◇Translation of Original Contribution◇

Noninvasive Measurement of Stiffness and Density of Bone for Its
Diagnosis Using Ultrasound
Hitoshi ASAI, EJSUM, and Hiroshi KANAI, EJSUM

Abstract
Because the acoustic and elastic properties of bone evaluated using ultrasound-based methods have proved so
useful in the direct evaluation of bone characteristics, many workers have developed methods and systems basedon the in vivo measurement of velocity, attenuation, or both, of ultrasound in bone. These include the acoustic
emission (AE), apparent velocity of ultrasound (AVU), and speed of sound-broadband ultrasound attenuation(SOS-BUA) methods. Bone stiffness is accepted as an effective index in the diagnosis of such bone diseases as
osteoporosis. The literature contains reports of the estimation of bone stiffness from velocity (speed of sound[SOS]) and attenuation (broadband ultrasound attenuation [BUA]). The physical explanation of the methodsof evaluating stiffness from the obtained values of BUA and SOS is still not clear, however. Here we propose a
new diagnostic method and system based on ultrasound measurement of the stiffness of bone. The proposed
method determines stiffness from the velocity of the leaky surface skimming compressional waves (LSSCWs)
obtained with the microdefocusing method and the acoustic impedance obtained with the reflectance method.
Thus this method can evaluate stiffness without exposing the patient to X-rays ; moreover, the physical basis of
the calculation of stiffness from velocity and impedance is well understood. We applied this system to the humantibia in vivo : stiffness and density in a young volunteer were successfully evaluated at 24.9 GPa and 2.01×103 kg
m3, respectively.
J Med Ultrasonics 2002 ; 29 (Autumn) : 129136.

Keywords
density, elastic constant, osteoporosis, microdefocusing method, reflectance method

1. Introduction
Ultrasonic diagnosis of bone is attracting attention
because of its merit in evaluation of the elastic
properties of bone and other bone characteristics
without exposing patients to x-rays.
An array of ultrasonic diagnostic methods andsystems have been devised to evaluate bone tissue :These include the acoustic emission (AE) method1)
and the apparent velocity of ultrasound (AVU)
method2) , both of which measure acoustic velocity ;the speed of sound-broadband ultrasound attenuation(SOS-BUA) method3)4) , based on the measured
acoustic velocity and the attenuation constant ; andthe impedance method5) , based on the measured
acoustic impedance. However, values determinedusing these methods are not basic physical values, andassociating these measured values with bone condition

may prove difficult. Accordingly, we propose
diagnosing bone by measuring stiffness and density,
which are more-basic physical values.
Abendschein et al6) measured acoustic velocity invitro using 100 kHz ultrasound and calculated bone
stiffness using the acoustic velocity thus obtained ; thedensity of the human tibia of the cortical bone wasmeasured separately. Stiffness calculated fromacoustic velocity and density correlates closely withstiffness obtained by the three-point bend method.
Stiffness has been shown to be less in patients withlongterm diabetes mellitus or osteoporosis than in
healthy individuals. Thus, if stiffness could bemeasured in vivo, the value could be used in the
diagnosis of osteoporosis.
Stiffness and bone density can be determined using
ultrasound by the measuring acoustic velocity

Department of Electrical Engineering, Graduate School of Engineering, Tohoku University, Aramaki-aza-Aoba
05, Sendai-shi 9808579, Japan
Original article is cited in J Med Ultrasonics 1999 ; 26 (10) : 10271033.
J Med Ultrasonics Vol.29 Autumn (2002)129

(stiffnessdensity )andacoustic
impedance(stiffness×density ) of the same local area.
However, measurement of bone acoustic velocity,which is routinely carried out, employs a transmission
technique and is difficult to combine withmeasurement of acoustic impedance.
Chubachi et al propose a method of measuring
acoustic velocity in bone on the basis of reflectance,
using leaky surface skimming compressional waves(LSSCWs), which propagate on the bone surface at avelocity close to that of longitudinal waves9)12) .
LSSCW is one of the elastic surface waves and is
excited through a line focus-beam acoustic lens, as
used in the ultrasonic microscope7) . Velocity cantherefore be measured by analyzing the () curve8).
The system used to measure the velocity of LSSCWs,
in which ultrasonic transmission and detection are
performed from a direction oblique to the surface ofthe bone, can be combined with the system used tomeasure acoustic impedance, in which ultrasoundsignals are transmitted and received perpendicular tothe surface of the bone. Because of the high level ofattenuation in bone, however, obtaining a ()curvewith a length sufficient for frequency analysis isdifficult. To analyze the () curve, we thus
developed14)15) a way to measure LSSCW velocity
using the microdefocusing method, which allows

measurement of the phase velocity of LSSCW using ashort moving distance  of the sample (thedefocusing quantity) that would be unlikely to
produce sufficient length of () curve. This method
uses a pair of focus-beam ultrasonic transducers tomeasure LSSCW velocity and is easily combined withthe acoustic impedance measurement16) . Here we
present both our method for measuring bone stiffnessand density on the basis of the principle previously

Fig. 1130

described and the results we obtained measuring thestiffness and density of the human tibia in vivo.
2. The measuring system of stiffness and density
Fig. 1 shows the system used to measure stiffness by
simultaneously measuring LSSCW velocity andacoustic impedance. The microdefocusing method13) isused to measure the acoustic velocity of LSSCWs14)15)
with two focus-beam transducers : a transducer fortransmitting (A) and a transducer for receiving (B).
Acoustic impedance, on the other hand, is obtainedusing a focus-beam transducer (C) for transmitting
and receiving. The ultrasound reflectance of the Pyrex
glass used as a reference standard in this system ismeasured, and acoustic impedance is calculated16)
using the acoustic impedance of water as the known
value. The operating center frequency of transducers(A), (B), and (C) is 1 MHz, and the focal distance is
95 mm.
LSSCWs are elastic surface waves that propagate
almost as fast as longitudinal waves on the bone
surface. These waves are excited by projecting
longitudinal waves at a critical angle on to the surfaceof the object. In the case of bone, longitudinal waves
propagate at between 2700 and 4100 m/s17), forming a
critical angle of 21 to 33 degrees. The use of LSSCWs
to measure acoustic velocity in bone allows
measurement without limiting the site ofmeasurement, because there is no need to place theultrasonic transducer on the far side of the bone, as
required by the conventional method of measuring
acoustic velocity using a transmission technique.
Further, our method, by which ultrasound istransmitted and received from a direction oblique tothe bone surface, is more easily combined withmeasurement of acoustic impedance, in which

Block diagram of the measuring system. The system can measure bothLSSCW velocity and acoustic impedance of the same local area of asample, and sample stiffness and density is calculated from the values
obtained.

J Med Ultrasonics Vol.29 Autumn (2002)

ultrasound is transmitted and received from thevertical direction.
3. Measuring method of LSSCWs
Analysis of the () curve, used in ultrasonicmicroscopy, is a method of measuring the phasevelocity of LSSCWs. Because of the greaterattenuation of bone, however, a () curve with alength sufficient for frequency analysis is difficult toobtain9)12) . To analyze the () curve we therefore
used14)15) a microdefocusing method13) that allows themeasurement of the phase velocity of LSSCWs usinga short moving distance  of the sample (thedefocusing quantity) and is thus unlikely to produce
sufficient length for frequency analysis.
Fig. 1 shows that the excitation of LSSCWs occurs
on the surface of the sample when a radio frequency(RF) burst pulse with a frequency of 1 MHz and apulse width of 50 s is applied to transducer A. The
LSSCWs propagate on the sample surface, reradiating
leaky waves in the water. While maintaining the angle
between the sample and the transducer, the distance between the surface of the sample and thetransducer is shortened, and the output waveformfrom transducer B at each after A/D conversion
using a digitizing oscilloscope is received by theworkstation, in which the phase spectrum is
calculated from the waveform at each , using the fast
Fourier transform (FFT). Water temperature at timeof measurement is determined using a resistancethermometer sensor and a digital multimeter, anddetermination of acoustic velocity in water at the timeof measurement is based on the temperaturedependence18) of acoustic velocity in water. Thechange () in the phase of the transmitted signal
when the sample is brought closer by to the pair oftransmitter-receiver transducers is expressed by Snell's
law as follows : 







 


(1)(2)

where  , ＝2  ,  ＝2   , ,and  are the critical angle of LSSCW at thewaterreference standard boundary, the acoustic
velocity of the longitudinal wave in water, the phasevelocity of LSSCW, the wave number of longitudinal
wave in water, and the wave number of LSSCWs,
respectively. Thus, obtaining the transmission signal
at each when the position of the sample is moved
closer to the transmitter-receiver transducers, thegradient in phase () of the sample relative to theposition () of the sample is determined as follows : 
(3)
From Eqs. (1) and (3), the phase velocity of
LSSCW,  is determined as follows :J Med Ultrasonics Vol.29 Autumn (2002)





 

 
1



(4)

4. Measuring method of acoustic impedance
Acoustic impedance is determined by measuringreflection coefficient at the surface of the bone. Using
Medium 1 with the known acoustic impedance , thereflection coefficient  can be obtained by verticallyprojecting plane ultrasound on the plane boundarywith Medium 2, of which the acoustic impedance is unknown. These values are used to obtain  fromthe following equation.

