Standardization of chromatographic systems

A primary chromatographic system is operated with a standard sample at several temperatures to generate primary retention times which are fitted to a function to determine thermodynamic constants to relate the times to temperature. A target chromatographic system is operated with the standard sample to generate secondary retention times. The function is used with these times to determine and an effective column parameter for the target system. The function then is used with this parameter and the primary times to determine a pressure program. Further operation of the target system with a application sample and the pressure program effects standardized retention times. A particular searching technique is utilized to apply the function. A temperature calibration technique with a selected sample measures column temperature for the target system, and a validation is done on the target system.

This invention relates to chromatographic systems and particularly to the 
standardization of such systems. 
BACKGROUND 
Chromatography involves physically separating constituents of a sample in a 
carrier fluid and measuring the separation. In gas chromatography (GC) the 
carrier is a gas or at least a supercritical fluid which acts similarly in 
the system. In liquid chromatography (LC) the carrier is a liquid. In 
either case a pulse of the sample is injected into a steady flow of the 
carrier, and the constituents are adsorbed or absorbed and desorbed by a 
stationary phase material in a column. At the end of the column the 
individual components are more or less separated in time. Monitoring the 
column effluent with a suitable detector provides a pattern of retention 
times which, by calibration or comparison with known samples, indicates 
the constituents of the sample qualitatively and quantitatively. The main 
components of such a system are the column, an injector with a mixing 
chamber for introducing the sample into the carrier, a detector at the 
outlet end of the column, fluid controls, and a computer for processing 
and displaying the output of the detector. The display is generally in the 
form of retention times. In GC an oven generally is used to elevate 
temperature to maintain the sample in a volatile state, and to improve the 
discrimination of constituents. Various gas chromatographic systems are 
disclosed in U.S. Pat. Nos. 5,405,432, 5,545,252 ("Hinshaw 1"), U.S. 
patent application Ser. No. 08/734,689 filed Oct. 21, 1996 ("Hinshaw 2"), 
and an article "The Effects of Inlet Liner Configuration and Septum Purge 
Flow Rate on Discrimination in Splitless Injection" by J. V. Hinshaw, J. 
High Resolution Chromatography 16, 247-253 (Apr. 1993). A liquid 
chromatographic system is disclosed in U.S. Pat. No. 4,579,663. 
Interpretations of retention time patterns in chromatography tend to 
require skill and experience, as different systems and particularly 
different columns behave differently so as to effect different patterns 
for the same sample material. An operator selects operating parameters, 
such as temperature and pressure, or may vary these parameters during a 
run, according to judgment. Thus uses of these systems for evaluating 
samples is dependent on the skills of the operators, and it has been 
difficult to compare results of different systems, columns and operators. 
When a chromatographic method is developed it is often desirable to 
transfer it to the same system at a later time, the same system with a 
different column, or another system. The task is made more complicated by 
other factors including different calibrations of temperatures and 
pressures, and different oven geometries resulting in different 
temperature gradients. Differing characteristics of columns include 
length, internal diameter, phase thickness and phase chemistry, and these 
characteristics are difficult to determine with precision without 
destroying the column. These variations in systems, particularly columns, 
cause the retention times to change for different systems and the same 
system at different times, even switching the order of some peaks. 
Recalibration is complex and can be time consuming. Standardization would 
be desirable, such as is done in optical spectroscopy, for example as 
disclosed in U.S. Pat. No. 5,303,165 (Ganz et al.) It would be 
particularly desirable to be able to provide a useful library of basic 
standards associated with specified types of columns, so that 
chromatographic results may be compared universally. 
Objects of the invention are to provide a novel method and a novel means 
for standardizing chromatographic systems so as to allow direct comparison 
of information generated from different systems and the same system at 
different times, including different chromatographic columns and the same 
column at different times. Particular objects are to provide a novel 
method and a novel means for establishing certain operating parameters for 
each chromatographic system such that retention times are substantially 
identical for different systems and the same system at different times. 
Other objects are to provide a novel method and a novel means for 
optimizing parameters for chromatographic systems. Additional objects are 
to provide a novel method and a novel means for measuring temperature of 
chromatographic columns, particularly to further standardization, and also 
to provide a novel method and a novel means for validating chromatographic 
systems. 
SUMMARY OF THE INVENTION 
The foregoing and other objects are achieved, at least in part, by a method 
and a means for standardizing a target chromatographic system with a 
primary chromatographic system. Each system includes carrier means for 
passing a fluid carrier through the column, injection means for injecting 
a pulse of sample into the carrier to effect a mixture passing through the 
column subject to characteristic retention times for constituents of the 
sample, detector means receptive of the mixture for effecting signals 
representative of the retention times, and processing means receptive of 
the signals for presenting corresponding retention indicators. Each system 
has system parameters and operating parameters, the operating parameters 
comprising a first parameter having selectable first programming and a 
second parameter having selectable second programming, each programming 
being with respect to time. The retention times are related to the system 
parameters and the operating parameters by a mathematical function having 
function parameters including thermodynamic constants associated with 
interactions of the constituents with the column. 
In preferred embodiments, the system is a gas chromatographic system with a 
gas carrier, the first parameter is column temperature and the second 
parameter is inlet pressure of the carrier to the column. Also, 
preferably, the retention indicators are retention times, and the system 
parameters include column dimensions. 
The primary system is operated with a standard sample, a selected primary 
second program (e.g. pressure) for the second parameter, and a plurality 
of selected primary first programs (e.g. temperature) for the first 
parameter, so as to generate corresponding primary retention indicators 
(e.g. times). The primary retention indicators and the first programs are 
fitted to the function, with the second program, so as to determine 
thermodynamic constants whereby the function is representative of a 
virtual chromatographic system. The thermodynamic constants are stored for 
future application with the target chromatographic system. 
A target chromatographic system is initially operated with the standard 
sample, substantially the primary second program, and a plurality of 
secondary first programs, so as to generate corresponding secondary 
retention indicators. Effective system parameters (e.g. column dimensions) 
are established for the target chromatographic system, by assumption, 
earlier measurement or a technique according to an aspect of the invention 
(explained below). A secondary second program then is determined for 
which, with the effective system parameters, the function yields 
substantially the primary retention indicators for the primary first 
programs. 
In an actual operation, the target chromatographic system is operated with 
a application sample, the secondary second program and a selected first 
program, so as to generate at least one corresponding test retention 
indicator. By use of such secondary program, each test retention indicator 
is standardized to the virtual chromatographic system. 
