J Med Ultrasonics (2004) 31:81–90
DOI 10.1007/s10396-004-0014-y© The Japan Society of Ultrasonics in Medicine 2004

ORIGINAL ARTICLE
Hideyuki HASEGAWA · Hiroshi KANAI
Nozomu HOSHIMIYA · Yoshiro KOIWA

Evaluating the regional elastic modulus of a cylindrical shell withnonuniform wall thickness

Received: June 9, 2000 / Accepted: October 19, 2000

Abstract
Purpose. For noninvasive diagnosis of atherosclerosis, we
attempted to evaluate the elasticity of the arterial wall bymeasuring small changes in thickness caused by the heartbeat. The elasticity of the arterial wall has been evaluatednoninvasively by measuring the change in diameter of theartery or the pulse-wave velocity; however, there is no
method for noninvasively evaluating the elasticity of thearterial wall from changes in its thickness.
Methods. Employing the phased tracking method that wedeveloped, changes in thickness of less than 100 µm were
measured in each regional area, which corresponded to thediameter of the ultrasonic beam.
Results. The elasticity of the arterial wall could be evaluated with better spatial resolution from the change in thickness than from the change in diameter of the artery or
pulse-wave velocity. We therefore propose a method forevaluating the elastic modulus of an arterial wall of nonuniform wall thickness.
Conclusions. In basic experiments employing silicone
rubber tubes with nonuniform wall thickness as arterial
models, the elastic moduli of silicone rubber tubes were
evaluated by measuring changes in wall thickness. These
results confirm the value of the proposed method.
Keywords atherosclerosis · change in thickness of arterial
wall · elastic modulus · small vibration on arterial wall

H. Hasegawa (*) · H. Kanai · N. Hoshimiya
Department of Electronic Engineering, Tohoku University Graduate
School of Engineering, 05 Aramaki-aza-Aoba, Aoba-ku, Sendai
980-8579, JapanY. Koiwa
Department of Cardiovascular Medicine, Tohoku University
Graduate School of Medicine, Sendai, Japan
This article is translated from the Japanese version, which was published in J Med Ultrasonics 2001;28:J3–13

Introduction
The steady increase in the incidence of myocardial infarction, cerebral infarction, and similar circulatory diseases
mainly caused by atherosclerosis has become a serious
problem. It is thus necessary to diagnose atherosclerosis in
its early stage. Noninvasive evaluation of the elasticity of
the arterial wall is useful for diagnosing atherosclerosis
because atherosclerosis significantly changes the elasticityof the arterial wall.1
Previously proposed methods for noninvasively evaluating the elasticity of the arterial wall included measuring
pulse-wave velocity and change in arterial diameter causedby the heartbeat. The elasticity of the arterial wall is evaluated using the Moens-Korteweg equation2 from measured
pulse-wave velocity or using the incremental elastic
modulus,3 the stiffness parameter,4 and the pressure elastic
modulus5 from the change in diameter.
We set out to evaluate the regional elasticity of the arterial wall by ultrasonically measuring the small change in its
thickness caused by the heartbeat.6 From measurements ofpulse-wave velocity and change in diameter, the average
elasticity between several 10-cm intervals in the axial direction of the artery and that of the entire circumference,
respectively, were evaluated. On the other hand, using themethod proposed here,6,7 the change in wall thickness canbe measured in each regional area, which corresponds to an
ultrasonic beam diameter of about 1 mm. If the elasticity of
the arterial wall can be evaluated using the change in wall
thickness, spatial resolution in measuring elasticity should
prove superior to measuring pulse-wave velocity andchange in diameter.
We propose a method for evaluating the elasticity of thearterial wall from the measured change in wall thickness
produced by the heartbeat. Basic experiments employing
silicone rubber tubes as arterial models show that the elasticity of the cylindrical shell can be evaluated using thismethod.

82

Measuring small changes in arterial wall thickness
caused by the heartbeat

Evaluating elastic modulus using small change inarterial wall thickness

To determine the change in thickness of the arterial wallcaused by the heartbeat, the velocity of the wall was measured as described below.6,7
As illustrated in Fig. 1, the received ultrasonic pulse hasa phase delay, θ(t), corresponding to the bidirectional propagation between the ultrasonic probe and the target. The
phase shift, Dθ(t)  θ(t  T)  θ(t), during one pulse repetition interval T, which corresponds to displacement of thetarget, is estimated from two consecutive echoes.
From the phase shift Dθ(t), the velocity of the target attime t  T/2 is obtained as follows:

When an artery can be assumed to be a cylindrical shellwith uniform wall thicknessTˆc Dθ(t )ÊvÁ t  ˜  
Ë
2¯
2ω0 T

(1)

where ω0 and c are the central angular frequency of
the ultrasound waves and the speed of sound, respectively. The speed of sound in tissue is assumed to be
1540 m/s.
The change in thickness, Dh(t), of the arterial wall can be
described by the difference between the displacement,
xin(t), of the intimal side of the arterial wall and displacement, xad(t), of the adventitial side. The change in thickness,
Dh(t), can thus be obtained by integrating the difference
between the velocity, vin(t), of the intimal side and the velocity, vad(t), of the adventitial side as follows.
Dh(t )  xin (t )  xad (t ){} Ú vin (t )  vad (t ) dt

(2)

Fig. 1. Method used to measure velocity on the arterial wall. RF,radiofrequency

An artery without atherosclerotic plaque can be assumed to
be a cylindrical shell with a wall of uniform thickness. Under
such conditions, the incremental strain, Dεr(t), in the radial
direction at time t resulting from the pressure increment,Dp(t), from the diastolic pressure, p0, is expressed as follows8
Dεr (t ) D σ r (t )Dσ θ (t )Dσ z (t )


Er
Eθ
Ez

(3)

where  is Poisson’s ratio. Er, Eθ, and Ez are elastic moduli
in the radial, circumferential, and axial directions, respectively; and Dσr(t), Dσθ(t), and Dσz(t) are incremental stresses
in the radial, circumferential, and axial directions, respectively. The second and third terms on the right side of Eq.
(3) show decrements in radial strain resulting from circumferential strain Dσθ(t)/Eθ and axial strain Dσz(t)/Ez, which
are determined by , Poisson’s ratio.
In vivo, the artery is strongly restricted in the axial direction.9 Deformation of the artery in the axial direction cantherefore be assumed to be negligible [Dσz(t)/Ez  0]. Equation (3) is thus approximated as follows:Dεr (t ) ª

D σ r (t )Dσ θ (t )

Er
Eθ

(4)

Because of the change in internal pressure, Dp(t), causedby the heartbeat, the arterial wall is deformed as shown in
Fig. 2. An enlarged view of the shaded area in Fig. 2 is shownin Fig. 3, in which the internal pressure and the externalpressure (atmospheric pressure) at time t are defined by
p1(t) and p2(t), respectively.
Considering the balance of forces within a small regionof the arterial wall, as shown in Fig. 3, forces on the x-axis,
act on the arterial wall balance, as indicated by the fact thatthe shaded region is symmetrical with respect to the y-axis.
The force in the y direction acting on a small region PP¢ iscaused by the internal pressure, p1(t), and is expressed asfollows.

Fig. 2. Change in diameter and thickness of the arterial wall caused bychange in internal pressure

83

ments of the internal radius, wall thickness, and pressure
from the inner radius, r0, and wall thickness, h0, at the diastolic pressure, p0, respectively, Eq. (10) can be rewritten as
follows.
σ θ (t ) 

r0  Dr (t )p0  Dp(t )h0  Dh(t )

{

}

(11)

In systole, the internal radius increases as a result of theincreased internal pressure [Dp(t)  0, Dr(t)  0].Wall thickness, on the other hand, decreases.
Stress, σr(t), in the radial direction is defined as the meanpressure between p1(t) at the inner surface of the wall andp2(t) at the outer surface.
σ r (t )  1p1 (t )  p2 (t )
2

