Jpn. J. Appl. Phys. Vol. 39 (2000) pp. 3257–3261
Part 1, No. 5B, May 2000
c
°2000
The Japan Society of Applied Physics

Reduction of Influence of Decrease in Signal-to-Noise Ratio in Measurement of Change
in Thickness of Arterial Wall Due to Heartbeat
Hideyuki H ASEGAWA ∗ , Hiroshi K ANAI, Nozomu H OSHIMIYA 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 18, 1999; accepted for publication January 20, 2000)

To diagnose early-stage atherosclerosis, for the local evaluation of the elasticity of the arterial wall it is necessary to increase
the spatial resolution in the axial direction of the artery to several millimeters, which corresponds to the size of the lesion on
the surface of the arterial wall. For this purpose, we previously proposed a method for measuring the small change in thicknessof the arterial wall during the cardiac cycle [H. Kanai et al.: IEEE Trans. UFFC 43 (1996) 791, H. Kanai et al.: IEEE Trans.
UFFC 44 (1997) 752 and H. Hasegawa et al.: Electron. Lett. 33 (1997) 340]. The change in thickness cannot be measured
from conventional B-mode or M-mode images because of its small amplitude of less than 100 micrometers. Though the change
in thickness is useful for in vivo assessment of the local elasticity of the arterial wall, in some cases successful measurements
based on only two consecutive echos fail because of the low signal-to-noise ratio (SNR) of the echos. To realize the practicaluse of the proposed method, this problem must be overcome. In this paper, we propose a method that is more robust in thepresence of noise. In this method, more than two echos are employed to estimate their phase shift for reducing the influenceof low SNR. Simulations provide an optimal value of the estimation period for each SNR. By evaluating the SNR of in vivo
experimental data, the optimal estimation period is determined. In in vivo experiments using an optimal estimation period of
20 ms, the change in thickness can be measured with good reproducibility even in the case of a low SNR of 28 dB.
KEYWORDS: atherosclerosis, change in thickness of arterial wall, phase shift, signal-to-noise ratio (SNR)

1. Introduction
The steady increase in the number of patients with myocardial infarction or cerebral infarction, which are consideredmainly to be caused by atherosclerosis, is becoming a serious
problem. Therefore, it is important to diagnose atherosclerosis in an early stage. However, the methods employed forthe diagnosis of atherosclerosis such as computed tomography (CT) and magnetic resonance imaging (MRI) cause great
physical and mental hardship to patients.
The pulse wave velocity (PWV) method has been developed as a technique for the noninvasive diagnosis ofatherosclerosis.1) In this method, the elasticity of the arterial wall is evaluated by measuring the velocity of the pressure wave propagating from the heart to the femoral artery. It
is useful in terms of the noninvasive evaluation of elasticity,
however, elasticity cannot be evaluated locally due to a low
spatial resolution of several tenths of a centimeter, which corresponds to the distance from the heart to the femoral artery.
To increase the spatial resolution in the measurement of
PWV, we previously proposed a method to accurately measure the propagation velocity of vibrations on the arterial wallusing ultrasound.2) Using this method, the PWV between two
adjacent points which are several centimeters apart is noninvasively evaluated by estimating the time delay between the
resultant vibrations at these two points.3)
In this study, we evaluated the elasticity in each local region of a few millimeters not only to diagnose early-stage
atherosclerosis but also to measure the spatial distribution of
the elasticity, which is useful to diagnose vulnerability of theatherosclerotic plaque. For this purpose, the small change inthe thickness of the arterial wall due to the heartbeat is accurately measured in each local region which corresponds tothe focal area of the ultrasonic beam.4, 5) From the resultant
change in thickness, the local strain and the elasticity of thearterial wall are noninvasively evaluated.

To obtain the change in thickness of the arterial wall in thepreviously proposed method,2) each displacement of the intimal side and of the adventitial side is obtained by estimating
the phase shift between two consecutively received echos, andthe phase shift can be estimated with high accuracy when theecho is measured under good conditions of SNR. From thebasic experiments, the accuracy in the estimation of the displacement is less than 1 µm.4, 6, 7)
However, under in vivo conditions, the SNR of the received
echo signal is not always high mainly because there is not alarge difference in the acoustic impedance between the arterial wall and blood. In the case of low SNR, the accuracy
decreases when only two consecutive echos are employed todetermine the phase shift between them. In this paper, we
propose a method for reducing the influence of a decrease in
the SNR of the echo in the estimation of its phase shift using
more than two consecutive echo signals.
2. The Previous Method for Measurement of Small
Change in Thickness of the Arterial Wall
As illustrated in Fig. 1, for accurate measurement of smallchanges in thickness, the phase shift of the echo due to its
propagation back and forth between the object and the ultrasonic transducer is estimated from two consecutive echos.2)
For this purpose, quadrature demodulation is applied to thereceived echo reflected by the object, and then the in-phaseand the quadrature signals are A/D converted. From the demodulated signal, z(t; d) reflected at a depth d at a time t,
the phase shift, 1θ(t), between two consecutive echos is obtained from the complex cross correlation function computedfor M + 1 samples in the depth direction as follows:
M/2
X

