1. Electrical Wave Propagation in a Minimally Realistic Fiber Architecture Model of the Left Ventricle Xianfeng Song, Department of Physics, Indiana University Sima Setayeshgar, Department of Physics, Indiana University March 17, 2006

2. This Talk: Outline  Motivation  Model Construction  Numerical Results  Conclusions and Future Work Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

3. 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 self- sustained waves of electrical wave activity in cardiac tissue are related to fatal arrhythmias.  Mechanisms that generate and sustain VF are poorly understood.  Conjectured mechanism: Breakdown of a single spiral (scroll) wave into a disordered state, resulting from various mechanisms of spiral wave instability. W.F. Witkowksi, et al., Nature 392, 78 (1998) Patch size: 5 cm x 5 cm Time spacing: 5 msec

4. Focus of this work Distinguish the role in the generation of electrical wave instabilities of the “passive” properties of cardiac tissue as a conducting medium  geometrical factors (aspect ratio and curvature)  rotating anisotropy (rotation of mean fiber direction through heart wall)  bidomain description (intra- and extra-cellular spaces treated separately) from its “active” properties, determined by cardiac cell electrophysiology.

5. From idealized to fully realistic geometrical modeling Rectangular slabAnatomical canine ventricular model 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, 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 bidomain description

6. LV Fiber Architecture  Early dissection results revealed nested ventricular fiber surfaces, with fibers given approximately by geodesics on these surfaces. Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore 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).  Peskin asymptotic model: first principles derivation of toroidal fiber surfaces and fiber trajectories as approximate geodesics

7. Model Construction  Nested cone geometry and fiber surfaces Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore  Fiber paths  Geodesics on fiber surfaces  Circumferential at midwall subject to: Fiber trajectory: Fiber trajectories on nested pair of conical surfaces: inner surfaceouter surface

8. Governing Equations  Transmembrane potential propagation  Transmembrane current, I m, described by simplified FitzHugh-Nagumo type dynamics* v: gate variable Parameters: a=0.1,  1 =0.07,  2 =0.3, k=8,  =0.01, C m =1 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore * R. R. Aliev and A. V. Panfilov, Chaos Solitons Fractals 7, 293 (1996) C m : capacitance per unit area of membrane D: diffusion tensor u: transmembrane potential I m : transmembrane current

9. Numerical Implementation  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. Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

10. Diffusion Tensor Local CoordinateLab Coordinate Transformation matrix R Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

11. Parallelization  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 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

12. Phase Singularities Color denotes the transmembrane potential. Movie shows the spread of excitation for 0 < t < 30, characterized by a single filament. Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore 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.

13. Filament-finding Algorithm Find all tips Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore “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

14. Filament-finding Algorithm Random choose a tip “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 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

15. Filament-finding Algorithm Search for the closest tip “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 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

16. Filament-finding Algorithm Make connection Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore “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

17. Filament-finding Algorithm Continue doing search “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 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

18. Filament-finding Algorithm Continue “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 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

19. Filament-finding Algorithm Continue “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 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

20. Filament-finding Algorithm Continue “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 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

21. Filament-finding Algorithm The closest tip is too far “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 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

22. Filament-finding Algorithm Reverse the search direction “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 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

23. Filament-finding Algorithm Continue “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 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

24. Filament-finding Algorithm Complete the filament “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 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

25. Filament-finding Algorithm Start a new filament “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 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

26. Filament-finding Algorithm Repeat until all tips are consumed “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 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

27. Filament-finding result FHN Model: t = 2 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore t = 999

28. Numerical Convergence Filament Number and Filament Length versus Heart size  The results for filament length agree to within error bars for three different mesh sizes.  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.  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. Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

29. Scaling of Ventricular Turbulence Both filament length 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) Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Log(total filament length) and Log(filament number) versus Log(heart size) The average filament length, normalized by average heart thickness, versus heart size

30. 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 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore