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Measurement of viscoelasticity of anisotropic viscoelastic phantom bydual ultrasound excitation
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REGULAR PAPER

Japanese Journal of Applied Physics 59, SKKE24 (2020)
https://doi.org/10.35848/1347-4065/ab8bc0

Measurement of viscoelasticity of anisotropic viscoelastic phantom by dual
ultrasound excitation
Hibiki Kawamura1, Shohei Mori1, Mototaka Arakawa1,2*, 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: arakawa@ecei.tohoku.ac.jp

Received December 4, 2019; revised March 20, 2020; accepted April 21, 2020; published online May 22, 2020
An estimation of anisotropic viscoelasticity is important for evaluating muscle lesions. In the previous study, we proposed a method for estimating
viscoelasticity in a local region by exciting a phantom specimen from both directions. In the present study, we observed the acoustic field having the
locality and directivity generated by dual ultrasound excitation. In addition, the displacement distributions for the isotropic and anisotropic
viscoelastic phantoms were measured, and the local generation of large displacement and its directionality was confirmed around the excitation
focal point. Furthermore, we applied our viscoelasticity estimation method to the anisotropic viscoelastic phantom. The estimated shear modulus
differed depending on the angle between the opposite direction of the ultrasound transducer projected on the phantom surface and longitudinal
direction of the cylindrical urethane rubber bundle in the anisotropic viscoelastic phantom. Therefore, the viscoelasticity estimation method in thepresent study could estimate anisotropic viscoelasticity locally. © 2020 The Japan Society of Applied Physics

1. Introduction

Muscle tissue is composed of bundles of elongated cells
called muscle ﬁbers with a diameter of 10–100 μm, so it
exhibits anisotropic viscoelastic properties.1–6) For example,
when shear wave elastography is applied to the brachialmuscle, the shear modulus estimated from the velocity of theshear wave propagating parallel to the muscle ﬁber direction
was approximately 4 times larger than that estimated from thevelocity of shear wave propagating perpendicular to themuscle ﬁber direction.6) The viscoelastic and anisotropic
properties of muscle are closely related to the progression oflesions. For example, the stiffness of the target tissue
increased as the disease progressed in Duchenne muscular
dystrophy.7) In addition, muscle ﬁber fragmentation due tonecrosis caused a decrease in anisotropy.8) Muscle tissue
became soft as an indication of decreased muscle tone in thecase of peripheral neuropathy and pyramidal tract
disorders.9,10) Athletes’ enlarged muscles and contracted
muscles were observed as increased stiffness.9) Diabetic
Pompe type showed an abnormal increase in muscle stiffnessand volume.9) In amyloidosis, there were cases of increased
stiffness and atrophy in muscle.9) Therefore, the evaluation of
the anisotropic viscoelastic properties of muscle has attracted
great clinical interests for the purpose of the diagnoses ofmuscular diseases and the planning for accurate treatment.11)
Conventional diagnostic methods for muscle tissues arecomputed tomography, magnetic resonance imaging, anddual-energy X-ray absorptiometry. However, there are drawbacks such as invasiveness, ﬁnancial burden, and inability tocarry the equipment.12) Therefore, ultrasound imaging is
considered as a reliable alternative as a non-invasive, lowcost, and portable method.6)
When the ultrasound elastography provided as a functionof the ultrasound diagnostic apparatus was applied tomuscles, the stiffness distribution of muscle tissue was
acquired.13,14) However, since this is qualitative, it isnecessary to estimate viscoelastic parameters quantitatively
in order to accurately evaluate the progression of the lesion.
The shear wave elastography is used as a quantitative
estimation method for elastic properties.6,7,15–17) This method

can estimate the shear modulus, from the shear velocity andthe tissue density. Furthermore, it is possible to estimate theanisotropic viscoelasticity of anisotropic elastic tissues bycombining shear wave elastography with shear wave
spectroscopy18) and rheological models.6,17) The acoustic
radiation force impulse (ARFI) imaging evaluates the elastic
properties from the magnitude of the displacement responseby exciting the tissue using pulse wave.19–24) It has beenstudied to estimate the anisotropic viscoelasticity of muscle
tissue.20–24) However, in order to generate a measurable
displacement by several ultrasonic pulses, high-intensity
pulsed ultrasound at 1000 W cm−2 was required.19)
According to the safety guideline for the use of diagnostic
ultrasound, it is recommended that the intensity of the spatial
peak-temporal average is below 240 mW cm−2 for pulsed
waves and the intensity is less than 1 W cm−2 for continuous
waves.25) The intensity of the pulsed ultrasound employed in
the ARFI imaging was therefore far greater than thatindicated by the safety guidelines. Anisotropic viscoelasticity
was estimated in the recent study, however, the intensity ofthe pulsed ultrasound was unclear.20)
We developed a method to locally excite a tissue by
applying dual AM acoustic radiation force.26,27) The direction of the main displacement to be generated can be ﬁxed to
one axis. In addition, the frequency characteristics of theacoustic radiation force and the displacement on the phantom
specimens were measured. Therefore, the relationship of thestress–strain transfer function was obtained indirectly. The
viscoelastic parameters of the phantom specimens were
estimated by ﬁtting a rheological model to therelationship.28) This method could estimate viscoelasticity
locally and quantitatively. However, it is necessary toinvestigate the possibility of estimating anisotropic viscoelasticity to apply this method to the evaluation of muscle.
The purpose of this study is to investigate whether theviscoelastic property estimation method adopted could locally estimate anisotropic viscoelasticity. In our previous
study, we investigated the acoustic ﬁeld formed by theultrasound excitation.29) Furthermore, we investigated thedisplacement distribution on an isotropic viscoelastic
phantom surface generated by ultrasound excitation.29) In

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the present paper, we investigated the displacement distribution on an anisotropic viscoelastic phantom surface generated
by dual ultrasound excitation. Furthermore, we showed that
anisotropic viscoelasticity could be estimated.
2. Principle and experimental methods
2.1. Acoustic radiation pressure generated on thetissue surface

Acoustic radiation pressure is generated when a travelingultrasound wave has an energy difference due to a reﬂector oran absorber.30) In the present study, the acoustic radiationpressure was given by applying a sinusoidal signal with two
frequency components, f0 and ( f0 + Df ) to two ultrasoundtransducers with the same speciﬁcation. Thus, two identicalacoustic radiation pressures were generated. When a plane
wave with frequency components f0 and ( f0 + Df ) is
incident at the angle of incidence qi on the tissue along thepath, the amplitude of the generated acoustic radiationpressure PR (t ) changes with the frequency Δf as shown in
Eq. (1).
P R (t ) »2r1p12 cos qi(r1c1 cos qt + r 2 c2 cos qi )2⎡ r2 c 2⎤⎛r ⎞2´ ⎢ 22 2 cos2 qi + ⎜1 - 2 2 ⎟ cos2 qt ⎥⎢⎣ r1 c1⎥⎦r1 ⎠⎝´ [1 + cos (2pDft )] ,(1 )where, p1 is the sound pressure amplitude of the ultrasoundjust before the incidence, r1 and c1 are the density and soundvelocity of the medium before the incidence, and r 2 and c2
are the density and sound velocity of the medium after theincidence, respectively. qt is the angle of refraction.
2.2. Generation of shear strain by dual ultrasoundexcitation and the influence of anisotropy
Figure 1 shows a schematic diagram of the acoustic radiation
force generated in a medium by dual ultrasound excitation.
As shown in Fig. 1(a), the focal points of two point-focusedultrasounds coincide and the ultrasounds are irradiateddiagonally to the medium surface. When acoustic radiation

