Japanese Journal of Applied Physics
Vol. 45, No. 5B, 2006, pp. 4706–4711#2006 The Japan Society of Applied Physics

Generation of Strain Inside Objects Using Dual Acoustic Radiation Force
Hideyuki H ASEGAWA, Mikito TAKAHASHI, Yoshifumi N ISHIO and Hiroshi K ANAI
Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan(Received November 30, 2005; accepted February 15, 2006; published online May 25, 2006)

Many studies have been carried out on the measurement of mechanical properties of tissues by applying an acoustic radiation
force induced by ultrasound to an object. However, one acoustic radiation force in a given direction (e.g., vertical direction)
does not generate strain in an object eﬀectively because it also causes changes in the object’s position, which has zero spatial
gradient in displacement (= no strain). Especially when the elastic modulus of the object is much higher than that of thesurrounding tissue (such as a tumor in breast tissue), one acoustic radiation force might generate only a change in the positionof the object unlike a strain in 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 in two diﬀerent directions(e.g., two opposite horizontal directions) to eﬀectively generate strain inside the object, even when the object is much harderthan the surrounding tissue. [DOI: 10.1143/JJAP.45.4706]
KEYWORDS: dual acoustic radiation force, strain, mechanical property

1.

Introduction

In recent years, some remote actuation methods based on
acoustic radiation force have been reported. Fatemi andcoworkers proposed an imaging method called ultrasoundstimulated acoustic emission (USAE).1,2) Their system
consists of two confocal ultrasonic transducers, and two
ultrasound beams with two slightly diﬀerent frequencies, fand ( f þ  f ), are transmitted. Acoustic radiation pressure,PR ðtÞ, exerted on the interface between two diﬀerent media
is a function of the energy density, eðtÞ, and the speciﬁc
acoustic impedances, Z1 and Z2 , of the media.3) The energy
density, eðtÞ, is proportional to the square of the sum of thesound pressures, p1 ðtÞ and p2 ðtÞ, generated by the twotransducers. In the intersectional area of these two beams,
therefore, an oscillatory radiation pressure PR ðtÞ at thefrequency diﬀerence,  f , is applied to the interface. The
radiation force produces acoustic emission which is closely
related to the mechanical frequency response of the medium.
By measuring the acoustic emission with a hydrophone, hardinclusions in soft material may be experimentally detected.
The spatial resolution in the depth direction corresponds tothe size of the intersectional area.
Nightingale et al. proposed an alternative imaging method
in which pulsed ultrasound is employed to apply theradiation force to a soft tissue during short durations (lessthan 1 ms). The viscoelastic properties of the tissue aremeasured from the magnitude and the transient response of
the displacement, dðtÞ, of the tissue.4–6) In order to generatemeasurable displacement, dðtÞ, by several ultrasonic pulses,
high-intensity pulsed ultrasound of 1,000 W/cm2 is employed. According to the safety guidelines provided by theJapan Society of Ultrasonics in Medicine (JSUM), however,
the intensity of ultrasound is recommended to be less than
240 mW/cm2 and 1 W/cm2 for pulsed and continuous
waves, respectively.7) Therefore, the intensity of the pulsed
ultrasound employed by Nightingale et al. is far greater thanthat indicated in the safety guidelines.
To improve the spatial resolution in measurements of the
response of an object, Michishita et al. used an ultrasoundcorrelation-based method, the ultrasonic phased trackingE-mail address: hasegawa@us.ecei.tohoku.ac.jp

