Japanese Journal of Applied Physics
Vol. 46, No. 7B, 2007, pp. 4900–4907#2007 The Japan Society of Applied Physics

Basic Study on Detection of Outer Boundary of Arterial Wall
Using Its Longitudinal Motion
Takanori N UMATA, Hideyuki H ASEGAWA, and Hiroshi KANAI
Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan(Received November 24, 2006; revised March 15, 2007; accepted March 22, 2007; published online July 26, 2007)

The longitudinal motion of the arterial wall can be observed in a B-mode image obtained by recent ultrasonic diagnostic
equipment. In this study, the longitudinal displacement was quantitatively estimated by the block matching of RF echoes using
cross correlation. However, the estimated longitudinal displacement is discrete depending on the spacing between two
ultrasound beams of about 0.1 mm. Such an accuracy is not suﬃcient for tracking the region of interest (ROI) in the arterial
wall. Therefore, the spacing of 0.1 mm is reduced using the interpolation of measured RF echoes to improve the tracking of
the ROI in the displacement estimation. In this study, the optimum parameter in the interpolation was investigated. The spatial
distribution of longitudinal displacements along the arterial radial direction was estimated using the optimum parameter.
There were signiﬁcant diﬀerences between the longitudinal displacements in the arterial wall and those in the region
considered to be tissue outside the arterial wall. These results show the possibility of using this method to identify the outerboundary of the adventitia, which has not been achieved by conventional ultrasonic imaging.
[DOI: 10.1143/JJAP.46.4900]
KEYWORDS: longitudinal displacement, complex interpolation, block matching, correlation function, adventitia position

1.

Introduction

Diagnosis of arteriosclerosis is important because it posesa serious risk of cardiovascular events. We developed anoninvasive method for measuring the elasticity of thearterial wall.1–7) In this method, the boundaries of the arterial
wall are assigned by manual inspection of a B-mode image.
Therefore, a method of identifying the boundaries is required
for the objective and reproducible estimation of theelasticity.
The carotid artery exhibits the expansion and contraction(radial movement) due to the change in blood pressureduring one cardiac cycle. At the same time, the carotid arterymoves in its longitudinal direction (longitudinal movement).
Recently, there have been some studies on the measurementof the arterial longitudinal displacement.8–12) In thesestudies, the displacement is estimated by the block matching8–14) of received echoes in two frames using the crosscorrelation function. The longitudinal displacement of thearterial wall is less than 1 mm during one cardiac cycle.11)
Generally, the measurement in the carotid artery is carriedout using a 10 MHz probe. The correlation function, which
is a function of the longitudinal displacement, is discrete
depending on the spacing between two ultrasonic beams of
about 0.1 mm in the 10 MHz probe case. Therefore, thelongitudinal displacement cannot be measured with anaccuracy of less than 0.1 mm. Such an accuracy is not
suﬃcient for tracking the region of interest (ROI) in thearterial wall.
In this study, the spacing of 0.1 mm is reduced by theinterpolation of measured RF echoes to improve the trackingof the ROI in the displacement estimation. The purpose ofthis study is to determine the optimum number of interpolated lines. This parameter was determined by evaluating thevariation in correlation coeﬃcient due to the inherent noiseof the measurement system. Using the determined number ofinterpolated lines, the longitudinal displacement and longitudinal stretch of the carotid artery were estimated. InE-mail address: hkanai@ecei.tohoku.ac.jp

addition, the spatial distribution of longitudinal displacements of the carotid arterial wall along its radial direction
was obtained, and the outer boundary of the arterial wall was
estimated using the displacement distribution.
2.

Methods

2.1 Complex linear interpolation of RF echoes
The ultrasound information was provided by an Aloka
SSD-6500 ultrasonic system. Using an ultrasonic probe of
10 MHz frequency, an ultrasound beam scans positions
along the carotid artery in its longitudinal direction withintervals of 0.1 mm. At each position, RF echoes are
acquired at a sampling frequency of 40 MHz. A B-mode
image can be constructed with a two-dimensional (radiallongitudinal) data set obtained by one scan, which is called aframe. The RF signal at beam position b and depth d in then-th frame is denoted by rf n ðd; bÞ.
To reduce the spacing of two beams, K lines areinterpolated between the original RF lines. Complex linear
interpolation was used in this study. In the case of theinterpolation of a real function, the interpolated waveform is
distorted by the phase diﬀerence of echoes along two
diﬀerent beams. Therefore, in this study, the complex linear
interpolation was applied to the quadrature demodulated
complex signal, zn ðd; bÞ, of RF echo, rf n ðd; bÞ, as follows:
 


0

z d; b þ knKþ1 ðK þ 1  kÞjzn ðd; bÞj þ kjzn ðd; b þ 1Þj
¼
;
ð2:1Þ
Kþ1
 

k
arg z0n d; b þ
Kþ1ðK þ 1  kÞ arg½zn ðd; bÞ þ k arg½zn ðd; b þ 1Þ
¼; ð2:2Þ
Kþ1
ðk ¼ 0; 1; 2; . . . ; KÞ;
where z0n ðd; b þ k=ðK þ 1ÞÞ is the interpolated complex
signal. Each interval of two beams is given by wðKÞ ¼
0:1=ðK þ 1Þ mm when K lines are interpolated between theoriginal RF lines.

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T. NUMATA et al.

2.2 Block matching using cross-correlation function
The radial and longitudinal displacements, xbr ðnÞ and b
xl ðnÞ,at the n-th frame are estimated using the correlation function
between two frames including the interpolated complex
signals.
First, the region of interest (ROI) is assigned in the initialframe. The center of the ROI at a time t ¼ nT (n-th frame)is deﬁned as d ¼ dn and b ¼ bn . Then, the complex correlation coeﬃcient, ðn; m; lÞ, between the n-th frame ata time t ¼ nT and the (n þ N)-th frame at a time t ¼
ðn þ NÞT is calculated as follows:

stage
controller

ultrasonic probe
radiallongitudinalwatersilicone tube
automatic XYZ stage

ðn; m; lÞwater tankFig. 1. Experimental system using a silicone tube.

2 mm
ð2:3Þ

5 mm

m¼M=2 l¼L=2Zm;l ¼ z0n ðd þ mD; b þ lwðKÞÞ;

ð2:4Þ

0
Zm;l
¼ z0nþN ðd þ mD; b þ lwðKÞÞ;ð2:5Þ
vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
u
M=2
L=2
X
X
u
1
ð2:6Þ
¼t
jZm;l j2 ;ðM þ 1ÞðL þ 1Þ m¼M=2 l¼L=2
vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
u
M=2
L=2
X
X
u
1
0
 ¼t
jZ 0
j2 ; ð2:7Þ
ðM þ 1ÞðL þ 1Þ m¼M=2 l¼L=2 mþm;lþlwhere  denotes the complex conjugate, D is the samplinginterval (19.25 mm) in the depth direction, m and l are theradial and longitudinal shifts between ROIs in the n-th and(n þ N)-th frames, respectively, T is the period between
two consecutive frames [T = 1/(frame rate) = 5.2 ms],and NT is the frame interval for the calculation of thecorrelation function. In this study, N was set to 10.
The complex correlation function ðn; m; lÞ between
the n-th and (n þ N)-th frames is obtained at each of theradial and longitudinal shifts, mn and ln , of the ROI in
the (n þ N)-th frame relative to the position of the ROI in
the n-th frame. The radial and longitudinal displacements aredn anddetermined from the radial and longitudinal shifts, mdln , which maximize the magnitude of the complex
correlation coeﬃcient, ðn; m; lÞ. From the estimateddn , the radial anddn and lradial and longitudinal shifts, mlongitudinal average velocities, vr ðnÞ and vl ðnÞ, between then-th and (n þ N)-th frames are respectively given by
dn
c0 m
;
ð2:8Þ
vr ðnÞ ¼
2 fs NTdnwðKÞl;ð2:9Þvl ðnÞ ¼
NT
where c0 and fs are the sound speed and sampling frequency,
respectively. Then, the radial and longitudinal displacements, b
xr ðnÞ and b
xl ðnÞ, at the n-th frame are estimated asfollows:b
xr ðn þ 1Þ ¼ b
xr ðnÞ þ vr ðnÞT;bxl ðn þ 1Þ ¼ b
xl ðnÞ þ vl ðnÞT:

ð2:10Þ
ð2:11Þ

The instantaneous position of the ROI in the (n þ 1)-thframe is determined by bxl ðnÞ and b
xr ðnÞ.

anterior wall

lumen

radial1¼
ðM þ 1ÞðL þ 1Þ 0
M=2
L=2
X
X
0

Zm;l Zmþm;lþl;

ROI
posterior wall

longitudinal
Fig. 2. B-mode image of the silicone rubber tube.

3.

Basic Experiments Using a Silicone Tube for
Determining the Number of Interpolated Lines

3.1 Displacement estimation
Figure 1 shows the experimental system. In the basic
experiment, an ultrasonic probe was moved with a constant
speed in the longitudinal and radial directions of a silicone
tube using an automatic stage.
Figure 2 shows a B-mode image of the silicone tube. In
this experiment, the luminal surface of the silicone rubber
was made rough because echoes from a smooth luminal
surface are uniform in the longitudinal direction of the tube
and there will be no distinct feature to be tracked. The ROI
size was set to 2.8 mm (longitudinal)  0.2 mm (radial). The
radial width was set to 0.2 mm since the size should besmaller than the intima-media thickness.
3.2 Evaluation of random error
Random error was deﬁned as the standard deviation of theestimated velocity. One of sources of random error is thevariation in correlation coeﬃcient due to noise. To evaluate
the eﬀects of noise, the stage was kept still, and the variance4901

t = 0 ms
t = 42 ms

amplitude

60
50
40
30

10
10

20

30
40
50
beam number

0

0

–1

(a)

K=0

0.5
0

–0.5
–1
0

0.1

0.2
0.3
time [s]

0.4

0.5

0.4

0.5

K=6

(b)
0.99
0.98
0.97
0.96
0.95

–0.1

–0.05

–9.6

–4.8

0
0.05
shift [mm]
0

4.8

0.1

mm/s

9.6

K=0

0.99
0.98
0.97
0.96
0.95

0.2
0.3
time [s]Fig. 5. (a) Estimated speed of silicone tube and (b) the means andstandard deviations of the correlation coeﬃcients for K ¼ 6. The frameinterval NT is 0.01 s.

1
(b)

0.1

1

1

–1.5

–1.37

0

60

correlation coefficient0

1.5
estimated speed [mm/s]

0.5

–1.5

Fig. 3. Amplitudes of RF echoes at t ¼ 0 and 42 ms.

correlation coefficient

1

1.37

K=6

(a)

–0.5

20

0

1.5

mm/s

70

T. NUMATA et al.

estimated speed [mm/s]

Jpn. J. Appl. Phys., Vol. 46, No. 7B (2007)

–0.1

–0.05

–9.6

–4.8

0
0.05
shift [mm]
0

4.8

0.1
mm/s
9.6

Fig. 4. (a) Estimated speed of silicone tube and (b) the means andstandard deviations of the correlation coeﬃcients for K ¼ 0. The frameinterval NT is 0.01 s.

of the correlation coeﬃcient at each shift was evaluated.
There should be no variance when there is no noise. Figure 3

shows the amplitudes of echoes from the silicone tube at
t ¼ 0 and 42 ms. Although the stage is still, there is adiﬀerence between echoes. Figure 4(a) shows the estimatedvelocity without interpolating RF signals (K ¼ 0). In
Fig. 4(b), plots and vertical bars show the means andstandard deviations of the correlation coeﬃcient, respectively. In Fig. 4, the frame interval NT is 10 ms. The
mean correlation coeﬃcient at a shift of 0 mm is much larger
than those at 0:1 mm in comparison with the standard
deviation. Therefore, there is no random error in estimatedvelocity as shown in Fig. 4(a).
However, as shown in Fig. 5, the mean correlationcoeﬃcient at a shift of 0 mm is not suﬃciently larger thanthose at 0:014 mm when the number K of interpolatedlines and frame interval NT are 6 and 10 ms, respectively. Therefore, there are random errors in the estimatedvelocity as shown in Fig. 5(a). The random error erandom inthe estimated displacement during the period of 2 frames isdeﬁned as follows:

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!2ﬃ
u N 1
N 1
u1 X
X
1
erandom ¼ t
vbl ðnÞNT 
vbl ðnÞNT :
N n¼0
N n¼0
3.3 Evaluation of bias error
Figures 6(a) and 6(b) show the correlation coeﬃcient at
each shift ln and the estimated and true (assigned stage
speed) longitudinal velocities at each time, respectively. In

ð3:1Þ

Fig. 6, the correlation coeﬃcient reaches its maximum value
at a shift corresponding to the true velocity. However, inmost cases, the true shift is located between echo lines(including interpolated ones), and the shift that is closest to4902

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0.3

Jpn. J. Appl. Phys., Vol. 46, No. 7B (2007)

7

K= 8

∆N = 2
∆N= 4
∆N= 6
∆N= 8
∆ N = 10∆ N = 12
∆ N = 14
∆ N = 16
∆ N = 18

0.2

random error [ µ m]

60true shift

2(b)theoretically estimated speed

1.5[mm/s]

estimated speed [mm/s]

shift [mm]0.1

estimated speed
stage speed

1.6

theoretical bias error

1

5
4
K= 4

3
2

(a)

1
K= 3

0.5

0

0

0

0

0

0.05

0.1

0.15

0.2

10

K=2

K=1

K=0

30

20

40

50

theoretical bias error [ µ m]

0.25

time [s]20–0.1∆N = 2
∆N= 4
∆N= 6
∆N= 8
∆ N = 10∆ N = 12
∆ N = 14
∆ N = 16
∆ N = 18

random error [ µm]

–0.2

0.92

0.93

0.95
0.94

0.97
0.96

0.98

1

0.99

(a)

correlation coefficientFig. 6. (a) Correlation coeﬃcient at each ROI shift ln . (b) Estimated andtrue longitudinal velocities at each time.

the true shift is selected to be the estimate when there is nonoise. This shift which is the closest to the true shift isdenoted by the theoretically estimated shift (velocity). Thus,
the theoretical bias error in the estimated velocity can bedeﬁned as the diﬀerence between the theoretically estimatedvelocity and the true one. The maximum value of thetheoretical bias error ebias is given by half of the intervalwðKÞ between interpolated echo lines as follows:
ebias ¼

wðKÞ
0:1
¼
:
2
ðK þ 1Þ  2

ð3:2Þ

15

10
K= 8

5
K= 4

(b)

K= 8
K= 4

0
0

5

10

K=2

K= 3

15

theoretical bias error [ µ m]

20

Fig. 7. (a) Random errors erandom in the estimated displacements at eachtheoretical bias error ebias . (b) Enlarged view of (a).

Theoretical bias error ebias will be reduced by increasing thenumber of interpolated lines, K.
3.4 Optimum number of interpolation lines
The optimum number of interpolation lines, Kopt , wasdetermined by theoretical bias error ebias and random error
erandom . Bias error ebias can be theoretically reduced byincreasing the number of interpolated lines, K. However, asmaller interval of echo lines (means a larger K) leads toincrease in random error erandom due to the variance incorrelation coeﬃcient. Therefore, the number of interpolation lines Kopt , at which random error erandom is similar tobias error ebias , is suﬃcient. Since erandom increases withincreasing K, the estimation error cannot be reduced by a K
larger than Kopt .
Figure 7(a) shows the random error erandom at eachtheoretical bias error ebias . Figure 7(b) is an enlarged viewof Fig. 7(a). In Figs. 7(a) and 7(b), the theoretical bias error
ebias is not signiﬁcantly reduced at a K that is larger than 8,
and ebias is reduced, similar to random error erandom at K ¼ 8.
Therefore, the number K of interpolation lines was determined to be K ¼ 8 (beam interval of 11 mm).
4.

