muscle Institute, HL-I5835), and the Muscular

Dystrophy Associations of America, Inc. (Henry M. Watts Neuromuscular Disease

Center, Philadelphia)

.

FIGURES

The figures show various stages in the reconstruction of yeast cell

mitochondria in two consecutive slices about O.h yaa.. thick from a series

of HVEM micrographs. The top four panels show HVEM micrographs of the two

slices, with fiduciary lines from distant objects drawn on the micrograph,

and the two corresponding data sets as displayed by the EDIT program (EDIT

supplies the square enclosing the profile data, and the lettering was added

by an artist) . Each display shows three mitochondrial profiles as well as a

tracing of the large vacuole in the cell, and one or two fiduciary lines.

The bottom four panels show stages in the editing and display process. Above at the left is the xmedited display of both slices superln5)osed. To

the right is the display after editing using one line and the vacuole for

alignment. Below on the left is the display from EDIT after all fiduciary

lines and profiles have been removed. At the bottom right is a three-dimensional display of the two slices after editing. In this display, only the

slab effect ajid the hidden lines routine have been used.

209
L. D. Peachey, C. H. Damsky and A. Veen

YCEL6

SLICE 6

210
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

DETERMINING THREE-DIMENSIONAL INTRACELLULAR STRUCTURE DATA FROM SCANNING PROTON MICRO- SCOPY LUMINESCENCE DATA

by Daniel G. Oldfield Department of Biological Sciences. De Paul University, Chicago, Illinois 60614, U.S.A.

INTRODUCTION

The spatial resolution presently attainable in cellular electron microscopy has improved visualization of specimen detail one or two orders of magnitude beyond the limits allowed by optical microscopy. However, the chemical resolution attained in electron microscopy—the ability to distinguish

various chemical species that may be present in the spatially resolved region

falls distinctly short of that permitted by optical microscopy (e.g., utilizing

cytochemlcal techniques)

.

Advances in scanning electron microscopy and, more particularly, the advent of scanning proton microscopy (Levi-Setti, 197A) prompt one to consider

the possibility of using small-diameter, minimally scattered charged-particle

beams to excite luminescence in small, beam-delimited regions of cells because

luminescence spectra so produced may provide at least a partial chemical

characterization of the cell region penetrated by the beam.

The purpose of this paper is (1) to formulate a basic theoretical model

for the use of heavy charged particles such as protons to excite luminescence

in localized regions of intact cells or organelles; and (2) to examine factors which affect the spatial resolution attainable using particle-induced lumi- nescence.

ANALYTIC BASIS OF THE METHOD

Beam and spearmen speoificatians

Consider an initially monoenergetic, monodirectional charged-particle beam

propagating in the direction of a positive z-axis fixed in space. If we

position the specimen in the beam, the intersection of the beam with the speci- men surface defines a point in space. The location of this point relative to a coordinate system fixed in the specimen is assumed to be determinable from the

(known) shape of the specimen via transformation equations such as Hilliard

(1972) has investigated for the analysis of SEM images. More generally, we are assuming that, given the coordinates x,y,z of any point in the specimen ex- pressed in the space coordinate system, we can determine via transformation

equations the coordinates of the same point expressed in the coordinate system

fixed in the specimen.

We limit consideration to dried cellular or organellar specimens having an average linear dimension of less than 100 microns. Assuming unit density for

the specimen, its average thickness would be 10 mg/cm^.

We asscnne that the specimen contains a luminor which is either an exogenously added substance known to specifically bind to, or be sequestered by, the specimen or an endogenous substance which is chemically extractible from the

cell. For example, the luminor might be one of the oxdiazoles used in fluorometric Feulgen techniques for the determination of chromosomal desoxyribonucleoprotein (Ruch, 1966); or the luminor might be a naturally occurring

nucleotide or protein or pigment such as riboflavin, globulin, or carotene

(Udenfriend, 1969).

Using the transformation equations above, the luminor concentration can

be expressed in the space variables x,y,z as C(x,y,z).

211
D. G. OldfieId

Interaotion specifications

Following the development employed by Fano (1963), we define: p(E)dE, mean pathlength (cm) of particles with energy in dE at E; X^j (E) , cross- section (cm^) for interaction (by inelastic collision) exciting an atom of type A to energy level Ej above its ground state per particle of energy E.

The function p(E) is a probability density function for mean pathlength.

Its integral over the energy interval 0 to Eq is the range of a particle

with initial energy E . If straggling is neglected, Pa(E) is simply the reciprocal of the specific energy loss (stopping power; -dE^/ds. Therefore,

assuming additivity for the various types of atoms present and summing, we

write for the overall interaction in the specimen -dE/ds = l/p(E).

Thus, p(E) can be estimated experimentally from specific energy loss measurements provided that data for low energy particles (0 to 10 mev per

nucleon) and thin specimens (0.001 to 10 mg/cm^) are obtainable.- Some progress

in this area has been reported (Jung, 1967; Dennis, 1972; Watt, 1972; Johnson,

et^ ad. , 1973). Also, theoretical efforts are continuing (Inokuti, 1971; Massey & Gilbody, 1974).

ANALYSIS OF LUMINESCENCE PRODUCTION

General

Energy is transferred from a beam particle to all atoms along the track

of the particle; but luminescence will occur only for energy transfers to atoms from which absorbed energy may migrate to the Iximinor molecule. Therefore, luminescence from a specimen will depend on: Itiminor concentration

C(x,y,2), the cross-section X(E) (crn^) for luminor excitation by a particle

of energy E, and the quantum yield q(k,l,E)dkdw of luminescent photons with

wave number k emitted into the solid angle w in direction 1 per liminor

molecule. Noting that X(E)p(E)dE is the average differential volume within

which excitation occurs, the luminescence produced by a particle which stops

in the specimen is determined by the (spectral and spatial) density function

Photon counting systems for luminescence detection are available in the wavelength range 400-700 nm.

Test and reference specimens

It is convenient to refer the analysis of the biological test specimen to luminescence parameters measured in an atomically similar but molecularly

simpler non-biological reference specimen containing a known uniform concentration of the luminor. For example, thin sheets of polystyrene or polyvinyltoluene containing the luminor might be suitable reference specimens. Thus, we

define a structure factor S = Xq/X^q , where the subscript e refers to the reference specimen. If the differential Ixtminescence from reference specimens

of known thickness and known luminor concentration are measured for various

initial beam energies and with only small energy losses per particle, the quantity Xg(E)qg(k,l,E) can be determined from the density equation (1). Also,

since the dependence of this quantity on reference specimen thickness is known, the quantity Xg(z)q (k,l,z) can be found (the independent variable now being

the penetration depth z into the specimen rather than the particle energy E)

.

E

(1)

212
On determining structure data from scanning proton microscopy luminescence

Finally, defining C, (z) as the average concentration of luminor over the cross-section of the Beam, the density function determining luminescence in

the test specimen is

G(k,l,R) = J

^ C^^(z)Xg(z)qg(k,l,z)S(k,l,z)dz, (2)

where R is the projected range.

SPATIAL RESOLUTION

Consider two beam particles with initial energies E' and E" which are successively incident on a test specimen and penetrate to projected ranges

R' and R". The difference in density functions is g = G(k,l,R")-G(k,l,R')

.

If R" - R' = r is very small, the functions within the integrals will change

only negligibly over this range of z. We then obtain an equation for the net

change in density function over the distance r in the specimen:

g(k,l,R) = rC^(R)X^(R)q^(k,l,R)S(k,l,R) ; R = 1/2 (R'+R") . (3)

We see from equation (3) that g(k,l,R) will refer to luminescence photons

produced only in the voltame b^r at z=R provided that the time-averaged photon

spectrum produced from z=0 to z=R' by beam particles with initial energy E' is exactly the same as that produced over the same distance by beam particles with

initial energy E^. We expect this would be true (Curran, 1953).

We conclude therefore that in the absence of range straggling and multiple

scattering, the spatial resolution limits fo_r determining the basic specimen

characteristics of luminor concentration C (R) and structure factor S(k,l,R2)

would be given by the range difference r along the beam axis and the beam area

b^ perpendicular to it. Neglecting practical considerations of instrumental

sensitivity and stability, specimen heating and damage, luminescence lifetimes

and attenuation in the specimen, r could in theory be made arbitrarily small,

and the beam area reduced to the cross-section of a single track (as determined

by impact parameter and lowest excited state considerations) . The irreducible

stochastic limitations on resolution would then be determined (1) along the z-axis, by range straggling; (2) in the x,y plane, by multiple elastic scattering.

For protons having an initial energy of a few mev, the increase in penetration due to range straggling would be estimated to be a few percent of the range; but the increase in beam cross-sectional width would be estimated to be

several tens of percent of the range (Fano, 1963). Therefore, for test speci- mens (such as organelles or flattened cells) in which moderate penetration

depths are sufficient for analysis, the resolution obtainable along the z-axis

would exceed that of optical microscopy for penetrations of 5 microns or less; for the equivalent resolution laterally, penetration depths would have to be restricted to less than 1 micron. It is possible, however, that these limitations could be eased by the use of stereographic image enhancement, pattern

recognition, or other techniques.

STEREOLOGIC ASPECTS

It la evident that the use of charged-particle beams at several discrete

energies in the manner described here constitutes a statistical sampling of specimen structure along the z-axis to which standard stereologic methods could

be applied. When heavy-charged-partlcle microscopy has been further developed,

stereologic analysis will undoubtedly be called upon to more rationally interpret the data.

213
D. G. Oldfield

REFERENCES

Curran, S. C. (1953) Lwninesoenoe & the Saintillation Counter. Academic Press, New York, p. 104.

Dennis, J. A. (1972) Interaction of low energy protons with ribonuclease.

Physios in Med. & Biol. 17, 304.

Fano, U. (1963) Penetration of protons, alpha particles, and mesons.

Ann. Rev. Nuclear Science, 13, 1.

Hilliard, J. E. (1972) Quantitative analysis of scanning electron

micrographs. J. Microscopy 95, 45.

Inokuti, M. (1971) Inelastic collisions of fast charged particles with

atoms & molecules—the Bethe theory revisited. Rev. Mod. Rhys., 43, 297.

Johnson, R, E. , Trevisani, E. T. 6> Harberger, J. H. (1973) Charge transfer

in bio-molecular damage by low energy protons. Physics in Med. & Biol.,

18, 287.

Jung, H. (1967) Inactivatlon of ribonuclease by elastic nuclear collisons.

Rad. Res. Suppl. , 7, 64.

Levi-Setti, R. (1974) Proton scanning microscopy: feasibility and promise.

Proceedings of the Seventh Annual Scanning Electron Microscopy Symposium,

pp. 125-134, IIT Research Institute, Chicago.

Massey, H. S. W. & Gilbody, H. B. (1974) Electronic & Ionic Impact Phenomena,

V. 4, Oxford Univ. Pr. , New York.

Ruch, F. (1966) Determination of DNA content by microfluorometry. In: Introduction to Quantitative Cytochemistry (Ed. by G. L. Wied) , p. 281. Academic Press, New York.

Udenfrlend, S. (1969) Fluorescence Assay in Biology and Medicine, Academic

Press, New York.

Watt, D. E. & Hughes, S. (1972) Bio-molecular damage by low energy heavy

particles. Physics in Med. & Biol., 17, 306.

ACKNOWLEDGMENTS

The author wishes to acknowledge support of the National Science Foundation

via a grant to De Paul University, and of the Faculty Research Participation

Program at Argonne National Laboratory of the United States Energy Research and Development Administration.

214
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg. Md., September 4-9, 1975 (Issued January 1976)

ARTIFACTS IN THREE DIMENSIONAL RECONSTRUCTION FROM MEDICAL RADIOGRAPHIC DATA

by E. L. Hall*, G. C. Huth**, R. A. Cans*', and I. S. Reed*

'Dept. x)f Electrical Engineering, University of Southern California, Los Angeles, California 90007, U.S.A. ''Laboratory of Nuclear Medicine & Radiation Biology, University of California-Los Angeles, Los Angeles,

California 90024, U.S.

A

ABSTRACT

Artificial objects (artifacts) produced in reconstructed images^'^ from medical radiographic porojection data are a serious problem. In the extreme case, an artifact in a diagnostic image could influence the medical treatment. More

commonly, artifacts produce a background noise which obscures the desired

image and creates confusion in interpretation of the images. The purpose of

this paper is to systematically describe the major sources of artifacts for the diagnostic radiology and nuclear medicine use of reconstruction techniques.

An awareness of the source and occurrence of artifacts can often lead to complete elimination of the artifacts through the use of correction techniques and

at least provides prior information to the viewer. For definiteness this study

will be restricted to the use of radiographic projection data and images reconstructed with the simple convolution algorithm, although many of the conclusions may be generalized to other situations.

INTRODUCTION

Artificial objects in reconstructed images may arise from four major

sources: 1) physical as sumptions, Z) engineering implementation in device

design, 3) computation algorithms, and 4) image display. Each of these sources will now be considered in detail. PHYSICAL ASSUMPTIONS

The physical assumption which forms the basis of the reconstruction technique is that the attenuation of photons passing through an object is described

by exponential decay:

I = Ijexp {-JJu(P, Z,E)dEds|

where Ig is the number of photons entering the object, ^i(P, Z, E) is the linear

attenuation coefficient at position P and dS is the elemental path length of the ray. The attenuation coefficient, |j, is a function of the composition of the object as indicated by its atomic number, Z. It is also a function of the energy

of the energy spectra, E, of the photon beam. The basic linear equation is obtained by simply taking the log of the previous equation:

D = -log j- = JJuiP, Z, E)dEdS.

The simplification of this linear equation for use in a reconstruction algorithm

requires several assumptions. For geometrical simplicity, one may assume

that the ray is positioned along the x axis and the position is x. The first major assumption is that the continuous position integral can be replaced by a summation:

D = E Ju(X;, Z,,E,)dE,Ax,.

i Note that this assumption is equivalent to assuming that the physical composition of the matter along the length, l\x.^, is homogeneous. This fact is denoted

by replacing Zj with an average atomic number, Zj. The consequence of this assumption in a reconstructed image is illustrated in Fig. 1 which shows that an element composed of a mixture of bone and air can appear identical to an element of tissue.

215
E. L. Hall, G. C. Huth, R. A Gans, and I. S. Reed

Bone Fat Air

Figure 1. % Bone and % Air

An element composed of a mixture of bone and

air appears identical to an element of tissue.

The continuous dependence of the attenuation coefficient upon the energy- spectra must also be simplified by replacing the integral by

U(Xi. Zi,E,) = fu(x,, Z,,Ei)dE,

where is the effective energy along the elemental area. The averaging ef- fect of this assumption is similar to the atomic number dependence, in that two

photons of low and high energy can produce the same result as a single photon

of medium energy.

A much more serious assumption is made in the reconstructed process.

This is that the attenuation coeffici_ent is only a function of the physical com- ' position, Z and not of the energy, E. This assumption is implicitly made when

the algorithm estimates the contribution of the projection data resulting from

an elemental area independent of the path length and object composition. The

variation of energy spectra passing through a material is a well known phenomena. Furthermore, the amount of attenuation coefficient variation can be significant.

Figure 2. Motion artifact resulting from subject motion in the reconstruction of a human skull. The two

circular objects are simulated tumors. The lin- ear object is the result of motion.

ENGINEERING IMPLEMENTATION

Several limitations in the accuracy of computer tomography are the result

of engineering implementation in the design of the mechanical motion portion of a scanner, the x-ray source and detector, the timing signals used for control

and the sampling or pulse counting of the project data. A serious limitation

is the resolution element size of the projection data. This resolution is typically

216
Artifacts from medical radiographic data

on the order of 1.0 to 3.0 mm and thus 140-160 elements are used to span approximately 50 cm. Another limitation is the signal to noise ratio or number

of counts collected at each data point. Typically, 50, 000 to 100, 000 counts are collected for each projection point although this time is controllable so

that any desired signal to noise ratio could be achieved. Subject motion, one

of the most serious artifacts, is also a function of the device design. An ex- ample of the linear artifact produced by subject motion is shown in Figure 2. The seriousness of this artifact is indicated by the fact that most n&vf scanners use a fan beam geometry rather than rectangular to minimize data collection time and thus the possibility of motion artifacts.

COMPUTATION ALGORITHMS

The computational algorithm also seriously influences the final reconstruction image. For definitenes s, only the popular convolution algorithm will

be considered. This method involves a log conversion of the projection data, a convolution for deblurring, and an interpolation to back project the data into the correct image element. The accuracy of log conversion and interpolation are considered well known. However, it is not well known that the convolution

operation can be computed with zero roundoff error using finite field methods.

This method will be briefly described.

Recently^ it has been shown that the convolution of two finite sequences of integers (a^) and (b,^) for k = 1, 2, . . . , d (d = 160, 256, 320, etc. ) can be obtained

as the inverse transform of the product of two transforms which were similar

to but different from the usual discrete Fourier transform.

The transforms were of the form

d-1

A^ = E a„2"''mod b

n = 0 where b is either a Mersenne number

b - 2^-1, p a prime

or b was the Fermat number

b = 2^'°+l, m an integer.

The usual DFT may be written as d-1

F^ = Z) a^jW"'', where w is the d**" root of unity. n = 0 The primary advantage of this type transform over the DFT is the fact that multiplication by powers of w are replaced by multiplication by powers of 2 which can be implemented by a simple binary shift. Thus, by proper scaling

a zero roundoff error computation may be obtained. Also, the method is potentially faster than the fast Fourier transform which is faster than direct convolution for more than 128 points.

This advantage must be weighed against the difficulty of computing the result modulo b, numeric constraints relating word length, length of sequence, d, and compositeness of d imposed by the two choices of b. IMAGE DISPLAY

The final source of artifacts is perhaps the most important since a diagnosis is made by visually observing a reconstructed image.

Two important aspects of computed tomography reconstructed images should

be mentioned. These are the large number of distinguishable gray levels and

the use of pseudocolor. Computed tomography has the capability of producing

images with about 1000 distinct shades of gray. The storage tube displays used

on the first computed tomography device''' had a capability of fewer than 30 gray

217
f. L. Mall. G. C. Huth. R. A. Cans, and I. S. Reed

levels. Also, the image was displayed at a temporal rate which produced noticeable flicker. More recently, digital refresh TV displays have been announced which present a flicker free display. These monochrome display devices can produce more than 30, but still fewer than 100 levels. Thus, the gray

level windowing technique must be used. This technique permits viewing of at most 10% of the information available. Therefore, the relation of small objects to the overall structure may be obscured by the limited viewing range. The limitation to fewer than 100 gray levels for monochrome image presentation has not prevented the widespread use of computed tomography. A method

for increasing the number of levels which may be perceived to the order of 1000 is by the correct use of pseudocolor. The human is capable of distinguishing several thousand colors. For example, a popular paint manufacturer

sells more than 700 colors of paint. Furthermore, even though the number of colors, reproducible on a TV display is significantly less than perceivable

colors, it is still on the order of 1000. The difficulty with the use of pseudocolor is the fact that a perceptual distance function in color space which could

be correlated with the numerical distance in the reconstructed images has not

yet been discovered. Thus, confusion in relationships and especially artifact

edges are easily produced. This effect may be minimized by the use of a fa- miliar color mapping such as the rainbow spectra.

SUMMARY

In this paper we have attempted to point out the importance of an awareness to artifacts in medical reconstructed images. The application of finite field transformations for the convolution can produce a zero roundoff error. The other sources of artifacts can at least be controlled by careful device design and usage. The revolutionary new area of computed tomography has been

a significant breakthrough in medical imaging; however, like all scientific in- struments the results must be carefully scrutinized.

REFERENCES

1. R. Gordon, G. T. Herman, and S. A. Johnson, "Image Reconstruction

from Projections," Scientific American, October 1975.

2. G. C. Huth and E. L. Hall, "Computed Tomography and Its Application

to Nuclear Medical Imaging," Computers in Medicine , (to be published).

3. I. S. Reed and T. K. Troung, "The Use of Finite Fields to Compute Convolutions," IEEE Transactions on Information Theory, vol. IT-21, no. 2, March 1975.

4. G. N. Hounsfield, "A Method of and Apparatus for Examination of a Body

by Radiation such as X or Gamma Radiation," Patent Specifications

1283915, Patent Office, London.

218
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaittiersburg, Md., September 4-9, 1975 (Issued January 1976)

ARTIFICIAL STEREO: A GENERALIZED COMPUTER ALGORITHM FOR COMBINING MULTI-CHANNEL IMAGE DATA

by William G. Pichel, R, L. Brower, D. R. Brandman and R. J. Moy

National Environmental Satellite Service, National Oceanic and Atmospheric Administration (NOAAj

Washington. D. C. 20233. U.S.A.

MULTI- CHANNEL INTERPRETATION PROBLEM

Recent advances in satellite scanning radiometer technology have produced

an increase in the number of channels employed to sense radiation simultaneously

in different spectral regions. However, advanced techniques for rapidly interpreting the vast number of resultant images have not been developed. An artificial stereo technique shows promise of being helpful in solving this problem.

The NOAA-4 polar orbiting satellite carries a two-channel imaging scanner having a .52- to .73-ym visible channel and a second channel sensitive to thermal radiation in the 10.5- to 12.5-pm region of the infrared (IR) spectrum.

Figures la and lb show, respectively, a visible and an IR image constructed from

NOAA-4 data. Orbital data are merged into mosaic image arrays on a polar ster- eographic map base with each image element representing an area approximately

10 km in width at 40° North Latitude. A 512 X 640 element subset of each mosaic

is depicted. An analyst interpreting the adjacent images will alternately refer

to one image, then the other, to develop a mental picture of the vertical structure of the clouds. The time required to perform this analysis could be decreased considerably if a stereogram of the visible channel were available.

Satellite images have occasionally been used to form stereograms. For instance, stereograms have been assembled from images in the overlap region of

two geostationary satellites (Bristor and Pichel 1974). Unfortunately, conven- tional stereograms have not been produced using scanner data from polar orbiting

satellites, since these scanners have not been designed to view the same geo- graphic area twice from different vantage points. However, it was found (Pichel

et al. 1973) that with a simple computer algorithm, one can create an artificial

stereogram of the polar scanner visible image in which the height of each cloud element is deduced from its corresponding IR channel temperature.

ARTIFICIAL STEREO ALGORITHM

The perception of depth in a fused stereo pair requires only the presence

of parallax. Parallax occurs if distances between corresponding image elements

measured horizontally between the two halves of a stereo pair are different for different pairs of elements. Those elements having the smallest relative dis- tance will appear the highest in the fused image. The results of a computer

algorithm which introduces parallax into the visible channel image using height

information obtained from the IR channel is shown in Figure 2. This stereogram

is created a row at a time from the visible image array. Each element in a row

of the IR image array is converted to a height "h" where "h" may vary from 0 (the surface level) to 20. The stereogram is then constructed by moving ele- ments within a row of the visible image. The position of each visible element

is moved "h" elements to the left in the right half and "h" elements to the right in the left half of the stereogram. Thus, the greater the value of "h" the smaller will be the distance between corresponding image elements of each half of the stereo pair and the higher the fused element will appear. During

the displacement process, if two visible elements compete for the same location

in the stereogram, then the higher element is retained. After all image ele- ments for a row have been processed, gaps in the stereogram are filled by the lower of the two image elements at the extremities of each gap.

APPLICATION TO QUANTITATIVE IMAGE INTERPRETATION

If the artificial stereogram is to be used quantitatively, IR image data

must first be normalized so that IR elements at the earth's surface are always

assigned to level zero and cloud elements are converted to an altitude above

this surface level. Ideally, vertical temperature profiles would be used for

219
W. G. Pichel, R. L Brower, D. R. Brandman. R. J. Moy

the normalization. However, as a first approximation, it is assumed that the decrease in temperature with height is the lapse rate of the U.S. Standard

Atmosphere, -6.5°C per kilometer of altitude. Each IR temperature is converted

to a height in kilometers by finding its deviation from the sea surface temper- ature and dividing by 6.5°C/km. Heights from 0.0 to 0.5 km are assigned to level 0. Level 1 is 0.5 to 1.0 km, level 2 is 1.0 to 1.5 km, and so forth, in 0.5 km increments up to level 20 (>10 km). Figure 2 is the stereogram resulting

from the application of the above normalization scheme. Accuracy is a function

of the deviation of: 1) the actual vertical temperature profile from the U.S. Standard Atmosphere, and 2) the actual emissivity of the cloud element from the assumed value of 1.0. However, if all errors were eliminated, the appearance of the corrected stereogram would probably not differ significantly from Figure 2.

REFERENCES

Bristor, Charles L., and Pichel, W. G., "3-D cloud viewing using overlapped

pictures from two geostationary satellites," Bulletin of the American

Meteorological Society , Vol. 55, No. 11, November 1974, pp. 1353-1355. Pichel, William G., Bristor, C. L., and Brower, R. L., "Artificial Stereo: A technique for combining multi-channel satellite image data," Bulletin of the American Meteorological Society , Vol. 54, No. 7, July 1973, pp. 688-691.

220
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

STEREOSCOPIC PHOTOMACROGRAPHY AND STEREOSCOPIC PROJECTION

by Yasumichi Fujimoto and Yoshikuni Ohta Department of Anatomy, Osaka Dental University, Osaka, 540, Japan

Stereoscopic photomacrography can be easily taken by placing a specimen

either on a sliding or tilting stage (Figs. 3 & 2), and using a single lens reflex camera (specimen sliding method and specimen tilting method, respectively ((Fig. l))). We have determined the optimum condition to obtain a correct stereoscopic image which is applicable to various magnifications (xl/6 to 6o) and subject-lens distances. In the case of the specimen sliding method, the correct value of the sliding distances (i.e. the stereo base) is one of D/lO,

D/20 or D/40, when the subject-lens distance is D. The value of D/10 or D/20

with a viewer, D/20 or D/40 with a screen projection is recommended. In the case of the specimen tilting method, the convergence angle of 12° with a viewer or 4-6° with a screen projection is considered proper. The relationship between the specimen tilting method and the specimen sliding method is expressed as follows: B = 2 sin9«D where, B = stereo base and 2© = convergence angle. A stereoscopic projection is facilitated by a polarization projection

method. "Stereo Black Screen" (Sun Screen Mfg. Co. , Ltd. ) has been found to be the far better screen than the aluminum pow^der-coated screen or the aluminum plate screen. The Stereo Black Screen is coated with scale-leaves of the scabbard fish. The projected images of a stereo pair must be superimposed on

the screen precisely. Since this procedure is difficult to align by hand, two different kinds of equipments have been designed. First, ordinary projectors

are placed on two different types of tables, one being inclined and the other

is able to be rotated (Fig. 4). Secondly, ordinary projectors are placed back

to back and the images are projected onto two mirrors, one inclined and the other is capable of being rotated (Fig. 5)» Both of these set-ups are driven

finely by reversible motors and can be remotely controlled by a speaker.

Therefore, in both cases, the projected images can be adjusted precisely to give an excellent stereo image. EXPLANATION OF FIGURES

Fig. 1 A schema of the specimen sliding method and the specimen tilting

method.

Fig. 2 The specimen tilting stage.

Fig. 3 The specimen sliding stage.

Fig. 4 Set-up of stereoscopic projection (l). Inclining stage IS, rotating

stage RS, remote control devise C, polarizing filters P and polarized

eyeglasses G. Fig. 5 Set-up of stereoscopic projection (2). Inclining mirror IM, rotating

mirror RM and power supply box S. Figs. 6 & 7« Stereo-photographs.

Fig. 6 A resin cast of the vascular pattern in the pig spleen. Specimen

sliding method (sliding 0.6mm, i.e. D/20). x43.

Fig. 7 A cleared specimen of the intestinal villi of the rat. The vascular

system is injected with India ink and red resin. Specimen tilting

method (convergence angle 12°). x 20. (Specimen courtesy of Assistant

Professor T. Tokioka.

)

BIBLIOGRAPHY

1) Fujimoto, Y. : Stereoscopic photomacrography and stereoscopic projection,

J. Med. Photogr., 11:43-59 (1975)

2) Martinsen, W.L.M. : Gross specimen photography, Med. biol. 111., 2:179-190,

(1952)

3) Morgan, W.D. and Lester, H.M. : Stereo realist manual, 283-287, 335-359,

(1954), The Fountain Press, London.

221
Y. Fujimoto and Y. Ohta

222
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg. Md., September 4-9, 1975 (Issued January 1976)

HOLOGRAPHIC METHODS FOR IN-DEPTH VIEWING OF MICROSCOPIC SUBJECTS

by R. J. Schaefer and J. A. Blodgett Naval Research Laboratory, Washington, D.C. . 20375, U.S.A.

ADVANTAGES OF HOLOGRAPHY

For the study of microscopic subjects, holography has two

primary advantages over conventional optical methods. First, the

hologram reconstructs an image of the subject in full depth, so that viewing is not confined to a single plane of focus. Second,

the reconstructed image contains phase information, and it can

thus be analyzed by interferometric techniques. For the study of

transient phenomena occurring at unpredictable locations these

characteristics are especially valuable. We have therefore used

holography to study the micromorphology of growing crystals,

which is frequently influenced by randomly distributed defects or erratic convection currents. Our apparatus and results have been

described in detail elsewhere (1)

.

HOLOGRAPHIC TECHNIQUE

In a holographic microscopy system, image resolution can be

traded off with volume of space recorded . Some systems have been

built (2) in which the image is first magnified by a microscope

and then recorded holographically . These systems can attain

excellent optical resolution, and the image is recorded in much

greater depth than that recorded by a conventional microscope

(3), but the field of view is no larger than that of the con- ventional microscope. For our studies of crystal growth, we wished to observe a volume of one or more cubic centimeters,

and therefore had to be satisfied with slightly diminished optical

resolution. Our approach was generally to reconstruct a real

image of the subject at unit magnification, and then to study

this image with a high resolution microscope. In some cases, a lens was used between the object and the hologram in both recording and reconstructing stages (4), but although this method gave

slightly improved resolution it was rather sensitive to the

alignment of the lens and was therefore infrequently used. More

often, the object was placed a few centimeters in front of the

holographic film or plate with no intervening optics. Holograms

of a high resolution test target indicated that 300 to 500 lines

per millimeter could typically be resolved in the reconstructed

image

.

A high quality undistorted real image for microscopic

examination was obtained by using a collimated reference beam

when recording the hologram. Reconstructing with a collimated

beam then produced either a real or a virtual image at unit

magnification, depending on the orientation of the hologram. For

recording holograms in rapid sequence, 35-mm film in a motorized

camera body was used, but for the highest quality images glass

plates were required

.

HOLOGRAPHIC INTERFEROMETRY

Interf erometric techniques were generally used for analysis

of the reconstructed images. In many cases a Mach-Zender interferometer arrangement was used to combine the reconstructed image

with a simple plane wave and thus obtain an interferogram showing

spatial variations in the specimen's optical thickness. This

technique requires good mechanical stability of the optical system

during reconstruction.

223
R. J. Schaefer and J. A. Blodgett

Double-exposure holography is a unique method which was used

for interferometr ic measurement of the changes in shape of an object. Two holographic exposures were recorded in sequence on

the same plate, and the hologram then reconstructed two images of

the object, with interference fringes delineating any change

which took place in the interval between exposures. The fringes

in this case defined contours of equal displacement of the object

surface

.

APPLICATIONS

We have used holography to study the growth of crystals of

certain transparent organic materials which exhibit crystal

growth forms similar to those found in solidifying metals,

thereby revealing processes which occur, but cannot be seen, in

metal ingots and castings. The phenomena which have been studied

include dendritic crystallization, morphological stability, and

crystalline anisotropy. The Mach-Zender interferometer technique

was frequently used in the reconstruction stage, so that fringe

spacings and orientations could be adjusted for convenient

analysis of the transient features recorded by the holograms.

We have also studied the electropolishing of copper in

phosphoric acid solutions, using double-exposure holography to measure the convective flow of copper-enriched electrolyte and

the resulting inhomogeneity in the rate of removal of metal from

the copper specimen. For this study we also used real-time

holography, in which a holographically reconstructed image of the

specimen is superimposed on the specimen itself, and moving

fringes are seen directly as the specimen changes.

SUITABLE SUBJECTS FOR HOLOGRAPHIC MICROSCOPY

With any imaging system using coherent light, objects which

are out of the plane of focus produce clearly visible diffraction

patterns which can interfere with observation of objects in the

image plane. Therefore the best holographic microscopy images

are obtained when the object under study is relatively isolated

from other objects in the field of view. In other words, whether

the specimen chamber contains growing crystals, protozoans, or particulate pollutants, the density of such objects should be

kept small.

Similarly, objects with a rough or diffuse surface are difficult to otjserve in detail with coherent light. In particular,

double-exposure interferometry effects are eliminated if the

irregular roughness features on the object surface change between

exposures, as occurred for example when the metal surface of an electropolishing specimen became etched.

In general, it is well to bear in mind that holography is not

the final step in the study of a microscopic subject. Analysis

and documentation of an experiment generally require that the

reconstructed hologram be photographed and that features in the

photographs be measured

.

REFERENCES

1. J. A. Blodgett, R. J. Schaefer, and M. E. Glicksman,

Metallography , 7, 453 (1974).

2. R. F. vanLigten and H. Osterberg, Nature , 211 , 282(1968).

3. K. C. Lawton and R. F. vanLigten, in Developments in

Holography , S.P.I.E. Seminar Proceedings, Vol. 25, B. J. Thompson

and J. B. DeVelis, editors, (1971).

4. L. Toth and S.A. Collins, Appl . Phys . Lett., 13, 7(1968).

224
National Bureau of Standards Special Publication 437 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg. Md., September 4-9, 1975 (Issued January 1976)

EVALUATION OF A NEW TECHNIQUE FOR DETERMINING THE VOLUME OF THE HUMAN LEFT VENTRICLE FROM ONE-PLANE RADIOGRAPHIC MEASUREMENTS

by Richard Moore, Ivo Beranek, Sung Kim, and Kurt Amplatz

Department of Radiology, University of Minnesota Hospitals. Minneapolis, Minnesota 55455, U.S.A.

In clinical medicine, especially in cardiovascular radiology, it is necessary to calculate the volume of the human left ventricle (3). The results

are useful to aid in the diagnosis of cardiac disease. They can suggest the best decision about therapy, either medical or surgical; and they can help to evaluate the effects of treatment (3).

The left ventricle is a three-dimensional object. In clinical practice,

a two-dimensional image is available from a single direction, or at most two directions, by means of cineangiography. This produces an X-ray movie of the contrast-filled left ventricle.

There are many methods in use to calculate the volume of the left ventricle

based on radiographic measurements made using the image (1,2,3,4). Most methods

are based on representing the heart by an assumed mathematical model, such as an ellipsoid of revolution and others (3,4). A cine frame is chosen, the silhouette

of the ventricle is traced, and the tracing is measured or planimetered. In order to calculate the volume from such measurements, besides a factor which

corrects for the nonparallelism of the X-ray beams, several assumptions are necessary (3) . However, it is not necessary to assume that the volume must be approximated by a regular geometric solid (1,2,3). One can use Simpson's rule

to approximate the volume of an irregular solid (1,2,3).

SCOPE OF THE STUDY

The aim of our contribution was to compare the results of calculations of left ventricular end diastolic and end systolic volume from single plane left ventriculographic measurements using two methods. The first method used, as a mathematical model, an ellipsoid of revolution. It was described by Sandler and Dodge (4) and has been most commonly used in clinical practice. The second

method used was a modification of Simpson's rule. It was described by a Swedish radiologist named Ahlberg (2), who verified its applicability only using

dogs. It is a new method, and its correctness has never before been verified

using humans. Both methods are based on the assumption that the left ventricle

has a circular cross-section perpendicular to its longest axis (3) . The Ahlbert

method appears intuitively to be a better one than that of Dodge, especially

calculating the volumes of the left ventricle if distorted by ischemic heart

disease.

METHOD

Both methods were used to calculate the end diastolic and end systolic

volume in 24 patients with ischemic heart failure. Left ventriculography was performed on all patients in the RAO (right anterior oblique) position, in which

the longest axis of the left ventricle is parallel to the film. Twelve of the patients had a normal ventricular shape, and twelve had abnormalities in the shape of the left ventricle. From each cine was taken one pair of silhouettes

representing the left ventricle in end diastole and in end systole. From each

of these tracings, the volume of the ventricle was calculated by both the Dodge and Ahlberg methods. This procedure eliminated several sources of variation due to the magnification factors, possible error in selecting the cine

frame traced, and subjective error in judging the border when the tracing was

225
R. Moore, I. Beranek. S. Kim, and K. Amplatz

made. All the measurements and calculations for one method were done by one

person, thus eliminating inter-obseryer variation. Separate persons performed

each method, thus eliminating biasing the results. In effect, a double blind

study was performed.

RESULTS

Table 1 shows the values of the coefficients of correlation between results,

from the area- length method and the modified Simpson's rule method, for the values of end systolic volume, la gives the coefficient of correlation, and its level of significance, for both groups, totaling 24 shapes, lb gives the values

for the twelve normal shapes, and Ic gives the values for the twelve abnormal

shapes. Table 2 shows the corresponding values for the end diastolic volume for

the same three groups, respectively.

DISCUSSION

It was surprising that there were no significant differences between the results from either method. We believe that this finding can be explained by an examination of the Dodge method. This method includes a kind of integration,

which is used for calculating the minor diameter from the planimetricallymeasured irregular area (3).

From the point of view of clinical practice, there are great differences

between the speed and ease of using these two methods. The Dodge method can be performed manually and produce results in a short time (ten-fifteen minutes/

case) . However, the Ahlberg method is very tedious for manual calculation

(requiring two or more hours/case). It is susceptible to subjective errors,

and it requires a dedicated or time-sharing computer to make its application

practicable. We have found that the Ahlberg method doesn't provide any improvement in the accuracy of the results for clinical practice. From the mathematical

point of view, both methods give almost the same results.

Davilla and San Marco (2) reported a comparative study of several roentgen

volumetric methods based on images made from two directions. Their study in- cluded the comparison of the results for ventricular volume calculated using

an ellipsoid of revolution as a model with the results calculated using other

methods based on Simpson's rule. They found a high correlation among the results from such methods. Consequently, their results are consistent with our

results, also showing a high correlation coefficient between measurements made

from one direction and comparing results using the Dodge method with a new method

based on a modification of Simpson's rule. However, an empirical correction

factor must be applied to the results from both methods (3).

SUMMARY

The determination of left ventricular volume is an important problem in clinical medicine. We investigated two methods for the determination of the three-dimensional left ventricular volume on the basis of single plane silhouettes. One method used an area- length measurement, and the other method used

a modification of Simpson's rule. We found that for the left ventricular volume,

both methods gave results which were not significantly different. However, one method is relatively easy and simple and can be done quickly by hand; and the other method is complex and requires either a computer or a lengthy man ual calculation.

This work was supported in part by Public Health Service Grant No. HL-16484, from the National Heart and Lung Institute.

226
New technique for determining the volume of the human left ventricle

REFERENCES

1. Ahlberg, N. E. ; Paulin, S. ; and Seaman, T. : Left Ventricular Changes

During Coronary Artery Occlusion in Dogs. Acta Radiologica Diagnosis 12:

798, 1973.

2. Davilla, J. C. and San Marco, M. E. : An Analysis of the Fit of Mathematical Models Applicable to the Measurement of Left Ventricular Volume. Amer.

J. Cardiol. 18: 31, 1966.

3. Mirsky, I.; Ghista, D. N. ; and Sandler, H. : Cardiac Mechanics : Physiological, Clinical, and Mathematical Considerations. New York, Wiley, 1974.

4. Sandler, H. and Dodge, H. T. : The Use of Single Plane Angiocardiograms for

the Calculation of Left Ventricular Volume in Man. Amer. Heart J. 75: 325, 1968.

VALUES OF THE COEFFICIENTS OF COREELATION BETWEEN

RESULTS FROM THE AREA-LENGTH METHOD (1) AND THE MODIFIED SWSON'S RULE METHOD (2)

TABLE 1 END SYSTOLIC VOLUME

Number of Patients

Coefficient of Correlation

Level of Significance

a. Both groups

b. Normals

c. Abnormals

24

1212

0.98

0.99

0.99

P

>0.05

>0.05 >0.05

TABLE 2 END DIASTOLIC VOLUME

Number of Patients

Coefficient of Correlation

Level of Significance

a. Both groups

b. Normals

c. Abnormals

2412

12

0.99 0.98

0.98

P

>0.05

>0.05 >0.05

227
R. Moore, I. Beranek, S. Kim, and K. Amp /at

z

FIGURE 1

Tracing of the left ventricular cavity. The straight line repre- sents the longest chord, L, through the tracing. The area. A, of the tracing is determined by planimetry. The volume, V, is determined using

V = (8/37r). A^/L

FIGURE 2

Tracings of the left ventricular cavity in end-diastole and end- systole with the midline (S) and perpendicular diameters (1) drau-n at inter- vals of 0.5 cm. •

,

228
National Bureau oi Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersbijrg. Md., September 4-9. 1975 (Issued January 1976)

MOVIE: PRIM-9 VIEWING HIGH DIMENSIONAL POINT CLOUDS

Narrated by J. W. Tukey. Department of Statistics, Princeton University, Princeton, NJ 08540, U.S.A.

Prcxiuced at SLAC by Bin 9 Productions Presented by Prof. G. S. Watson, Princeton University

Review

This movie demonstrates a special visual display terminal associated with

a special program for the Stanford computer. Multidimensional data are displayed to show any selected three dimensions simultaneously with the display constantly rotated about a selected axis in order that the 3-d form may be recognized. A manual control panel allows selection of the dimensions to be displayed and choice of the rotation axis, speed, and direction. Subsets of the data can be masked out in order that other sets

can be more clearly seen.

The demonstration shows data for 500 nuclear events. Each event originally

supplied 9 measured parameters. Conservation and symmetry rules allow

transformation to 5 new orthogonal parameters which were displayed. The

data grouped into several distinct and separate clouds in hyperspace. One cloud could be seen to have a triplet layered structure when viewed from

one specific direction.

This reviewer believes that the original Prim-9 system could be applied

directly to feature recognition or feature classification using several measured parameters of each feature of a microstructure. Such data are presently available from the quantitative television microscope systems.

Reasonable expansions of the program should permit display of functional

equation solutions in data systems. Three-dimensional display of recon- structed real solids should be possible when the necessary information is available from serial sections or projections.

The continuous rotation system proves to be a highly effective method of displaying 3-dimensional structure. Facilities to exploit this principle

should be useful in many stereological applications.

George A. Moore

APPLICATIONS IN MATERIALS

Chairpersons

:

F. N. Rhines

J. H. Steele, Jr.

H. E. Exner

J. Gurland

P. Stroeven

G. Ondracek

R. y[. Doerr

National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

APPLICATIONS OF QUANTITATIVE MICROSCOPY IN METAL SCIENCE

by F. N. Rhines Distinguished Service Professor, Department of Materials Science and Engineering, University of

Florida. Gainesville, Florida 32611, U.S.A.

Much of the daily usage of quantitative metallography is pragmatic in nature, being concerned mainly with the achievement

of precision in routine measurements. fly present interest is not

in such applications, but rather in uses wherein quantitative

microscopy provifies the mathematical basis upon which new science

is built. it has long been the dream of the physical metallurgist

to so associate microstructure with the various properties of

metals and alloys as to create an exact science. With the growth

of quantitative microscopy it seemed that we had been given all

that was needed to regularize the structure-property

relationships, but progress in this direction has not been rapid.

The geometry of microstructure has turned out to be far more

subtle than was generally realized. it is my present purpose to assess the current status and to suggest where further progress

may be forthcoming.

One of the first identifiable scientific uses of metal

microstructure was in the development of phase diagrams. B. H. Roozeboom (1), who first assembled the scattered knowledge of

phase equilibria in the form of phase diagrams, became interested

in metals through the works of Sir W. Roberts-Austen (2) and of A. Sauveur (3) on equilibria in the iron-carbon system. The

correspondence between microstructure and the phase

transformations that had been studied thermally became apparent.

Thenceforth, metallography became a standard tool of phase

equilibrium investigation. In this role it furnished exact

numerical data. The number of phases P, referred to in the Phase

Rule (i.e. P + F = C + 2), is subject to exact count in the

microstructure of the equilibrated alloy. No identification of

the chemical nature of the phases is required. The Phase Rule

asks only: "How many?" And metallography gives this information

directly and precisely. This was an auspicious start for physical

metallurgy and gave promise of a brilliant future.

Another early achievement of metallography came with the

studies of F. Osmond ( U ) v;ho, through the observation of the

symmetries of mi c ro- i nden ts , was able to deduce the crystal

structure of a phase seen with the microscope. His work was the more remarkable because he v/as able to apply it at high

temperature as v/ell as at room temperature. Again, Osmond's

observations provided exact numerical data of fundamental

scientific significance. Long before the advent of X-ray

crystallography he had established the crystal structures of both

alpha and gamma iron, as well as those of some non-ferrous metals.

The fact that World War I intervened, after which the X-ray

methods preempted the field, detracts in no way from the magnitude

of Osmond's scientific accomplishment. At that point in time, it must have appeared that physical metallurgy was about to join the

ranks of the exact sciences, chemistry and physics.

Early in this century, studies upon the grain structure of

single phased alloys v/ere pursued by Z. Jeffries (5) with vigor,

but limited success. He proposed two kinds of measurement of

grain size. One consisted in counting the number of grains, as seen in unit area of microstructure at standard magnification.

233
F. N. Rhines

The Other depended upon counting the number of grains crossed by

standard length of line laid upon a microsect ion . Neither did the

job to the satisfaction of all and ASTM, among others, complicated

the situation by adding a third method based upon a comparison of

the unknov;n mi c ros t rue tu re with a series of "standardized

photomicrographs." Each v/orker followed his own taste in the

matter of selecting a measuring technique. The results were considerably less than satisfactory from a scientific point of

view. This example has meaning for us today, because we are still

enmeshed in its web of confusion and our liberation seems to be a requirement for us to resume the dream of creating an exact

science of mi crostructure

.

Meanwhile physical metallurgy has been pursuing a qualitative

course of development. It is, for example, a familiar fact that

the grains of a pol ycrystal 1 ine aggregate grow at elevated

temperature, but the kinetics of the process, as they have been

seen until very recently, have appeared complex and debatable.

Similarly, cold worked metals are known to rec ry sta 1 1 i ze upon

heating, but the association betv/een the original and the

rec ry s ta 1 1 i zed structures remains incompletely resolved. Phase

changes, like the decomposition of Austenite, have been observed

since the time of Sorby. Much is known in a qualitative way of

the mechanisms and their rates, but the processes, as a whole,

still seem bewi 1 de r i ngl y complex. With the advent of the

transmission electron microscope, dislocations became observable

and there was hope, for a time, that the integrated processes of

plastic deformation would become simplified and understandable.

Instead, the observations have accumulated and the ultimate

synthesis of deformation remains out of reach. The field ion

microscope provided the ultimate in resolution in its ability to reveal the locations of individual atoms in the metal structure

and to reveal the detailed structures of such features as grain

boundaries. Yet it has given us no new exact science. Most

recently, the scanning electron microscope has provided the depth

of focus needed to view fracture surfaces of metals. By this

route we seem to have returned to the grounds of the primitive

metallurgist, with greatly refined vision, but what we see in this

way seems even less susceptible to expression in exact terms.

And, finally, the Quantimet has taken much of the labor out of the

measurements of quantitative microscopy, but has added n o new

metal science. We knov/ so much about structure-property

relationships in metals, but have been able to organize our knowledge so little. Surely, there is a basic element missing and

I suggest that it lies in our understanding of the geometry of

m i c ros t rue tu re

.

The long confusion over the measurement and expression of

grain size is indicative of the presence of a very real geometric

blockade, the nature of which has not generally been recognized,

but which has stood in the way of our making further progress with

mi cros t rue tu re as a science. The blockade began to yield about

twenty years ago, however, when C. S. Smith (5) and others

undertook the development of a system of relationships of

quantitative metallography. It was a crucial aspect of this

development that each of the relationships was based upon a sound

mathematical derivation. The grain size measurements, and others

that had proved unsatisfactory in the past, had not been so based.

Seven proven relationships of quantitative metallography have thus

far resulted from these efforts.

1, Length of 1 i ne in unit volume Ly is equal to twice the

234
Applications of quantitative microscopy in metal science

number of point intercepts in unit area of microsection

(Fisure 1 )

:

Ly = 2 (units cm"2)

2. Area of surface in unit volume Sy is equal to twice the

number of intercepts of unit length of test line with

lineal feature in the microsection (Figure 2):

Sy = 2 IJj^ (units cm" ^ )

3. Volume fraction of phase in unit volume Vy is equal to the area per unit area A^ in the m i c ros t rue tu re , or the

length in unit length of lineal traverse, or the point

fraction Pp of random points upon the two-dimensional

mi c ros t ructu re (Figure 3):

Vy = A^ = = Pp (uni tl ess)

1*. Total curvature of surface in unit volume fly summed over

all of the surface in unit volume (i.e., /g * l/r2}dSy), is equal to pi times the number of tangencies v;ith lineal feature in the m i c ros t rue ture, p ro- duced by line sv;eeping through unit area (Figure h):

My = IT (units cm"'^)

5. Total curvature of line in unit volume Cy is equal to

pi tines the number of tangencies Ty occurring between

lineal feature in space and unit area of surface swept

through unit volume of material. This measurement can

be made by the use of serial sections (Figure 5):

Cv " ''^ '''v (units cm"'')

G. Numbe

r

in unit volume Ny is a topological parameter,

usually counted directly in space by serial section

metallography (Figure 6). An exception exists in phase

counting, v/hich is performed upon the tv/o-d imens ional

section. (units cm"^)

7. Connect i v i t

v

in unit volume Gy is also a topological

parameter and is a property of networi<s. It is measured

directly in space, usually by counting the number of

branches By and nodes Ky of the net in unit volume, by

serial section metallography, (Figure 7) and by applying

the re 1 at i onsh i p

:

Gy ~ " '^V * ^ (units cm" 5)

These geometric properties have several characteristics that

distinguish them and that define a unique geometry of microstructure. It should be recognized, in the first place, that

microstructures do not have repeating patterns, in the manner of

crystals. It is not possible, for example, to identify a typical

unit cell of mi c ros t ructu re and then to reproduce it repeatedly to recreate the configuration of the mi crostructure in space. It is true, nevertheless, that the geometric properties defined by

quantitative microscopy are capable of statistical repetition

throughout the structure. For this reason, the total of any

geometric property in a specified volume of material can be

defined exactly, can be measured with precision and can be

expressed usefully in mathematical terms. All of the properties

235
F. N. Rhines

O O O O (t -^/o o o o

-irW-<i ° 4 •( 000°

^ 000

'<5'^^4j> 000 00000 Q/'^-JTb o o Q^^'i^ o <^sjt-c o

-4-V oV^ o

Pp =38/100 Vw : 38% Figure 3

m \

Cy = TT 2 = 6. 2 8/cm''

Figure 5

Ny : 2/cm

Figure 6

236
Applications of quantitative microscopy in metal science

of quantitative microscopy are thus expressed as totals in unit

vol ume

.

Another characteristic of the parameters of quantitative

metallography is that all are simple magnitudes, insensitive to shape in the sense that each can be measured and expressed without

any knowledge v/hatever of the shape of the subject being measured.

This means, for example, that the length of a line is independent

of whether the line is straight, curved, tied in a knot or branched (Figure 8). The area of a surface is likewise

independent of its outline, or whether the surface is curved or folded (Figure 9). The magnitude of a three-dimensional body is its volume and is independent of its height, thickness, v/idth,

surface area or any other measure of its shape (Figure 10).

Curvature of surface is in no way affected by hovi the surface is deployed in space (Figure 11). The curvature of line is not changed by putting the line together in various ways or by

segmenting the line (Figure 12). it is rather obvious that the

number of anything is independent of the shapes of the entities

being enumerated (Figure 13). And finally, the connectivity of

any netv/ork, including, for example, the edges of a polyhedron, is obviously independent of form (Figure \h) . Such i nsens i t i v i ty to shape is of great potential value in relating properties to

mi crostructure, as will be illustrated presently.

An obvious, but often overlooked, property of microstructure

is that it is continuously connected in space. Unlike a shower of

snov; flakes, or a heap of sand, the parts of the microstructure

are fitted together so as to fill all space. This remains true even of a porous material, if its total volume is defined and the

porosity is regarded as one of its phases, i.e. the gas phase. As

a consequence, any change in the microstructure, that affects its

volume, must bring about a compensating adjustment elsewhere, to preserve the continuity of the structure. The geometric ability

to deal with this kind of requirement is provided by the topological parameters, number and connectivity. All of the other parameters of quantitative metallography are compatible with con- nectivity, in that none is shape sensitive.

When the fundamental parameters of quantitative metallography

are combined, as ratios, they yield average properties. Thus, the

reciprocal of the number of grains in unit volume 1/Ny is the

average grain volume, an absolute measure of grain size. Also

the volume of a phase divided by the area of its enclosing surface

4Vy/SY mean free path through the phase. A special case of

the latter is the mean intercept of contiguous grains 1/N|_, v/hich

is the same as 2V\j/S\j when Vy = 1 . Also the ratio of the total

curvature to the total surface area My/Sy is the average mean curvature of the surface. Essentially every ratio that can be

constructed among the fundamental parameters has real geometric

meaning. This greatly enlarges the scope of application of these

parameters in describing metallurgical states and processes. A rather curious limitation that intrudes with the making of ratios,

hov/ever, is that shape i nsens i t i v i ty is sometimes lost by making

the ratio.

An interesting and important example of the loss of shape

i nsens i t iv i ty is found in the case of the grain mean intercept

1/Ni . This ratio is one of the Jeffries' measures of grain size.

It has been widely assumed that it is either the average grain

diameter, or is proportional to the grain diameter, C. S. Smith

and L. Guttman (5) long ago pointed out that this is an incorrect

assumption. The average grain diameter can be computed from the

237
F, /V. Rhines

Lines ol Equal Length

Figure 8

Surfaces of Equal Area

Figure 9

Bodies of Equal Volume

Figure 10

Surfaces of Equal Curvature Figure 11

Lines of Equal Curvature

Figure 12

Number of Objects

Figure 13

Figures wirh Equal Connectivity

of Edge

Figure 14

238
Applications of quantitative microscopy in metal science

average grain volume 1/Ny, however, which must be measured in three dimensions, as by serial section analysis, but cannot be

deduced from any measurement made exclusively in the twodimensional section. Because l/fJ^ = 2/Sy it is apparent that the mean intercept is equal to the reciprocal of one half of the grain

boundary surface area. It varies, therefore, with the area of the

grain boundary, rather than with the volume of the grain. Since

the total surface area of the grains can vary with the grain

shape, while the grains are maintaining constant volume, it is clear that the mean intercept must be shape sensitive. It is altogether possible for two specimens to have the same grain mean

intercept and yet have very different grain mean volumes and

average grain diameters, or vice versa. Small differences in grain shape appear to have large effects in this respect.

The other Jeffries' (5) measure of grain size, i.e. the

number of grains intercepted by unit area of m i c rosec t i on , is related to the measurement of the length of lineal feature, Ly = 2Ny\. It deals with the pol yc rys ta 1 1 i ne aggregate as though each

grain center v/ere a node, connected with neighboring grain centers

by lineal branches. The number of grains intercepted by the

counting area is then equal to half the total length of the

connecting branches. It can be shown that the average branch

length is equal to the average grain diameter, wherefore the

Jeffries' (5) grain size is seen to be half the length of chain

that could be made by stringing all of the grains in unit volume

beadv/ise upon their individual diameters. To ascertain from this

the true average grain diameter it is still necessary to evalute

Ny, because D = 2N^/N^. In short, none of the standard methods

for measuring grain size gives any index of grain volume, or provides access to any volume property of the grain structure,

such as average grain diameter. In order to get at the volume

properties it is necessary to resort to three-dimensional

measurement, as in serial sectioning. This is unfortunate,

because of the practical difficulty of serial section analysis.

There is nothing to be gained, however, by ignoring this

inexorable fact.

It may seem surprising, in viev/ of what has just been said,

that some v/orkers have found simple relationships between "grain

size" and certain mechanical properties of metals. Of particular

note is the supposed relation between "grain size" and the "yield

point" in metals. f^.any workers have reported a simple

relationship of this kind. When the data are carefully evaluated,

hov/ever, it appears that the relation weakens, the more carefully

the mechanical measurement is made. As the "yield point" is taken

closer to the real elastic limit, its sensivity to grain size

tends to vanish. This implies that the sensitivity to grain size

is a property of the degree of plastic flow, rather than of the

elastic limit. The finding of a simple relationship between

"grain size" and "yield point" is evidently fortuitous. Further,

since the measures of grain size that are in use are related to the grain boundary area, it seems likely that the resistance to plastic flov/ is, in fact, a function of grain boundary area. This

example serves to illustrate the peril inherent in using unproved

geometric relationships, as well as loosely defined properties of

materials, in any search for scientific truth.

The implication of the foregoing example has, in fact, been

borne out by other investigators v/ho have found the resistance to plastic flow proportional directly to the grain boundary area. Of

these, perhaps the first v/as II. T. Angus and P. F. Summers (7) who

found a nearly linear relationship between the Crinell Hardness

239
F. N. Rhines

Number of copper alloys and their grain boundary areas. Subsequent studies of this relationship, (8) using proven

quantitative me ta 1 1 ograph i c methods for the measurement of the

grain boundary area Sy, have shov/n repeatedly that the Brinell

Hardness Number (BHN), used as a measure of the load required to produce a specified deformation, increases in direct and precise

proportion to the increase in the surface area (Figure 15): BHN ( increase) = k where k is a constant of proportionality. This is what should be

expected if the grain boundary area v;ere to act as a barrier to the passage of dislocations. The resistance should, indeed, be

directly proportional to the total grain boundary area. In this

example, the shape of the grain boundary is not involved, because

Sy is shape insensitive.

F i gure 15 Brinell Hardness of annealed

70 - 30 brass versus grain

(and twin) boundary area.

Grain Boundary Area cm^/cm^

A more impressive case (9) has been found in the relation of

the sintering force to surface tension, through the mi c ros t rue ture

of the sinter-body. The sintering force is defined as that

tensile load which, v/hen applied to a sintering mass of metal

powder, will just prevent the shrinkage of the body in the

direction of loading. It is measured in the laboratory by spring

loading a bar composed of metal powder, suspended in a sintering

furnace at a sintering temperature. The length of the bar is monitored continuously and the load is read when the rate of

length change is zero. At the same time, the bar is removed from

the furnace and is sectioned for metal lograph ic analysis. The

analysis consists of reading Vy, Sy and My of the porosity in the

mi crostructure . It is then found that the sintering force per

unit area F/a is related to the surface tension y by the

expression (Figure 16):

F/a = Y Vy • My/Sy

This relation is unusual in several respects. In the first place,

it is an exact relation, involving no adjustment constant.

Further, its geometric analysis involves an evaluation of the

curvature of a multiply-connected surface of indescribable

intricacy, a feat which would be unapproachable by any

conventional geometric method and which becomes possible only

through the shape i nsens i t i v i ty of the total curvature parameter.

It is true, however, that the ratio My/Sy is not, of itself, shape

i nsens i t i tve . The expression is applicable, therefore, only as long as the local curvature of surface is the same as the average

curvature, a condition which is closely approximated during the

second stage of sintering. Especially, it is to be noted that the

solution has not made use of any unit cell type of model, as has

been the custom in the analysis of similar problems. In these

ways, major simplification, as well as precision, has resulted

240
Applications of quantitative microscopy in metal science

from an application of the special geometry of quantitative

mi croscopy.

/ 200 400 600 800 ^ I I I I I I I I L y • Vy • My/Sy

gms / cm ^

Figure 16 Directly measured sintering

force F/a for copper powder

in hydrogen at 1000°C versus

the sintering force calculated from neta 1 1 ograph i c measurements Vy'My/Sy and the

surface tension y. Three

ranges of copper pov;der size

are included, namely: 12,30

and 48 y.

A quite different case, one involving continuity of structure

was encountered in the analysis of steady state grain growth in single phased metals and alloys. (10) Here, the total volume of

the system remains fixed and grain growth occurs by the loss of

grains from the system. Continuity is maintained by the grain

boundaries adjusting, under the influence of surface tension, to the new topological state created by the change in the number of

grains. The topological state is thus described by the number of

grains Ny and the average grain volume by its reciprocal l/My.

The kinetics of grain growth are controlled by the rate of

movement of the grain boundary, which depends upon its mobility y, its surface tension y and its relative curvature a, which is equal

to tlySy/Nv During steady state grain growth the relative

curvature remains constant, so that the rate of migration of the

grain boundary is constant. Accordingly, grain growth is found to be proportional directly to the time t (Figure 17):

1/iJy^ = 1/riy q{1 + h]syat}

Once again, a very simple relationship has emerged through

the use of fundamental quantitative me ta 1 1 ograph i c parameters to frame the problem. There are no adjustable parameters. There is no unit cell model. The result has the characteristics of a natural law.

This brings my story to the present time. The progress that

has been quoted surely indicates a trend. It seems that we should

look forward to growth, perhaps accelerating growth, in the exact

science of physical metallurgy, through the application of the

fundamental geometry of quantitative metallography. Since my

purpose in v/r i t i ng this paper is, in part, to attempt a glimpse

into the future, I shall hazard a few guesses concerning areas

that may yield to meta 1 1 ograph i c analysis. Unfortunately, the

most interesting and important developments are almost certain to be ones which cannot be foreseen at this time, but I shall not allow that handicap to deter me.

241
F. N. Rhines

Figure 17 Grain Size of aluminum 1/Ny versus time at 635°C.

I i me - mi nutes

It is, of course, most probable that me ta 1 1 ograph i c laws will

arise from practical efforts to control mi crostructure . Such

efforts are constantly in progress in various forms, such as the

making of composite materials, the controlled freezing of

eutectics, controlled solid state transformations, and the

sintering of mixed powders. In poly-phased structures of this

kind, it is usually expected that the plastic properties of the

product will vary with the degree of refinement of the dispersion.

There have been a number of studies of the properties of such

aggregates, but none is known to this writer to have made full use of the tools of quantitative metallography. Intuitively, it seems that the plastic behavior of aggregates should be related to the area of phase interface. The application of the exact

geometry of quantitative metallography should result in the

development of simple and useful laws in this field.

It is to be expected, also, that exact science will arise

across the broad front of metallurgical properties and processes,

as old problems are recast in terms of the geometry of

quantitative metallography. The large body of l;nowledge of

structure related phenomena, that has accumulated during the past

half century, is mainly couched in terms of highly complex

equations, derived upon the basis of models that fail to conform

to the reality of the fundamental geometry of m i c ros t rue tu re . The

reconstruction of these in suitable m i c ros t rue tura 1 terms may not

be easy and in some cases may prove to be impossible, but those

\/hich do emerge in simple exact form will be of greatly enhanced

value. It seems, for example, that diffusion controlled

processes, 1 i Ice homogen i za t i on and some phase transformations,

vjhere the boundary conditions have been approximated by various

unit cell models, might respond to the use of a simple parameter

like mean free path. In so doing, it will be important to avoid

problems of shape, eiL!"ier by using a shape insensitive parameter,

or by finding conditions under which the process occurs with shape

constant. The latter state prevailed, it will be recalled, in the case of second stage sintering and also in steady state grain

growth

.

In dealing with the quantification of metallurgical

processes, the importance of the topological properties (number

242
Applications of quantitative microscopy in metal science

and connectivity) can scarcely be exaggerated. These are probably

the most fundamental of all geonetric properties. It will be

recalled that the application of mi crostructure to the Phase Rule was upon the basis of the topological property number. Again, the

solution of the grain growth kinetics was made simple by the use of the topological ratio 1/Ny. Unfortunately, fev/ scientists are accustomed to thinking of geometrical problems in terms of their

topological properties. Yet, a process such as sintering can scarcely be visualized successfully in any terms other than the

connectivity of the powder stack and its subsequent topological

evolution. Whenever a process is one which can be described in terms of changes in whole numbers, like the number of particles,

or the number of bonds between particles, it is probable that it will be best described in terms of topological parameters.

In addition to those familiar characteristics of metals which

may be amenable to scientific description in terms of an exact

geometry of metallography, there may be awaiting us new discoveries in the form of processes which are new both in their

physical character and in their geometry. One such seems to have

emerged in the course of the recent studies upon grain growth.

The new characteristic is amenability to grain growth. Some

metals are much more stable structurally than others at elevated

temperature. This property has often been ascribed to the

presence of pinning particles. The recent grain growth studies

indicate, however, that there is a much more fundamental reason

for such effects. The tendency to sustain grain growth increases

with the value of o (i.e. l\ • Sy / , which is to say that it increases with the relative curvature of the grain boundary. But,

it appears, the relative curvature is a function of the range of

topological forms of the grains existing in the metal and this

range is unaltered by grain growth. In some aluminum samples that were analyzed by serial sectioning, it was found that the

topological forms of the grains ranged from three to more than

fifty faces per grain in the same piece of metal. The broader the

range of topological forms, the greater the tendency to sustain

grain growth. The most recent addition to this discovery is that

the topological distribution can be established, more or less at will, by suitable control of the conditions of rec ry s ta 1 1 i za t i o n

.

Thus, the high temperature stability of a metal may be built in by

a deliberate and exact control of its grain topology.

Speculation of this kind could be extended indefinitely. It seems, though, that enough kinds of examples have been suggested

to make it clear that there exists a vast field for the

application of the geometry of quantitative microscopy in the

building of exact science. I believe that we are moving, slowly

but surely, tov/ard a full quantification of all of the structureproperty relationships in metal systems. Progress in this

direction may eventually provide us with the power to design and

to construct materials to property requirements. What more could

vie ask of the physical metallurgy of tomorrow?

243
F. N. Rhines

B i b 1 i ograph

V

1. B. H. Roozeboom, die Heterogenen

G 1 e i chgewi ch te vom Standpunkte der

Phasenlehre, Verlag von Friedrich Viev/ig und

Sohn, Braunschweig (1901).

2. IV. C. Roberts-Austen, Fourth and Fifth Alloy

Research Reports, Proc. Inst. Mech. Eng.

(1897) 31-100 and (1899) 35-102.

3. A. Sauveur, The Mi crostructure of Steel and

the Current Theories of Hardening, Trans.

Amer. Inst. Min. Eng. ( 1896 ) 26.863-906.

k. F. Osmond, and G. Cartaud, The

Crystallography of Iron, J. Iron-Steel Inst. (1906) 21 kkk-k<^l.

5. Z. Jeffries, E. A. H. Kline and E. 8. Zimmer,

The Determination of Grain Size in Metals,

Trans. Amer. Inst. Min„ Eng. (1916) Sjt 591*- 607.

6. C. S. Smith and L. Guttman, f-leasurement of

internal Boundaries in Three-dimensional

Structures by Random Sectioning, Trans.

Amer. Inst. Min. and Met. Egrs. (1953) 197

81-87.

7. H. T. Angus and P. F. Summers, The Effect of

Grain Size Upon Hardness and Annealing

Temperature, J. Inst. Metals (1925) H 115-

1 hi.

8. W. J. Babyak and F. N. Rhines, The

Relationship Between the Boundary Area and

Hardness of Recrystal 1 i zed Cartridge Brass,

Trans. Amer. Inst. Min. and Met. Egrs. (1960)

218 21-23.

9. R. A. Gregg and F. N. Rhines, Surface Tension

and the Sintering Force in Copper,

Metallurgical Transactions (1973) i 1365-

1 37it.

10. F. N. Rhines, K. R. Craig and R. T. DeHoff,

Mechanism of Steady State Grain Growth in Aluminum, Metallurgical Transactions (197U) 5. 1113-1125.

244
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

APPLICATION OF STEREOLOGICAL TECHNIQUES TO THE QUANTITATIVE CHARACTERIZATION OF WOOD

MICROSTRUCTURE

by James H. Steele, Jr.*. Geza Ifju and Jay A. Johnson

Forest Products,Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, U.S.A.

INTRODUCTION

Wood is the product of a complex biological system, the tree, and as such it has widely variable properties which depend upon its microscopic cellular structure.

The presently accepted practice of characterizing wood structure involves an ordered sequence of two-way decisions about size, distribution and shape of the anatomical elements (1) . These qualitative identification keys do not provide

quantitative information about the anatomical features which occur in wood

microstructure. Application of stereological counting measurements upon trans- verse sections of several wood species for quantitative characterization is described in this paper. The results which are presented represent a preliminary and exploratory study involved in evaluating the feasibility of using

stereological techniques for quantitative microstructural description and identification of wood.

Conventional wood anatomy distinguishes two types of elements, longitudinal and

transverse, with respect to the direction of the tissues within the tree. Longitudinal elements may be vessels, various types of fibers, tracheids or parenchyma cells. Transverse tissues which are called rays may contain tracheids, or parenchyma elements, or both. The relative size, and size distribution of these elements within growth increments or annual rings, are used for identification and for qualitative microstructural descriptions. Since size,

spatial distribution, and size distribution of the elements in highly complex

structures such as wood are difficult for the human eye to recognize, minute

features, or sculpturings on the elements are often used as secondary sources of identification or description.

It is often desirable to tabulate the distinguishing features and their presence

or absence in the various species in order to provide and aid to the microscopist. Certain distribution patterns have to be judged subjectively according

to these tables to be "gradual" or "abrupt", "diffuse-porous", or "semi-ringporous", "up to 4 seriate", "widest more than 10", "chains", "nested", "multiples", etc.(l). All these distribution patterns are quantifiable. These subjective decisions of the presence or absence of some elements may be answered

with either yes or no, or possibly not at all with any certainty. This points

out the need for a statistical approach. With proper sampling and statistical

treatment, the probability of the presence of a certain feature in a wood can be calculated. A quantitative approach would also provide the necessary basis for

the calculation of probabilities. Although conventional descriptive anatomy can

be used for distinguishing between species of wood, it does not provide any information for the prediction of properties. Description in qualitative terms

is a one-way street providing basically one result, an identification.

The history of wood identification started with the advent of plant anatomy in the late 17th century (2). During the 300 years history, only a few attempts of quantifying anatomical structure of wood are documented in the literature. Most of the work was done on coniferous woods that have simple microscopic

structure (3,4,5). Ladell (6) and later Kellog and Ifju (7) applied point

countings to microscopic images of angiosperm wood that had complicated anatomical structures. These attempts, however, did not result in the calculation of the important parameters necessary for the complete quantitative characterization

of the structures studied.

*Presently with Armco Steel Corporation, Middletown, Ohio

245
J. H. Steele, Jr., G. If/u and J. A. Johnson

PROCEDURES

A. Material

The material for this study consisted of the wood of sassafras, aspen, yellow

birch, paper birch, and sugar maple. The two birch species are diffuse-porous

and are known to be anatomically very similar, with sugar maple somewhat similar

to birches in its microstructure . While the birches and maple have wood of relatively high density, aspen is a low-density diffuse-porous wood of quite

uniform macros tructure . The wood of sassafras has a so-called ring-porous structure and therefore it is distinctly different from the diffuse-porous woods

included in this study.

Transverse microtome sections of 20-40 um in thickness were prepared from small blocks of the above species. The source of wood for the blocks was a random

selection from a large lumber supply. Thus, variability representative of wood

could be studied. The microtome sections were stained using a fast green- safranine double-staining technique. A minimum of ten slides from each block

studied were mounted on permanent glass slides for microscope projections.

B. Stereological Countings

The cross-sectional images of the transverse sections were projected through a microscope onto a 25 point square grid illustrated in Figure 1. Magnification

of the image was chosen for each species such that the size of the largest

longitudinal elements, namely the vessels, was approximately equal to the grid

spacing. The image was rotated so that the grid lines were parallel and perpendicular to the rays. A 10-digit blood-cell counter was used to tabulate the stereological counting information. The following data were obtained for each anatomical element:

1. Point counts with each grid intersection considered a sampling point on the lumens and cell walls of fibers, vessels, and rays.

2. Number of intercepts of grid lines with vessels in a direction perpendicular to the rays. From this, the intercept density in the tangential direction Nj^(T) was calculated.

3. Number of intercepts of grid lines with vessels in a direction parallel to the rays. From this, the intercept density in the radial direction N-^(K.) was calculated.

4. Number of vessels within the grid area, N .

Figure 1. 25 point grid on typical Figure 2. Five sampling fields within

transverse wood section. typical annual growth ring.

246
On the quantitative characterization of wood microstructure

The spatial distribution of structural elements across a growth increment or annual ring was also determined. Five zones along the radial direction within

each annual ring were selected to assess these variations. A typical sampling

field is shown in Figure 2 where it is seen that only the tangential locations were randomly selected. Average values for P , N (T) , Nj^(R) and were determined from 20 grid placements within each of Ihe five zones of the annual ring.

The first zone is associated with spring or earlywood, and the fifth zone with

the summer or latewood. An unfortunate aspect of using the annual ring as a sampling unit is the lack of width uniformity within individual trees, species,

or between species. Consequently a comparison of features within the zones is complicated by a scaling problem. However, the ring is laid down in a fixed

period of time and is a record of the tree's growing pattern which, it is presumed, remains reasonably invariant in time. Another reason for using growth

zone sampling is the possibility of correlating mechanical properties of samples

extracted from the growth zones with the stereological measurements. The tech- niques of testing within-growth-ring properties of wood have been developed by

Ifju (8).

RESULTS AND DISCUSSION

Although there are a number of different cell types found in hardwoods, the three types: vessels, fibers, and wood rays predominate the anatomical organization. The characteristics of these features as found in five species of North American hardwoods will be discussed in this section. Figure 3 shows transverse microtome sections of the five species studied.

Figure 3. Examples of transverse sections of the five hardwood

species used for this study.
J. H. Steele, Jr., G. /fju and J, A. Johnson

A. Size and Shape Characteristics of Mlcrostructural Elements

Average values and standard deviations (in parentheses) for the counting measurements are given in Table 1 for the five species studied. Each average is based on 100 countings, i.e. the combined value of 20 countings from each of the five growth zones. The vessel area fraction in sassafras is quite similar to that of aspen, however, this cannot be readily seen under the microscope because

of their relatively small size in aspen. The last column in Table 1 shows that

the reason for this similarity is that the number density of vessels in aspen

is approximately 8 times that of sassafras. Note also that the birches and sugar maple take an Intermediate position between the above two species. Also,

it should be noted that standard deviations for the point counts on vessels of the four diffuse-porous woods appear to be independent of species, whereas

sassafras, which is ring-porous, shows an appreciably larger variation in vessel

area fraction across its growth rings.

Table 1. STEREOLOGICAL MEASUREMENTS FOR FIVE NORTH AMERICAN HARDWOODS

Point Count Vessel Vessel

Species Pp(%) Intercept Density

Count

Subfeature* Vessel Fiber Ray Total R

L = Lumen T+

V! = Wall

Ring-Porous

Sassafras L 27(22) 21(11) 7(5) 55(14) R- -3 .2(1.1) 17(6) W 10 ( 6) 27(14) 8(6) 45(14) T- -3 .9(1.5)

Total 37(25) 48(23)15(9) 100(14)

Diffuse-Porous

Aspen (1)# L 28( 7) 22(12) 3(3) 53(10) R- -8 .6(2.9) 113(21) W 10 ( 6) 33( 9) 4(3) 47(10) T- 13 .4(3.1)

Total 38(13) 55(13) 7(5) 100(10)

Aspen (2)# L 34{ 7) 18( 7) 6(5) 58(10) R- 10 .9(2.1) 126(22) W 10( 7) 28( 9) 4(3) 42(10) T- 14 .1(2.2)

Total 44(11) 46(11)10(6) 100(10)

Sugar Maple L 14( 7) 23( 7)10(7) 47( 9) R- -5 .5(1.6) 52(12) W 7( 4) 42( 9) 4(4) 53(10) T- -6 .4(1.4)

Total 21 ( 8) 65( 9)14(8) 100(10)

Yellow Birch L 14( 7) 19( 7) 9(6) 42 ( 6) R- -3 .4(1.4) 30(13) W-

4( 4) 50(10) 4(4) 58( 4) T- -4 .2(1.3)

Total 18( 9) 69( 9)13(7) 100 ( 7)

Paper Birch L 16( 7) 16( 6) 6(5) 38( 9) R- -5 .7(5.1) 49(17) W 6( 4) 53( 9) 3(3) 62 ( 9) T- -6 .1(2.0)

Total 22( 8) 69 ( 8) 9(6) 100 ( 9)

* L stands for the lumen or hollow region of the cell and W. for the wall. The numbers in parenthesis are standard deviations.

# Two different blocks of aspen were used.

+ R indicates radial and T indicates tangential direction. All units are expressed in mm.

248
On the quantitative characterization of wood microstructure

The area occupied by fibers in the five species studied reveals several other

subtle differences. For example, sassafras, maple, and aspen fibers are relatively

thin-walled with less than 2/3 of the total area occupied by the wall. The birches, on the other hand, have fibers with very thick cell walls. This difference might be important from the point of view of utilization, since thin-walled

slender fibers are better suited for papermaking than stiff, thick-walled ones. Also, the permeation of fluids, such as chemicals in pulping or in preservative

treatments is facilitated by the larger volume fraction of cell cavity (lumen).

Ray cells occupy a relatively small percentage of the total area in the structure of all species studies. Large standard deviations in these measurements do not allow definite conclusions to be drawn in regard to their sizes. One {Joint,

however, which is indicated by the data is associated with wall thickness of the ray cells. Interestingly and rather unexpectedly, the two birches and sugar maple have ray cells with a large lumen to wall ratio when compared to aspen and sassafras. There is no apparent reason for a species like the birches with

thick-walled longitudinal fibers to develop ray-cells of lower relative wall

thickness than do species like the aspens that have inherently thin-walled

longitudinal elements.

The total area occupied by all elements may be used as an estimate of wood

density. If the density of the cell wall material is known, then the apparent

density of the wood tissue may be calculated. Because of the solvent exchange

procedure used in preparation of the microtome sections the accepted specific

gravity of 1.5 for the cell walls cannot be used. It is estimated that a specific gravity near unity is more representative of the swollen cell wall

material observed under the microscope (1). Thus the total cell wall area fraction gives a reasonable estimate of density values of published data.

This study has revealed quite clearly that there may be various reasons for differences in wood densities. Density may vary due to differences in the size

or in the number of thin-walled elements such as vessels. A comparison between

sassafras and aspen illustrates this point. Total cell wall area fraction of the two woods indicates that they are of similar density. Yet aspen contains some 8 times as many vessels per unit area as sassafras. Thus, the size of an average vessel in sassafras must be appreciably greater than in aspen, as shown by the data.

Intercept counts in the radial and tangential directions were used for the calculation of the size and shape parameters which are listed in Table 2. Mean chord intercept (^wj-.-]-) snd mean free path (^j.jjm) are used here to describe

the average length of chords within the features ana between the features,

respectively, which result from test lines oriented radially or tangentially

within transverse sectioning planes . These are calculated from the following

equations

.

^ MCI = 2Pp/\

^ MFP = 2(1-Pp)/N^

The mean chord intercept for sassafras is significantly greater than in the other four species studied. However, mean free path between vessels is greatest

for yellow birch. Aspen again shows that its vessels are very small and that

the mean free path between them is also small, suggesting that this anatomical

element is quite numerous in aspen. As expected, the aspect ratios of vessels

suggest an elongated shape in the radial direction, in all species studied

except paper birch which has an almost circular cross-section. This subtle

difference between the two birches may provide a possible diagnostic tool for identification.

249
J. H. Steele, Jr., G. Ifju and J. A. Johnson

TABLE 2. SHAPE & SIZE PARAMETERS FOR VESSELS OF FIVE NORTH AMERICAN HARDWOODS

Sampling* Mean Chord Mean Free Aspect Estimated Estimated

Direction Intercept Path Ratios First Second

\ (mm) X (mm) X (R) Moments Moments MCI MFP — MCI -T , s j2 , 2, d (mm) d (mm ) X (T) MCI

Sassafras R .23 .39 1.23 .075 .028

T .19 .32 .090

Aspen (1) R .10 .14 1.75 .031 .004 T .06 . ^ .09 .048

Aspen (2) R .08 .10 1.34 .035 .005 T .06 .08 .045

Sugar Maple R ,08 .29 1.22 .043 .005 T .07 .25 .049

Yellow Birch R .11 .48 1.36 .055 .009

T .08 .39 .069

Paper Birch R .10 .27 1.08 .048 .006

T ^09 ^26 ^052

*R stands for "radial", and T for "tangential" on a transverse section

The first two moments of the size distribution (denoted M^^ and M ) of vessel

cross sections were calculated for each species to give quantitative indication

of their size distributions. These may be obtained from the counting measurements since.

^1 =- I y=o

y F(y)dyE d = L/n = 4N^/Tr

and.

.y=<» 2 2 — — —

y F(y)dyE d = 4A/Tr = 4Pp / ttN^ y=o

Here F(y)dy represents^ the size distribution func^tion for spherical vessel

_transverse sections, d is the average diameter, L is the average perimeter, and A the average area. Although these two moments do not completely characterize

the size distribution they can be used to show if significant differences occur. These moments may also be used as diagnostic quantities for distinguishing

different species of wood, however, feasibility for this can only be assessed

after stereological analyses of many different woods from widely different sources are made.

250
On the quantitative characterization of wood microstructure

B. Variation in Microstructural Features Across Growth Increments

1. Within-Species Variation

The spatial variations (or patterns) , exhibited across an annual ring, for the vessel and fiber volume fractions of two aspen blocks are illustrated in Figure

4. No attempt was made in this exploratory study to systematically sample all possible forms of wood which might occur for a given species. Although there

are some statistical differences within individual growth zones, in general the patterns are similar. The variation patterns across the growth increment for the two sources are very similar; they show a decreasing vessel volume fraction

and an increasing fiber volume fraction while traversing from earlywood to latewood. Other parameters such as ray Pp, vessel N^, Nl, and XMFP have distinctive patterns which are similar with respect to the two sources of aspen despite differences in individual growth zones. Two observers were also used to obtain stereological data for aspen with only minor discrepancies found between

their data. Thus there appears to be consistency of the stereological analysis

within a species.

The spatial patterns of vessels and fibers are complimentary in the sense that

Pp (vessels) + Pp (fibers) = 1. This is based upon the observation that ray

cell volume fraction is small, and is approximately constant across the annual

ring. Any increase in vessel volume fraction thus corresponds to a proportionate decrease in fiber volume fraction. Hence the spatial patterns may be

described using either of these two elements. Vessel distribution was utilized

in this study.

100

earlywooo sourcel ^3

sourceU

r-1

.3 -T .7

relat

100 early wooc

7O

la tew

source I g

sourc.l S

1

•7 -9 ive po5i^ior> wiitViin annual rin^

Figure 4. Volume fraction variations for vessels and fibers within a growth

ring from two sources of aspen (in Figures 4-9) statistical scatter

is represented by + 2S.E. units, and consequently the total width

represents a 95% confidence interval.

251
J. H. Steele, Jr., G. Ifju and J. A. Johnson

2. Differences Between Anatomically Widely Different Species

Having established a reasonable consistency within a species, the next area of inquiry was the comparison between species. In Figure 5 patterns for vessel

volxame fraction are shown for two species belonging to grossly different structural categories (ring-porous versus diffuse-porous). Aspen, according to conventional classification of woods, belongs to the diffuse-porous group. The term "diffuse-porous" is defined in wood anatomy as having vessels or pores of uniform size evenly dispersed within growth increments. It was shown previously

that the size distribution of vessels for aspen is not uniform and also that

vessel volimie fraction varies from the beginning to the end of the growth in- crements. An examination of the growth ring patterns for other diffuse-porous

species reveals that virtually all such woods deviate in different degrees from

the ideal uniform dispersion of vessels. These results indicate that the term "Diffuse-porous" should either be redefined or perhaps deviations from its

classical form should be characterized preferably in_quantitative terms. Sassafras, a ring-porous wood, exhibits a maximum vessel Pp in the second growth

zone, whereas aspen, which is diffuse-porous, shows monotonically decreasing

values over the entire growth increment, as illustrated in Figure 5. The con- tribution of wall and Ixamen to the total Pp indicates that the cell wall voliame fraction (which is 10% for both species) is essentially uniform over the annual

ring. Thus, sassafras and aspen allocate about the same relative amovmt of wood

substance per growth ring to vessels, but each species varies the vessel shape,

size, location, and number to obtain a different microarchitecture.

Figure 5. Spatial variations for vessel volume fraction in aspen and sassafras.
On the quantitative characterization of wood mtcrostructure

The volume fractions of the cell wall

material for all elements across growth

zones for sassafras and aspen are com- pared in Figure 6. Since wall plus

lumen equals 100%, the region above the bars represents volume fraction of lumen in the wood. The cell wall vol- \me density for the two species is the same in the earlywood, but sassafras

is considerably denser in the latewood

where the wall volume fraction exceeds

the lumen volume fraction.

Another comparison between these two species is shown in Figure 7. The mean

chord intercepts of vessels in the tan- gential and radial directions for sas- safras show maxima in the second growth

zone. Aspect ratios for sassafras vary between 1.0 to 1.4 throughout the ring.

Aspen, on the other hand, shows a de- crease in size but a uniformity in shape (aspect ratio deviates only

slightly from a value of 1.5) across

its growth ring.

r- 100

4-

_0

sassafras Fl

J I

so6iti.»w witUiLn. avinuLixl ri

Figure 6. Volume fraction of the cell wall substance for all elements

throughout the growth increment of aspen and sassafras.

If quantification of microstructure is to be used for identithen clearly, two species which are very easy to distinguish

The patterns shown in Figures 5 through

7 indicate inherent differences between

the two species, and suggest that

stereological data may be used to distinguish species,

fication purposes. visually must be distinguishable on the basis of stereological measurements.

However, the two methods of identification need not be mutually exclusive since

spatial patterns of the stereological quantities can serve to focus the ob- server's attention on subtle differences which exist in the complex visual

images.

Figure 7. Shape and size

variations of vessels through- out the annual

ring for aspen

and sassafras.

.3

u

.1

r

t

-3 .1

sassafras radial g

i

1

-i

— S

S -1

rcla^i^vt jDO «i 't 1.01-1 vMitUif^ arinu.<j

253
J. /y. Steele, Jr., G. Ifju and J. A. Johnson

12.0

90

|Z fcO

<u

suga^ maple

jcllow btrchl

f -7 -7 relatwe jjosi'tLon wi"ty-nr, annual rinc

Figure 8. Variation in the density of vessels across the annual ring.

3. Differences Between Anatomically Closely Related Species

Within the diffuse-porous category, the structure of woods is similar, and thus

it is often very difficult to visually identify individual species. Also, one should expect that stereological data would indicate the similarities as well as the differences in diffiase-porous species. To illustrate this, the nimiber of

vessels per unit area, N^, for the four diffuse-porous species analyzed are plotted as a function of position in the growth increment in Figure 8. There is no problem in distinguishing aspen, however, only minor differences exist be- tween maple and the two birches. It appears that is more uniform throughout

the annual ring for sugar maple with a somewhat greater value than for the birches. However, to determine whether the patterns for the maples and the birches are significantly different, more data are required. This is an important point. Extra data acquired by visually scanning more areas will undoubtedly enhance the observer's ability to distinguish a particular species, however, there is no assurance that he will decipher the information correctly. A

greater statistical confidence for the assertions made can be obtained with

additional stereological data. Thus, stereological characterization of wood

microstructure for identification has a 'built-in' system of evaluation which is lacking in the visual method.

254
On the quantitative characterization of wood microstructure

Figxire 9. Spatial variation of vessel volume fraction within the growth increment of two birch species.

4. Comparison Between Two Species of the Same Genus

The final comparison involves differences between two species within the same

genus: yellow birch vs. paper birch. The spatial patterns of vessel volume

fraction across the growth ring are shown in Figure 9. As with aspen and sassafras, the cell wall volume fraction appears relatively uniform across the growth

zones. _The patterns for total vessel volume fraction are quite different. The vessel Pp for yellow birch in the last four zones drops to 1/2 its value in the first zone, whereas the same quantity in paper birch exhibits a variation which

is within a 10% band with a maximum at about the center of the growth ring.

These results led to visual examination of cross-sections of yellow birch and paper birch of different sources (1) . In all cases it was found that the largest

area occupied by vessels in yellow birch was at the beginning of the earlywood.

Paper birch sections consistently exhibited a higher relative vessel area at or near the middle of the growth rings. This example illustrates that subtle

distribution patterns of anatomical elements within growth increments are difficult if not impossible to recognize using conventional methods of wood

anatomy. This is probably a result of the complexity of the microstructure of

wood which does not allow the human eye to distinguish relatively small differences without some quantification.

The distinctly different vessel area distribution patterns for the two species

of birches, sugar maple and aspen which were studied points out again the in- adequacy of combining most of the known angiosperms or hardwoods into the cate- gory of diffuse-porous woods. Although this study is considered the initial

step toward numerical assessment of wood anatomy, it has shown differences among

a few of the diffuse-porous species.

255
J. fi. Steele, Jr., G, Ifju and J, A. Johnson

SUMMARY AND CONCLUSIONS

Microscopic transverse section of five North American hardwoods were subjected

to a quantitative analysis to test the feasibility of using stereology as a basis for characterization of wood microstructure and wood identification. The growth increment or annual ring served as a microscopic unit within which three anatomical elements (vessels, fibers, and ray cells) and their features (cell

wall and lumen) were sampled. The set of three volume fraction values for the elements of a given species were used to distinguish groups of species such as sassafras and aspen which were different from the birches and maple, for ex- ample. Within these groups vessel density, size, and shape were used to differentiate species.

The greatest possiblity for applying stereology as a diagnostic tool for identification of wood, as indicated by this study, would be in describing quantitatively the spatial variation of the microstructural elements through a growth

ring. In this regard two sources of aspen showed consistent patterns. Since

the volume fraction of ray calls is small and approximately xaniform over the annual ring, the volume fraction of vessels and fibers show an inverse relationship across the growth increment. Although more worlc needs to be done to standardize the effect of the size and variability of the growth increment

itself, the spatial patterns measured on the five species studied suggest the following conclusions:

1) None of the species classified as diffuse-porous showed a uniform

distribution vessels within their annual ring.

2) Spatial distribution patterns of anatomical elements within growth

increments appeared consistent and species specific.

3) Relative cell wall area measurements gave reasonable estimates for apparent density of the species studied.

4) Cell shapes, sizes and distributions in species of similar density were widely different suggesting that wood density alone is not a sufficient measure for wood properties.

5) Further studies are required to determine the effect of growth conditions on the various stereological parameters.

6) Quantitative treatment of radial and tangential planes of woods is needed to provide data for a complete three-dimensional characterization of their microstructure.

REFERENCES

1. Panshin, A. J. and de Zeeuw, C. (1970) Textbook of Wood Technology , Vol. 1. p. 705, McGraw-Hill, New Yor)c. 2. Kisser, J. G. (1967) History of wood anatomy. Wood Science and Tech. 1, 161.

3. Denyer, W. B. C. , Gerrard, D. J. & Kennedy, R. W. (1966) A statistical

approach to the separation of white and black spruce of the basis of xylem anatomy. Forest Science 12, 177.

4. Scallan, A. M. & Green, H. V. (1973) A technique for determining the transverse dimensions of the fibers in wood. Pulp s Paper Research

Inst. Can. , Pulp & Paper Rept. No. 88.

5. Smith, D. M. (1957) Microscopic method for determining cross-sectional

cell dimensions. U. S. Forest Service , FPL Rept. No. 79.

6. Ladell, J. L. (1969) A method of measuring the amount and distribution of

cell wall material in transverse microscope sections of wood,

J. Inst. Wood Sci . 3, 82.

7. Kellogg, R. M. and Ifju, G. (1962) Influence of specific gravity and

certain other factors on the tensile properties of wood. Forest

Prod. J. 12, 463.

8. Ifju, G. (1969) Wi thin-growth-ring variation in some physical properties

of southern pine wood. Wood Science 2, 11.

256
Nationa I Bureau of S tandards Specia I Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., Septemtjer 4-9, 1975 (Issued January 1976)

TEXTURE ANALYZER MEASUREMENTS AND PHYSICAL ANISOTROPY OF TROPICAL WOODS

by A. Mariaux, * 0. Peray** and J. Serra** "Centre Technique Forestier Tropica/. 94, Nogent-sur-Marne, France "'Centre de Morphologie MathSmatique. Ecole des Mines de Paris, 77300, Fontainebleau, France

SUMMAilYQuantitative analysis of wood structure by texture analyzer measurements,

using in particular the variogram method, gives new ways of research into the description of wood anatojny. The main interest of the variogram is its ability

to take in various scales of structures at the same time. We shall see in particular that it allows us to relate the punctual measurements in one field

(punctual variogram) with the measurements taken between different fields (regularised variogram). Then, together with classical regression techniques, the variogram demonstrates a strong relationship between the arrangement of fibres

in the wood and the anisotropy of shrinkage during drying.

INTRODUCTION

The following study asks of Mathematical Morphology a very particular question. Consider the wood transverse section of Fig. 1 : it was cut-off perpendicularly to the trunk axis and presents a complex structure. This wood, called "Au- coumea klaineana", is one of twenty tropical woods on which we want to relate

morphology to an important physical property : the anisotropy of shrinkage during

drying. We obtained this property by calculating the shrinkage ratio during drying in two directions, radial and tangential (Fig. 1). Let us briefly describe

the sequence of the wood structures, as we change our scale of observation. The

wood anatomy is made up of a set of cells that form a skeleton of vertical tubes made of rigid walls surrounding cavities that are different in dimensions and

shape. At a ten microns scale, we see the fibres (the smallest cells), lined up

in strings. They have a direction parallel to the rays (the long slim cells) ; this direction is the one which radiates from the trunk centre (Fig. 2) . Both

these series of cells induce a strong anisotropy in the wood structure. At a hundred microns scale, the vessels appear, round and isolated, big tubes conducting the sap. Then they are more or less regrouped in small clusters, and, at a millimetric scale, we can see their density slowly change (Fig. 1). At the same scale we can also observe slow variations of darkness in the fibres ; these variations, called "tree-rings", are due to a regionalized flattening of the fi- bres and a thickening of their walls. The difficulty of the study under consideration is that previous analyses have shown that neither the number, nor the shape, nor the dimension of the vessels play a role in the variations of the shrinkage anisotropy. Now it is only the vessels which appear as structures of isolated particles, the others, the rings and fibre strings, look much like a continuous medium. Thus we need an investigation method

i) independent from the notion of particles

ii) involving a directional character which is required by the structure anisotropy

iii) able to scan the continuous succession of scales from the fibre strings

to the rings, to be sure that no possibility of correlation with the physical parameter could escape.

In terms of Mathematical Morphology, the only structuring element able to satisfy these three requirements is the couple of two points, the associated measurement of which is the variogram.

THE PUNCTUAL VARIOGRAM.

By thresholding, the cavities and walls of Fig. 2 are transformed into a medium of two phases X and x'^. Let us consider_^the structuring element B, made

of two points, x and x+h, ends of the vector h ( |h|,a). We shall define the punctual variogram Y(h), to be the area percentage of the set X - X ®B (where

O represents the erosion symbol). In other words, Y(h) is the probability that

c . a random point x falls in the set X, and x+h falls in X . Notice that the defi257
A. Mariaux, 0. Peray and J, Sena

nition can be extended to a medium of n phases. In this study, we have computed

the variogram in two directions, for a = 0 (tangential) and ci = ^

(radial). For

h = 0, yih) is obviously equal to zero. The greater is h, the larger are the scales of structure that are investigated. Now, the possibility that the variogram

may grow indefinitely (starting from y(0) = 0) makes it more able to describe

the superpositions of structures than the closely related notion of covariance.

One can compute that the tangent of the variogram at the origin is equal to the number of intercepts of X per unit of test line, in the considered direction.

For each of the twenty woods, the two variograms were calculated by taking their average value on 36 fields of Fig. 2 type, the fields being set on the points of

a square grid with grid spacing of 1280;j . The maximal step h of the punctual

variogram is 80y .

STRUCTURAL INTERPRETATION OF THE VARIOGRAM.

One can observe three kinds of phenomena on the variogram of Fig. 3 : a) a strong growth from 0 to 15u , followed by an oscillation called "hole- effect", which corresponds to a pseudo-periodicity of the fibres. The abscissa

of the minimum gives the average distance between fibres in the described direction. In the case of the radial direction, the cavities are aligned in strings

and hence the abscissa is also equal to the average diameter of these fibres.

The interpretation of the maximum is more delicate : one needs to simulate the given structure from a probabilistic model, and proceed from there. The difference of ordinates between minimum and maximum explains the degree of regularity

in the periodicity of distances between the fibres.

b) The variograms do not reach a sill, but keep on growing slowly. This

Phenomenon proves the existence of macrostructures due here to the vessels. The initial slope of this second growth gives the number of intercepts of the vessels per unit of test line.

c) When comparing the two variograms, one may notice that they are nearly

Identical, up to an additive constant. This loss of ordinate in the radial direction, called "zonal anisotropy", simply means that when a couple of points

B falls in a ray, the two points are likely to stay in the cavities, even for high values of h.

THE REGULARISED VARIOGRAM,

Notice that the definition of the punctual variogram is equivalent to the following : „ Y(h) = J

[f(x+h)-f (x)]

where f is the indicator function of the phase X. Now let g(y) be the proportion

of the area of X in one of the 36 fields of the grid, centered on y :

g(y) = ^

ds y sy We define the regularised variogram (in the two directions) , as :

Yg(h) = [g(y+h)-g(y)']

where h is a multiple of the grid spacing ( 1 280P ). In the punctual variogram,

we obtain the step by translating a point within the field ; in the regularised

variogram, the step is obtained by moving the stage of the microscope. From these new definitions, we now see that the variogram quantifies, in terms of variance

of the indicator function f (or of the regularised function g) the average similarity which exists between a random point (or a field) and its translation at a distance h. With the regularised variogram, we may follow the structures on a

^'^ch larger scale (Fig. 4). The lack of sill in both directions again exhibits new macrostructures probably coming from the tree rings and the grouping of vessels. The parabolic growth denotes a continuity in the structure variations at the larger scale. Thus the variogram, given by the above mathematical definition,

exhibits a range of important structural information. By quantifying this information, we may now try to connect the structures and the physical properties.

258
Texture analyzer measurements and physical anisotropy of tropical woods

Fig. 3 : Punctual Variogram of Bridelia

Fig. 4 : Regularised Variogram of Aucoumea

259
A. Mariaux, 0. Peray and J. Serra

A STRUCTURAL EXPLANATION OF THE SHRINKAGE ANISOTROPY.

In each wood, a transverse section was cut-off in the neighbourhood of the test-pieces. Its dimensions were 1 square centimeter by 25u thickness. The sample was treated with Na CI 0 to eliminate cell residuals, then coloured in green

to improve the contrast required by the texture analyser. The number of samples

in each species was small, because at the beginning we only wanted to find a structural explanation for the shrinkage anisotropy, independent of the species. The study was done with regression techniques, using, on one hand the coefficient

T/R of the shrinkage anisotropy, and on the other hand some geometrical characteristics of the variograms (see the Greek letters on Fig. 3). The first regression tried told us that only 35% of the shrinkage anisotropy variations could be explained by the three most significant parameters of the variogram. But, after

the elimination of the four "Commiphoras" (and hence we no longer believe that

the shrinkage anisotropy is completely independent of the species) , the percentage of explanation went up to 55% without a loss of statistical significance.

According to these last results, the explanation is given by the following para- meters : A

r - the ratio — ; where A represents the degree of periodicity present in

r . . . the distances between the cavities and is the average value of these distances taken in the radial direction. The ratio is consistent with the fact that, the smaller j-he distance between the particles, the stronger is the regularity

in the periodicity. a^. - the ratio — , where is the average distance between the cavities in

a t the tangential direction. Here is the expression of the obtained relation : a A T/R = 4.5 - 1.56 ^ - 4.29 ^ + e r r

where ^ is in %, a and " in V , and e represents a residual,

r ' t r

^ Therefore it appears that more than half of the variance of the shrinkage

anisotropy is explained by the given characteristics of small scale structures

of wood anatomy. Now, if we recall the effect of removing the influence of the "Commiphoras", we expect that the remaining unexplained variance is due to the difference in species.

CONCLUSION.

In conclusion, we will emphasize two points. First, the variogram is the only theoretical tool which allows a scanning over structures of different scales. It permits with equal ease computations either from point measurements or from measurements between fields. Here, we covered scales from ly to 5 mm. Second,

to attain a very specific goal (here, the estimation of a physical property), we have used a structural analysis which is extensive and which is also of interest

by itself. In fact, both these aspects are not at all opposite. It is precisely

by its broadness of investigation that the second point of view allowed us to lo- cate the two structural characteristics that we needed for a good correlation.

REFERENCES. The same as in : J. SERRA, Stochastic models

in Stereology : Strengths and Weaknesses.- Proceedings

of the 4th International Congress for Stereology, 1975 Washington D.C., U.S.A.

260
national Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, ^W., September 4-9, 1975 (Issued January 1976)

DECISION CRITERIA IN THE QUALITY CONTROL OF STEEL WITH RESPECT TO NONMETALLIC INCLUSIONS

by Stig Johansson

Sandvik AB. Sandviken, Sweden

ABSTRACT

Quality Control is carried out to establish if a lot is within a certain standard regarding nonmetallic inclusions. The decision has to be

based on statistics and therefore the inclusion population has to be

described by the mean and the standard deviation when the metallurgical

treatment is constant within controllable limits. Critical features must

be recorded or predicted from something that can be recorded. Such critical

features can be either agglomerations of inclusion (segregations) or individual inclusions above a critical size. It is most important that

the analysis is based on a sufficiently large area to obtain even a moderate degree of acc\iracy in a quantitative assessment of steel cleanness, therefore modem scanning technique has to be used. Among all the various parameters that can be arrived at in image analysis three have

been selected as decision criteria. These are the proportion of fields

with an area fraction above a certain level, the average area fraction

A^ and the particle size distribution slope.

INTRODUCTICW

In the field of metallography and cleanness assessment of steel two dimensional automatic image analysis has been used for a long time. The

madln advantages of this technique is the time saving factor compared to manual quantitative stereological work. All basic descriptive parameters

arrived at in the latter type of work can easily be calculated from

automated image analysis provided the features of interest can be selected

by means of variations in optical reflectivity.

The important question is what is interesting to quantify. It is obvious that the principles in choosing descriptive parameters are different for research work and for quality control . In research and development

the metallurgical parameters usually are deliberately subjected to changes

for obtaining some wanted property. In many of these cases the result of

the change can be seen in any position of the bulk steel as the linked

change in for instance inclusion morphology and/or distribution is altered everywhere.

For quality control it is quite the opposite case. The metallurgical

parameters are specified beforehand and kept constant, the inclusion

morphology is known and the analysis is carried out to find unwanted

features in the distribution. These unwanted features will usually not

appear very frequently which means that statistical treatment is necessary

to be able to assure anything. Unwanted features in quality control are usually agglomerations of inclusion (segregations) or individual inclusions above a critical size. These features can be recorded either directly

or indirectly by establishing the nature of the distribution.

It has been pointed out earlier by Allmand''" that cleanness assess- ment must be based on a sufficiently large area to obtain even a moderate

degree of accuracy. He stated t^at at least 300 to 400 fields with an individual figld area of 0.5 mm , making a total covered surface area of

150 to 200 mm , was needed.
S. Johansson

In the present work the analysis has been carried out on a Quantime;

360 image analyzer. The covered surface area in each analysis was 160 mm' divided into 500 fields.

2 The technique is described elsewhere by Gibbons .

DECISION CRITERIA

Area measurement

For material release it is necessary to establish a fundamental

knowledge about the inclusion population when the metallurgical treatment

is constant within controllable limits. If the area fraction (A ) A percent of oxide is chosen as a descriptive parameter then one of

the possible measures of location for the population is described, others

would be the mode, the median, some centile or the maximum value. In quality control one usually would like to know the proportion or percentile of the distribution which exceeds a certain critical value (degree

of segregation) and the maximum value. In other words, some characteristic

of the tail of the distribution is generally of great importance in deciding whether the material is good or not.

Figure 1 shows a common frequency distribution of an oxide area measurement for a specific grade of steel. The distribution is recorded

as the specific field area percent of oxide inclusions. As can be seen

the distribution is very far from a standard normal distribution in a single specimen with approximately 95 percent of the fields below an oxide area of 0.1 percent. The remaining 5 percent form a tail which is of the greatest importance in quality control as this part is usually

responsible for the deterioration of the material properties. As men- tioned earlier the tail would be approximately characterized by a centile

at any two or more chosen positions.

Knowing the normal shape of an acceptable tail it is then possible

to set the acceptance limit with respect to a percentile optimising the "sellers" and "buyers risk" for any single specimen.

The spread calculated as the standard deviation would also give a good estimate of the tail. Unfortunately there are no commercial instru- ments available which can present standard deviation and at the same time

beeing fast enough to be used for quality control of steel. When dealing with the population mean obtained from recording a finite number of individual sample means, the standard normal distribution can be assumed and the population can be characterized by a grand mean and the standard deviation. Then each mean can be subjected to a gono go decision based on the criterion distance from the population mean.

We now have two criteria for a go-no go decision the percentile of segregated fields and the distance from the population mean.

Particle size distribution

It has been found that the particle size distribution normally can

be described in a log- log diagram as a straight line if the logarithmic

niomber of particles greater than a certain size is plotted against the logarithmic particle size as indicated. By establishing the nature of the distribution in the microscopic region it is possible to predict the probability for macroscopic features. We have in fact been able to link

together two size distributions obtained from two different methods. The first distribution was obtained from image analysis of microscopic features

extended from 8 ,um to 126 .um in length in the rolling direction and the

262
Quality control of steel with nonmetallic inclusions

40'

30-

20-

10 lb

4-t

3-

Tail:

Compressed and

enlarged, factor 10

ItL

0.05 0.1 0.5 1.0 /o Area fraction percent

Figure 1. Distribution of specific field area fraction for oxide inclusions in one analysis

Indicated logarithmic size mm

Figure 2. Particle size distribution for ductile

inclusions

second was obtaihed from mafcroscopic features extended from half a millimeter upwards in length on a step down test piece. Figure 2 shows this

relation for silicate inclusions in a rolled 18/8 stainless steel. The slope can be used as a decision criterion in quality control. The pre- dicted number ot macroscopic features from the slope coincides with the actual number provided the macroscopic features belong to the same population. Pieces of refractory materials or other exogeneous macroscopic

features are not constituents of this population.

This slope adds a third decision criterion to the previous two.

263
S, Johansson

CONCLUSION

It would be satisfactorily safe to use the percentile of segregated

fields, the average area fraction A and the particle size distribution

slope as decision criteria in quality control. All other information are details which seldom have any significant influence on material properties provided the right properties have been obtained by optimising

inclusion shape and acceptable level of inclusions on a research and development status of the product.

REFERENCES

1. Allmand, T.R., Coleman, D.S. The Microscope, Vol. 20, No 1, Jan 1972 pp 57-81

2. Gibbons, J., Soames, M.R. and Knowles, W.R. The Microscope, Vol. 20, No 1, Jan 1972 pp 1-20

264
National Bureau of Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Vti.. September 4-9, 1975 (Issued January 1976)

AUTOMATIC RECOGNITION OF NON-METALLIC STRINGERS IN STEEL

by D. Jeulin* and J. Serra** 'IRSID Station d'Essais MAIZIERES-les-METZ. France **Centre de Morphologie Math^matique,Ecole des Mines de Paris, France

1 - INTRODUCTION : Two different ways may be used to give quantitative cha- racteristics of non metallic stringers in steel : to compare stringers seen through a microscope to standard patterns, or to make measurements with an image analyzer. Standard patterns (S.P.)> well known from metallographists

,

are used to define criterions of steel quality. Image analyzers, working

automatically, allow to get objective and representative characteristics.

But measured parameters have not always an explicit meaning and their links

with the S.P. may be not obvious. So, it would be interesting to know if it

is possible to select few parameters correlated with the S.P. To answer it, we have to survey all the kinds of patterns. In this paper, we build the theoretical approach in a particular case, the German standard patterns for stringers in forged or rolled steel DIEGARTEN (1). The analysis includes two steps : a) study of each pattern to find which combination of 'parameters

enables to get its best characterization ; b) application of the results

to real steels jointly recognized in the same places by an observer and by

an analyzer, and test of the statistical equivalence of both estimations.

In this paper, we only carried out the first step, leaving the second one

to metallurgists.

2 - THE STANDARD PATTERMS FOR STRINGERS IN STEEL build a rectangular table.

Its columns are ten families, numbered from 0 to 9 (Threadlike sulfides for families 0 and 1 ; oxides alumina type : 2, 3, 4 ; threadlike oxides, silicate type : 5, 6, 7 ; globular shaped oxides : 8, 9). Each family contains

9 patterns (numbered from 0 to 8;: 02 is the third of family 0, ...). The

size of stringers increases from pattern 0 to pattern 8. Stringers are black

or grey (families 0 and 1) grains on a white circular ground. Shapes of grains change with the types of stringers (points, dashes, or globules of different sizes) . They are more or less numerous and scattered (n° 75) or gathered (n" 37). Beside that wide range of shapes, two preferential directions may be noticed : the "vertical" direction of elongation of stringers

after forging or rolling, and the "horizontal" direction^ giving the width

of details.

3 - MORPHOLOGICAL PARAMETERS : We use notions (star, erosion, closing) based

on Mathematical Morphology, exposed in (2), (3), and (5). To measure parameters defined in Morphology was conceived the texture analysis system (T.A.S.)

(6-7) manufactured by LEITZ firm (8) (Licence IRSID - Ecole des Mines). The

three more simple parameters we used are area (A), perimeter (L) , and oonnexity nimber (N) . Elongation and thickness of stringers are estimated with

parameters of mean intercept computed from the histograms of intercepts measured in the two directions, the stars D* (vertical) and d* (horizontal) and with the maximal vertical and horizontal intercepts (D, d) , of similar meanings

but more fluctuating. Clusters of stringers are characterized with two parameters : hexagonal dilation and closing (A-8) . Figure 1 shows the results

of these operations : the pattern (la) is dilated (lb) or closed (Ic) by an hexagon of size 1.6 mm. The dilation regroups near grains but enlarges the whole. The closing deletes small enclaves inside stringers and welds details

without enlarging them. After dilation is measured the aonnexity number, Nq,

giving information about the number of clusters. After closing are measured

the area Ay, the length of maximal horizontal intercept dp, and the aonnexity

number Np. At last, threshold variable (S) indicates whether stringers are grey (0 for families 0 and 1) or black (J for the others).

4 - CORRESPONDENCE ANALYSIS of J. P. BENZECRI (9) was chosen among methods

of multivariate analysis to compute the results of measurements, because it corrects redundancies of data : two parameters giving similar information

(as A and Ap, D and D*, or d and d*) can be replaced by one of them in a

265
D. Jeulin and J. Serra

further step such as a classification, without changing the structure of

the data put forward in the analysis. Three levels of correspondences can be seen : which S.P. are similar when reported to the measured parameters ? Which parameters are similar from the patterns point of view ? Which parameters give the best characterization of each family ? The correspondence

analysis enables to visualise these 3 types of relations in order to get

criterions of automatical recognition of each S.P. The method considers the

90 patterns, characterized by 12 morphological parameters, as points of a

12 dimensional space, and the parameters as points of a 90 dimensional

space. The system of main axis of inertia of these points, provided with

weights, is computed, and it is possible to give a simultaneous representation of the S.P. points and of the parameters points on these axis and on factor planes. Points of weight zero, called additional elements, can be projected on the axis when heterogeneous data could disturb the analysis,

or when a classification is required for a sample. This principle may be

used to recognize non metallic stringers in steel with the texture analyzer.

5 - RESULTS : The data, being somewhat heterogeneous (numbers, areas, perimeters), had to be balanced before processing with correspondence analysis.

The threshold variable S, having no morphological meaning, is not used at that step. The family 8, where stringers are globular shaped, is particular,

and thus was introduced as additional elements. The inertia ratio of the four first factors (93.89 % of the whole variation) warrants a good representation of the data in a four dimensional space. In each factor plane are noticed proximities of some parameters : A is always near from Ap (the

closing does not disturb the area of stringers but only deletes some enclaves). D is closed to D* and d to d* : stars and maximal intercept length

have the same meaning here, yet stars are better : computed by integration

of granulometry they are less sensitive to statistical fluctuations. The perimeter L, near from origin, is not important in this study. The description of the factors is given in (10) : factor 1 opposes vertical intercepts

(D* , D) to horizontal (d*, d) , and to numbers (N, Np , Njj) and families 5-6

to the others. Factor 2 , correlated with areas A and Ap, arranges families

from pattern 0 to pattern 8, as in the table of S.P. Factor 3 opposes horizontal intercepts (d*, d, dp) to numbers Np and Np (Fig. 2), and separates

family 8 (globular stringers) from the others. Factor 4 correlated with Np,

opposite to N, is the axis of clusters : on one side families 2, 3, A, 5 and 6 have numerous stringers gathered in one or two clusters ; on the opposite side, families 0, 1 , 7 and 9 have more clusters. Figure 2 shows factor

plane 3-A, where the localisation of families is caused by their number of clusters (Np) , of details (N) or their horizontal intercepts.

6 - AUTOMATIC CLASSIFICATION OF THE STANDARD PATTERNS : The results of the correspondence analysis showed that certain families are well detached in the different factor planes. We consider in the 2 dimensional space (axis

1-3-4) parallelipipeds parallel to the axis containing only one family,

the limits of which are determined with the help of factor planes. To class

a field of measurement, we compute his coordinates in the factor space, f j

,

f3, f 4 , from the measured values of parameters xj (j = 1, ... 11) with

formula (1) given below and coefficients aij given in table I ; execution

of tests of table II gives the family of stringers seen in the field, while

the area of stringers gives its position inside the family (axis 2)

.

7 - CONCLUSION : This way of automatic recognition can be performed easily

and instantaneously (on one field, the parameters are measured in a fraction

of second, and a small computer, connected with the T.A.S., can make the summation (1) and the discrimination of table II), and may be transposed

to real samples of steel. Otherwise, few parameters are sufficient to make

the recognition, since it was shown that it is not necessary to measure the perimeter L, the maximal intercepts (D,d) the area and the number of connexity after closing (Ap, Np) . In the case of other types of stringers, a preliminary correspondence analysis has to be done with the concerned table

of standard patterns, to determine new coefficients aij and new tests. About

the precision of the results, we have to notice that it is difficult to make

a discrimination between certain families ( (2 and 3), (1,0, 7), (5 and 6) ),

266
Automatic recognition of non metallic stringers in steel

certainly because of their great likeness. But if an objective analysis

pointed out such a likeness, it certainly means that human recognition may

sometimes confuse too similar families, such as 2 and 3. So, it would be

good to complete our study with a comparison between visual and automatic

analysis on the same samples.

BIBLIOGRAPHY

1 - Standard patterns of non metallic stringers

in steels. German standard DIEGARTEN, 1973.

2 - G. MATHERON : Elements pour une theorie des milieux poreux. Paris, MASSON, 1967.

3 - A. HAAS, G. MATHERON, J. SERSA : Morphologie

machematique et granulometries en place. Annales des Hines, dec. 1967.

4 - J. SERRA : Introduction a la Morphologie

Mathematique. Cahiers du C.M.M. , fascicule 3, 1969. Ecole des Mines de Paris.

5 - J. SERRA : Stereology and structuring element

Stereology 3 - 1971.

6 - Texture analyzer : irrench patents IRSID

n° 23.273 (1965), and ARMINES-SERRA

n° 70.21322 (1970).

7 - J.C. KLEIN, J. SERRA : The Texture Analyzer

Stereology 3 - 1971.

8 - LEITZ - Scientific and technical information

suppl. 1 - April 1974.

9 - J. P. BENZECRI : L'analyse des donnees PARIS

DUNOD 1973.

10 - D. JEULDJ : Caracterisation des chartes d'inclusions dans les aciers. These D.E.A. 1974.

Table H - CLASSIFICATION OF THE FAMILIES OF STANDARD PATTERNS

family 4 or S P 35 to 38

or 25 lo 28

267
0. Jeulin and J. Serra

TABLE I Coefficients aij

XI a 1 \ 3i

-'J a 4i

1 00 N 0 . 543 0 119 -0 274 A -0 .103 -0 051 0 108 2,5 d 0 . 147 -0 466 0 018

100 Nf 0 .520 0 304 0 076 Af -0 .044 -0 067 0 029

2,5 dp 0 .093 -0 305 0 016

2,5 L 0 .010 0 044 -0 030

2,5 D -0 .481 0 137 -0 066

100 Nd^ 0 .371 0 265 0 468

1000 d^ 0 .095 -0.415 0 014

100 D -0 .603 0 168 -0 036

(i = ! ,3,4)

268
National Bureau of Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

METALLOGRAPH I C CHARACTERIZATION OF FRACTURE SURFACE PROFILES ON SECTIONING PLANES

by J. R. Pickens and J. Gurland

Division of Engineering, Brown University, Providence, R.I. 02912, U.S.A.

It is the purpose of this work to investigate certain metallurgical and geometrical features of fracture surfaces by means of measurements on metallographic sections showing a fracture surface profile.

Quantitative measurements of the linear fracture profile of SAE U520 steels, in the quenched and tempered condition, were carried out by Shieh [1]

who detennined the relative concentrations of the constituents along the linear fracture profile and used them to calculate a fracture path preference

index and a fracture probability index for each constituent in each fracture

mode. It would be useful to extend these and other linear parameters to the characterization of the area features of the fracture surface. However, the sampling problem associated with nonrandom surfaces severely limits either

the ease of measurement or the generality of the results obtained from limited

measurements. Nevertheless, it will be shown that even under restricted

conditions, a number of interesting results may be obtained which may

contribute to the analysis and understanding of fracture problems.

I. THEORY AND DEFINITiaiS

1) Line and surface fractions of constituents

The trace of the fracture surface on a polished cross section is the lineage of interest. The basic relations between the mean length of the lineal featvire per unit area of randan plane sections, L. ; the mean area of

the surface feature per unit volume, S^; and the mean number of intersections

between the surface or lineal features etnd random test lines per unit length

of test line, Pj^, for a phase a, are:

Sv La J. Jl = Ji or fl- fl (1) ^ h h

where fg and f? are the surface and line fractions of the constituent a, respectively. Tne pi^eceding argument applies to any surface, random or oriented, if the test planes and test lines are randomly oriented, or if the surface

elements of the surface in question are randomly oriented in space.

The conditions of random orientation are generally not present in the analysis of linear fracture traces. Equation 1 does not hold true for preferentially-oriented surfaces and nonrandomly-oriented test planes, except for

the special case when the angular spatial distribution of the surface elements

is the same for all the constituents

.

2) Roughness

Lineal roughness is defined in terms of a reference ] ine which is the trace of a reference plane on a metallographic section plane. The lineal

roughness, R^, is defined as the ratio of the true length of the linear fracture surface profile to the projected length of this trace on the reference

line. The lineal roughness can be easily measured on metallographic sections,

although the apparent roughness is greatly affected by the magnification and

the resolution of the optical system. The orientation of the reference line in the specimen must be carefxilly defined.

269
J. R. Pickens and J. Gurland

3 ) Curvature

The curvature of planar curves and of curved surfaces is defined and their relationship is reviewed in Underwood [2]. For a planar c\irve , such

as the surface trace on a section plane, the local curvature is:

, dadT (2)

where a is the tangent direction angle and i. is the curve length. The numerical average of the local curvatures would be expected to differentiate

between flat and curved line segments, i.e.

= (3)

where n is the number of measurements . The local curvature may be determined

at selected points on the surface trace, by measuring the loccil radius of curvature r, where r is the radius of the circle of curvature tangent to the curve at a given point, and r = 1/k .

II. EXPERIMENTAL INVESTIGATION OF FRACTURE SURFACES OF 1018 STEEL

1) Procedures

Twenty Charpy V-notch specimens of cold-rolled 1018 steel were fractured

at temperatures ranging from -193. 5°C to +100°C. A plot of the impact energy

versus temperature of fracture is shown in Fig. 1. The transition temperature

lies between 15°C and '+5°C, which is in agreement with published work on the transition temperature of similar low-carbon steels [3,4]. The average hardness of the steel is 92.5, Rockwell B, with a grain size corresponding

approximately to ASTM no. 8.

The linear analysis of the fracture surface profiles was carried out on metallographic sections through the fracture surface of the broken specimens.

The sections will be referred to as parallel or perpendicular cuts. The plane

of the parallel cuts is perpendicular to the fracture surface and parallel to

the direction of fracture propagation. The perpendicular cuts are perpendicular to the fracture surface and to the direction of fracture propagation. The fracture surfaces were nickel plated before sectioning and the metallographic

sections were etched with 3% nital. Typical fracture sections of the steel

broken at high and low temperatures are shown in Fig . 2 . The pearlite volume

fraction of specimens selected for metallographic analysis was determined by

point counting on random fields well removed from the fracture. These values compare well with the calculated value of 20.5% pearlite for a nominal 0.18% carbon steel.

2) Pearlite fractions on linear fracture surface traces

The line segment lengths of the fracture traces were determined by

moving the cross hairs of a Hewlett-Packard Digitizer, programmed to measure

line length, across photographs of the fracture svirface profile on the metallographic sections. For each specimen, four contiguoxis fields were measured at each of four locations equally spaced along the length of the fracture surface.

The total length of line segments in pearlite divided by the total

fracture surface lineage measured gives the pearlite lineal fraction (Fig. i).

It is noted that the pearlite lineal fraction of the specimens which lie on the lower shelf of the impact energy curve, is not statistically different

from the pearlite volume fraction. The pearlite lineal fraction of specimens

which lie on the upper shelf is significantly less than the pearlite volume

fract ion

.

270
Fracture surface profiles on sectioning planes

Measurements of the orientation of fracture lineage segments of pearlite

and ferrite were made on parallel and perpendicular sections of 2 specimens.

The method consisted of measuring the angle between the surface tangent and a horizontal reference line at randcni points of the surface trace . Angular distributions were plotted and the data compared by statistical methods (see

Table A). As discussed in Part I, the equivalence of lineal fraction and sur- face fraction of one microccxistituent may be claimed only if the angular

distributions cire equal for the two microconstituents . Results show that this

equivalence holds for the low temperature specimen but not necessarily for the high temperature specimen. Therefore, only for the brittle fracture case is the pearlite surface fraction on the fracture surface given by the fraction of

pearlite on the lineal trace of the fracture surface. The relatively low pearlite lineal fractions on ductile specimens nevertheless indicate that the fracture path favors ferrite at the expense of pearlite in the ductile mode

and that void growth takes place preferentially in the ferrite . No preference

has been shown in the brittle mode

.

According to Shieh's nomenclature [1], the fracture path preference index

of the two constituents is equal for both constituents (and is therefore equal

to unity) in the brittle mode, but is not equal for both constituents in the ductile mode

.

3) Roughness

Lineal roughness was measured using the Hewlett-Packard digitizer. A line perpendicular to the specimen axis was used as the reference line for all roughness measurements. Lineal roughness values for 5 specimens are included

in Table B. No significant differences or trends in lineal roughness could be shown between samples broken at different temperatures or between parallel and

perpendicular cuts on the same sample.

H) Curvature

Local planar curvatuj?es were computed from the output of the HewlettPackard digitizer. For each of two specimens, the coordinates of the linear

fracture surface traces were collected on four photomicrographs at 0.01 inch

intervals . At each point , the curvature was computed for line segments varied

in length from 0.01 to 0.5 inch on both sides of the reference point. The radius of curvatiire was calculated from the equation of the circle of curvature based on the midpoint (i.e. the reference point) and the two end points of the line segments. Figure 3 shows graphically the variation of the averages of

the local radii of curvature as a function of the line segment lengths . The significant results are found at the shortest line segments since they represent

the true local curvature, limited only by the precision of the tracing device.

With long line segments, the local curvature is obscured by the large-scale

roughness of the surfaces

.

At a line segment length of 0.01 inch, the average radius of curvature of

the brittle fracture surface trace (Specimen No. 1) is found to be about 50% larger than the corresponding average of the ductile fracture surface trace

(Specimen No. 5). The actual values for the two specimens are: 19.4, 22.7,

23.8, 28.9 inches, and 30.8, 31.0, 31.1, 35."+ inches, respectively. The

lengths are reported in inches, as measured on the photomicrographs, and are not corrected for the magnification factor (lOOOx). These results indicate

that the local planar curvature of the fracture surface traces can perhaps be

used to characterize a fracture surface, although it remains to be shown

whether the values obtained are indicative of the fracture mode. Conclusion

The techniques of quantitative microscopy which enable quantitative

infoimation to be obtained from measurements made on metallographic sections

of the fracture surface, were applied to Charpy V-notch specimens of 1018

steel. Statistically significant restilts indicate that the fracture path

271
J. R. Pickens and J. Gurland

avoids pearlite in the ductile mode but appears to have little or no preference in the brittle mode . No statistically significant difference in rough- ness between the two modes of fracture could be detennined. However, the average local planar curvature of brittle and ductile fracture surfaces was found to be appreciably different.

Acknowledgements

The authors gratefully acknowledge the financial support of the U. S. Atomic Energy Commission, Division of Physical Research, and the invaluable

help of Professor S. Bums, who initiated this work, cind of Mr. Paul Williams,

undergraduate research assistant.

References

W. T. Shieh, Met. Trans., 5^(1974) 1069.

E. E. Underwood, Quantitative Stereology, Addison-Wesley Publ. Co.,

Reading, Mass. 1970.

F. W. Boulger and W. R. Hansen, Trans. Met. Soc. AIME 227 (1963) 212.

J. A. Rineholt and W. J. Harris, Trans. ASM 43 (1951) 1175.

Fracture Test

Temperature

°C

-193.5

+ 100

Table A Pearlite

^ 100

Ferrite b 75 Average Angular Average Angular cc Measurement Measurement

6.30°

1.74°

Table B

7.60°

11.8°

50 O 25 - 0

FIGURE 1 IMPACT ENERGY

Fracture Test Temperature

°C Linear Roughness

Parallel Cut -193.5 2.54

-73.5 2.18

-28.5 2.08

0 2.50 100.0 2.30

FIGURE 3 0

XO

o

"0 So 9

0

o o

X o

0 SPECIMEN NO. 5 0

X SPECIMEN NO. 1 X

X X

1

1 1 1 1 1

1 1

1

.02 .03 05 .1 .2 .3 .4.5 LINE SEGMENT LENGTH, (inch)

25 -

20

15 -

10 -

5

-

VOLUME

• AVERAGE

LINEAR PEARLITE FRACTION ALONG FRACTURE SURFACE

PROFILE

I I I -200 -100 0 100 TEMPERATURE, °C

a. Brittle b. Ductile

FIGURE 2

272
National Bureau of Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

THE EQUIVALENT FOIL THICKNESS PRINCIPLE

by S. M, El-Soudan

i

Department of Metallurgy and Materials Service. The University of Cambridge, Cambridge. England

SUMMARY

The introduction of stereological methods to quantitative fractography is deemed possible provided certain conventions are adopted and remain consistent

throughout any comparative analyses. The analogy between quantitative metallographic and fractographic relations is demonstrated using several examples

whereby the analyzed features are truncated outside the test volume. These

findings are formulated in terms of "the equivalent foil thickness principle."

INTRODUCTION

In a recent publication the author [l] introduced several applications of stereology to quantitative fractography. The region of a fractured metal swept

by the plastic zone of a propagating crack was modelled as a thin film that was

siibjected to a special technique of pro jected-image analysis. In view of the complexity of the fracture process, it may sometimes be convenient to confine

the test volume to a region of the plastic zone upon which the fractographer may

wish to focus his attention. In doing so, one must keep in mind that in order to obtain useful correlations using stereological methods a selected approach must

remain consistent from one specimen to another, and also, when comparing fracture

behaviors in different materials. Such an approach to a rather complex problem

is basically sound. However, the ultimate success of a stereological theory

characterizing fractures will be assessed in terms of its consistency and ability

to interpret the observed fracture behavior of materials.

In this paper we shall analyse a basic problem which simplifies the introduction of stereological principles to quantitative fractography. In singling out a test region of the plastic zone, the fractographer often finds himself con- fronted with the problem of truncation of features outside the region of interest.

In fact, metallographers would face the analogous situation in thin film microscopy of dispersed-particle structures. A useful principle was introduced [l] without derivation for dealing with these situations. This principle, heretofore

to be called "the equivalent foil thickness principle," may be formulated as follows :

(a) Metallography : "In order to account for the truncation of convex particles 6 , whose mean intercept length is g at the sxirf aces of a thin foil of thickn«ss t, we may without changing the volume fraction of particles replace the given foil with another having a thickness t + and solve the problem for

the new foil assximing no truncation of particles."

(b) Fractography ; "In order to account for the truncation of convex voids

6, whose mean intercept length is [l^Jg at the boundaries of the maximum-profileheight region of thickness, E, enveloping the fracture surface, we may without

changing the volume fraction of voids replace this region by its equivalent of thickness E + [l^Jd and solve the problem for the new fracture zone assuming no

trxincation of voids."

The analogy between the problems of metallography and fractography will

beccme apparent as we treat several cases of analysis. For simplification of

the notation to be used, we shall omit the subscript g throughout the derivations and include it only in the final expressions.

273
S. M. E/-Soudani

ANALOGY BETWEEN SEVERAL METALLOGRAPHIC AND FRACTOGRAP H I C PROBLEMS

A. Effect of Truncation Without Overlapping of Particles on their Pro- jected-Image Area Fraction (See Fig. 1) Suppose that we need to determine the total projected-image area fraction,

[a^] , of the features 6 of one size class r. We may classify the 6 particles

contributing to the pro jected-image area fraction into two groups: those whose

centers fall inside the foil, and those with their centers falling outside the

foil. For the former group we may write W = [N;r"-^ = N^twr2 (1)

where r is the sphere radius, t is the foil thickness, [n^]"'^'^'^ is the number of inner particles per unit area of the projected image, and N^ is their number per unit volume. For the outer particles, we may also write

u^r^' = [n^]°^" [^1°^^ = 2N^ [|l [jr""' (2)

where H is the mean tangent diameter and ['^^'J '^ number of outer B par- ticles per unit area of the projected image. _

The mean projected area of the outer particles [_A'] may be determined

by methods of geometrical probability as follows:

A' p (A')dA' (3) A'mm

where the integration is confined to the outer 2 particles. The product p(A')dA'

is the probability that a given outer particle will have a projected area between

A' and A' + dA'. Defining an angle 9 as shown in Fig. 2, and assuming that the intersections of g particles with the foil surface will be such that all values of

6 from O to ir/2 are equally probable, it can be shown [2] that p(A')dA'=sin 9 d9.

The integration in Eq. (3) may thus be carried out in terms of 9 as follows:

/•tt/2 f—^l out

IT (r sin 9 ) ^ sin 9 d 9 = j irr^ (4) C Substitution of Eq. (4) into Eq. (2) and the addition of Eqs. (1) and (2) yields

the total pro jected-image area fraction of the 3 particles as follows

where for the spheres 6 of one size, r, we note that the mean intercept length of particles {1^2% ^ ^^/^ [^1 '

• In the absence of truncation, we obviously have

l^kT"- -KV^f

Let us now assume that we have j size classes. The simunation may first be carried out over each size class individually, assuming all other classes to be

absent, followed by the summation over all size classes. It can be shown that the analogous expression to Eq. (5) takes the form

and in the absence of truncation, we also have

where F^' = Z [u^]] / [n^J^, and [I^]^ = 4 Z, [nJ . 3 E . [n^] . rj

Comparison of Eqs. (5) , (6) , (7) and (8) shows that the effect of truncation can

be accounted for by substitution of t + [L3]g instead of t.

274
The equivalent foil thickness principle

Figure 1 - A thin foil containing spherical particles of one size class truncating without overlapping at foil surfaces.

Figure 2 - Geometrical relations of spherical-particle

intersections with foil sxirfaces.

275
S. M. El-Soudani

The derivation in the case of randomly shaped convex particles may be car- ried out in a similar manner to the above cases and will, therefore, be omitted.

B. Effect of Truncation of Features on Volume Fraction Measurements

Several expressions have been derived [l] relating the volume and surface

fractions of convex voids appearing on the fracture surface as equiaxed dimples. The following cases are of interest:

(1) Volume fraction of voids which are neither truncating nor overlapping

and are intersected along their outermost perimeters by the fracture surface:

where, as before, E is the maximum profile height of the fracture surface and

[Sg]g is the surface fraction of dimples.

(2) Volume fraction of voids for the case of truncation without overlapping, and the fracture surface intersects all voids along their outermost perimeters except, for physical considerations, those with centers in regions I&V(Figl)

/UlJ, + E} (10)

t (12)

(3) Volume fraction of particles in a thin foil of thickness, t: The volume fraction of B particles for the case of "no overlapping or truncation" is given in Ref. [2] as follows

[v^lg = [a^Ib^sIb

For the case of truncation of particles without overlapping, the volume frac- tion is expressed by [2]

[^v^B = [^A^B

-

where [s^^g is the surface area of particles B per unit volume. Substitution

of the Tomkeieff Equation [s^lg = ^ [\]g / t^3^B arrangement yields

Comparison of Eqs. (9), (10), (11) and (13) indicates that the fractographic

analogues of the foil thickness, t, and pro jected-image area fraction, [^^Ig > are respectively, the fracture-surface roughness index, E, and the true surface fraction, [Sg]g

.

DISCUSSION AND CONCLUSIONS

In the previous section we have treated two different problems: the determination of the pro jected-image area and volume fractions of dispersed particles

which are truncating at the surfaces of a thin foil of thickness, t , and the measurement of volume fraction of voids intersected by a fracture surface having

a surface roughness index, E . The analogy between the two cases, which follows directly from comparing Eqs. (5), (6), (7), (8), (9), (10), (11) and (13) is quite clear, and appears to be of a general nature. This analogy may be used

primarily as a mathematical device in simplifying rather complex derivations.

In treating such cases, one hypothetical ly replaces the test volume of thickness

t or E with another of thickness (t + iT^) or (E + L^) » respectively, and treats

the problem ignoring truncation effects. The proposed quantitative fractographic

model outlined in detail in Ref. [l] is, therefore, based on the employment of stereological methods to the maximum profile-height region enveloping a fracture

surface. REFERENCES

1. S.M. El-Soudani, "Theoretical Basis for the Quantitative Analysis of Fracture

Surfaces," Metallography, Vol. 7, No. 4, pp. 271-311 (1974).

2. E.E. Underwood, Quantitative Stereology, Addison Wesley, Reading, Ma. (1970).

276
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

STEREOLOGICAL METHOD FOR ESTIMATING RELATIVE MEMBRANE SURFACES IIM FREEZE-FRACTURE

PREPARATIONS.

by E. R. Weibel, G. A. Losa and R. P. Bolender Department of Anatomy. University of Berne, Berne, Switzerland

INTRODUCTION

Freeze-fracture techniques (1,2) have become increasingly popular in the study of cellular membranes because they reveal some aspects of internal membrane structure, particularly a characteristic distribution pattern of protein particles (3) . This is due

to the tendency of membranes, in frozen samples {-100°C), to split

in half between the two phospholipid leaflets whenever they are hit by the fracture plane. From the point of view of stereology

this means, however, that the observable "fracture surface"

follows certain elements of structure, namely the membrane "midplane", and is hence not independent of the structure.

It is the purpose of this paper to show that under certain con- ditions a stereological analysis of the relative surface of membranes of various classes can nevertheless be performed. The method to be developed should allow us to estimate the relative surface of membranes characterized by a certain particle density.

Denoting particle density with a (measured in particle number per

pm^ ) we will attempt to estimate by counting procedures the relative surface of "a membranes"

Sc = s /S„ (1) Sa a T where S„ is the total membrane surface.

MODEL AND METHOD

Let the material be composed of spherical membrane vesicles as they occur, e.g., in microsomal fractions. If a horizontal fractiir?e plane through the ice matrix of the frozen suspension hits

the vesicles above the equator the fracture surface is deflected

upwards forming a convex profile; if the vesicle is hit below the

equator a concave profile is formed. The probability that any profile is formed is proportional to the vesicle diameter 2R; with a numerical density of vesicles N^ the numerical density of profiles

N^=N^.2R (2)

It is useful to restrict the analysis to concave profiles only.

The reason is that the two membrane halves are asymmetrically

loaded with particles; there is adequate evidence that a concave

profile of a specific type of membrane will mostly if not always

display the same membrane face. It can further be shown that the

density of protein particles per unit area of concave membrane

face is characteristic for various membrane classes. The density

of particles a on concave faces may therefore be used as a label

for membrane types

.

The particles are revealed by shadow casting the fracture surface with platinum at an angle of 45° to the normal to the frac277
f. R, Weibel. G. A. Losa and R. P. Bolender

ture plane. As a consequence many profiles will display a cast

shadow which obscures the particles. A certain fraction of profiles will however be devoid of cast shadow, namely those generated by a fracture plane hitting the vesicle in the region marked

with X in Fig. la, namely the cap between the lower pole and the

tangent point of a 45° line. Since

X = (1- Vl/2) -R (3) we find the numerical density of profiles without cast shadow to be

N * = (l-y[l/2) -R-N (4) A - » ' ' V

It can easily be shown that, if the sphere radius R is variable,

the numerical density of profiles without cast shadow depends on

the mean vesicle diameter:

N^* =

(1- VT72")R-N^ (5) The number of concave profiles without cast shadow hence depends

directly on the arithmetic mean radius of the vesicles. Since the

total profile number is

= 2R-N^ (6) the relative frequency of profiles without cast shadow is

(l--v/I/2)/2 2i 14.65% (7)

meaning that about 15% of all profiles should be concave without

cast shadow. This has been shown to be the case. This relationship

is independent of vesicle size, so that the study of concave shadowless profiles affords an unbiased sample of the vesicles in the

fraction

.

NUMERICAL FREQUENCY DISTRIBUTION OF VESICLES WITH RESPECT TO

PARTICLE DENSITY

The first measurement that can now be performed is the estimation of the relative number of vesicles with particle density a:

N., = N /N^ Na a T To this end, a small test circle of known area is centered onto concave shadowfree profiles and the particle number in the test

area is recorded. This leads immediately to a numerical frequency

distribution of vesicles with a as the variable.

As shown above the restriction of the sample to shadow-free

concave profiles has not introduced any bias. However, the use of

a test circle of finite size g to measure particle density (Fig.

lb) has eliminated from the sample all profiles whose radius is

smaller than g. This will affect small vesicles more severely than

larger ones (Fig. lb) and will thus introduce a bias. Eq. (5) be- comes , _ L/ i,2_„2 N^' = N^ [V^'-g" - R-vfi/s] (8)

It is hence not a constant function of vesicle size but depends on the difference between F and g. This may introduce a bias into the

numerical frequency distribution of particle density if membranes

of different values of a had a tendency to form vesicles of largely differing size. However, it can be shown that the choice of a small enough test circle (R/g > 6) reduces the sampling error to less than 5%.

278
Estimating relative membrane surfaces in freeze-fracture preparations

A further bias may have been introduced in the estimation of

particle densities on the projection of a curved surface onto a plane. Due to this a is an apparent particle density which is an overestimate of true density as a function of membrane curvature.

It can be shown, however, that with small test circles this over- estimation is inappreciable.

MEMBRANE SURFACE DISTRIBUTION OF VESICLES WITH RESPECT TO

PARTICLE DENSITY

The" surface freguency"or relative membrane surface of vesicles

with particle density a is

^Sa = V^"/^V-^

where sa and s are the mean vesicle surface in class a and in the

total population, respectively. The problem is to infer Sg from

the estimated numerical frequency of profiles

It is evident from the above that Nj^^ is an unbiased estimate

of Nyq(/Nv if the size of vesicles in class a does not deviate

greatly from that in the total population. If the distribution of

radii in a agrees well with that in the total population we also

find that the mean surface areas sa and s are equal. In this case

S„ = N^, (10) Sa Na

If a vesicles show a different size distribution from that in the

total population this will affect both the sampling bias and the mean surface of the vesicle, which is proportional to the second

moment of the distribution. Expressing the vesicle radius R as a multiple of the test circle radius g, such that R = q-g, we find

^Sa = ^N^j^ [l-E (g ijq^-l) /E (q^ )] / [ 1-Ea (q ^q'-D /Ea (q^ )] (11)

It can easily be shown that, if q > 3, i.e. if the test circle is

smaller than one third the vesicle radius, this can be approximated by

= N^^

• [Ea (q^ ) /E (q^ )] = N^^ • [Ea (R^ ) /E (R^ )] (12)

where Ea(R^) and E(R^) are the second moments of the size distribution in class a and in the total population.

The membrane surface distribution among the various particle

density classes hence follows from the numerical frequency distribution, but the ratio of second moments must be used to correct

for a bias due to unequal size distribution of vesicles derived

from various membrane types. Experimentally we have found that the

second moments of the size distribution in the subclasses investigated deviated no more than 5 to 10 percent from that of the total

population, which probably related to the fact that the microsomal

preparations were obtained by differential centrifugation which

selects a certain range of vesicle sizes.

DISCUSSION

The method here developed has enabled us to quantitatively re- partition the pool of membrane vesicles in a microsomal fraction

to the various classes of cellular membranes from which they were

279
f. R. WeibeJ. G. A. Losa and R. P. Bolender

derived; this gave a very close agreement with parallel cytochemical estimates of relative membrane area (4,5). This method permits

to exploit the technique of f reeze-fracturing for quantitatively

classifying membranes on the basis of their characteristic particle densities.

The only assumption made was that the vesicles were spherical,

a condition fulfilled in our specific preparation. In further development this type of analysis should be extended to become applicable to other vesicle forms.

Acknowledgement

This work was supported by grants from the Swiss National Science

Foundation.

References

(1) Moor, H,: Freeze-etching. Int. Rev. Cytol. 2_5' 1969

(2) Branton, D.: Exp. Cell Res. 45, 703, 1967

(3) Branton, D.: Phil. Trans. Royal Soc. London B, 261 , 133, 1971

(4) Losa, G.A., Bolender, R.P., and Weibel, E.R.: Quantitative

characterization of microsomal membranes by freeze-etching.

(in preparation)

(5) Bolender, R.P., Paumgartner, D., Losa, G.A., and Weibel, E.R.:

Correlated biochemical and morphometric analysis of liver cells,

(in preparation)

Fig. la Fig. lb

280
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

APPLICATION OF VARIOUS STEREOLOGICAL METHODS TO THE STUDY OF THE GRAIN AND THE CRACK STRUCTURE OF CONCRETE

by Piet Stroeven

Materials Science Group, Stevin Laboratory, Department of Civil Engineering.

Delft University of Technology, Stevinweg 4, Delft, Netherlands

Various statistico-geometrical as well as deterministic stereological methods have been applied to study the structure and the structural defects in plain

concrete, subjected to a compressive loading. Hitherto, use has been made of an artificial spherical single'-sized ceramic aggregate instead of coarse gravel for most of the investigations. Crack patterns which have been visualized with the help of the filtered particle method (based on fluorescence), have been analysed

by means of the method of directed secants. Apart from structural information

and results for engineering applications, this approach facilitated a mutual com- parison of various stereological techniques in this field of materials science.

ZNTRODUCnCN

At present, though still hesitatingly, a definite interest is being exhibited in the more fundamental background of concrete material behaviour. An in- creasing number of research projects is nowadays conducted in which the phenomenological approach is supplemented to a larger extent with results from structural investigations than ever before. It is found, of course, that the methods used

to tackle concrete problems are quite comparable to those used by investigators

(sometimes for decades already) in other diciplines. Analogous problems had to be solved in fields like metallurgy, petrography, mineralogy, ceramics technology,

etc. As yet, however, the use of stereology in concrete materials science is rather unique. As for concrete, such analyses have been handled exclusively for petrographic investigations by Larsen and for porosity measurements by Lauer, among others.

It is clear that many examples can be indicated where the investigators

could have taken full advantage of the stereological techniques, e.g. in the in- vestigation of grain anisotropy by Wu and Karl, in a study of surface cracking

by Bennet and Raju, and by the establishment of a semi-empirical relationship between A^ and for glas marble concrete by Pigeon!

In order to attain that stereology will be accepted, however, as a suitable

approach to the determination of the relevant features of the grain, the crack or

the pore structure in concrete, one should direct his attention to the development of labour-saving methods. For the time being, a manual step cannot be avoided in the evaluation procedures, which prevents reaping full benefit from

the usage of (automatic) image analysers (Stroeven, 1975).

Since various aspects of this work have been published elsewhere (Stroeven,

1973a, 1973b, 1973/74, 1975) I will concentrate at present on the newly obtained

results and the further elaborated primary data. For a good understanding, however, a rough sketch has to be presented of a more general framework for this particular application of stereology.

STRICTURAL EXPERIMENTS

An artificial aggregate was used in the applied mixes instead of the coarse

gravel. The weight fraction of this single-sized spherical material (diameter D =

16 mm) has been varied between 10 and 50% (with respect to the amount of sand in

the mix)

.

Cubical specimens with linear dimensions of 200 mm were obtained by removing a disturbed boundary layer of 25 mm from larger units. Then, approximately 11 mm

thick tiles were sawn from the specimens, the sawcut measuring about 5 mm. Finally, all sections were photographed, what resulted in 35 to 40 images per specimen,

all presenting an effective area element (AE) of 200 mm x 200 mm. The structural analysis has been performed in either of the following ways:

a. by random sampling, followed by statistico-geometrical analyses of the selected image(s), yielding average structural information;

281
p. Stroeven

b. by uniform sampling and by subjecting all successive images to statistico-geometrical analyses, a procedure which reveals structural gradients in addition to overall information;

c. by serial section reconstruction performed by a computer; this method offers insight into the complete configuration (Hammersly) apart from all information supplied by following procedure b. Concrete is a very complex, macroscopically heterogeneous material. As a consequence, being confronted with a coarse structure, one has to select an AE of relatively large size in order to be representative for composition. Therefore,

the AE will coincide usually with the sampled sub-area in our experiments. On

the contrary, the size of the AE will be generally inadequate to attain configuration homogeneity. This problem has to be solved by scanning a large quantity

of sub-areas.

STEREOLOGICAL ANALYSIS

a. Grain structure

An areal, a lineal, a features count and a random secants analysis have

been applied for the acquisition of volume fraction data. For the experiments

falling in category b, the last method was selected, since it is the least time

consuming operation in stereology. The other methods were used, according to procedure a, for the structural analysis of all three mixes and in accordance with

c, for the study of the grain structure of the intermediate mix. In the latter

case an IBM-1130 computer performed all operations.

Attention was also paid to characteristic distances like the nearest neighbour distance (in 2 and 3 dimensions) and the free spacing. Apart from average

stereo logical data, the serial section technique also yielded information with

respect to the complete configuration of the spheres.

The lineal analysis executed by the computer has proved to be convenient

to estimate the size of the RAE for composition as well as for conf iguration_

homogeneity. On the one hand, the mean intercept length covering the mortar L , is related to the mean free spacing of the spheres A, and by that, to the compo- sition of the mix, i.e. V,,. On the other hand, the distribution of L reflects

V . m . the complete range of gram distances. However, this part of our experiments was described previously (Stroeven, 1973b, 1975). In the near future computer generated random states of single-sized spheres

will be analogously analysed. The density will be gradually increased up to values met in the experiments. By transforming the growing amount of overlap into

displacements of the proper spheres according to prescribed rules, we will try

to approach "reality" as revealed by the stereological analyses.

b. Crack structure

The lineal features in a plane, cons-tituted by the crack pattern, have

been analysed with the help of Saltikov's method of uniformly spaced directed

secants. Sections situated parallel to the loading direction (the axis of sjrmmetry) will reflect the representative features of the spatial crack structure as indicated by Underwood (1968, 1970) and described by Milliard (1962). From an engineering standpoint this approach is promising, since the crack development process under increasing loads can be interpreted in tangible structural terms, like crack density, or total specific crack length and the com- plete orientation distribution (or, simply, the degree of orientation u)

.

RESULTS

The degree of experimental scatter has been compared with the calculated

values of the variance for methods of volume fraction analysis (Hilliard, 1968)

.

The data have, therefore, been subjected to a suitable regression analysis, in order to eliminate the effect of segregation in the "rich" mix and the edge

disturbance effect in the lean one. In accordance with systematic point count results, all applied methods have demonstrated to be more accurate than predicted (Table 1). Moreover, in the case

of the intermediate mix, the various methods showed about the same degree of scatter when based on the same subarea. This confirms a study of Hilliard & Cahn, who reported that the variance of an optimized analysis is primarily determined

by the number of observations, the type of observation being of secondary importance .

282
The study of the grain and crack structure of concrete

Table I. Comparison of results from various methods for volume fraction analysis

Reference Experiment (Z) CV Theory (Z) CV

e Experiment CV (D) e Weight Directed Areal Lineal Features Areal) Lineal) Directed Areal Features CV^(L)

f raccion secants count secants count

(Z) (D) (A) (L) (D (A) (L) (D) (A) (F) 50 36.89 8.14 6.50 0.80

30 22.60 20. 12 21.00 20.63 14.2 1 1.21 10.05)' 11.0 10.4 0.90

10 9.23 16.43 15.28 0.93

CV^(A) CV fA)

) value could be biased (Moran) , is confirmed, however, by an identical result (CV = 10.2) obtained from a random set of 11 out of 38 images. CV • coefficient of variation /c/^*^ 1600 mn (50Z) 2800 mn (30Z) 4000 ran (lOZ)

To be able to interpret these results in a more general way, one can com- pare the formulae for the variance of an areal (A), a lineal (L) and a systematic point count (P) analysis over the complete range of volume fractions (without

taking into account the possible violations of the assumptions which underlie

the formulae) . The following inequalities can be derived readily from Fig. 1 .

:

CV(P) < CV(A) < CV(L) for < 0.26

CV(P) < CV(L) < CV(A) for > 0.26 where CV is the coefficient of variation.

As a consequence, for practical values of V^, the systematic point count

is expected to show the lowest variance. It is worthy of mention that the mutual comparison is concerned with the intermediate mix, which is situated in the range where CV(A) * CV(L)

.

The line spacing 5 in the lineal analysis, however, has been based on the condition that on the average every_circle in the image plane should be cut by a single traverse (thus, 5 = 10 mm ^ x = /3 D). The calculated value of the variance of the lineal analysis will be only 70% of that of the areal analysis,

since, in doing this, the number of intercepts will be doubled as compared to the ntimber of features. The experimental scatter of L has as yet not been analysed. The variation in the material composition will manifest itself in a damped way in the scatter of the bulk modulus K. These stiffness fluctuations can be approximated by means of the Hashin & Shtrikman bounds for a two-phase model

(which is in this case a reasonable assumption according to Brown) . Accepting a CV of, sa^, 5% for mixes used in practice, an RVE with linear dimensions of 200. (3.6/5.0) ^ 100 mm, i.e. 6.5 times the grain size, is required, which corresponds to results of a study on the variance in strain gauge readings (Cooke & Seddon)

.

The configuration homogeneity, on the other hand, was studied by means of

the discrete intercept length distribution F(L^), where the index m refers to the mortar. Such histograms, as shown in Fig. 2 have been obtained by subjecting 18 successive sections of a specimen of the intermediate mix to a lineal analysis

with the help of uniformly spaced traverses; an operation which has been re- peated for 7 different orientations 6^ of the line array. The theoretical density

distribution of L (representative for a very large AE) was approximated by cal- culating the average frequency distribution with respect to the location of the section and the orientation of the line array. The differences between the "theoretical" distribution and a histogram re- presenting an increasing ^umber of subareas N for a given value of 6^ ^an be ex- pressed by means of the X -value. Mean and standard deviation of the y -values

fgr a given value of N facilitate the determination of the confidence limits of

X. Finally, a suitable regression analysis by means of hyperbolic functions pro- vided a series of continuously decreasing functions with rising values of N (Fig.

3. The size of the RAE for configuration homogeneity could be obtained by re-

^ quiring the various curves to attain different levels of significance for the x ~

283
p. Stroeven

value. The solutions for N were located in the range from 24 to 36. This means

that the linear dimensions of the RAE should be 5 to 6 times larger than the sampled AE (i.e. 1000 to 1200 mm). And, as a consequence, the linear dimensions

of the RVE's for structure insensitive (e.g. K) and structure sensitive properties, respectively, will differ one order of magnitude.

The final conclusion, drawn from data previously published on the distribution of the spheres in concrete, is that the largest deviations from the random

state occur locally. Particularly the nearest neighbour distribution in space was

shown to be strongly influenced by this tendency to strive for (local) order. On

the contrary, the free spacing was expected to be less governed by this tendency.

Therefore, the average value of the mean free spacing was calculated for 33 spheres included in a hypothetical cubical element, centrally located in the re- constructed part of a specimen, containing a total number of 455 spheres. A completely ordered system (based on a single-stagger configuration) will

give occasion to 40 unobstructed free spacings per sphere, with an average length

of 31.9 mm (V^ = 20.6%). From the random secants analysis a value of 38.8 mm

could be deduced. The "exact" value of X yielded 43.8 mm on the average, being

based on 97 unobstructed spheres. This latter result, particularly, demonstrates

that the average deviations from the ordered state are quite large (which is also confirmed by the size distribution of the circles in the image plane) . In the near future, it will become possible to analyse X in the computer-generated

quasi-random systems, as described before.

The analysis of the defect structure in concrete has been limited, hitherto,

to the zero-state and a state of uniaxial compression beyond discontinuity (say, 75% of ultimate) . The total specific crack length and the complete orientation

distribution have been determined in sections, situated parallel to the axis of symmetry, constituted by the loading direction. Typical roses have been obtained.

Indeed, perpendicular to the axis of symmetry an isometric system was revealed by

the circular shape of the rose. In addition, the dependence of the properties of the roses on the orientation of the section with respect to the axis of symmetry

is now being studied.

Finally, we are at the moment investigating axial sections over the complete stress-strain range in uniaxial compression in order to gain information with

respect to the size of the RVE, the onset of structural loosening and the mechanisms of cracking that govern the different stages of the mechanical behaviour of concrete. A single example is presented in Fig. 5. CaCLUSICNS

The stereological approach to the study of the grain structure and, particularly, the defect structure in concrete as a function of the loading history,

the mix proportions, the casting and densif ication techniques, etc., is very in- trigueing from a methodological standpoint, but also very promising with respect

to the engineering interest. The time-consuming operations of preparing large

(or a large number of) specimens and of executing the stereological analyses - accepting for the time being that a manual step cannot be avoided in the data acquisition - are a handicap for the investigator in this particular field of materials science.

284
Fig. 1. Comparison of the variance of optimized lineal (L) areal (A) and systematic point count (P) analyses

applied to a dispersion of singlesized spheres

I CV = 1.35(1 - V ),= CV(L)/CV(A)

II CV : = 0.91(1 - VJ;)^= CV(P)/CV(A)

rel V

Fig. 4. Combination of roses-of-thenumber of intersections in a polar

figure and according to the orientation of the relevant section in the specimen.

Different loading stages are indicated.

Fig. 2. Histograms for the intercept

length through the mortar.

A: subarea size 200 mn x 200 mm.

B: collected values representing 15 subareas

.

C: collected values for 18 subareas

and 7 different orientations of

the line array.

Fig. 3. Deviations between the in- tercept length distribution in the mortar for a total number of N subareas and for a representative area element with respect to configuration

(viz.^Fig. 2) quantified by means of

the X -tests. The regression curves represent the average vglue with re- spect to orientation, x » and the three upper confidence limits t for n = 7 and for the significant

levels a = 10%, 5% and 1%, respectively. For further explanation see

the text.

285
p. Stroeven

2500 2500

Fig. 5. Lineal features in a plane as constituted by the copied crack patterns.

The patterns represent the state of disrupture at the surface on an axial section of a cubical specimen shown in Fig. 11 of Stroeven (1975). The specimen has been subjected to a relatively high uniaxial compressive load. The pattern at

the right only reveals the more advanced cracks. The roses of the number of inter- sections at the bottom present the orientation distribution of both patterns. As can be observed, the microcracks which have joined under relatively high loadings

have caused the degree of orientation oj to increase

.

: 0)= 17% : 0)= 29%. Dimensions in picture points (Quantimet 720).

REFERENCES

flcnnet, E.U. i Raju. N.K. (1971) CiamilaCive fatigue damage of plain concrete in compreBsloD. In; Proc. Conf. Structure, Solid Mechanics and

Engineering Design, 1096. J. Wiley & Sns. Brown. C.B. (1965) Hiniraum volumes to ensure homo- geneity in certain conglomerates. J. Frankl.

Inst. 279. 189. Cooke. R.W. S Seddon. A.E. (1956) The Uboratory

use of bonded-wire electrical resistance strain-gauges on concrete at the Building

Research Station. Hag. Conor. Rea. 8, 31. Hashin, 2. & Shtrikman, S. (1963) A variational approach to the theory of elastic behaviour

of multiphase materials. J. Meah. Phya. Solids, 11. 127. Hammersly, J.M. (1972) Stochastic models for the distribution of particles in space. In: Proc.

Symp. Statistical and Probabilistic Problems

in Metallurgy, 17. Suppl. Adv. Appl. Prob.

Is el. Hilliard, J.E. (1962) Specification and measurement

of microstructural anisotropy. Tram AIME.

224, 1201. Hilliard, J.E. (1968) Measurement of volume in volume. In: Quantitative Microacopy, 45. McGrawHill Book Co. Milliard. J.E. & Cahn, J.U. (1969) An evaluation of procedures in quantitative metallography for volume-fraction analysis. Trans Met. Soc.

AJl'fE. 221, 344. Ursen. G. (1961) Microscopic point measuring. A quantitative petrographic method of determining the Ca(OH), content of the cement paste of concrete. Mag. Conor. Bee. 13. 71. Laucr. K.R. (1971) The effect of the boundary conditions on the air void system of air entrained

concrete. In: Proc. Ree. Conf. SO-years of

Building Institute, 76, Academia, Prague.

Moran, P. A. P. (1972) The probabilistic basis of etereology. In: Proc. Symp. Statistical and

Probabilistic Problems in Metallurgy, 69. Suppl. Adv. Appl. Prob. Israel.

Pigeon, M. (1969) The process of crack initiation

and propagation in concrete. Ph.D. Thesis,

Imp, Coll. London. Stroeven, P. (1973a) Structural investigations of concrete by means of stereological techniques.

In: Proc. RILEM Symp, Fresh Concrete, 5-2-1,

Stroeven, P. (1973b) Some aspects of the micromechanics of concrete. Ph.D. Thesis, Delft. Stroeven, P. (1973/74) Methods for the evaluation

of topological characteristics of the pore structure of concrete; application to crack

porosity. In: Proc. RILEM/IUPAC Syrnp. Pore Structure and Properties of Materials, C529. Academia, Prague. Stroeven. P. (1975) Stereometric analysis of structural inhomogeneity and anisotropy of concrete. In: Proc. Symp. Quantitative Analysis of Microstructures, 291. Practical Metallography Special Issue 5. Dr. Riederer Verlag.

Stuttgart.

Underwood, E.E. (1968) Surface area and length in volume. In: Quantitative Microscopy, 77. McGraw-Hill Book Co. Underwood, E.E. (1970) Quantitative Stereology, 48. Addision Wesley Publ. Co.

Wu, H. S Karl, F. (1970) Gefugeuntersuchungen an Betonen und Diskussion des Gef uge-einf lusses

auf die technischen Eigenschaf ten. Tonind.Z.

94. 449,

286
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

THE USE OF QUANTITATIVE METALLOGRAPHY AND STEREOLOGICAL METHODS TO DESCRIBE COMPOSITE

MATERIALS

by Jean-Louis Chermant and Michel Coster, Groupe de Cristallographie et Chimie du Solide. ERA 305. Laboratoire de Chimie Minirale Industrielle, Universite de Caen, 14032 Caen Cedex, France. .

ABSTRACT

This investigation indicates some examples of using quantitative metallography and stereology, to investigate physical phenomena related to the microstructure of materials. After a critical study of the different methods leading to the choice of the most suitable, we investigate the growth kinetics of tungsten carbide crystals in cobalt. It was shown that the proposed growth law

corresponds to a kinetic which is limited by a second order interface reaction,

and this result was obtained only after employing stereology. The study terminates in an extension of stereology to quantitative f ractography . It was possible to show that for tungsten carbide-cobalt composites, the rupture has a preferred propagation in the cobalt phase and, for the carbide phase, the transgranular fracture represents only a small percentage of the total surface of fracture.

INTRODUCTION

Composites are an interesting example for metallographic investigation

using quantitative and stereological methods (I) (2). There are several ways of analysing stereological problems : either by generalization of known methods to more complex geometry, or by the use of mainly experimental methods. We have

chosen in fact the latter methods to investigate a material of relatively simple structure : tungsten carbide crystals in a cobalt matrix.

We shall briefly give the reasons which led us to use the chosen stereological methods (3) , afterwards we shall give two kinds of application of these methods : the analysis of a growth kinetics and the analysis of fracture paths.

CRITICAL ANALYSIS OF METHODS

The crystals of tungsten carbide are prismatic crystals and their grain

size is close to a micron. It was therefore necessary to choose (i) the observation procedure which is the easiest to use without introducing too large systematical errors, (ii) the stereological method which allows the crystal size in three-dimensional space to be determined the most accurately.

The different methods of observation (optical, scanning and transmission

electron microscopy) were compared. A comparison between scanning and transmission electron microscopy, on replicas, shows that no error is introduced during

the sampling. The discrepancy is small between optical and electron microscopy

and is only significant for the smallest crystals. Optical microscopy is used

to investigate the largest particle size, while electron microscopy is used for

the smallest particle size. Point and linear analysis are comparable for measuring the volume fraction

according to similar kinds of observation. The electron microscopy gives results closer to those of chemical analysis than optical microscopy, but coincidence of

the results is obtained only if the volume ratios, found experimentally , are corrected so as to eliminate the Holmes effect (4), the importance of which is due to the large difference in hardness of the two phases.

The metallographic analysis gives a measure of crystal size in two-dimensional space. If we want know this crystal size in three-dimensional space, we must

use corrective methods which allow us to pass from two- to three-dimensional space (5)-(9). These methods have been established for spherical particles, which is

287
J.-L. Chermant and M. Coster

not the case of WC-Co. We therefore tried to find out if one of these methods

could be used for non-spherical crystals.

To investigate this problem, we have used chromium powder and spheroidal

bronze, as reference specimens. By microscope observations, the true particle

size distribution was established. These powders were mixed with a resin and

the polished surfaces of these resin composites were then analyzed. The results

of the analysis of these sections were transformed by means of the correction me- thods of S.A. Saltykov (5) and A.G. Spektor (9) in order to obtain the repartition in volume.

The Saltykov method is perfectly verified for spherical particles, in spite of the errors which accumulate for the smallest classes. The conformity is not so good for non spherical particles, but the existing errors are kept within

reasonable limits, and the result is a good representation of the space distribution. The Spektor method does not give as good results as the Saltykov method,

in particular, it is sensitive to the chosen class interval and to the non-sphericity of the particles. This is the reason why, subsequently, we used the Sal- tykov method whose geometric progression of classes is well adapted to the TGZ 3 Zeiss analyzer and to particle size of WC-Co. APPLICATION OF STEREOLOGY TO A GROWTH KINETICS

In order to apply stereological methods to the evolution of the microstructure of a material, we first investigated the growth kinetics of tungsten carbide

crystals in cobalt. Whatever the limiting mechanism, the growth of crystals is due to their dif- ference of solubility. G.W. Greenwood, I.M. Lifshitz and V.V. Slyozov (11) and C. Wagner (12) have proposed a kinetic model limited by the diffusion, where the dispersed particles had a diameter increasing as : = Dq^ + kt . Elsewhere H.E. Ex- ner and F.H. Fischmeister (13) and C. Wagner (12) proposed another model limited

by interface reactions with a diameter increasing as : = Dq^ + kt. Finally, E. Hanizsch and M. Kalhweit (14)-(16) have investigated the problem of the growth of dispersed particles from a general point of view. They indicated that for an infi- nite time, there exist a limit distribution curve of the diameter of spherical

crystals growing in a matrix. This limit curve is a function of the type of reaction. These authors have reported for a second order reaction controlled kinetic

that the diameter of the crystals increases as : = Dq^ + kt. The kinetic growth has been investigated on WC-Co materials with a cobalt range from 10 to 25 wt.%, and with two WC crystal diameters : 2.2 and 0.7 ym. The growing was carried out under vacuum of I0~^ torr in graphite crucibles at tempe- ratures between 1350 and 1700°C. The sintering time is between 2 h.30 and 100 h. After sintering the specimens are polished and etched by the standard techniques

used for cemented carbides, and analyzed by point and linear manual counts and with a TGZ 3 Zeiss analyzer. Afterwards the particle size distribution was transformed into the diameter of an equivalent sphere in three-dimensional space. The growth process by difference of solubility can be followed by the growth

path method of R.T. de Hoff (17) and H.J. Woodhead (18). In a first representation for different treatment times the reciprocal of the cumulative curve of the number of crystals per unit volume is plotted

as a function of their diameter. One notes

the large decrease in the number of crystals

and specially that of the smallest, during

treatment. This indicates that it is the smallest crystal that dissolves to the profit

of the large crystals. From this first se- ries of curves, a second type of representation is derived : one plots as a time function the diameter of the smallest of the lar- gest N'v crystals which allows to follow the evolution of families of crystals of different initial diameters (Fig. 1). This plot

puts in evidence a critical diameter

Fig. 1 : Growth-paths for the WC-Co Dj; = f(t) which corresponds to crystals which

15 wt.%, DyQ = 2.2 um, for diffe- neither increase nor decrease at a given morent treatment times at 1450°C. ment. The ensemble of the maxima of the cur288
Quantitative stereology in composite materials

ves at N'v = K gives the variation of this critical diameter as a function of time. This variation is given according to the law : Dq^ = + kt

.

With the growth path method the kinetic appears clearly, and in spite of

the uncertainty of its outline this method is the only one which permits a de- termination of the critical diameter. The use of this method is dependant on a knowledge of the number of crystals per unit volume. This explains the great im- portance which must be given to the use of Saltykov's method, which allows one

to obtain this parameter reasonably accurately.

The evolution of the microstructural characteristics was then investigated

as a function of time. It was noticed that contiguity (19) does not constitute

a determining factor of the kinetic as the following law is obtained : cwc = Co + k"t-o-i2 On the other hand the mean free path in cobalt phase and the mean diameter

evolve according to the laws : l^co = ^o^ ' '^^WC ~ ^o^ The plots relative to the two particle sizes are fairly parallel, the speed constants are therefore pratically constant. On the other hand, the fact that the law in is satisfied is not significant enough for the search for a model.

The evolution of the size distribution curves of equivalent sphere diameters

is given by the cumulative representaWC G-Col5% T:USO°C

"•,t = 2 h30

^

1 log normal m 1 f 1 diffusion

interfacial l^^order

-- '-^ L.

interfacial 2TMtrd«r

1 1 1 1 1 • 1 2 3 4 S D^^c yrr^^^

. 2 : Theoretical limit curves accortion. For each given treatment time, on the same graph were plotted (Fig.

2) : - the experimental data obtained after Saltykov corrections.

- the log-normal distribution curve having the same median and the same standard deviation as the experimental points.

- the limit curve in diffusion-controlled growth.

- the limit curve in reaction controlled growth of the first and second

ding to the different growth processes (K: orders. Kalhweit, R : Reaction) and experimental The comparison of the different

points for WC-Co 15 wt.%, D^^ = 2.2 um. curves with the experimental results

(Fig. 2) indicates that the limit curve of second order interface reaction is the closest to our experimental points. The evolution of the size distribution curves and of the main characteristics as

a function of treatment time indicates that these curves had the same limit form

for every treatment time. It should be noted that such a result can only be obtained by using a stereological corrective method. Indeed the surface analysis

without corrections gives a curve close to the log-normal curve and this is not interpretable from the kinetic point of view for dispersed systems.

The whole lot of obtained results (20) has allowed us, thanks to stereology,

to find that the growth kinetic of WC in Co is governed by a second order interface reation which is : W„ + ->• WC ^ . , and the rate is in accordance with

v = k(W^^).(C,„). ^°

APPLICATION OF STEREOLOGY TO FRACTURE INVESTIGATION

Most methods of quantitative metallography can be transposed to the investigation of fracture paths in scanning and transmission electron microscopies.

Nevertheless some of these methods must be adapted in order to define new para- meters describing the fracture feature. The measure of fracture surface occupied by each type of rupture (trans- or inter-granular) is achieved in the same way as the ratios of polished surface

occupied by each phase. In this case linear analysis seems to be the most effective (20) (22). The crystal distribution curve along the fracture path can also

be obtained by linear or surface analysis. Nevertheless it would be interesting

to calculate the size distribution at the surface of fracture path in terms of

the diameters of the equivalent spheres, rather than the size distribution in volume. It was shown that the Saltykov method can be used for this type of cal- culation after some small corrections (20) (22).

By way of example, we chose to investigate the fracture of WC-Co plates by

289
J.-L. Chermant and M. Coster

three-point bending. Two types of analysis were performed on each fractured spe- cimen : first a polished surface was analyzed in order to determine the grain

size and the principal microstruc tural parameters, and secondly the same kind of study was undertaken on the fractured faces. Point analysis has allowed us to define, on the one hand the fracture ratios in the cobalt and, on the other hand, the ratios of inter- and trans-granular

fracture in the tungsten carbide phase (Table I). Surface analysis allowed to esMicrostruc tural parameters Fractured surface ratios

pm trans-granular % inter-gra- nular Z

un-notched

plate

notched plate

(KjQ specimen)

20

20

20

20

9

2.26

1.05 0.67

2.25

2.25

0.37

0.34

0. 30

0.37

0.73

0.33 0.20

0.73

0.73

3645

46

30

12

6

10

10

12

58

54

54

60

76

Table I : Values of microstructural parameters and of ratios of fractured surface for different composites WC-Co fractured by three point bending : Vv(co) : cobalt volumic

ratio ; Due = mean diameter of tungsten carbide crystals ; Cyj : contiguity of carbide crystals ; : standard deviation. tablish the apparent outlines of the crystals concerned by an intergranular fracture surface (23). The results in this table indicate that the fracture travels preferentially through the cobalt phase and that the fracture is essentially in- tergranular along the carbide phase. A significant difference is observed between

the size distribution curves determined from the polished surfaces and those de- termined from the fracture path. The variation of this difference indicates that

the fractured surface contains larger crystals than the polished surface for a ra- ther small grain size, and the larger the difference is the smaller the grain size. We can thus conclude: for this kind of materials the fracture is predominantly in the cobalt and the essentially intergranular fracture travels along the largest

crystals of the grain size. CONCLUSION

We were not attempting to present, in an exhaustive way, all the possibilities offered by stereology. We wished to show that its use entails a judicious choice of methods and, once this achieved, it could help to solve many

problems, as much in the field of microstructural changes, as in the field of mechanical properties. Thus quantitative metallography and stereology can con- tribute largely to the characterization of materials, and to find correlations

between mechanical, physical and chemical characteristics as a function of microstructural parameters.

REFERENCES

1

.

E.E. UNDERWOOD - Quantitative Stereology 14. E. HANITZSCH, M. KAHLWEIT - Z . Phys. Chem. Addison Wesley 1970. N.F., 57, 1968, 145.

2. R.T. DE HOFF, F.N. RHINES - Quantitative Mi- 15. E. HANITZSCH, M. KAHLWEIT - Z . Phys. Chem. croscopy, Mc Graw Hill Book, 1968. N.F. , 65, 1 969, 290.

3. J.L. CHERMAfIT, M. COSTER, A. DESCHANVRES -

16. E. HANITZSCH, M. KAHUffilT - Symposium on Metallography

,

8, 1975, 121. Crystallization, 1969, p. 130.

4. J.W. CAHN, J. NUTTING - Trans. AIME, 215, 17. R.T. DE HOFF - Met. Trans., 2, 1971, 521.

1959, 526. 18. J.W. WOODHEAD - Metallography, 1, 1968, 35.

5. S.A. SALTYKOV - Stereometric Metallography, 19. J. GURLAND - Trans. AIME, 1958, p. 452. Metal lurgizdat , Moscou, 1958, 2"'^ Ed. 20. M. COSTER - These de Docteur es-Sciences,

6. E. SCHEIL - Z Metallkde, 27, 1935, 199. Caen, juillet 1974.

7. H.A. SCHWARTZ -- Metals Alloys, 5, 1934, 139. 21

.

J.L. CHERMANT, M. COSTER, G. HAUTIER, P.

8. W.A. JOHNSON - Metal. Progr., 49, 1946, 87. SCHAUFELBERGER - Fowd. Met., 17, 1974, 85.

9. A.G. SPEKTOR - Zavod. Lab., 16, 1950, 173. 22. J.L. CHERMANT, M. COSTER - Metallography,

10. G.W. GREENiroOD - Acta Met. , 4, 1956, 243. to be published.

1 1

.

I.M. LIFSHITZ, V.V. SLYOZOV - J. Phys. Chem. 23. J.L. CHERMANT, M. COSTER, A. DESCHANVRES,

Sol., 19, 1961 35. A. lOST - 4eme Symposium Europeen de Metal12. C. WAGNER - Z. Electrochem. , 65, 1961, 581. lurgie des Poudres, Grenoble, France, 13-

13. H.E. EXNER, F.H. FISCHMEISTER - Z. Metall- 15 mai 1975. kde, 57, 1966, 187.

290
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersburg, ^W., September 4-9, 1975 (Issued January 1976)

A DEFINITION OF THE TWO-DIMENSIONAL SIZE OF IRREGULAR BODIES

by G. M. Timiak

Mineral Research Laboratory. Kosice Technical University, Kosice, Czechoslovakia

ABSTRACT

A method of deriving 2D size ("shape") of irregular, curvi-angular bodies

is given. In this method a statistically adequate ninaber of random planar

sections subjected to threefold processing are used. The section shapes are

converted to "polygons" using the principle of Phines Tangent Count for the

generation of the "polygon" vertices; then the frequency distribution of the

sections with different number of vertices is determined; for the "polygons"

with the greatest frequency of occurrence the area distribution of their

parent sections is established. The most frequently occurring area value of

the "polygon" with the greatest frequency of occurrence is considered to be

the "true 2D size" of the given body.

The 2D size ("shape") of regular polyhedrons has been found to be definable as the most frequently observed area of the most frequently occurring

polygonal random sections (Tim^ak 1973/74) using a two-step sizing procedure

(of. Fig.l). This method has been suitable also for the determination of the true form of sectioned regular polyhedrons.

The above definition was, however, not directly applicable to irregular,

curvl-angular bodies (such as real mineral grains) due to the difficulty to get

a sufficiently distinctive description of the section shapes of such bodies.

It appears, however, that this problem could be overcome if the section shapes

could be approximated to some angular shapes or defined by descriptors of two- dimensional character. A method suitable for this purpose was found to be the Rhines Tangent Count (DeHoff 1967). In this counting, points of tangent of a imidirectionally sweeping line with the perimeter of the particle sections are generated at the loci where the character of the particle section curvature

allows such event. When joined within every particle section, these points

form a network which can be treated as polygons using the method suggested

earlier (cf. Fig. 2). In the first stage of this sizing procedure, the number

of tangent points (TP) can generate "polygons" with different number of edges. The generated "polygons" do not describe the original section shape fully,

but if the sectioning planes are randomly distributed, they provide an unbiased

approximation of the original shape. In case that bodies distributed in space

291
G. M. TimSSk

are considered, the random distribution of both, the bodies and the planes

is important (cf. Timcak 1975). The procedure of 2D sizing would thus consist of three steps: 1) The conversion of the original shape to a "polygonal" one (cf. Fig. 2a, 2b). This

procedure may be performed either using a suitable TV display system fitted

with a light pen, or an acoustic tracing system, linked to a digital computer

(cf. Dvorak et al. 1974, Cowan&Wane 1973) or on a suitably modified TV based

image analyser (cf. Fisher 1971, Muller 1973). 2) The determination of the frequency distribution of the generated sections with various number of TPs

(i.e. edges) (Fig. 2c). At present, this procedure can be performed semimanually or on a computer based image analysing system capable of complex pattern

recognition (cf. Grasseli 1959). 3) The area analysis of the "polygons" with

the most frequently occurring number of TPs (Fig. 2d). For this measurement,

the original areas of the sections are taken and not those covered by the generated "polygon". This type of measurement can be performed by any of the methods mentioned in point 2 or using a suitable TV-based image analyser

(cf. Gibbard et al. 1972, Muller 1973). The example in Fig. 2a shows pagioclasses from an East Slovakian andesite

(C.No. 5001/1) obtained after the image analysis of a thin section. The mineral is chemically eroded and has developed a hypidiomorphic form. The transformed image is shown in Fig. 2b. The frequency distribution of features

with different number of TPs is given in Fig. 2c. A dominance of features

with 4 TPs is evident, so the areas of the parent shapes of these features were analysed next (Fig. 2a). The result (Fig. 2d) shows that assuming that the grains belong to a monodispersed population, the grain section having the "true 2D size" of the analysed mineral is the one with 4 TPs and an area of approximately 40 units^. Grain sections satisfying these conditions were

marked by black triangles (Fig. 2a). As, however, in truth the sections arose

by sectioning a polydispersed population, the marked sections represent a weighted average 2D size of the sectioned minerals.

The stereological reconstruction of polydispersed systems would be analogous to that suggested for polyhedral bodies (TimcSk 1973/74).

Contrary to the polyhedral bodies, the determination of the true form of irregular bodies from the above determined data would be difficult. It may be described, however, in terms of largest and smallest 2D parameters and of partial surface curvatures. This limitation, however, is not critical in cases when only a statistical size (or the distribution of this size) of a body,

or a population of bodies is required.

292
A definition of the two-dimensional size of irregular bodies

LITERATURE:

COWAN W.M. , WANN D.F. (1973): A computer system for the measurement of cell

and nuclear sizes; J. Microscopy, Vol.99, Pt.3, Dec, pp. 331-348

DE HOFF R.T. (1967): The relationship between mean surface curvature and the stereologic counting measurements; In: H. Elias (Ed.): Stereology;

Sprinver V., Berlin; pp. 95-105

DVORAK J. A. et al. (1974): A simple video method for the quantification of microscopic objects; J. Microscopy, Vol.102, Pt.l, Sept., pp. 71-78

FISHER C. (1971): The new Quantimet 720; Microscope, Vol.19, 1st q. , pp. 1-20

GIBBARD D.W. et al. (1972) : Area sizing and pattern recognition on the Quantimet 720; Microscope, Vol.20, 1st q. pp. 37-50

GRASSELI A. (ed.) (1969): Automatic interpretation and classification of images;

Acad. Press, N. York, 430 pp.

MULLER W. (1973): Das Leitz Textur-Analyse-System; Leitz Mitteilungen; Suppl.

I., 4; pp. 101-116

TIMCAK G.M. (1973/74): A probabilistic definition of the 1 and 2 dimensional

"size"; Proc. Pore Symp. RILEM/IUPAC; Academia, Prague, Vol.3, pp. A 33-45

TImCaK G.M. (1975): Stereological analysis of the size and spatial distribution

of idealised random mineral systems; Manuscript; 120pp.

30

f

20

10 -

n—I—I—

r

3 4 5 6 —

I—

10

Fig. la. Frequency distribution of cube sections with different number

of sides. Quadrangular sections

showing greatest incidence.

Fig. lb. Area frequency distribution

of quadrangular cube sections. Quadrangular sections with approx. 13 sp. units represent the "true 2D size" of

the cube (1' is the face area of the

cube)

.

293
G. M. Timcak

Fig. 2a. Thin section of andesite with

discriminated plagioclasses . Dots show the generated TPs. Grains with

superimposed shading have 4 TPs, those marked with triangles represent the weighed "true 2D size" of the sectioned

grain population.

2S

20

v5- 234567890Nrp

Fig.2c. Frequency distribution of grain

sections with different number of TPs.

Fig. 2b. The same image with the natural boundaries supressed, displaying the "polygons" formed by the joining of the TPs of the individual

sections.

40-

30-

20-

10-

Areo

[units']

Fig. 2d. Area frequency distribution

sections with 4 TPs.

294
National Bureau of Standards Special Publication 437

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

STEREOMETRIC MICROSTRUCTURE AND PROPERTIES OF TWO PHASE MATERIALS

by Gerhard Ondracek

Institut fur Materia/- und Festkorperforschung, Kernforschungszentrum Karlsruhe, Germany

The properties of materials are governed by the variables of state, e.g. tem- perature and chemical composition, their jc-ray structure and their microstructure. It is the matter of the present paper to consider the effect of microstructural parameters of two phase matrix materials on their properties at constant state conditions and unchanged x-ray structure.

The dependence of the properties on the microstructure of a two phase matrix

material concerns the geometry and geometrical arrangement of its continuous

(matrix) and discontinuous (dispergent) phases. It leads to the variation of each property between a lower and upper bound, which correspond - stereologically - to the well-known cases of parallel and series array of the materials

phases. There are three stereometric factors completely describing the microstructure quantitatively which determine the special property value of given

real two phase materials in between the bounds: the shape factor, orientation

factor and concentration factor of the dispersed phase (fig. 1).- The width of

Oientatxr

Vo/ioiion

ShQpe^^

Vanationj*

Sze

*

s

" 2 c

c ^

Voriaticr 0 Oq Cooce Von a Case A

I Parallel

I A/roy

Property related

Preference Direction

Case B

ot- Phase 25 50 75 p- Phase Phase Concentration (Vot'/o)

fig.l: Property bounds and stereometric fac- tors of two phase materials

Molr.i- 20 iO 60 80 Einta9«<un. phose phase Concentration (Vol7o

)

fig. 2:Theoretical shape of the electrical resistivity of two phase matrix materials (pj^: pjj=102)

parallel array of the phases |

disc inclusions perpendicular to the electric field A

fiber inclusions parallel to the electric field X

disc inclusions statistically

oriented +

fiber inclusions stat . oriented

spherical inclusions ^ fiber inclusions perpendicular to the electrical field

disc inclusions parallel to the electrical field

series array of the phases ^

295
6. Ondracek

the variational region between the bounds increases with increasing difference

between the property values of the phases (compare phases A and B, fig.l). Ex- tended variational regions therefore occur in porous materials considering

pores as the dispersed phase. But also if both phases are solid^ this difference

may cover several orders of magnitude as for example in the case of cermet re- sistivities (o(Al202/o (Mo)>10''^). In such cases,- the calculation of the property values of a given two-phase material as a function of its microstructure

is technically relevant. Due to that statement^ the electrical resistivity of two phase materials and its dependence on stereometric factors will be the sub- ject treated as an example here. - Equations have been derived treating the two phase material as quasi- homogeneous continuum which provide the theoretical

tool to calculate the changes in the electrical resistivity by variation of the orientation, shape or concentration of the dispersed phase /1-5/. The most general form of the stereometric_function_of_the_elec^

Eh2§s_matrix_materials is

-p p D "^M f{F,cos a) (l-F)cos g + 2F(l-cos a)

(l-F)cos a + 2F(l-cos a)

IP,

ii)(F,cos a)

(1)

f (F , cos a) =

F (1-2F)

2 ? 1 - (l-F)cos a - 2F(l-cos a)

(2)

F(1-2F) 2F(1-F)

ii)(F,cos a) =

2 2 1 - (l-F)cos a - 2F(l-cos a)

2 2 (l-F)cos a + 2F(l-cos a)

- - 1 (3)

Simplifications of equation 1 follow for extreme difference between the resistivities of the phases

I -cos

2 g ^ cos2 g ,22 1-cos g _ cos g

/, X F 1-2F , ,, s F-1 2F

(5)

and also in particular cases of shape and orientation /5/. Using these equations^ resistivity curves as shown in fig. 2 are obtained for an assumed two phase material pointing out graphically the influence of stereometric microstructure. - It is the aim of the indirect stereometric factors introduced in equations 1-5 to take into account theoretically the effect of the microstructure on the properties concerned. Their definition therefore follows obligatory from the derivation of the equations. As usual, if real structures and relationships have to be described quantitatively, assumptions are required leading to approximate results only. Defining the stereometric factors due to equa- tion 1^ these assumptions include that

- the actual particles of the dispersed phase differentiated by size, shape and orientation can be substituted theoretically by the same number of average

sized, shaped and oriented particles, where the irregular particle shape is characterised by an adequate spheroid.

Following this line^

- the phase concentration factor is simply the volume content of the dispersed

phase

- the shape factor is directly related to the deelectrization factor which follows from fig. 3 as a function of the axial ratio /6-9/^ - the orientation factor is the average of the cosinus squares of those angles

which are formed by the rotation axes of the phase particle substituting

spheroids and the direction of the electric field. It is also a function of the mean axial ratio of the spheroids /8,10/:

z

2 P 2 (-) (fr) -1 X „ a

(r)

prolate

spheroids

2 cos g oblate

spheroids

(7) = (^>

A

(6)

296
Stereometric microstructure and properties of two ptiase materials

fig. 3: Shape factor and axial

ratio of spheroids /5,6/

fig. 4: Mean axial ratio of intercept ellipses in plane and axial

ratio by the respective spheroid (—)

The phase concentration factor immediately follows by measuring the mean areal concentration of the dispersed phase in section planes according to the principle of Delesse. In order to determine the shape and the orientation factor^

it is necessary to measure the following quantities stereometrically in plane

sections of the material investigated:

- the largest extension of each area of sectioned dispersed particles which is taken to be the large axis of the substituting sectional ellipse (a'), - the area of each section of dispersed particles itself (A'), - the mean area concentration of the dispersed phase (c^), - the average number of sectioned areas of dispersed particles per unit area of plane (nJ), - the total number of section areas of dispersed particles (N'),

Using fig. 3 and equation 6 as well as the following relationships /4 , 7 , 8 , 10, 1

1

,

Ml, these measurable stereometric quantities lead to the indirect shape and orientation factor:

(f)

=

(|)__ + nj^)^ - (7)

(f) = ; (^)^ = f(|J) (s.fig.4) (8) _c^N^_ 6cp[(|,) a__. (|T)bJ-2N' •^•a.,b^

'^=- V - V 1i[(7r)a„+(J^)b^][aJb^-a„b?,]

b' ^ 4A' ^

!I(a')^ I- = ,(,,)2 = b- - 4A'

(9)

(10)

' ^ b' lb' ,a\ 1 ^ a' 1 a' , ,

— 1 4r^-(-)

a' = r a- ; = ^ ; a„ = ^

(12)

Comparisons between measured electrical resistivities of cermets and calculated

curves^ taking into account the stereometric effects by using the above equa297
G. Ondracek

tions, are given in fig. 5 and 6. Additional information about such comparisons

including other properties as thermal conductivity. Youngs modulus and thermal expansion coefficients, is already available in the literature /4,5 , 12 , 13, lA/. Summarizing the results, the comparisons point out that the derived equations are not only a sufficient engineering approach to reality, but also

~ provide a better insight into the relationships between properties and microstructure,

- offer one way to taylor made materials by precalculat ing optimalized microstructures causing desired property values^

- may be used instead of data collections by calculating the properties of twophase materials, the phase properties of which are known,, - substitute the direct property measurement by a stereometric microstructural

analysis or extrapolation in such cases where the first is either less accurate, very difficult or even impossible.

An impressive example in which direct measurements have been substituted successfully by stereometric microstructural analysis is the determination of the thermal conductivity of porous nuclear fuels which cannot be measured directly by the disturbing irradiation influences / 14- 1 6/ . -The extrapolation of conductivities

of intermetallic compounds may serve as another demonstration in that context.

Due to their brittleness, pure intermetallic compounds often cannot be classified

satisfactorily by powder metallurgical methods and also cannot be prepared singlephased by melting techniques. This is the reason why properties of pure intermetallic phases are frequently unknown. But such phases are available in dense two-phase materials. Measuring the conductivities of these materials and their

nonintermetall ic phase the equations considered here allow the calculation of

the conducticity of the intermetallic phase /17/.

matrix U02-cermets [5] o UO2 Spheres [5]

List of symbols and literature: a' = larger axis of sectional ellipse in plane; A' = area in plane of sectioned dispersed particle; a = angle between the

rotation axisof a spheroid and the field direction;

(a'/b *) , (b '/a ' ) = mean values of axial ratios of

the sectional ellipses measured in planes taken statisti cally through the specimen; (a'/b'),^ , (b'/a');^ = mean values of axial ratios of the sec- tional ellipses measured in planes perpendicular

to the direction of the electric field; b' " smaller axis of the sectional ellipse in plane; c • phase content; C = index for composite, cermet, two phase material; cos^a = orientation factor; D - index for dispersed phase; F = (indirect) shape factor (concerning field properties); M • index for matrix material; n== relative amount of oblate spheroids; N' = total number of section areas of dispersed phase particles; NJy = average number of section areas of dispersed phase particles per

unit area of plane; p = electrical resistivity:

X = aaj = b,( - minor axis of spheroid; z ati b, = rotation axis of spheroid; ^

- axial ratio of prolate (index „) or oblate (index -) spheroids. Literature

~7T7 Nielel W. , Ann.Phys. 6-10(1952)336

121 Ondracek G. , Schulz B. , Ber.Dtsch.Keram.Ges.

48-10(1971)427

/3/ Ondracek G. , Schulz B. , Ber.Dtsch.Keram.Ges. 48-12(1971)525

/4/ Schulz B. , Dissertation Universitat Karlsruhe (1974)

/5/ Ondracek G. , Z. f .Uerkstof ftechnik 5-8(1974)

416

/6/ Stille II., Arch.Elektrotechnik 38-3/4(1944)91 ni OnHracek G. , Schulz B. , Practical Metallography (1973)16

/8/ Ondracek G. , Schulz B. , Practical Metallography (1973)67

79/ Ondracek G. , KFK-Ext. 6/73-2(1973)40

/lO/ Hotf R.T.de, Rhines F.N., Trans. Met. Soc. AIM

221(1961)975

/II/ BaJtke T. , Nazare S., nie Microscope Fourth

Quarter (1975) /12/ Nazare S., Ondracek G. , Powder Metallurgy Int. 6-1(1974)8

/13/ Nazare S., Ondracek G. , Ceramurgia (1975)

to be published

/14/ Ondracek G. , Schulz B. , J . Nucl . Mat . 46 3(1973)

253

/15/ Ondracek G. , Schulz B., KFK 1274/1(1974)1 12 /16/ Ondracek G. , Schulz B. , KFK 1999(1974)43

/17/ Schmitt R. , Diplomarbeit UoiversitSt Karlsruhe, Institut far Werkstof fkundc II (1975)

298
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

USE OF MATHEMATICAL MORPHOLOGY TO ESTIMATE COMMINUTION EFFICIENCY

by Ph. Cauwe

Societe Wild Leitz-Paris. 86, Avenue du 18 juin 1940. 92504 Rueil-Malmaison, France

By using the rigorous theoretical background of Mathematical Morphology of

G. MATHERON and J. SEREA and with the associated electronic device, the Texture Analysing System (LEITZ T.A.S.), we have tried to study comminution problems with

a new point of view. A first attempt of a model uses the net of Poisson flats in Euclidean spaces as described by R.E. MILES. Due to bad agreement with experimental data we have built a new model of Poisson twin flats in Euclidean spaces. Ex- perimental values performed on comminuted products by use of the Texture Analysing System are in good agreement with the model and lead to easy interpretations

of milling conditions.

1. DEFINITION : A comminution is a size reduction of large particles to finer

ones. The comparison of size distributions before and after milling is the more

usual way to estimate the action of the operation.

2. THE SIZE OF A PARTICULE : In the field of comminuted products, the size of a particule or grain is much more difficult to define than these of a circle or a cube. In practice, the mesh sizes of sieves are used: this method does not refer

to any true concept of size. Another manner is to build up a concept of size

independently of the technological ways used to obtain size distributions of particles. By doing so, each physical method can refer to a same principle and can

be compared. This way is the one used by the Mathematical Morphology developed

by G. Matheron and J. Serra /6/. Using the well known structuring element, the Mathematical Morphology gives the geometrical interpretation of the studied structure. Using the theoretical background and the associated Texture Analysing

System (LEITZ T.A.S.) we try to build up a model to estimate the efficiency of comminution processes.

3. THE WAY TO A MATHE>1ATICAL MODEL OF COMMIN-UTION

:

Like the electronics engineer looking for the equation describing the signal transform when going through

a "black box", we try to find the transfer function of a milling product going

through a comminution machine. We try to combine tools given by the Mathematical

Morphology to describe the transformation of a material of large particles to finer ones. By doing so, we are independent of experimental conditions and the parameters involved in the model have their own signification. We can expect to have a sharp description of a size distribution.

4. POISSON FLATS: A FIRST MODEL OF COMMINUTION. With reference to the literature,

one can admit that the fracture propagation within ores and minerals fragments

follows a net of existing cracks. We can imagine that comminuted particles have

statistically equivalent geometric properties as these of a net of polyhedra in

the three dimensional space. Our first attempt is to describe the fracture process by a net of Poisson flats yet described by R.E. MILES /5/. Using Poisson

flats in the 3-dimensional space we obtain a random partition of the space into

polyhedra. The parameter of the model is (independent of direction a ): the density of the Poisson law. Morphologically speaking, in the 1-dimensional space,

the erosion by a linear structuring element of a segment defined by two consecutive points of the Poisson process and containing the origin gives an eroded segment, if not empty, having the same properties as the original one. This fact is equivalent to the lack of ageing. G. Matheron generalises this property to Poisson

flats of space : the probabilist properties (volume,...) of Poisson polyhedra

containing the origin are not modified after erosion by a convex compact on condition the eroded polyhedra are not empty /4/. Knowing that Poisson flats in the 3-dimensional space induces Poisson lines and Poisson points on planes and lines

going through them, very interesting relations between model's parameters in the different spaces can be calculated /l/,/A/,/5/. For example, the relation between

the Poisson densities A. in space i : A = 2A /it = X /n (1).

299
Ph. Cauwe

With this kind of relation, experimental measures on lines and areas provide two different ways to compute the parameters of the model in the 3-dimensional space. One of the basic relation used is the linear granulometry as defined by

G. Matheron and J. Serra /6/. The practical equation is : 1-G(t) = (1+ttA t). Exp (-^'A 't) where t is the length of a segment included in a polyhedra and

1-G (t) the cumulative size distribution according t. Figure 1 shows equation

(2) computed for different values of X. where is a measure of the fineness.

Comparison of theoretical equation with experimental values performed by the Texture Analysing System on polish sections of comminuted products have shown general good agreement (figure 2) . But theoretical equation has shown also a systematic underestimation of the amount of fine particles. This fact led us to modify the model to a new original distribution of flats in Euclidean space.

5. THE MODEL OF POISSON TWIN FLATS. Roughly, one can say that each plane of the Poisson flats in the 3-dimensional space is replaced by two planes. The distance

between the planes of a twin is given by a probability law F^ (x) . Here also the model of the 3-dimensional space induces Poisson twin lines and twin points on planes and lines going through it. Same kind of relations between parameters of the model in different spaces can be calculated /I/. One can easily understand

that internal region of each pair of planes is cut by a higher number of planes

giving a greater proportion of small polyhedra. The equivalent relation to equation (2) can be calculated /I/. By comparing this equation with experimental

results we have found much better agreement, specially for the part of fine particles. To verify we have also used the probability P(Cj^) for having a circle Cr

of radius R inside a polygon of the Poisson twin flats in space 2.

6. CONFIRMATION OF THE MODEL OF POISSON TWIN PLANES . Using the linear and the two dimensional logics facilities of the Texture Analysing System (LEITZ T.A.S.) com- putation of equations leads to experimental values for parameters and d^

(i = 1,2) (di is the parameter of F^(x) ). SAMPLE PREPARATION: Experimental measures using optical microscope and TV camera

can only be performed on flat surfaces as polish sections. By mixing the comminuted products with a Teflon powder (10,um) homogeneous distribution of ores particles is obtained. After compression, sintering and polishing measures are performed on the grains sections. According that the properties of the Poisson polyhedra depend only on themselves and not on their mutual disposition in space; the absence of contact between two grains in a sample is the sole condition to reach valuable experimental data. 180 polished sections from 30 different tests of milling conditions have been done and measured by the Texture Analysing System. Two kinds of products have been studied: a copper-lead-zinc ore with a complex

mineralogic texture and a highly homogeneous magnetic concentrate of iron. MODEL'S PARAMETERS ESTIMATION. Experimental values of and P(Cr), obtained

by the Texture Analysing System (LEITZ T.A.S.) , have been smoothed (quadratic

equation) before calculations of logarithms and derivatives. These numerical

treatments lead to values of A^ and d^ (i=l,2) reported on Table I with the values of calculated from experimental values of A according equation (1). One

5t 1 can easily test that the ratio ^2^'^2 equal to one. More precisely it can be seen on table I that the mean value of the ratio is 1,023 with a standard de- viation of 0.085. This result is also presented on figure 3. Table II presents

similar results: estimation of d^ from experimental values of dj^ and d2. Tables

I and II are only the more demonstrative results proving the validity of the Poisson Twin Planes as a model for comminuted products. Other relations between

the 1 and 2-dimensional spaces have been verified as shown in /I/. USE OF PARAMETERS TO ESTIMATE COMMINUTION CONDITIONS. Model's parameters and com- minution conditions have been obtained from a Magnetite concentrate milled in a pilot ball mill under different experimental conditions: weight of concentrate,

size of balls. For tests I, II, III various samples have been taken at different

times of milling and then analysed. In the following, results are presented versus time (minute) and power (KWH) consumption.

THE PARAMETER d: On figure A it can be seen that the values of d stabilise at a constant value after a given time. As discussed in /I/, the parameter d is re- lated to the smaller size of particules and the steady stade value is a measure

of the smallest grain size obtainable for given grinding conditions.

THE PARAMETER A : For example figure 5 presents A and A„ versus time for Test I.

300
Use of mathematical morphology to estimate comminution efficiency

From this kind of diagrams one can appreciate how grinding action is going for gi- ven conditions. Comparison of A-curves of various milling condition leads to the better grinding effect. POWER CONSUMPTION: Knowing that the maximum power efficiency of a comminution is

1%, correlations between power and model's parameters must be studied. Diagrams

of power versus time of milling is of general use in comminution studies (fig. 6): very poor information is given by these diagrams. On the other hand, curves of

X versus power consumption lead to easy appreciation of the better comminution

conditions. From figure 7, it is obvious that the better conditions are these of test III: - up to 20 KWH/T tests I, II, III are equivalent; - for power higher

than 20 KWH/T, higher power consumption does not increase fineness of the products

in test I conditions; - this situation is better for test II but even better for

test III. Another kind of result can be read on this figure: for the same fineness

of product (same X) conditions of test III comparing to those of test II, save

20% of energy.

7. CONCLUSIONS . - For the first time, sharp relations have been found between geo- metrical characteristics of particules and comminution conditions. Using the powerfull "tools" given by the Mathematical Morphology this study gives easy and clear view of the comminution process.

- Knowing that fractures propagation follows pre-existing cracks it is remarkable

that to adjust a model to a milling process it is not necessary to know the net of cracks in situ. This fact is specially interesting when thinking of the not visible cracks! - It is well known that equations with many parameters can always fit experimental data but these parameters are hardly related to physical realities.

One can remark that our model has only two parameters. We have tested /I/ that

it enables to cover the wider range of comminution conditions: crushers, grinders,

ball mills minerals and ores of various types. - Furthermore, the model can help

for interpretating physical phenomena involved in comminution processes. As discussed in HI the parameter x can be interpretated as a direct measure of milling

conditions efficiency and parameter d can be correlated to a practical limit for

the minimum size of grains obtainable for given experimental conditions.

8. BIBLIOGRAPHY . HI CALTvE PH.: Morphologie Mathematique et Fragmentation-These-Univ . Cath . LouvainLab. Traitement des Minerais-Belgique.

HI HADWIGER H.: Vorlesungen iiber Inhalt, Oberflache und Isoperimetrie-Springer

Verlag-1957.

/3/ KLEIN J.C.-SERRA J.: The texture analyser-Stereology 3, 1971.

/4/ MATHERON G. : Random sets and integral geometry- Intersciences-Wiley & S. 1975.

151 MILES R.E.: A synopsis of Poisson flats in Euclidean space-Izv. Akad. Nauk. armianskoi SSR, 3, 1970, 263-285.

/6/ SERRA J. : Introduction a la Morphologie Mathematique-Cahiers du Centre de Morphologie Mathematique-Fascicule 3-1969.

Ft£. 1: linear Bize dletrlbutlon of PolsBon

planoB In space "i*

301
Ph. Cauwe

Fig. 3: Relation between a and A

O 10 20 30 40 50 60 MINUTES

Pig. 6: Power coneumptloc vs. time of mlUlsg.

0^ i 8 12 15 20 24 B 31 POWER CONSUMPTION - KWH/T

Pig. 7: ^ values ve. power consumption.

302
Use of mathematical morpho/ogy to estimate comminution efficiency

TABLE I TABLIE II

Test M A! A2

/ 1 2 3

1

d,-

32 d,,/d^-

number 1/ (Um

/ 1/ .um

/ 1/ ,um

/ yum ^um ^um ^um

6737 0. 182 0.091 0.091 1.001 760 595 507 505 1 . UU4

6738 0.266 0. 133 0. 144 0.923 760 595 507 505 1 . UU^ 6739 0.372 0. 186 0.202 0.920 760 595 507 505 1 . 004 6740 0.658 0.329 0.298 1 . 104 760 595 507 505 1 . 004

6741 1 .401 0.700 0.658 1.064 760 595 507 505 1 . 004 6742 2.399 1 .200 1.315 0.912 500 370 333 3 1

4

1 . 06

1

6743 4.008 2.004 2.054 0.975 220 130 147 1 1

0

1 . 336 6744 9.230 4.515 4.236 1.090 99 57 66 48 1 .375 6745 25.702 12.851 12.291 1.046 53 36 35 3 1 1 . 1 29 6746 36.436 18.218 17.077 1.067 35 24 23 20 1 • 1 50 6747 43.701 21 .851 20. 127 1 .086 28 21 19 18 1 .056 6748 51 .847 25.924 23.789 1 .090 26 20 17 1

7

1 .000

10701 8.229 4.115 4.115 0.999 83 60 55 5

1

1 . 784

10702 18.759 9.380 7.495 1.251 42 31 28 26 1 .071 10703 27.303 13.652 14.657 0.931 31 20 21 17 1.235 10704 34. 187 17.094 17.092 1.001 24 19 1 6 16 1 .000

10705 45.841 22.921 21 .072 1.088 24 19 16 16 1.000 10706 47.300 23.650 22.353 1.058 24 19 16 16 1.000

10707 20.353 10. 177 1 1. 127 0.915 32 24 21 20 1 .050

10708 26. 174 13.087 12.805 1 .022 24 19 16 16 1.000

10709 34.840 18.920 18.920 1 .000 24 19 16 16 1.000

10710 44.094 22.047 23.884 0.923 24 19 16 16 1.000

1071

1

57.351 28.675 27. 151 1 .056 24 1

9

1 D 1 0

1 nnn

10713 10.013 5.007 5.030 0.995 83 60 55 51 1.784

10714 23.294 1 1.647 10.715 1.087 32 24 21 20 1.050

10715 29.300 14.650 15.787 0.928 24 19 16 16 1.000

10716 36.830 18.415 18.650 0.987 24 19 16 16 1.000

10717 34.944 17.472 19.738 0.885 24 19 16 16 1 .000 10718 53.716 26.858 25.618 1 .049 24 19 16 16 1 .000 10719 78.290 39. 145 32.022 1 .222 24 19 16 16 1 .000

Mean value of the ratio 1 .023 1 .067 standard deviation 0.085 0.208

A values have to be multiplied by 10-3 A2 - 0.5A^ ; d 31 - 2^1/3 ;

303

National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersburg. Md.. September 4-9, 1975 (Issued January 1976)

MICROSCOPY, RADIAL DISTRIBUTION ANALYSIS, AND SORTABILITY

by Robert M. Doerr

U.S. Bureau of Mines, Rolla Metallurgy Research Center, Rolla, MO 65407. U.S.A.

A method, based on a radial distribution analysis, was developed for determining the spatial distribution of a phase in an aggregate. Two components of the determination are (1) the concentration of the phase and (2) the enclosed

proportion of the total quantity of the phase present, each as a function of the radius of an appropriately centered enveloping sphere. A measuring microscope

and computer analysis are used to obtain the radial distribution of the phase in plane sections of the aggregate. The method was developed to quantify statistically the dependency of the grade and recovery potentials on the fragment size

to which a copper ore is broken for sorting.

INTRODUCTION

This is a report on a method (1) to determine the sizes of regions of se- lected target grade in an ore and (2) to estimate the potential recovery of the desired mineral when the ore, broken to such sizes, is processed to yield a con- centrate of that grade. When coupled with cost data, mean grade of ore, and product value, such information is useful in deciding the feasibility of sorting

and the fragment size at which sorting would be most effective.

Modem digital electronic circuitry has made high speed serial automatic

sorting practical for various separations, including minerals beneficiation ,

In some methods of separation, such as flotation and by magnetism, the separating action is a direct response to the properties on which the separation is based. However, the two essential steps of sorting, detection and separation,

can occur at different times and/or places. Thus, for example, if fragments

that possess a characteristic (such as above-threshold radioactivity, induction,

fluorescence, or some combination) are readily and instrumentally identifiable

as different from those not exhibiting the characteristic, a distinct, powered

unit can be triggered to separate the individual fragments of one class from the stream. For minerals benef iciation , the significant difference is usually that

of grade, or proportion of a valuable species, between fragments. Effective ap- plication of such sorting depends on the spatial distribution of the valuable

species within the host rock and breakage to fragments to 'liberate' regions of sufficient grade difference. The economics of sorting are adversely affected by

over-breakage of the ore,'- The importance of fragment size on sortability has been recognized and the effects researched , ^

To visualize the radial distribution analysis, consider as a specimen a plane section of an ore fragment, with a vanishingly small circle centered on an appropriate copper particle in the plane surface. The grade of copper within

the circle is unity. As the radius of the circle is increased, areas of rock

devoid of copper, and other copper particles, are enveloped; the overall fraction of copper within the circle generally declines. With allowance for areas

of the circle that extend beyond the edges of the specimen, the grade within the circle ultimately reaches the specimen grade. The largest radius at which the grade within the circle is equal to a chosen target grade, taken to be greater

than the mean, is determined from an empirical equation relating the grade to the radius. The fraction of total specimen copper within that radius is taken

as the potential recovery for the target grade chosen. The most appropriate

central particle to choose is the one that leads to the highest potential re-- covery at the target grade, given an acceptable fit of the equation to the

data. It has been shown that grade (area fraction) analyses made in two dimensions are statistically applicable to three dimensions (volume fraction). 13-22

305
R. M. Doerr

PROCEDURE

The sample used was drawn from fragments of conglomerate native copper

ore that had been crudely preconcentrated by testing each fragment for copper by an induction bridge coupling method and retaining those for which the presence of metal was indicated. Raw ore would have necessitated too much data

taking and would not have helped elucidate the application of the method.

The ore fragments were sectioned by diamond sawing and the resulting plane

faces were polished. These faces constituted the sample. By use of a microscope with a stage vernier and a grid reticle, data on the size and location,

relative to an arbitrary origin, of each copper particle in the plane were

recorded, as were data on the location of the perimeter of the specimen and

of any pores therein. In the nine-specimen sample, totaling 7 260 mm^ , the plane area and location of 3'4 660 eopper particles were measured and recorded.

The grade vs. radius data for the radial distribution analyses (about

appropriate 'central' particles) were developed and a forward stepwise linear

multiple regression analysis routine^ ^ was modified and used to fit the radial distribution curve. For each specimen, the radius and potential re- covery for each of the concentrate grades 2 to 12 area % copper were tabu- lated and the radial distribution curve plotted.

The central particle for each specimen was chosen for reasons of economy

from among the approximately 50 largest copper particles in the specimen.

Temporarily ignoring all the smaller particles, the grade within a certain

distance of each particle was determined, and the particle chosen to be the central one was the one that led to the highest grade within that distance.

Two distances were tried for each specimen: half the distance to the most

remote of these largest particles and 5 mm. In most cases, the two distances

led to different central particles; the one used was the one that led to the higher indicated potential recovery. The 6- mm distance was found better

only for lean, probably reject, specimens and the half-maximum distance for the others. The model used for the radial distribution relationships was:

where y is In (grade within circle of radius r) , is the ith regression

^ coefficient, = 1, x^ = radius r , = x^^ , x^ = x^3^ = 1/x^, x^ = x^ , = x^'^, xy = In (x. ) , and e is random error. For another ore, a different

model may be required. A grade-vs , -radius point was calculated for each copper particle, not just for the approximately 50 largest.

RESULTS AND DISCUSSION

Characterization of the sample is presented in Table 1, in which are shown the rock and copper contributions of each specimen and selected totals and means. Most of the copper is contained in two rich specimens which ag- gregate 65% of the copper in 24% of the rock. These and all percentages in this report are area-percent. Owing largely to the two rich specimens, con- centrates with 5% copper and fairly high recovery are seen to be possible

from sorting the sample without further crushing. This occurrence presume?

ably stems from the small sizes of the specimens.

The results from fitting the data to the model are also shown in Table

1. For no specimen did all the terms of the model prove significant. The resulting radii for a target concentrate grading 5% copper, recoveries, and

a measure of the goodness of fit of the model to the data are presented.

Three specimens are seen to have regions of 5% copper with radius not less than 12 mm. Crushed to a 12- to 15-mm radius range, the sample should be sortable to a concentrate with a grade over 5% and with recovery of about 62% of the copper

in about 18% of the feed. All of specimen 1 would be included in a 5% concentrate, even without further crushing. Crushed pieces from specimens 6, 7, and

9, aggregating 34% of the feed, would be rejected if the value of the copper re- quires a concentrate grading 5%, on the basis of the low potential recovery at 5%, and the pieces from specimens 3 and 8 would probably be rejected if the cost of sorting at a 6-mm radius is uneconomic.

i=0

+ e

306
Microscopy, radial distribution analysis and sortability

TABLE 1. - Sample Data and Results from Radial Distribution Analyses

Specimen

Size

,

^ mm

L.U area %

In

C Correlation

coefficient)

Ra.dius for 5 area

/o target

concentrate, una

Recovery

At This

D I 1 I e ^/ Kd Q I U S , /o 1 946.2 7.79 .999 32.5*** 100.0***

2 757.9 4.48 .886 15.4 56.4

3 671.3 2.02 .983 3.8 17.7

X.J/ Q 1^ A O • U J / . D 5 1365.1 1.15 .946 12.0 71.4

6 1230.7 1.08 .991 2.7 8.6

7 655.5 0.927 .991 1.7 6.2

8 lll.U 0.868 .987 3.3 20.9

9 610.8 0.049 .998 0.1 2.1 Totals 7260.3 19.734 Means 806.7 2.193** ,971

* Arranged in order of descending grade.

** Weighted mean is 2.29 area %, *** Specimen grade is about 7.8%.

TABLE 2. - Radial Distribution TABLE 3. - R.adial Distribution

Analysis for Specimen 2 Analysis for Specimen 5

Target Radius for Target Radius for Concentrate Target Recovery Concentrate Target Recovery

Grade

,

Grade

,

At This Grade, Grade

,

At This

Aj;ea % Tnm Radius, % Area % mm Radius, %

5 15.4 56.4 2 25.2 94.7

6 2.7 1.9 3 18.9 91.2

7 2.3 1.9 4 15.0 83.4

8 2.1 1.5 5 12.0 71.4

9 1.9 1.2 6 9.4 67.7

10 1.8 1.2 7 7.0 51.5

11 1.7 1.2 8 3.8 10.5

12 1.6 1.2 9 1.6 3.8

10 1.3 3.6 11 1.1 3.6

12 1.0 1.8

As examples of the radial distribution analysis, tabular results are presented for two specimens. For specimen 2 these results are presented in Table

2. The recovery falls off abruptly from 56 to 2% upon changing the target concentrate from 5 to 6%, and strongly suggests that the specimen contains two

or more rich regions which, together with the intervening rock, all in one relatively large fragment, could be acceptable in the 5% concentrate but not

in a 6% concentrate. Thus, the potential recoveries indicated for the 6% and richer cases are low if one or more additional comparably large regions are of target grade. It is thus necessary to use specimens of size appropriate to the target grade and to the distribution of the valuable mineral. The most

remote copper particle is about 39 mm from the central one; this suggests that the central one is not near the center of the specimen, i.e., that fracture

occurred through or near the richest copper region. Results for specimen 5 are presented in Table 3. Here the recovery falls off slowly when the target concentrate grade is increased from 2 to 7%, while the radius drops from about

25 mm to about 7 mm. Thus, much of the specimen copper is in the region that includes the central particle. The method of radial distribution analysis enables the determination of key data needed for judging the feasibility of sorting an ore and the size at which to sort. However, manual recording of the voluminous data proved too tedious to permit working a sample comprised of enough specimens to perform

needed statistical analyses and provide confidence limits on the results. Specimens large enough to permit full determination of the copper distribu307
R. M, Doerr

tion were similarly precluded. Thus it is recommended that for implementation, the data be recorded by a computerized microscope^**"^^ with the computer controlling the stage drives. Such apparatus is within the state of the

art. Further, with an automated microscope, it would be feasible simply to assay each hexagonal field of view, record its position, and develop an assay-vs . -location data point for each such field rather than for each particle. This approach, which is not feasible with manual data acquisition,

would reduce computer memory requirements and enable simplified programming.

It should allow application of cluster analysis, and with it, treatment of the shape factor and independence from limitations on the shapes of rich

regions and from the problem of multiple rich regions, so that large enough

specimens can be evaluated.

CONCLUSIONS

The method of radial distribution analysis provides data that are useful, when combined with economic factors, for judging whether or not a mineral deposit lends itself to sorting as a means of concentration or pre- concentration, providing the valuable species is detectable. These data

relate the effect of the rock fragment size at which sorting is performed,

and selected target grades to the potential recoveries at such grades.

ACKNOWLEDGMENT

This report is modified from the author's thesis submitted to the University of Missouri-Rolla in partial fulfillment of the requirements

for the degree Master of Science in Ceramic Engineering.

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Sorting to 1972, Mines Branch Monogr. No. 878 , Canada, Dept. of Energy, Mines, and Resources,

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2. Newman, P. C. and P. F. Whelan, Photometric

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It. Harris, Virgil, Electro-Optics Sort Products by Color, Automation , Vol. 17, Jan. 1970,

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,

Optical Separation of Limestone in Southern

Finland by Lohjan Kalkkltehdas Oy, Trans

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White Marble, Rock Prod. , Vol. 74, June 1971,

pp. 52-56.

11. Zeising, Gordon F., William S. Hannan, and William A. Griffith, Concentration, Min. Eng. , Vol. 24, Feb. 1972, pp. 91-94.

12. Discrimination in Diamonds, Engineering , Vol. 207, May 9, 1968, p. 708.

13. X-rays Sift Diamonds From Gravel Mix, Mach. Design , Vol. 41, Aug. 7, 1969, p. 112.

14. Colborne, G. F. , Electronic Ore Sorting

at Beaver Lodge, Canada, Min. & Met. Bull. , Vol.

56, No. 616, 1963, pp. 664-668.

15. Carlson, David H., "Some Characteristics

Affecting the Mechanical Sortability of Native

Copper Ores", SME-AIME Preprint No. 70-B-47 . Feb. 1970, 9 pp.

16. Carlson, D. H. and M. E. Volin, "Electro- mechanical Sorting of Poor Rock From Michigan

Copper Mining Operations", Paper Presented at AIME Upper Peninsula Section Spring Technical

Meeting, May 22-23, 1968.

17. Miller, Vernon R. , Robert W. Nash, and Alfred E. Schwaneke, "Detecting and Sorting Disseminated Native Copper Ores." U.S. BuMines

Report of Invest. 7904, Washington, 1974, 14 pp.

18. Doerr, Robert M., Radial Distribution

Analysis For Sorting, Trans. SME/AIME , Vol.

255, 1974, pp. 4-9.

19. Underwood, Erwin E. , Quantitative Metallography, Metals Eng. Quart. , Vol. 1, Aug. 1961,

pp. 70-81.

20. Delesse, M. Precede Mechanique Pour Deter miner La Composition Oes Roches, Ann. Des Mines , Vol. 13, "tth Series, 1848, pp. 379-38B21. DeHoff, Robert D., and Frederick N. Rhines

Eds., Quantitative Microscopy , McGraw-Hill, N.Y.

1968, 422 pp.

22. Pellisier, G.E., and S. M. Purdy , Chairmen

Stereology and Quantitative Metallography , SIP

504, Amer. Soc . for Testing and Materials, Philadelphia. 1972, 182 pp.

23. Efroymson, H. A., "Multiple Regrtession Ana

sis", Chapt. 17 in Mathematical Methods for Digital Computers , Ed. by Anthony Ralston and Herbert S. Wilf, Vol. 1, John Wiley t Sons, Inc. New York, I960, pp. 191-203.

24. Cole, Michael, Image Analysis, Amer. Labor

tory . Vol. 3, June 1971, pp. 19-28.

25. Gray, R. J., "The Present Status of Metallography", Chapt. in Fifty Years of Progress in Metal lographic Techniques , SIP 430, Am. Soc. for Testing Materials, Philadelphia, 1968, pp. 17-62

26. Schuck, John, TV Analyzing Computer Expand Scope of Microscopic Studies, Res & Develop .

,

Vol . 17, Apri 1 1966, pp. 28-31

.

308
National Bureau of Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

SIZE DISTRIBUTION ANALYSIS IN SITU ON INDIVIDUAL OR INTERCONNECTED PHASES BY IMAGE ANALYSES

by C. Gateau and J. M. Prevosteau

Service Geologique National (B.R.G.M.j. B.P 6009. 45018 Orleans Cedex, France

ABSTRACT

We present the analytical conditions which must be respected for the dif- ferent size distribution possibilities by image analyser ; size distributions

in numbers or in measurements, all or nothing size distributions, sizing by

openings according to a structural element.

Using examples of applications taken from the study of natural materials,

and in particular of ores, we illustrate the choice of size distribution analysis modes selected which depends on the texture examined, the type of problem

posed, the stereological interpretation and, finally, on the analytical efficiency .

In addition, we show how several modes of size distribution analysis can

be used together in order to define supplementary morphological criteria which

are extremely interesting for the three-dimensional restitution of the distributions measured in one or two dimensions.

A-CHOICE OF A SIZE DISTRIBUTION TYPE

The idea of size distribution supposes a classification of different parts

of a whole according to size criteria. The nature of this geometric parameter

must obviously be adapted to the texture of the material to be analysed. The least favourable case corresponds to an interconnected medium (e.g.

fissural or porous network). Size distribution in such a material can only be attempted by the intermediary of its opening (defined by Matheron) according to

a structural element, of variable size and possibly form ; the dimension of this structural element then serves as a size distribution parameter.

In addition to the preceding analysis, which can always be applied, the individual particles may be characterised by new parameters. In particular, they may be classified according to their specific area, to their perimeter or their

vertical and horizontal projections. In addition to these size criteria, we have

also developed new modes of distribution analysis according to parameters of form and structure. When the particles are in the form of aggregates, it is necessary to use

the intermediary of structural elements, preferably b i-dimensional (octogon,

hexagon, rectangle)

.

When the size parameter has been chosen, then the method of representing

the size distribution naist be selected. This may be expressed either as a number

or as a measurement. In the later case the number of particles belonging to each

class is balanced by their average dimension.

The determination of all the above parameters can only be made rapidly and automatically by an image analysis procedure.

The choice of the type of measurement must therefore be adapted to the pro- blem posed. Several factors must be considered in this choice : .the type of texture : individual or interconnected medium-grains with or without inclusions - morphology of grains ; .the nature of the problem : size distributions in numbers or in measurements - need to measure certain parameters in view of a stereological interpretation ; .the analytical efficiency conditioned by the size of the analysis frame and

the magnification used.

309
C. Gateau and J. M. Prevosteau

B-SIZE DISTRIBUTION ANALYSIS BY LINEAR MEASUREMENTS OR BY OPENINGS ACCORDING TO A STRUCTURAL ELEMENT

1-RECALL OF THE MATHEMATICAL DEFINITION OF THE MOST USED SIZE DISTRIBUTIONS

P (h) =probability that a segment of length h is entirely included in the grains

(h)=prob ability that a chord of grains belongs to a chord of less than h (size

distribution in number)

Gj (h)=probability that a point of grains belongs to a chord of less than h (size

distribution in measurement)

Aj (h)=probability that an analysis point belongs to a chord of grains greater or equal to h. The quantities P (h) , (h) and A^ (h) are not independent parameters, but

are accessible by different measurement functions at the level of the image ana- lyser. Ai (h) = p {1-G2(h)} = P(h) - h P'(h) where p is the percentage of grains

1 Fa (h) -jtJqJ

^2 ) =probabi lity that an analysis point belongs to the area scanned by a structural element S, entirely included in the grains.

This structural element S may, in our case, have very varied forms : octogon, hexagon, square, rectangle, segment, etc. Grains size distribution analysis of the linear type or by openings accor- ding to a bi-dimensional structural element may be applied to all textures whether they be individual or interconnected.

2-RECALL OF THE ANALYTICAL CONDITIONS TO BE RESPECTED

The size distributions defined from one single image are the most frequently erroneous owing to the intersection of the phase considered with the edges of the field of analysis. If the number of the analysis field is high, this error

is statistically eliminated, on condition, nevertheless, that a guard region is used whose dimensions are greater than the largest erosion step employed : h^j^^- It is the case for an interconnected medium P (h) ; Aj (h) (fig- la)

Automatic linear opening analysis ^^(h) requires a double guard region : ^^max ib) • Size distribution analysis by bi-dimensional opening ^2^^) volves new restrictions on the dimensions of the guard region depending on the form and the maximum size of the structural element (fig. 2-3) . All these specific conditions for the QTM 720 are summarized in table 1.

3-STUDI OF THE SIZE DISTRIBUTION OF ALLUVIAL PRODUCTS

All the various possibilities of size distribution analysis may be applied

to this very simple case (isolated convex grains).

In the absence of any other restrictions, the method giving the best re- sults is therefore chosen. In this case, linear measurements from P(h) would be used, since it is this method which allows the largest analysis frame.

4-STUDY OF A MASSIVE MINERALIZATION

The size distribution study of an interconnected medium, whether this be a porous or fissural network, a gangue or a massive mineralization, obviously precludes the all or nothing methods. In this case, techniques of openings accor- ding to a structural element (linear ni(h) or oi-dimensional ^2^^)) will be used. More particularly, for some ore-dressing applications which are connected

with problems of comminution, of mineral liberation and of fissural networks

which may be clogged, it is necessary to take into account all the previsional

factors which are responsible for the quality of the products obtained and to acquire them by a bi-dimensional analysis.

C-ALL OR NOTHING SIZE DISTRIBUTIONS

1-DEFINITION

The all or nothing size distribution analysis, which can only be applied to individual media, can be constructed according to size parameters , and also form

(F) or structure (m) criteria.

The general definition of such a size distribution may be expressed as

310
Size distribution analysis of phases by image analyses

follows : A2(P)=P rob ability that an analysis point belongs to a grain whose parameter is more than a value P.

2-BECALL OF THE ANALYTICAL CONDITIONS TO BE RESPECTED

The all or nothing size distribution analysis requires that the guard re- gion be chosen so that its dimension is greater than the largest size of the parti cules : Fm^x (fig- 4 J .

3-STUDy OF A ZIRCON-RIEILE ORE

As in the first example, rutile has the form of individual particles. However, the presence of inclusions greatly disturbs the measurements of size distributions whether these be by linear methods or by bi-dimensional openings.

As these inclusions are much smaller compared with the grains, the all or no- thing area sizings A2(a) are perfectly adapted to the problem.

4-PARTICULAJi CASE OF THE QUANTITATIVE EVALUATION OF MIXED GRAINS

This new kind of all or nothing size distribution analysis makes it possible to establish a distribution of mixed grains (two phases) according to a pa- rameter of structure on which the relation may be calculated at the level of each particle of the area of a phase over the total area of the two phases : A2 (m) . The practical adaptation of such a study has made it necessary to work on

two phases X and Y which are simultaneously detected and differentiated. m = {area X} / {area (X + Y)} for each particle.

Using the example of an ore containing two sulphides : pyrite and chalcopyrite, it is possible to count the grains either as a whole in the usual way or

to isolate those which contain a certain percentage of pyrite. Also we can measure the area concerning only the mixed grains. Photograph 1 gives the number of mixed grains and photograph 2 shows us the area of mixed grains which are bigger in area than 800 and which contain more than 30 % of pyrite. It should be remembered that the grains measured may be identified by the presence of a luminous dash at their lower extremity.

The study of mixed grains of this ore is in fact translated by a distribution curve A2 (m) in number or in measurement according to the degree of mixture m which varies from 0 to 1

.

An important problem still remains for the stereological interpretation.

Certain causes of error in this domain are mentinned by JONES (5). The estimation of mixed grains is, in fact, biased, because the informations from the image analyser at the level of an analysis plan underestimates their importance

owing to the textural arrangement specific to each grain.

D-COMBINED USE OF SIZE DISTRIBUTION : CHARACTERIZATION OF THE ELLIPTICITY OF GRAINS

All size distributions depending on size parameters are obtained in one or

two dimensions. We have determined three-dimensional restitution formulae, par- ticularly for Aj (h) (ref. 1) and for A2(a) (ref. 2), and have applied Matheron's

results which hcwever are only applicable to spherical grains. Thus it is very important to determine with what degree of accuracy these formulae may be applied

.

Using the QTM 720 pattern recognition system we have carried out automatic

morphological analyses of particles by the combination of several size factors with which a form factor can be defined. The choice of this form factor is however limited to combinations of simple functions. In particular, it is possible

to establish a distribution of particles according to the relation : ^ area , , . , K = : r~r of each particle. (perimeter)^

This parameter, which in fact is an index of circularity, has the advantage

of being non-dimensional. Its use is nevertheless limited to convex particles.

Another disadvantage is seen with the magnification of the object studied in the revelation of certain details, in particular jagged border edges which increase

the value of the perimeter without much modifying the area. The system may be improved by the simultaneous use of an all or nothing

311
C. Gateau and J. M. Prevosteau

size distribution of the area A2(a) and an octogonal opening size distribution

for example Q,2{s) • From amongst the particles of a given area (chosen by the all

or nothing technique) it is possible to retain only those which are likely to contain a structural element also of a given size.

In practice, if the classes are fine enough, the selection of particles can be made according to values obtained by the relation : area of largest structural element contained in the grain

actual area of the grain

We have just defined a form coefficient which characterizes the difference

from a given geometric figure. The major disadvantage of this method is that it does not use non-dimensional parameters. If the distribution of particles according to their form is sought then the dimension of the structural element must be varied for each grain size. Such a technique would quickly lead to a prohibitive number of measurements . However, let us suppose that the grains are to be separated into two classes only : for example, those for which a > 0.75 (i.e. those which are the most

nearly octogonal) and all the others. In practice, at the QTM 720 scale and with

only one programmer it will be possible to obtain in one single operation, an

area size distribution in 16 classes for each of the 2 sub-divisions defined by

the value of a.

E-CONCLUSION

A large number of possibilities of size distribution analysis methods are now available so that all types of individual or interconnected textures may be dealt with. The success of these automatic analyses will be variable if the most

suitable mode of distribution is not chosen and the corresponding analytical

precautions are not taken. A particular and original type amongst these distribution modes is the size

distribution of mixed grains which we propose as a criterion of quality appreciation. This type of characterization of mixed grains has the advantage of offering new and very promising application perspectives for ore dressing but at present suffers from the stereological bias introduced into its estimation. It

is therefore important to develop a mathematical support which will make it possible to correct this bias by taking into account the effects of size, localisation and form of the components of a mixed grain spatially.

REFERENCES

1- J.L. PINAULT, J.M. PREVOSTEAU, D. G. WILLIAfB

Perspectives of application of quantitative

television microscopy to earth sciences. Journal of microscopy (1972), 95, 357-365.

2-C. GATEAU, J.M. PREVOSTEAU Exploitation d'un analyseur d'images a I'etude

de la structure des pores d'un materiau. Congrie RILEM-JUPAC (PRAGUE 19731. Comptesrendus IV, C 517-527.

3-C. GATEAU, J.M. PREVOSTEAU, F.X. VAILLANT Correlations between texture of rock samples

and induced polarization reaction. CangresB of Petrophyaics (NEWCASTLE 1974) .

Etude texturologique et mineralogique appliquee

a la valorisation des gres a zircon et rutile. JoumSee des sciences de la Terre (MONTPELLIER

1975).

5-M.P. JONES, J.L. SHAW Automatic measurement and stereological assessment of mineral data for use in mineral technology. Xe International mineral processing congress^ LONDON 1973.

6- G. BARBERY Determination of particle size distribution

from measurements on sections. Powder technology, 9 (1974) 231-240.

4-J.M. PREVOSTEAU. Z. JOHAN

312
Size distribution analysis of phases by image analyses

Position of the analysis zone according to the type of size distribution

J. Linear structural element

a) erosion b) opening

2 . Rectangular structural element T

»

W

a) erosion b) opening

3.0ctogonal structural

o— >

3 W

a; erosion b) opening

4. All or nothing area size distribution

greatest feret

TABLE 1

313
OaODDOa QOOOODO

C. Gateau and J. M. Prevosteau

1 - Detection of two phases (white : pyrite - grey : chalcopyrite)

- Counting mixed grains.

OO 11328 OODDOOl

2 Area measurement with both criteria for

each grain : ^ - size : more than 800 p - mixity : more than 20 7c, pyrite.

314
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

fteld at NBS. Gaittiersburg. ^W.. September 4-9, 1975 (Issued January 19761

STEREOLOGICAL DETERMINATION OF THE PROPORTIONAL VOLUME OF DENDRITES IN CAST IRON BY MEASURING PLAN MICROGRAPHS

by Robert Wlodawer

Sonnenblickstrasse 3, CH-8404 Wintertftur, Switzerland

SUMMARY

As is generally known, the determination of the true proportional

volume of oriented structural constituents, in this case of rectangular

dendrites, is a rather difficult task (i.e. see bibliography in

UNDERWOOD 1970 and in GAHM 1971).- The present paper describes a method

that allows the volume of rectangular dendrites to be determined, and

that even in the case of oblique and randomly cut sections.

By way of verbal simplification, dendrites of this kind may be described as "millimeter paper translated in space" (see Tab .

1

) . Oblique

sections through such cubes result in cutting triangles with a large

variety of angles. These angles may be finally described by spatial

superposition of trigonometric functions. By means of nomograms it is

possible to recalculate the true relations in the rectangular system

for the triangle angles measured in the micrograph and thus to determine the proportional volume of the dendrites.

First we determine the re- lations in rectangular

systems, considering the

proportional volume of the

rods (=cylinders) and the

junctions(^ spheres) .Intro- ducing the quotient Q = d/a

we determine the dendrite

volume Vy immediately

( Tab.l and Figs. 1 and 2). The relations, the terms

and calculations in randomly cut or all-over oblique

sections are shown in Fig.3i resulting in Fig.^.

The correctness of the

method has been verified:

(1) On dendrite model lattices with known dendrite

volume Vv( WLODAWER, 1970, 71) (2) by reciprocal checking

of the results with rectan- gular and all-over obliqueangled sections (BUCHI,

SOLTERMANN , WLODAWER , 1973

)

in actual structures. USING OF ,THE NOMOGRAM FOR OBLIQUE SECTIONS AND EXAMPLES

First we measure the angle of the cutting triangle and determine PJ. Then we count the number of sectional dendrite axes (N^) within the triangle and rectify this by means of the quotient PJ , i.e. we imaginarily

set the oblique lying dendrite lattice parallel to the basal planes. Con- sequently we determine the axis density DA in the rectangular system.

illustrates an all-over oblique section through dendrites in cast iron. The cutting ellipses are oriented in markedly different directions. The very simple computations are arranged in tabular form. There

resulted a proportional volume = 0,23 or 23 % dendrites etc. Illustrative is the comparison of the real state with the equilibrium state (in this alloy approx. only 3 % Dendrites). The actual properINT-RODUCTION

Table 1- I Derivation of oquationa and representation of simple geometrical relations.

Beispiei ExaTipfe

Filch.

Plan

, L•1e1i^^»^N^lO.

L, =

100 = {**/

3 ^'

„=1.73 {L, )

r • 1 . 1 .1 . 3 eTM

3 . 100

• 300/ein3

Oendfile*

( Cfn3/cm3)

,»1,6.( L,)"

315
R. W/odawer

ties of the material depend on the actual structure. Hitherto unexplained

or puzzling problems associated with defects may be consequently solved

by stereological methods (WLODAWER 1976).

b= ^= const.

/ '''''' l^^oO

Fig* J I Graphic elucldat'ion of the terra "projection density" PJ : Projection of dendrite axes on

the cutting triangle, sum of th« partial pictures.- The auramarj picture would be rather confusing

without construction aide. Example of the trigonometric computation of PJPJ [i + cos/3^.t, 59 + si„/5^..B y^c ]

0. 3090

0.9511

•9 fc 1+ ig >fc-[ ='"/^A+ "^/^aI .cos PJ

27 0.5095 1.64 30 o.aolo 1,463

30 0.5 7 74 1.7 280 0.8660 1,496

Fig. 1 Principle and secondary axes parallel to edges a,b and c Comparison of the perspective

sketch with real sectional

figures

,

Fig. 2

A secondary axis parallel to

edge b.- All other axes are oblique.- Comparison of the perspective sketch with real section figures.

316
On the proportional volume of dendrites in cast iron

4 : Nomograms for the determination of the projection PJ from

the angles of the ellipse chains. Triangle chain formed of cutting ellipses. The ellipses are from dendrite axes originating from different directions. Representation of PJ

as a spatial diagram and as a plane nomogram.

317
R. ^lodawer

Fig. ^ I Example for the ddtermlnatlon of an obliquely eectioned dendrltlo etruoture (caet iron, etching Marble, ; 10 X).The calculations are tabulated. Lamener.9taph.t

/ Untereutekhsi

Flake Graphite Hypoeutectic

Graphite lamallaira \ Hypoeulectiqi

FIdche

Plan A

=Dendr( Dendn

1 D

6,5 A

Orig- Opt. = 10 X 4 = 40 X

^7.8 -^'25.35

(cm2>

N=32B

* 13 ? N - 340 = DA

Rsctiricallen

i90V90°/90°: RiiumKch

Spatial

Dendrilen «

Dendrites T?^^ rt^^

=3x7,9x0^85 =20,2^-fA

(_cm/cm3) .^L«-

V3«2,02»T0 V = 0,228

(ll (cm3/cm ^ 23 7o

1,2x10"^ cm

©

DA=7,9/cm2

Wirhlrch

Really Effectivement

; = 9,25x10 mm

rlO"^ cm

miDEBWOOD, E.E. Quantitative Stsreology

( AddieoD-Wesley Publishing Company,

Reading, Mass., 1970 ).

QAHH, J. Handbuch fUr die vieuelle Messung mlt den quantitativea Fernseh-Mikroskop VIDEOHAT

(Flrmenaohrlf t

, Carl ZEISS, W-Germany, Oberkochen 1971)

bOcHI, R. ; SOLTERMANN, B. (1973) Basisversuche zur quant itat iven Eratarrungo- leokung in hochteraperierten keramischen Feingussformen fiir dick-und dlinnwandige StUcke aua Stahlen Kobaltbasie-Legierungen,

(Ingenieurarbelt , Fachhochschule Duisburg )

VLODAWER, R. Directional Solidification of Iron Castings

(book in preparation, Gieaaerei-Verlag QrabH. pUsaeldorf, 1976 ).

WLODAWER, R. (1971)

J. Microscopy 95 , 285

WLODAWER, R. (1970) aioeserei-Rundschau , 17 , 23 ACKNOWLEDGEMENTS

author wishes to express his thanks to Prof E.R.Weibel, DepartAnatomy, Berne University, for the valuable discussion bearing

The ment of on this paper.

318
APPLICATIONS IN BIOLOGY

Chairpersons

:

H. Elias

E, R. Weibel

W. S. Tyler

G. A. Meek

A. M. Carpenter

H. Haug

J. L. Binet

National Bureau of Standards Special Publication 431

Proceedings oi the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

MORPHOMETRY, STEREOMETRY AND STEREOLOGY APPLIED TO CANCER DEVELOPMENT IN THE LARGE

INTESTINE OF MAN

by Hans Elias, D. Bokelmann and R, Vogtle ——

.

Anatomic Institute, University of Heidelberg, Germany

The three terms which appear in the title of this presentation are often used as synonyms; but they are not. Morphometry

is the measurement and counting of structures, regardless of the

methods employed. Measuring tapes, tachometers, telemeters, etc. can be employed to obtain morphometric information. Stereometry

is the same as solid geometry. It is one of the methods employable for morphometry.

Stereology is three-dimensional interpretation of flat

images by criteria of geometric probability (extrapolation from

two- to three-dimensional space). As Professor Haug aptly put it,

stereology as contrasted with photogrammetry is based on uniaxial viewing in cases where parallax can not be employed. Stereology is the most frequently used method in the search for the

solution of problems in morphometry. It can also answer questions

of shape-determination, of orientation in space and of density

gradients

.

Anisotropic objects can usually be adapted to stereological

investigation by random distribution of fragments of the object

or by random sectioning.

In the following pages I will present a pathological object

which exhibits stages from normal intestinal mucosa, through

various transitional forms, each of specific properties, to the

fully developed cancer. In cases of this kind stereology can be

applied only to the terminal stage of transformation, namely to the fully developed malignant tumor.

Figure 1 is a schematical presentation of the marginal area

of a typical ring-shaped adeno-carcinoma of the colon. Beginning

with normal gut wall (A), we encounter in the direction toward

the tumor: an area of hypotrophy and hypoplasia (B), a trench

with strongly hypertrophic and hyperplastic branched glands (C), an area of renewed hypotrophy with nuclear multiplication and

hyperchromasia (D) and almost abruptly following the fullfledged

adenocarcinoma (E). Toward the center of the ring-shaped lesion

is an area of ulceration (P)

.
H. Elias, D. Bokelmann and R. Vdgtle

Confronted with specimens of this kind, randomization of

the entire piece can not lead to a significant result, since

morphometric information is needed for each specific area. Fig. 7 shows how heterogeneous the zone of transformation is.

Stereological approach

Only in the cancer area itself, in which the development

is completed, are stereological procedures apnlicable. For ex- ample, the nucleus/cytoplasmic volume ratio has been determined

by superimposing optically (see demonstration by Elias and Botz)

a grid of lines with their intersection points. This can not be

done in conventional paraffin sections, but only in semithin

sections, where the thickness of the slice is negligible (Pig. 2)

.

Result in case 1

:

^nuclei/ ^cytoplasm = 0.32 , using the formula V/V^= P/PrjThe surface area of the nuclear envelope is determined by

the intersection method, S/V,-n= 2 P/lm(Pig. 3). It was found to

2 0

^ 2 1 be approximately 69 cm /mm of cancer epithelium and 193 cm /mm

of nuclear volume.

Direct morphometry

To determine the lengths of the glands for each developmental area, the specimen must be oriented carefully to permit

cutting exact longitudinal sections through the tubular glands.

The length of individual glands is measured directly by an optically superimposed scale (see demonstration by Elias and

Botz, Fig. 4). This and the next step involve direct morphometry. And this next step is the measurement of the diameter

(2r) of the gland (Fig. 5, right). These two steps permit to determine the surface area of each gland.

Stereometry

From the length i of the gland and its diameter, its sur- face area can be determined by the well-known formula

S = i 2r1T

In tangential sections of the gland (and in serial sections

tangential sections can be found for every longitudinally cut

gland) one counts the number of nuclei per unit area A'p(Fig. 5,

left)

.

322
On cancer development in the large intestine of man

If the average number of nuclei per test area A-,is n, then

the number of cells in the entire gland is

N = / 2r tT 5 (Fig. 6)

-fim This numerical information enables the observer to determine whether a gland is enlarged by hypertrophy or hyperplasia

or by both, or if its size is reduced by hypoplasia or hypotrophy

or both.

Table 1 shows preliminary results in case 2.

323
H. Elias, D. Bokelmann and R. Vdgtle I •HoU I aJ X O 0) oC U

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(A 3

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dO (U H ^O -H" O

(U

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CO « I m M OJ

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p>ST-t "m O *H

CO

rA 1 H EJ

cs

a

•HH

tiO

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fH

O

CSe-p

(D cd

<D •H •

03 «H U o

0)

a

+= I^- •H

CD o S t3 0)

Cfl CJ ft U ni CQ 03

(U o •> -p

•H - o e (D o m W) ^ 0) ft C « U -H o i-q a • -p O C\J a <D o o ra

CQ o! (U H o

o3 CQ Ph OS t3 s a o o

r~

to

03 T— 1 a to O o o d cr, tr> •H to o

CM

ca:

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UPv oo CO to o CO C\) CM CM O o

CJ CM CM o oo o o oo oo

3. CM o oo

CM CO1 o o o LO o o o o CM o LO o

CM CM CM CO o c— a LO to

-p o ^c^ u o LPv ;i CO

03 CM o to

to

d

a LPv o ^ to CO o CM o CTi c— o to

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324
On cancer development in the large intestine of man

325
H. Elias. D. Bokelmann and R. Vdgtle

326
National Bureau of Standards Special Publication 431

Proceedings of tiie FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

tteld at NBS, Gaithersburg, NW.. September 4-9. 1975 (Issued January 1976)

ANALYSIS OF HETEROGENEOUS COMPOSITION OF CENTRAL NERVOUS TISSUE

by S. Eins and J. R. Wolff, Max-Planck-lnstitute for biophysical Cfiemistry, Dept. Neurobiology/Neuroanatomy, Gottingen, Germany

We should like to discuss two points of methodological importance

in stereological analysis of brain structures.

The first problem is connected with the calculation of numeric density parameters, which require the absolute values of that quantity

to which the density relates (volume, area, length). Especially in

biological objects dimensional changes occur which are caused by

preparative influences involving chemical and physical alterations

of the object. The resulting changes are strongly dependent on the

particular preparative procedure and small variations in the method

may result in serious differences of preparative behaviour. There - fore it seems indispensable to assess the effect of a given method

on the same object which has to be analysed by morphometry. For

this purpose, automatic image analysis is a very convenient instrumentation method. Area, perimeter and coordinates can be measured

separately for each brain slice out of a given number of samples by

an image analysing computer (Quantimet 720, IMANCO, Ltd.). The frequency of measurements may be varied by an appropriate program and

prints similar to that of figure 1 are obtained. Here the relative

volume changes of 8 frontal rat brain slices are compared, which

underwent exactly the same preparative procedure in a perfusion

chamber. Differences between the final volumes of individual slices

in the order of 10 % and more are usual. This may be partly due to the variable composition of slices according to the inhomogeneous

structure of central nervous tissue (gray and white matter, layers,

barrels, columns and not yet identified patterns). According to our experience (1) with central nervous tissue it is impossible to predict the final volume change of a given slice even after a systematic investigation of the particular preparative method used. This

is a strong argument for following the behaviour of single slices

instead of averaging larger samples and calculating mean correcting

factors. Nevertheless, there are typical effects of the various

preparative procedures on tissue volume.

Our second point is to demonstrate a further special aspect in the

analysis of heterogeneous material, i.e. the necessity of appropriate sampling methods . Random sampling, although a very useful

method for general statistics in homogeneous populations is less

appropriate, if the detailed structure of an object is blurred by

averaging over the inhomogeneous composition of the specimen. In these cases only measurements from comparable object regions should

contribute to a data set. As this pattern of homologous start

points may not be known before the measurements it must be detected

by systematic sampling. This first step leads to a defined space

standard for the distribution of homologous points in the object

to which further measurements should be related. Of course, within

homologous regions random sampling must be applied. We have found

that a special set of blood vessels vertically penetrating the ce- rebral cortex can be used as a marker for sampling. Applying this

sampling pattern a systematic variation of packing density of synapses and myelinated nerve fibres in Lamina I has been detected.

Vascular pattern : According to the ontogenetic development of the

cerebral cortex, the intracerebral vessels form a typical distribution and branching pattern (see fig. 2). This is due to the fact

327
S. Eins and J. R. Wolff

that the cortical plate (anlage) develops by consecutive apposition

of the more superficial layers on top of the deeper ones (2) . In addition to thickening the cortex also increases its area. Thickening is accompanied by elongation and terminal branching of preexisting vessels, while the increase in area evokes the formation

of new penetrating vessels which supply the superficial layers. The

complete (mature) vascular pattern is characterized by typical mean

distances between old, large penetrating vessels, which may be used

as starting points for the structure specific sampling in the cere- bral cortex.

Degenerating fibres : After small intracortical lesions degenerating

fibres appear in Lamina I (fig* 3). Since this type of degeneration

can also be induced after undercutting, these nerve fibres must

originate from local neurons. Their distribution is cluster-like

and cannot be related to the position of the inducing lesion. However, maxima and minima of degeneration are distributed in a manner

similar to that of the large penetrating cortical vessels.

If a regular (linear or 2-dimensional) sampling is applied instead

of random sampling, the mean synaptic density with large standard

deviations is transformed into a periodic variation which is

strongly correlated to the position of penetrating vessels (3).

In conclusion : Ontogenetically old, long penetrating vessels can be

used as starting points for systematic sampling in the cerebral

cortex. It can be demonstrated by the distribution of other structures (synapses and special nerve fibres) that the vascular system

in the cortex is not only a practical, but also a biologically re- levant spatial standard. Further studies are required to search for

similar ontogenetical spatial standards in other organs and

tissues

.

References

(1) S. EINS and E. WILHELMS, Fortschritte der automatischen Bildanalyse, IMANCO-Symposium, Frankfurt 1975.

(2) M. BERRY and A.W. ROGERS, J. Anat. 99, 691 (1965).

(3) J.R. WOLFF, this conference.

328
Composition of central nervous tissue

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THICKENING - NUMBER OF VASCULAR BRANCHINGS

fig- 2

329
S. Ems and J. R. Wolff

330
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

STEREOLOGICAL ANALYSIS OF THE HETEROGENEOUS COMPOSITION OF CENTRAL NERVOUS TISSUE: SYNAPSES OF THE CEREBRAL CORTEX

by J. R, Wolff Max-Planck-lnstitute for biophysical Chemistry, Dept. I^eurobiology/Neuroanatomy, Gottingen, Germany

Central nervous tissue represents a complex 3-dimensional network

which is heterogeneous, being composed of many interdigitating neu- ronal and glial cell processes. This inhomogeneity has several con- sequences: 1.) The typical "dendritic" shape of neurons is caused

by polar, arborised cell processes. Processes of different polarity

(axons and dendrites) make specific contacts (synapses) with each

other. Consequently the size, orientation and arrangement of the

processes of a neuron may define the probability of synapse formation with other neurons. 2.) Each neuron represents only a small

fraction of the space, in which its processes are distributed.

Therefore normal neurons may be forced to change their shape, if

the tissue shrinks after the degeneration of surrounding neurons.

3.) Random sampling is unsuitable for detecting variations in small

tissue fractions within complex tissues of heterogeneous composition. Consequently stereological analysis relies on sampling and

statistical methods which do not blur the observed heterogeneity

(1) . In the cerebral cortex various types of heterogeneities are known,

such as vertically oriented barrels, stripes and columns, as well

as bundles of dendrites and axons which penetrate a horizontal set

of layers (see fig. 3). Varying numbers of laminae and sublaminae

have been defined in relation to the distribution of different com- ponents. Obviously none of these structural characteristics can be

used as a space standard or as starting points for sampling in the

cerebral cortex.

All available data on the synaptic density in the cerebral cortex

show relatively high standard deviations of 20 - 25°6 about the mean

(2) . In the present study we tried to determine whether this high

variability is caused by non-random, systematic variations of the

synaptic density in the cortical tissue.

Therefore a systematic sampling has been applied in the present

study consisting of vertically and horizontally oriented equidistant sequences of electronmicrographs (magnification 15000 : 1). In addition, all structures which do not contain or make synapses

(cell nuclei, blood vessels) were subtracted from the micrograph

area

.

Thus the packing density of synapses has been estimated in comparable regions of the "neuropil".

In all vertical sample sequences the synaptic density decreases

from the superficial to the deeper cortical layers. However, the mean level of the absolute values may differ from one sequence to another (fig. 1). This difference increases with the horizontal

distance between the vertical lines of measurements.

Horizontal sequences of samples show a periodic variation (increase

and decrease) of the packing density of synapses. The maxima are correlated with the position of rather thin, vertically orientated

blood vessels (arteries?) branching in the superficial two-thirds

of the cortex. In contrast, the minima surround wide blood vessels

(venes) which penetrate most of the cortex.

331
J. R. Wolff

In agreement with the variable distance between the vertical blood

vessels in sections through the cortex, the periodicity of the synaptic density is not a regular one. However, in the visual cortex

of rats a minimum frequently occurs about 300-500 ^m from the next

maximum (fig. 2 and 3).

The distribution of synapses has been found to be similar in va- rious parts of the cortex and in different species: rats, cats,

rhesus monkeys. In addition, we compared the synaptic density in

the normal visual cortex with that after undercutting of the posterior lateral gyrus of cats (fig. 2) and of area 17 of rats. Although, after undercutting all afferent nerve fibres degenerate in

the cortex, the periodic variations were essentially retained(f ig

.

