52

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no. 1, january 2006

Improving Accuracy in Estimation ofArtery-Wall Displacement by Referring toCenter Frequency of RF Echo
Hideyuki Hasegawa, Member, IEEE, and Hiroshi Kanai, Member, IEEE
Abstract—Noninvasive measurement of mechanical properties, such as elasticity, of the arterial wall, is useful for diagnosis of atherosclerosis. The elasticity of the arterial wall
can be estimated by combining measurement of displacement of the arterial wall with that of blood pressure. In
general, the displacement of the arterial wall is estimatedfrom the phase shift of radio frequency (RF) echoes between
two consecutive frames using a correlation estimator withquadrature demodulated complex signals. Recently, digitized data of broadband RF echoes are available in modern
diagnostic equipment. The Fourier transform can be used
to estimate the phase of the RF echo at each frequency
within the RF frequency bandwidth. Therefore, the phaseshifts between RF echoes of two consecutive frames can be
estimated at multiple frequencies. In this estimation, due
to object displacement, the RF echo is time shifted in comparison with that of the previous frame. However, the position of the time window for the Fourier transform is notchanged between two consecutive frames. This change in
relative position between the RF echo and the time window
has a strong inﬂuence on the estimation of the artery-walldisplacement, resulting in error. To suppress this error, thephase shift should be estimated at the actual RF center
frequency. In this paper, this error suppression was investigated through simulation experiments and in vivo experiments on the human carotid artery.

I. Introduction
oninvasive measurement of mechanical properties of
the arterial wall, such as elasticity, is useful for diagnosing atherosclerosis because there are signiﬁcant diﬀerences between the elastic moduli of normal arterial walls
and those aﬀected by atherosclerosis [1], [2].
For assessment of mechanical properties, various methods have been proposed to measure the displacement of thearterial wall. Elasticity, based on estimation of the pulse
wave velocity (PWV) [3]–[6], and homogeneity of distensibility [7], which is evaluated by using the change in diameter obtained from displacements of the near and far walls,
are noninvasively evaluated by measuring displacements at
multiple points along the axial direction of the artery. By
deﬁnition, one-point measurement of the change in diameter is widely used for assessment of mechanical propertiesof the arterial wall [8]–[10].

N

Manuscript received December 28, 2004; accepted June 16, 2005.
H. Hasegawa and H. Kanai are with the Department of Electronic
Engineering, the Graduate School of Engineering, Tohoku University,
Sendai 980-8579, Japan (email: hasegawa@us.ecei.tohoku.ac.jp).

In addition to the displacement distribution in the axial direction of the artery, the distribution in the radial
direction of the arterial wall has been measured with intravascular ultrasonography (IVUS) [11], [12]. In the citedstudies, the elasticity distribution of coronary atherosclerotic plaque was obtained by the measured displacement
distribution in the radial direction. Comparison of suchmeasured elasticity distribution with the pathological image suggested the potential for tissue characterization ofatherosclerotic plaque by measuring its elasticity.
As a transcutaneous approach, the displacement andstrain around carotid atherosclerotic plaque have beenmeasured using tissue Doppler imaging [13]. The inhomogeneity in displacements measured upstream and downstream of atherosclerotic plaque suggested that artery-wallmotion has potential for use in the evaluation of plaquevulnerability.
We have been measuring the displacement and change
in thickness of the arterial wall caused by the heartbeatwith transcutaneous ultrasound [14]–[16]. Elasticity images of the human carotid artery have been obtained by
the measured displacement distribution, and the novel
potential for transcutaneous tissue characterization has
been shown by classifying the elasticity images using theelasticity reference data obtained by in vitro experiments[17], [18].
In the various methods described above, an autocorrelation technique [19] is widely used for measurementof velocities of blood ﬂow and tissue motion. This technique was designed to estimate the phase shift, ∆θ(f0 ),with respect to the center frequency, f0 , of a radio freˇ
quency (RF) echo using a complex signal, rf(t),which isobtained by quadrature demodulation at a demodulation
frequency, fdem . Therefore, use of a conventional autocorrelator requires knowledge of the actual center frequency,
f0 , of an RF echo to obtain unbiased velocity estimates.
To overcome this problem, both the mean phase shift,∆θ(f0 ), and the RF center frequency, f0 , are estimated byapplying the two-dimensional (2-D) [time (depth)–frame]
Fourier transform to RF echoes, rf1 (t) and rf2 (t), of twoconsecutive frames [20]. However, an autocorrelation technique with the quadrature demodulated complex signal,
ˇrf(t),
would be preferable because autocorrelation methods require lower sampling frequency in comparison withthe analysis of RF echoes. Therefore, use of a 2-D [time(depth)–frame] autocorrelator with quadrature demoduˇ
lated signals, rf(t),has been proposed to estimate the meanc 2006 IEEE
0885–3010/$20.00 hasegawa and kanai: estimation of artery-wall displacement

Doppler frequency and the RF center frequency. Estimation by the 2-D autocorrelator has been proven to be identical to that with the 2-D Fourier transform [21]. A 2-D
autocorrelator recently was applied to the measurement ofvessel wall displacement [22].
Recently, digitized data of broadband RF echoes havebecome available in modern diagnostic equipment. For velocity (displacement) estimation, the Fourier transform
can be used to extract the phase information of RF echoes
at multiple frequencies within the frequency bandwidth ofan RF echo. In the phase-shift estimation with the Fouriertransform, the position of the time window for the Fouriertransform relative to an RF echo changes frame by framedue to object displacement. Such a change in relative position inﬂuences the estimated phase shift of the RF echo
at frequencies other than the RF center frequency. In thepresent study, this problem was investigated through simulation experiments and in vivo experiments on the human
carotid artery.

(5)

∆d
n = dn+1 − dn =n (f )c0 (τn+1 − τn )c0 · ∆θ
=−
.
2
4πf

(6)

From the estimated displacement, ∆dn , between two
consecutive frames, the velocity of the object is obtainedusing the frame interval, T , as follows:vn =

n (f )∆dc0 · ∆θ
n
=−
.
T
4πf T

(7)

n


v
n · T,(8)1

Let us deﬁne the time series and the frequency spectrumof the transmitted ultrasonic pulse by s(t) and S(f ), respectively. The received RF echo has a speciﬁc time delay
depending on the distance between the ultrasonic probe
and the object. When the object is displaced, time delaysof RF echoes measured at two consecutive frames diﬀer.
By deﬁning time delays at the n-th and (n + 1)-th frames(n = 1, 2, . . . , N −1) by τn and τn+1 , received RF echoes at
the n-th and (n+1)-th frames are expressed as s(t−τn ) ands(t−τn+1 ), respectively. The frequency spectra of s(t−τn )and s(t−τn+1 ) are obtained by applying the Fourier transform, FT[·], as follows:(1)−j2πf τn+1S.
n+1 (f ) ≡ FT[s(t − τn+1 )] = S(f ) · e(2)Therefore, the change in time delay, τn+1 − τn , whichcorresponds to the object displacement between the n-thand (n + 1)-th frames, can be estimated from the phase,∆θn (f ), of the cross-spectrum, Sn∗ (f ) · Sn+1 (f ), at a frequency, f , as follows:∗= |S(f )|2 · e−j2πf (τn+1 −τn ) ,(4)

Describing object displacement between the n-th and(n+1)-th frames by ∆dn = dn+1 −dn , object displacement,∆dn , between two consecutive frames is obtained from then (f ), at a frequency, f , estimated by (3) asphase shift, ∆θfollows:d
n+1 = d0 +A. Velocity Estimation from the Phase Shift of
RF Echoes with Wall Trackingj∆θn (f )
n (f ) · SSn+1 (f ) = |Sn (f )||Sn+1 (f )| · e

c0 τn
,
2
c0 τn+1.
dn+1 =
2dn =In this process, the wall position, dn , is tracked by integrating the estimated velocity, vn , as follows [14]:II. Methods

n (f ) ≡ FT[s(t − τn )] = S(f ) · e−j2πf τn ,
S

53

(3)

where ∗ represents the complex conjugate.
By deﬁning the sound speed as c0 , distances dn anddn+1 between the ultrasonic probe and the object at n-thand (n + 1)-th frames are given by:where d0 is the initial wall position manually assigned at
the initial frame.
B. Principle for Simulating RF Echoes
Fig. 1 illustrates the time shift of the RF echo due toobject displacement. When a short time window is used toobtain the phase of the received signal, a portion of the duration in the period shown by the region with oblique lines
is excluded due to the time shift of the RF echo caused by
object displacement. This change in the relative position
between the echo and the time window causes an error inestimation of the time shift of the RF echo. In this study,
the error caused by the change in relative position between
the echo and the time window was investigated using RF
echoes simulated in the following way. A digitized ultrasonic RF echo, rfn (m), from an interface or a point scatterer of the n-th frame at a time t = mTs (m = 1, 2, 3, . . . ;Ts : sampling interval) is deﬁned by a sinusoidal wave at a
center angular frequency of ω0 = 2πf0 and an envelope,envn (m), as shown in (9) and (10) (see next page), where∆Tp and τn are the pulse length and the time delay at then-th frame, respectively. In the simulation experiments,
τn is deﬁned as the time delay from the beginning of thetime window as shown in Fig. 1. The time delay, τn , diﬀers
frame by frame when the object is displaced.
The change in time delay, τn+1 −τn , between two neighboring frames is estimated based on (3) by applying thediscrete Fourier transform to rfn (m) and rfn+1 (m).
A very simple model was used in the present study
because in cases of arterial walls without atherosclerotic
plaque, echoes can be modeled by such a simple model

54

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no. 1, january 2006

ejω0 (mTs −τn ) − e−jω0 (mTs −τn )rfn (m) = envn (m) · sin {ω0 (mTs − τn )} = envn (m) ·
,
2j
⎧
s −τn
s −τn

j2π mT∆T−j2π mT∆T
⎪
p
p
⎪
+e
mTs − τne2⎪⎪
= 0.5 1 −⎨sin π ∆T2penvn (m) =⎪(τn ≤ mTs ≤ τn + ∆Tp ),⎪⎪⎪⎩0 (mT < τ , mT > τ + ∆T ),snFig. 1. Illustration of the change in relative position between the RFecho and the time window due to object displacement using simulated RF echoes.

because dominant echoes are reﬂected from two interfaces(lumen-intima and media-adventitia interfaces) [23] andbecause the distance between these interfaces is greaterthan the pulse duration. The distance between these interfaces is greater than 0.5 mm for carotid arteries, andthe duration of the ultrasonic pulse used in this paper isabout 0.25 µs, which corresponds to 385 µm in the case of asound speed of 1540 m/s. Although a more complex model
also should be investigated, the error source described inthis paper inﬂuences even such a simple model. Therefore,a simple model was considered in this study to show anerror source in estimation of the time shift of RF echoes.

III. Results of Simulation Experiments
A. Power Spectrum of RF Echo Reﬂected from
Carotid Artery
Fig. 2(a) shows an ultrasonic pulse of the diagnostic
equipment used measured by a hydrophone placed in a

s

n

(9)

(10)

p

water tank. The hydrophone was placed at a distance of
20 mm from the surface of a linear-type probe, which corresponds to the electric focal depth. The signal received
by the hydrophone was acquired at a sampling frequencyof 1 GHz with a digital oscilloscope. The nominal center frequency of the linear-type ultrasonic probe used was
10 MHz.
An echo is expressed by the product of a sinusoidal wave
at a center frequency, f0 , and an envelope, env(t), in thetime domain. The frequency spectrum of a sinusoidal wave
at frequency f0 is expressed by a delta function, δ(f − f0 ),in the frequency domain. Therefore, the frequency spectrum of an echo is a convolution of the spectrum, Senv (f ),of the envelope and the delta function, δ(f − f0 ), thatshows the shifted version, Senv (f − f0 ), of the spectrum of
the envelope. When an envelope of an echo is expressed byfunctions such as a Hanning window or a Gaussian function, its power spectrum, |Senv (f )|2 , becomes maximum
at f = 0. Therefore, we can determine the actual center
frequency by referring to the power spectrum of the echo,|Senv (f − f0 )|2 , because |Senv (f − f0 )|2 becomes maximum
at f = f0 .
As shown in Fig. 2(a), the envelope of the ultrasonicpulse has a ﬁnite duration of ∆Tp , that is, the sampled
version of the envelope, env(m), is limited from m = 0 tom = Mp (Mp : number of samples in the duration of thepulse). Therefore, a period that corresponds to the duration of the pulse is suﬃcient for the length of the time
window for the Fourier transform. However, for example,
the time window of 0.5 µs, which is suﬃcient for echoes inthis paper, gives a sampling rate of 1/(0.5 µs) = 2 MHz forthe frequency spectrum estimated by the discrete Fouriertransform. To overcome such a low resolution of a frequency spectrum, (M0 − Mp ) zero points are added to thetime sequence extracted by the time window before application of the discrete Fourier transform. The estimatedfrequency spectrum is not changed by adding zero points
as explained below. The time sequences y(m) and y  (m),with and without zero points, respectively, are equivalent
to the time functions y(t) and y  (t), expressed as follows:y(t) = env(t) (0 ≤ t ≤ ∆Tp ),y  (t) =env(t) (0 ≤ t ≤ ∆Tp ),0 (∆Tp < t ≤ T0 ),

(11)
(12)

hasegawa and kanai: estimation of artery-wall displacement

55

Fig. 2. Ultrasonic pulse of the diagnostic equipment used. (a) Ultrasonic pulse received by a hydrophone placed in a water tank (sampling
frequency, 1 GHz). (b) Power spectrum of (a). (c) B-mode image of a human carotid artery. (d) RF echo from the posterior wall along thewhite vertical line shown in (c) (sampling frequency, 40 MHz). (e) Power spectrum (d).

where T0 = M0 Ts . There is no diﬀerence between the spectra Y (f ) and Y  (f ) of the time functions y(t) and y  (t):∞Y  (f ) =y  (t) · e−j2πf t dt

−∞
∆Tp

=

(13)
env(t) · e−j2πf tdt = Y (f ).

0Therefore, no distortion occurs by the addition of zero
points, and only the sampling rate of the spectrum, Y (f )(f = k/(M0 Ts ) (k = 0, . . . , M0 − 1)), obtained by thediscrete Fourier transform is improved from 1/(M Ts ) to1/(M0 Ts ).
The sampled version of the frequency spectrum,
Senv (k), of the envelope is obtained from the convolution of
Y (k) with the frequency spectrum of the Hanning windowused in the discrete Fourier transform.
A Hanning window with a length of 0.5 µs (512 points)
was applied to the data set shown in Fig. 2(a) to extract a time sequence, and 1536 zero points were added.
The power spectrum then was obtained by applying theFourier transform as shown in Fig. 2(b). The power spectrum was found to have the largest value at about 8 MHz.
In Fig. 2(b), the power spectrum at 20 MHz is suﬃciently
smaller (−30 dB) than that at the actual center frequencyof 8 MHz. Therefore, the sampling frequency of 40 MHz
of the ultrasonic diagnostic equipment used in this paper
is suﬃcient for measuring RF echoes.
Fig. 2(c) shows the B-mode image of a human carotid
artery. Along the white vertical line shown in Fig. 2(c),RF echoes reﬂected from the posterior wall were received
as shown in Fig. 2(d) by the ultrasonic probe, which bothtransmits ultrasound and receives echoes (sampling frequency: 40 MHz). Fig. 2(e) shows the power spectrumof the RF echo reﬂected from the lumen-intima interface, which was obtained from the time sequence (16points = 0.4 µs) extracted by a Hanning window at the position shown in Fig. 2(d). Before application of the Fouriertransform, 48 zero points were added. In Fig. 2(e), it canbe seen that the center frequency was slightly changed to
7.5 MHz due to the frequency-dependent attenuation intissue. Based on these results, the center frequency of thesimulated RF echo is set at 7.5 MHz in the following section.
B. Simulated RF Echo
Fig. 3 shows the simulated RF echos, rf1 (m) and rf2 (m),at the 1st and 2nd frames based on (9) (center frequency:
7.5 MHz; 3 wavelength; envelope: Hanning window). The
sampling frequency was set at 40 MHz. rf2 (m) was shiftedby 12.5 ns in comparison with rf1 (m) due to object displacement. This time shift corresponds to a displacementof 9.625 µm and a velocity of 2.753 mm/s at a sound speedof 1540 m/s and a frame rate of 286 Hz. Typical velocityof the human carotid arterial wall is less than 10 mm/s.
Therefore, the phase shifts between rf1 (m) and rf2 (m) were
estimated for the cases of τ2 − τ1 = 12.5 ns (2.753 mm/s),
25 ns (5.506 mm/s), and 37.5 ns (8.258 mm/s) in the following section.
C. Estimation of Phase Shift of RF Echoes by Discrete
Fourier Transform with a Hanning Window
Fig. 4(a) shows the signals after application of a Hanning window (length Mw : 10 points = 0.25 µs) to RF

56

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no. 1, january 2006

Fig. 3. Simulated RF echoes, rf1 (m) and rf2 (m), based on (9). The
time delay of rf2 (m) from rf1 (m) is 12.5 ns. The limits of the Hanningwindow are indicated by vertical dashed lines.

echoes, rf1 (m) and rf2 (m), as shown in Fig. 3 (diﬀerence
between time delays τ2 − τ1 : 12.5 ns). By applying theFourier transform, power spectra, |RF1 (k)| and |RF2 (k)|,at a frequency f = k/(Mw · Ts ) were obtained as shownin Fig. 4(b). Fig. 4(c) shows the phases of the crossspectra, {RF1 (k)}∗ · RF2 (k), and the dashed lines show
−2πf (τ2 − τ1 ) for the cases of τ2 − τ1 = 12.5 ns, 25 ns, and37.5 ns.
In Fig. 4(c), it is found that the phase shift estimatedat the nominal center frequency of 10 MHz is biased from
−2πf (τ2 − τ1 ). However, the change in time delay, τ2 − τ1 ,can be estimated precisely at the actual center frequencyof 7.5 MHz. Therefore, the actual center frequency should
be determined by referring to the power spectrum of thereceived RF echo, and the phase shift should be estimatedat the actual center frequency.
The magnitude, ∆τmax , of the change in time delay islimited by half of the wavelength at the actual (selected)
center frequency f0 (∆τmax ≤ 0.5/f0 ).
D. Considerations Regarding Error in Estimation of
Change in Time Delay
For mathematical simpliﬁcation, the envelope of anecho, envn (m), is assumed to be rectangular in this section. A Hanning window was used for the discrete Fouriertransform.
1. Estimation of Change in Time Delay at a Frequency
f = f0 : Frequency spectra, RF1 (k) and RF2 (k), at a frequency f = k/(Mw · Ts ) = f0 of RF signals, rf1 (m) andrf2 (m), which are illustrated in Fig. 1, are obtained byapplying the discrete Fourier transform with a Hanningwindow, w(m), of Mw points as follows:
mTs
2w(m) = sin 2π
Mw T s
(14)
mTs
mTs
j2π M
−j2π Mw Ts + e
w Ts
1 e
.
= −
2
4
M
w −1
RF1 (k) =w(m) · rf1 (m) · e−jωmTs
m=0

=

M
w −1

w(m) · rf1 (m) · e−jωmTs

m=M1

=

Mw −M
1 −1

w(m + M1 ) · rf1 (m + M1 ) · e−jω(m+M1 )Ts

m=0=

Fig. 4. (a) Signals after application of a Hanning window [0.25 µs(= 10 points)] to RF echoes shown in Fig. 3 (τ2 − τ1 = 12.5 ns).
(b) Power spectra, |RF1 (k)| and |RF2 (k)|, of rf1 (m) and rf2 (m) (τ2 −τ1 = 12.5 ns). (c) Phases of cross spectra, {RF1 (k)}∗ · RF2 (k), forthe cases of τ2 − τ1 = 12.5 ns, 25 ns, and 37.5 ns. The dashed line
shows −2πf (τ2 − τ1 ) for each case of τ2 − τ1 .

1
2

1
−
4
1
−
4

Mw −M
1 −1

rf1 (m + M1 ) · e−jω(m+M1 )Ts

m=0
Mw −M
1 −1 j2πe(m + M1 )TsMw T s· rf1 (m + M1 ) · e−jω(m+M1 )Ts

m=0
Mw −M
1 −1

e−j2π

(m+M1 )Ts
Mw Ts

· rf1 (m + M1 ) · e−jω(m+M1 )Ts

m=0= A1 (k) − B1 (k) − C1 (k). (15)hasegawa and kanai: estimation of artery-wall displacement

The ﬁrst term, A1 (k), in the right-hand side of (15), isrewritten as follows:

57

From (17), (18), and (19), (15) is expressed as follows:
1 −jω0 τ1 j(ω0 −ω)M1 Ts·ee
2
1 − ej(ω0 −ω)(Mw −M1 )Ts·
1 − ej(ω0 −ω)Ts
2π
1− e−jω0 τ1 · ej(ω0 −ω+ Mw Ts )M1 Ts
42π
1 − ej(ω0 −ω+ Mw Ts )(Mw −M1 )Ts·
2π
1 − ej(ω0 −ω+ Mw Ts )Ts
2π
1− e−jω0 τ1 · ej(ω0 −ω− Mw Ts )M1 Ts
42π
1 − ej(ω0 −ω− Mw Ts )(Mw −M1 )Ts·.
2π
1 − ej(ω0 −ω− Mw Ts )Ts

RF1 (k) ≈2 · A1 (k) =

M
w −1

rf1 (m) · e−jωmTs

m=0

=

M
w −1

rf1 (m) · e−jωmTs

m=M1

=

Mw −M
1 −1

rf1 (m + M1 ) · e−jω(m+M1 )Ts

m=0

=

1
2j

−e

Mw −M
1 −1 ejω0 {(m+M1 )Ts −τ1 }m=0−jω0 {(m+M1 )Ts −τ1 }
e−jω(m+M1 )Ts1= e−jω0 τ1 · ej(ω0 −ω)M1 Ts
2j−

1 jω0 τ1 −j(ω0 +ω)M1 Ts
·e
e
2j

Mw −M
1 −1

As well as (20), RF2 (k) is obtained as follows:ej(ω0 −ω)mTs

1 −jω0 τ2 j(ω0 −ω)M2 Tse·e
2
1 − ej(ω0 −ω)(Mw −M2 )Ts·
1 − ej(ω0 −ω)Ts
1 −jω0 τ2 j(ω0 −ω+ M2πT )M2 Ts
w s
− e
·e
4
2π
1 − ej(ω0 −ω+ Mw Ts )(Mw −M2 )Ts·
2π
1 − ej(ω0 −ω+ Mw Ts )Ts
2π
1− e−jω0 τ2 · ej(ω0 −ω− Mw Ts )M2 Ts
42π
1 − ej(ω0 −ω− Mw Ts )(Mw −M2 )Ts·
2π
1 − ej(ω0 −ω− Mw Ts )Ts

RF2 (k) ≈

m=0
Mw −M
1 −1

e−j(ω0 +ω)mTs

m=0

1
1 − ej(ω0 −ω)(Mw −M1 )Ts= e−jω0 τ1 · ej(ω0 −ω)M1 Ts ·2j
1 − ej(ω0 −ω)Ts1
1 − e−j(ω0 +ω)(Mw −M1 )Ts− ejω0 τ1 · e−j(ω0 +ω)M1 Ts ·,2j
1 − e−j(ω0 +ω)Ts(16)
where ω = 2πk/(Mw · Ts ) and M1 corresponds to the sampled point that is just after t = τ1 . In the ultrasound system used, the sampling frequency, 1/Ts , is equal to fourtimes the nominal center frequency (Ts ≈ 1/(4f0 ): f0 is
the actual center frequency). Therefore, the ﬁrst term in
the right-hand sidelarger than the sec of (16) is much
ond term because 1 − ej(ω0 −ω)Ts   1. Thus, (16) can beapproximated as follows:

A1 (k) ≈(20)

1 −jω0 τ1 j(ω0 −ω)M1 Tse·e
2
1 − ej(ω0 −ω)(Mw −M1 )Ts·. (17)
1 − ej(ω0 −ω)TsAs well as A(k), B(k) and C(k) are obtained as follows:
2π
1
B1 (k) ≈ e−jω0 τ1 · ej(ω0 −ω+ Mw Ts )M1 Ts
42π
1 − ej(ω0 −ω+ Mw Ts )(Mw −M1 )Ts·.
2π
1 − ej(ω0 −ω+ Mw Ts )Ts2π
1C1 (k) ≈ e−jω0 τ1 · ej(ω0 −ω− Mw Ts )M1 Ts
42π
1 − ej(ω0 −ω− Mw Ts )(Mw −M1 )Ts·.
2π
1 − ej(ω0 −ω− Mw Ts )Ts

where M2 corresponds to the sampled point that is justafter t = τ2 .
As shown in (20) and (21), in this case, the phase of{RF1 (k)}∗ · RF2 (k) does not only depend on the change
in time delay, τ2 − τ1 .
2. Estimation of Change in Time Delay at the Center
Frequency f = f0 : By assuming that a discrete frequency,
f = k0 /(Mw · Ts ), is identical to the actual center frequency, f0 , of the received RF echo, the frequency spectra, RF1 (k0 ) and RF2 (k0 ), of the RF signals, rf1 (m) andrf2 (m), are obtained by substituting f = k0 /(Mw ·Ts ) = f0
into (15). A1 (k0 ) is obtained as follows:
2 · A1 (k0 ) =

(18)

(21)

1 −jω0 τ1e2j· e−j2ω0 M1 Ts

Mw −M
1 −1
m=0
Mw −M
1 −1

1−

1 jω0 τ1e2je−j2ω0 mTs

m=0

(19)

(22)
1
1
=
· (Mw − M1 ) · e−jω0 τ1 − ejω0 τ1
2j2j−j2ω0 (Mw −M1 )Ts
1−e· e−j2ω0 M1 Ts.
1 − e−j2ω0 Ts

