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REGULAR PAPER

Optimization of Correlation Kernel Size for Accurate Estimation

of Myocardial Contraction and Relaxation

Yasunori Honjo, Hideyuki Hasegawa, and Hiroshi Kanai

Jpn. J. Appl. Phys. 51 (2012) 07GF06

Reprinted from

# 2012 The Japan Society of Applied Physics

http://dx.doi.org/10.1143/JJAP.51.07GF06

http://jjap.jsap.jp/



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Optimization of Correlation Kernel Size for Accurate Estimation

of Myocardial Contraction and Relaxation

Yasunori Honjo1, Hideyuki Hasegawa1;2, and Hiroshi Kanai1;2�

1Graduate School of Biomedical Engineering, Tohoku University, Sendai 980-8579, Japan2Graduate 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 200Hz, corresponding to the sampling period of 5ms. However,

myocardial function during short periods, as obtained by these conventional speckle tracking methods, remains unclear owing to low temporal and

spatial resolutions of these methods. Moreover, an important parameter, the optimal kernel size, has not been thoroughly investigated. In our

previous 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 cross-

correlation between RF echoes at a high frame rate of 860Hz, corresponding to a temporal resolution of 1.1ms. 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 the

evaluation 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 high

frame rate. In a basic experiment, the optimal kernel sizes at depths of 20, 40, 60, and 80mm yielded similar results: in particular, SNR was more

than 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 the

correlation 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 860Hz). In this study, transition

in 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 beuseful in estimating myocardial function, such like contrac-tion and relaxation.1–5) In recent years, there have beenmany studies on the measurement of two-dimensional (2D)myocardial strain or strain rate obtained by speckle tracking,which is direct tracking of backscattered echoes usingregional pattern-matching techniques.6–9) However, theframe rate (FR) in conventional speckle tracking methodsis limited to about 200Hz, corresponding to the samplingperiod of 5ms.10–13) Conventional speckle tracking methodsat low temporal resolution may fail in estimating the heartwall motion because it leads to large changes in echopatterns (decorrelation) owing to the motion in the elevationdirection.14) In addition, in the period of one transition inmyocardial contraction/relaxation, the heart wall movesrapidly during a short period of about 10ms.4) Therefore,the detailed analysis of the heart function has been limitedowing to the low temporal resolution of conventionalspeckle tracking methods. The continuous observation ofsuch rapid heart wall motion requires a high temporalresolution of more than 500Hz.1,4)

Moreover, the accuracy of the velocity and displacementestimated by the speckle tracking method depends on animportant parameter, i.e., the size of the correlation kernel.To track backscattered echoes accurately, it is necessary todetermine the optimal kernel size. Kaluzynski et al. reportedthat a silicone tube, with a diameter of 31mm could bemeasured by tracking echoes from scatterers by correlationwith 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 two-dimensionally at velocities up to 2.5m/s, with a correlationkernel size of (1:0� 2:0) mm2. The blood flow in the lateraldirection was imaged in in vitro experiments using acorrelation kernel size of (0:8� 0:8) mm2.16) By in vivomeasurements for estimating the velocity and displacementspontaneously caused by the heart wall motion, the kernelsize was determined to be (10:0� 10:0) mm2 by examiningseveral kernel sizes in comparison with manual observa-tion.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 signal-to-noise ratio (SNR), and the determined optimal kernelsize was applied to the in vivo measurement of 2Ddisplacements of the heart wall by block matching usinga normalized cross-correlation between RF echoes at ahigh frame rate of 860Hz, corresponding to a temporalresolution of 1.1ms.22) However, estimations under lowSNRs and the effects of the difference in echo character-istics, 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 strainrate was estimated so that the myocardial function can bemeasured at a high frame rate.

�E-mail address: [email protected]

Japanese Journal of Applied Physics 51 (2012) 07GF06

07GF06-1 # 2012 The Japan Society of Applied Physics

REGULAR PAPERDOI: 10.1143/JJAP.51.07GF06




2. Principles

2.1 Definition of correlation kernel

In this study, 2D velocities and displacements of the heartwall are estimated by speckle tracking using the normalizedcross-correlation function between RF echoes. Speckle sizedepends on the width of the ultrasonic beam and pulselength.14,23) In this study, the size of the correlation kernel inthe lateral direction WlðdÞ and that in the axial directionWdðdÞ were defined using the �20 dB widths of the lateralprofile of the ultrasonic field �lðdÞ and the envelope of theultrasonic pulse �dðdÞ at several depths, respectively. Asillustrated in Fig. 1, the shape of the correlation kernel was abivariate normal distribution. The variances �2

l ðdÞ and �2dðdÞ

of the normal distribution in the lateral and axial directionswere determined from the size of the point spread function(PSF) ðlateral� axialÞ ¼ ð�lðdÞ ��dðdÞÞ and the variablecoefficient �, respectively.22)

2.2 Calculation of 2D velocities and radial strain rate

The lateral velocity vlðl; d; nÞ and axial velocity vdðl; d; nÞbetween the n-th and (nþ�NF)-th frames at a spatial pointðl; dÞ (l: lateral, d: axial positions) in a heart wall are givenby the determined lateral shift �bmn and axial shift �bkn,respectively, which maximize the interpolated normalizedcross-correlation function as follows:22,24)

vlðl; d; nÞ ¼ �bmn � ð�l=MlÞ�NF ��T

ðmm=sÞ; ð2:1Þ

vdðl; d; nÞ ¼ �bkn � ð�d=MdÞ�NF ��T

ðmm=sÞ; ð2:2Þwhere �T and �l are the time interval between consecutiveframes and the lateral spacing �l ¼ 2d � sinð��=2Þ (��:angular interval) of beams at depth d, respectively, and�NF 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 displace-ment.22) The Ml and Md are the interpolation factors in thelateral and axial directions, respectively.

We introduced the strain rate to evaluate how thethickness of each layer changes with respect to time. Asshown in Fig. 2(a), the velocity in the radial direction,vrðri; � j; nÞ, which was calculated from the velocity in thelateral direction, vlðl; d; nÞ, and that in the axial direction,vdðl; d; nÞ, is given by

vrðri; �j; nÞ ¼ vlðl; d; nÞ � sin½�j þ �ðlÞ�þ vdðl; d; nÞ � cos½�j þ �ðlÞ� ðmm=sÞ; ð2:3Þ

where �j and �ðlÞ are =�OO0A and =�O0OA in Fig. 2,respectively. As shown in Fig. 2(b), strain rate in the radialdirection, Srði; �j; nÞ, between the i-th and (iþ 1)-th layer isgiven by

Srðri; �j; nÞ ¼ vrðriþ1; �j; nÞ � vrðri; �j; nÞR½ðmm=sÞ=mm�; ð2:4Þ

R ¼ riþ1 � ri; ð2:5Þwhere i and j are the numbers of layers and directions,respectively. The numerator shows the difference betweenvelocities at positions ðriþ1; �jÞ and ðri; �jÞ. The denominatorR shows the initial thickness of the layer for the normal-ization of the velocity difference. After the strain rates wereestimated at each radial position, they were superimposed onthe 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 fieldand the envelope of an ultrasonic echo obtained fromexperiments using a fine nylon wire (diameter: 31 �m)placed at depths of 20, 40, 60, and 80mm. RF data wereacquired using a 3.75MHz 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

σl

σd

d( )

d( )

d

d( )σ2 l d( )

Wl d( )

d( )

d( )

σl2

beam profile correlation kernel W

σ2

σ2

d

d

lateral

axial

W

d( )Δd

d( )lΔd( )

Fig. 1. (Color online) Schematic of correlation kernel (WlðdÞ �WdðdÞ).

nl( , ; )dlv

nl( , ; )ddv

θj

θ( )l

π/2− θj

θj

vr ri θ( , ; )n

(a)O

O’

A

θ( )+( )l

lθ( )+A

j

i−th

θj+1

θj

vr r i θj+1( , ; )nvr θj+1r i( , ; )+1 n

i

(b)O

O’

A( +1)−th

Fig. 2. (Color online) Schematic of calculation of radial strain rate.

Y. Honjo et al.Jpn. J. Appl. Phys. 51 (2012) 07GF06



6020Hz using a 3.75MHz 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 15MHz.Figure 3 shows the size of PSF (�lðdÞ ��dðdÞ) measured atdifferent depths. The error bars show the standard deviationfor 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 optimumkernel size (Wl0ðdÞ �Wd0ðdÞ). The phantom was madefrom silicone rubber (Momentive Performance MaterialsTSE3503) 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 rubberat 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 1.5mm from the upper surface of thephantom, where the depths fdg were set at 20, 40, 60, and80mm. The phantom was a silicone rubber plate (soundspeed c0: 990m/s) with a thickness of 15mm. The motionvelocity of the silicone rubber was controlled by anautomatic X–Y stage. The phantom was moved two-dimensionally. Both the lateral velocity vl0ðl; d; nÞ and axialvelocity vd0ðl; d; nÞ, which were set at a constant speed of

5mm/s, were smaller than those of the heart wall.26)

Therefore, the 2D displacements, xlðl; d; nÞ and xdðl; d; nÞ,of the silicone rubber between two frames �NF wererespectively set to be similar to those of the heart wall byincreasing 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 inthe in vivo measurement. The white noise �wðl; dÞ wascreated by MATLAB (MathWorks). SNR of an echo signalfrom the phantom, which contains noise �0ðl; dÞ originatingfrom the ultrasonic equipment and �wðl; dÞ, is given by

SNR ¼ 10 log10ðEðl;dÞ½ðEt½rfðl; dÞ�Þ2�Þ� 10 log10ðEðl;dÞ½ð�0ðl; dÞ þ � � �wðl; dÞÞ2�Þ

¼ 10 log10ðEðl;dÞ½ðEt½rfðl; dÞ�Þ2�Þ� 10 log10ðEðl;dÞ½�0ðl; dÞ2 þ ð� � �wðl; dÞÞ2�Þ; ð3:1Þ

where Et½�� and Eðl;dÞ½�� are the time and space (lateral andaxial directions) average operations, respectively, and � isthe amplitude coefficient controlling SNR. There is nocorrelation between the noise �0ðl; dÞ caused by theultrasonic equipment and white noise �wðl; dÞ, i.e.,Eðl;dÞ½�0ðl; dÞ � �wðl; dÞ� ¼ 0. The SNRs in the basic experi-ment in the regions of pj at depth d in Fig. 4(b) are given by

SNRðpj; dÞ ¼ 10 log101

A

Xl5l¼l0

1

N

XN�1

n¼0

ðrfnðl; dÞÞ( )2

24 35� 10 log10

"1

A

Xl5l¼l0

(ðEt½�0ðl; dÞ2�Þ

þ

�

N

XN�1

n¼0

ð�wðl; dÞÞ2!)#

; ð3:2Þ

ðA ¼ l5 � l0Þ; ð3:3Þwhere 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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4.0

10 20 30 40 50 60 70 80 90

leng

th [m

m]

depth [mm]d

Δd

Δl

( )d

( )d

pulse length

beam width

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 80mm.

controllerstage

lv vd

ultrasonicprobe

= 40 mmd

automatic X−Y stage water tank

water

silicone rubber

axial

lateral(a) (b)

position

...

d0l ( )5l d ( )

p 0

p 4

10 mm

Fig. 4. (Color online) Schematic of phantom experiment to determine the

optimal kernel sizes (Wl0ðdÞ �Wd0ðdÞ).

d 0 ( ) d m ( )

0

10

20

30

40

50

60

10 20 30 40 50 60 70 80 90 100 110depth [mm]

varia

tion

of th

e R

F s

igna

ls in

the

wat

er

Fig. 5. Variance of RF signals in water.




Fig. 5, the noise �0ðl; dÞ was estimated as the variance of theoutput signal from the equipment, and was measured withoutany objects in water, as follows:

�0ðl; dÞ2 ¼1

N

XN�1

n¼0

ðrfnðl; dÞ � rfðl; dÞÞ2; ð3:4Þ

rfðl; dÞ ¼ 1

N

XN�1

n¼0

rfnðl; dÞ; ð3:5Þ

where N is the number of frames used in the calculation ofthe variance.

3.4 Determination of the optimal size of correlation kernel

In this study, the sizes WlðdÞ and WdðdÞ of the correlation

kernel in the lateral and axial directions were defined by�2�l ð¼ �2� ��lðdÞÞ and �2�d ð¼ �2� ��dðdÞÞ, respec-tively. To determine the optimal value �optðd; SNRðpj; dÞÞ of� giving the optimal kernel size (Wl0ðdÞ �Wd0ðdÞ), the 2Ddisplacements of the phantom estimated using differentvalues of f�g and SNRs were examined using the root-mean-squared (RMS) error of the estimated displacement from theactual displacement. The RMS errors in the estimated 2Ddisplacements, f�ð�; SNRðpj; dÞ; dÞg, which were evaluatedfor each value of � and SNRðpj; dÞ at depth d, werecalculated from the lateral displacement xlðli; d; nÞ and axialdisplacement xdðli; d; nÞ estimated for each position ðli; dÞand the actual lateral displacement xl0ðnÞ and axialdisplacement xd0ðnÞ as follows:

�ð�; SNRðpj; dÞ; dÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

I � NX5i¼0

XN�1

n¼0

ð�xlðli; d; nÞ2 þ�xdðli; d; nÞ2Þvuut ; ð3:6Þ

�xlðli; d; nÞ ¼ xlðli; d; nÞ � xl0ðnÞ; ð3:7Þ�xdðli; d; nÞ ¼ xdðli; d; nÞ � xd0ðnÞ; ð3:8Þ

where N and I ð¼ 6Þ are the numbers of averaged framesand measurement points, respectively. In this experiment,the displacements at the position fðli; dÞg in the area of 2Dtracking were measured using normalized cross-correlationfunctions with different values of � ¼ 0:3� 2:0 at the fivepositions (p0–p4) at each depth of 20, 40, 60, and 80mm.The original sampled normalized cross-correlation functionwas interpolated with Ml ¼ 120 in the lateral direction and

Md ¼ 15 in the axial direction to obtain the cross-correlationfunction with similar resolutions in the lateral and axialdirections, corresponding to 2.2 �m.

Figure 6 shows the distributions of RMS errors,f�ð�; SNRðpj; dÞ; dÞg, estimated using different values off�g of the correlation kernels at the five positions [(a)p0 = 1.5mm � (e) p4 = 7.5mm]. As shown in Fig. 6,when the size of the correlation kernel is set to be small,

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(e)

(a) (b) (c)

(d) = 7.5 mm

= 3.0 mm

= 6.0 mm

= 1.5 mm = 4.5 mmRMS error in thedisplacement [dBm]

RMS error in thedisplacement [dBm]




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p0 p1

p3 p4

p2

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

SNR.




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 havesimilar echo patterns, induce misestimation of displacement.When the size of the kernel is set to be too large, the peak ofthe correlation coefficient is degraded because the specklepattern of the central position in the correlation kernel isdifferent from that of the edge in the correlation kernel,owing to deflection of the ultrasonic beam. In addition, itis rare that the echo patterns in the two frames exactlycoincide, which degrades the accuracy of the estimationand the spatial resolution.27–29) However, the displacementestimation is less susceptible to noise because the echopattern in the correlation kernel is more unique.

Figures 7(a)–7(d) show the relationships among theSNRs fSNRðpj; dÞg ( j ¼ 0; 1; 2; 3; 4), the values off�optðd;SNRðpj; dÞÞg, which are the minimum RMS errors,and the positions fpjg at each depth d from the upper surface.As shown in Fig. 7, the distributions of the values off�optðd;SNRðpj; dÞÞg at the positions of a boundary of thesilicone rubber (p0 and p1) were different from those insidethe silicone rubber (p2–p4). As shown in Fig. 4(b), thebrightness is high and relatively uniform in the lateraldirection at the boundary of the silicone rubber. In such

cases, there might be several regions in the next framethat have echo patterns similar to those in the correlationkernel 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 functionsobtained for points of interest at the boundary (p0) andinside (p4) the silicone rubber, respectively. As shown inFig. 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 ofthe silicone rubber (at p0). On the other hand, as shown inFig. 8(b-1), at � ’ 1:0, the profile around the peak of thenormalized cross-correlation function in the lateral directionbecame narrowest inside the silicone rubber (at p4). Theprofiles around the peaks of the normalized cross-correlationfunctions in the axial direction around the boundaryand inside the silicone plate were similar, as shown inFigs. 8(a-2) and 8(b-2), respectively. Therefore, the kernelsize at the boundary must be set larger than that inside.

As shown in Fig. 1, in our study, the correlation kernelwas defined using the size of PSF (�lðdÞ ��dðdÞ), andthe optimal sizes fðWl0ðdÞ �Wd0ðdÞÞg were determinedas �2 � �optðd; SNRðpj; dÞÞð�lðdÞ ��dðdÞÞ. As shown inFigs. 7(a)–7(d), when SNR is larger than 15 dB, the optimal

1

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

80mm).




value of �optðd;SNRðpj; dÞÞ at each depth d (¼ 20, 40, 60,and 80mm) is 1.6–1.8 around the boundary and 1.0 insidethe silicone rubber. From these results, the optimal kernelsizes fðbWl0ðdÞ � bWd0ðdÞÞg were determined to be about seventimes (�2 � �optð�lðdÞ ��dðdÞÞ (�opt ¼ 1:6{1:8)) and fourtimes (�2 � �optð�lðdÞ ��dðdÞÞ ð�opt ¼ 1:0Þ) the size ofPSF (�lðdÞ ��dðdÞ) around the boundary and inside thesilicone rubber, respectively. It was found that the optimalkernel sizes fðWl0ðdÞ �Wd0ðdÞÞg can be determined fromthe size of PSF (�lðdÞ ��dðdÞ), which is determined bythe lateral and axial widths at a �20 dB ultrasonic fieldmeasured in advance.

4. Basic Experiment of Estimation of Radial Strain

Rate

4.1 Experimental system

As illustrated in Fig. 9, in this experiment, the strain rate ofthe phantom owing to the change in the internal pressure,which was applied using a flow pump, was measured. Thehomogeneous cylindrical phantom with a wall thickness of2mm was made from silicone rubber. In this study, the RFdata were acquired using the 3.75MHz sector-type probe ofultrasonic diagnostic equipment (Aloka �-10). All param-eters, sound speed c0, the sampling frequency fs of the RFsignal, frame rate FR, and interpolation factors Ml andMd in

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

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

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

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

α α α α α α α

0

0.2

0.4

0.6

0.8

1.0

−0.02 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14

corr

elat

ion

coef

ficie

nt

shift in the axial direction [mm]

0

0.2

0.4

0.6

0.8

1.0

−0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

corr

elat

ion

coef

ficie

nt

shift in the axial direction [mm]


(1) (2)


0

0.2

0.4

0.6

0.8

1.0

−1.0 −0.5 0 0.5 1.0

corr

elat

ion

coef

ficie

nt

shift in the lateral direction [mm]

= 1.5 mmp0

p3= 6.0 mm


(1)

0.2

0.4

0.6

0.8

1.0

−1.0 −0.5 0 0.5 1.0

corr

elat

ion

coef

ficie

nt

0

shift in the lateral direction [mm]

(b)

(a)

(2)

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

boundaryinside

insideinside

boundary

1

32

45

i−th layer

10 mm

(a)

flow pump

front viewside view

phantom

(b)

phantom

Fig. 9. (Color online) (a) Cross-sectional image of silicone tube at a

depth of 40mm. 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 fSrðri; � j; nÞg.




the lateral and axial direction, were the same as those used inthe basic experiment using the silicone rubber. In this study,to obtain the actual velocity vr0ðri; �j; nÞ and strain rateSr0ðri; �j; nÞ in the phantom’s radial direction, a 1D phase-sensitive method (1D method) was used for the central beam(�j ¼ 0�). The silicone rubber was assumed to be anisotropic medium in the circumferential direction. Therefore,the actual strain rate Sr0ðri; �j; nÞ (�j ¼ 0�) obtained by the1D method and shown in Fig. 10 can be used as referencesfor all directions f�jg and layers frig.

4.2 Experimental results

Some features of speckle patterns, such as the size of aspeckle, depend on the PSF produced by the ultrasonicfield.14,23) In this study, the RF data were acquired using the3.75MHz sector-type probe of ultrasonic diagnostic equip-ment (Aloka �-10), which was the same as that used in thebasic experiment so that the features of speckle size werenot changed. Therefore, the size of a correlation kerneldetermined by the basic experiment was also used in thisexperiment. As illustrated in Fig. 9(a), the correlation kernelat the position of the boundaries (first and fifth layers) andinside (second, third, and fourth layers) the silicone rubberwere 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 ratesestimated using different kernel sizes were examined usingthe normalized mean-squared error (MSE) of the estimatedstrain rate Srðri; � j; nÞ from the actual one Sr0ðri; �j; nÞ. Thenormalized mean-squared error �ð�jÞ was calculated from theradial strain rate Srðri; �j; nÞ at each position ðri; �jÞ andactual Sr0ðri; �j; nÞ as follows:

�ð�jÞ ¼

XI�1

i¼0

XN�1

n¼0

ðSrðri; �j; nÞ � Sr0ðri; �j; nÞÞ2

XI�1

i¼0

XN�1

n¼0

ðSr0ðri; �j; nÞÞ2; ð4:1Þ

where I and N are the number of layers and the number offrames, respectively. Figure 11 shows the distribution ofMSEs, f�ð�jÞg, estimated using different correlation kernelsizes7,21) of (4:0� 4:0) mm2 and (10� 10) mm2 and pro-posed 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. Onthe other hand, the strain rate estimated using the previouslyexamined kernel sizes, (4:0� 4:0) and (10� 10) mm2, weremarkedly different from the actual one. As shown in Fig. 11,MSEs f�ð�jÞg became smaller at around f�jg of 0 and �180�.In contrast, MSEs f�ð�jÞg became larger at around f�jg of 90�because the component of the velocity in the lateral directionwas larger than the axial one at around f�jg of 90�, andhence, the accuracy of the estimation of the velocity in thelateral direction was low. Compared with conventionalkernel sizes, this result shows that the proposed correlationkernel size realizes more accurate measurement of the strainrate at the expense of the spatial resolution.

However, as shown in Fig. 11, the minimum MSE inthe strain rate was about 20% because the instantaneousdisplacement (velocity� 1=FR) of the silicone tube duringtwo frames was very small. The maximum instantaneousdisplacement was 12 �m. The sampling intervals �l and �d ofthe correlation function in the lateral and axial directionsafter interpolation were both 2.2 �m at the depth d of 40mm.Therefore, even if the error between estimated and actualdisplacement is minimum, corresponding to one point ofthe sampled correlation function, the error becomes about20% [= (2.2 �m)/(12 �m)]. In future work, it is necessaryto improve the spatial resolution of the ultrasound systemby narrowing PSF, because the size of PSF, ð�lðdÞ ��dðdÞÞ ¼ ð2:48� 1:96Þmm2, at the depth d of 40mm wasmuch larger than the instantaneous displacement of thesilicone tube.

d 0W0lW ( x ) mm (10.0 x 10.0) mm 19) (4.0 x 4.0) mm

6)

θj( )

ε

θ j

0

20

40

60

80

100

120

140

160

180

200

−180−150−120−90 −60 −30 0 30 60 90 120 150 180

norm

aliz

ed M

SE

[%

]

degree [deg]

Fig. 11. (Color online) Distribution of MSEs f�ð� jÞg using different sizes

of correlation kernels. (Red line) proposed optimal kernel size (Wl0 �Wd0).

(Blue and green lines) conventional kernel sizes.

0.05

−0.05[(m/s)/m]

thicknessdecrease in

thicknessincrease in

0

(b) Trigger pulse

(c) velocity [mm/s]

(d) strain rate [m/s/m]−0.05

0.05

−5.0

5.0

0 0.5 1.0 time [s]

10 mm

0

0

Fig. 10. (Color online) Actual strain rate Sr0ðri; � j; nÞ of silicone tube

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




5. In vivo Experimental Results

5.1 Acquisition of RF signals in interventricular septum

Figure 12 shows a typical cross-sectional image (short-axisview of left ventricle) of the heart obtained from a healthy22-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 860Hz. Asshown in Fig. 12(b), the instantaneous velocities and strainrates of the regions in the interventricular septum (IVS) andposterior wall (PW) of the left ventricle were estimated by2D tracking. The yellow points in this cross-sectional imageshow 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 3.75MHzsector-type probe of ultrasonic diagnostic equipment, whichwas the same as that used in the basic experiment so that thefeatures of speckle size were not changed. The correlationkernel sizes fðWl0ðdÞ �WdðdÞÞg at the depths of IVS and PWdetermined in the basic experiment were also used in thein vivo experiment. To determine the optimal kernel sizesfrom the data in Fig. 7, SNR in the in vivo experiment wasestimated.

The heart wall continuously contracts and relaxes duringa cardiac cycle. Therefore, it is difficult to calculate SNRdirectly in in vivo measurements. The SNRs in in vivomeasurements were estimated by measuring another staticmuscle (brachioradial muscle). The relationship betweenSNRs 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 �0ðl; dÞ of the RFechoes (N), as in eq. (3.4). Then, the regression linedescribing the relationship between the SNR and the powerof the echo is determined by the least-squares method. Usingthe determined regression line, the SNR in the in vivomeasurement 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 inthe measurements of brachioradial and heart muscles. Thepower PPSFði; j; nÞ of the RF echo rfnðl; dÞ ¼ rfnði � �l; j � �dÞin the region of the PSF (�lðdÞ ��dðdÞ) in an in vivomeasurement is given by

PPSFði; j; nÞ ¼ log101

M

XB=2b¼�B=2

XK=2

k¼�K=2

frfnððiþ bþ�bmnÞ � �l; ð jþ k þ�bknÞ � �dÞg2" #; ð5:1Þ

ðM ¼ ðBþ 1ÞðKþ 1Þ;�l ¼ ðBþ 1Þ � �l;�d ¼ ðKþ 1Þ � �dÞ; ð5:2Þ

LV

RV

APM

PPM

(a)

10 mm

8 mm

8 mm

=20 degθ

d( )dΔlΔ d( )

RV

PPM

LV

APM

O’

O(b)

=20 degθ(IVS)

(PW)

(5)

(2)

(1)

(4)(5)

(5)

(2) (3)

(6)

(2)

(1)(4) (6)

(3) (4)

(6)

(3)(1)(1’)(4’)

(3’)

(3)

