1. 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