Investigation on Elasticity-Based Tissue
Characterization of Arterial Wall
Hideyuki Hasegawa1,2 and Hiroshi Kanai2,1
1 Graduate School of Biomedical Engineering, Tohoku University
2 Graduate School of Engineering, Tohoku University
E-mail: hasegawa@ecei.tohoku.ac.jp, hkanai@ecei.tohoku.ac.jp

Abstract
Artery-wall motion due to the pulsation of theheart is often measured to evaluate mechanicalproperties of the arterial wall. Such motion is thought
to occur only in the arterial radial direction because themain source of the motion is an increase of bloodpressure. However, it has recently been reported thatthe artery also moves in the longitudinal direction.
Therefore, a 2-D motion estimator is required evenwhen the artery is scanned in the longitudinal direction
because the arterial wall moves both in the radial(axial) and longitudinal (lateral) directions. Methods
based on 2-D correlation of RF echoes are often used
to estimate the lateral displacement together with axial
displacement. However, these methods require muchinterpolation of the RF echo or correlation function toachieve a sufficient resolution in the estimation of
displacement. To overcome this problem, Jensen et al.
modulated the ultrasonic field in the lateral direction at
a designed spatial frequency to use the lateral phase forthe estimation of lateral motion. This method, namely,
the lateral modulation method, generates complex
signals whose phases change depending on the lateralmotion. Therefore, the lateral displacement can be
estimated with a good resolution without interpolation,although special beamformers are required. The present
paper describes a method that can be applied toultrasonic echoes obtained by a conventional
beamformer to estimate lateral displacements using thephases of lateral fluctuations of ultrasonic echoes. In
the proposed method, complex signals were generated
by the Hilbert transform, and the phase shift wasestimated by correlation-based estimators. The
proposed method was validated using a cylindricalphantom mimicking an artery. The error in the lateraldisplacement estimated by the proposed method was
13.5% of the true displacement of 0.5 mm with akernel size used for calculating the correlation functionof 0.6 mm in the lateral direction, which was slightly
smaller than the width at -20 dB of the maximum
lateral ultrasonic field (about 0.8 mm).

1. Introduction
Noninvasive measurement of mechanical propertiesof 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 [13]. In particular, mechanical properties of plaque areimportant because the rupture of plaque may cause
acute myocardial infarction and cerebral infarction [46]. Magnetic resonance imaging (MRI) andintravascular ultrasound (IVUS) are promisingtechnologies for directly imaging plaque morphology[7,8]. On the other hand, the dynamic change of arterydiameter due to the pulsation of the heart can bemeasured noninvasively by the previous method withultrasound [9-13]. Some parameters related to arterywall elasticity can be obtained by the measured change
in diameter of the artery [14-16]. However, in thederivation of these parameters, the artery is assumed to
be a cylindrical shell with an uniform wall thickness
and, thus, the elasticity of atherosclerotic plaque cannot
be evaluated.
For measurement of the mechanical properties of thearterial wall, including the case with atherosclerotic
plaque, we previously developed a method, namely, thephased tracking method, for measuring small
vibrations in the heart wall or arterial wall withtranscutaneous ultrasound [17,18]. For some years, wehave been measuring the displacement and smallchange in thickness of the arterial wall caused by theheartbeat using this method [19-22]. In our phased
tracking method, a set of two points is assigned along
an ultrasonic beam, and the change in thickness of thelayer between these two points is estimated.
Furthermore, by sliding the position of the layer along
the ultrasonic beam by intervals of the sampled points,
the spatial distribution of changes in thickness along
the ultrasonic beam can be obtained.
In the above-mentioned methods including our
method, the radial motion of the arterial wall, such aschanges in diameter [9-13] and radial strain [19-22],
are measured because it is considered that the source of
the artery-wall motion is the change in internalpressure (blood pressure) and that there is nolongitudinal motion. However, Cinthio et al. showedthat the artery also moves in the longitudinal direction[23]. Therefore, a 2-D motion estimator is required toestimate both the axial (radial) and lateral(longitudinal) displacements, even when an artery isscanned in the longitudinal direction.
de Korte et al. introduced a motion estimator basedon the correlation between RF echoes to measure the 2D displacement of the arterial wall in the crosssectional scan [24]. Such a motion estimator based on

correlation between RF echoes had previously beendeveloped and thoroughly investigated in the field oftissue elasticity imaging [25-28], results indicating thatit accurately estimates the 2-D displacement. However,much interpolation is required to realize an accurate
estimation.
Jensen et al. introduced a method, namely, the lateralmodulation method, in which the ultrasonic field is
modulated in the lateral direction at a designed spatial
frequency to realize a lateral displacement estimation
using the lateral phase induced by the modulated field[29]. This method generates complex signals whose
phases change depending on the lateral displacement
and, therefore, the lateral displacement can be
estimated with a good resolution because the phasechange can be directly converted into the lateraldisplacement. However, this method requires special
beamformers that are not available in conventional
equipment.
To overcome this problem, Chen et al. recently
proposed a method to estimate lateral displacements
using the lateral phases of ultrasonic echoes obtainedby conventional beamformers [30]. The accurate
estimation of lateral displacements would be very
useful, particularly when it could be done based onconventional beamformers. The present paper
describes a method, which also uses the lateral phasesof echoes obtained by conventional beamformers, forestimation of lateral displacements of arterial walls[31]. In the proposed method, complex signals are
generated by the Hilbert transform, and the phase shift
due to the lateral motion is estimated by a correlationbased estimator. The proposed method was validated
using a cylindrical phantom mimicking an artery andcompared with the lateral modulation method.

where * denotes convolution.
Let ux(n) be the lateral displacement of the object
between the n-th and (n + 1)-th frames. Reflection
coefficient r(x; n + 1) in the (n + 1)-th frame isexpressed by r(x; n + 1) = r(x - ux(n); n), where it canbe assumed that there is only the lateral motion when
the axial motion is compensated by an axial motion
estimator (in this study, the method proposed in [32]was used). In addition, it was assumed that there is no
distortion in r(x; n) between the nth and (n + 1)-thframes. Under such conditions, the amplitude s(x; n +1) is given by

2. Materials and Methods
2.1. Difficulties Encountered in Estimation of
Lateral Motion Using Phases of UltrasonicEchoesFig. 1. Geometry for measurement.

This section describes the fundamental theory ofultrasonic fields to show why it is difficult to use thephase information of ultrasonic echoes for estimationof lateral motion.
When the ultrasonic field is focused at a depth ofinterest z using a linear array probe, a point spreadfunction (PSF) h(x) is created as illustrated in Fig. 1,
where only its profile in the lateral direction x at depth
z is considered. By defining the spatial distribution of
the amplitude reflection coefficient of an object in then-th frame as r(x; n), the amplitude s(x; n) of an echo at
depth z obtained by an ultrasonic beam focused at the
point of interest (x, z) is expressed as follows:s ( x; n + 1) = h( x ) * r ( x; n + 1)
= h( x ) * r ( x − u x (n); n).

(2)

In the lateral modulation method proposed by Jensen
et al. [29], 2 point spread functions, hr(x) and hi(x),which oscillate at the same spatial frequency, fx0, butwith a phase difference of 90 degrees, are produced tocreate a complex signal g(x; n), whose phase changes
depending on the lateral displacement of the object.
Based on the relationship in (1), g(x; n) can beconsidered to be the complex version of s(x;n). In thiscase, the 2 point spread functions hr(x) and hi(x) areapproximately expressed by cos(2πfx0x) and−sin(2πfx0x), respectively. Therefore, complex signal
g(x; n) can be obtained based on (1) as follows:∞g ( x; n) = ∫ r ( x − ξ ; n) ⋅ ( hr (ξ ) + jhi (ξ )) d ξ
−∞

(3)

∞

= ∫ r ( x − ξ ; n) ⋅ exp( − j 2 π f x 0ξ ) d ξ .
−∞∞s ( x; n) = ∫ h(ξ ) ⋅ r (ξ − x; n) d ξ−∞∞= ∫ h(ξ ) ⋅ r ( x − ξ ; n) d ξ−∞= h( x) * r ( x; n),(1)

As can be seen in (3), the complex signal g(x; n)obtained by the lateral modulation method is theFourier coefficient R(fx0; n) of reflection coefficient r(x;n) at spatial frequency fx0. Therefore, g(x; n + 1) in the(n + 1)-th frame can be expressed by g(x; n + 1) = g(x;n)·exp{-2πfx0ux(n)}. Under such condition, the lateraldisplacement ux(n) can be estimated by the phase shift-2πfx0ux(n) from g(x; n) to g(x; n + 1) using theconventional correlation technique because the spatial
modulation frequency fx0 can be appropriately obtainedby designing the point spread functions hr(x), and hi(x).
In the present study, the Hilbert transform wasapplied to RF echoes obtained by conventional
beamforming to use the lateral phase. Complex
spectrum S(fx; n) of s(x; n) of (1) is expressed by H(fx;n) and R(fx; n) of the point spread function h(x) andreflection coefficient r(x; n) at spatial frequency fx as
S(fx; n) = H(fx)·R(fx; n). Complex spectra H(fx) and R(fx;n) are described as follows:∞H ( fx ) = ∫ h(ξ ) ⋅ e− j 2 π f xξ

−∞

(4)

dξ

∞

(5)

R( f x ; n) = ∫ r (ξ ; n) ⋅ e − j 2 π f xξ d ξ−∞

Analytic signal y(x; n) of s(x; n) is obtained by the
inverse Fourier transform of S(fx; n) in the range of
positive spatial frequencies as follows:∞y ( x; n) = ∫ H ( f x ) ⋅ R ( f x ; n) ⋅ e 2 π f x x d f x0∞= ∫ H ' ( f x ) ⋅ R ' ( f x ; n) ⋅ e 2 π f x x d f x

(6)

−∞

= h' ( x) * r ' ( x; n),where⎧ H ( f ) ( f x ≥ 0),
H '( fx ) = ⎨ x x⎩ 0 ( f x < 0),⎧ R( f x ; n) ( f x ≥ 0),
R ' ( f x ; n) = ⎨⎩ 0 ( f x < 0),

(7)
(8)

∞

h' ( x) = ∫ H ' ( f x ) ⋅ e j 2 π f x x d f x

(9)

−∞

∞

⇔ H ' ( f x ) = ∫ h' (ξ ) ⋅ e− 2 π f xξ

−∞

dξ,

∞

r ' ( x; n ) = ∫ R ' ( f x ; n ) ⋅ e j 2 π f x x d f x

(10)

