IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control , vol. 55, no. 9, September 2008

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Reduction of Influence of Variation in Center
Frequencies of RF Echoes on Estimation ofArtery-Wall Strain
Hideyuki Hasegawa and Hiroshi Kanai, Member, IEEE
Abstract—Atherosclerotic change of the arterial wall leads
to a significant change in its elasticity. For assessment of elasticity, measurement of arterial wall deformation is required.
For motion estimation, correlation techniques are widely used,
and we have developed a phase-sensitive correlation method,
namely, the phased-tracking method, to measure the regional strain of the arterial wall due to the heartbeat. Althoughphase-sensitive methods using demodulated complex signals
require less computation in comparison with methods using
the correlation between RF signals or iterative methods, thedisplacement estimated by such phase-sensitive methods arebiased when the center frequency of the RF echo apparently
varies. One of the reasons for the apparent change in the center
frequency would be the interference of echoes from scatterers
within the wall. In the present study, a method was introduced
to reduce the influence of variation in the center frequenciesof RF echoes on the estimation of the artery-wall strain whenusing the phase-sensitive correlation technique. The improvement in the strain estimation by the proposed method wasvalidated using a phantom. The error from the theoreticalstrain profile and the standard deviation in strain estimatedby the proposed method were 12.0% and 14.1%, respectively,
significantly smaller than those (23.7% and 46.2%) obtainedby the conventional phase-sensitive correlation method. Furthermore, in the preliminary in vitro experimental results, thestrain distribution of the arterial wall well corresponded withpathology, i.e., the region with calcified tissue showed verysmall strain, and the region almost homogeneously composedof smooth muscle and collagen showed relatively larger strain
and clear strain decay with respect to the radial distance fromthe lumen.

I. IntroductionPathological change of the arterial wall due to atherosclerosis leads to a significant change in its mechanical properties [1], [2]. Most conventional methods
for noninvasive measurement of elasticity are based on
measurement of the pulse wave velocity (PWV) [3]–[6] orthat of the change in diameter [7]–[10]. Although thesemethods are useful in terms of their ability to assess thevascular elasticity noninvasively, the spatial resolution inelasticity measurement is limited to the propagating distance of the pulse wave or the entire circumference.
Manuscript received February 3, 2007; accepted February 24, 2008.
This work was supported by the Ministry of Education, Science,Sports and Culture and was partly supported by the Suzuken Memorial
Foundation.
H. Hasegawa and H. Kanai are with Department of Electronic Engineering, Graduate School of Engineering, Tohoku University, Sendai,
Japan (email: hasegawa@us.ecei.tohoku.ac.jp).
Digital Object Identifier 10.1109/TUFFC.884

Elastography was introduced by Ophir et al. to measure
the strain distribution of biological tissue by cross-correlating 2 RF signals before and after deformation [11]–[14].
Elastography was applied to strain imaging of the arterial
wall with intravascular ultrasonography (IVUS), namely,
intravascular elastography [15], [16]. In the cited studies, the elasticity distribution of coronary atherosclerotic
plaque was obtained by the measured displacement distribution in the radial direction. This measured elasticity
distribution was compared with the pathological image,and the results suggested the potential for tissue characterization of atherosclerotic plaque by measurement of its
elasticity.
As a transcutaneous approach, the displacement andstrain around carotid atherosclerotic plaque during one
cardiac cycle have been measured using tissue Doppler imaging [17]. The inhomogeneity of displacements measured
upstream and downstream of atherosclerotic plaque suggests that artery-wall motion has potential for use in theevaluation of plaque vulnerability. Furthermore, some research studies on noninvasive vascular elastography havebeen conducted [18]–[20]. In most of the above-mentioned
methods, a correlation-based technique is used for the estimation of the displacement and strain of the arterial
wall [15]–[18], [21]. Also, in recent years, iterative methods have been introduced for estimating 2-D displacement[19], [22].
We have been studying the measurement of the displacement and radial strain (change in thickness) of thearterial wall based on the phase-sensitive correlation technique and have developed the phased-tracking method[23]–[27]. Elasticity images of the human carotid arteryhave been obtained based on the measured strain distribution, and the potential for transcutaneous tissue characterization has been shown by classification of elasticity
images using elasticity reference data obtained by in vitro
experiments [28]–[31].
The phase-sensitive correlation method using a demodulated complex signal, which was first introduced by
Kasai et al. for blood flow measurement [32], requires lesscomputation in comparison with the cross-correlation between RF signals. Methods based on the cross-correlation
between RF signals can estimate displacement larger thanhalf of the wavelength of ultrasound, but such estimation
requires calculation of correlation functions at multiple
lags between RF signals in 2 frames. On the other hand,phase-sensitive correlation methods estimate displacement
less than half of the wavelength, but require calculation of