Basic Experiments Using Silicone Tube for Detectionof Outer Boundary of Wall
Figure 8 shows the experimental system. The diﬀerence

ultrasonic probe

water

silicone tube
silicone board

water tank

automatic XYZ stage

Fig. 8. Experimental system for the boundary detection.

among the longitudinal displacements around the outerboundary of the wall was investigated using a silicone tube
on a silicone plate mimicking the external tissue. The stage
speed was set to 4 mm/s, and the longitudinal displacementof the silicone tube was estimated. The frame rate was
1=T ¼ 192 Hz.
Figure 9 shows a B-mode image of the silicone tube on
the silicone plate. The longitudinal displacements were
estimated at depth positions A, B, C, and D. Figure 10 shows4903

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T. NUMATA et al.

2 mmwall of tube

5 mm

0

anterior wallof silicone tube

ROI

A

posterior wallof silicone tube
B
C
D

correlation coefficientlongitudinal displacement [mm]

1.2
1

A

0.8

D

0.6

1.5

2

2.5

3

3.5

4

(a)
4.5

depth fromlumen–far wall boundary [mm]

0.5
0.4 A

D

0.3

B
C

0.2
0.1
0

(b)
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

depth fromlumen–far wall boundary [mm]

1
0.99
0.98
0.97
0.96

(c)
0.95

C

1

0.6

silicone board

Fig. 9. B-mode image of the silicone tube on the silicone plate.

0.5

0.7

longitudinal

depth

displacement [mm]

0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

depth fromlumen–far wall boundary [mm]

0.4
0.2

Fig. 11. (a) RF echo from the silicone tube and silicone plate. (b)Longitudinal displacement distribution along the ultrasonic beam. (c)Means and standard deviations of the maximum correlation coeﬃcientsaveraged during the period from 0 to 0.15 s.

B

0
0

0.05

0.1

0.15
time [s]

0.2

0.25

0.3

Fig. 10. Longitudinal displacements at depth positions A, B, C, and D ateach time.

8000
3.0 mm6000

5.

In vivo Measurement

The longitudinal and radial displacements of a carotid
artery of a 25-year-old male are estimated using the4000
amplitude

the estimated longitudinal displacements at each time at
depth positions A, B, C, and D. The longitudinal displacements were diﬀerent depending on the depth.
In Fig. 11, the longitudinal displacements at a time of
0.15 s are shown at each depth position along a line.
Figure 11 shows (a) RF echoes from the silicone tube andthe silicone board, (b) the longitudinal displacement distribution along the line, and (c) the means and standard
deviations of the maximum correlation coeﬃcients from 0 to
0.15 s. In Fig. 11(b), the longitudinal displacement rapidly
decreased at a depth position of 3.2 mm. Figure 12 shows the
RF echo from the same silicone tube without the silicone
plate. In Fig. 12, the thickness of the wall is found to be
3 mm. Therefore, the longitudinal displacement was found tochange at the boundary.

2000
0

–2000
–4000
–6000

–8000

8

9

10

11
12
depth [mm]

13

14

15

Fig. 12. RF echo from the silicone tube.

determined optimum number of interpolation lines Kopt ¼
8 and a frame interval NT ¼ 44 ms (N ¼ 4, T ¼
1=90 Hz ¼ 11 ms). Figure 13 shows (a) a B-mode image ofthe carotid artery, (b) an electrocardiogram, and (c) and (d)estimated longitudinal and radial displacements, respective-4904

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T. NUMATA et al.

intima–media

beam position 1
2 mm

5 mm

0

B

longitudinal displacement [mm]

(a)

A
C
(a)

systole

1.2

(c)

1
0.8

correlation coefficientheart direction
depthradial displacement [mm] longitudinal displacement [mm](b)depth Adepth Bdepth C

