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Ray Tracing Boundary Value Problems: Simulation and SAFT Reconstruction for Ultrasonic Testing
(2016)
Spectral Deferred Correction methods for adaptive electro-mechanical coupling in cardiac simulation
(2017)
We investigate spectral deferred correction (SDC) methods for time stepping
and their interplay with spatio-temporal adaptivity, applied to the solution
of the cardiac electro-mechanical coupling model. This model consists
of the Monodomain equations, a reaction-diffusion system modeling the cardiac
bioelectrical activity, coupled with a quasi-static mechanical model describing
the contraction and relaxation of the cardiac muscle. The numerical
approximation of the cardiac electro-mechanical coupling is a challenging
multiphysics problem, because it exhibits very different spatial and temporal
scales. Therefore, spatio-temporal adaptivity is a promising approach
to reduce the computational complexity. SDC methods are simple iterative
methods for solving collocation systems. We exploit their flexibility for combining
them in various ways with spatio-temporal adaptivity. The accuracy
and computational complexity of the resulting methods are studied on some
numerical examples.
Conduction velocity in cardiac tissue is a crucial electrophysiological parameter for arrhythmia vulnerability. Pathologically reduced conduction velocity facilitates arrhythmogenesis because such conduction velocities decrease the wavelength with which re-entry may occur. Computational studies on CV and how it changes regionally in models at spatial scales multiple times larger than actual cardiac cells exist. However, microscopic conduction within cells and between them have been studied less in simulations. In this work, we study the relation of microscopic conduction patterns and clinically observable macroscopic conduction using an extracellular-membrane-intracellular model which represents cardiac tissue with these subdomains at subcellular resolution. By considering cell arrangement and non-uniform gap junction distribution, it yields anisotropic excitation propagation. This novel kind of model can for example be used to understand how discontinuous conduction on the microscopic level affects fractionation of electrograms in healthy and fibrotic tissue. Along the membrane of a cell, we observed a continuously propagating activation wavefront. When transitioning from one cell to the neighbouring one, jumps in local activation times occurred, which led to lower global conduction velocities than locally within each cell.
The locality of solution features in cardiac electrophysiology simulations calls for adaptive methods. Due to the overhead incurred by established mesh refinement and coarsening, however, such approaches failed in accelerating the computations. Here we investigate a different route to spatial adaptivity that is based on nested subset selection for algebraic degrees of freedom in spectral deferred correction methods. This combination of algebraic adaptivity and iterative solvers for higher order collocation time stepping realizes a multirate integration with minimal overhead. This leads to moderate but significant speedups in both monodomain and cell-by-cell models of cardiac excitation, as demonstrated at four numerical examples.