2) . These results suggest that the periodic variations of the

packing density of synapses, which depend on the position of vertically orientated blood vessels, represent a pattern which is cha- racteristic for the neocortex.

The cortical vascular pattern becomes complete during the early

postnatal life (3) and is retained even after heavy degeneration in

the undercut cortex (unpublished) . Therefore the intracortical vas- cular system may be used as a space standard (number of penetrating

or vertical vessels per surface area X total thickness of the cortex), which indicates all variations of the tissue volume caused by

growth or retardation, swelling or atrophy. In addition, the vertical vessels can be used as starting points for systematic sampling

in the horizontal level, i.e"^ parallel to the surface.

A study on the distribution in lamina I of myelinated axons originating from neurons in the deep layers of the cortex demonstrates

that variations in their packing density seem also to be related to the position of long penetrating vessels (1). Further studies are required to determine whether these topological relations between

different structural components depend on common developmental con- ditions . By simultaneous estimation of the volume fractions of blood vessels, cell nuclei, glial processes and myelinated axons, the number

of synapses could be related to various types of tissue volumes(cortex volume; neural tissue volume; "neuropil"; neuronal neuropil;

collective volume of dendrites, unmyelinated axons and boutons)

.

These calculations show that neither the gradient from superficial

to deeper layers nor the periodic variations of the synaptic density parallel to the surface of the cortex vanish, if any of the vo- lumes is applied. Consequently, all the intracortical variations of

the synaptic density depend on real variations of the number

of -synapses per dendrite or axon. Since variations of about 100 % occur, there must be neurons, e.g. pyramid cells in the same Lamina, which are only about 300 to 500 ^m apart, but receive very

different amounts of synaptic input. These results may demonstrate

the dangers of random sampling and averaging in a stereological

analysis of complex and heterogeneously composed tissues.

References

(1) S. BINS and J.R. WOLFF (this meeting)

(2) B.C. CRAGG: Brain 98, 81 - 90 (1975)

G. VRENSEN and D. De GROOT : Brain Res. 58, 25 - 35 (1973)

(3) Th. BAR and J.R. WOLFF: Z. Anat . Entw.-^sch. 141 ,207-1 221 (1973)

332
The heterogeneous composition of central nervous tissue

40

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PIA MATER

P65 / 2

SUBCORTICAL fig. 1 The values of synaptic density are connected by lines along vertical sampling sequences, post. lat. gyr. of cat. Note: different

sample sequences may be separated over their total length, although

all show the same gradient (slope) towards the subcortical white

matter

.

^ SYNAPSENDICHTE IN LAMINA II BildHTSunfl ~ PTBa lunterschnittefi I

0 ;1 ^ J X 5 6 ,7 .8 .9 ID mm fig- 2 The density of synapses varies periodically in a sequence of samples which have been taken from lamina II of the rat's cerebral cor- tex (area 17) 3 months after undercutting. Gray-Type II synapses

show similar, but smaller variations.

333
J. R. Wolif

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334
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg. Md., September 4-9, 1975 (Issued January 1976)

THE DEVELOPMENT OF MYELINATED FIBERS IN THE VISUAL CORTEX OF CATS

by H. Haug

Medical School LUbeck-Department of Anatomy, D-2400 Liibeck, Ratzeburger Allee 160, Germany

The development of myelinated fibres has been studied with stereological

procedures. Earlier quantitative investigations of these fibres have been

carried out more in the peripheral nervous system (1, 3, 6, 9, 11, 12 etc.)

than in the central nervous system (2, 4, 5, lo) . The results have been

gained by veiry different procedures and samples of various sizes. In the following it will be reported on an investigation, which is based on 15 animals of different ages counting 44ooo myelinated fibres. The length per

unit volume has been measured by the nximber of intercepts per area (8) and

in the same step the diameters have been estimated with a TGZ 3 Particle

Size Analyser in low-power electron micrographs of the visual cortex (4)

.

From each electronmicrograph the distance from the surface of the cortex

has been registered in order to estimate the different developmental

speed in the different cortical layers. The procedure of evaluation in rows and layers was similar to the above mentioned method in light

microscopy (7 )

.

Our investigations have produced the following results:

1) The myelination in the visual cortex of the cat begins in the 3o*^ day

of life. This fact indicates that the function of the visual cortex begins

in about 3o days after birth. After the opening of the eyelids it must be

assumed that the visual function is directed by lower visual centers of

the brain.

2) The first fibres which appear are input fibres of greater diameters

(1,5 urn). They grow only up to the lo^r third of the cortex and have

developed nearly completely by the 5o day.

3) The average diameter will be smaller with increasing age (o,9 Mm) in an adult cat , but this is due to the predominant development of smaller

fibres and not with a loss of thicker fibres which are growing, as well,

but at a slower rate.

4) After the 6o day the first internal system of myelinated fibres can be

observed. It lies below the surface and has almost no connection to the deeper layer of fibres for in- and output.

5) gie size of brain of the cat has been fully developed with about the

loo day. Howgver, the last fibre systems of very small fibres are growing

after the 154 day. The oldest brain of an advanced adolescent cat has

only two third of the adult cat fibres

.

335
H. Haug

6) The small fibres which appear after the loo day belong to the internal systems of the visual cortex and are lying mostly in the middle

of the cortex. Their diameters are small, below 1 um.

7) Our evaluation shows that the storing of the original values of the fibre diameters has a great advantage in respect to different combinations

of results for different kinds of distribution eg. in rows perpendicular

to the surface or in layers parallel to the surface.

Summary

The development of the myelinated fibres was studied by means of optomanual stereological procedures (TGZ 3) . The f^lowing results are

important. The first input fibres ^pear at the 3o day of life. The last internal fibres grow after the 15o day at a time in which the cat is in the transition phase between adolescence and adult stage. Most of the fibres have a diameter smaller than 1 um.

References

1) Berthold, C.-H. and Carldstedt, T., Neurobiol. 3 (1973), 1 - 18.

2) Bishop, G. H. and Smith, J. M. , Exp. Neurol. 9 (1964) 483 - Sol.

3) Donovan, A., J. Anat. lol (1967) 1 - 11.

4) Haug, H. , Stereology, Proceedings of the Second International Congress

for Stereology, Chicago 1967, Springer New York Inc. 66 - 67.

5) Haug, H., Proceedings 25. Anniversary Meeting Electron Microscopy

Society of America 1967 b, Ed. C. Arceneaux, Claitoc's Book Store, Baton Rouge, 56 - 57.

6) Haug, H. und Rast, A., Microsc. Acta 72 (1972) 136 - 146.

7) Haug, H. , in Quantitative Analyse von Gefugen in Medizin, Biologie

und Materialentwicklung, Ed. H. E. Exner, Dr. Riederer GmbH,

Stuttgart (1975) 91 - lo2.

8) Hennig, A., Z. wiss. Mikrosk. 65 (1963) 193 - 194.

9) Landolt, J. P., Topliff, E. D. L. and Silverberg, J. D. Brain Res. 54 (1973) 31 - 42.

10) Samorajski, T. and Friede, R. L. , J. Comp. Neur. 134 (1968) 323 - 338.

11) Sunderland, S. and Roche, A. F., Acta anat. 33 (1958) 1 - 37.

12) Young, R. F. and King, R. B., J. Neurosiirgery 38 (1973) 65 - 72.

336
Nationat Bureau of Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

THE VOLUMES OF ELASTIC TISSUE AND SMOOTH MUSCLE CELLS IN THE ARTERIAL WALL

by Vladimfr Levicky

Institute of Normal and Pathological Physiology, Slovak Academy of Sciences, 884 23 Bratislava, Czechoslovakia

Most studies dealing with the amount of elastin, collagen

and Other components of the blood vessels wall have been performed by means of biochemical methods /Lov/ry et al.,1941; Neuman and

Logan, 1950; Harkness et al.,1957; Banga,1959/. Studies concerning

the volume of individual tissues and thus directly morphological

parts of the wall are substantially limited in number and concentrate mainly on aorta while to other arteries they pay only a little attention /Hurthle , 1920 j Lang, 1965/.

This work is intended to broaden and specify existing knowledge concerning the volume of two important structural components of the arterial wall-elastic tissue and smooth muscle cells

in the walls of arteries of different diameter. At the same time

it is an attempt at correlating the results with the function of

the arterial wall.

Method

For taking the blood vessels excisions in the experiment

were used 39 dogs.

The point-counting procedure /Underwood , 1970/ was used to

find out the data on the volume of both elastic tissue and

smooth muscle cells. The histological section of the arterial

wall was observed by means of a test grid inserted in the eyepiece of the microscope. The total magnification was 1250x.

Through moving the microscope stage the vessel wall section is

covered, beginning with the outside boundary of the adventitia

towards the intima, until the number of about 500 or more test

points is reached. The distance between test points was 100 pm.

The relative values of errors for the structures observed

were calculated from formulas and tables presented by Haug/1962/

and oscillated v/ithin 5-15

In the first stage of the observation the relative volume

relationship was determined between the smooth muscle cell and

its nucleus. The coefficient that was obtained served to calculate the volume of total smooth muscle on the basis of the volume

of nuclei of smooth muscle cells. This procedure had been chosen

in viev/ of the fact that the used method requires a precise delimitation of the evaluated structures that may not be achieved

through the simultaneous staining of elastic tissue and smooth

muscle cells. In the next stage the relative volumes of elastic

tissue and nuclei of smooth muscle cells in the arterial walls of

different areas in the direction from aorta towards the vascular

bed periphery were determined which enables making comparisons of

measured values in the arteries of different structure, diameter

and function. The following staining methods were used; elastic

tissue was stained by the resorcin - fuchsin, the smooth muscle

cells nuclei by the Mayer's haematoxylin and the smooth muscle

cells for the calculation of the cell to its nucleus volume

relation by the Heidenhain's haematoxylin.

Re suits

A. The ratio of nucleus volume to that of a whole smooth muscle

cell equals 1:14.5, i.e. the nucleus volume accounts for 5.9%

337
V. Levicky

of the total smooth muscle cell volume at the average. No

significant differences in this ratio were found on the sec- tions of arteries taken from different areas of arterial bed.

B. Figs. 1-6 represents the volume of elastic tissue and smooth

muscle cells nuclei in the aorta and the arteries of different

consecutive regions. These measurements could be summarized

in the following way:

1/ The elastic tissue volume decreases from the proximal part

of aorta where it achieves values from 22.0-25.0% in the

distal direction where its values decrease down to 4.0-6.0%

in the smallest arteries /palmar branch of radial artery,

the arteries of the arterial circle of the brain/ while in

the other arteries /dorsal pedal artery, facial artery, ar- cades of the jejunal arteries/ they are somewhat higher:

8.0-11.0%.

2/ The volume of smooth muscle cells nuclei is the same in all

the arteries. The average values move in the range of 3.0

to 4.0%. As the volume of smooth muscle cell is a 14.5

multiple of that of its nucleus, the total volume of smooth

muscle calculated on the basis of the nuclei volume fluctuates in most arteries between 45.0-55.0%.

Discussion

In our study we put under examination arteries of considerably differing size - from aorta with the diameter of 11.0 mm un- til arteries v/ith the diameter around 0.5 mm. In its course we examined the volume of smooth muscle cells in the arteries of

both elastic and muscular type. In all the cases we arrived at

equal values, in other words we found out that the relative

volume of smooth muscle cells is constant and does not depend on the arterial wall structure.

From the functional point of view the main function of elasri

tic tissue consists in effecting the elastic tension in response

to the vessel distension with lower and normal transmural pressures /Burton, 1965/. Its presence makes also possible a graded con- trol of vessel diameter exerced by smooth muscle. Because in the

smaller arteries the vessel wall tension decreases and the diameter is being more and more controlled by the smooth muscle

cells, the decrease in the elastic tissue volume in smaller arteries appears to be an accompanying phenomenon of functions

changed in this way.

On the basis of presented findings it is possible to draw

the conclusion that the property of arteries of muscular type to

actively change the blood vessel diameter by means of muscular

contraction - in comparison with arteries of elastic type - is

made possible by decreasing in volume of elastic tissue with

keeping the same relative volume of smooth muscle - i.e. active

contractile component of the vascular wall.

Bibliography

Banga,J., in Int. Symp.Biochem. Vascular

Wall, Fribourg 1968 Part II, pp. 18-92,'

Ed. by M. Gomel and L.Laszt, S.Karger, Basel-New York 1969

Burton, A. C, Physiology and Biophysics of the Circulation, Year Book Medical Publishers, Chicago, 1965 Harkness,M.L.R. , Harkness,E.D. , McDonald,!).

A., Proc.Roy .Soc.Lond.Ser.B. 146:541-551,

1957

Haug,H., in Med . Grundlagenforschung 4:299-

344, 1962 Hurthle,K., Pflugers Arch. 183:253-270,1920

Lang, J., Angiologica 2:225-284,1965

Lowry,O.H., Gilligan,D.R. , Katersky,E.M.

,

J.Biol.Chem. 139:795-804,1941

Neuman,R.E., Logan, M. A., J.Biol.Chem,

186:549-556,1950

Underwood ,E.E. , Quantitative Stereology,

Addison-Wesley Publ.Co., Heading, Mass, 1970

338
Volumes of elastic tissue and smooth muscle cells in arterial wall

MLZ

25

20

IS

10

UC THM MD1 *»2 IIE FEN POP IB PED Fig.l. Aorta and arteries of the hind-limb: ASC — ascendoaorta, THOR

-r thor. aorta, ABD 1 — abdomaaorta before the arising of cranoinesent

»

art., ABD 2 — abdom. aorta before the arising of renal art., ILE -

ext. iliac art,, FEM - femoral art., POP - popliteal art., TIB - crano tibial - arto-j PED - dors. pedal art. Symbols: • elastic tissue, *— —* smooth muscle cells nuclei, vertical lines = standard deviations, P = the degree of statistical

significancy^ NB = a non-significant difference

VOLZ

20

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0 ASC IRC SIC 1 SIC 2 AXIl IR RAD R.PAl Figo2, Brachiocephalic artery, subclavian artery and arteries of the

fore-limb: BEC - brachiocephal.art. , SBC 1 - subclav.art»/ the beginning /, SBC 2 - subclavo art ./ the end /, AXIL — axillary art.,

BR ~ brachial art., RAD - radial art,, R.PAL - palmar branch of radial art.

VOLJ I. VDLI

0- - . . - , .

, i ' 0 ilC UtCI CUC2 CUC3 CilEI CUE2 MCI FAC2 CUE2 IS 1 TS 2 Figo3o Common carotid artery, external carotid artery and its ex- tracranial branches: CAR CI- comm. carot. art./ the beginning /,

CAR C 2 - comm. carot. art. before the arising of cran. thyreoid. art »

,

CAE C 3 — comm. carot. art./' the end /, CAR El- extern. carot. art.

/' the beginning /', CAR E 2 - extern. carot. art. before the arising of

facial art., FAC 1 - facial art./ the beginning /,. FAC 2 - facial

art. before the arising of submental art., TSl- superfic . temporal

art./ the beginning /, TS 2 - superfic. temporal art./ the end /

339
V, Levicky

W12

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10

5

0 URC3 CARI-B MRU CARI2 CARI3 CERMI CERH2 CERM3 Figo4« Internal carotid artery and middle cerebral artery: CAE I-B - bulb of internocarotaart» , CAR X 1 - intern. carot. arto just after the

bulb, CAR 12- internocarotoartcbefore getting into carotocanal,

CAR 13- intern. carotoart«after getting out of carot.canalj GER M 1 - midd.cerebr. art. before the first branching, CER M 2 - midd.cerebro

artobefore the second branching, CER M 3 ~ midd.cerebroart.before

the third branching.

V01.Z 15

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u - ,

, , , ^

, . 0 CARC3 CAR El MAXI HAX2 HEN 1 MEN 2 NEN 3 Figo5o External carotid artery, maxillary artery and middle meningeal artery: MAX 1 - maxilloart./ the beginning /, MAX 2 - maxill.art.

before the arising of midd.mening. art, , MEN 1 - midd.mening.art.before getting into oval foramen, MEM 2 - midd.mening. art . after getting

out of oval foramen, MEN 3 ~ midd.mening.art.in the temporal region.

I p-ooa 1 VOLZ

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, r

, , ,

, - 0 ABD1 MESI HES2 JEJ ARC 1 ARC 2 Fig.6o Mesenterial arteries: MES 1 - cran.mesenter.art./ the beginniing /, MES 2 - cran.mesenter. art. before the arising of 6th jejunal

art., JEJ - 6th jejunal art,, ARC 1 and ARC 2 - primary and secondary arcades of jejun.art.

340
National Bureau of Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

PROGRESS, SUCCESS AND PROBLEMS IN APPLYING STEREOLOGY IN BIOLOGICAL RESEARCH

by Ewald R, Weibel Department of Anatomy, University of Berne, Berne, Switzerland

Exactly 50 years have passed since, for the first time, a stereological principle was derived with a biological application in mind:

in 1925 the Swedish mathematician S.D. Wicksell published his now classical memoir entitled "The corpuscle problem: a mathematical

study of a biometric problem" (1) . It is only slightly more than a decade, however, that the stereological approach is making progress

in becoming an established method of biological investigation. The

number of biological papers making successful use of stereological

methods has been rapidly increasing over the past years. However, one of the remaining problems is the lack of awareness for the potential of a stereological analysis among still a large fraction of

biomorphologists ; indeed, in perusing some of the leading biological

journals one is astounded to find a large number of papers in which

a quasi-quantitative analysis of structure was used to produce am- biguous data, whereas solid morphometric data could have been easily

produced - often with even less effort.

Inspite of considerable progress - and even some success - it is

clear, however, that the application of stereology in biological re- search still poses a number of very serious problems. These become

particularly prominent as one attempts to proceed from descriptive

miorphometry to the use of stereology in an analytical framework in

combination with a physiological or biochemical analysis. These

problems derive (a) from the nature and complexity of biological

specimens, (b) from the difficulty of controlling artefacts, (c) from the necessity to work with a variety of morphological methods

to reveal the properties of structural elements, and (d) , last but

not least, from the persisting lack of adequate technology to efficiently produce the large volume of data often required. I shall not

deal with this latter aspect as this has been done elsewhere (2,3),

but shall concentrate on the other problems, particularly focussing

on those arising in attempts to quantitate the structural make-up of

cells by electron microscopy.

1. Identification of cell constituents

Cells are subdivided into various functional compartments by membranes. These membranes are sheet-like aggregates of proteins and

lipids} in their basic constitution they are all alike and conse- quently look very similar in an electron micrograph (Fig. 1), but

they differ by the function of the proteins most of which are highly

specific enzymes. Thus various types of membranes can be distinguished; but they are recognised in electron micrographs essentially

by the configuration of the membrane-bounded space and by their context. The major problem resulting from this situation is that image

analysis depends on interpretation of the constituents by trained

investigators, and that automatic image analysis is still not feasible, except under very special circumstances. Hence, stereological

analysis of electron micrographs of cells remains a laborious enterprise, although the proper planning of point counting procedures.

341
f. p. Weibel

the use of a variety of test systems, and computer assistance can greatly alleviate the chore (4,5) . In recent years considerable progress has been made in specifically "tagging" membranes and cell compartments by cytochemistry,

i.e. by allowing an enzyme to react in situ with a suitable substrate so as to produce an insoluble electron-opaque precipitate.

Fig. 2 shows such a preparation in which the black reaction product

marks sites of glucose-6-phosphatase , a marker enzyme for endoplasmic reticulum. The problem with these methods lies in the difficulty

to adequately control the reaction quantitatively and to assure even penetration of the substrate. Some progress is currently being made

by using isolated cells in suspension (6).

h further new technique for tagging specific protein components

is that of immunocytochemistry (7) , but here the pitfalls of the

method are even greater. If these methods of specific tagging of

cell constituents are further improved automatic image analysis may

eventually become accessible on a broader scale.

The method of f reeze-fractur ing offers a further way of identifying different mem.brane types by the pattern, size, and density of

intramembranous particles; these are displayed on the fracture sur- face because the fracture has a tendency to follow the mid-plane of

membranes and hence splits them in half (8) . The main problem for

using stereology on such preparations is that we are dealing with a non-flat surface, and that the "section" (= fracture surface) leads

to a biased display of structure elements. We shall show elsewhere,

however, how non-conventional stereology can be used to quantify

membrane systems on f reeze-fracture preparations (9).

2. Biological organisation and sampling for stereology

Stereological methods are derived on the basis of probabilistic

models: the structural elements are supposed to be randomly distributed, randomly oriented, and to be randomly intercepted by a stereological test procedure (section, test lines, test points, etc.).

Biological structure is, however, characterised by an extraordinarily high level of organisation: cells of different function are clearly separated and often arranged in a strictly hierarchical order. Many cells, tissues or organs also show a pronounced polarity

or anisotropy. This appears to exclude a stereological approach to

quantitation of such structural systems.

Indeed, this is one of the more fundamental problems in biomorphometry. We shall consider three situations.

2.1. Gland-like structures

A gland-like structure is characterised by a system of branching

ducts which are organised according to a strict hierarchy which es- tablishes a strong order and polarity of the elements (Fig. 3) . A

morphometric analysis of such a hierarchical system cannot be based

on classical stereological methods, but must use systematic "hierarchical" sampling procedures (10,11).

In most gland-like structures the "parenchyma" or functional com- plex of cells is found at the termination of the last branches of

the duct system (Fig. 4) . The units attached to each terminal branch

show a remarkable uniformity: they occur in very large number, and

342
Progress, success and problems in applying stereology in biological research

appear furthermore rather "randomly" oriented in space because the

duct system branches in all directions like a tree. A plane section

will therefore hit these parenchymal units randomly; a stereological

analysis hence becomes possible, but it must exclude "duct elements"

from the sample. In general, gland-like structures must therefore be

analysed by multiple stage sampling procedures. Such an approach has

been successfully used for example in studying lung (10,12), liver

(13,14,15), Salivary glands (16), and pancreas (17).

As the demands on the reliability of the stereological analysis

increase it will however be mandatory to examine whether a probabilistic structural model is indeed satisfactory or whether more

stringent model conditions must be introduced.

2.2. Polarised stratified structures

Such structures are exemplified by skin or mucous membranes: sev- eral layers of cells extend from the base to the surface, and there

is a definite gradient in the structural and functional property of

the cells. Such systems clearly call for systematic stratified sam- pling procedures; whilst some progress has been made (18) a large

nximber of problems remain to be solved.

2.3. Anisotropic and periodic structures

The extreme case of anisotropic and periodic structure is certainly skeletal muscle (Fig. 5): the strict parallelism of the straight

myofibrils renders these cells anisotropic; the subdivision of the

myofibril into sarcomeres of identical length makes a perfect periodic structure. It is found that all other cell constituents comply to this strict order and are likewise anisotropic and periodic. Eisenberg et al. (19) have proceeded to a thorough analysis of the conditions for performing a stereological analysis on such cells.

An anisotropic system could actually be analysed on a strict

cross-section, perpendicular to the anisotropy axis; periodicity of

the elements however would introduce considerable sampling errors.

It has been shown that an oblique section yields a representative

sample and allows measurement of both Vv and Sv of various components (20) . Alternatively, one can use sections parallel to the anisotropy axis and orient a test line system at a fixed angle (19,21) . By this approach one has however abandoned to a large measure the

fundamental probabilistic basis on which stereological methods have

been derived.

3 . Preparation artefacts

3.1. Processing artefacts

Before a section can be viewed in the (electron) microscope the

tissue must be fixed, stained and processed through a considerable

number of steps. Cells are osmometers and may swell or shrink, depending on the osmotic properties of the processing solutions. Compression of the section by the microtome knife is another important

artefact. Although some studies of such effects have been made (22)

,

in most instances control of these artefacts is still rather intuitive .

343
E. R. Weibel

3.2. Section thickness and Holmes effect

Basic stereological methods are developed for true two-dimensional sections of no thickness. The systematic errors introduced by using them on sections of finite thickness are well-known (23-26) but

nevertheless difficult to control in practice: they depend on the

relative dimension of section thickness and object size, but are heavily dependent on the shape of the objects. As the demands on the

reliability and reproducibility of stereological data increase with

the sophistication of biomorphometric studies, such effects - or rather errors - must find a reliable compensation. One way is to develop correction coefficients which adequately consider the properties of the specimen and the measuring condition. In view of cor- recting volume and surface density estimates on cellular membrane

systems in the framework of an analytical study (27) we have recently re-examined this problem systematically.

The cytoplasmic membrane system of cells occurs in essentially

three forms (Fig. 1): (a) as spherical vesicles, (b) as narrow cisternal spaces (rER) , and (c) as tubular networks. Approximating

these shapes with (a) spheres, (b) thin flat discs, and (c) thin

cylindrical tubules we derived a set of correction coefficients for

Vy and Sy estimates which consider the total projection of the sec- tion content, leading to overestimation , as well as the loss of profiles produced by grazing sections, leading to underestimation. In the case of spherical vesicles the effect of size distribution on

the error due to finite section thickness was also considered.

Table I presents a synopsis of the formulas developed for the three

basic shapes; these correction factors, by which the raw estimates

must be multiplied, are easily determined by means of graphs, as shown in Figs. 6-9.

The analysis of these three basic shapes has shown that section

thickness leads to very different errors in the estimation of Vy and

Sv of various cell components. For endoplasmic reticulum of the rat

liver, for example, it was found that raw estimates of Sy had to be

corrected by a factor of 0.95 for cisternal rER, whereas a factor of

0.70 was required for the tubular sER. The surface area estimate of

spherical vesicles as they are found in microsomal fractions from

the same tissue required correction by a factor of 0.72. Whilst the

surface area estimates of rER cisternae was affected by a small er- ror only, the estimate of Vy of rER required correction by a factor

of 0.54; this is due to the fact that the length of the membrane

trace, on which the estimation of Sy is based, is little affected

by varying inclination of the cisterna to the section plane, whereas

the area of the projected band, the basis for the Vy estimate, in- creases both with section thickness and with inclination of the cisterna . As this example shows, consideration of section thickness effects

can significantly improve the validity of stereological data. The use of correction factors together with basic stereological methods

is, however, only partially satisfactory. It is to be hoped that the

development of new stereological principles will allow us, in the

future, to obtain our measurements with methods that account a priori

for finite section thickness.

344
Progress, success and problems in applying stereology in biological research

4. Stereology as a tool in analytic cell biology;

the beginning of a future

The foundations of modern cell biology have essentially been laid

by Albert Claude, Christian de Duve, and George Palade, last year's

Nobel Laureates. Their concept was that the many functions of the

cell must be associated with a particular structural element, and

that it should be possible to identify that association (28) . They

proceeded to a combined use of cell fractionation (based on size and

density of the components), biochemistry, and electron microscopy,

partly combined with cytochemistry and autoradiography.

In this work biochemistry was quantitative, allowing the establishment of balance sheets. The structural analysis however remained

qualitative, so that a true quantitative structure-function correlation could not be established. Instead of relating the activity of

membrane-bound enzymes to the area of membrane present, they used

the protein content as a biochemical reference for structure.

Only rather recently have Baudhuin (29) and Wibo(30) from de

Duve's group proceeded to a stereological analysis of the membrane

vesicles present in the various fractions by using a Wicksell analysis; this resulted in a good correlation between structural and biochemical data.

Over the past years, Bolender et al. (27) have conducted a systematic study in which they attempted a quantitative correlation

between biochemical determinations of enzyme activities measured in

fractions and a stereological analysis of intact cells; this re- quired an intermediate step, namely the stereological analysis of

the fractions. The difficulty of identifying the origin of membrane

vesicle in a fraction required the use of methods of -"tagging" by

cytochemistry and a stereological evaluation of f reeze-fracture

preparations. The final attempt to establish a correlation further

necessitated sound correction of artefacts, such as compression and

section thickness. It had further to consider that the cell population of the liver is heterogeneous, and that the various cell types

may contribute differently to the homogenate and to the stereological analysis which focussed predominantly on hepatocytes - hence a sampling problem was prominent (31)

.

The approach developed in this context is very involved and laborious; if it can be rendered more efficient this may however prove

to be a profitable field for applying stereology.

5. Conclusions

The application of stereological methods in biological research

has undoubtedly made considerable progress. The biologist now possesses a set of methods that lend themselves to a large variety of

applications, from basic research in cell biology to more practical

aspects of pathology (32-34). In many fields the use of stereology

has been undoubtedly successful. Nevertheless, a large number of

problems persist or even increase as the stereological approach becomes more sophisticated. The main residual problems are (a) an improvement in the efficiency of the methods, (b) a better control of

artefacts, and (c) development of methods that ensure the validity

of stereological measurements inspite of non-ideal conditions of the

specimen. These problems need to be solved in the near future.

345
E. p. Weibel

References

Wicksell, S.D.: Biometrica 17, 84, 1925

Haug, H.: Experiences with opto-manual automated evaluation

systems. (This volume)

Weibel, E.R., Fisher, C, Gahm, J., and Schaefer, A.:

In: Stereology 3. J. Microscopy 9_5, 367, 1972

Weibel, E.R.: Stereological techniques. In: Principles and

techniques of electron microscopy. M.A. Hayat, ed..

Van Nostrand-Reinhold , New York, 1973

Ratz, H.U., Gnagi, H.R., and Weibel, E.R.: J. Microscopy 101 , 267, 1974

Wanson, J.C., Drochmans, P., May, C., Penasse, W. , and

Popowski, A.: J. Cell Biol. £6, 23, 1975

Kraehenbiihl, J. P., and Jamieson, J.D.: Review of Exp.

Pathology 13_, 1, 1974

Branton, D.: Phil. Trans. Royal Soc. London B, 261, 133, 1971

Weibel, E.R., Losa, G.A., and Bolender, R.P.: Stereological

method for estimating relative membrane surfaces in

f reeze-f racture preparations. (This volume)

Weibel, E.R.: Morphometry of the human lung. Berlin, Springer,

1963

Horsfield, K., and Gumming, G.: J. appl. Physiol. 2^, 373, 1968

Weibel, E.R.: Physiol. Reviews 5^, 419, 1973

Loud, A. v.: J. Cell Biol. 37, 27, 1968

Weibel, E.R., Staubli, W. , Gnagi, H.R., and Hess, F.A.:

J. Cell Biol. _42, 68, 1969

Bolender, R.P., and Weibel, E.R.: J. Cell Biol. 56, 746, 1973

Albegger, K.W., and Miiller, O.: Arch. klin. exp. ONK-Heilk.

204 , 27 , 1973

Bolender, R.P.: J. Cell Biol. £1, 269, 1974

Schroeder, H.E., and Miinzel-Pedraz zoli , S.: J. Microscopy 92 , 179, 1970

Eisenberg, B.R., Kuda, A.M., and Peter, J.B.: J. Cell Biol. _60, 732, 1974; J. Ultrastr. Res. 51, 176, 1975

Weibel, E.R.: J. Microscopy 229 , 1973

Sitte, H. : In: Quantitative Methods in Morphology, p. 167,

E.R. Weibel and H. Elias, eds., Berlin, Springer, 1967

Weibel, E.R., and Knight, B.W.: J. Cell Biol. 21, 367, 1964

Holmes, A.H.: Petrographic Methods and Calculations, London,

Murby, 1927

Chayes, P.: Mineral Magazines 3_1 , 276 , 1956

Hennig, A.: Z. wiss. Mikr. 6_3, 67, 1956 ; Mikroskopie 12,7,1957

Cahn, J.W.: Am. Mineralogist 44, 435, 1959 25,154,1969

346
Progress, success and problems in applying stereology in biological research

(27) Bolender, R.P., Paumgartner , D., Losa, G.A., and Weibel, E.R.:

Correlated biochemical and stereological analysis of

liver cells (in preparation)

(28) De Duve, Ch.: Science 189 , 186, 1975

(29) Baudhuin, P., and Berthet, J.: J. Cell Biol. 3^, 361, 1967

(30) Wibo, M. , Amar-Costesec , A., Berthet, J., and Beaufay, H.:

J. Cell Biol. 51, 52, 1971

(31) Blouin, A., Bolender, R.P., and Weibel, E.R.: The distribution

of organelles and membranes between hepatocytes and non- hepatocytes in the rat liver parenchyma (in preparation)

(32) Thurlbeck, W.M. : Thorax 22, 483, 1967

(33) Kapanci, Y., Weibel, E.R., Kaplan, H.P., and Robinson, F.R.:

Lab. Invest. 20, 101, 1969

(34) Bachofen, M. , and Weibel, E.R.: Chest £5, 14, 1974

Acknowledgements

This work was supported by grants from the Swiss National Science

Foundation

.

Note added after the Congress ;

The reader may be intrigued by the fact that in Fig. 7 the curves intersect the dashed line for discs of infinite diameter, implying that as the section thickness increases truncation leads to greater overprojectlon, which is evidently

not well possible. Using the formulas for overprojection developed by Miles

(1975) and presented at the Congress, one finds, however, that the curves for both Ky and Kg become steeper as the relative diameter 6 decreases. They start

at 1.0 and converge with the curves presented in Fig. 7 as section thickness in- creases.

Reference :

Miles, R. E. : On estimating aggregate and over-all characteristics from thick

sections by transmission microscopy. This volume p. 3, 1975.

347
f. R. Weibel

TABLE I

CORRECTION COEFFICIENTS FOR SECTION THICKNESS

AND TRUNCATION

NO TRUNCATION

SPHERES

(2 3g

T/D = g d„/D = p imi - + sin'^Vl - p^ + 2gj

2 + 3g

IT + 4g

SPHERES

F(D)

E (D") m = " E"(D)

T = h^/D

2m, 2m,

2 3 2m2 + Sgm^ - 3y + y 2in2 + 3gin2

2|(p + sin """Yl - P^)^2 + 2g - p - p ^1 -

|

IT + 4gin

DISCS

^/C~~\ D - 1 + g

T/d = g D/d = 6

^ I- I^)

4* + 6 + 1

CYLINDERS

A - 2 + -q/^A - 2g +

4

T/d = g L/d = A

TiA +

1 + g

TT + 2g

348
Progress, success and problems in applying stereology in biological research

4 "^ik-.

Fig. 1: Electron micrograph of rat li- ver hepatocyte showing various forms

of cytoplasmic membrane system. rER: ribosome-studded cisternae of endoplasmic reticulum, sER: tubules of

smooth ER among glycogen granules

(g) , MI: mitochondria, N: nucleus.

Fig . 2

:

Comparable region of hepatocyte cytoplasm after glucose-6-phosphatase reaction in situ. ER profiles

are marked by black precipitate. Pre- paration and micrograph courtesy of

Dr. D. Paumgartner.

Fig. 3: Resin cast of peripheral air- way tree from human lung. Note branching of ducts in gland-like structures. Preparation Dr. h.c. W. Weber.

Fig. 4: Scanning electron micrograph

of human lung parenchyma . Alveoli (A) as the terminal units of airways are densely packed. Courtesy Dr. P. Gehr.

J. - ,

T

Fig. 5: Longitudinal section of skeletal muscle showing anisotropy and periodicity of myofibrils and organelles. S: Sarcomere, SR: sarcoplasmic reticulxim.

349
E. R. Weibel

04 0.6 08 10 12 U 16 1.8 2.0 RELATIVE SECTION THICKNESS

. 08 1.0 1.2 U 15 18 20 RELATIVE SECTION THICKNESS

u w

•H o

s: w

-p H

T) co U

0

-pu

0) c w 0

•rH 4Ju

0)

S-l tn u

•rH ou

RELATIVE SECTION THICKNESS

04 0.6 QB 10 12 14 16 16 20 RELATIVE SECTION THICKNESS

350
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

HEXACHLOROBENZENE-INDUCED SMOOTH ENDOPLASMIC RETICULUM IN RAT LIVER: A CORRELATED

STEREOLOGIC AND BIOCHEMICAL STUDY.

by T. Kuiper-Goodman, D. Krewski, H. Combley, M. Doran and D. L. Grant Health Protection Branch, Bureau of Chemical Safety, Toxicology Research Division, Tunney's

Pasture, Ottawa, K1A 01^. Canada

Introduction

Hexachlorobenzene (HCB) is a fungicide and an industrial pollutant.

It causes porphyria cutanea tarda in man (1) and rats (2,3). In the rat HCB

also induces the activity of drug metabolizing enzymes (3) and a proliferation

of smooth endoplasmic reticulum (4) . Because of its low biodegradability the accumulation of HCB in the biochain is of concern to health and environmental

scientists. The present study was conducted to relate the morphological and biochemical changes induced by HCB in order to better assess their toxicological

significance

.

Methods

Groups of six-week old male and female Charles River (COBS) rats

(70 per group) were fed diets containing HCB at daily doses of 0, 0.5, 2.0, 8.0 or 32.0 mg per kg body weight in ground Master Fox cubes with 5% corn oil. Four rats from each dose group were sacrificed at 21, 42, 62 and 89 days for measurement of morphological and biochemical parameters and for electron

microscopy. To assess the reversibility of the changes noted the remaining rats were returned to an HCB-free diet after 103 days, and 4 rats per group were

sacrificed at 7, 14, 28, 49, 112 and 231 days after cessation of treatment.

The rats were anesthetized with ether and a small slice of the large

lobe of the liver was excised for electron microscopy. Small blocks of less than

1 mm-^ were fixed for 3 hrs in 2% phosphate buffered glutaraldehyde (ph 7.3, 425 mOsm) at 4°C, post fixed in 1% osmium tetroxide, dehydrated through alcohols and propylene oxide, and embedded in Epon. Sections stained with uranyl acetate

and lead citrate were examined at 60 kV (for better contrast) in a Siemens

Elmiskop lA. Photographs were printed on Kodak Polycontrast rapid RC paper, which does not stretch during processing. For comparison with biochemical data,

stereologic analyses were performed on 4 treated animals and 2 out of 4 control

animals per group. For each animal 2 blocks were selected at random, and 9 photographs were taken both at 1500 x (one grid square of 300 mesh uncoated grid) and at 10,000 x (specified comer of each grid square) magnification, from which

7 at each magnification were selected at random. For each block, estimates of the morphometric parameters for the liver as a whole were made by applying the point counting procedures described by Weibel (5) to the seven photographs

treated as one unit (a set of IBM Fortran IV computer programs was developed for this purpose) , and estimates for each animal were obtained by averaging the two block values. The size of the block was such that usually 2 out of 3 zones of the liver acinus were included (6). Drug metabolizing enzyme activity and porphyrin levels were measu-red using methods described previously (3). Mean values of morphometric and biochemical parameters for each treated

group were compared with those of the control group at each time point. A pooled

estimate of error variance was used where appropriate with an overall level of significance of less than 0.05 (7).

Results and Disaussion

Relative liver weight was significantly increased (p < 0.05) at the two highest dose levels for both sexes, and slowly returned to normal after the animals were returned to an HCB-free diet (Fig. 1). Examination of semi-thin

sections of liver by light microscopy showed enlargement of hepatocytes in zones 2 and 3 at the two highest dose levels in both males and females (Fig. 2 a,b). In the enlarged cells there were large pale areas, which by electron

microscopy, were found to consist of proliferated smooth endoplasmic reticulum

(SER) (Fig. 3). In males large whorls of compacted membranes surrounding lipid

droplets often accompanied the proliferated SER. The nuclei of enlarged hepato351
T, Kuiper-Goodman, D, Krewski, H, Combley, M, Doran and D. L, Grant

X o

5i§

RELATIVE LIVEfl WEIGHT FEMALES

V-''

-l-JL^L

aWTHOL • • 0.5 MC/KG/0*r • •

2 MG/KG/OAY o- - -o 6 yC/KG/OAY o o 32 MG/KG/DAY * •

, Hce

Figure 1. Relative

liver weight in rats

fed HCB in the diet

for 15 weeks followed

by control diet . The standard error of the difference between

two means is indicated

on the abscissa for each

time interval.

Figure 2. Semi-thin sections of

rat liver stained with toluidlne

blue: a. zone 2 of a control rat

b . zone 2 and 3 of a male rat fed HCB at 32 mg/kg for 15 weeks. Pale SER area (single arrow)

,

membranous whorls (double arrow)

.

X 400.

Figure 3. Electron micrograph of

a zone 3 cell from a male rat fed

32 mg/kg HCB for 12 weeks showing

proliferated SER and membranous

whorls . (length of bar = ly).

HEPATOCYTE AVERAGE SIZE

9.0-

8.0-

7.0-

6.0-

5.0- 4.0-

i HCB FEEDING STOPPED

-i.. o o 0.0 MG/KG \ » A 0.5 MG/KG \. --a 2.0 MG/KG \ • > 8.0 MG/KG \. * 4 32 MG/KG

28 56 84 112 140 168 196 224

DAYS ON EXPERIMENT

Figure 4. Average size of hepatocytes in male rats fed HCB for 15 weeks followed by a recovery period. Data not corrected for zone distribution.

352
Hexachlorobenzene-induced smooth endoplasmic reticulum in rat liver

cytes also appeared enlarged. Mitochondria in enlarged hepatocytes were very

small and sparse. Cells in zone 1 appeared unaffected by HCB. Stereologic analysis of the liver showed some changes with time in control animals. These changes were probably related to changes in the relative

liver weight shown in Figure 1. In general the results agreed well with those reported by others (5,8). In control males the total volume of liver per 100 gm body weight declined from 3.62 ± 0.01 ml at 9 wks of age to 2.09 ± 0.13 ml

at 37 wks of age; females declined from 3.07 ± 0.06 ml to 2.62 ± 0.07 ml at 28 weeks. Total SER volume varied between 0.06 and 0.25 ml/100 gm body weight in males and between 0.04 and 0. 37 ml in females throughout the experiment. The total volume of liver per 100 gm body weight (V]^) was significantly

increased (p < 0.05) at the two highest dose levels in males and at the highest

dose level in females mainly due to an increase in total hepatocyte volume (Vjj)

.

There was no apparent change in the number of hepatocytes per 100 gm body weight

(Nil), so that the increase in Vh was only due to an increase in hepatocyte size

(Fig. 4). Total nuclear volume (Vq) increased slightly at the highest dose level in males and females; this appeared to be due to an increase in nuclear size, as evidenced by an increase in nuclear diameter. Fig. 5 shows that the percentage

of SER in hepatocyte cytoplasm increased in a dose related manner until the cytoplasm in males at the highest dose level contained about 60% SER. Most of the increase in Vjj was, therefore, due to an increase in Vser- Similar changes

of a lesser magnitude were observed in females. Approximately 80% of the hepatocytes were. found in zones 2+2,3+3,

where the effects of HCB were most pronoimced (53-77% SER in cytoplasm at the highest dose level). Cells in zone 1+1,2 contained about the same amount of SER (10-12% of cytoplasm) as control hepatocytes. On this basis post-stratified

estimates by zone (9) could be prepared and an estimate for the liver as a whole obtained by weighting the zone estimates in the above proportions. Because

of the high degree of homogeneity within zones the stratified estimates will be

considerably more precise than estimates which ignore zone differences.

Assuming that the drug metabolizing enzymes are located on the surface

of the SER membranes, the surface area of SER should be related to drug meta- bolizing enzyme activity. There was a dose related increase in SER surface

area/gm liver (Sger) (up to 10-fold at the highest dose-level in males) . In males the increase in Sggj- was associated with the increase in drug metabolizing

enzyme activity (Fig. 6), but at the highest dose level there was an excess of

SER, suggesting the formation of membrane without enzyme. This was also accompanied by the formation of whorls of excess membrane. In females at the highest dose level, Sser "as significantly increased (p < 0.05) at 3 weeks and subsequently declined somewhat. This was accompanied by an initial increase in drug metabolizing enzyme activity followed by a 50% inhibition. At this time the liver porphyrin level was greater than 700 nM/g liver (10)

.

Most parameters had recovered to normal values after males had been

placed on a recovery diet for 16 weeks. Females also tended to recover, but

to a lesser extent due to the shorter recovery period of only 7 weeks. It is likely that the recovery could be attributed to a gradual removal of excess membranes either by autophagy as suggested by Bolender et ai (11) or through

myelin whorl formation or both, rather than to a replacement of hepatocytes.

There was no evidence of cell division, and in the normal liver there is no turnover of liver cells. The changes in SER, with the accompanying increase in drug metabolizing

enzyme activity appear, therefore, to be adaptive rather than toxic, in that liver cells seem to recover to normal hepatocytes, without morphological signs

of permanent injury due to proliferated SER. The long term implications of chronically elevated levels of these membranes and enzymes are, however, not

known.

353
T. Kuiper-Goodman, D. Krewski, H. Combley, M. Doran and D. L Grant

SIZE OF CELLULAR COMPONENTS AND

TOTAL CYTOPLASM PER CELL

3W 6W I2W I6WR Figure 5. Size of cellular components

and total cytoplasm per cell in male

rats, based on an estimate of cell number pooled over all groups and

time period. (W = weeks).

40 80 120 160 200 240 280 320 DAYS ON EXPERIMENT Figure 6. Aniline hydroxylase

activity in male rats. Values

are expressed relative to control

values

.

References :

1. Cam, C. and G. Nigogosyan. Acquired toxic porphyria cutanea tarda due

to hexachlorobenzene. J. Amer. Med. Assoc. 183 , 88, 1963.

2. Ockner, R.K. and R. Schmid. Acquired porphyria in man and rat due to hexachlorobenzene intoxication. Nature 189 , 499, 1961.

3. Grant, D.L., F. Iverson, G.V. Hatina and D.C. Villeneuve. Effects of hexachlorobenzene on liver porphyrin levels and microsomal enz5Tnes in

the rat. Environ. Physiol. Biochem. 4_, 159, 1974.

4. Medline, A., E. Bain, A.I. Menon and H. Haberman. Hexachlorobenzene

and rat liver. Arch. Pathol. 9b_, 61, 1973.

II II 5. Weibel, E.R., W. Staubli, H.R. Gnagi and F.A. Hess. Correlated

morphometric and biochemical studies on the liver cell. I. Morphometric model, stereologic methods, and normal morphometric data for rat liver. J. Cell. Biol. 42^, 68, 1969.

6. In this study we used the 3 circulatory zones of liver acini as suggested by Rappaport . Rappaport, A.M., Z.J. Borowy, W.M. Lougheed

and W.N. Lotto. Subdivision of hexagonal liver lobules into a structural and functional unit; role in hepatic physiology and pathology. Anat. Record 119, 11, 1954.

7. Miller, R.G. Simultaneous Statistical Inference. McGraw Hill, Toronto,

1967, p67, p76.

8. Loud, A.V. A quantitative stereological description of the ultrastructure

of normal rat liver parenchymal cells. J. Cell. Biol. J7, 27, 1968.

9. Raj, D. Sampling Theory. McGraw Hill, Toronto, 1958.

10. Grant, D.L., G. Hatina, I.C. Munro. Environmental Quality and Safety.

Thieme, Stuttgart, 1975. in press

11. Bolender, R.P. and E.R. Weibel. A morphometric study of the removal of phenobarbital - induced membranes from hepatocytes after cessation of treatment. J. Cell Biol. 56, 746, 1973.

354
National Bureau of Standards Special Publication 437 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

AUTOMATED MEASUREMENT OF AIRSPACES IN LUNGS FROM DOGS CHRONICALLY EXPOSED TO AIR POUUTANTS

by D. Hyde,* A. Wiggins,* D. Dungworth,* W. Tyler,** and J. Orthoefer*** ^School of Veterinary Medicine, University of California, Davis, California 95676. U.S.A.

^'California Primate Research Center, University of California, Davis, California 95616. U.S.A. ""U. S. Environmental Protection Agency. 1055 Laidlaw Avenue. Cincinnati, Ohio 45237, U.S.A.

SUMMARYMorphometric data were obtained by automated analysis from the lungs of 66 dogs which had been subjected to long-term exposure to a variety of air pollutants. FORTRAN IV (level H) programs allowed data integrity checking, editing,

and reduction before computing various parameters. A distribution of percent

cumu lative frequency of airspace chords in 14 size groups was computed and fitted

by a two parameter cumulative probability law of the form F(x) = P(X - x) =

(1 - e~a^)5. The estimated mean chord length (Ox) was obtained by calculating

the area above the curve (1 - e~'^x) 3 . The yx was found to be a sensitive indicator of group differences in sizes of the distal airspaces. Lungs of dogs

exposed to oxides of nitrogen, irradiated automobile exhaust, or oxides of sulfur

alone and with irradiated automobile exhaust had significantly enlarged distal

airspaces by all methods of evaluation.

INTRODUCTION Stereological methods were used to evaluate lungs from groups of dogs exposed to a variety of air pollutants. Quantitation of histological sections by

manual methods is time-consuming and the probability of operator error increases

with the number of slides processed. To avoid these limitations, we employed

automated analysis in which optical images were analyzed by a television scanner coupled to a computer (Cole, 1966) . Thurlbeck (1974) commented that automated

measuring microscopes offer a reasonable compromise bet\jeen the tedium and error

of the human mode and the complexity and great expense of computer pattern

recognition techniques which utilize binary image digitization and analysis

(Levine et al., 1970). Our primary objective was to determine whether there were significant in- creases in sizes of airspaces distal to the end bronchi in the lungs of any of the groups of dogs and thus provide quantitative assessment of changes indicative

of early emphysema. This was achieved by using a mathematical model based on the distribution of chord lengths of distal airspaces.

METHODS

Sixty-six dogs, distributed among control and treatment groups (Table 1), were exposed to a variety of air pollutants for 61 months (Hinners et al., 1966). A wide variety of physiological parameters were determined before, during, and after exposure (Lewis et al. , 1974; Bloch et al. , 1972, 1973; Vaughan et al.,

1969). Two years after cessation of exposures, the dogs were weighed, measured,

and killed by an intravenous injection of pentobarbital. The lungs, trachea,

and attached structures were removed from the thorax and trimmed of extraneous

tissue. The lungs were weighed, and subsequently tfie left lung was removed at the left primary bronchus for biochemical analysis. The trachea was cannulated

and the right lung was fixed in the normal dorsoventral orientation by intratracheal perfusion with dilute Kamovsky's fixative (Kamovsky, 1965) at 30 cm H2O

pressure in a fixative bath for 16 to 18 hours. The volume of the fixed right

lung was then measured by fixative displacement. After samples were taken for scanning electron microscopy (SEM) and transmission electron microscopy (TEM)

,

the lobes of the right lung were cut into 1 cm thick slices (Fig. 1) . The fraction of non-parenchyma (bronchi and blood vessels down to 2 mm diameter)

and parenchyma were determined using the point-count method of Dunnill (1962)

.

Nine 2 to 3 cm^ blocks of tissue were cut from each right lung (Fig. 1) and measured. Paraffin sections 7u thick were cut, stained with hematoxylin and

eosia, and measured microscopically using a calibrated eyepiece reticle. The linear correction factor (p) for shrinkage of the fixed tissue due to processing

was calculated from these measurements

.

355
D. Hyde, A. Wiggins. D. Dungworth, W. Tyler and J. Ortlioefer

The stained slides were examined at a magnification of 125X on a Quantimet

720 image analysing computer (Cole, 1966). The instrument was set to detect the airspaces, and the detection level was standardized by use of a preset level of detection on a wire mesh grid. Measurements were made at 20 stratified random

points on each slide (Thurlbeck, 1967) . At each point the volume fractions of airspace and tissue, the number of alveolar intercepts, and the number of intraalveolar chord lengths in 14 size classes were determined, printed on a teletype, and punched on papertape. The punched papertape was read into a Burroughs 6700

for computation of the 201,960 measurements. FORTRAN IV (level H) programs allowed data integrity checking, editing, and reduction before computing and graphically displaying various stereologic parameters

.

The fractions of tissue and airspace were corrected for the Holmes effect

(Weibel, 1963). The total length of scanning lines multiplied by the number of fields and by the fraction of the corrected mean airspace per field, divided by

the number of intra-alveolar chords >12p gave the mean chord length of the pro- cessed tissue (Lcj) . Those chords <12y were rejected because they represented

mostly structures not normally air-filled, such as capillaries and post-capillary

venules

.

The estimated mean chord length (L^,) of unfixed tissue was calculated from

the processed to unfixed tissue by formula 1 (Thurlbeck, 1967)

:

/TLC \l/3

K - ^c, <P) [^) (1)

The total lung capacity of the right lung (TLCp^) was determined by multiplying

TLC by 0.59 which is the fractional volunn of the right lung (Cree et al . , 1968). TLC was measured on these dog lungs by pie thysmo graphic methods (Lewis et al.,

1974) . The volume of the fixed right lung (Vl^) was determined by fixative displacement . The internal surface area of the unfixed right lung (Slj^) was calculated as described by Weibel (1963) and Thurlbeck (1967)

.

A distribution of percent cumulative frequency of airspace chords in 14 size groups was computed for each dog lung. A two parameter cumulative probability law of the form F(x) = P(X ^ x) = (1 - e-«x)g was found to provide the best fit to that distribution. The parameters a and B were estimated using a combination of minimum modified chi-square (Cramer, 1946) and the Newton-Raphson

procedures (Froberg, 1965). The estimated mean chord length jj^ was obtained by

calculating the area above the curve (1 - e-o'X)g using the 6th Order Newton-Cote

formula for numerical integration, formula 2 (Froberg, 1965):

C(X) = fll - F(x)]dx (2)

•'o The variance of X was determined by formula 3 (Froberg, 1965) :

h\ = 2jx[l - F(x)]dx - [Jl - F(x)dx]^ (3)

Ux Ox were corrected for the Holmes effect and to unfixed dimensions by for- mula 1. Since the average n per group was large (1347 fields), the differences

between nx s^nd of the control group and exposure groups were tested using

the standard normal test statistic (Snedecor and Cochran, 1967) .

RESULTS

The Ux calculated by formula 2 had a standard error of 2.76, while L^, calculated by formula 1 had a standard error of 4.14. Because of the lower standard

error for all groups, Ux was used to calculate Slj^. The corrected °f groups

4, 6, 7, and 8 demonstrated significant increases at the 0.1% level while group

3 demonstrated a significant increase at the 1% level relative to the control

group (Table 1) . The Slj^/TLC ratio of groups 2-4 and 6-8 were also significantly

decreased at the 0.1% level when compared to the control group (Table 1).

llie most to least significant increase in occurred in groups 7, 4, 6, 8, and 3, respectively. The group order of significant differences determined by

Slr/TLC was 4, 7, 6, 3, 8, and 2. The TLC to body weight (BW) ratio was significantly increased at the 1% level only in group 4.

356
Automated measurement of airspaces in lungs of dogs

DISCUSSION

The automated analysis method of estimating L^, was compared to the manual

method of linear integration for estimating (Weibel, 1963) in a study of the distal airspaces in a series of young and aged dogs (unpublished) . The slopes

of the regression lines of age plotted against L^, by automated analysis and Lj, by the manual method were tested to be equal at the 5% level of significance

(Snedecor and Cochran, 1967) . The SE of Lq by automated analysis was less than

that of Lc by the manual method. Bie y-intercept of the regression line of Lc by automated analysis was slightly smaller than the regression line of L^, by the

maniial method, because automated analysis counted some blood vessel walls as interalveolar septa.

The Ux values of our distribution formula are sensitive to subtle shifts in airway chord lengths. There were highly significant increases in the sizes of

the distal airspaces in lungs of dogs in groups 3, 4, 6, 7, and 8 with complete

agreement between the two parameters used to indicate enlargement (Px. ^lr/TLC) . The degree of enlargement was greatest in groups 7 and 4 and slightly less in groups 6 , 8 , and 3

.

Scanning electron microscopic observations of these same dogs provided con- firmation that destruction of tissue accompanied dilatation; therefore, the lesion represented a true emphysematous process. However, in group 4, hyperinflation accomts for the significant increase in the TLC/BW ratio in this group.

The conclusions with regard to the effects of the various mixtures of air

pollutants tested are that 32-36 months after the end of an approximate 5-year

exposure period, mild emphysema was present in the lungs of dogs exposed to oxides of nitrogen, irradiated automobile exhaust, or oxides of sulfur alone or with irradiated automobile exhaust. Evaluation of possible regional differences

among various lobes and regions of the lung is underway, as is correlation of

the total spectrum of morphologic changes observed in lungs of exposed dogs with

pulmonary function data.

3

FIGURE 1. Blocks of tissue were selected from sites in the cranial and middle (transverse plane) , caudal (sagittal

plane), and accessory lobes (frontal plane). The cranial,

middle, and cavidal lobes are depicted from the lateral

view, while the accessory lobe is depicted from the caudal

view

.

357
D. Hyde, A. Wiggins, D. Dungworth, W. Tyler and J. Orthoefer

EXPOSURE GROUP

1 2 3 4 5 6 7 8

CA R I SOx R + SOy. I + SO^ NO2 high NO hi]

No . Dogs 11 9 A 7 9 11 6 9

X 102 1.A4 1,.50 1.58 1.69 1.45 1.64 1.77 1.60

(microns)

SE 2.06 2,.50 3.90 2.90 2.20 2.31 3.73 2.51

P 1.0% 0.1% 0.1% 0.1% 0.1%

Slr/TLC X loi 1.85 1,.69 1.59 1.49 1.79 1.60 1.50 1.68

(m2/l)

SE X 10-1 2.60 2,.84 3.83 2.51 2.66 2.22 3.09 2.60

P 0,.1% 0.1% 0.1% 0.1% 0.1% 0.1%

TABLE 1. The dogs were distributed among a control group (CA) and treatment

gro'j-ps of raw automobile exhaust (R) , irradiated automobile exhaust (I), oxides

of sulfur (SOx) > automobile exhaust and oxides of sulfur (R + SOjj) , irradiated automobile exhaust and oxides of sulfur (I + SO^) , nitrogen dioxide high/

nitric oxide low (NO2 high) , and nitric oxide high/nitrogen dioxide low (NO

high) . The mean chord length (ux) and internal surface area of the unfixed

right lung to total lung capacity ratio (Slr/TLC) are recorded along with their

standard errors (SE) and tested level of significance (P) in eight groups of dogs.

ACKNOWLEDGEMENTS

This study was supported by EPA Contract No. 68-02-1732. We would like to thank Dott Morrissey for typing and editing the manuscript. We acknowledge the assistance of Don Hallberg and Mike Vaida for computer programming, and Janet Mumford for sketching Figure 1.

REFERENCES

Bloch, W.N., Lewis, T.R. , Busch, K.A. , Orthoefer, J.G., and Stara, J.F. (1972) Cardiovascular status of female beagles exposed Co air pollutants. Arch. Environ. Health 24, 342. Bloch, W.N. , Lassiter, S.G., Stara, J.F., and

Lewis, T.R. (1973) Blood theology of dogs exposed

to air pollutants. Toxicol. Appl. Pharmacol. 25, 576. Cole, M. (1966) The metals research Quantimet. Microscope 15, 148. Cramer, H. (1946) Mathematical Msthods of

Statistics. Princeton Univ. Press, Princeton, New

Jersey.

Cree, E.M., Benfield, J.R. , and Rasmussen,

H.K. (1968) Differential lung diffusion, capillary

volume, and compliance in dogs. J. Appl. Physiol.

25, 186. Dunnill, M.S. (1962) Quantitative methods in the study of pulmonary pathology. Thorax 17, 320. Froberg, C.E. (1965) Introduction to Numerical Analysis. Addison-Wesley , Reading, Massachusetts.Hinners, R.G., Burkart, J.K., and Conner, G.L.

(1966) Animal exposure chambers in air pollution

studies. Arch. Environ. Health 13, 609.

Kamovsky, M.J. (1965) A formaldehyde-glutaraldehyde fixative of high osmolality for use in electron microscopy. J. Cell Biol. 27, 136A.

Levine, M.D. , Reisch, M.L., and Thurlbeck,

W.M. (1970) Automated measurement of the internal surface area of the human lung. IEEE Trans. Biomed. Eng. BME-17, 254. Lewis, T.R., Moorman, W.J. , Yang, Y. , and

Stara, J.F. (1974) Long-term exposure to auto ex- haust and other pollutant mixtures. Arch. Environ. Health 29, 102. Snedecor, G.W., and Cochran, W.G. (1967) Sta- tistical Methoas. The Iowa State Univ. Press, Ames,

Iowa. Thurlbeck, W.M. (1967) The internal surface

area of nonemphysematous lungs. Am. Rev. Respir.

Dis. 95, 765. Thurlbeck, W.M. (1974) Measuring tissues. Chest 65, 1. Vaughan, T.J. Jr., Jennelle, L.F., and Lewis,

T.R. (1969) Long-term exposure to low levels of air pollutants : Effects on pulmonary function in

the beagle. Arch. Environ. Health 19, 45. Weibel, E.R. (1963) Morphometry of the Human

Lung. Academic Press, New York.

358
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

VOLUMETRIC DETERMINATIONS OF CELLS AND CELL ORGANELLES FROM TWO-DIMENSIONAL

TRANSSECTIONS

by L. G. Lindberg Cytodiagnostic Department, University Hospital of Lund, Lund, Sweden

One of the most important problems in stereology is to estimate how weli a purely mathematical model used in calculations conform to the actual problem

when it is applied e.g. in biological work. To calculate volumes of cells or cell organelles from area measurements in transsections has since long been a frequent problem. - Sections of many cells

or cell organelles fairly well conform to the shape of wellknown geometrical

figures and presumedly their three-dimensional counterpart to the shape of bodies that can be described in relatively simple mathematical terms.

In general terms the formula 2/2

V = ;3 X a

describes how the mean transsectioned area a can be used to calculated the volume V of a body from which the sections were obtained. There is one difficulty, however, and that is to estimate /3 in the expression above - a coefficient which is dependent on the shape of the transsected body only.

The most common shapes of cells in mammalian tissues are ellipsoid or cylinderlike shapes and generally there is a rotational symmetry, i. e. two of the three axis are the same. In this case a relation between /5 and the eccentricity

e (the relation between the perpendicular axis to the axis of rotation) can be found. The general formula is , /

a = V // • H„ sine • de

where a is the average transsectional surface area of the body and Hg is the projected length of the body and 6 is its angle of inclination and V is the volume. When the volume of the ellipsoid body or cylinder is used the following

two relations are found between the eccentricity e and the coefficient /5 (fig 1).

^ellipsoid =^'/^- h(^)-Vh77)/

wnere h(e ) = 0.5 + (0.5 e^/ V 1 - e^) ®log ((i +Vl - e^)/e) fore<i (i)

h( e ) = 1 fore =1 (2)

(t ) = 0.5 + (0.5 e^/ Vi - e ^) (7r/2 - arctg (1 /Ve^ - D) fore-1 (3)

d

^cylinder = (2 -^-6) • V7T;r7/ e.e-V^)

h

and

Thus the practical problem of volume estimation from transsections has been

reduced to find a good method of obtaining the numerical value of e

.

Theoretically this is very simple - make enough transsections and calculate

the ratio between the shortest and longest axis - the smallest of these values can be taken as a good approximation of e.

However, in practical work it is impossible to make transsections of biological

material v/i thout a section compression i.e. the cutting knife compresses the

359
L. G. Lindberg

section as it cuts through the embedding media and tissues - this causes a shortening of all lengths along the cutting direction (in the order of 10-301). Thus length measurements cannot be used. Another way can be used - the largest possible area of a transsectioned ellipsoid body is TT b2 /£ (if the axes are a, b and b). This area can be approximated by the largest observed area, and thus e can be calculated.

However, the use of extreme values can be dangerous as these are most lil<ely to be influenced by methodological errors in processing or preparation of tissues, e.g. swelling or shrinking due to osmotic effects, toxic agents,

traumatic damages etc.

It was found that the distribution of the transsected areas is dependent only

on e (fig 2). (2) (For e<l the density is increasing in (0,1), decreasing

in (1,1 ^ e) with a peak in y = 1).

From this figure it can be seen that using two different moments of the distribution of the areas, ecan be calculated. A requirement that must be fulfilled

is that the size of the transsected bodies is fairly constant. One approach is too choose the average (expected) area E(A) and the variance V(A) and compute

the square of the coefficient of variation

= V(A) / E (A)^

2 (2

)

The exact mathematical relation between fi { 6 ) and R (e) can thus be given .

E (A) = 2 TTb^/ (3 h ( e ))

and

V (A) = I6 7r2 • (h (e) - 0.5)/ (I5e^ • h (e)) - E (A)^

where h {€) is given above (1). Figure 3 gives a diagram of this relation.

2 2-2 R (e) is estimated by S / a , where S and a are observed standard deviation

and mean area respectively. For values of € over 0.3 up to 1 a simple approximation of /5 (f ) can be used. This approximation is exact for € = 1 and having an error that is slowly

increasing as e decreases. (Fore>0.3 this error is below 3 per cent).

Thus V ^ = (1.382... + 3.'* (S^/a ^ - 0.2)) • 'a ^''^ est / A practical application of the present principles has been undertaken when different varieties of a malignant experimental tumor - Rous sarcoma - induced

in rats - were studied. This tumor is composed of elongated cells irregular! ly arranged. - The cells fairly well conform to ellipsoid bodies as seen under the light microscope and even the el ect ron mi croscopi c picture is in agreement with

this assumption. Transsect i ons of cells and their nuclei are arranged according to size in a histogram and superimposed upon this the theoretical distribution of areas

is given (fig h) . As is seen the practical application conform rather well to the theoretical basis although there must be some variation both in size and eccentricity of the transsected cells. Thus the sampling of tissues, the measurements and all calculati.on methods seem to be reliable as the obtained

results when control led against the theoretical model fairly welt conform

to what is expected . ^

360
Volumetric determinations of cells and cell organelles

Ellipsoids Cylinders Fig 1 Relation between

the coef f i cient /3 and the eccentricity 6 of rotatory

ellipsoids or cylinders.

2

-

1.0 2.0 O

20

361
L. .6. Lindberg

1 400 R2(€)

Fig 3 Relation between the coefficient /J 2 and the coefficient of variation R (6).

Z--0. 35

Fig 4 Observed distribution of areas (histogram) plotted against

theoretical distribution

of areas for ellipsoid

tunnor cells from a Rousvirus induced rat sarcoma.

REFERENCES

1. Lindberg, L.G. 6 Vorwerk, P.: On calculating

volumes of transsected bodies from two- dimensional micrographs.

Lab Invest 23:315, 1970

2. Lindberg, L.G. S Vorwerl<, P.: Estimation of volumes of ellipsoid bodies in conglomerates

through random plane sections.

Lab Invest 27:38'), 1972

3. Lindberg, L.G., Mitelman, F. 6 Vorwerk, P.: A comparative morphologic study of Rous rat sarcoma with normal and abnormal chromosomal

pattern.

Lab Invest 27:387, 1972

k. Lindberg, L.G. S Mitelman, F. : Morphometric

analysis of sarcomas with different karyotypes. Lab Invest 31:90, 197*1

362
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

MORPHOMETRIC ANALYSIS OF THYMIC CELLS DURING MURINE RADIOLEUKEMOGENESIS

by J. Boniver, R. Courtoy and L. J. Simar

Institute of Pathology, University of Liege, Belgium

Summary : By using stereological methods, the following quantitative changes of the blastic population in the thymic subcapsular

zone during radiation induced lyirphomagenesis in C57BL mice %vere demonstrated : 1) Apparent accumulation of blast cells, mainly due to a simultaneous decrease of the small lymphocyte number;

2) Modifications of the number of the three subcapsular blast cell

aspects ( lymphoblasts, X cells, RSN cells) : accumulation of RSN

cells and of X cells during post-KX regeneration in the late atrophic preleukemic thymuses;

3) A few variations of the cellular structure, except for a reduction of the surface density of the Golgi membranes in X cells and

RSN cells of the atrophic and lymphoraatous thymuses.

The relation between these changes and kinetic and functional modifications of thyrdc blast cells during leukemogenesis is discussed,

RSN : ring-shaped nucleolus cells.

Introduction : Murine radiation-induced lymphomas result likely

from the cancerous transformation of thymic subcapsular blast cells

by an oncornavirus (11). At the ultrastructural level, this morphologically heterogeneous blastic population contains two peculiar

cell aspects respectively named "X cells" (13) and "ring-shaped nu- cleolus cells" (5), besides typical lym.phoblasts (fig. 1). A mor- phometric analysis (4) was performed to determine quantitatively

the eventual morphological changes of this population during radioleukemogen esis (5).

Results : 1. Numerical density : The numerical density of blast

cells and small lymphocytes in the subcapsular zone, determined by

using the Abercrombie ' s method (1), and the absolute blast cell num

ber, based on a evaluation of the subcapsular volume (4) show parallel variations (fig. 2) . After each X ray induced thy^^.ic involution, regeneration is characterized by a transient accumulation of

blast cells, the numier of which does not depend on the number of

X ray doses. However, after the fourth irradiation, the number of

small lymphocytes does not return rapidly to the normal level;

then, the blast cell proportion becomes higher than normally. Later

on, the single changes occur on day SO, in the late atrophic thymuses in 2Cf/o treated mice; the small lymphocyte nuiiiber is statistically smaller than in the other thymuses, inducing an apparent accu- mulation of blast cells. The numerical density of each blast cell

type in the subcapsular zone is determined by multiplying their re- spective percentage determined with electron microscope by the total

numerical density described above (fig. 3). Each 150 R dose induces

a transient increase of the density of lymphoblasts and X cells.

After the last irradiation, X cells and ring-shaped nucleolus cells

becom.e more numerous than normally. In the late atrophic thymuses,

the numerical density of lymphoblasts and ring-shaped nucleolus

cells is higher than in non atrophic and in untreated thyiTiUses.

363
J. Boniver, R, Courtoy and L. J. Simar

2 . Nuclear and cytoplasmic constituents : The volumetric and sur- face densities are obtained by point and intersection counting me- thods (6, 14, 15, 16) and the numerical density by VJeibel and

Gomez method (22)

.

Dense chromatin (fig. 4) : Its volu-.etric density in the nucleus is

higher in ring-shaped nucleolus cells and mainly in X cells than

in lymphoblasts . It does not change during leukemogenesis

.

Nucleolus (fig. 5, 6) : There is no significant difference of its volumetric density in the three blast cell types; the respective proportions of fibrils, granules and fibrillar centre inner the nu- cleolus are not identical since the fibrillar centre is peculiarly

well developed in ring-shaped nucleoli.

Cytoplasmic organelles : The cell content in mitochondria, endoplasmic reticulum and dense granules is similar in the three cell

types and shows few variations during leukemogenesis. The surface

density of Golgi membranes (fig. 7) is decreased in X cells and

mainly in ring-shaped nucleolus cells in late atrophic and lymphomatous thymuses.

Discussion : The apparent accumulation of blast cells in the subcapsular zone of the thymus during radioleukemogenesis is mainly

due to a decrease of the nimiber of small lymphocytes. This phenomenon is simultaneous with changes of the thymic blast cell composition. As according to recent kinetic studies (5) , the three blast

cell aspects correspond to different stages in the cell cycle, it

is likely that these morphological modifications are related to

changes of the kinetics of thymic lymphopoiesis at some important

phases of radioleukemogenesis as regeneration (1) and late atrophy

(10, 12) . The single cellular change observed during the preleukemic period is a reduction of the Golgi zone in some blast cells.

This modification could be correlated with a variation of the polysaccharide production since it is well admitted that the Golgi zone plays a role in the synthesis of the cell coat material (2, 8, 9)

which is frequently modified in cancerous cells (7).

References : ABERCROMBIE, M., Anat . Rec, 1946, 94, 239-251.

BENNETT, G., LEBLOND, CP., HADDAD, A., J. Cell Biol,, 1974, 60. 258-234.

BONIVER, J., COURTOY, R., SIMAR, L.J., J. Microscopie, 1975, 22 . 57a.

BONIVER, J., COURTOY, R., SIMAR, L.J., BETZ, E.H., in preparaticn

BONIVER, J., DELREZ, M,, SIMAR, L.J. and HAOT, J., Beitr . Path.,

1973, 150, 229-245.

910

11

12

13

14

DE HCFF, R.T. and RHINES, F.N., Quantitative microscopy, Mc Graw

Hill Book Co, New York, 1968.

EMMELOT, P., Europ. J. Cancer, 1973, 9, 319-334.

FAVARD, P., The Golgi apparatus. In : Handbook of Molecular Cytology, A. Lima de Faria, Ed., North Holland, Amsterdam, 1969,

1130-1155.

ITO, S., Fed. Proc, 1969, 28, 12-25.

JARPLID, B., Acta Radiol., supplt. 279, 1968.

KAPLAN, H.S., Nat. Cancer Inst. Monographs, 1964, 14' 207-217.

SIEGLER, R., HARRELL, W. and RICH, M.A., J. Nat. Cancer Inst.,

1966, 37, 105-121.

SIMAR, L.J., HAOT, J. and BETZ, E.H., Europ. J. Cancer, 1968, 4,

529-535.

UNDERWOOD, E.E., Quantitative stereology, Addison Wesley Ed.,

Reading, Mass, 1970.

364
Morphometric analysis of thymic cells

15) WEIBEL, E.R., Int. Rev. Cytol., G.H. Bourne and J.F. Danielli

Ed,, 1969, 2&, 235-302.

15) WEIBEL, E.R. and ELIAS, H., Quantitative Methods in Morphology,

Springer Verlag, Berlin, 1967.

17) TOIBEL, E.R. and GOMEZ, D.M., J. Appl . Physiol., 1962, 12, 343-

365.

Figure 1 : Ultrastructural aspects of thymic siibcapsular blast cells

in C57BL mouse (x6650) : a) typical lymphoblast : few abundant dense chromatin and compact nucleolus (Nu) ; b) ring-shaped nucleolus

cell : granular and fibrillar components encircle the fibrillar cen

tre; c) X cell : abundant dense chromatin irregularly distributed,

compact nucleolus.

Days

Figure 2 : Variations of the thymic weight (in mg) , nxamerical density (Nv(zsc) of blast cells (• •) and small lymphocytes (•- -•)

and absolute number of blast cells (Nzsc) in the subcapsular zone

of the thymus of C57BL mice which have received 1, 2, 3 or 4 irradiations of 150 R (arrows)

.

^

control; o o o : evolution to late atrophy (A)

.

365
J. Boniver, R. Courtoy and L. J. Simar

contirox

35 d. old 1 Rx D.4 |4 Rx D.8

1j4 Rx J.c

[ atrophy {

L-ontir ox

5 months

Lblasts 438-r98 7134- 111 1 251-:-90 ! 101-^-25 ! 230^15 ! 104+2 7 RSN cells 99+26 68" 26 { 194+42 j 70+42 1 235~34 ! 119"41

X cells 94-^2 3 2 61+ 121 j 380+48 j 155"35 j 90-1-2 3 j 75-~18

Fig .3 : Numerical density of blast cells in the subcapsular zone during radioleukemogenesis

.

Dense chromatin Vhino)

0.6O

Q20-

Q60-

0,20-

0.60-

0,20-

Lymphoblasts

Ring-shaped nucleolu

t t I I t t I ®

.u{no)

Q06-

0.02-

Q04-

1+1

Lymphoblasts

1+1

Ring-shoped nucleolus nnnnnnn

Q08-1

004-

t t t I t t t ®

Nucleolar composltio

1,00- 0,80- 0,60- Q40-

020-

0,60- Q40-

0,20-

Lymphoblasts

Ring-shaped nucleoli i i A

100- 080-

0£0- Q40-

Q20-

X Cells .

Jad bMi ksU L^J i»J LaJ ki^

V'lu- 020-

Q20-

Golgi zone

Lymphoblasts ^

^ n n

rh

Ring-shaped

rh

0,10-

t t t t f t t

r+i

I i f i t t t

iFibrlU Fibrillar cenire

fniGranule. (b) ®

Figures 4-7 : Quantitative changes of thyrnic blast cells during ra- dioleukemogenesis. Volumetric density of dense chromatin (fig .4) and

nucleolus ( fig .5) in the nucleus; volumetric density of various com- ponents in the nucleolus (fig. 6) and surface density of Golgi mem- branes in the cytoplasm (fig. 7)

.

366
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9. 1975 (Issued January 1976)

STUDIES ON THE PARTIALLY-ORIENTED SURFACES OF SKELETAL MUSCLE MITOCHONDRIA

by N. T. James and G. A. Meek

Department of Human Biology & Anatomy, The University, Sheffield S70 2TN, United Kingdom

The mitochondria of skeletal muscle fibres are usually elongated and lie

with their long axes parallel with that of the fibre. Consequently, their

surfaces fom a partially orientated system. In a partially orientated system

there is a linear component which is formed by the surfaces of mitochondria

parallel with the long axis of the fibre and an isometric ccmponent which is formed by the rounded ends of the mitochondria.

Several quantitative techniques are available for analysing partially

orientated systems (Underwood, Quantitative Stereology, Addison-Wesley , 1970).

They are widely used in non-biological subjects, for example, in analysing the

partially orientated crystals in metal alloys. They have not previously been

used to analyse the properties of orientated organelles in markedly anisotropic

muscle fibres. The use of such techniques could provide much data on the

organization of muscle fibres which might otherwise be unobtainable.

The general relationship of any system of organelles , including mitochcmdria, is

2 -3

S = 2 P. ym um

v L

viiere is the number of intersections of organelle surface per unit length of

test line in a stereological grid. This formula could be used to analyse

orientated structures provided that a sufficient number of section planes were

used in order to satisfy the necessary criterion of complete randomness. The

data so obtained, however, cannot be used to derive further information on the

surface areas of organelles.

The theory for analysing partially orientated surfaces conveniently requires

that only one plane of section be used for analysis. On each longitudinal

section of a skeletal muscle fibre a linear test array is applied sequentially,

first in a plane parallel with the long axis of the fibre and then perpendicular

to this axis. When the lines of the test array are parallel with the long axis

of the fibre (Fig. 1) only the terminal isometric surfaces of the mitochondria

intersect the test lines. The surface area of the isonetric regions (S ). v isom

is given by

^Visom = 2(Pl)„ m^m' (1)

where CP]^),| is the number of surface intersections per unit length of the parallel test array.

V/hen the test array is perpendicular (Fig. 2) to the long axis of the

fibre both the linear orientated and the isometric surfaces of the mitochondria

367
N. T. James and G. A. Meek

intersect the test lines. The surface area of the orientated regions of mitochondria (S ), . is given by

Total surface area of the mitochondria (S^) is given by the fomula:

S = (S ) . + (S ), (3) V V isom V lin

Addition of (2) and (3) gives

Sv ' 2(P^)„ - I(Pl)„ pm-3

The degree of linear orientation of any partially orientated system of surfaces

( u) T

. ) can be defined as (S ), . / S whereby

lin V lin I ^

to ^

.

lin

V lin

(S + (S ). V Im V isom

which reduces to:

0)

(P, ), - (P, ), L J- L II ^

(R ), + - (P, ) - (P, )„ Li IT L II L "

The pectoralis major muscle of the pigeon was selected for analysis since

it contains two highly different types of muscle fibre. These are arbitrarily

designated type 1 and type II (George & Berger, Avian Myology, Academic Press,

1966). Type I fibres are relatively narrow in diameter {"^ 30 ym) and contain many mitochondria. They are slew twitch fibres and they derive their energy aerobically by the oxidation of lipids. Type II fibres are relatively broad

(,'\' 70 Mm) and contain few mitochondria. They are fast twitch fibres and they

derive their energy by anaerobic glycolysis. Seme typical values for and w ^ . are recorded in Table 1. Im

It is possible that estimates of and to

-j^^^^ could be useful in quantifying alterations in structure which occur in skeletal muscles following

experimental procedures, for example, during muscular hypertrophy.

368
Partially orientated surfaces of skeletal muscle mitochondria

Table 1

Stereological parameters calculated for

avian type I and II fibres

Parameters Calculated

Muscle Fibres

Type I Type II

-1

1.75 0.06

-1

0.64 0.04

S ym ym

-3 3.0U 0.11

lin 0.58 0.29

Values obtained by applying 500 ym test line length to each of 20 type I

fibres and 20 type II fibres.

A value of <^ -, close to zero indicates there is no linear orientation of

lin

the mitochondrial surfaces . ffeximum orientation occurs at (Pj^) =0 when u

- 1.

Fig. 1. A randomly selected longitudinal section of a type I muscle fibre in

the pectoralis major muscle of the pigeon. A linear test array has been applied

parallel with the long axis of the fibre. The number of intersections of

mitochondrial surfaces with the test lines provide values for (Pj^) . x 35 000

Fig. 2. The same electron micrograph as in Fig. 1. The same linear test array

has been applied perpendicular to the long axis of the fibre. The number of

intersections of mitochondrial surfaces with test lines provide values for (P, ) .

369
N. T. James and G. A. Meek
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

DETERMINATION OF THE NUMBER OF CELLS WITH MULTIPLE NUCLEOLI IN HISTOLOGICAL SECTIONS

by A. Schleicher, H. -J. Kreschmann, F. Wingert, and K. Zilles Department of Anatomy, Medical University Hannover, and Institute of Medical Information Science and

Biomathematics, University of Miinster, Germany

For certain neuroanatomical investigations the number of neurons in different parts of the nervous system has to be estimated. The neuronal cell body, its nucleus, or its nucleolus

may be used as the unit to indicate the presence of a cell in the histological section, but the larger the unit used,

and the thinner the section, the greater the counting error due

to multiple counting of units split by cutting. Therefore, the

most widely used method is that of counting the nucleoli of the neurons as the smallest, but still well distinguished cellular

constituent ( 1 )

.

The method for counting neuronal nucleoli is based on the as- sumption that each neuron has one nucleus with only one nucleolus. Counting samples in the brain stem of Tupaia belangeri, we found mononucleated neurons with one, two and three nucleoli.

In order to estimate the total number of neurons as well as the

number of neurons with one, two and three nucleoli, two aspects

must be considered:

When cutting the structure, nuclei may be split in such a way

that each fragment has one or more nucleoli. Thus nucleoli belonging to the same neuron may appear in adjacent sections

(Fig. la).

Smaller fragments of split nucleoli are not seen or recognized, but fragments exceeding a certain diameter are also

counted as nucleoli (Fig. lb).

Counting nuclei or nuclear fragments with at least one nucleolus

as neurons in the sections we find x' neurons with one nucleolus,

x'2 neurons with two and x^ neurons with three nucleoli:

(1) n = x^ + x^ + x^

The true numbers are denoted by x^ , yi^ and x^:

(2) N = x^ + x^ + x^

The relations between the numbers of neurons counted and the true

numbers are formulated as a system of linear equations:

371
A. Schleicher, H. -J. Kretschmann, F, Wingert, K. Zilles

(3) a^,.x^ + a2i-X2 + ^^yx^ = x

^22'^2 * *32'^3 ' *2

*33'^3 " ^3

For instance, the number of neurons counted with one nucleolus is

the sura of nuclei with one nucleolus, of nuclear fragments with

one nucleolus from nuclei with two nuleoli and of nuclear fragments with one nucleolus from nuclei with three nucleoli.

a b

cutting plane

Fig, 1 Schematic illustration of counting errors caused

by split nuclei and nucleoli.

The calculation of the coefficients is based on the following

assumptions

:

Both the nuclei and the nucleoli are spheres.

The nuclear diameter is smaller than or equal to the thickness of

the section.

The nuclei are uniformly distributed perpendicular to the cutting

plane.

By using geometrical probability assumptions one finally obtains

the following expressions:

372
The number of cells with multiple nucleoli

(4) (1-(l-2p^^).2).x^ + ^PzTf'^Z * ^^31*1*^3 "

^'l

32 T 3 (1-(1-2P33).^).X3 = X3

D is the nuclear diameter and T is the thickness of the sections.