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IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control , vol. 55, no. 9, September 2008

only one complex correlation function at no lag between
demodulated complex signals in 2 frames. The displacement of an object can be estimated using phase-sensitive methods by keeping the frame rate sufficiently highso that the displacement between 2 frames becomes lessthan half of the wavelength of ultrasound. However, thedisplacements estimated by phase-sensitive methods arebiased when the center frequency of the RF echo apparently varies. One of the reasons for the apparent change incenter frequency would be the interference of echoes fromscatterers within the wall. The center frequency could beapparently changed depending on the scatterer spacing.
Konofagou et al. proposed a method to estimate tissue
strain from the shift of the centroids in power spectra of
RF echoes by assuming that the change in center frequency is proportional to the magnitude of strain. As describedin [33], [34], this method is robust when the decorrelationof RF signals in 2 different frames is significant due to alarge strain of up to 10% between frames. However, theaccuracy of this method is worse than those of methods
based on cross-correlation of RF signals when the signal
correlation is assured.
In methods based on the cross-correlation of RF signals in 2 frames, companding, termed temporal stretching, is applied to the RF signals before correlating themto compensate for the change in center frequency distributions between 2 frames (called scaling) [12], [35]–[42].
Such difference between the center frequency distributionsin 2 frames is not negligible when the strain that arises
between 2 frames is large because a large deformation significantly changes the spacing of scatterers. Although thecenter frequency “difference” between 2 frames is compensated by companding, the center frequency distributions
themselves are not spatially homogeneous, even after theapplication of companding. Nevertheless, such methods
yield unbiased displacement estimates by compensating
for only the center frequency difference between 2 frames[21] because the displacement estimation by the cross-correlation between RF signals does not require knowledge ofthe center frequency. However, as described in [21], suchmethods require considerable computation because boththe scale factor in companding and time delay between
RF signals must be determined.
In the phase-sensitive correlation-based methods, a subsample small displacement between 2 consecutive frames
is estimated using the phase of the complex correlation
function. In these methods, the center frequency distributions in 2 consecutive frames are assumed to be the same
when the frame rate is sufficiently high and the deformation between 2 consecutive frames is small. Therefore,
companding is not applied. However, knowledge of thecenter frequency at each location is necessary to obtain
an unbiased displacement estimate using the phase of thecomplex correlation function. Although Loupas et al. proposed a method for estimating the center frequency distribution by correlation of the demodulated complex signals[43], [44], center frequency distributions in 2 frames arestill assumed to be the same.

In this study, a new method using the phase-sensitive
correlation technique, which does not require the assumption that center frequency distributions in 2 different
frames are the same, was examined with the aim of reducing the influence of center frequency variation on the estimation of artery-wall strain. The improvement of accuracy
by the proposed method was validated using a cylindricalphantom. Furthermore, preliminary in vitro experimental
results were obtained and are herein presented.
II. Principles
A. Conventional Phase-Sensitive Correlation Method for
Estimating Displacement Distribution in the Arterial Wall
The ultrasonic beam scans I (= 46 in this study) positions along the longitudinal direction of the artery. As
shown in Fig. 1, with respect to each beam position i (=1,2,…,I), RF pulses are transmitted with a time interval
T from an ultrasonic probe, and the echo reflected by thearterial wall is received by the same probe. By referring toa B-mode or M-mode image constructed based on the received RF echoes, the depths, x1(0) and x M T (0) (corresponding to the distances from the probe), of the lumenintima and media-adventitia boundaries of the posteriorwall are manually assigned in the initial frame, where MT
is the number of sampled points between these assigned
boundaries. Depth xm(n) at the m-th sampled points (m= 1,2,…,MT) in the nth frame is defined as follows:x m(n) = x 1(n) + (m - 1) × DX,(m = 1, 2,..., M T ) (1)where ΔX = c0/(2fs) (fs: sampling frequency) is the interval of sampled points in the direction of depth. After
depths x1(0) and x M T (0) in the initial frame n = 0 are
manually assigned, the displacements {um(n)} (m =1,2,…,MT) at multiple points along an ultrasonic beam
are obtained by estimating the instantaneous displacement, {Δum(n)}, between nth and (n + 1)th frames (n =0,1,2,…,N) at the respective points m (= 1,2,…,MT) as
described below.
Let us define the RF signal at depth x and beam position i in the nth frame by si(n;x). The instantaneous displacement, Δum(n), at xm(n) between the nth and (n + 1)
th frames (corresponding to the average velocity between
2 frames) is estimated based on the time delay, Δτm(n),between RF signals, si(n;x) and si(n + 1;x), at x = xm(n)because displacement Δum(n) between the nth and (n +1)th frames is expressed by Δum(n) = c0Δτm(n)/2, wherec0 is the speed of sound [23], [32]. Time delay Δτm(n)at the mth sampled point is related to the phase changeΔθm(n) of si(n + 1;x) from si(n;x) as follows:
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Dq m(n),
2p f c,m(n)(2)

Hasegawa and Kanai : reduction of influence of variation on estimation of artery-wall strain

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Fig. 1. Illustration of variables.

where fc,m(n) is the center frequency of the RF echo at
depth xm(n) in the nth frame. As described in Section
II-B, center frequency fc,m(n) is not spatially constantalong the ultrasonic beam and temporally varies because
the frame is different. Using the quadrature demodulated
signal, zi(n;x), of si(n;x), phase shift Δθm(n) between 2frames at depth xm(n) can be estimated by the complex
correlation rm,n(m′,n′) as follows [32]:r m,n(m ¢, n ¢) =i + I av M c -1

å å z *j (n; x m(n) + l × DX )j =i -I av l =0

(3)× z j(n + n ¢; x m(n) + (l + m ¢) × DX ),where * represents the complex conjugate and rm,n(m′,n′)defines the complex correlation function with lags, m′ andn′, in the directions of depth and frame, respectively. Although the correlation function is usually normalized by
the average amplitude of the complex signal in the correlation window, normalization was not applied in thisstudy because only the phase of the complex correlation
function was used for estimation of the displacement. The
number of sampled points in the direction of depth used
for calculating correlation is defined by Mc, and the correlation window extends from the mth point to the (m + Mc− 1)th point because the leading edge of the echo from thepoint of interest (= mth points) should correspond to mth
point. Phase angle Ðrm,n(0,1) of rm,n(0,1) corresponds tothe phase shift Δθm(n) between RF echoes in 2 consecutive frames [32], where rm,n(0,1) means the correlation
between signals at the same depth (m′ = 0) in 2 consecu-

tive frames (n′ = 1). In blood flow measurement using thecomplex correlation, the correlation function rm,n(m′,n′)is temporally averaged in the direction of the frame. Thiscan be done because the RF signals are acquired at avery high frame rate (several kHz) in the measurement ofblood flow. Although the frame rate in this study (286 Hz)
was higher than that in conventional B-mode imaging (atmost 60 Hz), it was much lower than that in blood flow
measurement, and temporal averaging in the direction of
the frame was not applied in (3).
In this study, a 2-D kernel was used in calculation of thecomplex correlation function to reduce the variance in theestimated displacement distribution in the arterial wall.
The 2-D kernel was used even when the displacement wasestimated based on the conventional methods described inthis section to compare the estimated displacements withthose by the proposed method. The phase angle Ðrm,n(0,1)
represents the spatially averaged phase shift of RF echoes
in the correlation window consisting of (Mc samples) ×((2Iav + 1) beam positions). In this study, Iav was set
to 1 because the number (2Iav + 1) of averaging (beampositions) is significantly increased from 1 to 3 (200% increase) when Iav changes from 0 to 1, but the number (2Iav+ 1) is not increased so much (67%: Iav = 1 → 2, 40%:
Iav = 2 → 3) when Iav is further increased. In addition, alarger Iav worsens the lateral spatial resolution.
In the conventional phase-sensitive correlation method, (n) = Ðr (0,1), inusing the estimated phase shift, Dqmm,nstantaneous displacement Δum(n) between 2 consecutive
frames is obtained as follows:

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IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control , vol. 55, no. 9, September 2008

q (n) c D

c 0D
c0
m
0 q m(n)Du m(n) =»=Ðr m,n(0, 1).
4p f c,m(n)4p f dem
4p f dem2) error reduction of the estimated displacement distribution using a newly proposed function.

(4)

1. Estimation of Displacement Distribution in Arterial Wall with Tracking of Global Motion: In conventional
phase-sensitive correlation methods, the center frequency
distributions in 2 different frames are assumed to be thesame. Therefore, it is desired that the frame rate be as
high as possible so that the deformation of the arterial
wall becomes negligible, and thus the displacement shouldbe estimated for every 2 consecutive frames.
On the other hand, the modified method does not require the above assumption. Therefore, the displacement
distribution in the arterial wall need not be estimated forevery 2 consecutive frames. Thus, several key frames aresparsely assigned, and the displacement distribution between 2 consecutive key frames is estimated. This procedure significantly reduces the number of calculations
because the displacement of the arterial wall between 2
consecutive frames (4.2 μm) is smaller than the spacingof sampled points (19.25 μm), and thus there are multiple
frames between 2 consecutive key frames, where the displacement of 4.2 μm is derived from the mean wall velocity of 1.2 mm/s during one cardiac cycle (measured in the
carotid artery of a healthy subject) at a frame rate of 286
Hz. In this procedure, only the displacement at the luminal boundary is estimated using the conventional phasesensitive correlation method described in Section II-A for
every 2 consecutive frames so that the correlation window
follows the position of the arterial wall (tracking of global
motion).
Details of the above procedures are as follows: In thisstudy, the global motion of the arterial wall is tracked
using the displacement u1(n) of the luminal boundary estimated by the conventional method described in Section
II-A because the echo from the luminal interface is usually
dominant in comparison with those from scatterers in thewall and is robust against interference. The displacementu1(n) of echo from the luminal interface is estimated forevery 2 frames at a high frame rate of 286 Hz. Therefore,
the instantaneous displacement Δu1(n) between 2 consecutive frames is less than the spacing ΔX of the sampled
points in the direction of depth x.
Fig. 3 shows an example of the accumulated displacement u1(n) at the luminal interface of the phantom usedin this study. In this study, several key frames are defined
as shown in Fig. 3. The frame number of the kth key frame
is denoted by nk, and key frames are set so that the difference between the accumulated displacement u1(nk + 1) inthe (k + 1)th key frame and that u1(nk) in the kth key
frame reaches spacing ΔX of the sampled points. Then,the global motion of the wall is compensated by shifting
the RF signal in the (k + 1)th key frame by ΔX of thesampled points before calculating the phase shift of RF
echoes to obtain displacement Δum(nk) between 2 consecutive key frames. This is done by setting lags m′ and n′in (3) to be Dˆ(n k +1) - Dˆ(n k ) and nk + 1 − nk, respectively,
as follows:

In (4), knowledge of the center frequency fc,m(n) is
necessary to obtain an unbiased estimate. It is, however,unknown. Therefore, in conventional phase-sensitive correlation methods, the frequency of the reference sinusoid,
fdem, in the quadrature demodulation is used as the frequency term in the denominator of (4), whereas it shouldbe the actual center frequency fc,m(n).
The accumulated displacement, um(n), is obtained byaccumulating the estimated instantaneous displacementΔum(n) between 2 consecutive frames as follows:uˆ m(n + 1) = uˆ m(n) + D
u m(n)(u m(0) = 0). (5)Depth x1(n) of the first point (luminal interface of theposterior wall) in the nth frame is determined by accumulated displacement û 1(n) as follows:xˆ 1(n) = x 1(0) + Dˆ(n) × DX,(6)

where D̂(n) is an integer that satisfies min D(n) |uˆ 1(n) −D(n) · ΔX | because the position of a correlation window
is set depending on the spacing of the sampled points. For
other points {m}, their depths {xm(n)} in the nth frame aredetermined by (1).
B. Modified Method for Estimation of
Displacement Distribution
Fig. 2(a) shows a B-mode image of the phantom usedin this study. The RF echo along the white vertical line in
Fig. 2(a) is shown in Fig. 2(b). As can be seen in Fig. 2(b),the center frequency is not constant along the ultrasonicbeam. This change in center frequency does not occur due
to the well-known frequency dependent attenuation of ultrasound because the center frequency in a deep region—
region B in Fig. 2(b)—is higher than that in a shallow
region (region A).
When a target contains multiple scatterers within
the duration of an ultrasonic pulse, interference between
echoes from these scatterers occurs. An ultrasonic pulse
has a certain frequency bandwidth, and thus such interference could be one of the reasons for the apparent change
in center frequency (see the appendix). This subsection
describes the displacement estimation method proposedin this study for reduction of the influence of center frequency variation.
The modified method comprises 2 main procedures: 1)
estimation of displacement distribution in the arterial wallwith tracking of the global motion of the arterial wall andAuthorized licensed use limited to: IEEE Xplore. Downloaded on November 6, 2008 at 03:52 from IEEE Xplore. Restrictions apply.

Hasegawa and Kanai : reduction of influence of variation on estimation of artery-wall strain

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Fig. 2. (a) B-mode image of the phantom used in this study and (b) waveform of received RF echo along white vertical line in (a).

c 0Df m(n k )D
u m(n k ) =4p f demc0
Ðr m,n(Dˆ(n k +1) - Dˆ(n k ), n k +1 - n k ),=4p f dem(7)

ery 2 consecutive key frames. This procedure significantly
reduces the number of computations because there are
multiple frames between 2 consecutive key frames.
2. Error Correcting Function: In the conventional method described in Section II-A, the frequency fdem of thereference sinusoid in the quadrature demodulation is used(n ) - D
(n ) |= 1 . The depth positions of corwhere | D
k +1
krelation windows in the kth and (k + 1)th key frames in(7) are different, whereas those in (4) are the same. Thisdifference in the window positions corresponds to the compensation of global motion between these 2 key frames asshown in Fig. 4. Accumulated displacement um(nk) at
depth xm(nk) in the kth key frame is obtained by accumulation of the instantaneous displacement Δum(nk) during
the period between 2 consecutive key frames as in (5).
uˆ m(n k +1) = uˆ m(n k ) + D
u m(n k )(u m(0) = 0). (8)In this procedure, it is only necessary to calculate thedisplacement u1(n) at the luminal interface for every 2
consecutive frames (nth and (n + 1)th frames). For otherpoints {m}, displacements {um(nk)} are calculated for ev-

Fig. 3. Illustration of key frames.

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IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control , vol. 55, no. 9, September 2008

Fig. 4. Illustration of displacement estimation with global motion compensation.

instead of the actual center frequency fc,m(n), as shown in(4), to estimate the displacement Δum(n). This assumption is also used in the modified method shown by (7).
However, fdem does not often match the actual center frequency fc,m(n) of the RF echo. Loupas et al. proposed amethod for estimating the mean frequency, f c,m(n) , of the
RF echo at each depth xm(n) in frame n [43]. In the method described in [43], [44], the estimated mean frequency

Fig. 5. (a) RF echoes and (b) center frequency distributions in 2 consecutive frames.

f c,m(n) is used in (4) instead of the demodulation frequency fdem to obtain an unbiased displacement estimate. The
mean frequency, f c,m(n) , at the mth point in the nth frame
is estimated as follows:Ðr m,n(1, 0)
,
(9)
2pT s
where Ts is the sampling interval of the demodulated signal. By assuming that the mean frequency distributions{f c,m(n)} (m = 1,2,…,MT) in 2 consecutive frames are thef c,m(n) = f dem -same, the mean frequency distributions {f c,m(n)} estimated by (9) are used in the estimation of instantaneous displacements {Δum(n)}(m = 1,2,…,MT) between 2 consecutive frames.
Fig. 5(a) shows RF echoes from the phantom used inthis study (the same as in Fig. 2) in 2 consecutive frames.
The frame interval is 3.5 ms (= 1/286 Hz). Fig. 5(b) shows
the ratio of the mean frequency f c,m(n) estimated by (9)—Mc in (3) was set to 20—to the demodulation frequency
fdem of 6.5 MHz. There is a difference in the mean frequency distributions {f c,m(n)} in 2 consecutive frames eventhough a high frame rate of 286 Hz was employed in thisstudy.
Figs. 6(a) and 6(b) show RF echoes and mean frequency
distributions estimated by (9) in 2 consecutive key frames.
The interval between these 2 key frames is 69.9 ms. As
shown in Fig. 6(b), there is a difference between the meanfrequency distributions in 2 key frames. This difference islarger than that between 2 consecutive frames (3.5 ms)
due to a longer frame interval.
In this study, an error correcting function, which does
not require the assumption that the center frequency distributions in 2 different frames are the same, is introduced
as follows: Let us consider the correlation functions

