Jpn. J. Appl. Phys. Vol. 41 (2002) pp. 3563–3571
Part 1, No. 5B, May 2002c
2002
The Japan Society of Applied Physics

Modified Phased Tracking Method for Measurement of Change in Thickness of Arterial Wall
Hideyuki H ASEGAWA∗ , Hiroshi K ANAI and Yoshiro KOIWA1
Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan
1 Graduate School of Medicine, Tohoku University, Sendai 980-8575, Japan(Received November 20, 2001; accepted for publication January 25, 2002)

In this study, the change in thickness of the arterial wall caused by the heartbeat was measured by the phased tracking method[IEEE Trans. UFFC. 43 (1996) 791] for noninvasive assessment of the regional elasticity of the arterial wall. In the phased
tracking method, the change in thickness of the arterial wall is obtained from the difference between displacements of two points
set along an ultrasonic beam. The displacement during the pulse repetition interval is determined by the phase of the complex
correlation between the quadrature modulated ultrasonic waves. For suppressing noise components, the complex correlation
function is spatially averaged in the region, which corresponds to the ultrasonic wavelength. However, spatial averaging ofdisplacements is not desirable for measurement of the change in thickness, because the change in thickness is caused by thespatial inhomogeneity of displacements. In this paper, the phased tracking method was modified for direct estimation of thechange in thickness without spatial averaging of displacements. [DOI: 10.1143/JJAP.41.3563]
KEYWORDS: phase shift, least-squares method, change in thickness of arterial wall, strain, elasticity, atherosclerosis

1. Introduction
The steady increase in the number of patients with myocardial infarction or cerebral infarction, both of which areconsidered to be mainly caused by atherosclerosis, is becoming a serious problem. Therefore, it is important to diagnose atherosclerosis in the early stage. Computed tomography(CT) and magnetic resonance imaging (MRI) are employed
for the diagnosis of atherosclerosis. Although they subject
patients to great physical and mental hardship, they provide
information only on the shape of the artery such as the diameter of the lumen. The diameter of the lumen, however, is notchanged by early-stage atherosclerosis.1)
Since there are significant differences between the elastic moduli of the normal arterial wall and that affected byatherosclerosis,2, 3) evaluation of the elasticity of the arterial
wall is useful for diagnosis of early-stage atherosclerosis.4)
In addition to diagnosis of early-stage atherosclerosis, it isalso important to diagnose vulnerability of the atherosclerotic
plaque, because rupture of the plaque causes acute myocardial infarction and cerebral infarction.5–7) Evaluation of themechanical properties such as elasticity of the atherosclerotic
plaque is useful for these purposes.
To measure the elasticity of the arterial wall, the pulse wave
velocity (PWV) method has been developed as a technique forthe noninvasive diagnosis of atherosclerosis.8) In this method,the elasticity of the arterial wall is evaluated by measuring thevelocity of the pressure wave propagating from the heart tothe femoral artery. Though it is useful in terms of the noninvasive evaluation of elasticity, the regional elasticity cannot
be evaluated due to a low spatial resolution of several tenths ofa centimeter, which almost corresponds to the distance fromthe heart to the femoral artery.
To increase the spatial resolution in measurement of PWV,
we have previously proposed a method to accurately measure
the propagation velocity of vibrations on the arterial wall using ultrasound.9–11) Using this method, the PWV between two
adjacent points, which are separated from each other by several centimeters, is noninvasively measured by estimating thetime delay between the resultant vibrations at these two points
set along the axial direction of the artery.
Methods for measurement of the change in artery diameter
have been proposed12–16) in order to obtain the circumferential distensibility of the arterial wall in the plane which is perpendicular to the axial direction of the artery. By assuming
the artery to be a cylindrical shell, the average elasticity ofthe entire circumference in the plane has been evaluated.17–19)
However, the regional elasticity of the atherosclerotic plaquecannot be obtained by these methods, because an artery withatherosclerotic plaque cannot be assumed to be a cylindricalshell with uniform wall thickness.
In this study, we attempted to evaluate the elasticity of amore local region, even in the case of atherosclerotic plaque.
Such a technique for measurement of the spatial distributionof the regional elasticity would be useful for diagnosis of thevulnerability of atherosclerotic plaque as well as for the diagnosis of early-stage atherosclerosis. For this purpose, thesmall change in the thickness of the arterial wall due to theheartbeat is accurately measured in each local region whichcorresponds to the focal area of the ultrasonic beam.10, 20, 21)
From the resultant change in thickness, the regional strain
and the elasticity of the arterial wall are noninvasively evaluated.22)
In the field of ultrasonic tissue characterization (such as
evaluation of the elasticity of the arterial wall and detectionof tumors), various methods have been proposed to measure
the tissue displacement from the peak of the autocorrelation
function between echoes15, 23–27) or the phase of the complex
autocorrelation function12, 13, 16, 28) for measurement of the tissue strain. However, measurement of the small change inthickness (< 100 µm) during one heartbeat of the dynamically moving arterial wall has not been reported. In the phased
tracking method,9) the instantaneous position of the arterial
wall is tracked by the velocity estimated from the phase shift
between two consecutively received echoes. The change inthickness of the dynamically moving arterial wall can be obtained from displacements, which are obtained by the integration of velocities, of the intimal side and the adventitial sideof the arterial wall. In basic experiments, the accuracy in the∗ E-mail address: hasegawa@us.ecei.tohoku.ac.jp

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H. H ASEGAWA et al.

estimation of the change in thickness is less than 1 µm using
the phased tracking method.10, 29, 30)
In these methods (including the phased tracking method)
for measurement of tissue displacement, the spatial averaged
displacement is estimated in order to obtain the strain of
the tissue. However, in case of measurement of the strain(change in thickness) of the tissue, spatial averaging of thedisplacements is not desirable because the change in thickness is caused by the spatial inhomogeneity of displacements.
In this paper, we propose a method for direct estimation of
the change in thickness (strain) of the arterial wall rather thanestimation based on the difference of spatially averaged displacements.

caused by the displacement of the object, is estimated fromtwo consecutive echoes.9) For this purpose, quadrature demodulation is applied to the received ultrasonic waves reflected by the object, and then the in-phase and the quadrature signals are A/D converted. From the demodulated signal,z(t; d + x(t)), reflected at a depth d + x(t) at a time t, where dand x(t) are the initial depth set at t = 0 and the displacementof the object in the depth direction, the phase shift, θ(t), between two consecutive echoes is obtained from the complex
cross correlation function calculated for M + 1 samples in thedepth direction as follows:

2. Principle
2.1 Measurement of the small change in thickness of the arterial wall by the phased tracking method
As illustrated in Fig. 1, for measurement of the smallchange in thickness, the phase shift of the echo, which isM/2z(t + T ; d + x(t) + m D) · z ∗ (t; d + x(t) + m D),
e j θ (t) = 

 

 M/2
∗


z(t
+
T
;
d
+
x(t)
+
m
D)
·
z
(t;
d
+
x(t)
+
m
D)



m=−M/2
m=−M/2where D and T are the interval of sample points in the depth
direction and the pulse repetition interval, respectively, and∗ represents the complex conjugate. In measurement of thechange in thickness, M + 1 in eq. (1) is set at 5 (= 0.4 µs)
in consideration of the pulse length of 0.46 µs. In estimationof the phase shift by eq. (1), the object position is tracked by
the integration of the average velocity, v(t + T /2), during thepulse repetition interval, T , as follows:


T
×T

x (t + T ) = 
x (t) + 
v t+
2
c0 
=
x (t) +
θ (t),(2)
2ω0Fig. 1.

Schematic diagram of the phased tracking method.

(1)

where ω0 and c0 are the center angular frequency of the ultrasonic pulse and the speed of sound, respectively.
From displacements x 1 (t) and x 2 (t) of two points, which
are set in the arterial wall along the ultrasonic beam, the smallchange in thickness, h(t), of the arterial wall is obtained asfollows:h(t) = x 1 (t) − 
x 2 (t) t{v1 (t) − v2 (t)}dt.
=

(3)

0

2.2 Modified phased tracking method for direct measurement of small change in thickness of the arterial wall
In the phased tracking method, the change in thickness isobtained by subtraction of the two spatially averaged displacements. In this section, the phased tracking method is
modified to directly estimate the change in thickness.
As illustrated in Fig. 2, the complex signal, β(t; d, d12 ),Fig. 2.

Estimation of the change in thickness by the least-squares method.

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H. H ASEGAWA et al.

which represents the phase difference between two points set
along the ultrasonic beam, is expressed by eq. (4).
β(t; d, d12 ) = z(t; d + x 1 (t)) · z ∗ (t; d + x 2 (t) + d12 ), (4)where d12 is the distance between two points initially set at
t = 0, and x 1 (t) and x 2 (t) are displacements of these twopoints, which are estimated by the phased tracking method as
described in §2.1.
Let us assume that the amplitude of β(t + T ; d, d12 ) andβ(t; d, d12 ) are equal because the pulse repetition interval,

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T , is sufficiently short. Thus, the complex signal, β(t +T ; d, d12 ), at the time t + T is estimated from β(t; d, d12 )as follows:(t + T ; d, d12 ) = β(t; d, d12 ) · e j θh (t) ,β(5)where θh (t) is the phase shift which represents the change
in thickness between these two points set along the ultrasonic beam. Then, the normalized mean squared difference,(t + T ; d, d12 ) isα(θh (t)), between β(t + T ; d, d12 ) and β
defined by

M/2 
2


(t + T ; d + m D, d12 )
β(t + T ; d + m D, d12 ) − βα(θh (t)) =m=−M/2
M/2
2 
2 
1  (t + T ; d + m D, d12 )
β(t + T ; d + m D, d12 ) + β
2 m=−M/2
M/2 
2



β(t + T ; d + m D, d12 ) − β(t; d + m D, d12 ) · e j θh (t) =m=−M/2
M/2
2 
2 
1  β(t + T ; d + m D, d12 ) + β(t; d + m D, d12 )
2 m=−M/2,(6)where M + 1 is the number of samples in the depth direction for calculating the difference α(θh (t)). By replacing thedenominator of eq. (6) by A, eq. (6) can be rewritten as follows:
A · α(θh (t)) =

M/2 
2



β(t + T ; d + m D, d12 ) − β(t; d + m D, d12 ) · e j θh (t) m=−M/2=M/2 
2β(t + T ; d + m D, d12 )
m=−M/2− β(t + T ; d + m D, d12 ) · β ∗ (t; d + m D, d12 ) · e− j θh (t)


2 


− β ∗ (t + T ; d + m D, d12 ) · β(t; d + m D, d12 ) · e j θh (t) + β(t; d + m D, d12 ) .

(7)

By taking the partial derivative of eq. (7) with respect to θh (t), the following equation is obtained:
A·

M/2 

∂α(θh (t))
jβ(t + T ; d + m D, d12 ) · β ∗ (t; d + m D, d12 ) · e− j θh (t)=∂θh (t)m=−M/2− jβ ∗ (t + T ; d + m D, d12 ) · β(t; d + m D, d12 ) · e j θh (t) .

(8)In order to determine the phase shift, θh (t), which gives the minimum value of α(θh (t)), by setting the right hand side ofeq. (8) to be zero,M/2β(t + T ; d + m D, d12 ) · β ∗ (t; d + m D, d12 ).
e j θh (t) = 

 

 M/2
∗


β(t
+
T
;
d
+
m
D,
d
)
·
β
(t;
d
+
m
D,
d
)
12
12



m=−M/2
m=−M/2

In measurement of the change in thickness, M + 1 in eq. (9)is set at 5 (= 0.4 µs) in consideration of the pulse length of
0.46 µs.
From eq. (9), velocity, vh (t), of the change in thickness isobtained by the following equation:Tc0 θh (t)vh t +
=
.
2
2ω0 T

(9)

(10)

The change in thickness, h(t), of the arterial wall is obtained by integrating the velocity, vh (t), of the change in3566

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H. H ASEGAWA et al.

Fig. 3. Spatial inhomogeneity in displacement as a cause of the change inthickness.

thickness as follows:h(t) =

 tvh (t)dt.

(11)

0

As illustrated in Fig. 3, there are two components of displacements, namely, parallel motion and the change in thickness, at two points along the ultrasonic beam. The change inthickness is obtained by subtracting the parallel motion fromthe displacement when the displacements of these two points
do not coincide with each other, that is, when they are spatially inhomogeneous.
In the phased tracking method, two correlation windows
are set to obtain displacements of two points along the ultrasonic beam. Though there are many reflected pulses underin vivo conditions, influences of the spatial averaging of displacements can be illustrated from the simple model as shownin Fig. 4. In this model, there are three different pulses reflected by three different scatterers which have different displacements. In order to precisely obtain the change in thickness around the region where correlation windows are set, itis necessary to obtain differences in displacements of thesepulses. However, in the phased tracking method, the difference in displacements cannot be obtained precisely due tothe spatial averaging of displacements of different pulses, andthe spatial averaging of displacements of different pulses iscaused by inclusion of different pulses in the correlation window. On the other hand, the change in thickness can be estimated precisely by the modified phased tracking method,
because the difference between displacements of each two
points around the correlation window is obtained as shownin Fig. 5.
In the phased tracking method, the spatial averaging ofdisplacements causes reduction of the amplitude of the estimated change in thickness in comparison with its true value
when the same pulse is included in both correlation windows.
In Fig. 4, pulse e2 is included in both correlation windows.
Therefore, spatially averaged displacements obtained fromboth correlation windows become close to each other because
the same displacement of pulse e2 is added to the spatial averaging in both correlation windows. As a result, the amplitudeof the estimated change in thickness is reduced.
In this paper, influences of the interference between pulses
is not taken into account. Further investigation is necessary to
discuss influences of such interference.

Fig. 4. Illustration of estimation of the change in thickness by the phased
tracking method.

Fig. 5. Illustration of estimation of the change in thickness by the modified
phased tracking method.

3. Basic Experiments Using a Rubber Plate
3.1 Experimental system for basic experiments
The experimental system for basic experiments is illustrated in Fig. 6.30) In this system, the change in thickness ofJpn. J. Appl. Phys. Vol. 41 (2002) Pt. 1, No. 5B

Fig. 6.

Experimental system for basic experiments.

H. H ASEGAWA et al.

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3.3 Basic experimental results
Figure 8 shows the results of measuring the change inthickness of the rubber plate. Figure 8(a) shows the amplitude of the quadrature modulated signals at a time of t = 0,
which is plotted as a function of depth. Figure 8(c) shows theoutput of the position sensor for detection of the origin angleof the motor. By setting two points A and B at the near and far
surface of the rubber plate in the M-mode image of Fig. 8(b),velocities, v A (t) and v B (t), of points A and B are obtained by
the phased tracking method as shown in Figs. 8(d) and 8(e).
By integrating the difference between the velocities of point
A and B, the change in thickness, h(t), is obtained as shownin Fig. 8(f) using the phased tracking method. The change inthickness, h(t), is also obtained by the proposed method
as shown in Fig. 8(h) by integrating the velocity, vh (t), of thechange in thickness (Figs. 8(g)). In Fig. 9, the same procedure
except for the setting of initial positions of tracked points is
applied to the same data as in Fig. 8.
As shown in Fig. 8(a), two correlation windows are set at
positions of different echoes, respectively. In this case, signals included in correlation windows A and B correspondto reflected echoes from the near and far surface of the rubber plate, respectively. Therefore, the change in thickness canbe obtained precisely by the phased tracking method and thea rubber plate is generated by expanding it in the longitudinal
direction by a motor. The theoretical value of the change inthickness is obtained from the change in length by assumingthat the rubber is incompressible (Poisson’s ratio ≈ 0.5). The
error in the measured change in thickness is evaluated using
this theoretical value.
3.2 Measurement system
The measurement system employed in this paper is illustrated in Fig. 7. The ultrasonic pulse (center frequency:
7.5 MHz) is transmitted and received by the ultrasonic probe
in the standard ultrasonic diagnostic equipment. The received
signal is amplified and demodulated. The resultant in-phaseand quadrature signals are simultaneously A/D convertedwith a 12-bit A/D converter at a sampling frequency of
10 MHz. Measured digital signals are transfered to a computer, and the change in thickness of the rubber plate is obtained by applying two methods, the phased tracking method
and the modified phased tracking method, mentioned in §2, tothese digital signals.

Fig. 7. Measurement system employed for basic experiments.

Fig. 8. Results of measurement of the change in thickness of a rubber plate
under a setting of correlation windows. (a) Amplitude of the quadrature
modulated signal. (b) M-mode image. (c) Output of sensor for detectionof the angle of origin of the motor. (d) Velocity, v A (t), of point A. (e)Velocity, v B (t), of point B. (f) Change in thickness, h(t), measured by
the phased tracking method. (g) Velocity, vh (t), of the change in thickness.
(h) Change in thickness, h(t), measured by the proposed method.

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

modified phased tracking method. On the other hand, in caseof Fig. 9, displacements in correlation window A are inhomogeneous because both pulses, which are reflected at the nearand far surface of the rubber plate, are included in correlation
window A. Therefore, displacement of the near surface cannot be obtained precisely from correlation window A due tospatial averaging of displacements of the near and far surface
in correlation window A.
Figure 10 shows the mean, maximum, and minimum of dis-H. H ASEGAWA et al.

placements within correlation window A for cases of Figs. 8
and 9, which are obtained from the phased tracking methodby setting M + 1 in eq. (1) at 1. In Fig. 10, means are shownby diamonds, and maximum and minimum of displacements
are indicated by vertical bars. From Fig. 10, inhomogeneity
in displacements within the correlation window in the caseof Fig. 9 is larger than in the case of Fig. 8. Therefore, spatial averaging of displacements has a greater influence on the
precision of the measured change in thickness for the case of
Fig. 9.
4. In vivo Experimental Results for the Human Common
Carotid Artery

Fig. 9. Results of measurement of the change in thickness of a rubber plate
under another setting of correlation windows. (a) Amplitude of the quadrature modulated signal. (b) M-mode image. (c) Output of sensor for detection of the angle of origin of the motor. (d) Velocity, v A (t), of point A. (e)Velocity, v B (t), of point B. (f) Change in thickness, h(t), measured by
the phased tracking method. (g) Velocity, vh (t), of the change in thickness.
(h) Change in thickness, h(t), measured by the proposed method.

Fig. 10. Mean, maximum, and minimum of displacements within the correlation window.

The proposed method was applied to in vivo measurementof the human carotid artery of a 28-year-old male subject.
Changes in thickness of the arterial wall were measured at two
different positions, p1 and p2 , on the arterial wall as shownin the B-mode image of Fig. 11. Figure 12 shows the results
measured at the beam position p1 . Initial positions of the intimal side and the adventitial side (points A and B in Fig. 12(a))
are set in the M-mode image of Fig. 12(a) at the timing of
the R-wave of the electrocardiogram shown in Fig. 12(b). Velocities, v A (t) and v B (t), of the intimal and adventitial sides
were obtained by the phased tracking method as shown in
Figs. 12(c) and 12(d), respectively. The change in thickness,
h(t), was obtained as shown in Fig. 12(e) by integrating thedifference between these two velocities. By directly integrating velocity, vh (t), of the change in thickness (Fig. 8(f)), thechange in thickness, h(t), was also obtained by the modified
phased tracking method as shown in Fig. 12(g). In results forthe position p1 , there was almost no difference in amplitudeof the measured changes in thickness by both methods.
Figure 13 shows results of measurement at position p2 . In
Fig. 13, the amplitude of the change in thickness obtainedby the phased tracking method decreases in comparison withthat obtained by the proposed method, the same as is the basic
experiments.
Figures 14(a) and 14(b) show the amplitude of quadrature
modulated signals at positions p1 and p2 , respectively. In Fig.
14(a), two reflected echoes, which are reflected by the intimalside and the adventitial side, respectively, are clearly sepa-

Fig. 11. B-mode image of the human carotid artery of a 28-year-old man
obtained by standard ultrasonic diagnostic equipment.

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Fig. 12. Results of measurement of the change in thickness of the arterial
wall at beam position p1 (28-year-old man). (a) M-mode image. (b) Electrocardiogram. (c) Velocity, v A (t), of the intimal side. (d) Velocity, v B (t),of the adventitial side. (e) Change in thickness, h(t), measured by thephased tracking method. (f) Velocity, vh (t), of the change in thickness. (g)Change in thickness, h(t), measured by the proposed method.

H. H ASEGAWA et al.

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Fig. 14. Amplitude of the quadrature modulated signal. (a) Beam position
p1 . (b) Beam position p2 .

Fig. 15. Mean, maximum, and minimum of maximum value of displacement during one heartbeat at each point in the correlation window.

Fig. 13. Results of measurement of the change in thickness of the arterial
wall at beam position p2 (28-year-old man). (a) M-mode image. (b) Electrocardiogram. (c) Velocity, v A (t), of the intimal side. (d) Velocity, v B (t),of the adventitial side. (e) Change in thickness, h(t), measured by thephased tracking method. (f) Velocity, vh (t), of the change in thickness. (g)Change in thickness, h(t), measured by the proposed method.

rated from each other, and each correlation window is set atthe position of each echo. In this case, two displacements,
which are obtained from correlation windows A and B by thephased tracking method, correspond to displacements of theintimal side and the adventitial side, respectively.

On the other hand, in Fig. 14(b), echoes which are reflectedby the intimal side and the adventitial side are not clearly
separated. In such a case, it is supposed that there are somereflected echoes besides those from the intimal side and theadventitial side. Under such condition, displacements in thecorrelation window are inhomogeneous because these echoes
are reflected by different scatterers, which have different displacements. Therefore, the precision in measurement of thechange in thickness (inhomogeneity in displacements) worsen
due to spatial averaging of displacements by the phased tracking method. As a result, the change in thickness, which is
obtained from the difference between spatially averaged displacements, becomes small in comparison with that by the
proposed method, because the spatial inhomogeneity of displacements (which corresponds to the change in thickness) is
suppressed due to spatial averaging.
As in the basic experiments, Fig. 15 shows mean, maxi-

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Jpn. J. Appl. Phys. Vol. 41 (2002) Pt. 1, No. 5Bmum, and minimum of displacements within correlation window A for the cases of Figs. 12 and 13, which were obtainedby the phased tracking method by setting M + 1 in eq. (1) at
1. In Fig. 15, means are shown by diamonds, and maximum
and minimum of displacements are shown by vertical bars.
From Fig. 15, inhomogeneity of displacements in the case of
Fig. 13 is seen to be greater than in Fig. 12. Therefore, it is
supposed that spatial averaging of displacements has a greater
influence on the measured change in thickness in the case of
Fig. 13.
5. Discussion
A question throughout this paper is whether the spatial averaging of displacements by the phased tracking method canbe prevented by setting M + 1 in eq. (1) (width of the correlation window) at 1. However, spatial averaging is necessary
to obtain adequate results as described below.
In Fig. 16, the same procedure as described in Fig. 9 except for width of the correlation window (M + 1 = 1) is applied to the same data shown in Fig. 9. In Fig. 16, the change
in thickness is obtained by the phased tracking method (Fig.
16(f)) with large error. It is supposed that the phase of thesample point at correlation window A does not represent the

H. H ASEGAWA et al.

displacement of the near surface of the rubber plate because
this sample point is located on the edge of the pulse, whichis reflected by the near surface. The quadrature modulation
can extract the phase information of the stationary sinusoidal
signal with a single frequency. Due to finite duration of theultrasonic pulse, however, the phase information around theedge of the ultrasonic pulse cannot be obtained precisely because the ultrasonic pulse cannot be assumed as a stationary
sinusoidal wave with a single frequency around its edge.
From this fact, spatial averaging during pulse duration ineq. (1) is necessary to prevent the displacement from being
estimated only from the sample point around the edge of theultrasonic pulse. The influence of the sample point around theedge of the ultrasonic pulse is suppressed by summation ofeq. (1), which is weighted by the amplitude of sample points,
because the amplitude of the sample point around the edge
is smaller than that around the center of the ultrasonic pulse.
In the proposed method, the influence of the incorrect phase
information around the edge is suppressed by the same mechanism as that of the phased tracking method, spatial averaging
in eq. (9). In the modified phased tracking method, by applying spatial averaging to the change in thickness rather than tothe displacement, spatial inhomogeneity in displacements is
not suppressed.
In consideration of these facts, spatial inhomogeneity ofdisplacements in the correlation window in Fig. 9 is causedby the following two factors.
(1) Actual change in thickness (two echoes which represent
displacements of the near and far surface of the rubber
plate are included in the correlation window. There is adifference between displacements of these two echoes.).
(2) Incorrect phase information around the edge of the ultrasonic pulse.
The above-mentioned influence of the phenomenon of item
(2) is suppressed by the spatial averaging in both the phased
tracking method and the modified phased tracking method
proposed in this paper. Thus, the spatial averaging is necessary for both methods to obtain adequate results. However,there is a difference between these two methods with regard towhat spatial averaging is applied to. Worsening of the precision in measurement of the change in thickness by the phased
tracking method is thought to be caused by the spatial averaging of different displacements in the correlation window.
In this paper, though the procedure for estimation of thechange in thickness is modified, determination of the object position depends on the phased tracking method. In eq.
(4), displacements, x 1 (t) and x 2 (t), which are still estimatedby the phased tracking method, determine the sample points
which are employed for estimation of the phase shift. Consequently, the precision in measurement of the change in thickness is not still completely independent from error in estimation of the displacements, x 1 (t) and x 2 (t).
6. Conclusions

Fig. 16. Results of measurement of the change in thickness of a rubber
plate with the narrower correlation window. (a) Amplitude of the quadrature modulated signal. (b) M-mode image. (c) Output of sensor for detection of the angle of origin of the motor. (d) Velocity, v A (t), of point A. (e)Velocity, v B (t), of point B. (f) Change in thickness, h(t), measured by
the phased tracking method. (g) Velocity, vh (t), of the change in thickness.
(h) Change in thickness, h(t), measured by the proposed method.

In this paper, a method has been proposed for reductionof the estimation error in measurement of the small change
in thickness of the arterial wall. The proposed method wasevaluated by basic experiments using a rubber plate, and thenapplied to in vivo measurement of the human carotid artery.
From these results, it was shown that the modified phased

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

tracking method proposed in this paper has potential for measurement of the change in thickness with higher precision incomparison with the phased tracking method.

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