NUMERICAL SIMULATION OF ELECTROMECHANICAL DYNAMICS IN PACED CARDIAC TISSUE
Henian Xia
Xiaopeng Zhao
Department of Mechanical, Aerospace and
Biomedical Engineering
University of Tennessee
Knoxville, TN 37996
xiahenian@gmail.com
xzhao9@utk.edu
ABSTRACT
We study electromechanical dynamics in paced cardiac
tissue using numerical simulations of a mathematical model
that accounts for excitation-contraction coupling as well as
mechanoelectrical feedback. A previously developed finite
element based parallel platform is adopted. Extensive
numerical simulations are carried out on a 2d tissue and a 3d
tissue to investigate the influences of various parameters on the
stability of propagating cardiac waves, including conduction
velocity, pathological scars, contraction, and stretch activated
channels.
INTRODUCTION
Sudden cardiac arrest (SCA), a leading cause of death in
the industrialized world, kills over 350,000 Americans each
year. SCA can often result from fatal arrhythmias such as
ventricular fibrillation. Much research has demonstrated the
connections between the fatal arrhythmias and the instabilities
of the cardiac electrophysiology. Alternans of action potential
duration (APD) is a well-known indicator of the instabilities.
Discordant alternans, a phenomenon of two spatially distinct
regions exhibiting APD alternans of opposite phases, amplifies
dispersion of repolarization and can lead to conduction block
and reentry [1][1]. However, the mechanisms of discordant
alternans are not fully understood.
METHODS
Models from multiple physics fields are integrated,
including
electrophysiology,
electromechanics,
and
mechanoelectrical feedback. A computational algorithm is
developed based on finite element methods.
Kwai Wong
National Institute for Computational Sciences
University of Tennessee
Knoxville, TN 37996
kwong@utk.edu
Cardiac Electrophysiology
Cardiac dynamics of electrophysiology in extended tissue
are modeled using a reaction-diffusion equation:
∂v
1
= div(D∇v) −
( I ion + I ext + I s )
Cm
∂t
(1)
where v represents the transmembrane voltage, D represents the
diffusion matrix, operator div[•] denotes the divergence with
respect to spatial coordinates x, Cm represents the
transmembrane capacitance, Iion is the total ionic current, Iext is
the excitation current and Is is the stretch activated channel
(SAC). The equation is supplemented by a number of ordinary
differential equations, describing the variation of membrane
conductance and ionic concentrations. In this study, the FMG
model [3] is adopted to describe the ionic currents.
Cardiac Mechanics and Mechano-Electric Feedback
The quasi-static stress equilibrium of the heart is described
by the following equation of balance of linear momentum:
div[ τ ] + B = 0
(2)
where B represents the given body force per unit reference
volume. τ represents the Eulerian Kirchhoff stress tensor, which
is composed of the passive stress τpas and the excitation induced
contraction τact [4]: τ = τpas + τact.
The resulting partial differential equations are solved using
finite element method (FEM). The time integration of the
reaction term is solved using forward Euler method with a time
step of 0.005ms. The time integration of the diffusion term is
solved using backward Euler method with a time step of 0.1ms.
Simulation Details
A square tissue (5cmx5cm) is stimulated at the leftbottom corner with a basic cycle length of 200ms. The tissue is
represented by a finite element hexahedral mesh with 20402
nodes in total and size of each element is about 0.05cm. A
square scar with a size of 0.05cm is presented at the center of
the tissue for some simulations. The electrical propagation is
isotropic along both x and y directions when not considering
contraction.
of the tissue on the APD alternans distribution. The scar is
formulated in the model by making the cells in the scar area
always at resting potential, and making the electrical
propagation impossible in such area.
RESULTS AND DISCUSSION
Influence of conduction velocity on formation of
discordant alternans in a 2d tissue
Previous studies have shown that discordant alternans can
be caused by interaction of conduction velocity and APD
restitution [5]. In this study, we found that, when the conduction
velocity was reduced by half in both directions of the tissue, the
concordant distribution of APD alternans would change to
discordant alternans.
Figure 1 shows the APD alternans distribution when conduction
is normal. The APD in the entire tissue alternates concordantly
between about 118ms and 162ms. The APD alternans at the 29th
beat in Figures 1-3 are equal to d29 - d28, where d28 and d29 are
the APDs at the 28th and 29th beats, respectively. Figure 2
shows the APD alternans distribution in the tissue when the
conduction is isotropic but slow in both directions. The nodal
line remains still as the simulations lasts. This study gives
another mechanism of formation of the discordant alternans: it
can arise from the coupling of the slow conduction and the high
pacing rate.
FIGURE 2. DISTRIBUTION OF THE MAGNITUDE OF APD
TH
ALTERNANS AT THE 29 BEAT IN THE TISSUE AT SLOW
CONDUCTION VELOCITY. TISSUE SIZE IS 5CM AND MESH
SIZE IS 0.05CM
FIGURE 3. DISTRIBUTION OF THE MAGNITUDE OF APD
ALTERNANS IN THE TISSUE WITH A SCAR IN THE CENTER
AT NORMAL CONDUCTION VELOCITY AT 29TH BEAT.
TISSUE SIZE IS 5CM AND MESH SIZE IS 0.05CM
0
0.1
-80
0.2
0.3
-60
0.4
-40
0.5
0.6
-20
0.7
0
0.8
0.9
20
1
40
FIGURE 1. DISTRIBUTION OF THE MAGNITUDE OF APD
TH
ALTERNANS AT THE 29 BEAT IN THE TISSUE AT NORMAL
CONDUCTION VELOCITY. TISSUE SIZE IS 5CM AND MESH
SIZE IS 0.05CM
Influence of scar on cardiac electrophysiology in a 2d
tissue
Myocardial scarring has been considered an important
anatomic component of the substrate for arrhythmias like
ventricular tachycardia and fibrillation [6]. In this study, we
investigated the influence of the presence of a scar at the center
FIGURE 4. WAVE BREAK IN THE TISSUE WITH A SCAR IN
THE CENTER AT SLOW CONDUCTION VELOCITY. FROM
LEFT TO RIGHT: 850MS, 2500MS, 6137MS.
When the conduction is normal, we found that, the part of
the tissue near to the scar and far away from the stimulation site
was stabilized. Fig. 3 shows that, in the area near the
stimulation site, the APD alternates between about 116ms and
165ms, while the green area that is far away from the
stimulation site remains an APD of about 145ms. Although the
discordant alternans is not formed, this heterogeneous
distribution of APD over the tissue may play a role as a
substrate to arrhythmias too.
When the conduction is slow, a sustained spiral wave
breakup is observed, as shown in Fig. 4. The breakup of spiral
waves may induce fatal cardiac arrhythmias such as ventricular
fibrillation. Thus the scar obviously disturbs the stability of the
cardiac electrophysiology.
Influence of excitation induced contraction and
mechano-electric feedback in a 3d tissue
The tissue is seen as a bunch of fibers, with fiber
orientation along vertical direction. The stress at the left-top and
right-bottom corner is fixed. A scar is presented at the center of
the tissue. Fig. 5 shows that the wave break happens in both top
and middle panels, but is absent in bottom panels. Because of
the occurrence of SAC at the left-top corner and the rightbottom corner, two electrical waves sweep horizontally in
opposite directions and collide with each other, thus prohibiting
the happening of spiral wave breakup. The SAC thus shows a
capability of stabilizing the cardiac electrophysiology.
0
0.1
-80
0.2
0.3
-60
0.4
-40
0.5
0.6
-20
0.7
0
0.8
0.9
20
1
40
CONCLUSION
We have studied dynamics of cardiac waves in two
tissues using numerical simulations through a parallel finite
element platform. Special focus is on spatial patterns of
alternans and on the development of spiral waves. A slow
conduction speed and the presence of a scar can both disturb the
stability of the cardiac electrophysiology. Moreover, the stretch
activated channel is shown to be able to prevent the spiral wave
breakup.
ACKNOWLEDGEMENT
This work was in part supported by the NSF under grant
number CMMI-0845753. This research was supported by an
allocation through the TeraGrid Advanced Support Program.
Henian Xia is a graduate research assistant at the National
Institute for Mathematical and Biological Synthesis, an Institute
sponsored by the National Science Foundation, the U.S.
Department of Homeland Security, and the U.S. Department of
Agriculture through NSF Award #EF-0832858, with additional
support from The University of Tennessee, Knoxville.
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FIGURE 5. MEMBRANE POTENTIAL DISTRIBUTION AT
277MS, 500MS AND 950MS. TOP: ONLY
ELETROPHYSIOLOGY, MIDDLE: ELECTROMECHANICAL
COUPLING WITHOUT SAC, BOTTOM:
ELECTROMECHANICAL COUPLING WITH SAC
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