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r m,n k (0, n k +1 - n k ) and r m,n k (Dˆ(n k +1) − D̂(n k ), n k +1 − nk)
between 2 consecutive key frames. The absolute differenceΔdm(nk) between the displacements given byÐr 0,n k (0, n k +1 - n k ) and Ðr m,n k (Dˆ(n k +1) − D̂(n k ), n k +1 −nk) by substituting them into (4) instead of Ðr m,n(0,1)
should be the spacing of sampled points ΔX because|D̂(n k +1) − D̂(n k )| = 1 , i.e., the difference between thedisplacement estimated by the phase change of RF echo, (n ) , and the true displacewhich corresponds to Ddmkment (corresponding to ΔX) is evaluated. Thus, the ratio (n )γ (n ) of the estimated displacement difference Dd
m

k

m

k

to the true displacement difference (= ΔX) between thek-th and (k + 1)th key frames is obtained as follows:c0|Ðr m,n k (0, n k +1 - n k )(10)
4p f demDX- Ðr m,n k (Dˆ(n k +1) - Dˆ(n k ), n k +1 - n k )| .

g m(n k ) =By defining fs as the sampling frequency of the demodulated signal, the numerator of (10) can be modified using
the relation ΔX = c0/(2fs) as follows:c0| Ð{r m*,n k (Dˆ(n k +1)
c
4p f dem × 2f0s- Dˆ(n k ), n k +1 - n k ) × r m,n (0, n k +1 - n k )} |g m(n k ) =kfs=| Ð{r m*,n k (Dˆ(n k +1)
2p f dem- Dˆ(n k ), n k +1 - n k ) × r m,n k (0, n k +1 - n k )} | .
(11)
Fig. 6(c) illustrates an example of the error correcting
function γm(nk) for the RF signals shown in Fig. 6(a).
 (n ) , is obThe error corrected displacement, Dumktained by using γm(nk) as the error correcting function as
follows: (n ) =
Du
m k

1
c0
g m(n k ) 4p f dem
Ðr m,n k (Dˆ(n k +1) - Dˆ(n k ), n k +1 - n k ).
(12)

Fig. 6. (a) RF echoes and (b) center frequency distributions in 2 consecutive key frames; (c) error correcting function γm(nk).

sampled point (depth xm) is estimated by minimizing themean squared difference, αm, between the measured regional displacements {û m} and the linear model, Um =εr,m ⋅ x + bm, as follows:Ms

a m = å uˆ m + j - e r,m × DX × j + b m

2(m £ M T - M s ),

j =1

(13)
C. Estimation of Radial Strain from the
Displacement Distribution
Fig. 7(a) shows an example of a received RF echo. Fig.
7(b) shows an example of the estimated accumulated displacement, uˆ m, at each sampled point m along an ultrasonic beam. Radial strain is obtained by estimating theregional slope of the displacement distribution {um} (m =1,2,…,MT). The regional slope, εr,m, around the mthwhere Ms defines the width of region R for estimating theregional slope.
To obtain the strain distribution {εr,m} in the directionof depth x, the region (layer) R in which the regional slope
εr,m is estimated is slid in the direction of depth x along
an ultrasonic beam by intervals of the sampled points.
Therefore, several layers overlap at the mth sampled point
because the layer thickness Ms ⋅ ΔX is greater than theinterval ΔX of the sampled points. Thus, the compounded

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Fig. 8. Schematic diagram of measurement system.

to be longer than the duration of the employed ultrasonicpulse.
III. Materials and Methods

Fig. 7. Illustration of strain estimation: (a) RF echo from the posteriorwall of the phantom used in this study and (b) displacement distribution
along an ultrasonic beam.

radial strain, e r,m [45], at the mth sampled point is obtained by simply averaging the estimated regional slopese of the layers, which overlap at the mth point as folr,mlows:ïìï 1 L mïïeˆr,m - j +1
ï L m j =1
e r,m = ïíïï 1 L mïï
eˆr,M T -M s - j +1
ïï L m j =1îå(m £ M T - M s ),

å(m > M T - M s ),(14)

where Lm is the number of overlapping layers at the mth
point.
As shown in Fig. 7(a), a dominant echo from the luminal interface is found in the period from 0 to about 0.13
μs. The phase shift of sampled points within this echocorresponds to the displacement of the luminal interface.
Therefore, a constant displacement is found during theduration of that echo as shown by the region surroundedby the dashed line in Fig. 7(b). As a result, strain is hardly
detected during that period (from 0 to 0.13 μs) when thewidth Ms for estimation of the regional slope in (13) isless than the pulse duration. However, such a decrease in
the estimated strain is just an artifact due to the finite
pulse duration. To reduce this artifact, in this study, thewidth used for estimating the regional slope εr,m was set

In this study, a cylindrical phantom and an excised
artery were measured by ultrasound. The outer and innerdiameters of the phantom, made of silicone rubber, were
10 and 8 mm, respectively. The phantom contained 5%
carbon powder (by weight) to obtain sufficient scatteringfrom inside the wall.
Fig. 8 shows a schematic diagram of the measurement
system. A change in pressure inside the phantom or artery
was induced by circulating a fluid using a flow pump. Thechange in internal pressure was measured by a pressure
sensor (9E02-P16, NEC, Tokyo, Japan).
In ultrasonic measurement, the phantom and artery
were measured with a linear-type ultrasonic probe (SSD6500, Aloka, Tokyo, Japan). The nominal center frequency was 10 MHz. RF echoes were acquired at 40 MHz at aframe rate of 286 Hz.
Pressure-diameter testing was applied to the phantom
to obtain the elastic modulus of silicone rubber. In thepressure-diameter testing, the change in external diameterof the phantom was measured with a laser line gauge (VG035, KEYENCE, Osaka, Japan).
In the in vitro experiment, a femoral artery excisedfrom a patient with arteriosclerosis obliterans was measured. The artery was placed in a water tank filled with
0.9% saline solution at room temperature. A needle was
attached to the external surface of the posterior wall using strings to identify the sections to be imaged during
the ultrasonic measurement. After the ultrasonic measurement, pathological images of the measured sections wereobtained with reference to the needle.
IV. Basic Experimental Results
A. Pressure-Diameter Testing of the Phantom
Fig. 9(a) shows the trigger signal for driving the flow
pump. Figs. 9(b) and 9(c) show waveforms of the internal pressure and change in the external diameter of thephantom, respectively. In Fig. 9, the waveforms of 10 measurements are superimposed. From the measured internalpressure, pi, and the change in external diameter, 2⋅Δro,the elastic modulus, E, is obtained as follows [46]:Authorized licensed use limited to: IEEE Xplore. Downloaded on November 6, 2008 at 03:52 from IEEE Xplore. Restrictions apply.

Hasegawa and Kanai : reduction of influence of variation on estimation of artery-wall strain

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in Fig. 10(a). In this section, the length of the correlation window determined by Mc and that for estimatingstrain determined by Ms are set to 0.5 μs (corresponding to the length at −20 dB of the maximum magnitudeof the quadrature demodulated signal of the ultrasonicpulse, Mc = 20) and 1.0 μs (twice the length at −20 dB,
Ms = 40), respectively, for both the conventional and proposed methods. In Fig. 10(a), plots and vertical bars show
the means and standard deviations of the radial strain
in the radial direction (along the ultrasonic beam). The
mean value and standard deviation at each radial position
are obtained by the individual strain distributions along
46 ultrasonic beams. The solid curve in Fig. 10(a) shows
the theoretical radial strain εr of a homogeneous tube ateach radial position r, which is obtained using the elastic
modulus E measured by pressure-diameter testing and themeasured internal pressure as follows [46]:er = -

3r i2r o2pi.
2
2 (r o - r i2)r 2 E(16)

In Fig. 10(a), mean values tend to follow the theoretical profile. However, the standard deviation is large. Mean
error emean and standard deviation SDmean evaluated by(17) and (18) were 23.7% and 46.2%, respectively.
M

Fig. 9. Waveforms of internal pressure pi and change in external diameter
2Δro of the phantom: (a) trigger signal for the flow pump; (b) internalpressure pi; (c) change in external diameter 2Δro; and (d) relationshipbetween internal pressure pi and change in external diameter 2Δro of
the phantom.

E =3 r i2 p i
3 r i2pi
3 r i2r o p i,==
2 r o2 - r i2 e q
2 r o2 - r i2 Dr o
2 r o2 - r i2 Dr oro(15)
where ri and ro are the original internal and external radii,
respectively.
In Fig. 9(d), the measured internal pressure pi is plotted as a function of the change in external diameter 2Δro.
The slope, pi/Δro, was estimated by applying the leastsquares method to the measured data shown in Fig. 9(d).
The estimated slope pi/Δro is shown by the solid line in
Fig. 9(d). The elastic modulus E of the phantom was determined to be 749 kPa.
B. Ultrasonic Measurements of the Phantom
1. Comparison of Conventional and Proposed Methods:
The radial strain distribution inside the wall of the phantom was estimated by the conventional phase-sensitive
correlation method described in Section II-A, as showne mean =T E [ê1
i m,i ] - e r = x m,
M T m =1e r =x m

å

(17)

M

SD mean =T
E i[(eˆm,i - E[eˆm,i ]) 2]
1,
M T m =1e r =x må(18)

where Ei[⋅] represents the averaging with respect to thebeam position i.
Fig. 10(b) shows the strain distribution obtained by
the conventional phase-sensitive correlation method withmean frequency estimation (Loupas’s method) [43]. Thereis no distinct improvement in the strain estimates. The
strain distribution obtained by Loupas’s method withspatial compounding [45] shown in Fig. 10(c) is in good
agreement with the theoretical strain profile, and the standard deviation is significantly reduced. Note that verticalscales in Figs. 10(c) and 10(d) are different from those in
Figs. 10(a) and 10(b).
Fig. 10(d) shows the radial strain distribution obtainedby the method proposed in this study. The difference between the mean values and the theoretical profile shownby the solid curve was further reduced, but the standard
deviation was similar to that obtained by Loupas’s method with spatial compounding. From this result, it can besaid that the variance was reduced mainly by the spatial
compounding. Mean error emean and standard deviation
SDmean of the strain distribution estimated by the proposed method were emean = 12.0% and SDmean = 14.1%,
respectively.

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IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control , vol. 55, no. 9, September 2008

Fig. 11. Error in the strain distribution estimated by the proposed method: (a) error at each combination of the period for estimating strain, Ms
in (13), and length of correlation window, Mc in eq. (3); (b) error distribution at the period for estimating strain of 1.0 μs.

Fig. 11(b) shows the error distribution at the period
for estimating strain of 1.0 μs. The strain estimates are
not much influenced by the correlation window length incomparison with the period for strain estimation, and theerror reduction is almost complete at the correlation window lengths of 0.5 μs. Therefore, the length of 0.5 μs(Mc = 20 in (3) and (7)) was selected to achieve a better
spatial resolution, whereas the error at 0.6 μs is slightly
smaller. The correlation window length of 0.5 μs (Mc =20) and the period for estimating strain of 1.0 μs (Ms =40) correspond to the length at −20 dB of the maximum
magnitude of the quadrature demodulated signal of theultrasonic pulse and twice that length, respectively.
V. In Vitro Experimental Results
Fig. 10. Radial strain distributions of the phantom obtained by (a) conventional phase-sensitive correlation method (phased-tracking method)[23]; (b) conventional phase-sensitive correlation method with mean frequency estimation (Loupas’s method) [43]; (c) Loupas’s method withspatial compounding [45]; and (d) proposed method.

2. Consideration of the Size of Correlation Window andPeriod for Estimating Strain: Fig. 11(a) shows error emeanin the strain distribution of the phantom estimated by the
proposed method at each combination of the period for estimating strain and the length of the correlation window.
As shown in Fig. 11, error emean is reduced by lengthening
the period for estimating strain, and this effect of error
reduction is almost complete at 1.0 μs (Ms = 40 in (13)).

Figs. 12(a-1) and 12(b-1) show the B-mode images of 2
different regions in the excised femoral artery. The region
shown in Fig. 12(a-2) shows very low strain. On the other
hand, the region shown in Fig. 12(b-2) shows relatively
larger strain, and there is a clear strain decay with respect
to the distance from the lumen [46]. Figs. 12(a-3) and12(b-3) show the elasticity images obtained by (16) using the measured internal pressure and estimated strain
distributions. By comparing the elasticity images andpathological images of the corresponding sections shownin Figs. 12(a-4), 12(b-4), 12(a-5), and 12(b-5), the very
hard region surrounded by the green line in Fig. 12(a-3)
corresponds to calcified tissue. On the other hand, theregion composed of smooth muscle and collagen showsa relatively homogeneous and lower elastic modulus dis-

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Hasegawa and Kanai : reduction of influence of variation on estimation of artery-wall strain

1931

tribution. These results show that the proposed method
successfully reveals the difference in the deformation properties resulting from different tissue compositions.
VI. Discussion

Fig. 12. In vitro experimental results for 2 regions: (a) that with calcified tissue and (b) that without calcified tissue and composed of smoothmuscle and collagen; (1) B-mode images, (2) strain images, (3) elasticity
images, and (4) and (5) pathological images with elastica-Masson andhematoxilin-eosin staining, respectively.

In the method proposed in this study, the interval offrames between which the phase shift of echoes is estimated
was increased to reduce the number of computations. However, aliasing does not occur because the large global motion is tracked (compensated) by the conventional method
at a high frame rate of 286 Hz. The global motion between
2 key frames is the distance corresponding to an intervalof sampled points, 19.25 μm in this study (speed of sound:
1,540 m/s). Therefore, the residual displacements at otherradial positions after global motion compensation are lessthan 19.25 μm after global motion compensation. Whenthe center frequency of the received RF echo is exactly
the same as the nominal center frequency (10 MHz) of theemployed ultrasonic probe, a displacement of 19.25 μm
produces a phase shift of RF echoes of π/2. The residual
displacement at each radial position after global motion
compensation is smaller than 19.25 μm, which is smaller
than the aliasing limit of 38.5 μm at 10 MHz.
The proposed method does not remove the global motion perfectly because the displacement at the luminalinterface estimated by the conventional phase-sensitive
correlation method is used to compensate for the global
motion, and, in usual cases, the frequency of the reference
sinusoid used in the quadrature demodulation does not exactly match the center frequency of the RF echo from theluminal interface. However, even in such situations, theproposed method can compensate for most of the global
motion, and it is one of the simplest ways to track the
global motion.
On the other hand, by using companding to find theexact match between the precompression and postcompression RF signals [11], [12], [35]–[40], more accurate
displacement estimates would be achieved because suchmethods do not require information on the center frequency and enable better correction of the difference between
the center frequencies of 2 RF signals. In the present study,
however, RF signals were sampled at a frequency only
4 times the transmit center frequency, whereas a higher
sampling frequency (or interpolation) is desirable in suchother methods. The objective of this study was to develop
a method that can be applied to an RF signal sampled ata lower frequency with less computation but reasonableaccuracy, and the scientific meaning of this study is theintroduction of an error correcting function that reduces
the influence of the center frequency variation on the displacement estimation without requirement of the assumption that the center frequency distributions in 2 different
frames are the same.
As described above, in this study, improvement of thestrain estimates by the proposed method was shown bybasic experiments using a cylindrical phantom. The true