58

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no. 1, january 2006

When the time window is suﬃciently long (Mw  1),
the second term in the right-hand side of (22) can be neglected:
A1 (k0 ) ≈1· (Mw − M1 ) · e−jω0 τ1 .
4j

(23)

B1 (k0 ) is obtained as follows:
4 · B1 (k0 ) =1 −jω0 τ1
e
2j

Mw −M
1 −1
m=0

Mw −M
1 −1

· e−j(2ω0 + Mw Ts )M1 Ts
2π

m+M1

ej2π Mw −1 jω0 τ1e2je−j(2ω0 + Mw Ts )mTs . (24)2πm=0

As well as (22), the second term in the right-hand side of(24) can be neglected.
1
4 · B1 (k0 ) ≈ e−jω0 τ1
2j
=−

Mw −M
1 −1

m+M1

ej2π Mw

m=0
M1
j2π Mw

1 −jω0 τ1 1 − e
.
e
2π
2j
1 − ej Mw(25)

When the time window is properly applied to the echo(the diﬀerence between the beginnings of the window andthe echo, which corresponds to M1 , is suﬃciently small,that is, M1  Mw ), B(k0 ) ≈ 0. As well as B(k0 ), C(k0 ) ≈0 under such a condition.
Therefore, RF1 (k0 ), can be expressed as follows:
RF1 (k0 ) ≈Fig. 5. Phase shift estimated at 7.5 MHz is plotted as a functionof window length (with respect to simulated RF echoes, τ2 − τ1 =12.5 ns).

1· (Mw − M1 ) · e−jω0 τ1 .
4j

(26)

depends on the magnitude of object displacement between
two consecutive frames, i.e., magnitude of velocity. The
velocity of the arterial wall is large during expansion andsmall during contraction. Therefore, the accumulated error
in the displacement, which is obtained by integrating theestimated velocity, does not become zero. From these results, although a discrete frequency often does not exactly
correspond to the actual center frequency, the phase shift
should be estimated at the frequency that is the nearest
to the actual center frequency to avoid the inﬂuence of thechange in relative position between the RF echo and thetime window.
E. Considerations Regarding Required Window Length

As well as (26), RF2 (f0 ) is obtained as follows:1
RF2 (k0 ) ≈· (Mw − M2 ) · e−jω0 τ2 .
4j

(27)

From (26) and (27):{RF1 (k0 )}∗ · RF2 (k0 )1= − (Mw − M1 )(Mw − M2 ) · e−jω0 (τ2 −τ1 ) . (28)16
As shown in (28), the phase shift, which corresponds
to the change in time delay, τ2 − τ1 , can be estimated at
the RF center frequency, f0 = k0 /(Mw · Ts ), even when
the relative position between the RF echo and the time
window is changed due to object displacement.
In Fig. 4(c), for example, the error in the estimatedchange in time delay at the nominal center frequency of
10 MHz is 2.3 ns in comparison with the true value of
12.5 ns, and it is not extremely large. Therefore, the measurement of velocity at a constant time (= depth) is not
signiﬁcantly inﬂuenced (for example, in blood ﬂow measurement). However, to obtain the artery-wall displacement, the estimated velocity must be integrated with respect to time, and the error between each adjacent frame isaccumulated. As shown in Fig. 4, the error described above

In Fig. 5, the phase of the cross spectra, {RF1 (f0 )}∗ ·
RF2 (f0 ), at the actual center frequency, f0 , which is obtained from the simulated RF echoes (f0 : 7.5 MHz, shownin Fig. 3), is plotted as a function of window length.
The dashed line shows 2πf0 (τ2 − τ1 ). When the window
length is less than 10 points (= 0.25 µs), the variationof the estimated phase shift is large. The Hanning window has a main lobe of ±2/(window length) in the frequency domain. When the frequency spectrum is estimated at the RF center frequency, f0 , the main lobe of
the Hanning window ranges from Bw1 = f0 − 2/(window
length) to Bw2 = f0 + 2/(window length). In the caseof (window length) < 2/f0 , Bw1 becomes less than zero,and the estimated frequency spectrum is inﬂuenced by
the negative-frequency components. Therefore, the window length should be equal or greater than 2/f0 . In Fig. 5,
the window length of 0.25 µs (= 10 points) = 2/(8 MHz)almost corresponds to 2/f0 = 2/(7.5 MHz).
In this paper, inﬂuences of random noise were not considered. The noise variance would be reduced using alonger time window. However, the window length is considered to be limited by the pulse duration. For example,
when the window length is too long, both echoes from thelumen-intima and media-adventitia interfaces are includedhasegawa and kanai: estimation of artery-wall displacement

59

Fig. 7. In vivo experimental results for a carotid artery withoutatherosclerotic plaque (30-year-old healthy male). Change in thickness of the posterior wall due to the heartbeat estimated from thephase shift at 7.5 MHz. (a) M-mode image. (b) Electrocardiogram.
(c) Velocity of the intimal side. (d) Velocity of the adventitial side.
(e) Change in thickness of the posterior wall.

Fig. 6. (a) RF echoes reﬂected by the posterior wall of the human
carotid artery. (b) Power spectra, |RF1 (k)| and |RF2 (k)|, obtainedby applying a Hanning window with a length of 0.25 µs = 10 points
to waveforms shown in (a). (c) Phase of cross spectrum, {RF1 (k)}∗ ·
RF2 (k).

in the same time window. In this case, the noise variancewould be reduced in comparison with a shorter time window. However, the change in time delay of the echo fromthe lumen-intima interface cannot be estimated because
the estimated change in time delay is biased by that of theecho from the media-adventitia interface.

IV. In Vivo Measurement for Human Carotid
ArteryFig. 6(a) shows RF echoes reﬂected from the posterior wall of the carotid artery obtained at two consecutive frames (frame rate: 286 Hz). With respect to the echo

from the lumen-intima interface, the frequency spectrum
was estimated with a Hanning window having a length of
0.25 µs = 10 points.
Fig. 6(b) shows the power spectra, |RF1 (k)| and|RF2 (k)|, obtained by extracting the time sequences fromrf1 (m) and rf2 (m) using the time window. From the estimated frequency spectra, RF1 (k) and RF2 (k), the phaseshift between echoes was obtained from the phase of theircross-spectrum, {RF1 (k)}∗ · RF2 (k), as shown in Fig. 6(c).
The arterial wall then was tracked using the phase shifts
estimated at both the nominal center frequency of 10 MHz
and the actual center frequency of 7.5 MHz.
Figs. 7 and 8 show the results of artery-wall tracking
using the phase shifts estimated at the actual center frequency of 7.5 MHz and the nominal center frequency of
10 MHz, respectively. Figs. 7(a) and (b) show the M-mode
image and the electrocardiogram, respectively. The velocities shown in Figs. 7(c) and (d) were obtained at the intimal side and the adventitial side of the posterior wall,
respectively. As shown in Figs. 7(a) and 8(a), the intimalside and the adventitial side were tracked by integrating
the estimated velocities, v
n , of (7), and the tracked positions, dn+1 , of (8) were superimposed on M-mode imagesby a white line for the intimal side and a black line forthe adventitial side. With respect to the intimal side, as
shown in Fig. 8(a), an obvious tracking error was foundwhen the phase shift was estimated at the nominal center frequency of 10 MHz. As for the adventitial side, thetracking error is not so obvious even at 10 MHz. However, the tracking shown by the black line is not in good
agreement with the M-mode image in comparison with theresult obtained at 7.5 MHz. For assessment of the relative
elasticity, the change in wall thickness, which corresponds

60

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no. 1, january 2006

Fig. 8. In vivo experimental results for a carotid artery withoutatherosclerotic plaque (30-year-old healthy male). Change in thickness of the posterior wall due to the heartbeat estimated from thephase shifts at 10 MHz. (a) M-mode image. (b) Electrocardiogram.
(c) Velocity of the intimal side. (d) Velocity of the adventitial side.
(e) Change in thickness of the posterior wall.

to the strain caused by the change in blood pressure, isobtained as shown in Fig. 7(e) [16], [17] from the diﬀerence between displacements at the intimal and adventitialsides. However, such a change in wall thickness as shown in
Fig. 8(e), which was estimated at the nominal center frequency of 10 MHz, does not correspond to the wall strain(in cardiac systole, the artery diameter expands, and thewall thickness should become smaller). From these results,
the phase-shift estimation at the actual center frequency
was shown to be important even for the data sets measured
for the human carotid artery.
The change in thickness would be dependent on thepositions of points A and B, and thus changes in thickness were estimated at ﬁve diﬀerent depths by shifting
points A and B in the depth direction by 77 µm at a pitchof 19.25 µm. The distance between points A and B was
kept constant. Mean and standard deviation of maximum
changes in thickness during a heartbeat were estimated to
be 97.6 µm and 6.9 µm. Slight inﬂuences of the positionsof the assigned points were found. However, in cases of arteries with atherosclerotic plaque, there are many echoesin addition to those from the lumen-intima and mediaadventitia interfaces. Under such a situation, the change
in thickness would be more dependent on the position of
the assigned points. Such a complex condition is not considered in this paper and should be investigated in a future
work.
Fig. 9 shows the results of tracking in the anterior wallof the same subject as that shown in Fig. 7. The phaseshift was estimated at 7.5 MHz. As in the case of the posterior wall, tracking was successfully performed for the anterior wall. A relatively smaller displacement was detectedin comparison with that of the posterior wall.

Fig. 9. In vivo experimental results for a carotid artery withoutatherosclerotic plaque (30-year-old healthy male). Change in thickness of the anterior wall due to the heartbeat estimated from thephase shift at 7.5 MHz. (a) M-mode image. (b) Electrocardiogram.
(c) Velocity of the adventitial side. (d) Velocity of the intimal side.
(e) Change in thickness of the anterior wall.

V. Comparison with Autocorrelator Using
Quadrature Demodulated Complex Signal
A correlation estimator with a quadrature demodulated
complex signal is widely used for assessing artery-wall displacement. It is desirable that the demodulation frequency
be the actual center frequency of the received RF echo because the problem mentioned below is the same as that
already discussed.
In the quadrature demodulation, the received RF echo
is multiplied by a complex sinusoidal function at a demodulation frequency, fdem , as follows:
1  j{(ω0 +ωdem )mTs −ω0 τ1 }e2j
− e−j{(ω0 −ωdem )mTs −ω0 τ1 } , (29)1  j{(ω0 +ωdem )mTs −ω0 τ2 }
rf2 (m) · ejωdem mTs =e2j
− e−j{(ω0 −ωdem )mTs −ω0 τ2 } , (30)
rf1 (m) · ejωdem mTs =where ωdem = 2πfdem .
When an ideal low-pass ﬁlter, LPFideal [·], is used, thequadrature demodulation gives the quadrature demoduˇ 1 (m) and rf
ˇ 2 (m), at a time, t = mTs , aslated signals, rf
follows:
ˇ 1 (m) = LPFideal [rf1 (m) · ejωdem mTs ]rf
1= − e−j{(ω0 −ωdem )mTs −ω0 τ1 } ,2jˇ 2 (m) = LPFideal [rf2 (m) · ejωdem mTs ]rf
1= − e−j{(ω0 −ωdem )mTs −ω0 τ2 } .
2j

(31)

(32)

hasegawa and kanai: estimation of artery-wall displacement61

In this case, as described in [21], the phase of the 2D autocorrelator, r(m , n ) (m : lag for the time (depth)
axis, n : lag for the frame axis), with an M ×N correlation
window gives the phase shift, ω0 (τ2 −τ1 ), due to object displacement and the diﬀerence, ω0 − ωdem , between the demodulation frequency and the actual center frequency. For
the simplest case (M = N = 1), to obtain object displacement at a depth that corresponds to t = mTs = 0 (= thebeginning of the time window as illustrated in Fig. 1),r(0, 1) and r(1, 0) are expressed using demodulated sigˇ 1 (m) and rf
ˇ 2 (m), as follows:
nals, rf

ˇ 2 (0) = LPFMA [rf2 (m) · ejωdem mTs ]rf
Mw −M
2 −1 1 1=
ejω0 {(m+M2 )Ts −τ2 }
2j Mw
m=0
(39)

−jω0 {(m+M2 )Ts −τ2 }−e· ejωdem (m+M2 )Ts .

r(0, n = 1) =N=1ˇ ∗ (0) · rf
ˇ n+n (0)
rf
n
(33)

n=1

ˇ ∗ (0) · rf
ˇ 2 (0) = 1 ejω0 (τ2 −τ1 ) ,
= rf
1
4

r(m = 1, 0) =M=1ˇ ∗ (m − 1) · rf
ˇ 1 (m − 1 + m )
rf
1

m=1

ˇ ∗ (0) · rf
ˇ 1 (1) = 1 ej(ωdem −ω0 )Ts . (34)
= rf
1
4
From the phase ∠r(1, 0), the actual RF center frequency can be estimated as follows:∠r(1, 0)ωdem −= ω0 .
Ts(35)

Therefore, the unbiased change in time delay, τ2 − τ1 ,which corresponds to object displacement, can be estimated as follows [22]:∠r(0, 1).
τ2 − τ1 =∠r(1, 0)ωdem −
Ts

