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A Model for Measured Traveling Waves at End-Diastolein Human Heart Wall by Ultrasonic Imaging Method
Naoaki Bekki, Seine A. Shintani, Shin’ichi Ishiwata, and Hiroshi Kanai
J. Phys. Soc. Jpn. 85, 044802 (2016)© 2016 The Physical Society of Japan

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Journal of the Physical Society of Japan 85, 044802 (2016)
http://doi.org/10.7566/JPSJ.85.044802

A Model for Measured Traveling Waves at End-Diastole in Human Heart Wallby Ultrasonic Imaging Method
Naoaki Bekki1+, Seine A. Shintani2, Shin’ichi Ishiwata1, and Hiroshi Kanai3
1

Department of Physics, Faculty of Science and Engineering, Waseda University, Shinjuku, Tokyo 169-8555, Japan2
Department of Physics, The University of Tokyo, Bunkyo, Tokyo 113-0033, Japan3
Department of Electronic Engineering, Tohoku University, Sendai 980-8579, Japan(Received December 24, 2015; accepted January 28, 2016; published online March 4, 2016)

We observe traveling waves, measured by the ultrasonic noninvasive imaging method, in a longitudinal beam
direction from the apex to the base side on the interventricular septum (IVS) during the period from the end-diastole tothe beginning of systole for a healthy human heart wall. We present a possible phenomenological model to explain partof one-dimensional cardiac behaviors for the observed traveling waves around the time of R-wave of echocardiography(ECG) in the human heart. Although the observed two-dimensional patterns of traveling waves are extremely complex
and no one knows yet the exact solutions for the traveling homoclinic plane wave in the one-dimensional complex
Ginzburg–Landau equation (CGLE), we numerically ﬁnd that part of the one-dimensional homoclinic dynamics of thephase and amplitude patterns in the observed traveling waves is similar to that of the numerical homoclinic plane-wave
solutions in the CGLE with periodic boundary condition in a certain parameter space. It is suggested that part of the
cardiac dynamics of the traveling waves on the IVS can be qualitatively described by the CGLE model as a paradigm forunderstanding biophysical nonlinear phenomena.

1.

Introduction

It is important to measure the rapid motions of the
myocardium in the human heart wall during systole anddiastole since information on them are useful for the
diagnosis of regional myocardial motility.1) The propagation
speed of mechanically excited traveling waves around theaortic-valve closure (AVC) is higher than that of electrically
excited waves along the cardiac muscle and the Purkinje
ﬁber around the beginning of the ejection period in thehuman heart wall.2) Therefore, a study of the measured
nonlinear waves due to rapid motions in the cardiac muscle
during systole by the ultrasonic imaging method is oneof the most interesting subjects in physics and biophysics
because there are very few experimental investigations
related to biophysical nonlinear phenomena in cardiac
systems.3,4)
A phase tracking method with high temporal and spatial
resolutions has been developed to measure the rapid velocity
in the heart wall by accurately tracking the movement of apoint in the heart wall by the constraint least-squares method
applied to both the phase and amplitude of quadraturedemodulated signals.5) This method that enables the precise
measurement of a target velocity for the myocardium plays
an important role in characterizing myocardial tissue in termsof systolic properties in vivo, which cannot be obtained byconventional echocardiography (ECG), tissue Doppler imaging, computer tomography, or magnetic resonance imaging.2)
The velocity components toward the ultrasonic probe as
waveforms and their instantaneous phases were observed byvisualizing the propagation of the myocardial response ofelectric excitation in the human heart wall during systole,
which may be modeled by the asymmetric zero-order modeof the Lamb wave.2) These components corresponding to thecontraction were generated on the interventricular septum(IVS) at the time of the T-wave of ECG (end-systole), andpropagated slowly in the clockwise direction along the left
ventricle circumferential direction (x-axis) and beam direction (y-axis) of the ultrasonic probe.

Since small-amplitude pulsive waves on the IVS can beapproximated by the Lamb-wave as guided waves6) with
nonstress-free surfaces, taking into account the eﬀect that thewave energy leaks into the surrounding blood, the Lambwave model for traveling pulsive waves on the IVS excitedby the AVC at end-systole has been investigated.2) It has
been shown that the fundamental properties of smallamplitude traveling waves during systole can be explainedby the simple dispersion equation based on the Lamb-wave
model, and that the viscoelastic constant of the myocardium
in the human heart wall has been estimated by the simple
dispersion equation7) without using the nonlinear optimization method.2)
In the case of large-amplitude pulsive waves on the IVS by
the AVC at end-systole, they are nonlinearly modulated inamplitude and phase, and locally generate a phase defect in
the IVS of the human heart wall. Strongly modulated pulsive
waves with the amplitude dip are created by the generated
phase jump, which heteroclinically connects two diﬀerent
patterns back and forth speciﬁed by two asymptotic wavenumbers in the longitudinal direction of the IVS from thebase to the apex.
However, there are very few nonlinear theoretical models
for large-amplitude pulsive waves excited by the rapid
motions of the cardiac muscle in the human heart wall.
Therefore, to explain the nonlinear behaviors of largeamplitude pulsive waves at the time of the T-wave of ECG(end-systole) in the human heart wall, the one-dimensional
complex Ginzburg–Landau equation (CGLE) model has been
applied to large-amplitude pulsive waves measured by theultrasonic noninvasive imaging method.8,9)
The behaviors of modulated propagating pulsive waves at
the time of the T-wave of ECG have been observed by theultrasonic imaging method.10–13) One of the observed phasejumps for one-dimensional traveling waves speciﬁed byðx; tÞ with a ﬁxed y can be described by the Bekki–Nozaki(BN) hole solutions8,9,14,15) with moving sources in the
CGLE,16–26) which is written as

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J. Phys. Soc. Jpn. 85, 044802 (2016)@ ¼  þ ðpi þ ipr Þr2   ðqi  iqr Þjj2 ;@tN. Bekki et al.

ð1Þ

where ðx; y; tÞ denotes a complex scalar function of twodimensional space ðx; yÞ and time t, and the Laplacian r2 
@2 =@x2 þ @2 =@y2 , pi < 0, and  > 0 are assumed.
By analyzing the data of traveling pulsive waves on the
IVS at end-systole in the lateral direction speciﬁed by the xaxis (from the left ventricle to the right ventricle) for a ﬁxed
y-axis, it is found that the BN hole solutions15) in the (1-D)
CGLE heteroclinically connect two diﬀerent patterns speciﬁed by the asymptotic wavenumbers (q~1 ≠ q~2 ≠ 0) near thephase jump of excited waves. A homoclinic hole,24) on theother hand, is deﬁned in the narrow sense as q~1 ¼ q~2  0.
Traveling pulsive waves on the IVS have been measured
by the ultrasonic imaging modality with high spatial andtemporal resolutions for healthy young males.10–12) However,
the one-dimensional dynamics of large-amplitude travelingwaves at the time of the R-wave of ECG (end-diastole) has
not been investigated yet. To understand part of the onedimensional behaviors of the traveling waves in the humanheart wall, therefore, a phenomenological model of explanation is at least needed on the basis of the direct measurementof traveling waves at end-diastole by the ultrasonic imaging
method.
We show that the behaviors of large-amplitude waves in
the longitudinal beam direction (y-axis) at the time of the
R-wave of ECG (end-diastole) are diﬀerent from those ofpulsive waves based on the data that were measured in thelateral beam direction of the ultrasonic probe at the time of
the T-wave of ECG (end-systole),2,8) and that part of the
cardiac dynamics of traveling waves on the IVS at enddiastole can be qualitatively explained by the CGLE model
as a paradigm for understanding biophysical nonlinear
phenomena.
We present here the one-dimensional CGLE model in the
one-dimensional cardiac dynamics for the above waves,which were measured by the ultrasonic imaging method in
the longitudinal beam direction (y-axis) from the apex tothe base around the time of the R-wave of ECG (enddiastole) on the IVS in the human heart wall, as shown in
Fig. 1.
2.

