Advisor: Sima Setayeshgar Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities Jianfeng Lv Advisor: Sima Setayeshgar May 15, 2009

Outline Motivation Numerical Implementation Numerical Results Conclusions and Future Work

[1] W.F. Witkowski, et al., Nature 392, 78 (1998) Motivation: Patch size: 5 cm x 5 cm Time spacing: 5 msec 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. [1] W.F. Witkowski, et al., Nature 392, 78 (1998)

Spiral Waves and Cardiac Arrhythmias Transition from ventricular tachycardia 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. [1] Tachychardia Fibrillation Courtesy of Sasha Panfilov, University of Utrecht Goal is to use analytical and numerical tools to study the dynamics of reentrant waves in the heart on physiologically realistic domains. [1] A. V. Panfilov, Chaos 8, 57-64 (1998)

Cardiac Tissue Structure Cells are typically 30 – 100 µm long 8 – 20 µm wide Propagation Speeds = 0.5 m / s = 0.17 m / s Guyton and Hall, “Textbook of Medical Physiology” Nigel F. Hooke, “Efficient simulation of action potential propagation in a bidomain”, 1992

Cable Equation and Monodomain Model Early studies used the 1-D cable equation to describe the electrical behavior of a cylindrical fiber. transmembrane potential: intra- (extra-) cellular potential: transmembrane current (per unit length): axial currents: ionic current: conductivity tensor: capacitance per unit area of membrane: resistances (per unit length): Adapted from J. P. Keener and J. Sneyd, Mathematical Physiology

Bidomain Model of Cardiac Tissue 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.[1-3] From Laboratory of Living State Physics, Vanderbilt University [1] J. P. Keener and J. Sneyd, Mathematical Physiology [2] C. S. Henriquez, Critical Reviews in Biomedical Engineering 21, 1-77 (1993) [3] J. C. Neu and W. Krassowska, Critical Reviews in Biomedical Engineering 21, 137-1999 (1993)

Bidomain Model Transmembrane current: Ohmic axial currents: Conservation of total currents:

Conductivity Tensors Bidomain: Monodomain: The ratio of the intracellular and extracellular conductivity tensors; Bidomain: Monodomain: Cardiac tissue is more accurately described as a three-dimensional anisotropic bidomain, especially under conditions of applied external current such as in defibrillation studies.[1-2] [1] B. J. Roth and J. P. Wikswo, IEEE Transactions on Biomedical Engineering 41, 232-240 (1994) [2] J. P. Wikswo, et al., Biophysical Journal 69, 2195-2210 (1995)

Monodomain Reduction By setting the intra- and extra-cellular conductivity matrices proportional to each other, the bidomain model can be reduced to monodomain model. (1) Substitute (1) into If , then we obtain the monodomain model.

From Streeter, et al., Circ. Res. 24, p.339 (1969) Rotating Anisotropy Local Coordinate Lab Coordinate From Streeter, et al., Circ. Res. 24, p.339 (1969)

Coordinate System

Governing Equations

Perturbation Analysis

Scroll Twist Solutions Scroll Twist, Fz Twist Twist Rotating anisotropy generated scroll twist, either at the boundaries or in the bulk.

Significance? In isotropic excitable media (a = 1), for twist > twistcritical, straight filament undergoes buckling (“sproing”) instability [1] What happens in the presence of rotating anisotropy (a > 1)?? Henzi, Lugosi and Winfree, Can. J. Phys. (1990).

Filament Motion

Filament motion (cont’d)

Filament Tension Destabilizing or restabilizing role of rotating anisotropy!!

Phase Singularity Tips and filaments are phase singularities that act as organizing centers for spiral (2D) and scroll (3D) dynamics, respectively.

Focus of this work Analytical and numerical works[1-5] have been done on studying the dynamic of scroll waves in monodomain in the presence of rotating anisotropy . Rotating anisotropy can induce the breakdown of scroll wave; Rotating anisotropy leads to “twistons”, eventually destabilizing scroll filament; The focus of this work is computational study of the role of rotating anisotropy on the dynamics of phase singularities in bidomain model of cardiac tissue as a conducting medium. [1] Biktashev, V. N. and Holden, A. V. Physica D 347, 611(1994) [2] Keener, J. P. Physica D 31, 269 (1988) [3] S. Setayeshgar and A. J. Bernoff, PRL 88, 028101 (2002) [4] A. V. Panfilov and J. P. Keener, Physica D 84, 545 (1995) [5] Fenton, F. and Karma, A. Chaos 8, 20 (1998):

Numerical Implementation of the Bidomain Equations with Rotating Anisotropy

Governing Equations Governing equations describing the intra- and extracellular potentials: Transmembrane potential propagation Conservation of total current : transmembrane potential : intra- (extra-) cellular potential : ionic current : conductivity tensor in intra- (extra-) cellular space

Ionic current models Ionic current, , described by a FitzHugh-Nagumo-like kinetics [1] These parameters specify the fast processes such as initiation of the action potential. The refractoriness of the model is determined by the function . [1] A. V. Panfilov and J. P. Keener, Physica D 84, 545-552 (1995)

Boundary conditions No-flux boundary conditions: Let For a rectangular, Normal vector to the domain boundary: Conductivity tensors in natural frame:

Numerical Implementation Numerical solution of parabolic PDE (for Vm ) Forward Euler scheme: Crank-Nicolson scheme: The spacial operator is approximated by the finite difference matrix operator

Numerical Implementation cont’d Numerical solution of elliptic PDE (for Ve ) Direct solution of the resulting systems of linear algebraic equations by LU decomposition. ai , bi , ci , mi are coefficients of terms after discretization of LHS. denotes the extracellular potential Ve on node (x=i, y=j, z=k). denotes the corresponding RHS after discretization.

Numerical Implementation cont’d Index re-ordering to reduce size of band-diagonal system Elements ai, bi, ci … are constants obtained in finite difference approximation to the elliptic equation.

Numerical Convergence A time sequence of a typical action potential with various time-steps. The figures show that time step δt = 0.01 is suitable taking both efficiency and accuracy of computation into account.

Filament-finding algorithm “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface Search for the closest tip

Filament-finding algorithm “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface Make connection

Filament-finding algorithm “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface Continue doing search

Filament-finding algorithm “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface Continue

Filament-finding algorithm “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface Continue

Filament-finding algorithm “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface Continue

Filament-finding algorithm “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface The closest tip is too far

Filament-finding algorithm “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface Reverse the search direction

Filament-finding algorithm “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface Continue

Filament-finding algorithm “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface Complete the filament

Filament-finding algorithm “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface Start a new filament

Filament-finding algorithm “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface Repeat until all tips are consumed

Numerical Results

Numerical Results Filament dynamics of Bidomain Examples of filament-finding results used to characterize breakup. Total filament length(grid points) Filament number Time (s) Time (s) Filament number Total filament length(grid points) Time (s) Time (s)

Numerical Results of previous work in Monodomain Previous study has shown rotating anisotropy can induce the breakdown of scroll wave.[1] Model size : 60x60x9 for 10mm thickness No break-up while the fiber rotation is less then 60o or total thickness is less than 3.3mm. Iso surfaces of 3D view of scroll wave in the medium with = 0.1111 [1] A. V. Panfilov and J. P. Keener, Physica D 84, 545-552 (1995)

Numerical Results Bidomain/Monodomain Comparison Results of computational experiments with different parameters of cardiac tissue. Twist Thickness (layer) Irregular behavior Monodomain[1] Monodomain Bidomain ∆x=0.5 ∆x=0.2 0.3 120o 9 No 0.1 Yes 0.06 60o 40o 5 3 For ∆x=0.5, the size of rectangular grid is 60x60x9 points For ∆x=0.2, the size of rectangular grid is 150x150x23 points [1] A. V. Panfilov and J. P. Keener Physica D 1995

Numerical Results: Larger Domain Size Result Contour plots of transmembrane potential selected tissue layers at t = 750 time units. Scroll wave breakup is evident in the middle layers. Filament number Time (s) Total filament length(grid points) Model size: 140x294x48; ∆x = ∆y = ∆z = 0.25 (space units) Time step: ∆t = 0.01 (time units) ; Time (s)

Conclusions so far … 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. In the finer monodomain model and bidomain model, the boundaries of irregular behavior shift; Numerical Limitation: Large space step in previous study causes mesh effect; Model size is too small. Increasing model size in bidomain model is limited by the physical memory;

Multigrid Techniques: Multigrid Hierarchy Relax Restrict Interpolate Relax Relax Relax Relax Dragica Vasileska, “Multi-Grid Method”

Multigrid Techniques: Multigrid method Coarse-grid correction Structure of multigrid cycles Compute the defect on the fine grid; Restrict the defect; Solve exactly on the coarse grid for the correction; Interpolate the correction to the fine grid; Compute the next approximation Relaxation “Numerical Recipes in C”, 2nd Editoin S denotes smoothing; E denotes exact solution on the finest grid. Descending line \ denotes restriction, each ascending line / denotes prolongation. William L. Briggs, “A Multigrid Tutorial”

Multigrid Techniques: Full Multigrid Algorithm Multigrid method starts with some initial guess on the finest grid and carries out enough cycles to achieve convergence. Efficiency can be improved by using the Full Multigrid Algorithm (FMG) FMG with the exact solution at the coarsest level. It uses V-cycles (W-cycles) as the solver on each grid level. “Numerical Recipes in C”, 2nd Editoin

Multigrid Techniques: Interpolation Trilinear interpolation between the grids 3D interpolation 2D interpolation The arrows denote the coarse grid points to be used for interpolating the fine grid point. The numbers attached to the arrows denote the contribution of the specific coarse grid point. Dragica Vasileska, “Multi-Grid Method”

Multigrid Techniques: Restriction 3D Restriction 2D Restriction In 3D, A 27-point full weighting scheme is used. The number in front of each grid point denotes its weight in this operation. Dragica Vasileska, “Multi-Grid Method”

Multigrid Results Convergence in 2D Typical action potential with various Pre and Post Relaxation-steps. The figures show that in 2D relaxation step 200 is suitable taking both efficiency and accuracy of computation into account. The domain is 127x127

Multigrid Results Convergence in 3D Typical action potential with various Pre and Post Relaxation-steps. In the case of 3D, relaxation step 200 is also an appropriate number taken both efficiency and accuracy into account. The domain is 127x127x7, the convergence plot and density plot are taken at Z=4.

Future Work Improve numerical efficiency, optimize the multigrid code to reduce the computation time; Systematic exploration of the role of cell electrophysiology in rotating anisotropy-induced scroll break-up in the Bidomain model;

Thank you

Ionic current models cont. Ionic current described by a FitzHugh-Nagumo-like kinetics[1] It is a simple model to study the dynamics of spiral wave in the excitable media. The e is chosen small so that the time scale of Vm is much faster than that of w. The local kinetics in the absence of spatial derivatives, has a stable but excitable fixed point at the intersection of the nullclines Im = 0 and dv/dt = 0; At any instant in time, almost all spatial points are within the dash line boundary layer. Uth is the excitability threshold for the fixed point. I.C. near the fixed point and to the left of the threshold decay to the fixed point. I.C. to the right of the threshold undergo a large excursion before returning to the fixed point. [1] Barkley D. (1991) "A model for fast computer simulation of waves in excitable media". Physica 49D, 61–70.