method,8,9) to measure the minute displacement, dðtÞ, causedby the acoustic radiation force.10) The accuracy in thedisplacement measurement by the phased tracking method
was evaluated to be 0.2 mm by basic experiments using arubber plate,11) and inﬂuences of the focal position of theultrasonic beam and the change in center frequency of RF
echo due to the frequency-dependent attenuation in tissue on
the measurement accuracy were also investigated.12–14)
Furthermore, the phased tracking method was modiﬁed tobe more robust against noise15) and to have a better spatial
resolution.16) The phased tracking method has been alreadyapplied to measurements of spontaneous or externallyapplied vibrations and viscoelasticity of the arterial wall17–26)
and heart wall.27–33)
In addition, to suppress the sound pressure of theultrasound employed, a CW ultrasound, which consists ofthe sum of the frequency components at f and f þ  f , wasemployed to cyclically actuate the acoustic radiationpressure, PR ðtÞ, at a low frequency  f to increase thesignal-to-noise ratio in the estimated displacement.10)
However, one acoustic radiation force in a given direction(e.g., vertical direction) does not generate strain in an object
eﬀectively because it also causes a change in the object’s
position, which has zero spatial gradient in displacement(= no strain). Especially when the elastic modulus of theobject is far greater than that of the surrounding media, one
acoustic radiation force may generate only a change in theposition of the object as illustrated in Fig. 1(a). In suchcases, the mechanical properties of the object cannot be
estimated. In this study, for eﬀective generation of strain
inside the object even in such cases, two acoustic radiation
forces with phases of  and  þ  were applied to two
diﬀerent positions in the object from two diﬀerent directions(e.g., two opposite horizontal directions) as shown in
Fig. 1(b). The displacements of the region inside the object,
where the two acoustic radiation forces were applied, were
measured by the ultrasonic phased tracking method.
2.

Principles

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 ðtÞ, is deﬁned as the4706

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H. HASEGAWA et al.

The energy density, eðtÞ, of the incident wave is proportional to the square of the sound pressure, pðtÞ, of theultrasound beam. When two ultrasonic beams with slightly
diﬀerent frequencies, f and f þ  f , cross each other, thesound pressure, psum ðtÞ, at the intersectional area is expressed by the sum of the sound pressures of two ultrasonicbeams as follows:

acoustic radiation
change inforce P R0 (t)position
focal point

psum ðtÞ ¼ p0 cos !t þ p0 cosð! þ !Þt;objectwhere p0 , !, and ! are the amplitude of the sound pressureof each ultrasound beam, the angular frequency of theincident wave (! ¼ 2 f ), and the diﬀerence in angularfrequency (! ¼ 2 f ), respectively. For this case, theenergy density, eðtÞ, is given by(a)acoustic radiation acoustic radiation
force P R1 (t)force P R2 (t)strain1
fpsum ðtÞg2
1 c211¼
fp0 cos !t þ p0 cosð! þ !Þtg2
1 c21eðtÞ ¼

focal points

p20
f1 þ cos !t þ cosð2! þ !Þt1 c211
1
ð2:5Þ
þ cos 2!t þ cos 2ð! þ !Þtg:2
2
From the second term on the right-hand side of eq. (2.5), it isfound that the energy density, eðtÞ, of the incident ﬁeld has acomponent at the frequency diﬀerence  f . Therefore, thecyclically oscillating radiation pressure, PR ðtÞ, at thefrequency diﬀerence  f is given by¼

object(b)
Fig. 1. Illustrations of (a) the change in position of an object caused byonly one acoustic radiation force, and (b) the strain inside an object
caused by two acoustic radiation forces.

PR ðtÞ ¼ ð1 þ R2 Þ

acoustic radiation force per unit area of the interface asfollows:1,2)
PR ðtÞ ¼ ð1 þ R2 ÞeðtÞ;

ð2:4Þ

ð2:1Þ

where R and eðtÞ are the pressure reﬂection coeﬃcient andthe energy density at the interface, respectively. In eq. (2.1),
the transmitted wave is assumed to be perfectly absorbed in
the object. Using the densities, 1 and 2 , and sound speeds,
c1 and c2 , of the medium and the object, the reﬂection
coeﬃcient, R, and the energy density, eðtÞ, are deﬁned byZ2  Z1
R¼
Z2 þ Z1
2 c2  1 c1
¼
;
ð2:2Þ
2 c2 þ 1 c1
1eðtÞ ¼fpðtÞg2 ;ð2:3Þ
1 c21
where pðtÞ is the sound pressure at the interface. The Japan
Society of Ultrasonics in Medicine (JSUM) provides safety
guidelines, in which the intensity of the CW ultrasoundshould be less than 1 W/cm2 . By assuming that the density
1 and sound speed c1 of water are 1:0  103 kg/m3 and1,500 m/s, respectively, the acoustic radiation pressure,PR ðtÞ, 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 1,568 and1,465 m/s, respectively.34) By assuming densities of thesetissue to be 1:0  103 kg/m3 , the reﬂection coeﬃcient, R, isdetermined to be 0.034. Thus, the reﬂection coeﬃcient canbe assumed to be zero.

p20
ð1 þ cos !tÞ:1 c21

ð2:6Þ

Thus, using an ultrasound beam generated by the sum of
signals at slightly diﬀerent frequencies, f and f þ  f , anoscillatory radiation force can be applied to the focal area ofthe beam.
In this study, in order to generate regional strain inside theobject, we use two phase-controlled acoustic radiationpressures, PR1 ðtÞ and PR2 ðtÞ, given byp2
PR1 ðtÞ ¼ ð1 þ R2 Þ 012 ð1 þ cos !tÞ;
ð2:7Þ
1 c1
p2
PR2 ðtÞ ¼ ð1 þ R2 Þ 022 f1 þ cosð!t þ Þg; ð2:8Þ
1 c1
where  is the phase diﬀerence between PR1 ðtÞ and PR2 ðtÞ.
These two acoustic radiation forces at  f , whose phase
diﬀerence is , are applied by setting focal points at two
diﬀerent sites inside the object. The insoniﬁcation angles are
assigned to 1 and 2 for respective transducers.
3.

Experimental Setup

An experimental setup is illustrated in Fig. 2. In order tomeasure the strain, we employed ultrasonic diagnostic
equipment (Toshiba SSH-160A) with a sector-type probe(center frequency: 5 MHz). The equipment was modiﬁed todetect the minute displacement of the object by the ultrasonic phased tracking method. An object made of gel (45 
45  17 mm3 , containing carbon powder to provide suﬃcient scattering) was placed in a water tank as shown in
Fig. 2. For the application of radiation pressures, PR1 ðtÞ andPR2 ðtÞ, two concave ultrasonic transducers were employed.

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H. HASEGAWA et al.

synchronization
signal generator

f1+∆f

(a)

signal generator

f2+∆f

f1

f2

adder

adder

mixer

divider

dividermixer

rectangular
A/D converter
pulsesfunction
generator

electronic controlled switch
power AMP

17 mm

ultrasonictransducer
1
θ1

block
water

ultrasonicprobe

power AMP

real

TRIG

imag

diagnostic
ultrasoundsystem

θ2
α

LPF

2

GPIB

β

workstation

gel

gel

block

a b c

water tank

PR1( t )Fig. 2. Experimental setup for measurement of the strain inside an object
cyclically actuated using two concave ultrasonic transducers.

PR2( t )
6 mm
a

4.

17 mm

15 mm
d

gel

45 mm
Fig. 3. Illustration of positions measured with a high-speed video camera.
(a) Top view. (b) Side view.

(a)
(b)

acoustic
radiation
force [Pa]

7 µm

(c)
(d)

7 µm

0.5

0

(e)
0.05

0.10

0.5

0

Results

4.1 Displacement measurement with video camera
As illustrated in Fig. 3, four diﬀerent sites, a, b, c, and d,
at the surface on the side of the object were measured with ahigh speed video camera at a frame rate of 100 Hz. In thisexperiment, beam angles, 1 and 2 , were set to 22 and 35 ,respectively, and the distance between  and  was set to
6 mm. As shown in Fig. 3, focal areas of the two transducers
were set at the edge of the top surface of the object. Thesefour sites were measured for one cycle (100 ms) of acoustic
radiation forces. Timing of the measured pictures relative toacoustic radiation forces is shown in Fig. 4. Bright spots inmeasured pictures correspond to the light reﬂected from

b c

(b)

acoustic
radiation
force
[Pa]

A concave ultrasonic transducer (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 þ 10 Hz. The resultant ultrasound beam was focused
50 mm away from the surface of the transducer, and the
focal point was set at a point, , on the top of the object witha beam angle, 1 . Another concave transducer (Tokimec
1Z20I-PF50-C; center frequency: 1 MHz) was driven by thesum of CWs at 1 MHz and 1 MHz þ 10 Hz. The focal point,
which is 50 mm away from the surface of the transducer, wasset at a point, , on the top of the object with a beam angle,
2 . The phase diﬀerence, , between PR1 ðtÞ and PR2 ðtÞ was
set to zero.
In this study, the spatial distributions of displacement (1)at the surface of the object and (2) inside the object aremeasured with a high speed video camera and ultrasound,respectively.
In the ultrasonic measurement, the CW ultrasound for
actuation interferes with the pulsed ultrasound for themeasurement of displacements. In order to avoid thisinterference, it is necessary to stop actuation during transmission and reception of the ultrasonic pulse for themeasurement. Therefore, we used an electrical switch tocontrol the cessation of the CW ultrasound for actuation.10)

time [s]

(f)
0.05

0.10

time [s]Fig. 4. Results obtained at point (a) a, (b) b, (c) c, and (d) d as indicated in
Figs. 3. (e) and (f) Acoustic radiation forces radiated by two ultrasonictransducers.

carbon powder contained in the object due to the irradiation
with the white light at a halogen lamp. At points a and c (twofocal points), displacements in the directions of respective
ultrasonic beams were obtained as shown in Figs. 4(a) and4(c). As shown in Fig. 4(b), point b moved upward when
acoustic radiation forces increased. This result shows thatthe region between two focal points was vertically thickened

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ultrasonic probe
for measurementgel (17 mm)acoustic radiation
force PR1(t)ABC

1.8 mm acoustic radiation
0.3 mmforce PR2(t)abcde f g

3.68 mm

(a)

x

17 mm

(b)

0
Pa 0.50.25

5.5 mmy

Pa 0.5
0.25

(c)

0
µm 3
0
-3
µm 3
0

measuredregion

gel

(d)

(e)

-3
µm 3
0

2.1 mm
15 mm
45 mm

(f)

-3
0

Fig. 5. Illustration of positions measured with ultrasound.

due to horizontal compression by application of dual
acoustic radiation force.
4.2 Displacement measurement with ultrasoundThe spatial distribution of displacements inside the object
was measured with ultrasound. In this experiment, angles 1and 2 of both beams were set to 25 , and the distance
between  and  was set to 15 mm. The positions measured
are illustrated in Fig. 5. Figure 6(a) shows an M-mode
image of the object obtained at measured position d.
Acoustic radiation pressures, PR1 ðtÞ and PR2 ðtÞ, shown in
Figs. 6(b) and 6(c) were calculated based on eq. (2.6) as
follows: The density, 2 , and the sound speed, c2 , of theobject were measured as 1:1  103 kg/m3 and 1:47  103
m/s, respectively. By assuming the density, 1 , and thesound 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 , respectively were
calculated as 0.038 and 0.0014 using eq. (2.2). Therefore, inthis study, by assuming the object to be a totally absorbent
material (R ¼ 0), the acoustic radiation pressures, PR1 ðtÞ andPR2 ðtÞ, exerted on two points,  and , in the object were
calculated based on eq. (2.6). In this calculation, we
obtained the amplitude of the focused sound pressures of
p01 and p02 beforehand by measuring the acoustic ﬁelds ofeach ultrasound for actuation with a hydrophone (ForceInstitute, MH-28-10).
In Fig. 6(d), the displacement, dðtÞ, at the surface(point A) of the object actuated by the acoustic radiation
forces was measured by the ultrasonic phased tracking
method. Tracking lines were superimposed with gray lines
on the M-mode image in Fig. 6(a). In Fig. 6(d), it is foundthat the object was cyclically actuated with an amplitude of afew micrometers,10) which was similar to that measured withthe high speed video camera.

0.5

time [s]Fig. 6. (a) M-mode image of an object. (b) Acoustic radiation pressure,PR1 ðtÞ. (c) Acoustic radiation pressure, PR2 ðtÞ. (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).

Two acoustic radiation forces were applied by setting
focal points at the upper surface of the object, and angles 1and 2 of both ultrasonic beams for actuation were set to 25 .
Therefore, directions of horizontal components of these two
acoustic radiation forces opposed each other by setting
 ¼ 0. Therefore, the region between the two focal points
was cyclically compressed along the horizontal axis by theseacoustic radiation forces, and the thickness of the region
between two focal points increases along the vertical axis.
Figures 6(d)–6(f) show displacements at multiple points
along an ultrasonic beam. When the acoustic radiation forces
shown in Figs. 6(b) and 6(c) increase, the surface of theobject and a point which was 1.5 mm deeper than the surface
moved upward [Figs. 6(d) and 6(e)]. On the other hand, apoint which was 3.0 mm deeper than the surface moved
downward as shown in Fig. 6(f). This result shows that the
thickness of the region between two focal points increased
along the vertical axis, which shows that strain was
generated inside the object.
The two-dimensional displacement distribution was measured by moving the ultrasonic probe for the measurement in
the horizontal direction as illustrated in Fig. 5. The displacement, dðtÞ, at every sampled point along each ultrasonicbeam was obtained, and the amplitude of the displacement,dðtÞ, at the frequency diﬀerence,  f ¼ 10 Hz, was estimatedby applying the Fourier transform to the displacement, dðtÞ,and the acoustic radiation pressure, PR1 ðtÞ, with a Hanningwindow of 0.68 s (1024 points). In addition, the phase delayof the displacement, dðtÞ, from the acoustic radiationpressure, PR1 ðtÞ, was also estimated. In Fig. 7, phase delaysof displacements, dðtÞ, were color-coded. Amplitudes ofdisplacements, dðtÞ, were indicated by the brightness of the4709

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x
2.1 mmultrasonictransducer

5.5 mmπ 10sound pressure [dB]

π/2
1µm

y

1 0

1
−π/2

sound pressure [dB]Fig. 7. Spatial distribution of displacements inside an object.

color. A phase diﬀerence of zero shows that that point
moves upward when the acoustic radiation force increases.
In Fig. 7, the region near the top surface tends to move
upward when the radiation force increases. On the other
hand, the region which is 3.0 mm deeper than the top surface
tends to move downward. From these results, it was foundthat strain inside the object was successfully generated using
two phase-controlled acoustic radiation forces.

0
z

75

at x=0 mm

70
65
60
55
20

5.

(a)