e j 1θ(t) =z(t + T ; d + m D) · z ∗ (t; d + m D)

m=−M/2
M/2
X,∗|z(t + T ; d + m D) · z (t; d + m D)|

m=−M/2∗ E-mail address: hasegawa@us.ecei.tohoku.ac.jp

(1)
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Jpn. J. Appl. Phys. Vol. 39 (2000) Pt. 1, No. 5B

pulse repetitioninterval T
trigger

H. H ASEGAWA et al.

timearterial wallskinsurface

depth

3258

v (t)
received
signalultrasonictransducer
original
sinusoidal

Zr(t)

M+1
samplestime (depth)0lowpass filter

2 echosignals........

M+1
samplesT

Zi(t)previous
method2T

A/D convertor
3T

workstation

N+1 echosignalsFig. 2. Difference in the length of time used in the estimation between the
new method and the previously proposed method.

4T

time

proposedmethodreceived signal for eachtransmitted RF pulsev (t)Fig. 1.

Illustration of measurement of a small vibration on the arterial wall.

where D and T are the interval of sample points in the depth
direction and the pulse repetition interval, respectively, and ∗represents the complex conjugate. From the phase shift accurately estimated by eq. (1), the small velocity signal at time
t + T /2 is obtained as follows:¶µc0 1θ(t)
T
=−
,
(2)
v t+
2
2ω0 T
where ω0 and c0 are the angular frequency of the ultrasonicpulse and the speed of sound, respectively.
By subtracting the displacement, xi n (t), of the intimal sideof the arterial wall from that of the adventitial side, x ad (t), a
small change in thickness, 1h(t), of the arterial wall is obtained as follows:
1h(t) = x ad (t) − xi n (t)Z t{vad (t) − vi n (t)}dt.
=

(3)

−∞

3. A Method for Reducing Influence of Decrease in SNR
in Measurement of Small Change in Thickness of theArterial Wall2)

In the previously proposed method, the accuracy in the
N/2X

α(1θ(t)) =M/2
Xmeasurement of the displacement is less than 1 µm from thebasic experiments.4, 6, 7) However, under in vivo condition, the
SNR of the received echo is not always high in comparison
with that in the basic experiments. In the case of the lower
SNR, the accuracy in the estimation decreases when only twoconsecutively received echos are employed for the estimationof their phase shift.
In this paper, we propose a method for reducing the influence of the decrease in the SNR in the estimation of the phaseshift of the echo signals using more than two echos as illustrated in Fig. 2. An accuracy is improved in the case of thelower SNR using the proposed method. The principle is asfollows.
Let us assume that the amplitudes of two consecutively received echo signals are equal because the pulse repetition interval, T , is sufficiently short. In addition, we assume thatthe phase shift, 1θ(t), of the echos due to the wall motion isconstant during the short period 1T = (N + 1)T . Thus, theestimator, bz(t + nT ; d), of the quadrature modulated signal,z(t +nT ; d), at the time t +nT expected by z(t +(n −1)T ; d)is defined as follows:bz(t + nT ; d) = z(t + (n − 1)T ; d) · e j 1θ(t) .

(4)

The normalized mean squared difference, α(1θ(t)), betweenz(t + nT ; d) and bz(t + nT ; d) is defined as follows:|z(t + nT ; d + m D) − z(t + (n − 1)T ; d + m D) · e j 1θ(t) |2n=−N/2 m=−M/2
N/2
X

M/2
X

,
|z(t + nT ; d + m D)|2

n=−N/2 m=−M/2(5)where M and N are the number of samples in the depth direction and the number of echos used for calculating the differenceα(1θ(t)). By replacing the denominator of eq. (5) by A, eq. (5) can be rewritten asAα(1θ(t)) =

N/2
X

M/2
X

{|z(t + nT ; d + m D)|2

n=−N/2 m=−M/2− z(t + nT ; d + m D) · z ∗ (t + (n − 1)T ; d + m D) · e− j 1θ(t)− z ∗ (t + nT ; d + m D) · z(t + (n − 1)T ; d + m D) · e j 1θ(t)+ |z(t + (n − 1)T ; d + m D)|2 }.
By taking the partial derivative of eq. (6) with respect to 1θ(t), the following equation is obtained.

