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

This Talk: Outline  Goal  Model Construction  Results  Conclusions Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

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

Model Construction - Background  Anatomical structure  Picture goes here  Peskin Asymptotic Model  C. S. Peskin, Communications on Pure and Applied Mathematics 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 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

Model construction – Nested Cone Approximation  Nested cone geometry and fiber surfaces Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Fiber paths on the inner sheet Fiber paths on the outer sheet  Fiber paths  To be geodesics  To be circumferential at the mid wall

Governing equations  Governing equation Cm: capacitance per unit area of membrane D: diffusion tensor u: transmembrane potential  Transmembrane current Im was described using a simplified excitable dynamics equations of the FitzHugh-Nagumo type (R. R. Aliev and A. V. Panfilov, Chaos Solitons Fractals 7, 293(1996)) 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

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

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

Parallelization  The communication can be minimized when parallelized along azimuthal direction  Computational results show the model has a very good scalability CPUsSpeed up ± ± ± ± ± 0.85 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

Tips, Filaments  Tip: The point around which the spiral wave (in 2 dimensions) are generated  Filament: The core around which that the scroll wave (in 3 dimensions) rotates Color denotes the transmembrane potential. The movie shows the spread of excitation in the cone shaped model from time=0-30. Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

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

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

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

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

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

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

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

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

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

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

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

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

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

Filament finding algorithm Repeat until consuming all tips “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

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

Numerical 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=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 normal heart size is less than 1 hours of cpu time using our electro-physiological model Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

Agreement with fully realistic model Both filament length The results agree with the simulation on the fully realistic model using the same electro-physiological model (A. V. Panfilov, Phys. Rev. E 59, R6251(1999)) Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore 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 normalized by average heart thickness versus the heart size. It clearly show that the this average tends to be a constant

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