Reprinted from

REGULAR PAPER

Optimization of Correlation Kernel Size for Accurate Estimationof Myocardial Contraction and Relaxation
Yasunori Honjo, Hideyuki Hasegawa, and Hiroshi Kanai

Jpn. J. Appl. Phys. 51 (2012) 07GF06# 2012 The Japan Society of Applied Physics

Person-to-person distribution (up to 10 persons) by the author only. Not permitted for publication for institutional repositories or on personal Web sites.

REGULAR PAPER

Japanese Journal of Applied Physics 51 (2012) 07GF06
DOI: 10.1143/JJAP.51.07GF06

Optimization of Correlation Kernel Size for Accurate Estimationof Myocardial Contraction and Relaxation
Yasunori Honjo1 , Hideyuki Hasegawa1;2 , and Hiroshi Kanai1;2 
12Graduate School of Biomedical Engineering, Tohoku University, Sendai 980-8579, Japan
Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan

Received November 18, 2011; accepted April 19, 2012; published online July 20, 2012
For noninvasive and quantitative measurements of global two-dimensional (2D) heart wall motion, speckle tracking methods have been developed
and applied. In these conventional methods, the frame rate is limited to about 200 Hz, corresponding to the sampling period of 5 ms. However,myocardial function during short periods, as obtained by these conventional speckle tracking methods, remains unclear owing to low temporal andspatial resolutions of these methods. Moreover, an important parameter, the optimal kernel size, has not been thoroughly investigated. In ourprevious study, the optimal kernel size was determined in a phantom experiment under a high signal-to-noise ratio (SNR), and the determined
optimal kernel size was applied to the in vivo measurement of 2D displacements of the heart wall by block matching using normalized crosscorrelation between RF echoes at a high frame rate of 860 Hz, corresponding to a temporal resolution of 1.1 ms. However, estimations under low
SNRs and the effects of the difference in echo characteristics, i.e., specular reflection and speckle-like echoes, have not been considered, and theevaluation of accuracy in the estimation of the strain rate is still insufficient. In this study, the optimal kernel sizes were determined in a phantom
experiment under several SNRs and, then, the myocardial strain rate was estimated such that the myocardial function can be measured at a highframe rate. In a basic experiment, the optimal kernel sizes at depths of 20, 40, 60, and 80 mm yielded similar results: in particular, SNR was morethan 15 dB. Moreover, it was found that the kernel size at the boundary must be set larger than that at the inside. The optimal sizes of thecorrelation kernel were seven times and four times the size of the point spread function around the boundary and inside the silicone rubber,respectively. To compare the optimal kernel sizes, which was determined in a phantom experiment, with other sizes, the radial strain rates
estimated using different kernel sizes were examined using the normalized mean-squared error of the estimated strain rate from the actual one
obtained by the 1D phase-sensitive method. Compared with conventional kernel sizes, this result shows the possibility of the proposed correlation
kernel to enable more accurate measurement of the strain rate. In in vivo measurement, the regional instantaneous velocities and strain rates in
the radial direction of the heart wall were analyzed in detail at an extremely high temporal resolution (frame rate of 860 Hz). In this study, transitionin contraction and relaxation was able to be detected by 2D tracking. These results indicate the potential of this method in the high-accuracy
estimation of the strain rates and detailed analyses of the physiological function of the myocardium.
# 2012 The Japan Society of Applied Physics

1. Introduction

Myocardial strain and strain rate have been shown to be
useful in estimating myocardial function, such like contraction and relaxation.1–5) In recent years, there have been
many studies on the measurement of two-dimensional (2D)
myocardial strain or strain rate obtained by speckle tracking,which is direct tracking of backscattered echoes using
regional pattern-matching techniques.6–9) However, theframe rate (FR) in conventional speckle tracking methods
is limited to about 200 Hz, corresponding to the sampling
period of 5 ms.10–13) Conventional speckle tracking methods
at low temporal resolution may fail in estimating the heart
wall motion because it leads to large changes in echopatterns (decorrelation) owing to the motion in the elevation
direction.14) In addition, in the period of one transition in
myocardial contraction/relaxation, the heart wall moves
rapidly during a short period of about 10 ms.4) Therefore,
the detailed analysis of the heart function has been limited
owing to the low temporal resolution of conventional
speckle tracking methods. The continuous observation ofsuch rapid heart wall motion requires a high temporal
resolution of more than 500 Hz.1,4)
Moreover, the accuracy of the velocity and displacement
estimated by the speckle tracking method depends on an
important parameter, i.e., the size of the correlation kernel.
To track backscattered echoes accurately, it is necessary to
determine the optimal kernel size. Kaluzynski et al. reported
that a silicone tube, with a diameter of 31 mm could bemeasured by tracking echoes from scatterers by correlation
with a correlation kernel size of (1.0 laterally  0.5

axially) mm2 .15) Similarly, Bohs et al. accurately determinedthe velocity of a string phantom, which moves twodimensionally at velocities up to 2.5 m/s, with a correlation
kernel size of (1:0  2:0) mm2 . The blood ﬂow in the lateral
direction was imaged in in vitro experiments using acorrelation kernel size of (0:8  0:8) mm2 .16) By in vivo
measurements for estimating the velocity and displacement
spontaneously caused by the heart wall motion, the kernel
size was determined to be (10:0  10:0) mm2 by examining
several kernel sizes in comparison with manual observation.17–21) However, the optimal kernel size has not yet beenthoroughly investigated.
In our previous study, the optimal kernel size wasdetermined in a phantom experiment under a high signalto-noise ratio (SNR), and the determined optimal kernel
size was applied to the in vivo measurement of 2D
displacements of the heart wall by block matching using
a normalized cross-correlation between RF echoes at ahigh frame rate of 860 Hz, corresponding to a temporal
resolution of 1.1 ms.22) However, estimations under low
SNRs and the eﬀects of the diﬀerence in echo characteristics, i.e., specular reﬂection and speckle-like echoes, havenot been considered, and the evaluation of the accuracy ofan estimation of the strain rate is still insuﬃcient. In thisstudy, the optimal kernel sizes were determined in aphantom experiment under several SNRs and positions.
The determined optimal kernel size was compared withconventional sizes in the evaluation of the accuracy in theestimation of the strain rate and, then, myocardial strain
rate was estimated so that the myocardial function can bemeasured at a high frame rate.

