Spiral-wave Instability in a Medium with a Gradient in the Fibroblast Density: a Computational Study

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Spiral-Wave Instability in a Medium with a Gradient in the Fibroblast Density:
a Computational Study
Soling Zimik1, Rahul Pandit1
1Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science,
Bangalore, 560012, India
Abstract
Fibrosis, a process of fibroblast proliferation in cardiac
tissue, is a major concern for patients with diseases like is-
chemia, heart failure, and cardiomyopathy, because of its
arrhythmogenic effects. Fibroblasts, in appreciable den-
sities, are known to affect the electrical-wave dynamics in
cardiac tissue, because the coupling of fibroblasts with my-
ocytes modulates the electrophysiological properties of the
myocytes. Furthermore, in fibrotic hearts, the distribution
of fibroblasts can be heterogeneous, so the density of fi-
broblasts can vary from wounded regions (like infarcted or
ischemic zones) to normal regions of the heart. Such gra-
dients in the fibroblast density (GFD) induce spatial varia-
tion in the electrophysiological properties of the tissue, and
the latter can initiate and affect the dynamics of patholog-
ical waves like spiral waves. We study the effects of GFDs
on the dynamics of spiral waves by using a state-of-the-
art mathematical model for human-ventricular tissue. We
find that, in the presence of GFDs, spiral waves are un-
stable, i.e., a stable spiral wave breaks into multiple spiral
waves. We find that GFDs induce spatial variations in the
local spiral-wave frequency ωin the medium. Such a vari-
ation in ωleads to anisotropic thinning of the spiral arm
that gives way to spiral-wave breaks. We also study the
factors that enhance the instability of the spiral waves in
the medium with GFD. Finally, we show that the presence
of GFD can spontaneously lead to spiral waves, via high-
frequency pacing, in the medium.
1. Introduction
The occurrence of abnormal spatiotemporal patterns of
electrical waves in cardiac tissue, in the form of spiral
waves, has been associated with cardiac arrhythmias [1].
The existence of a single spiral wave is associated with
tachycardia [2], and a multiple-spiral-wave state is asso-
ciated with a life-threatening arrhythmia known as fibril-
lation [3]. A single-spiral-wave state can transition into
a multiple-spiral-wave state if the spiral becomes unsta-
ble and breaks up, giving rise to many daughter spirals [1].
Given that fibrillation is a more lethal condition than tachy-
cardia, it is important to understand the mechanisms of the
transition process from a single- to a multiple-spiral-wave
state. Here, we present a mechanism of spiral-wave in-
stability in a medium with a gradient in fibroblast density
(GFD).
Fibroblasts are passive cells that are needed for the
proper functioning of a heart, because they, along with
other cells, form the extracellular matrix and ensure the
structural integrity of the heart. However, the abnormal
proliferation of fibroblasts because of diseases like is-
chemia, heart failure, and cardiomyopathy is considered
to be arrhythmogenic [4]. Fibroblasts, if they form gap-
junctional coupling, can change the electrophysiology of
myocytes in cardiac tissue [5]. This, in turn, can affect the
dynamics of electrical waves in cardiac tissue. Many stud-
ies have been performed to investigate the effects of fibrob-
lasts on wave dynamics in cardiac tissue [6, 7]; however,
most of the studies, which consider fibroblast-myocyte
(FM) coupling, deal with a homogeneous distribution of
fibroblasts in the domain. But, the density of fibroblasts in
a diseased cardiac tissue may not necessarily be homoge-
neous [8]. Therefore, it is important to study the effects of
FM coupling on wave dynamics in a domain with hetero-
geneous distributions of fibroblast density.
We investigate the effects of GFD on spiral-wave dy-
namics in a mathematical model for human-ventricular tis-
sue. We find that GFD induces a spatial variation of the lo-
cal spiral-wave frequency ωin the domain. This variation
of ωin the domain leads to spiral-wave instability, and the
degree of instability depends on the magnitude of the vari-
ation of ω. We also investigate the factors that modulate
the variation of ωin the domain. We find that, for a given
GFD, the resting membrane potential Efof the fibroblasts,
and the number of fibroblasts Nfthat are coupled to a my-
ocyte can change the spatial variation of ωin the domain.
Finally, we also show that GFD can spontaneously initiate
spiral waves via high-frequency pacing.
Computing in Cardiology 2017; VOL 44 Page 1 ISSN: 2325-887X DOI:10.22489/CinC.2017.034-043

2. Methods And Materials
We use the O’Hara-Rudy dynamic model [9] for our
human-ventricular myocyte cell. The fibroblasts are mod-
elled as passive cells; for these fibroblasts, we use the
model given by MacCannell, et al. [10].
In our two-dimensional (2D) simulations the fibroblasts
are attached atop the myocytes; thus, our 2D simulation
domain is a bilayer. The spatiotemporal evolution of the
membrane potential (Vm) of the myocytes in tissue is gov-
erned by a reaction-diffusion equation, which is the fol-
lowing partial-differential equation (PDE):
∂Vm
∂t +Iion +Igap
Cm
= D∇2Vm,(1)
where Dis the diffusion constant between the myocytes.
