Jpn. J. Appl. Phys. Vol. 41 (2002) pp. 3613–3618
Part 1, No. 5B, May 2002c
2002
The Japan Society of Applied Physics

Optimization of Condition of Ultrasonic Beamfor Measurement of Small Change in Thickness of Arterial Wall
Masaru WATANABE∗ , Hideyuki H ASEGAWA† 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 20, 2001; accepted for publication January 22, 2002)

We previously developed a method for measuring small changes in thickness of the arterial wall during one cardiac cycle.
Knowledge of this change in thickness is useful for in vivo assessment of the regional elasticity of the arterial wall. In thisstudy, from computer simulations, it is found that measurement error depends on the distance of the ultrasonic beam from thecenter of the artery and it can be reduced by optimally setting the focal position. In basic experiments using a silicone rubber
tube and in in vivo experiments with a human carotid artery, it is found that by optimizing the focal position, measurement of thechange in thickness becomes more robust against mispositioning of the ultrasonic beam. From these results, it is demonstrated
that optimum focal positioning provides more robustness in measurement, even if there is arterial wall motion causing theposition of the ultrasonic beam to deviate from the center of the artery. [DOI: 10.1143/JJAP.41.3613]
KEYWORDS: atherosclerosis, change in thickness of arterial wall, computer simulation, acoustic lens, focus

1. Introduction
The increase in the number of individuals suffering fromacute myocardial or cerebral infarction, both of which aremainly caused by atherosclerosis, has become a serious clinical problem. It is, therefore, important to diagnose atherosclerosis at an early stage. However, to date, there are no methods
available for the detection of minute changes in the elasticityof the arterial walls due to early-stage atherosclerosis.
We have developed a method for evaluating the elasticity
within about 400 µm of each local region in the arterial wall1)
to diagnose vulnerability of atherosclerotic plaques. In our
method, the small change in thickness of the arterial wall dueto the heartbeat is accurately measured in each local region
around the focal area of the ultrasonic beam. From the resultant change in thickness, the regional strain and the elasticityof the arterial wall are noninvasively evaluated.2) By scanningthe ultrasonic beam, the spatial distribution of the elasticity isalso obtained.
In this measurement, however, accuracy is influenced by
the setting of the electric focal depth. In our previous paper,the optimum ultrasonic focal position of electric focusing of
the linear array probe was investigated for accurate measurement of the change in thickness.3) Another effective method
for reducing this influence is setting multiple focal points
along one scan line, as is done in commercial ultrasonic diagnostic equipment. In multiple focusing, however, the timeinterval between successive scans for the same focal points
increases. In our measurement of the change in thickness of
the arterial wall, the time interval between successive scansmust be less than 5 ms to prevent aliasing due to the assumption that maximum velocity of the arterial wall is 5 mm/s. In
this study, therefore, in vivo measurement of small changes inthickness is simulated for single focusing in order to prevent
aliasing.
In addition to the electric focal position, another factor
which decreases the accuracy in measurement is mispositioning of the ultrasonic beam. Mispositioning means that the ul-

trasonic beam does not pass through the exact center of theartery, which occurs when the sonographer does not correctly
maintain the position of the ultrasonic probe and/or movement of the arterial wall causes the position of the ultrasonicbeam to deviate from the center of the artery. In this study, ultrasonic propagation in 3-D space is simulated in order to investigate the optimum ultrasonic focal position of the acoustic
lens of the linear array probe in order to reduce the influenceof the deviation of the position of the ultrasonic beam from thecenter of the artery. From basic experiments and in vivo experiments, by optimizing the focal position, it is demonstrated
that the change in thickness can be measured accurately even
if the position of the ultrasonic beam slightly deviates fromthe center of the artery.
2. Principle of Computer Simulation for Measurementof Small Change in Thickness
In the simulation, an arbitrarily shaped reflector in the x −y − z space, which corresponds to a tube, is divided into manysmall scatterers as shown in Fig. 1. An electric scanning linear
probe with (2N +1) elements is employed. The distance fromyx

element

ultrasonic probe-N0(n, l)

scatterer ipxNdelay line

∗ E-mail address: watanabe@us.ecei.tohoku.ac.jpzFig. 1. Geometry of simulation model.

† E-mail address: hasegawa@us.ecei.tohoku.ac.jp‡ E-mail address: hkanai@ecei.tohoku.ac.jp

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M. WATANABE et al.

yelement pitch = pyn

( n,L )
( n,L-1 ) Ele(n,l)

xdelay line
for electric
focusingz

Sci Ele(n,l)( n,l )zdelay line
0Sci
for simulating
acoutsic lens( n, -L+1 )

nl-i

i

( n, -L )i th scatterer

Fig. 2. Simulation of acoustic lens.

the nth element (−N ≤ n ≤ N ) to the electric focal point

at a depth of z = d f is given bypx denotes the element pitch along the x-axis. In addition,
to simulate characteristics of the acoustic lens, each element
is divided into (2L +1) point-sources along the y-axis and the
distance between the point-source is p y , as shown in Fig. 2.
Thus, in order to focus the ultrasonic pulses transmitted from
(2N + 1) elements, where each element consists of (2L + 1)
point sources, on each focal point (electric focal point: z =d f , acoustic focal point: z = d fac ), the following delay time
τnl is added to the delay line connected to the lth point-sourcein the nth element:d f 2 + (N · px )2 − d f 2 + (n · px )2
τnl =
c


d fac 2 + (L · p y )2 − d fac 2 + (l · p y )2,+c
d 2f + (n · px )2 , where(−N ≤ n ≤ N , − L ≤ l ≤ L)

(1)

where c is the sound speed. The first term on the right-handside simulates the electric focal system along the x-axis, andthe second term simulates the acoustic lens characteristics
along the y-axis.
The incident angle, θnl−i , from the (n, l)th element of theprobe to the ith small scatterer on the surface of the reflector is determined by the regional shape of the reflector andthe geometrical relationship between the (n, l)th element andthe scatterer. The incident angle, θnl−i , is described by the po−−−−→
sition vector, Ele(n,l) , of the (n, l)th transmitting element of−→
the probe, the position vector, Sci , of the ith scatterer, and−→the normal vector, i , of the surface of the reflector at theposition of the ith small scatterer as follows:→−→ −−−−→ −Sci − Ele(n,l) · i  .
(2)cos θnl−i = −→
 → −−−−→ −
 Sci − Ele(n,l)   i 
By assuming a mirror reflection, the ultrasonic pulse is reflected with the same angle θnl−i . The receiving angle, ϕi −n l ,from the ith scatterer to the receiving (n , l )th element is determined in the same manner, as follows:−−−−→ −−−−→ −→(0, 0, 1) · Ele(n ,l ) − Scicos ϕi −n l = −−−−→ −−−−→ −
(3)
→ .


(0, 0, 1)  Ele(n ,l ) − Sci 
The amplitudes of the incident wave and the received waveySci Ele(n,l)
from (n,l)thelementii th scatterernl-iEle(n’,l’)Sci

received
element
i-n’l’(n’,l’)(0,0,1)

Fig. 3.

z

Receiving angle, ϕi −n l .

depend on these values of cos θnl−i and cos ϕi −n l , respectively.
Each traveling time, τnl−i , from the (n, l)th element to theith small scatterer, and τi −n l , from the ith small scatterer tothe (n , l )th element is determined by dividing the distanceby sound velocity, c, as follows:→
−−−−→ − Ele(n,l) − Sci ,(4)
τnl−i =c→ −−−−→− Sci − Ele(n ,l ) τi −n l =.
(5)c
The RF ultrasonic wave yn l (t) received at the (n , l )th 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 {(n, l)} as
follows:4, 5)
yn l (t) =

L
N



W (t − τnl ) · sin ωo (t − τnl )n=−N l=−L∗i

δ(t − τnl−i ) · cos θnl−i∗ δ(t − τi −n l ) · cos ϕi −n l ,(6)where W (t) is the Hanning window employed to describe thetransmitted pulse waveform, ∗ is the convolution operation,

Jpn. J. Appl. Phys. Vol. 41 (2002) Pt. 1, No. 5B

M. WATANABE et al.

3615

and δ(t) is Dirac’s delta function. The output RF signal y(t)of the probe is given by the delayed summation of the RF
ultrasonic wave received at all elements {(n, l)} as follows:y(t) =

LN

yn l (t) ∗ δ(t − τn l ).

(7)n =−N l =−L

By applying quadrature demodulation to the output signal, y(t), the change in thickness, ĥ(t), is estimated by thephased tracking method.6)
3. Experiments by Computer Simulation
3.1 Simulated object
The simulated object is a tube with an inner radius of
3.4 mm, an outer radius of 4 mm, and a length of 20 mm. The
object is divided into 28,800 scatterers with a circumferential
pitch of 2 degrees and a longitudinal pitch of 250 µm. Let us
assume that the inner surface and the outer surface have velocity signals, vi (t) and vo (t), in the direction away from thecenter of the tube as follows:(8)vi (t) = (α + 1)dmax · 2π f rot sin(2π f rot t),(9)where α is a constant that determines the difference betweenvi (t) and vo (t). The actual change in thickness h(t) is obtained by integrating the difference between vo (t) and vi (t) as
follows:
 t
h(t) =
{vo (t) − vi (t)}dt.
(10)0

In the following experiments, the parameters are set as follows: α = 0.5, f rot = 1.5 Hz, f 0 = 7.5 MHz, c = 1, 480 m/s,
and the pulse repetition frequency (PRF) = 1 kHz.
During one cardiac cycle T , the RMS error, erms , of theestimated change in thickness, ĥ(t), from the actual change
in thickness, h(t), is defined as follows:
1 T
erms ={h(t) − ĥ(t)}2 dt.
(11)
T 0
3.2 Investigation of bias error caused by mispositioning of
the ultrasonic beam
Figure 5(a) shows the M-mode image for the posterior wallof the tube obtained at a distance, dl , of 1 mm from the center
axis of the tube to the ultrasonic beam as shown in Fig. 4. The
time interval, T , between the transmission timing pulse train
in Fig. 5(b) shows one cycle of the change in thickness. By
tracking the two points set in the posterior wall at the peakof the timing pulse, velocity signals, vi (t) and vo (t), are estimated by the phased tracking method,6) as shown in Figs.
5(c) and 5(d). The change in thickness, ĥ(t), is estimatedby integrating the difference between these velocity signals,
as shown in Fig. 5(e). The actual change in thickness, h(t),of Eq. (10) is superimposed on the estimated one in Fig. 5(e).
Figure 6 shows the estimation error, erms , plotted as a function of the distance, dl , from the beam position to the center
axis of the tube for five different settings of the focal depth,
d fac , of the acoustic lens. In these experiments, the electrical
focal depth, d f , is set precisely at the upper surface of theposterior wall of the tube. At the same dl , the estimation error, erms , changes with the focal depth, d fac , and is determined

Fig. 4. Definition of the distance, dl , from the ultrasonic beam to the centerof the tube/artery.

1.875 mmvo (t) = αdmax · 2π f rot sin(2π f rot t),

tracking line
rubber tube(a) M mode image(b) transmission timing pulse train
0.1(c) velocity of upper surface
0[mm/s]-0.1
0.1(d) velocity of lower surface

0[mm/s]-0.110(e) change in thickness
measured

0[ m]
-10

theoretical
0

0.667

time [s]Fig. 5. Result of estimation of the change in thickness. (a) M-mode image; (b) the transmission timing pulse train; (c) (d) velocity signals, vi (t)and vo (t), of the tube surface; (d) the measured and theoretical changes inthickness, ĥ(t) and h(t).

by the characteristics of the acoustic lens in the case of longitudinal measurement. When dl is shorter than 0.4 mm, highaccuracy is achieved with every setting of the focal depth,
d fac . When dl is greater than 0.4 mm, estimation error, erms ,is suppressed by setting the focal depth, d fac , farther from theobject (d fac is set at 27 mm and 34 mm). For the focal depth
setting of 41 mm, however, measurement error cannot be suppressed. From these results, it is found that measurement error
depends upon the position, dl , of the ultrasonic beam and that
it is reduced by setting the focal position, d fac , of the acoustic
lens farther from the object. As a result, accurately measur-

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Jpn. J. Appl. Phys. Vol. 41 (2002) Pt. 1, No. 5B

M. WATANABE et al.

0.70

rms error, e rms [ m]

0.65
0.60

probe

focal depth, d fac ,of acoustic lens [mm]11.6 mm

12 mm
19 mm
27 mm
34 mm
41 mm

0.55
0.50
0.45

tube
12 mm
8 mm
19 mm

dl

0.40

27 mm

0.35
34 mm
0.30

wall depth = 19 mm
electric focal depth, d f , = 19 mm

0.25
0.20

0

0.2

0.4

0.6

0.8

41 mm
focal depth, d f

ac

1.0

distance, d l , from center of tube
to center of element [mm]Fig. 6.

Result of computer simulation for each characteristic of the acoustic lens.

able area of dl is extended by optimizing the focal depth, d fac ,of the acoustic lens.
4. Results of Basic Experiments Using a Silicone Rubber
Tube
Measurement accuracy has been evaluated for various values of the focal position, d f , in experiments using a silicone
rubber tube, in which the pulsatile flow is induced by an artificial heart.7) The change in thickness, ĥ(t), of the tube wallis generated by the pressure change caused by the arrival of
the pulsatile flow.
Figure 7(a) shows the M-mode image of the tube obtained
for the case where the distance, dl , from the ultrasonic beam
to the center axis of the tube is 0.3 mm. Figure 7(b) shows thedriving pulse of the artificial heart. Figure 7(c) shows the inner pressure upstream of the ultrasonic measurement position.
By tracking the two points set in the posterior wall when thedriving signal rises, velocity signals, vi (t) and vo (t), are estimated by the phased tracking method, as shown in Figs. 7(d)and 7(e). Figure 7(f) shows the change in thickness, ĥ(t),which is obtained by integrating the difference between thesevelocity signals. The maximum amplitude, h max , of thechange in thickness, ĥ(t), is about 4 µm for every threeheart beats. The elastic modulus of the silicone rubber tube
employed in basic experiments is 2.4 MPa.2) The actual maximum amplitude of the change in thickness of the tube is obtained as 4.1 µm from the elastic modulus, the thickness of
the wall, the inner radius of the tube and the change in innerpressure.
In Fig. 8, the maximum amplitude of the change in thickness is plotted as a function of distance, dl , from the beam
position to the center of the tube for four different settings ofthe electric focal depth, d f . When the electric focal depth, d f ,is set at 19 mm, which is the same depth as that of the innersurface of the posterior wall, the change in thickness, h(t),can be measured with high accuracy only around the center
axis of the tube. The measurement accuracy decreases significantly when distance, dl , from the ultrasonic beam to theFig. 7. Results of measurement of the change in thickness of the tube wall
obtained at the center of the tube. (a) M-mode image; (b) artificial heart
driving signal; (c) inner pressure; (d) (e) velocity signals, vi (t) and vo (t),of both surfaces of the tube; (f) estimated change in thickness, ĥ(t).

center of the tube is greater than 0.5 mm. On the other hand,when the electric focal depth, d f , is set deeper than that of theouter surface of the posterior wall, high accuracy is achievedin a broader region up to 1.5 mm from the center axis.
Using a broadened ultrasonic beam, the change in thickness is accurately estimated from the ultrasonic wave reflectedfrom the region around the the center axis on the posteriorwall even when the beam position, dl , is farther from the center axis. It is found that the measurement error dependent
upon the position, dl , of the ultrasonic beam is reduced bymaximum change in thickness [ m]Jpn. J. Appl. Phys. Vol. 41 (2002) Pt. 1, No. 5B

M. WATANABE et al.

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probe
0
9.5 mm
tube

2.0

11 mm
19 mm

4.0

dl
6.0

8.0

27 mm

electric focal depth, d f
19 mm
27 mm
34 mm
41 mm

34 mm
41 mm
focal depth, d f

0

-0.5

0.5

1.5

1.0

2.0

2.5

3.0

4.0

3.5

distance, d l , of beam axis from center of tube [mm]Fig. 8. The maximum change in thickness is plotted as a function of distance of the beam from the center of the tube.

8 mm
19 mm

dl

focal depth,dfgeometry ofin vivo experimentchange in thinckness (in vivo) [ m]

12 mm

60

10.0

50

9.5

40

9.0

30

in vivo

8.5

20
8.0
10

computer simulation
7.5

0

-1.5

Fig. 9.

-1.0

-0.5
0
0.5
dl : distance from center of tube (or artery)to center of element

1.0

1.5

probe
11.6 mmchange in thinckness (simulation) [ m]probe
center of
the artery

tube
8 mm
19 mm

dl

focal depth, d fComparison between results of in vivo measurement and computer simulation.

ratio ofmaximum change in thickness

0.2
0.3
0.4

ac

geometry ofsimulation experiment

probe

focal depth, d f
5 mm

0.5

12 mm

0.6

19 mm

center of
the artery

5 mm 12 mm
12 mm
8 mm
19 mm

0.7

27 mm

0.8

dl

27 mm

0.9
1

depth of posterior wall = 19 mm

1.1
1.2

0

0.15

0.30

0.45

0.60

0.75

0.90

focal depth, ddistance, d l , from center beam position [mm]Fig. 10.

f

Results of in vivo measurement for each setting of focal position.

setting the focal position, d f , farther from the object, whichis also estimated in the simulation experiments.

5. In vivo Experimental Results
Measurement accuracy is evaluated in in vivo measurement
at the human common carotid artery under the condition of anelectric focal depth, d f , of 19 mm.8)

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Jpn. J. Appl. Phys. Vol. 41 (2002) Pt. 1, No. 5B

Figure 9 shows the maximum amplitude, ĥ max , of thechange in thickness obtained as a function of the distance, dl ,of the ultrasonic beam from the center axis of the artery. The
maximum amplitude, h max , of the change in thickness obtained by computer simulation is also plotted in Fig. 9. From
Fig. 9, it is found that in vivo experimental results are welldescribed by computer simulation. Figure 10 shows the ratioof the maximum amplitude, ĥ max , of the change in thickness measured at each beam position to that measured at thecenter axis. It is plotted for other settings of the focal depthof 5 mm, 12 mm and 27 mm. From Fig. 10, it is also shownby in vivo measurement that the change in thickness can be
accurately measured in a broader region by setting the focal
position farther from the object.
6. Conclusions
In this study, ultrasonic propagation in 3-D space is simulated to investigate the optimum focal position, which makes
measurement of the change in thickness more robust against
lateral change, dl , in the position of the ultrasonic beam. From

M. WATANABE et al.

basic and in vivo experiments, it is confirmed that the region
where the change in thickness, h(t), is accurately measured
is extended to ±0.45 mm around the center of the artery by
setting the electric focal position, d f , deeper than the posterior wall of the artery. This result provides successful in vivo
measurement of the change in thickness, which is less sensitive to the position, dl , of the ultrasonic beam.

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