1. Numerical Implementation Diffusion Tensor Governing Equations From Idealized to Fully- Realistic Geometrical modeling Phase Singularities Model Construction Conclusions and Future Work 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. 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. 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 ParallelizationMotivation Ventricular fibrillation (VF) is the main cause of sudden cardiac death in industrialized nations, accounting for 1 out of 10 deaths. Experimental evidence strongly suggests that self- sustained waves of electrical wave activity in cardiac tissue are related to fatal arrhythmias. Electrical Wave Propagation in a Minimally Realistic Fiber Architecture Model of the Left Ventricle Xianfeng Song, Sima Setayeshgar Department of Physics, Indiana University, Bloomington W.F. Witkowksi, et al., Nature 392, 78 (1998) Patch size: 5 cm x 5 cm Time spacing: 5 msec 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. Rectangular slabAnatomical canine ventricular model J.P. Keener, et al., in Cardiac Electrophysiology, eds. D. P. Zipes et al. (1995) Courtesy of A. V. Panfilov, in Physics Today, Part 1, August 1996 Construct 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  Easily parallelizable and with good scalability  More feasible for incorporating realistic electrophysiology, electromechanical coupling Fibers on a nested pair of surfaces in the LV, from C. E. Thomas, Am. J. Anatomy (1957). LV Fiber Architecture Early dissection results revealed nested ventricular fiber surfaces, with fibers given approximately by geodesics on these surfaces. inner surfaceouter surface subject to: Fiber trajectory: Transmembrane potential propagation C m : capacitance per unit area of membrane D: diffusion tensor u: transmembrane potential I m : transmembrane current v: gate variable Parameters: a=0.1, m 1 =0.07, m 2 =0.3, k=8, e=0.01, C m =1 Local CoordinateLab Coordinate Transformation matrix R The communication can be minimized when parallelized along azimuthal direction. Computational results show the model has a very good scalability. CPUsSpeed up 21.42 ± 0.10 43.58 ± 0.16 87.61 ±0.46 1614.95 ±0.46 3228.04 ± 0.85 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. Finding all tips Add current tip into a new filament, marked as the head of this filament Find the closest unmarked tip End Choose an unmarked tip as current tip Is the distance smaller than a certain threshold? Set the closest tip as current tip Mark the current tip set reversed=0 Add current tip into current filament Set the head of current filament as current tip Is revered=0? Are there any unmarked tips? Set reversed=1 Definition: Distance between two tips (1)If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity (2)Otherwise, the distance is the distance along the fiber surface Yes No Yes No t = 2 t = 999 Fiber trajectories on nested pair of conical surfaces Fiber paths as:  geodesics on fiber surfaces  circumferential at midwall Transmembrane current, I m, described by simplified FitzHugh- Nagumo type dynamics Working in spherical coordinates, with the boundaries of the computational domain described by two nested cones, is equivalent to computing in a box. Standard centered finite difference scheme is used to treat the spatial derivatives, along with first-order explicit Euler time-stepping. Scaling of Ventricular Turbulence Log(total filament length) and Log(filament number) versus Log(heart size ) The average filament length, normalized by average heart thickness, versus heart size 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) Filament-finding Algorithm The left images show the simulation at time t=2 and t=999 units. The right images show the filament finding results, corresponding to the scroll waves. Peskin asymptotic model: first principles derivation of toroidal fiber surfaces and fiber trajectories as approximate geodesics. Fibers on a nested pair of surfaces in the LV, from C. E. Thomas, Am. J. Anatomy (1957). Fiber angle profile through LV thickness: Comparison of Peskin asymptotic model and dissection results, from C. S. Peskin, Comm. in Pure and Appl. Math. (1989). Cross-section along azimuthal direction Numerical Convergence The results for filament number agree to within error bars for spatial mesh size 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. 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. Filament Number and Filament Length versus Heart size The phase transition between fibrillation state to stable state 3D simulation tends to be more unstable than 2D simulation On the boundary between instable and stable parameters, the decrease in domain size could increase the instability