(6’)(6)

(2)(2’)

(5)(5’)

(4)

(1)

Fig. 12. (Color online) (a) Acquisition area of RF signals in the short-

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).

brachioradial muscle

(a)

0

10

20

30

40

50

60

70

3 4 5 6 7 8 9 10

SN

R [d

B]

amplitude of RF signal (log scale)

(b)

Fig. 13. (Color online) (a) Cross-sectional image of the brachioradial

muscle. (b) Relationship between amplitude of RF signals and SNR.




where B and K are the numbers of beams and samplingpoints in the PSF (�lðdÞ ��dðdÞ), respectively. The �bmn

and �bkn are the shifts in the lateral and axial directions,respectively, which maximize the interpolated normalizedcross-correlation function. As shown in Fig. 14, theestimated SNRs in the in vivo measurement were more than20 dB. This result shows that the sizes fðbWlðdÞ � bWdðdÞÞg ofthe 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.

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., specularreflection and speckle-like echoes, was not considered.Therefore, 2D displacements at the boundary of the heartwall estimated using the previous22) and proposed correla-tion kernel sizes were compared.

The yellow points in Figs. 15(a) and 15(b) show theinstantaneous displacement obtained using the previous andproposed correlation kernels. The previous and proposedoptimal kernel sizes (Wl0ðdÞ �Wd0ðdÞ) at the boundary ofthe heart wall correspond to four and seven times the size ofthe PSF (�lðdÞ ��dðdÞ). Figures 15(a-1) and 15(b-1) showthe same image, which is the first frame at the beginning ofthe R-wave. As shown in Figs. 15(a-2) and 15(b-2), thedisplacement in the axial direction was similar. However,

PCG

ECG

0

0

IVS

PW

(b)

(a)

(c)

10

20

25

30

35

40

15

45S

NR

[dB

]

10

15

20

25

30

35

40

45

−0.2 0 0.2 0.4 0.6 0.8 1.0 1.2time [s]

SN

R [d

B]

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.

0 0.2 0.4 0.6 0.8 1.0 times [s]

(1) (2)

(2)(b) (1)

(2)(1)

10 mm10 mm

10 mm10 mm

(a)

0.38 s

0.38 s

Fig. 15. (Color online) Results obtained using previous (a) and proposed

(b) correlation kernels. The yellow points show the instantaneous positions

of points of interest.




the displacement in the lateral direction was different, and itwas found that two yellow points in the interventricularseptum overlapped, as shown in Fig. 15(a-2). In the presentstudy, the characteristics of echoes were considered so that2D motion could be estimated accurately at any position.The previous correlation kernel (� ¼ 1:1) could not providean 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 rateSrðri; �j; nÞ and velocity vrðri; �j; nÞ in the radial direction ofthe 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 86Hz,because the effective components in the heart wall vibrationsare known to be measured at frequencies less than 90Hz.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 2ms. Blue indicates thickening layer caused bymyocardial contraction, and red indicates a thinning layercaused by myocardial relaxation. Roughly, it was found thatIVS and PW moved complicately around the R-wave afterthe R-wave. These motions could not be observed byconventional speckle tracking because of the low frame rateof less than 50Hz. At the beginning of the first heart sound,the myocardial contraction was found to begin at around themiddle of the IVS and propagate along the IVS even duringa very short period of 20ms.

Figures 17(a) and 17(b) show the distributions of strainrates fSrðri; �j; nÞg and velocities fvrðri; �j; nÞg at every 2msin the radial direction between the end of the ejection periodand the start of the second heart sound. The white curves inFig. 17(a) indicate the manually identified positions wherethe polarities of the strain rates change. Roughly, in theejection period, the LV contraction is associated with thedownward displacement (to the posterior wall) in the IVS

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

26 msafter R−wave

(9)

(4)

(10)

(5)

(8)

(3)

(6)

(1)

propagationposition

2 ms2 ms

2 ms 2 ms 2 ms 2 ms

10 mm 10 mm 10 mm 10 mm 10 mm

10 mm 10 mm 10 mm 10 mm 10 mm

2 ms 2 ms 2 msthicknessdecrease in


(b)

(a)

(7)

(2)

0

1.0

−1.0[(mm/s)/mm]

(1) (4)

(2) (5)

(3) (6)

(3) (6)

(2) (5)

(1) (4)

(6)−(6’)

(5)−(5’)

(4)−(4’)

(3)−(3’)

(2)−(2’)

(1)−(1’)

(6)−(6’)

(5)−(5’)

(4)−(4’)

(3)−(3’)

(2)−(2’)