The function relating retention times to the parameters such as temperature 
and pressure are based preferably on theoretical relationships of a 
chromatographic system. As such a function generally is complex, special 
techniques may be required for its application, particularly in the 
determination of an effective column dimension and a secondary second 
(pressure) program. In an aspect of the invention, a method and a means 
are provided for determining values for one or more specified parameters 
for a chromatographic system. There are system parameters (e.g. column 
dimensions) and operating parameters (e.g. temperature and pressure) 
related to retention times by a mathematical function having function 
parameters including these parameters as well as others such as 
thermodynamic constants related to interactions of the sample with a 
stationary phase in the column. The function parameters have predetermined 
or assumed values except for the specified parameters. A specified 
parameter may be column inlet pressure, or column length. 
The system (e.g. target system) is operated so as to generate retention 
indicators. An initial data base is provided, defining ranges of potential 
values of the specified parameter or parameters. Theoretical retention 
indicators are computed with the function for the potential values and the 
predetermined or assumed values, differences are computed between the 
theoretical retention indicators and the secondary retention indicators, 
and the differences are searched for a minimum therein, such that the 
minimum establishes an effective value for each specified parameter. 
In another aspect a method and a means are provided to determine column 
temperature of the target chromatographic system relative to that of the 
primary chromatographic system. A temperature standard is provided 
comprising a calibration compound having temperature dependent retention 
time, and a plurality of homolog standards having a homolog relationship 
between corresponding retention indicators and retention times. The 
primary chromatographic system is operated with the temperature standard, 
a selected primary pressure program and a plurality of selected 
calibration temperatures so as to generate a primary set of retention 
times for each calibration temperature, each primary set comprising 
homolog retention times for the homolog standards and a compound retention 
time for the calibration compound. The homolog relationship and the 
primary set of retention times are first utilized for each calibration 
temperature to determine calibration constants for a temperature 
relationship relating retention indicator for the calibration compound to 
column temperature for the primary system. The target chromatographic 
system is operated with the temperature standard and a measured column 
temperature so as to generate a test set of retention times, the test set 
comprising test retention times for the homolog standards, and a test 
retention time for the calibration compound. The homolog relationship and 
the test set of retention times are secondly utilized to determine a 
secondary retention indicator for the calibration compound. The 
temperature relationship is applied with the calibration constants and the 
secondary retention indicator to determine a calibrated temperature 
corresponding to the measured temperature. 
In a further aspect a method and a means are provided for validating a 
target gas chromatographic system having a calibrated temperature 
relationship between its column temperature and the column temperature for 
a primary gas chromatographic system. A validation standard is provided 
comprising selected validation constituents and a plurality of homolog 
standards having a homolog relationship between corresponding retention 
indicators and retention times. The primary chromatographic system is 
operated with the validation standard, a selected primary pressure 
program, and a primary validation temperature for the column, so as to 
generate primary validation retention times for the validation 
constituents and homolog retention times for the homolog standards. The 
target chromatographic system is operated with the validation standard, 
substantially the primary pressure program, and the measured column 
temperature so as to generate secondary validation retention times for the 
validation constituents and test retention times for the homolog 
standards. The homolog retention times are utilized to determine primary 
homolog parameters for the homolog relationship, and the test retention 
times are utilized to determine secondary homolog parameters for the 
homolog relationship. The homolog relationship, the primary homolog 
parameters and the primary validation retention times are utilized to 
effect preliminary retention indicators. The homolog relationship, the 
secondary homolog parameters and the secondary validation retention times 
are utilized to effect secondary validation retention indicators. The 
preliminary retention indicators are adjusted with the temperature 
relationship to a calibrated temperature corresponding to the secondary 
validation temperature so as to effect primary validation retention 
indicators. Differences between corresponding primary validation retention 
indicators and secondary validation retention indicators are calculated, 
and it is determined whether the differences are less than a predetermined 
limit corresponding to whether the target chromatographic system is valid.

DETAILED DESCRIPTION 
The invention is utilized in an otherwise conventional or other desired gas 
chromatographic (GC) system such as described in the aforementioned U.S. 
Pat. No. 5,545,252 ("Hinshaw 1") and U.S. patent application Ser. No. 
08/734,689 ("Hinshaw 2"), each being of the present assignee and 
incorporated herein in its entirety by reference. A suitable system is a 
Perkin-Elmer Autosystem XL (trademark). 
A preferred type of GC system 10 (FIG. 1) utilizes split flow with back 
pressure regulation in the manner illustrated in the aforementioned 
Hinshaw article, FIG. 2(b) thereof. A carrier gas from a 
pressure-regulated source 12 is supplied through a gas flow controller 14 
to an injector device 16, each of which may be essentially any 
conventional or other desired type. For example, the flow controller is of 
the type taught in Hinshaw 2, and the injector is of the type taught in 
Hinshaw 1. A portion of the carrier is passed from the injector 16 into 
and through a chromatographic column 18 formed of a long tube, e.g a fused 
silica tube 25 m long and 0.25 mm inside diameter, having a selected 
stationary phase on the inside column wall such as methyl silicone 0.25 
.mu.m thick. 
Most of the remainder of the inlet flow passes out to the ambient space 
(normally atmosphere) through a back pressure regulator 13, for example as 
taught in Hinshaw 2, so as to maintain a constant, selected pressure of 
carrier into the column passage. The back pressure regulator is connected 
to a split flow outlet 15 from the injector, preferably with a charcoal 
filter 17 in the line to protect a downstream component from clogging. In 
one form of pressure controller, a variable flow restrictor 19 follows the 
filter. A pressure transducer 20 is connected to measure pressure at the 
split flow outlet which is the pressure at the inlet to the column. An 
electronic feedback device 23 connected from the transducer to the 
restrictor is utilized, preferably under control by computer 30. A 
conventional purge gas outlet from the injector comprises, for example, a 
fixed pressure regulator 27 tapped into the injector and connected to a 
fixed gas flow resistor 29. 
A sample material is formed of chemical constituents, generally organic 
molecules including those containing other elements besides carbon and 
hydrogen, such as chlorine, oxygen, nitrogen and/or sulphur. A pulse of 
the sample is injected from a sample source into the carrier in the 
injector device where a mixture is formed with the carrier gas. The pulsed 
mixture passes through the column during a time period which typically is 
several minutes after the sample injection. In the column 18 (FIG. 2) a 
stationary phase of a suitable substance on the tube wall adsorbs from the 
carrier gas 21 and then desorbs the chemical constituents of the sample. 
Different constituents have different affinities for the stationary phase 
and thereby exit the column at different characteristic times, known as 
retention times, associated with different times for retention in the 
stationary phase. The velocity of the carrier gas ("mobile phase") 
contributes to the total retention time; the term "retention time" means 
the total time from injector to detector in the stationary and mobile 
phases. A detector 26 at the column outlet measures a physical property of 
the carrier and mixture, the magnitude of the property changing with each 
constituent passing through. Various types of detectors are used, such as 
hot wire, flame ionization, electron capture, thermionic and flame 
photometric. The detector effects signals on a line 28, the signals being 
representative of the retention times as well as concentrations. 
A computer 30 receives and processes the signals into a series of peaks 
(called "components") representative of the sample constituents, the 
plotted locations of the components representing corresponding retention 
times. The computer presents (e.g. on a monitor) corresponding retention 
indicators which may be the retention times directly or other indicators 
computed from the times such as retention indices (explained below). The 
peak components may be identified by an operator or the computer to known 
chemical constituents, and peak heights provide a quantitative measure. 
The computer system 30 is conventional and actually may be a combination of 
processing units including a main computer such as a DEC PC LP433 
incorporated into the GC by the manufacturer thereof. Auxiliary processing 
units may include one for automatic sample selection, another for 
controlling the oven, and another for communications and pneumatic 
controls. These units communicate to the main computer via an interface 
processor. Each unit may include appropriate firmware. As this computer 
system is conventional, and the details are not important to the present 
invention, except for an oven controller 46 it is depicted as a single 
computer 30 in FIG. 1. Thus the computer generally includes a central 
processing unit 32 (CPU) with associated memory 34 (RAM); appropriate 
analog/digital converters (in and/or out as required); disk memory 
sections (more generally a computer readable storage medium) typically 
including a hard disk 36, laser disk (CD-Rom) and/or means for accessing a 
floppy disk 38, a keyboard 40 for operator input, and a monitor 42 and/or 
a printer for presentation of the retention indicators. 
The computer programs for the standard GC operations and the present 
invention are written in a conventional language such as "C", "C++", 
Visual Basic.TM. and data is managed by a spreadsheet program such as 
Excel.TM.. Programming required for the present invention will be 
recognized readily from the flow charts and descriptions herein, and can 
be achieved by those of ordinary skill in the art. 
The column 18 is enclosed in an oven 44 or the like with the controller 46 
to set and regulate the temperature of the column. The temperature is 
measured with a platinum resistance thermometer 48 (or other precision 
temperature sensor) with a temperature signal being passed on a line 51 to 
the computer 30. Retention times are temperature dependent, so data 
usually are taken at one or a series of known temperatures, optionally 
with ramping or other temperature program during a run. Similarly, the 
retention times are pressure dependent, and the data also may be taken at 
one or a series of known pressures, optionally with ramping or other 
pressure program. With ramping, the associated parameters include start 
and finish values as well as ramping rate and start or finish time for the 
ramping. 
The term "program", as applied to temperature, pressure or other such 
parameter, and as used herein and in the claims, means a fixed level (e.g. 
isothermal or isobaric) or a varying of such parameter with time during a 
run with an injected sample. "Ramping" is typically but not necessarily a 
linear change, usually increasing, and a program may combine fixed levels 
and ramping, and may include several rampings. 
Thus operating parameters for the system typically include isothermal or 
other programmed temperature of the chromatographic column, constant or 
other programmed inlet pressure to the column, and composition of carrier 
gas (which may be a fixed or variable mixture, for example, of 
methane/argon or N.sub.2 /Ar). Other program parameters may include ramp 
rates, starting and final temperatures and/ or pressures, times at each 
level, and/or initial and final times for the ramping. A program may be 
more complex, such as with several fixed levels with ramping between, or 
non-linear changes. 
Pressure at the column outlet generally is atmospheric, or may be vacuum 
where the GC is used, for example, in conjunction with a mass 
spectrometer. The outlet pressure P.sub.o is measured with a barometer 49 
but ordinarily is not regulated. However outlet pressure could be reset or 
varied as part of a pressure program. Another possible operating parameter 
may be column length taken from its categorization as a column dimension 
(described below), as the column length is readily measured and the column 
may be cut successively for a series of system runs, particularly with the 
primary system discussed below. 
Use is made of theoretical relationships that describe operation of a GC, 
in the form of a mathematical function. A suitable function is expressed 
by or derived from an integral: 
EQU .intg..sub.0.sup.t.sbsp.R t.sub.0.sup.-1 
.multidot.(1+a/.beta.e.sup.b/T+cT).sup.-1 .multidot.dt=1 Eq. 1 
where: 
EQU t.sub.0 =32.andgate.(T)/3 L.sup.2 /r.sub.c.sup.2 (p.sub.i.sup.3 
-p.sub.o.sup.3)/(p.sub.i.sup.2 -p.sub.o.sup.2).sup.2 Eq. 1a 
and t.sub.R is retention time, T is column temperature, .andgate.(T) is 
carrier gas viscosity as a known function of temperature, L is column 
length (FIG. 2), r.sub.c is column radius, p.sub.i is inlet pressure and 
p.sub.o is outlet pressure. The term t.sub.0, variously called dead time, 
mobile phase time or gas holdup time, represents the time of the pulse in 
the carrier gas. The term .beta., called phase ratio, is the ratio of 
volume of the mobile phase (carrier gas) to that of the stationary phase, 
such that .beta.=r.sub.c /2d.sub.f where d.sub.f is thickness of the 
stationary phase on the column tube wall. Column dimensions for the column 
geometry are in the function as L, r.sub.c and .beta.. The thermodynamic 
constants a and b are related to enthalpy and entropy and, without the 
constant c, were discovered to be slightly temperature dependent. To 
substantially remove this dependency, the additional thermodynamic 
constant c is introduced, and all of a, b and c are deemed to be constant 
for each sample component, (but generally are different for different 
components and stationary phases. However, c may be quite small and even 
assumed to be zero if resulting accuracies are sufficient. Eq. 1 is used 
conventionally without the c term which is added according to an aspect of 
the present invention. Temperature and/or pressure may vary with time 
during a run, so solution or application of the integral depends on which 
and how these parameters are so varied during the integrating time from 0 
to t.sub.R. 
