J Med Ultrasonics (2013) 40:91–98
DOI 10.1007/s10396-012-0400-9

ORIGINAL ARTICLE

Measurement of regional pulse wave velocity using very highframe rate ultrasound
Hideyuki Hasegawa • Kazue Hongo •Hiroshi Kanai

Received: 1 May 2012 / Accepted: 1 August 2012 / Published online: 10 October 2012 The Japan Society of Ultrasonics in Medicine 2012

Abstract
Purpose Pulse wave velocity (PWV) is the propagation
velocity of the pressure wave along the artery due to the heartbeat. The PWV becomes faster with progression of arteriosclerosis and, thus, can be used as a diagnostic index ofarteriosclerosis. Measurement of PWV is known as a noninvasive approach for diagnosis of arteriosclerosis and is widely usedin clinical situations. In the traditional PWV method, the average PWV is calculated between two points, the carotid andfemoral arteries, at an interval of several tens of centimeters.
However, PWV depends on part of the arterial tree, i.e., PWVs
in the distal arteries are faster than those in the proximal arteries.
Therefore, measurement of regional PWV is preferable.
Methods To evaluate regional PWV in the present study, theminute vibration velocity of the human carotid arterial wall
was measured at intervals of 0.2 mm at 72 points in the arterial
longitudinal direction by the phased-tracking method at a hightemporal resolution of 3472 Hz, and PWV was estimated byapplying the Hilbert transform to those waveforms.
Results In the present study, carotid arteries of threehealthy subjects were measured in vivo. The PWVs in shortsegments of 14.4 mm in the arterial longitudinal direction
were estimated to be 5.6, 6.4, and 6.7 m/s, which were ingood agreement with those reported in the literature. Furthermore, for one of the subjects, a component was clearlyfound propagating from the periphery to the direction of theheart, i.e., a well known component reflected by the
H. Hasegawa  K. Hongo  H. Kanai
Graduate School of Biomedical Engineering, Tohoku University,
6-6-05 Aramaki-aza-Aoba, Aoba-ku, Sendai 980-8579, Japan
H. Hasegawa (&)  H. Kanai
Graduate School of Engineering, Tohoku University,
6-6-05 Aramaki-aza-Aoba, Aoba-ku, Sendai 980-8579, Japane-mail: hasegawa@ecei.tohoku.ac.jp

peripheral arteries. By using the proposed method, thepropagation speed of the reflection component was also
separately estimated to be -8.4 m/s. The higher magnitudeof PWV for the reflection component was considered to be
the difference in blood pressure at the arrivals of the forward
and reflection components.
Conclusion Such a method would be useful for moresensitive evaluation of the change in elasticity due toprogression of arteriosclerosis by measuring the regional
PWV in a specific artery of interest (not the average PWV
including other arteries).
Keywords Regional pulse wave velocity  Parallel beam
forming  High frame rate

Introduction
The number of patients suffering from arteriosclerosis is
rapidly increasing with the aging of society and dietary
westernization. It is important to make an early diagnosis
of arteriosclerosis to prevent serious diseases such as
myocardial and cerebral infarction.
Recently, there are many methods for diagnosis of arteriosclerosis, such as angiography, intra-vascular ultrasound(IVUS), optical coherence tomography (OCT), computed
tomography (CT), magnetic resonance imaging (MRI), andpositron emission tomography (PET) [1–3]. These methods
are invasive and/or not easily applicable. Therefore, they are
unsuitable for repetitive diagnosis to monitor progression ofarteriosclerosis. On the other hand, diagnosis with ultrasoundis noninvasive, easy to use at bedside, and suitable for repetitive diagnosis.
Ultrasound B-mode imaging is widely used for the morphological diagnosis of the arterial wall, particularly for

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measurement of intima-media thickness [4–6]. In addition,
methods for evaluating the elasticity and viscosity of thearterial wall have recently been developed [7–13]. For diagnosis of early-stage arteriosclerosis, it is useful to evaluate theelastic property of the arterial wall because arterial-wallelasticity changes with progression of arteriosclerosis [14].
For noninvasive clinical evaluation of arterial elasticity,
the pulse wave velocity (PWV) method was developed for
diagnosis of arteriosclerosis nearly 100 years ago [15, 16].
This method measures the pulse wave, which is the pressure
wave generated by the heartbeat, that propagates in thelongitudinal direction of the artery. Since PWV cPWV
becomes faster with progression of arteriosclerosis, it is used
as a diagnostic index of arteriosclerosis and is expressed by
the Moens–Korteweg equation as follows [16]:sﬃﬃﬃﬃﬃﬃﬃﬃ
Eh hcPWV ¼;
ð1Þ
qD
where Eh is the Young’s modulus in the circumferential
direction of the artery, h is the thickness of the arterial wall,
q is the blood density, and D is the diameter of the artery.
The assumptions underlying the Moens–Korteweg equation
are that the artery has a thin wall and is filled with anincompressible inviscid liquid. Since PWV is traditionallymeasured between the carotid artery and femoral artery(cfPWV (carotid-femoral PWV) method) [17, 18] it is notsuitable for early diagnosis of arteriosclerosis because
lesions of early-stage arteriosclerosis are several millimetersin size. In addition, PWV depends on a part of the arterial
tree, i.e., PWV in the distal arteries is faster than that in theproximal arteries. Thus, measurement of regional PWV
would be useful for diagnosis of early-stage arteriosclerosis.
We previously studied regional PWVs by measuring
small vibration velocities of the arterial wall at two points
along the arterial longitudinal direction [19]. To estimate
regional PWV more accurately, in the present study, vibration velocities at multiple points in a region of about 14 mmalong the arterial longitudinal direction were measured at ahigh temporal resolution of 3472 Hz. Vibration velocities of
the arterial wall have frequency components mostly up to
30 Hz. However, a high temporal resolution is desirable toestimate a small time delay between vibration velocities due
to pulse wave propagation, i.e., the pulse wave propagates aregion of about 14 mm in 2.8 ms when PWV is 5 m/s.

J Med Ultrasonics (2013) 40:91–98

measured by the phased-tracking method [20] at 72 points
in the arterial longitudinal direction with intervals of
0.2 mm (total length of 14.4 mm) at a high temporal resolution of 3472 Hz. Modified ultrasonic diagnostic equipment (Aloka, a-10) was used to acquire ultrasonic RF
echoes received by each transducer element [21, 22]. The
center frequency and fractional bandwidth of an emitted
ultrasound pulse were 8 MHz and 61 %, respectively. The
received RF echo was sampled at 40 MHz with 16-bit
resolution. The method for high frame rate imaging
developed by our group [21] was used to obtain beamformed ultrasonic RF echoes at 72 beam positions at a very
high frame rate of 3472 Hz.
Typical PWV is several m/s and passes through a small
region of 14.4 mm in a very short period of a few milliseconds. In such a situation, a small random error in estimation of the time delay between velocity waveforms
would lead to a significant error in estimation of PWV
when PWV is estimated using velocities measured at only
two points along the arterial longitudinal direction.
Therefore, in the present study, vibration velocities were
measured at multiple points to reduce random errors, suchas electrical noise and arterial motion estimation error due
to random interference of echoes from scatterers in thearterial wall [13].
In the phased-tracking method, ultrasonic pulses atangular frequency x0 ¼ 2pf0 were transmitted from anultrasonic transducer placed on the skin surface at a timeinterval of DT ¼ 1=fFR (fFR: frame rate) at each ultrasonicbeam position. The instantaneous distance between themoving arterial wall and the ultrasonic transducer isdenoted by zðtÞ ¼ c0 su ðtÞ=2, where c0 is the velocity ofultrasound, and su ðtÞ is the time delay required for two-way
transmission between the ultrasonic transducer and thearterial wall. The phase h(z;t) of the quadrature demodulated signal of the received ultrasonic echo is given by
hðz; tÞ ¼ x0 su ðtÞ ¼ 2x0 zðtÞ=c0 . The phase differenceDhðz; tÞ between time t and ðt þ DTÞ is given bycarotid artery 0.2 mm......

