Spatial Heart Simulation and Adaptive Wave Propagation

Sudden cardiac death, caused mostly by ventricular fibrillation, is responsible for at least five million deaths in the world each year. Despite decades of research, the mechanisms responsible for ventricular fibrillation are not well understood. As most computational studies are limited primarily to planar simulations, experiments so far have not elucidated the mechanisms responsible for spatial phenomenon (Janse, Wilms-Schopman, & Coronel, 1995) of ventricular fibrillation. It would be important to understand how the onset of arrhythmias that cause fibrillation depends on details such as heart size (Winfree, 1994), geometry (Vetter & McCulloch, 1998; Panfilov, 1999), mechanical and electrical state, anisotropic fiber structure (Fenton & Karma, 1998), and inhomogeneities (Antzelevitch et al., 1999; Wolk, Cobbe, Hicks, & Kane, 1999). The main difficulty in development of a quantitatively accurate simulation of an entire three-dimensional human heart is that human heart muscle is a strongly excitable medium whose electrical dynamics involve rapidly varying, highly localized fronts (Cherry, Greenside, & Henriquez, 2000). In ventricular tissue (which is the most important to study) the width of a depolarization front is usually less than half mm and a simulation that approximates the dynamics of such a front requires a spatial resolution of ∆x ≤ 0.1 mm. Forasmuch the muscle in an adult heart has a volume of 250 cm3, and so a uniform spatial representation requires at least 2.5∙108 nodes. Taking into account that each node’s state is described with at least 50 floating number (with at least 4 byte resolution) the necessary storage space rises higher than 50GB (personal computers have no opportunity to handle such a huge amount of data in memory). It is known that the rapid depolarization of the cell membrane (this is the fastest event in heart functioning) blow over in few hundred microseconds. A reasonable resolution of this event requires a time step ∆t ≤ 25 microseconds. Since dangerous arrhythmias such as ventricular fibrillation may require several seconds to become established, the 1010 floating point numbers associated with the spatial representation would have to be evolved over 105 106 time steps. Such a huge uniform mesh calculation currently exceeds all existing computational resources (Cherry, Greenside, & Henriquez, 2003). Previous efforts to improve the efficiency of simulations have followed several approaches. The most common way of simplification is based on a reduced mathematical model that can reproduce some of the behavior observed in more complex models but with only a few coupled fields. One widely used example is the two-variable FitzHugh–Nagumo model (FitzHugh, 1961), which describes behavior of a general excitable medium. This model can be modified to approximate various types of cardiac dynamics (Berenfeld & Pertsov, 1999). Another example is a three-variable model developed by Fenton and Karma (Fenton & Karma, 1998) that is designed to reproduce the restitution curves of more complex cardiac models. Albeit these and other simplified models can reproduce many known features of cardiac electrical propagation, some electronic (Fenton, 2000) and drug effects cannot be handled properly. Accordingly, efficient algorithms for more quantitatively-based models are still desirable. The spatiotemporal structure of wave dynamics in excitable media suggests an automatically adjustable resolution in time and space. The basic idea of this improvement (Cherry et al., 2000, 2003) is deducted Sándor Miklós Szilágyi Hungarian Science University of Transylvania, Romania