Chapter 3

STRAIN INDUCED BY DUAL
ACOUSTIC RADIATION FORCE
AND ITS ULTRASONICMEASUREMENT
Hideyuki Hasegawa and Hiroshi Kanai
Graduate School of Biomedical Engineering, Tohoku University, Sendai 980-8579, Japan

ABSTRACT
There are many studies on measurement of tissue mechanical properties by applying an
acoustic radiation force induced by ultrasound to an object. However, one acoustic radiation force along a direction (e.g., vertical direction) does not generate the strain of an object
effectively because it also causes change in the object position, which has zero spatial
gradient in displacement (= no strain). Especially, when the elastic modulus of the object ismuch larger than that of the surrounding tissue (such like a tumor in the breast tissue), one
acoustic radiation force might generate only the change in position of the object and thestrain of the object is hardly generated. In such cases, mechanical properties of the object
cannot be evaluated. In this study, two cyclic acoustic radiation forces are simultaneously
applied to an object along two different directions (e.g., two opposite horizontal directions)to effectively generate the strain inside the object even when the object is much harder than
the surrounding tissue.

3.1 INTRODUCTION
In recent years, some remote actuation methods based on acoustic radiation force have beenreported. Fatemi and coworkers proposed an imaging method called ultrasound stimulated
acoustic emission (USAE) [1, 2]. Their system consists of two confocal ultrasonic transducers, and two ultrasound beams with two slightly different frequencies of f and (  f + D f  )
are transmitted. Acoustic radiation pressure, PR(t), exerted on the interface between two
different media is a function of the energy density, e(t), and the specific acoustic impedances, Z1 and Z2, of the media [3]. The energy density, e(t), is proportional to the square of
the sum of the sound pressures, p1(t) and p2(t), generated by the two transducers. In the intersectional area of these two beams, therefore, an oscillatory radiation pressure PR(t) at thefrequency difference, Δ D f, is applied to the interface. The radiation force produces acoustic
emission that is closely related to the mechanical frequency response of the medium. By
41

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measuring the acoustic emission with a hydrophone, hard inclusions in soft material wereexperimentally detected. The spatial resolution in the depth direction corresponds to thesize of the intersectional area.
Nightingale et al [4] proposed an alternative imaging method in which the pulsed ultrasound is employed for applying the radiation force to a soft tissue during short durations(less than 1 millisecond). The viscoelastic properties of the tissue are measured from themagnitude and the transient response of the displacement, d(t), of the tissue [4-6]. In orderto generate measurable displacement by several ultrasonic pulses, high-intensity pulsed
ultrasound of 1000 W/cm2 is employed. According to the safety guideline shown by theJapan Society of Ultrasonics in Medicine (JSUM), however, the intensity of ultrasoundis recommended to be less than 240 mW/cm2 and 1 W/cm2 for the pulsed and continuous waves, respectively [7]. Therefore, the intensity of the pulsed ultrasound employed by
Nightingale et al [4] is far greater than that indicated in the safety guideline.
To improve the spatial resolution in measurement of the response of the object, Michishita et al used an ultrasound correlation-based method, the ultrasonic phased tracking
method [8-10], to measure the minute displacement, d(t), caused by the acoustic radiation
force [11]. In addition, to suppress the sound pressure of the employed ultrasound, a continuous wave (CW) ultrasound, which consists of the sum of frequency components at f and
f + D f, was employed to cyclically actuate the acoustic radiation pressure PR(t) at a lowfrequency D f to increase the signal-to-noise ratio in displacement estimation [11]. However,
one acoustic radiation force along a direction (e.g., vertical direction) does not generate thestrain of an object effectively because it also causes change in the object position, whichhave zero spatial gradient in displacement (= no strain). In particular, when the elastic
modulus of the object is far greater than that of the surrounding media, one acoustic radiation force may generate only the change in position of the object as illustrated in Fig. 3-1a.
In such cases, the mechanical properties of the object cannot be estimated. In this study, foreffective generation of the strain inside the object even in such cases, two acoustic radiation forces with phases of r1 and r2 were applied to two different positions in the object
from two different directions (e.g., two opposite horizontal direction) as shown in Fig. 3-1b[13]. Furthermore, the displacements of the region inside the object, where the two acoustic
radiation forces were applied, were measured by the ultrasonic phased tracking method.

Figure 3-1. Illustrations of (a) the change in position of an object caused by only one acoustic radiation force and (b) the strain inside an object caused by two acoustic radiation forces

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Strain Induced by Dual Acoustic Radiation Force      

    43

3.2 GENERATION OF FLUCTUATING ACOUSTIC RADIATION FORCE
When a continuous plane-wave ultrasound is incident on an interface between two different media, a constant force, which is called an acoustic radiation force, is exerted on theinterface. Acoustic radiation pressure, PR(t), is defined as the acoustic radiation force perunit area of the interface as follows [1, 2]:
PR (t) = (1+ R2) e(t)(3 – 1)

where R and e(t) are the pressure reflection coefficient and the energy density at the interface, respectively. In Eq. (3-1), the transmitted wave is assumed to be perfectly absorbed
in the object. Using the densities, r1 and r2, and sound speeds, c1 and c2, of the media andobject, the reflection coefficient, R, and the energy density, e(t), are defined byR=ρ 2 c2 − ρ 1 c1
Z2 − Z1=
ρ 2 c2 + ρ 1 c1
Z2 + Z1e(t) =

1
r 1c21{ p(t) }2(3 – 2)(3 – 3)where p(t) is the sound pressure at the interface. JSUM shows the safety guideline in which
the intensity of the CW ultrasound should be less than 1 W/cm2. By assuming that the density r1 and sound speed c1 of water are 103 kg/m3 and 1500 m/s, respectively, the acoustic
radiation pressure, PR(t), exerted on the interface is calculated to be 6.67 Pa when the ultrasound intensity is 1 W/cm2. In this calculation, the reflection coefficient, R, is assumed to be
zero. For example, sound speeds of muscle and fat are 1568 m/s and 1465 m/s, respectively[13]. By assuming densities of these tissues to be 103 kg/m3, the reflection coefficient R isdetermined to be 0.034. Thus, the reflection coefficient can be assumed to be zero.
As shown in Eq. (3-3), the energy density e(t) of the incident wave is proportional to
the square of the sound pressure p(t) of the ultrasound beam. When two ultrasonic beams
with slightly different frequencies, f and f + D f, are crossed each other, the sound pressure,psum(t), at the intersectional area is expressed by the sum of sound pressures of two ultrasonic beams as follows:
psum (t) = p0 cos(ωt + ϕ ) + p0 cos(ω + ∆ω)t

(3 – 4)

where p0, w, and Dw, are the amplitude of the sound pressure of each ultrasound beam,
angular frequency of the incident wave (w = 2p f  ), and difference in angular frequency(Dw = 2p Df  ), respectively, and r is the initial phase of the CW at frequency f. In this case,the energy density e(t) is given by12
r 1c21 { psum (t)}

1

= r 2
c
1 1
=

p20
r 1c21{ p0 cos(w t + j ) + p0 cos(w + Dw )t}2]e(t) =

1 + cos(Dw t - j) + cos {(2w + Dw )t + j }+ 1 cos 2(w t + j ) + 1 cos 2(w + Dw)t2
2

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44          Biomedical Applications of Vibration and Acoustics for Imaging and Characterisations
From the second term of the right-hand side of Eq. (3-5), it is found that the energy densitye(t) of the incident field has a component at the frequency difference D f. Therefore, thecyclically oscillating radiation pressure PR(t) at the frequency difference D f is given by
PR (t) = (1 + R2)

p02

r 1c12{1 + cos(Dw t - j )}(3 – 6)

Thus, using an ultrasound beam generated by the sum of signals at slightly different frequencies f and f + D f, the oscillatory radiation force can be applied to the focal area of thebeam.
In this study, in order to generate the regional strain inside the object, we use two phasecontrolled acoustic radiation pressures, PR1(t) and PR2(t), given by
PR1(t) = (1 + R2)

p201{ 1 + cos(Dw t + j1)}
r 1c12(3 – 7)PR2(t) = (1 + R2)

p202{ 1 + cos(Dw t + j2)}
r 1c22(3 – 8)where r1 and r2 are the initial phases of CWs at frequency f for generating PR1(t) andPR2(t). These two acoustic radiation forces at D f, whose phase difference is r2 - r1, are applied by setting the focal points of two transducers at two different sites inside the object,
and the insonification angles are assigned to q1 and q2 for the respective transducers.

3.3 DISPLACEMENT OF OBJECT INDUCED BY FLUCTUATING
ACOUSTIC RADIATION FORCE
3.3.1 Experimental Setup
An experimental setup is illustrated in Fig. 3-2. An object made of gel was placed in a water tank. For the application of radiation pressures PR(t), a concave ultrasonic transducer
was employed. To generate a fluctuating acoustic radiation force, the concave ultrasonictransducer (Tokimec 1Z20I-PF50-C; center frequency: 1 MHz) was driven by a sum of two
continuous waves at two slightly different frequencies of 1 MHz and 1 MHz + D f Hz. The
resultant ultrasound beam was focused at 50 mm away from the surface of the transducer,
and the focal point was set at a point on the top surface of the object with beam angle
q = 60 deg. In this experiment, the displacement at the focal point was measured by a laser
velocimeter (ONO SOKKI, LV-1300).
In the following ultrasonic measurement, the CW ultrasound for actuation will interfere in the pulsed ultrasound for the measurement of displacements. In order to avoid thisinterference, it is necessary to stop actuation during transmission and reception of the ultrasonic pulse for the measurement. The application of ultrasound for actuation is controlledusing a custom-made electrical switch [11]. As shown in Fig. 3-2, the electrical switch
is controlled by the trigger signal from a function generator that synchronizes with thetrigger signal from ultrasonic diagnostic equipment (Toshiba, SSH-160A). The time chart
for controlling the application of ultrasound for actuation is shown in Fig. 3-3.

3.3.2 Experimental Results
Figure 3-4 shows the displacements d (t) on the top surface of the object, which are obtainedby integrating the velocities measured by the laser velocimeter. In Figs. 3-4a, b, c, and d,

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Strain Induced by Dual Acoustic Radiation Force           45Figure 3-2. Experimental setup for measurement of the displacement on the surface of the object
cyclically actuated using a concave ultrasonic transducer

Figure 3-3. Time chart for controlling the application of ultrasound for actuation. (a) Clock signal fromultrasonic diagnostic equipment. (b) Signal for controlling the application of ultrasoundfor actuation. Ultrasound is applied when this signal is high. (c) Sound pressure psum(t)consisting of the sum of two CWs at slightly different frequencies f and f + D f appliedintermittently only during actuation period (A). (d) Acoustic radiation pressure PR(t)generated by the sound pressure shown in (c). (e) Sampling gate for acquiring ultrasonicpulses for measurement, which determines the period for measurement (M)ASME_Fatemi_chap03.indd 45

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46          Biomedical Applications of Vibration and Acoustics for Imaging and Characterisations
frequency difference D f was set to 5 Hz, 10 Hz, 15 Hz, and 20 Hz, respectively. As shownin Fig. 3-4, the fluctuating displacements at the corresponding frequency difference D f wasdetected. From these results, it was confirmed that the object was actuated by applying thefluctuating acoustic radiation force. In Fig. 3-5, the amplitude of displacement is plotted asa function of frequency difference D f. As shown in Fig. 3-5, the amplitude of displacement
decreases with frequency difference (= actuation frequency) D f. When the object is assumed
to be a viscoelastic material, which is explained by the Voigt model, the displacement d (t)is expressed using static elastic modulus Es and viscosity constant h as follows:
PR(t) = Esd(t) + hḋ (t)(3 – 9)PR(t) = PR0ej2π∆ft(3 – 10)d(t) = d0ej(2π∆ft+φ )(3 – 11).
where d(t) is the time derivative of displacement. In this experiment, acoustic radiationpressure PR(t) and resulting displacement d (t) can be expressed by sinusoidal waves atfrequency D f as follows

By substituting Eqs. (3-10) and (3-11) into Eq. (3-9), Eq. (3-9) is modified as follows:
d0 =

PR0

√

E + (2π∆fη )2(3 – 12)2sFigure 3-4. Displacement d(t) on the surface of the object measured by the laser velocimeter. Object
was actuated at (a) D f = 5 Hz, (b) 10 Hz, (c) 15 Hz, and (d) 20 Hz

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Strain Induced by Dual Acoustic Radiation Force           47Figure 3-5. Amplitude of displacement plotted as a function of actuation frequency D f

In this experiment, the amplitude of radiation force PR0 was set to be constant. The amplitudeof displacement d0 decreased with actuation frequency D f, as shown by Eq. (3-12).

3.4 DEFORMATION INDUCED BY DUAL ACOUSTIC RADIATIONFORCE CAPTURED BY VIDEO CAMERA
3.4.1 Experimental Setup
An experimental setup is illustrated in Fig. 3-6. As in the experiments described in Section
3.3, the object made of gel (45 mm × 45 mm × 17 mm) was placed in a water tank. For
effective generation of the object deformation, two concave ultrasonic transducers were
employed to apply two acoustic radiation pressures PR1(t) and PR2(t). A concave ultrasonictransducer (Tokimec 1Z20I-PF50-C; center frequency: 1 MHz) was driven by a sum of
two CWs at two slightly different frequencies of 1 MHz and 1 MHz + 10 Hz. The resul­tant ultrasound beam was focused at 50 mm away from the surface of the transducer, andthe focal point was set at a point, a, on the top of the object with beam angle q1. Anotherconcave transducer (Tokimec 1Z20I-PF50-C; center frequency: 1 MHz) was driven by asum of CWs at 1 MHz and 1 MHz + 10 Hz. The focal point, which is 50 mm away from thesurface of the transducer, was set at a point, b, on the top of the object with beam angle q2.
The phase difference, j 2 – j1, between PR1(t) and PR2(t) was set to be zero.
In this experiment, the application of acoustic radiation forces was not stopped because
the amplitude of displacement is reduced when acoustic radiation forces are appliedintermittently. The resolution of the video camera in detection of the displacement is poorer
in comparison with the laser velocimeter, and the amplitude of displacement is desired to
be as large as possible. As indicated in Fig. 3-6, four different sites (a, b, c, and d) wereseparately measured. Therefore, the image acquisition was triggered by a rectangular signalthat synchronizes with the signal applied to transducers for actuation.

3.4.2 Experimental Results
As illustrated in Fig. 3-6, four different sites (a, b, c, and d) at the surface on the side of theobject were measured with a high-speed video camera at a frame rate of 100 Hz. In thisASME_Fatemi_chap03.indd 47

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Figure 3-6. Experimental setup for measurement of deformation of the object using a video camera

experiment, beam angles q1 and q2 were set to be 22 deg. and 35 deg., respectively, and the
distance between points a and b was set to be 6 mm. Focal areas of two transducers wereset at the edge of the top surface of the object. The four sites, a, b, c, and d, were measured
for one cycle (100 milliseconds) of acoustic radiation forces. The period for acquiring pictures in relation to acoustic radiation forces is shown in Fig. 3-7. In Figs. 3-7a to d, brightspots in measured pictures correspond to the light reflected from carbon powder, whichis contained in the object, due to the irradiation of the white light using halogen lamp. At
points a and c (two focal points), displacements along directions of respective ultrasonicbeams were found as shown in Figs. 3-7a and c. As shown in Fig. 3-7b, point b was foundto move upward when acoustic radiation forces increased. This result shows that the region
between two focal points was vertically thickened (deformed) due to horizontal compression by application of dual acoustic radiation force.

3.5 ULTRASONIC MEASUREMENT OF DISPLACEMENT
DISTRIBUTION INSIDE AN OBJECT INDUCED BY DUAL
ACOUSTIC RADIATION FORCE
3.5.1 Experimental Setup
An experimental setup is illustrated in Fig. 3-8. In order to measure the strain using
ultrasound, we employed ultrasonic diagnostic equipment (Toshiba SSH-160A) with a sector-type probe (center frequency: 5 MHz). The equipment was modified to detect the minutedisplacement of the object by the ultrasonic phased tracking method. As in Section 3.4,
an object made of gel (45 mm × 45 mm × 17 mm, containing carbon powder to obtain sufficient scattering) was placed in a water tank as shown in Fig. 3-8. For the application ofradiation pressures PR1(t) and PR2(t), two concave ultrasonic transducers were employed.
The settings for the application of acoustic radiation forces were the same as described in
Section 3.4.1.

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Strain Induced by Dual Acoustic Radiation Force      49

Figure 3-7. Results obtained at point a (a), b (b), c (c), and d (d) as indicated in Fig. 3-6 (e) and (f).
Acoustic radiation forces radiated by two ultrasonic transducers

Figure 3-8. Experimental setup for measurement of the strain inside the object cyclically actuatedusing two concave ultrasonic transducers

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50     Biomedical Applications of Vibration and Acoustics for Imaging and Characterisations
As described in Section 3.3.1, in the ultrasonic measurement, when the CW ultrasounds
are used for actuation, they will interfere in the pulsed ultrasound for the measurement of
displacements. In order to avoid this interference, it is necessary to stop actuation duringtransmission and reception of the ultrasonic pulse for the measurement. Therefore, we used
an electrical switch to control the cessation of the CW ultrasound for actuation [11].

3.5.2 Experimental Results
The spatial distribution of displacements inside the object was measured with ultrasound.
In this experiment, angles q1 and q2 of both beams were set to be 25 deg., and the distance
between a and b was set to be 15 mm. The measured positions were illustrated in Fig. 3-9.
Figure 3.10a shows an M-mode image of the object obtained at measured position d.
Acoustic radiation pressures PR1(t) and PR2(t), shown in Figs. 3-10b and c, were calculated
based on Eq. (3-6) as follows: The density, r2, and the sound speed, c2, of the object were
measured as 1.1 × 103 kg/m3 and 1.47 × 103 m/s, respectively. By assuming the density,
r1, and the sound speed, c1, of water to be 1.0 × 103 kg/m3 and 1.5 × 103 m/s, respectively,
the pressure reflection coefficient, R, and the energy reflection coefficient, R2, were respectively calculated as 0.038 and 0.0014 using Eq. (3-2). Therefore, in this study, by assuming
the object to be totally absorbing material (R = 0), the acoustic radiation pressures PR1(t)and PR2(t) exerted on two points, a and b, in the object were calculated based on Eq. (3-6).
In this calculation, we obtained the amplitude of the focused sound pressures of p01 andp02 beforehand by measuring the acoustic fields of each ultrasound for actuation with ahydrophone (Force Institute, MH-28-10). In Fig. 3-10d, the displacement d(t) at the surface(point A) of the object actuated by the acoustic radiation forces were measured by the ultra-

Figure 3-9. Illustration of positions measured with ultrasoundASME_Fatemi_chap03.indd 50

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Strain Induced by Dual Acoustic Radiation Force      51sonic phased tracking method. The tracking lines were superimposed by the red lines on the
M-mode image in Fig. 3-10a. In Fig. 3-10d, it is found that the object was cyclically actuat­edwith the amplitude of a few micrometers [11], which was similar to that measured withthe high-speed video camera. Two acoustic radiation forces were applied by setting focal
points at the upper surface of the object, and angles q1 and q2 of both ultrasonic beams for
actuation were set to be 25 deg. Therefore, directions of horizontal components of these
two acoustic radiation forces were opposite each other by setting r2 - r1 = 0. Therefore,
the region between two focal points was cyclically compressed along the horizontal axis
by these acoustic radiation forces, and the thickness of the region between two focal points
becomes thick along the vertical axis.
Figures 3-10d, e, and f show displacements at multiple points along an ultrasonicbeam. When acoustic radiation forces shown in Figs. 3-10b and c increase, the surfaceof the object and a point that was 1.5 mm deeper than the surface moved upward (seeFigs. 3-10d and e). On the other hand, a point that was 3.0 mm deeper than the surface
moved downward, as shown in Fig. 3-10f. This result shows the thickness of the region
between two focal points increased along the vertical axis, which shows that the strain was
generated inside the object.

Figure 3-10. Example of ultrasonic measurement of displacement inside the object. (a) M-mode image of the object. (b) Acoustic radiation pressure, PR1(t). (c) Acoustic radiation pressure,PR2(t). (d) Displacement at point A (surface) indicated in (a). (e) Displacement at point
B (1.5 mm from the surface). (f ) Displacement at point C (3.0 mm from the surface)

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52     Biomedical Applications of Vibration and Acoustics for Imaging and Characterisations

Figure 3-11. Spatial distribution of displacements inside the object. (a) Amplitudes and (b) phasesof displacements

Then, the two-dimensional displacement distribution was measured by moving theultrasonic probe for the measurement in the horizontal direction as illustrated in Fig. 3-9.
The displacement d(t) at every sampled point along each ultrasonic beam was obtained, andthe amplitude of the displacement d(t) at the frequency difference f = 10 Hz was estimated,
as shown in Fig. 3-11a, by applying the Fourier transform to the displacement d(t) and theacoustic radiation pressure PR1(t) with a Hanning window of 0.68 second (1024 points).
In addition, the phase delay of the displacement d(t) from the acoustic radiation pressure
PR1(t) was also estimated as shown in Fig. 3-11b. In Fig. 3-11, amplitudes and phase delays of displacements d(t) were shown in gray scale. The phase difference of zero showsthat that point moves upward when the acoustic radiation force increases. In Fig. 3-11, theregion that is 2 mm to 3 mm from the top surface tends to move upward when the radiation force increases. On the other hand, the region that is over 3.0 mm deeper than the top
surface tends to move downward. From these results, it was found that the strain inside theobject was successfully generated using two phase-controlled acoustic radiation forces.

3.6 DISCUSSION
In this study, the spatial distribution of displacements caused by two acoustic radiation
forces was measured with ultrasonic phased tracking method. The applied acoustic radiation forces and resulting displacement distribution are discussed below. Figure 3-12a shows
the sound pressure distribution generated by the ultrasonic transducer used for actuation.
From Fig. 3-12a, it was supposed that the width at half maximum of the acoustic radiation
force is about 5 mm in the x-axis, and the acoustic radiation force was almost constant along
the z-axis in the object. Therefore, the region in which the acoustic radiation force was overthe half maximum is roughly considered as shown by the region with shadow in Fig. 3-12b.
In Fig. 3-12b, the measured region is surrounded by the dashed line. Two ultrasound beams
crossed each other at a depth of about 4 mm from the top surface. At each point in the region
closer than 4 mm from the top surface in the measured region, the sum of horizontal components caused by two acoustic radiation pressures PR1(t) and PR2(t) are not zero. (Magnitudes of these components are different.) Therefore, the region nearer than 4 mm from the
top surface was compressed in the horizontal axis, and, as shown in Fig. 3-11, the resulting
upward displacement during the increase of acoustic radiation forces was found.

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Strain Induced by Dual Acoustic Radiation Force      53Figure 3-12. (a) Sound field generated by the ultrasonic transducer used for actuation. (b) Consideration of displacement distribution generated by two acoustic radiation forces

Around 4 mm from the top surface, horizontal components of two acoustic radiation
forces were almost compensated, and only vertical components remain. Therefore, the region around 4 mm from the top surface moves downward during the increase of acoustic
radiation force because of the downward displacement caused by vertical components ofacoustic radiation forces and that caused by the horizontal compression in the region nearer
than 4 mm from the top surface.
As shown in Fig. 3-11, the measured displacement distribution showed that the strain
inside the object was successfully generated. However, at this moment, it is difficult to estimate the elastic modulus distribution precisely due to the problems described below.

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54     Biomedical Applications of Vibration and Acoustics for Imaging and Characterisations     (1) Measurement of Strain Distribution    The stress applied by the acoustic radiation force is not the plane stress. Therefore,
the region, where the acoustic radiation force is applied, is displaced along not only
one direction. However, in this study, only the beam-axis component of displacement was measured. To obtain the precise strain and elastic modulus distributions, amethod for measuring three dimensional components of displacement is needed.
     (2) Estimation of Magnitude of Applied Stress    Equation (3-6), which used for estimation of the magnitude of the acoustic radiation force, was derived by assuming the object to be a perfectly absorbing material.
However, the attenuation coefficient of soft tissue has a finite value. For example,
the mixture of muscle and fat (abdominal wall), is typically 3 dB/cm/MHz [13]. To
estimate the magnitude of the applied acoustic radiation force, the attenuation coefficient of tissue should be considered.
     (3) Estimation of Stress Distribution    Figure 3-12a shows the sound pressure distribution, which is directly related to theacoustic radiation force as shown by Eq. (3-6), generated by the ultrasonic transducer used for application of the acoustic radiation force. As shown in Fig. 3-12a,
the acoustic radiation force is a function of the geometry, and the stress distribution
must be considered in addition to the strain distribution.
In addition, the distance between two focal points, beam angles, and depth of the region of interest will be restricted by the factors described below.
When focal areas of two ultrasonic transducers overlap, the horizontal components of
two acoustic radiation forces are compensated each other in the intersectional area. Therefore, for effective generation of the strain, two focal points are desired to be assigned sothat sound fields generated by two transducers do not overlap within the object (= regionof interest). On the other hand, when the distance between two focal points is very large incomparison with the size of the focal area, the displacement distribution is not so different
from that caused by one acoustic radiation force. Therefore, the size of the region of interest(between two focal areas) should be comparable to the width of the ultrasonic beam.
Furthermore, to keep beam angles q1 and q2 similar at every depth, the distance between two focal points should be changed in relation to the depth of the region of interest.
This will be done using a phased array probe.
As described above, at this moment, it will be difficult to estimate the “absolute” elastic
modulus precisely using the proposed method. Therefore, the elasticity of different individuals measured by the proposed method are difficult to be compared each other. However,for example, the temporal change or frequency dependence of the strain distribution within
an individual is directly related to the change in the elastic property. The proposed method
can detect such a relative change in the elastic property of an object.

3.7 CONCLUSIONSIn this study, in order to generate the regional strain inside the object, we constructed anexperimental setup for application of cyclic remote actuation in the object using two focused ultrasonic transducers. The phase difference between two acoustic radiation forces,
which were respectively applied at two different positions in the object, was controlled to
be 0 deg. The resultant displacements inside the object were successfully measured by theultrasonic phased tracking method. These results show potential of the proposed method
for generation of the regional strain inside the object.

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Strain Induced by Dual Acoustic Radiation Force      55

Acknowledgments
The authors thank Mr. Michishita, Mr. Nishio, Mr. Takahashi, and Mr. Odagiri for theirefforts in the experiments.

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