Japanese Journal of Applied Physics
Vol. 44, No. 6B, 2005, pp. 4609–4614#2005 The Japan Society of Applied Physics

Frequency Analysis of Strain of Cylindrical Shell for Assessment of Viscosity
Hideyuki H ASEGAWA and Hiroshi K ANAI
Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan(Received November 11, 2004; accepted March 30, 2005; published June 24, 2005)

For tissue characterization of atherosclerotic plaque, we have developed a method, namely, the phased tracking method, [H.
Kanai et al.: IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43 (1996) 791] to measure the regional strain (change in wall
thickness) and elasticity of the arterial wall. In addition to the regional elasticity, we are attempting to measure the regional
viscosity for a more precise tissue characterization. Previously, we showed that the viscosity can be obtained by measuring thefrequency dependence of the elastic modulus using remote actuation [H. Hasegawa et al.: Jpn. J. Appl. Phys. 43 (2004) 3197].
However, in this method, we need to apply external actuation to the subject. To simplify the measurement, we instead to
obtain the frequency dependence of the elastic modulus from the change in arterial wall thickness spontaneously caused by theheartbeat because this change in thickness consists of frequency components up to 20–30 Hz. In this paper, the frequency
dependence of the elastic modulus of a silicone rubber tube was investigated by applying frequency analysis to the change inwall thickness caused by the change in internal pressure simulating the actual arterial blood pressure.
[DOI: 10.1143/JJAP.44.4609]
KEYWORDS: change in wall thickness, frequency characteristics, viscoelasticity, atherosclerosis

1.

Introduction

There are signiﬁcant diﬀerences between the elastic
moduli of a normal arterial wall and those aﬀected byatherosclerosis.1,2) Therefore, evaluation of the elasticity of
the arterial wall is useful for diagnosis of atherosclerosis.3) In
particular, mechanical properties of plaque are importantbecause the rupture of the plaque may cause acute
myocardial infarction and cerebral infarction.4–6) However,
the mechanical properties, such as elasticity, of plaque itself
cannot be noninvasively measured by previous methods7–9)
on the basis of the measurement of the change in diameter of
the artery caused by a heartbeat.10–14)
For assessment of the regional elasticity of plaque itself,
we have developed a method, namely, the phased tracking
method, for measurement of the small change in thickness of
the arterial wall (less than 100 mm) due to the heartbeat.15–23)
From basic experiments, the accuracy in measurement of thechange in thickness has been found to be less than 1 mm
using the phased tracking method.16,19,20) From the change inthickness measured by our method, the regional strain andthe elasticity of the arterial wall can be noninvasively
evaluated.22,24)
From the measured elastic property, it should be possible
to discriminate tissues in the atherosclerotic plaque, such as
ﬁbrous tissue and lipids, due to large diﬀerences in theirelasticities. However, it is diﬃcult to discriminate sometissues because of their small diﬀerence in elasticity. One of
the additional mechanical properties of tissues which has apotential to provide useful information on discrimination oftissues is viscosity.
We showed that the viscosity can be estimated based on
the measurement of the elastic moduli at multiple frequencies by applying external actuation.25,26) To obtain theregional elastic moduli at multiple frequencies, it isnecessary to generate the change in diameter (=change ininternal pressure) instead of the bending vibration of theartery. However, it is not easy to stably generate the change
in internal pressure using remote actuation. To overcomeE-mail address: hasegawa@us.ecei.tohoku.ac.jp

this problem and simplify the measurement procedure,measurement of the frequency characteristics without external actuation is desired. It is reported that the rapid
increase in blood pressure in the early systole, which is
caused by contraction of the left ventricle, has frequency
components up to 20–30 Hz.27) However, there is a large
increase in blood pressure of typically 50 mmHg. Due tosuch a large increase in blood pressure, the viscoelasticity
estimation is largely inﬂuenced by the nonlinearity of thestress–strain relationship of the wall. To overcome thisproblem, we focused on the rapid decrease in blood pressurein the late systole. This pressure decrease is caused by thefollowing mechanism: Blood supply is stopped by closure of
the aortic valve. However, blood tends to ﬂow away to theperiphery due to the inertia of blood. Then, blood pressure
decreases. In this stage, the absolute value of the change inblood pressure is much smaller than that in the early systole,
and the inﬂuence of the nonlinearity can be assumed to be
negligible. In this paper, basic experiments using a silicone
rubber tube were conducted to measure the viscosity of acylindrical shell by applying frequency analysis to thechange in wall thickness caused by a sophisticated ﬂow
pump that can simulate the change in arterial blood pressure.
2.
2.1

Principles

Frequency analysis of change in wall thickness
measured with ultrasoundThe change in wall thickness, hðtÞ, caused by the change
in internal pressure, pðtÞ, can be measured by the phased
tracking method,15) which uses the phase shift of echoes
reﬂected by a moving object. For assessment of thefrequency dependence of the elastic modulus of the wall,
frequency analysis was applied to the measured changes inwall thickness, hðtÞ, and internal pressure, pðtÞ, to obtain
their amplitudes, h0 ð f Þ and p0 ð f Þ, at a frequency f .
Figure 1(b) shows the velocity of the carotid arterial wall of
a 30-year-old male for seven heartbeats measured by thephased tracking method. Figure 1(c) shows the displacementof the posterior wall obtained by integrating the measured
velocity. Figure 1(d) shows the power spectra of thevelocities during periods (A) and (B) obtained by applying

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H. HASEGAWA and H. KANAI

velocity [mm/s]

(a)

6
(b)
4
2

(B)

0

displacement [mm]

–2

(A)

–4
0.3
(c)
0.2

0.1

0
0.2

power [dBmm/s]

0

0.4

0.6

20
(A)

0.8

1
time [s]
(d)

0
–20
(B)
–40
–60
–80

–100
1
(e)

(A)
0.8
(B)

0.6
0.4
0.2
0
0

20

40

60

100
80
frequency [Hz]Fig. 1. Vibration of human carotid arterial wall caused by heartbeat.
(a) Electrocardiogram. (b) Velocities of wall for seven heartbeatsmeasured by phased tracking method.15) (c) Wall displacements obtainedby integrating velocities. (d) Power spectra of velocities during periods(A) and (B). (e) Reproducibility functions.

the Fourier transform. Plots and vertical bars show means
and standard deviations for seven heartbeats, respectively.
Furthermore, Fig. 1(e) shows the reproducibility function.15)
If the reproducibility function is one at a certain frequency,
f , the amplitude and phase of the frequency component at f
are identical for seven heartbeats. As shown in Fig. 1(e), therapid motion in the early systole, which is indicated asperiod (A) in Fig. 1(b), was measured with respect tofrequency components up to 20 Hz with good reproducibility. However, there is a large increase in internal pressure of
typically 50 mmHg. For the subject shown in Fig. 1, theblood pressure was increased from 60 mmHg to 110 mmHg
during period (A) of about 100 ms. To obtain the frequency
spectrum from 5 Hz to 20 Hz, the required length of the time
window for the frequency analysis is at least 200 ms, whichcorresponds to one wavelength of the 5 Hz components.
Therefore, the rapid increase in blood pressure from

60 mmHg to 110 mmHg for 100 ms is included in theanalyzing period. It was reported that the stress-strain
relationship of the arterial wall is nonlinear.28) This factmeans that the viscoelasticity of the arterial wall itself is
modiﬁed by the change in internal pressure corresponding to
the stress acting on the wall. The viscoelasticity of the wallis expressed by the complex elastic modulus. The real andimaginary parts correspond to the static elastic modulus, Es ,and the viscous term, 2 f , respectively. No data on thepressure dependence of the arterial-wall viscosity, , has yet
been presented in the literature. The pressure dependence ofthe static elastic modulus, Es , of the carotid artery wasreported to be from 0.4 MPa to 1.1 MPa (from 60 mmHg to
110 mmHg).28) The viscosity of the carotid arterial wall isreported to be about 5 kPas,28) and the viscous component,
2 f , changes from 0 to 0.6 MPa (from direct current to
20 Hz). The change in static elastic modulus due to thechange in blood pressure is comparable to the change in theviscous component with respect to frequency. Therefore,
from the measured elastic modulus, which corresponds tothe absolute value of the complex elastic modulus, it isdiﬃcult to distinguish the frequency dependence from thepressure (or stress) dependence during period (A).
We can ﬁnd another pulsive velocity in the late systole
shown by period (B) in Fig. 1(b). From the wall displacement shown in Fig. 1(c), the decrease in blood pressure is
expected to be from 110 mmHg to 100 mmHg during period(B). This change in internal pressure is much smaller thanthat in the early systole, and the change in elasticity wasreported to be from 1.1 MPa to 1.0 MPa.28) Therefore, it canbe assumed that only the frequency dependence of the elastic
modulus can be obtained in this timing. As shown in
Fig. 1(e), it was found that frequency components up to atleast 10 Hz can be measured with good reproducibility. Fromthese results, the pulsive velocity of the wall in the latesystole has a potential for measurement of the frequency
dependence of the elastic modulus.
In this study, a change in internal pressure similar to thatat the human carotid artery was generated by a sophisticatedﬂow pump. The resultant change in internal pressure and thewall thickness of a silicone rubber tube were measured witha pressure transducer and ultrasound. The frequency dependence of the elastic modulus was obtained by applying
frequency analysis to the vibration corresponding to that of
the arterial wall in the late systole.
2.2

Elastic modulus obtained by measured change in wall
thickness
From the measured change in wall thickness, the circumferential elastic modulus is obtained as follows:22) Under invivo conditions, the artery is strongly restricted in the axial
direction. Therefore, a two-dimensional stress-strain relationship can be assumed.
Under such conditions, the radial incremental strain,"r ðtÞ, which is deﬁned by dividing the change in thickness,hðtÞ, by the original thickness of the wall, h0 , at the end
diastole, is expressed by the radial and circumferential
incremental stresses, r ðtÞ and  ðtÞ, as follows:"r ðtÞ ¼

hðtÞ
h0

Jpn. J. Appl. Phys., Vol. 44, No. 6B (2005)¼

r ðtÞ ðtÞ

;ErE

H. HASEGAWA and H. KANAI

∆σ(t)

ð2:1Þ

where Er , E , and  are the radial and circumferential elastic
moduli and Poisson’s ratio, respectively.
From the change in internal pressure, pðtÞ, the circumferential and radial incremental stresses, r ðtÞ and  ðtÞ,are respectively expressed as follows:r0 ðtÞ ¼ pðtÞ;
ð2:2Þ
h0
1
r ðtÞ ¼  pðtÞ;ð2:3Þ
2

1 pðtÞ
r0 pðtÞ

:2 Er
h0 E 

∆ε(t)

η

Es

η: coefficient ofviscosity
Es: static elastic
moduluswhere r0 is the inner radius at the end diastole.
By substituting eqs. (2.2) and (2.3) into eq. (2.1), eq. (2.1)
is rewritten as follows:"r ðtÞ ¼ 

4611

Fig. 2. Voigt model.

ð2:4Þ

By assuming that the arterial wall is incompressible(  0:5) and elastically isotropic (Er  E ),22,29) the elastic
modulus, Eh , obtained from the change in wall thickness isdeﬁned as follows:
r 0 1 ErpðtÞhE ¼  þ
h0 2 E "r ðtÞ


1 r0
pðtÞ

þ1
:
ð2:5Þ
hðtÞ
2 h0

h0
When we describe a frequency component of changes inwall thickness and internal pressure at a frequency f by
h0 ð f Þ  e j½2 ft ð f Þ and p0 ð f Þ  e j2 ft , eq. (2.5) is rewritten as the complex elastic modulus as follows:
1 r0
p0 ð f Þ j ð f ÞEh ð f Þ ¼
þ1
e
;
ð2:6Þ
h0 ð f Þ
2 h0h0
where h0 ð f Þ and p0 ð f Þ are the amplitude of the change inwall thickness and that of the change in internal pressure,respectively, and ð f Þ is the phase delay of the change inwall thickness from the change in internal pressure at
frequency f .
The imaginary part of the complex elastic modulus, which
depends on the viscosity, has a frequency dependence.
Therefore, both the absolute value, jEh ð f Þj, and the phase,ð f Þ, of the complex elastic modulus have frequency
dependences. However, particularly under an in vivo condition, it is diﬃcult to measure the change in internalpressure, pðtÞ, at the exact same point where the change inwall thickness, hðtÞ, was measured. Therefore, the phase,ð f Þ, of the complex elastic modulus cannot be estimated. In
this paper, the absolute value, jEh ð f Þj, of the complex elastic
modulus deﬁned by eq. (2.6) is obtained from h0 ð f Þ andp0 ð f Þ, which are obtained at each frequency f .
2.3 Viscoelasticity estimation using the Voigt modelIn our previous study,25) it was found that the elastic
modulus, jEh ð f Þj, of the arterial wall increases withfrequency f . The reason for this is explained as follows.

When we assume the Voigt model, which is illustrated inFig. 2, as a viscoelastic model of the wall, the relationshipbetween the stress, ðtÞ, and the strain, "ðtÞ, is expressed
as follows:ðtÞ ¼ Es  "ðtÞ þ  d"ðtÞ;dt

ð2:7Þ

where Es and  are the static elastic modulus and theviscosity constant, respectively, and Es and  are assumed
not to be changed by frequency.
By deﬁning the frequency components at a frequency
f by ðt; f Þ ¼ 0 ð f Þ  e j2 ft and "ðt; f Þ ¼ "0 ð f Þ
e j½2 ft ð f Þ , eq. (2.7) can be rewritten asðt; f Þ ¼ Es  "ðt; f Þ þ j2 f   "ðt; f Þ;

ð2:8Þ

where 0 ð f Þ and "0 ð f Þ are the amplitudes of the stressand strain, respectively.
From eq. (2.8), the complex elastic modulus, EVoigt ð f Þ ¼
jEVoigt ð f Þje j ð f Þ ¼ ðt; f Þ="ðt; f Þ, is given by
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ð2:9Þ
jEVoigt ð f Þj ¼ Es2 þ ð2 f Þ2 ;
2 f 
ð f Þ ¼ tan1:ð2:10Þ
Es
In this paper, the viscosity constant, , and the static elastic
modulus, Es , were determined so as to minimize thediﬀerence between the measured jEh ð f Þj and the modelbased jEVoigt ð f Þj.
3.

Experimental Setup

3.1 Experimental system for basic experiments
The experimental system for basic experiments using asilicone rubber tube is illustrated in Fig. 3. In this system,
the change in pressure inside the silicone rubber tube was
generated by a sophisticated ﬂow pump (Shelly Medical
Systems, CompuFlow1000). The change in wall thickness
due to the resulting change in internal pressure wasmeasured with ultrasound, and the internal pressure wasalso measured by a pressure transducer (NEC, 9E02-P16)
placed inside the tube. From measured changes in the wall
thickness and internal pressure, the elastic modulus was
obtained using eq. (2.6).

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flow pump(CompuFlow1000)

H. HASEGAWA and H. KANAI

1 cm

ultrasonicprobe
specimenpressure sensor(9E02–P16)

anterior
wall
lumen
1 cm

measurement
systemFig. 4. B-mode image of silicone rubber tube.
Fig. 3. Experimental setup.

3.2 Measurement system
An ultrasonic pulse (center frequency: 7.5 MHz) wastransmitted and received by the ultrasonic probe of astandard ultrasonic diagnostic apparatus (Toshiba SSH140A). The received signal was ampliﬁed and demodulated.
The resultant in-phase and quadrature signals were simultaneously A/D converted by a 12-bit A/D converter at asampling frequency of 10 MHz. The measured digital signals
were transfered to a computer, and the change in wall
thickness was obtained by applying the phased tracking
method15) to these digital signals.
4.

Basic Experiments Using a Silicone Rubber Tube

Figure 4 shows the B-mode image of a silicone rubber
tube. The M-mode image was obtained along an ultrasonicbeam, as shown in Fig. 5(a). Figure 5(c) shows the measured
internal pressure. From Fig. 5(c), it was found that thechange in internal pressure, which is similar to that in thehuman artery, was generated by the ﬂow pump. By setting
two points, A and B, along the ultrasonic beam at a time
t ¼ 0 in the M-mode image, the velocities, vA ðtÞ and vB ðtÞ,of these points were obtained by the phased tracking
method, as shown in Figs. 5(e) and 5(f), respectively. Thechange in thickness of the posterior wall was obtained byintegrating the diﬀerence between vA ðtÞ and vB ðtÞ, as shownin Fig. 5(h).
Then, the amplitude, h0 ð f Þ, of the measured change inwall thickness at a frequency f was obtained by applying theFourier transform to the rate of the change in wall thickness,h_ðtÞ ¼ vB ðtÞ  vA ðtÞ, shown in Fig. 5(g). A Hanningwindow with a length of 200 ms was applied to the period
shown in Fig. 5(g). Together with the change in wall
thickness, the amplitude, p0 ð f Þ, of the measured change ininternal pressure shown in Fig. 5(c) was obtained from the‘‘rate’’ of the change in internal pressure  p_ ðtÞ. The direct
current component of the change in internal pressure is muchlarger than the alternating current components, and theHanning window with the length of 200 ms has a main lobe
with a half bandwidth of 5 Hz in the frequency domain.
Therefore, the large direct current component aﬀects theestimated frequency spectra at the other frequency. To
remove this direct current component, the change in internalpressure, pðtÞ, was diﬀerentiated with respect to time
before applying the Fourier transform. From the amplitudes,
h_0 ð f Þ and  p_ 0 ð f Þ, of h_ðtÞ and  p_ ðtÞ estimated at eachfrequency f , h0 ð f Þ and p0 ð f Þ were obtained as follows:h_0 ð f Þ;2 f p_ 0 ð f Þ:
p0 ð f Þ ¼
2 fh0 ð f Þ ¼

