Long-Time Simulations on High Resolution Meshes to Model Calcium Waves in a Heart Cell

Abstract

A model for the flow of calcium on the scale of one heart cell is given by a system of time-dependent reaction-diffusion equations coupled by nonlinear reaction terms. Calcium ions enter into the cell at release units distributed throughout the cell and then diffuse. At each release unit, the probability for calcium to be released increases along with the concentration of calcium, thus creating a feedback loop of waves regenerating themselves repeatedly. The validation of this model requires simulations on the time scale of several repeated waves and on the spatial scale of the entire cell. This requires long-time studies on spatial meshes that need to have a high resolution to resolve the positions of the calcium release units throughout the entire cell. We detail the development of a special-purpose numerical method and parallel implementation for this problem. Parallel performance studies demonstrate the scalability of the implementation on a distributed-memory cluster with low-latency interconnect. Convergence studies verify convergence to analytical expectations and confirm the appropriateness of all numerical parameters. Application studies on the desired time and length scales confirm that the model exhibits the desired feedback mechanism for calcium currents through the release units at suitable high levels, but the long-time studies demonstrate also that the current model with its present parameters leads to excessive calcium concentrations over time. This phenomenon could only be observed using a computational method able to reach laboratory scale final times for a domain on the scale of a complete cell.