REGULAR PAPER

Japanese Journal of Applied Physics 53, 07KF17 (2014)
http://dx.doi.org/10.7567/JJAP.53.07KF17

Measurement of shear viscoelasticity using dual acoustic radiation pressure
induced by continuous-wave ultrasounds
Kaori Tachi1, Hideyuki Hasegawa1,2, and Hiroshi Kanai1,2*
1Graduate School of Biomedical Engineering, Tohoku University, Sendai 980-8579, Japan
Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan
E-mail: kanai@ecei.tohoku.ac.jp
2

Received November 29, 2013; revised January 31, 2014; accepted February 13, 2014; published online June 19, 2014
It is important to evaluate the viscoelasticity of muscle for assessment of its condition. However, quantitative and noninvasive diagnostic methods
have not yet been established. In our previous study, we developed a method, which used ultrasonic acoustic radiation forces irradiated from two
opposite horizontal directions, for measurement of the viscoelasticity. Using two continuous wave ultrasounds, an object can be actuated with anultrasonic intensity, which is far lower (0.9 W/cm2) than that in the case of the conventional acoustic radiation force impulse (ARFI) method. In thepresent study, in vitro experiments using phantoms made of polyurethane rubber and porcine muscle tissue embedded in a gelatin block wereconducted. We actuated phantoms by ultrasonic radiation force and measured the propagation velocity of the generated shear wave inside thephantoms using a diagnostic ultrasound system. The viscoelasticities of phantoms were estimated by ﬁtting a viscoelastic model, i.e., the Voigt
model, to the frequency characteristic of the measured shear wave propagation speed. In the mechanical tensile test, a softer polyurethane
phantom exhibited a lower elasticity and a higher viscosity than a polyurethane phantom with a higher elasticity and a lower viscosity. The
viscoelasticity measured by ultrasound showed the same tendency as that in the tensile test. Furthermore, the viscoelasticity of the phantom withporcine muscular tissue was measured in vitro, and the estimated viscoelasticity agreed well with that reported in the literature. These results show
the possibility of the proposed method for noninvasive and quantitative assessment of the viscoelasticity of biological soft tissue.
© 2014 The Japan Society of Applied Physics

1.

Introduction

Viscoelastic properties of muscle tissue are closely related tothe pathological state. For example, owing to pyramidal
tract disorders or peripheral neuropathy, the elastic modulus
of muscle decreases. In amyloidosis, atrophy and elevationof the hardness of muscle occur. Also, polymyositis andmyoglobinuria lead to muscle weakness (decrease in muscle
elasticity).1,2) Therefore, it would be valuable to measure theviscoelasticity of muscle for early detection and quantitative
diagnosis of muscle disorder.
Over the past decade, some remote actuation methods
based on acoustic radiation forces have been reported. Fatemi
and coworkers proposed an imaging modality that uses theacoustic responses of an object, which are closely related tothe mechanical properties of the medium. By measuring theacoustic emission with a hydrophone, hard inclusions, suchas calciﬁed tissues in soft materials, were detected experimentally.3,4) However, the spatial resolution was limited by
the size of the intersectional area of ultrasound beams at twoslightly different frequencies.
Nightingale and coworkers proposed an alternative imaging method (acoustic radiation force impulse: ARFI), in
which focused ultrasound is employed to apply a radiation
force to soft tissue for a short duration (less than 1 ms). The
viscoelastic properties of the tissue were investigated fromthe magnitude of the transient response, which was measured
with ultrasound as the displacement of tissue.5–7) However,in order to generate a measurable displacement by several
ultrasonic pulses, high-intensity pulsed ultrasound at 1,000
W/cm2 was required. According to safety guidelines for theuse of diagnostic ultrasound, it is recommended that theintensity be below 240 mW/cm2 (ISPTA) for pulsed waves
and 1 W/cm2 for continuous waves.8) The intensity of thepulsed ultrasound employed by Nightingale and coworkers
was therefore far greater than that indicated by the safety
guidelines. Later many groups studied methods for actuation

using high-intensity pulsed ultrasound and the measurementof viscoelastic properties of tissue.9,10)
To decrease the ultrasonic intensity for actuation of soft
tissue, we chose continuous-wave ultrasound, as in the work
carried out by Fatemi and coworkers. The maximum intensityof 1 W/cm2 for continuous waves given by the safety
guidelines generates an acoustic radiation force of 6.67 Pa,
which is very small. Therefore, to generate a measurable
displacement by acoustic actuation, a method for effective
application of acoustic radiation forces should be developed.
However, a single acoustic radiation force does not generate
deformation in an object effectively because it primarily
produces a change in object’s position. In our previous
study, we developed a method, in which two cyclicalradiation forces were simultaneously applied to a phantom
from two opposite horizontal directions to cyclically compress the object in the horizontal direction. Furthermore, theresultant regional displacement and strain in the tissue were
measured using a different ultrasonic probe. This method
enables the effective generation of deformation by acoustic
actuation.11,12)
There is another study using two sources of shear
waves,13) in which a method for evaluating the mechanicalproperties of an object by making two shear waves interferewith each other is presented. However, this method requiresthat two shear wave sources are perfectly opposite, and thatshear waves are insoniﬁed from the skin surface by an
external vibrator. On the other hand, our method does nothave these requirements. Also, the shear wave sources(corresponding to focal spots of two ultrasonic transducers)
can be placed inside an object and can be localized using
focused transducers.
In the present study, the propagation of the shear wave
generated by our ultrasonic actuation method was measured
also by ultrasound to evaluate the viscoelastic properties ofan object.

07KF17-1© 2014 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 53, 07KF17 (2014)

2.

K. Tachi et al.

þ 2 cosf2kz  2ð2f0 þ fÞtg

Materials and methods

kp20 2ze½sin 2ðkz  2f0 tÞ22 c22

2.1 Acoustic radiation pressure at frequency difference of
two continuous ultrasoundsþWhen the ultrasound at a single frequency propagates in a
medium, a constant force is generated in the direction of
propagation. This force is called the acoustic radiation force.
The acoustic radiation pressure is deﬁned as the acoustic
radiation force per unit area.14) When inserting an object(density µ2, sound speed c2) in the medium (density µ1, soundspeed c1), the ultrasound wave, which is incident perpendicularly on the surface of the object and progressing in theobject, is given by Euler’s ﬂuid motion equation:15)

þ sin 2fkz  2ðf0 þ fÞtg@vðz; tÞ@vðz; tÞ1 @pðz; tÞþ vðz; tÞ
¼
;
@t
@z
2 @z

ð1Þ

þ 2 sinð2kz  2ð2f0 þ fÞtÞ p0 ez ½ cosðkz  2f0 tÞþ  cosfkz  2ðf0 þ fÞtg þ k sinðkz  2f0 tÞ
þ k sinfkz  2ðf0 þ fÞtg:

Since the displacement of the object generated by the highfrequency component is negligible because it is sufﬁciently
smaller than that generated by the low-frequency components
at DC and ¦ f, the acoustic radiation pressure PR ðz; tÞ isapproximately given by

where vðz; tÞ and pðz; tÞ are the particle velocity and the soundpressure in an object, respectively. Equation (1) is modiﬁedto2@vðz; tÞ@ 1
@pðz; tÞþ
2 v2 ðz; tÞ ¼ 
;
@t
@z 2
@z

ð2Þ

where 2 @vðz; tÞ=@t and 2 v2 ðz; tÞ=2 correspond to theacoustic radiation pressure and the kinetic energy of theultrasound, respectively. Therefore, the acoustic radiationpressure is given byPR ðz; tÞ ¼ @
ðeðz; tÞ þ pðz; tÞÞ:@z

þ p0 cosfkz  2ðf0 þ fÞtg;PR ðz; tÞ 

Therefore, the following equation is obtained from
Eqs. (3)–(5).
@1PR ðz; tÞ ¼ ez ½p0 cosðkz  2f0 tÞ@z 22 c22