Nuclear fragments with j nucleoli or visible nucleolar fragments

from nuclei with i nucleoli have the probability p. . to be counted in the section, assuming that all nuclear splits are equally

probable and that the nucleoli are equally distributed throughout

the nucleus. The values of these probabilities depend on

(5) b = ^ and c = min

where d is the nucleolar diameter and d . is the diameter of the

smallest nucleolar fragments counted.

Table of the probabilities p^j for c

b P1I P2I P22 P3rP32 P33

0.00 0.500 0.257 0.371 0.193 0.307

0.05 0.500 0.245 0.377 0.1 83 0.317

0.10 0.500 0.234 0.382 0.1 76 0.325

0.15 0.500 0.224 0.387 0.169 0.331

0.20 0.500 0,218 0.390 0.163 0.337

0.25 0.500 0.216 0,391 0.162 0.338

0.30 O.SOO 0.216 0,392 0.161 0.329

0.35 0.500 0.218 0.390 0.160 0.340

0.40 0.500 0.225 0.386 0.164 0,336

The probabilities depend on c by

(6) Pj-i(b,c) = p..(b,c=o) + i-bc for i = j

The calculation of some of these probabilities may be done explicitly (Pi^; ViA ai^'i <i = 0) » others are calculated using a Monte-Carloprocedure on a computer where the probabilities are estimated by

their frequencies occurring in a simulation procedure (2)

.

The following example demonstrates the differences between counted

and corrected numbers. In the nucl .n,oculomotorii of a Tupaia belangeri we measured: thickness of section 20 ym, nuclear diameter

10.6 urn, nucleolar diameter 2.6 urn and the diameter of the smallest

nucleolar fragments counted 2.0 ym. The results are as follows.

373
A. Schleicher, H. -J. Kretschmann, F. Wingert, K. lilies

counted numbers corrected numbers

/ ^1

== 7426 ^1 = 6176

= 3078 ^2 = 3168

/ ^3 = 74 ^3 81

n == 10578 N = 9425

References

1. B.Wo KONIGSMARK, "Methods for the counting of neurons",

Contemporary Research Methods in Neuroanatomy,

edited by W.J.H. NAUTA and S.O.E. EBBESSON.

New York: Springer-Verlag [1970) 315

2. A. SCHLEICHER, K. ZILLES, H.-J. KRETSCHMANN, F. WINGERT,

"Bestimmung der Anzahl der Zellen mit mehr als einem

Nucleolus im histologischen SchnittprSparat" , Acta

microsc, (in press).

374
National Bureau of Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, tM.. September 4-9, 1975 (Issued January 1976)

COMBINED LIGHT AND ELECTRON MICROSCOPY OF SERIALLY SECTIONED CELLS AND TISSUES

by Yrjo Collan, Second Department of Pathology, University of Helsinki, Finland

ABSTRACT By studying herring eggs sectioned in series it was shown that corresponding

images of cells in sections could no more be identified reliably after the distance between sections increased above 0.13 times the diameter of the egg (psychological safety limit) . For rotind cells the probability for a cell to be present in one section if it is found in the other is 2r-t / 2r4-si+S2+t,

where 2r is the diameter of the cell, t the distance between the sections,

and S"! and s^ the thicknesses of the sections. To pick up those cells common

in both sections two types of criteria were recommended, one based on the occurrence of the nucleus within the images of the cell in both sections, the

other on the diameter of the cell or nuclear image. Formulas for the application of these criteria were developed. Applicability of the method was studied in the laboratory mouse. Brain glial cells (ample material between

cells), liver parenchymal cells and kidney proximal tubule cells (large cell

size) could easily be studied with this method. Small intestinal gland cells

(tall but thin cells), and spleen lymphocytes or spermatogonia (small cell

size) needed special precautions that when taken sec\ired reliable matching.

Based on experiments on the sensitivity of autoradiography it was concluded

that ccanbined light microscope autoradiography and electron microscopy \ander

s\xitable conditions will be about 50 times more sensitive than conventional

electron microscope autoradiography.

INTRODUCTIOH

In electron microscope laboratories sections stained with various basic dyes are found useful in determining the area under cutting. However, the expansion of methodology offered by sectioning electron and light microscope sections in series is far from exploited. Aspects related to this question have

been studied in the author's laboratory (l, 2).

Rir/IEW OF PEDICIPLES When combining light and electron microscopy the object to be studied must be

present in both li^t microscope (m) and electron microscope (em) sections.

In the present method 1 - 5 EK sections are cut first and immediately after

these a light microscope section, 0.5-1.0 fm thick, 'rfhile studying these sections it is important to minimize the risk of erroneously matching two different cells, one present in the EM- and the other in the LM-section, but in

relation to other cells in corresponding location in each section. It was

shown that both psychological and geometrical factors are involved.

The psychological factors were studied by cutting densely packed paraffin

embedded herring eggs in series. Pairs of sections were cut 4 - 1200^ apart

frco each other. Each section was photographed and the pairs of photographs

were shown to 15 persons. These were asked whether they were able to identify

corresponding cells in the picture pairs. The results showed that above the distance of 100 ^m matching was found difficult or impossible. The limiting

distance was called "the psychological safety limit (PSL)". For general use

on round objects it was presented as the ratio between the distance of the sections and the diameter of the eggs, giving the value 0.13. Application of the PSL does not guarantee that the imsiges of cells in cor^

responding location in sections are images of the same cells. Even immediately adjacent sections contain a certain number of cells that are present in only one of the sections. The probability for a cell or any tissue structure to be present in two sections provided it is found in one of them can be deteinined by the fonmila

375
Yrjd ColIan

2r - t

I

. ' ~ 2r + Si + + t

where 2r is the diameter of the tissue structure perpendicular to the plane

of sectioning, t is the distance between the sections, and s-| and S2 are

thicknesses of the sections. To find the cells common to both sections three parameters have been investigated: l) the occurrence of the nucleus in the cell in the section,

2) the diameter of the cell in the section, 3) the diameter of the nucleus

in the section.

Of these, the first one is that most easily applicable to practical situations; only cells are studied that show nucleus in both sections. When

applying this criterion for round cells, to avoid mismatching one of the following formulas should hold

2r (1 - sin a) > t ; a<60'' (2) 2v - 2r (l - sin a) > si + S2 + t (3) (where r is the radixis of the cells under study, a is the angle between the cutting plane and the line connecting the centres of the two cells, t is the distance of the nucleus from the cell surface, si and S2 are thicknesses of the sections).

After characterization of the cell population vinder study, the conditions necessary for applying the method can be determined. This is most easily

done by drawing the two sine curves presented by the left sides of the formulas and adjusting the parameters si, S2 and t so that tmder any given

test conditions at least one of the formulas holds. The two other parameters are applied in a corresponding fashion (2), APPLICATION TO VARIOUS TISSUES

To determine to what extent the method is applicable, tissues of an adult

white laboratory mouse weighing 20 g were investigated. Samples were taken

from the small intestine, testis, kidney cortex, liver, spleen and brain,

fixed in 5/'' glutaraldehyde and embedded in Epon. The small intestinal gland

cells, spermatogonia, proximal tubule cells, liver parenchymal cells, lymphocytes and glial cells were studied. Cell diameter, nuclear diameter, cell surface — nuclear surface distance sind the distance between the nuclei of two adjacent cells were measured (Table l).

The table also contains other data, such as values of the formula (1) and

the PSL for the smallest cell diameters in each group for immediately adjacent li^t and electron microscope sections. Under these conditions the most unfavourable situation in tissues is combatted by the most favourable way of matching sections. In the following various tissues are discussed

in detail; the presence of the nucleus of the cells in both sections being

the criterion of sameness. Small intestinal fclands If studied in sections cut perpendicular to their long axis the cells cause no problems. However, gland cells often appear longitudinal in sections.

The cell nuclei are long and the distance between the nuclei is a limiting

parameter (si + S2 + t <1 0.7 ^) because mismatching is improbable when the cells are cut with a plane not roughly parallel to their long axis.Further,

less than 90 % of cells are present in both sections and the PSL reaches

high values easily. This is why the use of immediately adjacent sections

only is recommended. The thickness of the ligjit microscope section should not exceed 0.5 ^»

Testis

These cells vary in size the smallest ones being the size of the lymphocytes

and the largest about double the diameter.More than 90 % of cells are common

to both sections. Also the PSL is small allowing comparison of 1 - 4 EM sec- tions with an 0.5 LM section. The formulas (2) and (3) recommend the use

376
Serial sectioning in /ight and electron microscopy

TABLE 1. Dimensions (pm) of cells in various organs in a laboratory mouse after

fixation in cacodylate buffered 5 glutaraldehyde and embedding in Epon.

Small Sperm- Kidney Liver Spleen Brain

intestine ato- proximal paxench. lympho- glial

glajid cells gonia tubules cells cytes cells

Cell diameter

Kinimun 4.3 6.8 10.2 20.3 5.1 10.5 Kaximum 51.0 15.5 25.7 39.0 11.0 15.5 Nuclear diameter

Kinimum 5.4 5.1 5.1 10.2 4.2 6.9

Maximum 15.5 11.8 9.5 15.3 8.5 11.2 Cell surface - nuclear stirface distance

tlinimum 0,5 0.5 0.8 5.4 0.2 1.0 Kaximum 15 .

5

5.9 6.8 16.9 4.2 5.4 Distance between nuclei in ad.-jacent cells XX Average 0.9 5.4 4.2 16.9 0.7 Minimum 0,7 1.7 1.7 8.5 0.5 — Maximum 3.0^ 6.8 8.5^ 30.5 5.9 Material between cells

+_ + + + Probability from the formula (l)^^^

0.88 0.92 0.94 0.97 0.89 0.94 PSL (si + S2 + t / 2r)^

0.14 0.08 0.06 0.03 0.12 0.06

^luclei of ajiother tubulus or gland often are still farther, up to 10 jm

not detennined

2r = smallest measured, t = 0, s-\ =0.5 ^2 ~ 0.1 pm.

of 1 - 3 EM sections with an 0.5 LM section. This reconmiendation should

be followed to secure reliable matching of the smallest cells also. Proximal tubule of the kidney Here the cells are cubical and have a large amount of cytoplasm around the nucleus. The formulas (2) and (3) show that with an 1 jm LM section three

most adjacent HI sections can be reliably matched, with an 0.5

^ LM section more than ten such sections.

Liver parench:nnal cells These cells were large and had so rich cytoplasm that no problems could develop in applying the method.

Spleen l;mphocytes

The distance between the nucleus and the cell surface is so small that the use of a thin (0.3 - 0.5 _pm) Ii4 section and only the adjacent SI4 section is recommended. Under these circumstances the comparison is reliable also for

the smallest cells. Both the PSL and the values from formulas (2) and (5) are critical if these limits are exceeded. This is due to the dense packing

of lymphocytes in the spleen. For these cells one could also use nuclear or

cell diameter as a criterion. Then, only cells with nuclei above a certain

diameter in both sections are matched. Under those conditions care must be taien not to rule out a true cell population with a small nuclear diameter.

377
Yrid Co/Ian

Brain glial cells

These cells gave data comparable to those of the kidney proximal tubule cells. However because of the thick layer of cytoplasm aroxmd the nucleus they allowed liberal application of the method.

Table 1 also reports interstitial material between cells. In liver and spleen

cells are practically without any intercellular material at all. If the same cells are found more loosely distributed - e.g. lymphocytes in the connective

tissue - the applicability of the method increases. Small intestine, kidney

and testis form concentrates of cells sxirrounded by a basement membrane. Between glands and tubules there is some connective tissue and this is why there

is little danger in mixing data between two glands or kidney tubules. In the brain intercellular material is ample and the method is even more applicable

than appears from the tabulated size estimates. SPECIAL APPUCATIONS

In addition to allowing comparison of various light microscopic staining characteristics (5»5) with ultrastructure and to bridging the gap betvfeen paraffin

sections and electron microscopy (4) the method is applicable to autoradiography (2). iiecause autoradiography is done on the LI'i section the sensitivity

of the method is better than that of EM autoradiography. To test this samples

of mouse testis were studied. The animal was injected with 1 ^Ci/g body weight

of tritiated thymidine intraperitoneally and killed one hour later. From one Epon block ntunerous silver EM sections and 0,5, 1,0, 1,5 thick LH sections were cut, LM autoradiography was carried out by using Ilford K5 emulsion as described earlier (2). For electron microscope samples the ssime emulsion, diluted 1:4 was used and spread on the grids with a steel wire loop. Exposure

time was 3 weeks, and samples were developed with Kodak D 19b for 2 minutes.

In screening the EM sections only occasional cells were found labelled with

one grain above the nucleus. The H'l sections gave following data

Section thickness Mean grain count of 100 cells + SD 0.5pm 7.7 + 2.7 1.0 " 10.2 + 3.8 1.5 " 12.1 + 5,5

Although definite counts could not be made from the EH sections there is great

difference in sensitivity. Difference in the thickness of sections may be 12 fold (or even more) in these methods {1,0 }m and 0,08 jun). In addition, the emulsion being thin and diluted in Eli autoradiography may cause further 1,5 -

5 fold decrease in sensitivity. So it turns out that under suitable conditions

the combined autoradiography method will be about 50 times more sensitive than EM autoradiography,

HEFEEEI\ICSS

1. Collan Y, Collan H: Interpretation of serial sections, Z Wiss Mikrosk 70: 156-167, 1970

2. Collan Y: Combining light and electron microscopic findings on individual

cells: A theoretical and methodological study exemplified by combined electron microscopy and light microscope autoradiography. Microscopica Acta

75:48-60, 1973

3. Kaufmann P, Stark J: Enzymdarstellungen im Semidtlnnschnitt, Acta Histochemica (jena) 42:178, 1972

4. Lynn JA: "Adjacent" sections - a bridge in the gap between light and elec- tron microscopy. Hum Pathol 6:400-402, 1975

5. Snodgree AB, Dorsey CH, Bailey GWH, Dickson LS: Conventional histopathologic staining methods compatible with Epon-embedded,osmicated tissue. Lab Invest 26:329, 1972

378
National Bureau of Standards Special Publication 437

Proceedings of tfte FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

field at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

MORPHOMETRIC "ORGANELLE PROFILES" AS METABOLIC INDICATORS

by Sasha Malamed and Lawrence C. Zoller Department of Anatomy, College of Medicine and Dentistry of New Jersey, Rutgers Medical School, Piscataway, New Jersey. 08854. U.S.A. .

SUMMARY

Two systems are under morphometric study in order todetect possible differences

in cell organelles pointing to specific metabolic alterations. Electron micrographs of normal and experimental or diseased cells are compared by using morphometric

"organelle profiles" (MOP's), i.e. , histograms of volume density, numerical density,

etc. of mitochondria, smooth endoplasmic reticulum (SER) , etc. MOP's of dissociated adrenocortical cells : The lack of an ACTH effect on mitochondrial volume and surface

density is consistent with the view that ACTH Increases steroid output by increasing

the activity, but not the amount of the mitochondrial enzyme which participates in pregnenolone synthesis. In contrast, the increases in the SER parameters induced by

ACTH suggest a concurrent increase in the amount of enzymes associated with the SER:

SP -hydroxysteroid dehydrogenase and 21 -hydroxylase. The roughly equal decrease in lipid droplet volume density and numerical density indicates a depletion almost entirely due to disappearance of lipid droplets, rather than to their diminution in size. MOP's

of cultured skin fibroblasts from normal and homozygotic familial hypercholesterolemic

(FH) patients : The almost twofold increase in the SER volume and surface density of FH cells agrees with reports of their increased cholesterol production.

INTRODUCTION

Our recent work has been based on the rationale that structural changes in cell organelles that can be detected by morphometry may reflect quantitative changes in metabolic activities characteristic of those organelles. We hope that such implied

metabolic alterations will aid in the understanding of certain cellular mechanisms and

thus lead to the design of therapeutic measures for diseased cells. We have used morphometry as an indirect metabolic probe in our studies of the

effect of ACTH on rat adrenocortical cells and our studies on the possible changes

associated with familial hypercholesterolemia (FH) in human skin fibroblasts. RAT ADRENOCORTICAL CELLS

Trypsin-dissociated (Sayers , Swallow and Giordano, 1971) rat adrenocortical cells were incubated for two hours with and without ACTH and then preparedfor transmission

electron microscopy and morphometry (Zoller and Malamed, 1975; Zoller, Gibney and Malamed, 1975). A section through part of a typical cell incubated without ACTH is shown in Fig. 1. The cells appeared qualitatively the same whether or not ACTH was

present.

We studied three cell organelles associated with steroidogenesis : The mitochondria

and the smooth endoplasmic reticulum (SER) both of which bear enzymes and the lipid droplets which supply cholesterol from which the steroid hormones are made (cf. Malamed, 1975). The results of morphometric analyses (Weibel, 1966, 1969) are shown

in Fig. 2. Of the eleven properties measured , five showed significant (Student's T test; P<0.05) ACTH effects. Diameter measurements in the light microscope and similar

methods usingelectron micrographs showed that the volume of cells with ACTH was no

different than the volume of the cells without ACTH. Thus the ACTH-induced per-unitvolume changes of the cell organelles were due to changes in the organelles themselves.

ACTH increased mitochondrial numerical density by 20 percent over the control

(no ACTH) value, but had no effect on the volume density or surface density of their inner

379
S. ^alamed and C. Zoller

or outer membraaes. The SER of ACTH-treated cells was 41 percent higher than

controls in volume density and 68 percent higher than controls in surface density. In contrast, lipid droplets were depleted by ACTH: 41 percent in volume density and 34 percent in numerical density.

What do these results mean in terms of the secretory function of the adrenocortical

cell? Because ACTH markedly enhances steroid hormone production in preparations

such as ours (Malamed, Sayers and Swallow, 1970), the following interpretations are

possible

:

Mitochondria: The increase in number of mitochondria may be necessary for later increases in mitochondrial volume as reportedfor long termACTH-treatment of adrenocortical cells in situ (Nussdorfer, Mazzocchi and Rebonato, 1971). The lack of an ACTH

effect on mitochondrial volume and surface density is consistent with the view that ACTH

increases steroid outputby increasing the activity, but not the amount of the mitochondrial

enzyme which participates in pregnenolone synthesis (Koritz and Kumar, 1970). SER: The increases in volume density and surface density induced by ACTH suggest

a concurrent increase in the amount of enzymes associated with the SER: SP -hydroxysteroid dehydrogenase and 21 -hydroxylase (Inano, Inano and Tamaoki, 1969). Lipid droplets: The 41 percent decrease in volume density and numerical density induced by ACTH agrees well with earlier qualitative observations with the light

microscope (cf. Malamed, 1975). Because 75-80 percent of the adrenocortical cholesterol in the rat is in the lipid droplets (Moses, et al. , 1969) and cholesterol is the raw

material from which steroid hormones are made (cf. Malamed, 1975), this acute depletion in lipid droplet volume was expected.

HUMAN NORMAL AND FAMILIAL HYPERCHOLESTEROLEMIC (FH) FIBROBLASTS

The cells used in this study were provided by Dr. Avedis K. Khachadurian from

his cultures of skin fibroblasts obtained from four jjatients with FH and from four normocholesterolemic siblings of these patients or unrelated individuals. FH is a

hereditary disorder characterized by elevation of plasma cholesterol and beta lipoproteins. The FH cells we studied were from FH homozygotes who characteristically

succumb to atherosclerotic heart disease at 21 years of age. The cells were cultured in monolayers in a lipid -rich conventional medium containing

10 percent calf serum and harvested by trypsLnization (Khachadurian and Kawahara,

1974). Preparations for transmission electron microscopy and morphometric and

statistical methods were similar to those used for adrenocortical cells as described

above.

Fig. 3 showsa section through part of a tj^ical skin fibroblast from a normocholesterolemic individual. Qualitatively it is indistinguishable froma typical skin fibroblast from an FH patient. Morphometric analyses, however, detected differences between

normal and FH cells in three of the thirteen organelle properties measured: volume

density of the mitochondria, volume density of the SER, and surface density of the SER.

Because the total volumes of normal and FH cells were the same, the mitochondrial and

SER differences in volume and surface were absolute as well as relative to cell volume.

The mitochondrial volume density of FH cells was 59 percent higher than that of normal cells, thus suggesting an increase in any of the metabolic activities performed

by these organelles as for example, electron transport and the synthesis of ATP. Hence

it would be tempting to conclude that the FH cells are "burning more fuel" than are normal cells. This may turnout to be true, but other morphometric data argue against

this view because all the other mitochondrial properties measured were the same in normal and FH cells. Thus, the significance of the increase in mitochondrial volume

of the FH cells is unclear. The changes in the SER are easier to interpret. FH cells overproduce cholesterol (Khachadurian and Kawahara, 1974) and cholesterol is synthesized by the SER (cf. Malamed, 1975). Thus the 71 percent higher volume density and 90 percent higher surface density of FH cells as compared to normal cells was expected. Of course.

380
Morphometric "organelle profiles" as metabolic indicators

these data alone tell us nothing about any possible causal relationships between the structural and functional changes in the SER of FH cells. VALUE OF MORPHOMETRIC "ORGANELLE PROFILES" AS

METABOLIC INDICATORS

Because the rate -limiting enzymes for steroidogenesis (Karaboyas and Koritz

,

1965) are thought to be mitochondrial (cf. Malamed, 1975), ACTH might be expected to produce some increase in mitochondrial properties. Yet our morphometric data show

no increase in mitochondrial parameters associated with ACTH-enhanced steroid output. Thus it appears that the "resting" or unstimulated adrenocortical cell has adequate

mitochondrial "apparatus" to respond to acute ACTH stimulation and the mechanism of the acute ACTH effect does not appear to involve the production of new mitochondrial

material, i.e. , enzymes. Rather, our findings in agreement with those of Koritz and

Kumar (1970), indicate that ACTH stimulates steroid secretion by activating already

available mitochondrial enzymes, perhaps by removing some inhibitory factor(s). The ACTH-induced changes in the SER and in the lipid droplets as shown by

morphometry are interpreted as indices of chemical events. The increases in SER

parameters indicate synthesis of steroidogenic enzymes such as 3p -hydroxysteroid

dehydrogenase and 21 -hydroxylase (Inano, Inano and Tamaoki, 1969)and the decreases

in lipid droplet parameters reflect the depletion of cholesterol used for the formation of the steroid hormones.

The similar decreases in lipid droplet volume density (41 percent) and numerical

density (34 percent) induced by ACTH permit an interesting inference about how the cell mobilizes its lipid droplet cholesterol when called upon to produce steroid hormones.

It appears that some lipid droplets are depleted entirely and disappear before the cholesterol in other droplets is drawnupon. Were this not the case , there would be no decrease

in number of lipid droplets along with the decrease in volume density. Why the cell should use up some droplets completely before starting to use others is puzzling and

raises other questions as, for example, how the cell chooses which lipid droplets to use

first.The results of studies on the FH cells so far tell us far less than do those on the adrenocortical cells. At first we thought that FH cells might differ from normal cells

in that they were less able to destroy excess cholesterol via action by lysosomes.

Accordingly we studied spheromembranous bodies, one form of these cell organelles.

However, we found nodifference between FH cells and controls in volume or number of spheromembranous bodies. The large increases in SER parameters of FH cells is to be expected in cells producing excess cholesterol, but the significance of the increase in mitochondrial

volume density is unclear. This mitochondrial change revealed by morphometry does,

however, suggest that mitochondrial oxidative phosphorylation and other functions be

investigated in the FH cells. Although our morphometric studies with these two cell systems are still at their earliest stages, the results so far indicate to us that morphometry is useful as a kind

of indirect cytochemistry. Ultimately it is hoped that this use of morphometry may have

value as a diagnostic tool in early detection of specific diseases.

This work was supported by NIH grant AM 16833, AHA grant GR-74-SOM-2, and

National Foundation March of Dimes grant 1-352. Ms. Jean Gibney provided expert

technical assistance.

381
S. Malamed and C. Zoller

REFERENCES

Inano, H. , Inano, A. & Tamaoki, B. (1969) Submicrosomal distribution of adrenal enzymes and cytochrome P-450 related to corticoidogenesis. Biochim . Biophys . Acta 191, 257. Karaboyas, G. C. & Koritz , S. B. (1965) Identity of the site of action of 3', 5'- adenosine monophosphate and adrenocorticotrophic hormone in corticosteroido- genesis in rat adrenal and beef adrenal cortex slices. Biochemistry 4, 462. Khachadurian, A. K. & Kawahara, F. S. (1974) Cholesterol synthesis by cultured

fibroblasts: decreased feedback inhibition in familial hypercholesterolemia.

J. Lab. Clin. Med. 83, 7. Koritz, S. B. & Kumar, A. M. (1970) On the mechanism of action of the adrenocorticotrophic hormone. J. Biol . Chem. 245, 152. Malamed, S. (1975) Ultrastructure of the mammalian adrenal cortex in relation to secretory function. In: Handbook of Physiology; Endocrinology VI (Ed. by G. Sayers, H. Blaschko and A. D. Smith), pp. 25-39. American Physiological

Society, Washington.

Malamed, S. , Sayers, G. & Swallow, R. L. (1970) Fine structure of trypsindissociated rat adrenal cells. Z. Zellforsch . 107, 447. Moses, H. L. , Davis, W. W. , Rosenthal, A. S. & Garren, L. D. (1969) Adrenal

cholesterol: localization by electron microscope autoradiography. Science

163, 1203. Nussdorfer, G. , Mazzocchi, G. & Rebonato, L. (1971) Long term trophic effect of A CTH on rat adrenocortical cells. Z. Zellforsch . 115, 30. Sayers, G. , Swallow, R. L. & Giordano, N. D. (1971) An improved technique for

the preparation of isolated rat adrenal cells. Endocrinology 88, 1063. Weibel, E. R. (1966) Practical stereological methods for morphometric cytology.

J. Cell Biol. 30, 23. Weibel, E. R. (1969) Stereologicalprinciplesfor morphometry in electron microscope

cytology, hit. Rev. Cytol . 26, 235. Zoller, L. C. & Malamed, S. (1975) Acute effects of ACTH on dissociated adrenocortical cells: quantitative changes in mitochondria and lipid droplets. Anat . Rec. 182, 473.

Zoller, L. C. , Gibney, J. & Malamed, S. (1975) Acute effects of ACTH on dissociated adrenocortical cells: quantitative changes in mitochondria and smooth endoplasmic reticulum. Anat. Rec . 181, 517.

382
Morphometric "organelle profiles" as metabolic indicators

SER

Fig. 1. Electron micrograph of a section through a trypsin -associated rat adrenocortical cell. M, mitochondrion; SER, smooth endoplasmic reticulum; L, lipid droplet;

N, nucleus. X23,500.

zoo - 190 -

PLASMIC DROP- CELL MITOCHONDRIA RETICULUM LETS * STATISTICALLY SIGNIFICANT (P <0 05) V =VOLUME Nv = NUMERICAL DENSITY

Vv = VOLUME DENSITY Sv- SURFACE DENSITY

Fig. 2. Morphometric organelle profiles of trypsin-dissociated rat adrenocortical

cells incubated with ACTH (1000 microunits; 2 hours).

383
S. Malamed and C. Zoller

^SB

SER M

Fig. 3. Electron micrograph of a section through a human skin fibroblast. M,

mitochondrion; SER smooth endoplasmic reticulum; L, lipid droplet; SB, spheromembranous body. X 7200,

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Fig. 4. Morphometric organelle profiles of cells from homozygotes with familial hypercholesterolemia.

384
National Bureau of Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg. Md., September 4-9, 1975 (Issued January 1976)

A QUANTITATIVE ANALYSIS OF SOME ULTRASTRUCTURAL ASPECTS OF SEED DEVELOPMENT

by Colin E. Hughes and Lewis G. Briarty Botany Department, School of 'Biological Sciences, Untversity of Nottingham, Nottingham NG7 2RD, England

SUMMARY

The application of straightforward stereology techniques in the analysis

of developing seed structure provides accurate information on morphogenesis.

Studies on the endosperm of developing wheat seeds show that the amyloplasts,

the organelles in which starch granules are deposited, divide only during the very early development of the tissue, and that subsequently the amyloplasts

are partitioned among the still dividing cells. Some information on starch

granule synthesis is provided by the correlation of starch build-up with

membrane changes within the amyloplasts.

Changes are also noted in the S/V of the RER in relation to reserve protein synthesis in both wheat and French bean; a high value for S/V is achieved before protein synthesis begins, this then drops fairly rapidly as the protein is produced. Nuclear volume remains a constant proportion of the cell

volume, even though the latter changes markedly.

The significance of these points is discussed in relation to the use of

stereology in the early analysis of the effects of genetic manipulation.

INTRODUCTION

The fine structure of developing seeds of economic importance, in particular cereals and legumes , is a research area receiving increasing

investigation as the importance of such seeds as primary sources of nutritional

protein, carbohydrate and lipid is realised. In this context we have been

working on two seed types, the French bean (Phaseolus vulgaris: Briarty, 1973) and wheat (Tritioum aestivum) in an attempt to quantify the changes in ultra- structure which occur in the cells of the seed storage tissues throughout the entire period of seed development. Both of these seeds produce significant

amounts of starch and protein, and it was with the hope of obtaining a better

understanding of the mechanism by which these materials are synthesised that

the studies were undertaken.

The results to be described were all obtained using material fixed in glutaraldehyde/OsO^ and embedded in epoxy resin using standard procedures

:

quantitative data was obtained by point- counting with light and electron

microscopy using techniques described by Weibel (1969) , and all data was processed using the Pocoster program (GnSgi et dl. , 1970)

.

WHEAT GRAIN DEVELOPMENT

In both bean and wheat the storage tissue development follows a similar

pathway, there is an initial phase of cell division during which the cells are fairly vacuolate, and this is followed by a phase in which the bulk of the reserves is laid down. Starch is deposited in amyloplasts, membrane-bound

organelles similar to chloroplasts , while protein is deposited in small vacuoles in the cytoplasm. During this phase of reserve deposition there is massive cell enlargement and the phase is terminated at maturity by dehydration

of the seed. Starch is the major reserve in the cells of the wheat grain, and the greater part of the starch is present at maturity in the form of large,

lenticular granules (A-type) , each granule lying within its own amyloplast . Thus

the number of granules present in the mature grain , and the consequent yield

of the seed, is a function of the number of amyloplasts produced in each cell, and of course the number of cells in the grain. One of the factors affecting

starch yield in the mature grain is therefore the number of amyloplasts

present in the individual cells during the phase of cell division, when the cell contents are divided up between the daughter cells. It is generally accepted

that amyloplasts themselves undergo division, thus restoring the organelle

385
C. E. Hughes and L. G. Briarty

numbers to some extent after cell division.

A plot of the number of A- type starch granules per cell (Fig.l) shows a steady drop to a plateau after the seed reaches an age of 12 days. This

corresponds to the age at which cell division stops in the grain: the indication is therefore that after this time there is no increase in the number

of A- type arayloplasts , and thus of starch granules, in each cell of the grain - a suggestion which has been made on other evidence by Evers (1974) . At what

stage of development, then, do the amyloplasts stop dividing? Do they them- selves divide as the cells divide, maintaining the number per cell to some extent, or are they distributed passively to the dividing cells?

A plot of the total number of amyloplasts present per grain (Fig. 2) shows

a fairly constant value as early in development as our data so far extends,

some 6 days after fertilisation, even though cell division is continuing fairly

rapidly. It therefore appears that, over this period at least, there is no multiplication of the amyloplasts, but that these organelles are simply

distributed to the daughter cells as the latter divide. There is little A-type

starch granule initiation after about 5 days after fertilisation: the number

of A-type starch grains which a cell produces is presumably fixed in the very

early stages of grain development.

Starch Synthesis

The mechanism of starch granule deposition is the subject of a great deal

of debate (Evers, 1974: Geddes , 1969) . While some of the biochemistry is well

known, the sites of action of the responsible enzymes within the amyloplast are unknown. A study of the changes in volume of the various components of the amyloplast during starch synthesis provides indications on this latter point.

The major component of the organelle, apart from the starch, is a system

of tubules which run around the periphery of the granule (Fig. 3) . A plot of starch granule volume against the volume of the tubules shows a straight line

relationship throughout the developmental stages examined (Fig. 4) , and it is tentatively suggested that some of the enzyme systems responsible for starch

synthesis may be associated with the tubular membrane system of the amyloplast.

(The S/V of the tubules remains virtually constant over the period examined.)

Protein Synthesis

In both wheat and bean seeds the synthesis of the protein reserves begins

somewhat later than starch synthesis. It is likely that in both species the reserve protein is synthesised on the rough endoplasmic reticulum (RER) membranes, and this idea is strengthened by the apparent increase in surface area of the RER which parallels the development of storage protein (Fig. 5). A parameter which has not previously received attention, however, is the S/V of the RER in relation to its role in protein synthesis. The data

presented above (Fig. 5) indicate that, prior to the onset of protein

synthesis, the S/V increases rapidly then drops as protein synthesis commences. The increase in S/V indicates a flattening of the RER cisternae while the decrease denotes a swelling, perhaps as a result of the accumulation of synthesised protein within the membranes. The path by which synthesised

protein moves to its site of deposition within the cell is unknown in these species, and this work provides at least circumstantial evidence that part of

that path lies in the RER cisternae.

The similarity of the conformational changes in the RER during protein

synthesis in these widely different species suggests that the phenomenon is perhaps more widespread, and deserving of further study.

Another parameter of interest in these developing cells is the V of the cell occupied by the nucleus. In wheat the cell volume increases 6-fold, and

in bean 43-fold over the periods studies. In spite of these large variations,

however, the V ^ccupied by the nucleus remains fairly constant at 3 -1% in wheat and 4.5 -0.7% in bean: the nuclear volume thus maintains a fairly

close relationship with the cell volume. The immediate interpretation of this is that an increase in nuclear volume, and therefore presumably in activity, is required to produce and/or to maintain control over a cell of increasing

volume. A similar relationship between nuclear DNA content and cell volume has been noted by Smith (1973)

.

386
A quantitative analysis in seed development

CONCLUSIONS

The results described above provide data on a number of parameters of developing seeds, relating in particular to the mechanisms of cell growth. They

also suggest possible points in the growth cycle at which outside influences

might effect some degree of crop improvement. It appears that the overall

starch yield in wheat, for example, is a function of many variables including

the number of cell divisions occurring in the seed, the number of times that

individual amyloplasts divide , and the number of amyloplasts inherited by the gametes from the parents , as well of course as the nutritional state of the parent plant during seed development. Morphometric stereology, providing

quantitative data on cellular morphogenesis, could be used to monitor changes

produced by genetic manipulation at a level relatively close to the controlling

DNA.

ACKNOWLEDGEMENTS

Financial support from the Science Research Council and the Flour Milling

& Baking Research Association is gratefully acknowledged. The authors wish to thank Dr.H-R.GnSgi for material help in application of the Pocoster program, and Tony Evers (F.M.B.R.A.) for valuable advice and discussion.

REFERENCES

Briarty, L.G. , 1973. Stereology in Seed Development studies: Some preliminary

work. Caryologia 25 supp., 289-301.

Evers, A.D., 1974. The development of the grain of wheat. Proc. 4th Int. Cong.

Food Science and Technology. Madrid, 1974. (In press.)

Geddes, R. 1969. Starah Biosynthesis. Quarterly Reviews. Chem. Soc. London 23 , 57-72. GnMgi, H-R. , Burri , P.H. & Weibel, E.R. , 1970. A multipurpose computer program

for automatic analysis of stereologicdl data obtained on electron

micrographs. 1 eme Cong. Intemat. de Micr. Electronique , Grenoble, Vol.1, 443-444. Smith, D.L. , 1973. Nucleic add, protein and starch synthesis in developing

cotyledons of Pisum arvense L. Ann. Bot. 37, 795-804. Stamberg, O.E., 1939. Starch as a factor in 'Sough formation. Cereal Chem. 16, 769.

'

Weibel, E.R. , 1969. Stereological principles for morphometry in electron

microscopic cytology. Int. Rev. Cytol. 26_, 235-302.

FIGURE LEGENDS

Fig. 1 Starch granule numbers per cell in developing wheat seed.

Fig. 2 Total number of cells {•) and amyloplasts (o) in developing wheat seed.

Fig. 3 Amyloplast in developing wheat seed showing starch and membrane

tubules (Scale = 0.5vim).

Fig. 4 Starch granule volume per amyloplast plotted against tubule volume per amyloplast in developing wheat seed. (Numbers = age, days after

fertilisation)

Fig. 5 Changes in S/V (•) and area (a) of RER, and protein content (!) in developing seeds of (a) bean and (b) wheat.

387
C. E. Hughes and L. G. Briarty

Fig. 3 Fig.

4

388
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9. 1975 (Issued January 1976)

THE BRAIN AS A "STEREOLOGICAL DEVICE"

by T. Radii-Weiss and J. Radilova

Institute of Physiology, Czechoslovak Academy of Sciences, Prague, Czechoslovakia

It Is probable that stereological Ideas will be soon adopted

In the study of visual perception as tlie brain is functioning

in a sense as a "stereological device" by which three-dimensional judgments can be i>orformed on the basis of two-dimensional

representations of the external world upon the retina. The aim

of the present paper is to report on the results of experiments

in which three-dimensional perceptual interpretation of plane

geometrical figxires by humans (i.e. a subjective phenomenon)

is studied by means of objective psychophysiological and qiiantitative techniques.

M£IHODS

Reversible figures (Fig.l) well known in experimental

psychology are used as stimulus material. Iheir basic feature

is that they may be interpreted perceptually in two different

ways. In case these patterns are constantly illuminated both

interpretations alternate spontaneously (the instant of change

is signaled by the subject by operating a switch). When they

are shown for a very short time, for instance by means of a flash light, the pattern is seen in any trial in one of the two

possible ways only (signaled by pushing one of a pair of knobs).

Time series analysis is performed with this data. Xn the case

stimtilus pattern is illuminated by flash electroencephalographlc

evoked potentials from the surface of the skull can be also

recorded, classified into two groups according to the subjective

interpretation of the stimulus (which remains the same physically) and separately processed for increasing the ratio between

"signal " - the evoked brain potential and the "noise" - the

spontaneous electroencephalographlc activity.

RESULTS

The time covcrse of alternation of pairs of perceptual

interpretations of the constantly Illuminated reversible figure

shown in Fig.l C have been analysed under two different conditions (Radilova et al. , 197^) • either when perceived as a re- versible three-dimensional concave or convex ol^ct, or when it was seen (in another experimental session after a different

verbal instruction) as a two-dimensional reversible figure - background pattern. Although the perceptual process develops

Unconsciously, the reversal rate was about two time slower in

case the drawing was interpreted as three-dimensional (Pig. 2).

Three-dimensional interpretation of a plane figure requires

probably more elaborate nervous processing needijag more time.

Rational explanation of the principle of three-dimensional

interpretation of a more complex two-dimensional reversible

object (Schroder staircase - Fig.l B) makes the spontEineous

reversal rate faster showing that learned cognitive factors play

an important role in the development of three-dimensional interpretation of plane objects (Radilovi, Radii-Weiss ,1975)

•

When a simple reversible figure the Necker cube (Fig.l a) was constantly illuminated (Radilovd et al. , 1972) both types

389
T. Radii-Weiss and J. RadilovS

of intervals were of the same length and the types of interval

distributions were almost equal demonstrating that both perceptual interpretations having in this case little symbolic meaning

are equivalent. Neighbouring intervals either of the same type

or of different type tend to be of similar duration what means

that the actual mode of perception depends to some degree from

the previous one. In the case of repeated exposure of the same pattern by flash subjective interpretations of the stimulus in

consecutive trials were not independent and underlying process

could be described on the basis of Markov chain model (Badilovd

et al. , 1973 a)« Thus in the case when both three-dimensional

interpretations of the two-dimensional pattern are equivalent,

the actual way of perceiving the object depends very much from

the mode of seeing it in the previous trial.

Evoked brain potentials have been recorded after the repeated presentation of this pattern and separately averaged in two

groups according to its actual subjective interpretation (Fig.3)«

The curves of averaged evoked brain potentials have been found

different in a part of subjects showing that different brain

processes might be responsible for the two types of three - dimensional interpretations of the same two-dimensional geometrical pattern (Radilovi et al, , 1973 b, 1975).

R£F£R£NC£S

Radilova,J. , & Radii-Weiss, T. (1975) Two and three dimensional

figure reversals. Activ. nerv. super, in press.

Radilovd,J. , Radii-Weiss, T. & Krekule, I. (1972) Psychophysiological analysis of the perception of reversible figures.

Physiol, bohemosl. 21, 429.

Radilovd, J,, Radii-Weiss, T. & Havranek, T. (1973 a) Qxiantitative

description of the Necker cube. Physiol, bohemosl. 22, 427-

428.

Radilovd,J., Radil-Welss ,T. & Spunda, J. (1973 b) Evoked responses induced by presentation of the Necker cube. Physiol,

bohemosl. 22, 427.

Radllova, J., Radii-Weiss, T. & Spunda, J. (197^*) Objective

analysis of subjective perception of reversible figures.

Activ. nerv. super. I6 . 297.

Radilovd, J., Radii-Weiss, T. , Spunda, J. & Indra, M. (1975)

Electroencephalographlc correlates of the perception of

visual pattern. Xn: Biokybemetik (Eds by M.Orischel, &

N. Tiedt), pp 268-272. Gustav Fischer Verlag, Jena.

TEXT TO FIGURES:

Fig.l: Two-dimensional reversible visual patterns used in psychophysiological experiments.

Fig. 2: Histograms of intervals corresponding to pairs of three

(a) and two-dimensional (B) interpretation of the same reversible figure. Histograms 1 and 2 represent pairs of intervals

of both types. On the horizontal axis intervals of different

length, on the vertical axis their incidence.

Fig. 3: Typical example (at regular stimvilus presentation with

5 sec intervals) of the average EEG activity. A - 5OO msec

before and B - 5OO msec after the presentation of the pattern

of the Necker cube by means of flash light. Curves (l) and

(2) represent averaged bioelectrical activity in cases of the

two different subjective three-dimensional perceptual interpretations of the same two-dimensional geometrical pattern.

Curve (3) is the arithmetical difference between both.

390
The brain as a "stereological device"

Radii -Weiss fig. I

391

National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

STEREOLOGICAL ANALYSIS OF NEURAL ORGANOGENESIS IN THE CHICK EMBRYO

by 0. Mathieu and Paul-Emil Messier Departement cfAnatomie, Universite de MontrSal, CP. 6128, Montreal, Canada

ABSTRACT

Stereological methods were used to study the organogenesis of the neural tube in chick embryos at stages 5 to 10+. Transverse sections from various le- vels along the cephalo-caudal axis were analysed. The point-counting method

was used to quantify, in light microscopy, the nucleo-cytoplasmic ratio and the volumetric density of the intercellular spaces, nuclei and cytoplasm of cells in the neural tube. Using the electron microscope we have studied the volumetric density and the surface ratio of mitochondria as well as the surface density of the endoplasmic reticulum in the cytoplasm.

INTRODUCTION

During embryonic development undifferentiated cells associate to form definite cellular layers, which are precisely shaped to form specific organs. Stereology is useful in the study of cytodif f erentiation as it permits to quantify cellular structures at various stages of development and to follow

their fate in time. This paper presents the morphometric analysis of cells of the neurulating chick embryo.

MATERIAL AND METHODS

Chick embryos ( Gallus domesticus ) were prepared for electron microscopy

using usual procedures. Several stages of development from the neural plate

to the closed neural tube (stage 5 to 10"^ according to Hamburger and Hamilton,

1951) were analysed. Transverse sections were cut at defined levels along the

neural tube of each embryo, considering the sequential development from cephalic to caudal region. For embryos at stage or older these levels were:

level 1: anterior to the head level 2: posterior to the head level 3: the first somite level 4: the last somite

level 5: the node of Hensen

Table 4 shows which levels were analysed in our younger less differentiated specimens.

The section area of the neuro epithelium was determined by cutting and weighting drawings made by projections of the sections on a constant density

paper. The point-counting method of Glagolev (1933), applied on micrographs

taken at a magnification of 1000 X with the light microscope, was used to de- termine l) the nucleo-cytoplasmic ratio 2) the volume fraction of intercellular spaces, nuclei and cytoplasm in the neural tube.

In electron microscopy, we determined l) the volume density of mitochondria in the cytoplasm 2) the surface density of the granular endoplasmic reticuliun in the cytoplasm 3) the surface to volume ratio of the mitochondria
0. Mathieu and P.-E. Messier

These parameters were determined according to Glagolev (l933) for volumetric

density and according to Saltykov (1958) for surface density.

RESULTS

Table 1 shows the value of the mean t standard error obtained at each le- vel of the six embryos of each stage for the area of the section of the neural tube. The statistical analysis (Friedman's m ranking and variance analysis) of these data has shown that there are differences (p ^ .05) l) along the cephalo-caudal axis of embryos of the same stage, stage 5 excepted

2) from early to later stages of development, at any level along the cephalocaudal axis except for level 3. Table 1 illustrates that at stage 8"*" -

10"*" the section area of the neural tube decreases from level 1 to level 4 and then

increases as the node of Hensen is approached. Between these stages the sec- tion area l) increases anterior to the head (level l) 2) decreases both

posterior to the head (level 2) and in the last somite region (level 4)

3) does not change in the first somite region (level 3).

Furthermore our results indicate that

1) the nucleo-cytoplasmic ratio (Table 2) does not change at any stage a- long the cephalo-caudal axis, except at stage 7 where it increases (p < .01) from the back of the head to the node of Hensen. Also, the value of the nucleocytoplasmic ratio changes (p ^ .05) from early to later stages, at each of the five levels studied.

2) the volume fraction of the intercellular spaces in the neural tube

(Table 3) decreases from the cephalic region to the node of Hensen at stages 6 to 8~. It also decreases from level 1 to level 4 whereupon it increases ap- proaching level 5 in embryos aged 8^ to 10+. Moreover the volume fraction of the intercellular spaces in the neural tube does not change from younger to older specimens at level 1 and 5, while from stage 8~ to 10+ it decreases

(p < .01) at levels 2, 3 and 4.

3) limited space does not allow us to present our results in a table

form, but we found that the volumetric densities of cytoplasm and nuclei both

increase (p < .05) from the head to the node of Hensen at each stage. More- over from stage 7 to 10" the volumetric density of nuclei in the neural tube

increases (p ^ .01) at all levels, while the volumetric density of the cytoplasm in the neural tube does not change, except at level 5. 4) none of the parameters considered change along the cephalo-caudal axis

of the stage 5 embryos.

Table 5 shows the results obtained at level 2 for the volume fraction of mitochondria in the cytoplasm (Vy mito/cyto)> "the surface density of the en- doplasmic reticulum in the cytoplasm (Sy R.E./(-y^Q) and the surface to volume

ratio of mitochondria (Rmito)* mentioned above the volume fraction of the cytoplasm in the neural tube did not change at level 2 from stage 5 to lOt The volume fraction of mitochondria and the surface density of the endoplasmic

reticulum in the cytoplasm did not change at level 2 from one stage to another. The surface to volume ratio of mitochondria decreases at level 2 along the

stage series. This decrease indicates that the size of the mitochondria in- creases from stage to stage, as there is a negative correlation between the

size of a particle and its surface to volume ratio. However, the volumetric

density of mitochondria does not change from one stage to another suggesting

that mitochondria are less numerous from stage 5 to 1(^.

DISCUSSION

Our results point to changes in relative volume of some cellular struc- tures as they evolve during neurulation. We find that all parameters remain

constant along the cephalo-caudal axis of young specimens (stage 5). When com- paring data of a given level from one stage to another, it must be kept in mind

that the position of level 2, 4 and 5 moves posteriorly within the embryos

from stage 5 to 10+. At level 3, we show that from stage 8~ to 10+ the variations in the total volume of the intercellular spaces does not interfere with

the unchanging area of the section of the neural tube. Yet as the neural tube

394
Stereological analysis of neural organogenesis in the chick embryo

Table 1: Area (mm^) of the neural tube section at different stages

Level 5 6 7

8" 8+ 9 10" 10+

• UZOD • UZl / > Ul /o • UoUO • Uo /O • U4dD

.0034 « .0025 .0019 .0019 .0011 .0050 .0068 .0056

o MA . noQA UzoD n/i no • U'4-Uz • UoUD

.0036 .0014 .0021 .0024 .0014 .0017 .0017

3 .0293 .0362 .0220 .0218 .0174 .0204 .0173 .0184

.0020 .0021 .0026 .0007 .0005 .0019 .0006 .0016

4 NA .0329 NA NA .0135 .0108 .0082 .0075

.0021 .0011 .0004 .0005 .0007

5 .0248 .0291 .0218 .0219 .0228 .0183 .0156 .0174

.0044 .0033 .0029 .0018 .0022 .0014 .0018 .0020

Table 2: Nucleo--cytoplasmic ratio at different stages Level 5 6 7 8- 8+ 9 10- lO^-

1 OA . zo .27 .28 .33 .32 .38 . 32

.01 .03 .02 .01 .02 .01 .02 .01

2 NA . 28 .26 .28 .31 .36 .34 . 35 .02 .0.2 .01 .02 .01 .02 .02

3 .28 .28 .29 .31 .31 .35 .37 .36

.01 .01 .02 .00 .01 .01 .01 .01

4 NA .27 NA NA .33 .35 .39 .34 .02 .02 .02 .03 .02

5 .28 .28 .32 .29 .35 .36 .36 .39 .02 .02 .02 .02 .02 .03 .02 .02

Table 3: Volume fraction of the intercellular spaces in the neural tube at different stages Level 5 6 7 8" 8+ 9 10" 10+

1 .18 .19 .23 .21 .21 .20 .21 .21 .02 .03 .03 .01 .01 .01 .01 .01

2 NA .17 .20 .22 .18 .17 .14 .13

.01 .03 .01 .01 .02 .01 .01

3 .13 .16 .17 .15 .13 .14 .10 .11 .00 .02 .01 .01 .01 .01 .01 .01

4 NA .14 NA NA .14 .11 .08 .09

.01 .01 .01 .01 .01

5 .12 .13 .13 .11 .12 .14 .11 .14

.01 .01 .01 .01 .01 .01 .01 .02 *In each of the above table, the first line of figures= mean of 6 embryos;

second line= ± standard error; NA= value not available (level not considered)

395
0. Mathieu and P.-E. Messier

closes, neuroepithelial cells get closer and closer so as to potentiate sur- face interactions that may be modified from stage to stage. Concomitantly

the nucleo-cytoplasmic ratio increases as a consec|uence of the increase in the

volume fraction of the nuclei. This may reflect an enlargement in euchromatin

content and therefore point to an increase in DNA replication process during

neurulation.

At level 2 the decrease in the neural tube section area is approximately

50^ between stages 8" and lO"*". This decrease cannot be caused only by the

reduction of the volume fraction of the intercellular spaces as these occupy respectively 22?^ and 13)(> of the volume of the neural tube at stage 8~ and 10+. Then it would seem that fewer cells may be implicated in the maturation of the

neural tube at level 2 along the stage series studied. Moreover the nucleocytoplasmic ratio and volumetric density of the nuclei increase in the neural

tube. The increase in DNA replication referred to earlier is not coupled with

an increase in endoplasmic reticulum-related synthesis of proteins since we

have shown that the surface density of the endoplasmic reticulum in the cytoplasm remains unchanged.

In conclusion, our data allow us to draw a curve of the characteristics

of the epithelial cells in the chick embryo undergoing normal neural organogenesis. Use of this curve will enable us to estimate the effect of various treatments on the chick embryo neuro epithelium provided we know l) the stage

of the embryo 2) the level which is analysed along the embryo's axis. This work was supported by the Medical Research Council of Canada and the Department of Health and Welfare, RODA division.

Table 4: Transverse levels analysed in embryos of stage 5 to 8"

Level 5 6 7

8-

1 middle of the

head anterior to the head

2 NA posterior to the head

3 equidistant be- tween 1 and 5 1/3 the distance

between 2 and 5 somite second somite

4 NA

2/2 the distance

between 2 and 5 NA NA

5 node of Hensen

NA: not available (level not considered)

Table 5: Volumetric density of the mitochondria in the cytoplasm,

surface density of the endoplasmic reticulum and surface to volume

ratio of the mitochondria 'at level 2 in embryos of different stages Parameter 5 6 7 8- 8+ 9 10- 10+

5.8 6.2 6.2 6.1 6.1 7.2 6.6 6.7

.6 .4 .3 .4 .2 .3 .3 .1

S R.E./ .37 .39 .37 .35 .35 .51 .37 .48

V cyto .06 .05 .04 .05 .03 .04 .02 .11

R 10.3 9.4 8.7 8.9 8.7 8.4 8.6 8.0 mito .4 .4 .2 .5 .5 .1 .2 .3

For each parameter, first line of figure = mean of 6 embryos, second line = - standard error BIBLIOGRAPHY

Glagolev, A. A. (1933): Trans. Inst. Econ. Min. Moscow, 59 Hamburger, V. and H.L. Hamilton (l95l): J. Morphol., 88: 49 Saltykov, S.A. (1958): "Stereometric Metallography", second edition, 446 p., Moscow

396
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg. Md., September 4-9. 1975 (Issued January 1976)

APPLICATION OF STEREOLOGICAL METHODS TO THE STUDY OF PRE-IMPLANTATION EMBRYOGENESIS IN MICE

by Russell L. Deter, Department of Cell Biology, Baylor College of Medicine, Houston. Texas 77025. U.S.A. ,

ABSTRACT

Application of stereol ogical methods to the study of mammalian preimplantation embryogenesis requires control of the morphogenic process, visual access

to the embryos during development, identification of individual embryos and adequate evaluation of sampling procedures. In preparation for a study of early mouse embryogenesis in vitro , methods have been developed which achieve these objectives. Control of embryogenesis through the blastocyst stage has been ob- tained by use of a perfusion culture system in which the parameters affecting

development are defined. The culture chamber of this system permits continuous

observation of embryo development with any type of optical microscope. In this

system, up to 97% of 2-4 cell embryos have developed into morulae or blastocysts

in 48 hours. Culturing of embryos in compartments formed by fibers of a nylon mesh has permitted identification of individual embryos by position. A method

for embedding embryos in plastic while they are still in the mesh has been developed. Because individual embryo identification and orientation are conserved, evaluation of the response of two cell embryos to the preparative procedure

has been possible. This in turn has led to selection of conditions which minimize dimensional distortion. Specification of section samples has been made

possible by the definition of a coordinate system based on the structure of the

nylon mesh. This coordinate system has been used to design a sectioning procedure which permits location of individual sections within the embryo. These sections contain a complete cross-section of the embryo and can be studied in their

entirety with either the light or electron microscope. Analysis of any level of

structural organization and evaluation of sample adequacy can be carried out using the information contained in these sections.

INTRODUCTION Mammalian embryogenesis prior to the time of implantation in the uterus is characterized by multiple cell divisions, cavitation of the resulting cell mass

(blastocyst formation) and differentiation of the first specific cell population

the trophoblasts. These events, which usually occur within the lumen of the oviduct, have been the subject of many morphological studies with both the light

and electron microscopes. Such studies have been greatly facilitated by the development of methods which can support embryogenesis to the blastocyst stage in

V i troAs a consequence of previous investigations, an outline of the morphogenetic events responsible for the transformation of a fertilized zygote into an im- planting blastocyst has been obtained. However, this outline is essentially descriptive as there has been no previous attempts to quantitate morphological

changes or assess their variability. To make possible a quantitative morphological analysis of embryo development, the following series of investigations were

undertaken.

OBJECTIVES Quantitative morphological studies of pre-implantation embryogenesis require that the following four objectives be realized:

1) embryogenesis must occur in a controlled environment to reduce the number of variables affecting development

2) visual access to embryos during their maturation must be possible so that the morphogenic process can be monitored and appropriate stages selected for further study

397
R. L. Deter

3) individual embryo identification must be maintained during all stages

of the investigation to allow assessment of variability in the developmental process and correlation of information obtained from both living

and sectioned embryos

4) evaluation of the adequacy of samples used in stereological analyses must be possible

Procedures developed in our laboratory have achieved these objectives.

METHODS

Embryos

The mouse embryos used in the development of the system to be described were obtained by inducing superovul ation in the immature BALB/C mice with

pregnant mare's serum (PMS) and human chorionic gonadotropin (HCG), followed by

mating with adult 129 males (Gates, A.H. Methods in Mammalian Embryology , 1972

p. 62). Two to four cell embryos were collected by flushing the oviduct with

Whitten's defined medium (Whitten, W.K., Adv. Biosci. 6:131, 1970), equilibrated

with 5% O2 - 5% CO2 - 90°Z N2, 44 hours after HCG injection.

Culture System

To obtain embryo development under controlled conditions and constant

visual access, a perfusion culture system built around the chamber designed by

Dvork and Stotler (Dvork, J. A. and Stotler, W.F., Exptl . Cell Res. 68:144, 1971)

has been developed. Embryo culture is carried out in a small space^Tvolume :0.2 ml) formed by two coverglasses separated by a spacer ring which is located in the center of the chamber. This space is perfused (1 ml/hr) with Whitten's medium by means of a peristaltic pump. The chamber is placed on the stage of bright

field microscope and its temperature maintained at 37^0 using a Nicholson air- stream stage incubator (Model C300).

As embryos can be swept out of the chamber by the moving medium, a means for restraining them must be used. For this purpose a piece of nylon mesh

(1.0 X 0.5 cm), bonded to a polystyrene or polyester coverslip by heating, has been found to be most satisfactory. The mesh fibers form small compartments

(100 X 100 u) into which individual embryos fall when placed over the mesh in suspension. Friction between the zona pellucida and the fibers of the mesh pre- vent both rotation and mechanical displacement if the mesh is stretched sufficiently.

Preparation of Embryos for Light and Electron Microscopy

High resolution light and electron microscopy can be carried out only

on specimens considerably thinner than single embryos or even one blastomere. Therefore, microscopic study of the embryogenic process requires sectioning.

This is possible if embryos are fixed and embedded in plastic. Fixation of mouse embryos has been carried out using 4% gl uteraldehyde followed by 1% osmium

tetroxide, both buffered in 0.1 M cacadylate buffer, pH 7.3. Embryos were then stained with uranyl acetate, dehydrated in ethanol and embedded in the low viscosity epoxy resin described of Spurr (Spurr, A.R. J. Ultrastruc. Res. 26:31 , 1969). These procedures were carried out on embryos confined to compartmetits of

a nylon mesh bonded to a polyester coverslip. An evaluation of the effects of this procedure on two cell embryos was made by determining the perpendicular dis- tance from the center of the cleavage plane to the periphery of one blastomere

after each processing step. These measurements, made on photomicrographs, were compared and the percent change calculated.

Sectioning

The key to sample evaluation lies in the process of sectioning. An important requirement for such evaluations is the ability to specify section

position within the intact embryo. This can be done with the following sectioning procedure by definition of an appropriate coordinate system. If one

398
Pre-implantation embryagenesis in mice

mesh fiber of the square containing the embryo is considered to lie on the X axis, a line perpendicular to that fiber (usually parallel to another mesh fiber) will represent the Y axis. This X-Y plane can be made congruent with the coverslip underlying the mesh and the origin placed under one corner of the

square. A line perpendicular to this plane and passing through the origin is then the Z axis. Using an LKB Pyramitome (Model 11800), it is possible to con- struct a cube whose face lies in the X-Z plane and whose right side lies in the

Y-Z plane. The other three sides are perpendicular to the face and cut to the point where sections parallel to the face (X-Y plane) are just large enough to contain the maximal embryo profile and the polyester-epoxy resin junction. For electron microscopy, ribbons of such sections are first picked up on an uncoated slot grid (sections float on a water droplet) which is then inverted and touched to the Formvar coated surface of a second grid. This procedure avoids

section folding which is often found following section transfer with usual techniques. Sections on the coated slot grid can be seen in their entirety if the slot has dimensions of 0.2 x 1.5 rran.

Microscopy

Sections prepared as just described can be studied with both the light

and electron microscopes following staining. A RCA-3G electron microscope, modified to operate between 700X and 10,000X and having a 350 negative roll film

camera, has been used to obtain negatives of sections at both low and high modifications. Stereological procedures are carried out on the projected image of these negatives using a specially designed projection system (Deter, R.L., J. Microsc. 100:341 , 1974).

RESULTS AND DISCUSSION

Embryo Development

With this perfusion culture system, approximately 80% of the 2-4 cells embryos develop into morulae or blastocysts when all conditions investigated are considered together. This value can be increased to 97% with selection of ap- propriate parameters such as duration in culture, embryo stage at beginning of culture, etc. No attempt has been made to maximize blastocyst development but

70°^ of the embryos reached this stage in some instances. The use of a defined

medium, which is replaced completely every 12 minutes, and an accurately con- trolled environmental temperature permit pre-implantation embryogenesis to proceed in a very reproducible manner.

Visual Access

Because of the special design of the Dvork-Stotler chamber, any type

of optical microscopy can be used to study embryos as they develop in culture. Polystyrene coverslips do not interfer with phase, interference contrast or bright field microscopy. The polyester coverslips, on the other hand, cannot be used with phase objectives. Because the spacer ring and the two coverglasses

are less than 1.2 mm thick, high magnification objective lens can be used.

Individual Embryo Identification

The sequestration of single embryos within compartments formed by the fibers of nylon mesh provides a means for permanently separating embryos from

one another. As the embryo's position in the mesh usually remains unchanged

throughout the culture period and even during fixation and embedding, data on individual embryos can be collected by both direct observation and from studies

on sections. This attribute allows an evaluation of variability in the morphogenic process at all levels of organization.

Sampl inq The ability to keep track of individual embryos throughout their prep399
R. L. Deter

aration for sectioning allows correlation of information obtained from studies

of sections with that obtained from living embryos. However, sectioning is a sampling procedure and is carried out after the embryo has been subjected to extensive chemical treatment. The usefulness of such correlations therefore depends upon knowledge of the effects of the preparative procedure (which could introduce distortion) and the representativeness of the section sample taken for

study. Examination of our procedure for fixing and embedding embryos has revealed a consistent embryo shrinkage during uranyl acetate staining and alcohol dehydration. This effect could only be counteracted by allowing swelling to occur during fixation. With proper balance of the swelling and shrinking artifacts,

embryos having dimensions similar to those in the living state {% change for un- fixed vs. embedded : -3 .2[+2 .4 S.D.]%) can be obtained. The variability among em- bryos indicates that evaluation of the response of each embryo to the preparative procedure is required if major distortions are to be avoided.

Evaluating the representativeness of samples requires information a- bout the size, shape, orientation and spacial distributions of the objects being studied and therefore must be decided in each individual case. However,

specification of the location of sections taken for study is an essential part

of this process. With the fixed coordinate system and sectioning procedure

previously described, such specification is possible because each individual

section can be related to the Y axis. By keeping a record of section numbers

and thicknesses (determined from interference colors) starting with the first

section to contain a part of the embryo, a Y coordinate value can be obtained

for each section in the sample. These can be located on a Y axis superimposed

on a micrograph made of the embedded embryo prior to sectioning. Perpendiculars

drawn through each of these points intersect the image of the embryo and indicate

where the embryo was sampled. If some specific distribution of sections is desired, appropriate Y coordinates could be chosen prior to sectioning.

With the section sample selected, subsequent sampling and measurement

problems become those common to all stereological studies carried out on sec- tions. Since the entire cross section of the embryo can be seen in the electron

microscope, any level of embryo organization can be sampled, with the added ad- vantage of high magnification for identification of boundaries or specific structures. X and Z coordinate values can be determined for any part of the section

by measuring the perpendicular distance to the right hand side of the section

(X) or to the polyester-epoxy resin junction (Z). With the flexibility in sampling provided by these procedures, many types of quantitative morphological studies should be possible.

400
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

THE DETERMINATION OF SIZE DISTRIBUTION ON LYMPHOBLASTS IN ACUTE LEUKEMIA

by E. Feinermann and G. A. Langlet Centre d'Etudes Nucleaires de Saclay, Division de Chimie, Service de Chimie-Pliysique. 8. P. 2

9 1 190-Gif-sur-Yvette, France

Stereology is the study of the structure of matter in

three dimensions at the microscopic level, based on the examination of two-dimensional sections through the material. According

to this definition (E. Underv;ood), we have entered upon the size

distribution of leucocytes within the study of acute lymphoblastic 1 e ukemi a

.

The s t e r e o 1 o gi c a 1 study of the leucocytes assumed spherical, for a group of patients suffering from this disease has ena- bled the population distribution of the diameter to be determined. Thin sections obtained by classical procedures of embedding, cutting and staining , have been examined with a Quantimet.

Several procedures have been suggested for determining

true-size distributions of spherical particles. Many biologists

and metallurgists concerned with the third dimension of cells

and materials, approached this question with s tereological parameters (Saltykov, 1958; Fisher and Cole, 1968; Underwood,

1 969; Weibel , 1969) . First of all, cells having diameters inferior to Di = 5.2

microns are not counted. Let Dr be the real diameter and Dm the

apparent mean diameter of spherical cells.

FIG. 1

401
f. Fe /'nermann and G. A. Langlet

Dm is given by ratio S/z if S is the area of the circular segment limited by Dr and Di on Fig. 1 , and z the height of

this s e pmen t

.

Area S is the sum of SI , S2 and S3 ;Di z

S 1 = S3 =

4 2 TT

S and : Dm = - = 0.5 (Di +

z 2 z

If one sets : x = Di/Dr , one gets : a = cos'^x

and : z = 0.5 Dr sina= 0.5 Dr

so that the formula becomes :

o -1 D r cos X Dm = 0 .5 (x Dr + Dr Vl-x^

i. e, : Dm = 0.5 Dr (x + cos'^ x / V'-x^ )

From Dm and Di, one can deduce Dr. The analysis of this

function by a program written in APL (A Programming Language)

shows that a good approximation is obtained when using the following empirical formula : Dr=(Dm-0.24Di)/0.73

with 6.5 < Dm < 10.5 and 4.5 < Di < 6.5 (mi- crons ) , so that : Dm = 0.73 Dr + 0.24 Di

Di would be equal to zero if all the cells could be counted. Then the formula v7ould become : Dr = 4 Dm / ir From the apparent diameters which have been measured, one may get the histogram of frequencies. A triple smoothing on three values at each time provides a histogram which obviously

exhibits a bi-modal distribution. The observed maxima of the

peaks are supposed to be close enough to mean apparent diameters

to allow the derivation of the real diameters and hence of the

volumes of the cells. The bi-modal distribution of the volumes

seems to consist in the sum of two gaussians. It has been shown

that, for all sufferers of Acute Lymphoblastic Leukemia whose

cells have been examined, the apparent diameters of the majority

of cells correspond to one of the two peaks at 9.2 and 10.9 microns, which leads to real volumes of 740 and 1280 cubic microns

and real diameters of 11.2 and 13.4 microns respectively.

402
Size distribution on lymphoblasts in acute leuken. REFERENCES

1. S.D. WICKSELL

"The Corpuscle Problem. I. Case of Spherical Corpuscles".

Biometrika, 18, (1926).

2. S.A. SALTYKOV

Stereometrie Metallography, second edition, Moscow

Metallurgizdat (1958) .

3. S.A. SALTYKOV

"The Determination of the size Distribution of Particles

in an opaque Material from a Measurement of the size

Distribution of their Sections".

Stereology, edited by H. Elias, Proc . Second. Int. Cong, for Stereology, New York. Springer - Verlag

(1967) , 163 .

4. E.E. UNDERWOOD, Particle - Size Distribution, Chapter 6 p. 149, Quantitative Microscopy (DeHoff and Rhines,

eds) 1970.

5. C. FISHER and M. COLE

The metals research Image analyzing computer, the

Microscope 16 p. 81 (1968) .

6. E.R. WEIBEL

Stereological principles for morphometry in electron

microscopic stereology

Int. Rev. Cytol. 26, 235.

403
E, Feinermann and G. A, Langlet

ACUTE LYMPHOBLASTIC

LEUKEMIA

OG nrnm

nrnoD nn mnnrTTrnnnn nppm

non ron mGnnmrrrp nm nnnnn rmnnnrnrnnn rnnnri nonmrmrnnrrrnprnmnpnr rrnrnr

fTirFTTTin rrinrrmT'iT'rTnTfTTin^rrin

rrriTiiPrnnrrPITTixnnrrirTpnnn nnnn

nnnnnnrrnrTPPnpriTV'^r'nTTipppr'rTin rTnnriPrr'nmpprtnrprrTTppprppprTPrm nrrinnPT nnprr^ppppppn rprrppprrprrn

rrirnipnrrnpprriTTTTnrnnrrpnrT"rTPprn mnnnrrppprppnrpprrppnpprpnrpnppnn

npnrprnpnpnpprTrnnprpnT^rT'rrpniTTT'r)

rrrinpprnnniT^n^ppnmrrprr^priTT^riT^nnrp mppnpnrpppprPppnpppmnrpnnrppprnnniT' rrn

nrTiriniTrrnirrnrrprnt^pppnnT'prrr rrrnrnrnrri^p

^ rnnnnpj'nnprT]n[|nDnannnijrnn:inn^|FPnrnnrypnpnnmniTtn 7 8 9 10 11 12

^ ACUTE LYMPHOBLASTIC LEUKEMIA

rm rm

nnn rnnm r ^^^^

____ nrTTTT} m

liTitipr iTTi^rrr'^nrrTPrrr'T'rTTTTi rprrrrrp cm

[riJT!rnnrrnrnrrrTrnrTirrrnrrprr! un i a

i

rr mn

fn rrrrTTnTTTrrpp^n^nrmpppprrnii*Tnrp mr

nnpnnrT rcri tttppptp"nnrnxrn^npTPrpnpnrri noro

ronfTTirTTfTnpnrriprTTiprnrrTTiro'r 'Tif! in n t to nmn

fT?nTT7rrTT'r^n^rrprrnpn"ppnnrn^rcnL'ut i .n ! 'p pitot nDfrprrrrnrnrnrTprrprrrmrPTi i oit ti n vp nnpnn

nprrnrPTtprpprrrrT'i nim i i'iiniin'ii(i'? "i.iipp cppftp p pnrprrnrnnnnnnnppnn^rTTrnnrprpprTTTinpnnD nrprm pp nnpnnpppppnppprpnrpprppnmrpppr

npppppppprrpnprpnrnn

PnnppppppppptjTTimppnnr^nf 00 80 100 120 1B0

ACUTE LYMPHOBLASTIC

LEUKEMIA

nnrprrrrn

pngmrnpp n

nnopppTPn r nnopnTtppn np

nirrfTiiTrfi m

PGPrprrTPnrti mr r,

pprmmnprnrrp rrpp r

ppni ..n rpnnpnrr rairi r rpnrrmcprpnrrn rmn nr pnnpnrnnpppnnrp nppn m

0 rpppprmprrprrpp ppnr rr n n pppppppopppprrpTi n rrrp t

n nci pppnpTPrTiriTPPrT'P p pfrr nn n

pn ppp pprrirr^rprTnnTT'rTi n ppnr rn r

rrip pppppnmppppppnrropppppp n rppr pp n pnmnPTTpmrrpprrnrpppppprppp m rrvp pp p rTirrtpppprrpprnnprrnrrprnprTPP p<ti mpn m rr pnnpppprpnpprnnnpppprTipnpmrr rrpn rrm pn pp PPnrrrTTTT'nppnnnppnrrnrrnprp pppp itp'^ pp pp npnppppppnnrnpnpprrpppnppppprp rrpn ppm rn ppp

nnpnrrppp mnnrnrTnrm''npprpppp mnn pppr np ppp p nppnnprppppppppppnppprrrrppppp ttp pppp nn rrr r ppmrppTPPPPrTPPrrpprppprppprp pppn pppn pp ppp pt n

nnrprPprrnpprrppnprnrrppprrrT'P rnnn rrm rn ppp pp rn nnppnrprnrpppppp"rrppnppprpprp rppp nrnn m rrn m pp ' ppppmpprnprppprpppppppppmrm rvrr> pppp rF rpp pn nn r rnpnnrnrrnqpnnnprrrpnnrirnrngrn ^.rnrjpr ppm pnnn rarrnmnnnpocnop

500

1 I 1000

I 1900

Fig. 2, Diameter, Surface and Volume

Distribution Curves

.

404
National Bureau of Standards Special Publication 437

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

A MORPHOMETRIC STUDY OF HUMAN PLACENTAE OF DIFFERENT GESTATIONAL AGES FROM NORMAL AND

TRIPLET PREGNANCIES

by Indra Bhargava,* K. Kamashki** and Yadollah Dodge,*

*Jundi Shapur University, Ahwaz, Iran '"Jawaharlal Institute of Postgraduate Medical Education and Research, Pondichierry, India

Bhargava (1969) has postulated a selective elongation of either component

of the fetal blood vessels of the placenta, as a common denominator in many

abnormal states of gestation and development. The present study is directed

towards the search of a morphometric confirmation of the above postulate.

MATERIAL AND METHODS:- The material for study comprises 10 placentae

consisting of early premature, - 10 weeks - 2, premature, non viable, - 24 weeks - 2, premature, viable, - 26 weeks - 2, fullterm 38 weeks - 2 and full- term triplets 38 weeks - 2.

These placentae were collected at cesarean section and fixed in 10 % formaline for 24 hours. Each placenta was divided into 1 c.c. blocks - 250

to 300. Out of these, 20 blocks were selected at random and refixed for further 48 hours. These were embedded and sectioned at 5 u. 25 sections were

randomly selected from each block and were stained with H & E.

Observations regarding volume, area and length per unit volume - c.mm.

,

of different structural elements of fetal placenta, namely villi and their

components - blood vessels, trophoblast and stroma, intervillous space and fibrin, were recorded on 5 fields from each section by point count volumetry

and analysis of linear intercepts, using coherent multipurpose test system of Weibel (1966).

Significance of differences between placentae of different groups, in relation to various parameters of each structural element, was assessed by means of the Least Significant Difference Test (Miller 1966), consisting of

two stages

1. Testing Null hypothesis by appropriate F test

2. Testing each single comparison with significant

F value, by appropriate T test.

Presence of a significant correlation between the measurements of different structural elements of the same placenta was investigated by

computing the coefficient of correlation 'r' from covariance and variance of the two variables. The critical value of r for rejection of the null hypothesis was taken as 0.445 at 5 %, and 0.564 at 1 % level of significance.

OBSERVATIONS AND COMMENTS:- The mean values for volume, area and length

of different structural components of placentae at different stages of a normal pregnancy and in fullterm triplet pregnancy, per unit volume - c.mm.

of placental tissue, are presented in Table I. These values will provide the base line for further studies on placentae from abnormal gestation.

A statistical evaluation of the differences between the parameters of different structural elements of placentae of different groups shows that significant differences are shown by:-

1. 10 week placentae in relation to S^ villi, S^ intervillous

space and L^ blood vessels;

2. 10 and 24 week placentae in relation to villi, stroma,

S-^ fibrin and trophoblast. ;

405
/. Bhargava, K. Kamashki and Y. Dodge

TABLE I SHOWING MEM VALUES OF PARAMETERS OF PLACENTAE IN DIFFERENT STAGES OF GESTATION

PARAMETER AGE

10 Wk 24 Wk 26 Wk 38 \^ Triplet 38Wk Villi V 0. 57+ 0.04 0.,52+ 0,,03 0,, 65+ 0.,04 0.52+ 0,,03 0.. 50+ 0,,04 \ 12 . 86+ 2 . 09 26,.99+ 4

,

7

1

24

,

, 69+ 2, 24 .31+ 4,,18 29,. 72+ 4

,

, 38 V 3.48+ 1.04 11.,27+ 1,,92 14,,74+ 1.,60 15 .08+ 2,,29 22 .22+ 4.,44

R1D-LUUU nnri V 0.41+ 0. 05 0,.36+ 0., 03 0.,42+ 0, 03 0 .43+ 0,,07 0 .33+ 0,,04 Vessels 47 . 38+ 6.90 53..66+ 5.,44 49,,82+ 3,,19 30 .51+ 4,,39 48,.78+ 9.,09 LV 87.30+ 11.76 71,.10+ 12,,81 70,,84+ 10,,48 55 .99+ 15,,87 78,.53+ 18.,14 Troph. V^ 0.14+ 0.02 0,. 14+ 0,,02 0,, 14+ 0,,04 0 .09+ 0,,01 0 .18+ 0.,03 Stroma V^ 0.19+ 0.05 0.,21+ 0,,03 0,,21+ 0,,03 0 .39+ 0,,02 0..17+ 0,.07 56.49+ 15.06 70,.84+ 6,,87 60,,68+ 3.,88 33 .83+ 3.44 85 .22+ 15,.09 I.V. V^ 0.20+ 0.03 0.27+ 0,,04 0,,19+ 0,,02 0 .30+ 0,,03 0 .25+ 0,.05 Space 12.91+ 2.09 26,,74+ 4,,38 24,,09+ 2,,42 24 .31+ 4,.18 29 .72+ 4,.38 Fibrin V^ 0.04+ 0.02 0,,06+ 0,,02 0,,05+ 0,,01 0.093+ 0,.03 0.063+ 0,.03 Sv 2.17+ 1.09 3,,72+ 1,,15 1,,86+ 0.,55 5 .18+ 1,.72 4,.74+ 2,.24

V^ - c .mm. /c .mm. , S,^ - sq .mm. /c .mm. , L^ - mm./c.mm.

From the preceding analysis, the trend of morphometric changes in placentae

with advance of gestation, can be stated as:

A. Increase up to 24 weeks, thereafter

1. Maintained - S^ intervillous space

2. Decreased - villi, S^ blood vessels, S,^ stroma

3. Increased further - L^ villi, V,^ stroma, fibrin, V^ blood vessels

B. Decrease up to 24 weeks, and decrease further - L^ blood vessels

and V^ trophoblast.

The trend of changes becomes rather abrupt at 26 weeks, the age of viability of the fetus and correspondingly the period of considerable

adjustments for the placenta. This is shown by significant differences of 26 week placentae in relation to:-

1. An increase in V,^ intervillous space, V,^ stroma, S^ fibrin

and 1,^ villi.

2. A decrease in V,^ villi, trophoblast, stroma, blood

vessels and L,y. blood vessels.

Out of these, the differences in relation to villi, S,^ stroma, Sv fibrin and V^ trophoblast are not shared by 24 week placentae.

The abruptness of these changes is associated with a marked decline in correlations between different structural elements, as well as their magnitude,

thereby indicating that at this stage, the coordinated process of maturation

of placenta has started making way for the process of senescence.

Correlations among different structural components of placentae of various

groups show a complex pattern and are shown in Table II. It appears that these correlations and consequently coordinated growth reaches a peak by 24 weeks,

followed by a gradual decline. At 10 weeks, outstanding correlations are seen

in relation to V,y. villi, V,^. blood vessels, V,^ stroma, blood vessels and intervillous space. At 24 weeks, these are seen in relation to V,^. blood

vessels, V,^ intervillous space, villi, L,^ villi, S^ blood vessels, Ly blood

vessels and 3,^. stroma. Regarding V,^ fibrin, fibrin and Vv trophoblast,

significant correlations are seen at 10 and 24 weeks.

406
Study of human placentae of different gestational ages

TABLE II SHOWING ' r '-COEFFICIENT OF CORELATION, BETWEEN PARAMETERS OF STRUCTURAL ELEMENTS OF PLACENTAE OF DIFFERENT GROUPS

VILLI BLOOD VESSELS TROPH. STROMA I. V. SPACE FIBRIN

Vv \ S^ L^ V^ V^ S^ V^ S^ S^

1 2 3 4 5 6 7 8 9 10 II 12 13

1 B0.45 AO. 65 A. 045 AO. 72 BO. 57 AO. 52

CO. 55

2 BO. 56 CO. 46 BO. 55 AO. 50 AO. 45 AO. 99

CO. 50 DO. 48 BO. 48 BO. 84 BO. 59 E0.58 CO. 49 E0.74 DO. 89

3 BO. 56 AO. 51 BO. 63

CO. 50 BO. 55 EG. 60 4 AO. 51 AO. 45 BO. 70 AO. 57 BO. 55 BO. 55 DO. 51 CO. 50

DO. 50 E0.49

5 AO. 65 AO. 45 AO. 46 AO. 74

CO. 46 BO. 55 BO. 74 DO. 49

DO. 48 CO. 50 E0.52 CO. 49 E0.58

E0.58 DO. 50 E0.90

E0.49

6 BO. 55 E0.52 BO. 56 BO. 56 DO. 45

7 AO. 45 AO. 50 AO. 57 AO. 51 BO. 55 BO. 57 BO. 48 E0.64 BO. 75

8 AO. 72 AO. 45 AO. 74 AO. 57 AO. 46 BO. 73 BO. 65 E0.64 E0.45 CO. 46

9 BO. 74 BO. 52 E0.74

CO. 50 E0.74 E0.90

10 BO. 57 E0.60 BO. 70 DO. 60 BO. 47

CO. 55 DO. 51

11 AO. 99 AO. 51 AO. 46

BO. 84 BO. 63 BO. 56 BO. 75 BO. 47 BO. 69 BO. 59 CO. 49 E0.60 DO. 48

DO. 89 E0.58

12 AO. 52 BO. 59 BO. 56 BO. 55 BO. 73 BO. 69 AO. 58

BO. 87 DO. 45 E0.45 DO. 65 E0.88

13 AO. 57 BO. 57 BO. 59 AO. 58

BO, 87 DO. 65 E0.88 A - 10 weeks, B - 24 weeks, C - 26 weeks, D - 38 weeks single and E - 38 weeks

triplet placenta

407
/. Bhargava, K. Kamashki and Y. Dodge

Throughout the gestation, a significant correlation is observed between

1. and of blood vessels,

2. Sy villi, and S intervillous space, and

3. v., fibrin and S fibrin.

V T

. villi and blood vessels show significant correlations only in the later part of gestation.

Thus, in the placenta, villi and their blood vessels appear to have

the most coordinated growth. In general, 24 week placenta shows the most coordinated growth, with 10 week, triplet, 26 and 38 week placentae following

in a descending order.

Triplet placentae of 38 weeks show significant differences in relation

to all the structural elements, when compared to a single placenta of similar

age. These placentae have similarities with

1. 10 week placenta in relation to villi, intervillous

space and blood vessels.

2. 10 and 24 week placentae in relation to trophoblast,

Sy fibrin and villi.

Different parameters of various structural elements of a triplet placenta

display more significant correlations of a larger quantum among themselves,

when compared with their counterparts in a placenta from single pregnancy of a similar duration. These facts indicate that in comparison to a placenta from a single pregnancy a triplet placenta shows a more coordinated growth of its elements, and continues to be in the process of maturation for a longer time.

A study of the interrelationship of different parameters of a structural

element indicates that after the initial bulk or volume is laid down in the first trimester, area becomes the most significant parameter, on account of its key role in governing the physiological exchanges.

The above study, and the inferences therein, can be regarded as a preliminary study of the quantitative aspects of growth, maturation and senescence of the structural elements of the placenta, which may define the norms for further studies in this direction.

REFERENCES

:

1. Bhargava Indra and P.T.K. Raja,

Fetal blood vessels on the chorial surface of the human placenta in abnormal development, Experientia, 25: 520-522; 1969.

2. Miller, R.G. , Jr.

,

Simultaneous statistical inference, McGraw Hill, New York, 1966.

3. Weibel, E.R. , G.S. Kistler and W.F. Scherle,

Practical stereologicalmethods for morphometric cytology, J. Cell.