For constant temperature and pressures the function may be integrated to a 
form: 
EQU t.sub.R =t.sub.0 .multidot.1+a/.beta.e.sup.b/T+cT ! Eq. 2 
This may used for constant (isobaric and isothermal) conditions or fixed 
portions of programs. Otherwise Eq. 1 is solved by a conventional 
computational technique such as with Simpson's rule using, for example, 20 
steps. 
Any other suitable function that describes chromatography may be used in 
place of these equations. Such function preferably is based on the physics 
of chromatography but may include or be based on empirical factors. For 
example a modification of the function may be made to compensate for 
slight leakage of the carrier gas through the column wall, such as taught 
in copending provisional U.S. patent application Ser. No. filed Apr. 15, 
1997 Docket No. ID4531! entitled "Method and Apparatus to Compensate for 
Chromatograph Column Permeativity", by inventors Jerry E. Cahill and David 
H. Tracy of the present assignee and incorporated herein in its entirety 
by reference. 
The function (e.g. Eq. 1 or 2) is stored in computer memory in the form of 
program code (for the function itself) and data code (for the parameter 
data). With either form of the function, the independent variable of the 
function preferably (and in the present example) is the column temperature 
T (or program thereof), with function parameters including the inlet and 
outlet pressures P.sub.i and P.sub.o, the column geometry .beta., L and 
r.sub.c, and the thermodynamic constants a,b,c. Alternatively, the inlet 
pressure may be useful as the independent variable in place of temperature 
which becomes a function parameter. More broadly, any of the operating 
parameters may be used for the independent variable, and there may be more 
than one independent variable such as temperature and its ramp rate, or 
temperature and pressure. 
To implement the invention, with reference to the flow chart FIG. 3, a 
primary chromatographic system 52 is provided which should be of the same 
general type as a target system (discussed below) including substantially 
the same type of column. The primary system has established (assumed or 
known) primary column dimensions for its chromatographic column. Such 
dimensions include the average thickness d.sub.f of the stationary phase 
on the column wall (FIG. 2), the column length L, and the column radius 
r.sub.c, thereby establishing the phase ratio .beta.=r.sub.c /2d.sub.f. 
The thickness may be measured, for example, by weighing of the tube during 
manufacture (before and after packing). Alternatively, the column 
dimensions may be determined after initial measurements with the primary 
system, by destruction of the column for measurement of the thickness and 
radius, as this column will no longer be needed. For a packed column, the 
volume of the stationary phase may be used as a geometry dimension. 
A standard sample 54 is selected to contain suitable constituents to span 
the range of expected interactions of actual samples with the stationary 
phase. A standard with about 8 to 10 compounds is useful. The compounds 
should be selected for suitability with the stationary phase, for example 
in a manner taught in an article "Characterization of Some Liquid Phases" 
by W. O. McReynolds, J. of Chromatographic Science 8, 685-693 (December 
1970), incorporated herein by reference. A suitable standard for a 
stationary phase of methyl silicone contains the following: n-nonane, 
2-octanone, n-decane, 1-octanol, n-undecane, 2,6-dimethyphenol, 
2,4-dimethylanaline, naphthalene, n-dodecane, and 2-propanol as solvent. 
The primary system 52 is operated 53 with the standard sample 54, a primary 
inlet pressure 56 (or, more broadly, a primary pressure program), and with 
a successive plurality of selected temperatures 58 for the temperature. 
Each temperature program may simply be an isothermal temperature level, or 
may consist of programming parameters for a run such as initial and final 
temperatures, ramping rate and initial and final times for the ramping; 
any one or more of these parameters may be varied for successive runs. 
Selection of temperature programs should depend on such factors as 
intended types of application samples and intended temperature ranges and 
programming. Examples of four programs are as follows; the first also 
shows how the selected programs can be useful for auxiliary purposes 
explained below: 
1) 120.degree. C. for temperature calibration, phase ratio, selectivity 
validation; 250.degree. C. for temperature calibration; Ramp 80.degree. C. 
to 250.degree. C. at 5.degree. C./min, for effective column length. 
2) Isothermals at 80.degree. C. to 100.degree. C. in 10.degree. C. steps. 
3) Isothermals at 250.degree. C. to 300.degree. C. in 10.degree. C. steps. 
4) Ramp from 80.degree.C. to 250.degree. C. at 5.degree. C./min; ramp from 
80.degree. C. to 250.degree. C. at 10.degree. C./min; ramp from 80.degree. 
C. to 250.degree. C. at 15.degree. C./min. 
This operation generates a set of primary retention times (RT's) 60 (which 
may be converted to other related retention indicators) for each 
temperature program, which may be plotted as a primary chromatogram (e.g. 
FIG. 4) with a component 61 (peak) for each constituent in the standard 
sample. These components are identified 62 by operator or a computer 
program in the conventional manner by comparison with a list of expected 
times in a pre-established order, accounting for temperature and rejecting 
noise peaks. It is advantageous to pick one standard time and ratio the 
other times to that for the selection process. 
For each component from the standard, the primary retention times and the 
primary temperatures (or programs) are fitted 64 to the function 66 (Eq. 1 
or 2). The computations for the fitting determine the thermodynamic 
constants a,b,c (68), such that the function relates retention time to 
column temperature, pressure and column geometry. All other function 
parameters are known, including the primary column dimensions 70. The 
thermodynamic constants are different for each component, i.e. each 
constituent of the sample, and are specific to the chemistry of the 
stationary phase. The number of temperature programs needed for a fit is 
at least as many as there are number of thermodynamic constants. (A 
fitting technique--"Application of Function"--is set forth below.) 
The function with the computed set of thermodynamic constants may be 
identified to a virtual (hypothetical) chromatographic system 69, with the 
pressure and column dimensions being adjustable according to variations in 
target systems. The primary system and its column are no longer needed. 
This virtual system is deemed to be a standard to compare with other GC 
systems (herein designated "target chromatographic systems") that are 
similar to the primary system, particularly with the same type of column 
including stationary phase. 
A combined plot from the functions for all components yields a simulated 
chromatogram of the virtual system which essentially will look like FIG. 
4. (Peak heights may be selected arbitrarily to be different for the 
components to aid in identification. Peak width is programmed to be 
similar to that of actual peaks.) A floppy disk 71 (or other computer 
readable storage medium such as a CD-ROM or tape) containing the 
thermodynamic constants may be provided along with an associated column 
and a standard sample. The storage medium may also contain the program 
base for the function if this is not already in the instrument computer. 
A target chromatographic system 72 is operated 73 with the standard sample 
54 (meaning the original or a substantial duplicate thereof), and with a 
pressure program setting 56' substantially as the same primary inlet 
pressure 56 (or other pressure program); this pressure may not be quite 
the same as the primary due to variations in system and settings. However 
a pressure calibration step is desirable for example by fully opening the 
flow valve 19 (FIG. 1) at the split flow exit and stopping carrier flow 
with the flow control 14 so as to expose the pressure gage to atmospheric 
pressure and use this as a zero calibration point 75 (gage pressure). 
A set of temperature values 74 is selected, which do not need to be the 
same as the primary temperatures. Suitable temperature programs are two 
fixed (isothermal) levels at 120.degree. C. and 250.degree. C., and a 
ramping from 80.degree. C. to 250.degree. C. at 5.degree. C./minute. 
Corresponding secondary retention times 76 (or other retention indicators) 
are determined for the selected temperature programs. The isothermal 
retention times have several uses including standardization of systems, 
temperature calibration, validation and determination of phase ratio. 
The temperature scale of the target system should be calibrated 78 to 
effect calibrated temperatures 79, for example in a manner described below 
using a secondary retention time for an isothermal for a specified 
component. Also, at this stage, a validation 80 of the target system 
(primarily to validate the stationary phase composition) is desirable, 
also as described below. If validation does not pass, further procedures 
are terminated to locate and fix 82 the problem, e.g. change columns. 
It is necessary to determine the parameters associated with column 
dimensions of the target system. There may be circumstances where the 
column dimensions for the target system are already established, e.g. in a 
prior run or by measurements during manufacture such as measuring the 
exact amount of stationary phase retained in the column. In this case the 
following procedure to determine column dimensions with the function may 
be skipped. 
The secondary retention times 76 are identified as target times for the 
function (Eq. 1 and/or 2). The target system (particularly the target 
column) is characterized 84 by reverse application of the function 66, to 
determine effective column dimensions 86 for which a computation with the 
function yields substantially each secondary retention time for the 
temperature at which the target system was run, using the previously 
determined thermodynamic constants 68 and the primary inlet pressure 56. 
The phase ratio may be determined with the function; however, as explained 
below, the phase ratio .beta. advantageously is determined from retention 
times, so only a parameter associated with length L (viz. L itself or 
aspect ratio L/r.sub.c) needs to be determined with the function. 
Next, an effective secondary program for inlet pressure 88 is determined 
90, again by reverse application of Eq. 1 and/or 2, for which, with the 
effective column dimensions 86 and the previously determined thermodynamic 
constants 68, the function yields substantially the primary retention 
times 60 for any temperature program, preferably one of the nominal 
selected temperature programs. A fixed pressure may be suitable, or a 
pressure program such as ramping may be advantageous to achieve suitable 
equality of retention times. This secondary pressure program 88 may be 
used 87 in subsequent system operations, or a selected program may be used 
by calibration. 
For such a selected pressure program, a fixed pressure for the program 88 
also provides a second calibration point for effective inlet pressure of 
the target system, compared to the pressure setting for the target system. 
With this point and the first calibration "zero" point 75 determined as 
described above, and with assumed linearity, a pressure calibration 87 is 
established. Any operating pressure 89 (P.sub.i, fixed or otherwise) may 
be selected for subsequent operations of the target system. The proper 
pressure program setting corresponding to the operating pressure program 
is ascertained from the calibration. Using the same procedures with other 
chromatographic systems operated for the same selected, calibrated 
pressure, retention times may be compared directly The actual pressure 
settings for the other systems would be determined, calibrated and scaled 
in the same manner as in the present case. 
In the foregoing, an ideal goal is to determine the effective column 
dimensions and the secondary inlet pressure such that the function yields 
retention times exactly equal respectively to the secondary and primary 
retention times. As this generally is not quite attainable, the 
clarification "substantially" is intended to mean within practical limits 
of attainability. Details for application of the function are provided 
below. 
The target chromatographic system 72 then is operated 91 for sample 
analysis, using an application sample 92 (usually unknown). For operating 
parameters, the selected pressure program 89, and any selected program 
(fixed or ramping) for the temperature 94 are used, preferably with 
temperature calibration 78. Such operation generates at least one test 
retention time 96 for each component and each temperature program. By use 
of the secondary pressure program (fixed or ramping), the test retention 
times are thereby standardized to the virtual chromatographic system 69, 
and may be utilized for analysis 98 of the application sample. Similar 
operations with other application samples and other target systems provide 
retention times that, after normalization to a selected temperature by use 
of the function, may be compared directly. This also allows computer 
comparison and identification with a library of such times for selected 
chemical constituents. 
In the foregoing, temperature is selected conveniently as the independent 
variable with the pressure program as a parameter in the function. These 
roles could be reversed, with pressure as the independent variable. More 
broadly, any of the other operating parameters could be used in these 
roles, namely outlet pressure, ramping rates and times (or other program 
parameters), carrier gas composition (affecting viscosity .andgate.), 
column length, and even another column parameter such as stationary phase 
composition or thickness if such can be varied controllably for a set of 
runs. Moreover, more than one of these variables could be used in each 
role at the same time, e.g. adding ramping to temperature. As used herein 
and in the claims the term "first parameter" means the independent 
variable (temperature in the above example), and "second parameter" means 
the parameter (e.g. inlet pressure) that is adjusted to standardize the 
target system to the virtual system. 
As the function (Eq. 1 or 2) of the present embodiment includes inlet 
pressure, it is preferable that the system utilize back pressure 
regulation of the split flow so that inlet pressure be controlled and 
known directly. However, the invention could be utilized with flow 
regulation of the split flow such as disclosed in Hinshaw 1, provided 
inlet pressure to the column is measured and preferably is reproducible. 
Alternatively, with such a flow regulation system, a function may be 
derived with column flow rate as a first or second parameter in place of 
pressure, such flow rate being reproducible and measured directly or 
ascertained by subtraction. 
The invention may be used with a supercritical fluid for the carrier. In 
this case the term "gas" herein includes such fluid and the procedures are 
substantially the same as described herein including use of the same or 
other suitable function that describes the chromatography. The invention 
also may be utilized in a liquid chromatographic (LC) system with a liquid 
carrier such as the type described in the aforementioned U.S. Pat. No. 
4,579,663. For LC additional consideration is given to interactions of the 
sample with the liquid carrier. 
Column Dimensions 
Characterizing 84 the effective column dimensions 86 (FIG. 3) for the 
target column conveniently has two aspects. The phase ratio .beta. may be 
determined directly from a retention time. Other column dimensions are 
ascertained as described below by application of the function. 