G ¼ c2s ;cs ¼

p20 2ze
ð1 þ cos 2ftÞ
2 c22þp20 2ze½cos 2ðkz  2f0 tÞ22 c22

þ cosf2kz  2ðf0 þ fÞtg2fl;

ð9Þ

where ¦l and ¦ª are the interval of ultrasonic beams formeasurement and the phase difference between displacements of the object measured in neighboring ultrasonicbeams, as illustrated in Fig. 1, respectively.
On the basis of Eq. (8), only the shear modulus can be
estimated from the shear wave propagation speed. In thepresent study, the viscoelastic properties of an object were
estimated as described below: The propagation of the shear
wave is expressed by the wave equation as16)@2 U
þ k2 U ¼ 0;
@x2
!
;
cs ¼
ReðkÞ

þ ½p0 cosðkz  2f0 tÞ¼

ð8Þ

where µ is the density in the object.
To determine the shear wave propagation speed cs,
displacements of the object produced by acoustic radiation
forces were measured also with ultrasound at multiple points
along the propagation path of the shear wave. From themeasured displacements, the shear wave propagation speed cs
is estimated asþ p0 cosfkz  2ðf0 þ fÞtg2

þ p0 cosfkz  2ðf0 þ fÞtg

ð7Þ

When acoustic radiation pressures are applied to an object, ashear wave is generated and propagates. The shear modulus
G is estimated from the measurement of the shear wave
propagation speed cs expressed asð4Þ

where ¡ and k are the attenuation coefﬁcient and the wave
number, respectively. The kinetic energy eðz; tÞ is expressed
as


1
1
pðz; tÞ 21
eðz; tÞ ¼ 2 v2 ðz; tÞ ¼ 2¼
p2 ðz; tÞ: ð5Þ
22
 2 c222 c22

p20 2ze
ð1 þ cos 2ftÞ:
2 c22

2.2 Estimation of shear modulus from propagation
velocity of shear wave using viscoelastic model of medium

ð3Þ

When two ultrasounds with the same sound pressure, p0,
at slightly different frequencies, f0 and ( f0 + ¦ f ), cross eachother, an acoustic radiation pressure which ﬂuctuates at thefrequency difference, ¦ f, is generated in the intersectional
space. The sound pressure, psum(t), in the intersectional spaceis given by
pðz; tÞ ¼ ez ½p0 cosðkz  2f0 tÞ

ð6Þ

ð10Þ
ð11Þ

where U, k, and ½ are the temporal Fourier transform ofdisplacement, wave number, and angular frequency, respectively. When the Voigt model is used as a viscoelastic modelof an object, the shear wave velocity is modeled as17)sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2ð21 þ !2 22 Þ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ;
cs ¼
ð12Þ
ð1 þ 21 þ !2 22 Þ
where µ, ®1, and ®2 are the density of an object, shear
elasticity, and shear viscosity, respectively.
07KF17-2© 2014 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 53, 07KF17 (2014)

K. Tachi et al.

(a)Δθ , Δtbeam number
ultrasoundxtransducer

height23 mm

tdistanceΔl

porcine muscletx+ Δ xfocal point... 2,1,0
beam number(b) ultrasound beam
5.0 mmshear wave propagation

focal point

gelatin25

porcine muscle

30
35

11 mm
measured region

porcine muscle
gelatinFig. 1. (Color online) Schematic diagram of the method of estimating
shear wave propagation speed.

40

0

16.5
width [mm]

(a)

z

y

12

z

16

6

8

power AMP

Low-Pass Filter

ultrasound
probe

(b)

ultrasoundtransducer
ultrasoundtransducer
oil

Diagnostic
ultrasoundsystem

z

24
16
8
0
0
dista 5.5 11
nce f
16.5
22
rom
foca 27.5 33 5.0
l poi
nt [m
m]

shear wave
Focal pointwater
tissueblock

Fig. 2. (Color online) Schematic diagram of the experimental setup.