Biol. 30: 23-38; 1966.

408
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

THE EFFECTS OF OPTICAL RESOLUTION ON THE ESTIMATION OF STEREOLOGICAL PARAMETERS

by H. J. Keller, H. P. Friedii, P. Gehr, M. Bachofen and E. R. Weibel Department of Anatomy, University of Berne, Berne, Switzerland

It is generally assumed that the estimation of parameters such

as volume and surface densities should be independent of the optical magnifications at which the measurements are performed. In comparing estimates of alveolar surface area of human lungs obtained by light microscopy (1) and by electron microscopy (2) systematic differences were however observed whose magnitude could

not be explained by the differences in specimen preparation alone.

It was concluded that the higher resolving power of the electron

microscope, together with the smaller section thickness, allowed

more surface detail to be sensed by the test system. Such systematic differences became particularly critical since the use of automatic image analysis was envisaged, and Bignon and Andre-Bougaran

(3) had already noticed systematic differences to point counting

methods to occur with these instrioments

.

The resolution of fine detail of an object is limited by two

effects

:

1) An object point is "blurred" by the analytic system to a circle

of non-zero diameter on the image.

2) The dimensions of the analyzing structure (a picture point on a TV tube or the thickness of a test line on a grid) are finite.

Overall resolution is expected to be roughly the s\im of these

effects. Its influence on the measurement of area and boundary

length should be different: the resolution of greater contour detail results in an increased boundary length whereas the enclosed

area should remain the same, at least as long as the resolved detail is small compared to the dimensions of the object.

To test this hypothesis we have performed comparative measurements of 40 samples from a human lung using (a) the Quantimet 720

combined with a light microscope operated at magnifications of 45x,

llOx, 440x, and llOOx, and (b) point counting on electron micrographs magnified lOOOx, 4300x, and ll'OOOx using established methods (4) . The lung had been fixed in situ by instillation of glutaraldehyde (4) ; samples from various regions were obtained following a systematic scheme and processed for embedding in Epon 812. Sections

about 2iJm thick were cut with glass knives and stained with hematoxy lin-eosin for light microscopy. From the same block 70nm sec- tions were cut for electron microscopy.

Fig. 3 shows that the estimation of volume density of these

septa in the unit lung volume is a stable parameter, showing no systematic differences with magnification, except at very low

powers, where the diameter of a picture point approaches the septum thickness indicating poor overall resolution. From Fig. 3 it

is seen that the surface density systematically increases with

magnification, both in the Quantimet-light microscope combination,

and in the electron microscope preparations evaluated by intersection counting. There are also systematic differences at the

same magnification between Quantimet measurements and intersection

409
H. J. Keller. H. P. FriedI i, P. Gehr. M. Bachofen and E. R. Weibel

counting estimations; one possible reason is that the two methods

should be compared at the same resolution rather than magnification, but it is difficult to establish the former for the point

counting method. An upper limit for our system is indicated in

Fig. 3. It is concluded that systematic differences in the estimation

of certain stereological parameters, such as surface densities,

may result from the use of varying magnifications. This is of particular concern if absolute values of a certain parameter are to

be obtained, or if results from various studies are to be compared.

Studies of this kind need to be extended to establish resolutiondependent factors for correlating stereological estimations of

various parameters.

Fig. 1 Fig. 2 Representative lung fields as light and electron micrographs

Fig. 3 Tissue volume density

VvT si^d alveolar sur- face density Sva of

human lung at different

magnifications

.

(o=light microscopy,

A=electron microscopy)

Acknowledgement

This work has been supported by a grant from

the Swiss National

Science Foundation.

References

(1) Weibel, E.R.: Morphometry of the human lung. Springer, Berlin,

1963

(2) Weibel, E.R., Bachofen, M. and Gehr, P.: Prog. Resp. Res. 1975

(in press)

(3) Bignon, J., Andre-Bougaran , J.: C.R. Acad. Sci . 269 , 409, 1969

(4) Weibel, E.R.: Respir. Physiol. 11, 54, 1970

410
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg. Md., September 4-9, 1975 (Issued January 1976)

QUANTIFICATION OF RARELY OCCURRING STRUCTURES IN ELECTRON MICROSCOPY

by Ulrich Pfeifer Pathologisches Institut der Universitat Wurzburg, Germany

In electron microscopy micrographs have always played a more important

role as a vehicle of information than in light microscopy. This is particularly clearly seen in the case of electron microscopic morphometry. Here the random samples in the form of micrographs are evaluated by the superimposition of various types of lattices - a procedure which allows the determination of a great number of diverse parameters (1). There is no need to elaborate on the merits of this method. Certain limitations, however, must not be disregarded. They stem from the fact that some structures, which are known

to occur regularly, are found only rarely, for instance once in a hundred or more micrographs taken at random. Under these circumstances the number of micrographs required for a statistically significant evaluation can exceed

the limits of practicability.

In such a situation one would do well to recall that in light microscopy

quantitative evaluations are performed mainly by counting and measuring the structures directly with the microscope. The mitotic index in the normal liver, for instance, is known to be in the range of one or two per ten thousand. Nobody would have considered determining it by taking light micrographs

at random, and then evaluating them. The question now arises whether the method of direct evaluation with the microscope can and should be adapted for electron microscopy. In the present paper some methods are described which we have found of use in our work on cellular autophagy (2,3) for the quantitative evaluation of rarely occurring structures with the electron microscope.

These methods have been developed for the Siemens Elmiskop I A, but may have

to be modified in some details, when other types of microscopes are used.

As a first step for the evaluation desired one has to define area units

which can be examined for the occurrence of the scantily represented structures of interest. As area units in our work the square openings of the copper grids supporting the specimen (100 mesh grid, Veco, Solingen, Germany) were used. It was, however, found that the size of the squares was not constant; measurements of the area units had, therefore, to be included in the evaluation. Two different methods can be used for this purpose. In the first

one the lengths of a certain number of squares were measured in the light

microscope by means of a micrometer screw in the ocular. Since the actual

squares, which had been evaluated in the electron microscope, can subsequently not be found again in the light microscope, only the mean values of the area of squares for one grid, or for a whole experimental series, can be determined. This disadvantage is avoided by the second method, in which each

square area unit can be measured directly in the electron microscope by means

of a special driving mechanism for the specimen stage (Hilde Haag, Heppenheim, Germany) . The digital counters of this driving mechanism indicate the actual position of the specimen stage. A calibration of the movement of the stage was performed in the following way (fig. 1):

The distance d^, when the stage is moved from one digit of the counter

to the next one, was measured by using a carbon grating replica (28 800

lines per inch, Fullam, Connecticut, U.S.A.) with the distance between

This work was supported by Deutsche Forschungsgemeinschaf t

,

Sonderforschungsbereich 105, Dr. K. Aterman gave valuable help

in preparing the English text.

411
u. Pieiier

1

>i

f —

1

n

i

y

n.

f

Fig. 1 Showing the geometrical basis for determining

the distance d^ by which the specimen stage is moved,

when going from one digit

of the counter to the next

one (see text). The thin

vertical lines represent

the lines of the carbon

grating replica with the distance i.

a

Fig. 2 Showing the squares of

the copper grid situated within a system of rectangular ordinates. These ordinates are represented by the two directions of movement of the spe- cimen stage. For each of the four angle points of the square outlined by thick

lines a pair of values on these ordinates can be determined.

two lines. The stage was moved in one direction by x digits and the number

(n^j) of replica interspaces passing in center of the screen during this movement was recorded. The desk was then moved in the other direction by y_ digits and again the number (fly) of replica interspaces was recorded. By

a geometrical calculation derived from fig. 1 it can be shown that

In the replica used, i_ was 0.883 fim. The distance d^ was found to be 2.975 pm;

it remained constant, whether the central or the more marginal areas of the specimen were examined.

The two directions in which the specimen stage is moved can be viewed as rectangular ordinates. For each of the four angles of the square area unit the values on the ordinates for the point of the angle were determined

(fig. 2) by bringing the four points successively into the center of the screen; the two values on the ordinates were then read off the counters. The area of the square was then calculated from these four paired values by means

of a computer program for rectilinear surface areas (Monroe 1766, punched

card program 1001 ENQS)

.

Since the total area evaluated is not always the appropriate parameter of reference, in most instances fractional areas have to be determined. Hence

412
Quantification of rarely occurring structures

Fig. 3 Showing the fluorescent screen of the electron

nicroscope equipped with an imprinted point lattice for

the determination of fractional areas. In addition

there is present a crossed mm scale for the determination of two diameters of a sectioned profile.

Fig. 4 Showing the distortion of the two diameters of a profile

because of the oblique position

of the screen. The diameters a' and b' read off the crossed

scale of the screen (see Fig. 3) are divided by the factors f^ and

f^ in order to obtain the true diameters a and b.

conventional morphometry with micrographs taken at random was used in our

studies as an indispensable additional method. A rough estimate of certain

parameters, however, can also be performed directly with the electron microscope. The fractional areas of parenchymal and non-parenchymal components of

a tissue, for ex^nrple, were determined by the point counting method, by pro- jecting the square area unit as a whole at low magnification onto the screen

which, as shown in fig. 3, was equipped with an imprinted point lattice. In

a study of the exocrine pancreas of the rat specimens of 24 animals were evaluated in this way, six square area units from each specimen. Without taking

any micrograph, the fractional area of parenchymal cells could be determined

with sufficient exactness; the mean was found to be 0.705, the standard de- viation 0.043, and the standard error of the mean 0.009.

After these determinations the scantily represented structures of in- terest could be searched for. Each area unit of the specimen was now viewed

at an appropriate primary magnification, which was calibrated by means of

the carbon grating replica mentioned earlier on. A stereoscopic lens, magnifying 9 times, was used in addition. The electron beam was maximally focussed

by the double condenser in such a way that in the main only that area of the screen was illuminated which was covered by the field of vision of the stereoscopic lens. Under these conditions a well illuminated image is obtained,

even if the current of the beam is kept very low. This is important both for

careful treatment of the specimen and of the cathode during the long periods

needed for the examination. When structures of interest were encountered

during the systematic scanning of the specimen by a meandering movement, the number, the type, and the size of these structures had to be registered. This

can be done by taking a micrograph from each structure at a magnification

413
U. Pfeifer

suitable for determining later on the area of the sectioned profile by means

of. the point counting method. Since, however, the number of even these se- lected micrographs may be considerable, it was thought advantageous to note

down the required parameters directly from the image on the screen. For measurements of the area of sectioned profiles having an approximately ovoid or round shape, two diameters were taken by means of a crossed mm scale printed

onto the screen as shown in fig. 3. Due to the oblique position of the screen during the measurement under the stereoscopic lens (fig. 4) the two values

a_' and b^' obtained had to be corrected by factors f^ and f^ in order to de- termine the true diameter a and b^. By measuring a set of profiles first in

the horizontal, and then in the oblique, position of the screen f^ was found empirically to have a value of 0.906, whereas the value of f^ was 1.357. The

area A of a profile of interest can now be calculated from the following formula:

il . ^1 . X =a'-b'. ^ =a'-b'-0.639

fa h ^ ^-fa-fb

In practice it was, therefore, not necessary to register every time the two individual diameters, but only the product of these two diameters. In order

to get the area of the profile in mm^ at the chosen magnification, this product had to be multiplied by the factor of 0.639.

In a study of the diurnal rhythm of cellular autophagy in the proximal

tubules of the rat's kidney (3) it has been shown that it is possible, with

the help of the method outlined here, to examine in one experiment specimen

areas amounting to 100 . 10^ /im2 and more, in order to search for scantily

occurring autophagic vacuoles. Their fractional volume, for example, ex- pressed as the mean value of the diurnal cycle was found to be 2.1 . 10 ^.

As always, when quantitative data in morphology are required the evaluation remains laborious. By avoiding taking micrographs, it is possible, however, with the method described here to venture into fields morphometry which

so far have been hardly accessible.

References

1) Weibel, E.R.: Stereological principles for morphometry in electron microscopic cytology. Int. Rev. Cytol. £6, 235-302 (1969)

2) Pfeifer, U. : Cellular autophagy and cell atrophy in the rat liver during

long-term starvation. Virchows Arch. Abt. B Zellpath. !£, 195-211 (1973)

3) Pfeifer, U. , Scheller, H. : A morphometric study of cellular autophagy, in- cluding diurnal variations, in kidney tubules of normal rats. J. Cell Biol.

64, 608-621 (1975)

414
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

THE MORPHOMETRY AND STEREOLOGY OF CEREBRAL ARTERIAL BIFURCATIONS

by Peter B. Canham, James G. Walmsley and J. F. Harold Smith Department of Biophysics, University of Western Ontario, London, Ontario, Canada

SUMMARY

Segments of human middle cerebral arteries were fixed under pressure in

10'^ formalin. Serial sections were prepared and stained either with hematoxylin

and eosin in order to study the arrangement of smooth muscle in the tunica media

or with silver inpregnation in order to study the arrangement of collagen in the tunica adventitia. Composite photomicrographs of whole arterial segments

were prepared. The darkly stained muscle nuclei were used as vector indicators

of cellular orientation. The nuclei were digitized. Trigonometric transformation, modelling and projection techniques were used. In contrast to the simple cell pattern in straight portions, the cells in the bifurcation region

were mul ti -di rectional and arranged in bundles. Collagen was found to be arranged circumferential ly (for the single specimen studied) using the Leitz

polarizing microscope and universal stage.

INTRODUCTION Saccular aneurysms, which are balloon-like enlargements of the vascular

wall, are found in the major cerebral arteries associated with the circle of Willis at the base of the brain. The fact that aneurysms occur nearly always

at the apex of the bifurcation (Hassler, 1961) has stimulated us to study the architecture of the blood vessel wall. In the region of the apex (Fig. 1) where the curvature in the plane of the bifurcation does not help support

transmural pressures, there seems to be no increased wall thickness (Stehbens,

1974; Macfarlane, 1975). However, the composition of the wall is different at

the apex with a reduced amount of medial smooth muscle and an increased amount

of adventitial collagen compared to the non-branching regions. We undertook

to see if the organization of the structural elements might also show differences in the region of the apex. Vasospasm, characterized by an active or

passive decrease in arterial caliber, causes reduction in regional cerebral

blood flow. Our working hypothesis is that the organization of fibers of both

smooth muscle and collagen are important in relation to saccular aneurysm

formation, vasospasm and atherosclerosis.

Two complementary studies are in progress, the one focussing on the organization of the smooth muscle, and the second, the organization of collagen.

Smooth muscle cells are long tapered cylinders, each with one cylindrical

darkly staining nucleus (for sections stained with hematoxylin and eosin).

Collagen is the main acellular component of the adventitia of cerebral arteries.

SMOOTH MUSCLE

We used the technique of formalin fixation of isolated vessel segments

under a maintained transmural pressure of 100 ± 10 mm Hg, paraffin embedding,

serial sectioning and staining with hematoxylin and eosin. We have concentrated on middle cerebral arteries, a common site for aneurysm formation in the anterior circulation. Vessels were obtained at autopsy. Certain specimens

were sectioned longitudinally and others transversely. Because bifurcations

are frequently not planar, a system of three cylindrical nerve fibers were

embedded into the paraffin block (method of Burston & Thurley, 1957) and sectioned along with the vessel material, the nerve fibers serving as a reference coordinate system. This method involves infusion of the nerves with

a higher melting point wax to allow implantation at right angles to the plane

of sectioning.

Our findings from longitudinal sections cut in the bifurcation region, but

away from the midplane, were suggestive of smooth muscle cell groups. A test

of the continuity of the cell groups, or bundles, was made using the technique

of counting the number of nuclear intersections with a line grid superimposed

415
p. B. Canham, J. G. Wa/ms/ey and J. F. Harold Smith

TrunkApex—

Branch

Nerve

Fig. 1. A bifurcation in sectioning Fig. 2. Section from branching region

block with reference nerves. with bundle outlined (#4 of Table 1)

on a composite photomicrograph (described as the method of roses. Underwood,

1970). A digitized and replotted sample composite is shown in Figure 2. Table 1 summarizes the results for five cell bundles which were identifiable

through SOpm of sectioning (each slide had three sections sliced at 6.9pm).

The parameter 6, is the mean angle (in the section plane) of the nuclei, and

9. is a measure of the extent of orientation, being zero for complete randomness. Similar analysis repeated on other serial sections showed similar

results and we concluded that the medial smooth muscle above and below the apex of middle cerebral arteries was arranged in groups, each group being

identifiable for several adjacent sections. The total effect was that of a coarsely woven cellular fabric. However the boundary between the bundles

was not always distinct, and the lack of quantitative data for the third

dimension led us to examine histologically the straight portion of the vessel segments for the purpose of identifying the basis of the pattern of muscle

bundles.

For the study of the straight portions, composite photomicrographs were

constructed using medium power optics. (The objective of the Nikon microscope

used was a lOx which had a suitable depth of focus to include the total 7.3pm

thickness of the section.) Subsequent detailed identification was done on each nucleus to determine if it traversed the section and which end was down, this

being done under oil immersion. The digitized result of a sample composite

is shown in Figure 3 with only those nuclei shown which completely traversed

the section depth. The plot in Fig. 3 is a corrected true cross section,

corrected for the obliqueness of cutting (17.5°), cutting strain in the X-Y plane, cutting strain in the Z plane which results in thickening of the section,

and rotation. The knobs are at the bottom ends. Attention is drawn to the approximate uniformity of the nuclei in any region, to the nuclei being more

angled into the section at the sides (shorter projection) and being more within the section at the top and bottom of the figure where they reverse

direction. In addition to analyzing actual histological sections we generated model oblique sections of cylindrical vessels and nuclei of uniform

helical pitch. We measured the dimensions of the nuclei for this study and

found their actual length to be 37 pm ± 6.4 (S.D.) and diameter 2.0 pm ± .83 (S.D.), a sample of 50 nuclei. We concluded that for the straight portions

of middle cerebral arteries (5 specimens from 2 autopsies) the smooth muscle

is arranged in a circular pattern in the tunica media, and that the pitch of

the helix is not different from 0°. Our finding, exampled in Figure 3, that

the projection of the nuclei in the section plane is not uniform around the circumference is interpreted as an error in correcting for the obliqueness of

the section. In straight arterial segments the media was found to be 75%

smooth muscle.

416
Cerebral arterial bifurcations

',' ''

short

nuclei

reversal ~ v,->-- region

+

2 mm

V.J.,

short

nuclei'''

'

1 1,1

reversal

'jj reg>Dn_'v'

18cf

1 8 O 200 Tangent Angle

Cf »ji

Fig. 3. Nuclear projections onto transformed cross-sectional olane.

Fig. 4. The planar and non planar

components of collagen orientation.

COLLAGEN

The orientation of collagen in the adventitia of straight vessel segments

has been done using a Leitz polarizing microscope with a four axis rotating

universal stage. After sectioning as above, the established birefringence of collagen (Schmitt, 1934) was intensified using silver impregnation stain

(Mello & de Campos-Vidal , 1972) and Permount was used as an immersion medium.

The universal stage permits one to rotate the section in order to make

a parallel fiber region aligned with, or perpendicular to the axis of the microscope. The results are obtained and recorded as two angles: a planar

section angle and an elevation angle. Figure 4 shows data obtained from one section cut at 17° from a true cross section. Collagen bundles measured are

of the order of 3 micrometers in any linear dimension. Attention is drawn

to the correlation between the projected angle of the collagen bundle within

the plane and the angle of the tangent to the vessel wall at that point.

In contrast, the elevation angle (angle of the bundle out of the section plane)

shows no correlation with tangent angle. The data in Fig. 4 is consistent with

a circumferentially arranged outer collagen sheath around the cylindrical

vessel

.

Our plan is to complete the 3 dimensional orientation measurements on straight vessel sections, which to our knowledge is an application of the polarizing microscope and universal stage not previously reported. The method

was learned from Professor John Starkey, a colleague in Physical Geology at

the University. Armed with the data on collagen and muscle in the straight segments we

are returning to investigate quantitatively the arrangement in the region of

vessel junctions.

ACKNOWLEDGMENT

This work was supported by the Ontario Heart Foundation.

417
p. B. Canham, J. G. Wa/ms/ey and J. F. Harold Smith

Slide 72 Slide; 74 Slide 76

NUMBER e e e

1 80 53% 76 45% 83 1*57,

2 190 64% 176 80% 163 62%

3 20 69% 21 59% 33 51%

4 120 50% 133 56%

5 170 39% 141 92%

Table 1. e and n for bundled regions in front of apex (cf. Fig. 2).

REFERENCES

Burston, W.R. & Thurley, K. (1957) A technique for the orientation of

serial histological sections. J. Anat. 91_, 409. Hassler, 0. (1961) Morphological studies on the large cerebral arteries

with reference to the aetiology of subarachnoid haemorrhage. Acta Psych. Neurol. Scand. 36, Supp. 154, 1-140. Macfarlane, T.W.R. (1975) The Geometry of Cerebral Arterial Bifurcations

and its Modification Dith Static Distending Pressure. M.Sc. Thesis, University of Western Ontario.

Mello, M.L.S. & de Campos-Vidal , B. (1972) Evaluation of dichroism and anomalous dispersion of the birefringence on collagen subjected

to metal impregnations. Aym. Histochem. J_7> 333. Schmitt, W.J. (1934) Polarisation optische Analyse des Submi kroskopischen

Baues von Zellen und Geweben. In: Ahderhalden's Handbuch der

Biologischen Arberts methoden (Ed. by Schmitt, W.J.),

pp. 435-665. Springer, Berlin.

Stehbens, W.E. (1974) Changes in the cross-sectional area of the arterial

fork. Angiology 25, 561. Turner, F.J. & Weiss, L.E. TT963) Structural Analysis of Metamorphic

Tectonites, pp. 197-199. McGraw-Hill, N.Y. Underwood, E.E. (1970) Quantitative Stereology, Chapt. 3. Addison-Wesley,

Reading, Mass.

418
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg. Md.. September 4-9, 1975 (Issued January 1976)

KARYO-INTERKARYOMETRY AS A PERIODIC CX)NTROL FOR CLINICAL PROGNOSIS

by E. C. Craciun and C. Tasca,

Institute of Endocrinology, Bucharest, Romania

Karyo-interkaxyometry gradually developed as a quantitative branch of research in medicine and biology. Its aim is to give an exact description of cell nucleus in terms of quantitative and, therefore, well comparable data. Tnese facts are correlated with the functional cellular aspects to allow a dynamic interpretation of cell physiology and physiopathology . Changes in cell function affect the cellular metabolism and consequently the volume of nucleus, nucleolus and cytoplasm. Nuclear size modifications gave rise to the principle of "clear intumescence" as described by Craciun, "nuclear functional

oedema" (Benninghoff) and "working hypertrophy" (Eichner) , each one of these

cellular appearances with its definite nosological trends.

MATERIAL AND MEIHOD

One case is presented of benign hepatitis epidemica patient A.N. , male

32 years, repeatedly studied by liver needle biopsy at 8, 50, and 180 days from the apparition of jaundice, on formalin fixed specimens.

Nuclear volumes, considered as rotation elipsoids were calculated by means

of a manual method. The relation between statistical distribution of nuclear

volumes and the distribution of nuclear surfaces was estimated. By calculating

the error of nuclear volume starting from nuclear surface, the probability is that only in 50% we can determine the "true" nuclear volume. For karyometry

the volumes of one thousand nuclei were calculated according to the formula of Puff : nuclear volume: 8s2 /SttA (S = surface of the section, A = major diameter). Secondly we measured the internuclear distance (interkaryometry) as an indirect proof to quantify the nucleo-plasmic ratio. The major reason is that

such a parameter allows a quantitative expression of the intercellular and tissular milieu. Moreover it brings information on spatial distribution of nuclei. The logarithmic system was applied for calculating and grouping the observed values, which were then submitted to a statistical study.

RESULTS

The first needle biopsy, eight days after the beginning of jaundice, showed

striking pictures of cytolysis with trabecular demolition. Hepatocytic chords

can usually be identified by the neighboring blood capilaries. Hepatocytes are most irregular as to size of nuclei, their chromatin content, their reciprocal

intervals and number of nuclei (Fig. 1). Many blood vessels are dilated without hemorrhage, as provided by the continuous reticulin net on Gomory silver

impregnation and the absence of macrophages and siderophages . No necrotic

elements and therefore no perinecrotic leucocytes. Histiocytic and lymphocytic polyb lasts are present, but no polymorphonucleares

.

Needle biopsy II (50 days) showed the same, but more accentuated features,

especially from the point of view of cytolysis and trabecular disorganization.

Needle biopsy III (180 days) presented a rather normal looking hepatocytes in

as much nuclei and cytoplasma do not evoke any more the previous stages; some

inf lanmatory changes still persist (Fig. 2) . Clinical recovery followed, while

hepatic tests and biochemical results were not quite normal six months after

the beginning of icterxis.

It is striking to compare lesions, figures, and graphics with clinical

evolution.

419
f. C. Craclun and C. Tasca

From the point of view of the mathematical analysis it can be stated that

the karyo-interkaryometric data correspond to a Gauss-Laplace curve as observed

in biological processes. Bimodal and especially plurimodal curves point to a nonhomogenous cell population as observed in our histological slides at 8 and

50 days (Fig. 3). For the 180 days the graphic still showed a quite miscellaneous hepatocytic population as nuclear volumes are concerned (Fig. 4). Most frequent are the nuclear volumes of 132-250 and 70-95 at 8 and 50 days,

while at 180 days the most frequent are those of 90-145 y^. Accordingly the mean volume is reduced from 240 y^ at 8 days to 182 y^ at 50 days and finally

to 110 y^ at 180 days.

Intemuclear distances reach a mean top value at 50 days of 10.6 y, as com- pared to 8 y at 8 days and 9 y at 180 days. A certain role is played by

anisocytosis and by incomplete recovery of distrophic processes.

The graphical appearance of different stages of hepatitis epidemica cor- responds to the histological picture. In the first needle biopsy a "feverish" sequence of high arrows with "saw teeth" appearance could be seen. Inter- karyometry revealed a predominance of large intervals corresponding to cytolysis and trabecular demolition. In the second biopsy the irregular aspect

still persists, but after 180 days the karyogram is almost completely normalized. A tendency exists to a reduction of the nuclear volumes and intemuclear intervals with the elapsed time. Excepting for the 50 days stage the distribution

of nuclear volumes is asymetric showing a dextrodeviation with the corresponding negative coefficient of asymmetry approaching null point. For the internuclear distances there is an asymmetry meaning a levodeviation . This reverse

type of correlation between frequencies of nuclear volumes and intervals seems due to the favourable evolution of illness. This situation corresponds to an increased density of hepatocytic population in good biological condition. It means a positive situation concerning the chances of histological recovery.

DISCUSSION

Karyometry if looking for extrapolation is limited referring to the whole

organ, given the relatively small number of measured cells. Such a situation

cannot annihilate neither karyometry nor its current use. From our data it results that nuclear dynamic parallels the anatomo-clinical picture. This

synergism appeared in all stages of hepatitis we studied (Craciun; Craciun and Tasca) . Quantitative variations of nuclear volumes could be therefore expressed

as a morphopathological nuclear syndrome corresponding to a quantitative

nuclear formula, typical for a certain stage of disease. From this point of view one could speak of a quantitative cytoprognosis (Tasca et al., Tasca). In the same way more than 1500 cases with various hepatic, endocrine, nervous, lymphoganglionary and neoplastic diseases were studied. A parallelism was always found between karyo-interkaryograms and clinical evolution. Karyometric

methods could be used like leukograms or thermic curves as a periodic control

for clinical prognosis. Moreover these methods need only routine laboratory

equipment easily appliable to each hospital.

REFERENCES

BENNINGHOFF, A.: Funtionnelle Kernschwellung und Kernschrumpfung . Anat. Hachr., 1949, 1, 50-53. CRACIUN, E. C: Specif icitatea leziunilor progressive si regresive din

hepatita epidemica icterigena. An. Inst. V. Babes, 1943, 16 , 178. CRACIUN, E. C, TASCA, C: La caryometrie et 1 ' intercaryometr ie dans I'hepatite

epidemique evolutive. Rev. Intern. Hepatol., 1968, 18, 8, 967-986. EICHNER, D.: Uber funtionelle Kernschwellung in den Nuclei supraoptici und paraventriculares des Hundes bei exper imentellen Durstzustanden . Z. Zillforsch., 1952, 37, 406-414.

\

420
Karyo-interkaryometry in clinical prognosis

TASCA, C, ENACHESCU, D. , CEACIUN, E. C. : A mathematical study of associated

karyo-interkaryo grams in Hepatitis Epidemica. Beitr. Path. Anat., 1968, 137 , 316-329.

IASCA, C: Cariocitometria ca detector cantitativ in diferite afectiuni. MD

Tnesis Bucuresti, 1967.

Fig. 1. Needle biopsy I (8 days). Distrophic cytoplasmic lesions and anisokarya. HE stain, x 200.

Fig. 2. Needle biopsy III (180 days). Some inflammatory changes still persist

but the distrophic lesions are no more visible HE stain, x 100.

421
f. C. Craciun and C. Tasca

Fig. 3. Karyo-interkaryogram at 8 days shows a "feverish" sequence with high

arrows

.

Fig. 4. Karyo-interkaryogram at 180 days shows a normal looking aspect with -a single major peak.

422
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

f}eld at NBS, Garthersburg, Md., September 4-9, 1975 (Issued January 1976)

NUCLEIC ACIDS CONTENT-NUCLEAR VOLUME CORRELATION AS A QUANTITATIVE PARAMETER IN CELLULAR MALIGNANCY.

by C. Tasca and E. C. Craciun

Institute of Endocrinology, Bucharest, Romania

The histopathological diagnosis is still based on personal experience, the possibility to express it in quantitative terms being very limited. Cytophotometrical methods developed by Caspersson (1936) and later by Sandritter, in order

to quantify by direct measurements the nucleic acids, brought interesting data

on the normal and pathological nucleo-cy toplasmic metabolism. Unfortunately the cytophotometry alone was not able to confirm the malignant growth. As known,

there are malignant tumors with either hyperchromatous or hypochromatous nuclei,

i.e., with high or small quantities of nucleic acids. On the other side, the increase of nuclear volume is not an exclusive characteristic of neoplastic tissues because it may be found in many other etiological conditions. The correlation between nucleic acids content and nuclear volumes seemed to us to better

quantify the "stages of malignancy."

MATERIAL AND METHODS

We selected two opposite kinds of tumors with respect to DNA content: skin

hyperchromatous tumors induced in mice and liver adenoma in rats as hypochromatoiis tumors. The skin carcinoma were obtained in NMRI mice using 3,4 benzpyren;

the hepatoma were induced in Sprague-Dawley rats fed for 8 months with 3 mg/kg

weight di-ethyl-nitrosamine. As a model of not tumorous tissular growth we used

the skin reversible hyperplasia obtained in mice by means of the co-carcinogen

Phorbol-ester Al (0.02ij Mol in 0.1 ml acetone). DNA was stained by the Feulgen method. We also used the gallocyanin-chromalum staining method after treatment with 1% ribonuclease.

The cytophotometric measurements were accomplished by means of the Univeralmicrospectrophotometer (UMSP I) of the Firma Carl Zeiss Oberkochen with scan- ning table and rapid measurement device. The mean extinction and the projected

surface of the nucleus were concomitantly measured. The Feulgen stained nuclei were measured in monochromatous light by 560 mp; the gallocyanin-chromalum

stained nuclei were measured by 500 my. By means of the computer RPC 4000 the nuclear volume and the content of DNA were calculated. To calculate the volume

we used the formula: V = 0.75/ (S = surface). The content (M) of DNA was calculated by the formula: M = ES/e' (S = nuclear surface, E = mean extinction,

e' = specifical decadic extinction coefficient in cm^g"-*). The e' was calculated after the formula of Sandritter.

The correlation between the nuclear volume and the DNA content was expressed as a stochastic equation. The most probable dependence between DNA content

and nuclear volume is represented as: DNA content = average DNA concentration x nuclear volume + const. In this equation the DNA concentration (gcm~^) represents the direct equivalent of correlation.

RESULTS Correlation DNA-Nuolear Volvme in Hypoohromatous Heptaoellular Tumors

The nuclei of hepatoma were irregular with fine chromatin granules adherent

to the nuclear membrane but with a clear center, i.e., with a low DNA concentration (Fig. 1) . The results of our cytophotometric measurements are summarized

in Fig. 2. The territories where most of the pair values (nuclear volumes and DNA content) were found, are represented as colijmns. The slopes of columns cor- respond to the regression equations. As compared to normal hepatic nuclei, the not yet tumorous nuclei around the neoplastic zones, showed a stronger correlation between DNA and nuclear volumes. The reason is that more nuclei contain a higher DNA content on one side and that the DNA synthesis is greater on the other

side because an increased liver regeneration. The regression lines of the neo- plastic liver nuclei showed a diminution of DNA concentration by an excessive

423
C. Tasca and E. C. Craciun

dyscorrelated nuclear volume increase. The correlation DNA-nuclear volume is therefore decrease in hepatoma.

Correlation DNA-Nualear Volume in Hyperahromatous Skin Tumors

The results of cy tophotometric measurements in hyperchromatic tumors induced

by 3,4 benzpyren are summarized in Fig. 3. The surfaces where DNA concentrations

and nuclear volumes were found, are represented as ellipses. The dotted lines

correspond to the regression equations. By comparing the values of normal not treated mouse skin with those of the benzpyren treated skin but without morphological modifications, it results that DNA concentrations and nuclear volumes are similar. The regression lines showed however a smaller angle of the slope in to benzpyren exposed skin. The same diminished correlation DNA-nuclear volume

appeared in all skin neoplasia.

Correlation DNA-Nuolear Volume in the Skin Beversible Hyperplasia after the Co-Carcinogen Phorbol-ester Al

In the reversible skin hyperplasia after Phorbol-ester Al treatment the regression line parallels the regression line of the normal skin (Fig. 4). On

the contrary, the regression line of the with benzpyren-treated skin without

morphological modifications showed a diminution of DNA-nuclear volume correlation.

DISCUSSION From our data it results that the concentration of DNA is a sensible indi- cator for neoplastic mechanisms. In previous work (Tasca et al.) it was shown

that the linear regression analysis disclosed for the nuclei of the experimentally induced papillomata a diminished correlation between nuclear volume and content of nucleic acids and histones. The diminution of the correlation could

therefore be the first quantitative phenomenon in the course of cancerogenesis

before other morphological modifications in routine histological methods are visible. Our results referred only to post-mitotic cell-collectives. The general value of our findings must be also further investigated.

REFERENCES

Caspersson, T. : Uber den chemischen Aufbau der Strukturen des Zellkerns. Scand. Arch . Physiol .73, Suppl . 8 , 19 36

.

Sandritter, W. : Die Nachweismethoden der Naukleinsauren. Z.Wiss .Mikr .Technik.

^,283-304,1954/1955.

Sandritter, W.: Ultravioletmikrospectrophotometrie. In: Graumann.W; Neumann,

K. H.: Hdb. der Histochemie, pp. 220-338, Stuttgart, Fischer , 1958. Tasca, C, Haag, D. , Goerttler, Kl. : Cy tophotometrische Befunde in der Trachealschlelmhaut und in DANA-behandelten Trachealpapillomen beim syrischen

Goldhamster. Z. Krebsforsch. ,^4, 335-367,1970.

LIST OF FIGURES

Fig. 1: Hepatoma with large and hypochromatous nuclei. HE stain. x500.

Fig. 2: Diagram of cy tophotometric measurements in DENA-hepatoma Increase of nuclear volumes and diminution of DNA-concentration.

Fig. 3: DNA-concentration as a function of nuclear volumes. Diminution of DNA-nuclear volume correlation of the skin without morphological modifications and of all stages of malignancy.

Fig. 4: Diminution of DNA-nuclear volume correlation in the mouse skin exposed

to benzpyren as a sign of incipient cancerogenesis.

424
On a correlation for a quantitative parameter in cellular malignancy

Fig. 1: Hepatoma with large and hypochromatous nuclei. HE stain. x500.

ONA -Content

50

30

20

10

200 BOO iooo mo

yolume(fj'^)

Fig. 2: Diagram of cytophotometric measurements in DENA-hepatoma Increase of nuclear volumes and diminution of DNA-concentration.

425
C. Tasca and E. C. Craciun

Fig. 3: DNA-concentration as a function of nuclear volumes. Diminution of DNA-nuclear volume correlation of the skin without morphological

modifications and of all stages of malignancy.

DM-Confent

- Epidermis

(36h)

'Benzpyren - Epidermis

(morph 0)

m 200 Volume (p3)

Fig. 4: Diminution of DNA-nuclear volume correlation in the mouse skin exposed

to benzpyren as a sign of incipient cancerogenesis

.

426
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

IS THERE AN UNRECOGNISED SYSTEMATIC ERROR IN THE ESTIMATION OF SURFACE DENSITY (SD) OF

BIOMEMBRANES?

by Albrecht Reith and Tudor Barnard Norsk Hydro's Institute for Cancer Research, The Norwegian Radium Hospital, Oslo, Norway, and Wenner Gren Institute, Stockholm. Sweden

A systematic error in the estimation of biomembrane SD is caused by the

invisibility of membrane profiles oriented obliquely to the axis of the

electron beam. This has been investigated quantitatively by Loud and a correction factor of 1.5 has been proposed for cristae membranes. SD

estimates made by different groups on a model biomembrane system (membranes of liver mitochondria) vary by a factor of 2 from 23.5 to more

than 53 m2/cm3 mitochondria. The theoretical SD of mitochondrial mem- branes (6 nm thick), tightly packed but including an intermembrane space

(6 nm) between adjacent pairs of membrane sheets was calculated to be 117

m2/cm3. This means that liver mitochondria with a total membrane SD of 53 m2/cm3 would have a "membrane volume fraction" of nearly 0.5 cm^, leaving

only about 0.5 cm^ for the matrix compartment. This would fit for types

of mitochondria richly endowed with cristae such as heart and brown fat

but not for liver. In this context the lower values are more reasonable.

The fact that even SD estimates that are uncorrected for loss of oblique

membrane profiles seem to be too high indicates that an as yet unrecognised systematic error seems to be affecting SD estimates, and therefore

a more rigorous standardisation of the criteria used for defining membrane profile intersections would offer a first step towards decreasing

the effects of this error.

It is agreed that there Is a systematic error in the estimation of biomembrane SD by counting intercepts between membrane profiles and a test

line. This is caused by the invisibility of membrane profiles oriented

obliquely to the axis of the electron beam.

Loud, using the nuclear membrane of adult rat liver, estimated the critical angle of"50^ visibility" to be about 60°. On this basis, correction

factors of 1.5 and 1.25 have been applied to estimates of cristae and

endoplasmic reticulum by several groups (2,3,4,6). Whether in fact estimates of biomembrane SD are routinely underestimated may be questionable,

as we shall try to show by the following report:

The table shows the result of SD estimates obtained by several groups over 0 model system, mitochondria in adult rat livers.

The cristae values are corrected by 1 .5 (uncorrected values are given in parenthesis): the outer membrane (and consequently the inner boundary

membrane) values were not corrected, as the presence of these membranes

can be assumed on the basis of other criteria, so that these intersection

counts should be less sensitive to underestimation (3) . The table shows

that the estimates of total membrane SD falls into two groups, one of about 53 m2 per cm3, the other of 20-30 m2 per cm^.

In an attempt to decide which of the results was the most "reasonable",

the theoretical SD of mitochondrial membranes (6 nm thickness) that could

be accommodated in 1 cm^ was calculated under two conditions: a) maximally tightly packed and b) tightly packed but with an intermembrane space

(6 nm) , intercalated between adjacent membrane pairs. The results were respectively 167 and 111 m^ per cm"^. Seen in this light, a total membrane SD of about 53 m^ per 1 cm^ would

thus occupy nearly 0.5 of the mitochondrial volume, leaving only 0.5 for

the matrix compartment. This might be expected for types of mitochondria

richly endowed with cristae such as heart and brown fat, but not for liver

(Fig.). Even if the uncorrected cristae SD estimates are used, the results still seem very high. In Reith' s study, only intersections with strictly cross-sectioned cris427
A. Reith and T. Barnard

tae profiles were recorded, and these corrected by 1.5. Under these conditions, the total membrane SD was 23.5 m^ per cm^, which gives a volume

fraction of membranes and the space between adjacent pairs of membranes

( intracristae and intermembrgne space) of 0.14 or 0.07 cm^/cm^ respectively. This leaves 0.79 cm^/cm^ for the matrix volume fraction which is

a more reasonable result for liver mitochondria. For comparison, estimates

of total mitochondrial SD of heart and brown adipose tissue mitochondria

gave values from 42 to 61 m^/cm^ (4,5,2). These figures appear to have correct relationships with the dense packing of cristae in such mitochondria compared to the relatively sparse cristae packing of liver mitochondria (see also Fig.). A somewhat different approach, based on reconstructed particle size distribution, was used by the Louvoin group(l), and their value of 32 m^ per cm3 is closer to 23 than to 53 m2 per cm3.

CONCLUSION: SD estimates made by different groups on a model biomembrane

system vary by a factor of 2. The lower values obtained correspond better

than the higher values with the matrix volume fraction as estimated by

point-counting. The fact that even SD estimates that are uncorrected for

loss of oblique membrane profiles can be unreasonable indicates that an as yet unrecognised systematic error seems to be affecting SD estimates.

The error apparently counteracts the tendency to loss of membrane profile

intersections and varies between laboratories. A more rigorous standardisation of the criteria used for defining membrane profile intersections

offers a first step towards decreasing the effects of this error.

fronn the rat:

SURFACE DENSITIES OF MEMBRANES IN 1 CM'' CUBE OF LIVER MITXHOrCRIA 2/ 3 - m /cm

(in parenthesis values uncorrected For oblique sectioning).

I i ver

weibel'*' reith'^' lowain'' '

Outer membrone: 8.96 8.07 7.0 7.3

Cristae membrane: 36.44(24.29) 36.63(24.42) 9.5 17.3

Totol inner membrane: 45.40(33.25) 44.70(32.49) 16.5 24.7 heart

^of oil membranes 54.35(42.20) 52.77(40.56) 23.5 32.0

1. Baudhuin,P. « Berthet,J .(1967) . Electron mi- croscopic examination of subcellular fractions.

J .Cell Biol. 35, 631 . 2. Lindgren,C. & Barnard, T. (1972) . Changes in interscapular brown adipose tissue of rat during perinatal and early postnatal development

and after cold acclimation. Exptl.Cell Res. 70, 81. 3. Loud, A. V. (1968) . A quantitative stereological description of the ultrostructure of normal

rot liver parenchymal cells. J. Cell Biol. 37, 27.

4. Page,E. (1973). Morphometry of mitochondrial

membranes and profile size distribution in rot

heart. J. Cell Biol. 59, 256a.

5. Reith,A.(1973) . The influence of triiodothyronine and riboflavin deficiency on the rot liver with special reference to mitochondria. Lab. Invest. 29, 216. Reith, A. & Fuchs,S . ( 1 973) . The heart muscle

of the rat liver under influence of triiodothyronine and riboflavin deficiency with special reference to mitochondria. Lab. Invest. 29,229.

6. Weibel E.R., Staubli,W., Gnogi,H.R. & Hess, F.A.(1969). Correlated morphometric and biochemical studies of the liver cell. J. Cell Biol. 42, 68.

428
National Bureau of Standards Special Publication 437

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

QUANTITATIVE STUDIES WITH THE OPTICAL AND ELECTRONIC MICROSCOPE OF THE PATHOLOGICAL CELLS OF THE ACUTE LYMPHOBLASTIC LEUKEMIA AND CHRONIC LYMPHOCYTIC LEUKEMIA

by J. L. Binet, P. Debre, P. D'Athis, D. Dighiero, and F. de Montaut. Department cfHematologie, U.E.R. Pitie Solpttriere, 91 boulevard de I'Hopital, 75013 Paris, France

ABSTRACT

The stereological method gives new results in acute lymphoblastic leukemia

and in chronic lymphocytic leukemia. In A.L.L. there are two cell populations;

in C.L.L. there is only one volumetric population.

After 2.5% glutaraldehyde fixation and further fixation in 2% osmium

tetroxide solution in phosphate buffer, the abnormal blood cells were dehydrated

and embedded in epon. Then they underwent a double treatment:

First, they were cut in semi-fine (0.5 y) sections, stained with Toluidine

blue, and the diameter of 1500 cells was measured on the Leitz Classimat. The histogram constructed from these diameters was submitted to mathematical analysis in order to derive the volume according to the d'Athis method;

Second, the same block was cut in fine sections and sixty pictures were

taken at random by the electron microscope. The volume density of the cytoplasm,

nucleus, and the two types of chromatin was also determined. Ten patients with acute lymphoblastic leukemia (ALL) were studied and the volumetric distribution showed a double population . The first population included

the majority of cells (69 to 94%) and its mean volume (Vj) varied from 170 to 340 P^. The mean volume was variable from one case to another. In 8 out of the

10 cases cytologists agreed with the sub-classification of the type of ALL. In

two cases there was disagreement. For 7 of the 8 patients, a good correlation

was found between morphological classification and volumetric determinations.

The mean volume of the second population (V2) was much higher (from 330

to 789 U^) and the percentage of cells also varied considerably from one patient

to another (6 to 31%). The mean of the ratios V2/VJ was 2.1, and due to the precision of the measurement, this ratio is not significantly different from 2. Autoradiographic studies of the same cells after incubation with tritriated

thymidine demonstrated a correlation between cells in the synthesis phase and cells having a large volume.

In 16 cases of chronic lymphocytic leukemia (CLL) , the mean volume varied

from one patient to another, and the distribution demonstrated a single population. The ratios of the volume of the cytoplasm, the nucleus, the heterochromatin, euchromatin and mitochondria varied little from one patient to another. A

sphericity index of the cytoplasmic membrane differentiated the smooth from the villous cells and permitted one to place 12 patients into these two categories.

429

National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 19761

SIMPLE DEVICES FOR STEREOLOGY AND MORPHOMETRY

by Hans Eli as and Erich Botz Anatomic Institute, University of Heidelberg, Germany

Measuring and Point Counting Device

A light "box with low voltage and a variable transformer is

placed under the end of a focusable camera lucida (drawing tube ) of a light microscope (Fig. 1).

At first, the drawing tube is installed on the microscope

body, immediately above it the photo attachment, while the binocular viewing piece takes the highest position. Using this

arrangement, a luminous scale or grid consisting of bright lines on a black background placed on the light box is reflected, by

the drav/ing tube, into the microscope so that its image appears

simultaneously in the viewing eye pieces and on the film. Using

scales or calibrated grids on the light box, direct measurements

and point counts are independent of the choice of oculars, since

the drav;ing tube produces an image of the scale or grid in the

plane of the primary, real image.

The determination of a volume ratio

is independent of the dimensions of the point grid, since we are searching, in this case, for a dimensionless ratio. For all other

stereological operations, such as ^At- f

' sAt= f , itAt= a(d f i - 2h') , the length of test lines and the size of the test area must be

known.

Also for direct morphometric measurements such as the length

of a longitudinally sectioned tubular gland or its diameter or the thickness of a mucous membrane, the length of the luminous

scale in relation to the object must be knovm,

A separate scale and a separate grid must be prepared for

each specific objective. The scales or grids are calibrated by means of a stage micrometer. Figure 2 shows an example. The primary magnification of the objective is written on the left side

and the optical length of the lines is written on the right side.

If photomicrographs are taken with this device, scales and

grids automatically projected on the film partake in any secondary enlargement and remain invariant in regard to the image, (Pig. 4)

The Particle Size Classifier

A perfected model of the optico-electrical particle size

classifier, originally described by Schwartz and Elias (1970),

will be demonstrated.

An illuminated iris diaphragm is reflected into the microscope by the focusable zoom drav/ing tube.

431
H. Elias and E. Botz

This device permits, together with the standard curves by

Hennig and Elias (1970) , to determine rapidly size distribution

and the mean diameter of spheroid particles from their sections.

The observer searches, at first, for the largest section

through a particle. The image of the fully open diaphragm is

superimposed on the image of that largest section (profile) , so that the area of the light circle (i.e. the image of the illuminated diaphragm) equals, as close as possible, the image of the

largest particle section. This can be accomplished by the proper

choice of the objective, the correct position of the zoom device

and, if necessary, by varying the distance of the diaphragm from

the drawing tube. If the particles to be measured are very small,

the distance of the diaphragm can be increased by inserting a pair of mirrors between the camera lucida and the diaphragm.

Fig. 5 shows the open diaphragm superimposed on the largest

glomerular profile in a kidney.

The lever of the diaphragm (Fig. 3) is provided with two

contact points. When depressed, one contact point closes a circuit with a continuous metal arc. It activates the totaling

counter (on the right of the counter battery)

.

The other contact point closes a circuit by touching a small metal disc and activates an individual counter for a specific size class.

Ten positions of the iris diaphragm permit the classification

of sectional circles into ten size classes, a subdivision which

is sufficient and realistic.

On the counter battery one can read the number of sectional

circles for each size class, and from these numbers a histogram

is constructed. Often, the two smallest size classes are empty or deficient because polar or near polar "sections" through the particles are frequently unidentifiable, since they are too thin to be visible, or they fall out of the slices.

These lowest size classes must be reconstituted by drawing a smooth curve through the upper midpoints of the bars of the graph.

This curve, which often ends with size class 0.3 Dmax, must be

smoothly drawn through to origin of the coordinate system.

The next step is to match the resulting curve with one of our standard curves (Hennig and Elias, 197 0) for the size distribution

of sectional circles.

To each standard curve x for the distribution of sectional

circles there exists a coordinated curve Xq which indicates the

size distribution of the spheres in space. For each such curve

for the spheres there exists, in our curves, an indication of the mean diameter of all spheres in the mixture.

If the observed, constructed curve does not exactly match one

of the standard curves, it will fall between two of them, and the

specific distribution can be found by interpolation.

References

Schwartz, E. and H. Elias, 1970, J. Microsc. 91^: 57-59

Hennig, H. and H. Elias, 1970, J. Microsc. 93^: 101-107

432
Simple devices for stereology and morp/iometry

1. Light microscope.

3. Circuit arrangement.

433
H. Elias and E. Botz

Fig. 4 : "Section" of colon carcinoma photographed with objective 20 x. (100

micron)^ counting grid superimposed.

Fig. 5 : "Section" of kidney. Illuminated, open diaphragm superimposed on largest glomerular profile.

434
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9. 1975 (Issued January 1976)

THE EDEMA SYNDROME IN MOTION PICTURES

by Oscar C, Jaffee Department of Biology, University of Dayton, Dayton, Ofiio 45469, U.S.A.

The early embryo heart tube has a characteristic form

(Jaffee, '67; Patten, '68) which was illustrated in motion

pictures of 60 hr (incubation) chick embryo hearts. Moderate

hypoxia (Jaffee, '74) administered at this stage was followed

by an osmotic upset coupled with a depression of cardiac function

(chief features of the edema syndrome) . Under these conditions

a swelling of the heart tube and some of the great vessels along

with a distortion of the form of the heart tube was noted.

A majority of the treated embryos revert to normal cardiac

function when returned to normal oxygen concentrations; more

than 25% of these survivors still display distorted heart tubes even though normal function was restored. The arterial outflow

tract, a major site of cyanotic congenital heart disease was a portion of the heart tube frequently malformed. In a following

series shown embryos had been treated as stated above but now incubated until five days and then examined and photographed.

Heart tube form was still abnormal in many of those coupled

with abnormal blood flow patterns suggestive of ventricular

septal defects and double outlet right ventricles; both of these

malformations had been previously shown to follow hypoxic

exposure of embryos (Jaffee, '74)

.

The stereological form of the heart tube and of the

patterns of blood flow through the tube are major factors in

normal cardiogenesis (cf . Jaffee, '67) . changes in heart tube

form and flow patterns, produced by environmental changes, have

been traced into recognizable cardiac malformations, emphasizing

the morphogenetic role of these factors.

References

Jaffee, O.C., 1967

Patten, B.M. , 1968

Jaffee, O.C., 1974

Anat. Rec. 158: 35-42.

Human Embryology, McGraw-Hill.

Teratology 10: 275-282.

435

PART II. WORKSHOP SECTIONS

MATHEMATICAL FOUNDATIONS OF STEREOLOGY

Chairperson

:

R. T. De Hoff

National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

THE MATHEMATICAL FOUNDATIONS OF STEREOLOGY

The Proceedings of a Workshop held at the Fourth International Congress for Stereology

The inference of geometric information about three dimensional

structures from observations made in lower dimensions (i.e., on plane sections, test lines, or point grids) involves the

amalgamation of a number of components of the broad field of

mathematics, including measure theory, topology, differential and

integral geometry, probability theory, and statistical inference.

In this workshop, four of the basic relations of stereology were presented in a series of derivations which began with the

mathematical foundations and v/ere developed into the simple,

working relationships of stereology. A generalization of the

intuitive notion of size was also presented. Some general comments

on the use of statistics in the practical application of these

simple results made the program for the workshop self-contained.

I . OVERVI EW

by R. T. DeHoff, Department of Materials Science and Engineering,

University of Florida, Gainesville, Florida, USA

A mi crostructure consists of points, lines, surfaces and

volumes distributed in three dimensional space. Usually the

surfaces are defined by the juxtaposition of two distinguishable

kinds of volumes; lines by three, and points by four.* The

microstructure may be defined descriptively, i.e., qualitatively;

or it may be quantified by applying stereology, or some alternative

techniques (DeHoff, (1975)). Thus, it is possible to define a

A. Qual i tat i ve Mi crostructural S tate . In its most rudimentary form, this state is simply a list of

all of the zero, one, two and three dimensional features that exist

in the structure. if the number of distinguishable kinds of

volumes in the structure exceeds tv/o, such a listing will not be a trivial matter, as can be seen from Tables 1.1 through 1.3. The

qualitative mi crostructural state may be further specified by

descriptive statements about the features contained: e.g., "fine",

or "coarse"; "small" or "large" amount; "equi-axed" or "elongated";

"simple" or "complex". If further specification of the structure

is required, one may define its

B. Quant i tat i ve Microstructural State . Each of the features listed in Tables 1.1 through 1.3 have

associated v/lth them geometric properties. The quantitative

microstructural state of a structure is defined by assigning

numerical values to one or more of these properties. The level of

quantification is related to the number of different geometric

properties evaluated for the structure. The properties themselves

can be classified into two categories:

1. The Quant 1 tat i ve Topological State is specified by

evaluating the topological properties of the structure. Table l.h.

Topological properties include primarily (a) the number of

Exceptions to these restrictions require a very high degree of

homogeneity and order in the structure.

439
R. T' DeHoff

disconnected parts of the feature and (b) the connect i v i tv , or number of redundant connections, of a connected feature (DeHoff,

Aigeltinger and Craig, (1972)). These properties are listed in Table l.i». Topological properties are not further developed in this workshop, because they are inaccessible to measurements made

on sections, and can only be estimated from reconstructions of the

three dimensional structure.

TABLE 1.1

Features that Exist in Single Phnse Structures

D i mens i on Feature Descr i pt i on

3 Vol umes (Grains)

a

2 Surfaces

(Gra i n Boundar i es

)

aa

1 Curves

(Grain Edges) aaa

0 Pol nts

(Grain Corners)

aaaa

Tota 1 : h Featu res

Features that

TABLE 1.2

May Exist in Two Phase Structures

D i mens i on Feature Descr i pt i on

Vol umes

(Grains, Particles)

Surfaces

(Grain Boundaries,

1 nterfaces

)

aa

a3

Curves

(Grain Edges,

Triple Li nes

)

aaa

aag

a66

On a6

I nterf ace

Po i n t s (Quadruple Points) aaaB

aPBB

On aB

I nter face

Tota 1 : Ih Features

440
The mathematical foundations of stereology

TABLE 1.3

Features that May Exist in Three Phase Structures

D i mens i on Feature Descr i pt i on

3 Vol umes a (Grains, Particles) 6y

e. Surfaces aa 66 YY (Grain Boundaries, oi6 6y oy

1 nte rf aces )

1 Curves aaa. 666, YYY

(Grain Edges, Triple aa3. a66

L I nes

)

aay. ctYY 66y, 6yy

ciBy

0 Po i n t s aaaa. 6666, YYYY

(Grain Corners, aaaB, aa66, a666

Quadruple Points) aaay. aaYY, cxyYY

66yY, 6yYY

ctaBY, a66Y, a6YY

Total : 3U Features

TABLE 1.1* Topological Properties of Features in Two Phase Structures

Property

Dimension Feature Designation Number Connectivity

3 Vol umes a /

6 /

2 Surfaces aa * 66 * a6 /

1 Curves aaa * ** 666 * ** aa6 * ** a66 * * *

0 Poi nts aaaa /

6666 /

aaa6 /

aa66 /

a666 /

Total Topological Properties: 21

Number of individual elements and number of disconnected networl<s

may be separately defined.

Connect i vi ty applies to network only, since the elements of the

network are simply connected.

441
R. J. peHoff

2. The Quantitative

ting the metric properties of

Rhines, (1968)).

Metric State is specified by evaluathe structure. Table 1.5 (DeHoff and

TABLE 1.5

Metric Properties of Features in Two Phase Structures

Property

Dimension Feature Designation Symbol Description

Vol umes

Surfaces

Vv"/ VyS Volume Fraction

Sv«^

K.'aB

Surface Area Per Unit Volume

Total Curvature

Per Unit Volume

Curves

age

Poi nts

. Vo O D

L aa6

LjaBB

Line Length Per Unit Volume

Total Curvature

Per Unit Volume

No Metric

Propert i es

Total Metric Properties: 11

The most obvious of these properties are those that report the

extent of the feature:

(a) Vy, the volume fraction , reports the total volume

of three dimensional features, per unit volume of structure;

ly, the surface area per un i t vol ume , reports the

kind of two dimensional feature

(b) Sy, the surface

total area of each specific

in the unit volume;

(c) L\j, the 1 i ne length pe r un of any lineal feature in unit

additional parameter is def

t. vol ume , reports the length

volume of structure.

One additional parameter is defined and derived in this wori<shop;

Ky, the tota 1 curvature i n uni t vol ume > is the integral

the local mean surface curvature over the area of surface

the vol ume, i.e..

(d)

of

i n

(1.1)

where r^ and r2 are the principal normal radii of curvature on the

surface. Their reciprocals, and k^' ^re the principal normal

curvatures. These properties are called me t r i c properties.

442
The mathematical foundations of stereology

because, in contrast to the topological properties, their values

depend upon the dimensions or sizes of the features in the

structure.

The essence of stereology is the demonstration that these

geometric properties of features that exist in a three dimensional

mi crost ructu re can be estimated from simple measurements, in fact

from counts of statistical events that occur, made upon

representative sections through the structure. The connections

between the higher and lov/er dimensions are founded upon integral

geometry, as is now developed.

I I PROBABI LiSTIC FOUNDATIONS OF STEREOLOGY

by R. E. Miles and Pamela Davy, Department of Statistics,

Institute of Advanced Studies, Australian National University,

P.O. Box h, Canberra, A.C.T. 2600, Australia

Points Eq, lines Ej^ and planes E2 in space E3 are parametrized

by (x, y, z), (6, 4); 5, n) and (6, 0; p), respectively, where (6,

<j)) are spherical polar coordinates of the line or normal to the

plane, n

)

are cartes 1 an coordi nates in the pl ane

(c)

Figure 2.1 The parameters that

specify (a) a point, (b) a must be assigned values

1 i ne, and (c) a pl ane.

to

443
R. E. Miles and P. Qavy

through the origin perpendicular to the line, and p is the length

of the perpendicular from the origin to the plane. The standard

isotropic invariant volume elements

dEo = dxdydz, {2.1a)

dEj = sine d6d())dCdn/ (2.1b)

dE2 = sine ded(i)dp (2.1c)

of Integral Geometry for points, lines and planes are discussed,

and the resulting measures

Mo(Y) = /dEg = V(Y), (2.2a)

Mi(Y) = /dEj = S(Y), (2.2b)

M2(Y) = /dEj = K(Y) (2.2c)

of the sets of points, lines and planes hitting a convex domain Y are derived; here V, S and K denote volume, surface area and

Integral of mean curvature, respectively.

The corresponding normalized (and therefore probability)

e 1 ements

dEQ/Mo(Y), dEi/Mi(Y), dE2/M2(Y)

govern the distribution of un i form random (UR) points In Y and

i sotropi c un i form random (lUR) lines and planes hitting Y. Next a classic result of Integral Geometry/Geometrical Probability, the

Crof ton / Hos t i nskv Theorem . Let the distance between two

independent UR points of Y be R, and let the length of the secant

determined by an lUR line hitting Y by L. Then, with | denoting

expectat i on,

(i + 3)(i+i»)V(Y)^E(R ) = irS(Y) Ed'*"*) (1 t -3, -h) . (2.3)

Is presented, since it exhibits the essential elementary features

of these two disciplines.

In the stereological context, the specimen X Is taken as an opaque bounded connected (but not necessarily convex) domain which

contains embedded sub-volumes, surfaces, curves and/or points.

(a) (b)

Figure 2.2 The set X is the sample; the set Y consists of 0,

1, 2 and 3 dimensional features that make up the "internal

structure" of X.

444
The mathematical foundations of stereology

To these correspond the stereological 'fractions'

'V V

vTx)

vTx)

V

p vXx) (2.1*)

for volume, area, length and number, respectively. Also of

importance for embedded surfaces are the integral of mean curvature

and integral of gaussian curvature fractions

.V = i/(Ki+K2>dS/V(X),

Gy = f<l<2 dS/V(X).

It is shown that, for an lUR plane section of X,

(2.5a)

(2.5b)

E (area of section) = 2-n V(X)/K(X) _ (2.6)

E (area of a phase sectioned) = 2tt V(Xa)/K(X) (2.7a)

I (length of intersection curve wit.h embedded surface) = (7r2/2) S(embedded surface)/K( X) (2.7b)

E (total curvature of intersection curve lii th embedded

surface) = 2t\ K(embedded surface )/ K( X ) , and (2.7c)

E (number of intersection points with embedded curve) =

TT Kembedded curve)/K(X), (2.7d)

v/here X denotes the convex hull of X. Note that each of (2.7a - d)

is of the form

(2) Ed) = 2TTa J(phase, surface or curve)/K(X),

It follows from (2.6) and (2.7) that

(3) Ed;^) = aJv - {K(X)/2TrV(X)} C[l^, A],

(2.8)

(2.9)

where C denotes covariance. That is, the customary stereological

estimates for lUR plane sections are biased. The biases may be

positive or negative, depending upon the spatial distribution of

the phase, surface or curve within the specimen X. However, the corresponding estimates 1^ for a different type

of random plane section of X, viz. an lUR plane section weighted by

its cross-sectional area with X or, equ I va 1 ent 1 y, an isotropic

plane through a UR point of X, are unb I ased . Moreover, the mean

square error of 1;^ minus the variance of 1^ is

{K(X)/27rV(X)} g[lA(2aJv-I/\), A] (2.10)

which may be expected to be positive in general.

In fact, these results extend to specimens and sections of all

dimensionalities. In this respect, this type of 'weighted' section

possesses a further practical advantage not possessed by the

corresponding lUR sections: such a line section

specimen is probabilistically equivalent to such a such a plane section of the specimen.

of

1 i ne

a spatial

section of

445
H. E. ^xner

111. DERIVATIONS OF THE WORKING RELATIONSHIPS OF STEREOLOGY

A. Vol ume Fraction Anal vs i

s

*

by Hans Eckert Exner, Max-Planck, I nsti tut fur Meta 1 1 forschung,

Stuttgart, W.-Gernany

Volume fraction analysis is the oldest and, apart from grain

size measurements, by far the most common procedure in practical

stereology. It is particularly simple to obtain the volume

fraction Vy from plane sections, since volume fraction, area fraction, lineal fraction, and point fraction are all equal. The

derivations of these equalities were first given by Delesse (18Ij8),

Rosiwal (1898 ) and Glagolev (igsit), respectively, and have more recently been demonstrated by Hilliard (1961, 1968), Weibel (1963),

and Underwood (19 70).

~0

or Ao = 1/L j'"A„dx

Vv =Va/V, = Aa L/L^

V„ =A„/L? =A./A=A.

dAc = Lady

Ao = 4'-dA„ = /^'-L,^x U = 1/L /''Lcdx

Aa = U L

Ai = A„/A| = La LIL'

Aa = La/L = L^

Figure 3.1 Derivation of Vv=Aa. Figure 3.2 Derivation of Aa=Ll.

A simple integration of the area fraction over the volume of

the sample yields the equality

Vv = Aa

(Figure 3.1). Similarly, the equality

Aa = Ll

is obtained (Figure 3.2). For proving the equality

Vv = Pp

(3.1)

(3.2)

(3.3)

a different approach is taken relating the probability of hitting

an a-feature by a point in the volume (Figure 3.3).

The experimental errors for a fixed number of section areas, linear intercepts, or hits on the a-phase are shown in Figure 3.k.

This v/orkshop paper consists entirely of information already

available in the literature. Only random structures and^ plane

random cross sections through opaque samples are considered.

(Mathematical rigidity was thought to be less pertinent than

simplicity in this context.)

446
The mathematical foundations of stereology

Type o( analysis

Figure 3.3 Schematic derivation of the

equal i ty Vy = and number of

and the total

that

a-phase )

.

PP ^^a' ^t' ^'a' ^^t: volume

elements of a-features

probab i 1 i ty

fall on the

cube Wa = po i n ts

Figure "S.k Experimental

error due to statistical variations for 100

observations according

to Hill iard (1961, 1968),

Systematic point counting (using a grid the spacing of which is

large enough to prevent two points falling Into the same a-sectlon)

has the smallest error due to the fact that no variations in size

and no measuring errors are involved, in contrast to lineal and

areal analysis. Therefore, systematic point counting is usually

the preferred method. The estimated range for the true volume

fraction is

Vy ± AVy = Pp ± /Pp • (1 - Pp)//Pt

P

(3„it)

where is exper i menta

1

square root seen from thi

points must

relative accu fractions of

respecti vel y

.

Figure 3

the total number

error decreases in of the total number

s useful relationsh

be counted for

racy i s requ i red

.

10 and 1%, 900

).5 shows an example

of points examined. Thus, the

proportion to the inverse of the

of points counted. It also can be

ip that a large total number of

small volume fractions if a high

(For AVy/Vy = 10% and volume

and 9 900 points must be counted,

of a volume fraction analysis by

Sysiematic point count with Weibel-grid

One grid count 100 grid counts

P, = 50 P, = 5000

Pa = 20*6/2 = 23 Pa = 2138 + 642/2 = 2460

Pp = 23/50 = 046 Pp =2460/5000 = 0492 Pp = V0 460.54/50' ±Pp =V0492 0.508/5000'

= 0.07 = 0 0071

Vv AVv = i6 t 7 % VvlAVv = 49 2 1 0.7%

Figure 3.5 Volume fraction analysis by point counting.

Points falling on phase boundaries are counted 1/2.

447
E. R. Weibel

point counting. For one grid count, the result Is too Inaccurate

to be useful. For 100 grid counts, the experimental error Is reduced to 1 absolute pet or 2 relative pet.

If automatic Instruments are used, the tedious counting and

measuring procedures are speeded up. The experimental error, however, is reduced only If a larger number of a-sections Is sampled than in point counting, and If no systematic errors are Introduced due to insufficient phase detection. Therefore, there

is no point In putting some hundred thousand points on a few

features in a small field of the sample since these will

characterize only the area fraction of this particular field. This

will usually be a poor estimate of the average area fraction and,

therefore, the volume fraction.

B . Surface Area

by Ewald R. Weibel, Department of Anatomy, University of Bern,

Bern, Swi tze r 1 and

Consider the measurement of surface area in a simple structure

made of tv^o phases: an object phase embedded in a matrix. The

interface betv^een these two phases is measured by its surface area

S, a stereol ogl ca 1 1 y measurable quantity being the surface density

Sy In the unit containing volume. I.e. In the volume V of the

entire structure:

Sv = S/V (3.5)

Stereology offers tv/o basic methods by v;hich Sy can be

determined (Cornfield (1951), Smith (1953), Saltykov nsSk),

Tomkeleff (19ii5): (a) the structure Is sectioned by a plane, E2,

(Figure 3.6a); the containing volume Is represented on this section

Figure 3.6 Fundamental relationships between surface,

boundary length, and intersection densities.

by the area A of Its profile, whereas the surface leaves a linear

trace of length L. The following relationship holds:

V V TT * A TT '^A (3.6)

(b) The structure is intercepted by a test line, Ei, (Figure

3.6b); the containing volume Is represented by the length of test line lying within the structure, L, and the surface by the number

of point intersections, P, formed as the line traverses the

surface, with the following relationship pertaining:

448
The mathematical foundations of stereology

Sv=7=^=2Pl (3.7)

These two relations shall now be derived, starting with a very

simple case and then proceeding to more generally valid

derivations. For reasons of simplicity we shall test the structure

v/i th a set of parallel test lines or planes; due to this we must

require that the surface within the entire structure is isotropic,

i.e. all orientations of surface elements must be equally likely.

If this does not hold, the direction of the test lines must be

suitably randomized.

1 . L i ne i nte rsect i on measurement

a. I sot rop i c model ]_: sphere i n cube

Let us consider first a simple structure of a cube of side

I containing a sphere of radius R (Figure 3.7). With V = Jl^ and

Figure 3.7 Isotropic model I.

S = Utt r2 the surface density is

V ^^ (3.8)

This structure is now sampled with a "beam" of parallel vertical

Lines; some of these will pass through the cube, call their number

n, and some will also intersect the sphere, call this nQ. What is the probability that such a line intersects the sphere? From

Miles' presentation [Section II], it follows that this probability

is the ratio of the measure of the enclosed object Y to the measure

of the enclosing space X*: M |Y| P^^'SCV} = (3.9)

These measures are the orthogonal projections of object and

structure, if you v/ant the "shadows" projected by the beam of lines

onto a plane perpendicular to the lines. For the simple model of

Figure 3.7 the projected area of the cube is and that of the

sphere ttR^, so that

p{E -^Sc=V} = — = -° (3.10)

_j I' " The left side of equation (3.9) is read "the probability that a test line (E^) intersects (+) a surface (S) contained in (c) a sample volume (V)".

449
f. R. Weibe I

This probability can evidently be empirically estimated by the

ratio of the number of lines crossing the sphere to those crossing

the cube.

Mow notice the following relationships: the number of

intersections formed between test lines and sphere surface is

P = 2 (3.11) one for the entrance and one for the exit, and the total test line

length within the structure is

1 = 1 n The density of intersections per unit test line length

p = P .

I n Substituting np/n from eq. C3.10), we find

P = IjrRi

(3.12)

s hence

(3.13)

(3.1ij)

Comparing this with the surface density defined by equation (3.8),

vye find that Pl is just half the surface density; it follows

immediately that

Sv = 2Pl

v/hich is equation (3.7).

b. I sotropic mode 1 I I : random! v or i ented surface el ement

We now relax the restrictions on the shape of the objects

(Figure 3.8); they may have any arbitrary shape, and we consider

tJA = dS cos 0

Figure 3.8 (a) Isotropic model 11 intersected by lines;

(b) Orientation hemisphere; (c) Surface element and

projected area.

their surface to be broken up into a very large number N of small

surface elements of equal area dS so that the total object surface

i s

450
The mathematical foundations of stereology

/dS N • dS (3.15)

Asofcontaining space we retain, for

side I, so that the surface densi reasons

ty i s of simplicity, the cube

Sv = 'dS N-dS

and we sanple again with a Isotropicity of the

orientations of the

0 between the normal

the test lines (Figure

The derivation of

the same reason!

n

intersects one surface element is

beam of parallel vertical lines,

enclosed surface is ensured if all

surface elements, measured by the angle

to the surface element and the direction of

3.8), are equally likely.

the expected number of intersections follows

as above: the probability that a line

p{E +dScV} = — (3.16)

where dA is the mean orthogonal projection of the surface element

and £2 jg ^he projection of the cube, as above; dA evidently

depends on dS and on the angle <b, such that (Figure 3.8c)

dkiif) = dS • cose (3.17)

Averaging must be performed over all orientations 0, whereby we must note that not all angles 6 are equally likely! fJote that, in this formulation, we have already integrated out the angle

6 measuring orientation v/ith respect to one of the axes in the

horizontal plane. Because of this, the probability of having an orientation betv/een 6 and (0 + dcj) is

p{$, do) = sin(}>d(> (3.18)

which is the area of a belt of width d<) and length sintj) divided

the area of the orientation hemisphere 2tt (assuming radius

(V/eibel, ( 1963 )). It is evident from Figure 3.8c that the area

this belt increases as 0 increases. The average projection area

dS is hence obtained by

by

1)

of

of

tt/2 HA = cosO-sin^-dO

__ ^

n/2 2 /q sin0-d$

By equation (3.16) this yields for the probability of obtaining an intersection with one surface element

p{E MSc: V} = i ^ . (3.20)

If the object surface is made of N surface elements and if n test lines intersect the cube, the expected number of intersections in the entire structure is

E(P) = fl • n • p{Ei + dSc:V} (3.21)

Noting that N = S/dS and n • J- = L, the total test line length, we obta i n

451
E. R. Weibel

TTfT,^ SLdS_lS, i^P^ = dS'l-—2- (3.22)

which finally leads us, by rearrangement, to

Pl = i(P)/L = \ (3.23)

a result identical to equation (3.7). Note that in this model the

object surface can have any shape — as lonn as it is isotropic on the whole — and that the surface need not be closed; indeed, it can be fragmented into as small elements as is possible.

We could now also relax the condition on the shape of the

containing space; it need not even be convex. In this general case

the test line length Lj enclosed in this space is the sum of all

intercept lengths

= Z £. (3.2U)

Eq. (3.23) would then read, after slight rearrangement

2.EP

= = TTiy = 2Pl (3.25)

It is important to note that in the case where the individual test line intercept with the structure has variable length, the unbiased

estimator of S\j is 2 times the ratio of the sum of intersection

points to the sum of the test line lengths.

2 . Boundary length measurement

To derive equation (3.6) we shall use the second model

introduced in the preceding section (Figure 3.8a): a surface which

can be fragmented into a large number of small surface elements dS

(Figure 3.9).

f b Figure 3.9 (a) Isotropic model II intersected by plane;

(b) Caliper diameters of containing space and surface element.

The problem is now to find the probability that a test plane

E2 intersects a surface element. In this case the measure M2 is the caliper diameter, i.e. the distance betv>feen the two tangent

planes parallel to the section plane (= the orthogonal projection

onto a line perpendicular to the section plane). For the cube this measure is £, and for the surface element (Figure 3.9b)

h = d • sin<l) (3.26)

452
The mathematical foundations of stereology

where d is the diameter of the element. A/^ain we must average over

all or i en ta t i ons

:

/g d-sin2$-d<j,

^ ^ = -^71 = (3.27)

/q siiKj-di})

so that

p{E2fdScV} = 1= J4^ (3.28)

The average length of line intercept betv/een surface element and

section plane is

where

A(dS) = J

- (3.30)

B(dS) = TTd (3.31)

are area and circumference of the element, respectively [Section

II].

Assuming the structure to contain a large number

M = S A(dS3 (3.32)

of such elements, the total length of intercepts on a test plane of area A = 2

E dL = fJ • E(dL) • p{E2 + dSc:V}

= {N.J.d^.J.l (3.33)

where the term in square brackets is the total object surface.

Dividing both sides by 5,2 obtain

ZdL ^ TT S ^1 ^ (3.3IJ)

This amounts to

L = --S

v/hich is equal to equation (3.6).

Here again the containing space need not be a cube. in analogy to equation (3.25) we find as an unbiased estimator of

surface density

453
E. R. Weibel

tt'zA

4 E(L)

tt'eTAT (3.35)

3 . Surface dens i t v est i mated by tv/o - stage procedure

The usual practical procedure for estimating is to cut a section and then to count intersections formed by a set of test lines on the plane of section with the linear trace of the surface.

The final question is, hence, whether this also yields the

relationship Sy = 2Pl.

By a reasoning analogous to that used in Section 2 only

reduced to a two-dimensional problem, one can show that

La = r^i (3.36)

Break the linear (surface) trace into N short segments (Figure

3.10),

Figure 3.10 Length of line on plane.

so that

La = /dL/A = n • dL/A

Deposit a test line grid of spacing d, so that

Lt = A/d

(3.37)

(3.38)

The probability that such a segment is intersected by a test line

is (Buffon (1777), Weibel (1967))

pIEifdLc: A}

TT 3~ (3.39)

which is obtained again by allowing all orientations 9 between the

segment and the test lines to be equally likely (VJeibel, ( 1967))

(Figure 3.10). The average number of intersections is

E(P) N-p{EitdLc:A} 2 L

7r*d (3.t»0)

or, by appropriate substitution for d from equation (3.38) and

rearrangement

E(P)

l.h.

tt'a (3. Ill) which is equivalent to equation (3.36). Allowing again for

irregular shape of the enclosing area, the unbiased estimator for

La 's

454
The mathematical foundations of stereology

IT E(P)

" 2" JTL^) (3.1*2)

In conclusion, we find that by substituting equation (3.36)

into equation (3.6) we end up with equation (3.7):

Q = T T = — P

t: , a ^A , 2 '-V ' (3.ii3)

If we allow for variable containing space and section area and

hence properly consider the ratio of means as given in equations

(3.35) and (3.U2), we find that we still obtain equation (3.25).

therefore, i"s that the two-stage

practice does not invalidate the

twice the density of intersections on

an unbiased estimator of surface

The final conclusion,

sampling procedure used in fundamental relationship that

the length of test line is dens i ty

.

In equation (3.25) the ratio of surface area to volume is expressed as the ratio of mean intersection number to mean line

intercept length. It can easily be shov/n that this formulation

applies also to the estimation of the so-called surface to volume

ratio of closed objects, as derived by Tomkeieff, (19U5) and

Cornfield and Chalkley, (1951), in v;hich case the reference volume

is taken to be the mean volume of the objects rather than the

volume of an enclosing space.

C. Length of L i nea 1 Features i n Space

by R. T. DeHoff, Department of Materials Science and Engineering,

University of Florida, Gainesville, Florida, USA

Space curves exist in microstructures; e.g. dislocations in crystals observed in transmission electron microscopy are space

curves, as are the edges (triple lines) in a cell structure. In addition, features that have one dimension very large in comparison

to the other two (e.g., tubules) may be considered to be space curves. Such features possess a length. A stereol ogi cal measure

of the length of linear features (Y) in a three dimensional

structure (X) is derived in this section.

Figure 3.11 shows such a linear feature Y in a sample volume,

X. Figure 3.11b focusses upon an element of length, dX, of the

feature. According to equation (2.1c), the measure of the set of

planes in space is: dE2 = dpsinededcj) (2.1c)

For the orientation shown in Figure 3.11b, event that a plane intersects the element, d\, is satisfied by those planes with

positions lying within the projection of dp on the direction normal

to the sectioning plane. That is, the value of dp satisfying the

event of interest is:

dp dXcose (3.1+ If)

455
Figure 3.11 A sample containing a set of lineal features (space

curves) Y is shown in (a). The measure of the set of planes

that intersect an element of this set dX is dp, the projection

of dX on the direction normal to the test plane.

A count of the total number of such intersections divided by the

total area sampled by a set of test planes yields an estimate of

the total length of the set of lineal features in unit volume of

the structure, X (Saltykov, (I95k), Smith, (1953)). This area point count, Pf^, is given by the ratio of the measure of the set of

planes that intersect lineal features to the measure of the area of

such planes. The numerator and denominator in this ratio are thus

integrals of the measure given in equation (2.1c):

/// dE /// dp-sineIntersections Intersections

= = (3.45)

/// AdE /// A-dp-sineAll Planes All Planes

2-17 TT

/f^ f /,dX •cose • sine •d(t)de

Ztj it /q /q /pA-dp-sine-d(j)de

27T 7T/2 /, dX-/„ d(})'2/„ cose-sine'de

2-n 7T/2 / Adp-/„ d(t>-2/ sine-de

456
The mathematical foundations of stereology

Note that fd\ is the total length A of the feature in the volume,

and that /Adp is the total volume V of the sample X, no matter what

orientation is chosen. Carrying out the integration gives

p ^ A'2tt-2(1/2) ^ V-2TT-2-1 -

vvh i ch s i mpl i f i es to

P = i.A = i L A 2 ^ 2 V (3.[*6)

Thus, in a series of plane sections taken through a three

dimensional structure, a count of the number of points of emergence

of the lineal feature on the planes of observation, divided by the

total area sampled, gives an unbiased estimate of the length of the

lineal features per unit volume of microstructure.

D . The Area Tangent Count and the Total Curvature

by R. T. DeHoff, Department of Materials Science and Engineering,

University of Florida, Gainesville, Florida, USA

Figure 3.12a shows a sample volume, X, that contains a set of embedded surfaces, Y. If this volume is sampled with test planes, intersections v/ith the surfaces will appear as linear

traces on the plane of observation. Figure 3.12b. Suppose the

sectioning plane is sampled by sweeping lines across it, each sweep

covering the plane area, A. The event of interest in the present

derivation is the formation of tangents between such sweeping test

lines and the traces of the surfaces on the planar section, Cahn,

(1967), DeHoff, (1967). The tangent count records the number of

times such sweeping lines form tangents in their sweep. A count is registered as pos i t i ve if the arc at tangency is convex; a negative

count is registered if the arc is concave. flote that v/hether an arc is "convex" or "concave" is a matter of convention, to be

decided before counting begins.

457
/?. T. DeHoff

Recall that the measure of the set of planes in space Is given

by equat ion (2.1c).

dE2 = dpsined(l)de (2.1c)

The measure of the set of sweeping lines in a plane is the same as the measure of the set of orientations in a plane, and is thus an angular interval, dct. Thus, the measure of the set of sweeping

test lines in space is

dT = da dE = da dp sined(l>de (3.tt7)

if A is the area of a sectioning plane, then the sweeping line

traverses an area A for each sweep.

The 1^ count (number of tangents formed per unit area traversed) is thus given by

//// dT

Tangent Events

//// AdT

All Values

//// da'dp-sine-d(t)de

Tangent Events

//// Adp-da-sine -dcfde All Values

For any plane, /da is or tt; for any orientation, /Adp = V, the volume

of the sample X. Finally, //sin6d6d(}) is the area of the

orientation sphere, i+tt. Hence, the denominator may be evaluated:

//// da'dp • sine •d(})de

Tangent Events

= (3.l»8)

It remains to evaluate the numerator in equation (3.48).

Focus upon an element of length, d£, produced by intersecting a surface element dS with a sectioning plane. This element has a local curvature, k, which is related to principal normal curvatures

of the surface being sectioned in the neighborhood of by

theorems of fleusnier and Euler (Struik, (1951)). This relationship

depends upon the orientation of the sectioning plane, (defined by

its normal), relative to the coordinate system on the surface

defined by the surface normal and the principal directions:

1 2 2 k((}>,e) = (k cos (|) + K sin (}> ) sine o.tJ;

A sv/eeping line in the plane will form a tangent with dC if its

direction of sweep lies vnthin the interval of normal directions

along dP., Figure 3.15a.

458
The mathematical foundations of stereology

/

(a) (b)

Figure 3.13 The measure of the set of sweeping test lines that

form tangents with a trace element dl is related to the local

curvature of the element, (a). The curvature of the element of

trace is in turn related to the principal normal curvatures

of the surface element from v;hich it derives (b).

