J Med Ultrasonics (2006) 33:143–151
DOI 10.1007/s10396-005-0095-2© The Japan Society of Ultrasonics in Medicine 2006

ORIGINAL ARTICLE

Hideyuki HASEGAWA · Hiroshi KANAI

Measurement of nonlinear property of artery wall using remote
cyclic actuation

Received: October 31, 2005 / Accepted: December 26, 2005

Abstract
Purpose. For tissue characterization of the arterial wall, wedeveloped a “phased tracking” method to measure thestrain (change in wall thickness) and elasticity of the arterial wall. To improve the accuracy of tissue characterization,
we are now attempting to measure other mechanical properties in addition to elasticity.
Methods. In this study, the change in elasticity during the
cardiac diastole was measured with ultrasound by generating a change in internal pressure using remote cyclic
actuation.
Results. From the measured change in elasticity during
cardiac diastole, the nonlinear property in the stress–strain
relationship of the artery wall was estimated. In basic experiments using a silicone rubber tube and in vivo experiments
in human carotid arteries.
Conclusion. The proposal method enables the noninvasive
measurement of the nonlinear mechanical property in addition to the elasticity of the arterial wall.
Keywords remote actuation · internal pressure · wall thickness · nonlinear property in stress–strain relationship ·atherosclerosis

Introduction
The steady increase in the number of patients with myocardial infarction or cerebral infarction, both of which aremainly caused by atherosclerosis, is becoming a serious
problem. Therefore, it is important to diagnose atherosclerosis in the early stages of the disease. Computed tomography (CT) and magnetic resonance imaging (MRI) are
employed for the diagnosis of atherosclerosis. Although
H. Hasegawa (*) · H. Kanai
Department of Electronic Engineering, Graduate School ofEngineering, Tohoku University, 6-6-5 Aoba, Aoba-ku, Sendai
980-8579, Japan
Tel. +81-22-795-7079; Fax +81-22-263-9230
e-mail: hasegawa@us.ecei.tohoku.ac.jp

they subject patients to physical and mental hardship, they
provide information mainly on the shape of the artery, suchas the diameter of the lumen. However, the diameter of thelumen is not altered during early-stage atherosclerosis.1
Since there are significant differences between the elastic
moduli of the normal arterial wall and those of arterial
walls affected by atherosclerosis,2,3 evaluation of the elasticity of the arterial wall is useful for diagnosis of atherosclerosis.4 In addition to diagnosis of early-stage
atherosclerosis, it is also important to diagnose vulnerability to atherosclerotic plaque, because the rupture of plaques
may cause acute myocardial infarction and cerebral infarction.5–7 Evaluation of mechanical properties, such as theelasticity of the atherosclerotic plaque, would be useful forthese purposes.
To obtain the circumferential distensibility of the arterial wall in the plane perpendicular to the axial direction of
the artery, methods for measurement of the change in arterydiameter have been proposed.8–12 By assuming the artery to
be a cylindrical shell, the average elasticity of the entire circumference in the plane has been evaluated.13–15 However,
the regional elasticity of atherosclerotic plaques cannot be
obtained by these methods because an artery with atherosclerotic plaque cannot be assumed to be a cylindrical shellwith uniform wall thickness.
To characterize tissues in atherosclerotic plaques, amethod for measurement of the regional elastic modulus of
the arterial wall in patients with or without atherosclerotic
plaque is needed. Such a technique for measurement of thespatial distribution of the regional elasticity would be usefulfor diagnosis of the vulnerability of the atherosclerotic
plaque as well as for the diagnosis of early-stage atherosclerosis. For this purpose, we developed the “phased tracking” method, for measurement of the small change in the
thickness of the arterial wall (less than 100 µm) due to theheartbeat.16–26 From basic experiments, the accuracy of measurement of the change in thickness has been found to beless than 1 µm using the phased tracking method.17,20,21 Fromthe change in thickness measured by using this method, theregional strain and the elasticity of the arterial wall can be
noninvasively evaluated.23,26