forces are applied diagonally from both sides, the x-axis
direction components of the acoustic radiation forces cancelout each other in the excitation area, so the acoustic radiation
force is applied to the medium surface only along the z-axis
direction. The magnitude of the z-axis direction componentof the acoustic radiation force applied to the medium
decreases as the distance from the focal point increases
because the focused ultrasounds are irradiated. Figure 2(a)shows a schematic diagram of the distribution of the z-axis
direction component of the acoustic radiation force generated
by the dual ultrasound excitation, taking up the y–z plane.
Focusing on the local area surrounded by the red dotted line
in Fig. 2(a), since the magnitude of the z-axis direction
component of the acoustic radiation force applied to two
adjacent points is different, the difference of the acoustic
radiation force is considered to be the shear force in z-axis
direction applied to the y–z plane as shown in Fig. 2(b).
In addition, the propagation path length of each excitation
ultrasound coincides at the point apart from the focal point in
the y-axis direction, however, it differs at point apart from theexcitation focal point in the x-axis direction as shown in
Fig. 3. The phase difference occurs between two excitation
ultrasounds along the x-axis direction on the medium surface.
Due to the effect of this phase difference, the magnitude of
the z-axis component of the acoustic radiation force on the x–
z plane variates periodically as it moves away from the focal
point in the x-axis direction as shown by the blue line in
Fig. 4. When the speed of sound in water is 1500 m s−1, thefrequency of the excitation ultrasound is 1 MHz, and theincidence angle of the excitation ultrasound on the object
surface is 35°, the magnitude of the acoustic radiation force
exhibits maxima and minima every 0.92 mm away from the
focal point in the x-axis direction. Therefore, it is consideredthat the z-axis component of the acoustic radiation force
decreases greatly when moving away from the focal point inthe x–z plane compared to the y–z plane as shown in Fig. 4.
Then, it is considered that the generated shear force in the zaxis direction increases because the difference between theforces in the z-axis direction of the two adjacent points islarger in the x–z plane. Therefore, the magnitude of shear

Fig. 1. (Color online) Schematic diagram of the acoustic radiation force generated by dual ultrasound excitation.

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(b)

(a)

Fig. 2. (Color online) (a) Schematic diagram of the z-axis direction component of acoustic radiation force and the shear force in z-axis direction generated bydual ultrasound excitation. (b) The shear force generated by the difference in the magnitude of the z-axis direction component of acoustic radiation force
between two adjacent points in the area surrounded by the red dotted line in (a).

Fig. 3. (Color online) Schematic diagram of the propagation path length ofeach excitation ultrasound at points equidistant from the focal point in dual
ultrasound excitation. d y: distance from a point away in y-axis direction fromthe focal point to both ultrasound transducers. dx1: distance from point awayin x-axis direction from the focal point to the left ultrasound transducer. dx2:
distance from point away in x-axis direction from the focal point to the rightultrasound transducer.

force in the z-axis direction in the x–z plane can be generated
larger than that in the y–z plane by dual ultrasound excitation.
We consider that dual ultrasound excitation can apply shear

force in the z-axis direction having directivity on the medium
surface.
Next, we consider the elastic modulus of an anisotropic
viscoelastic material having a structure in which elongatedﬁbers are bundled like muscle. It has a large elastic modulus
in the axial direction of the ﬁber bundle due to the strong
binding force among tissues. By contrast, the elastic modulus
in the radial direction of the ﬁber bundle is small because thebonding force among ﬁbers is weaker than the bonding force
between tissues. Therefore, it is considered that the anisotropy of viscoelasticity is caused by the difference in thebonding force depending on the structure. Figure 5 shows aschematic diagram of the shear strain in the z-axis direction
generated when an acoustic radiation force having a distribution of z-axis direction components as shown in Fig. 2(a) is
applied to an anisotropic viscoelastic material with a ﬁber
bundle structure. Figures 5(a) and 5(b) show the schematic
diagrams when the acoustic radiation force is applied in theaxial and the radial directions of the ﬁber bundle, respectively. Due to the difference in elastic modulus caused by thebonding force depending on the structure, the shear strain in
the z-axis direction generated by applying the acoustic

Fig. 4. (Color online) Schematic diagram of the distribution of the z-axis direction component of acoustic radiation force generated on each plane by dual
ultrasound excitation. Red line: on the y–z plane, and blue line: on the x–z plane.

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(a)

(b)

Fig. 5. (Color online) The shear strain in the z-axis direction in an anisotropic viscoelastic body with a ﬁber bundle generated by applied the acoustic
radiation force having the distribution of the z-axis direction component. Directions in which acoustic radiation force is applied are (a) the axial direction of the
ﬁber and (b) the radial direction of the ﬁber.

