Japanese Journal of Applied Physics
Vol. 47, No. 5, 2008, pp. 4193–4199#2008 The Japan Society of Applied Physics

Ultrasonic Measurement of Strain Distribution Inside Object
Cyclically Compressed by Dual Acoustic Radiation Force
Yoshitaka O DAGIRI, Hideyuki H ASEGAWA, and Hiroshi K ANAI
Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan(Received November 26, 2007; revised February 23, 2008; accepted March 3, 2008; published online May 23, 2008)

One possible way to evaluate acupuncture therapy quantitatively is to measure the change in the elastic property of muscle
after application of the therapy. Many studies have been conducted to measure mechanical properties of tissues using
ultrasound-induced acoustic radiation force. To assess mechanical properties, strain must be generated in an object. However,a single radiation force is not eﬀective because it mainly generates translational motion when the object is much harder than
the surrounding medium. In this study, two cyclic radiation forces are simultaneously applied to a muscle phantom from two
opposite horizontal directions so that the object is cyclically compressed in the horizontal direction. By the horizontal
compression, the object is expanded vertically based on its incompressibility. The resultant vertical displacement is measured
using another ultrasound pulse. Two ultrasonic transducers for actuation were both driven by the sum of two continuous
sinusoidal signals at two slightly diﬀerent frequencies [1 MHz and (1 M + 5) Hz]. The displacement of several micrometersin amplitude, which ﬂuctuated at 5 Hz, was measured by the ultrasonic phased tracking method. Increase in thickness inside
the object was observed just when acoustic radiation forces increased. Such changes in thickness correspond to verticalexpansion due to horizontal compression. [DOI: 10.1143/JJAP.47.4193]
KEYWORDS: acoustic radiation force, phased tracking method, acupuncture therapy, strain

1.

Introduction

Acupuncture was ﬁrst used more than 2,000 years ago andhas been developed based on the results of experience.
However, the mechanism by which acupuncture therapy
works is not clearly understood. Therefore, the quantitativeevaluation of acupuncture therapy is important and themeasurement of changes in elastic properties of muscle due
to treatments is one of the strategies for quantitativeevaluation of the therapy. When acupuncture is eﬀective,the elastic modulus of muscle after acupuncture therapy
decreases compared with that before therapy.
Recently, some remote actuation methods based on
acoustic radiation forces have been reported. Fatemi andGreenleaf proposed an imaging modality that uses theacoustic response of an object to a localized dynamic
radiation force.1,2) The radiation force produces an acoustic
emission which is closely related to the mechanicalfrequency response of the medium. By measuring theacoustic emission with a hydrophone, hard inclusions suchas calciﬁed tissues in soft material were detected experimentally. The spatial resolution in the depth direction
corresponds to the size of the intersectional area of ultrasound beams at two diﬀerent frequencies.
Nightingale et al. proposed an alternative imaging
method in which focused ultrasound is employed to apply
the radiation force to soft tissue for short durations (lessthan 1 ms). The viscoelastic properties of the tissue were
investigated from the magnitude of the transient response,
which was measured as the displacement, dðtÞ, of thetissue.3–5) In order to generate measurable dðtÞ using
several ultrasonic pulses, high-intensity pulsed ultrasound(1,000 W/cm2 ) was employed. According to safety guidelines for the use of ultrasound, the intensity is recommended to be below 240 mW/cm2 for pulsed waves and1 W/cm2 for the continuous waves.6) Therefore, theintensity of the pulsed ultrasound employed by NightingaleE-mail address: hkanai@ecei.tohoku.ac.jp

et al. would be far greater than that indicated in the safety
guidelines.
Acoustic radiation force has possibilities for use in themeasurement of mechanical properties of tissues. However,a single acoustic radiation force does not generate strain in
an object eﬀectively because it primarily produces a change
in the object’s position which has zero spatial gradientin displacement (no strain). Speciﬁcally when the elastic
modulus of the object is far greater than that of thesurrounding medium, a single acoustic radiation force may
generate only a change in the position of the object. To
eﬀectively generate strain in the object, Hasegawa et al.
used two acoustic radiation forces.7) Two sets of cyclicallyoscillating acoustic radiation forces induced by two ultrasounds at frequencies f and f þ  f were simultaneously
applied to two diﬀerent positions in the object from two
diﬀerent directions. In this case, two ultrasound beams
crossed each other at certain depth from the surface asshown in Fig. 1, and the shadowed region that compressedhorizontally corresponded to the generation of strain.
In our previous studies,7,8) ultrasound at 1 MHz and
5 MHz was used for actuation and measurement, respectively. Therefore, irradiation of ultrasound for actuationneeded to be stopped during transmission and reception ofultrasound for measurement because the frequencies ofthese two ultrasounds is insuﬃciently diﬀerent, and it isdiﬃcult to separate them by ﬁltering. In this study,
continuous ultrasound for actuation was used to obtain alarger amplitude of displacement in comparison with thatin previous studies. In order to separate the pulsed
ultrasound for the measurement from the continuous
ultrasound for actuation by ﬁltering, high-frequency ultrasound was employed (center frequency: 16 MHz) for
measurement of displacements. Using this newly developed
experimental system, an object was stably actuated by two
acoustic radiation forces with a larger amplitude, and thechange in thickness (= strain) inside the object was imaged
at a high spatial resolution by the ultrasonic phased
tracking method.9,10)4193

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acoustic radiation
force PR1(t)acoustic radiation
force PR2(t)the region compressedhorizontally

focal point

2.

Y. ODAGIRI et al.

mediumobjectFig. 1. (Color online) Illustration of acoustic radiation forces generated by two ultrasound beams
focused at two points separated by a given distance.

focal point

Principles

When ultrasound propagates in a medium (density: 1 ,sound speed: c1 ), a constant force is generated in the
direction of propagation. This force is called the acoustic
radiation force.11) The acoustic radiation pressure, PR ðtÞ, isdeﬁned as the acoustic radiation force per unit area asfollows:
PR ðtÞ ¼ ð1 þ R2 ÞeðtÞ;ð2:1Þ

where R and eðtÞ are the pressure reﬂection coeﬃcient andthe energy density at the interface between the medium andobject, respectively. In eq. (2.1), the transmitted wave isassumed to be perfectly absorbed by the object. Using thedensities, 1 and 2 , and sound speed, c1 and c2 , of themedium, 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 ¼½pðtÞ2 :ð2:3Þ
1 c21For example, the speed of sound in muscle and fat are 1,568
and 1,465 m/s, respectively. By assuming densities of thesetissue to be 1:0  103 kg/m3 , the reﬂection coeﬃcient R is
0.034. Thus, the square of the reﬂection coeﬃcient, R2 , isassumed to be zero. The energy density, eðtÞ, of the incident
wave is proportional to the square of the sound pressure,pðtÞ, of the ultrasound beam. When two ultrasound waves
with the same sound pressure, p0 , at slightly diﬀerent
frequencies, f and f þ  f , are crossed each other, an
acoustic radiation pressure, PR ðtÞ, which ﬂuctuates at thefrequency diﬀerence,  f , is generated in the intersectional
area. In this case, the sound pressure, psum ðtÞ, in theintersectional area is expressed as follows:
psum ðtÞ ¼ p0 cos 2 ft þ p0 cos 2ð f þ  f Þt;

From the second term on the right-hand side of eq. (2.5),
it is found that the energy density, eðtÞ, has a component at
the frequency diﬀerence  f . With respect to the lowfrequency component, the acoustic radiation pressure, PR ðtÞ,acting on the interface is given byp2
PR ðtÞ ¼ ð1 þ R2 Þ 0 2 ð1 þ cos 2 ftÞ 1 c12p
ð2:6Þ
’ 0 2 ð1 þ cos 2 ftÞ: 1 c1
To improve the spatial resolution in measurements of the
response of an object, an ultrasound correlation-based
method, the ultrasonic phased tracking method, is used tomeasure the minute displacement, dðtÞ, caused by the
acoustic radiation force.8) The accuracy of the displacement
measurement by the phased tracking method was determined
to be 0.2 mm by basic experiments using a rubber plate.12) In
addition, the phased tracking method has already been
applied to measurements of vibrations and viscoelasticity of
the arterial wall13,14) and heart wall.15,16)
3.

The experimental setup is illustrated in Fig. 2. In order tomeasure the displacement distribution at a high spatial
resolution, we employed ultrasonic diagnostic equipment(Dynamic Imaging DIASUS) with a linear probe (center
frequency: 16 MHz). The equipment was modiﬁed to detect
minute displacement in the object by the ultrasonic phased
tracking method. A homogeneous object made of gel(45  45  17 mm3 , static elastic modulus Es : 6.3 kPa,
containing 5% carbon powder by weight to provide

synchronization

signal generator signal generator
f+Δ f
f
f+Δ f
f

ð2:4Þ

and the energy density, eðtÞ, is given by
1½psum ðtÞ2
1 c211¼½p0 cos 2 ft þ p0 cos 2ð f þ  f Þt2
1 c21p20¼
1 þ cos 2 ft þ cos 2ð2 f þ  f Þt
1 c21
1
1
þ cos 4 ft þ cos 4ð f þ  f Þt ;2
2

Experimental MethodseðtÞ ¼

adder

adder

divider

divider

power AMP

power AMP

ultrasonic
probe

17 mm

ultrasonictransducer

water

φ =25°

block

mixer
mixer

ultrasonictransducer

LPF

workstation
diagnostic
ultrasoundsystemφ =25°
objectblock

ð2:5Þ
Fig. 2. Experimental setup for measurement of the displacement of an
object cyclically actuated by ultrasound.

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ultrasonicultrasonic probetransducer

oil
water

φ =25°Y. ODAGIRI et al.

ultrasonictransducer

ultrasonictransducer

φ =25°x0zFig. 3. Measurement of the interface between oil and water actuated byradiation forces.

100

(a)

at x =0 mm

sound pressure [kPa]90

suﬃcient scattering) was placed in a water tank as shown in
Fig. 2. To apply radiation pressures PR1 ðtÞ and PR2 ðtÞ, twoconcave ultrasonic transducers were employed. Each concave ultrasonic transducer (center frequency: 1 MHz, focal
length: 52 mm) was driven by the sum of two continuous
waves (CWs) at two slightly diﬀerent frequencies of 1 MHz
and 1 MHz + 5 Hz. The resultant ultrasound beam was
focused 50 mm away from the surface of the transducer, andthe focal point was set at the top surface of the object at 25
in beam angle from the horizontal direction.
Ultrasound signals for actuation (around 1 MHz) received
by the linear-array probe were ﬁltered to estimate thedisplacement distribution using only the pulsed ultrasoundfor measurement (center frequency: 16 MHz).

60

(b)

50
40
30
10
35

100
90
80

Results

Displacement measurement at the boundary of oil andwater
As illustrated in Fig. 3, the interface between oil andwater was measured with ultrasound because the interface ofliquids is easily actuated in large amplitude, and the actuatedregion can be easily identiﬁed. In this experiment, beam
angles of two ultrasound waves for actuation were set to 25 ,and the distance between two focal points was set to 16 mm.
Figure 4 shows the sound ﬁeld of the ultrasonic transducer
used for actuation. Figures 4(b) and 4(c) show the measured
sound pressure distribution in the depth and radial directions,
respectively. From these results, the width, zp , at half
maximum of the sound pressure is determined to be about
2.5 mm along the x-axis (z ¼ 52 mm).
Figure 5 shows the displacement of each beam formeasurement during a cycle of acoustic radiation forces.
The time of the measured displacement relative to acoustic
radiation forces is shown in Fig. 5(b). In Fig. 5(a), measured
displacements are superimposed on the B-mode images
using a color code shown in the bottom of Fig. 5. Blue andred correspond to downward and upward displacement of theinterface, respectively. In Fig. 5, the downward displacement around focal points of ultrasound for actuation during
the increase in acoustic radiation forces is clearly visualized,
and the regions through which acoustic radiation forces passcan be easily recognized. The regions showing downward
displacement during the increase of applied acoustic
radiation forces are wider than zp of the sound pressure
distribution shown in Fig. 4(c) because ultrasound beams for
actuation were insoniﬁed at an angle of 25 between theinterface and beam, and the beam diameter on the interface
was wider than zp .

70

20

sound pressure [kPa]

4.

80

4.1

40

45

50
55
60
distance, z [mm]6570

at z =52 mm

70
60

(c)

50
40
30
20
10
-2.5 -2- -1.5 -1 -0.5 0 0.5 1
distance, x [mm]

1.5

2

2.5

Fig. 4. (a) Measurement axis of sound pressure of an ultrasonic transducer
used for actuation. (b) Sound pressure distribution in the depth direction
at x ¼ 0 mm. (c) Sound pressure distribution in the radial direction atz ¼ 52 mm.

4.2 Displacement measurement inside the object
When two acoustic radiation forces were applied to two
diﬀerent positions in the object from two diﬀerent directions, acoustic radiation forces overlap inside an object madeof gel. The spatial distribution of displacements inside theobject was measured with ultrasound. In this experiment, thedistance between two focal points was set to 0 and 8 mm.
Figure 6(a) shows an M-mode image of the object along thez-axis at the center (x ¼ 0) of two focal points. In Fig. 6, thedistance between two focal points was set to 0 mm. 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 the object were
measured as 1:1  103 kg/m3 and 1:47  103 m/s, respectively, and the pressure reﬂection coeﬃcient, R, and theenergy reﬂection coeﬃcient, R2 , were calculated as 0.038
and 0.0014 ( 0), respectively, using eq. (2.2). In this study,
by assuming the object to be a totally absorbing material, theacoustic radiation pressures, PR1 ðtÞ and PR2 ðtÞ, were calculated based on eq. (2.6). By measuring the acoustic ﬁeld of

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0

(a)

x

acoustic radiationpressure [Pa]
0
10
0

z

1 mm

16 mm

0

(b)

0.1

0.2
time [s]
-4.0

-0.5

upward0.5

4.0 [μ m]

downward

Fig. 5. (Color online) (a) Displacement distribution at the interface between oil and water successfully obtained every 50 ms during
one cycle of the actuation from the top panel to the bottom panel. (b) Acoustic radiation force during the same period.

0depth 3 mm

1.5 mm

z

(a)
30

Pa

(b)
0
30

(c)

Pa
0
-0.1
mm/s 0
0.1
-1
μm 0
1

(d)
(e)
0

0.5
time [s]1.0

Fig. 6. (Color online) (a) M-mode image of an object. (b) Acoustic
radiation pressure, PR1 ðtÞ. (c) Acoustic radiation pressure, PR2 ðtÞ. (d)Vibration velocities at 0 and 1.5 mm from the surface. (e) Displacements
at 0 and 1.5 mm from the surface.

the ultrasound for actuation with a hydrophone (ForceInstitute MH-28-10), the amplitude of the focused soundpressure of p0 was obtained.
The vibration velocities, fvz ðtÞg, at beam position x ¼ 0
and depths z ¼ 0 mm (surface) and 1.5 mm within the object
actuated by the acoustic radiation force were measured by
the ultrasonic phased tracking method, as shown in
Fig. 6(d). By integrating velocities fvz ðtÞg, the displacements, fdz ðtÞg, at the depths z ¼ 0 mm and 1.5 mm wereobtained as shown in Fig. 6(e). It was found that the object

was cyclically actuated at the frequency diﬀerence f ¼ 5 Hz at an amplitude of 1.5 mm.
By applying the ultrasonic phased tracking method atevery sampled points from the surface to 3 mm deep within
the object, the spatial distributions of displacements,
fdx;z ðtÞg, were measured. Moreover, changes in thickness(strain), fhx;z ðtÞg, were obtained as the spatial gradient of
displacements calculated by applying the least-squares
method to the displacement distribution in every region
with a width of 1.5 mm in the depth direction. Spatiallyrandom variation of displacements was large, and it isdiﬃcult to obtain the change in thickness from the diﬀerence
between the displacements of two points.
Figures 7 and 8 show the spatial distributions of displacements, fdx;z ðtÞg, and changes in thickness, fhx;z ðtÞg, during acycle of acoustic radiation forces at distances between two
focal points of 0 mm and 8 mm, respectively. The dashedlines in Figs. 7(c) and 8(c) show the time when each
distribution is measured. In Figs. 7(a) and 8(a), the measured
displacements, fdx;z ðtÞg, are superimposed on the B-mode
images using a color code shown at the bottom of Figs. 7(a)and 8(a). Blue and red correspond to the upward anddownward displacements, respectively. In Figs. 7(b) and
8(b), the measured changes in thickness, fhx;z ðtÞg, aresuperimposed on the B-mode images using a color code
shown in the bottom of Figs. 7(b) and 8(b). Blue indicates athinning region and red indicates a thickening one.
In Fig. 7(a), the downward displacements were generated
at the surface of the object during the period when the
acoustic radiation forces increased. The region around thesurface of the object was displaced downward slightly morethan a deep region because two acoustic radiation forcesintersected at the surface. Therefore, in Fig. 7(b), the region
around the surface became somewhat thinner when theapplied acoustic radiation forces increased because theobject was not compressed horizontally.
In Fig. 8(a), in the region (around a depth of 1.8 mm)
where two acoustic radiation forces crossed, amplitudes of4196

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Y. ODAGIRI et al.

(a)

(b)
0

z

(c)
acoustic radiationpressure [Pa]
0
30
0

x

3 mm

0

0

x

5 mm

0.1

0.2
time [s]-1.00upward-3

1.0 [μ m]-10 -5.0 5.0
10 [×10 %]decrease in increase inthicknessthickness

downward

Fig. 7. (Color online) Experimental results from the phantom at a distance between two focal points of 0 mm. (a) Distribution of
displacements, fdx;z ðtÞg, inside the object. (b) Distribution of strain (changes in thickness), fhx;z ðtÞg, inside the object. (c) Acoustic
radiation force during one cycle.

(a)

(b)
0

z

x

3 mm

5 mm
0

0

x

(c)
acoustic radiationpressure [Pa]
0
30
0

0.1

0.2
time [s]-1.0

upward0-31.0 [μ m] -10-5.0 5.0
10 [×10 %]decrease in increase inthicknessthickness

downward

Fig. 8. (Color online) Experimental results from the phantom at a distance between two focal points of 8 mm. (a) Distribution of
displacements, fdx;z ðtÞg, inside the object. (b) Distribution of strain (changes in thickness), fhx;z ðtÞg, inside the object. (c) Acoustic
radiation force during one cycle.

displacements were larger than those in the other region.
Although the region around the surface of the object wascompressed in the horizontal direction by the horizontal components of the applied acoustic radiation forces, the region
was displaced upward or downward unidirectionally because

the applied radiation forces have vertical components. The
horizontal compression can be revealed by estimating the
spatial distribution of the strain (change in thickness),fhx;z ðtÞg, as shown in Fig. 8(b). In Fig. 8(b), the region
around the midpoint between two focal points thickened

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(a)

(b)

(d)

(c)
0

5 mm

x

0

x

z

5 mm

0

acoustic radiationpressure [Pa]
6
0
0

0.1

0.2
time [s]-2.0

upward02.0 [μ m]

downward-3-10 -5.0 5.0 10 [×10 %]decrease in increase inthicknessthickness

Fig. 9. (Color online) (a) B-mode image of porcine liver. (b) Distribution of displacements, fdx;z ðtÞg, inside the porcine liver.
(c) Distribution of strain (changes in thickness), fhx;z ðtÞg, inside the porcine liver. (d) Acoustic radiation force during one cycle.

when the applied acoustic radiation forces increased. This
result suggests that the object is actually compressed in thehorizontal direction and is expanded in the vertical direction
due to its incompressibility. From these results, it was foundthat strain inside the object was successfully generated
speciﬁcally around the midpoint between two focal points.
4.3 In vitro experiments using porcine liver
An in vitro experiment using porcine liver, which is softer
than silicone gel, was conducted. In Figs. 7 and 8, thedistance between two focal points is set at 8 mm. In thisexperiment, acoustic radiation forces were applied at a low
intensity of 6 Pa compared with the basic experiments using
silicone gel (about 1/5 of the intensity). This intensity
complies with the safety standards for ultrasound.
Figure 9(a) shows a B-mode image of the porcine liver.
There was an artery in the region which is surrounded by the
red rectangle in Fig. 9(a). Figures 9(b) and 9(c) show thespatial distributions of displacements, fdx;z ðtÞg, and strains(changes in thickness), fhx;z ðtÞg, along each beam during
a cycle of acoustic radiation forces shown by color codesindicated in the bottom of these ﬁgures. The dashed lines inFig. 9(d) show the time when each distribution is displayed.
In Fig. 9(b), in spite of the fact that acoustic radiation

forces were applied at a low intensity, amplitude ofdisplacements were larger than those of the silicone gel. In
addition, as shown in Fig. 10, it was observed that a shear
wave propagated to the outside from the region where
acoustic radiation forces were radiated. The measured regionin Fig. 10 was moved 10 mm in the horizontal direction incomparison with that in Fig. 9. The propagation of the shear
wave was visualized because the liver is much softer than
the gel.
In Fig. 10, the propagation velocity of the shear wave, cs ,is calculated as 1.3 m/s. By assuming the density of liver, ,
to be 1:0  103 kg/m3 and the liver to be incompressible,
shear modulus, G, is calculated by
G ¼ c2s :ð4:1Þ

The shear modulus, G, was determined to be 1.7 kPa, which
is similar to that reported in literature.17) This result showsthat it is possible to evaluate elasticity from the propagation
speed of a shear wave.
In Fig. 9(c), large changes in thickness were generated in
the region over the blood vessel. The region became thinwhen overall downward displacements were caused byacoustic radiation forces because the blood vessel was hardcompared with liver tissue.

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(a)

5 mm

water
object
8 ms
(b)

water
object

1.3 m/sFig. 10. (Color online) Propagation of shear wave in the porcine liver.

5.

Discussion

In this study, the spatial distributions of displacements andchanges in thickness caused by two acoustic radiation forceswere measured with the ultrasonic phased tracking method.
In this section, the applied acoustic radiation forces andresulting changes in thickness distribution are discussed. The
region where the acoustic radiation force was over the half
maximum is roughly illustrated in Fig. 11(a). Figure 11(a)shows the result when the focal points of two ultrasoundbeams for actuation were set at the same position. In thiscase, the resultant force from two acoustic radiation forces
has almost only a downward component at the vicinity of thesurface, and the resulting downward displacement during theincrease of acoustic radiation forces is shown in theexperimental results in Fig. 7. Figure 11(b) illustrates theacoustic radiation force when the distance between focal
points at the surface was set to 8 mm. Two ultrasound beams
crossed each other at a depth of 1.8 mm from the surface. In
this case, the region between two ultrasound beams for
actuation was compressed along the horizontal axis, and theregion which vertically thickens during the increases of
acoustic radiation forces is shown by the basic experimentusing a phantom.
As shown in Fig. 8, in a homogeneous object, strain inside
the object was successfully generated by applying two
acoustic radiation forces from two opposite horizontal
directions. Therefore, the elastic properties of a homogeneous object, such as large muscle, can be assessed from themagnitude of the deformation generated. However, in an
inhomogeneous object, for example, a blood vessel in livertissue as shown in Fig. 9, the strain is not generated in thesame manner as in the basic experiment because of thecomplex deformation generated. The displacement distribution generated also depends on the elasticity and elastic
inhomogeneity of an object, and such dependence should befurther investigated in future work.
6.

Conclusions

In this study, the spatial distributions of displacements andchanges in thickness inside an object were generated by two
acoustic radiation forces and they were measured by theultrasonic phased tracking method. It was shown that strain
could be generated using two acoustic radiation forces.
These results show the potential of the proposed method forthe region compressedhorizontally
Fig. 11. (Color online) Illustrations of (a) resultant radiation forces when
focal points were set at the same position and (b) resultant radiation
forces when the distance of focal points between ultrasound beams for
actuation was set to 8 mm.

evaluation of the change in elastic spatial properties ofmuscle. Moreover, this method could be useful for themeasurement of elastic properties of tissues, for example,
liver and breast tissue, which require an external force to be
deformed.

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