−∞

∞

⇔ R' ( f x ; n) = ∫ r ' (ξ ; n) ⋅ e− j 2 π f xξ−∞In actual measurements, s(x; n) is sampled at theinterval of scan lines Δx, and the sampled version ofs(x; n) is denoted by s(mΔx; n) ≡ s(m; n) (m = -M / 2,
-M / 2 + 1, …, -2, -1, 0, 1, 2, …, M / 2), where M + 1 isthe number of scan lines (lateral length of the scanned
region L = M·Δx). In such a discrete system, (6) and(11) are denoted in the digital system as follows:y ( m; n) = h' ( m) * r ' ( m; n),y (m; n + 1) = h' (m) * r ' (m; n)*

1 − e j ( 2 π/ M )( M / 2+1)( m −( u x ( n ) / Δx )),1 − e j ( 2 π/ M )( m −u x ( n ) / Δx ))

(12)
(13)

where (12) and (13) are obtained from (6) and (11),respectively, by replacing the integration, spatial
frequency fx, and lateral spatial position x by thesummation, k / (M·Δx) (discrete spatial frequency), andmΔx (discrete spatial position), respectively. As shownin (12) and (13), the phase shift from y(m; n) to y(m; n+ 1) actually depends on the lateral displacement ux(n)of the object between the n-th and (n + 1)-th frames.
The phase shift of complex signal g(x; n) of (3)obtained by the lateral modulation method is simply
related to lateral displacement ux(n) such as g*(x;n)·g(x; n + 1) = |g(x; n)|2·exp{j2πfx0ux(n)}, where *denotes complex conjugate. However, it is difficult torelate the phase shift from y(m; n) to y(m; n + 1) to thelateral displacement ux(n) because h(x) cannot beassumed to be a sinusoidal wave fluctuating at a known
spatial frequency of fx0 when a conventional
beamformer is used. This is a major difficulty for
utilization of the lateral phase with conventional
beamformers. In the present study, a new method wasintroduced to overcome this problem.

2.2. Principle of Lateral Displacement
Estimation Using the Lateral Phase
Let us define the complex correlation function γ(Δm;m; n) between y(m; n) of (12) and y(m; n + 1) of (13) atlateral lag Δm·Δx asMc

γ(Δm; m; n) = ∑ y * ( M + k ; n) ⋅ y (m + Δm + k ; n + 1), (14)dξ.

k =− M c

In the integration of (6), in a strict sense, the direct
current component (at fx = 0) should be multiplied by
0.5. However, in this study, the direct current
component in the measured signal s(x; n) was removed
before applying the Fourier transform to s(x; n). In thiscase, the Hilbert transform can be expressed by (6).
Similarly, the analytic signal y(x; n + 1) of s(x; n + 1)
in the (n + 1)-th frame is expressed as follows:

where Mc determines the number of sampled points
used for calculating the correlation function.
The phase shift Δθ(m; n) from y(m; n) to y(m; n + 1)
induced by lateral displacement ux(n) between the n-thand (n + 1)-th frames can be obtained by setting laterallag Δm at 0 (conventional correlation technique [33])as follows:Δ θ( m; n) = ∠ γ( Δm; m; n) Δm→0∞y ( x; n + 1) = ∫ H ' ( f x ) ⋅ R ' ( f x ; n) ⋅ D( f x ) ⋅ e − j 2 π f x u x ( n )(15)= ∠ γ(0; m; n).

−∞⋅ e j2 π fxx d f x

In this study, as illustrated in Fig. 2, it was assumed
that there is a linear relationship between the lateral∞= h' ( x) * r ' ( x; n) * ∫ e j 2 π f x ( x −u x ( n )) d f x .
0(11)

displacement ux(n) and the change in the lateral phaseΔθ(m; n), as expressed byu x (n) = a (n) ⋅ Δ θ( m; n)(16)= a (n) ⋅ ∠ γ(0; m; n),α (n) =Δm2

∑ {ΔmΔx − a(n)∠ γ' (Δm; m; n)}2Δm = Δm1=∑ [Δm Δx − 2ΔmΔx ⋅ a(n)∠ γ' (Δm; m; n)
Δm2

2

2

Δm = Δm1(22)]+ a 2 (n){∠ γ' (Δm; m; n)} .
2

where a(n) is a constant (corresponding to the slope ofthe linear relationship), which linearly relates the phaseshift to the lateral displacement.
When Δm is set at 1, it can be considered that y(m; n+ 1) of (13) is artificially displaced by Δx ( = aninterval of scan line) relative to y(m; n) of (12).
Therefore, the following relationship holds:

To determine aˆ ( Δm ,Δm ) (n) which minimizes the mean
12squared difference α(n), the partial derivative of (22)with respect to α(n) is set to be zero:(17)

u x (n) + Δx = a(n) ⋅ ∠ γ(1; m; n).

By solving simultaneous equations consisting of (16)and (17), slope a(n) and lateral displacement ux(n) are
estimated as follows:Δx∠ γ(1; m; n) − ∠ γ(0; m; n) (18)Δx=,
∠ γ * (0; m; n) ⋅ γ(1; m; n)Δxuˆ x ,( Δm1 ,Δm2 ) ( n)=Δm1 → 0 , Δm2 →1∠ γ * (0; m; n) ⋅ γ(1; m; n) (19)⋅ ∠ γ(0; m; n),aˆ ( Δm1 ,Δm2 ) (n)Δm1 → 0 , Δm2 →1

=

(

)

(

)

where aˆ ( Δm ,Δm ) (n) and uˆ x ,( Δm ,Δm ) (n) are the slope and
1

2

1

2

the lateral displacement, respectively, which are
estimated by correlation functions at lateral lags fromΔm1 to Δm2.
Although more computation is required, othercorrelation functions at different lateral lags can beused for displacement estimation. As in (17), thefollowing relationship holds:
u x (n) + ΔmΔx = a(n)∠ γ(Δm; m; n),(20)

where it should be noted that ΔmΔx is an artificial
displacement.
Using more than 2 correlation functions, lateraldisplacement ux(n) of a target can be estimated by theleast-squares method. To do that, let us consider the
relationship that is obtained by subtracting (16) from(20) as follows:
ΔmΔx = a(n){∠ γ(Δm; m; n) − ∠ γ(0; m; n)}≡ a(n)∠ γ' (Δm; m; n),Fig. 2. Assumed relationship between phase shift andlateral displacement. Complex correlation function
γ(Δm; m; n) is shown by γ(Δm).
Δm2∂α= −2Δx ∑ Δm∠ γ' (Δm; m; n)∂aΔm =Δm1
Δm2

2

Δm = Δm1= 0.

By solving (23), aˆ ( Δm ,Δm ) (n) is obtained as follows:
1

2

Δm2

(21)

where ∠γ’(Δm; m; n) = ∠γ (Δm; m; n) - ∠γ (0; m; n).
By considering the left and right sides of (21) to be
the actual and model lateral displacements, respectively,
the mean squared difference α(n) between the actualand model displacements is expressed as follows:(23)+ 2a (n) ∑ {∠ γ' (Δm; m; n)}aˆ ( Δm1 ,Δm2 ) (n) =Δx ∑ Δm∠ γ' (Δm; m; n)Δm = Δm1
Δm2∑ {∠ γ' (Δm; m; n)}

2

(24)

Δm = Δm1
Δm2=

Δx ∑ Δm{∠ γ(Δm; m; n) − ∠ γ(0; m; n)}Δm = Δm1
Δm2∑ {∠ γ(Δm; m; n) − ∠ γ(0; m; n)}Δm = Δm1

2.

By substituting (24) into (16), lateral displacementux(n) is estimated as follows:Δm2uˆ x ,( Δm1 ,Δm2 ) (n) =Δx ∑ Δm{∠ γ(Δm; m; n) − ∠ γ(0; m; n)}Δm = Δm1
Δm2∑ {∠ γ(Δm; m; n) − ∠ γ(0; m; n)}2Δm = Δm1⋅ ∠ γ(0; m; n).

(25)

received by each array element could be acquired(frame rate: 289 Hz), was employed together with a 10MHz linear array probe (UST-5545, Aloka, Tokyo,
Japan) [34]. With this system, plane waves weretransmitted and the receive beamforming was
performed with the apodization and delay factors
shown in Fig. 4 [29]. Figures 5(a) and 5(b) show abeamformed RF echo from a fine wire (diameter: 16μm) placed about 20 mm away from the ultrasonicprobe, and Fig. 5(c) shows the spectrum of the echo
signal shown in Fig. 5(a) obtained by the 2-D Fouriertransform. There were 72 scan lines at lateral intervals
Δx of 0.2 mm, and the lateral modulation frequency fx0
was set at 0.89 mm−1.

Fig. 3. Schematic of measurement system.
In a subsequent section describing experiments usinga phantom, accuracies in estimation of lateraldisplacements achieved by the 2 estimators, i.e., thecomputationally efficient version given by (19) and theestimator given by (25) consisting of 3 correlation
functions at lags {Δm} of -1, 0, and 1 (Δm1 = -1, Δm2 =1), are compared.

2.3. Experimental System
In this study, a cylindrical phantom (inner diameter: 8mm; external diameter: 10 mm) made from silicone
rubber (elastic modulus: 750 kPa) containing 5%
carbon powder (by weight) was measured in theexperimental system illustrated in Fig. 3. The radial
motion ( = axial motion) of the phantom was inducedby changing the internal pressure using a flow pump(pulse pressure: about 60 mmHg, theoretical resulting
radial strain: about 4%). The longitudinal motion ( =lateral motion) was simulated by moving a ultrasonicprobe using an automatic stage. The maximum lateraldisplacements were controlled to be 0.1, 0.25, and 0.5
mm by the automatic stage. The stage was triggered bya signal from the flow pump, which shows the
beginning of ejection.
In ultrasonic measurements for the method proposedin this study, RF echoes from the phantom wereacquired at a frame rate of 286 Hz with a 10-MHz
linear array probe (UST-5545, Aloka, Tokyo, Japan)
equipped with conventional ultrasonic diagnostic
equipment (SSD-6500, Aloka, Tokyo, Japan). The
phantom was scanned in the longitudinal direction atintervals Δx of 0.15 mm (46 scan lines), and RF echoes
were sampled at 40 MHz at a 16-bit resolution.
For the lateral modulation method, a scanner (itsfront end is same as that of α-10 [Aloka, Tokyo,
Japan]), which was modified so that RF echoes

Fig. 4. (a) Amplitude apodization and (b) time delay(for in-phase [solid line] and quadrature [dashed line]
beamformers) values for creating the lateralmodulation during receive beamforming.

3. Results of Experiments Using a Phantom
3.1. Estimation of Lateral Displacements by the
Proposed Method
Fig. 6 shows the procedure to obtain analytic signals{y(m; n)}. Fig. 6(a) shows RF echoes from thephantom that was scanned in the longitudinal direction
using the linear array probe with a conventional
beamformer. Envelopes {s’(m; n)} of the RF echoes
were detected as shown in Fig. 6(c). The direct current
component contained in the envelope signal s’(m; n)
has no phase information and, therefore, it wasremoved. In this study, the envelope signal withoutbias was used as echo amplitude s(m; n), which is
given bys(m; n) = s' (m; n) − E m [s' (m; n)],(26)

where E[·] denotes the averaging operation withrespect to lateral position m·Δx. The Hilbert transformwith a Tukey window shown in Fig. 6(b) was thenapplied to s(m; n) to obtain the analytic signal y(m; n).
The real and imaginary parts of analytic signal y(m; n)were obtained as shown in Figs. 6(e) and 6(f).

The method proposed in Section II-B was applied toanalytic signal y(m; n) to estimate lateral displacementux(n). Twenty points of interest were assigned in theposterior wall at axial intervals of 50 μm along eachscan line, and lateral displacements {ux(n)} of thesepoints were estimated. In Figs. 7(1) and 7(2), plots andvertical bars show means and standard deviations of
the maximum lateral displacements {ûx,(01),max(n)} and{ûx,(-11),max(n)} along each scan line estimated by theestimators given by (19) and (25), respectively, with 4
different sizes of kernels used for calculatingcorrelation function γ(Δm; m; n) defined by (14). The
dashed lines in Fig. 7 show the actual assigned
maximum lateral displacements. In calculation of thecorrelation function, 2-D kernels were used, the axial
size of the kernel being fixed to be the optimum value(0.5 μs) determined in [32], and the lateral size of akernel being changed. In Figs. 7(b), 7(c), 7(d), and 7(e),the lateral sizes of kernels were set at 0.6 mm (Mc = 2),
1.2 mm (Mc = 4), 2.4 mm (Mc = 8), and 3.6 mm (Mc =12), respectively. As shown in Fig. 7, standard
deviations were reduced by the estimator consisting of
3 correlation functions given by (25).
Fig. 8 shows rms errors of the estimated lateraldisplacements obtained by the estimator given by (19)from the actual displacement, where the errors were
evaluated from all the estimates except for thoseobtained at 10 scan lines at each edge of the scanned
region because the estimates at these scan lines were
influenced by the shape of the Tukey window (not flat).
As shown in Fig. 8, the errors were reduced byincreasing the size of the correlation kernel, and similarerrors resulted from the kernel sizes greater or equal to
2.5 mm.

{g(x; n)} are shown in Fig. 10(2) for different
correlation kernel sizes. There were no improvements
by using the proposed motion estimator together withthe lateral modulation method. As shown in Fig. 11,
rms errors of the lateral displacements obtained by the
motion estimator in [29] were calculated from all theestimates shown in Fig. 10(1).

3.2 Comparison with Results Obtained by the
Lateral Modulation Method

Fig. 5. Radio frequency echo from a fine wire obtainedby the lateral modulation method. (a) Radio frequency
echo Re{g(x; n)} obtained by inphase receive
beamforming, where Re{·} means the real part. (b)Lateral profiles at a depth indicated by the arrow in (a)obtained by in-phase and quadrature receive
beamforming. (c) Magnitude of spectrum of the echo
signal Re{g(x; n)} shown in (a) obtained by the 2-D
Fourier transform.

Fig. 9(a) shows RF echoes from the phantom obtainedby the lateral modulation method (in-phasebeamformer), and Fig. 9(b) shows the lateral profiles of
RF echoes at a depth indicated by the arrow in Fig. 9(a)obtained by in-phase and quadrature beamformers. As
in the displacement estimation by the proposed method,changes in the phases of complex signals {g(x; n)}obtained by in-phase and quadrature beamformers were
estimated by the correlation technique described in [29]to obtain lateral displacements {ux(n)} at 20 points ofinterest assigned along each scan line at axial intervalsof 50 μm.
As in Section III-A, the use of different lateral sizesof kernels used for calculation of correlation functions
was examined. Lateral displacements estimated byapplying the motion estimator in [29] to complex
signals {g(x; n)} obtained by the lateral modulation
method are shown in Fig. 10(1). In Figs. 10(1-a), (1-b),(1-c), and (1-d), lateral displacements {ux(n)} were
estimated using kernel sizes of 0.6 mm (Mc = 2), 1.2
mm (Mc = 4), 2.4 mm (Mc = 8), and 3.6 mm (Mc = 12),
respectively. As in Fig. 10(1), lateral displacements
obtained by applying the proposed motion estimator to

5. Discussion
In this study, lateral displacements were estimatedusing the phases of complex signals generated by theHilbert transform applied to ultrasonic echoes obtainedby a conventional beamformer. The phase shift due tothe lateral displacement was estimated by theconventional correlation technique, and a larger
correlation kernel size in the lateral direction was
found to achieve better accuracy. Errors in theestimated lateral displacements were reduced byincreasing the kernel size, and similar errors wereobtained with kernel sizes greater or equal to 2.5 mm,
which roughly corresponds to 3 times the width at −20
dB of the maximum lateral ultrasonic field (about 0.8mm). In a previous study, the estimation of axial

displacements, the optimal kernel size in the axial
direction corresponded to a pulse duration defined by
the width at -20 dB of the envelope of an ultrasonicpulse (about 0.4 mm) [32]. In the axial displacement
estimation, a smaller kernel relative to the point spreadfunction (0.4 mm) yielded good estimates because
there were 2 to 3 oscillations during pulse duration. On
the other hand, there was no oscillation in the lateral
direction (or more exactly, one oscillation because
there was a peak in the lateral profile of the ultrasonicfield). Therefore, a kernel size of about 3 times thepoint spread function was required to minimize theerror.