radiation force having a distribution of z-axis direction
components to the radial direction of the ﬁber bundle isgreater than that by applying it to the axial direction of the
ﬁber bundle. When acoustic radiation force is applied to ananisotropic viscoelastic material with a ﬁber bundle structureby dual ultrasound excitation, the magnitude of the shear
strain in the z-axis direction is mainly related to the angle
between the x–z plane and the ﬁber bundle since the shear
force in the z-axis direction generated on the x–z plane islarger than that on the y–z plane. Therefore, since the shear
force in the z-axis direction generated by dual ultrasoundexcitation has directivity, the anisotropy of viscoelasticity of
the anisotropic viscoelastic material with a ﬁber bundle
structure could be detected by measuring the shear strain in
the z-axis direction.
2.3. Experimental method
Figure 6 shows the experimental arrangement in the present
study. Two concave transducers (TR1, TR2) with a center
frequency f0 of 1 MHz, an aperture half-angle of 22.6°, aneffective aperture diameter of 50 mm, and a focal length of
60 mm were used for ultrasound excitation. They were
opposed with an irradiation angle of 35° to the phantom.
The focal points of the two ultrasound transducers coincidedwith the center of the phantom surface, and this point was
treated as the center of distribution measurements of theacoustic ﬁeld and the displacement. Sinusoidal signals withfrequencies of f0 and ( f0 + Df ) generated by a function
generator (Tektronix, Inc., AFG2020) were summed up,
ampliﬁed, and applied to the transducers. The applied voltage
to each transducer was 37.5 V. The sound pressure of thecontinuous ultrasound generated at the focal point by dual
ultrasound excitation was 280 kPa, which corresponds to
2.6 W cm−2 in ultrasound intensity. This intensity is 2.6
times higher than the recommended intensity in the safety
guideline.25) However, this study is a basic study aimed at
establishing a method for estimating anisotropic viscoelasticity, and the intensity does not affect the estimation result.

Therefore, this intensity was used for the experiment. For themeasurement of the acoustic ﬁeld, we replaced the phantom
with a hydrophone (Toray Industries, Inc., NH8133) asshown in Fig. 6(b). It was installed parallel to the z-axis tomeasure the z-axis direction component of the soundpressure. The output signal of the hydrophone was input tothe FFT analyzer (Ono Sokki Co. Ltd., CF-930), and thefrequency spectrum was acquired to extract the excitation
frequency Δf component of the sound pressure. The excitation frequency Df was set as 20 Hz.
The phantom simulating biological soft tissues with a sizeof 40 × 40 × 43 mm3 was prepared by Asker-C hardness 7
urethane resin with 2 wt% graphite. In addition, an anisotropic viscoelastic phantom was prepared by bundling 100
cylindrical urethane rubbers with a diameter of 300 μm andhardening them with Asker-C hardness 0 urethane resin. The
displacement perpendicular to the phantom surface wasmeasured by the laser displacement meter (Keyence Corp.,
LK-G80). The measurement range of the laser displacement
meter is 75.2 ± 14 mm. The phantom, ultrasound transducers,
and laser displacement meter shown in Fig. 6(a) have thesame dimension ratio as the experimental arrangement. Only
the phantom caused light reﬂection within the measurable
range. Therefore, the displacement of the phantom surface
could be measured by the laser displacement meter. The
sampling frequency of the laser displacement meter was set
to 5000 Hz. The output signal of the laser displacement meter
was input to the FFT analyzer to extract the Df component of
the displacement. The excitation frequency Df was set as
20 Hz.
The viscoelasticity estimation method of our researchgroup28) was applied to the measured frequency characteristics of the stress–strain relationship for the anisotropic
viscoelastic phantom. The direction opposing the ultrasoundtransducer on the x–y plane was set as θ = 0°. Cylindricalurethane rubber bundles in the phantom were placed at three
angles of θ = 0°, 45°, and 90°, and the displacement at each

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(a)

(b)
Fig. 6. (Color online) Experimental arrangement employed in the present study. (a) Displacement distribution measurement. (b) Acoustic ﬁeld measurement.

excitation focal point was measured. The frequency characteristics of the displacement were measured by changing
the excitation frequency Df in the range of 5–2000 Hz.
Similarly, the frequency characteristics of acoustic radiation
force were measured by exciting an accelerometer (YamaichiElectronics, 111BW). The relative compliance was calculated
from the frequency characteristics of the displacement andthe acoustic radiation force. Viscoelastic parameters were
estimated by ﬁtting a rheological model to the calculated
relative compliance.
3. Results
3.1. Acoustic field