2.3

0

0
0
dista 5.5 11
nce f
rom16.5 22
foca 27.5
5.0
l poi
nt [m33m]

Experimental methods
Figure 2 shows a schematic diagram of the experimental
system. To measure the displacement distribution at a highspatial resolution, we employed ultrasonic diagnostic equipment (Hitachi-Aloka ProSound F75) with a 10 MHz linear
probe (Hitachi-Aloka UST-5415). The frame rate and interval
between ultrasonic beams were set at 486 Hz and 2.75 mm,
respectively. The number of beams used in the measurement
was 13 (beam number: 0–12). As shown in Fig. 1, the 0-th
beam was located at the center of the two ultrasonic focal
points for actuation.
To improve the spatial resolution in the measurement of
the response of an object to the acoustic radiation force, anultrasound correlation-based method, the phased-tracking
method,18,19) was used to measure the distribution of minutedisplacements. The accuracy of the displacement measurement by the phased-tracking method has already been
evaluated to be 0.2 µm by basic experiments using a rubber
plate20) and, also, has already been applied to measurementsof vibrations and viscoelasticities of the arterial wall21,22) andheart wall.23,24)
We used two phantoms made of two different polyurethane
rubbers that simulate soft biological tissues and anotherphantom containing porcine muscle tissue to measure the

1.25
2.5
3.75
]
m
[m
depth

0

y

20
10
0

1.25
2.5
3.75
m]
depth [mamplitude [μm]xsignal generater

power AMP

33

Fig. 3. (Color online) Illustration of porcine muscle phantom.

ysynchronization
signal generater

gelatin

20

amplitude [μm]ultrasoundtransducer(c) 15

diameter
90 mm

1
Δf

depth [mm]ultrasound probe

0

Fig. 4. (Color online) Spatial distributions of amplitudes of displacements
at ¦ f of 5 Hz estimated by the Fourier transform. The results were obtainedfor phantoms of ASKER C hardnesses with (a) 0 and (b) 5.

propagation of the shear wave and estimate the viscoelasticity. The dimensions of the two polyurethane rubber
phantoms were 90 mm in diameter and 20 mm in height,
and their hardnesses were determined to be 0 and 5 using the
ASKER C-type durometer. The porcine muscle phantom
used in the present study is illustrated in Fig. 3. As illustrated
in Fig. 3, the phantom was obtained by embedding a porcine
muscle tissue in gelatin to prevent the deterioration of the
porcine muscle by immersing in water. The position of themeasured region is indicated by the blue dashed line.
3.

Experimental results and discussion

3.1 Basic experiments using polyurethane rubber
phantoms

Figures 4(a) and 4(b) show the spatial distributions ofdisplacements in polyurethane rubber phantoms with ASKER
C hardnesses of 0 and 5, respectively, estimated at ¦ f = 5 Hz
using the Fourier transform. Focal points of ultrasonic beams
for actuation were set at a depth of approximately 5 mm. The
intensity of each ultrasound for actuation was 0.9 W/cm2.
As can be seen in Fig. 4, the shear wave is attenuated inaccordance with its distance from the focal point.

07KF17-3© 2014 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 53, 07KF17 (2014)

K. Tachi et al.
y(a)beam number(distance from focal point)z

4 (11 mm)
5 (13.75 mm)
6 (16.5 mm)
7 (19.25 mm)
8 (22 mm)

0
−4

(a)

0.8
5.38

depth [mm]

4

−8

5.5

−0.4

0.2

0.4

0.6

0.8

1.0

−1.2

time [s]

11

(b)