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H. H ASEGAWA et al.

Jpn. J. Appl. Phys. Vol. 39 (2000) Pt. 1, No. 5B

A·

3259

M/2
N/2
X
X
∂α(1θ(t)){ j z(t + nT ; d + m D) · z ∗ (t + (n − 1)T ; d + m D) · e− j 1θ(t)=∂1θ(t)
n=−N/2 m=−M/2− j z ∗ (t + nT ; d + m D) · z(t + (n − 1)T ; d + m D) · e j 1θ(t) }.

(7)

By setting the right hand side of eq. (7) to zero, the phase shift of the echo is determined by the following normalized complex
cross-correlation function.
N/2
X

M/2
X

z(t + nT ; d + m D) · z ∗ (t + (n − 1)T ; d + m D)

n=−N/2 m=−M/2¯.
e j 1θ(t) = ¯
¯
¯ X
M/2
X
¯
¯ N/2
∗
¯
¯
z(t
+
nT
;
d
+
m
D)
·
z
(t
+
(n
−
1)T
;
d
+
m
D)
¯
¯
¯
¯n=−N/2 m=−M/2

In the case of the lower SNR, accuracy in the estimationof the phase shift is improved by increasing the time period,
1T = (N + 1)T , used for the phase estimation. However,
the time resolution worsens when the estimation period, 1T ,becomes long. In this section, in the following simulations,the accuracy of the proposed method is evaluated for different
SNRs and estimation periods, 1T , to determine the optimal
estimation period for each SNR.
Figure 3 shows the waveform of the ultrasonic pulse used
for simulating the echos. Its center frequency is 7.5 MHz andthe pulse duration is 5 wavelength. In this simulation, thereceived echo signal is described by the summation of two
pulses reflected by the interfaces of the intima and adventitiaof the arterial wall, and the distance between these two interfaces is 0.6 mm at the R-wave of an electrocardiogram (ECG).
The pulse repetition frequency and the sampling frequency ofthe simulated echo are 1 kHz and 1 GHz, respectively. The
phase shifts due to the wall motion, which corresponds tothe displacements of intima and adventitia shown in Fig. 4,
are introduced into the echos reflected by the interfaces of
the intima and adventitia. After quadrature demodulation, themodulated signals are extrapolated to a sampling frequency

amplitude

1
0
(a)
0
Fig. 3.

2time [ µs]4

Waveform of the ultrasonic pulse used for simulating the echos.

From Fig. 5(b), it can be observed that the error curves haveminima at different locations, depending on the SNR.

displacement [ µm]4. Accuracy Evaluation by Simulationsof 10 MHz, which is used in the actual measurement system,
and Gaussian white noise is added to them to adjust the SNRof the demodulated signal.
The phase shifts between consecutively transmitted and received echo signals at the intimal side and adventitial side
are estimated by the proposed method for each SNR andeach estimation period 1T . The change in thickness, 1h(t),obtained by estimating the phase shift between two consecutively received echo is shown in Fig. 5(a) for different
SNRs. The time axis in Fig. 5(a) corresponds to one heartbeat. Under each condition, the root mean squared (RMS)
error, eRMS (snr; 1T ), of the estimated instantaneous change
vad (t; 1T ) − b
vi n (t; 1T ), from its
in thickness, b
vh (t; 1T ) = b
true value, vh (t; 1T ), is computed to evaluate the accuracyof the estimation of the phase shift as follows:
v
u
N
u1 X
|b
vh (t; 1T ) − vh (t; 1T )|2 , (9)eRMS (snr; 1T ) = t
N n=10
intimaadventitia
-100
(a)
-200

change in thickness [ µm]Such averaging has been employed in conventional pulsed
Doppler systems.8) Though the accuracy is improved using
multiple echo signals, when the estimation period becomes
long, the accuracy worsens because the time resolution ispoorer. For this reason, the estimation period should be set
to an optimum for each SNR condition. However, in conventional pulsed Doppler systems, the estimation period is fixed
to a certain length. In this paper, the optimal estimation period is determined by simulations, and in in vivo experiments
the optimal estimation period is used by evaluating SNR of invivo experimental data.

