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Electrical Wave Propagation in a Minimally Realistic Fiber Architecture Model of the Left Ventricle Xianfeng Song, Sima Setayeshgar Department of Physics, Indiana University Model Construction (cont.) Motivation 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 selfsustained waves of electrical wave activity in cardiac tissue are related to fatal arrhythmias. 2 L f ( , 1 d , )d d f d f L 0 d ' z subject to: 0 0 Fiber trajectory: 1 a sec 1 1 2 Mechanisms that generate and sustain VF are poorly understood. One conjectured mechanism is: Breakdown of a single spiral (scroll) wave into a disordered state, resulting from various mechanisms of spiral wave instability. 1 inner surface Speed up 2 1.42 ± 0.10 4 3.58 ± 0.16 8 7.61 ±0.46 16 14.95 ±0.46 32 28.04 ± 0.85 Fiber trajectories on nested pair of conical surfaces The computation time for dr=0.7 for one wave period in a normal heart size is less than 1 hour of CPU time using FHN-like electrophysiological model. The communication can be minimized when parallelized along azimuthal direction. Computational results show the model has a very good scalability. Governing Equations Transmembrane potential propagation W.F. Witkowksi, et al., Nature 392, 78 (1998) u Cm ( Du ) I m t From idealized to fully realistic geometrical modeling Anatomical canine ventricular model Cm: capacitance per unit area of membrane D: diffusion tensor u: transmembrane potential Im: transmembrane current Phase Singularities Tips and filaments are phase singularities that act as organizing centers for spiral (2D) and scroll (3D) dynamics, respectively, offering a way to quantify and simplify the full spatiotemporal dynamics. Courtesy of A. V. Panfilov, in Physics Today, Part 1, August 1996 Minimally realistic model of LV for studying electrical wave propagation in three dimensional anisotropic myocardium that adequately addresses the role of geometry and fiber architecture and is: Simpler and computationally more tractable than fully realistic models Scaling of Ventricular Turbulence Finding all tips Transmembrane current, Im, described by simplified FitzHugh-Nagumo type dynamics v: gate variable Parameters: a=0.1, m1=0.07, v v 1 v ku(u a 1 m2=0.3, k=8, e=0.01, Cm=1 t 2 u Choose an unmarked tip as current tip Add current tip into a new filament, marked as the head of this filament set reversed=0 Set reversed=1 Is the distance smaller than a certain threshold? Yes The average filament length, normalized by average heart thickness, versus heart size Set the closest tip as current tip Definition: Distance between two tips Is revered=0? (1) If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity No Diffusion Tensor Yes More feasible for incorporating realistic electrophysiology, electromechanical coupling, Mark the current tip Log(total filament length) and Log(filament number) versus Log(heart size) No Set the head of current Yes filament as current tip Easily parallelizable and with good scalability Add current tip into current filament Find the closest unmarked tip I m ku(u a)(u 1) uv J.P. Keener, et al., in Cardiac Electrophysiology, eds. D. P. Zipes et al. (1995) CPUs The results for filament number agree to within error bars for dr=0.7 and dr=0.5. The result for dr=1.1 is slightly off, which could be due to the filament finding algorithm. outer surface Fiber path equation Patch size: 5 cm x 5 cm Time spacing: 5 msec Rectangular slab Numerical Convergence Parallelization Are there any unmarked tips? These results are in agreement with those obtained with the fully realistic canine anatomical model, using the same electrophysiology. A. V. Panfilov, Phys. Rev. E 59, R6251 (1999) (2) Otherwise, the distance is the distance along the fiber surface No End Transformation matrix R Filament finding algorithm Conclusions and Future Work Model Construction Local Coordinate Dlocal D// 0 0 0 D p1 0 0 0 D p 2 We have constructed and implemented a minimally realistic fiber architecture model of the left ventricle for studying electrical wave propagation in the three dimensional myocardium. Lab Coordinate Dlab R 1 Dlocal R Early dissection results revealed nested ventricular fiber surfaces, with fibers given approximately by geodesics on these surfaces. t=2 Our model adequately addresses the geometry and fiber architecture of the LV, as indicated by the agreement of filament dynamics with that from fully realistic geometrical models. Numerical Implementation Fibers on a nested pair of surfaces in the LV, from C. E. Thomas, Am. J. Anatomy (1957). Working in spherical coordinates, with the boundaries of the computational domain described by two nested cones, is equivalent to computing in a box. Our model Adopted Nested cone geometry fiber surfaces the fiber paths are both geodesics on fiber surfaces and circumferential at midwall. Crossection along azimuthal direction Standard centered finite difference scheme is used to treat the spatial derivatives, along with first-order explicit Euler time-stepping. t = 999 The filament finding results. The left pictures show the simulation at time=2 and time=999. The right pictures show the filament finding results, corresponding to the scroll waves. Our model is computationally more tractable, allowing reliable numerical studies. It is easily parallelizable and has good scalability. As such, it is more feasible for incorporating Realistic electrophysiology Biodomain description of tissue Electromechanical coupling