The contraction of the heart is preceded and caused by a cellular electro-chemical reaction, causing an electrical field to be generated. Realistic models for this process include a set of complicated ordinary differential equations (ODEs) to describe the flow of ions through the cell membrane. Numerical simulations of the electrical activity of the heart involves solving a set of partial differential equations (PDEs) on a computational grid, in addition to solving one such ODE system for each node in the grid. For realistic simulations the number of nodes is about 200000 in 2D and more than 60 million in 3D, making the solution of the ODE systems an extremely CPU intensive task. For our simulations, we have chosen to use a model developed Winslow et al becauce it is well documented and accurate. This model consists of 31 coupled non-linear ODEs. However, if the individual equations are treated one by one, holding all other variables constant, 25 of the equations are in fact linear. In order to utilize this we developed a number of sequential solvers involving steps based on the analytical solution of linear ODE's in combination with a Crank-Nicholson discretization. We present numerical results for a number of different combinations of these methods, and compare their performance to more traditional methods. Finally, we present the utilization of a Singly Diagonally Implicit Runge-Kutta (SDIRK) method to solve the ODE systems. Because the ODEs are coupled to a set of PDEs, the implementation of the SDIRK solver needs to be customized for our specific application. Furthermore, because of our strong focus on computational efficiency, some of the most CPU intensive tasks of the solution method are optimized to the specific ODE system.

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Helge Holden <holden@math.ntnu.no>

2000-10-06 15:22