−10

5.12

−1
−1.3

5.25

−20
0

0.2

0.4

0.6

0.8

11

1.0

13.75
16.5
19.25
22
distance from focal point [mm]time [s](a) 0.4
0.3
phase [rad]Figure 5 shows the displacement waveforms measured in
the respective polyurethane rubber phantoms located 5.5 to22 mm away from the focal point. The depths, where thedisplacements were measured, were (a) 5.5 and (b) 5.1 mm. It
can be seen that the phase lag occurs between the waveforms
in Fig. 5, which is caused by the wave propagation. For
example, the peak of the waveform at beam number 4(distance from the focal point is 11 mm) is formed slightly
before that at beam number 5 (distance from focal point is13.75 mm).
Figures 6(a) and 6(b) show spatial distributions of phasesof displacements measured in the phantoms with ASKER C
hardnesses of 0 and 5, respectively, estimated at ¦ f = 5 Hz
using the Fourier transform. As shown in Fig. 6, it is
conﬁrmed that the phase increases with the distance, which isshown by the red arrow in the ﬁgure. Means and standard
deviations of phases at ¦ f evaluated for three measurements
are shown in Fig. 7. The depths, where the phases in
Figs. 6(a) and 6(b) were estimated, were 5.5 and 5.1 mm,
respectively. The phases were obtained as the values relative
to the phase measured at beam number 4 (distance from focal
point: 11 mm). From Fig. 7 and Eq. (9), it can be seen thatthe shear wave propagates in the direction away from the
focal point. Shear wave velocities of polyurethane rubber
phantoms with hardnesses of 0 and 5 were estimated to be
1.2 and 1.7 m/s, respectively, using Eq. (9) with the slopes
corresponding to ¦ª/¦l in Eq. (9) of the regression lines(green lines in Fig. 7) determined by the least-squares
method. To discuss the accuracy of the estimated values,
we measured the shear wave velocity of the polyurethane
rubber phantom with an ASKER C hardness of 0 for threetimes. The mean and standard deviation were 1.76 and0.06 m/s (at ¦ f of 5 Hz), respectively. Therefore, in thepresent paper, the number of digits after the decimal point
was set at one. There is a relatively large variation in theestimated phase. The reason for the variation is not perfectly
clear, but the generated displacement is very small and might
be affected by an undesirable vibration from the environment.

−1.6

Fig. 6. (Color online) Spatial distributions of phases of displacements at¦ f of 5 Hz obtained for phantoms with ASKER C hardnesses of (a) 0 and(b) 5.

0.2
0.1
0
−0.1
−0.2
−0.3

11

13.75
16.5
19.25
distance from focal point at y −axis [mm]

22

11

13.75
16.5
19.25
distance from focal point at y −axis [mm]22(b) 0.4
0.3
phase [rad]Fig. 5. (Color online) Displacements of polyurethane phantoms with
ASKER C hardnesses of (a) 0 and (b) 5 at ¦ f of 5 Hz. Displacements were
measured at (a) 5.5 and (b) 5.1 mm in depth.

24.75

phase [rad]4 (11 mm)
5 (13.75 mm)
6 (16.5 mm)
7 (19.25 mm)
8 (22 mm)

0

24.75

−0.7
depth [mm]

10

13.75
16.5
19.25
22
distance from focal point [mm](b) 5.0

beam number(distance from focal point)

20
displacement [μm]

−0.8

5.62
0

−30

0.4
0

phase [rad]displacement [μm]

8

0.2
0.1
0
−0.1
−0.2
−0.3

Fig. 7. (Color online) Averaged phases of displacements at ¦ f of 5 Hz
plotted as functions of distance obtained for phantoms with ASKER C
hardnesses of (a) 0 and (b) 5. The green lines show the regression lines
determined by the least-squares method.

We need to clarify the reason for the variance and to develop
a method for separating the displacement induced by acoustic
radiation force from the environmental vibration because we
aim to actuate an object within the maximal acoustic output
suggested by the safety guideline.
The propagation velocities of shear waves in the phantoms
with hardnesses of 0 and 5 are plotted as a function of ¦ f in
Fig. 8. The measured frequency characteristics of shear wave
propagation velocities ﬁt well to those obtained using the
Voigt model shown by Eq. (12). The estimated elastic moduli
and viscosity constants of the phantoms with hardnessesof 0 and 5 were (01 ¼ 2:2 kPa, 02 ¼ 48:1 Pa0s) and (51 ¼

07KF17-4© 2014 The Japan Society of Applied Physics

hardness : 0
hardness : 5

5
4
3
2
1
0

10

20

30

40
50
frequency [Hz]

60

70

40

80

30

60
20
40
10
0

80

Fig. 8. (Color online) Shear wave propagation speeds in phantoms plotted
as functions of ¦ f. The solid lines show the frequency characteristics
obtained using the Voigt model.

