Japanese Journal of Applied Physics
Vol. 44, No. 6B, 2005, pp. 4588–4592#2005 The Japan Society of Applied Physics

Cross-Sectional Elasticity Imaging of Arterial Wall by Comparing Measured Change
in Thickness with Model Waveform
Jiang TANG, Hideyuki HASEGAWA and Hiroshi K ANAI
Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan(Received November 12, 2004; accepted March 13, 2005; published June 24, 2005)

For the assessment of the elasticity of the arterial wall, we have developed the phased tracking method [H. Kanai et al.: IEEE
Trans. Ultrason. Ferroelectr. Freq. Control 43 (1996) 791] for measuring the minute change in thickness due to heartbeats andthe elasticity of the arterial wall with transcutaneous ultrasound. For various reasons, for example, an extremely small
deformation of the wall, the minute change in wall thickness during one heartbeat is largely inﬂuenced by noise in these cases
and the reliability of the elasticity distribution obtained from the maximum change in thickness deteriorates because themaximum value estimation is largely inﬂuenced by noise. To obtain a more reliable cross-sectional image of the elasticity of
the arterial wall, in this paper, a matching method is proposed to evaluate the waveform of the measured change in wall
thickness by comparing the measured waveform with a template waveform. The maximum deformation, which is used in thecalculation of elasticity, was determined from the amplitude of the matched model waveform to reduce the inﬂuence of noise.
The matched model waveform was obtained by minimizing the diﬀerence between the measured and template waveforms.
Furthermore, a random error, which was obtained from the reproducibility among the heartbeats of the measured waveform,
was considered useful for the evaluation of the reliability of the measured waveform. [DOI: 10.1143/JJAP.44.4588]
KEYWORDS: change in wall thickness, elasticity, matching, reliability

1.

Introduction

Recently, the increasing number of patients suﬀering frommyocardial infarction or cerebral infarction has become aserious social problem. Therefore, it is important to diagnose
atherosclerosis at an early stage because such circulatory
diseases are mainly caused by atherosclerosis. Since theelasticity of the arterial wall changes as atherosclerosis
develops,1) the evaluation of the regional elasticity of thearterial wall using ultrasound is useful for the diagnosis ofatherosclerosis. For the assessment of the elasticity of thearterial wall, we developed a method, the phased tracking
method,2) of measuring the change of several tens of microns
in wall thickness due to heartbeats with transcutaneous
ultrasound.3–7) However, particularly in a hard region, thechange in thickness becomes several microns, and themeasurement of such a minute change in thickness may beinﬂuenced by noise. Although the elastic modulus of thewall is obtained from the maximum change in thickness, themaximum value estimation is easily inﬂuenced by noise. In
this paper, by employing a model waveform of the change inwall thickness which is considered to be noise free, amatching method to evaluate the reliability of the elasticity
image is proposed. The model waveform was ﬁtted to themeasured waveform using the least-squares method to obtainthe matched model waveform. The maximum deformationof the wall was obtained from the amplitude of the matchedmodel waveform to reduce the inﬂuence of noise. Furthermore, the reproducibility among heartbeats of the measured
waveform will be useful for the evaluation of the reliability.
2.

Principles

2.1 Elasticity estimation
An ultrasonic beam was sequentially scanned at 60
positions with a linear-type ultrasonic probe of 7.5 MHz
using conventional ultrasound diagnostic equipment(Toshiba SSH-140A), and multiple (Nm þ 1) points were

E-mail address: hkanai@ecei.tohoku.ac.jp

preset from the luminal surface to the adventitia along themth ultrasonic beam (m ¼ 1; . . . ; 60) with constant intervalsof h0 ¼ 375 mm at time t0 just before the ejection period. By
dividing the arterial wall into multiple layers, we deﬁned thenth layer (n ¼ 1; . . . ; Nm ) as being between two contiguous
points, n and n þ 1, along each beam. For the measurementof the change in the thickness of each of the Nm layers, theinstantaneous depth xm;n ðtÞ of the nth point along the mthbeam was simultaneously tracked by applying the phased
tracking method2) to the received ultrasound. The minutedecrease of several tens of a micrometer in the thickness ofthe nth layer resulting from the arrival of a pressure wave atthe beginning of the ejection period was determined usinghm;n ðtÞ ¼ xm;nþ1 ðtÞ  xm;n ðtÞ  h0 .
From the ratio of the maximum decrease in thickness
during one heartbeat, hm;n;max ¼ maxt jhm;n ðtÞj, to theinitial thickness h0 of the nth layer, the maximum deformation of the nth layer was obtained using "m;n;max ¼
hm;n;max =h0 . Because the deformation was suﬃciently
small and was in the linear regime, it showed an incremental
strain in the radial direction. By assuming that the arterial
wall is incompressible and that the blood pressure is applied
perpendicular to each layer, the elastic modulus of the nthlayer along the mth beam, E;m;n , is approximately given by8)p
 1 m;n;0 þ Nm  n þ 1
E;m;n ¼
2 h0  N m
Nm
"m;n;max
ðn ¼ 1; . . . ; Nm ; m ¼ 1; . . . ; MÞ;ð2:1Þ

where m;n;0 is the initial inner radius of curvature of the nthlayer along the mth beam at time t0 . We assumed that thepressure in the arterial wall decreases linearly with the
distance from the intimal side to the adventitia and that thearterial wall is almost isotropic.9)
For the region with a length of 18 mm along the axis of theartery, the regional elasticity E;m;n was estimated on the
cross-sectional image. Because the reﬂected ultrasound wasreceived at a sampling interval of 100 ns (¼ 75 mm along
depth direction) after the quadrature demodulation, we

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(a)
intimaaveragemedia

amplitude [µm]Jpn. J. Appl. Phys., Vol. 44, No. 6B (2005)

0

J. T ANG et al.

tNmeasured y (n)model Hk (n)model media
–20
–30

4 MPa

...

t2

t1

–10

model adventitia

adventitia
0 MPa

(b)

model intima

0

0.2

0.4

0.6

0.8

1.0

after extension
after linear interpolation h k (n)1.2time [s]
T1

...

T2

Fig. 1. Preparation of model waveforms. (a) Regions assigned as intima,
media, and adventitia in elasticity image showing typical layered
structure. (b) Model waveforms obtained by averaging changes inthickness in respective assigned regions.

TN 0

Fig. 2. Linear interpolation.

model

further divided each layer with a thickness of h0 into 5
points, shifted the initial depth of each layer by one ﬁfth ofh0 , and applied the above procedure to each depth. Thus,E;m;n was estimated at intervals of 75 mm in the depth
direction and 300 mm in the axial direction. Using a silicone
rubber tube with two layers set in an artiﬁcial circulation
system,8) the accuracy of the measurement of the regionalelasticity of each layer has already been validated to be0:1 MPa8)—that is, the error is 8% of the elasticity
obtained by a separate static pressure-diameter test.

measured

before equalization h k (n)after equalization Ak

mean waveform y (n)h k(n)

Fig. 3. Amplitude equalization.

2.2 Matching process
In our matching process, template waveforms wereprepared as model waveforms, Hk ðnÞ (n ¼ 1; 2; . . . ; N0 ), of
4 MPa

(a)
original
elasticity
image
3 mm
3.75 mm–40–40

ym (n) −y (n)(d)

amplitude [µm]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]
0

(1)

–20
–40

y (n)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]amplitude [µm]–20ym (n)

200

(2)

100
0

ym (n) −y (n)–100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]amplitude [µm]amplitude [µm]

(c)

(1)

–40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]
0

–20

0

(2)

–20
–40

y (n)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]amplitude [µm]ym (n)

0

r=32.8%
(2)

400
300

s3

r=87.6%
(3)

200
100

ym (n)

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]
200
(3)
100
0

–100

ym (n) −y (n)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]amplitude [µm]–20s2amplitude [µm]amplitude [µm](b)0r=10.7%(1)

amplitude [µm]

0 MPa

s1

4589

200
(3)
150
100
50
0
y (n)
–50
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]Fig. 4. In vivo measurement of mean waveform of changes in thickness of human carotid artery of 32-year-old subject. (a) Elasticity
image obtained from maximum change in wall thickness. (b) Measured change in thickness waveform, ym ðnÞ, for each heartbeat.
(c) Diﬀerences, ym ðnÞ  yðnÞ, between mean waveform and measured waveforms, ym ðnÞ. (d) Measured mean waveform, yðnÞ.

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

J. T ANG et al.

the change in wall thickness. A model waveform is ﬁtted tothe measured waveform as follows: A model waveform,
Hk ðnÞ, is ﬁtted to a mean waveform, yðnÞ (n ¼ 1; 2; . . . ; N),of changes in thickness, ym ðnÞ (m ¼ 1; 2; . . . ; M; n ¼ 1;
2; . . . ; Nm ), measured for M heartbeats. The mean waveform,yðnÞ, is obtained usingM
1 XyðnÞ ¼ym ðnÞ ðn ¼ 1; 2; . . . ; NÞ;
M m¼1

N
X

hk ðnÞyðnÞ ¼ 0:

ð2:6Þ

n¼1

Then, the constant Abk that minimizes k is determinedusing
hk ðnÞ  yðnÞn¼1:
Abk ¼ NX2
jhk ðnÞj
n¼1

n¼1

where Ak is a constant that expresses the diﬀerence inamplitude between the measured and model waveforms. To
determine the constant Abk that minimizes k , the partial
derivative of k with respect to Ak is set to zero.
!
N
@2k
@ X
2
¼
jAk hk ðnÞ  yðnÞj ¼ 0:
ð2:5Þ@Ak n¼1@Ak

ð2:7Þ

The constant Abk determined by eq. (2.7) is substituted intoeq. (2.4) to calculate the residual diﬀerence k;min , whichshows the diﬀerence between the measured and model
waveforms expressed asvﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
u
2

u
N

X
u


u
hk ðnÞyðnÞ

N
uX


n¼1
:
ð2:8Þ
k;min ¼ ujyðnÞj2 
u
N
X
u n¼1
2
t
jhk ðnÞj
n¼1

The bias error bk is deﬁned asvﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
u
2

u
N

X
u


u
hk ðnÞyðnÞuk;minn¼1:
bk ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ u
u1  N
N
N
X
X
u
X
2
2
t
2
jhk ðnÞj jyðnÞj
jyðnÞj
n¼1

n¼1

n¼1

ð2:9Þ
The amplitude of the model waveform, hk ðnÞ, is preset so
as to give an elastic modulus Eh of 1 MPa. The elastic
modulus E;m;n is determined using one of the estimatedconstants, Abk , with the least bias error using

ð2:3Þ

where tn and Tn are the times at the nth point after and before
linear interpolation.
Then, as shown in Fig. 3, the amplitude of the interpolated
model waveform, hk ðnÞ, is equalized to the measured mean
waveform, yðnÞ, by minimizing the root mean squared
diﬀerence, k , between the model waveform, hk ðnÞ, and themeasured waveform, yðnÞ, using
vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
u N
uX
ð2:4Þ
k ¼ tjAk hk ðnÞ  yðnÞj2 ;Equation (2.5) is rewritten as

fhk ðnÞg2  2

n¼1

ð2:2Þ

Hk ðnÞ  Hk ðn  1Þ
 ðtn  Tn1 ÞTn  Tn1

ðn ¼ 2; 3; . . . ; N; hk ð1Þ ¼ Hk ð1ÞÞ;