(5)
The reflection coefficient  is determined as theratio between the amplitude of the incident wave andthat of the reflected wave. In this system, instead of
measuring the amplitudes of the incident and thereflected waves, the amplitude of the wave reflectedfrom a reference standard of which the reflection
coefficient is known and that from the measurement
sample were determined16). The Pyrex glass was usedas the reference standard.
A 1 MHz center-frequency ultrasonic pulse isradiated from the focus-beam transducer (C). The
wave () reflected from the measured object is
detected by the same transducer (C) and transmitted
to the workstation after AD conversion using a
digitizing oscilloscope. Frequency components
distributed in a frequency bandwidth determined by
the frequency characteristics of the measuring device
and by the frequency dependence of the acoustic
characteristics of the propagating medium arecontained in (). Thus the wave reflected from thereference standard of known reflection coefficient andthe amplitude of the reflected wave from themeasured sample are compared at each frequency.
The calculated mean of the frequency bandwidths
showing favorable signal-to-noise ratios (S/Ns)
determine the reflection coefficient. First, for each of
the waves reflected from the reference standard
() and from the measurement sample (), thepower spectra ( ) and ( ) , respectively, areobtained by fast Fourier transform (FFT). The
reflection coefficient ( ) at each frequency is thencalculated from Eq. (6) using an assumed value of , the reflection coefficient of acoustic pressure at
the water-reference standard boundary :
( 
)

)
 ( 

( 
)

(6)

Further, Eq. (7) shows mean ( ) at thefrequency bandwidth with favorable SN calculated
as the measured value of the reflection coefficient.

131 ( )

 


(7)



The center frequency of the transducer is 1 MHz, andthe frequency bandwidth is approximately 1 MHz±
0.2 MHz.
5. Measuring method of stiffness and density
Equations (8) and (9) show the relationshipamong acoustic impedance , acoustic velocity of
longitudinal wave  , stiffness , and density .
 

(8)



 

(9)

Therefore, using the ratio
between LSSCW
velocity  and the velocity of longitudinal waveare obtained from Eqs.

 , stiffness , and density
10 and 11, with acoustic impedance and acoustic
velocity of the longitudinal wave  .







(10)






(11)

6. Noninvasive measurement of the stiffness anddensity of human tibia
The tibia was chosen as the site of measurement
because of its thin soft-tissues and relatively planesurface. Fig. 2 shows a block diagram formeasurements of stiffness and density of bone.
The effect on the soft tissues surrounding the bonemust be investigated for measurements in vivo.
To measure LSSCW velocity after exciting
LSSCWs on the surface of the bone, the excitation

Fig. 2132

and detection of LSSCWs has to be carried out fromoutside the body via the soft tissues around the bone.
In this measurement, the phase velocity of LSSCWs is
likely to vary with acoustic velocity and density of thesoft tissue. Also, refraction may occur at the boundary
between water and soft tissue, and the refraction angle
will vary with acoustic velocity of the soft tissue.
Here, we investigated the effect of soft tissues
surrounding bone on measurement method using
LSSCW using the model shown in Fig. 3. The
refraction angle at the boundary between water andsoft tissue shown in Fig. 3 is determined by Snell'slaw. Setting the  and  as shown in the figure,
Snell's law can be described as follows :


 












(12)


, 
, and  indicate the velocity of thelongitudinal wave in water, the velocity of thelongitudinal wave in soft tissues, and the velocity of
LSSCWs propagating through the soft tissue-boneboundary, respectively. The gradient of the phaserelative to distance of the measurement signal isexpressed as follows :





 


(13)













































(14)

The difficulty of obtaining LSSCW velocity by transforming the equation, required us to repeat
the calculation to find the value of  at which

Block diagram showing the system used to measure the tibia.

J Med Ultrasonics Vol.29 Autumn (2002)
(15)

Assuming that the direction of propagation in themeasurement of LSSCW velocity is the axial directionof bone (33 Direction), taking the actual
measurements into consideration, LSSCW velocity to
be measured is expressed by Eq. (16), using density
and stiffness  of the axial direction :Fig. 3

A refraction model for measuring LSSCW velocity inbone in vivo.

percent change in phase calculated from Eq. (14) isequal to the measured value, in order to determine from the gradient in phase, taking into
consideration the effect of refraction. The acoustic
velocity of the soft tissue must be measured orassumed.
Reflection on the surface of soft tissue andattenuation in soft tissue must be considered whenmeasuring acoustic impedance. To reduce the effectsof these factors, we have proposed16) a method toestimate the acoustic impedance taking ultrasonicreflection and attenuation into account by using an
expression of the site of measurement using the
propagation model, and measuring the refrection
waves from both the surface of the soft tissue and thesurface of the bone. In the tibia, chosen in this study
as the site of measurement, however, discrimination
between the reflected waves from the soft tissue
surface and those from the bone surface iscomplicated by the thinness of the soft tissue aroundthis bone. Thus, this method could not be applied in
the present study.
Anisotropy of bone must be considered whencalculating stiffness and density. Katz et al19) haveindicated that anisotropy of human bone can be
expressed in a hexagonal model using the long axis ofbone as the Z-axis. The direction of ultrasonic-wave
propagation in the measurement of acoustic
impedance is the radial direction of the bone (11Direction), and is therefore expressed as follows,
using density and stiffness of the radial direction :J Med Ultrasonics Vol.29 Autumn (2002)










 





(16)

 is the ratio between the velocity of axial
longitudinal wave  and the axial LSSCW velocity.
Using ＝ as the ratio between  and , can be calculated from the data obtained in the invitro measurement. With the value reported by Katz
et al20) , is calculated at approximately 1.17.
Stiffness  and  and density are determinedusing the following formulas.



(17)

 















(18)









(19)

“Stiffness” is used as an index alternative to stiffness(determined21) at the Achilles, the,),transmission diagnostic equipment for osteoporosis by
Lunar Corporation.
Stiffness＝0.67(BUA−50)＋ 0.28(SOS −1380)(20)
Broadband ultrasound attenuation (BUA) is definedas the ultrasound attenuation coefficient, and the
speed of sound (SOS) is the apparent velocity of theultrasound transmitted through the calcaneal region.
The physical explanation of the Stiffness value
calculated from Eq. (20) remains obscure, however ;on the other hand, stiffness, ,  and density 
calculated from Eqs. (17), (18), and (19) has aphysical basis.
7. Experiments and results
LSSCW velocity and acoustic impedance were
measured and stiffness and density were calculatedusing phenolic resin (Risholite; Risho Kogyo Co,Ltd) and acrylic resin (Acrylite ; Mitsubishi Rayon
Co, Ltd), as samples. Estimated values werecompared with separately measured acoustic velocityof the longitudinal wave using the ultrasoundtransmission method and with the density calculated
from the mass and volume. Table 1 shows the results.
Stiffness and density estimated by this method are133

Table 1 Experimental Results of the Measurement of the Stiffness and Density of Phenol
Resin, Acryc resin, and the Human Tibia
Proposed Method





kgm2･smskgm3
Phenol resin
Acryc resin
Tibia (a)Tibia (b)5.31×106
3.31×106
7.08×106
7.08×106

3515
2696
3522
3638

1.51×103
1.22×103
2.01×103
1.95×103

almost equal to those measured by the transmission
method, making the proposed system useful inestimating stiffness and density.
This method was used for in vivo measurement of
LSSCW velocity and reflection coefficient of the righttibia of a 25-year-old man to estimate stiffness anddensity. Values in columns (a) of the table exclude
the effect of refraction in the calculation of the
LSSCW velocity ; those in column (b) include it.
Acoustic velocity of the soft tissues was assumed to be
1540 ms. Estimates of LSSCW velocity, stiffness, anddensity obtained excluding the effect of refraction
caused by the presence of soft tissues did not differ
substantially (approximately three percent) fromthose including it.
Estimates of stiffness and density obtained using
the proposed method were compared with thefollowing results obtained from in vitro measurementof the tibias of healthy volunteers by Abendschein etal6) : LSSCW velocity, 3526 ms ; density, 1.97×103kgm3 ; and stiffness calculated from these values, 24.5
GPa. Thus the proposed method also produced
reasonable estimates in the tibia.
8. Conclusion
A noninvasive ultrasonic diagnostic method for
determining stiffness and density of bone bymeasuring the LSSCW velocity and acoustic
impedance in almost the same local area of a sample
is described, and the method produced reasonable
results when applied in human tibia. Further studies
must determine the accuracy and detectability of themeasurements obtained by accumulating data fromindividuals in a broad range of age groups before thismethod can be used to diagnose osteoporosis andother bone diseases. The method will also have to bemodified to reduce the effect of the soft tissues aroundthe bone.
We are grateful to the Nikka Densoku Company
for their support regarding the experimental
equipment. This study was partially supported by the
Fellowship Grant in Aid for Scientific Research of
the Ministry of Education (Japan).

1)

134

×

GPa

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