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On the basis of the measured elasticity, it should be possible to discriminate tissues in atherosclerotic plaques, suchas fibrous tissue and lipids, because of the large differencesin their elasticity.26 However, it is difficult to discriminate
some tissues, such as lipids and blood clots, because of thesmall differences in their elasticity.27 Therefore, we areattempting to measure other mechanical properties.
In the literature, it has been reported that the nonlinear
property in the stress–strain relationship of the artery wall
is different in normal and atherosclerotic artery walls.28,29
Thus, such a nonlinear property has the potential to
improve tissue characterization based on the mechanicalproperties of the artery wall.
There have been several studies on measurement of thenonlinear elasticity of soft tissue. Catheline et al. investigated the measurement of stress-dependent change in the
propagation speed of shear waves generated by ultrasoundinduced acoustic radiation force.30 Erkamp et al. and Nitta
and Shiina investigated measurement of the nonlinear relationship between strain and applied stress.31,32
In the present study, a method for assessing the nonlinear property in the stress–strain relationship of the arterywall was investigated. For this purpose, external cyclic actuation was applied from the skin surface on the brachialartery.33,34 The resultant change in internal pressure propagates along the artery, causing changes in the diameter andwall thickness at the carotid artery. By measuring thechange in wall thickness of the carotid artery with transcutaneous ultrasound and the change in blood pressure at theradial artery with an applanation tonometer, the change inelasticity during the cardiac diastole and the nonlinear
property were obtained noninvasively.

where D and T are the spacing of sampled points inthe depth direction and the pulse repetition interval,
respectively, and * represents the complex conjugate. In
measurement of the change in thickness, M + 1 in Eq. 1 isset at 5 (= 0.4 µs) in consideration of the pulse length of
0.46 µs.
From the estimated phase shift, ∆ϕ̂(t), velocity, v(t), of theobject is obtained as follows:vˆ (t ) = -c0 Djˆ (t ),2w 0 T(2)

where ω0 and c0 are the center angular frequency of theultrasonic pulse and the speed of sound, respectively.
In estimation of the phase shift by Eq. 1, the object position is tracked by integrating the average velocity, v(t),during the pulse repetition interval, T, as follows:xˆ (t + T ) = xˆ (t ) + vˆ (t ) ¥ T= xˆ (t ) -c0
Djˆ (t ).
2w 0

(3)

From displacements, xA(t) and xB(t), of two points, which
are set in the arterial wall along an ultrasonic beam, thesmall change in thickness, ∆h(t), between these two points
is obtained as follows:Dhˆ (t ) = xˆ A (t ) - xˆ B (t )t{}= Ú vˆ A (t ) - vˆ B (t ) dt.
0(4)

The accuracy in measurement of the change in thickness
was validated to be 0.2 µm by basic experiments using arubber plate.21

Principles
Measurement of the change in wall thickness using thephased tracking method

Elastic modulus obtained by measuring change inwall thickness

For measurement of the small change in wall thickness, thephase shift of an echo, which is caused by the displacementof an object, is estimated from two consecutive echoes.16 For
this purpose, quadrature demodulation is applied to thereceived ultrasonic waves reflected by the object, and thenthe in-phase and the quadrature signals are A/D converted.
From the demodulated signal, z(t; d + x(t)), reflected at
depth d + x(t) at time t, where d and x(t) are the initial depth
set at t = 0 and the displacement of the object in the direction of depth, respectively, the phase shift, ∆ϕ(t), between
two consecutive echoes is obtained from the complex crosscorrelation function calculated for M + 1 samples in the
direction of depth as follows:

From the measured change in wall thickness, the circumferential elastic modulus is obtained as follows:23 under a
two-dimensional stress–strain relationship in the plane thatis perpendicular to the axis of the artery, the radial incremental strain, ∆εr(t), which is defined by dividing the change
in thickness, ∆h(t), by the original wall thickness, h0, isexpressed by the radial and the circumferential incremental stresses, ∆σr(t) and ∆σθ(t), as follows:

M 2

e jDj( t ) =Â z(t + T ; d + x(t ) + mD) ◊ z* (t; d + x(t ) + mD)

m =- M 2
M 2,

Â z(t + T ; d + x(t ) + mD) ◊ z* (t; d + x(t ) + mD)

m =- M 2(1)

De r (t ) =Dh(t )h0

Dsr (t )Dsq (t )
=
-v
,
Eq
Er

(5)

where Er, Eθ, and v are the radial and the circumferential
elastic moduli and Poisson’s ratio, respectively.
From the change in internal pressure, ∆p(t), the circumferential and the radial incremental stresses, ∆σr(t) and∆σθ(t), are expressed as follows:

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r0
Dp(t ),h01
Dsr (t ) = - Dp(t ),2
Dsq (t ) =

(6)
(7)

where r0 is the inner radius at the end diastole.
By substituting Eqs. 6 and 7 into Eq. 5, Eq. 5 is rewritten
as follows:
De r (t ) = -r Dp(t )1 Dp(t )
-v 0
.
h0 Eq
2 Er

(8)

By assuming that the arterial wall is incompressible(v ≈ 0.5), Eq. 8 can be rewritten as follows:
Eq =1 Ê r0 Er ˆ Dp(t ).
Á + ˜
2 Ë h0 Eq ¯ - De r (t )(9)Furthermore, by assuming isotropy (Er ≈ Eθ), the elastic
modulus, Eθh, obtained from the change in wall thickness isdefined as follows:Eqh =ˆ Dp(t )1 Ê r0.
Á + 1˜2 Ë h0¯ Dh(t )h0(10)

The accuracy in measurement of Eθh was validated to beabout 8% by basic experiments using silicone rubber tubes
with uniform and non-uniform wall thicknesses.23,24

Fig. 1. Basic experimental setup

Basic experiments using a silicone rubber tube
Experimental setup
Basic experiments using a silicone rubber tube (inner
radius: 6 mm; outer radius: 8 mm) were conducted to confirm
that the relationship between the strain and elastic modulus
is measured by application of remote actuation. Figure 1
shows a schematic diagram of the basic experimental setup.
In this setup, the pulse pressure was generated by a flow
pump, and the cyclic remote actuation at fac = 10 Hz wasapplied at a position 30 cm away from the measured position. Resultant changes in internal pressure and wall thickness were measured with a pressure transducer placed
inside the tube and with ultrasound, respectively.

Estimation of the nonlinear parameter
The stress–strain relationship of the arterial wall is nonlinear, and this nonlinearity between the pressure increment,∆p, (incremental stress) and the incremental strain, ∆εr, canbe modeled using an exponential function as follows:15
Dp = p0 ◊ ea◊De r ,(11)where p0 and α are the diastolic blood pressure and a parameter showing the degree of nonlinearity, respectively.
The elastic modulus, Eθh, in Eq. 10 can be expressed asfollows:
Eqh = K∂Dp
= a ◊ K ◊ p0 ◊ ea◊De r ,
∂De r

(12)

where K = 0.5(r0/h0 + 1). Equation 12 is rewritten as follows:( )ln Eqh = ln(a ◊ K ◊ p0 ) + a ◊ De r
= a ◊ De r + ln(E0 ),(13)

where E0 is the elastic modulus at diastolic blood pressure(∆εr = 0).
It is found that the nonlinear parameter, α, can beobtained from the slope of the relationship between theincremental strain, ∆εr, and the natural logarithm of theelastic modulus, ln(Eθh).