Figure 7 shows the measurement result of the acoustic ﬁeld
by the dual excitation acquired at 0.2 mm intervals. The
results were normalized by the maximum sound pressure.
The contour lines at −8, −13 and −18 dB were ellipses withthe y-axis as the major axis. Since these contours are denser
in the x-axis direction than in the y-axis direction, it was
clearly conﬁrmed that the sound pressure decreased rapidly inthe x-axis direction as the distance from the focal point

increased. It caused an increase of the difference between theforces at two adjacent points and the generated shear force inthe z-axis direction in the x-axis direction. Therefore, a large
shear force in the z-axis direction could be generated in the xaxis direction in dual ultrasound excitation. The area within
the 3 dB contour line in Fig. 7 was 0.8 mm2, so it was also
conﬁrmed that sound pressures were locally applied.
Therefore, the local acoustic ﬁeld was observed near theexcitation focal point.
3.2. Displacement distribution in the isotropic viscoelastic phantom
Figure 8 shows the measurement result of the displacement
distribution on the isotropic viscoelastic phantom by the dual
excitation acquired at 0.2 mm intervals. The elliptical contour
lines with the major axis in the y-axis direction wereconﬁrmed similar to Fig. 7. Since these contours are densein the x-axis direction, it was considered that a large shear
strain in the z-axis direction was generated in the x-axis
direction. Therefore, the displacement distribution generated
by dual excitation was affected by the directivity of theacoustic ﬁeld. However, compared to the measurement result

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Fig. 7. (Color online) The measurement result of the acoustic ﬁeld generated by dual ultrasound excitation. Measurement range: 10 × 10 mm2, measurement
interval: 0.2 mm.

Fig. 8. (Color online) The measurement result of displacement distribution on the isotropic viscoelastic phantom generated by dual ultrasound excitation.
Measurement range: 10 × 10 mm2, measurement interval: 0.2 mm.

of the acoustic ﬁeld, the displacement measurement result
conﬁrmed that the decrease in displacement amplitude whenmoving away from the focal point in the x-axis direction was
gentler. However, the area within the 7 μm contour line in
Fig. 8 was 2.8 mm2, so it was conﬁrmed that the displacement generated by dual excitation was local.

3.3. Displacement distribution in the anisotropic viscoelastic phantom

Figures 9(a) and 9(b) show the displacement distribution on
anisotropic viscoelastic phantom by the dual excitation. They
were measured at 0.5 mm intervals. The directions of thecylindrical urethane rubber bundle in the phantom were

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(b)

(a)

(c)

(d)

Fig. 9. (Color online) Displacement distribution on the anisotropic viscoelastic phantom generated by dual ultrasound excitation. (a) Δf = 20 Hz, ﬁber
direction: parallel to y-axis. (b) Δf = 20 Hz, ﬁber direction: parallel to x-axis. (c) Δf = 55 Hz, ﬁber direction: parallel to y-axis. (d) Δf = 200 Hz, ﬁber direction:parallel to y-axis.

installed parallel and perpendicular to the y-axis in Figs. 9(a)and 9(b), respectively. Compared with Fig. 8, the contour
lines spread in the x-axis direction in Fig. 9(b) although there
was no signiﬁcant change in Fig. 9(a).
Furthermore, Figs. 9(c) and 9(d) are measurement results
under the same conditions as Fig. 9(a) except for Df . Df
were set as 55 Hz and 200 Hz in Figs. 9(c) and 9(d),respectively. In both Figs. 9(c) and 9(d), displacement
distributions similar to that in Fig. 9(a) were measured.

Therefore, there was almost no difference in the displacement
distribution measured by changing the excitation frequency
Df .
3.4. Estimation of viscoelasticity of anisotropic viscoelastic phantom
Figure 10 shows the calculated relative compliance forθ = 0°, 45°, and 90°. According to the results shown in
Fig. 10, the relative compliance did not depend on frequency
when low-frequency stress was applied. Its behavior followed

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Fig. 10. (Color online) Relative compliance on the anisotropic viscoelastic
phantom. Blue line: θ = 0°, purple line: θ = 45°, and red line: θ = 90°.

the frequency characteristics of the Voigt model which is atypical viscoelastic model. Therefore, inﬂection point G /hand relative shear modulus G were estimated by ﬁtting thefrequency characteristics of the Voigt model. The estimatedinﬂection points G /h were 8 Hz at θ = 0°, 6 Hz at θ = 45°,
and 5 Hz at θ = 90°. Furthermore, assuming that G estimatedfrom the relative compliance values at the low-frequency
region where they became constant was 1 at 90°, they were
1.2 and 1.6 at 45° and 0°, respectively. When viscosity η wasestimated from the estimated G /h and G, η was estimated to
be almost constant. It is considered that η hardly changes bychanging the direction of the cylindrical urethane rubber
bundle of the phantom. Therefore, the estimated shear
modulus for the direction of the cylindrical urethane rubber
bundle parallel to the opposing direction of the excitation
ultrasound was explicitly larger than that for the perpendicular direction.
4. Discussion

At ﬁrst, we discuss the factor that the directivities of thedisplacement distribution in Fig. 8 were smaller than those of
the acoustic ﬁeld in Fig. 7. Since the phantom was elasticallycontinuous, by locally applying the strong acoustic radiation
force, displacements of frequency Df were generated not
only at the applied point but also around it caused by the
propagation of the shear wave. Therefore, the measured
displacement distribution became the sum of the shear
displacement directly generated by the acoustic radiation
force and the displacement indirectly caused by the propagation of shear waves generated by the large acoustic radiation
force near the focal point. The propagation attenuation ofshear waves depends on the frequency, and it was
1.1 dB mm−1 when Df was 20 Hz.28) The decrease of thedisplacement of the shear wave at a point 5 mm away fromthe focal point in the x-axis direction was estimated as 6 dB.
In Fig. 7, the sound pressure at that point was more than
20 dB lower than the sound pressure at the focal point.
Therefore, we considered that the shear wave generated by alarge acoustic radiation force near the focal point had agreater inﬂuence indirectly on the measurement of displacement than the shear displacement generated directly by thesmall acoustic radiation force. As a result, it is consideredthat the directivity of the displacement distribution was
smaller than the directivity of the acoustic ﬁeld.
Next, we consider the reason that the contour lines of thedisplacement distribution in Fig. 9(b) spread in the directionof the cylindrical urethane rubber bundle of the phantom
compared to that in Fig. 8. Figures 11(a) and 11(b) show thedisplacement distributions on the x-axis and y-axis of Figs. 8
and 9(b), respectively. The black and red lines show thedisplacement distribution of the isotropic (Fig. 8) andanisotropic [Fig. 9(b)] phantoms, respectively. Each displacement distribution in the range of ±1.5 mm was shown. In
Fig. 11(a), the differences in displacement for each phantom
were 11 and 13 μm at the focal point and −1 mm, respectively. The displacement distribution of the isotropic phantom
was affected by the directivity of the acoustic ﬁeld. Since theacoustic ﬁeld generated by dual ultrasound excitation decreased rapidly in the x-axis direction, the generated displacement on the isotropic phantom decreased as the distancefrom the excitation focus increased. In contrast, since thedirection of the cylindrical urethane rubber bundle wasparallel to the measurement direction, it is considered thatthe propagation of shear wave generated near the focal point
along the cylindrical urethane rubber bundle was observed.
Therefore, the displacement measured on the anisotropic
phantom has a large amplitude even when it was away fromthe focal point. In Fig. 11(b), the displacement differences foreach phantom were 11 and 6 μm at the focal point and−1 mm, respectively. Since the acoustic ﬁeld generated bydual ultrasound excitation decreased gently in the y-axis
direction, the generated displacement on the isotropic
phantom was almost unchanged as the distance from theexcitation focus increased. In contrast, since the direction of
the cylindrical urethane rubber bundle was perpendicular to
the measurement direction, it is considered that the amplitudeof the shear wave attenuated every time as it crossed thecylindrical urethane rubber caused by the difference of
acoustic impedances between urethane rubber and urethane

(a)

(b)
Fig. 11. (Color online) Displacement distributions on the isotropic
phantom (Fig. 8) and the anisotropic phantom [Fig. 9(b)] with a ﬁber
direction parallel to x-axis. (a) On x-axis. (b) On y-axis. Black lines: isotropic
phantom, and red lines: anisotropic phantom.

