The Role of the Bidomain Model of Cardiac Tissue in the Dynamics
of Phase Singularities Jianfeng Lv and Sima Setayeshgar Department
of Physics, Indiana University, Bloomington, Indiana 47405
Motivation Patch size: 5 cm x 5 cm Time spacing: 5 msec [1] W.F.
Witkowksi, et al., Nature 392, 78 (1998) Bidomain Model of Cardiac
Tissue Spiral Waves and Cardiac Arrhymias Transition from
ventricular tachychardia, characterized by single spiral wave, to
fibrillation, characterized as a spatiotemporally disordered state:
Tachychardia Fibrillation Paradigm: Breakdown of a single spiral
wave into disordered state, resulting from various mechanisms of
spiral wave instability Courtesty of Sasha Panfilov, University of
Utrecht Focus of our work Computational study of the role of the
rotating anisotropy of cardiac tissue within the Bidomain model.
Transmembrane potential propagation : capacitance per unit area of
membrane : transmembrane potential : intra- (extra-) cellular
potential : transmembrane current : conductivity tensor in intra-
(extra-) cellular space The coupled governing equations describing
the intra- and extracellular potentials are: Ionic current,,
described by a neurophysiological model adopted for the
FitzHugh-Nagumo system [1] [1] A. V. Panfilov and J. P. Keener
Physica D 1995 Conductivity Tensors The intracellular and
extracellular conductivity tensors are proportional The ratios of
the diffusion constants along and perpendicular to the fiber
direction in the intra- and extra- cellular spaces are different.
Monodomain: Bidomain: Dissection results indicate that cardiac
fibers are arranged in surfaces, where fibers are approximately
parallel in each surface while the mean fiber angle rotates from
the outer (epicardium) to inner (endocardium) wall. Rotating
Anisotropy Numerical Implementation Numerical simulation for the
parabolic PDE Forward Euler scheme: Crank-Nicolson scheme: is
approximated by the finite difference matrix operators Numerical
simulation for the elliptic PDE Direct solving the system of linear
algebraic equations by LU decomposition [1] Roth, B.J. IEEE
transactions on Biomedical Engineering, 1997 The monodomain result
is obtained from reduced Bidomain model by allowing the
conductivities tensors in the intra- and extra-cellular
proportional. We use filament-finding algorithm to determine the
break-up behavior of the spiral wave. If there are more than 2
filaments, the spiral wave breaks up. Numerical Results Fiber
Rotati on Thickness Break-up MonodomainBidomain 0.70.4120 o 10mmNo
0.50.4120 o 10mmNoYes 0.30.4120 o 10 mmNoYes 0.10.4120 o 10 mmYes
0.060.4120 o 10 mmYes 0.30.460 o 10mmNoYes 0.10.460 o 10 mmYes
0.30.440 o 10mmNoYes 0.10.440 o 10 mmNoYes 0.10.460 o 5 mmYes
0.10.440 o 3.3 mmYes Convergence Results Conclusion From
Laboortatory of Living State Physics, Vanderbilt University In the
bidomain model, the complex microstructure of cardiac tissue is
treated as a two-phase conducting medium, where every point in
space is composed of both intra- and extracellular spaces and both
conductivity tensors specified at each point. [2] J. P. Keener and
J. Sneyd, Mathematical Physiology [3] C. S. Henriquez, Critical
Reviews in Biomedical Engiineering 21, 1-77 (1993) Governing
Equations Conservation of total current The elements a, b, c.. in
the matrix depend are coefficients depending on discretized
equations In our rectangular model, we have, by re-ordering the
indices, we reduce the size of the compactly stored band-diagonal
matrix Activation map in cardiac tissue using bidomain model
EpicardiumMidmyocardiumEndocardium
EpicardiumMidmyocardiumEndocardium t = 0 s t = 5 s t = 10 s t = 50
s t = 100 s t = 150 s t = 200 s t = 250 s Comparison of bidomain
model and monodomain models Reduced Bidomain Bidomain In both
models, Thickness =10mm ; Twist = 120 o ; The fiber direction is 0
o at the epicardium and 120 o at the endocardium. The conductivity
tensors using in the Reduced Monodomain is t = 0 s t = 5 s t = 10 s
t = 50 s t = 100 s t = 150 s t = 200 s t = 250 s The conductivity
tensors using in the Bidomain model is We developed various
numerical methods to solve the Bidomain equations in both 2D and 3D
models with modified Fitz- Nagumo models as an ionic model. We
studied the break-up of the spiral wave in both Monodomain and
Bidomain models with fiber rotation incorporated. In our Bidomain
model, the anisotropy of coupling plays an important in the
break-up of spiral wave, the fiber rotation has a less prominent
role. While fiber rotation is important in Monodomain model.
Results of computational experiments with different parameters of
cardiac tissue Future Work Acknowledgements Ventricular
fibrillation (VF) is the main cause of sudden cardiac death in
industrialized nations, accounting for 1 out of 10 deaths. Strong
experimental evidence suggests that self-sustained waves of
electrical wave activity in cardiac tissue are related to fatal
arrhythmias. Goal is to use analytical and numerical tools to study
the dynamics of reentrant waves in the heart on physiologically
realistic domains. And the heart is an interesting arena for
applying the ideas of pattern formation. A time sequence of a
typical action potential The convergence result in three-dimension
Bidomain model Time (s) Time step (s) Transmembrane potential u
(mv) Rectangular grid 60 x 60 x 9; Spatial steps dx = 0.5mm,
dy=0.5mm, dz=0.5mm; Filament-finding result for Bidomain Model with
Twist = 120; Thickness = 10mm; Time (s) Filament number column is
for Bidomain model only; Spatial step dx=0.5mm, dy=0.5mm, dz=0.5mm,
dt=0.01s System size: Thickness=10 mm; Fiber Rotation = 120 o
Rectangular grid: 60 x 60 x 9 Numerical parameters: dx=0.5 mm,
dy=0.5 mm, dz=0.5 mm; dt=0.01s Develop Semi-implicit Algorithm to
eliminate time step limitation. Reduce the computational cost of
the linear solves by developing more efficient numerical methods.
The linear system Ax = y could benefit from applying multigrid
methods. We acknowledge support from the National Science
Foundation and Indiana University. We thank Xianfeng Song in our
group for helpful advice on various aspects of the numerical
implementation. Monodomain results are from monodomain code. Work
in progress includes: Conductivity Tensors