Basic experimental results
Figure 2 shows the changes in internal pressure and wall
thickness. By referring to the M-mode image shown in Fig.
2a, two points, A and B, were set at the outside and insideof the anterior wall at t = 0. Then, velocities, vA(t) and vB(t),of these two points were measured by the phased tracking
method, as shown in Fig. 2d,e. By integrating the difference
between these two velocities, vA(t) and vB(t), the change inthickness, ∆h(t), of the anterior wall was obtained, as shownin Fig. 2f.
To obtain instantaneous amplitudes, A∆p(t = nT; fac) andA∆h(t = nT; fac) (n = 1, 2, . . . , N; where T is the samplinginterval for blood pressure and change in thickness, and N
is the number of samples), of changes in blood pressure andwall thickness caused by remote actuation, discrete Fouriertransform was applied to the change in wall thickness, ∆h(t),and blood pressure, ∆p(t), at each time t with a Hanningwindow with a length of ±1/fac. From the estimated amplitudes, A∆p(t; fac) and A∆h(t; fac), the elastic modulus at eachtime was obtained based on Eq. 10. During the period from
0.6 s to 1.0 s after each trigger for the flow pump, which simulates the cardiac diastole, the elastic modulus was plotted
as a function of the mean strain during the correspondingtime window, as shown in Fig. 3. In Fig. 3, solid curves areshown for ten beats of the flow pump. It was found that the

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Fig. 2. a M-mode image ofthe silicone rubber tube.
(A: external surface, B: luminal
surfac). b Drive signal of theflow pump. c Internal pressure.
d Velocity, vA(t), at the outside of
the anterior wall. e Velocity,vB(t), at the inside. f Change inthickness, ∆h(t), of the anterior
wall

mmHg/s, and the resultant change in wall thickness wasmeasured with ultrasound. The elastic modulus at eachstrain level was determined, as shown by the dashed line in
Fig. 3, by estimating the slope of the stress–strain relationship using the least-squares method. The nonlinear parameter, α, obtained using this quasi-static experiment was
97.3. Nonlinear parameters, α, obtained using the measurement with remote actuation and the static experiment were
found to be in good agreement. In Fig. 3, it was supposedthat the elasticity measured using remote actuation wasslightly biased relative to that measured using the static
experiment due to the difference between the dynamic andthe static elastic modul.34,35
Fig. 3. Relationships between the incremental strain and elastic
modulus measured using remote actuation (solid line) and the static
experiment (dashed line)

In vivo experiments at the human carotid artery
Investigation of effective actuation frequency

elastic modulus increased with the mean incremental strain,
and the nonlinear parameter, α, was estimated to be 98.6
from the mean curve of the relationship between the strain
and elastic modulus.
For comparison with the ultrasonic measurement, thenonlinear parameter, α, was also measured using a quasistatic experiment. In this static experiment, the internalpressure was increased with an air pump by about 15

In in vivo experiments, remote actuation was applied at theskin surface on the brachial artery. The change in internalpressure caused by remote actuation propagates along theartery, and it causes a change in wall thickness at the carotid
artery. In this section, an effective actuation frequency for
actuating the artery wall was investigated. For in vivo experiments, an actuation system designed to change the actuation frequency heartbeat by heartbeat was constructed.

147

In this system, a custom-made computer with A/D and
D/A converters controlled the actuation frequency andtiming. The R-wave of the electrocardiogram was used as a
trigger. The actuation was started at the time of the R-wave.
At 0.6 s after the R-wave, the system stopped actuation andchanged actuation frequency before the next R-wave.
As shown in Fig. 4, an actuator (MS-VE-01N; IMV,
Osaka, Japan) was attached to the skin surface above thebrachial artery using a cuff and a belt, and it was tightly fixedby pressurizing the cuff. The actuator was heavy (about
15 kg) to reduce undesired vibration of the entire arm.
Figure 5 shows the results measured at a carotid arteryof a 30-year-old man with a linear-type probe of 7.5 MHz.

Fig. 4. Actuator used in the in vivo experiments
Fig. 5. a M-mode image of acarotid artery of a 30-year-oldman. b Electrocardiogram.
c Velocity of the posteriorwall

At a time t = 0, a point to be tracked was manually assignedby referring to the M-mode image shown in Fig. 5a. Then,the velocity of the assigned point was obtained by using thephased tracking method, as shown in Fig. 5c. In this measurement, the actuation frequency was cyclically changed
from 10 Hz to 20 Hz heartbeat by heartbeat. It was foundthat the velocity component of the arterial wall caused byremote actuation in cardiac diastole peaks at a frequency of
13 Hz. Therefore, in the following section, the nonlinear
parameter, α, was investigated at an actuation frequency of
fac = 13 Hz.
In Fig. 6, the velocities of the anterior and posterior walls
were measured at an actuation frequency of 13 Hz. The
phase difference between velocity components of the anterior and posterior walls caused by remote actuation was
about 180°, which shows that the change in diameter wascaused by remote actuation. From this result, it was supposed that the change in internal pressure at the carotid
artery was successfully generated by remote actuation.

Fig. 6. Velocities of the anterior (dashed line) and posterior (solid line)
walls measured at an actuation frequency of 13 Hz

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Fig. 7. a M-mode image of acarotid artery of a 30-year-oldman. b Electrocardiogram.
c Blood pressure measured at
the radial artery with an applanation tonometer. d Velocity,vin(t), at the inside of the posterior wall. e Velocity, vad(t), at theoutside. f Change in thickness,∆h(t), of the posterior wall

Measurement of the nonlinear parameter
Figure 7a shows the M-mode image of the carotid arteryof the same 30-year-old man. Figure 7c shows the bloodpressure measured at the radial artery with an applanation
tonometer. At the time of the R-wave of the electrocardiogram shown in Fig. 7b, two points were assigned at the
intimal and adventitial sides. Then, velocities, vin(t) andvad(t), of these two points were estimated using the phased
tracking method, as shown in Figs. 7d and 7e, respectively.
By integrating the difference between these two velocities,
the change in thickness, ∆h(t), of the posterior wall was
obtained, as shown in Fig. 7f. For comparison, measurements were also performed at actuation frequencies of fac =12 Hz and 14 Hz.
As was the case in the basic experiments, discrete Fouriertransform was applied to the change in wall thickness, ∆h(t),and blood pressure, ∆p(t), at each time t with a Hanningwindow with a length of ±1/fac to obtain instantaneous amplitudes, A∆p(t = nT; fac) and A∆h(t = nT; fac) (n = 1, 2, . . . ; where
T is the sampling interval for waveforms of blood pressure
and change in thickness), of changes in blood pressure andwall thickness caused by remote actuation. From the estimated amplitudes, A∆p(t; fac) and A∆h(t; fac), the elastic

modulus at each time was obtained based on Eq. 10. In Figs.
8a,b, the elastic modulus and mean incremental strain during
the Hanning window at each time t were plotted as functionsof time t from the R-wave of an electrocardiogram.A similartendency was found in the results obtained at frequencies of12, 13, and 14 Hz. The rapid increase in the elastic modulus
around t = 0.6 s at an actuation frequency of 13 Hz was supposed to be influenced by the stoppage of actuation for
preparing for the next heart cycle. Therefore, the measured
elasticity from 0.58 s to 0.6 s after the R-wave was eliminatedfor the actuation frequency of 13 Hz. Figure 9 shows the relationship between the elastic modulus and mean incremental
strain. It was found that the elastic modulus increased withmean incremental strain and that there were similar tendencies in the results obtained at actuation frequencies of12, 13, and 14 Hz. Learoyd and Taylor reported that theelastic modulus of the human carotid artery changes in therange from 0.5 MPa (80 mmHg) to 1.0 MPa (100 mmHg)
in in vitro experiments.35 The change in elasticity noninvasively measured by the proposed method was in a similar
range.
In Fig. 10, elastic moduli measured at the carotid arteries of healthy 30-year-old and 22-year-old men were plotted
as a function of mean incremental strain (actuation fre-

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Fig. 10. Relationship between incremental strain and logarithm of theelastic modulus, ln(Eθh), of the carotid arteries of healthy 30-year-old
and 22-year-old men (actuation frequency: 13 Hz)

nonlinear stress–strain relationship of the arterial wall.
Finally, nonlinear parameters, α, for the 30-year-old and22-year-old men were determined to be 19.1 and 32.3,
respectively.