Thus, the event occurs if

dct = kd£ (3.50)

If All element of length in the surface being sectioned that

is perpendicular to dS, (Figure 3.13b), then the event that the

sectioning plane intersects the surface element defined by dfc and

d£i requires (Figure 3.12b)

dp = dli sine (3.51)

The numerator in equation (3.tf8) may nov; be evaluated by combining

equations (3.U0), (3.50) 12and (3.51):

2 -(kiCOS K-,sin (|))-d£(d£ sinB) -sinB -dttde

i / 1 ////

sine

Events

Carrying out the integration over 6 and tf gives:

//(1/2) (k^ + K^) -dS

ttV The numerator is the integral of the mean curvature, K [see

equati on (1.1)]:

T = - 1 ^ A ttV TT (3.52)

In summary, sweeping test lines in a sectioning plane form

tangents \/ith traces of surfaces in proportion to the angle subtended by such traces. This subtended angle is related to the

curvature of the trace. The curvature on a trace is in turn

related to the principal normal curvatures on the surface from

which the trace is derived. This combination of circumstances

makes the tangent count on a section related to the mean curvature

of the surfaces being sectioned, integrated over the surface area

in the system.

459
J. Serra

IV. GRAfJULOMETRY FOR I SOLATED PARTI CLES OR COriNECTED MEDIA

by Joan Serra, Center de Morphologie Mathemat i que,

Ecole do Mines de Paris, Fonta i nbl eau, France

A . Ax i orna tics

Starting from the idea that for unconnected, distinguishable

grains, size dispersions are easily obtainable through the physical

operation of sieving, and that more complex systems ought to be

made amenable to similar procedures, G. Matheron has axiomatized

the concept of granulometry in the following way.

Let Y be a population, or set, to be sieved by a family of

sieves having mesh sizes A > 0. Applying sieve A to Y will yield

an oversize ^\ (Y) depending upon the shape and dimensions of the

sieve used as well as those of Y itself. It should thus be

possible to use various kinds of sieve shapes yielding different

results. If (Y) is to define a granulometry, it must satisfy

the following conditions:

i>X (Y) cz Y and ti^o (Y) = Y (it.l)

if ZC Y then (Z) cz <I'A (Y) {h.D

i.e. the oversize from sieve A of any subset Y is contained in the

oversize from sieve A of Y itself.

Successive sieving through various mesh sizes v/ill yield a final result identical to that given by the sieve having the

highest mesh number,, i.e.:

ii^X o (X) = i>p o (Y) = *sup(A,M) (Y) - ih.l,)

where o designates the combination of sieving operations.

B . Exampl es

1. Connected component vol ume ( or area ) Let Y be a set of distinguishable grains or particles. If (Y) is the subset consisting of all particles having a volume

greater than A, then ij;^ (Y) satisfies i.h.1) to (U.3) above and thus

defines a granulometry. Similar granulometr ies may be obtained

v^ith a cross section area in E^.

2 . Maximum chord 1 ength

Again, let Y be a set of distinguishable grains. The

collection \\)\ (Y) of all grains having a maximum chord length

greater than A in any direction, defines a granulometry onto Y (i.e. when the chord direction is fixed, or normal to a fixed one).

These two examples assume the grains to be distinguishable,

which is not alv/ays the case. They also ignore the s te reo 1 og i ca 1 meaning of the granu 1 ome t r i es on polished sections which will be

considered later.

46,0
The mathematical foundations of stereology

3 . Open i riKS by convex s t ructu r i np: el emnnts

Let B be a convex conpact set. The "opening of Y according to

B" (noted Yg) is defined as the volume of space swept by B under

the condition that B remains entirely included in Y. Given B(o),

let B(A) be the family of all sets homothetic to B(o). Then YgQ)

defines a granulometry. There are thus as many different

granu 1 ome t r i es as basic shapes B(o). In planar measurements, two such shapes are of particular interest because of their

morphological significance. They are the line segment and the

circle, i.e. the least and the most isotropic convex bodies.

C. We i ghts and Stereol ogy

1. With each ii;;^ (Y), we associate its volume (or area). The

quant i ty

V[*, (Y)]

G(X3 = 1 vrvT (~h.k)

is then

2. section of Z; d i s t r i bu t i on

(f vary (i.e.

a distribution

Let Zi, ... Z f unct ion

.

be sets in E-" Let us

a rea A [Z • ( p, granulometry defined in B.l).

by an 1 UR

f unct ion

convex

pi ane E 2

.

of the

z;(p,

denote by

and 0) be

Gi (X)

when p , e

the

the

and

G(X) = ZK. G- (A) ZK.

(It. 5)

where G(X) is the granulometry of Z and Kj the integral of

the mean curvature of Z;. ' 3. Let Y be the union of disjoint subsets Y| (the Y; not necessarily convex, nor connected), and tjjx (Yj ) the granulometry

according to the family of homothetic convex sets B;^^. G(X) is defined as formerly. Then:

G(X) = ZV. G. (X)

ZV,

(V; vol ume of X ; ) ( t* . 6 )

REMARK : The openings nay be spherical,

intercepts, the weighting remains unchanged.

c i rcul ar, or 1 i near

V. STAT I ST I CS OF MEASUREMENTS

by V/. L. Nicholson, Pacific Uorthv/est Laboratories,

A Division of BATTELLE MEMORIAL INSTITUTE,

Richland, Washington 99352, USA

A. I ntroduct i on The main topic of the v/orkshop has been "what can be

estimated?". How v/e consider the topic "hov/ many measurements must

be taken?" or more generally "how precise is the estimate?". Basic

s te reo 1 og i cal relationships describe global or average value

geometric properties of a three-dimensional structure. Actual

measurements will be distributed about such a global property. The

precision of such a global property estimate depends upon the

method of taking measurements and the complexity of the structure

as v/ell as the number of measurements. If the structure is spatially heterogeneous, precision is highly dependent on the

sampling plan used to select the sections for examination.

461
W. (Vicholson

B. Theoret i ca 1 Approach

A specimen is examined s tereol og i cn 1 1 y to estimate a threedimensional global structure property P3. A random section of the

specimen is selected for examination. Observation is limited to a randomly selected field or subsection u'\th a two-dimensional

structure property P2 which is an unbiased estimate of P3. A scheme for making measurements of the subsection produces an unbiased estimate P2 of P2. The relative variance of P2 is expressible as a2(^ )/p2 = E[a2(p IP )]/p2 + o2(p )/p2 (5.1)

^ 5 = lit) I ^ where a2(P2l P2)^ is the conditional variance of P2 given P2 (i.e.,

the variance of P2 as an unbiased estimate of P2 ) and £ is the

expectation operator over the randomization process which selected

the field of observation;^ Formula (5.1) expresses relative

variance as variability of P2 about P2 plus variability of P2 about

P3. In statistical jargon these variabilities are "within field"

and "between field", respectively. Any investigation of precision

of a global property estimate should consider both variabilities.

The number of measurements should be thought of as a product

(measurements/field) x fields. Formula (5.1) is primarily

conceptual and of limited practical value without strong

assumptions on the three-dimensional structure.

A simple structure for which working formulas have been

developed is randomly distributed alpha phase particles in a threedimensional beta phase matrix. Hilliard and Cahn, (19G1) consider

the relative advantages of areal analysis, random and systematic

point counting, and lineal analysis for volume density estimation.

They conclude that coarse grid systematic point counting is most

economical. Formula (5.1) for coarse grid systematic point

counting reduces to 1/Mct, where is the expected number of points

in the grid intersected by alpha phase particle profiles.

Nicholson and Merckx (1969) develop variance formulas for

estimation of particle size distributions, IVhile outside the realm

of basic property estimation, this work shows the type of variance

formulas that can be developed once sufficient assumptions are made

about the geometry and spatial distribution of the threedimensional structure. Scheaffer, (1975) determines an approximate

within field variance formula for a line intersection count

estimate 7tN/2L of the boundary between a tv/o phase Poisson field

generated planar mosaic. A simple upper bound for this relative

variance is 2/N.

C . Empi r i ca 1 Approach

When the structure of the specimen is not modeled, the number

of measurements question must be determined empirically. Volume

density data from rat lung specimens illustrates the empirical

approach. The author is indebted to Professor Ewald Weibel for

these data. To estimate volume density of alveoli in rat lung, six

planar sections were prepared from each of seven lungs. On each

section 12 fields were selected for analysis. For each field, a single random application of a regular triangular lattice of 168

test points gave a point count ratio estimate Pa/168 of area density in the alveoli phase. The 7 x 6 x 12 = 50ft estimates range

from 0.1tt3 to 1. 000 in a bimodal distribution, a composite of a smooth bell -shaped frequency function (mode of 0.70 and upper

extreme of 0.86), and a spike at 1. 000 consisting of 11*2 (28°j)

estimates. The bimodal ity suggests heterogeneity in alveoli

structure. To learn more about the heterogeneity, the two

462
The mathematical foundations of stereology

subdi s t r i bu t i ons of alveoli volume density estimates (partial

tissue fields and embedded in a single larre alveolus cross-section

fields) v/ere looked at separately. A nested analysis of variance

run on the 362 partial tissue fields is summarized below.

Table 5.1

Analysis of Var i ance Tab 1 e for Al veol i Vo 1 ume Dens i tv Data

Sou rce De,^rees of Freedom Mean Square S i ,":n i f i cance

Lungs 5 0. 07lf07 0. 5°o Sections 35 0. 01778 10 ?c, Fields 320 0.01235

The betv/een-f i el ds mean square is too large to be purely withinfield sampling variability. Thus, real structural changes between

fields on the same section are indicated. The slightly significant

between-sect i ons mean square suggests some global heterogeneity not evident locally from field variability within a section. More data

is needed to establish the magnitude of such an effect. The highly

significant between-lung mean square is primarily due to a single

lung with a partial tissue alveolar volume density of 66"; compared

to a median lung estimate of 73%. An independent analysis was done of the frequency per section

of alveolus embedded fields. Comparison with theoretical binomial

frequencies suggests some degree of systemat i cness in the

positioning of large alveoli features on the section.

Summing the variabilities in the two distributions (assuming

no section effects, neither added variability nor systemat icness

)

gives 0.02U0 as the variance of a single field alveoli density

estimate. Nov/ a partial answer is available to the number of

measurements question. The basic sampling unit is the field and

the measurement a point count ratio from a single application of

the IGo point systematic grid. The 95°i confidence level precision

of an averaged over K fields single lung alveoli volume density

estimate is

23a = 2/oTo2To7k = 0.310//!. (5.2)

For the present study K = 72 \/hich gives a precision of 0.037.

With alveoli volume density of 0.3 this is a two s i gma coefficient

of variation of about h-^. The precision of alternative sampling

plans follows directly from formula (5.2).

REFERENCES

1. Buffon, G. S. ( 1777 ): Suppl . g L'Histoire IJaturelle . Vol. h.

2. Cahn, J. U. (1967): Trans. Het. Soc. AlflE , 239 , 610.

3. Cornfield, J. and Chalkley, H. I,'. (1951): J. ',.'ash. Acad.

Sci .

,

hi, 226.

k. Delioff, R. T. ( 1967 ): Trans. Met. Soc. AlflE , 239 , 617.

5. DeHoff, R. T. and Rhines, F. f!.: Quant i tat i ve Mi croscopv , fIcGraw Hill Co., Inc., Mew York.

6. DeHoff, R. T., Aigeltinger, E. H. and Craig, K. R. (1972):

J. Microscopy , 95 Pt. 1, 69.

7. DeHoff, R. T, ( 1975 ): fleta 1 1 ographv , S., 71.

463
W. L. I^icholson

8. Milliard, J. E. and Cahn, J. W. (1961): Trans, f^et. Soc.

AIME . 221 , l>lk.

9. Miles, R, E.: Probab i 1 i s t i c Foundat i ons of Stereol ogy , this volume.

10. rJicholson, W. L. and llerckx, K. R. (1969 ): Technomet r i cs . n, 70 7.

11. Saltykov, S. A. (195it, 1958 and 1972 ): Stereometr i c

tleta 1 1 ORraphy , Moscow, State Publishing Mouse for Hetals

Sc i ences

.

12. Scheaffer, R, L. ( 1975 ): J. flicroscopv , in press.

13. Smith, C. S. and Guttman, L. (1953): Trans. AiflE , 197 , 81.

\k. Struik, D. J. (1950): Lectures on Classical Differential

Geomet rv . Add i son-Wes 1 ey Press, Cambridge, Mass., p. 76

and 81.

15. Tomkeieff, S. I. (191(5): Mature , 155 , 2h

.

16. Weibel, E. R. (1963): Lab. Investigation , 12, 2.

17. l.'eibel, E. R. and Elias, H. ( 1957 ); Quantitative flethods

i n Mor phol Qgy , Berlin, Springer Verlag, p. 12.

464
PARTICLE SCIENCE

Chairperson

:

B. H. Kaye

National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at N8S. Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

WORKSHOP SESSION- PARTICLE SCIENCE

by Brian H. Kaye, Chairman

Institute for Fineparticle Research, Laurentian University, Sudbury. Ontario, Canada

INTRODUCTION REMARKS

Fineparticle science embraces those systems in which at least one component exists in a finely divided state such that the physical

properties and behaviour of that component are determined by the

competition between surface and microscopic properties. (Note:

Classical physics deals in general with systems in which surface

forces can be neglected, colloid science is concerned with those

systems in which surface forces are dominant)

.

Stereology has been concerned historically with those fineparticle

systems in which the dispersed particles are trapped in an inaccessible matrix such as particle of ore in a piece of rock. When

characterizing f ineparticles in a rock matrix the analyst has a- vailed himself of the classic work of Rosiwal and Chayes

.

From a fundamental scientific point of view pores in a sintered

ceramic or a piece of coke also constitute a dispersed phase of zero density. Again the analyst in this situation has used stereological technology to measure the inaccessible pores of this

type of system by making a section through the system and conducting an examination of the two dimensional system exposed in the

section. One paper of this session deals with a fineparticle problem which up until now has remained intractable , that of determining the size distribution of pigment particles as actually dispersed in a paint film.

Wherever possible the fineparticle analyst has chosen to free

particles from an inaccessible matrix and study them in an alternative medium because of the difficulty of obtaining enough measurement from sections to gain sufficiently high levels of confidence in the measured parameters . The advent of the new automated

iconometric systems such as the Bausch and Lomb Omnicon, the Leitz

Texture Analyzer and the Imanco systems, each of which can

be interfaced directly to large digital computers, open up the possibility that the analyst will turn more and more to the technology

of stereology in order to characterize his fineparticle systems.

In general when discussing a fineparticle one talks about its ex- ternal morphology , that is its gross shape characteristics, its

surface topography which concern itself with the humps and hollows

that are visible on the surface of the particle, the particle tex- ture that is any obvious regular structure discernable as being

superimposedpon the topography of the particle and the topology

of the partifile, that is the relationship of holes within a particle to its total structure. Thus a particle which contains no pores is said to be Genus 0 one topologically speaking, any particle which contains a pore which does not reach the surface of the

fineparticle is termed a Genus 1 solid and so on. The fineparticle

analyst does not have good techniques for determining external morphology of very f ineparticles and one of the areas of progress may

well be that of developing techniques for placing f ineparticles in

a cast resin and giving them a preferred orientation by the application of electrostatic forces.

467
B. M- K-aye

In summary, in the past the fineparticle analyst has avoided

stereological measurement wherever possible because of the difficulty of making enough measurements on a section. As we will

probably find out in this workshop session the use of new automated iconometrics technology will probably shift the balance of

preference of the fineparticle analyst in his choice of techniques

to include more often stereological examination.

468
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg. Md.. September 4-9, 1975 (Issued January 1976)

STEREOLOGY AND PARTICLE TECHNOLOGY

by Hans Eckart Exner

Max-Planck-lnstitut fur MetalIforschung, Stuttgart. Germany

Quantitative applications of microscopy are common features

in the evaluation of microstructures and in the testing of particulate matter. Much of the progress in one field would be useful

in the other as well.

Figure 1 shows, for example, the evaluation of size distributions. Graticule sizing as developed for particle technology (P/T)

is now used in stereology (ST) in many other fields of research

(e.g. biology, anatomy, geology, materials technology) as well.

Lineal analysis, on the other hand, has been developed in ST and,

though infrequently, is adopted for P/T. Mathematical evaluation by

special functions (e.g. the RRS distribution) is more common for

powder particles but is often applied to particle and grain size

distributions as well. Television microscopy is utilized in both

fields

.

Figure 2 gives a review of some geometric quantities and

their importance in particle and materials technology, respectively.

Some of them are very important in both fields (++) , e.g. specific

surface, others cannot be used (-) , e.g. the degree of patterness

for characterizing a powder. As in Fig. 1, the arrows show in

which direction parameters were adopted.

Field P/T ST

Graticule Sizing

+ —- ()

Television Microscopy

Lineal Analysis

-- ++

Mathematical Treatment

+ + ^ +

Field — Particle Technology Materials Technology Features — Particles or Pores Grains or Particles Parameters

Vv

Sv5 + * U) Ul

f(D) Shape () - U)

D1/D2

Orientation

Patterness 7

Fig. 1 Size distribution analysis Fig . 2 Geometric quantities

in ST and P/T, in ST and P/T.

Powder particles are usually studied in projection. The first

part of Figure 3 shows some relationships derived in mathematical

stereology between quantities which can be measured on projected

images and some useful parameters for describing the geometric

nature of powder particles. Mean surface curvature is not yet used

for powder characterization. Even more interesting is the adoption

of parameters calculated from quantities measured from random

cross sections (Fig. 3) . Random cross sectioning of a powder is

possible by means of dispersion in high-viscosity, fast hardening

resins. A concise characterization of specific surface and shape

is possible in this way.

For determining particle size distributions, stereological relationships are only available for spherical particles (see Fig. 3) For discrete, insoluble particles in opaque materials, etching away

the matrix would allow the use of all the methods developed for particle size distribution analysis in P/T. In Figure 4, some of these

469
H. E. Exner

methods are evaluated with

with physical properties a tungsten carbide powders

pic analysis using a grati

tive and accurate methods

the accuracy of a size dis

(Fischmeister and co-worke

the calculated and measure

the method is not useful i

respect to sensitivity, correlation

nd accuracy according to an analysis of

Exner and Fischmeister, 1966). Microscocule proved to be among the most sensiFigure 5 shows a way to characterize

tribution analysis developed for P/T

rs, 1961, 1963, and 1966). Deviations of

d curves for a 1:1 blend show that

n this size range.

Proiection

1 S = A A'

3 L'p =eNl/N4

5' n(d,)=[a„n(l;)-Ia n(l,.,ii/l,

1=1

'

Section

1 S„ =4N/LtVv

3 f, -- pI m|/Vv Na M,M3

5*N(D..„2)=-^I2N(L,)-N1L,.,11/L^,

2 D' =Nl/n;

4 Km =TlD'/2A' = f/L'

6* NlD.l =i N IL',lA,/&

2 L = 4/Sv= Lt Vv/N

4 f2 = 12S^'^/TtV =M4 (L)/L'

6'NID,.„2l=-|-IN(U/l2L,-A)-

NlL,.,)/(2L,*All/A JmM

Correlation Fit

I

I

Fig. 3 Useful stereological re- Fig . 4 Efficiency of

lationships for P/T. sizing methods.

Accuracy particle

Particle size [m^]

Fig. 5 Additivity test for measured

size distributions (dashed curves calculated, curve 5 measured for 1:1 blend

of powders of size distributions 1 and

4) .

Fig . 6 Comparison of

size distributions of

WC particles before and

after sintering (a) coarse powder , (b) blend 1:1 of a and c,(c)fine powder.

An example of the application of ST to P/T is shown in Figure 6. Here, the size distributions of particles of WC-powders are compared

with those of the WC-grains in cemented carbide samples sintered

from these powders (Fischmeister, Exner, Lindelof, 1966). Useful con- clusions on sintering kinetics can be drawn if the methods used for

the analysis are comparable.

In conclusion, stereology can be usefully applied to particle

technology. However, more effort is needed to make stereological me- thods more widely known in all fields of possible application.

470
National Bureau of Standards Special Publication 437

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

REPRESENTATION OF PARTICLE SIZE AND SHAPE

by B. Scarlett Department of Chemical Engineering, University of Technology, Loughborough. Leicestershire, England

1. Introduction

The subject of stereology is concerned with the measurement and quantification of the geometry of a inhomogeneous medium and it is fitting that, at a conference on stereology, we should consider the purpose of making such measurements. The ultimate objective of making such measurements must be to relate

them in a quantitative manner to the behaviour of the medium. In a three dimensional world it is possible to measure only length, area or volume. All of these parameters are scalar parameters and thus, in that form, they can only be related to other scalar parameters. On the other hand, it is known that many of

the physical properties of particles are tensor quantities of varying rank. For

example, the viscous resistance to motion of an irregular particle has been shown

by Brenner (1 ) to be described by three tensors of the second rank. The question to be asked is whether the scalar measurements of length, area and volume

can ever be related to such complicated parameters. The answer to this question

lies in the realisation that, in making the scalar measurements of length,

information has already been discarded about the relative position and orientation of the extremities of the length. Another way of stating the same fact is that, in making the scalar measurements, the associated unit vectors have been

disregarded

.

In this paper the philosophy is proposed that, in all problems which

attempt to relate the geometry and the behaviour of a set of particles, it should be assumed that the complete vector locus of the particle array is known. Although the paper is addressed to particulate systems, the formulation proposed

is equally applicable to other studies.

If the complete vector locus is assumed, it can then be reduced to the form necessary for the solution of a particular problem. Only at that stage, is it necessary to consider how to actually make the measurements. It should be emphasised that each physical problem requires a different parameter of varying

complexity and that there is no universal particle parameter, there are only the basic building blocks of length, area and volume. Furthermore, it should be recognised that, in regarding these measurements as the basic building blocks,

a set of Cartesian axes has been subconsiously imposed on the problem. The basic vector locus is completely invariant of axes and it would be possible to devise a different stereological system such as the polar coordinate approach

as proposed by Cheng and Sutton ( 2 )

.

2. Particle Vector Locus

A particle shape is itself a vector locus. The particle outline can be considered, with respect to any arbitrary origin to be the three dimensional

graph of its own particle size and shape. This is illustrated in Figure 1, where the vector r_ denotes the position vector of any element of surface, dS

.

An array of particles is similarly a unique vector locus. The vector locus is independent of axes or co-ordinate systems. It can. however, be changed by

reorienting the particle or moving the particle. Thus, it is immediately apparent that the locus contains more information than simply the size and shape of

the particle, it also contains information about the position and orientation of the particle. It is clear that this total amount of information can be discarded

471
B. Scarlett

Figure 1

and reduced to parameters which are independent of these factors if it is required to isolate the particle parameters from those of the array.

In considering the vector locus of a single particle, or of an array of particles, as shown in Figure 1 we could consider that, in principle, the position vector of each element of the volume is recorded. Alternatively, the outline is recorded if each element of the surface of the particle is known. In fact, neither of these sets of data is sufficient to generate meaningful parameters about the particle. Not only must the position of each element of the surface be known, but also the direction of some of the vector quantities

associated with that element of surface. The most important of these is the unit vector, ii, which lies at right angles to the surface. The other vectors

are those associated with the curvature of the surface. These vectors corres- pond to first and second differentials of the position vector and it would be possible to use also even higher orders whose physical significance is less apparent. The assumption that these vectors are known, therefore, is quite

compatible with total knowledge of the vector locus, it means that the locus can also be differentiated. The use of the volume position vector would require a similar autocorrelation to be useful but is not necessary for particles which,

although they may be hollow, have a homogeneous solid texture.

3. Particle Parameters

From a set of vector quantities, tensor quantities of any order may be formed. Thus, two vectors form a dyadic which is a completely general form of a second rank tensor. This form of tensor algebra is well described by Gibbs and Wilson (3). It should be emphasised again that the purpose of stereological measurement is to determine a strategy of breaking the particle vectors into

constituent scalars which can be measured. However, from the particle vectors, tensors of any rank can be formed.

The tensors may then be integrated or averaged over the whole particle

surface to form particle parameters. Alternatively, the limits of integration

may be restricted, for example to the exterior surface of a hollow particle.

In this way, it may be possible to describe mathematically the most complex

physical behaviour.

For the purpose of illustration, it seems worthwhile to list all the possible combinations of vectors and dyadics, integrated over the whole particle

surface, which can be formed from the position vector, r^, and the unit normal,

n. The properties of each are described although the description of physical

properties usually require more complex combinations of the basic vectors.

(a) rJ ds r^ is a vector quantity

472
Representation of particle size and shape

It expresses only the position of the centre of gravity of a thin shell which

has the same shape as the body with respect to the origin 0. This is easily

proved. If r is the position of the centre of gravity with respect to the —oc _ arbitrary origin, 0:-

f ds r = f ds r + / ds r

' — op ^ — oc — cp

s s s .'.fdsr ' —

= Sr +0 op — oc

s where S is the total surface.

The second term is zero by definition.

(b) / ds n^ is a vector quantity and is zero by definition.

s Thus, the projection of the surface in any direction, is also zero:

i.e. /dsn.i^ = 0

s where is the unit vector in some fixed direction.

However, the integral of the modulus of the projected surface is, of course,

a commonly used stereological property.

/ ds |n . i^l = twice the total projection in direction i^ and is the

s quantity relating the mean chord and the volume.

^ = i / ds |n.i|

L

2 s

where i^ can be any direction

It should be emphasised that this parameter, total projection, is not a resolvable quantity. If a mutually perpendicular set of axes, i , i , i are formed and the projection measured in each direction, the result- ^

^ ant is three scalar values of the projection. Because of the modulus sign

around the direction cosines, it is impossible to rotate these values to another

set of axes. It is not, therefore, in any sense a reliable measurement of anisotropy. Consequently, the mean chord is also not a means of detecting

anisotropy

.

(c) / ds r^ r^ is a second rank tensor.

It is of limited interest. Its scalar invariant represents the moment of iner- tia of a thin shell of the same shape.

Thus:- / ds r.r = S. M.

s

(d) J ds n^ 11 is a second rank tensor.

This tensor is a most convenient means of expressing anisotropy in a particle

array. The tensor can be resolved in any direction, i^.

Thus:- / ds n^ n^ . is a vector quantity which may lie in any direction,

s

473
B, Scarlett

It expresses the direction of the normal to the surface averaged with respect to the projected surface in direction i^.

If the particles are all randomly orientated the tensor reduces to a simple form:- /dsnn=-|si^

s

where X is the idemf actor (unit tensor).

Even if the system is not randomly orientated, the scalar invariant of the tensor is the total surface, S. The tensor can thus be made dimensionless by

dividing by S

.

It should be emphasised that this tensor expresses anisotropy due to particle orientation, as illustrated in Figure 2a, not that due to particle

position as illustrated in Figure 2b.

Figure 2a Figure 2b

This tensor is resolvable in the sense that if it is determined in one

set of axes it can be rotated into an alternative set. This statement is really

an expression of the fact that anisotropy is a tensor quantity and cannot, as shown in section (b) be expressed as a vector.

(e) ! '^^ E. IL reduces to a very simple parameter.

_s

Thus :- /ds£n = I^V

s where V is the volume of the particle or particles.

This is easily proved by considering the particle locus in Cartesian form

and multiplying by the idemf actor.

474
Representation of particle size and shape

/dsrn = /dsr^n.Ci^i^ """ii ^k)

s s

.'. Jdsrn = /(ix + J.y +kz){ (n. i_) ds + (n- j.) ds

s s + (n.k) ds k}

The diagonal elements of the tensor are each seen to be equal to the volume of

the particle i.e.

J X ds (n..i^) = / y ds (n.j^) = / z ds (ii-k) = V

s s s

The other elements of the tensor are zero because the projected surface at a fixed coordinate forms a thin slice, as shown in Figure 3, and its value is zero parallel to the coordinate. Thus, each of the other six terms in the tensor is zero

.

Thus, the scalar invariant of this tensor is also related to the volume.

/ ds i^.n = 3V

s

This can be most easily illustrated by considering Figure 4 . The volume of the element of particles is the difference in volume of the two cones which have 0

as the apex.

475
B. Scarlett

It follows also that the vector invariant of the tensor is zero.

Thus :- /ds£xii = 0

s

It also follows that the tensor is symmetric.

Thus :- ! L B. ~ J'^^ B E. s s

These generalised expressions of the particle size and shape are particularly convenient because they can be manipulated mathematically. This is in contrast to the statistical distributiors of ten produced by measurements such as chord lengths. Furthermore, although the various moments of these distributions

can be used in a similar manner, their relationship to a physical problem is not obvious. In contrast, the generalised expressions can arise naturally as the solution to a problem. The ease of manipulation of the expressions can easily

be appreciated by the well known transformations for translation and rotation

of the axes.

a) The particle can be moved with respect to the origin, 0, to a new

origin, p, by subtraction of the transposition vector:-

- - -op

b) The particle may be rotated from axes i^, j^, k to axes i^ j , k''' by

multiplying by the idemfactor.

1 11 , 1 , In

Conclusions

In considering the characterisation of a particle array, the number of possible parameters is infinite. The choice of the appropriate parameter for a particular problem is most logically made by assuming a knowledge of the complete vector locus and then mathematically reducing that data to the minimum

amount necessary for the problem. At that stage, the measurement strategy should

be considered. This philosophy is in contrast to postulating a parameter and attempting to relate or correlate it with any convenient property of the system.

Complex parameters can best be formed from the position vector of an element of surface and its various differentials juxtaposed to constitute tensors

of varying rank. The tensor quantity is usually integrated over part or the whole of the surface.

References

1. Happel J., Brenner H. , Low Reynolds Number Hydrodynamics (Noordhoff) 1973

2. Cheng D.C.H., Sutton H.M., Nature (Physical Science) 232 No. 35, p. 192 (1971) Sutton H.M. , Bunalli M., Proc. Soc. Anal. Chem. p. 13, 1973

3. Gibbs J.W. , Wilson E.B., Vector Analysis, (Dover) ( 1960)

Acknowledgements :

The author is grateful to the British Steel Corporation for the award of a Fellowship during the academic year 1970/71. The ideas contained in this paper were formulated at that time.

476
National Bureau of Standards Special Publication 437

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

THE APPLICATION OF STEREOLOGY TO PARTICLE TECHNOLOGY

by B. Scarlen and P. J. Lloyd Department of Chemica/ Engineering, University of Technology, Loughborough, Leicestershire, England

Introduction

One of the major problems facing the Particle Technologist in formulating the basic science of particulate systems which are found

in the form of powders, slurries, pastes, emulsions or aerosols is how to describe the discreteness of the particles. With the exception

of spheres, no particle can be described by a single parameter and throughout the development of the subject use has been made of the concept

of an equivalent sphere, that is to say the diameter of a sphere which would

have the same physical property as the true particle. Thus an irregular

particle may be described by a number of equivalent spheres diameters with

respect to its volume, its surface area, its settling velocity, its ability

to pass through a sieve mesh etc. This situation is in itself difficult

but it is made more complex by the fact that real particulate systems consist

of distributions of particle diameters and possible also of particle shapes.

This problem has not been tackled by many investigators who prefer to ignore

the problem and use a mean size which is often dependent both on the analytical technique used and the material used in the study. This is not very satisfactory and there is a need for a method to characterise

distributions of particles both uniquely and in a way such that the characterisations can be used to describe the physical properties of the particle system.

In 1966 Rumpf 8c Debbas (1) used many of the principles of stereology

in a study of the packings of size distributed spheres and irregular

particles. They did not use any of the stereology standard references but did refer to a paper by an astronomer Wicksell (2) who used stereology

principles in the analysis of photographs of stars taken through a telescope. Rumpf & Debbas were interested in discovering if size segregation of spheres occurred in the packing and analysed the size of the exposed areas at a series of sections through the packing. The size

distribution of the areas were related back to the original sphere

distributions by relatively simple formulae. They also attempted to use

the same method with irregular particles but the result was not as elegant

or convincing. Other workers notably Dullien & co-workers (3) (4) have

followed a similar approach but in each case an attempt is made to relate back the distributions of the areas or laminae or of the chords, or filaments, back to the original particle size distribution. This method

is severely limited since this relationship is only valid if the shape of the particles is known and is the same for all particles. In 1970,

Scarlett (3) proposed that the sub-particle characteristics of laminae and filaments together with the volume of the particle were a sufficient

description of a system of particles and moreover could be used to describe

some of the physical properties of particle systems.

Sub-Particle Characteristics

A particle may be considered as being cut into a large number of laminae

each with an identical but small finite thickness. The area of each of the laminae is clearly well defined and easily measured. If the same particle was sliced in all possible directions, the resulting distribution of the areas

477
B. Scarlett and P. J. Lloyd

would uniquely characterise the particle . It may not be possible to reconstruct the particle but the distribution will contain all the information

concerning the particle diameter and shape. The distribution is also easily

measured. A large number of identical particles are scattered onto the surface of a cold setting resin close to setting point. More resin is poured

on top and after the resin has set the block is cut through the plane of particles. After polishing the plane the areas of the particles are measured

using an automatic microscope. Each of the laminae can now be considered cut into filaments of small but uniform cross-section. The resulting distribution

of the lengths will also be characteristic of the particle. This procedure

is shown in figure 1.

Particle Laminae Filaments

Fig. Distribution function of a number of laminae against area

Of course it is not necessary to consider just a single particle as a distribution of particles of different sizes and shapes can be analysed in exactly the same way. In fact the filaments having only a length dimension

can be considered as the primary sub-particle characteristic. The filaments

can be linked together to form laminae and a distribution of areas, or to form complete particles and a distribution of volumes. This is illustrated

in figure 2.

Particles

n{v)

Laminae

n{a]

Filaments

Fig. 2 Distributions that uniquely characterise a set of particles

The distributions of filament lengths will not however uniquely define a particle system since they can be linked together in many different ways. A similar argument applies to the distribution of areas, but the three distributions together will contain sufficient information to define a

478
The application of stereology to particle technology

system of particles. The difficulty of defining shape has been avoided but the definition will be implicit in the three distributions.

Fortunately these three distributions can be measured relatively easily.

The distributions of the filament lengths and laminae areas are very easily

measured on an automatic microscope of the Quantimet or Time type. Indeed these distributions are easier to measure than the conventional microscope

equivalent sphere diameters such as Feret's or Martin's. The distribution of volumes can easily be measured with a Coulter Counter since it is the distribution that is always measured but which is always converted using an assumption of constant shape factor into a weight versus the diameter of a sphere of equivalent volume. Thus techniques are available to measure the distributions. Fortunately, however, as the following sections indicate,

only one of the distributions may be needed to describe a physical property

of the particle system.

Critical Porosity

A bed of particles can only be sheared easily when the porosity has reached a value known as the critical porosity. If the density of the packed

bed is too great the bed must dilate to the critical porosity before any relative movement of the particles is possible. This property is peculiar to particulate systems and was named by Osborne Reynolds as the property of dilatancy. This critical porosity is a function of the size, size distribution and shape of the particles. Scarlett and Todd (6) measured the critical

porosity of the sheared plane in an annular shear cell (7) by scanning the cell with an x-ray beam. The attenuation of the x-ray beam is a function of

the solid material in the beam and was calibrated directly against each

material used.

Using the concept of sub-particle characteristics, Scarlett et al (8) are able to predict the critical porosity. Two assumptions are made initially:

(1) The bed is completely random (i.e. any particle has an equal chance

of being at any point in the bed).

(2) The bed is initially isotropic (i.e. any line drawn at random through

the bed will indicate both the same filament size distribution and the same ratio of void filaments to the total length of line (i.e. the same porosity )

.

When the bed is sheared the motion of the particles is very complex

but three distinct components can be detected:

(a) the particles translate perpendicular to plane of shear in order to pass over one another;

(b) the particles may have motion relative to one another in the plane of shear because they may overlap;

(c) the particles may rotate in the plane of shear to present a smaller

aspect perpendicular to the plane of shear;

A random distribution of particles (Figure 3) can be represented by

filaments selected as those perpendicular to the plane of shear. Movement of the particles relative one to another is now represented by the relative movements of the filaments. Two planes AA-'- and BBl can be drawn through the filaments such that AA^ moves relative to BB-'-. These planes are drawn parallel

479
B. Scarlett and P. J. Lloyd

Fig. 3 Random chord distribution

liiI'lH"

|i|>ii I 'I' I I

Fig. 4 Representation of a bed of particles by random chords

A —

W

##1

mm

Fig. 5 Representation of sectioned chords contained in a critical volume

Fig. 6 Condition for flow

480
The application of stereology to particle technology

but in fact must be tortuous and will cut each filament at some random

position. The separation between AA ' and BB ' will be the critical value when movement occurs and will depend on the randomly sectioned filament

distribution g(y) which is related to the filament distribution f (x) by the formula

r x=x

I max

g(y) =

I dx Jx=y

Clearly if the separation is greater than x max flow can occur but the critical porosity is less than this maximum value. By considering

conditions (b) and (c) Scarlett & Todd concluded that the separation should equal twice the medium value of the sectioned filament distribution Ym. This

is demonstrated in Figure 6, and the critical porosity ec found to be given

bv the relation

fx fj

/ max /

2 1 y g(y) dy

I •'o Jo

2 1 y g(y) dy / y g(y) dy

1 - ec =

m /•'o

max

2 ^\ g(y) dy Ym

Experimental results are compared with the predicted for three samples

of sand and the results shown in Table 1. As is seen the predicted results

are all low but the agreement is close.

Table 1 Critical Porosity Measurements

Mesh Experimental Critical Predicted

Size Porosities Critical Porosity

B.S.S.

14-18 .481 .475 .472 .452 36-52 .455 .452 .446 .444 60-85 .459 .458 .454 .443

Structure of Packed Beds

In a recent investigation, Ward et al (9) examined the structure of packed beds of sand particles formed in by number of different methods:

by filtration, by dry compaction and by sedimentation. In each case the final

compact was set in resin and sectioned. After careful polishing the compacts

were examined with an Quantiraet image analysing system. The purpose of the investigation was to find out if there were detectable differences in the structure. Since the study was primarily concerned with filtration attention

was focussed on the void space and the particles selected were rounded sand particles which would show little orientation effects. The compacts were analysed in two directions both in the direction of flow or applied force and perpendicular to that direction. Each time the filament length distribution

of both the solids and the voids was determined. A one way analysis of variance was used to compare the variance of filament distribution in the direction of scan with that perpendicular to the scan. The results presented

showed the value of the technique and showed that the structure of the beds of particles depended on the method of construction.

481
B. Scarlett and P. J. Lloyd

Transmission of Force through a Powder Compact

In 1972 De Silva (10) explored the possibility of using a Monte Carlo

technique to explore the transmission of force through a bed of particles.

The basic concept was to consider the equilibrium of each particle of the bed

and the transmission of force through the points of contact. To do this it was necessary to be able to predict the positions of point contacts on each

particle. The maximum distance in the direction of the applied force will be one of the Feret diameters. If the distribution of Feret diameters is sectioned randomly, as described previously, a point A is obtained. A second sectioned filament at right angles will give B. The position r and direction

0 of the contact B from the input of force 0 is now known. In the azimuthal

direction (() is assumed to be another random variable.

By representing the sectioned Feret diameters and the filament lengths

as cumulative fractional number undersize v. length, a representative length/

diameter could be obtained by selecting a random number from 0 to 1 and referring this fraction to the distribution.

De Silva traced a uiit input force through the bed of particles by

considering the equilibrium on a large number of particles. Only one force was traced to the container wall. The process was repeated many times until

the distribution of force on the container was obtained. The predicted

values agreed well with those measured with small pressure transducers.

Conclusion

This paper has put forward the suggestion that the distributions of the sub particle characteristics, filaments and laminae together with the distribution of volumes are a complete description of a particle system which

are easily measured and avoid the difficulties of defining shape since such information is implicit in the differences in the distributions. Furthermore

it has been shown in three cases that these sub-particle characteristics can

be used in describing and/or predicting some physical properties of a system

of particles. Clearly more research into this concept is required but it is believed to have sufficient promise for an extended research programme.

BIBLIOGRAPHY

1. Debbas, S., & Rumpf, H., (1966) Chem. Eng. Sci., 2_1, 583.

2. Wicksell, S.D., (1925) Biometrika, 2/7, 84.

3. Dullien, F.A.L, & Mdita P.N., (1971/72) Powdr Technol, 5^, 179.

4. Dullien, F.A.L. , & Dhawan, E.G.K., (1973) Powdr Technol, 1, 305.

5. Scarlett, B., Particle Size Analysis 1970

(Pub. Soc. for Anal. Chem., London).

6. Scarlett, B., & Todd A.C. (1969) Trans. ASME 91 478

7. Scarlett, B, & Todd A.C. (1968) J. Sci. Inst. 1 655

8. Scarlett, B., Akers, R.J., Parkinson, J.S., Todd, A.C, (1969/70

Powdr Technol 3^ 299.

9. Ward, A.S., Lloyd P.J., Smith I., (1973) Chemical Engineer. 584

10. De Silva (1972) Ph.D. Thesis Loughborough University of Technology

482
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

TECHNIQUES FOR PARTICLE MEASUREMENT USING IMAGE ANALYSIS

by Roger R, A. Morton Bausch & Lomb. Rochester. New York 14625. U.S.A.

Particle analysis has long been recognized as one of the important

fields of application for image analysis. Accurate measurements

of size distributions for a wide variety of particulates and powders are routinely obtained using image analysis equipment.

Certain precautions must, however, be taken to ensure that the

desired accuracy of measurement is attained. It is a purpose of

this paper to discuss some of the precautions which should be

taken.

Sampling - The first step, one of the most important in obtaining

data which accurately represents a powder sample, is to ensure

that the image presented to the instrumentation accurately represents the parent sample from which it was derived. Very rarely

in particle measurements is it possible to measure eveiry particle

in a sample to be characterized. Thus, it is generally inferred

that the portion of the sample actually measured does indeed represent the entire sample from which it was derived. To ensure

that this inference is indeed valid, care must be taken in selecting and preparing a specimen from the parent sample. (See, for

example, Herdan, Ch, 5.)

The optimum sampling technique will,' of course, depend on the

nature of the particulates, whether, for example, the parent

sample is in suspension in a gas or liquid, or whether it is only

accessible as it passes along a conveyor belt, or is held in a storage bin.

A broad range of sampling techniques are available for use in

these various situations. To be successful, such techniques must

be able to independently select every particle which is to form

the selected sample. Furthermore, in general, this sample must

be selected from throughout the entire parent sample.

Overlap - The methods of mounting a selected sample on a microscope slide are numerous and well documented. A major requirement is that individual particles are made to lie on the microscope slide at random positions, uniformly across the slide, and

that the position of each particle be independent of the position

of other particles. These precautions will, of course, not ensure

that overlap does not occur but they do permit us to predict the

degree of overlap which we can expe^ct. Today's image analysis instrumentation, such as the Omnicon Pattern Analysis System, provides the user with a number of alternatives and solutions to the

overlap problem.

1. Based on a prior knowledge of the expected shapes of the particles to be measured, it is possible, using shape and size discrimination to simultaneously identify agglomerates from individual

features and to simply measure the individual objects. If the particular distribution has a large spread of size ranges, statistical

correction can be applied to correct for the bias arising from ex- cluding the agglomerates. If the distribution of particle size has

a narrow spread, no such correction need be applied.

483
R. p. A. Morton

2. Based on prior knowledge of the shape of the particles, agglomerates may be identified as above and separated into their component

particles using boundary fitting techniques based on the prior shape

information

.

3. In cases where a more advanced image analysis instrumentation

is not justified, or where the shape of the particles is quite

arbitrary so that agglomerated particles may look just like a single particle, a sample may be prepared with a sufficiently low

concentration (expressed as fraction of area of the image covered

by particles Ap^) and the effect of any of the residual overlap

upon results may be predicted. Thus, for example, the reduction

of count, due to overlap for low concentrations will for circular

particles of identical size be given by Armitage as 4A^.

If the sample has large spread in size, then this reduction in

count may increase to as high as 8Aj^ and if the sample has both

a large spread in size and irregular shapes the multiplier may go

higher. Similarly, the total area of such a sample can be ex- pected to be reduced by the fraction A;^ and the mean size increased

by 3A;^ or more.

If the sample is prepared so that the value of Aa is between 0.5%

and 0.2% and stage motion together with automatic focus options

employed in the image analysis system to provide rapid field to

field analysis; accurate results may be obtained quickly without

significant overlap error. How Many Particles ? - Another question frequently asked when using

image analysis for particle sizing is, "How many particles must be

counted before the results of the specimen can be taken as accu- rately corresponding to the results of the parent sample?" The answer to such a question, of course, depends on the type of re- sults to be extracted. As a general guideline, the accuracy with

which a count within a certain classification can be expected to

correlate, on a normalized basis, to the count for the entire

parent sample is given by (with 9 5% confidence)

X 100%

where N is the number of particles counted in the classification.

A classification may be a particular size range classification or

a classification based on shape, density or other criterion. Thus

classification applies, for example, to size ranges in a size

histogram, where the number in each interval is independent of the

number in any other subinterval.

In an oversize or undersize cumulative distribution, however, this

independence between each classification does not hold and a different expression is required for the estimated error. In a oversize or undersize distribution normalized to 100%, the error (with

95% confidence) is given by Kolmogorov (see Walsh, Ch. 10) as

X 100%

where N is the total number of particles in the distribution.

Other results, of course, apply to measurements such as means,

standard deviations, and so on. Unfortunately, space does not

permit their inclusion at this time.

484
Techniques for particle measurement using image analysis

Selection of Measurements - A wide range of measurements is now available to the image analysis user from systems such as the

Omnicon Pattern Analysis System. The choice of the appropriate

measurement to be used for particular situations is an important

one. It depends largely on characteristics which it is desired to

analyze. Again, space does not permit a detailed coverage of the

correlation of different measurements to the characteristics. However, it should be noted that measurements may be correlated to

surface area, equivalent sedimentation diameter, equivalent filter

of blocking diameter, equivalent weight distribution as well as correlated to a wide range of shape factors, and orientation

measures

.

Data Processing - Measurement data extracted from particulate

sample may be presented in many different ways. A list of the

measurements made for each particle is the most extensive and

probably least informative of these. On the other hand, presenting the data as a size distribution is both compact and yet highly

informative. Many different statistical parameters may be ex- tracted from the distribution including mean, standard deviation,

various higher order moments, median, mode, and percentile measurements, which may be used to indicate the fraction of particles in

the sample outside specified bounds.

Figure 1 shows display of size distribution as a sideways plot on

a television monitor. The upper part of the display shows the

commands xised to obtain the measurements and resulting size distribution. Next comes the statistical parameters computed from the

distribution, and finally, the distribution presented as a size

distribution plot rotated through 90°. The left column represents

the size intervals, while the right column represents the number

of particles in each size interval.

MC LP MKHMtKU.SUmZt.EXCLUK.KHSITV.MITOIlS

SELECTIOH P80CESSINC lie.«.4( OUTPUT: SCE NEASURENENT A/C mm-J IMX'37 MN FEATUSES-81888 ICMN13 4312 ST KV •C 93848 FIELK'8M28 iM mm HICINM239

STACT: • IIEtCMT«M«4

4 MM 88238 m^^^m

8 IMM 8889« ^^^^^

12 MM M123

1( MM 88W7 n

24 MM

MM

889S1 88829 ^1^^^^^^

28 MM 8884S

32 88M 88822 r 3( tM« 88828 1

48 MM

Figure 1: An example of data processing output as presented on

the Omnicon PAS display. The measurement area/longest

dimension (A/C) was measured for 1880 features over 20

fields of view. The distribution is shown over 10

intervals in the range 0 to 40 microns.

485
R. R, A. Morton

REFERENCES

ARMITAGE, P. An overlap problem arising in particle counting.

Biometrika, V.36, pp. 256-266. (1949).

HERDAN, G.Small particle statistics. 2nd ed.

Academic Press Inc., New York, N.Y. (1960).

WALSH, J.E.

Handbook of non-parametric statistics.

Van Nostrand, New Jersey, (1962)

.

486
National Bureau of Standards Special Publication 431

Proceedings of the FOUFTTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

MICROSCOPIC CHARACTERIZATION OF FINE POWDERS

by P. Ramakrishnan Department of Metallurgical Engineering, Indian Institute of Technology, Bombay-400076, India

ABSTRACT

Problems associated with the characterization of fine flaky powders in the subsieve range have been considered. Although water covering area can be iised to characterize the inass of powder, special attention is focused on scanning

electron microscopy for the study of agglomeration, particle morphology and surface characteristics. Since the chemical reactivity of the particles is also related to the internal structure, transmission electron microscopy can provide

very valuable information. The importance of microscopic observations in the characterization of these particles has also been discussed.

INTRODDCTION

Although characterization of powders is of fundamental importance, the factors which characterize an individual particle and a mass of powder are difficult to determine. Further different test methods used to determine a single property, say particle size of a powder, do not have identical

results. At the same time these characteristics are very important in the compacting and sintering of metallic and ceramic powders, the hiding power of paints and pigments or the sensitivity of a chemical reaction. Though the shape of a powder is an important characteristic, it is difficult to define.-^ By using the surface contour concept as a guide line it may be possible to distinguish rounded, irregular, angular, etc., particles. Many attempts have

also been made for shape characterization on a quantitative basis by the use

of various mathematical treatments including the dynamic shape factor of par tides. But these are based on simplified models and are far away from

the real characteristics of the particles. When the particle size of the powder

is reduced the powder becomes more cohesive and sticking and the flow is restricted. The problems related to handling and dispersion become more acute. At the same time large quantities of these powders are required in many industrial applications. In this study the characterization of fine flaky aluminum

powders used in the paint, pigment and explosive industries have been attempted.

EXPERIMENTAL WORK

The powders were prepared by the atomization of a thin stream of molten

metal passing through a nozzle with compressed air under pressure. 5 Since

the characteristics of powders are different for different applications^

the atomized powders were wet ball milled to achieve the required characteristics. The powders were filtered, dried and the -325 mesh fraction (44 microns) was used for subsequent investigation. A simple sieve analysis is not sufficient to characterize the powders and hence other methods have been

tried. For fine flaky powders, covering area on water can give reasonably good

reproducible results. But this again does not provide much information about

the size, shape and surface topography of the particles. One of the best

approaches will be the direct observation of the particles microscopically by

using suitable techniques. Hence subsequent characterization studies were made

by using optical microscopy, transmission electron microscopy and scanning

electron microscopy. Only few micrographs are presented to illustrate the salient features.

487
p. Ramakrishnan

RESULTS AND DISCUSSION

Optical microscopy is simple and has been of considerable value in the characterization of certain powders. But there are many limitations when ap- plied to fine particles. This is because of the limitation in the resolution

at the highest available magnifications. Fig. 1 shows the optical micrograph

of the powders. Although one can observe a comparatively large number of particles which may give some idea about the distribution, this micrograph

does not give much information about the shape and other surface characteristics. Scanning electron microscopy offers many advantages^'^>9 because of its better resolution, depth of focus and range of magnifications (about 20 to 50,000). In the scanning electron microscope one can achieve resolutions

approaching 200 A and a depth of focus of about 300 times the optical microscope. The scanning electron micrograph fig. 2 can provide considerable

information about the aggregation, particle morphology and surface characteristics. Fig. 3 is the micrograph of the same material at a higher magnification showing more details about particle morphology and surface topography.

Particles which appear to be single at a low magnification really consist of an aggregate of smaller particles. Another interesting feature is that the information obtained from scanning electron microscopy regarding the particle

characteristics can be used for the interpretation of the results from the water covering area of the powders. Fig. 4 and fig. 5 are the scanning

electron micrographs of the two different powders with water covering area of 12,000 cm^/gm and 9000 cm2/gm respectively. The lower water covering area

can be attributed to the segregation and the associated morphology of the particles as shown in fig. 5 which is not favorable for spreading, while fig. 4 shows the well developed particle morphology which can cover a larger area.

The reactivity of these powders are also related to their internal structure. Transmission electron microscopy can provide considerable information

about the internal structure. ''^ The powders are flaky in nature and many

particles have thicknesses less than 1000 A, favorable for direct electron

transmission. Fig. 6 is a direct electron transmission micrograph showing a considerable amount of dislocations, while fig. 7 is comparatively free from such defects. Powders shown in fig. 7 are found to be less sensitive in a chemical reaction than the powders shown in fig. 6.

CONCLUSIONS

Microscopic observations are playing a predominent role in the characterization of fine flaky powders. By selecting suitable techniques such as optical, electron scanning and electron transmission microscopy, considerable

information could be collected regarding the distribution, agglomeration,

particle morphology, surface topography and internal structure of these

powders. These are valuable tools for the characterization of powders where

conventional methods are inadequate and do not provide the necessary

information.

REFERENCES

1. Kaye, B. H. , Chemical Engineering _7, 239 (1966).

2. Fischmeister, H. F., et al. , Powder Met. 7_, 82 (1961).

3. Hausner, H. H., Planseeber. Pulvermet. 14_, 2, 75 (1966).

4. Medalia, A. I., Powder Technology 4, 3, 117 (1970/71).

5. Ramakrishnan, P., et al. , Trans. Ind. Inst. Metals 22_, 3, 48 (1969).

6. Ramakrishnan, P., Compaction (1973), ed. A. S. Goldberg, Powder

Advisory centre, 23.

488
Characterization of fine powders microscopically

7. Johari, 0., Metal Progress 94, 147 (1968).

8. Hung-Chi Chao, Modem developments in powder metallurgy _5, 369 (1971),

ed. H. H. Hausner, Plenum Press.

9. Nixon, W, C, J. Sci. Instr. 7_, 8, 685 (1974).

10. Brimhall, J. L., et al., A.S.T.M., S.T.P., 48, 97 (1970).

489
p. Ramakrishnan

Fig. 1. Optical micrograph.

Fig. 2. Scanning electron micrograph.. Fig. 3. Scanning electron micrograph.

490
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, IVW., September 4-9. 1975 (Issued January 1976)

STEREOLOGY OF PAINT FILMS

by Brian H. Kaye and Ian Robb

Institute for Finepanicle Research, Laurentian University, Sudbury, Ontario, Canada

Classical studies of the performance of pigment particles in an organic film matrix have concerned themselves with the particle

size of pigment as measured in very dilute systems before being

incorporated in the paint film. There is a growing realization

that this approach has severe limitations since any section through

a real paint film soon demonstrates the highly agglomerated structure of the pigment dispersed in the paint film. A stereological

examination of paint particles set in the paint film poses severe problems for the analyst. Not the least of his problem is the

development of an adequate sectioning technique. Since the organic

matrix is orders of magnitudes softer than the f ineparticles any

attempt to cut through the film usually ends in the distortion of

the film or the popping out of pigment particles in the sectioning

technique. Determination of the average pigment content in any ex- posed section is relatively simple however the determination of the

agglomeration state is very complex. The complexity of the field

of view is such that automated iconometric systems are not able to

deal with them without extensive interference from the executive

analyst with his light pen. And even then the analyst is taking

decisions as to what constitutes a homogeneous or an existential

agglomerate (i.e. one that exists from the point of view of the

physical process being studied in the experiment) . Two approaches

are being considered at Laurentian University to overcome these

problems on analysing any section through a paint film. The first

technique involves a new optical information processing technology

known as "spatial filtering". In this technology a diffraction

pattern is generated by passing a laser beam through a transparency

of the whole field of view. This diffraction pattern is filtered

in the Fourier transform plane, the spatial filter used being con- structed by studying a single particle with laser light. It is then

possible to reconstruct a field of view in which all the individual

particles which are the same size as the particles used to construct

the spatial filter can be studied as single points of light in the

plane of reconstruction. There is some difficulty in reconstructing

images when the particles are non-symmetrical and there are many

technical problems to be overcome, however the reconstruction of the

field as a set of bright spots enables its size and number information of the pigment to be collected very easily by iconometric devices. Also this process is a parallel processing technology.

In general it should be noticed that stereology workers have tended

to ignore the tremendous power of optical parallel processing for

image analysis and probably one of the coming developments in stereology will be the further use of optical image processing rather than

sequential iconometric analysis based on a linescan logic.

491
B. H. Kaye and I. Robb

REFERENCES

1. Kaye, B.H., Naylor, A.G. : "Spatial Filtering for Detection of

Particles of a Particular Shape". Journal of Colloid and Interface Science, Vol. 39, no. 1, 1972.

2. Kaye, B.H., Naylor, A.G. : "Optical Information Processing in

Characterization by Microscopy". American Laboratory, April,

1974 .
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersburg, Md.. September 4-9, 1975 (Issued January 1976)

WORKSHOP SESSION- PARTICLE SCIENCE

by Brian H. Kaye, Chairman

Institute for Fineparticle Research, Laurentian University, Sudbury, Ontario, Canada

CONCLUDING REMARKS

Fron the papers that have been presented in this workshop we can see the

development of three lines of endeavour. Dr. Exner has focussed attention

on the fact that we can obtain information on fineparticle systems by embedding particles in a matrix followed by section of the matrix. It is probably

generally true that the fineparticle analyst is not avare of the power and the

nunber of relationships that have been developed within stereology for abstracting information from sectioned fields of view. One of the values of a conference such as this is the bringing together of fineparticle analysts with

people in the field of stereology. I am s ure that as a result of this conference we will see a wider diffusion of stereological relationships into the

technology of the fineparticle analyst. We also see developments of mathematical relationships for evaluating fineparticles as viewed through iconometric

systems. In the past the tremendous information level required ha^ tended to

deter the analyst from developing these relationships. Now that we have the

automated iconometric technology available we can anticipate increasing

development of studies such as those presented by Drs. Scarlett and Lloyd

and Dr. DuUien.

What is missing from iconometric technology however is a realization that often

iconometric devices work in the opposite direction from the point of view of

experimental efficiency. They gather all the information in a field of view

and then discard, intelligently, information not required. More efficient

iconometric devices must work the other way; they must be able to discriminate

in a field of view what they need to know and abstract that information for

fut\ire use. Perhaps a name for this approach to iconometrics device would be

"minimal iconometrics" as distinct from present forms of automated iconometrics.

So far in these remarks we have focussed on what the stereologist has to offer

to the fineparticle analyst. There is the other side of the story. It is

obvious that the art of optical information processing of images is more

advanced in fineparticle science than in stereology. The powerful methodology

for the parallel processing of images using holographs, spatial filtering and

Fourier transforms will obviously be a great interest to the stereologists as they become more widely known. In conclusion one can say that this workshop

has been a fruitful encounter between two groups, both of which are interdisciplinary and it should stimulate the diffusion of exciting technology into

both areas of endeavour.

493

SIZE DISTRIBUTIONS

Chairpers on

:

. H. Steele, Jr.

National Bureau of Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg. Md., September 4-9, 1975 (Issued January 1976)

ESTIMATION OF LINEAR PROPERTIES OF SPHERICAL BODIES IN THIN FOILS FROM THEIR PROJECTIONS

by Frank Piefke

Institut B fur Mathematik. Technische Universitat Braunschweig, 33 Braunschweig, Pockelsstr. 14, Germany

We investigate the size distribution of spheres in a given

unit volume V=1 . Consider opaque spherical bodies with the radii

X. (i=1 , 2 , . .

. ,N ) , which are distributed at random in a transparent

matrix; the centers of them may be distributed homogeneously.

A linear property G is defined by (NICHOLSON 1970): ^

e = I H{x ) (1) i=1

where H(x) is a given real function. Two examples of linear

properties are:

N^ with H(x) =1 and with H{x) = 4 tt x .

We take a slice of thickness T and area A from the structure.

This slice contains a certain number AN of sections of the

spheres. We measure the radii of the circular projections of

these sections. These radii may be y. (i=1 , 2 , . . . , AN ). We suppose

that a positive resolution point y exists; particles

with radii less than y cannot be seen.

1

Now we can formulate the estimators of linear properties using the

theory of NICHOLSON. An unbiassed estimator of the linear property

S given by ( 1 ) is

:

AN A A

0 = I h(y ) i=1 ^

(2)

where h(y) is the solution of a certain integral-equation. In our

case h(y) is given by;

(3)

h(y) = ^^^^ ^^'"^o '"^^o' +

J /y2-n2)H'(n) dn)

^o

H(y) must be dif f erentiable . The function f is given by:

497
F. Piefke

f (w) = exp( ) j exp( -t^/2 ) dt (4) w

For example an unbiased estimator of the number is:

AN

1 = 1 This is an old result of my teacher G. BACH. I have calculated

estimators of Ly the sum over all radii, the total surface

of the particles, the total volume of the particles and of the

variances of these estimators. An explicit formula for h(y) is

known if the function H(x) is of the form:

H(x) = M x^ with r=0,1,2,...

Now I have tested the estimation theory by a Monte-CarloSimulation on a computer. A cube of volume 1 is constructed in

which the centers of the spheres are distributed homogeneously.

Then a theoretical density function g(x) of the radii of the

spheres is chosen; a radius given by a random number corresponds

to each center. A real model of a distribution of spheres in space

is constructed by this method. For the function g(x) I have chosen

three cases:

2 1) triangle-distribution, 2) g(x) = B x exp ( -Bx /2 ) and

3) two-point-distribution.

Secondly the slice of thickness T is taken from the cube and the

radii y. are measured. Now the estimators can be calculated and

be compared with the true values of the linear properties.

I have chosen N^ = 1000 spheres in the cube. A few hundred of

them dependent on T and y are seen in the slice. Table 1 shows some results of my calculations.

Table 1

T

^o

A

N

V

A distribution

5 0 1000 1005 2 .26 10^ 2 .28 10^ 1) 5 1000 971 2 .26 10^ 2 .28 10^ 1) 2 1000 101 2 2 .26 10^ 2 .29 10^ 1) 5 1000 1000 4 .05 10^ 3 .99 10^ 2) 5 1000 986 4 .05 10^ 3 .99 10^ 2) 2 1000 992 4 .05 10^ 3 .93 10^ 2) 5 0 1000 1062 2 .38 10^ 2 .42 10^ 3)

5 1000 1051 2 .38 10^ 2 .41 10^ 3)

2 1000 1085 2 .38 10 2 .42 10 3)

REFERENCES NICHOLSON, G.S., Biometrika 57_, 2, 273 (1970)

BACH, G., Dissertation, Selbstverlag des Mathematischen Seminars

Giessen (1959)

PIEFKE, F., Dissertation, Technische Universitat Braunschweig (1975)

498
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9. 1975 (Issued January 1976)

ON SIZE DISTRIBUTION METHODS

by F. A. L. Dullien and K. S. Chang,

University of Waterloo, Waterloo, Ontario, Canada

I. NOTE ON SECTION DL|METER THEORIES Kendall and Moran in their little book on "Geometrical Probability" pre- sented a derivation of the integral equation for the probability distribution

i}i(5)d<S of the diameters of the circle of intersection when it is known that the sphere does intersect the plane. The final expression is

(j.(6)d5 = /" -^^"^ dDdS (1) D 6 /D^ - 5^

o They commented that an incorrect theory, which was apparently widely used,

neglected the fact that larger spheres have a greater probability of being

included in the section. In an atten^jt to clarify matters it will be shown

below that a different widely used theory (e.g. Scheil ) in which there is no explicit reference to this fact, is equivalent to (1). First the derivation in Kendall and Moran is given.

The expected ntimber of spheres whose diameters lie in the range D D + dD

and which intersect an arbitrary plane in imit area of the latter is N^F(D)dD

where N is the mean number of centers of spheres in unit volume and F(D)dD is the probability of a sphere chosen at random to have diameter in the range

D -i- D + dD. The probability distribution of diameters of spheres which intersect the plane will be

DF(D)dD DF(D)dD

f(D)dD = — = (2)

/ DF(D)dD D

o o where is the mean sphere diameter. Note that is the nijmber of spheres

intersected by \mit area of the plane.

If a sphere of diameter D does Intersect the plane, t^e probability that its center lies at a distance X X + d^ from2t^a2plane is 2D dX and the diameter

of the circle of intersection is (D - 4X )

Hence the probability of intersecting a sphere of diameter D and obtaining

a circle of diameter 6 is

d"""" 6 d6 6F(D)dDd6 ——— = === (3) /D^ - 6-^ D v^2~r^

o where (2) has been used.

The probability distribution (})(5)d6 of the diameters of the circle of intersection when it is known that the sphere does intersect the plane is therefore given by (1)

.

According to the other theory, the expected number of spheres of diameter

D -> D + dD and which intersect an arbitrary plane in a unit area of the latter at

a distance X X + dX from the plane is 2N F(D)dDdX. Notice that N^F(D) dD is equal to N(D)dD, the number of spheres with diameters D D + dD in unit volume. Hence the expected number of circles of diameter 6 in unit area of plane,

obtained by intersecting spheres of diameter D is

499
f. A. L. DullI en and K. S. Chang

F(D) dDdS

. (4)

The expected number of circles of diameter 6 in unit area of plane, n(6)d6,

obtained by the intersecting spheres of any diameter (D >_ 6) is

F(D) dD n(5)d5 = N 5d6 /" ———; (5) V

6 /d2 _ 62

The total expected number of circles in unit area of plane is equal to the number of spheres intersected by unit area of plane

N„ /" D F (D) dD = N D (6) V o V o Hence the probability distribution of the diameters of the circle of intersection also by this theory is

n(6)d6 5 F(D)dDd6

= — ^ (7) N„D D 6 /d-^ - 62 V o o The lefthand side of (7) is nothing but (f)(6) d6 and thus (5) agrees with (1). Therefore the two theories are equivalent.

II. NOTE ON THE SECTION - ^I^TER METHOD FOR NON-SPHERICAL OBJECTS Dullien and co-workers have devised stereological methods for non-spherical

objects which they have used successfully for void size distribution determination in sandstones and bead packs. In the original presentation of the theory there was no clear line drawn between rigorous mathematical theory and the simplified procedure used. An attempt is made here to clarify some of these problems

.

The size of an object is characterized by the magnitude of the longest chord,

or intercept length, ^2^^' '^^ convenient to choose this chord as axis A of

the object. The objects may have any shape, but for increasingly non-convex

objects the longest chords will represent the true size of the object with de- creasing accuracy. All objects are assumed to have the same shape and they may differ only in size.

The orientation of an object in a cartesian coordinate system is given by

the polar angle 6 enclosed by the axis A with the positive z-direction, and the angle <)) enclosed by the projection of A onto the x-y plane with the positive

x-direction.

Each object is thought to be placed in the coordinate system in such a way

that the x-y plane is exactly half way the distance D^ between the tangent planes

to the two extremities of the object which are parallel to the x-y plane.

The expected number of objects of diameter D^^,.„ D_^,.^ + dD„_,,^ of any

. , . , . J , 1 2MAX . . 2MAX ^, 2MAX , given orientation which are intersected by a plane parallel/ uo the x-y plane m

unit area of the latter at a distance z z + dz from the x-y plane, giving rise

to two-dimensional features of diameter 5 6 + d6 in the intersecting plane, is

2V(W'^°2MAX^^ (sineded^)/4. = 2N^F(D2^) dD2^ d6(sin6ded(j)) /Att (8)

500
On size distribution methods

The diameter 6 of a two-dimensional feature is defined as 4ie longest

chord of intersection with the feature in the x-direction.

The total number of two-dimensional features B(6)d6dD. with diameter

obtained by intersecting objects of diameter D2j,j^, which nasbeen assumed

random, is therefore

sined6d<}>/4Tr

2N,:^(D,

2MAX' If ^^2MAX^^ (9)

The operation in (9) presumes that z = g(6) is known as a function of the orientation of the object as given by the pair of numbers (e,(})).

Determination of z = g(<5) by direct observation would involve making

successive parallel sections of individual objects in various random, but known

orientations and me asuring 6 in each section as a function of z. Could the derivative | 3z/35 | be obtained by direct observation then this would be some fun c- which would take the place of the expression (1/2) {S / t^B^-S^) , The expected number of two-dimensional features of diameter

then be

tion of ^2KAY

used for spnerical objects

obtained in unit area of sectioning plane, regardless of ^2}^AX

n(6)d5 = 2N d6 /

6

The question arises what happens when the object is not convex and g(6) is, in general, not unique and 3z/36 has singularities. Evidently the probability

of an event cannot be infinite and therefore at the points where 3z/36 has singularity do must diminish to zero. It is worthwhile to note that even for

the case of spherical objects 3z/35 is infinite at the equator. Therefore singularities which may arise in the case of non-convex objects do not present a real problem. |3z/35|d6 merely provides a measure of the number of two-dimensional features with diameter 5 ->• 5 + d5 obtained when sectioning an object of

a specified orientation with equidistant parallel planes. It is immaterial

whether features of a given diameter 6 originate when sectioning the object only

at one or at several different values of z. Therefore lack of convexity of the objects does not invalidate the theory.

For practical purposes the function g(6) mtist be determined from observations made on sections through many objects of random orientation which are imbedded in a solid matrix. Moreover the variotis objects never have exactly the same shape. In view of these complications and in order to simplify the procedure y vs . 5 has been measured in the two-dimensional features obtained by

sectioning the objects of random orientation imbedded in the solid matrix. The

assumption has been made that the average of y = g(5) taken over two-dimensional

features of various sizes and orientations in a sample is the same as the average

of z = G(6) taken over the corresponding three-dimensional objects.

The dimensionless form has been introduced

(11)

where D_ is the largest value of 6 for a given feature of specified orientation.

501
F. A. L. Dullien and K. S. Chang

Differentiating (11) gives

"1

2D^

(12)

whence, by averaging over a large number of two-dimensional features of various

sizes, shapes, and orientations

(13)

Using (13) and the assumption |3y/3S| = |3z/36| in (9), the following

approximate expression is obtained for the expected number of two-dimensional

features of diameter 6 in unit area of sectioning plane, regardless of D, and orientation of the objects 2MAX

n(6)d6

D2 / 2MAX

(lA)

This is the integral equation to solve for the unknown N F(D. JdD- in, the case of non-spherical objects. The solution has been worked ou? by S^wan and

it has been successfully applied to the void size distribution determination of various sandstones and bead packs. In certain cases where the relationship

(6/D2) (15)

holds |if)'(5/D2)| may be replaced by \^ (6/D2) | in (14).

REFERENCES

1. Kendall, M.G. , and Moran, P. A. P.: "Geometrical Probability". Hafner

Publishing Co., New York, 1963.

2. Scheil, E.: Die Berechnung der Anzahl und GrBssenverteilung kugelflirmiger

KBrper mit Hilfe der durch ebenen Schnitt erhaltenen Schnittkreise. Zeit. anorg. allgem. Chem. , 201 (1931), 259-264.

3. Dullien, F.A.L. and Dhawan, G.K. : "Photomicrographic Size Distribution Deter- mination of Non-Spherical Objects," Powder Tech. ]_ (19 73), 305-313.

4. Dullien, F.A.L. and Dhawan, G.K. : "Characterization of Pore Structure by a Combination of Quantitative Photomicrography and Mercury Porosimetry ," J. Coll. Int. Sci., 47 (1974), 337-349.

5. Dullien, F.A.L.: "Photomicrographic Pore Size Distributions Using Quantitative Stereology and Application of Results in Tertiary Petroleum Recovery,"

in "Pore Structure and Properties of Materials" RILEM-IUPAC Proceedings of

the International Symposium, Prague, September 18-21, 1973, Preliminary

Report - Part I. Academia Prague 1973.

6. Dhawan, G.K. : "Photomicrographic Investigation of the Structure of Sandstones

and Other Porous Media by the Methods of Quantitative Stereology: With Application to Oil Recovery," Ph.D. Thesis, University of Waterloo, 1972.

502
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

SIZE DISTRIBUTION OF RETAINED AUSTENITE PHASE IN A QUENCHED STAINLESS STEEL

by Anant V. Samudra and Cm Johari ITT Research Institute. Chicago. Illinois 60616. U.S.A.

INTRODUCTION Retained austenite is a metastable phase in quenched steel . The properties of the steel and their behavior toward operating stress and temperature

variations depend on the resistance of the retained austenite to martensite

formation. This resistance, called the austenite stability, has been quantitatively correlated to the microstructural features of retained austenite.'

The portion of work on characterizing the size distribution of retained aus- tenite is presented here.

MATERIAL AND PROCESSING A high-purity Fe-28Ni-0.1C alloy was austenitized and quenched to a range

of temperatures from +10''c to - 196°C to achieve different volume fractions of

martensite. The specimens were prepared and studied by optical metallography.

RESULTS

As martensitic transformation progresses, the austenite gets partitioned

into compartments. The austenite and martensite distribution is nonuniform,

and the austenite compartment formation is not well developed until the trans- formation exceeds 80 percent. At higher extents of transformation the retained

austenite is finely distributed in the martensite matrix. The data on two- dimensional size distribution in retained austenite in six samples are presented in Table 1.

DISCUSSION

Quantitative characterization of three-dimensional size distribution in retained austenite requires a quantitative metallography method. At present,

most methods depend on the assumption that particles approach a spherical or

spheroidal shape. ^

^ In the present study, the Johnson-Saltykov method based

on size distribution of two-dimensional areas was employed. Retained austenite

was assumed to be dispersed uniformly as discrete, spherical particles in a martensite matrix. The original Saltykov table had to be extended to smaller

sizes. There were great practical difficulties in exact counting and classifying of very small austenite areas, in spite of the lOOOX magnification em- ployed.^' ''^ Also, the apparent zero count in size group (-1) resulted in negative numbers because of the sequential dependence of the derivation. The

analyses therefore exclude data beyond size group (0). The analyses summarized

in Tables 2 and 3, allow the following conclusions:

1. A size distribution in the retained austenite phase exists

for each sample.

2. The maximum size and the mean size of retained austenite

particles decrease with lower volume fraction of retained

austenite

.

3. The number of large size austenite particles decreases

with lower volume fraction of retained austenite.

The particle size distributions are approximate, and no special significance is to be attached to secondary peaks and valleys. The metallurgical in- terpretations of these size distributions and their correlation with the stability of retained austenite will be published elsewhere.

503
A. V. Samudra and Om Johari

I,

As mentioned earlier, the accuracy of area counts and their classification

is limited in the small areas. In the analyses presented, the very small areas

(size group (-2)) were excluded. To estimate how the results would be affected

if these data were included, the counts from size group (-1) and (-2) were lumped

together to facilitate the sequential analysis. The new estimates for sample B are presented in parentheses in Tables 2 and 3.

Clearly, the new particle counts in these small size groups are very large

and, therefore, the average particle size has decreased to half its value. Also,

the relative proportion of the particles in different size groups changes significantly (see Table 3)

.

It is well known that the mean value of the particle volume depends on the magnification employed and that the estimate of mean particle volume decreases

as magnification increases . ^ ' ^ Thus, comparisons of mean particle volume must be

made at the same magnification. It should be realized that the relative contribution to particle counts by very small size areas increases and there is a greater and greater error in the estimate of mean austenite particle volume, as the volume fraction of austenite decreases.

For improving the accuracy of measurement in this type of study, techniques

that can give higher magnifications and better resolution than optical microscopy are necessary. Scanning electron microscopy is not suitable for studying

smooth surfaces and requires a somewhat deeper etch (for metallographic samples), and the attendant broadening of the phase boundaries makes quantitative measurements difficult. Transmission electron microscopy would require large sample

areas of uniform thickness to obtain statistically usable data.

Underwood^ has described the new simplified method suggested by Saltykov for deriving particle size from size distribution of areas. If the data from Table 1 are analyzed by this new formulation, estimates of particle counts result are higher by 15-20 percent than those by the Johnson-Saltykov method. Table 4 com- pares the analyses for the two methods. The reason for the discrepancy is not

clear.

CONCLUSION

The particle size distribution in a practical case of austenite-martensite

mixture is not easily determined. With the currently available experimental

techniques and mathematical formulations, the chances of improving these results appear small.

REFERENCES

1. A.V. Samudra, "Stability and Structure of Retained Austenite in a Quenched

Stainless Steel," Ph.D. Thesis, IIT, Chicago, 1975.

2. E.E. Underwood, "Particle Size Distribution," in Quantitative Microscopy , R.T. DeHoff and F.N. Rhines (eds.), McGraw-Hill, New York, 1968, pp. 149-200.

3. E.E. Underwood, "The Mathematical Foundations of Quantitative Stereology,"

in Stereology and Quantitative Metallography , ASTM STP 504, 1972, pp. 3-38.

4. J.W. Cahn and R.L. Fullman, "On the Use of Lineal Analysis for Obtaining

Particle Size Distribution Function in Opaque Samples," Trans. AIME, 1956,

vol. 206, no. 5, pp. 610-612.

5. R.T. DeHoff, "The Determination of the Size Distribution of Ellipsoidal Par- ticle from Measurements Made on Random Plane Sections," Trans. AIME, 1962, vol. 224, no. 6, pp. 474-477.

6. J.E. Hilliard, "Measurement of Volume in Volume," in Quantitative Microscopy , R.T. DeHoff and F.N. Rhines (eds.), McGraw-Hill, New York, 1968, pp. 45-74.

504
Table 1

Size distribution of retained austenite

- Distribution o£ Section Areas of Austenite Phase

Section Number of Section Areas Area A B C D E F roup Limits

,

(.5.0-6 (.o . 4-6 ri

l^i . o-s

f ^ HQ, (1 .S-o ri 10- (1.1-6 NO

,

ym^ Aust ,

)

Aust .

)

Aus t .

)

Aust .

)

Aust .

)

Aus t .

)

12 124-197 1 11 78.5-124 2 3 1 10 49.6-78.5 9 8 2 1 9 31.3-49.6 23 15 12 2

8 19.7-31.3 43 26 22 9

7 12.4-19.7 49 33 35 15

u / D

C 1 oi C7O / JO Qo 4

cD

A Q A 7 Q c: 7 A oU DO A70 / QO o 4 3.13-4.96 60 59 44 55 16 10

3 1.97-3.13 83 141 102 108 42 41

2 1.24-1.97 33 52 31 36 26 45

1 0.785-1.24 24 59 50 45 26 33

0 0.496-0.785 15 41 46 25 10 12

-1 0.313-0.496 0** 0 0 0 0 0 -2 0.197-0.313 87** 215 NC* NC* NC* NC* Total area

examined , mm 0.0301 0.0507 0.0507 0.0608 0.0608 0.0608

*NC stands for not counted. **The reason for zero count in group C-1) and a large count in group (-2) is due to practical difficulties in exact measurement of small areas

and in their accurate classification.

Table 2 - Number of Austenite Particles per mm

Number of Austenite Particles Per mm in Samples (X 10^) A B C D E F (17% (8.6% (3 . 4% (1.8 % (1.5% (1.1%

Group No. Aust. ) Aust .

)

Aust

,

.) Aust .) Aust..) Aust.

)

12 1.9

11 8. 0 6.5 2.,4 10 43. 3 21.4 5,,0 2 .5

9 107. 9 47.7 42,,7 5 .5

8 295. 8 101.8 86.,0 33 .3

7 371. 3 149.9 168.,4 62 .2 6 778. 7 525 346 208 .5 50..1 25.0

5 779. 3 529 473 441 46..7 62.9

4 620 374 251.,5 341 131.,2 73.3

3 1378 1767 1254 1100 520 470

2 87.2 26 .5 195..3 519

1 802 516 511 308,.5 366.5

0 515 173 (7070)*

Total 4382. 3 4928.4 3145.,0 2904 .7 1251,.8 1516.7 Avg Vol/

(11998)*

Particle,

X 10"^ mm^ 3. 93 1.94 2..10 1 .15 0.,51 0.3S (0.81)*

*The numbers in parentheses show the new values after including the counts of very small particles in the analysis.

505
A. V. Samudra and Om Johari

Table 3 - Relative Percentages of Austenite Particles

Austenite Particles per imn in Samples, Relative Percentages A B C D E F Group (17% (8.6% (3. 4% (1.8% (1.5% (1.1%

No. Aust:.) Aust.) Aust.) Aust.) Aust .

)

Aust.)

12 0 . 04

11 0

.

18 0,.13 0.,07 10 0. 99 0,.43(0.18)* 0.,16 0. 08

9 2. 46 0,.97(0.39) 1. 36 0. 19

8 6. 75 2,.07(0.85) 2,,73 1.15

7 8. 47 3,.04(1.24) 5.,36 2.14

6 17. 77 10,.65(4.37) 11. 0 7.15 4.0 1.65

5 17. 78 10..74(4.41) 15. 0 15.2 3.73 4.15

4 14. 15 7..6(3.11) 7. 99 11.73 10.45 4.84

3 31. 44 35..9(14.73) 39. 9 37.9 41.5 31.0

2 1..77(0.73) 0.91 15.6 34.25

1 lb..3(6.68) 16. 4 17.6 24.6 24.2

0 10..5(4.29) 5.96

-1 (58.93)

Total 100 100 (100.0) 100 100 100 100

*The numbers in parentheses show the new values after including the counts of very small particles in the analysis.

Table 4 - Comparison of Particle Distributions According to Two Methods

Group

No.

Sample A Sample B Sample C New Saltykov J-S* New Saltykov J-S New Saltykov J-S

12 2..3 1. 9 11 9..7 8. 0 7,.8 6. 5 2..9 2. 4 10 51,.5 43. 3 25,.7 21. 5 6..2 5. 0 9 156..3 107. 9 57..2 47. 7 51,.7 42. 7 8 347..3 295. 8 121.,4 101. 8 105,.8 86. 0 7 428..4 371. 3 176.,9 149. 0 201..6 168. 4 6 899..1 778. 7 629.,5 525. 0 412..1 346. 0

5 927.,5 779. 3 623..0 529. 0 559..8 473. 0 4 784..4 620 430.,8 374. 0 287.,0 251. 5 3 1786.,3 1378 2111,,4 1767. 0 1502..0 1254. 0 2 105..3 Neg 127.,8 87. 2 Neg Neg

1 298..7 Neg 914.,1 802. 2 917..6 516. 0 0 163..5 Neg 584. 9 515. 0 953.,1 Neg

*J-S refers to the Johnson-Saltykov Method.

506
National Bureau of Standards Special Publication 431

Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS. Gaithersburg, Md., September 4-9. 1975 (Issued January 1976)

ANALYSIS OF A SET OF SPHERICAL CELLS RELATIVE TO THEIR VOLUME

by Philippe cf Athis Department cfHSmatologie. U.E.I^. Pitie Salpetridre, 91 boulevard de I'Hopital, 75013 Paris, France

DEFINITIONS

Given a set E of cells, let g be the density of frequency for the cellular

volume in this set; g is a positive function such as:

I g(v)dv = frequency of cells, the volume of which lies between the values a

J a and b

,

and consequently gives the equality:

J

+00 g(v)dv = 1 0

Let (t -> r^) a family of positive functions such as:

f r (v)dv = 1

J 0

(Each function may be looked at as a density of frequency.) We say the set E is homogeneous relatiye to the family (t r^) if a yalue t can be found such as

:

and is heterogeneous relative to that family if values (t^,Pj^), (t2,P2)

(t ,p ) (where n > 2) can be found, such as: n n

1) > 0, > 0, . . . > 0

2) + + . . . + 1

3> Pi^, + Pa'^t = 8

1 n

(Demonstrating a set of blood cells to be heterogeneous leads to presuming it

to contain physiological different cells.)

PROBLEM

A sample being randomly extracted from the set E, every cell of this sample is cut randomly and therefore gives a plane section.

The problem is to analyze E, that is to say, to appreciate with the help

of measures on sections whether the set E is homogeneous or heterogeneous

relative to a given family (t r^) of densities.

