The Luo-Rudy phase I bidomain model is numerically studied with the adaptive mesh refinement (AMR) algorithm. It is well-known that the bidomain model consists of a singularly perturbed reaction-diffusion system coupled with a set of nonlinear stiff ordinary differential equations (Luo-Rudy phase I membrane dynamics). The AMR algorithm first uses an operator splitting technique to separate linear diffusion from nonlinear stiff reactions in the bidomain model. The reactions are integrated adaptively with a singly diagonally implicit Runge-Kutta method, which is both accurate and absolutely stable. The decoupled linear diffusion is implicitly discretized with a conforming finite element approximation on adaptively refined grids, which are locally, dynamically and automatically created by the AMR algorithm. The resulting composite grid equations are solved by a standard multigrid iteration technique.
In the simulation studied, the computational domain is the unit square (1cm x 1cm). The fibers are aligned with the positive diagonal. The intracellular conductivity in the principal axis is 6.0 mS/cm; the extracellular conductivity in the principal axis is 24.0 mS/cm. The intracellular conductivity has an anisotropy ratio 6.0 : 0.6 = 10; and the extracellular conductivity has an anisotropy ratio 24.0 : 12.0 = 2. The membrane capacitance is chosen to be 1.0 muF/cm^2. The surface-to-volume ratio in the bidomain equations is set as 4036.5 cm^{-1}. We fix the extracellular potential at the lower-left corner of the domain to be ZERO, in order to get a unique solution for the bidomain equations. The coarsest level grid in the AMR algorithm consists of 16 elements. That is, the unit square domain is initially partitioned into a 4 by 4 grid. Totally, seven AMR levels with refinement ratio two (r=2) are used in the simulation.
Some snapshots after a stimulus is applied at the the center of the domain are presented below. The first five snapshots are the adaptively refined grids (on the left) and the transmembrane potential (on the right). Other snapshots are the transmembrane potential (on the left) vs. the extracellular potential (on the right). The same hot-cold color map is used for both transmembrane and extracellular potentials. Blue color represents a low voltage (-90 mV). Red color denotes a high voltage (20 mV). White color in the colormap means voltage values greater than 20 mV. Only those iso-contours with values in the range of -20 mV to 20 mV are plotted.
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