Computational and Applied Math Proseminar

Department of Mathematics and Statistics
Arizona State University

Thursday, January 23, 2003, 12:15 p.m. in GWC 604

Matthias Gobbert

Department of Mathematics and Statistics
University of Maryland, Baltimore County

Parallel Numerical Simulation of Calcium Waves in Human Heart Cells

Abstract The release of calcium ions in human heart cells is modeled by a system of non-linear reaction-diffusion equations, which describe the interaction of calcium ions with other chemical species and the effects of various cell processes on them. The release of calcium ions is modeled by a probabilistic forcing term involving Dirac delta functions. Several releases might self-organize into a wave of increasing calcium concentration throughout the cell. Improperly triggered calcium waves can lead to severe medical conditions, including irregular heartbeat.

A specialized finite element method is developed that allows for the use of memory-efficient matrix-free linear solves without assembling a system matrix. A coarse-grained parallel algorithm is obtained by decoupling the three equations through a semi-implicit time discretization.

Numerical results confirm that the method is second order convergent for a scalar test equation with smooth forcing term, as predicted by classical theory. Convergence of the method is also demonstrated for the full model including Dirac delta functions, for which the classical theory does not apply. Results on memory usage show that the coarse-grained parallelism allows for the solution over meshes with a finer resolution than possible on a single-processor machine.

At the heart of the talk will be a derivation of the spatial discretization of a parabolic reaction-diffusion equation using the finite element method. This general derivation is designed to be accessible to all graduate students.

For further information please contact: mittelmann@asu.edu