METHOD OF ANALYSIS

This abstract only considers a set of SPHERICAL cells; then a section is

a spherical zone, the thickness of which is equal to the thickness h of the slice and the function of repartition F of the APPARENT (= larger) diameter

can be defined:

507
p. d'Athis

F(z) = frequency of sections, the apparent diameter of which is lower than the value z.

When It can be assumed that

1) the random sampling uniformly probabllizes the set E; and

2) every cell from the sample is cut uniformly, relative to one of Its diameters;

it is then demonstrated that:

2 ,h +v6c-z, /TT 3, J ^ ( hTl^—^ 8 ^2^ ^'^^

Then, with the theoretical equality

g = Pi^^ + P,r + . . . + p r

1 t.122 t„ n t n

and the experimental function Fj^, we can estimate the parameters ti,t2, . . . tj^; Pi,P2> • • • Pn '-'^^ "least square method," that is, by minimizing the function

:

Q (t,p) =

I I F(z) - F* z| I where

t = (t^,t2, . . . t^) and p = (p^,P2, . . . p^)

+°° fl 2

„j. / V , " I 2 ,h + vx - z , J. ^TT 3, , F*(z) = 1 - - I X ( Y^^^r^—) g* (y X )dx

8* = Pl'^t, + P2^, + • • • + Pn^t 12 • n

APPLICATION

The two most frequently used families (t r^) are

1) the normal family:

I \ 1 . / 1 rV-mi2. ' ' '^t^^^ = exp ( -

2 {—}

)

t = (m,s)

which can only be an approximation because

r (v)dv =1

0

^

The quality of this approximation Improves with the ratio — and may be con- sidered admissible from the value — = 3.

s

2) The log-normal family:

!,

, 1 . / I rln(v)-m-,2, .1

^t^^) = 1^ exp ( -

2

} ^ V -

,

t = (m,s)

508
National Bureau of Standards Special Publication 431 Proceedings of the FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

held at NBS, Gaithersburg, Md., September 4-9, 1975 (Issued January 1976)

APPENDIX: BASIC STEREOLOGY

by Ervin E. Underwood

President of the International Society for Stereology

Since its inception in I96I, the International Society for Stereology has been active in promoting a uniform system of nomenclature for the new science

of stereology. Because of the broad scope of stereology, no attempt is made

to define all possible terms. Rather, only the basic geometrical quantities

are defined, affording instant recognition to scientists anyi^here, regardless

of language. In this way, stereological symbols, like music and mathematics,

have attained an equally international acceptance.

The following brief review has been prepared with a dual purpose in mind. One is to familiarize readers of stereological papers with this system of nomenclature. The other objective is to encourage those that publish stereo- logical papers to use these basic symbols, thereby broadening their audience.

Thus, we present the basic system of symbols, measurements, definitions and

equations commonly used in the stereological analysis of microstructures.

Table 1. List of Basic Symbols and Their Definitions

Symbol Definitions

p Number of point elements. Point fraction. Number of points (in areal features) per test point

^L

-1 m Number of point intersections per unit length of test line p"A

-2 m I'lumber of points per unit test area Number of points per unit test volume

L m

m/m

Length

Lineal

of lineal elements. Mean length: L = L^/^^

fraction. Length of intercepts per unit length of test line

/ 2 m/m Length of lineal elements per unit test area

m/m~' Length of lineal elements per unit test volume

A

2m Planar area of intercepted features. Mean area: A = A /N

S 2 m

2 , 2 m /m

S\irface or interface area (not necessarily planar) . s" = Sy/Ny

\ Area fraction. Area of intercepted features per unit test area

2/ 3 m /m Surface area per unit test volume

V

\

Volume

Volume

of three-dimensional features. Mean volume: V = V^/I^y

fraction. Volume of features per unit test volume

N

^^L -1 m

Number

Number

of features (as opposed to points)

of interceptions of features per unit length of test line

\

-2 m Number of interceptions of featiires per \mit test area m-3 Number of features per unit test volume

Dimensions arbitrarily shown in meters.

509
£. E. Underwood

Basically, P is associated with points, L with lines, S with surfaces, and V

with volumes. The quantity A represents the flat area cut by the random

section plane through a volume element, and thus varies from zero to a maxi- mum value. S, on the other hand, is the external surface or interface

(usually curved) and has a fixed value for any particular microstructural

featijre. N is reserved for the number of objects or features , as opposed to the number of points (P). The compound symbols represent fractions, and are ratios of a microstructural quantity to a test quantity. Thus, for example,

S^, or S/V^, is the surface area per unit volume of the microstructure, where

is the test volume. Similarly, A^, L^j, and P^ are the test area, length of test line, and number of test points.

These combined symbols are merely a convenient way of writing the fraction.

Each symbol has a definite geometrical meaning and an associated dimension.

Thus the dimensionality of the combined terms is readily apparent, as well as the dimensional consistency of the equations. Note that descriptive names are avoided because the symbol changes for each name. For example, if B is used

for boundary, or P for perimeter, or I for interface, etc., this instantly creates confusion. Basically, they are all lines, L, in the section plane.

Appropriate identifying subscripts can be used as required.

The triangular matrix of Table 2 is arranged to show the interrelationship of measured quantities on the section plane to the calculated spatial quantities.

The equations listed in Table 2 represent the basic relationships of stereology. The top line reveals the direct equality of volume ratio, areal ratio,

lineal ratio and point ratio of an adequately sampled and measured second

phase. The second line includes two equations, = 2 P^ and = (t'/2) P^, in which the surface area per \init volme and the line length per unit area

are both related to the P measurement. The equation = 2 P^ is for a system of lines in space where P^ represents the points per unit area created

by the random section plane cutting the lines. P^ in the last line cannot be

obtained except for special cases. If the points can be approximated by small particles, then N is the quantity sought.

Table 2. Relationship of Measured (Q) and Calculated () Quantities

Micros tructural Feature

Dimensions of Symbols

0 -1 -3 -3 mm mm mm mm

Vo lumes

Lines

Surfaces

Points

Basic Equations

o mm

-1 mm

-2 mmmm

-3

510
Basic stereo/ogy

Although this is not the place to go into the derivations of the basic equations, it may be

of interest to indicate the approach adopted.

Figure 1 shows a model used to derive the re- lationship between and A^. (l) Randomly

oriented and positioned volume elements

(e.g., a-phase) in a test volume are cut by a thin test slice of thickness Sx. By letting

Sx become sufficiently smaZJL, we can express

the volume of the a-phase in terms of the area

fraction. Then, in the limit, the total volume

of a-phase is obtaining by integrating from

0 to S.. Dividing through by the test volume

Vrj, gives V„ = A. , which states that the average value of A^ represents an estimate of the volume fraction, V^.

Figure 1. Model for Deriving

the Relationship = A^.

Figure 2 is used in deriving the relationship = 2P^. (l) A cube containing randomly-oriented sur- faces is penetrated by many ran- domly-positioned vertical test

lines. An expression can be set up for the density of intersections with elementary surface

areas having all orientations in

space. Those elementary surface elements oriented normal to the test lines wi2JL have a larger

number of intersections than ele- ments oriented parallel to the test

lines. Then, by adding the individtial contributions, we obtain the total number of intersections with

the total surface area. Figure 2. Model for Deriving the Relationship = 2P^.

A similar idea is employed in Figure 3 to obtain the relationship = 2P„. (l) A cube containing a randomly-oriented

system of lines is cut by a large number

of vertical and parallel test planes.

Since the probability of intersection of the test planes with the elementary line

segments is proportional to their pro- jected lengths normal to the sectioning

planes, it can be shown that the pro- portionality constant is equal to 2.

Figure 3. Model for Deriving

the Relationship = 2P^.

511
E. E. Underwood

Basic measurements are shown graphically in Figure ^4. The six combined symbols

in the top line all represent simple counting measurements. Examples of each are given in the microstructures below. Pp represents points that fall within

a selected phase divided by the total number of test points; P^^ is the number

of points of intersection with linear elements in the microstructure per unit

length of test line; and means the number of point features in the microstructure per unit test area. Note that the use of P in all three cases in- volves no ambiguity when used in the combined term. In any case, if additional

specification is necessary, the basic terms are modified by appropriate sub- scripts; for example, (Pp)ci would mean the point fraction for the a- phase. Note that the T^, or tangent count measurement, is made with a sweeping test line instead of a stationary test line. The N^ and N^ counting measurements

are made on objects (such as particles or second-phase regions) rather than

points. They represent the number of objects per unit length and per unit

area, respectively. The three remaining combined terms require measurements

of area and length. Note that = N for V„ = 1, and = 2N^ for V„ < 1. L L V L L V

BASIC MEASUREMENTS

Pp Pi Pa Nl Na Ta

Li La

Figure h. Basic measurements in stereology.

It should be emphasized that adequate sampling and sufficient measurements on

the samples must be made to achieve the desired accuracy. These questions are discussed in the literature of stereology. For a selection of source books on

stereology see the references (1-5) listed below.

REFERENCES

1. Quantitative Stereology by E, E. Underwood, Addison-Wesley Publ. Co.,

Reading, Mass. (1970).

2. Stereometric Metallography by S, A. Saltykov, Metallurgizdat , Moscow (1970), 3rd Ed.

3. Stereology 3 ; Proceedings of the Third International Congress for Stereology,

Berne, Switzerland (1972) edited by E. R. Weibel, et al. , Blackwell Scientific

Pubis. , Oxford.

h. Quantitative Microscopy edited by R. T. DeHoff and F. N. Rhines, McGraw-Hill

Book Co., New York (1968)

.

5. Stereology, Proceedings of the Second International Congress for Stereology,

Chicago, 111. (1967), edited by H. Elias, Springer-Verlag, New York.

512
ABBREVIATED GLOSSARY OF EQUIVALENT SYMBOLS

In order to assist the reader with the symbollsn In some of the more mathematical

papers, we have assembled this glossary of equivalent symbols. Note that all terms are

not included (see p. 509 for a complete list of basic terms). Here we show only those

terns that differ from or duplicate others.

The Editors

Description of Symbol Equivalent

I.S.S. Symbol

BASIC TZBiS

Number of objects, particles, etc. per unit length Number of points per unit length, area or volume Number of particle centroids per unit volume

Surface area per unit volume Mean caliper, or tangent, diameter

Number of tangent points per unit area or volume Length of test line test area

test volume Length of line segment

Foil thickness

T L,l,dl

Miles

(p. 3)

1 > III

M

DeHoff & Gehl

(p. 29)

V

o dX.dp

Rhlnes

(p. 233)

SPECIAL TERMS

-(c)

(d)

Curvature at a point of a curve, d9/dl Integrated curvature of curves, J<dl

per unit length, - k//di

per unit volume, " k////dv

Integrated mean curvature of surfaces'" per unit surface area, = K///ds

per unit volume, = K////dv

Integrated Gaussian curvature of surfaces

per unit surface area, « G///ds

per unit volume, " G////dv

Nunber of inflection points per unit volume

Hunger of Intersection points per unit volume

Length of Intersection curves per unit volume

Number of branches per unit volume Nuaber of nodes per unit volume Connectivity per unit volume Constant of proportionality Random convex set Coaplementary cone at I-polnt

K

G

(P)infl

^ V'l-points

-(b)

.(b)

(b)

G

V

,(a)

,(a)

/k(X)dX E

M

"v

I

V

.(a)

(All quantities refer to the projection plane)

Number of points

per unit length or area

P'

P' ,P* L' A

N' Lproj

Nuiid>er of Inflection points per unit area 7;>infi

^ A^infl

I

J proj A proj Nun^er of tangent points

per unit area

T'^;

^proj

Aproj Curvature at a point of a curve

Integrated curvature of curve C" per unit area "A

Length of line segment dl' dp

Ad hoc notation. Notation not well-established, !i//(Kj+K2)ds, J!>: iK2i3

S13

DIRECTORY OF MEMBERS OF THE ISS

September 1975

C = Conferee

Ch = Chairperson

Dr. Shora Agajanyan

Academy of Science of the USSR Pushchino Moscow Region, U.S.S.R.

Cia = Conferee in absentia

A = Author

Cia Heinz Barthel Cia

Veitscher Magnesitwerke A.-G. Forschunsinstitut Leoben

Magnesi tstrasse 2 A-8707 Leoben-Goss, Austria

Dr. R. V. Ambartzumian Cia Institute of Mathematics

Academy of Science of the Armenian

S.S.R. Erevan, USSR

Mevr. Drs. W.H.M. Amesz-Voorhoeve C Department of Anatomy

69 Oostersingel

Groni ngen THE NETHERLANDS

Prof. Dr. G. C. Amstutz, Direktor

Mineralog.-Petrographischen

Institutes der Universitat Heidelberg

Postfach 840, Berl i nerstrasse 19 D69 Heidelberg, Germany

Dr. R. S. Anderssen C, Australian National University A, Computer Centre, P. 0. Box 4 Ch Canberra ACT 2600 Australia

Mr. R. H. Atkinson Cia

861 Tice Place Westfield, NJ 07090

U.S.A.

J. Baak

Vrije Universiteit de Boelelaan

Amsterdam

The Netherlands

Dr. Gunter Bach

Oppel nstrasse 28

33, Braunschv/eig

Germany

Prof. Lida K. Barrett Cia

Dept. of Mathematics

Ayres Hall University of Tennessee

Knoxville, TN 37916

U.S.A.

Ian Bartky C Scientific Assistant

Institute for Applied Technology

National Bureau of Standards

Washington, DC 20234

U.S.A.

Dr. Rolf Baur Anatomisches Institut

Pestalozzistr. 20

CH 4056 Basel Switzerland

Marie Bellemare

Department of Anatomy

University of Montreal, CP 5128

Montreal, PQ Canada

Mr. G. Bernroider C, Universitat Salzburg Ch,

Zoologisches Institut A

Akademiestrasse 26 A-5020 Salzburg, Austria

Dr. Martin Berry Cia,

Anatomy Dept., Medical School A University of Birmingham

Vincent Drive

Birmingham B15 2TJ , England

Dr. Felix D. Bertalanffy

Professor of Anatomy, Medical College

750 Bannatyne Avenue

Winnipeg 3, Manitoba

Canada

Prof. Michael B. Bever Cia

Massachusetts Institute of Technology

Room 13-5066 Cambridge, MA 02139

U.S.A.

Dr. Indra Bhargava • C>

P. 0. Box 1059 A

Ahwaz

Iran

515
Directory

Prof. Jacques-Louis Binet

Faculte Pitie-Salpetriere

91 Boulevard de L'Hopital

75013 Paris France

C, Mr. R. D. Buchheit

Ch, Battelle Memorial Institute

A 505 King Avenue

Columbus, OH 43201 U.S.A.

Dr. Edwin S . Boatman C F561 Health Sciences SC-34

University of Washington

Seattle, WA 98195

U.S.A.

Dr. Robert P. Bolender Cia University of Washington

Seattle, WA 98195

U.S.A.

Dr. Jose Bosch-Figueroa

Dept. of Crystallography & Mineralogy

University of Barcelona

Av. Jose Antonio 585 Barcelona 7, Spain

Dr. E. A. Bossen C Department of Pathology

Duke University Medical Center

Durham, NC 27710

U.S.A.

Dr. Mary Blair Bowers C Bldg. 3, Room 318 National Institutes of Health

Bethesda, MD 20014

U.S.A.

Dr. S. Bradbury Cia

Department of Human Anatomy

South Parks Road Oxford 0X1 3gx

England

Dr. A. Braun Laboratoire Suisse de Res. Horlogeres

Rue A. L. Breguet 2 CH-2000 Neuchatel Switzerland

Dr. L. G. Briarty

Botany Department

Nottingham University

University Park Nottingham NG7 2RD England

David Brown A-205 Chemistry Bldg.

National Bureau of Standards

Washington, DC 20234

U.S.A.

C,A

Dr. G. Colin Budd Cia

Medical College of Ohio at Toledo

P. 0. Box 6190 Toledo, OH 43614

U.S.A.

H. C. Burnett C B-264, Bldg. 223 National Bureau of Standards

Washington, DC 20234

U.S.A.

Peter H. Burri , M.D.

Dept. of Anatomy

University of Berne Buhlstrasse 26, CH-3012 Berne Switzerland

Dr. Robert J. Buschmann

Veterans Administration West Side

Hospital

MP 113, P. 0. Box 8195

Chicago, IL 60680, U.S.A.

Dr. Romulo Luis Cabrini

Departamente de Radiobiologia

Comison Nacional de Energia Atomila

Avda. Libertador 8250

Buenos Aires, Argentina

Massimo Calabresi, M.D. 300 Ogden St. New Haven, CT 06511 U.S.A.

Dr. Cisar Ccinepa Cia

Division de Exploraciones MINERO-PERU

1649, Av. Arcquipa

Lima, Peru

Dr. Peter B. Canham Cia,

Department of Biophysics A The University of Western Ontario

London N6A 5C1 Canada

Nancy V. Carlson New Eng. Deaconess Hospital Cancer Research Institute

Electron Microscopy

194 Pilgrim Road

Boston, MA 02215

U.S.A.

516
Directory

Dr. Anna-Mary Carpenter C, University of Minnesota Ch

Dept. of Anatomy

254 Jackson Hall Minneapolis, MN 55455, U.S.A.

Dr. Wayne A. Cassatt C A-121, Bldg. 222 National Bureau of Standards

Washington, DC 20234, U.S.A.

Dr. Philippe R. L. Cauwe C, Wild Leitz-France A

86, avenue du 18 juin 1940, B.P. 107 92504 Rueil-Malmaison

France

Dr. James P. Chalcroft Cia

Meat Industry Res. Inst, of New Zealand

P. 0. Box 9002 Hamilton North, New Zealand

Dr. Harold W. Chalkley

Bethesda, MD, U.S.A. Deceased, September 25, 1975

Dr. Chi-Hau Chen C, Electrical Engineering Department Ch,

Southeastern /^ass. University A

N. Dartmouth, MA 02747, U.S.A.

George C. Cheng Ch,

Rogers Road, M-102 A Athens, GA 30601 U.S.A.

Dr. J. -L. Chermant Cia University de Caen A Laboratoire de Chimie Min^rale

Industrielle

CAEN-Cedex 14032

France

Dr. Robert B. Chiasson

Dept. of Biology Sciences

University of Arizona

Tucson, AZ 85721

U.S.A.

Dr. Antonio Chizzola

Institute de Clinica Med. Gen.

dell Universita de Parma

Ospedali Riuniti 43100 Parma

Italy

Dr. William P. Clancy

Materials Research Lab. Army Mat. & Mech. Research Center

Watertown, MA 02172

U.S.A.

Charles I . Cohen C Owens Corning Fiberglass Corp.

Technical Center, P. 0. Box 415 Granville, OH 43023, U.S.A.

Dr. Morris Cohen

Room 13-5046

Mass. Institute of Technology

Cambridge, MA 02139

U.S.A.

Docent Yrjb Collan, M.D. C, Second Dept. of Pathology A

University of Helsinki

00290 Helsinki 29 FINLAND

J. Stanley Cook C Roy C. Ingersoll Research Center Wolf & Algonquin Roads

Des Plaines, IL 60018, U.S.A.

Dr. M. Coster C, Laboratoire de Chimie A

Min^rale Industrielle

University de Caen

CAEN Cedex 14032, France

Prof. Emil Craciun, M.D. Cia,

Inst, of Endrocri nology A

Str. Dionisie Lupu 62 Bucarest I (22) Rumania

Dr. Ian Crain Cia

Rm. 226, Health Protection Bldg.

Dept. of Health & Welfare

Tunney's Pasture

Ottawa, Ontario

Canada

Dr. Luis-M. Cruz-Orive C, Dept. of Probability & Statistics A The University

Sheffield, S3 7RH England

Prof. Robin L. Curtis

Department of Anatomy

Marquette School of Medicine

561 North 15th Street

Milwaukee, WI 53233

U.S.A.

517
Directory

Mr. Ronald K. DeFord

University Station

Box 7609 Austin, TX 78712, U.S.A.

Dr. Robert T. DeHoff C, Dept. of Materials Sci. & Eng. Ch,

University of Florida A Gainesville, FL 32611, U.S.A.

Dr. Felix A. de la Iglesia

Warner-Lambert Research Institute

of Canada Sheridan Park Mississauga L5K 1B4, Ontario

Canada

Robert M. Doerr C,

U.S. Bureau of Mines Ch,

P. 0. Box 280 A

Rolla, MO 65401, U.S.A.

C. G. Dodd Cia

Schick Safety Razor Company

Division of Warner-Lambert Company

Milford, CT 06460, U.S.A.

Prof. F. A. L. Dullien C, University of Waterloo A

Waterl oo Ontario N2L 3G1 • , . Canada

Prof. Dr. Andre G. DeWilde C Anat. Emb. Lab. Oostersingel 69 ' . Groningen

Netherlands

Dr. Roland deWit C, National Bureau of Standards Ch,

B-120, Bldg. 223 A Washington, DC 20234, U.S.A.

J. Leland Daniel C Battel le - Northwest

PSL-1610

Richland, WA 99352

U.S.A.

Dr. Sven 0. E. Ebbesson

Dept. of Neurological Surgery

University of Virginia

School of Medicine

Charlottesville, VA 22901, U.S.A.

Cammander George T. Eden, DC, USN C, A-145, Polymers Bldg. (311.04) A National Bureau of Standards

Washington, DC 20234, U.S.A.

Dr. Siegfried Eins Cia Max-Planck-Institut fCir Biophysikalische Chemie

D 3400 Gottingen- Nikolausberg

Am Fassberg, Germany

Prof. Per E . Danielsson

Dept. of Electrical Engineering

Linkoping University

Linkoping S-58183

Sweden

Dr. Paul Davies Beth Israel Hospital

330 Brookline Avenue

Boston, MA 02215, U.S.A.

Dr. L. Delaey

Dept. of Metallurgy

De Croylaan 2

B 3030 Heverlee

Belgium

Dr. Brenda Eisenberg

Department of Medicine

U.C.L.A. Medical School

Los Angeles, CA 90024

U.S.A.

Dr. Farouk El-Baz

National Air & Space Museum

Smithsonian Institution

Washington, DC 20560, U.S.A.

Prof. Dr. phil. Hans Elias

463 Marietta Drive

San Francisco, CA 94127

U.S.A.

C,CH, A

Dr. Russell L. Deter

Dept. of Cell Biology

Baylor College of Medicine

1200 Moursund Blvd. Houston, TX 77025

U.S.A.

C, Dr. Peter M. Elias

A Dept. of Pathology

School of Medicine

University of California

San Francisco, CA 94143

U.S.A.

518
Directory

Dr. David Bertram Ell son Cia Division of Tribophysics, CSIRO

Box 4331 Melbourne 3001 Australia

Sami M. El-Soudani Cia,

Dept. of Metallurgy & Mat. Sci. A University of Cambridge

Cambridge CB2 3QZ

England

Dr. Edgar S. Etz C Rm. A-121, Bldg. 222

National Bureau of Standards (312.02)

Washington, DC 20234

U.S.A.

Dr. Hans Eckart Exner C, Ch, A Max-Planck-Institut fiir Metallforschung

175 Buesnauer Str. 7000 Stuttgart

Germany

Dr. D. G. Fagan Cia Hospital for Sick Children

555 University Avenue

Toronto, Ontario M5G 1X8 Canada

Dr. Colin Fisher

Metals Research Ltd.

Mel bourn, Royston

Hertfordshire, SG8 6EJ England

D. E. Fleet C Dept. of Metallurgy & Materials Sci. University of Cambridge

Pembroke Street Cambridge CB2 3QZ

England

Mr. Peter G. Fleming, Admin.

The Royal Microscopical Society

Clarendon House

Corn Market Street

Oxford, 0X1 3HA England

Dr. Eva Foh Anatomisches Institut der Universitat

Olshausenstrasse 40-60

23 Kiel Germany

J. B. Forrest C McMaster University Medical Centre

1200 Main Street West Hamilton 16, Ontario L8S 4J9 Canada

J. K. Fawell General Toxicology Division

Inveresk Research International

Musselburgh, Midlothian, EH21 7UB Scotland

Cia Mr. Richard H. Foster Cia

The City University

St. John Street

London ECIR 4TR England

Dr. Micheline Federman Cia Cancer Research Institute

N.E. Deaconess Hospital

185 Pilgrim Road

Boston, MA 02215, U.S.A.

Dr. Enmanuel Fei nermann C,A Centre d' Etudes Nucl^aires de Saclay

SCP, BP No. 2 91190 GIF/Yvette

France

R. E. Filipy C Battel le-Northwest

Box 999 (LSL II, 3000)

Richland, WA 99352

U.S.A.

Dr. Vernon W. Fischer

St. Louis University

Med. School, Anatomy Department

1402 S. Grand Avenue

St. Louis, MO 63104

U.S.A.

Dr. William G. Fricke, Jr. Physical Metallurgy Division

Alcoa Technical Center Alcoa Center, PA 15069

U.S.A.

Dr. med. habil. Siegfried Fritsch Cia Zentral institut fiir Mi krobi ol ogi

e

und Experimental le Therapie

Beutenbergstr. 11, 69 Jena Germany

Yasumichi Fujimoto, Lecturer C, Dept. of Anatomy A

Osaka Dental University

Osaka, 540 Japan

Dr. R. Buckminster Fuller 3500 Market Street Philadelphia, PA 19104

U.S.A.

519
Directory

Dr. Ralph Gander

Wild Heerbrugg Co. 9435 Heerbrugg

Swi tzerland

Dr. James Q. Gant, Jr. C 1835 Eye Street, N.W., Suite 201 Washington, DC 20006

U.S.A.

Dr. Peter Gehr C, Department of Anatomy A University of Bern BUhlstrasse 26 3012 Bern, Switzerland

Dr. H. D. Geissinger

University of Guelph

Dept. of Biomedical Science

O.V.C. Research Station

Guelph NIG 2W1 Ontario, Canada

Dr. Juan Gi

1

University of Berne Anatomisches Institut

Biihlstrasse 26, 3012 Berne, Switzerland

Dr. Pierre Gilles

Ingenieur Service Chimie Physique

Conmissariat Energie Atom. CEN/Saclay

BP #2, 91190 GIF-sur-Yvette

France

Dr. Walter Good C, Swiss Avalanche Research Institute A CH-7260 Weissfluhjoch/Davos

Swi tzerland

Dr. Richard Gordon Ch,

Image Processing Unit A

Bldg. 36, Room 4D-28

National Institutes of Health

Bethesda, MD 20014, U.S.A.

Dr. Hilmar Grimm

Inst. Mikrobiologie u. exp. Therapie

Beuthenbergstrasse 11 Jena Germany

Dr. Sverker Griph C Dept. of Anatomy

University of Uppsala, Box 571 S-75123 Uppsala

Sweden

Dr. Miguel Guirao-Perez

Director .Institute "Foloriz"

Facultad de Medicinia de Granada

Granada

Spai n

Prof. J. Gurland C, Brown University Ch,

Division of Engineering A Providence, RI 02912

U.S.A.

Prof. Ernest L. Hall C, Univ. of Southern California A

Dept. of Electrical Engineering

Los Angeles, CA 90007

U.S.A.

Dr. Gilbert F. Hamilton

Department of Anatomy

Marshal College University

Aberdeen

Scotland

Dr. Paul Hamosh C Dept. of Physiology & Biophysics

Georgetown University School

of Medicine

Washington, DC 20007, U.S.A.

Prof. John W. Harman Diagnostic Laboratory

Department of Pathology

University College

Dublin 2, Ireland

Prof. Earl E. Gose University of Illinois, Chicago

Box 4348

Chicago, IL 60680

U.S.A.

Cia Dr. Bruce S. Hass C Argonne National Laboratory

Bio-Medical Division

7900 Cass Avenue

Argonne, IL 60439, U.S.A.

Dr. Tony Greday

Centre de Recherches Metal lurgiques

Rue du Val-Benoit, 69 B-4000 Liege

Belgium

Prof. Dr. med. H. Haug C, Medizinischen Akademie Lubeck Ch,

Ratzeburger Allee-160 A D-2400 Lubeck Germany

520
Directory

Dr. P. W. Hawkes Cia

Laboratoire d'optique Electronique

29 Rue Jeanne Marrig 31055 Toulouse Cedex

France

Dr. August Hennig

Department of Anatomy

University of Munich

Pettenkoferstrasse 15 D-8 Munchen 11 Germany

Dr. Jean-Pierre Herveg

Rm. 4B-47, Bldg. 10 National Institutes of Health

Bethesda, MD 20014, U.S.A.

Dr. Peter Hill Carl Zeiss, Inc. 444 Fifth Avenue

New York, NY 10018, U.S.A.

Prof. John E. Hilliard

Dept. of Materials Science

and Engineering

Northwestern University

Evanston, IL 60201, U.S.A.

Jan Hinsch C

E. Leitz, Inc. 3930 knowles Avenue

Kensington, MD 20795, U.S.A.

Robert W. Hinton C Homer Research Lab Bethlehem Steel Corp.

Bethlehem, PA 18016

U.S.A.

Miss Clara S. Hires

Mistaire Laboratories

152 Glen Avenue

Millburn, NJ 07041, U.S.A.

Dr. John D. Hoffman, Director

Institute for Materials Research

National Bureau of Standards

Washington, DC 20234

U.S.A.

Dr. Emanuel Horowitz C Deputy Director

Institute for Materials Research

National Bureau of Standards

Washington, DC 20234, U.S.A.

Dr. Hans P. Hougardy Ch, C, A Max-Planck-Inst. fiir Eisenforschung

Max-Planck Strasse 1 4000 Dusseldorf

Germany

C,A

C,A

C,A

Dr. Geza Ifju C, Forestry Products Dept. A

V.P.I, and S.U. Blacksburg, VA 24061 , U.S.A.

M. Indra Cia,

Inst, of Physiology A Czechoslovak Academy of Sciences

Praha 4 - Krc Budejovicka 1083 Czechoslovakia

Oscar A. Iseri, M.D. C

V. A. Hospital (Univ. of Maryland)

3900 Loch Raven Blvd. Baltimore, MD 21218, U.S.A.

C, Dr. M. I. Ismail Cia

A Chemical Engineering Dept.

Alexandria University

Alexandria

Egypt

Cia Prof. Genrich R. Ivanitsky, D.Sc. Cia

Biophysical Inst, of Ac. Sci . USSR

Dept. of Applied Mathematics, 142292 Pushchino Moscow Region

U.S.S.R.

C Werner Hunn

E. Leitz, Inc. Link Drive

Rockleigh, NJ 07647, U.S.A.

C Dr. Otto Hunziker

Basic Medical Research Dept.

Sandoz Ltd. CH-4002 Basel, Switzerland

C, Dal Hyde

Ch, School of Veterinary Medicine A University of California

Davis, CA 95616

U.S.A.

Ing. Vratislav Horalek, Dr. Sc. Mechenicka 14/2558

141 00 Praha 4 - Sporilov II Czechoslovakia

521
Directory

Prof. Oscar C. Jaffee

University of Dayton, Biology

300 College Park Avenue

Dayton, OH 45469, U.S.A.

C, Prof. Laveen Kanal Cia A Dept. of Computer Science

University of Maryland

College Park, MD 20742, U.S.A.

C, Dr. Shirley L. Kauffman C A State University of New York

450 Clarkson Avenue

Brooklyn, NY 11203 U.S.A.

Dr. A. J. Jakeman

Computer Center

Austral ian National University

P. 0. Box 4 Canberra, A.C.T. 2600

Austral ia

Dr. Nigel T. James Cia,

Dept. of Human Biology and Anatomy A University of Sheffield

Sheffield SIO 2LA England

Dominique Jeulin C, A Institut de Recherches de la Sid^rurgie

IRSID Maizieres-l§s-Metz 57210 France

Mr. Stig Johansson C, Sandvik AB A TMP Process Metallurgy

S-81101 Sandviken 1 Sweden

Dr. Jay A. Johnson C, Forestry Products Division A V.P.I, and S.U. Blacksburg, VA 24061 U.S.A.

K. A. Johnson Cia

P. 0. Box 1663, MS-328

Los Alamos, NM 87544

U.S.A.

Ronald B. Johnson C Executive Officer

Institute for Materials Research

National Bureau of Standards

Washington, DC 20234, U.S.A.

Meurig P. Jones, Senior Lecturer Cia,

Dept. of Mining and Mineral Tech. A Imperial College

London SW7 England

Prof. dr. Miroslav Kalisnik

Institute for Hist. -Embryol . , Med. Fac. University of Ljubljana

Zaloska 4/1 Ljubljana 61105 Yugoslavia

Dr. Brian H. Kaye, Director Ch, Inst, for Fine Particle Research A Laurentian University C Sudbury, Ontario P3E 2C6 Canada

Dr. Hausjorg Keller C, Department of Anatomy A

University of Bern

BLihlstrasse 26 3012 Bern, Switzerland

Thomas Kelly C, Image Analysing Computers A

40 Robert Pitt Drive

Monsey, NY 10952

U.S.A.

Dr. M. D. Kendal C St. Thomas's Hospital Medical School Lambeth Palace Road London, SEl 7EH England

Dr. Charles R. Key Cia

Dept. of Pathology

University of New Mexico

915 Stanford N.E. Albuquerque, NM 87131, U.S.A.

Russell A. Kirsch C, National Bureau of Standards Ch A-317, Administration Bldg.

Washington, DC 20234

U.S.A.

Dr. Gonzague S. Kistler

Anat. Institut der Universitat

Gloriastrasse 19 8006 Zurich

Switzerland

Prof. Victor Klee University of Washington

Department of Mathematics

Seattle, WA 98105

U.S.A.

522
Directory

Dr. Andres J. P. Klein-Szanto

Abteilung fur Orale Strukturbiologie

Plattenstrasse 11 8028 Zurich

Switzerland

Prof. Dr. Ekkehard Kleiss

Head, Dept. of Embrology

University de Los Andes Apartado 38, Merida

Venezuela

Prof. Andre G. Laurent

Wayne State University

Department of Mathematics

Detroit, MI 48202

U.S.A.

Dr. Katheryn E. Lawson Organization 5833 Sandia Laboratories

Albuquerque, NM 87115

U.S.A.

Prof. Dr. Wayne Kraft Cia

Dept. of Met. and Mat. Science

Lehigh University

Bethlehem, PA 18015

U.S.A.

John Krc, Jr. C 26669 Hunting Road Huntington Woods, MI 48070

U.S.A.

Priv. Doz. Dr. med. Ortwin Leder

Anatomisches Inst. d. Albert-Ludv/igs-U.

Albertstrasse 17 Freiburg im Breisgau

Germany

VladimTr Levicky, M.D., Ph.D. C, Inst, of Normal & Pathological A Physiology

Slovak Academy of Sciences

Sienkiewiczova 1 884 23 Bratislava, Czechoslovakia

Ing. Ivan Krekule C, Inst, of Physiology A Czechoslovak Academy of Sciences

Praha 4 - Krc Budejovicka 1083 Czechoslovakia

Dr. Karl G. Lickfeld

Institute of Med. Microbiology

55 Hufelandstr.

D-43 Essen 1 Germany

Dr. David E. Kuhl Dept. of Radiology

Hospital of the Univ. of Pa. Philadelphia, PA 19104

U.S.A.

Ass. Prof. Lars Goran Lindberg

Cytodi agnostic Department

University Hospital of Lund

Lund S-22185

Sweden

Dr. Tine Kuiper-Goodman C, National Health & Welfare A Health Protection Branch

Tunney's Pasture Ottawa KIA 0L2

Canada

Prof. Dr. Willi Lindemann

Lehrstuhl fur Kristal 1 strukturlehre

Konigsbergerstrasse

87 Wiirzburg ^ Germany

Mr. P. Lamb

I CI Plastics Division

Bessemer Road Welwyn Garden City Herts AL7 IHD, England

Cia Dr. Lewis E. Lipkin

National Institutes of Health

8000 Rockville Pike

Bldg. 36, Rm. 4D24

Bethesda, MD 20014, U.S.A.

P. D. Lambert Cia,

Coal Preparation Branch A

National Coal Board, South Notts Area

Bestwood, Nottingham N96 SUE England

Dipl. Math. Reinhard Lang Cia

Uni vers i tat Bielefeld

Fak. Mathematik

Kurt-Schumacher-Str. 6 D-48 Bielefeld, West Germany

Mr. P. J. Lloyd

Dept. of Chemical Engineering

University of Technology

Loughborough, Leicestershire

England

Dr. Arthur L. Loeb

Dept. of Visual and Environmental

Studies

Harvard University

Cambridge, MA 02138, U.S.A.

523
Directory

Gerald H. Lolmaugh

7721 Bristow Drive

Annandale, VA 22003

U.S.A.

Prof. Alden V. Loud C

Dept. of Pathology

New York Medical College

Walhalla, NY 10595

U.S.A.

John Ludbrook

Department of Surgery

Royal Adelaide Hospital

Adelaide 5000 South Australia

Mr. C. J. McCarthy C Bausch & Lomb

820 Linden Avenue

Rochester, NY 14625

U.S.A.

Walter C. McCrone McCrone Associates, Inc. 2820 S. Michigan Avenue

Chicago, IL 60616, U.S.A.

Bruce McCollough, D.V.M. , Ph.D. C Southwest Foundation for Res. and Education

P. 0. Box 28147

San Antonio, TX 78284, U.S.A.

Prof. Paul J. McMillan C Dept. of Anatomy

School of Medicine

Loma Linda University

Loma Linda, CA 92354, U.S.A.

Dr. Manuel Miranda Magalhaes

Lab. de Histologia

Faculdade de Medicina de Porto Porto

Portugal

Philippe Mai re CEA Cen-Sacley

91 Gif-Yvette BP No. 2 France

Dr. J. R. Manning C B-264, Bldg. 223 National Bureau of Standards

Washington, DC 20234, U.S.A.

Jacques Marilleau C CEA Commissariat a L'Energie

Atomique

Boite Postale N° 7 93270 Sevran, France

Dr. R. Bruce Martin C, Division of Orthopedic Surgery A West Virginia Univ. Medical Ctr. Morgantown, WV 26506

U.S.A.

Robert Martin C Administrative Officer

Institute for Materials Research

National Bureau of Standards

Washington, DC 20234, U.S.A.

Prof. Luigi Marti no

No. Via Carulli

5 Bari Italy

Dr. Tadeusz B. Massalski

Head, Met. Phys. Group

Carnegie-Mellon University

4400 Fifth Avenue

Pittsburgh, PA 15213, U.S.A.

Gloria D. Massaro, M.D. C Veterans Administration Hospital

50 Irving Street, N.W. Washington, DC 20422

U.S.A.

E. D. Massey C Medical Research Council Laboratories

Woodmansterne Road Carshalton, Surrey SMS 4EF England

Odile Mathieu C, Dept. d'Anatomie, Faculty de Mgd. A Univ. de Montreal , CP. 6128 Montreal 101 , P.Q.

Canada

Dr. Sasha Malamed C, Rutgers Medical School A College of Medicine and Dentistry

of New Jersey

Piscataway, NJ 08854

U.S.A.

Dr. T. M. Mayhew

Dept. of Human Biology & Anatomy

Univ. of Sheffield, Western Bank Sheffield SIO 2TN England

524
Directory

Dr. Geoffrey A. Meek C, University of Sheffield Ch,

Dept. of HLsnan Biology and Anatomy A Sheffield SIO 2TN England

Dr. Wolfgang J. Mergner C Dept. of Pathology

University of Maryland

22 South Green Street

Baltimore, MD 21201, U.S.A.

Dr. Paul-Emil Messier C, Dept. of Anatomy A University of Montreal

B. 0. 6128 Montreal Quebec, Canada

Dr. Roger E. Miles C Dept. of Statistics Ch,

Australian National University A

P. 0. Box 4 Canberra A.C.T. 2600 Australia

R. Moy C, Nat'l. Environmental A

Satell ite Service

NOAA, FOB #4, S1124 Washington, DC 20233, U.S.A.

Prof. Pieter Muije

Georgia Institute of Technology

School of Chemical Engineering

Atlanta, GA 30332

U.S.A.

Dr. Timo Nevalainen

Dept. of Pathology

University of Turku

Kiinamyllynkatu 10 Turku, Finland

Dr. Wesley L. Nicholson C,

Battelle - Northwest Ch,

P. 0. Box 999 A

Richland, WA 99352

U.S.A.

Bert Mobley

1234 19th Street, #13 Santa Monica, CA 90404

U.S.A.

Dr. Gastone G. Nussdorfer

Istituto Anatomia Umana Normale

Lab. Micr. Electr.

Via Gabelli , 37 35100 Padova, Italy

Fernand Moliexe

Inst, de Rech. de la SidSrurgie Franc.

185 rue de President Roosevelt

Saint Germai n-en-Laye, 78104

France

Dr. George A. Moore C, B-118, Bldg. 223 Ch,

National Bureau of Standards A Washington, DC 20234

U.S.A.

Dr. Yoshikuni Ohta

Dept. of Anatomy

Osaka Dental University

Osaka, 540 Japan

Dr. Daniel G. Oldfield

Dept. of Biological Sciences

DePaul University

1036 West Bel den Avenue

Chicago, IL 60614, U.S.A.

C,A

Prof. Richard Moore C, Box 701, Mayo Memorial Bldg. Ch,

University of Minnesota A Minneapolis, MN 55455

U.S.A.

Mr. Richard E. M. Moore Cia Anatomy Dept.

Guy's Hospital

London SEl 9RT England

Dr. Roger R. A. Morton C, Analytical Systems Division A Bausch & Lomb

820 Linden Avenue

Rochester, NY 14625

U.S.A.

Dr. Gerhard Ondracek C, Institut fur Material Ch,

u. Festkorperforschung A

Kernfors Chung szentrum D-7500 Karlsruhe, Postfach 947 Germany

Prof. P. Darrell Ownby Cia,

Dept. of Ceramic Engineering A School of Mines and Metallurgy

Roll a, MO 65401 U.S.A.

Prof. Monique Pavel Cia

145, Avenue de Malakoff A

75116 - Paris France

525
Directory

Paul Parniere

I.R.S.I.D.

185 rue President Roosevelt

St. Germain-en-Laye, 78104 France

Prof. Lee D. Peachey

Department of Biology

University of Pennsylvania

Philadelphia, PA 19174, U.S.A.

E. C. Pearson Alcan International

Research Centre

Box 8400

Kingston, Ontario

Canada

Ltd.

C Dr. Jan Pitha Cia

Laboratory Service

VA Hospital

921 N.E. 13th Street

Oklahoma City, OK 73104, U.S.A.

C, Floyd W. Preston Cia

Ch, Dept. of Chem. and Petr. Eng. A University of Kansas Lawrence, KS 66046, U.S.A.

C Dr. J. M. Prevosteau C, B.R.G.M. A

B.P. 6009 45018 - Orleans G^dex France

Prof. Dr. med Tomas Pexieder Cia

Inst. D'Histologie et D' Embryologie

9 Rue du Bugnon

1011 Lausanne/Sui sse Switzerland

Dr. Ulrich Pfeifer C, Pathologisches Institut

Universitat Wurzburg

Lui tpoldkrankenhaus

87 Wurzburg, Germany

Dr. Victor A. Phillips

Martin Marietta Laboratories

1450 S. Rolling Road Baltimore, MD 21227 U.S.A.

William G. Pichel Cia,

National Environmental Satellite A Service

NOAA, S11212

Washington, DC 20233, U.S.A.

Prof. T. Radii-Weiss Cia,

Czechoslovak Academy of Sciences A Institute of Physiology

Budejovicka 1083,

Prague 4, Czechoslovakia

Edward Rae C

E. Leitz, Inc. 1123 Grand View Drive

S. San Francisco, CA 94080

U.S.A.

Dr. Brian Ralph C, Dept. of Met. & Materials Sci. Ch,

University of Cambridge A Pembroke Street Cambridge CB2 3QZ

England

Dr. P. Ramakrishnan C, Indian Institute of Technology A

Dept. of Metallurgical Engineering

Powai, Bombay-400076

Indi a

Dipl. Phys. Frank Piefke

Institut B flir Mathematik

TU Braunschweig

Pockelstrasse 14

33 Braunschweig, Germany

Dr. Ruggero Pierantoni

Laboratorio de Cibernetica

Biofisica del C.N.T. Cammogli 16032 Italy

Dr. Hermann Pinkus

Dept. of Dermatology

Wayne State University

540 E. Canfield

Detroit, MI 48201 U.S.A.

Cia

Cia

Dr. J. Rathje,

c/o Bayer AG Postfach 1140, 4047 Dormagen West Germany

Cia

Geb. Da 5

Dr. William Reichel

Franklin Square Hospital

9000 Franklin Square Drive

Baltimore, MD 21237

U.S.A.

Dr. Curt W. Reimann

Scientific Assistant

Institute for Materials Research

National Bureau of Standards

Washington, DC 20234

U.S.A.

526
Directory

Dr. Albrecht Reith

The Norwegian Radium Hospital

Montebello, Oslo-3

Norway

Dr. F. N. Rhines

Dept. of Materials Sci. and Eng.

University of Florida

Gainesville, FL 32611, U.S.A.

Dr. Manfred Rink Technical University Clausthal Institut fur Geophysik

Adolf Rdmer-Str. 2A D-3392 Clausthal-Zellerfeld

Germany

Cia, Prof. Alvin H. Rothman

A California State College

Department of Biology

Fullerton, CA 92631, U.S.A.

C, Mr. Richard R. Rowand

Ch, Tech. Mgr. for NOT

A Air Force Material Lab. (MAMN)

Wright-Patterson AFB, OH 45433, U.S.A.

C, Dr. Letty Salentijn

A Dept. of Anatomy

College of Phys. & Surgeons

Columbia University

630 West 168th Street

New York, NY 10032, U.S.A.

Ian Robb

Inst, for Fine Particle Research

Laurentian University

Sudbury, Ontario P3E 2C6 Canada

C, Prof. Dr. S. A. Saltikov

A October Ave. 27, Apt. 16 Erevan 18, Armenia

U.S.S.R.

Max Robinowitz, M.D. C Georgetown University Medical School 3900 Reservoir Rd. , N.W. Washington, DC 20007, U.S.A.

Dr. Otto Rohm

Ch. des Mouetts 16 CH-1007 Lausanne Switzerland

Prof. Dr. med. Hanspeter Rohr C, Pathologisches Institut A University of Basel Schonbeinstrasse 40 CH-4056 Basel, Switzerland

Dr. Cesar Romero-Sierra

Anatomy Dept.

Queens University

Kingston, Ontario K7L 3N6 Canada

Dr. Anant V. Samudra Cia

IIT Research Institute A

10 West 35th Street

Chicago, IL 60616, U.S.A.

Prof. Dr. Masao Satake C Dept. of Civil Engineering

Tohoku University

Aoba, Aramaki

Sendai 980, Japan

Mr. Brian Scarlett C, Dept. of Chemical Engineering A University of Technology

Loughborough, Leicestershire

England

Robert J. Schaefer C, Naval Research Lab. A

Code 6350 Washington, DC 20375

U.S.A.

Mark Rosenblum C University of Cincinnati

339 Brookside Dr. Dayton, OH 45406

U.S.A.

Azriel Rosenfeld Cia

Computer Science Center University of Maryland

College Park, MD 20742

U.S.A.

Axel Schleicher C, Medizinische Hochschule A

3000 Hannover 61 Karl-Wiechert Allee 9 Germany

Dr. Holger Schmeisser C, IMANCO GmbH A

D605 Offenbach

Schreberstrasse 18 Germany

527
Directory

Dr. Renate Schmidt-Koschel

Bundesanstalt fiir Material priifung

Dienststelle 0.12 Unter den Eichen 87 Berlin 45, Germany

Dr. Charles C. Schock

Henry Ford Hospital

2799 W. Grand Blvd. Detroit, MI 48202

U.S.A.

Dr. G. C. Sornberger

Beth Israel Hospital , KB-23

330 Brookline Avenue

Boston, MA 02215

U.S.A.

Dr. Ian J. Spark

Gippsland Inst, of Advanced Ed. Box 42, Churchill

Victoria, 3842 Australia

Cia

C,A

Cia

Dr. Jurgen R. Schopper

Institut fur Geophysik

Technische University

Adolf-Rbmer-Strasse 2A D-3392 Clausthal-Zellerfeld

Germany

Prof. Dr. H. E. Schroeder

Department Oral Structural Biology

Dental Institute

University of Zurich

11 Plattenstrasse

Zurich 8028, Switzerland

Dr. Bernard Schwartz

Tufts - New England Med. Center

171 Harrison Avenue

Boston, MA 02111, U.S.A.

Prof. Jean Serra Centre de Morphologie Math^matique Ch,

Ecole des Mines A

Fountainebleau, Paris France

Cia

C,

Dr. Michael B. Sporn

Bldg. 37, Rm. "3009 National Institutes of Health

Bethesda, MD 20014

U.S.A.

Dr. James H. Steele, Jr. C, Armco Steel Research Center Ch, 703 Curtis Street A Middletown, OH 45043

U.S.A.

Dr. Sherman F. Stinson C

Bldg.. 37, Room 3C02

National Cancer Inst. Bethesda, MD 20014, U.S.A.

Dr. Piet Stroeven Ch, C, A Dept. of Civil Eng., Mat. Sci. Group

Delft Univ. of Technology

Stevinweg 4, Delft

Netherlands

Dr. Joseph Sherrick

1128 Jeffrey Court West

Northbrook, IL 60062

U.S.A.

Dr. Leon Simar

Lab. d'Anatomie Pathologique

Inst, de Patholo^ie

University de Liege

1, rue des Bonnes Villas

B-4000 Li&ge, Belgium

Dr. Jack Sklansky

School of Engineering

University of California

Irvine, CA 92664

U.S.A.

C,A

Cia

Dr. Jen Sturgess Cia

Dept. of Biochemistry

Hospital for Sick Children

555 University Avenue Toronto 2, Ontario, Canada

Mr. Louis Sutro Cia

Mail Station 20, DL-15, Room 218 Draper Laboratories

68 Albany Street

Cambridge, MA 02139

U.S.A.

C. Tasca, M.D. cia,

Inst, of Endocrinology A

Str. Dionisie Lupu 62 Bucuresti, 1(22)

Romania

Dr. Arne Sollberger

Life Science I Southern 111. Univ., School of Med. Carbondale, IL 62901 U.S.A.

Dr. Maurice M. Taylor

Dept. Nat. Defence, Perception Group

D.C.I.E.M. , Box 2000 Downsview, Ontario

Canada

528
Directory

A. C. Terrell C, Metals Research Limited A Moat House, Mel bourn Royston, Herts. SG8 6EJ England

Dr. Thomas Thorsen

Division of Math. & Eng.

El Camino College

16007 Crenshaw Blvd. Torrance, CA 90506, U.S.A.

Dr. G^za M. TimcSk Cia,

Mineral Research Lab A Faculty of Mining

Kolice Technical University

Svermova 5c KoSice, Czechoslovakia

Mr. Herold A. Treibs C Westinghouse Hanford Co. Box 1970

Richland, WA 99352

U.S.A.

Or. Oleh J. Tretiak

Biomedical Engineering & Science

Drexel University

Philadelphia, PA 19104, U.S.A.

Dr. Heinrich E. Tuchschmid

Metallurgist

Franzosenweg lOA Ch 850 Frauenfeld

Switzerland

George A. Uriano C Scientific Assistant to the Director

Institute for Materials Research

National Bureau of Standards

Washington, DC 20234, U.S.A.

H. B. M. Uylings Cia

Central Institute for Brain Research

Amsterdam

The Netherlands

Prof. dr. ir. c. J. D. M. Verhagen

Delft University of Technology Cia

Lorentzweg 1 Delft 2208 The Netherlands

Dr. Werner Villiger

Lab. fiJr Elektronenmikrospie

Pestalozzistrasse 20 4000 Basel

Swi tzerl and

Dr. Robin T. Vollmer C Department of Pathology

Duke University

Durham, NC 27705, U.S.A.

Mr. James G. Walmsley C, Dept. of Biophysics A University of Western Ontario

London, Ontario

Canada

Dr. Donald Ruomi

Roy C. Ingersoll Research Ctr. Borg-Warner

Wolf and Algonquin Roads

Des Plaines, IL 60018, U.S.A.

Cia Francis J. Warmuth Cia

Special Metals Corporation

Middle Settlement Road, Rt 5B New Hartford, NY 13413

U.S.A.

Mr. Allan F. Turpin

Bausch & Lomb

847 Azalea Drive

Rockville, MD 20850

U.S.A.

Dr. Nancy E. Warner

Dept. of Pathology

Univ. of Southern California

2025 Zon^l Avenue

Los Angeles, CA 90033, U.S.A.

Walter S. Tyler, D.V.M., Ph.D. C, Director, Calif. Primate Res. Ctr. Ch,

University of California A

Davis, CA 95616

U.S.A.

Dr. Ervin E. Underwood C, School of Chemical Engr. (Met.) A Georgia Institute of Technology

Atlanta, GA 30332

U.S.A.

Dr. Bruce A. Warren Associate Prof., Dept. of Pathology

The University of Western Ontario

London, Ontario

Canada

Dr. Richard Warren Cia,

Dept. Engineering Metals A Chalmers Univ. of Technology

S-402 20 Gothenburg

Sweden

529
Directory

Prof. Geoffrey S. Watson C Dept. of Statistics

Fine Hall Princeton University

Princeton, NJ 08540, U.S.A.

Dr. John H. L. Watson C, Dept. of Physics & Biophysics A Edsel B. Ford Inst, for Med. Research

2799 W. Grand Blvd. Detroit, MI 48202, U.S.A.

LCDR Robert A. Waugh

U.S. Navy

Box 129, Naval Hospital

Bethesda, MD 20014, U.S.A.

Prof. Dr. Ewald R. Weibel C, Dept. of Anatomy Ch,

University of Bern A Buhlstrasse 26 3012 Bern, Switzerland

D. G. Williams

Joyce, Loebel & Co. Ltd. Princes Way Team Valley-Gateshead NEll OUJ England

Nornian J . Wilsman C University of Minnesota

1342 Keston Street

St. Paul, MN 55108

U.S.A.

Dr. Robert Wlodawer Cia, Sonnenbl ickstrasse 8 A

CH-8404 Winterthur

Switzerland

Dr. Lowell A. Woodbury C Radiobiology Lab

Uni versi ty of Utah

Salt Lake City, UT 84132

U.S.A.

Robert F. Whitman C

I MAN CO 5209 Edmondson Avenue

Baltimore, MD 21229

U.S.A.

Robert L. Willes C, Image Analysing Computers, Inc. A 40 Robert Pitt Drive

Monsey, NY 10952

U.S.A.

Dr. Ch. Zaminer Cia

Bundesansalt fur Material prufung Unter den Eichen 87

1 Berlin 45 Germany

Dr. Johan Zwaan Cia The Children's Hospital Medical

Center, Dept. Ophthalmology

300 Longwood Avenue

Boston, MA 02115

U.S.A.

530
CONTRIBUTOR INDEX

Page nimiber for Chairmen in italics

Amplatz, K. , 225 Anderssen, R.S., 1, 13, 69 Bachofen, M. , ^09 Barbery, G., 189 Barnard, T. , 427 Beranek, I. , 225 Bernroider, G. , 106, 127 Berry, M. , 49 Bhargava, I. , 405 Binet, J.L., 319, 429 Blodgett, J. A., 223 Bokelmann, D. , 321 Bolender, R.P. , 277 Bollmann, W. , 93 Boniver, J. , 363 Botz, E., 431 Brandman, D.R. , 219 Briarty, L.G. , 385 Brower, R.L., 219 Canham, P.B., 415 Carpenter, A.-M., 319 Cauwe, P., 299 Chang, K.S., 499 Chen, C.H., 105, 113

Cheng, G.C., 105, 107 Chermant, J.-L., 287 Collan, Y. , 375 Combley, H. , 351 Coster, M. , 287 Courtoy, R. , 363 Craciun, E.C., 419, 423 Cruz-Orive, L.-M. , 79 d'Athis, P., 429, 507 DeHoff, R.T., 1, 29, 437, 439,

455, 457

de Montaut, F. , 429

de Wit, R., 1, 45 Damsky, C.H., 207 Davy, P. , 443

Debre, P., 429 Deter, R.L., 397 Dietrich, M.R.*

Dighiero, D. , 429 Dodge, Y. , 405 Doerr, R.M. , 93, 231, 305 Doran, M. , 351 Dullien, F.A.L. , 499 Dungworth, D. , 355 Eden, G.T., 87 Eins, S., 327

Elias, H., vii, 319, 321, 431 El-Soudani, S.M., 273

Exner, H.E., 231, 446, 469 Feinermann, E. , 401 Frledli, H.P., 409 Fujimoto, Y., 221 Cans, R.A., 215 Gateau, C, 309 Gehl, S.M. , 29 Gehr, P., 409

Good, W. , 75 Gordon, R. , 199, 201 Grant, D. , 351 Gurland, J., 231 Hall, E.L., 215 Haug, H., 139, 167, 319, 335 Heil, R.H.*

Hilliard, J.E., 1, 59, 173 Hoffmann, J.D., xi Hougardy, H.P., 139, 141 Hughes, C.E., 385 Hunn, W. , 149 Hunziker, 0., 203 Huth, G.C., 215 Hyde, D. , 355 Ifju, G., 245 Indra, M. , 121 Jaffee, O.C. , 435 Jakeman, A.J., 13, 69 James, N.T., 367 Jeulin, D. , 265 Johansson, S., 261 Johari, 0., 503 Johnson, J. A., 245 Jones, M.P., 189 Kamashki, K. , 405 Kaye, B.H., 465, 467, 491, 493 Keller, H.J. , 409 Kelly, T.*

Kim, S., 225 Kirsch, R., 139 Krekule, I., 121 Kretschmann, H.-J., 371 Krewski, D. , 351 Kuiper-Goodman, T. , 351 Lambert, P.*

Langlet, G.A. , 401 Lawson, K. E. , 123 Leahy, W.D., 99 Levick^, V. , 337

Paper presented at Congress but not printed in Proceedings.
Contributor index

Lindberg, L.G., 359 Lloyd, P.J. , 477

Losa, G.A., 277 Malamed, S., 379 Mariaux, A. , 257 Martin, R.B., 55 Mathieu, 0., 393 Meek, G.A., 519, 367 Messier, P.-E., 393 Miles, R.E., 1, 3, 443 Moore, G.A. , 41, 87, 139, 197 Moore, R. , 199, 225 Morton, R.R.A. , 159, 483

Moy, R.J., 219 Nicholson, W.L., 1, 19, 461

Ohta, Y. , 221 Oldfield, D.G., 211 Ondracek, G. , 231, 295 Orthoefer, J., 355 Ownby, P.D., 93 Pavel, M., 133 Peachey, L.D., 199, 207

Peray, 0., 257 Pfeifer, U. , 411

Pichel, W.G., 219 Pickens, J.R., 269 Piefke, F. , 497 Prevosteau, J.M. , 309 Radii-Weiss, T. , 389 Radilova, J., 389 Ralph, B., 139, 177 Ramakrishnan, P., 487 Reed, I.M., 215 Reith, A. , 427 Rhines, F.N. , 231, 233

Rink, M. , 117, 155

Robb, I., 491

Rohr, H.P., 193 Samudra, A.V. , 503 Scarlett, B. , 471, 477 Schaefer, R.J. , 223 Schleicher, A., 371 Schmeisser, H. , 163 Schopper, J.R., 185 Schulz, U., 203

Serra, J., 1, 83, 203, 257, 265, 460

Simar, L.J., 363 Slater, J., 177

Smith, J.F.H., 415 Steele, J.H., Jr., 99, 231, 245, 495 Stroeven, P. , 231, 281 Tasca, €., 419, 423 Terrell, A.C., 181 TimcSk, G.M. , 291 Tyler, W.S., 319, 355 Tukey, J.W., 229 Underwood, E.E., 91, 509 Veen, A., 207 Vbgtle, R. , 321 Walliser, €., 203 Walmsley, J.G., 415 Warren, R. , 25 Watson, G.S., 229 Watson, J.H.L.*

Weibel, E.R., 277, 319, 341, 409, 448 Wiggins, A. , 355 Willes, R.J., 181 Wingert, F. , 371 Wlodawer, R. , 315 Wolff, J.R., 327, 331 Zilles, K., 371 Zoller, L.C., 379

532
SUBJECT INDEX

A

Abnormal distributions, 197 ACTH, 379 Acute leukemia, 401 Adenocarcinoma, 321 Adrenocortical cells, 379 Aerodynamic equivalent diameters, 123 Agglomerations of inclusions, 261 Aggregate size, 507 Ahlberg method, 225 Air pollutants, 355 Aluminum powders, 487 Amyloplasts, 385 Anisotropic materials, 55, 357 Architecture, of blood vessel wall,

415 Area distribution, of sections, 291 Area frequency, of cube sections, 291 Arterial wall, composition, 337 Elastic tissue, 337 Smooth muscle walls , 337 Aspect ratio factors, 117 Autofocus, 181 Automated image analyzing computer,

123 Automated measurements, 197 Analysis, 355 Automatic comparative feature selection, 127 Automatic feature analysis, 185 Linear analysis, 189 Automatic image analysis, 177, 327 Accuracy, 141 Central nervous tissue, 327 Instruments, 141 Autoradiographic studies, 375, 429 Axons, 331

B

Ball models, contactivity , 41, 45 Hard spheres, 41, 45 Soft spheres, 41, 45 Banded structure, 163 Basic equations, 509 Derivations , 509 Measurements , 509 Biased samples, 79 Biases, 59 Bifurcation ratios, 49 Bifurcations, cerebral arteries, 415 Bimodal distribution, 401, 429, 463 Biological research, 341 Biomembranes, 427 Blastocyst formation, 397 Blood vessels, 337 Bone, human, 55

Boolean scheme, 83 Boundary length in a plane, 452 Derivation, 454 Brain, human, 167, 389 Structure analysis, 327 Branching patterns , 49 Brinell hardness, 233

vs. grain-boundary area, 233

Cancer, in human intestine, 321 Cancerous transformation, 363 Canonical patterns, 133 Cardiac disease, 225 Cardiogenesis , 435 Cardiovascular radiology, 225 Cast iron, 315

Cat, brain cortex, 203 Cerebral cortex, 203 Categorical setting, 133 Category theory, 133 Oriented approach, 107 Cell constituents, identification 341

Biology, 341 Cell Organelles, 359 Volumes, 359 Cellular orientation, 415 Malignancy, 423 Structure, 211 Central nervous tissue, 327, 331

Automatic image analysis, 327 Heterogeneous composition of, 327 Sampling, 327 Cerebral cortex, 203, 331 Synaptic density distribution, 331 Characterization of powders, 487 Chemical resolution, 211 Chick embryo, 393, 435 Heart, 435 Cholesterol, 379 Chord sizing, 163 Cineangiography, 225 Classifier - collector, 123 Cleanness assessment, of steel, 261 Closing, 151, 157 Clustering, network model of, 49 Clusters, of stringers, 265 Coefficient of variation, 59 Coimninution, 299 Composite materials, 287, 295 Computational methods, 69 Computer-assisted-by-man, 197 Computer input and processing, 121 Computer-plotted projection, 93 Concentric spheroids, 79 Concrete, crack structure, 281 Grain structure, 281

533
Subject Index

Pore structure, 281 Configurations, 133 Connectivity, 49, 55, 233 Of dispersed particles, 41, 45 Network simulation, 99 Connexity number, 265 Consistent estimate, 127 Contact states, distribution of, 41, 45 Contactivity , 41, 45 Contacts, per particle, 41 Hard contacts, 41 Soft contacts, 41 Contextual information, 107 Contour, of planar structures, 121 Convex particles, 3, 91, 509 Grains, 309

Op aque , 3 Truncation, 273 Correction coefficients, 341 Functions, 79 Correlation matrix, 75 Count, area point, 31 Area tangent, 35, 291, 457 Inflection point, 35 Line intercept, 34 Volume tangent, 33 Counting measurements, 31, 59, 509 Statistics, 59 Covariogram, 149 Covariance, 19, 83 Matrix, 19 Crack orientation distribution, 281 Critical porosity, 479 Crofton's theorem, 69 CRT display, 121 Cursor and scaling, 121 Crystal growth, 223 Cubo-spherical particles, 25 Mean intercept length, 25 Surface area, 25 Volume, 25 Curvature, 3, 29, 233, 442, 445 Average, 33 Mean, 35 Principal normal, 35 Total, 33, 235, 457 Curvature, of edges, 38

Of space curves, 30, 235 Of surfaces, 3, 35, 235 Curvature density, 3, 29, 233 Integrated, 3 Integrated mean, 3, 36, 235, 442,

445, 457 Integrated Gaussian, 3, 445 Cutting, 155 Cytochemistry, 341 Cytophotometry , 423

D

Data analysis, stereological view of,

87

Data compression, 113 Data preprocessing, 107 Definitions, basic terms, 509 Deformations, 133 Degree of orientation, 415 Dendrites, in cast iron, 315 Dendritic trees (sic) , 49 Density, of planes, 36 Of sweeping test lines, 36 Dental amalgams, 87 Brittle fracture of, 87 Derivations, basic equations, 509 Digiscan, 173 Digital computer, 117 Image storag;e, 141 Dihedral angle, 29, 36 Dilatation (dilation), 149, 155, 163, 265 Dimensionally-reduced modelling, 55 Directed secants, method of, 281 Discriminant analysis, 75 Discrimination, between phases, 189 Dispersed particles, connectivity of,

41, 45 Display, 159 Distal airspace, 355 Distribution, of contact states, 41, 45 Abnormal, 197 Normal, 197

Of inclusions, 197 Dodge method, 225 Dog, lung of, 355 Driving mechanism, specimen stage, 411 Drug metabolizing enzymes, 351

Dust particles, 123

Eccentricity, 359

Of cylinders, 359 Of ellipsoids, 359 Of spheres, 79 Edema syndrome, 435 Edges, 29 Electrical resistivity, 295 Electron probe microanalyzer , 189 Ellipticity, 309 Embryo, mice, 397 Emphysema, 355

End diastolic volume, 225 End systolic volume, 225 Endoplasmic reticulum, 351, 393 Endosperm, 385 Erosion, 149, 155, 163 Errors, observational, 59 Evoked brain potentials, 389 Exocrine pancreas, in rat, 411

Eye, human, 193

F

Faceted spheres, 25

534
Subject Index

Model, 177 Factorial analysis, 75 Feature extraction, 113 Classification, 229 Descriptors, 159 Measurements, 159, 181 Modification, 141 Recognition, 141, 229 Selection, 127 Sizing, 163 Specific data, 141 Specific measurements, 163 Feret (tangent or caliper) diameter,

117, 123 Statistical, 117 Fibrin, 405 Fibroblasts, 379 Field ion image, 93 Field specific data, 141 Figure morphometry, 117 Figure parameters, 117 Figure-specific comparison, 117 Fine particle science, 467 Fixed coordinate system, for sections,

397 Form factors, 91 Function, 181 Fraction of particles, in contact

states, 41, 45 Fracture, in composite materials, 287 Fracture surface, 269, 277, 341 Profiles, 269 Of steels, 269 Line fraction of constituents, 269 Surface fraction of constituents, 269 Fragment size, 305 Freeze fracturing, 341 Preparations, 277 Functional significance, 87

G

Geometrical probability, 3, 444 Gestalt, 87 Gland-like structures, 341 Global properties, 461 Golgi membranes, 363 Grains, 59 Contiguous, 59 Convex, 309 Growth, 233 Mean count, 375

Size, 189, 233

vs. time, 233

Snow, 75 Granulometry , 460 Isolated particles, 460 Connected media, 460 Gray steps, 167 Levels, 141, 159 Growth kinetics, in composite materials, 287 Growth models, 49

Growth path method, 287 Growth rings, in wood, 245 Patterns, 245

H

Hardwoods, North American, 245 Heart, failure, 225 Motion pictures of, 435 Hemispherical projection, 93 Hepatitis epidemica, 419 Herring eggs, 375 Hexachlorobenzene, 351 Heywood factor, 117 High density boxes, 127 Holmes effect, 69 Holographic interferometry , 223 Holography, 223 Double-exposure, 223 Hydraulic radius, 117 Hyperchromatic tumors, 423 Hyperplasia, 321 Hypertrophy, 321 Hypoplasia, 321

H3rpotrophy, 321 Hypoxia, 203

1

Identification, of wood, 245 Identification process, 133 Image analysis, 117, 155, 159, 163,

193, 309, 483, 491

In human eye, 193 Optical parallel processing, 491

Of particles, 483 Procedure, 117 Image categories, 133 Preprocessing, 113 Representation, 113 Retract, 133 Image digitalizing points , 117 Image resolution, 223 Analyzers, 181

Colors, 215 Display, 215 Immunocytochemistry , 341 Inclusion morphology, 261, 265 Inclusions, distribution of, 197 Quality control, 197 Statistical analysis, 197 Inflection point count, 35 Information content, 127 Infrared spectrum, 219 Integral geometry, 444 Intercept, distribution function, 83 Data, 117 Intercept length distribution, 173 Intercommected phases, 309 Interferometric techniques, 223 Intergranular relations, 75 Intemuclear distance, 419 Intervillous space, 405

535
Subject Index

Intestinal mucosa, 321 Intragranular relations, 75 Invariance lifting properties, 133 Inversion formulas , 13 Ischemic heart disease, 225 Isolated convex grains, 309 Isomorphisms, 133 Iterative sizing procedures, 177

K

Karyo-Interkaryometry , 419 Keyboard instructions, 159 Krumbein diameter, 117

L

Large sampling problems, 197, 411 Leukemogenesis , 363 Light pen, 141 Line drawings, 107 Lineal features, 29, 91, 509 Average curvature , 33 Curvature of edges, 38

In a plane, 452

In space, 455 Torsion, 35 Total curvature, 33 Lineal roughness , 269 Linear functionals , 19 Linear properties, of spherical

bodies, 497 Lipid droplets, 379 Liver, rat, 427 Mitotic index, 411 Local curvature, of surface trace, 269 Luminescence spectra, 211

Lung, human, 409 Dog, 355 Lymphoblastic leukemia, 401, 429 Lymphocytic leukemia, 429

M

Magnification, effect on resolution,

409 Magnification effect, 409 Manual interaction, 141 Manual optical picture analyzer, 193 Mapping, 133 Matrix materials, 295 Maximum likelihood, 19 Mean chord length, 117 Mean conditional uncertainty, 127 Mean contact coordination number, 41,

45 Mean free path, 59 Measure, 36, 444 Membrane, cellular, 277 Relative surface, 277 Metabolic alterations, 379 Metal science, 233

Metallography, 233 Metric state, 442

Mice, 375, 423 Embryo, 397 Skin carcinoma, 423 Microstructural state, 439 Minerals, 189, 299, 305 Phases, 189 Minimal iconometrics , 493 Minkowski functional, 69 Mitochondria, 367, 393 Mitochondrial membranes, 427 Mitotic index, in liver, 411 Models, in stereology, 83 Modular construction of system, 121 Moire patterns, 93 Moments, method of, 189 Monodispersed phase, 79 Systems, 91 Monte Carlo procedure, 371 Simulation, 99, 497 Motion pictures, of heart, 435 Multidimensional, 127 Data, point clouds, 229 Feature space, 127 Observations, 87 Multivariate data analysis, 75 Muscular hypertrophy, 367 Myelinated fibers, 335 Development, 335 Length per unit volume 335

N

Needle biopsy, 419 Neoplastic tissue, 423 Network analysis, 49 Network models, 99 Of clustering, 99

Of connectivity, 99 Of percolation, 99 Neural organogenesis, 393 Neurobiology, 127 Neuronal specificities, 127 Cell body, 371 Nucleoli, 371 Neurons, 49

In cat's brain, 203 Dendritic structure, 331 True numbers, 371 Nomenclature, I.S.S., 509, 513 Nonmetallic inclusions, 197, 261, 265 Stringers, 265 Normal distribution, of set averages,

197 Normal plane, 31 Vector, 31 Nuclear volume, 79, 385, 419, 423 Nucleated particle phases, 79 Nucleo-cy toplasmic ratio, 393 Nucleus, 79 Number density, of spheres, 69

536
Subject Index

Of particle centroids, 3 Number, of measurements, 462 Numerical analysis, 13 Stability, 69

0

Oblate spheroids, 79 Observations, in hyperspace, 87 'Opaque' convex particles, 3 Opening, 149, 155 Optical character recognition, 107 Optical resolution, effects of, 409 Optimal properties, 69 Of linear data, 69 Of planar data, 69 Optimum (Wiener) filtering, 113 Optomanual instruments, 167 Procedures, 335 Systems, 193 Organelle profiles, 379 Orientation, degree of, 415 Distribution, of cracks, 281

Of lineage segments, fracture, 269 Orthogonal coordinate frame, optimum,

75 Orthographic projection, 93 Osculating plane, 30 Circle, 30 Spherical image, 34

P

Packed beds, of particles, 481

Paranuclear profiles, 79 Partially-oriented surfaces, 367 Particle distributions, 478

By filament lengths, 478

By lamina areas, 478

By number, 478 Particle shape, 25, 91, 476 Particle size distribution, 185, 299, 427 Saltykov area method, 503 Saltykov Intercept method, 287 Johnson-Saltykov method, 503 Particle size analyzer, 167, 335 Classifier, 431 Particle technology, 469, 477 Particles, image analysis of, 483 Data processing, 485 Measurements, 484 Number, 484 Overlap, 483 Sampling, 483 Particles, non-contiguous, 59 Particulate systems, 471, 477 Pattern classification, 113 Analysis, 159 Pattern recognition, 107, 123, 127, 133 Heuristic approach, 107

Heuristic and structural approaches,

107 Mathematical-statistical approach,

107 Pictorial, 107, 113 Statistical approach, 107 Structural or linguistic approach,

107 Visual, 107 Perfusion culture system, 397 Percolation, network model of, 99 Probability, 99 Picture matrix, 107 Pigeon, pectoralis major muscle, 367 Pigment particles, in paints, 491 Placenta, human, 405 Pocoster program, 385 Point clouds, 229 Point counting device, 431 Light box, 431 Stage micrometer, 431 Poisson field, 19 Poisson process, 3 Flat process, 83 Point process, 83 Polarization projection method, 221 Polarizing microscope, 415 Poly crystalline aggregate, 233 Microstructure , 99 Polydispersed phase, 79 Systems, 91, 291 Population, double, 429 Single, 429 Overlapping, 107 Pores (see also Voids), 19, 155 Near-spherical, 19 Porosity, 55 Porous bodies, 55 In nuclear fuels, 295 Position-invariant operations, 113 Powder particles, analysis, 469 Projection, 470 Random section, 470 Size distributions, 470 Pregnancies, 405 Normal, 405 Triplet, 405 Preimplantation embryogenesis , 397 Preparation artefacts, 341 Probability, 45, 177 Geometrical, 3, 444

Of particle contacts, 45 Probabilistic foundations, of stereology, 443 Probes, 69, 133 Linear, 69 Test, 59 Product integration, 13 Profile size analyzer, 167 Projected image, 33, 273, 509 Projection geometry, 93 Projections, 133, 202, 497

537
Subject Index

Of particle overlap, 483 Projectors, 133 Prolate spheroids, 79 Protein synthesis, 385 Psychophysiological techniques, 389

Quality control, of steel, 261 Quantitative image interpretation, 219 Quantitative f ractography , 273

R

Radial distribution analysis, 305 Radiation, 363 Random sectioning, in a computer, 177 Random spatial structure, 3 Random spheres model, 41, 69 Rarely-occurring structures, 197, 411

Rat, brain, 121, 127 Exocrine pancreas, 411 Kidney, 411

Liver, 351 Liver adenoma, 423 Rous sarcoma, 359 Recognition categories, 133 Function, 133 Reconstructed images, 223 Artifacts in, 215 By holography, 223 By high-voltage electron microscopy,

202, 207

By X-ray scanners, 202 Reconstruction , three-dimensional,

from

Projections, 202, 215 Serial slices, 207 Scanning proton microscopy luminescence data, 211 Medical radiographic data, 215 Relative surface coverage, 87 Membrane surface, 277 Surface fraction of constituents,

269 Of dimples, 273 Resolution, 59, 141 Reversible figures, 389 Rose, of the number of intersections,

281

Of direction, 149 Rounded cubes , 25

S_

Sampling, 341 Anisotropic structures, 341 Periodic structures, 341 Stratified structures, 341 Sampling errors, 59 Boundary area analysis, 59 Lineal feature analysis, 59 Volume fraction analysis, 59

Satellite images, 219 Scan instruments, 167 Scanning methods, limitations, 193 Scanning optical microscope, 173 Self-instructional, 173 Semi-automatic, 173 Statistical summary, 173 Scanning proton microscopy, 211 Section thickness, 341, 375 Effect of, 341 Seed development, 385 Semi-automatic systems, 193 Scanning optical microscope, 173 Serial sectioning, 203, 375, 415 Frozen sections, 203 Metallic specimens, 233 Shading corrector, 159 Shape, 91

Of povjders, 485 Shape factor, dynamic, 485 Shape parameters, 91 Coefficients, 359 Discrimination, 181 Distribution, 185 Function, 123, 181 Simultaneous regression analysis, 87 Single cell characteristic, 127 Sintering force, 233 Size analysis, of regular polyhedrons,

185, 291

Of irregular bodies, 291 Size distribution, of Leucocytes, 401 Non-spherical objects, 499 Ores, 309 Retained austenite, 503 Size distributions, 13, 19, 69, 123,

177, 181, 185, 189, 299, 427, 431,

497, 499, 507 Log-normal, 91 Saltykov area method, 503 Saltykov intercept method, 287 Johnson-Saltykov method, 503 d'Athis method, 429 Sizing distributor, 123 Skeletal muscle, 367 Snow grains, 75 Software, 141, 159 Analysis, 141 Sortabllity, of ores, 305 Spatial distribution, 245, 305 Wood Annual ring, 245 Spatial filtering, of paint film, 491 Spatial smoothing, 113 Specific surface, 55 Specimen tilting method, 221 Spectral differentiation, 13 Derivative, 69 Spherical image, 34 Sphericity index, 429 Stable estimates, 13 Stainless steel, 503 Starch synthesis, 385

538
Subject Index

Statistical decision theory, 113 Statistical relationships, 59 Significance, 87

Sunnaary, 159 Statistics of measurements, 461 Global properties, 461 Number of measurements , 462

Steel, inclusions in, 197, 261, 265 Non-metallic stringers, 265 Standard patterns for, 265 Quality, 265 Stereograms, 219 Stereographic projection, 93 Stereological quantities, 3, 79, 509 Fundamental, 3 Parameters, 79, 91 Stereometric factors, 295 Stereoscopic image, 221 Magnifications, 221 Photomacrography , 221 Stochastic processes, 83, 113 Stroma, 405 Structural elements , correlation with

mechanical properties, 75 Structure of clouds , 219 Structure-property relationships, 91, 233 Structuring element, 149, 155 Surface area per unit volume, 448 Derivation, 449 Surface roughness index, 273 Symbols, stereological, 509, 513 Syntax-directed recognition, 107 Systematic error, 427 Critical angle, 427 Point count, 447

T

Tangent coiint , 35, 233, 291, 457 Areal, 35, 233, 291, 457 Volume, 33, 233 Templates, 133

Test line, 3, 34

Sweeping, 33, 235, 458

Test planes, 3, 36 Sweeping, 33, 235 Texture analyzer, 203, 257, 265, 299 Analysis, 149 Thermal conductivity, 295 Thick sections, 3 Thin sections, 13, 75, 507 Model, 69 Foils, 273, 493 Three-dimensional reconstruction, 121, 207 Display, 207 Shape, 177 Threshold of detection, 123 Setting, 141 Thymic blast cells, 363 Time series analysis, 389 Tomography , 202

Topological analysis, 49 Properties, 185, 233 Topological state, 439 Connectivity, 439 Number, 439 Torsion, 29, 35 Total curvature, 29 Of lineal features, 33

Of surfaces, 35 Toxicology, 351 Tracing instruments, 167 Tracing stylus, in reconstruction, 207 Transmission microscopy, 3 Transnuclear profiles, 79 Transparent space, 3 Trophoblast, 405 Truncated moments, 13 Truncation, of convex particles, 273

Of convex voids , 273 TV display, 215 Two-dimensional transformation, 149 Amendment, 163 Two-line principle, 117 Two-phase nuclear materials, 295

U

Ultrasonic device, 167

V

Variance, 59 Vector, 29 Binormal, 31 Normal, 31 Tangent, 30 Vector locus, particle, 471 Ventricule, human, 225 Ventriculography, 225 Villi, 405 Visual cortex, cats, 335 Visual display terminal, 229 Visual perception, 389 Voids (see also Pores),

In dental amalgams , 87 In size distribution, 495 Truncation of, 273 Volume fraction, 446 Derivation, 446 W

Wood, 245 Cells, 257 Diffuse-porous, 257 Fibers, 245, 257 Lumen, 257 Rays, 245, 257 Ring-porous, 257 Shrinkage, 257 Vessels, 245, 257 Wood microstructure, 245 Anisotropy, 257

539
Subject Index

Distribution pattern, 245 Quantitative analysis, 2A5, 257 Stereological characterization, 245 Woods,

North American, 245 Tropical, 257 Wulff construction, 25

540
NBS.IMA (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

K PLiBLirATlON OR RKPORT NO. NBS SP-4312. Gov't Accession

No.

3. Recipient's Acccs.sion No.

4. TITLt" AND SUBTITLt

FOURTH INTERNATIONAL CONGRESS FOR STEREOLOGY

5. Publication Date January 1976

6. Performing Organization Code

7. AUTHOR(S) Ervin E. Underwood, Roland deWit, and George A. Moore

8. Performing Organ. Report No.

». PERFORMING ORGANIZATION NAME AND ADORES.^ NATIONAL BUREAU OF STANDARDS DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Prcject/Task/Work Unit No.

11. Contract /Grant No.

12. Sponsoring Organization Name and Complere Address (Street, City, State, ZIP) 13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 76-8668

16. ABSTRACT (A 200-word or less factual summary ol most signilicant inlormation. It document includes a signilicant bibliography or literature survey, mention it here.) These proceedings contain most of the papers presented at the technical sessions and workshops of the Fourth International Congress for Stereology, held at the National Bureau of Standards, Gaithersburg,

Maryland, from 4-9 September 1975. Of the 113 papers recorded here,

10 are invited lectures, 66 are submitted papers, and 37 come from the

six workshops. Three of these workshops containing 17 papers were

integrated into the technical sections for these proceedings. Topics covered in the Technical Sections are Principles and Mathematical

Developments, Pattern Recognition, Instrumentation, Three-dimensional Reconstruction, and Stereological Applications in Materials and

Biology. Additional topics covered in the Workshops are Mathematical

Foundations of Stereology, Particle Science, and Size Distributions.

17_ KEY WORDS: Anatomy, automated microscopy; biology; convex particles; curvature; geometrical probability; image analys is ; materials science; mathematics; microscopy; microstructure ; morphometry; oriented structures ; particles; pattern recognition; probability theory; quantitative

microscopy; reconstructions; sections, serial sectioning; shape parameters;

size distributions; spacing distributions, statistical analysis; statistics;

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the tirst letter of the first key word unless a proper name; separated by semicolons) Stereology; stereometry; stereoscopy; stochastic models; structures; three - dimensional reconstruction.

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