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  Goal  Model Construction  Results  Discussion and future plan Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

3. Minimally Realistic Model: Goal  Construct a minimally realistic model of the left ventricle for studying electrical wave propagation in the three dimensional anisotropic myocardium.  Adequately addresses the role of geometry and fiber architecture on electrical activity in the heart  Simpler and computationally more tractable than fully realistic models  More feasible to incorporate contraction into such a model  Easy to be parallelized and has a good scalability Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

4. Anatomical Heart  A nested layered geometry for the left ventricle  A single macroscopic fiber bundle starting at the basal plane outside the midwall traverses down toward the apex on an outer surface, and at some point before reaching the apex, changes direction, traverses back along an inner surface reinserting at the basal plane inside the midwall. Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

5. Nested Cone Approximation A simple nested cone geometry, represents the left ventricle which does not incorporate the valves.  i =8   e =16  Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

6. Fiber construction  Construction principles  Peskin Asymptotic Model (Derived by Peskin in 1996) The fiber paths are approximate geodesics on the fiber surfaces.  Requiring the fibers to be circumferential where the double sheets meet at midwall  Euler-Lagrange equations (f: fiber trajectory):  Result Fiber paths on the inner sheet Fiber paths on the outer sheet Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

7. Governing equations  Governing equation (a conventional parabolic partial differential equation)  Transmembrane current Im was described using a simplified excitable dynamics equations of the FitzHugh-Nagumo type (R. R. Aliev and A. V. Panfilov, 1996) 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

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 finite differencing is used to treat the spatial derivatives, along with explicit Euler time-stepping Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

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

10. Parallelize the code  The communication can be minimized when parallelized along the theta direction  Computational results show the model has a very good scalability CPUs Speed up 21.40 43.65 87.80 16 15.50 32 29.20 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

11. Finding the filament 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

12. Result - Simulation FHN Model: Color denotes the u variable in FHN model. The movie shows the spread of excitation in the cone shaped model from time=0-30. Filament initially time=2 The filament after breaking up time=999 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

13. Result - Convergence Filament number and Filament length vs Heart size  The results of filament length agree within error bar for three different mesh sizes  The results of filament number agree within error bar between dr=0.7 and dr=0.5. The result for dr=0.5 is slightly off, which could be due to the filament finding algorithm  The computation time for dr=0.7 for one cycle in normal heart size is approxamately 3 hours of cpu time using our electro-physiological model Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

14. Result - Filaments Both filament length Scaling of ventricular turbulence. The log of the total length and the log of the number of filaments both have linear relationship with log of heart size, but with different scale factor. The average filament length/avearge heart thickness versus the heart size. It clearly show that the this average tends to be a constant

15. Discussion and 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 very good scalability Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore