Jpn. J. Appl. Phys. Vol. 42 (2003) pp. 4608–4612
Part 1, No. 7A, July 2003
#2003 The Japan Society of Applied Physics

Ultrasonic Measurement of Minute Displacement of Object Cyclically Actuatedby Acoustic Radiation Force
Kazuaki M ICHISHITA, Hideyuki H ASEGAWA and Hiroshi K ANAI
Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan(Received November 18, 2002; accepted for publication March 12, 2003)

Recently, several researchers have investigated the potential of the use of acoustic radiation force for imaging the mechanicalproperties of tissue. However, these methods have some problems as to safety and the spatial resolution. Therefore, analternative ultrasonic remote actuation and measurement method is proposed in this paper, in which the intensity of theapplied continuous ultrasonic wave is suppressed to be lower than the safety guideline (1 W/cm2 ) recommended by the Japan
Society of Ultrasonics in Medicine (JSUM). A minute displacement with amplitude of less than 1 mm is cyclically generated
by the radiation force with a low frequency f of several hertz. For simultaneous measurement of the minute displacement,
cyclic radiation force is applied intermittently by maintaining its envelope waveform of the low-frequency component of
f Hz. At the same time, an ultrasonic correlation-based method, namely, the phased tracking method, is employed tomeasure the minute displacement. In basic experiments, the minute displacement of several micrometers was generated insilicone rubber by applying radiation force, and it was successfully measured by the phased tracking method.
[DOI: 10.1143/JJAP.42.4608]
KEYWORDS: acoustic radiation force, noninvasive measurement, measurement of minute displacement, viscoelasticity
measurement, tissue characterization

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 diﬀerent media is afunction of the energy density, eðtÞ, and the speciﬁc acoustic
impedances, Z1 and Z2 , of the medium.3) The energy density,eðtÞ, is proportional to the square of the sum of the soundpressures, p1 ðtÞ and p2 ðtÞ, generated by the two transducers.
In the intersectional area of the two beams, therefore, anoscillatory radiation pressure PR ðtÞ with the frequency
diﬀerence, f , is applied to the interface. The radiation
force produces acoustic emission which is closely related tothe mechanical frequency response of the medium. By
measuring the acoustic emission with a hydrophone, hardinclusions in a soft material were experimentally detected.
However, the spatial resolution in the depth direction
corresponds to the size of the intersectional area.
Nightingale et al. proposed an alternative imaging method
in which the focused ultrasound is employed for applying
the radiation force to a soft tissue during short durations (lessthan 1 ms). The viscoelastic properties of the tissue areinvestigated from the magnitude and the transient responseof the displacement, dðtÞ, of the tissue.4) In order to generatemeasurable displacement, dðtÞ, using several ultrasonicpulses, high-intensity pulsed ultrasound of 1,000 W/cm2 isemployed. According to the safety guideline shown by
JSUM, however, the intensity of ultrasound is recommended
to be less than 240 mW/cm2 and 1 W/cm2 , for the pulsed
wave and the continuous wave, respectively.5) Therefore, theintensity of the pulsed ultrasound employed by Nightingale
et al. is far greater than that indicated in the safety guideline.
E-mail address: hkanai@ecei.tohoku.ac.jp

In this study, in order to apply ultrasound within the rangeof the safety guideline and to detect the minute displacement, dðtÞ, generated by the resultant acoustic radiation
force, the two continuous-wave (CW) ultrasound beams of fand f þ f are employed to cyclically actuate the acoustic
radiation pressure at low frequency (f ) to increase thesignal-to-noise ratio. For the measurement of the resultant
minute displacements, an ultrasound correlation-based method, the phased tracking method,6) is employed. By sweeping
the low frequency f , the frequency characteristics of theobject are determined. Moreover, from the amplitude
characteristics of the displacement, dðtÞ, continuouslygenerated by the acoustic radiation pressure, PR ðtÞ, theviscoelasticity of the object can be determined. We
constructed an experimental setup and conﬁrmed themeasurement of the minute cyclic displacement by thephased tracking method using several micrometers.
2.

Principle

When continuous plane-wave ultrasound is incident on aninterface between two diﬀerent media, as illustrated in Fig. 1,
a constant force, which is called an acoustic radiation force,is exerted on the interface. Acoustic radiation pressure,PR ðtÞ, is deﬁned as the radiation force per unit area of theobject, as follows:PR ðtÞ ¼ ð1 þ R2 ÞheðtÞi;ð2:1Þ

where R and heðtÞi are the reﬂection coeﬃcient of soundpressure and the temporally averaged energy density at theFig. 1. The acoustic pressure pðtÞ of the incident wave exerts the acoustic
radiation pressure, PR ðtÞ, on an object.
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Jpn. J. Appl. Phys. Vol. 42 (2003) Pt. 1, No. 7A

K. MICHISHITA et al.

interface, respectively. In eq. (2.1), the incident wave
through the interface is assumed to be perfectly absorbed
into the object. Using the density, 1 and 2 , and soundspeed, c1 and c2 , of each medium, the reﬂection coeﬃcient,R, and temporally averaged energy density, heðtÞi, aredeﬁned byZ2  Z1
R¼Z2 þ Z1
ð2:2Þ
2 c2  1 c1¼;
2 c2 þ 1 c1
1 1
heðtÞi ¼
1 c21 T

ZTfpðtÞg2 dt;

ð2:3Þ

0

where pðtÞ is the sound pressure at the interface and T is theaveraging period. JSUM shows the safety guideline, inwhich the intensity of the CW ultrasound should be less than
1 W/cm2 . By assuming that density 1 and sound speed c1 ofwater are 103 kg/m3 and 1,500 m/s, respectively, the acoustic
radiation pressure, PR ðtÞ, exerted on the interface of thetotally absorbing object is calculated as 6.67 Pa when theultrasound intensity is 1 W/cm2 .
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 interface is expressed by thesum of sound pressure of two ultrasonic beams as follows:
psum ðtÞ ¼ p0 cos !t þ p0 cosð! þ !Þt;ð2:4Þ

where p0 , !, and !, are the amplitude of the soundpressure pðtÞ of each ultrasound beam, angular frequency of
the incident wave (! ¼ 2f ), and diﬀerence in angularfrequency (! ¼ 2f ), respectively. For this case, theenergy density eðtÞ is expressed by21 eðtÞ ¼
psum ðtÞ2
1 c1
21 p0 cos !t þ p0 cosð! þ !Þt
2
1 c1

p20
¼
1 þ cos !t þ cosð2! þ !Þt1 c21
11
þ cos 2!t þ cos 2ð! þ !Þt :2
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displacements dðtÞ generated by the radiation force, theviscoelasticity of the object can be characterized.
3.

Experimental Setup

An experimental setup is shown in Fig. 2. In order tomeasure the displacement, dðtÞ, we employed ultrasonicdiagnostic equipment (Toshiba SSH-160A) with a sector
array probe (center frequency: 5 MHz). The equipment was
modiﬁed to detect the minute displacement dðtÞ of an object
by the phased tracking method. The object(45 40 10 mm silicone rubber; static elastic modulus
ES was measured as 20 kPa) was placed at the bottom of awater tank, and the ultrasonic probe was ﬁxed 5 cm awayfrom the object. For the application of radiation pressure,PR ðtÞ, an ultrasonic transducer (Tokimec 5Z10I, centerfrequency: 5 MHz) was driven by a sum of two CWs with
two slightly diﬀerent frequencies (5 MHz and 5 MHz + f )generated by a function generator (Sony Tektronix
AFG2020).
When CW ultrasound is employed for actuation, interference occurs between the CW ultrasound for actuation andthe pulsed ultrasound for displacement measurement. In
order to avoid this interference, it is necessary to stop
actuation during transmission and reception of this ultrasonicpulse for measurement. Therefore, we made an electricalswitch to control the cessation of the CW ultrasound foractuation. Figure 3 shows a time chart for actuation andmeasurement. Only for actuation period (A) when thecontrol signal in Fig. 3(b) was turned on, did the electricalswitch pass the signals for actuation. Then, the signal wasampliﬁed and the ultrasound for actuation was transmitted
from the transducer during period (A). It is necessary to set
the length of period (A) so that the successively transmitted

¼

ð2:5Þ

From the second term of the right-hand side of eq. (2.5), it isfound that the energy density eðtÞ of the incident ﬁeld has acomponent with the frequency diﬀerence f . Therefore, thecyclically oscillatory radiation pressure, PR ðtÞ, at thefrequency diﬀerence f of eq. (2.1) is given byPR ðtÞ ¼ ð1 þ R2 Þp20cos !t:1 c21

ð2:6Þ

Thus, using the two crossed ultrasound beams at the slightly
diﬀerent frequencies, f and f þ f , the oscillatory radiation force can be cyclically applied to the intersectional areaof the beams. The amplitude and phase characteristics of thedisplacement, dðtÞ, which is generated as a response to theradiation force, relate to the viscoelasticity of the target. By
sweeping the frequency diﬀerence f and detecting theFig. 2. Experimental setup for measurement of the displacement of anobject cyclically actuated by ultrasound.

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Jpn. J. Appl. Phys. Vol. 42 (2003) Pt. 1, No. 7A

Fig. 3. Time chart for actuation and measurement. (a) Timing clock fromthe diagnostic scanner. (b) Period (A) of actuation controlled by theelectrical switch. (c) The sound pressure, psum ðtÞ, consisting of the sum oftwo ultrasound beams with slightly diﬀerent frequencies, f and f þ f ,applied intermittently only during actuation period (A). (d) The acoustic
radiation pressure, PR ðtÞ, generated by the sound pressure in Fig. 3(c). (e)Timing of the transmission of the ultrasonic pulse for measurement fromthe probe. (f) The sampling gate of the RF echo data in probe period (P).

measurement pulse is prevented from interfering withreverberations of the ultrasound for actuation. Therefore,
as shown in Fig. 3(c), the ultrasound for actuation wastransmitted intermittently, and then the radiation pressure
PR ðtÞ became a series of pulses whose envelope consisted of
the sinusoidal wave with the frequency diﬀerence f , asshown in Fig. 3(d).
At the rising edge of the trigger pulse in Fig. 3(e), on theother hand, an ultrasonic pulse was transmitted from theultrasonic probe in Fig. 2. In order to track the displacementdðtÞ of the object, the quadrature-demodulated signals of theRF echo data were A/D converted during the gates shown in
Fig. 3(f). In order to avoid interference between theultrasound for actuation and that for measurement, the data
which were stored in the periods other than probe period (P)were not used for the measurement of the displacement dðtÞof the object.
4.

K. MICHISHITA et al.

Fig. 4. The results of the experiment with the frequency diﬀerencef ¼ 7 Hz. (a) M-mode image of the object. The black line at the depthof x ¼ 7:725 mm shows the tracked displacement. (b) The estimatedacoustic radiation pressure, PR ðtÞ, given by eq. (2.6). (c) The vibration
velocity of the object, vA ðtÞ, obtained by the phased tracking method.
(d) The displacement of the object, dðtÞ.

By measuring the acoustic ﬁeld of the ultrasound foractuation with a hydrophone (Force Institute, MH-28-10),
the amplitude of the spatially averaged sound pressure, p0 ,was obtained beforehand and plotted as a function of thevoltage applied to the transducer, as shown in Fig. 5. In thisstudy, the amplitude of the applied voltage, V0 , was 32 V.
Therefore, by neglecting the attenuation of the medium
surrounding the object, the component of the sound pressure,p0 , which is perpendicular to the interface of the object, wascalculated as 61 kPa.
The absorption coeﬃcient of the object was also measured
beforehand. Figure 6(a) shows the measurement systemused. The ultrasonic transducer was driven by a sum of twoCWs with slightly diﬀerent frequencies (5 MHz and 5 MHz+ 10 Hz, the amplitude of the applied voltage, V0 , for each
CW: 30 V). The sound pressure, p0 , of the ultrasound wasmeasured with the hydrophone which was ﬁxed at 5 cm
away from the transducer. As shown in Fig. 6(b), in the caseof interposing the object between the transducer and thehydrophone, the measured sound pressure, p1 ðtÞ, wasExperimental Results

Figure 4(a) shows the M-mode image of the object. InFig. 4, the diﬀerence frequency f ¼ 7 Hz was employed.
Acoustic radiation pressure, PR ðtÞ, shown in Fig. 4(b) wascalculated based on eq. (2.6) as follows: The density, 2 , andthe sound speed, c2 , of the object were measured as
0:98 103 kg/m3 and 1:0 103 m/s, respectively, in a
separate experiment. By assuming the density, 1 , and thesound speed, c1 , of water to be 1:0 103 kg/m3 and1:5 103 m/s, respectively, the pressure reﬂection coeﬃcient, R, and the energy reﬂection coeﬃcient, R2 , wererespectively calculated as 0:21 and 0.044 using eq. (2.2).

Fig. 5. Spatially averaged sound pressure p0 transmitted from thetransducer as a function of the voltage applied to the transducer.

Jpn. J. Appl. Phys. Vol. 42 (2003) Pt. 1, No. 7A

Fig. 6. (a) The experimental setup for measurement of the absorption
coeﬃcient of the object. (b) The sound pressure, p1 ðtÞ, was measured
using a hydrophone with the object interposed between the transducer andthe hydrophone. The sound pressure, p2 ðtÞ, was measured without theobject. (c) Ultrasonic intensities, I1 ðtÞ and I2 ðtÞ, calculated from p1 ðtÞ andp2 ðtÞ, respectively.

smaller than that, p2 ðtÞ, without the object due to attenuationby the object. Figure 6(c) shows the ultrasonic intensities,
I1 ðtÞ and I2 ðtÞ, calculated from sound pressures, p1 ðtÞ andp2 ðtÞ, respectively. Temporally averaged intensities of I1 ðtÞand I2 ðtÞ were calculated as 0.23 W/cm2 and 8:8 103 W/
cm2 , respectively. The absorption coeﬃcient, , is deﬁned
by the ratio of the transparent intensity, I2 ðtÞ, to the incident
intensity, I1 ðtÞ. Thus, the absorption coeﬃcient, , wascalculated as 0.038. Since the acoustic intensity wasattenuated by about 96% by the object, the object can beassumed to be a totally absorbing material.
From these results, by assuming the object to be a totallyabsorbing material (R ¼ 0), the acoustic radiation pressure,PR ðtÞ, exerted on the interface of the object was calculated as
1.7 Pa, using eq. (2.6) as shown in Fig. 4(b).
The vibration velocity, vðtÞ, of the object actuated by theacoustic radiation force was measured by the phased
tracking method, as shown in Fig. 4(c). By integrating thevelocity vðtÞ, the displacement of the object, dðtÞ, wasK. MICHISHITA et al.

4611

Fig. 7. For each frequency diﬀerence f , three experimental results areshown: the estimated acoustic radiation pressure PR ðtÞ given by eq. (2.6),
the vibration velocity vðtÞ of the object measured by the phased tracking
method, and the displacement dðtÞ of the object. (a) f ¼ 7 Hz.
(b) f ¼ 9 Hz. (c) f ¼ 11 Hz. (d) f ¼ 13 Hz.

obtained as shown in Fig. 4(d). In Fig. 4(d), it is seen that theobject was cyclically actuated at the frequency diﬀerence,f , of 7 Hz with an amplitude of the order of a few
micrometers.
For the frequency diﬀerence, f , swept from 7 Hz to13 Hz, the results are shown in Fig. 7. For each f , analmost cyclic component with f can be found in themeasured displacement.
By assuming the Voigt model to be a viscoelastic modelof the object, the absolute value of the complex elastic
modulus, jE j, of the object is given by
PRmax
jE j ¼ jES þ j!j ¼
;
dmax
ð4:1Þ
d0
where PRmax and dmax are the amplitude of the radiationpressure PR ðtÞ and that of the change in thickness from thethe initial thickness, d0 , of the object (d0 ¼ 10 mm),
respectively. ES and  are the static elastic modulus andthe viscosity coeﬃcient, respectively.
In Fig. 8, the absolute value of the complex elasticmodulus, jE j, of the object is plotted as a function of thefrequency diﬀerence, f . Circles and diamonds show theresults measured by the proposed method and using the laser

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Jpn. J. Appl. Phys. Vol. 42 (2003) Pt. 1, No. 7A

Fig. 8. The absolute value of the complex elastic modulus, jE j, of theobject as a function of the frequency diﬀerence f , obtained by eq. (4.1).
Circles and diamonds show the results measured by the proposed method
and using the laser Doppler velocimeter, respectively.

Doppler velocimeter, respectively. In eq. (4.1), the absolutevalue of the complex elastic modulus, jE j, gradually
increases with the frequency diﬀerence, f , which reﬂects
the viscoelastic properties of the object.
5.

Conclusion

In this study, we constructed an experimental setup for
cyclic remote actuation of the object and showed theK. MICHISHITA et al.

measurement results of minute displacements, dðtÞ, in the
micrometer order actuated by the acoustic radiation pressure,PR ðtÞ, by using the phased tracking method. A single
transducer was driven by the sum of the two continuouswaves having slightly diﬀerent frequencies, f and f þ f .
The duration of the ultrasound for remote actuation wascontrolled to prevent ultrasonic displacement measurementfrom the interference. The frequency characteristics of theabsolute value of the complex elastic modulus jE j calculated from the resultant displacement, dðtÞ, generated by theradiation pressure, PR ðtÞ, showed the viscoelastic propertiesof the object. From these results, it was shown that theproposed method has the potential to accurately measure theviscoelasticity of a regional area of the object.
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