Observation of Traveling Waves in Human HeartWall

By the ultrasonic measurement technique for rapid
myocardial motions in vivo, we observed nonlinearly excitedwaves speciﬁed by ðx; y; tÞ on the IVS around the time of
the R-wave of ECG for a healthy young male. We obtained
two-dimensional patterns of phase ðx; y; tÞ and amplitudejðx; y; tÞj of excited waves on the IVS, and their observation
time corresponded to 99:2  t  99:2 ms during the period
from the T-wave to the R-wave of ECG, for at most 0.2 (s).
Figure 1 shows a typical snapshot of two-dimensional spatial
patterns of phase ðx; yÞ with t ¼ 93:25 ms from the T-waveof ECG in the beam direction from the apex to the basedeﬁned by the y-axis set 1,869 points on the IVS and a lateral(scanning) direction to the beam by the x-axis set about 10
points. We can observe a number of phase defects in the twodimensional spatial phase patterns ðx; yÞ with a ﬁxed timet ¼ 93:25 ms.

Fig. 1. (Color online) Snapshot of two-dimensional spatial pattern of thephase ðx; yÞ with a ﬁxed time (t ¼ 93:25 ms from the T-wave of ECG) in
the 40 Hz component of excited waves on the IVS for a healthy young male,using the ultrasonic measurement technique for myocardial motions in vivo.
Point O denotes the origin of the measured 2-D phase plane transformed
from the polar coordinate system to the Cartesian coordinate system ðx; yÞ.
This spatial pattern is cross-sectional image of the color-coded phase values
just after the time of the ﬁrst heart sound. A phase value at a local point on
IVS, for example, changes from cyan (+180°) near a certain phase defect, togreen (+90°), and to red (0°) at a diﬀerent point and a diﬀerent time. The left
inset shows the scanning range of the ultrasonic beams in this measurement:
LV, left ventricle; LA, left atrium; RV, right ventricle; RA, right atrium; USprobe, ultrasonic probe; IVS, interventricular septum; Ao, aorta; ECG,
electrocardiogram; PCG, phonocardiogram (heart sound).

Figure 2 shows a series of snapshots (a)–(f ) of twodimensional spatial patterns of phase ðx; y; tÞ for excitedwaves on the IVS; (a) t ¼ 5:95 ms, (b) t ¼ 13:89 ms, (c) t ¼
33:73 ms, (d) t ¼ 53:71 ms, (e) t ¼ 73:41 ms, and (f ) t ¼
93:25 ms, from the T-wave of ECG. Many phase defects (yh )are also observed along the beam line.
Figure 3 shows typical amplitude patterns jðy; tÞj with aﬁxed x on the IVS in the region of 69  y  95 mm and29:8  t  93:25 ms; Region A, which is shown by arectangle: 90  y  95 mm, 29:8 ðT1 Þ  t  45:7 ms ðT2 Þ;
Region B: 70  y  80 mm, t ¼ 53:71 ms; Region C: 89 y  95 mm, t ¼ 83:4 ms. From these patterns, we can clearlyobserve traveling amplitude holes in Regions A, B, and C.
Since the resolution in the x-axis is lower than that in the
y-axis, thereafter, the x-axis (beam number) is ﬁxed. For aﬁxed x, let us deﬁne the measured data of one-dimensional
traveling waves around the time of the R-wave of ECG as
follows:ðy; tÞ ¼ jðy; tÞj exp½iðy; tÞ;

ð2Þ

where y (mm) is directed to the longitudinal axis from theapex to the base on the IVS and the observed time t (T1 <t < T2 , T2  T1 ¼ 15:9 ms).
The position of a phase defect is speciﬁed by y ¼ yh . By
using two diﬀerent patterns speciﬁed by the wavenumbers q~1
and q~2 near yh , a heteroclinic phase pattern is deﬁned as
q~1 ≠ q~2 ≠ 0 at y ¼ yh . For a heteroclinic phase pattern, apair of asymptotic wavenumbers q~j ( j ¼ 1; 2) at y ≠ yh isgiven byZ T2
1
ðy2 ; tÞ  ðy1 ; tÞ
dt;
ð3Þ
q~j ¼
T 2  T 1 T1
y2  y1

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N. Bekki et al.