{

}

(12)

As with Eq. (10), adding stress, p2(t), that balances theatmospheric pressure gives us stress, σr(t), in the radial
direction.
Fig. 3. Balance of forces in a small region of the arterial wall

p1 (t )PP ¢  2 p1 (t )r1 (t ) sin Dθ

σ r (t )  (5)Similarly, the force in the y direction acts on the small
region QQ¢ because the external pressure, p2(t), is2p2(t)r2(t)sin Dθ. Forces in the y direction caused by theinternal pressure, p1(t), and the external pressure, p2(t),balance the y-axis component of tension T.
2 p1 (t )r1 (t ) sin Dθ  2 p2 (t )r2 (t ) sin Dθ  2T sin Dθ  0{}

D σ θ (t ) 

{

}

(8)

When tension, T, in Eq. (7) is divided by the wall thickness h(t), stress, σθ(t), in the circumferential direction isexpressed as follows.
σθ 

r1 (t )T
p1 (t )  p2 (t )  p2 (t )

h(t )
h(t )

{

}

(9)

Stress p2(t) expands the arterial wall to counterbalance
the atmospheric pressure p2(t). Therefore, by adding p2(t) to
Eq. (9), stress, σθ(t), in the circumferential direction is
obtained byσ θ (t ) r1 (t )
p1 (t )  p2 (t )
h(t )

{

}

(10)

Measured internal pressure, p(t), at time t corresponds top1(t)  p2(t). By defining Dr(t), Dh(t), and Dp(t) as incre-

{

{

}

(13)

}

r
r0
p0  Dp(t )  0 p0h0h0r
 0 Dp(t )  0
h0

{

}

1
1
p0  Dp(t )  p0
2
2
1
 Dp(t )  0
2

D σ r (t )  

When h(t)  0, Eq. (7) is approximated by the following
Laplace equation.
T ª p1 (t )  p2 (t ) r1 (t )}r0  Dr (t )rp0  Dp(t )  0 p0h0h0  Dh(t )

ª

(6)

(7)

{

Given Dr(t) and Dh(t) sufficiently smaller than r0 and h0,
incremental stresses, Dσθ(t) and Dσr(t), in the circumferential and radial directions become increments from stresses
r0p0/h0 and p0/2, respectively, at diastolic pressure.

Defining r2(t) as r1(t)  h(t) using the wall thickness, h(t),tension, T, is expressed as follows.10
T  p1 (t )  p2 (t ) r1 (t )  p2 (t )h(t )11p1 (t )  p2 (t )   p0  Dp(t )
2
2

{

(14)

}

(15)

When Eqs. (14) and (15) are substituted into Eq. (4), theratio Dp(t)/Dεr(t) of the incremental pressure to the incremental strain in the radial direction can be described asfollows, with the arterial wall assumed to be incompressible(  0.5).11
Dp(t )20
Dεr (t )Ê 1 r0
1ˆ

Á
˜
Er ¯
Ë Eθ h0(16)Equation (16) can then be expressed in the following form.
Eθ E ˆ Dp(t )1 Ê r0 θ˜ÁEr ¯ Dεr (t )2 Ë h0(17)

The circumferential elastic modulus, Eθ, in Eq. (17)cannot be derived when Eθ/Er is unknown. To obtain Eθ84

Fig. 4. Relation between elastic modulus, Eθ, obtained by Eq. (17) andanisotropy Er/Eθ. Broken line shows the elastic modulus, Eθh, obtainedby Eq. (18), which is derived by approximating Er by Eθ

from incremental strain, Dεr(t)  Dh(t)/h0, in the radial
direction, the arterial wall must be assumed to be elasticallyisotropic (Er  Eθ). Approximating Er as Eθ has the following effect. The circumferential elastic modulus, Eθh, isdefined by the following equation by approximating Er as
Eθ using incremental strain Dεr(t).
Eθh ˆ Dp(t )1 Ê r0 1˜
Á
2 Ë h0¯ Dεr (t )(18)