E-mail address: kanai@ecei.tohoku.ac.jp

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lateral

(a)

axial

2 σl (d )2 σd (d )σl (d )

O

2 σl (d )vd ( l , d ; n )θ( l )Δ l (d )σd (d )
2 σd (d )A
Wd (d )A

Wl (d )
correlation kernel Wθj O’

π/2−( θ( l )+ θj )(b)Fig. 1. (Color online) Schematic of correlation kernel (Wl ðdÞ  Wd ðdÞ).

vl( l , d ; n )θ( l )+ θjΔd (d )

beam profilevr ( r i ,θj ; n)O

2. Principles

θj+1

2.1 Definition of correlation kernel

In this study, 2D velocities and displacements of the heart
wall are estimated by speckle tracking using the normalized
cross-correlation function between RF echoes. Speckle size
depends on the width of the ultrasonic beam and pulse
length.14,23) In this study, the size of the correlation kernel inthe lateral direction Wl ðdÞ and that in the axial direction
Wd ðdÞ were deﬁned using the 20 dB widths of the lateral
proﬁle of the ultrasonic ﬁeld lðdÞ and the envelope of theultrasonic pulse dðdÞ at several depths, respectively. As
illustrated in Fig. 1, the shape of the correlation kernel was abivariate normal distribution. The variances  2l ðdÞ and  2d ðdÞof the normal distribution in the lateral and axial directions
were determined from the size of the point spread function(PSF) ðlateral  axialÞ ¼ ðlðdÞ  dðdÞÞ and the variable
coeﬃcient , respectively.22)( i +1)−th
i −th

θj O’

vl ðl; d; nÞ ¼

bmn  ðl=Ml ÞNF  Tbkn  ðd=Md Þðmm=sÞ;ð2:1Þ

ðmm=sÞ;ð2:2Þ
NF  T
where T and l are the time interval between consecutive
frames and the lateral spacing l ¼ 2d  sinð=2Þ (:angular interval) of beams at depth d, respectively, andNF is the frame interval for the calculation of correlation.
In this study, the normalized cross-correlation function wasinterpolated by reconstructive interpolation24) to improve theaccuracy in the estimation of the velocity and displacement.22) The Ml and Md are the interpolation factors in thelateral and axial directions, respectively.
We introduced the strain rate to evaluate how the
thickness of each layer changes with respect to time. As
shown in Fig. 2(a), the velocity in the radial direction,
vr ðri ;  j ; nÞ, which was calculated from the velocity in thelateral direction, vl ðl; d; nÞ, and that in the axial direction,
vd ðl; d; nÞ, is given by
vd ðl; d; nÞ ¼

vr ( r i +1,θj +1;n )vr ( r i ,θj +1 ;n )Fig. 2. (Color online) Schematic of calculation of radial strain rate.

vr ðri ;  j ; nÞ ¼ vl ðl; d; nÞ  sin½ j þ ðlÞþ vd ðl; d; nÞ  cos½ j þ ðlÞ
= OO0 A

ðmm=sÞ; ð2:3Þ

= O0 OAandin Fig. 2,
where  j and ðlÞ arerespectively. As shown in Fig. 2(b), strain rate in the radial
direction, Sr ði;  j ; nÞ, between the i-th and (i þ 1)-th layer isgiven bySr ðri ;  j ; nÞ ¼

2.2 Calculation of 2D velocities and radial strain rate

The lateral velocity vl ðl; d; nÞ and axial velocity vd ðl; d; nÞbetween the n-th and (n þ NF )-th frames at a spatial point
ðl; dÞ (l: lateral, d: axial positions) in a heart wall are givenby the determined lateral shift b
mn and axial shift bkn ,respectively, which maximize the interpolated normalized
cross-correlation function as follows:22,24)A

vr ðriþ1 ;  j ; nÞ  vr ðri ;  j ; nÞR
½ðmm=sÞ=mm;ð2:4Þ

R ¼ riþ1  ri ;
ð2:5Þ
where i and j are the numbers of layers and directions,
respectively. The numerator shows the diﬀerence between
velocities at positions ðriþ1 ;  j Þ and ðri ;  j Þ. The denominator
R shows the initial thickness of the layer for the normalization of the velocity diﬀerence. After the strain rates were
estimated at each radial position, they were superimposed on
the B-mode image in every frame using a color code toobtain the 2D spatial distribution of the strain rate.
3. Determination of Optimal Size of Correlation
Kernel by Basic Experiment
3.1 Measurement of the size of point spread function

In the present study, the kernel size was deﬁned as a multipleof the size of PSF (lðdÞ  dðdÞ) at each depth d, as(lðdÞ; dðdÞ). The size of PSF is determined by thewidths, at 20 dB, of the lateral proﬁle of the ultrasonic ﬁeld
and the envelope of an ultrasonic echo obtained fromexperiments using a ﬁne nylon wire (diameter: 31 m)placed at depths of 20, 40, 60, and 80 mm. RF data wereacquired using a 3.75 MHz sector-type probe of ultrasonicdiagnostic equipment (Aloka -10). In this study, planewaves were transmitted in Nt ¼ 7 diﬀerent directions atangular intervals of 6 at a pulse repetition frequency fPRF of

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Y. Honjo et al.

4.0

3.6
3.4
3.2
length [mm]60variation of the RF signals in the water

beam width Δl (d )pulse length Δd (d )

3.8

3.0
2.8
2.6
2.4
2.2
2.0

50
40
30
20
10
0

1.8
1.6

10

20

30

40
50
60
depth d [mm]

70

80

10
(d 0 )

90(a)lateralstage
controller

(b)
axial

ultrasonic
probe

10 mm
d = 40 mm
positionp0

silicone rubber

...

p4
vl
automatic X−Y stage

vd

water

l 5 (d )l0 (d )

water tankFig. 4. (Color online) Schematic of phantom experiment to determine theoptimal kernel sizes (Wl0 ðdÞ  Wd0 ðdÞ).