Phase ratio .beta.=r/2d in the primary dimensions 70 for the primary column 
(.beta..sub.p) preferably is determined from actual measurements on the 
column, by destruction if necessary. Although the measurements should be 
as accurate as practical, absolute accuracy is not necessary because, in 
the characterization 84 for target column dimensions 86, the phase ratio 
for target columns (.beta..sub.t) is determined relative to a known 
.beta..sub.p. The phase ratio also has a relationship .beta.=K/k where k 
is a capacity factor and K is a partition coefficient that is constant for 
a given component, stationary phase and temperature, so that .beta. is 
inversely proportional to k. The latter is calculated from k=(t.sub.R 
-t.sub.o)/t.sub.o where t.sub.R and t.sub.o respectively are retention 
time and mobile phase time as defined above. The capacity factor k can be 
calculated from any of the isothermal retention times taken with the 
primary and secondary system in the course of the other procedures. The 
phase ratio for the target column is related to that of the primary column 
by .beta..sub.t =.beta..sub.p .multidot.(k.sub.p /k.sub.t). This is used 
to compute the phase ratio for the column of each target system and is 
entered into the function prior to determination of the aspect ratio. The 
latter is determined from the function as explained below. 
Application of Function 
The integral function (Eq. 1) is sufficiently complex for there to be no 
apparent analytical solution, so that special techniques generally are 
required for its application. Any standard or other desired mathematical 
techniques may be used. In one preferred approach, the dead time t.sub.0 
is first determined by using a set of certain homologous standards such as 
n-alkanes, the set advantageously being included in the standard sample 54 
(FIG. 3). Each homologous standard has a unique homolog number, such 
number being an integer number C.sub.n of carbon atoms for the n-alkanes. 
About five or six such alkanes with contiguous numbers are suitable, such 
as those having known C.sub.n numbers from 6 to 10. Retention time t.sub.R 
is related to this number by a homolog relationship: 
EQU ln (t.sub.R -t.sub.o)=g.multidot.C.sub.n +h Eq. 3 
where t.sub.o is time in the mobile phase ("dead time"), and g and h are 
homolog parameters that are potentially temperature dependent. Other 
homologous standards may be used, provided they have identifiable 
equivalent numbers (not necessarily integers) in a similar relationship. 
Determination of standard retention times t.sub.R for the alkanes is 
included in the operations of the primary system, for at least one 
selected temperature. The dead time and constants are determined by 
fitting the equation to the measurements of t.sub.R. 
To do this (FIG. 5), the primary system 52 is operated 53 as before with a 
sample 200 containing the n-alkane standards (or other homologous 
standards) using the primary pressure 156 and one of the selected 
temperatures 201 to generate alkane retention times 158. Utilizing Eq. 3 
(164) an algorithm inserts a selected initial value 202 for t.sub.0 and 
performs a linear least squares (or other statistical) computation to fit 
204 the data to generate the constants and a statistical error factor 206 
for the fit. The dead time is changed incrementally 207 to a new t.sub.0 
208 and the process is iterated 210 until a first value of t.sub.0 214 for 
each selected temperature is found 212 that minimizes the error within a 
preset limit and thereby gives a "best" statistical fit. This also 
determines the parameters g and h (163). 
As pressures are the same for the several isothermal operations, it may be 
seen from Eq. 1a that t.sub.0 is proportional to carrier gas viscosity 
.andgate.(T) which is temperature dependent. A data base is stored 216 in 
the computer for the viscosity over the desired temperature range, 
conveniently in the form of parameters for a function relating viscosity 
to temperature. Values for other dead times t.sub.0 are obtained for the 
other relevant temperatures in proportion to the viscosities at the 
original and the other temperatures to effect the temperature dependent 
t.sub.0 (T). With these dead times, Eq. 1 is integrated (e.g. with 
Simpson's method) over the three temperature programs for the standard 
compounds, to provide three equations to solve 218 for the three 
thermodynamic constants a,b,c. 
For the dimensions 86 (FIG. 3) of the target column, the phase ratio .beta. 
is determined as explained above. The column length appears in Eqs. 1 and 
2 via Eq. 1a as an aspect ratio .alpha.=L/r.sub.c. Thus either this ratio 
may be determined as a length parameter, or r.sub.c may be estimated and 
an effective value for L determined (which corrects for any inaccuracy in 
r.sub.c). Conveniently the length L is taken to be the parameter. 
A searching technique may be used for solving the function to determine one 
or more parameters such as the length L. A suitable technique (FIG. 6) for 
solving the function involves utilizing a stored initial parameter data 
base 102 defining tentative values of the length (or other parameter) 
within a predetermined range over expected operating conditions, such as 
from 20 m to 40 m in 1 m increments for a column having a nominal length 
of 30 m. (The data actually stored may be the lowest and highest lengths 
plus increment value.) Other parameters 104 are known, namely the 
thermodynamic constants 68 (FIG. 3), set pressure 56, calibrated 
temperature 79 and phase ratio .beta.. With the function 66 (e.g Eq. 1), 
theoretical retention times 106 are computed 108. Differences 110 
("errors" or "residuals") between the theoretical times and measured times 
76 are calculated 112. This is done for each value in the length base and 
for each of the sample components, and is presented advantageously in the 
form of root-mean-square ("rms") residuals. 
The residuals may be plotted against the parameter if desired, or as 
contours if there are several parameters in the search, using conventional 
techniques. Such plot may be useful in visualizing a search, but is not 
important to the present invention. 
A search for the minimum may be done manually (e.g. by pointing and 
clicking an appropriate monitor display of a plot of the residuals vs. 
length) or with any available or other desired computer program. An 
initial coarse search 122 is advantageous, if not done previously 124, to 
find the region containing the lowest minimum. There may be mathematically 
forbidden areas in the range ("fractal space") which, if found, are 
assigned an arbitrarily high value such as 1000. The minimum residual 120 
then is determined. (Although not likely for the length L, in other 
applications for the searching there may be several minima, and the coarse 
search should find the lowest.) A revised (narrowed) length data base 126 
with a smaller range such as 2 m around the minimum residual is selected 
129, Eq. 1 is applied again to compute 108 theoretical retention times 
106, and residuals 110 from the measured values 76 are recalculated 112. 