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resin. Therefore, the displacement measured with the anisotropic phantom decreased as the distance from the excitation
focus increased. At the point where the acoustic radiation
force was signiﬁcantly smaller than the focal point, considering that the propagation of shear wave had a larger
inﬂuence on the displacement measurement than the shear
displacement generated directly by the acoustic radiation
force, it is considered that the contour lines spread in thelongitudinal direction of the cylindrical urethane rubber
bundle due to the propagation of the shear wave along thecylindrical urethane rubber bundle rather than the directivityof the acoustic ﬁeld.
The relative shear modulus was greater when the directionof the cylindrical urethane rubber bundle in the phantom wasplaced parallel to the opposite direction of the ultrasoundtransducer on the x–y plane than when it was placed
perpendicular to that direction in Fig. 10. This result indicatesthat the shear modulus in the axial direction of the urethane
rubber bundle was larger than the shear modulus in the radial
direction since the shear force in the z-axis direction
generated on the x–z plane is larger than that on the y–z
plane. Similar results were obtained by other measurement
methods.6,17,20) Therefore, it is considered that our method
was also able to obtain anisotropic viscoelasticity. In addition, from the results in Figs. 6(a), 6(c) and 6(d), the tendencyof the displacement distribution does not greatly depend on
the excitation frequency Df . Therefore, our method has thepossibility to evaluate the two-dimensional distribution ofanisotropic viscoelasticity.
In the phantom experiment, it is considered that thedisplacement related to compressive strain was measured inaddition to the displacement related to shear strain since thedisplacement of the phantom surface was measured.
Therefore, the estimated anisotropic viscoelasticity might
include the effect of displacement related to compressive
strain. Considering that the shear modulus and the compressive modulus are proportional to the square of the soundvelocity of the shear wave and longitudinal wave in thephantom, respectively, the compressive modulus is muchlarger than the shear modulus since the velocity of longitudinal waves (approximately 1500 m s−1) is much fasterthan that of shear waves (approximately 10 m s−1).
Therefore, the displacement related to the compressive strain
generated by dual ultrasound excitation is small and negligible for the displacement related to the shear strain.
Therefore, we considered that the viscoelasticity was estimated based on the assumption that the measured displacement was mainly related to shear strain. Furthermore, since
the displacement on the phantom surface was measured, it is
difﬁcult to discuss the events occurring inside the ﬁber
bundle structure. It is unclear how much the effects ofreﬂection and refraction at the boundaries inside the ﬁber
bundle structure affect the estimation results. In the future, we
discuss this effect by measuring the displacement inside theﬁber structure using ultrasonic waves.
5. Conclusions

We measured the acoustic ﬁeld generated by dual ultrasoundexcitation. The formed acoustic ﬁeld in dual ultrasoundexcitation had directivity in the direction perpendicular tothe opposing direction of the transducer and the excitation

region was local. We measured the displacement distribution
generated by dual ultrasound excitation for an isotropic
viscoelastic phantom. The displacement distribution was
directional in the same direction as directivity of the acoustic
ﬁeld, however, the inﬂuence of the directivity of the acoustic
ﬁeld was small at the point where the sound pressure wassmall. We considered that this was indirectly caused by the
propagation of shear waves generated by a large soundpressure near the focal point has a greater inﬂuence on themeasurement at the point where the sound pressure wassmall. We also measured the displacement distribution for ananisotropic viscoelastic phantom. In the anisotropic phantom,
the contour lines of the displacement distribution spread in
the direction of the cylindrical urethane rubber bundle
compared with the result for the isotropic phantom. We
considered that this result was caused by the effects of the
propagation of shear waves along the direction of thecylindrical urethane rubber bundle rather than the directivityof the acoustic ﬁeld. We applied the viscoelasticity estimation
method of our research group to the anisotropic viscoelastic
phantom. The estimated shear modulus differed dependingon the angle between the direction that shear force generated
by dual ultrasound excitation is large and the longitudinal
direction of the cylindrical urethane rubber bundle. This was
similar to the literature.6,17,20) Therefore, our viscoelasticity
estimation method can be applied to the evaluation ofanisotropy of the anisotropic viscoelastic tissues.
Considering that the tendency of the displacement distribution did not change with the excitation frequency, the methodof our research group may be able to evaluate the twodimensional distribution of anisotropic viscoelasticity.
In the future, we plan to examine the applicability of
biological tissues by estimating anisotropic viscoelasticity
such as chicken breast.
Acknowledgments

This work was partially supported by JSPS KAKENHI
19K22943.

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