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Frequency characteristics of vibration generated by dual acoustic
radiation force for estimating viscoelastic properties of biological tissues
To cite this article: Ryoichi Watanabe et al 2018 Jpn. J. Appl. Phys. 57 07LF09

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REGULAR PAPER

Japanese Journal of Applied Physics 57, 07LF09 (2018)https://doi.org/10.7567/JJAP.57.07LF09

Frequency characteristics of vibration generated by dual acoustic radiation force
for estimating viscoelastic properties of biological tissues
Ryoichi Watanabe1*, Mototaka Arakawa2,1, and Hiroshi Kanai1,2
1Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan
Graduate School of Biomedical Engineering, Tohoku University, Sendai 980-8579, Japan2*E-mail: ryoichi.watanabe@ecei.tohoku.ac.jp
Received November 6, 2017; revised January 1, 2018; accepted January 15, 2018; published online June 5, 2018
We proposed a new method for estimating the viscoelastic property of the local region of a sample. The viscoelastic parameters of the phantoms
simulating the biological tissues were quantitatively estimated by analyzing the frequency characteristics of displacement generated by acoustic
excitation. The samples were locally strained by irradiating them with the ultrasound simultaneously generated from two point-focusing transducers
by applying the sum of two signals with slightly diﬀerent frequencies of approximately 1 MHz. The surface of a phantom was excited in thefrequency range of 20–2,000 Hz, and its displacement was measured. The frequency dependence of the acceleration provided by the acoustic
radiation force was also measured. From these results, we determined the frequency characteristics of the transfer function from the stress to thestrain and estimated the ratio of the elastic modulus to the viscosity modulus (K/η) by ﬁtting the data to the Maxwell model. Moreover, the elastic
modulus K was separately estimated from the measured sound velocity and density of the phantom, and the viscosity modulus η was evaluated bysubstituting the estimated elastic modulus into the obtained K/η ratio. © 2018 The Japan Society of Applied Physics

1.

Introduction

It is well known that the progression of a lesion isaccompanied by changes in the hardness and stickiness ofthe biological tissues, speciﬁcally internal living tissues suchas muscle, liver, and breast. Therefore, it is important toevaluate the viscoelasticity of the internal living tissues for the
rapid detection and quantitative diagnosis of their disorder.1,2)
Ultrasound imaging is a noninvasive diagnostic tool highly
suitable for viewing living soft tissue.3–5) Various methods
using ultrasonic imaging have been proposed for characterizing tissues6–8) and estimating their displacements.9–11) An
improvement in image quality has also been reported. In
recent years, ultrasonic methods have been studied for thenoninvasive evaluation of the elastic properties of living soft
tissues. In these methods, the speed of the shear wave or the
relationship between the pulse pressure and the change inthe diameter of the radial artery was estimated.12–14) Methods
based on the hysteresis between the applied force and thestrain have also been proposed.15)
Acoustic radiation force is used for the noninvasive
characterization of living tissues. It can generate a displacement inside the living body without damaging the surface.
Thus, it is possible to measure the displacement response
inside a tissue. Recently, various elastic property measurement methods using acoustic radiation force have been
proposed.16) Nightingale and coworkers proposed the acoustic radiation force impulse (ARFI) method that applied apressure to internal tissues using pulsed ultrasonic waves, andevaluated the elastic properties of such tissues from themagnitude of the displacement response.17–19) However, ahigh-intensity pulsed ultrasound of 1,000 W=cm2 was applied
to generate measurable displacement. According to the safety
guidelines20) for the application of diagnostic ultrasound, it
is recommended that the intensity be less than 240 mW=cm2
for pulse waves and 1 W=cm2 for continuous waves. Thus,
the ultrasonic intensity used by Nightingale and coworkers
was markedly beyond the recommended value deﬁned by
the safety guidelines. Several other research groups havereported methods using high-intensity pulsed ultrasound tomeasure the elastic properties of tissues.21,22) The use of othermethods using a static pressure has also been suggested toestimate the elastic properties.23) However, these methods
cannot evaluate viscosity because they cannot acquire thefrequency characteristics.
To solve these problems, a method of measuring tissue
responses to a dynamic pressure was proposed. Fatemi andcoworkers excited tissues by using two ultrasound waves
with slightly diﬀerent frequencies. The acoustic radiationpressure applied with the frequency diﬀerence of the two
ultrasonic waves was generated in the region where two
acoustic radiation pressures crossed inside the tissues. The
authors experimentally detected hard tissues, such as
mineralized tissues, in a soft tissue by measuring theamplitude and phase obtained from the vibration detectedby a hydrophone.24,25) However, in this method, the directionof the displacement is unknown, and the elastic properties
cannot be measured quantitatively.
Another research group has presented the harmonic motion
imaging method. This method uses the acoustic radiation
force generated by an amplitude-modulated (AM) ultrasonicsignal to excite the tissues and measures the displacement
response.26) The displacement direction, i.e., the excitation
direction, is restricted to the irradiation direction because
only one transducer is used for the excitation. However,
because of a single acoustic radiation pressure, the tissues
are forced to move, and the local displacement cannot be
generated eﬀectively.
Thus, our group developed a method to locally excite atissue by applying dual AM acoustic radiation force. The
direction of the main displacement to be generated can beﬁxed to one axis, and it is possible to change the direction of
excitation. In previous studies, the ﬁber structure of muscles
as anisotropic samples was excited by dual acoustic radiation
force.27,28) When the dual transducer was installed parallel tothe muscle ﬁber direction and excited it, local strain was not
generated. This showed that local strain was generated byinstalling dual transducers vertically to the muscle ﬁber andexciting it. This method can lead to an eﬀective displacement
and strain by utilizing the acoustic radiation force.
In another 29) of our previous studies, the velocity of theshear wave generated by excitation was employed for the07LF09-1© 2018 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 57, 07LF09 (2018)

R. Watanabe et al.

estimation of viscoelasticity. Since an AM ultrasonic wave
was used for excitation, it became possible to change theexcitation frequency. By scanning the excitation frequency,
the generated shear wave at each excitation frequency wasobserved by the ultrasound diagnostic equipment, and theshear wave velocity was measured for each frequency. By
ﬁtting a viscoelastic model to the obtained frequency
characteristic of the shear wave velocity, the viscoelasticityof the sample was estimated. Moreover, the visualization of
the displacement and propagation of a shear wave in thethree-dimensional space inside a tissue was demonstrated.30)
However, since the maximum frame rate of the ultrasounddiagnostic equipment employed in these previous studies
was 521 Hz, the excitation frequency was restricted to 100 Hz
or less.
In the present study, we focused on the fact that thefrequency characteristics of the transfer function from stress
to strain of the viscoelastic samples depend on theviscoelasticity of the samples. In the present study, therefore,
a method of estimating the viscoelasticity of the local regionof the samples from the measured frequency characteristicsof the local displacement generated by acoustic excitation
was proposed. The local region of the phantom simulating
the soft tissues was excited at an arbitrary frequency bygenerating an acoustic radiation force with AM continuous
sinusoidal ultrasonic waves irradiated from two directions.
The frequency of the applied ultrasound was scanned in thefrequency range of 20–2,000 Hz, and the frequency characteristics of the displacement of the phantom were recorded.
The viscoelastic parameters were evaluated by ﬁtting theobtained frequency characteristics using the Maxwell model.
2.

Experimental methods

2.1

TheoryIn the present study, we generated a signal with two
frequency components, f0 and ( f0 + Δf ), and applied it to
two ultrasonic transducers with the same speciﬁcations. Thus,
we generated two identical acoustic radiation forces. The
acoustic radiation force PR ðd; tÞ applied by a single transducer is given byPR ðd; tÞ 

p20 2de
ð1 þ cos 2ftÞ;
2 c22

ð1Þ

where α, ρ2, c2, p0, and d are the attenuation coeﬃcient,density, and sound speed of the sample, the sound pressure
on the sample surface (d = 0), and the distance from the

y
x

x
y
z
f0

f0

f

water
tank

Fig. 1. (Color online) Experimental setup employed in the present study.

sample surface, respectively.29) As shown in Fig. 2, thedistance d is diﬀerent from the distance z. The axis along
d shows the irradiation direction of ultrasound, and the axis
along z shows the direction perpendicular to the sample
surface.
If the medium around the target sample is very softcompared with the sample, a single acoustic radiation force
can primarily change the position of the sample. Thus, astrain cannot be generated in the local region of the sample,
and the mechanical properties of the sample cannot be
evaluated. As shown in Fig. 1, in the present study, twotransducers are installed obliquely to the surface of thespecimen. A force is generated by each transducer in an
oblique direction. When the two acoustic forces cross eachother, an acoustic pressure of frequency diﬀerence Δf isgenerated in the intersectional region. The forces thatsynchronize to push the surface of the target sample fromboth sides are generated using two transducers, andsubsequently, a strain can be produced in the sample.
Figure 2 depicts a schematic of the process of straining thesample by bidirectional ultrasonic excitation. On the basis of
Eq. (1), the acoustic radiation force acquires the minimum
value of 0 when cos 2πΔft = −1, as shown in Fig. 2(b). At

(a)

(b)

z

d'

d

Fig. 2. (Color online) Mechanism of the periodic strain generation by two acoustic radiation forces when (a) cos 2πΔft = −1 in Eq. (1) and(b) cos 2πΔft = 1 in Eq. (1).

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Jpn. J. Appl. Phys. 57, 07LF09 (2018)

R. Watanabe et al.

Fig. 3. Experimental setup employed for acceleration measurement.

5.0
displacement [μm]that time, no force is applied from both sides. In contrast,
when cos 2πΔft = 1, the acoustic radiation force takes the
maximum value of ð2p20 =2 c22 Þe2d , as shown in Fig. 2(a).
Thus, the sample is largely strained along the z-axis directionin the focal region. Therefore, a periodic strain of frequency
Δf can be locally generated eﬀectively in the verticaldirection on the sample surface.30) At the focal point set
inside the sample, the sample is locally strained because it islocally compressed in the horizontal direction and the force isapplied in the vertical direction by two acoustic radiation
forces.27)
2.2 Methods
To excite the sample using ultrasonic waves, two continuous
sinusoidal waves with frequencies of 1 MHz and 1 MHz +Δf Hz were generated using a function generator (Tektronix
AFG2020). The two signals were added together and thenampliﬁed using a power ampliﬁer (Yokogawa 7058=10-0).
The ampliﬁed signal was applied to two point-focusing
concave ultrasonic transducers, and the sample was irradiatedwith ultrasonic waves. In the present study, two transducers
with the same speciﬁcations namely a center frequency of
1 MHz, an eﬀective aperture width of 50 mm, and a focal
length of 60 mm, were installed to intersect their focal points
on the sample surface.
In the following experiments, the signal f0 was ﬁxed to
1 MHz, and Δf was varied from 20 to 2000 Hz. By measuring
the displacement generated on the sample surface with a laser
displacement meter (Keyence LK-G80) having a resolutionof 0.2 µm, we measured the frequency characteristics of thesample displacement caused by the dual acoustic radiation
force. We obtained the frequency power spectrum by
inputting the measured displacement waveforms to a fast
Fourier transform (FFT) analyzer (Ono Sokki CF-940) for
each excitation frequency Δf. The power spectra, averaged
eight times by the FFT analyzer, were acquired 30 times,
from which the average and standard deviations of the powercomponent with Δf were obtained. To eliminate the lowfrequency components and electrical noise contaminating thedetected displacement signal, frequency analysis was performed to detect only the component of the vibration
frequency Δf in the FFT analyzer.
The phantoms mimicking biological soft tissues wereprepared by mixing graphite in a ratio of 2% to a urethane
resin. Three phantoms with diﬀerent hardnesses wereprepared using gels (Exile H0-100, H5-100, and H15-100)

2.5
0
-2.5
-5.0

0

0.02

0.04

0.06

0.08

0.1

time [s]Fig. 4. (Color online) Displacement waveform when the phantom(hardness: 5) was excited at Δf = 50 Hz.

with Asker-C hardnesses of 0, 5, and 15, respectively. In
the experiment, an ultrasonic excitation was also applied togranite, a rigid body, for which the viscous component can beignored. Using the three phantoms with diﬀerent hardnesses
and granite, the diﬀerences in the frequency characteristics ofthe displacement were measured and discussed.
We also measured the frequency dependence of theacoustic radiation force applied by the ultrasonic transducer.
According to the equation of motion F = ma, where F isthe force, m is the mass, and a is the acceleration, theacceleration is proportional to the force. In the present study,
the frequency characteristic of the stress was indirectly
evaluated by measuring the frequency characteristic of theacceleration because the stress, which is the force per unit
area, and the force are in a linear relationship. For themeasurement of the acceleration, a piezoelectric accelerometer (Yamaichi Electronics 111BW) was employed, as shownin Fig. 3. Its resonant frequency was 4 kHz. The acceleration
was measured by varying the ultrasonic excitation frequency
Δf from 20 to 2,000 Hz. The acceleration signal detected foreach Δf was input to the FFT analyzer, and the frequency
power spectrum was obtained by averaging 128 power
spectra.
3.

Results

3.1 Acquisition of frequency characteristic ofdisplacement

The displacement of a phantom was acquired by the laser
displacement meter. Figure 4 shows the displacement waveform obtained when the excitation frequency Δf was set to

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R. Watanabe et al.

5.0

voltage of accelerometer [mV]

displacement [μm]Jpn. J. Appl. Phys. 57, 07LF09 (2018)

0.5
0.05
0.005
0

50

100
150
frequency [Hz]

200

10
5.0
0
-5.0
-10
-15
0

Fig. 5. (Color online) Power spectrum obtained by inputting thedisplacement waveform in Fig. 4 to the FFT analyzer. The phantom(hardness: 5) was excited at Δf = 50 Hz.

0.02 0.04 0.06 0.08 0.1
time [s]Fig. 7. (Color online) Acceleration waveform when the accelerometer
was excited at Δf = 100 Hz.

hardness0

10

hardness5-20
amplitude |A( f )| [dB]displacement |D( f )| [μm]15

hardness15
granite1.0-20 dB/decade-60

lower limit of
0.1 laser displacement meter
10

100
1000
vibration frequency [Hz]

-40

10
3000

100
1000
vibration frequency [Hz]Fig. 8. (Color online) Frequency characteristics of the acceleration
amplitude.

Fig. 6. (Color online) Frequency characteristics of the displacement for
the three phantoms with diﬀerent hardnesses and granite.

50 Hz. The frequency spectrum was acquired by inputting
the displacement waveform to the FFT analyzer. Figure 5
displays the average power spectrum when the excitation
frequency Δf was set to 50 Hz. By scanning the excitation
frequency Δf from 20 to 2,000 Hz, the displacement at eachfrequency was measured for the three phantoms and granite.
The obtained frequency characteristics are shown in Fig. 6.
In each phantom, the displacement decreased with a slopeof −20 dB=decade in the frequency range of 20–200 Hz as
the vibration frequency increased. In contrast, the displacement became almost constant in the frequency range of
700–2,000 Hz. The frequency at which the displacement
became constant increased as the phantom became harder.
For the granite sample prepared as a rigid body, thedisplacement decreased with a slope of −20 dB=decade in thefrequency range of 20–400 Hz. The displacement becamealmost constant in the frequency range of 400–2,000 Hz. In
this frequency range, the displacement was smaller than
0.2 µm, which is the resolution of the laser displacement
meter.
3.2 Acquisition of frequency characteristic of stress
The stress at the focal point applied by the dual transducer
was acquired by the piezoelectric accelerometer. Figure 7
displays the response of the accelerometer when the
excitation frequency Δf was set to 100 Hz. The acquired
waveform was a sinusoidal signal.
Figure 8 displays the power spectra of the acquired
accelerations shown in Fig. 7 at each excitation frequency
Δf. By assuming that the force and acceleration are in a
proportional relationship, the frequency characteristic ofacceleration A(ω) in Fig. 8 can be regarded as the frequency
characteristic of the ultrasonic excitation stress. From thisﬁgure, it can be assumed that there is almost no frequency
dependence of the acceleration, that is, practically the samestress was applied to the sample by the acoustic radiation
force, regardless of Δf.
3.3 Estimation of viscoelastic properties
The viscoelastic model to be used for ﬁtting was selected.
Maxwell and Voight models are typical viscoelastic models.
According to the results shown in Fig. 6, the displacement
amplitude did not depend on frequency when high-frequency
stress was applied. Its behavior follows the Maxwell model.
Thus, we ﬁtted the Maxwell model to the experimental
results of the frequency characteristics, as shown in Fig. 9.
The spectrum magnitude ∣H(ω)∣ in Fig. 9 exhibits the transfer
function when the input is the dynamic pressure and theoutput is the strain, where ω, K, and η are the angularfrequency, elastic modulus, and viscosity modulus, respectively. The frequency characteristic of acceleration ∣A(ω)∣ in
Fig. 8 is used as the frequency characteristic of the dynamic
stress, which corresponds to the input of H(ω) in Fig. 9(a).
The strain is linearly related to the displacement; therefore,
the frequency characteristic ∣D(ω)∣ of the displacement in
Fig. 6 is the same as the frequency characteristic of the strain
corresponding to the output of H(ω) in Fig. 9(a). Figure 10
shows the frequency characteristics of ∣H(ω)∣ obtained by
dividing ∣D(ω)∣ in Fig. 6 with ∣A(ω)∣ in Fig. 8. The
characteristic ∣H(ω)∣ in Fig. 10 corresponds to the theoreticalgraph of the Maxwell model in Fig. 9(b).
By ﬁtting the experimental results of the frequency
characteristics in Fig. 10 to the inﬂection point in theMaxwell model, the K=η ratio between the elastic modulus

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R. Watanabe et al.

and its viscosity modulus is very small compared with theelastic modulus. Therefore, the inﬂection point K=η would
exist at a very high frequency for granite. Thus, only thefrequency characteristic of granite with a lower frequency
than the apparent inﬂection point was measured in theexperiments.
Moreover, we estimated the viscosity modulus η bysubstituting the elastic modulus K (= ρv2, ρ: density, v:
velocity) into the obtained K=η ratio, where the density ρ
was calculated from the ratio of the mass to the volume ofthe phantom and the sound velocity v was measured by theultrasonic pulse–echo method. The elastic modulus K was
calculated to be 1.95, 2.08, and 2.30 GPa for the threephantoms with Asker C hardnesses of 0, 5, and 15,
respectively. By substituting these K values to the obtained
K=η ratio, the viscosity modulus η was estimated to be 1.90,
1.48, and 0.89 MPa·s for the three phantoms with Asker C
hardnesses of 0, 5, and 15, respectively. The results indicatethat the phantom with the hardness of 15 has a higher elastic
modulus and the phantom with the hardness of 0 has a higher
viscosity modulus. Similar tendencies of the viscoelastic
modulus were found in the present and previous studies.29)

(a)

(b)
Fig. 9. (Color online) Features of the viscoelastic tissues assuming theMaxwell model. (a) Two elements of the Maxwell model. (b) Frequency
characteristics of the strain.

Hardness0
Hardness5
Hardness15

amplitude |H( f )| [dB]04.