Discussion
Fig. 8. Elastic modulus (a) and incremental strain (b) plotted as functions of time

Fig. 9. Relationship between incremental strain and logarithm of theelastic modulus, ln(Eθh), of a carotid artery of a healthy 30-year-old man
at frequencies of 12, 13, and 14 Hz

quency: 13 Hz). The range of mean incremental strain of
the 22-year-old man was lower than that of the 30-year-old
man because the pulse pressure of the 22-year-old man(≈30 mmHg) due to the heartbeat was smaller than that
(50 mmHg) of the 30-year-old man. As shown in Fig. 10, it
was found that the elastic modulus of the 22-year-old man
also increased with mean incremental strain as well as thatof the 30-year-old man. These results show the potential of
the proposed method for noninvasive assessment of the

In this study, an applanation tonometer was used in in vivo
experiments for measuring blood pressure waveforms. Thechange in blood pressure at the carotid artery caused byremote actuation could be estimated by the blood pressure
waveform measured at the radial artery because the distance between the brachial artery (actuated point) and the
carotid artery is similar to that between the brachial arteryand the radial artery (measured with an applanation
tonometer). However, the distance between the heart andthe carotid artery is very much different from that between
the heart and the radial artery, and the change in blood pressure at the carotid artery caused by the heartbeat cannot be
estimated by the blood pressure waveform measured with
an applanation tonometer. Therefore, in this study, elastic
moduli were plotted as a function of the strain measured at
the carotid artery.
Sugawara et al. reported that the change in diameter ofthe carotid artery almost coincided with the blood pressure
waveform at the carotid artery.36 Therefore, there is a possible alternative way in which the change in diameter of the
carotid artery is used for the blood pressure waveform.
Using the waveform of the change in diameter, which wasmeasured using the phased tracking method and was calibrated by the maximum and minimum blood pressure, theelastic modulus at each time was estimated and plotted asa function of strain, as shown in Fig. 11. In Fig. 11, the elastic
modulus became almost constant. One of the reasons forthis result might be as follows: in the transcutaneous measurement of the blood pressure waveform, the strain of an
employed sensor, which is caused by the blood pressure,must be calibrated by the maximum and minimum blood

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parameter was estimated from the slope of the relationshipbetween the incremental strain and elastic modulus. Such amethod for assessment of the nonlinear property in the
stress–strain relationship of the artery wall will be useful forthe diagnosis of atherosclerosis. In future work, more in vivo
data should be investigated to show the clinical feasibilityof the proposed method.
Acknowledgment This research was supported by the Research andDevelopment Committee Program of the Japan Society of Ultrasonics
in Medicine.

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Fig. 11. a Blood pressure waveform obtained by calibrating the change
in diameter measured by the phased tracking method using maximum
and minimum blood pressure measured with a cuff at the upper arm.
b Relationship between incremental strain and logarithm of elastic
modulus, ln(Eθh), with respect to the same 30-year-old man shown in
Fig. 10. The elastic modulus was calculated using the blood pressure
waveform obtained from the change in diameter shown in Fig. 11a. The
actuation frequency was 13 Hzpressure. An applanation tonometer calibrates the strainof a transducer tightly attached to the skin surface above
the radial artery. In this case, the nonlinearity in the
stress–strain relationship of the transducer material is negligible. However, the nonlinearity cannot be neglected when
the strain of the artery wall (change in diameter) is calibrated. Slight nonlinearity was found in the stress–strain
relationship of the carotid artery wall.36

Conclusion
In this paper, a preliminary study was conducted for assessment of the nonlinear property in the stress–strain relationship of the artery wall. A change in internal pressure
was caused by remote cyclic actuation applied at thebrachial artery to generate a change in the thickness of thearterial wall. From the measured changes in wall thickness
and blood pressure, the instantaneous elastic modulus
during diastole was obtained. It was found that the elastic
modulus decreased in relation to the decrease in the incremental strain during cardiac diastole, and the nonlinear

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