Iion is the sum of all the ionic currents of the myocyte [9].
Igap is the gap-junctional current between a myocyte and
a fibroblast in an FM composite. Igap =0 if no fibroblast is
attached to a myocyte.
We use the forward-Euler method for time marching
with a five-point stencil for the Laplacian. We set D=
0.0012 cm2/ms. The temporal and spatial resolutions are
set to be δx= 0.02 cm and δt=0.02 ms, respectively. We
use a domain size of 960 ×960 grid points. We initiate
the spiral wave by using the conventional S1-S2 cross-field
protocol.
3. Results
The coupling of fibroblasts to a myocyte can change the
electrophysiological properties of the myocyte. We show
in fig. 1 the action potentials (APs) of a myocyte coupled
to fibroblasts with different values of Ef: The black curve
indicates the AP of an isolated myocyte, and the red, and
blue curves indicate the APs of a myocyte coupled to fi-
broblasts of Ef= -25 mV, and -15 mV, respectively. We can
see that the action potential duration (APD) of the myocyte
increases if it is coupled to fibroblasts, and the increase in
the APD depends on the value of Efof the fibroblasts.
The changes in the APDs of myocytes because of fi-
broblast coupling can affect the electrical-wave properties
in a tissue. For example, changes in the APD of the con-
situent myocytes affect the frequency ωof a spiral wave as
follows. Consider a stably rotating spiral wave, if we ne-
glect curvature effects, dimensional analysis gives ω≃θ
λ,
where θis the conduction velocity and λis the wavelength.
Furthermore, λ≃θ×APD, and, therefore,
ω≃1
APD .(2)
In a domain with, on average, a homogeneous distri-
bution of fibroblasts (see fig. 2 (a)), the spiral-wave fre-
quency ωdecrease with the increase of fibroblast percent-
age pf(see fig. 3), because as fibroblast coupling increase
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
−100
−50
0
50
Vm (mV)
time(sec)
Ef= −15mV
Ef= −25 mV
No fibroblast
Figure 1. Plots showing the action potentials of an isolated
myocyte (black curve), and a myocyte attached to Nf= 2
fibroblasts of Ef= -15 mV (blue curve), and Ef= -25 mV
(red curve).
the APD of the myocytes (see fig. 1), the increase in pf
increases, on average, the APD of the myocytes in the
medium; and, thus, eqn. 2 implies that ωdecreases with
the increase of pf. Figure 3 shows the variation of ωwith
pf.
Figure 2. Pseudocolor plots showing two types of fibrob-
last distributions. (a) the density distribution of fibroblasts
is uniform throughout the domain, on average. The cyan
color indicates myocyte density with no fibroblasts, and
the magenta color indicates fibroblast-myocyte compos-
ites. The value of pfis 40%. (b) There is a gradient of
fibroblast density (GFD) along the vertical y axis, where
pfvaries linearly from 10% (at the bottom of the domain)
to 100% (at the top of the domain).
3.1. Effects of GFD on the stability of spiral
waves
The presence of gradient in fibroblast density (GFD) in-
duces spiral-wave instability. Figure 2 (b) shows the GFD
in a domain, where the fibroblast density increases from
the bottom (pf=10%) to the top (pf=100%) of the domain.
The magenta and cyan colors indicate fibroblast-mycoyte
composites and myocytes, respectively. Figure 4 shows the
break-up of a spiral wave in the domain shown in fig. 2 (b).
We see that the spiral wave breaks in the top region where
the density of fibroblasts is high. The reason for the spiral-
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0 50 100
3.5
4
4.5
5
5.5
pf(%)
ω (Hz)
(a)
Ef= −15mV
Ef= −25mV
0 50 100
1
2
3
4
5
pf(%)
ω (Hz)
(b)
Nf=1
Nf= 2
Nf= 3
Figure 3. Variation of the spiral-wave frequency ωwith
the percentage of fibroblasts pf, in a domain with a homo-
geneous fibroblast distribution. (a) Plots showing the vari-
ation of ωwith pffor Ef= -15 mV (black curve), and Ef=
-25 mV (blue curve). (b) Plots showing the variation of ω
with pffor three different values of Nf=1 (black curve),
Nf=2 (blue curve), and Nf=3 (red curve).
wave instability is as follows. The presence of GFD in the
medium induces a spatial variation of the local value of ω
(refer fig. 3). In fig. 4 the local value of ωdecreases from
the bottom to the top of the domain. This anisotropic varia-
tion of ωleads to an anisotropic thinning of the wavelength
of the spiral arms (see fig. 4 2.14 s), which gives way for
spiral-wave break-up. The wave-thinning occurs in the up-
per region, because it has large APD. As the upper region
has a large APD, the conduction velocity is lower in that re-
gion (conduction-velocty restitution property [11]); there-
fore, in order to support the high-frequency wave trains
coming from the lower region, the wavelength of the waves
thins as it propagates towards the top region.
Figure 4. Pseudocolor plot of Vmshowing the spatiotem-
poral evolution of the break-up of a spiral in the presence
of GFD with pfgoing from 10%-100%, and Ef=-25 mV.
The spiral arm breaks up in the upper region, which is the
low-ωregion.
As the instability of the spiral waves arises because of
the spatial variation of ωin the medium, the degree of in-
stability of the spiral waves depends on the magnitude of
the gradient in ωthat is induced by GFD. We, therefore,
study the factors that modulate the variation of ωwith pf.
We find that high values of Efinduce high variation of ω
with pf. Figure 3 (a) shows that the variation of ωwith pf
is more drastic for Ef= -15 mV (black), as compared to
Ef= -25 mV (blue). Moreover, we also find that the varia-
tion of ωwith pfis enhanced if we couple more fibroblasts
number Nfto a myocyte. Figure 3 (b) shows the variation
of ωwith pffor three different values of Nf:Nf=1(black),
Nf=2(blue), and Nf=3(red). We see that variation of ωin-
creases as we go from Nf=1 to Nf=3. These results tells
us that, for a given GFD, a spiral wave is more vulnerable
to break-up for high values of Efand Nfas compared to
the low values of Efand Nf.
3.2. Spontaneous initiation of spiral waves
The presence of GFD in the medium can also initiate
spiral wave spontaneously if we pace the medium at a high
frequency. Figure 5 shows the initiation of spiral waves in
a medium with GFD with pfgoing from 10%-100%. We
apply periodic line stimuli at the bottom edge of the do-
main at a pacing cycle length PCL=250 ms. The reason
for the spiral-wave initiation is as follows. The top region
of the medium, because it has a large APD, repolarizes
slowly. Now, if we pace the medium, at high-frequency,
there is a prominent waveback-wavefront interactions of
the waves in the top region. This waveback-wavfront inter-
actions induce corrugations in the wavefronts of the waves
(because of the random distribution of fibroblasts), which
eventually leads to the initiation of spiral waves.
Figure 5. Pseudocolor plot of Vmshowing the the sponta-
neous initiation of a spiral wave, via high-frequency pac-
ing, in a medium with GFD as shown in fig. 2 (b). The
pacing stimuli are applied at the bottom of the domain at
PCL= 250 ms.
Such pacing-induced spiral waves occur at high-
frequency pacing and not at low-frequency pacing, be-
cause the wavefront-waveback interaction between the
consecutive waves is more prominent at high-frequency
pacing. Figure 6 shows a stability diagram in the EF-P CL
plane indicating the regions where spiral waves (magenta
color) occur and no spiral waves (black color) occur.
In conclusion, we have shown how fibroblast-myocyte
(FM) coupling can affect the the electrical-wave properties
in a tissue. We see that FM coupling changed the APD of
the myocytes, and this, in turn, modulated the properties
of the spiral waves in a tissue, like its frequency ω. We
show that ωdecreases with the increase of the percentage
of fibroblasts pf, because the FM coupling increases the
APD of the constituent myocytes. We then study how the
gradient in the fibroblast density (GFD) affects the dynam-
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−20 −15 −10 −5
200
220
240
260
280
300
320
No spiral waves
Spiral waves
Ef (mV)
PCL (ms)
Figure 6. Stability diagram in the Ef-PCL plane. Figure
showing the regions where we observe spiral waves (black
color) and no spiral waves (magenta color) in the Ef-PCL
plane.
ics of the spiral waves. We find that GFD induces spiral-
wave instability in the medium. We show that this insta-
bility arises because of the spatial variation of the local
value of ωin the medium induced by the GFD, and the
spiral wave breaks in the low-ωregion. Our finding of
spiral-wave break in the low-ωis consistent with the ex-
perimental results of Campbell, et al., in monolayers of
neonatal-rat ventricular myocytes, where the gradeint in ω
is induced by varying the IKr ion-channel density. We also
investigate the factors that affect the variation of ωwith
pf. We find that the variation of ωwith pfis more for high
values of Efand Nfas compared to when their values are
low. And thus, for a given GFD, the degree of spiral-wave
instability is higher at high values of Efand Nfas com-
pared to their low values. Finally, we show how GFD in the
medium can initiate spiral waves spontaneously via high-
frequency pacing. We hope our results will lead to detailed
studies of GFD-induced spiral-wave instability at least in
in vitro experiments on cell-cultures. At the simplest level,
we suggest fibroblast analogs of the experiments of Camp-
bell, etal. [12].
Acknowledgements
We thank the Department of Science and Technology
(DST), India, and the Council for Scientific and Industrial
Research (CSIR), India, for financial support, and the Su-
percomputer Education and Research Centre (SERC, IISc)
for computational resources.
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Address for correspondence:
Soling Zimik
Centre for Condensed Matter Theory, Department of Physics, In-
dian Institute of Science, Bangalore, 560012, India
solyzk@gmail.com
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