Xianfeng Song, Department of Physics, Indiana University

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 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 for understanding VF: 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. From idealized to fully realistic geometrical modeling Rectangular slab Anatomical 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 a minimally realistic model of left ventricle for studying electrical wave propagation in the 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 contraction

5. Model Construction Early dissection revealed nested ventricular fiber surfaces, with fibers given approximately by geodesics on these surfaces. Peskin Asymptotic Model C. S. Peskin, Comm. on Pure and Appl. Math.42, 79 (1989) Conclusions: The fiber paths are approximate geodesics on the fiber surfaces When heart thickness goes to zero, all fiber surfaces collapse onto the mid wall and all fibers are exact geodesics Fibers on a nested pair of surfaces in the LV, from C. E. Thomas, Am. J. Anatomy (1957). Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

6. Model construction (cont’d) Nested cone geometry and fiber surfaces Fiber paths on the inner sheet Fiber paths on the outer sheet • Fiber paths • To be geodesics • To be circumferential at the mid wall Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

7. Governing Equations • Transmembrane potential propagation • Transmembrane current, Im, described by simplified FitzHugh-Nagumo type dynamics* • Cm: capacitance per unit area of membrane • D: diffusion tensor • u: transmembrane potential • Im: transmembrane current v: gate variable Parameters: a=0.1, m1=0.07, m2=0.3, k=8, e=0.01, Cm=1 * R. R. Aliev and A. V. Panfilov, Chaos Solitons Fractals 7, 293 (1996) Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

8. 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

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

10. Parallelization • The communication can be minimized when parallelized along azimuthal direction. • Computational results show the model has a very good scalability. Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

11. 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. 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

12. 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 Find all tips Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

13. 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 Random choose a tip Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

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

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

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

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

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

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

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

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

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

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

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

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

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

27. Numerical Convergence • 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 Filament Number and Filament Length versus Heart size Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

28. Scaling of Ventricular Turbulence The average filament length, normalized by average heart thickness, versus heart size Log(total filament length) and Log(filament number) versus Log(heart size) Both filament length The 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

29. Conclusion • We constructed a minimally realistic model of the left ventricle for studying electrical wave propagation in the three dimensional myocardium and developed a stable filament finding algorithm based on this model • The model can adequately address the role of geometry and fiber architecture on electrical activity in the heart, which qualitatively agree with fully realistic model • The model is more computational tractable and easily to show the convergence • The model adopts simple difference scheme, which makes it more feasible to incorporate contraction into such a model • The model can be easily parallelized, and has a good scalability Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore