Jpn. J. Appl. Phys. Vol. 40 (2001) pp. 3918–3921
Part 1, No. 5B, May 2001c
2001
The Japan Society of Applied Physics

Optimization of Focal Position of Ultrasonic Beamin Measurement of Small Change in Arterial Wall Thickness
Masaru WATANABE∗ and Hiroshi K ANAI†
Department of Electrical Engineering, Tohoku University Graduate School of Engineering, Aramaki-aza-Aoba 05, Aoba-ku, Sendai 980-8579, Japan(Received November 23, 2000; accepted for publication February 5, 2001)

We have previously developed a method for measurement of a small change in thickness of the arterial wall during a single
cardiac cycle [H. Kanai, M. Sato, Y. Koiwa and N. Chubachi: IEEE Trans. UFFC 43 (1996) 791]. The resultant change inthickness is shown to be useful for the in vivo assessment of the regional elasticity of the arterial wall. Although the accuracyof the measurement of the change in thickness is found to be within 1 µm, it is affected by the interference of ultrasonic pulses.
In this study, we simulate the propagation of ultrasonic pulses transmitted and received by a linear probe. In the simulation
experiments, the ultrasonic pulses generated by a computer are reflected by a tube, which has a small change in wall thicknessof 10 µm. The optimum focal position of the ultrasonic beam is determined by evaluating the root-mean-square (rms) error in
the measured change in thickness.
KEYWORDS: atherosclerosis, change in thickness of arterial wall, computer simulation, interference, focus

1. Introduction
The increase in the number of individuals suffering frommyocardial or cerebral infarction, both of which are mainly
caused by atherosclerosis, has become a serious clinical problem. It is therefore important to diagnose atherosclerosis at
an early stage. However, to date, there is no method available
for the detection of minute changes in the elasticity of arterial
walls due to early-stage atherosclerosis.
We have developed a method for evaluating the elasticity
in each local region within about 400 µm from the arterial
wall1, 2) to diagnose its vulnerability to atherosclerotic plaque.
The small change in thickness of the arterial wall due to theheartbeat is accurately measured in each local region aroundthe focal area of the ultrasonic beam. From the resultant
change in thickness, the local strain and the elasticity of thearterial wall are noninvasively evaluated. By scanning the ultrasonic beam, the spatial distribution of the elasticity is also
obtained.
In this measurement, however, the accuracy is affected by
the interference of ultrasonic pulses. Figure 1 shows the measured waveform reflected from a rubber plate in a water tank.
The employed transmitted ultrasonic pulse is 7.5 MHz and1.0 µs long. In Fig. 1, the thicknesses of the rubber plates
are 0.40 mm and 0.65 mm. The anterior surface and posterior surface can be separated only as shown in Fig. 1(b). In
Fig. 1(a), the two surfaces cannot be separated due to the interference of reflective waves. If the optimum focal position is
employed, the effects of the interference will be reduced. Currently, in commercial ultrasonic diagnostic equipment, multiple focal locations are employed for multiple transmissions,but the pulse repetition frequency (PRF) is reduced. Our research objective is to determine the spatial distribution of theelasticy for a wider region and in more detail. For this purpose, the PRF cannot be reduced. In this study, therefore, we
simulate an in vivo measurement of the small change in thickness in the arterial wall during a single cardiac cycle. Fromthe simulation experiments, the focal position which is optimized to reduce the effects of the interference of ultrasonic∗ E-mail address: watanabe@us.ecei.tohoku.ac.jp(a)0.40 mm(b)

0.65 mmtime

Fig. 1. The measured waveform reflected from a rubber plate in a watertank. (a) The thickness of the rubber plate is 0.40 mm. (b) The thicknessof the rubber plate is 0.65 mm.

pulses is chosen.
2. Principle of Computer Simulation for Measurementof Small Change in Thickness
In the simulation, an arbitrarily shaped reflector in the x–
z plane of Fig. 2, which corresponds to a tube, is divided
into small scatterers. An electrical scanning linear probe with
(2N +1) elements in Fig. 2(a) is employed. The distance fromthe nth element (−N≤ i ≤ N ) to the focal point at a depth ofz = d f is given by d 2f + (n · h)2 , where h denotes theelement pitch. Thus, in order to focus the ultrasonic pulses
transmitted from (2N + 1) elements on the focal point, thefollowing delay time tn is added to the delay line connectedto the nth element:d f 2 + (N · h)2
d f 2 + (n · h)2
tn =
−
,
(1)
c0
c0
where c0 is the sound speed. The incidence angle, θni , fromthe nth element of the probe to the ith small scatterer on the† E-mail address: hkanai@us.ecei.tohoku.ac.jp3918

Jpn. J. Appl. Phys. Vol. 40 (2001) Pt. 1, No. 5B

M. WATANABE and H. K ANAI

surface of the reflector is determined by the regional shapeof the reflector and the geometrical relationship between thenth element and the reflector shown in Fig. 2(b). The ultrasonic pulse is reflected with the same angle θni , as shown in
Fig. 2(b). The receiving angle ϕi m from the ith small scatym (t) =W (t − tn ) · sin ωo (t − tn ) ∗−N ≤n≤N

terer to the mth element in Fig. 2(c) is determined in the samemanner. Thus, the RF ultrasonic wave ym (t) received at themth element is given by the summation of the reflective waves
at all scatterers {i} for the transmitted signal with an angularfrequency ω0 = 2π f 0 radiated from all elements as follows:δ(t − tni ) cos θni ∗ δ(t − ti m ) · cos θi m ,(2)iwhere tni and ti m denote the traveling time from the nth element to the ith small scatterer and that from the ith small
scatterer to the mth element, respectively, W (t) is the Hanning window which is four times the wavelength λ, δ(t) is
the Dirac delta function, and ∗ denotes the convolution operation. The terms cos θni and cos θi m denote the widths of theith small scatterer and the mth element, respectively, observed
from the propagating ultrasonic beam. The output signal y(t)of the probe, with the delayed summation of {ym (t)}, is givenbyym (t) ∗ δ(t − tm ).
(3)y(t) =

outer surface, vo (t), are given byvi (t) = (α + 1)π dmax f rot sin(2π f rot t)(4)vo (t) = απ dmax f rot sin(2π f rot t),(5)where α determines the ratio of vi (t) to vo (t).
In the following experiments, α = 0.5, f rot is 1.5 Hz, f 0 =7.5 MHz, c0 = 1, 480 m/s, and PRF = 1 kHz. The change
in thickness, d(t), is obtained from the integration of thedifference between the estimates of v̂i (t) and v̂o (t) as follows: td(t) =(vi (t) − vo (t))dt.
(6)0−N ≤m≤N

3. Results of Simulation Experiment on the Measurement of a Tube
Figure 4 shows a B-mode image reconstructed from thesimulated RF signal y(t) for each position k of the employed

12.0
13.0

depth [mm] (z axis)

m

m

14.0
15.0

0

The instantaneous position of the object is changed according
to the assumed change in thickness. By applying the phase
tracking method3) to the quadrature-demodulated signal ofy(t), the change in thickness d(t) of the object is estimated.
In this simulation experiment, the number of elements,
2N + 1, is 17 with the element pitch of h = 150 µm. The
spacing of small scatterers on the reflector is 10 µm between
points. The reflector is a tube with an inner radius of 3.4 mm
and an outer radius of 4 mm. Its cross-sectional view on thex–z plane is shown in Fig. 3.
The maximum value dmax of the change in arterial wall
thickness during a single cardiac cycle with the period 1/ f rot
is assumed to be 10 µm. Under this condition, the velocityof the inner surface, vi (t), of the tube and the velocity of the

4.



3919

3.4 mm

16.0
17.0
18.0
19.0
20.0

element pitch =-5.0 -4.0 -3.0 -2.0 -1.0

probe surface(skin surface)

h
N
N-1

electricalscannedprobe(linear-type)

1.0 2.0 3.0 4.0 5.0

Fig. 3.

The shape of a tube employed in the computer simulation.

Kndelay linexz = d f [mm]z0

focal point

employedelements

1
-N+1
-N

1

ultrasonictransducer

(a)
to
m th
element

fromn thelement

sound speed = cwater or bodyni

i th scattererm thelement

ni

im

from
i th scatterer(c)depth from the probe in z-axis [mm]

k

(b)

0

lateral distance [mm] (x axis)

objectposition k of the employed elements
15

11.5
13.0
14.5

a

16.0

a

17.5
19.0
20.5

-6

Fig. 2. The electric scanned linear probe is employed by the computer simulation. (a) The simulation system, (b) the reflection on the ith scatterer,and (c) the reception on the mth element.

30

-4

-2

0

2

4

6

lateral distance from the center of the probe
in x-axis [mm]Fig. 4. The B-mode image reconstructed from the simulated RF signals
y(t) in the simulation experiment.

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Jpn. J. Appl. Phys. Vol. 40 (2001) Pt. 1, No. 5B

M. WATANABE and H. K ANAI

elements in Fig. 2(a). The distance between the adjacent positions is 0.5 mm. The received signal y(t) reflected at bothsides (a and a in Fig. 4) of the tube is small in amplitude because the values of cos θni in eq. (2) are small at these points.
These phenomena are also shown in the actual B-mode image in Fig. 5 in detected by commercial ultrasonic diagnostic
equipment (Toshiba SSH-140A, f 0 = 7.5 MHz).
Figure 6 shows the actual values and their estimated resultsin the simulation experiments, where the distance d f of the
focal point from the probe is assumed to be 15 mm. The estimates of the change in thickness, d̂(t), of the anterior wallin Fig. 6(e) and the posterior wall in Fig. 6(f) are similar to
the assumed waveforms.

10.6mmdepth [mm]

10

20

0.5mm

3.7125 mm

Fig. 5. The actual B-mode image of a silicone rubber tube detected by theultrasonic diagnostic equipment. The actual cross-sectional shape is shownby broken lines. The silicone tube is almost the same size as that employedin the simulation experiment.

4. Optimum Focal Position Determined by Simulation
Experiments
For the comparison between the change in thickness,ĥ(t), estimated from the RF signal y(t) generated in thesimulation experiment described above and the actual change
in thickness, h(t), which is assumed in the simulation experiment, the following rms error, erms , is defined during one
cardiac cycle T:
1 Terms ={h(t) − ĥ(t)}2 dt.
(7)
T 0
For various values of the focal position d f , the rms error erms
is obtained for each of the anterior and posterior walls, as
shown in Fig. 7, where the distance from the probe to the outersurface of the anterior wall is fixed at 11 mm. It is found thatfor both walls the optimum focal position d f is about 12 mmfrom the probe, which corresponds to the position between
the anterior wall and the center depth of the tube.
Moreover, the characteristics of the rms error for the anterior wall differ from those of the posterior wall. The reason
for these differences is considered to be as follows. There arethree main types of walls, as shown in Fig. 8, that is, convex(anterior), concave (posterior), and flat walls. For various values of the focal position d f , the rms error erms is obtained forthe three kinds of walls shown in Fig. 9.
In the measurement of the anterior wall, high accuracy is
obtained by setting the focal position d f just on the reflector
surface. In the measurement of the posterior wall, on the other
hand, higher accuracy is obtained by setting the focal positiond f nearer than the depth of the reflector surface, which is similar to the phenomenon where the focal point is situated at thefront of the wall if the wall has parabolic curvature.
Thus, the optimum focal position d f in the posterior wallis set to be slightly nearer than the reflector surface. The dis-

tracking line of anterior wall

0.14
(a)

1

(b)

0

0.1

(c)

[mm/s] 0

anterior wall
posterior wall

0.12

rms error [ m]tracking line of posterior wall

0.10
0.08
0.06
0.04

-0.1

0.02

0.1
[mm/s] 0

(d)
-0.1
10.0

0

0

5

10

15

20

25

30

35

focal depth [mm](e)[ m] 0
-10.0
10.0

(f)

[ m] 0
-10.0
0

0.667

Fig. 7. The rms error in the estimation of the change in thickness of the
tube with the center depth of 15 mm, the outer radius of 4.0 mm, and theinner radius of 3.4 mm.

time [s]Fig. 6. The result of the measurement with a tube by computer simulation.
(a) M-mode image, (b) timing pulse train showing one cardiac cycle, (c)velocity estimates, v̂i (t) (dashed line) and v̂o (t) (solid line), of the anteriorwall, (d) velocity estimates, v̂i (t) (dashed line) and v̂o (t) (solid line), of
the posterior wall, (e) the actual value d(t) of the change in thickness of
the anterior wall, and its estimate d̂(t), (f) the actual value d(t) of thechange in thickness of the posterior wall, and its estimate d̂(t).

ultrasonicprobe

flat wall
anterior wall

posterior wall

aiming point

Fig. 8. Three reflector types: the convex (anterior), flat, and concave (posterior) walls.

Jpn. J. Appl. Phys. Vol. 40 (2001) Pt. 1, No. 5B

M. WATANABE and H. K ANAI

0.12

wall thickness for various focal positions. Based on the results, we have determined the optimum focal position for eachmeasurement.

depth of objects =15 mm

rms error [ m]0.10
posterior wall

0.08
0.06

flat wall

0.04
0.02
anterior wall
0

5

10

15

20

25

30

35

40

45

focal depth [mm]Fig. 9.

3921

The rms error of the three types of reflectors with a depth of 15 mm.

tance from the reflector position to the optimum focal point
depends on the curvature of the tube.
5. Conclusions
From the simulation experiments, we have evaluated theaccuracy of the measurement of the small change in arterial

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