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IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control , vol. 55, no. 9, September 2008

Fig. 13. RF echoes and their power spectra (center frequency: 7.5 MHz): (a) from a single scatterer; (b) from 2 scatterers with a spacing of 162 μm;
and (c) from 2 scatterers with a spacing of 240 μm.

strain distribution in an actual arterial wall is difficult
to know, particularly in the case of a diseased region,
as shown in Fig. 12(a). Therefore, in this study, we justdemonstrated that the strain and elasticity distributions
estimated by the proposed method were in good agreement with the pathology. The usefulness of the proposed
method in clinical situations should be further investigated by applying it to more arteries in in vitro and in vivo
experiments.
Finally, in the estimation of the elastic modulus, determination of the stress distribution is a major problem. In
this study, the stress distribution defined by (16) was used
to obtain the elasticity images shown in Fig. 12(a-3) and12(b-3). However, it agrees with the actual stress distribution only in the case of a homogeneous cylindrical shell.
Such a difference between the actual stress distribution
and that used in elasticity estimation leads to a significant
error. Recently, several studies have been conducted toreconstruct elastic modulus distributions [47]–[49]. However, the method employed has not yet been applied toboth longitudinal and cross-sectional scans of the arteries,
and determination of the stress distribution remains animportant problem in vascular elasticity imaging.
VII. Conclusions
In the displacement estimation based on the phasechange of echoes, the displacement estimates are biased
when the center frequency of the RF echo changes. Such an
apparent change in the center frequency could be causedby the interference of echoes from scatterers. To reduce theinfluence of the center frequency variation on the estimation of artery-wall strain, in this study, an error correcting
function, which does not require the assumption that thecenter frequency distributions in 2 different frames are thesame, was introduced. As a result, the proposed method

provides better strain estimates in comparison with conventional phase-sensitive correlation methods.

Appendix
Influences of Interference on Center
Frequency
In the case of a target containing multiple scatterers,
the interference between echoes from these scatterers occurs because there are multiple scatterers within the duration of an ultrasonic pulse. An ultrasonic pulse has acertain frequency bandwidth, and thus the interference of
echoes could be one of the reasons for the change in center
frequency.
Fig. 13(a-1) shows a simulated RF echo from a single
point scatterer, and Fig. 13(a-2) shows its power spectrum. The center frequency is set to 7.5 MHz, and the envelope is a Hanning window with a length corresponding
to 5 cycles of the center frequency. Figs. 13(b-1) and 13(c1) show the RF echoes from 2 scatterers that exist along
an ultrasonic beam. The spacings of scatterers in Figs.
13(b-1) and 13(c-1) are 162 μm and 240 μm, respectively.
As shown in Figs. 13(a-2), 13(b-2), and 13(c-2), dependingon the spacing of scatterers, even when there are only 2
scatterers, the dominant frequency changes in a complex
way due to the interference of echoes.

Acknowledgments
We are grateful to Panasonic Co., Ltd. for providing
the phantoms used in this study. This work was supportedby the Ministry of Education, Science, Sports and Culture and was partly supported by the Suzuken Memorial
Foundation.

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Hasegawa and Kanai : reduction of influence of variation on estimation of artery-wall strain

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Hideyuki Hasegawa was born in Oyama, Japan, in 1973. He received the B.E. degree fromTohoku University, Sendai, Japan, in 1996. He received the Ph.D. degree from Tohoku University
in 2001. He is presently an associate professor in
Graduate School of Engineering, Tohoku University.
His main research interest is medical ultrasound, especially diagnosis of atherosclerosis
based on measurement of mechanical propertiesof the arterial wall.

Dr. Hasegawa is a member of the IEEE, the Acoustical Society ofJapan, the Japan Society of Ultrasonics in Medicine, and the Institute ofElectronics, Information and Communication Engineers.

Hiroshi Kanai (M’98) was born in Matsumoto,Japan, on November 29, 1958. He received a B.E.
degree from Tohoku University, Sendai, Japan in
1981, and M.E. and the Ph. D. degrees, also fromTohoku University, in 1983 and in 1986, both inElectrical Engineering.
From 1986 to 1988 he was with the Education
Center for Information Processing, Tohoku University, as a research associate. From 1990 to 1992
he was a lecturer in the Department of ElectricalEngineering, 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 Acoustical Society of Japan, the Institute of Electronics Information and Communication Engineering ofJapan, the Japan Society of Ultrasonics in Medicine, Japan Society of
Medical Electronics and Biological Engineering, and the Japanese Circulation Society.

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