0.6
0.4
0.2

1.2
1
0.8
A
0.6
0.4

B

0.2

C

0
0
1

0.2

0.4

0.6

0.8

(b)
1

0.9

0.8

(c)

0
–0.2
0
0.2

0.7
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
(d)

0.1

0.2

0.4

0.6

0.8

1

1.2

Fig. 14. (a) RF echo from the region from intima (0 mm) to a depth of
1.3 mm. (b) Longitudinal displacement distribution along the ultrasonicbeam. (c) Means and standard deviations of the maximum correlationcoeﬃcients for the period from 0 to 0.13 s.

–0.1

0

0

depth from intima position [mm]depth Adepth Bdepth C

0

–0.2

1.2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
time [s]Fig. 13. (a) B-mode image of the carotid artery. (b) Electrocardiogram.
(c) Estimated longitudinal and (d) radial displacements of the carotid
arterial wall.

ly, of the carotid arterial wall during one cardiac cycle atpositions A, B, and C. The waveforms of the longitudinal
and radial displacements are similar at each position.
However, the amplitudes of the longitudinal displacements
are diﬀerent among the measured positions. The diﬀerencein amplitude between depth positions B and C is larger thanthat between depth positions A and B.
Figure 14 shows (a) RF echo in the range from 0 mm
(luminal surface) to 1.3 mm along an ultrasonic beam, (b)the longitudinal displacement at a time t ¼ 0:13 s in
Fig. 13(c), and (c) the means and standard deviations of
the maximum correlation coeﬃcients for the period from 0
to 0.13 s. This time of t ¼ 0:13 s was determined by the ﬁrst

peak of the longitudinal displacement shown in Fig. 13(c).
The correlation values are suﬃciently large at points A, B,
and C. However, there are diﬀerences in longitudinal
displacements among these points.
Figure 15 shows a B-mode image of the carotid artery of
the same subject. The longitudinal displacements were
estimated at two diﬀerent ultrasonic beams. Figure 16 shows
the longitudinal displacements at beam positions 1 and 2 atthe depth A plotted as a function of time.
As shown in Fig. 13, the carotid artery moves toward theheart when the heart ejects blood. The longitudinal displacements decreased between 0.1 and 0.2 s. It is supposed thatthe carotid artery is pulled by the movement of the base of
the heart toward the apex during systole from 0 to 0.1 s.
Then, the longitudinal displacement is decreased by theshear stress of blood ﬂow from 0.1 to 0.2 s.
In Fig. 14, the longitudinal displacement decreased
rapidly in the region of around 0.6 mm. It is thought thatthe longitudinal displacement decreased around the outerboundary of the arterial wall, which is similar to the basic
experimental results in Fig. 11(b).
The longitudinal displacements at diﬀerent beam positions were compared to evaluate the stretch of the carotid

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longitudinal displacement [mm]Jpn. J. Appl. Phys., Vol. 46, No. 7B (2007)

4 mm

5 mm

2 mm

depth A1

longitudinal displacement [mm]beam position 1
beam position 2

1
0.8
0.6
0.4
0.2
0

0

0.2
0.4
0.6
0.8
1
1.2
depth from intima position [mm]Fig. 17. Longitudinal displacement distributions at beam positions 1 and2.

2

beam position
heart direction

0.4

is detected by the curve ﬁtting of the correlation function15)
because the variance in correlation coeﬃcient also leads toerror in the curve ﬁtting. The estimation accuracies of the
proposed method and the curve ﬁtting method should becompared in future work.
In addition, there are parameters whose optimums were
not investigated in this study. For example, the optimum
frame interval depends on the velocity of an object whereas
the frame interval NT was set to be constant in thisstudy. The detectable displacement xl between two frames
over a line interval is shown using the longitudinal velocity
vl and frame interval NT as follows:

0.2

xl ¼ vl NT  wðKÞ:Fig. 15. B-mode image of the carotid artery.

ECG

1.2

beam position 1
beam position 2

1
0.8
0.6

ð6:1Þ

Therefore, N should be

0

–0.2

0

N 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
time [s]Fig. 16. Longitudinal displacements at beam positions 1 and 2.

arterial wall. Figure 17 shows the longitudinal displacement
distributions along the two beams. In this ﬁgure, thelongitudinal displacements at a time of 0.13 s are shown ateach depth position along the corresponding lines.
Although the longitudinal displacements on two beam
positions were diﬀerent in Fig. 16, the longitudinal displacement at beam position 1 is biased from that at beam position
2 as typically shown from 0.4 to 0.7 s, and this diﬀerence isconsidered to be a bias error. Therefore, it is considered thatthe diﬀerence between the longitudinal displacements was
small and the carotid arterial wall between two beams was
not stretched in the longitudinal direction. However, therapid decreases in longitudinal displacements around theouter boundary of the arterial wall were found in bothbeams.
6.

1.2

Discussion

In this study, the optimum number of interpolation lines,
Kopt , was investigated by evaluating the variance incorrelation coeﬃcient due to noise. Such an investigationis also important for methods in which the correlation peak

wðKÞ
:
vl T

ð6:2Þ

However, in in vivo measurement, the correlation coeﬃcientis reduced by increasing the frame interval because the echopattern in a ROI is changed by the arterial wall deformation.16) Therefore, the frame interval needs to be as short as
possible, and N should be wðKÞ=ðvl TÞ.
In this study, a constant frame interval (N ¼ 4 or 10)
was employed. However, the following procedure will
provide a superior displacement estimate in in vivo measurement, although it requires more computation. First, thevelocity of longitudinal movement is roughly estimated by aconstant frame interval, and the optimum frame interval
NT is determined at each time using the estimatedvelocity. Then, the longitudinal displacement is estimatedmore accurately using the optimum frame interval at eachtime.
In addition, the optimum ROI size was not investigated.
Figure 18 shows the amplitudes of RF echoes from the
carotid artery along the longitudinal direction. The longitudinal displacement cannot be detected when there is novariation in echo signals in the ROI. Therefore, the ROI size
should be larger than the cycle of the variation of echo
signals along the artery. As shown in Fig. 18, the ROI size
employed in this paper is suﬃciently larger than the cycle ofthe variation in echo signals. On the other hand, the ROI size
should be smaller than the inverse of the spatial frequency of4906

Jpn. J. Appl. Phys., Vol. 46, No. 7B (2007)

T. NUMATA et al.

along the arterial radial direction was estimated in in vivo
measurement in the carotid artery. The longitudinal displacement rapidly decreased around the position that is
considered to be the outer boundary of the arterial wall, andthis result was similar to that in the measurements in asilicone tube on a silicone plate mimicking an arterysurrounded by external tissue. These results show that the
proposed method has the potential to be used in detecting theouter boundary of adventitia on the basis of the measured
displacement distribution.

3500
ROI size

amplitude

3000

2.8 mm

2500
2000
1500
1000
500
0

0

20

40

60

80

100 120 140

beam number
Fig. 18. Amplitude of RF echo along the longitudinal direction in in vivo
measurement.

the deformation of the arterial wall because the signal
pattern in the ROI will be signiﬁcantly changed by the
deformation with a larger ROI. The carotid artery measured
in this study showed almost no deformation in its longitudinal direction, and a large ROI was considered not toaﬀect the estimation accuracy so much. However, thecarotid arteries of other subjects should be measured toinvestigate the deformation of the arterial wall in its
longitudinal direction. Furthermore, the silicone rubber tube
used in the basic experiments of this study shows nodeformation. The deformation of the wall may aﬀect the
optimization of the number of interpolated echo lines.
As described above, although some remaining problems
require further investigation, the proposed method in thispaper is useful for a detailed investigation of vessel wall
movements.
7.

Conclusions

In basic experiments using a silicone tube, the number ofinterpolation lines for the estimation of the longitudinal
displacement was determined by considering bias andrandom errors. Using the optimum number of interpolationlines, the spatial distribution of longitudinal displacements

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