(1)−(1’)

0 PCG

0 ECG S first

R

Q

posterior

posterior

radi

al d

irect

ion

[mm

/s]

vel

ocity

in th

era

dial

dire

ctio

n [m

m/s

] v

eloc

ity in

the

anterior

−80

−40

0

80

40

anterior

−100 −50 50 100

−80

−40

0

40

80

time from R−wave [ms]

0

PW

IVS

radi

al d

irect

ion

[mm

/s]

str

ain

rate

in th

e

0 ECG S first

R

Q

PCG 0

−3.0

−2.0

−1.0

0

1.0

2.0

3.0

radi

al d

irect

ion

[mm

/s]

str

ain

rate

in th

e

PW−3.0

−2.0

−1.0

0

2.0

3.0

1.0

−50 50 100−100 0


IVS

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 2ms. (b) Velocities in the radial direction in the IVS and PW obtained by 2D tracking.




and the upward displacement in the PW. As shown inFigs. 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 PWaround the R-wave and the second heart sound could beobserved at a high temporal resolution by the proposedmethod. These results indicate the potential of this method inthe high-accuracy estimation of 2D velocities and strainrates 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 myocardialfunction can be measured at a high frame rate. In a basicexperiment of the determination of the optimal kernel size,the optimal sizes of the correlation kernel at depths fdg of

20, 40, 60, and 80mm were similar: in particular, SNRswere more than 15 dB. Moreover, it was found that thekernel size at the boundary must be set larger than that insidethe wall. The optimal kernel sizes were seven times and fourtimes the size of the PSF (�lðdÞ ��dðdÞ) for the boundaryregion and inside the silicone rubber, respectively. Tocompare the optimal kernel size, which was determined inthe phantom experiment, with other sizes, the radial strainrates 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 ofthe strain rate. However, even if the error between estimatedand actual value is the minimum, corresponding to onesampling point, a 20% error of strain rate occurs [= (2.2 �mof sample spacing)/(12 �m of maximum instantaneousdisplacement)]. In future work, it is necessary to improvethe spatial resolution and PSF should be made more narrow

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

10 mm 10 mm10 mm330 msafter R−wave

10 mm

2 ms 2 ms 2 ms 2 ms

10 mm

2 ms2 ms

10 mm 10 mm10 mm 10 mm 10 mm thicknessdecrease in


(b)

(a) (1) (2) (3) (4) (5)

2 ms 2 ms

(6) (7) (8) (9) (10)

2 ms

0

1.0

−1.0[(mm/s)/mm]

(6)−(6’)

(5)−(5’)

(4)−(4’)

(3)−(3’)

(2)−(2’)

(1)−(1’)

(6)−(6’)

(5)−(5’)

(4)−(4’)

(3)−(3’)

(2)−(2’)

(1)−(1’)

(2) (5)

(1) (4)

(3) (6)

(2) (5)

(1) (4)

(3) (6)

T

0

0

PCG

ECG

second

−3.0

−2.0

−1.0

0

1.0

2.0

3.0

250 350 400 450 500 300

−3.0

−2.0

−1.0

0

1.0

2.0

3.0

radi

al d

irect

ion

[mm

/s/m

m]

str

ain

rate

in th

era

dial

dire

ctio

n [m

m/s

/mm

] s

trai

n ra

te in

the

posterior

posterior

anterior

anteriorPW

radi

al d

irect

ion

[mm

/s]

vel

ocity

in th

e

IVS

radi

al d

irect

ion

[mm

/s]

vel

ocity

in th

e

T

0

0

PCG

ECG

second


−80

−40

0

40

80

250 350 400 450 500 300

−80

−40

0

40

80

IVS

PW


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 2ms. (b) Velocities in the radial direction in the IVS and PW obtained by 2D tracking.




by wringing the ultrasonic beam because the PSF, ð�l��dÞ ¼ ð2:48� 1:96Þmm2, at the depth d of 40mm, wasmuch larger than the instantaneous displacement of thesilicone tube.

In the in vivo measurement, the regional instantaneousvelocity and strain rate in the radial direction of the heartwall were measured at an extremely high temporalresolution (frame rate of about 860Hz) without significantincrease in the lateral spacing of scan lines (beam interval�s ¼ 0:375�). The estimated velocities of IVS and PWcorresponded well with typical heart wall motion. Further-more, the regional strain rate in the radial directions at IVSand 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 myocardialcontraction/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 measuredby conventional tracking methods with a low temporalresolution.

Acknowledgment

This work was supported by a Grant-in-Aid for JSPSFellows.

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