Materials and methods
Measurement of small vibration velocity for the carotid
arterial wall
In the present study, as illustrated in Figs. 1 and 2, minutevibration velocities of the human carotid arterial wall were123

proximalbeam number 0, 1, ...

distal, 71

Fig. 1 Illustration of measurement of minute vibration velocity
waveforms of the human carotid arterial wall at 72 points at intervalsof 0.2 mm

J Med Ultrasonics (2013) 40:91–98

93Fig. 2 Illustration of themethod for estimation of timedelay between acceleration
waveforms a(xm; t) and{a(xm?M/2; t ? si)} using theiranalytic signals..
δτ+τ −1 a .
( x m+M/2 ; t+τ −1)δτphase shiftγ( t 0 ; τ i )regressionlinea ( x m+M/2 ; t )0lag τ ia ( x m ; t)δτ+τ 1

Dhðz; tÞ ¼ hðz; t þ DTÞ  hðz; tÞ ¼

2x0 DzðtÞ
;
c0

ð2Þ

where DzðtÞ ¼ zðt þ DTÞ  zðtÞ is the displacement of an
object during the period DT. By dividing the displacement
Dz by the period DT, the average velocity, denoted byvðt þ DT=2Þ, of the object during the period DT is given by
the phase difference Dhðz; tÞ between echoes as follows:


DT
DzðtÞ
c0 Dhðz; tÞ
v tþ
¼
;
ð3Þ
¼
2
DT
2DT x0In the present study, parallel beam forming (PBF) [21]was applied to realize a high frame rate of 3472 Hz. The
minute vibration velocity waveforms were measured at 72
points on the human carotid artery wall at intervals of
0.2 mm.
Estimation of PWV by complex correlation approach
In the present study, the acceleration waveforms {a(xm; t)}were obtained by applying time differentiation to thevibration velocity waveforms {v(xm; t)}, where xm is themth ultrasonic beam position, to enhance high frequency
components contained in the measured velocity waveforms
because the use of high frequency components improves
the time resolution in estimation of time delays among thevibration waveforms due to the propagation of the pulse
wave. To estimate PWV, the phase /ðxm ; t0 Þ of acceleration waveform aðxm ; t0 Þ at a time of interest t0 (=frame)measured at the mth ultrasonic beam position xm is
obtained by applying the Hilbert transform to the acceleration waveforms {a(xm; t)}. The Hilbert transform wasapplied to the acceleration waveforms during just an entire
cardiac cycle. The phase /ðxm ; tÞ is determined from thecomplex analytic signal g(xm; t) of a(xm; t).
To determine the PWV from the phases of the multiple
analytic signals, a method based on the complex crosscorrelation function was developed in the present study.
The complex cross correlation function cðt0 ; si Þ at t0 at laga ( x m+M/2 ; t +τ 1)..
.

intercept qslope psi ¼ i  DT, where i = -Nl, -Nl ? 1, …, -1, 0, 1, …,
Nl - 1, Nl (Nl denotes the number of lags used in estimation of PWV), is defined as follows:bc ðt0 ; si Þ ¼

M=2 X
Nc
X

g ðxm ; t0 Þ  gðxmþM=2 ; t0 þ si Þ;ð4Þ

m¼0 i¼Ncwhere si ¼ i  DT, M is the number of ultrasonic beam
positions, and Nc determines the number of frames used for
calculation of the complex correlation function.
By denoting the original time delay of g(xm?M/2; t0)
from g(xm; t0) due to the propagation of the pulse wave inthe length of dx  M=2 (dx: lateral spacing of ultrasonicbeams) by ds and denoting the dominant frequency of{g(xm; t0)} at t0 by fa, ds is expressed as follows:ds ¼ 

\cðt0 ; 0Þ
:
2pfa

ð5Þ

For other lag si, the model \bc ðt0 ; si Þ of the phase of c(t0, si)
is expressed as follows:\bc ðt0 ; si Þ ¼ 2pfa ðsi  dsÞ¼ 2pfa  ði  DTÞ  2pfa ds :ð6Þ

 p  i þ q:
To estimate the dominant frequency fa and time delay ds
due to propagation of the pulse wave, the coefficients p andq are determined by fitting a linear function defined by
Eq. (6) to measured f\cðt0 ; si Þg using the least squares
method as follows: the difference a(t0) between the angleof the measured complex correlation function \cðt0 ; si Þand the model \bc ðt0 ; si Þ is expressed as follows:NlXj\cðt0 ; si Þ  ðp  i þ qÞj2 :ð7Þ
aðt0 Þ ¼
i¼Nl

To determine the coefficients pb and qb, which minimizea(t0), partially differentiated a(t0) with respect to p and q
are set to zero. By solving the resultant simultaneous
equations, the coefficients p and q are determined asfollows:

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pb ¼

qb ¼

J Med Ultrasonics (2013) 40:91–98

ð2Nc þ 1Þ

nP
Nc

i¼Nc i  \cðt0 ; si Þ

o



P
Nc

i¼Nc inP

Nc
i¼Nc \cðt0 ; si Þ

2
P
 P
Nc
Nc
2 
ð2Nc þ 1Þ
i
i
i¼Nc
i¼Nc

pb

P Nc

i¼Nc i 

P Nc

i¼Nc \cðt0 ; si Þ

ð2Nc þ 1Þ:ð9Þ

Based on Eq. (6), the dominant frequency fa is determinedusing the estimated coefficient pb as follows:
fba ¼

pb
:
2pDT

ð10Þ

Finally, PWV cPWV(t0) is estimated based on Eq. (6) using
the estimated dominant frequency fba and coefficient qb as
follows:
M

 dx
c
ds
p fb  M  dx
:
¼ a
qb

cPWV ðt0 Þ ¼ 2

ð11Þ

In Eq. (4), the complex correlation function between analytic signals {g(xm; t0)} and {g(xm?M/2; t0)} was usedbecause the distance between each pair of the analytic
signals should be as large as possible to increase the timedelay between the analytic signals and also all the M analytic signals can be used to reduce random errors.
Results
Figure 3a–c shows the B-mode images of 24-(subject A),
23-(subject B), and 38-(subject C) year-old males,
respectively. Figures 4, 5, and 6 show in vivo experimental
results of subjects A, B, and C, respectively. The shortsegments of their common carotid arteries of 14.4 mm ((72beam positions) 9 (0.2 beam spacing)) were measured. In
each of Figs. 4, 5, and 6, electrocardiogram (ECG) (a); thedisplacement waveform (b), which is obtained by integrating the vibration velocity in (c) with respect to time;
vibration velocity measured at the 0th beam position (the0th beam was nearest to the heart) (c); and the acceleration
waveform (d), which is obtained by temporal differentiation of the vibration velocity in (c), are shown. Velocities at
the media-adventitia boundaries of the posterior walls were
measured as the vibration velocities of the arterial walls.
The vibration of the arterial wall is generated by theejection of blood from the left ventricle of the heart, andexpansion of the artery corresponds to the acceleration
waveform around 0.1 s. For example, the enlarged view of
the acceleration waveforms of subject A measured at the

123

o
:

ð8Þ

0th (proximal) and 71st (distal) beams during a period
0–0.3 s is shown in Fig. 7. As can be seen in Fig. 7, it isrecognized that the positive peak around 0.07 s at the 0th
beam precedes that of the 71st beam very slightly. In thepresent study, the large positive peak around t = 0.1 s (t0for estimation of PWV is shown by the vertical dashed line
in Fig. 4), which was caused by ejection of blood from theleft ventricle, was analyzed for estimation of PWV.
To estimate PWVs, the Hilbert transform was applied tothe acceleration waveforms. The complex cross correlation
function defined in Eq. (4) was obtained by setting t0 to be
the time of the peak in the acceleration waveform measured at the 0th beam position around 0.1 s. The assigned t0
is shown by the vertical dashed lines in Figs. 4, 5, and 6.
The length of the correlation window was set at 10 ms(corresponding to Nc = 17), which roughly corresponded
to the width of the peak in the acceleration waveform. Plotsin Fig. 8 show the measured phase \cðt0 ; si Þ of the complex correlation function plotted as a function of the lag si.
The PWV was estimated from the slope pb and intercept qb
of the regression line (shown by the dashed line in Fig. 8)
based on Eq. (11). By applying the same procedure to theacceleration waveforms of subjects B and C, regional
PWVs of subjects A, B, and C were estimated to be 5.6,
6.4, and 6.7 m/s, respectively. The estimated pulse wave
velocities were similar to those reported in the literature[23–25].
By observing the acceleration waveform of subject A(shown in Fig. 4d), there is a small but distinct peak aroundthe time of 0.2 s. In the literature, such a peak detected in
the acceleration waveform is reported to be a pulse wave
component reflected in the distal position of the arterial
tree [26–28]. As can be seen in Fig. 7, the acceleration
waveform at the 71st beam (nearer to the head) precedesthat at the 0th beam very slightly. The PWV for thiscomponent was estimated by setting t0 to the time of thepeak in the acceleration waveform measured at the 0th
beam position around 0.2 s (t0 for this reflection component and is shown by the vertical dashed line in Fig. 4).
The estimated PWV for the reflection component was-8.4 m/s, where the negative velocity means the propagation from the periphery to the heart. As described above,
the regional PWV of the reflection component could also
be estimated separately using the proposed method.
Furthermore, for evaluation of the reproducibility ofmeasurements using the proposed method, five additional

J Med Ultrasonics (2013) 40:91–98

95

2.0 mm

2.0 mm

2.0 mm

1.9 mm

(a)

(b)

(c)

Fig. 3 B-mode images of carotid arteries of 23-(subject A), 24-(subject B), and 38-(subject C) year-old healthy males

(a)

(a)
0.3

0.3

0.2

0.2

mm 0.1mm 0.1
0

0

(b)

−0.1
8

8

4

mm/s 4

mm/s
0

0

(c)

−4

(c)

−4
0.4

0.4

m/s2

(b)

−0.1

t 0 for forward component

0.2

m/s

2 0.2

0

0
t 0 for reflection component

−0.2
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(d)
0.8

0.9

−0.2

(d)

t0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

time [s]time [s]Fig. 4 In vivo experimental results of subject A. a Electrocardiogram,b displacement, c velocity v(xm; t), and d acceleration a(xm; t) of theposterior wall measured at the 0th ultrasonic beam

Fig. 5 In vivo experimental results of subject B. a Electrocardiogram,b displacement, c velocity v(xm; t), and d acceleration a(xm; t) of theposterior wall measured at the 0th ultrasonic beammeasurements were performed with respect to subject C on
the same day. Figure 9a shows acceleration waveforms in
the 0th scan lines for the five measurements. Using thephase \cðt0 ; si Þ of the complex correlation function shownin Fig. 9b, the regional pulse wave velocities were estimated. The mean and standard deviation for the fivemeasurements were 7.0 and 0.72 m/s (10.4 % of the meanvalue), respectively.

Discussion
For one of the subjects in the present study, a pulse wave
component, which was considered to be reflected by the
distal arterial tree, was found. By applying the Hilberttransform and the proposed complex cross correlation
technique to the acceleration waveforms obtained by differentiating the measured velocity waveforms, the forward

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J Med Ultrasonics (2013) 40:91–98

(a)
0.3
0.2

mm 0.1
0

(b)

−0.1
8

phase of correlation function [rad]

96

0

−0.1

−0.2

−0.3
−1

4

0

1

lag τi [ms]

mm/s
0

(c)

−4
0.4
2

m/s

Fig. 8 Phase of complex correlation function \cðt0 ; si Þ plotted as afunction of lag si. Plots and the dashed line show measured phases
and the regression line

0.2
0

0

0.1

0.1

(d)

t0
0.2

0.3

0.4

0.5

0.6

0.7

0.8

time [s]Fig. 6 In vivo experimental results of subject C. a Electrocardiogram,b displacement, c velocity v(xm; t), and d acceleration a(xm; t) of theposterior wall measured at the 0th ultrasonic beam

m/s 2

−0.2

0

−0.1

(a)

0.4

0
0.3

0−th beam

0.1
0

−0.1
−0.2
0

0.1

0.2

0.3

time [s]Fig. 7 Enlarged view of the acceleration waveforms {a(xm; t)} of
subject A during 0–0.3 s measured at the 0th (solid line) and 71th
(dashed line) ultrasonic beams

and reflection components were separately analyzed. The
estimated propagation speed of the forward component was
lower (5.6 m/s) than that (8.4 m/s in the opposite direction)of the reflection component. One of the reasons for thisdifference was considered to be the difference between
blood pressures at the arrival times of the forward andreflection components in the carotid artery.
It was reported that the arterial wall distension waveform can be assumed to be the waveform of blood pressure[29–31]. Figure 4b shows the displacement of the posteriorarterial wall obtained by integrating the measured velocity

123

0.4

0.6

0.8

0
lag [ms]

1

1time [s]phase of correlation function [rad]mm/s 2

0.2

−0.3

0.2

71−th beam

0

−0.1

−0.2

−0.3

(b)
−1

Fig. 9 a Acceleration waveforms in the 0th scan lines measured forthe five additional measurements with respect to subject C. b Phasesof complex correlation functions f\cðt0 ; si Þg plotted as a function oflag si. Plots and lines show measured phases and the regression lines

waveform with respect to time. At the time (around 0.1 s)of the peak of the forward component in the acceleration
waveform shown in Fig. 4d, the magnitude of the displacement is nearly zero. This fact suggests that bloodpressure at the arrival of the forward component corresponds to the diastolic blood pressure. On the other hand,the displacement at the time of the arrival of the reflection
component is nearly maximum, which suggests that blood

J Med Ultrasonics (2013) 40:91–98

97

pressure at the time of arrival of the reflection component isnearly the systolic blood pressure. It is also reported thatthe stress–strain relationship of the arterial wall is nonlinear [32–35], and the elastic modulus of the arterial wall at
higher blood pressure is larger than that at lower bloodpressure. Therefore, the PWV of the reflection component(presumably at higher blood pressure) was considered to belarger than that at the forward component.
In the present study, the propagation of pulse waves in a
local region in the carotid artery was analyzed in detail bymeasuring vibration velocities of the arterial wall at a very
high temporal resolution of 3472 Hz. To measure theregional PWV in an arterial longitudinal segment of about
15 mm, a small time delay of a few milliseconds between
vibration waveforms of the arterial wall needs to be estimated. For this purpose, in the present study, acceleration
waveforms, which contain higher frequency components
compared with displacement waveforms, were measured at
a high temporal resolution, and a method for estimation of asmall time delay using the phase of the acceleration waveform was developed.
Figures 10b–d show the displacement, velocity, andacceleration waveforms sampled at a frame rate of 35 Hz(waveforms in Fig. 4 were down sampled), which is atypical frame rate of conventional ultrasonic diagnostic
equipment. As shown in Fig. 10, it is difficult to measure
the velocity and acceleration waveforms at a temporal

(a)

mm

(b)
8
4

mm/s
0

(c)

−4
0.4

m/s2

resolution of 35 Hz. Using the displacement waveformshown in Fig. 10d, which was sampled at 35 Hz, the pulse
wave velocity was also estimated by the proposed method
described in ‘‘Estimation of PWV by a complex correlation
approach’’. The estimated PWV was -37.2 m/s, which
was significantly different from that reported in the literature. Therefore, it is necessary to measure the vibration
waveform at a high temporal resolution.
As described above, in the present study, regional pulse
wave velocities were noninvasively measured in vivo by
the proposed method and compared with those reported in
the literature. In future studies, it will be necessary toevaluate the accuracy of estimation of regional PWV using
the proposed method by basic experiments using phantoms.
In addition, the conditions of the spacing and number ofscan lines and frame rate, etc., required for measurement ofregional PWV also need to be investigated by phantom
experiments. However, the results obtained in the present
study show the possibility of the proposed method formeasurement of regional PWV, and such a method would
be useful for diagnosis of arteriosclerosis.

Conclusion
For detailed analyses of pulse wave propagation in a local
region of 14.4 mm in the carotid artery, in the present
study, small vibrations on the arterial wall were measured
with ultrasound at a very high frame rate of 3472 Hz. By
applying the Hilbert transform to the acceleration waveforms of the arterial wall, which were obtained by differentiating the measured vibration velocity waveforms, pulse
wave velocities of three healthy subjects were estimatedusing the proposed complex cross correlation approach.
Furthermore, in one of the subjects, there was a component
propagating from the heart to the periphery and from theperiphery to the heart. Even in such a case, pulse wave
velocities for these forward and reflection components
were separately estimated, and the magnitudes of theestimated pulse wave velocities were similar to thosereported in the literature. Such a method for measurementof regional PWV would be useful for more sensitive
evaluation of the change in elasticity due to progression ofarteriosclerosis.

0.2

Conflict of interest

0
−0.2

None.

(d)
0 t 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

time [s]Fig. 10 Vibration waveforms of subject A (waveforms in Fig. 4 aredown sampled at 35 Hz). a Electrocardiogram, b displacement,c velocity v(xm; t), and d acceleration a(xm; t) of the posterior wallmeasured at the 0th ultrasonic beam

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