x

40
60
top surface

80
100
120
distance, z [mm]75at z=50 mm

70
65
60
55
–10

–5

0

5
10
distance, x [mm](b)PR2 (t)PR1 (t)gel

Discussion

In this study, the spatial distribution of displacements
caused by two acoustic radiation force was measured withultrasonic phased tracking method. The applied acoustic
radiation forces and resulting displacement distribution are
discussed in the following. Figure 8(a) shows the soundpressure distribution generated by the ultrasonic transducer
used for actuation. From Fig. 8(a), it was assumed that thewidth at half maximum of the acoustic radiation force isabout 5 mm along 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 over thehalf maximum is roughly considered as shown by the
shadowed region in Fig. 8(b). In Fig. 8(b), the measured
region is surrounded by a dashed line. Two ultrasoundbeams crossed each other at a depth of about 4 mm from the
top surface. At each point in the region within 4 mm of the
top surface in the measured region, the sum of horizontal
components caused by two acoustic radiation pressures,
PR1 ðtÞ and PR2 ðtÞ, is not zero (Magnitudes of thesecomponents are diﬀerent.). Therefore, the region within
4 mm of the top surface was compressed along the horizontal
axis, and, as shown in Fig. 7, the resulting upward displacement during the increase in acoustic radiation forces wasfound.
Around 4 mm from the top surface, horizontal components of two acoustic radiation forces almost compensatedfor each other, and only vertical components remained.

17 mm
45 mm

Fig. 8. (a) Sound ﬁeld generated by the ultrasonic transducer used for
actuation. (b) Consideration of displacement distribution generated by
two acoustic radiation forces.

Therefore, the region around 4 mm from the top surface
moves downward during the increase in acoustic radiation
force because of the downward displacement caused byvertical components of acoustic radiation forces as well asthat caused by the horizontal compression in the region
nearer than 4 mm from the top surface.
As shown in Fig. 7, the measured displacement distribution showed that strain inside the object was successfully
generated. However, at this moment, it is diﬃcult to estimate
the elastic modulus distribution precisely because of the
problems described below.
. Measurement of strain distribution
The stress applied by the acoustic radiation force is notplane stress. Therefore, the region where the acoustic
radiation force is applied is displaced in not only one
direction. However, in this study, only the beam-axis
component of displacement was measured. To obtain4710

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H. HASEGAWA et al.

precise strain and elastic modulus distributions, amethod for measuring three dimensional components
of displacement is needed.
. Estimation of magnitude of applied stress
Equation (2.6) 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 attenuation coeﬃcientof the mixture of muscle and fat (abdominal wall) is
typically 3 dBcm1 MHz1 .34) To estimate the magnitude of the applied acoustic radiation force, theattenuation coeﬃcient of tissue must be considered.
. Estimation of stress distribution
Figure 8(a) shows the sound pressure distribution,
which is directly related to the acoustic radiation force
as shown by eq. (2.6), generated by the ultrasonictransducer used for application of the acoustic radiation
force. As shown in Fig. 8(a), the acoustic radiation
force is a function of the geometry, and the stress
distribution must be considered in addition to the strain
distribution.
Furthermore, the distance between two focal points, beam
angles, and the depth of the region of interest is restricted by
the factors described in the following.
When focal areas of two ultrasonic transducers overlap,
the horizontal components of the two acoustic radiation
forces compensate for each other in the intersectional area.
Therefore, for eﬀective generation of strain, two focal points
are assigned so that sound ﬁelds generated by the twotransducers do not overlap within the object (= region ofinterest). On the other hand, when the distance between two
focal points is very large in comparison 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 the beam angles 1 and 2 similar at
every depth, the distance between two focal points should bechanged in relation to the depth of the region of interest.
This is done using a phased array probe.
As described above, it is currently diﬃcult to estimate the‘‘absolute’’ elastic modulus precisely using the proposed
method. Therefore, the elasticities of diﬀerent individuals
measured by the proposed method are diﬃcult to compare.
However, for example, the temporal change or frequency
dependence of the strain distribution within an individual is
directly related to the change in elastic properties. The
proposed method can detect such a relative change in theelastic property of an object.
6.

Conclusions

In this study, in order to generate regional strain inside an
object, we constructed an experimental setup for theapplication of cyclic remote actuation in an object using
two focused ultrasonic transducers. The phase diﬀerence
between the two acoustic radiation forces, which wereapplied at two diﬀerent positions in the object, wascontrolled to 0 . The resultant displacements inside theobject were successfully measured by the ultrasonic phased
tracking method. These results show the potential of the
proposed method for the generation of regional strain inside
an object.

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