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 Spiral Waves and Cardiac Arrhymias Patch size: 5 cm x 5 cm Time spacing: 5 msec W.F. Witkowksi, et al., Nature 392, 78 (1998) Bidomain Model of Cardiac Tissue Big Picture Transition from ventricular tachychardia to fibrillation 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: Transmembrane 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 conductivities are treated proportionally The ratios of the diffusion constants along and perpendicular to the fiber direction in the intra- and extra- cellular spaces are different. In Monodomain: In Bidomain: The orientation of fibers in succe- ssive layers of cardiac tissue rotates through the thickness. 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 Numerical Results [1] Roth, B.J. IEEE transactions on Biomedical Engineering, 1997 The thickness of the layer is 10 mm The fiber rotation is linear gradient with total rotation 120 o We can see that in all models the spiral waves can not maintain their regular propagations and create new wave breaks. The fiber rotation plays less important role in Bidomain model than in Monodomain model, as we can see that the Monodomain figures column shows a more complicated wave- breaks pattern than the Bidomain figures. Numerical Results D2/D1TwistThickness Irregular behavior MonodomainBidomain o 10 mmNo o 10 mmYes o 10 mmYes o 10 mmYes o 10 mmNo o 5 mmYes o 3.3 mmNo [1] A. V. Panfilov and J. P. Keener Physica D 1995 The Monodomain results obtained from reduced Bidomain model by setting the conductivities in the intra- and extra-cellular spaces proportional. The figures show that time step δt = 0.1 is suitable taking both efficiency and accuracy of computation into account. Numerical Results Conclusion Live state physics, Vanderbilt University In Bidomain model, we view cardiac tissue as a two-phase medium, as if every point in space is composed of a certain fraction of intracellular space and a fraction of extracellular space. [1] [1] “Mathematical Physiology”, James Keener, James Sneyd Governing Equations Conservative 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 The conductivity tensors for the Bidomain model are The comparison of Bidomain model and Monodomain model Monodomain Reduced Bidomain Reduced Bidomain In all the models, the thickness of the layers is 10mm with total 120 o fiber rotation. The fiber direction is the same as what we shown in the Bidomain activation maps with 0 o at the epicardium and 120 o at the endocardium. The conductivity tensor using in the 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 tensor 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-down of the spiral wave in both Monodomain and Bidomain described models with fiber rotation incorporated. The results obtained from our reduced Bidomain model agree with the previous results from Monodomain.[1] In our Bidomain model, the anisotropy of coupling and the thickness of the tissue play an important in the break-down 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 [1] [1] A. V. Panfilov and J. P. Keener Physica D 1995 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 with various time-steps. The convergence result in three-dimension Bidomain model