6020 Hz using a 3.75 MHz sector-type probe. For eachtransmission, Nr ¼ 16 receiving beams were created atangular intervals of  ¼ 0:375 . The FR obtained byparallel beam forming (PBF)25) was FR ¼ fPRF =Nt ¼ 860
Hz. The sampling frequency of the RF signal was 15 MHz.
Figure 3 shows the size of PSF (lðdÞ  dðdÞ) measured at
diﬀerent depths. The error bars show the standard deviation
for ﬁve measurements at each depth d.
3.2 Basic experimental system

Figure 4 shows the basic experimental system. The motionof a phantom was measured to determine the optimum
kernel size (Wl0 ðdÞ  Wd0 ðdÞ). The phantom was madefrom silicone rubber (Momentive Performance Materials
TSE3503) containing 5% graphite powder by weight toprovide suﬃcient scattering. The phantom was set at thedepths fdg where the size of the PSF were measured. In thiscross-sectional image of silicone rubber, the silicone rubber
at the red points was analyzed. There were six beams li set atintervals from 41 (i ¼ 0) to 71 (i ¼ 5). Positions (p0 –p4 )were set at intervals of 1.5 mm from the upper surface of thephantom, where the depths fdg were set at 20, 40, 60, and
80 mm. The phantom was a silicone rubber plate (soundspeed c0 : 990 m/s) with a thickness of 15 mm. The motion
velocity of the silicone rubber was controlled by anautomatic X–Y stage. The phantom was moved twodimensionally. Both the lateral velocity vl0 ðl; d; nÞ and axial
velocity vd0 ðl; d; nÞ, which were set at a constant speed of30

40

50 60 70
depth [mm]8090

100 110(d m )

Fig. 5. Variance of RF signals in water.

Fig. 3. (Color online) Distribution of 20 dB widths of lateral amplitude

proﬁle l and pulse envelope d of ultrasonic pulse at depths of 20, 40, 60,
and 80 mm.

20

5 mm/s, were smaller than those of the heart wall.26)
Therefore, the 2D displacements, xl ðl; d; nÞ and xd ðl; d; nÞ,of the silicone rubber between two frames NF wererespectively set to be similar to those of the heart wall by
increasing the frame interval NF for calculating thecorrelation function.22,26)
3.3 Calculation of signal-to-noise ratio

SNRs from the phantom placed in water are higher thanthose from the heart wall in in vivo measurement. Therefore,
in this study, white noise was added to the RF signalsfrfðl; dÞg measured in the basic experiment so that SNR ofechoes in the basic experiment became similar to those in
the in vivo measurement. The white noise w ðl; dÞ wascreated by MATLAB (MathWorks). SNR of an echo signal
from the phantom, which contains noise 0 ðl; dÞ originatingfrom the ultrasonic equipment and w ðl; dÞ, is given by
SNR ¼ 10 log10 ðEðl;dÞ ½ðEt ½rfðl; dÞÞ2 Þ 10 log10 ðEðl;dÞ ½ð0 ðl; dÞ þ   w ðl; dÞÞ2 Þ
¼ 10 log10 ðEðl;dÞ ½ðEt ½rfðl; dÞÞ2 Þ 10 log10 ðEðl;dÞ ½0 ðl; dÞ2 þ ð  w ðl; dÞÞ2 Þ; ð3:1Þ
where Et ½ and Eðl;dÞ ½ are the time and space (lateral andaxial directions) average operations, respectively, and  isthe amplitude coeﬃcient controlling SNR. There is nocorrelation between the noise 0 ðl; dÞ caused by theultrasonic equipment and white noise w ðl; dÞ, i.e.,
Eðl;dÞ ½0 ðl; dÞ  w ðl; dÞ ¼ 0. The SNRs in the basic experiment in the regions of p j at depth d in Fig. 4(b) are given by
2
(
)2 3
l5
N
1
X
X
1
1
SNRðp j ; dÞ ¼ 10 log10 4
ðrf n ðl; dÞÞ 5
A l¼l N n¼0
0
"
(
l5
1X
 10 log10ðEt ½0 ðl; dÞ2 Þ
A l¼l
0
!)#
N
1
X

2
ð ðl; dÞÞ
þ
;
ð3:2Þ
N n¼0 w
ðA ¼ l5  l0 Þ;ð3:3Þ

where j is the number of depths ( j ¼ 0; 1;    ; 4) set in thesilicone rubber, as shown in Fig. 4. In this study, as shown in

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2

−9

1.6

0

1.4

−8

1.2

−2

1.0

−4

0.8

−6

0.6

−8

0.4
0

5 10 15 20 25 30 35 40 45 50
SNR [dB]

4

1.8

2

1.6

−8
0

1.4
1.2

−8

1.0

2.0

−6

0.6

−8

0.4
0

5 10 15 20 25 30 35 40 45 50
SNR [dB]

4

1.8

−9
−8
−8
−9
−10

1.6
1.4
1.2

2
0
−2

1.0

−4

0.8

−6

0.6

−8

0.4
0

−2
−4

0.8

RMS error in thedisplacement [dBm](d) p3 = 6.0 mm
alpha (parameter of correlation kernel)

−10

2.0

5 10 15 20 25 30 35 40 45 50
SNR [dB]−10(e) p4 = 7.5 mm
2.0

−10

RMS error in thedisplacement [dBm](c) p2 = 4.5 mm
alpha (parameter of correlation kernel)

4

1.8

RMS error in thedisplacement [dBm](b) p1 = 3.0 mm

alpha (parameter of correlation kernel)

2.0

alpha (parameter of correlation kernel)

RMS error in thedisplacement [dBm](a) p0 = 1.5 mm
alpha (parameter of correlation kernel)

Y. Honjo et al.

2.0

4

1.8
1.6
1.4

−9

2

−8

0
−2

1.2
1.0

−4
−9
−8

0.8
0.6

−6
−8

0.4
0

5 10 15 20 25 30 35 40 45 50
SNR [dB]−10

RMS error in thedisplacement [dBm]
4

1.8

2

1.6
1.4

−8

0

1.2

−9
−10

−2

1.0

−4

0.8

−6

0.6

−8

0.4
0

5 10 15 20 25 30 35 40 45 50
SNR [dB]−10

Fig. 6. (Color online) RMS errors fð; SNRðp j ; dÞ; dÞg at the depth of 40 mm obtained as a function of the correlation window width coeﬃcient  and

SNR.

Fig. 5, the noise 0 ðl; dÞ was estimated as the variance of theoutput signal from the equipment, and was measured without
any objects in water, as follows:0 ðl; dÞ2 ¼
rfðl; dÞ ¼

1
X
1N
ðrf n ðl; dÞ  rfðl; dÞÞ2 ;
N n¼0

ð3:4Þ

1
X
1N
rf n ðl; dÞ;
N n¼0

ð3:5Þ

where N is the number of frames used in the calculation of
the variance.
3.4 Determination of the optimal size of correlation kernel

In this study, the sizes Wl ðdÞ and Wd ðdÞ of the correlation

kernel in the lateral and axial directions were deﬁned by2 l ð¼ 2  lðdÞÞ and 2 d ð¼ 2  dðdÞÞ, respectively. To determine the optimal value opt ðd; SNRðp j ; dÞÞ of giving the optimal kernel size (Wl0 ðdÞ  Wd0 ðdÞ), the 2D
displacements of the phantom estimated using diﬀerent
values of fg and SNRs were examined using the root-meansquared (RMS) error of the estimated displacement from theactual displacement. The RMS errors in the estimated 2D
displacements, fð; SNRðp j ; dÞ; dÞg, which were evaluatedfor each value of  and SNRðp j ; dÞ at depth d, were
calculated from the lateral displacement xl ðli ; d; nÞ and axial
displacement xd ðli ; d; nÞ estimated for each position ðli ; dÞand the actual lateral displacement xl0 ðnÞ and axial
displacement xd0 ðnÞ as follows:

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
u
5 N
1
u 1 X
X
ðxl ðli ; d; nÞ2 þ xd ðli ; d; nÞ2 Þ;ð; SNRðp j ; dÞ; dÞ ¼ t
I  N i¼0 n¼0

ð3:6Þ

xl ðli ; d; nÞ ¼ xl ðli ; d; nÞ  xl0 ðnÞ;ð3:7Þ

xd ðli ; d; nÞ ¼ xd ðli ; d; nÞ  xd0 ðnÞ;

ð3:8Þwhere N and I ð¼ 6Þ are the numbers of averaged frames
and measurement points, respectively. In this experiment,
the displacements at the position fðli ; dÞg in the area of 2D
tracking were measured using normalized cross-correlation
functions with diﬀerent values of  ¼ 0:3  2:0 at the ﬁve
positions (p0 –p4 ) at each depth of 20, 40, 60, and 80 mm.
The original sampled normalized cross-correlation function
was interpolated with Ml ¼ 120 in the lateral direction and

Md ¼ 15 in the axial direction to obtain the cross-correlation
function with similar resolutions in the lateral and axial
directions, corresponding to 2.2 m.
Figure 6 shows the distributions of RMS errors,fð; SNRðp j ; dÞ; dÞg, estimated using diﬀerent values offg of the correlation kernels at the ﬁve positions [(a)p0 = 1.5 mm  (e) p4 = 7.5 mm]. As shown in Fig. 6,
when the size of the correlation kernel is set to be small,

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Jpn. J. Appl. Phys. 51 (2012) 07GF06(a) 2.0(b) 2.0

alpha (parameter of correlation kernel)

alpha (parameter of correlation kernel)

Y. Honjo et al.

1.8
1.6
1.4
1.2
1.0
0.8

p 1 = 1.5 mm
p 2 = 3.0 mm
p 3 = 4.5 mm
p 4 = 6.0 mm
p 5 = 7.5 mm

0.6
0.4
0

10

15

20

25 30
SNR [dB]

35

40

45

1.8
1.6
1.4
1.2
1.0
p 1 = 1.5 mm
p 2 = 3.0 mm
p 3 = 4.5 mm
p 4 = 6.0 mm
p 5 = 7.5 mm

0.4
0

5

10

15

20

1.2
1.0
0.8

p 1 = 1.5 mm
p 2 = 3.0 mm
p 3 = 4.5 mm
p 4 = 6.0 mm
p 5 = 7.5 mm

0.6

0

(d)

0.6

1.4

50

2.0

0.8

1.6

0.4

alpha (parameter of correlation kernel)

alpha (parameter of correlation kernel)

(c)

5

1.8

5

35

40

45

20

25 30
SNR [dB]

35

40

45

50

25 30
SNR [dB]

35

40

45

50

1.8
1.6
1.4
1.2
1.0
0.8

p 1 = 1.5 mm
p 2 = 3.0 mm
p 3 = 4.5 mm
p 4 = 6.0 mm
p 5 = 7.5 mm

0.6

0

50

15

2.0

0.4
25 30
SNR [dB]

10

5

10

15

20

Fig. 7. (Color online) Optimal values of opt ðd; SNRðp j ; dÞÞ plotted as a function of SNR for ﬁve positions (p0 –p4 ) at each depth d (d ¼ 20, 40, 60, and

80 mm).

there may be several regions in the next frame that haveecho patterns similar to those in the small correlation kernelin the previous frame. These multiple regions, which have
similar echo patterns, induce misestimation of displacement.
When the size of the kernel is set to be too large, the peak of
the correlation coeﬃcient is degraded because the speckle
pattern of the central position in the correlation kernel is
diﬀerent from that of the edge in the correlation kernel,
owing to deﬂection of the ultrasonic beam. In addition, it
is rare that the echo patterns in the two frames exactly
coincide, which degrades the accuracy of the estimationand the spatial resolution.27–29) However, the displacement
estimation is less susceptible to noise because the echopattern in the correlation kernel is more unique.
Figures 7(a)–7(d) show the relationships among the
SNRs fSNRðp j ; dÞg ( j ¼ 0; 1; 2; 3; 4), the values offopt ðd; SNRðp j ; dÞÞg, which are the minimum RMS errors,
and the positions fp j g at each depth d from the upper surface.
As shown in Fig. 7, the distributions of the values offopt ðd; SNRðp j ; dÞÞg at the positions of a boundary of thesilicone rubber (p0 and p1 ) were diﬀerent from those inside
the silicone rubber (p2 –p4 ). As shown in Fig. 4(b), thebrightness is high and relatively uniform in the lateral
direction at the boundary of the silicone rubber. In suchcases, there might be several regions in the next framethat have echo patterns similar to those in the correlation
kernel in the previous frame. Such a situation leads tothe misestimation of displacement. Figures 8(a) and 8(b)show proﬁles of the normalized cross-correlation functions
obtained for points of interest at the boundary (p0 ) andinside (p4 ) the silicone rubber, respectively. As shown in
Fig. 8(a-1), at large  ( ¼ 1:8; 2:0), the proﬁles around thepeak of the normalized cross-correlation functions in thelateral direction became narrower around the boundary of
the silicone rubber (at p0 ). On the other hand, as shown in
Fig. 8(b-1), at  ’ 1:0, the proﬁle around the peak of thenormalized cross-correlation function in the lateral direction
became narrowest inside the silicone rubber (at p4 ). The
proﬁles around the peaks of the normalized cross-correlation
functions in the axial direction around the boundary
and inside the silicone plate were similar, as shown in
Figs. 8(a-2) and 8(b-2), respectively. Therefore, the kernel
size at the boundary must be set larger than that inside.
As shown in Fig. 1, in our study, the correlation kernel
was deﬁned using the size of PSF (lðdÞ  dðdÞ), andthe optimal sizes fðWl0 ðdÞ  Wd0 ðdÞÞg were determinedas 2  opt ðd; SNRðp j ; dÞÞðlðdÞ  dðdÞÞ. As shown in
Figs. 7(a)–7(d), when SNR is larger than 15 dB, the optimal