Fig. 2. (Color online) Series of snapshots (a)–(f ) of two-dimensional spatial patterns of phase ðx; y; tÞ for excited waves on the IVS; (a) t ¼ 5:95 ms,(b) t ¼ 13:89 ms, (c) t ¼ 33:73 ms, (d) t ¼ 53:71 ms, (e) t ¼ 73:41 ms, and (f ) t ¼ 93:25 ms, from the T-wave of ECG. A number of phase-defects ( yh ) are
observed along the beam line. Figure 2(f ) corresponds to Fig. 1.

q~h /Fig. 3. (Color online) Typical amplitude patterns jðy; tÞj with a ﬁxed x
on the IVS measured by the ultrasonic imaging method in the region of
69  y  95 mm and 29:8  t  93:3 ms. Region A, which is shown by arectangle: 90  y  95 mm, 29:8  t  45:7 ms; Region B: 70  y  80
mm, t ¼ 53:7 ms; Region C: 89  y  95 mm, t ¼ 83:4 ms.

where q~1 is wavenumber for y1 < y2 < yh and q~2 iswavenumber for yh < y1 < y2 during T1 < t < T2 . We alsodeﬁne the phase jump
~ ob ¼ lim sup jðy  yh  ; tÞ  ðy  yh þ ; tÞj;!þ0 y2R

ð4Þ

where the phase ðy; tÞ is linearly extrapolated for a ﬁxedtime.
Let us also deﬁne the position y0 of minimum amplitude at
the time t0 and the position y00 of minimum amplitude at thetime t00 ; then, in a uniform linear motion of phase defects, wehave its average velocity c~h (mm=ms) deﬁned by
c~h ¼

00

0

y y
:
t00  t0

ð5Þ

As shown in Fig. 3, from Eq. (5), we can obtain that the
average velocity of the amplitude hole is about c~h  0:04 
0:01 mm=ms.
For a non-heteroclinic phase pattern with a ﬁxed time, on
the other hand, from Eq. (3), we obtain a local peak ofasymptotic wavenumber q~h at y ¼ yh@ðy; tÞ:@y

ð6ÞUsing Eqs. (3)–(6) for the observed data, we obtained thefollowing characteristics: (i) a pair of asymptotic wavenumbers q~1 ¼ q~2  0 mm−1 at y ≠ yh , which is called ahomoclinic hole solution in the CGLE,24) (ii) a local peakof wavenumber jq~h j ≠ 0 at y ¼ yh from Eq. (6), (iii) a
ﬁnite velocity of moving phase defect jc~h j ≠ 0 mm=ms,
and (iv) a phase jump of the homoclinic hole ~ ob (rad)diﬀers from that of the heteroclinic BN hole at y ¼ yh in
the CGLE.
As shown in Fig. 4, we obtained a local phase proﬁleðy; tÞ (rad) near the local point yh ¼ 92:4 mm in Region A:(a) t1 ¼ 29:8 ms, (b) t2 ¼ 33:7 ms, (c) t3 ¼ 37:7 ms, (d) t4 ¼
41:7 ms, (e) t5 ¼ 45:7 ms. The proﬁle (f ) denoted by thebroken line is the derivative of the phase proﬁle (e), whichcorresponds to a local wavenumber obtained from Eq. (6) at
t5 ¼ 45:7 ms. We also observed a growing phase defect nearyh in the longitudinal direction for 29:8  t  45:7 ms andthe maximum phase jump ~ ob  2:9 rad for the proﬁle (e). It
is clear that two asymptotic wavenumbers q~1 and q~2 at y ≠ yh
are small in this region A, that is, q~1 ¼ q~2  0 at y ≠ yh , andthere exists a local peak of wavenumber q~h ≠ 0 at y ¼ yh fora homoclinic phase pattern, as shown in Fig. 4(f ). In fact,
as shown in Fig. 4(g), the phase proﬁle of Fig. 4(e) (90 y  95) at t5 ¼ 45:7 ms can be approximated byðy; t5 Þ  1:0 þ 1:4 tanh½ðy  yh Þ:

ð7ÞThe insertion of Eq. (7) into Eq. (6) yields the normalized
local wavenumber qh as
qh ðyÞ  sech2 ½ðy  yh Þ:

ð8ÞThis local normalized wavenumber qh ðyÞ coincides with thecurve shown in Fig. 4(f ) generated by the phase proﬁleshown in Fig. 4(e). Then, for a traveling plane-wave model in
the CGLE,24) we havepﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃjðy; t5 Þj  1  qh ðyÞ2  1:
ð9Þ

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J. Phys. Soc. Jpn. 85, 044802 (2016)

N. Bekki et al.

Fig. 4. (Color online) Local phase proﬁle ðy; tÞ (rad) near the local pointyh ¼ 92:4 mm in Region A: (a) t1 ¼ 29:8 ms, (b) t2 ¼ 33:7 ms, (c) t3 ¼
37:7 ms, (d) t4 ¼ 41:7 ms, and (e) t5 ¼ 45:7 ms (circle). Proﬁle (f ) is thenormalized derivative of the phase proﬁle (e), which corresponds to a local
wavenumber at t5 ¼ 45:7 ms. We can observe a growing phase defect nearyh in the longitudinal beam direction for 29:8  t  45:7 ms and obtain the
maximum phase jump ~ ob  2:9 rad from Eq. (4). Two asymptotic wavenumbers q~1 and q~2 (broken line) at y ≠ yh are small in Region A: q~1 ¼
q~2  0 [see (f )]. The curve (g) (solid line) is given by ðy; t5 Þ  1:0 þ
1:4 tanh½ðy  yh Þ. The phase proﬁle of the measurements in the y-axis isquite diﬀerent from that of the heteroclinic BN hole solutions.

This suggests that the approximated phase proﬁle (7) for
Fig. 4(e) qualitatively explains the creation of amplitude
holes, which are called homoclinic holes. These have been
known only as numerical solutions.24,27)
Since the characteristics of these phase proﬁles are quite
diﬀerent from those of the heteroclinic BN hole solutions in
the CGLE, another phenomenological physical model istherefore needed on the basis of the present measurement oftraveling waves on the IVS around the time of the R-waveof ECG (end-diastole) for a healthy human heart. These
characteristics (i)–(iv) for measured waves on the IVS lead totraveling homoclinic plane waves based on the numericalsolutions in the one-dimensional CGLE. Note that we do notknow yet the exact solutions for them in the CGLE.27)
3.

CGLE Model for Traveling Homoclinic Plane Waves

In
our
case,
the
scaledCGLE16,18,19,21–24,26) is given by

one-dimensional@@2ðy; tÞ ¼  þ ð1 þ ic1 Þ 2   ð1  ic3 Þjj2 : ð10Þ@t@yHere, for simplicity, we put the coeﬃcients in Eq. (1) as
follows: c1 ¼ pr , c3 ¼ qr , pi ¼ 1, qi ¼ 1, and  ¼ 1. A setof two parameters, i.e., c1 and c3 , in the CGLE (10)determines complex spatiotemporal dynamics due to the
Benjamin–Feir instability.21)
We are interested in the coherent localized structures of
excited waves, which are measured noninvasively by theultrasonic noninvasive imaging method.2) We assume theexistence of uniformly propagating solutions of the form in
the variable  ¼ y  vt for the CGLE;^  vtÞei!t ;ðy; tÞ ¼ ðy^ðÞ¼ aðÞeiðÞ ;ð11Þ

Fig. 5. (Color online) Typical numerical solution of traveling plane wave
in the CGLE at t ¼ 210: jðyÞj is amplitude (square) and ðyÞ (rad) thephase (red dotted line) for a local space 1  y  6:5 (system length L ¼ 10)
near the position of phase-jumps yhk ( yh1 ¼ 2:2, yh2 ¼ 4:2, yh3 ¼ 5:4). The
broken line corresponds to the phase gradient jq^hk j Eq. (15) for thehomoclinic phase pattern. The thick solid line can be obtained using Eq. (16)with ^ k ¼ 1:2.

^
where aðÞ  jðÞj,ðÞ, v, and ω denote the amplitude,
phase, velocity, and angular frequency, respectively.
The insertion of Eq. (11) into Eq. (10) leads to thefollowing coupled ﬁrst-order ordinary diﬀerential equations:daðÞ¼ ðÞaðÞ;ddðÞ¼ 2 þ q2dþ

ð1  c1 c3 Þa2  1  c1 !  vð þ c1 qÞ;
1 þ c21

ð12Þ

dqðÞ
¼ 2qdðc1 þ c3 Þa2  c1 þ !  vðc1   qÞ:
1 þ c21

Here, we deﬁne ðÞ ¼ @ a=a and qðÞ ¼ @ .22) A family oftraveling plane-wave solutions known as coherent structures
has been numerically studied using Eq. (12).22,24,27) A
homoclinic hole has been deﬁned as an unstable homoclinic
orbit associated with a saddle-node bifurcation.24,27) Note thathomoclinic holes are not exact solutions of the CGLE (10)and that their numerical solutions are extremely delicate instability for a parameter space ðc1 ; c3 ; !; vÞ.22,24,27)
Homoclinic holes are parameterized by the angularfrequency ω and the velocity v, and are numerically observed
from Eq. (12) in a certain parameter space.24,27) All theamplitude patterns of the measured excited waves in Fig. 3,
however, cannot be explained by homoclinic holes basedon Eq. (12) at all. Since numerical solutions of the CGLE in
the case of c1 c3  1 show chaotic patterns and modulated
amplitude waves instead of BN holes and homoclinic
holes,24,26,27) therefore, we carried out direct numericalsimulations of the CGLE (10) with the condition c1 c3 < 1.
Figure 5 shows a typical snapshot of numerical solutionsin the CGLE with c1 ¼ 0:6 and c3 ¼ 1:4 at t ¼ 210 for the
local space 1  y  6:5 near the position of three phasejumps yhk (yh1 ¼ 2:2, yh2 ¼ 4:2, yh3 ¼ 5:4), which may

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J. Phys. Soc. Jpn. 85, 044802 (2016)

N. Bekki et al.

create almost homoclinic amplitude holes for traveling planewaves. The dotted line denotes the typical spatial phaseproﬁle ðyÞ at t ¼ 210. The broken line is the absolute valueof the phase gradient jq^hk j given by Eq. (15), that is, a local
peak of wavenumber for the homoclinic phase pattern.24,27)
We can clearly observe three amplitude holes near yhk(k ¼ 1; 2; 3) since each amplitude hole is governed by thephase gradient near the phase jumps. It is suggested that thehomoclinic amplitude holes obtained numerically can beapproximated by the homoclinic plane-wave solutions.
We have used the fourth-order Runge–Kutta scheme fortime and the central-diﬀerence scheme for space withperiodic boundary condition. The time and space resolutions
were t ¼ 0:001 and y ¼ 0:1, respectively, and the system
length L ¼ 10{500. The typical initial condition was given asðy; 0Þ ¼ ie4yi tanh½ðy  y0 Þ;
where the constant  ¼ 0:08 and y0 ¼ L=4.
One-dimensional phase distribution for a ﬁxed time may
be approximately decomposed into small intervals 2 k (> 0)in y containing the position of the phase-defect yhk (1 k  n):n[ðyÞ ¼
k ;
ð13Þ
k¼1

k ¼ k H1 H2 tanh½ k ðy  yhk Þ þ ð0Þ
k ;

ð14Þ

where H1  Hðy  yhk þ k Þ and H2  Hðyhk þ k  yÞ are
the Heviside step functions, and k , k , and ð0Þ
k are constant.
From Eq. (14), each normalized local peak of the wavenumber q^hk near y ¼ yhk may be approximately given asq^hk ¼@k¼ ~ k sech2 ½ k ðy  yhk Þ;
@y

ð15Þ

where ~ k is a constant that depends on the nonlinear
interactions between phase defects. Then, from Eq. (15), we
can qualitatively estimate the homoclinic amplitude hole neary ¼ yhk ,sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
 2ﬃ
q^jh ðyÞj  1  hk ;ð16Þ^kwhere ^k is a ﬁtting constant. In fact, as shown in Fig. 5,
Eq. (16) can explain the creation of almost homoclinic
amplitude holes jh ðyÞj due to the phase jumps at yh1 , yh2 ,and yh3 when time is ﬁxed, that is, the numerically obtained
amplitude jðyÞj (square symbol) of the CGLE coincides
with the homoclinic amplitude hole jh ðyÞj with ^k ¼ 1:2(thick solid line).
However, we cannot analytically obtain each magnitude ofstrongly modulated amplitudes jðyÞj of excited waves since
we do not know yet the exact traveling plane-wave solutionsof the CGLE [see Fig. 6(f )]. We have shown that Eq. (16)may qualitatively explain the creation of homoclinic planewave amplitude holes for measured nonlinear excited waves
on the IVS in human heart wall during the period fromend-diastole to the beginning of systole by the ultrasonicnoninvasive imaging method.
4.

Discussion

As shown in Fig. 4, we observed the growing phase defectnear yh ¼ 92:4 mm in the longitudinal direction (y-axis) for
29:8  t  45:7 ms. Then, we obtained the maximum phaseFig. 6. (Color online) Typical amplitudes jðy; tÞj near the local pointyh ¼ 92:4 mm in Region A, which correspond to Figs. 4(a), 4(b), 4(d), and4(e). Figure 4(c) at t3 ¼ 37:7 ms was omitted to avoid confusion created by asimilar curve. A ﬁtting curve for the amplitude proﬁle (e) is shown in (f ). We
can clearly ﬁnd homoclinic-type holes near the yh since the phase defect nearyh creates the amplitude holes [see Fig. 4(f )].

jump from Eq. (4): ~ ob  2:9 rad for Fig. 4(e). The observed
two asymptotic wavenumbers q~1 and q~2 at y ≠ yh are small
in Region A (Fig. 3): q~1 ¼ q~2  0 [see Fig. 4(f )], whichis called a homoclinic hole24) of jðyÞj at t5 ¼ 45:7 ms[Fig. 6(e)]. Figure 6 shows typical amplitudes jðy; tÞj near
the local point of phase defect yh in Region A, whichcorrespond to Figs. 4(a), 4(b), 4(d), and 4(e). Figure 4(c) at
t3 ¼ 37:7 ms was abbreviated to avoid confusion with asimilar curve. We can clearly observe homoclinic-type holes
near yh since the phase jump near yh creates the amplitude
hole jh ðy; tÞj, which decreases and forms a dip shaped like ahole. Namely, the amplitude holes of the measured waves canbe qualitatively explained by Eq. (16) with the local wavenumber q^hk . Note that the phase proﬁle of measured excitedwaves on the IVS in the y-axis is quite diﬀerent from that ofheteroclinic BN hole solutions.
We show, as in Fig. 7, a snapshot of four phase defects
and four non-heteroclinic amplitude holes in Region B (seeFig. 3) near the local points yk (mm) from left to right (k ¼1; 2; 3; 4); (y1 ¼ 71:4) mm, (y2 ¼ 74:9) mm, (y3 ¼ 76:9)mm, and (y4 ¼ 79:3) mm. The red dotted line denotes thephase ðyÞ (rad), the black square symbol stands for theamplitude, and the blue broken line stands for the normalized
wavenumber jq^hk j of the homoclinic phase proﬁle for
Region B with a ﬁxed time t ¼ 53:71 ms. Since the phase
defect at yk creates the local peak of the wavenumber from
Eq. (15) and decreases its amplitude according to Eq. (16),we can ﬁnd homoclinic-type amplitude holes at (y ¼ yk ) mm.
Amplitude patterns in Region C can be similarly explainedby the same homoclinic plane wave as in Region A andRegion B. As shown in Figs. 5 and 7, we ﬁnd that the phaseand amplitude patterns of the measured waves are similar to
those of the numerical solutions in the CGLE with periodic
boundary condition. This strongly suggests that the dynamics
of the measured waves in the longitudinal beam direction on
the IVS by the phase tracking method can be qualitatively
explained by homoclinic plane-wave numerical solutions in
the CGLE for a certain parameter space.
The statistical properties of local structures in defectturbulence for the CGLE have been studied and revealed

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J. Phys. Soc. Jpn. 85, 044802 (2016)

N. Bekki et al.

(5) We expect that the CGLE model for measured travelingwaves in the human heart wall plays an importantrole in understanding the phenomenon of sarcomericoscillations such as spontaneous oscillatory contraction(SPOC)29) and hyperthermal sarcomeric oscillations(HSOs)30) occurring in cardiac muscles at an intermediate activation level.
(6) We hope that our fundamental study may be useful forcharacterizing and quantifying normal and pathologicalhearts (ischemia and infarction) in vivo, using myocardial elastography, since the propagation speed ofcontraction waves in a healthy heart is diﬀerent fromthat in a pathological one.31)

Fig. 7. (Color online) Snapshot of measured four phase jumps and fourhomoclinic amplitude holes near the local points yk (mm) from left to right(k ¼ 1; 2; 3; 4); y1 ¼ 71:4, y2 ¼ 74:9, y3 ¼ 76:9, and y4 ¼ 79:3. The red
dotted line denotes the phase ðy; tÞ (rad), the black square symbol stands forthe amplitude jðyÞj, and the blue broken line is the normalized phasegradient jq^hk j for Region B with a ﬁxed time t ¼ 53:71 ms. We can observe
four homoclinic-type amplitude holes at y ¼ yk since each phase jump at yk
creates its corresponding local peak of four wavenumbers.

to be sub-Poisson statistics in the context of birth–deathprocesses.28)
5.

Conclusions

By the phase-tracking method10) in ultrasonic diagnostic
equipment, we have obtained signiﬁcant data of excitedwaves due to mechanical motions during the period fromend-diastole to the beginning of systole. We have observed
the following main properties of excited waves in thelongitudinal beam direction from the apex to the base: (i)two asymptotic wavenumbers q~1 ¼ q~2  0 mm−1 for locallyhomoclinic phase patterns,24,27) (ii) a local wavenumber
satisﬁes the condition 0 < jq^hk j < 1, (iii) the velocity of themoving phase defect jc~h j ≠ 0 mm=ms, and (iv) the phasejump j~ob j ≠ 0 as in Fig. 4.
Let us list the principal results of the present investigation:(1) We have observed the two-dimensional phase andamplitude patterns of excited waves on the IVS in ahuman heart wall.
(2) We have found that the dynamics of excited waves in
the longitudinal beam direction are quite diﬀerent fromthose of heteroclinic BN holes in the CGLE since theasymptotic wavenumbers satisfy q~1 ¼ q~2  0 at y ≠ yh .
(3) We have carried out numerical simulations of travelingplane-wave solutions in the CGLE with c1 c3 < 1 under
the periodic boundary condition.
(4) We have shown that the observed cardiac dynamics ofthe traveling waves in the longitudinal beam direction
on the IVS can be qualitatively explained by thenumerical homoclinic plane-wave solutions in the
CGLE.

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n.bekki@kurenai.waseda.jp
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