ð4:1Þ
ð4:2Þ

Figure 6(a) shows the power spectra of rates of changes inwall thickness, h_ðtÞ, and internal pressure,  p_ ðtÞ. Plots andvertical bars show means and standard deviations for elevenbeats, respectively. Figure 6(b) shows the reproducibility
functions. In Fig. 6(b), it was found that both changes inwall thickness and internal pressure were measured withgood reproducibilities up to 30 Hz. The elastic modulus,jEh ð f Þj, was obtained from h0 ð f Þ and p0 ð f Þ for elevenbeats, as shown in Fig. 7. In Fig. 7, the frequency characteristics of the elastic modulus were measured three times forthe same silicone rubber tube (eleven beats for eachmeasurement). Plots and vertical bars show means andstandard deviations of the elastic moduli measured foreleven beats, respectively. In the frequency range under
10 Hz, where the reproducibility function is almost one, it
was found that the elastic modulus, Eh ð f Þ, was measured
with suﬃcient reproducibility and Eh ð f Þ increased withfrequency f . Such frequency characteristics can be explained by the Voigt model. Therefore, from the frequency
characteristics under 10 Hz, the viscosity constant, , and thestatic elastic modulus, Es , were determined by applying the
Voigt model. In Fig. 7, the estimated Voigt models areshown by solid curves, and the estimated parameters,  andEs , for each measurement were the following: ﬁrst measurement, 7.2 kPas and 2.2 MPa; second, 6.8 kPas and 2.1 MPa;and third, 7.6 kPas and 2.1 MPa. These values were in good
agreement with  ¼ 6:46 kPas and Es ¼ 2:26 MPa obtainedby the diﬀerent mechanical test (Shimadzu, Tritech2000) in
which the amplitude and phase of the strain generated by theapplied sinusoidal stress was measured for a string extracted
from the silicone rubber tube. In this mechanical testing, Es
was not obtained from the static experiment, and a sinusoidal
stress at a frequency of 5 Hz, which corresponds to the centerof the frequency range for the estimation of Es and  in the

H. HASEGAWA and H. KANAI

A
B

(a)

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Jpn. J. Appl. Phys., Vol. 44, No. 6B (2005)

(b)
80
40
mmHg 0(c)

600
mmHg/s 0–600
0.6mm/s 0–0.6
0.6mm/s 0(d)

Hanning window

(e)

(f)

–0.6
0.2
mm/s 0Hanning window

(g)

–0.2
20
µm 0
–20

(h)

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14
time [s]

50

(a)

0
pressure–50
change in thickness

reproducibility

–100
1

(b)

0.8
0.6

elastic modulus [MPa]rate of pressurechange [dBmmHg/s]rate of change inthickness [dBmm/s]

Fig. 5. (a) M-mode image of silicone rubber tube. (b) Drive signal of ﬂow pump. (c) Internal pressure. (d) Rate of internal pressure.
(e) Velocity at A. (f) Velocity at B. (g) Rate of change in thickness of anterior wall. (h) Change in wall thickness.

2.4
1st measurement
2nd measurement

2.3

3rd measurement

2.2

2.1

2.0

0.4

pressurechange in thickness

0.2

1.9

0
0

10

20

30
frequency [Hz]Fig. 6. (a) Power spectra of rates of changes in internal pressure and wall
thickness. (b) Reproducibility functions.

0

2

4

6

8

12
10
frequency [Hz]Fig. 7. Elastic moduli at each frequency measured for three trials. Verticalbars show standard deviations of elastic moduli measured for elevenbeats.

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Jpn. J. Appl. Phys., Vol. 44, No. 6B (2005)

ultrasonic measurements, was applied. Therefore, Es obtained from the ultrasonic measurements and that from themechanical test was slightly diﬀerent. The string was
produced by cutting the tube along the axial direction, andthe sinusoidal stress was also applied along the axial
direction by assuming isotropy.
From these results, the proposed method was shown tohave a potential for estimating the viscosity of a cylindricalshell by applying frequency analysis to the change in wall
thickness caused by the change in internal pressure.
5.

Conclusions

In this study, we aimed to assess the viscosity of thearterial wall by frequency analysis of the change in wall
thickness caused by the heartbeat in the late systole. Basicexperiments using a silicone rubber tube were conducted tovalidate this approach. The change in internal pressure,which is similar to that at the human artery, was generated
by a sophisticated ﬂow pump, and the frequency analysis
was applied to the measured changes in wall thickness andinternal pressure. In the frequency range up to 10 Hz, inwhich the measured waveform had high reproducibility, theelastic modulus increased with frequency, and the viscosity
constant was estimated from such measured frequency
characteristics. These results show the potential of the
proposed method for assessment of the viscosity of thearterial wall with ultrasound.

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