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 • 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 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 scalable 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. fi=8 fe=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, m1=0.07,m2=0.3,k=8,e=0.01, Cm=1

8. Diffusion Tensor Transformation matrix R Local Coordinate Lab Coordinate

9. Numerical Implementation • Working in spherical polar coordinates, with the boundaries of the computational domain described by two nested cones, reduces the numerics 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

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

11. Finding all tips Choose an unmarked tip as current tip Add current tip into a new filament, marked as the head of this filament set reversed=0 Find the closest unmarked tip Add current tip into current filament Mark the current tip Is the distance smaller than a certain threshold? Set the closest tip as current tip set reversed=1 Is revered=0? Set the head of current filament as current tip Are there any unmarked tips? End Finding the filament • Definition: 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

12. Finding the filament Finding all tips Choose an unmarked tip as current tip Add current tip into a new filament, marked as the head of this filament set reversed=0 Add current tip into current filament Find the closest unmarked tip Mark the current tip Is the distance smaller than a certain threshold? Yes Set the closest tip as current tip Set reversed=1 No Is revered=0? • Definition: 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 Set the head of current filament as current tip Yes No Are there any unmarked tips? Yes No End

13. Result - Simulation FHN Model: Filament initially Color denotes the u variable in FHN model. The movie shows the spread of excitation in the cone shaped model. The filament after break up

14. Result - Convergence • As the mesh size decreases, the quantitive behavior convergent to a certain value • The result for dr=0.7 agree with the result for dr=0.5 within the error Filament number and Filament length vs Heart size

15. 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.

16. 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