Fig. 6. (a) Radio frequency echoes from the phantom.
(b) Tukey window used for frequency analysis ofenvelope signals of RF echoes. (c) Envelope signals of
RF echoes. (d) Envelope signals after bias removal. (e)Imaginary and (f) real part of analytic signals {y(x; n)}.
To estimate the artery-wall motion using the phasesof ultrasonic echoes, it is necessary to avoid thealiasing effect. Therefore, the frame rate fFR should be
kept as high as possible. In general, motion of thearterial wall in the radial direction is larger than that in
the longitudinal direction, and the oscillation frequency
in the axial direction ( = radial direction) is higher thanthat in the lateral direction ( = longitudinal direction).
Therefore, the aliasing limit for the radial motion ( =axial motion) should be considered.
Basically, using the change in the axial phase Δφ,the axial velocity va is estimated as follows [33], [35]:va =c0 Δ φf FR ,
4 π f0(27)

where c0 and f0 are the speed of sound and the center
frequency of ultrasound, respectively. In addition, atypical maximal velocity of the carotid arterial wall in
the radial direction is about 10 mm/s. The frame raterequired for the measurement of the axial motion at 10mm/s is obtained by substituting va = 10 mm/s, Δφ = πrad, c0 = 1500 m/s, and f0 = 10 MHz into (27) as

follows: fFR = 4 × (10 × 10−3) × (10 × 106) / 1500 ≈ 267
Hz. To achieve a frame rate higher than 267 Hz at a
fixed pulse repetition frequency of 13156 Hz, thenumber of scan lines was reduced to 46 (this is thelowest available number of scan lines of the employedultrasonic equipment).
As can be seen in Fig. 8, there is a trade-off between
the accuracy and the kernel size (spatial resolution).
When we want to estimate the global motion of an
object, a large kernel can be used for higher accuracy.
On the other hand, it is necessary to use a smaller
kernel to obtain the spatial distribution of lateraldisplacement, although the accuracy will be degraded.
Therefore, the method should be optimized dependingon the purposes, for example, a greater number ofcorrelation functions in an estimator would reduce
standard deviations, as shown in Fig. 7, at the expenseof computational efficiency. In addition, further
improvements would be required to achieve better
accuracy with a smaller kernel. In this study, a linear
relationship between the lateral displacement and thechange in the phase of the complex signal obtained by
the Hilbert transform was assumed. Fig. 12(a) shows a
B-mode image of a fine wire (same as that in Fig. 5),
and Fig. 12(b) shows a lateral profile of envelopes of
RF echoes at a depth indicated by the arrow in Fig.
12(a). By applying the Fourier transform to the lateralprofile shown in Fig. 12(b), a power spectrum, which isshown by the solid line in Fig. 12(c), was obtained. As
shown by the solid line in Fig. 12(c), in general, thedirect current component is largest (central spatial
frequency is zero). Therefore, it is difficult to use thelateral phase with conventional beamformers. The
dashed line in Fig. 12(c) shows a power spectrum
obtained by applying the Fourier transform to thelateral profile after removing the direct current
component based on (26). In this case, the power
spectrum is largest at a certain spatial frequency (≠ 0),
and it can be considered that the lateral profile isfluctuating at the central spatial frequency (≠ 0), as in
the lateral modulation method. Therefore, a linear
relationship was assumed in the present paper. In
addition, correlation functions in a range from -1 to 1,
which corresponded to a range from -150 μm to 150
μm in the lateral direction were used for estimation oflateral displacements. This region is much smaller than
the size of the width at -20 dB of the maximum lateralultrasonic field. Therefore, the assumption of the linear
relationship between the change in the lateral phaseand the lateral displacement in such small range wasconsidered to be appropriate, even when the overall
relationship was not perfectly linear. To use a greaternumber of correlation functions, identification of abetter function describing this relationship would be
required.
In the present study, although it was assumed that
there was no deformation, the proposed method could
estimate the lateral displacements of a phantom under aspecific degree of deformation. However, a method, inwhich the distortion of r(x; n) due to deformation of an
object is taken into account, should be developed toimprove the accuracy, and such method should bevalidated under the existence of various degrees ofdeformation.
In the present study, the results obtained by the
proposed method were compared with those obtainedby the lateral modulation method. The results obtainedby the lateral modulation method with the motion
estimator described in [29] were worse than thoseobtained by the proposed method, and the combinationof the lateral modulation method with the motion
estimator proposed in the present paper did not achieve
any improvements. It could be considered from theseresults that some optimization might be necessary toobtain correctly modulated fields. Therefore, it cannot
be concluded at present that the proposed method isbetter than the lateral modulation method. Nevertheless,the proposed method would be useful because it can beapplied to the ultrasonic echoes obtained byconventional beamformers.

displacement was estimated by correlation-based
estimators. The proposed method was validated using acylindrical phantom mimicking an artery. As a result,the lateral displacements could be measured with anerror of 13.5% of the true displacement of 0.5 mm, andthe proposed method would be useful because it can beapplied to ultrasonic echoes obtained by conventional
beamformers.