Mw −1
1 
.
Mw m=0M
w −1 
ˇ 1 (0) = 1 1
rfej{(ω0 +ωdem )mTs −ω0 τ1 }2j Mw m=0
− e−j{(ω0 −ωdem )mTs −ω0 τ1 }1 1 jω0 τ1 1 − e−j(ω0 −ωdem )Mw Ts
≈
e
·
,
2j Mw
1 − e−j(ω0 −ω)TsMw −1 1 1 
ˇrf2 (0) =ej{(ω0 +ωdem )mTs −ω0 τ2 }2j Mw m=0
− e−j{(ω0 −ωdem )mTs −ω0 τ2 }1 1 jω0 τ2 1 − e−j(ω0 −ωdem )Mw Ts
≈
e
·
.
2j Mw
1 − e−j(ω0 −ω)Ts(36)

Next, let us consider a realistic case in which the moving average operator is used as a low-pass ﬁlter in thequadrature demodulation. The moving average estimator,
LPFMA [·], can be expressed as follows:
LPFMA [·] =ˇ 1 (0) shown by (38) is identical to
It is found that rf
A1 (k) at a frequency f = k/(Mw · Ts ) = fdem as shownby (16). Therefore, when k/(Mw · Ts ) = fdem = f0 , thechange in time delay, τ2 − τ1 , cannot be estimated by(36) because r(0, 1) corresponds to the cross spectrum,{RF1 (k)}∗ · RF2 (k), and the phase of {RF1 (k)}∗ · RF2 (k)does not give the change in time delay, τ2 − τ1 , at a frequency other than the actual center frequency as describedin Section III-D.
The 2-D autocorrelator gives the desirable results by(36) when rf1 (m) and rf2 (m) are stationary sinusoidal
waves. If rf1 (m) and rf2 (m) are stationary sinusoidal waves
ˇ 1 (0) and rf
ˇ 2 (0) can be exat an angular frequency ω0 , rfpressed as follows:

(37)



r(0, n = 1) =

1 1
2j Mw

−e

ejω0 {(m+M1 )Ts −τ1 }m=0−jω0 {(m+M1 )Ts −τ1 }· ejωdem (m+M1 )Ts ,ˇ ∗ (0) · rf
ˇ n+n (0) = rfˇ ∗ (0) · rf
ˇ 2 (0)
rf
n
1

n=1


2
1  1 − e−j(ω0 −ω)Mw Ts · ejω0 (τ2 −τ1 ) ,=−4Mw2  1 − e−j(ω0 −ω)Ts 
(42)
M=1


ˇ ∗ (m − 1) · rf
ˇ 1 (m − 1 + m )
rf
1


2
1  1−e−j(ω0−ω)Mw Ts =−· ej(ω0−ωdem )Ts .
4Mw2  1−e−j(ω0−ω)Ts 
(43)

(38)

Mw −M
1 −1 

N
=1


m=1

ˇ 1 (0) = LPFMA [rf1 (m) · ejωdem mTs ]rf
Mw −1 1 1 =
ejω0 (mTs −τ1 )2j Mw m=0
− e−jω0 (mTs −τ1 ) · ejωdem mTs=(41)

The terms at the summed frequency, ω0 + ωdem , as well as(17), were neglected. Thus,r(m = 1, 0) =Therefore:(40)

In this case, from (42) and (43), it is found that thechange in time delay, τ2 − τ1 , can be estimated by (36).
From these facts, the 2-D autocorrelator is considered to besuitable for the narrow-band signal that can be considered
as a continuous wave in an integration period of a low-passﬁlter.

62

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no. 1, january 2006

Most conventional ultrasound systems use relatively
narrowband ultrasonic pulses for the Doppler mode incomparison with that for the B-mode. Under the conditionof a narrowband pulse and a high frame rate of several
kilohertz, the 2-D autocorrelator provides precise results
at a given demodulation frequency based on (36). However, a broadband pulse is preferable for improving the
spatial resolution in displacement estimation. Under sucha condition, the estimation should be performed at thecenter frequency of the received RF echo. In the case of
an autocorrelator with quadrature demodulated signals,
the demodulation frequency should be set at the center
frequency of the RF echo.

VI. Conclusions
Digitized data of broadband RF echoes are available inmodern ultrasonic diagnostic equipment. The phase shiftsof RF echoes due to object displacement can be obtainedat multiple frequencies by frequency analysis of RF echoes.
This phase-shift estimation is inﬂuenced by the change in
relative position of the RF echo in relation to the time
window due to object displacement. Through simulations
and in vivo experiments, this study showed that the phaseshift should be estimated at the actual center frequency ofan RF echo.