(8)

0

-15

(b)
-30
0

0.5

1
time [s]Fig. 4. (a) Displacements of the intimal side and the adventitial side, whichcorrespond to the phase shifts of the simulated echos. (b) Change in thickness (difference between the displacements shown in (a)).

H. H ASEGAWA et al.

Jpn. J. Appl. Phys. Vol. 39 (2000) Pt. 1, No. 5Bchange in thickness [µ m]

3260

1 cm

0

-10

(a)

-30

0

0.5

time [s]

(a)

1 cm

B-mode0.375 mm

-20

RMS error e RMS (snr; ∆ T) of estimatedinstantaneous change in thickness [mm/s]

beam position

actual
SNR=45dB
SNR=30dB
1

M-mode(b)

0.2
SNR=45dB
SNR=40dB
SNR=35dB
SNR=30dB

ECG

0.1

(b)
0

0

100
50
estimation period ∆T [ms]Fig. 5. (a) The change in thickness, 1h(t), estimated by the previously
proposed method for different SNRs. (b) The RMS error of the estimatedinstantaneous change in thickness from the true one as a function of SNR
and the estimation period.

3
mm/s 0-3
3mm/s 0
-3
40
µm 0
-40
3
mm/s 0-3
3mm/s 0
-3
40
µm 0
-40

vin (t)(d)vad (t)(e)∆h(t;0.3 ms )(f)vin (t)(g)vad (t)(h)∆h(t; 10 ms )
(i)
1

0

5. In vivo Experimental Results for the Human Common
Carotid arteryTwo sets of data were measured from the common carotid
artery of a 50-year-old male subject and a 25-year-old malesubject as shown in Figs. 6 and 7, respectively. The data sets
are acquired along the ultrasonic beam shown in Figs. 6(a)and 7(a). The velocity signals, vi n (t) and vad (t), on the intimal side and adventitial side are obtained by tracking points
preset at the beginning of the M-mode image as shown in
Figs. 6(d) and 6(e). Velocity signals are also obtained in thesame manner as shown in Figs. 7(d) and 7(e). The change inthickness, 1h(t), of the arterial wall is obtained by integrating the difference between vi n (t) and vad (t) as shown in Figs.
6(f) and 7(f). There are some differences between the change
in thickness, 1h(t), in Fig. 6 and that in Fig. 7. In Fig. 6,
1h(t) is reproducible from heartbeat to heartbeat. This is not
the case in Fig. 7. One reason for these differences is the decrease in the SNR from Fig. 6 to Fig. 7.
To determine the optimal estimation period 1T , the SNRof these two sets of data are evaluated. For this purpose, theecho signal reflected by the interface of the rubber plate fixedin the water tank is measured many times. The difference between the average and each echo represents the noise. The
noise estimate is extracted this way: First, the average, µ,of the absolute amplitude of the received signal, z(t; d), reflected at the depth d, is obtained by
1 X|z(t + nT ; d)|,
N n=1
N

b
µ=

(10)

where the period for averaging is 1 second, that is, N = 1000
(T = 1 ms). The noise power, σn2 , is evaluated by

(c)

2

3
time [s]Fig. 6. In vivo experimental results for the human carotid artery of a
50-year-old male subject (SNR = 39 dB). (a) B-mode image. (b) M-mode
image. (c) Electrocardiogram. (d), (e), and (f) Velocity signals, vi n (t) andvad (t), on the intima and adventitia, and the change in thickness, 1h(t),estimated with an estimation period of 333 µs (2 frames). (g), (h), and (i)Velocity signals, vi n (t) and vad (t), on the intima and adventitia, and thechange in thickness, 1h(t), estimated with an optimal estimation periodof 10 ms.

1 X{|z(t + nT ; d)| − b
µ}2 .
N n=1
N

b
σn2 =

(11)

Second, the signal power, σs2 , in an in vivo experiment isobtained by
1 X|z(t + nT ; d + b
x (t + nT ))|2 ,
N n=1
N

b
σs2 =

(12)

where bx (t +nT ) is the estimated displacement of the wall due
to its motion estimated by the phased tracking method.2)
Finally, the SNR in an in vivo experiment is obtained by
SNR = 10 log10

b
σs2
[dB].
b
σn2

(13)

For the data in Figs. 6 and 7, the SNRs are evaluated as
39 dB and 28 dB, respectively. The optimal estimation periods for Figs. 6 and 7 are 10 ms and 20 ms, respectively,
according to the results of the simulation experiment in
Fig. 5(b).
Using the optimal estimation period 1T , the change inthickness is obtained as shown in Figs. 6(i) and 7(i). In thecase of the relatively high SNR (Fig. 6), the waveforms are
hardly changed due to this new method in Figs. 6(f) and 6(i).