Discussion

4.1

-20

-40
10

100
1000 3000
vibration frequency [Hz]Fig. 10. (Color online) Frequency characteristics of H(ω) calculatedusing D(ω) in Fig. 6 and A(ω) in Fig. 8.

K and the viscosity modulus η for each phantom was
obtained. K=η [s−1] was estimated to be 1.0 × 103, 1.4 × 103,
and 2.6 × 103 for the three phantoms with Asker C
hardnesses of 0, 5, and 15, respectively. The elastic modulus
K increased as the phantoms became harder; therefore, it
can be considered that the estimated results of K=η arereasonable. An inﬂection point also existed in the result of
granite as shown in Fig. 6. However, its inﬂection point
would be due to the resolution of the laser displacement
meter, not the characteristics of granite, since granite is rigid

(b)

(c)

normalized
displacement input
signal(a)Eﬀect of standing wave
As in these experiments, when pressure was applied to thesample surface, shear waves were generated, and they would
be reﬂected at the edge of the sample surface, producing
standing waves. These standing waves could aﬀect theaccuracy of the displacement measurement at the focal point.
How much they aﬀected the accuracy was conﬁrmed by thefollowing experiments by measuring the behavior of theshear wave propagation.
When acoustic pressures of Δf = 20, 100, and 500 Hz for
one wavelength were applied to the phantom surface, theresultant displacement waveforms were measured at threepoints on the phantom surface. The phantom with an Asker C
hardness of 5 was used as a sample. From the results, theamplitude of reﬂected waves from the edge of the phantom
surface was evaluated.
Figure 11 shows the displacement waveforms measured atthree points on the phantom surface when it was irradiatedwith acoustic radiation pressures of Δf = 20, 100, and 500 Hz
for one wavelength. We measured the displacements atthree points of coordinates ðx; y; zÞ ¼ ð0; 0; 0Þ; ð0; 6:5; 0Þ;ð0; 13:0; 0Þ in Fig. 1 (in mm). The excitation focal point of
the phantom surface was at the origin ð0; 0; 0Þ, and the edge

1
0
-1
1
0
-1
1
0
-1

y = 0.0 mm
y = 6.5 mm
y = 13.0 mm
0

200

400

0

50
time [ms]

100

0

10

20

Fig. 11. (Color online) Displacement waveforms at the three points of the phantom (hardness: 5) excited at (a) Δf = 20 Hz, (b) Δf = 100 Hz, and(c) Δf = 500 Hz.

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R. Watanabe et al.

of the phantom was located at ð0; 15:0; 0Þ. The red line in
Fig. 11 shows how the shear wave propagates, the solid line
shows the traveling wave, and the broken line shows thereﬂected wave. For the vertical axis of Fig. 11, the measured
displacement was normalized by the displacement peak value
at the focal point.
From these results, the propagation velocities of shear
waves generated by 20, 100, 500 Hz excitations were roughly
1.2, 3.8, and 5.5 m=s, and the attenuation coeﬃcients were1.1, 0.96, and 1.1 dB=mm, respectively. There was 30 mm forthis shear wave to reciprocate between the focal point andthe edge of the phantom surface. Therefore, when the shear
waves returned to the focal point, their amplitudes would be
attenuated by −33, −29, and −33 dB, that is, 0.02, 0.03, and0.02 times the amplitude of the original displacement at the
focal point, respectively. The amplitudes of the shear waves
which returned to the focal point were much smaller than thestandard deviation of the displacements measured in Fig. 6.
In the present study, therefore, the standing wave had no fatal
eﬀect on the estimation results of the displacements in Fig. 6.
In the displacement waveform at y = 0 mm of the phantom
excited at 20 Hz, a wave like ringing continued after apositive peak in about 0.15 s. A similar phenomenon wasobserved for excitation frequencies of 100 and 500 Hz. At
any excitation frequency, the phenomenon regarding thewave with a frequency lower than the excitation frequency
was observed. In the proposed method, the displacement
wave was frequency-analyzed and the power component at
the excitation frequency was measured. Therefore, the wave
like ringing did not aﬀect the power of the excitation
frequency. However, it is still necessary to investigate thecause of this wave in the future.
4.2 Advantages of the proposed method
In the present study, by scanning the excitation frequency
more widely than in the previous study, the frequency
characteristics of the stress-strain were obtained. From theresults, the viscoelastic property of the phantom wasestimated. When the Maxwell model was applied, there
should only be one inﬂection point in the frequency
characteristics. If there are two inﬂection points, however,the Zener model using three viscoelastic elements should beapplied. Therefore, when the measured frequency range
becomes wider and the number of inﬂection points observed
is more than one, a more appropriate viscoelasticity model
should be selected.
As a typical method of measuring the viscoelastic
property, a rheometer, which sandwiches the sample between
two plates and gives dynamic strain, is used to measure thestress generated from the sample by the contacting plate.
Thus, it measures the viscoelastic property of the entire
sample, which highly depends on the various conditions of
the contact between the sample and the plate. In the proposed
method, however, by changing the position of the focal pointof the acoustic excitation, it is possible to estimate the local
region of an internal tissue at each depth separately.
Moreover, the spatial resolution of the measurement is higher
than that of the rheometer. When measuring the viscoelastic
properties of a solid sample with a rheometer, a cylindricalsample with a diameter of about 10 mm or more should be
prepared. In the proposed method, on the other hand, theexcited region depends on the wavelength of the ultrasonicwave. When an ultrasonic wave of 1 MHz is employed as in
the present study, it is possible to excite and estimate a local
region of about 2 mm or less. Thus, the proposed method
can be applied to lesions of 10 mm or less found in breast,
thyroid, and the like.
In the present study, a laser displacement meter was
employed for the measurement of the displacement waveform. However, by using ultrasonic diagnostic equipment in
the future, the proposed method can be applied to internal
tissues that are the main target of measurement in our study.
For this purpose, it is necessary to increase the frame rate of
the ultrasonic diagnostic equipment. The frame rate can beincreased by reducing the depth of ﬁeld of measurement andincreasing the pulse repetition frequency. Therefore, lesionsof muscles, breasts and thyroid glands located in a shallow
area from the skin are suitable as actual targets of the
proposed method. Moreover, by parallel beam forming,31)
which is a technique for increasing the frame rate of theultrasonic diagnostic equipment, the application of the
proposed method to deeply positioned tissues such as liveris anticipated.
5.

Conclusions

In the present study, we excited samples by dual acoustic
radiation force. We proposed a new method of estimating theviscoelasticity of the excited local region of samples by using
the frequency characteristics of the obtained dynamic
displacement amplitude. We generated a periodic displacement on the surfaces of phantoms mimicking living soft
tissues by irradiating them with the ultrasound generated
by the sum of two continuous sinusoidal waves with slightly
diﬀerent frequencies from two directions. The generated
displacement was measured using a laser displacement meter.
From the frequency characteristics of the displacement andacceleration obtained by varying the excitation frequency Δf,
the frequency characteristics of the stress–strain transfer
function of the sample were estimated, through which it was
conﬁrmed that the ultrasonic excitation stress has almost no
frequency dependence. The frequency characteristics of thedisplacement depended on the hardness of the phantom.
Currently, vibrations at frequencies below 20 Hz arecontaminated with large noise in the measurement. If we
can accurately measure the frequency characteristics forthe lower frequency components, the frequency model will
be widely applicable. Furthermore, it is necessary to measure
the phase diﬀerence between the applied stress waveform andthe obtained displacement waveform. If phase informationis stably obtained, we can ﬁt the result to a theoretical modelsuch as the Maxwell model in a complex domain including
the amplitude and phase. Then, we can select a more
appropriate model and improve the ﬁtting accuracy. The
establishment of the phase measurement method is one of our
future tasks.

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