1. Break-up (>= 2 filaments)
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
Numerical Results
Rotating Anisotropy
Comparison of break-up in bidomain and monodomain models:
[1] W.F. Witkowksi, et al., Nature 392, 78 (1998)
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.
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.
And … the heart is an interesting arena for applying the ideas of pattern formation.
Monodomain
Bidomain
Fiber Rotation
Thickness
Break-up (>= 2 filaments)
1.0
0.4
120o
5mm No
0.9
0.8
0.7
0.6
0.5
Yes
0.3
0.1
0.06
60o
40o
Focus of This Work
Patch size: 5 cm x 5 cm Time spacing: 5 msec
Computational study of the role of the rotating anisotropy of cardiac tissue on the dynamics of phase singularities in the bidomain model of cardiac tissue.
Rectangular grid: 60 x 60 x 9; dx=0.5 mm, dy=0.5 mm, dz=0.5 mm; dt=0.01s
Spiral Waves and Cardiac Arrhythmias
Governing Equations
Example of filament-finding results used to characterize breakup ( ):
Transition from ventricular tachychardia to fibrillation is conjectured to occur as a result of breakdown of a single spiral (scroll) into a spatiotemporally disordered state, resulting from various mechanisms of spiral (scroll) wave instability. [2]
Governing equations describing the intra- and extracellular potentials:
Ionic current, , described by a FitzHugh-Nagumo-like kinetics [9]
Transmembrane potential propagation
Filament length(grid points)
Tachychardia
Fibrillation
Filament number
Conservation of total current
Time (s)
Courtesty of Sasha Panfilov, University of Utrecht
Time (s)
Goal is to use analytical and numerical tools to study the dynamics of reentrant waves in the heart on physiologically realistic domains.
: capacitance per unit area of membrane
: transmembrane potential
: intra- (extra-) cellular potential
: transmembrane current
: conductivity tensor in intra- (extra-) cellular space
[4] A. V. Panfilov, Chaos 8, (1998)
Filament length(grid points)
Filament number
Bidomain Model of Cardiac Tissue
[9] A. V. Panfilov and J. P. Keener, Physica D 84, (1995)
Numerical Implementation
The bidomain model treats the complex microstructure of cardiac tissue as a two-phase conducting medium, where every point in space is composed of both intra- and extracellular spaces and both conductivity tensors are specified at each point.[3-5]
Numerical solution of parabolic PDE (for um )
Time (s)
Time (s)
Conclusions
Forward Euler scheme:
We have numerically implemented electrical wave propagation in the bidomain model of cardiac tissue in the presence of rotating anisotropy using FHN-like reaction kinetics.
Preliminary numerical results indicate that in the bidomain model, scroll wave breakup is more sensitive to the anisotropy ratio than the fiber rotation rate, in contrast with the monodomain model.
From Laboortatory of Living State Physics, Vanderbilt University
Crank-Nicolson scheme:
Conductivity Tensors
is approximated by the finite difference matrix operator
Numerical solution of elliptic PDE (for ue )
Bidomain:
Monodomain:
Direct solution of the resulting systems of linear algebraic equations by LU decomposition.
Future Work
The ratios of the diffusion constants along and perpendicular to the fiber direction in the intra- and extra-cellular spaces are different.
The intracellular and extracellular conductivity tensors are proportional.
Improve numerical efficiency, using recent approaches such as multigrid methods for elliptic equation [8]
Systematic exploration of the role of cell electrophysiology in rotating anisotropy-induced scroll break-up in the bidomain model
Analytical and numerical perturbation analysis of scroll waves in the bidomain model, in the limit that intra- and extracellular conductivity tensors are approximately proportional
Index re-ordering to reduce size of band-diagonal system
Cardiac tissue is more accurately described as a three-dimensional anisotropic bidomain, especially under conditions of applied external current such as in defibrillation studies.[6-7]
However, unlike the monodomain, analytical and numerical studies based on the bidomain model remain technically challenging. [8]
Acknowledgements
[3] J. P. Keener and J. Sneyd, Mathematical Physiology
[4] C. S. Henriquez, Critical Reviews in Biomedical Engineering 21, 1-77 (1993)
[5] J. C. Neu and W. Krassowska, Critical Reviews in Biomedical Engineering 21, (1993)
[6] B. J. Roth and J. P. Wikswo, IEEE Transactions on Biomedical Engineering 41, (1994)
[7] J. P. Wikswo, et al., Biophysical Journal 69, (1995)
[8] J. P. Keener and K. Bogar, Chaos 8, (1998)
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.
Elements ai, bi, ci … are constants obtained in finite difference approximation to the elliptic equation.