Fig. 8. Root mean square errors of lateraldisplacements {ux,(0,1)(n)} obtained by the proposed
estimator given by (19) are plotted as a function ofcorrelation kernel size in the lateral direction. The 3
curves show the errors in cases of actual lateraldisplacements of 0.1, 0.25, and 0.5 mm.

Fig. 7. Means and standard deviation of lateraldisplacements obtained by the estimators given by (1)Eq. (19) and (2) Eq. (25). (a) Tukey window used in
the Hilbert transform. Lateral displacements estimatedby the proposed method with correlation kernel sizes of(b) 0.6 mm, (c) 1.2 mm, (d) 2.4 mm, and (e) 3.6 mm.
The 3 horizontal dashed lines in each figure (b)–(e)show the actual assigned displacements.

5. Conclusion
In this study, a method was developed to estimate
lateral displacements using the lateral phase, which canbe applied to ultrasonic echoes obtained by aconventional beamformer. In the proposed method,
complex signals were generated by the Hilberttransform, and the phase shift due to the lateralFig. 9. Radio frequency echo from the phantom
obtained by the lateral modulation method. (a) Radio
frequency echo Re{g(x; n)} obtained by in-phasereceive beamforming. (b) Lateral profiles of {g(x; n)}
at a depth indicated by the arrow in (a) obtained by inphase and quadrature receive beamforming.

Fig. 10. Means and standard deviations of lateraldisplacements obtained by the lateral modulation
method with (1) motion estimator given in [29] and (2)that given by (19). Means and standard deviations
obtained with correlation kernel sizes of (a) 0.6 mm, (b)1.2 mm, (c) 2.4 mm, and (d) 3.6 mm in the lateral
direction. The 3 horizontal dashed lines in each figure
show the actual assigned displacements.

Fig. 12. (a) B-mode image of a fine wire obtained by aconventional beamformer. (b) Lateral profile ofenvelopes of received ultrasonic RF echoes at a depth
indicated by the arrow in (a). (c) Power spectra of lateralprofiles of envelopes. The solid line shows power
spectrum of the envelope shown in (b). The dashed line
shows power spectrum of the envelope that was obtainedby removing the direct current component from (b)based on (26).

Acknowledgments
We are grateful to Panasonic Co., Ltd., and Panasonic
Shikoku Co., Ltd., for providing the cylindrical phantom
used in this study.

References[1] Lee RT, Grodzinsky AJ, Frank EH, Kamm RD and

Fig. 11. Root mean square errors of lateral displacements
estimated by the lateral modulation method with the
motion estimator in [29] are plotted as a function ofcorrelation kernel size in the lateral direction. The 3
curves show the errors in cases of actual lateraldisplacements of 0.1, 0.25, and 0.5 mm.

Schoen FJ. Structure-dependent dynamic mechanicalbehavior of fibrous caps from human atherosclerotic
plaques. Circulation 83, 1764–1770, 1991.
[2] Loree HM, Grodzinsky AJ, Park SY, Gibson LJ
and Lee RT. Static circumferential tangential modulusof human atherosclerotic tissue. J Biomech 27, 195–
204, 1994.
[3] Simons PCG, Algra A, Bots ML, Grobbee DE andvan der Graaf Y. Common carotid intima-media
thickness and arterial stiffness. Circulation 100, 951–
957, 1999.
[4] Falk E, Shah PK and Fuster V. Coronary plaquedisruption. Circulation 92, 657–671, 1995.
[5] Davies MJ. Stability and instability: Two faces ofcoronary atherosclerosis. Circulation 94, 2013–2020,
1996.
[6] Golledge J, Greenhalgh RM and Davies AH. The
symptomatic carotid plaque. Stroke 31, 774–781, 2000.
[7] McConnell MV, Aikawa M, Maier SE, Ganz P,
Libby P and Lee RT. MRI of rabbit atherosclerosis in
response to dietary cholesterol lowering. Arterioscler
Thromb and Vasc Biol 19, 1956–1959, 1999.
[8] Potkin BN, Bartorelli AL, Gessert JM, Neville RF,
Almagor Y, Roberts WC and Leon MB. Coronary
artery imaging with intravascular high-frequency
ultrasound. Circulation 81, 1575-1585, 1990.

[9] Hoeks APG, Ruissen CJ, Hick P and Reneman RS.

[24] Ribbers H, Holewijn S, Blankensteijn JD, and deTranscutaneous detection of relative changes in arterydiameter. Ultrasound Med Biol 11, 51–59, 1985.
[10] Hoeks APG, Di X, Brands PJ and Reneman RS.
Comparison of the performance of the RF crosscorrelation and Doppler autocorrelation technique toestimate the mean velocity of simulated ultrasoundsignals. Ultrasound Med Biol 19, 727–740, 1993.
[11] Länne T, Stale H, Bengtsson H, Gustafsson D,
Bergqvist D, Sonesson B, Lecerof H and Dahl P.
Noninvasive measurement of diameter changes in the
distal abdominal aorta in man. Ultrasound Med Biol 18,
451–457, 1992.
[12] Brands PJ, Hoeks APG, Rutten MC and Reneman
RS. A noninvasive method to estimate arterial
impedance by means of assessment of local diameterchange and the local centerline blood flow velocity
using ultrasound. Ultrasound Med Biol 22, 895–905,
1996.
[13] Meinders JM, Brands PJ, Willigers JM, Kornet L
and Hoeks APG. Assessment of the spatialhomogeneity of artery dimension parameters with highframe rate 2-D B-mode. Ultrasound Med Biol 27, 785–
794, 2001.
[14] Bergel DH. The static elastic properties of thearterial wall. J Physiol 156, 445–457, 1961.
[15] Peterson LH, Jensen RE and Parnel J. Mechanicalproperties of arteries in vivo. Circ Res 8, 622–639,
1960.
[16] Hayashi K, Handa H, Nagasawa S, Okamura A
and Moritake K. Stiffness and elastic behavior ofhuman intracranial and extracranial arteries. J Biomech
13, 175–184, 1980.
[17] Kanai H, Sato M, Koiwa Y and Chubachi N.
Transcutaneous measurement and spectrum analysis ofheart wall vibrations. IEEE Trans Ultrason FerroelectFreq Contr 43, 791–810, 1996.
[18] Kanai H, Hasegawa H, Chubachi N, Koiwa Y and
Tanaka M. Noninvasive evaluation of local myocardial
thickening and its color-coded imaging. IEEE TransUltrason Ferroelect Freq Contr 44, 752–768, 1997.
[19] Hasegawa H, Kanai H, Koiwa Y and Chubachi N.
Noninvasive evaluation of Poisson’s ratio of arterial
wall using ultrasound. Electron Lett 33, 340–342, 1997.
[20] Hasegawa H, Kanai H and Koiwa Y. Modified
phased tracking method for measurement of change inthickness of arterial wall. Jpn J Appl Phys 41, 3563–
3571, 2002.
[21] Kanai H, Koiwa Y and Zhang J. Real-timemeasurements of local myocardium motion and arterial
wall thickening. IEEE Trans Ultrason Ferroelect Freq
Contr 46, 1229–1241, 1999.
[22] Kanai H, Hasegawa H, Ichiki M, Tezuka F and
Koiwa Y. Elasticity imaging of atheroma withtranscutaneous ultrasound -preliminary study-.
Circulation 107, 3018–3021, 2003.
[23] Cinthio M, Ahlgren AR, Jansson T, Eriksson A,
Persson HW and Lindström K. Evaluation of an
ultrasonic echo-tracking method for measurements ofarterial wall movements in two dimensions. IEEE
Trans Ultrason Ferroelect Freq Contr 52, 1300–1311,
2005.

Korte CL. Non-invasive two dimensional elastographyof the carotid artery. Proc 2005 IEEE Ultrason Symp,
1113–1116, 2005.
[25] Ophir J, Céspedes I, Ponnekanti H, Yazdi Y andLi X. Elastography: A quantitative method for imaging
the elasticity of biological tissues. Ultrason Imaging 13,
111–134, 1991.
[26] Céspedes I and Ophir J. Reduction of image noisein elastography. Ultrason Imaging 15, 89–102, 1993.
[27] Konofagou EE, Ophir J, Varghese T and Kallel F.
A new elastographic method for estimation andimaging of lateral displacements, lateral strains,corrected axial strain and Poisson’s ratios in tissues.
Ultrasound Med Biol 24, 1183–1199, 1998.
[28] Ophir J, Garra B, Kallel F, Konofagou E,
Krouskop T, Righetti R, and Varghese T. Elastographic
imaging. Ultrasound Med Biol 26, S23–S29, 2000.
[29] Jensen JA and Munk P. A new method for
estimation of velocity vectors. IEEE Trans Ultrason
Ferroelect Freq Contr 45, 837–851, 1998.
[30] Chen X, Zohdy MJ, Emelianov SY and O’Donnell
M. Lateral speckle tracking using synthetic lateralphase. IEEE Trans Ultrason Ferroelectr Freq Contr 51,
540–550, 2004.
[31] Hasegawa H and Kanai H. Phase-sensitive lateralmotion estimator for measurement of artery-walldisplacement – phantom study. IEEE Trans Ultrason
Ferroelect Freq Contr 56, 2450-2462, 2009.
[32] Hasegawa H and Kanai H. Reduction of influenceof variation in center frequencies of RF echoes on
estimation of artery-wall strain. IEEE Trans Ultrason
Ferroelect Freq Contr 55, 1921–1934, 2008.
[33] Kasai C, Namekawa K, Koyano A and Omoto R.
Real-time twodimensional blood flow imaging using
an autocorrelation technique. IEEE Trans Son Ultrason
SU-32, 458–464, 1985.
[34] Hasegawa H and Kanai H. Simultaneous imaging
of artery-wall strain and blood flow by high frame rateacquisition of RF signals. IEEE Trans Ultrason
Ferroelect Freq Contr 55, 2626–2639, 2008.
[35] Kanai H, Sato M, Koiwa Y and Chubachi N.
Transcutaneous measurement and spectrum analysis ofheart wall vibrations. IEEE Trans Ultrason FerroelectFreq Contr 43, 791–810, 1996.