Jpn. J. Appl. Phys. Vol. 39 (2000) Pt. 1, No. 5B

H. H ASEGAWA et al.

On the other hand, clear improvement is obtained in the caseof lower SNR as shown in Figs. 7(f) and 7(i).
In Fig. 8, the improvement in reproducibility, γ , between
heart cycles is plotted as a function of SNR. The improvement in reproducibility is defined by the difference between
standard deviations, σ2 f r and σopt , evaluated for each estima-tion based on two echos and the optimal estimation period as
follows:
γ = σ2 f r − σopt .

(14)

The standard deviations, σ2 f r and σopt , between M heartbeats
are computed during one cardiac cycle, which corresponds tothe time period N T , as follows:

1 cm

3

(15)

(16)

21-year-old man
23-year-old man
25-year-old man
31-year-old man
33-year-old man
50-year-old man

2
1
0
26

30

34

38
SNR [dB]

Fig. 8. Improvement in the reproducibility of the change in thickness plotted as a function of SNR.

beam position
1 cm

B-mode0.375 mm5mm/s 0-55mm/s 0
-5
40
µm 0
-40
5
mm/s 0-55mm/s 0
-5
65
µm 0
-65

Improvement inreproducibility, γ [ µm]

v
N u
M
M
X
1 Xu
1 X
t1
σ2 f r ={1h(nT + Tr (m); T ) −1h(nT + Tr (m); T )}2 ,N n=1 M m=1
M m=1
v
N u
M
M
u1 X
X
1
1 X
t
{1h(nT + Tr (m); 1Topt ) −1h(nT + Tr (m); 1Topt )}2 ,σopt =
N n=1 M m=1
M m=1
where Tr (m) and 1Topt are the time of mth R-wave of theelectrocardiogram and the optimal estimation period. Figure 8 shows that the proposed method is effective, especiallyin the case of the lower SNR.

3261

(a)

6. Conclusions
M-mode(b)

ECGvin (t)(c)(d)

vad (t)(e)∆h(t;0.3 ms )(f)vin (t)(g)

We have already proposed a method for the accurate measurement of displacement by estimating the phase shift of twoconsecutive echo signals. However, the accuracy decreases
with the SNR of the received signal. In this paper, a method
is proposed for reducing the influence of the decrease in the
SNR using multiple echos. Using this method, a small change
in thickness of the arterial wall is obtained with good reproducibility even in the case of low SNR.
Acknowledgement
This work was supported in part by a Grant-in-Aid for Scientific Research (10650400, 10555134, 61011900) and Intelligent Cosmos Academic Foundation.

vad (t)(h)∆h(t; 20 ms )
(i)
0

1

2

3
time [s]Fig. 7. In vivo experimental results for the human carotid artery of a
25-year-old male subject (SNR = 28 dB). (a) B-mode image. (b) M-mode
image. (c) Electrocardiogram. (d), (e), and (f) Velocity signals, vi n (t) andvad (t), on the intima and adventitia, and the change in thickness, 1h(t),estimated with an estimation period of 333 µs (2 frames). (g), (h), and (i)Velocity signals, vi n (t) and vad (t), on the intima and adventitia, and thechange in thickness, 1h(t), estimated with an optimal estimation periodof 20 ms.

1) P. Hallock: Arch. Int. Med. 54 (1934) 770.
2) H. Kanai, M. Sato, Y. Koiwa and N. Chubachi: IEEE Trans. UFFC 43(1996) 791.
3) H. Kanai, K. Kawabe, M. Takano, R. Murata, N. Chubachi and Y. Koiwa:
Electron. Lett. 30 (1993) 534.
4) H. Kanai, H. Hasegawa, N. Chubachi, Y. Koiwa and M. Tanaka: IEEE
Trans. UFFC 44 (1997) 752.
5) H. Hasegawa, H. Kanai, N. Chubachi and Y. Koiwa: Electron. Lett. 33(1997) 340.
6) H. Hasegawa, H. Kanai, N. Hoshimiya, N. Chubachi, Y. Koiwa: Jpn. J.
Appl. Phys. 37 (1998) 3101.
7) H. Kanai, K. Sugimura, Y. Koiwa and Y. Tsukahara: Electron. Lett. 35(1999) 949.
8) C. Kasai, K. Namekawa, A. Koyano and R. Omoto: IEEE Trans. Sonics& Ultrason. 32 (1985) 458.