N
X

N
X

where the least number of sampled points in a heart cycle
among M heartbeats is used for the number of sampled
points N of the mean waveform.
In this paper, three types of model waveform, Hk ðnÞ(k ¼ 1; 2; 3), were prepared for the intima, media, andadventitia regions because mechanical properties of theselayers are diﬀerent from each other.10) In a measured data
showing a typical layered structure in the elasticity image,
three layers were manually assigned, as shown in Fig. 1(a).
As shown in Fig. 1(b), three model waveforms for the
intima, media, and adventitia regions were obtained byaveraging changes in thickness within the respective
assigned regions.
Since the cardiac cycle diﬀers among the subjects and thesampling frequency diﬀers among the types of equipments
used, the number of sampled points, which is related to thechange in thickness waveform for one heart cycle, also
diﬀers. Therefore, as shown in Fig. 2, the model waveform,
Hk ðnÞ, was expanded or compressed to the same time length
as the measured mean waveform, yðnÞ, in the time axis, andthen, linearly interpolated. The model waveform after
interpolation, hk ðnÞ (n ¼ 1; 2; . . . ; N), can be determinedusing
hk ðnÞ ¼ Hk ðn  1Þ þ

2Ak

Eh ¼

1
b
Ak

½MPa:

ð2:10Þ

Furthermore, the normalized random error r between themean waveform yðnÞ and the measured waveforms ym ðnÞ ofM heartbeats is deﬁned asvﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!
u X
M
u1 N 1 X
2ujym ðnÞ  yðnÞjuN
u n¼1 M m¼1:
ð2:11Þ
r¼u
N
u
1X
t
2
jyðnÞj
N n¼1
r shows the average amplitude of random errors during M
cardiac cycles.
3.

In vivo Experiments

Figures 4 and 5 show the measurement of the meanchange in thickness waveforms and the matching process fora 32-year-old male subject, respectively. Figure 4(a) shows
the cross-sectional elasticity image of the carotid arterial
wall obtained from the maximum change in thickness. In
systole, the artery diameter is expanded by an increase ininternal pressure, and the wall thickness becomes smaller asshown by the measured mean waveforms yðnÞ in the regions

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

J. T ANG et al.

amplitude [µm]amplitude [µm]

(a)

(1)

0

amplitude [µm]

s2

s1

(2)

0

–20

–20
–40

ym (n)–40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]amplitude [µm]ym (n)

4591

s3

200
150

(3)

100
50
0
–50

ym (n)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]model intima

0–10model adventitia

–20

–20

amplitude [µm]

0

(2)

–20

(1)

0

–20
–40

A in =0.5–40
b in =46.5%
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]amplitude [µm]amplitude [µm](1)0A me =3.8–40
b me =41.7%
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]

(e)

(2)

–20

A in =3.1–40
b in =44.0%
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s](d)0

A me =0.5–40
b me =50.7%
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]
0

(2)

–20

A ad =3.4
b ad =53.3%
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]E=1/A meA ad =0.7–40
bad =48.9%
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]E=1/Ain

200

(3)

150
100
50

A in = 11.3
b in =49.2%
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]
200
(3)
150
100

amplitude [µm](1)0amplitude [µm]0.2 0.4 0.6 0.8 1.0 1.2
time [s]amplitude [µm]0amplitude [µm]amplitude [µm]

–30

(c)

matching

model media–20

amplitude [µm]

(b)

0

50

A me = 14.4
b me =40.7%

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]
200
(3)
150
100
50
A ad = 12.9
b ad =51.6%

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time [s]A<0be deleted

3mm
(f)

4 MPa

new
elasticity
image
3.75 mm0 MPa
Fig. 5. Matching process using three model waveforms for 32-year-old subject. (a) Measured mean waveform, yðnÞ. (b) Model
waveforms, Hk ðnÞ. (c) Measured mean waveform, yðnÞ, and matched model for intima region, h1 ðnÞ. (d) Measured mean waveform,yðnÞ, and matched model for media region, h2 ðnÞ. (e) Measured mean waveform, yðnÞ, and matched model for adventitia region, h3 ðnÞ.
(f) Elasticity image obtained from estimated constant, Abk , with least bias error.

s1 and s2 in Figs. 4(d-1) and 4(d-2) obtained from the
waveform of each beat, ym ðnÞ, shown in Fig. 4(b). In suchcases, the estimated constant Abk for matching the model
waveform hk ðnÞ to the measured one is positive as shown in

Figs. 5(c), 5(d), and 5(e). The elasticity of such a region was
calculated using the estimated constant Abk for the model thatgives the least bias error. Figures 4(d) and 5(a) show thesame measured mean waveform yðnÞ. In the region s3 inFig. 4(a), the measured waveform shows an increase in wall
thickness, as shown in Fig. 5(a-3). In such a region, thechange in wall thickness is considered not to be measured
successfully. The reason for such a region showing an
increase in the wall thickness can be explained as follows: Achange in wall thickness is obtained from the diﬀerence
between displacements at two points along an ultrasonicbeam. When this set of two points very closely approaches tothe lumen, the point on the luminal side is located in thelumen. Because ultrasonic beams are set to be perpendicular
to the wall, these beams are also perpendicular to thedirection of blood ﬂow. Therefore, the velocity detected in
the lumen becomes zero or, if detected, becomes small andrandom because echoes cannot be obtained from the same
blood particle due to blood ﬂow and the particle cannot be
tracked during an entire heart cycle. Thus, the time integral
of velocity (= displacement) at the point on the luminal side
becomes almost zero. On the other hand, the displacement at
another point in the posterior wall is positive (the distance
between the ultrasonic probe and the posterior wall
increases). Therefore, the change in thickness obtained bysubtracting the displacement at the luminal point from that at
the point in the posterior wall becomes positive (showing an
increase in thickness). In such cases, the estimated constant
Abk for each matched model is negative, as shown in
Figs. 5(c-3), 5(d-3), and 5(e-3). Therefore, the elasticity insuch a region was not displayed. Furthermore, the random
error r was obtained from the diﬀerence ym ðnÞ  yðnÞ shownin Fig. 4(c). Random errors are described at the top of
Fig. 4(b).
Figure 5(f) shows the elasticity image obtained from theestimated constant Abk with the least bias error. In Fig. 5(f),the elasticity is color-coded only when the constant Abk islarger than zero.
4.

Discussion

In Fig. 6, bias error is plotted as a function of the distancefrom the lumen along each ultrasonic beam in the region
surrounded by a green dashed line in Fig. 5(f). Figure 6(a)shows bias errors obtained using only one model waveform
for the media region. Figure 6(b) shows the least bias error
obtained using three model waveforms for the intima, media,
and adventitia regions. In this study, three model waveforms
were prepared because the mechanical properties of theintima, media, and adventitia are diﬀerent. Using these three
types of model, the bias error, which shows the diﬀerence
between the model and measured waveforms, can be
reduced as shown in Fig. 6. However, in this paper, we do
not consider diﬀerences between a healthy subject and apatient. Further investigation is necessary to optimize themodel waveforms.
The main purpose of this study is to evaluate the measured
elasticity image on the basis of the measured waveform of
the change in wall thickness. In this study, a region showing
an increase in wall thickness in systole was rejected not to be
displayed in an elasticity image because such a change inwall thickness is not reliable. In our future work, bias andrandom errors should be used for evaluating the reliability ofa measured elasticity image quantitatively.

J. T ANG et al.

(a)

100%

bias error [%]Jpn. J. Appl. Phys., Vol. 44, No. 6B (2005)

80%

using only one model for media

60%
40%
20%
0

(b)

0

600

distance from lumen [ µ m]

1200

100%
using three models for intima,
media, and adventitia

80%

bias error [%]

4592

60%
40%
20%
0
0

600

distance from lumen [ µ m]

1200

Fig. 6. Bias error distribution in region surrounded by green dashed line
in Fig. 5(f). (a) Case of using only one model for media. (b) Case of using
three models for intima, media, and adventitia.

5.

Conclusions

In this paper, a method for evaluating the reliability of theelasticity image by calculating the error between themeasured and model waveforms of the change in wall
thickness is proposed. Using the proposed method, a region
showing an increase in wall thickness in systole (considered
not to be reliable) was automatically rejected not to be
displayed in an elasticity image. Further investigation isnecessary to utilize bias and random errors for thequantitative evaluation of a measured elasticity image.

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