The solid line in Fig. 4 shows the elastic modulus, Eθ,
defined by Eq. (17) plotted as a function of Er/Eθ. The
broken line shows the circumferential elastic modulus, Eθhdefined by Eq. (18). Dp(t)/Dεr(t) is assumed to be 200 kPa,
and the ratio, r0/h0, of the inner radius to wall thickness isdetermined as follows. Figure 5 shows the ratio, r0/h0, of 68
carotid arteries without atherosclerotic plaque measured
using B-mode ultrasonography. The mean value of 4.7 isused as a typical value when calculating Eθ and Eθh in Fig. 4.
Er/Eθ has been reported to equal 0.8 in vivo.9 The error
in Eθh obtained by Eq. (18) from Eθ obtained by Eq. (17) isabout 3% when Er/Eθ  0.8. Approximating Er as Eθ thus
does not significantly influence the diagnosis of atherosclerosis because the change in elasticity caused by atherosclerosis exceeds 200%.1 This shows that Eq. (18) evaluates theelastic modulus of an arterial wall of uniform thickness
using incremental strain, Dεr(t), in the radial direction.

When an artery cannot be assumed to be a cylindricalshell of uniform wall thickness
Wall thickness becomes nonuniform and greater than thatof the normal arterial wall when the artery contains atherosclerotic plaque. We divided atherosclerotic plaque into
several layers (N layers) and calculated the elastic modulusof each layer to assess the spatial distribution of the elastic
modulus.
We divided an arterial wall of thickness h0 into N layers
as shown in Fig. 6 to study the balance of forces resulting
from internal pressure, p1(t), and external pressure, p2(t), in
the shaded region of Fig. 6. As shown in Fig. 7, we assume

Fig. 5. Ratio of wall thickness, h0, to inner radius, r0, for 68 humancommon carotid arteries without atherosclerotic plaques. a Wall thickness h0. b Inner radius r0. c Ratio, r0/h0, of the internal radius to wall
thicknessthat pressure in the arterial wall changes linearly from p1(t)to p2(t) proportional to the distance from the internalsurface of the arterial wall. Pressure, pn(t), at the internal
radius of the nth (n  1, 2, . . . , N) layer is therefore
described as follows.
pn (t ) Nn1
p1 (t )  p2 (t )  p2 (t )
N

{

}

(19)

As indicated by Eq. (6), the balance of forces in the y
direction in the shaded region of Fig. 6 is expressed as
2 pn (t )rn (t ) sin Dθ  2 pn1 (t )rn1 (t ) sin Dθ  2Tn sin Dθ  0(20)
where rn(t) and Tn are the internal radius and the tensionof the nth layer, respectively. By expressing rn1(t) by rn(t) h(t)/N using the thickness, h(t)/N, of a layer, the following equation is obtained in the same manner as was Eq. (7).

85

σ θn (t) 

rn 0  Drn (t)p0  Dp(t)

N
Ï h0
¸
 Dhn (t)˝
Ì
N
˛
Ó
Nn
p0  Dp(t)

N

{

(23)

}

As with Eq. (13), the radial stress, σrn(t), of the nth layer
can be expressed as follows.
1pn (t )  pn1 (t )  p2 (t )22N  2n  1
p0  Dp(t )

2N

{

σ rn (t )  

}

{

(24)

}

Equations (23) and (24) are confirmed in the appendix.
Given Dhn(t)  h0/N and Drn(t)  rn0, incremental
stresses, Dσθn(t) and Dσrn(t), in the circumferential and theradial directions of the nth layer can be obtained using thefollowing equations as with Eqs. (14) and (15).
Dσ θn (t) 

Fig. 6. Small region within the arterial wall

rn 0  Drn (t)p0  Dp(t)

N
Ï h0
¸
 Dhn (t)˝
Ì
˛
ÓN
r
p0
Nn
Nn
p0
p0  Dp(t)  n 0NNÊ h0 ˆ N
Á ˜
Ë N¯

{

}

Ï
¸
ÔÔ r
ÔÔ Dp(t)ª Ì n 0  (N  n)˝Ô ÊÁ h0 ˆ˜Ô NÔÓ Ë N ¯Ô˛

2N  2n  1
2N  2n  1p0p0  Dp(t ) 
2N
2N
2N  2n  1
Dp(t )
2N(26)

Dσ rn (t )  
Fig. 7. Distribution of internal pressure in the radial direction{}

Tn  p n ( t )  p n1 ( t ) rn ( t )  p n1 ( t )

h(t )
N

(21)

The circumferential stress, σθn(t), of the nth layer isobtained by dividing the tension, Tn, by the thickness, h(t)/N,of a layer.
σ θn (t) 

{

(22)

}

Increments in internal radius, layer thickness, and pressure from the internal radius rn0 and the layer thickness,
h0/N, at the diastolic pressure, p0, are defined by Drn(t),Dhn(t), and Dp(t), respectively. Adding the stress, p2(t), tocounterbalance atmospheric pressure to Eq. (22), the following equation obtained, as with Eq. (11).

{}As with Eq. (4), the incremental strain, Dεr(t)  N ·Dhn(t)/h0, in the radial direction of the nth layer can be
described as follows.
Dεrn (t ) rn (t) p1 (t)  p2 (t)NÏÔ h(t) ¸Ì˝ÔÓ N Ô˛Nn
p1 (t)  p2 (t)  p2 (t)N(25)

Dσ rn (t )Dσ θn (t )

Ern
Eθn

(27)

where Ern and Eθn are elastic moduli in the radial and thecircumferential directions, respectively. Assuming  equal
0.5 and substituting Eqs. (25) and (26) into Eq. (27) yields
the following equation as with Eq. (17).
Ï
¸
ÔÔ r
E ÔÔ
Eθn  Ì n 0  ( N  n)  (2 N  2 n  1) θn ˝Ern ÔÔ ÊÁ h0 ˆ˜ÔÓ Ë N ¯Ô˛Dp(t )
2 NDεrn (t )

(28)

86

By approximating Eθn/Ern as 1, the circumferential elastichmodulus, Eθn, of the nth layer is obtained as follows :Ï¸ÔÔ Dp(t )1 ÔÔ rn 0
 ( N  n  1)˝
E ª Ì
2 Ô Ê h0 ˆÔ NDεrn (t )Á ˜ÔÓ Ë N ¯
Ô˛
h
θn

(29)

(n  1, 2 . . . . . . , N )Substituting rn0  r10  (n  1)h0/N into Eq. (29) yields
Eq. (30) when the incremental strain, Dεrn(t), is uniform in
the radial direction (Dεrn(t)  Dεr0(t)).
Ï¸ÔÔ Dp(t )1 ÔÔ r10 1˝
E ª Ì
2 Ô Ê h0 ˆÔ NDεr 0 (t )Á ˜ÔÓ Ë N ¯
Ô˛
h
θn

(30)

(n  1, 2 . . . . . , N )Equation (30) shows that the elastic moduli of all layers
are equivalent when the incremental strain is uniform. Furthermore, substituting r10  r0 into Eq. (29) when there is
only one layer makes it identical to Eq. (18).
Eθh1 

ˆ Dp(t )1 Ê r0 1˜
 EθhÁ2 Ë h0¯ Dεr 1 (t )Fig. 8. Experimental setup for measuring the elastic modulus of thesilicone rubber tube

(31)

Basic experiments using silicone rubber tubes
Using tubes with uniform wall thickness
The following basic experiments were carried out toconfirm that the elastic modulus of a cylindrical shell of
uniform wall thickness can be evaluated by Eq. (18). As
illustrated in Fig. 8, the change in the wall thickness of thesilicone rubber tube brought about by a change in internalpressure generated by an artificial heart was measured using
ultrasound. The internal pressure and the drive signal of theartificial heart were acquired simultaneously. The size ofthe tube and the speed of sound in tube A are shown in
Table 1.
The B-mode image of the silicone rubber tube obtainedusing standard diagnostic ultrasonic equipment is shown at
the top of Fig. 9. The M-mode image shown in Fig. 9a was
obtained by echoes reflected from a location on the tube.
Figure 9b,c shows the drive signal of the artificial heart andinternal pressure, respectively. Velocities, vad(t), and vin(t), at
the outside and inside of the anterior wall were derived, as
shown in Fig. 9, from the received echoes, (Figs. 9d and 9e,
respectively); and the change in wall thickness, Dh(t), shownin Fig. 9f was obtained by integrating the difference
between these two velocities.
Incremental strain, Dεr(t)  Dh(t)/h0, in the radial direction was obtained from the change in wall thickness, Dh(t),and thickness, h0, of the wall shown in Table 1. Figure 10

Fig. 9. Measuring change in thickness and the internal pressure. Top
B-mode image of the silicone rubber tube obtained by standard ultrasonic diagnostic equipment. a M-mode image. xin(t) and xad(t) show
traces of tracking for the inner and external surfaces of the tube. b
Drive signal of the artificial heart. c Internal pressure. d Velocity signalof the external surface vad(t). e Velocity signal of the internal surface
vin(t). f Change in wall thickness Dh(t)87Table 1. Size, speed of sound, and elastic modulus of three silicone rubber tubes
Tube

Inner
radius (mm)Wallthickness (mm)Speed ofsound (m/s)

Incremantal elastic
modulus Einc (MPa)

Rubber tube A
Rubber tube B
Rubber tube C

4.0
7.5
5.1

1.5
2.5
0.7

992
1291
947

5.7
1.9
1.2

10

5

pressure 0
[kPa]
–5
(a)
–10
–0.010–0.0050.000
strain

0.005

0.010

(b)
5
kPa 0
(c)

Fig. 11. System used to test the relation between the internal pressure
and the external diameter

0.002
0
(d)
0

1

2

3

4

5

6
time [s]Fig. 10. Relation between internal pressure and strain in the radial
direction in tube A. a Relation between internal pressure and strain in
the radial direction. b Drive signal of the artificial heart. c Internal pressure. d Strain in the radial direction

shows the relation between incremental strain, Dεr(t), in theradial direction and incremental pressure Dp(t). From themeasured relation between incremental stress and incremental strain shown in Fig. 10a, the slope, Dp(t)/Dεr(t), of thestress-strain relation was obtained using the least-squares
method; and the elastic modulus, Eθh, was calculated as
5.8 MPa by Eq. (18).
The elastic modulus of the tube was also measured by
testing the relation between internal pressure and external
diameter to validate the measured elastic modulus Eθh. The
experimental setup is shown in Fig. 11. This experimenttested the relation between internal pressure and external
diameter by increasing the internal pressure by about 5 kPa
using an air pump. The internal pressure and external diameter were measured using a pressure detector (NEC 9 E 02P 16) and a laser line gauge (KEYENCE VG-035). Using
the measured relation between internal pressure and external diameter, the elastic modulus in the circumferential
direction was evaluated by the incremental elastic modulus,
Einc, as follows3

Fig. 12. Results of testing the relation between internal pressure andexternal diameter (tube A)

Einc 

3 r02re Dp(t )2 re2  r02 Dre (t )(32)

where re and Dre(t) refer to the original external radius andthe change in external radius [Dre(t)  0].
Figure 12 shows the measured relation between incremental pressure, Dp(t), and incremental strain Dre(t)/re.
From the slope, re · Dp(t)/Dre(t), obtained using the least-

88

squares method, the incremental elastic modulus, Einc, determined as 5.7 MPa by Eq. (32), agrees well with the elastic
modulus, Eθh, of 5.8 MPa measured by ultrasound.
Similar experiments were carried out on another silicone
rubber tube with a different radius and wall thickness. The
size and speed of sound of tube B are shown in Table 1.
From experiments on tube B, Eθh and Einc were calculated tobe 2.0 and 1.9 MPa, respectively, and are in close agreement.
These results show that the elastic modulus of a cylindricalshell with uniform wall thickness can be evaluated by Eq.
(18) when Er can be approximated by Eθ.