When a coarse search cycle 123 is determined 124 to have been done a set 
number of times (once should be sufficient), a fine search 128 is effected 
in the revised matrix 126 for the selected region so as to zero in on the 
minimum in the selected well. This may be done conventionally such as with 
linear programming, simulated annealing or, advantageously, an adaptive 
non-parametric search such as an algorithm for a downhill simplex method 
described in "Numerical Recipes in C" by W. H. Press, S. A. Teukolsky, W. 
T. Vetterling and B. P. Flannery, The Art of Scientific Computing, 2nd 
ed., Cambridge University Press (1992). A conventional simplex search 
program determines the average of the residuals for two proximate points 
that define a short line. The program flips the line over one of the 
points, redetermines the average and whether it has reduced; if not the 
line is flipped over to the other way. The procedure is repeated in search 
of lower residuals 129. An advantageous modification to the simplex search 
shortens the point separations by a preset factor when the residual 
average is reduced, or lengthens the separations by such a factor when a 
residual average increases, for example by a factor of two in each case. 
The starting points may the previously determined minimum and the next 
best point. 
When a low residual 129 is found reflecting a "well", a test 130 for a 
nearly flat bottom of the well is made for the rms residual not to change 
more than a preset limit such as 0.00001. If this is not met, the simplex 
cycle is repeated 133 with a revised data base 126 of lengths selected 129 
in smaller intervals around the latest region. If the number of simplex 
cycles exceeds a limit 134 such as 500 cycles, a problem is assumed to 
exist and the program is terminated 136. Otherwise the last low residual 
is selected as the minimum 131 and this determines 137 the corresponding 
column length L or other parameter. 
A similar procedure may be used to apply the function with coarse and fine 
searching to determine 90 a secondary pressure program 88 (FIG. 3). If 
this is a fixed pressure, an initial data base of a range of potential 
pressures is used for the initial parameter base 102 (FIG. 6) in place of 
the initial length base, the length L and radius r.sub.c replace the 
initial pressure as predetermined parameters, and the measured retention 
times are the primary retention times. Otherwise the procedures of FIG. 6 
are substantially the same. In the case of pressure or some other 
parameters, there may be several minima from which the coarse search 
serves to select a lowest minimum before the simplex search. 
In the case of a pressure program with ramping, the procedures are effected 
with a matrix of parameters associated with the program such as ramp rate 
and initial and final pressures (thus a 3-dimensional matrix). More 
broadly, the matrix has as many axes as variables being considered, e.g. 
one, two, three or more. Residuals are computed for all of the compatible 
combinations in the matrix. For the coarse search, the residuals are 
searched by computer program to find the low in the same manner as 
described above. For a simplex search with a two-axis matrix, three 
proximate points are used in place of two for the averaging of residuals, 
and a triangle is visualized in place of the short connecting line. In the 
search the triangle is flipped over one of its sides for recomputation of 
an average. For a three-axis matrix, a pyramid is visualized with similar 
flips over an edge. 
The plotting and searching technique may be used for broader purposes, for 
example for an operator of a chromatographic system to optimize selected 
operating parameters without necessarily being for the forgoing 
standardization to a virtual system. In the broader case, the system has 
operating parameters including selected parameters for optimization and 
remaining parameters, and operation of the chromatographic system is 
represented by a mathematical function having function parameters 
including the operating parameters. The primary chromatographic system is 
operated with a sample and selected values for the operating parameters so 
as to generate corresponding measured retention indicators. A data set or 
matrix is provided comprising potential values of the selected operating 
parameters over predetermined ranges of such parameters in predetermined 
increments, the data set representing combinations of such parameters. 
Theoretical retention indicators are computed with the function for the 
combinations of such parameters and for the remaining parameters which are 
known or assumed for the purpose of the computation. Differences 
(residuals) between the measured retention indicators and the theoretical 
retention indicators are computed. The residuals are searched for a 
minimum in the differences, such that the minimum establishes optimized 
selected parameters. The system then is operated with the optimized 
parameters. 
It may be desirable for a plot of the retention times (actual and 
simulated) at each stage to be displayed on the monitor for operator 
viewing. Operator instructions for proceeding may be entered by way of pop 
up menus. Software (or firmware) with the function and the residual 
plotting and searching means for applying the function, along with matrix 
data, may be incorporated into the computer programming of the system, or 
may be provided separately such as on a floppy disk. 
It is intended, as an aspect of this invention, that the foregoing 
searching technique may also be used directly for determining one or more 
optimum operating parameters for a chromatographic system, independently 
of any standardizing. 
Column Temperature 
Temperatures for the primary chromatographic system should be measured as 
accurately as practical by conventional means, such as with several 
thermocouples distributed in the oven near the column and allowing the 
system to stabilize at each temperature. Ultimately, however, the primary 
system temperatures may be considered to be standard, and absolute 
accuracy is not critical, as long as temperatures of subsequent system 
columns are accurate relative to the original temperature scale of the 
primary. 
The operating temperatures of the target column should be determined with 
precision relative to the temperature scale of the primary system. 
Calibration of temperature for the target column, according to an aspect 
of the invention, is made with use of a selected calibration compound. For 
this, it is advantageous to express the retention indicator in an 
alternate form "retention index" RI, also known as "Kovats Index", as for 
example in the following references: E. Kovats, Helv. Chim. Acta 41, 
1915-1932 (1958); E. Kovats, Z. anal. Chem. 181, 351-366 (1961); P. Toth, 
E. Kugler, and E. Kovats, Helv. Chim. Acta 42, 2519-2530 (1959); A. Wehrli 
and E. Kovats, Helv. Chim. Acta 42, 2709-2736 (1959); L. S. Ettre, Anal. 
Chem. 36 (8), 31A-47A (1964); E. Kovats, in Advances in Chromatography 
Vol. 1 (J. C. Giddings and R. A. Keller, eds.), M. Dekker, Inc., New York, 
1965; pp. 229-247. Retention index is defined as RI=100.multidot.C.sub.n, 
where C.sub.n is a number associated with n-alkanes (or other standards) 
described above with respect to the homolog relationship Eq. 3 which 
thereby becomes: 
EQU RI=(100/g).multidot.ln (t.sub.R -t.sub.o)-h! Eq. 4 
Any arbitrary compound (other than an n-alkane) has a retention index 
corresponding to a generally non-integer C.sub.n determined from Eq. 4 by 
measurement of retention time. The retention index for such a compound 
thus is relative to the alkane standards, and is substantially independent 
of most parameters except temperature. This allows the retention index to 
be used in systems with varying parametric conditions while determining 
temperature dependence. To the extent that the retention index has a minor 
dependence on such parameters as pressure, such parameters should be 
repeated as closely as practical for successive runs. 
For an aspect of the present invention, at least one temperature 
calibration compound is selected, the compound preferably having a 
retention index that has a relatively strong dependence on temperature. 
This compound is included in a temperature standard sample with the 
several homologous standards (e.g. alkanes). To cover a desired 
temperature range it may be desirable to utilize two or more such 
compounds such as naphthalene and anthracene, each being most effective in 
a separate, narrower range for the temperature calibration, e.g. 
120.sub..degree. C. and 250.sub..degree. C. respectively. 
Advantageously the temperature standard sample (with alkanes and 
calibration compounds) is included with in the standard sample with the 
constituents used to define the virtual system, so only one set of runs is 
necessary, and temperature is calibrated simultaneously with test 
operations. All or some of the calibration compounds and alkanes may even 
be used for such constituents, except to define the virtual system it may 
be advantageous to use other constituents that have less temperature 
dependence. Moreover, such other constituents may better simulate the 
range of application sample materials likely to be tested. 
To establish temperatures, (FIG. 7), the primary chromatographic system 52 
is operated 53 with the temperature standard 152 at a plurality of 
selected calibration temperatures 154 for the column and with a selected 
pressure program 56. (For convenience these are included in the same 
conditions as for the standardizing runs, with the temperatures used here 
being one of the isothermal runs. The number of temperatures depends on 
the number of constants in Eq. 5 below, being three in the present case.) 
This generates a primary set of retention times for each temperature, 
comprising homolog (e.g. n-alkane) retention times 158 for each of the 
calibration compounds and a compound retention time 160 for the 
temperature calibration compound. After peak identification (not shown) 
the homolog retention times and the known retention indices C.sub.n for 
the standards are used to determine 162 homolog parameters g and h (163) 
for the established relationship 164 (Eq. 3), relating homolog numbers to 
retention indicators (e.g. indices), these parameters being temperature 
dependent. A primary retention index 166 for the calibration compound is 
calculated 168 from the relationship 164 (Eq. 4) with the parameters g, h 
and the compound retention time 160 for the each calibration temperature, 
thereby relating a homolog number at each temperature for the calibration 
compound to its retention index. The term t.sub.0 in Eq. 4 is determined 
as described above. This homolog number is associated with the calibration 
temperature T.sub.c (154). A temperature relationship 170 between 
retention index and temperature is close to being linear with temperature, 
but a quadratic fit may be used for accuracy: 
EQU RI=u+vT.sub.c +wT.sub.c.sup.2 Eq. 5 
where u, v and w are calibration constants 174 that are calculated 172 from 
the retention indices and temperature data. Several (three in the present 
case) primary temperature runs with different calibration temperatures are 
needed to get these constants; again these may be combined with the 
original runs. These constants may be included in a data disk (or other 
such medium) along with the program base for Eq. 5 if necessary. 
Advantageously this is the disk that also contains the data base for the 
virtual system. 
The target chromatographic system then is operated 73 with the primary 
pressure program 56 and a selected secondary temperature 180 (or two such 
temperatures if two calibration standards are used), and with the sample 
152, so as to generate a corresponding test set of retention times. The 
temperature 180 is measured with the scale (which may be arbitrary) 
associated with the target system. This operation is a temperature 
calibration run that for convenience could be the same as one of the runs 
for the standardization. These times comprise alkane retention times 182 
and a compound retention time 184. The alkane retention times are used to 
redetermine 162 new homolog parameters g and h (188) for the established 
relationship (Eq. 3), and calculate 190 a secondary compound retention 
index 192 from the relationship 193 (Eq. 4) and the new parameters. The 
calibration relationship Eq. 5 (170, FIG. 7), with its earlier-determined 
constants 174, is applied with the calculated retention index 192 to 
determine 194 the calibrated column temperature 79 for continuing with 
other procedures (FIG. 3), related to the primary system, that existed at 
the time of operation the target chromatographic system. If desired, a 
series of these temperatures may be determined to calibrate the 
temperature sensing system on the target system, so that the sensor may be 
used directly thereafter. 
Although retention index is a preferred form of retention indicator for the 
temperature calibration, as it simplifies the computations, other forms 
could be used. The retention indicator is advantageously in a form that is 
substantially independent of system parameters and operating parameters 
other than temperature, the homologous standards each having a 
predetermined retention indicator in such form. Also, the temperature 
calibration may be achieved with one or more other homologous standards in 
place of the n-alkanes described above, provided such standards have a 
known, established relationship to their retention indicators. 
It is intended, as an aspect of this invention, that the foregoing 
technique for temperature calibration may also be used directly for 
calibrating a chromatographic system, independently of any standardizing. 
Validation 
It is desirable to validate the target system, to ensure particularly that 
the target column is of the type intended and in satisfactory condition, 
and more particularly that the stationary phase chemistry ("selectivity") 
is satisfactory. Such validation (80 in FIG. 3) may be effected with 
reference to FIG. 8. A validation sample 250 has a set of selected 
validation constituents which may be included in the standard sample, and 
advantageously are the same as the constituents used for standardizing. 
Thus, as before, and conveniently during respective operations 53, 73 of 
standardizing runs with the primary and target system 52, 72, the primary 
validation retention times 60 and the secondary validation retention times 
76 are obtained and identified 62 for the validation sample constituents 
250. The temperature programs 58, 74 each preferably includes an 
isothermal run (advantageously one of the original runs) with a primary 
validation temperature for the present case. 
Preliminary retention indices 252 are calculated 168 with Eq. 4 (193) (in 
the same manner as for the temperature calibration with reference to FIG. 
7) and, similarly, secondary validation retention indices 258 are 
calculated 190. The primary indices are adjusted 254 to the secondary 
temperature 74 with Eq. 5 (170) to effect primary validation retention 
indices 256. 
The differences 260 between the primary and secondary validation indices 
are calculated 262. The test for validation 80 is whether all of the 
differences are within predefined limits; if so, procedures are continued 
from the validation 80 with respect to FIG. 3 or, if not, the procedures 
are stopped to investigate and fix 82 the problem. 
It is intended, as an aspect of this invention, that the foregoing 
validation technique may also be used directly for validating a 
chromatographic system, independently of any standardizing. 
While the invention has been described above in detail with reference to 
specific embodiments, various changes and modifications which fall within 
the spirit of the invention and scope of the appended claims will become 
apparent to those skilled in this art. Therefore, the invention is 
intended only to be limited by the appended claims or their equivalents.