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(a) p0= 1.5 mm
(2)
1.0

0.8

0.8

correlation coefficientcorrelation coefficient

(1)
1.0

0.6
α = 0.8
α = 1.0
α = 1.2
α = 1.4
α = 1.6
α = 1.8
α = 2.0

0.4

0.2

0

−1.0

−0.5
0
0.5
shift in the lateral direction [mm]α = 0.8
α = 1.0
α = 1.2
α = 1.4
α = 1.6
α = 1.8
α = 2.0

0.6

0.4

0.2

0
−0.02

1.0

0

0.02 0.04 0.06 0.08
0.1
shift in the axial direction [mm]

0.12

0.14

(b) p3= 6.0 mm
(2)
1.0

0.8

0.8

correlation coefficientcorrelation coefficient

(1)
1.0

0.6
α = 0.8
α = 1.0
α = 1.2
α = 1.4
α = 1.6
α = 1.8
α = 2.0

0.4

0.2

0

−1.0

−0.5
0
0.5
shift in the lateral direction [mm]α = 0.8
α = 1.0
α = 1.2
α = 1.4
α = 1.6
α = 1.8
α = 2.0

0.6

0.4

0.2

0
−0.02

1.0

0

0.02 0.04 0.06 0.08 0.10
shift in the axial direction [mm]

0.12

0.14

Fig. 8. (Color online) Distribution of correlation coeﬃcient in the lateral (1) and axial (2) directions. (a) p0 ¼ 1:5 mm. (b) p3 ¼ 6:0 mm.

value of opt ðd; SNRðp j ; dÞÞ at each depth d (¼ 20, 40, 60,
and 80 mm) is 1.6–1.8 around the boundary and 1.0 inside
the silicone rubber. From these results, the optimal kernel
bl0 ðdÞ  W
bd0 ðdÞÞg were determined to be about sevensizes fðWtimes (2  opt ðlðdÞ  dðdÞÞ (opt ¼ 1:6{1:8)) and fourtimes (2  opt ðlðdÞ  dðdÞÞ ðopt ¼ 1:0Þ) the size ofPSF (lðdÞ  dðdÞ) around the boundary and inside thesilicone rubber, respectively. It was found that the optimal
kernel sizes fðWl0 ðdÞ  Wd0 ðdÞÞg can be determined fromthe size of PSF (lðdÞ  dðdÞ), which is determined by
the lateral and axial widths at a 20 dB ultrasonic ﬁeld
measured in advance.

i−th layer

(a)

1
2
3
4
5

10 mm

boundary
insideinsideinsideboundary(b)

4. Basic Experiment of Estimation of Radial Strain
Rate
4.1 Experimental system

As illustrated in Fig. 9, in this experiment, the strain rate of
the phantom owing to the change in the internal pressure,which was applied using a ﬂow pump, was measured. The
homogeneous cylindrical phantom with a wall thickness of
2 mm was made from silicone rubber. In this study, the RFdata were acquired using the 3.75 MHz sector-type probe ofultrasonic diagnostic equipment (Aloka -10). All parameters, sound speed c0 , the sampling frequency fs of the RF
signal, frame rate FR, and interpolation factors Ml and Md inphantom
phantomside view

flow pumpfront viewFig. 9. (Color online) (a) Cross-sectional image of silicone tube at a
depth of 40 mm. The yellow points in this cross-sectional image show
positions fðl; dÞg for displacement estimation. (b) Schematic of phantom
experiment to estimate the radial strain rates fSr ðri ;  j ; nÞg.

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increase inthickness( W l 0 x W d 0 ) mm

19)(10.0 x 10.0) mm

6)(4.0 x 4.0) mm

0.05

200

10 mm

180

decrease inthickness(b) Trigger pulse
5.00(c) velocity [mm/s]

−5.0
0.05

normalized MSE ε ( θ j )−0.05[(m/s)/m]

[%]

0

0
0

140
120
100
80
60
40
20

(d) strain rate [m/s/m]

−0.05

160

0.5

1.0

time [s]0−180−150−120 −90 −60 −30 0
degree θ jFig. 10. (Color online) Actual strain rate Sr0 ðri ;  j ; nÞ of silicone tube

along the central beam ( j ¼ 0) obtained by 1D phase-sensitive method.

the lateral and axial direction, were the same as those used in
the basic experiment using the silicone rubber. In this study,
to obtain the actual velocity vr0 ðri ;  j ; nÞ and strain rate
Sr0 ðri ;  j ; nÞ in the phantom’s radial direction, a 1D phasesensitive method (1D method) was used for the central beam( j ¼ 0 ). The silicone rubber was assumed to be an
isotropic medium in the circumferential direction. Therefore,
the actual strain rate Sr0 ðri ;  j ; nÞ ( j ¼ 0 ) obtained by the
1D method and shown in Fig. 10 can be used as references
for all directions f j g and layers fri g.
4.2 Experimental results

Some features of speckle patterns, such as the size of a
speckle, depend on the PSF produced by the ultrasonicﬁeld.14,23) In this study, the RF data were acquired using the3.75 MHz sector-type probe of ultrasonic diagnostic equipment (Aloka -10), which was the same as that used in thebasic experiment so that the features of speckle size were
not changed. Therefore, the size of a correlation kernel
determined by the basic experiment was also used in thisexperiment. As illustrated in Fig. 9(a), the correlation kernel
at the position of the boundaries (ﬁrst and ﬁfth layers) andinside (second, third, and fourth layers) the silicone rubber
were set at seven times [(18:2  16:9) mm2 ] and fourtimes [(10:4  9:7) mm2 ] the size of the PSF (l  d),respectively.
To compare the optimal kernel size determined in thephantom experiment, with other sizes, the radial strain rates
estimated using diﬀerent kernel sizes were examined using
the normalized mean-squared error (MSE) of the estimatedstrain rate Sr ðri ;  j ; nÞ from the actual one Sr0 ðri ;  j ; nÞ. The
normalized mean-squared error ð j Þ was calculated from theradial strain rate Sr ðri ;  j ; nÞ at each position ðri ;  j Þ andactual Sr0 ðri ;  j ; nÞ as follows:I 1 N1
X
XðSr ðri ;  j ; nÞ  Sr0 ðri ;  j ; nÞÞ2ð j Þ ¼ i¼0 n¼0I1 N1XX
i¼0 n¼0;
2

ðSr0 ðri ;  j ; nÞÞ

ð4:1Þ

30 60 90 120 150 180[deg]

Fig. 11. (Color online) Distribution of MSEs fð j Þg using diﬀerent sizesof correlation kernels. (Red line) proposed optimal kernel size (Wl0  Wd0 ).
(Blue and green lines) conventional kernel sizes.

where I and N are the number of layers and the number offrames, respectively. Figure 11 shows the distribution ofMSEs, fð j Þg, estimated using diﬀerent correlation kernel
sizes7,21) of (4:0  4:0) mm2 and (10  10) mm2 and proposed optimal kernel sizes corresponding to the sizes of(18:2  16:9) mm2 and (10:4  9:7) mm2 , which were set atthe boundary and inside the silicone rubber, respectively.
The strain rates in the radial direction estimated with theproposed correlation kernel size yielded the least MSE. On
the other hand, the strain rate estimated using the previously
examined kernel sizes, (4:0  4:0) and (10  10) mm2 , weremarkedly diﬀerent from the actual one. As shown in Fig. 11,
MSEs fð j Þg became smaller at around f j g of 0 and 180 .
In contrast, MSEs fð j Þg became larger at around f j g of 90
because the component of the velocity in the lateral direction
was larger than the axial one at around f j g of 90 , andhence, the accuracy of the estimation of the velocity in thelateral direction was low. Compared with conventional
kernel sizes, this result shows that the proposed correlation
kernel size realizes more accurate measurement of the strain
rate at the expense of the spatial resolution.
However, as shown in Fig. 11, the minimum MSE in
the strain rate was about 20% because the instantaneous
displacement (velocity  1=FR) of the silicone tube during
two frames was very small. The maximum instantaneous
displacement was 12 m. The sampling intervals l and d ofthe correlation function in the lateral and axial directions
after interpolation were both 2.2 m at the depth d of 40 mm.
Therefore, even if the error between estimated and actual
displacement is minimum, corresponding to one point of
the sampled correlation function, the error becomes about
20% [= (2.2 m)/(12 m)]. In future work, it is necessary
to improve the spatial resolution of the ultrasound system
by narrowing PSF, because the size of PSF, ðlðdÞ dðdÞÞ ¼ ð2:48  1:96Þ mm2 , at the depth d of 40 mm wasmuch larger than the instantaneous displacement of thesilicone tube.

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(a)

(a)

RV

LV
APM
PPM

O

(b)

(IVS)
θ=20 deg

10 mm

(1)
RV
(2)

(1)
(4)

(5)

(1’)
(4’)

8 mm

(2)
(2’)
(5’)
(5)

(4)

(3)
(6)

brachioradial muscle

(3)
(3’)
(6’)
(6)

(b) 70Δ d (d )

PPM
(1)

O’
APM
(2)

(4)
(5)

60

Δ l (d )

(PW)
(1)
8 mm

(2)

(3)

(4)

(6)

θ=20 deg

(5)

50

(3)
SNR [dB]

LV(6)axis view. (b) 2D velocities and strain rate are estimated at the yellow points
on the cross-sectional image of IVS and PW. The intervals of the points
were set to be equivalent to the size of the point spread function(lðdÞ  dðdÞ).

5.1 Acquisition of RF signals in interventricular septum

Figure 12 shows a typical cross-sectional image (short-axis
view of left ventricle) of the heart obtained from a healthy
22-year-old male. Figure 12(a) shows the RF acquisitionarea. In the acquisition of RF signals, the ultrasonic beamscanned 112 diﬀerent directions densely to maintain a highspatial resolution. The frame rate was set at 860 Hz. As
shown in Fig. 12(b), the instantaneous velocities and strain
rates of the regions in the interventricular septum (IVS) andposterior wall (PW) of the left ventricle were estimated by
2D tracking. The yellow points in this cross-sectional image
show the positions fðl; dÞg of the points of interest, whichwere set at intervals of the PSF (lðdÞ  dðdÞ).
5.2 Calculation of SNR in in vivo experiment

In this study, the RF data were acquired using the 3.75 MHz
sector-type probe of ultrasonic diagnostic equipment, which
was the same as that used in the basic experiment so that thefeatures of speckle size were not changed. The correlation
kernel sizes fðWl0 ðdÞ  Wd ðdÞÞg at the depths of IVS and PW
determined in the basic experiment were also used in thein vivo experiment. To determine the optimal kernel sizes
from the data in Fig. 7, SNR in the in vivo experiment wasestimated.
"
PPSF ði; j; nÞ ¼ log1030
20Fig. 12. (Color online) (a) Acquisition area of RF signals in the short-

5. In vivo Experimental Results

40

10
0

3

4

5
6
7
8
amplitude of RF signal (log scale)

910Fig. 13. (Color online) (a) Cross-sectional image of the brachioradialmuscle. (b) Relationship between amplitude of RF signals and SNR.

The heart wall continuously contracts and relaxes during
a cardiac cycle. Therefore, it is diﬃcult to calculate SNR
directly in in vivo measurements. The SNRs in in vivo
measurements were estimated by measuring another static
muscle (brachioradial muscle). The relationship between
SNRs and the power of the RF echo is obtained by themeasurement of brachioradial muscle. As shown in Fig. 13,
the SNR was calculated as the ratio of the power of the RFecho from the static brachioradial muscle (S), which wasaveraged for frames, and the variance 0 ðl; dÞ of the RF
echoes (N), as in eq. (3.4). Then, the regression line
describing the relationship between the SNR and the powerof the echo is determined by the least-squares method. Using
the determined regression line, the SNR in the in vivo
measurement of the heart is estimated from the power of anecho from the heart wall, assuming that the relationshipbetween the SNR and the power of the echo is the same in
the measurements of brachioradial and heart muscles. The
power PPSF ði; j; nÞ of the RF echo rf n ðl; dÞ ¼ rf n ði  l; j  dÞ
in the region of the PSF (lðdÞ  dðdÞ) in an in vivo
measurement is given by

#
B=2
K=2
1 X X
frf n ðði þ b þ bmn Þ  l; ð j þ k þ bkn Þ  dÞg2 ;M b¼B=2 k¼K=2

ðM ¼ ðB þ 1ÞðK þ 1Þ; l ¼ ðB þ 1Þ  l; d ¼ ðK þ 1Þ  dÞ;
07GF06-8

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ð5:2Þ

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(a)
ECG 0
PCG 0

45

(b)

40
SNR [dB]

35
30
25
20
15

IVS

10
45
40

PW

SNR [dB]

35
30
25
20
15

(c)

10
−0.2

0

0.2

0.4

0.6

0.8

1.0

1.2

time [s]Fig. 14. (Color online) Calculation of the average SNR in a cardiac cycle in in vivo measurement. The red region is composed of many lines that show
standard deviations in the respective frames. (a) Electrocardiogram and phonocardiogram. (b) IVS. (c) PW.

(1)

where B and K are the numbers of beams and sampling
points in the PSF (lðdÞ  dðdÞ), respectively. The bmnand bkn are the shifts in the lateral and axial directions,
respectively, which maximize the interpolated normalized
cross-correlation function. As shown in Fig. 14, theestimated SNRs in the in vivo measurement were more thanbl ðdÞ  W
bd ðdÞÞg of
20 dB. This result shows that the sizes fðW
the correlation kernel determined by the basic experimentwere also used in the in vivo measurement, i.e., they wereseven times [opt ¼ 1:6 (IVS), 1.8 (PW)] and four times(opt ¼ 1:0) the size of the PSF (lðdÞ  dðdÞ) around theboundary and inside the heart wall, respectively.

0(2)

0.2 0.4 0.6 0.8 1.0(a) (1)times [s]

(2)

10 mm

10 mm
0.38 s

5.3 Comparison of previous and proposed correlation
kernels

As shown in Figs. 7 and 8, the correlation kernel size at theboundary must be set larger than that inside. However, in theprevious study, the characteristics of echoes, i.e., specular
reﬂection and speckle-like echoes, was not considered.
Therefore, 2D displacements at the boundary of the heart
wall estimated using the previous22) and proposed correlation kernel sizes were compared.
The yellow points in Figs. 15(a) and 15(b) show the
instantaneous displacement obtained using the previous andproposed correlation kernels. The previous and proposed
optimal kernel sizes (Wl0 ðdÞ  Wd0 ðdÞ) at the boundary of
the heart wall correspond to four and seven times the size of
the PSF (lðdÞ  dðdÞ). Figures 15(a-1) and 15(b-1) show
the same image, which is the ﬁrst frame at the beginning of
the R-wave. As shown in Figs. 15(a-2) and 15(b-2), thedisplacement in the axial direction was similar. However,(b) (1)

(2)

10 mm

10 mm
0.38 sFig. 15. (Color online) Results obtained using previous (a) and proposed(b) correlation kernels. The yellow points show the instantaneous positionsof points of interest.

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(1)

(2)

(3)

(4)

(5)

10 mm

10 mm

10 mm

10 mm

10 mm

2 ms

2 ms

26 ms
after R−wave

2 msincrease inthickness
1.0

2 ms

propagation
position
0

(7)

(6)

(8)

(9)

10 mm

10 mm

10 mm

2 ms

2 ms

2 ms

(b)

10 mm

10 mm

2 ms

2 ms

(1) (2) (3)(4)(5)(6)(7)(8) (9) (10)(1) (2) (3)(4)(5)(6)(7)(8) (9) (10)

R
ECG 0

Q

R
ECG 0

S

40
0
−40
−80

(1)−(1’)(4)−(4’)3.0(2)−(2’)(5)−(5’)2.0(3)−(3’)(6)−(6’)anterior
strain rate in theradial direction [mm/s]

velocity in theradial direction [mm/s]

IVS

(1)

(4)

(2)

(5)

(3)

(6)

posterior

1.0
0
−1.0
−2.0
IVS
−3.0
3.0

PW

anterior

40
0
(1)

(4)

(2)

(5)

(3)

(6)

−100

posterior−50

0time from R−wave [ms]50

strain rate in theradial direction [mm/s]

80
velocity in theradial direction [mm/s]

first

PCG 0

80

−80

S

Q

first

PCG 0

−40

−1.0
[(mm/s)/mm]decrease inthickness

(10)

2.0
1.0

(4)−(4’)(2)−(2’)(5)−(5’)(3)−(3’)(6)−(6’)

0
−1.0
−2.0
PW
−3.0

100

(1)−(1’)

−100

−50

0

50

100

time from R−wave [ms]Fig. 16. (Color online) (a) Distributions of the strain rate in the radial direction around the R-wave. The cross-sectional images of the short-axis view of
LV (1)–(10) were obtained every 2 ms. (b) Velocities in the radial direction in the IVS and PW obtained by 2D tracking.

the displacement in the lateral direction was diﬀerent, and it
was found that two yellow points in the interventricular
septum overlapped, as shown in Fig. 15(a-2). In the present
study, the characteristics of echoes were considered so that
2D motion could be estimated accurately at any position.
The previous correlation kernel ( ¼ 1:1) could not provide
an accurate estimation of the displacement at the boundaryof the heart wall (p0 ). On the other hand, the displacementsin the lateral direction estimated using the proposedcorrelation kernel were in good agreement with typicalheart wall motion.
5.4 Estimation of myocardial strain rate in radial direction

Figures 16(a) and 16(b) show the distributions of strain rate
Sr ðri ;  j ; nÞ and velocity vr ðri ;  j ; nÞ in the radial direction of
the IVS and PW of the left ventricle around the R-wave.
Low-pass ﬁltering was applied to strain rate waveforms inthe direction of the frame at a cutoﬀ frequency of 86 Hz,
because the eﬀective components in the heart wall vibrations
are known to be measured at frequencies less than 90 Hz.5)

Figures 16(a-1)–16(a-10) show the cross-sectional imagesof the short-axis view of the left ventricle (LV) obtained atintervals of 2 ms. Blue indicates thickening layer caused bymyocardial contraction, and red indicates a thinning layer
caused by myocardial relaxation. Roughly, it was found that
IVS and PW moved complicately around the R-wave after
the R-wave. These motions could not be observed byconventional speckle tracking because of the low frame rateof less than 50 Hz. At the beginning of the ﬁrst heart sound,the myocardial contraction was found to begin at around the
middle of the IVS and propagate along the IVS even during
a very short period of 20 ms.
Figures 17(a) and 17(b) show the distributions of strain
rates fSr ðri ;  j ; nÞg and velocities fvr ðri ;  j ; nÞg at every 2 ms
in the radial direction between the end of the ejection period
and the start of the second heart sound. The white curves inFig. 17(a) indicate the manually identiﬁed positions where
the polarities of the strain rates change. Roughly, in theejection period, the LV contraction is associated with the
downward displacement (to the posterior wall) in the IVS

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(1)

(2)

(3)

(4)

(5)

10 mm
330 msafter R−wave

10 mm

10 mm

10 mm

10 mm

2 ms

2 ms

2 ms

2 ms

increase inthickness
1.0

0

(6)

(7)

(8)

(9)

(10)

10 mm

10 mm

10 mm

10 mm

10 mm

2 ms

2 ms

2 ms

2 ms

2 ms

(b)

−1.0
[(mm/s)/mm]decrease inthickness(1) (2) (3)(4)(5)(6)(7)(8) (9) (10)(1) (2) (3)(4)(5)(6)(7)(8) (9) (10)

T

T

ECG 0

ECG 0

PCG 0

PCG 0
secondsecondvelocity in theradial direction [mm/s]

80

(1)

(4)

(2)

(5)

(3)

40

anterior

(6)

0
−40
posterior

IVS

strain rate in theradial direction [mm/s/mm]

3.0

1.0

(1)−(1’)(4)−(4’)(2)−(2’)(5)−(5’)(3)−(3’)(6)−(6’)(1)−(1’)(4)−(4’)(2)−(2’)(5)−(5’)

0
−1.0
−2.0
IVS
−3.0
3.0

PW

anterior

40
0
−40
−80
250

(1)

(4)

(2)

(5)

(3)

(6)
300

posterior
350
400
time from R−wave [ms]

450

500

strain rate in theradial direction [mm/s/mm]−80
80
velocity in theradial direction [mm/s]

2.0

PW
2.0
1.0
0
−1.0
−2.0
−3.0

(3)−(3’)250(6)−(6’)
300

350
400
time from R−wave [ms]

450

500

Fig. 17. (Color online) (a) Distributions of the strain rate in the radial direction around the second heart sound. The cross-sectional images of the short-axis
view of left ventricle (LV) (1)–(10) were obtained every 2 ms. (b) Velocities in the radial direction in the IVS and PW obtained by 2D tracking.

and the upward displacement in the PW. As shown in
Figs. 17(a-1)–17(a-10), the transition from contraction torelaxation was not spatially uniform, and propagated in theclockwise direction.
In this study, the velocities and strain rates in IVS and PW
around the R-wave and the second heart sound could beobserved at a high temporal resolution by the proposed
method. These results indicate the potential of this method in
the high-accuracy estimation of 2D velocities and strain
rates and in detailed analyses of the physiological functionof the myocardium.
6. Discussion and Conclusion

In this study, the optimal kernel sizes were determined ina phantom experiment with several SNRs and, then, themyocardial strain rate was estimated so that the myocardial
function can be measured at a high frame rate. In a basic
experiment of the determination of the optimal kernel size,
the optimal sizes of the correlation kernel at depths fdg of20, 40, 60, and 80 mm were similar: in particular, SNRs
were more than 15 dB. Moreover, it was found that thekernel size at the boundary must be set larger than that inside
the wall. The optimal kernel sizes were seven times and fourtimes the size of the PSF (lðdÞ  dðdÞ) for the boundary
region and inside the silicone rubber, respectively. To
compare the optimal kernel size, which was determined in
the phantom experiment, with other sizes, the radial strain
rates estimated using diﬀerent kernel sizes were examinedusing the normalized MSE of the estimated strain rate fromthe actual one obtained by the 1D phase-sensitive method.
Compared with conventional kernel sizes, the proposedcorrelation kernel realized a more accurate measurement of
the strain rate. However, even if the error between estimatedand actual value is the minimum, corresponding to one
sampling point, a 20% error of strain rate occurs [= (2.2 mof sample spacing)/(12 m of maximum instantaneous
displacement)]. In future work, it is necessary to improve
the spatial resolution and PSF should be made more narrow

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Y. Honjo et al.

by wringing the ultrasonic beam because the PSF, ðl dÞ ¼ ð2:48  1:96Þ mm2 , at the depth d of 40 mm, wasmuch larger than the instantaneous displacement of thesilicone tube.
In the in vivo measurement, the regional instantaneous
velocity and strain rate in the radial direction of the heart
wall were measured at an extremely high temporal
resolution (frame rate of about 860 Hz) without signiﬁcant
increase in the lateral spacing of scan lines (beam intervals ¼ 0:375 ). The estimated velocities of IVS and PW
corresponded well with typical heart wall motion. Furthermore, the regional strain rate in the radial directions at IVS
and PW around the R-wave and the second heart sound wereanalyzed in detail. Although the accuracy in estimation ofstrain rates should be further improved in future work toquantitatively assess the propagation speed of myocardial
contraction/relaxation, the present results indicate thepotential of this method in the high-accuracy estimation ofstrain rates and in detailed analyses of the transition incontraction and relaxation, which has not been measured
by conventional tracking methods with a low temporal
resolution.

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Acknowledgment

This work was supported by a Grant-in-Aid for JSPS
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