100

20
0

5

10
15
20
Elasticity [kPa]25

viscoelasticity measured
using universal tester

Viscosity (ultrasound) [Pa·s]velocity [m/s]6

K. Tachi et al.
Viscosity (universal tester) [kPa·s]

Jpn. J. Appl. Phys. 53, 07KF17 (2014)

hardness : 0
hardness : 5
viscoelasticity measured
by ultrasoundhardness : 0
hardness : 50Fig. 10. (Color online) Comparison of viscoelastic constants ofpolyurethane phantoms measured using precision tensile tester with thosemeasured by ultrasound.

z

4.5

y

3
1.5

40

18

0
0

30

14
10

20

dis 5.5
tan 11ce
fro 16.5m
foc 22
al 27.5
point 33[m
m]

Viscosity η[kPas]

Elasticity E [kPa]

22

9.5

7.6

5.7

3.8

1.9

amplitude [μm]Elasticity
Viscosity0m]depth [mFig. 11. (Color online) Spatial distribution of amplitudes ofdisplacements at ¦ f of 5 Hz measured in porcine muscle phantom.

6

10
0
5
hardness of phantom determinedusing ASKER C-type durometerof the phantom with the hardness of 0) were found in bothmeasurements with ultrasound and the universal tester.

Fig. 9. (Color online) Viscoelastic constants of polyurethane phantoms
measured using precision tensile tester.

5:1 kPa, 52 ¼ 18:8 Pa0s), respectively. The root mean squaredifference (RMSE) between the measured values and thevalues estimated using the model shown in Fig. 8 was also
evaluated. RMSEs with respect to phantoms with hardnessesof 0 and 5 were 7 and 13%, respectively.
Figure 9 shows the elastic moduli and viscosity constantsof the respective phantoms measured using a precision
universal tester (Shimadzu AG-X 10kN). In Fig. 9, plots andvertical bars show means and standard deviations for 10
measurements, respectively.
As shown in Fig. 9, viscoelastic properties of the polyurethane rubber phantoms estimated using the precision^ 01 ¼ 5:92  0:24 kPa, ^ 02 ¼ 33:40 universal tester were (5^ 1 ¼ 23:32  0:08 kPa, ^ 52 ¼ 14:54 
5:68 kPa0s) and (1:58 kPa0s).
There were signiﬁcant differences between the absolutevalues obtained by ultrasound and those obtained using the
precision universal tester. In the measurement using theuniversal tester, the actuation frequency was about 1 Hz, andthe maximum strain in the object was about 20% due tolimitations of the speciﬁcations of the tester. In general, theviscoelastic properties depend on the magnitude of strain, andthe different levels of deformation may be one of the reasonsfor the differences in the absolute values of the viscoelastic
constants. However, as shown in Fig. 10, similar tendenciesof the viscoelastic constants (higher elastic modulus of thephantom with the hardness of 5 and higher viscosity constant

