1. 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 Arrhythmias 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. Tachychardia Fibrillation Courtesty of Sasha Panfilov, University of Utrecht Transmembrane potential propagation : capacitance per unit area of membrane : transmembrane potential : intra- (extra-) cellular potential : transmembrane current : conductivity tensor in intra- (extra-) cellular space Governing equations describing the intra- and extracellular potentials: Ionic current,, described by a FitzHugh-Nagumo-like kinetics [1] [1] A. V. Panfilov and J. P. Keener Physica D (1995). 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: Numerical Implementation Numerical solution of parabolic PDE (for u m ) Forward Euler scheme: Crank-Nicolson scheme: is approximated by the finite difference matrix operator, Numerical solution of elliptic PDE (for u e ) Direct solution of the resulting systems of linear algebraic equations by LU decomposition. Numerical Results MonodomainBidomain Fiber Rotation Thickness Break-up (filaments number) MonodomainBidomain 1.0 0.4120 o 10mm11 0.9 0.4120 o 10mm11 0.8 0.4120 o 10mm11 0.7 0.4120 o 10mm11 0.6 0.4120 o 10mm11 0.5 0.4120 o 10mm12 0.3 0.4120 o 10 mm13 0.1 0.4120 o 10 mm>=9>=8 0.06 0.4120 o 10 mm>=20>=16 0.3 0.460 o 10mm12 0.1 0.460 o 10 mm>=10>=6 0.3 0.440 o 10mm13 0.1 0.440 o 10 mm>=8>=7 0.1 0.460 o 5 mm>=8 Conclusion From Laboortatory of Living State Physics, Vanderbilt University The bidomain model treats the complex microstructure of cardiac tissue is 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. [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 Elements a i, b i, c i … are constants obtained in finite difference approximation to the elliptic equation. Index re-ordering to reduce size of band-diagonal system 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. Computional results 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. Filament-finding result with rotation =120; Thickness = 10mm; 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. Work in progress includes: Conductivity Tensors Focus of This Work 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. Time (s) Filament number Time (s) Filament length(grid points) Insert refs. 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 Time (s) Filament number Time (s) Filament length(grid points) Filament number is greater than 2 with Filament number is 1 with