CNCS Graduate Certificate Recipient

Elizabeth Cherry

Thesis Title: A space-time Adaptive Mesh Refinement Method for Simulating Complex Cardiac Electrical Dynamics

Ph.D. Final Defense Date:  October 10, 2000

Ph.D. Dissertation Committee:

Henry S. Greenside (Chair)
William K. Allard
Craig S. Henriquez
Donald J. Rose
Xiaobai Sun

Abstract:

The electrical dynamics of heart muscle and associated diseases like fibrillation are extremely difficult to simulate efficiently because of the fast time and short length scales of the propagting action potentials. This thesis summarizes resaerch that should improve quantitative simulations of three-dimensional cardiac tissue. Our adaptive mesh refinement algorithm (AMRA), based on one originally developed for shock dynamics in fluids by Berger and collaborators, represents the solution to partial differential equations specifying the cardiac tissue dynamics on Cartesian grid patches using a simple geometry of d-dimensional boxes. Finite differences are used to approximate spatial derivatives in space and time, and an explicit time-integration method is used to advance in time. The AMRA automatically and dynamically increases or decreases the spatial and temporal resolutions locally to achieve a specified truncation error. This allows regions away fromthe highly localized wave fronts that form in highly localized wave fronts that form in highly excitable media to use much coarser resolution, thereby reducing computation time and memory requirements.

Our results show that for plane-wave and many-spiral states of the quantitatively-based Luo-Rudy 1 model in a large (8 cm x 8 cm) two-dimensional domain, the AMRA can achieve a factor of five reduction in computational effort and memory when compared to a comparable method using a uniform space-time mesh at the finest resolution. For point-stimulated waves in three dimensions, a speedup of 15 and a savings in memory of a factor of 10 is obtained using the AMRA. Analysis of temporal profiles and conduction velocities in 1d and spiral core trajectories in 2d show that the AMRA maintains the accuracy of a uniform mesh at the AMRA's highest resolution despite the decreased computational effort required. Applications to domains with anisotropy and inhomogeneities demonstrate that the AMRA handles these cases with the same accuracy as a Cartesian uniuform mesh at the AMRA's finest resolution. A detailed description of the AMRA and its associated data structures is given, along with a discussion of issues associated with the time-integration schemes available. Limitations associated with our algorithm and with our implementation and areas for further improvements are identified are discussed. Overall, adaptive mesh refinement is shown to be a promising technique for reducing the time and memory requirements associated with large-scale, long-time simulations of three dimensional cardiac muscle, and hence it deserves further study and improvement.