3.2

In vitro experiment using porcine muscle tissue
In the present study, an in vitro experiment using a gelatinphantom containing porcine abdominal muscle tissue wasconducted to investigate whether the proposed method could
generate and measure the presumed shear wave. Figure 11
shows the spatial distribution of amplitudes of displacements
at ¦ f = 5 Hz estimated by applying the Fourier transform todisplacement waveforms measured by ultrasound. The focal
points of ultrasonic beams for actuation were set at a depth ofapproximately 5.7 mm.
Figure 12(a) shows the displacement waveforms measured
at a depth of 5.7 mm in the porcine muscle at ¦ f = 5 Hz. The
spatial distribution of phases of the measured displacement
waveforms is shown in Fig. 12(b), and the averaged phase at¦ f of 5 Hz is shown in Fig. 12(c), where plots and verticalbars show means and standard deviations, respectively. In
Fig. 12(a), a frequency component of about 100 Hz is clearly
seen. The amplitude of this component increases when the
acoustic radiation force increases (corresponding to thedownward displacement of the object). Therefore, thiscomponent might be the high-frequency component of theacoustic radiation force, which was neglected in the present
study. Although the high-frequency component of theradiation force is nearly 2 MHz, the sampling frequency ofthe displacement waveform is 486 Hz (corresponding to theframe rate). Thus, an aliasing effect might be one of thereasons that the component is seen as a 100 Hz component.
As shown in Fig. 12(a), the displacement caused byultrasonic actuation is attenuated during propagation. Also, in

07KF17-5© 2014 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 53, 07KF17 (2014)8

beam number(distance from focal point)

displacement [μm]62.5

4 (11 mm)
5 (13.75 mm)
6 (16.5 mm)
7 (19.25 mm)
8 (22 mm)

4
2
0
−2

velocity [m/s](a)K. Tachi et al.

−4
−6
−8

0

0.2

(b)

0.4

0.6

0.8

1.0

1.5
1

time [s]

y

2

0
4.6

−1.7

5.0

−2

5.4

−2.3

5.8

(c)

phase [rad]depth [mm]

z

20

30
40
50
frequency [Hz]6070

Fig. 13. (Color online) Shear wave propagation speeds in porcine muscle
phantom plotted as a function of ¦ f. The solid line shows the frequency
characteristic obtained using the Voigt model.

−2.6
11

13.75
16.5
19.25
22
distance from focal point [mm]24.75

possibility of the proposed method for the noninvasive andquantitative assessment of the viscoelasticity of biological
soft tissue.

1.2
1
0.8

phase [rad]

10

0.6
0.4
0.2
0
−0.2
−0.4

11

13.75
16.5
19.25
distance from focal point at y−axis [mm]22Fig. 12. (Color online) (a) Displacement waveforms, (b) spatial
distribution of phases, and (c) average phases of displacements at ¦ f of 5 Hz
obtained for porcine muscle phantom.

Figs. 12(b) and 12(c), phase delays due to the propagation of
the shear wave are observed. The position of the analyzed
region is indicated by the green square in Fig. 3. By
determining the regression lines, as shown by the green line
in Fig. 12(c), the shear wave propagation speed at ¦ f of 5 Hz
was estimated using the slope of the regression line.
In Fig. 13, the shear wave propagation speeds are plotted
as a function of the frequency difference ¦ f. By ﬁtting thefrequency characteristic of the shear wave propagation speed
obtained using the Voigt model to the frequency characteristic measured by ultrasound, the viscoelastic constants of the
porcine muscle tissue were estimated to be (p1 ¼ 1:6 kPa,
p2 ¼ 10:9 Pa0s), which were similar to the viscoelastic
constants of cow muscle (®1 = 30–400 kPa, ®2 = 30–200
Pa0s),25) beef muscle (®1 = 20 kPa, ®2 = 23 Pa0s),26) andhuman muscle (®1 = 2.5 kPa, ®2 = 15 Pa0s)27) reported in theliterature. From these results, the proposed method could
generate and measure the shear wave propagation and was
also applicable to the estimation of the viscoelasticity of
biological tissue.
4.

Conclusions

In the present study, we actuated phantoms simulating soft
biological tissue using dual acoustic radiation pressure andmeasured the propagation velocity of the generated shear
wave as a function of the actuation frequency ¦ f. In
polyurethane phantoms, the viscoelastic properties measured
by ultrasound tend to be similar to those measured using aprecision universal tester, though there were differences intheir absolute values. Furthermore, an in vitro experimentusing porcine abdominal muscle was conducted and theviscoelastic properties were also estimated as in the measurements of polyurethane phantoms. These results show the1) S. Shoji, Kin Shikkan no Shindan to Chiryo (Nagai Shoten, Osaka, 1988)
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