mental pressure, Dp(t), and incremental strain Dre(t)/re. Itshows that the incremental elastic modulus, Einc, was evaluated to be 1.2 MPa by Eq. (32), agreeing closely with thehhmeasured values of elastic moduli, Eθ1and Eθ2, of tube C.
These results are summarized in Table 2.
These results show that the elastic modulus in the circumferential direction can be evaluated approximately by
Eq. (29), even when the shell is cylindrical and the thicknessof its wall is not uniform.

Conclusions
When the thickness of the tube is not uniform
Experiments similar to those previously described wereconducted with a two-layered silicone rubber tube (tube C)
with a nonconcentric cross section. Tube C served as amodel of an artery with nonuniform wall thickness. The
speed of sound of the silicone rubber tube was 947 m/s, andits size is described in Fig. 13. The two layers were made of
the same material.
Figure 14c,d shows measured strains, Dεr1(t) and Dεr2(t),in the radial direction in the inner and the outer layers,respectively. Figure 15a shows the relation between incremental pressure, Dp(t), and incremental strain, Dεr1(t), in thehradial direction of the inner layer. The elastic modulus, Eθ1,of the inner layer was determined to be 1.4 MPa by Eq. (29)from the slope of the relation between Dp(t) and Dεr1(t) andthe size of the tube. Similarly, Fig. 15b shows the relation
between incremental pressure, Dp(t), and incremental
strain, Dεr2(t), in the radial direction of the outer layer. Thehelastic modulus, Eθ2, of the outer layer was determined tohbe 1.2 MPa by Eq. (29), and similar values of Eθnwereobtained in these two layers.
To measure the incremental elastic modulus, Einc, definedby Eq. (32), the relation between internal pressure andexternal diameter was tested of a silicone rubber tube withuniform wall thickness (tube D) made of the same material as tube C. The size of tube D is described in Table 1.
Figure 16 shows the measured relationship between incre-

Fig. 13. Size of the cross section of silicone rubber tube C. The measureregion is shaded

We propose a method for evaluating circumferential elastic
modulus of a cylindrical shell by measuring change in its
wall thickness resulting from change in its internal pressure.
An artery without atherosclerotic plaques can be assumed
to be a cylindrical shell of uniform wall thickness. However,wall thickness becomes nonuniform when the artery contains plaques. The proposed method provides an approach
to measuring the regional elastic modulus of the arterial
wall, even when the artery contains atherosclerotic plaques.
In basic experiments, elastic moduli of silicone rubber tubes
were evaluated using the proposed method and indicatedthat the elasticity of the arterial wall can be evaluatedfrom ultrasonically measured changes in wall thickness.
Such a method for noninvasively evaluating the elasticityof the arterial wall should prove useful for diagnosing
atherosclerosis.

Table 2. Evaluation of elastic moduli of three silicone rubber tubes
TubeEθh (MPa)

Rubber tube A
Rubber tube B
Rubber tube X (first layer)
Rubber tube X (secound layer)

5.8
2.0
1.4
1.2

Fig. 14. Strain in the radial direction (tube C). a Drive signal of theartificial heart. b Internal pressure. c Strain in the radial direction of
the inner layer. d Strain in the radial direction of the outer layer

89Fig. 15. a Relation between
internal pressure and strain in
the radial direction of the inner
layer in tube C. b Relation
between internal pressure andstrain in the radial direction of
the outer layer of tube Cab

In the same manner, from circumferential stresses,σθn(t) (n  1, 2, . . . , N), of N layers defined by Eq. (23),the mean circumferential stress, σθ(t), of the entire wallN

can be obtained by dividing the tension T  Â Tn n1NÂ h(t )σ (t ) N of the entire wall by the wall thickness, h(t).

n1

θn

1 N h(t)Â σ θn (t)h(t) n1 N

Fig. 16. Relation between internal pressure and external diameter intube D

Appendix: confirmation of definition of stressesσθn(t) and σrn(t)Even when the arterial wall has uniform thickness, it can bedivided into N layers. Mean radial stress obtained from thesum of the stresses of N layers should be identical to theradial stress of the entire wall defined by Eq. (13) as follows.
σ r (t )  1p1 (t )  p2 (t )
2

{

}

(33)

The mean radial stress, σr(t), of N layers can be obtained
as follows using Eq. (24).
1 N Ï 2N  2n  1¸
p1 (t)  p2 (t) ˝Â Ì2NN n1 Ó˛{}

1 2 N  N (N  1)  N
p1 (t)  p2 (t)N
2N1  p1 (t)  p2 (t)
2
2

{



{

}

}

Equation (34) turns out to be identical to Eq. (13).

(34)

¸
Ï
Ô
Ô
1 N h(t) Ô rn (t)Ô p1 (t)  p2 (t)
N
n



(
)
˝
Â
Ì
h(t) n1 N Ô Ê h(t) ˆNÔÔ ÁË N ˜¯Ô˛Ó¸ÏÔ r (t)  h(t) (n  1)
Ô
1 N h(t) Ô 1
N



N
n
(
)Ô˝
Â Ì
h(t) n1 N ÔÊ h(t) ˆ
Ô
Á N ˜
Ô
Ô
Ë
¯
˛
Ó
¸
Ï
Ô
Ô
p1 (t)  p2 (t)h(t) Ô r1 (t)1Ô (N  2 n  1)˝Â ÌNh(t) n1 N Ô Ê h(t) ˆÔÔ ÁË N ˜¯Ô˛Óp1 (t)  p2 (t)r1 (t)
p1 (t)  p2 (t)
h(t)
N
N

{

}

(35)
Equation (35) also turns out to be identical to Eq. (10).
Thus the radial and circumferential mean stresses of theentire wall defined by Eqs. (13) and (10) can be derived
from stresses, σrn(t) and σθn(t), of each layer, theoreticallyconfirming Eqs. (24) and (23).

90

References
1. Lee LT, Grodzinsky AJ, Frank EH, et al. Structure-dependent
dynamic mechanical behavior of fibrous caps from human atherosclerotic plaques. Circulation 1991;83:1764–70.
2. Hallock P. Arterial elasticity in man in relation to age as evaluatedby the pulse wave velocity method. Arch Intern Med 1934;54:
770–98.
3. Bergel DH. The static elastic properties of the arterial wall. J
Physiol 1961;156:445–57.
4. Hayashi K, Handa H, Nagasawa S, et al. Stiffness and elastic
behavior of human intracranial and extracranial arteries. J
Biomech 1980;13:175–84.
5. Peterson RH, Jensen RE, Parnell R. Mechanical properties ofarteries in vivo. Circ Res 1960;8:622–39 (in Japanese).

6. Hasegawa H, Kanai H, Chubachi N, et al. Evaluation of the elastic
modulus of the arterial wall by accurate noninvasive measurementof change in wall thickness. J Med Ultrasonics 1997;22:851–60 (inJapanese).
7. Kanai H, Sato M, Koiwa Y, et al. Transcutaneous measurement andspectrum analysis of heart wall vibrations. IEEE Trans UFFC 1996;
43:791–810.
8. Patel DJ, Janicki JS, Vaishnav RN. Dynamic anisotropic viscoelastic properties of the aorta in living dogs. Circ Res 1973;32:93–
107.
9. The Japan Society of Mechanical Engineers. Mechanical Engineers’ Handbook, Engineering, C6 Biotechnology and Medical
Engineering. Tokyo: Maruzen; 1988. p. 148 (in Japanese).
10. Oka S. Rheology. Tokyo: Shokabo; 1974. p. 308–36 (in Japanese).
11. Carew TE, Vaishnav RN, Patel DJ. Compressibility of the arterial
wall. Circ Res 1968;23:61–8.