References[1] R. T. Lee, A. J. Grodzinsky, E. H. Frank, R. D. Kamm, and F. J.
Schoen, “Structure-dependent dynamic mechanical behavior ofﬁbrous caps from human atherosclerotic plaques,” Circulation,
vol. 83, pp. 1764–1770, 1991.
[2] H. M. Loree, A. J. Grodzinsky, S. Y. Park, L. J. Gibson, and
R. T. Lee, “Static circumferential tangential modulus of humanatherosclerotic tissue,” J. Biomech., vol. 27, pp. 195–204, 1994.
[3] M. Benthin, P. Dahl, R. Ruzicka, and K. Lindström, “Calculation of pulse-wave velocity using cross-correlation—Eﬀects ofreﬂexes in the arterial tree,” Ultrasound Med. Biol., vol. 17, no.
5, pp. 461–469, 1991.
[4] H. Kanai, K. Kawabe, M. Takano, R. Murata, N. Chubachi, andY. Koiwa, “New method for evaluating local pulse wave velocity
by measuring vibrations on aortic wall,” Electron. Lett., vol. 30,
pp. 534–536, 1993.
[5] P. J. Brands, J. M. Willigers, L. A. F. Ledoux, R. S. Reneman,and A. P. G. Hoeks, “A noninvasive method to estimate pulse
wave velocity in arteries locally by means of ultrasound,” Ultrasound Med. Biol., vol. 24, no. 9, pp. 1325–1335, 1998.
[6] A. Eriksson, E. Greiﬀ, T. Loupas, M. Persson, and P. Pesque,“Arterial pulse wave velocity with tissue Doppler imaging,” Ultrasound Med. Biol., vol. 28, no. 5, pp. 571–580, 2002.
[7] J. M. Meinders, P. J. Brands, J. M. Willigers, L. Kornet, and
A. P. G. Hoeks, “Assessment of the spatial homogeneity of artery
dimension parameters with high frame rate 2-D B-mode,” Ultrasound Med. Biol., vol. 27, no. 6, pp. 785–794, 2001.
[8] A. P. G. Hoeks, C. J. Ruissen, P. Hick, and R. S. Reneman,“Transcutaneous detection of relative changes in artery diameter,” Ultrasound Med. Biol., vol. 11, no. 1, pp. 51–59, 1985.
[9] T. Länne, H. Stale, H. Bengtsson, D. Gustafsson, D. Bergqvist,
B. Sonesson, H. Lecerof, and P. Dahl, “Noninvasive measurementof diameter changes in the distal abdominal aorta in man,” Ultrasound Med. Biol., vol. 18, no. 5, pp. 451–457, 1992.
[10] P. J. Brands, A. P. G. Hoeks, M. C. M. Rutten, and R. S. Reneman, “A noninvasive method to estimate arterial impedanceby means of assessment of local diameter change and the local center-line blood ﬂow velocity using ultrasound,” UltrasoundMed. Biol., vol. 22, no. 7, pp. 895–905, 1996.
[11] C. L. de Korte, E. I. Céspedes, A. F. W. van der Steen, andC. T. Lancée, “Intravascular elasticity imaging using ultrasound:
Feasibility studies in phantoms,” Ultrasound Med. Biol., vol. 23,
no. 5, pp. 735–746, 1997.
[12] E. I. Céspedes, C. L. de Korte, and A. F. W. van der Steen, “Intraluminal ultrasonic palpation: Assessment of local and crosssectional tissue stiﬀness,” Ultrasound Med. Biol., vol. 26, no. 3,
pp. 385–396, 2000.
[13] O. Bonnefous, “Blood ﬂow and tissue motion with ultrasoundfor vascular applications,” Comptes Rendus de l’Acadmie desSciences—Series IV—Physics, vol. 2, no. 8, pp. 1161–1178,
2001.
[14] H. Kanai, M. Sato, Y. Koiwa, and N. Chubachi, “Transcutaneous measurement and spectrum analysis of heart wall vibrations,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 43,
pp. 791–810, 1996.
[15] H. Kanai, H. Hasegawa, N. Chubachi, Y. Koiwa, and M. Tanaka,“Noninvasive evaluation of local myocardial thickening and its
color-coded imaging,” IEEE Trans. Ultrason., Ferroelect., Freq.
Contr., vol. 44, pp. 752–768, 1997.
[16] H. Kanai, Y. Koiwa, and J. Zhang, “Real-time measurements oflocal myocardium motion and arterial wall thickening,” IEEE
Trans. Ultrason., Ferroelect., Freq. Contr., vol. 46, pp. 1229–
1241, 1999.
[17] H. Hasegawa, H. Kanai, N. Hoshimiya, and Y. Koiwa, “Evaluating the regional elastic modulus of a cylindrical shell withnonuniform wall thickness,” J. Med. Ultrason., vol. 31, pp. 81–
90, 2004.
[18] H. Kanai, H. Hasegawa, M. Ichiki, F. Tezuka, and Y.
Koiwa, “Elasticity imaging of atheroma with transcutaneous
ultrasound—Preliminary study,” Circulation, vol. 107, pp. 3018–
3021, 2003.
[19] C. Kasai, K. Namekawa, A. Koyano, and R. Omoto, “Real-time
two-dimensional blood ﬂow imaging using an autocorrelation
technique,” IEEE Trans. Sonics Ultrason., vol. SU-32, pp. 458–
464, 1985.
[20] T. Loupas and R. W. Gill, “Multifrequency Doppler: Improving
the quality of spectral estimation by making full use of the information present in the backscattered RF echoes,” IEEE Trans.
Ultrason., Ferroelect., Freq. Contr., vol. 41, no. 4, pp. 522–531,
1994.
[21] T. Loupas, J. T. Powers, and R. W. Gill, “An axial velocity estimator for ultrasound blood ﬂow imaging, based on a full evaluation of the Doppler equation by means of a two-dimensional
autocorrelation approach,” IEEE Trans. Ultrason., Ferroelect.,Freq. Contr., vol. 42, no. 4, pp. 672–688, 1995.
[22] S. I. Rabben, S. Bjærum, V. Sørhus, and H. Torp, “Ultrasoundbased vessel wall tracking: An auto-correlation technique with
RF center frequency estimation,” Ultrasound Med. Biol., vol. 28,
no. 4, pp. 507–517, 2002.
[23] A. D. M. van Swijndregt, S. H. K. The, E. J. Gussenhoven, C. T.
Lancée, H. Rijsterborgh, E. Groot, A. F. W. van der Steen, N.
Bom, and R. G. A. Ackerstaﬀ, “An in vitro evaluation of theline pattern of the near and far walls of carotid arteries using
B-mode ultrasound,” Ultrasound Med. Biol., vol. 22, no. 8, pp.
1007–1015, 1996.

Hideyuki Hasegawa was born in Oyama,
Japan, in 1973. He received the B.E. degreefrom Tohoku University, Sendai, Japan, in
1996. He received the Ph.D. degree from Tohoku University in 2001. He is presently a research associate in Graduate School of Engineering, Tohoku University. His main researchinterest is medical ultrasound, especially diagnosis of atherosclerosis based on measurementof mechanical properties of the arterial wall.
Dr. Hasegawa is a member of the IEEE, theAcoustical Society of Japan, the Japan Society of Ultrasonics in Medicine, and the Institute of Electronics,
Information and Communication Engineers.

hasegawa and kanai: estimation of artery-wall displacement
Hiroshi Kanai (M’98) was born in Matsumoto, Japan, on November 29, 1958. Hereceived a B.E. degree from Tohoku University, Sendai, Japan in 1981, and M.E. and the
Ph.D. degrees, also from Tohoku University,
in 1983 and in 1986, both in electrical engineering.
From 1986 to 1988 he was with the Education Center for Information Processing, Tohoku University, as a research associate. From1990 to 1992 he was a lecturer in the Department of Electrical Engineering, Faculty of Engineering, Tohoku University. From 1992 to 2001 he was an associate
professor in the Department of Electrical Engineering, Faculty of Engineering, Tohoku University. Since 2001 he has been a professor in
the Department of Electronic Engineering, Graduate School of Engineering, Tohoku University.
His present interests are in ultrasonic measurement and digital
signal processing for diagnosis of heart diseases and arteriosclerosis.
Dr. Kanai is a member of the IEEE, the Acoustical Society ofJapan, the Institute of Electronics Information and Communication
Engineering of Japan, the Japan Society of Mechanical Engineers,
the Japan Society of Ultrasonics in Medicine, the Japan Society of
Medical Electronics and Biological Engineering, and the Japanese
Circulation Society.

63