Because the simplicity of states and superior computational speed resulted from rules, cellular automata have been applied in simulations of myocardial electrical activity in the heart, but such simplistic and rule-based approaches cannot always properly capture the shape of transmembrane potentials. Every cell has the same rule for updating, based on the states in its neighborhood. A cellular automaton is a discrete model which usually consists of a regular grid of cells, each in one of a finite number of states. A number of computational models have been developed to simulate the macroscopic electrical propagation process, such as cellular automata and reaction-diffusion systems. There have been efforts in simulating myocardial electrical activations using computational models with known physical parameters, including the source intensities and locations, material properties, and boundary conditions, because these simulations can help to understand the measurement data, suggest new experiments, and provide insights into the basic mechanism of electrical activity in the heart. Propagation of electrical activations inside the myocardium is a highly complicated process mainly due to the fibrous structure of myocardium, as shown in many experiments. Myocardial contraction is driven by a sequence of propagating electrical activations throughout the myocardium. The derivation of this solution technique is presented along a series of numerical experiments and a solution of monodomain model using a FitzHugh-Nagumo (FHN) membrane model in a canine ventricular model and a human-heart model which is constructed from digitized virtual Chinese dataset. After the monodomain equations are converted to their Galerkin weak form and solved using EFGM, the propagation of myocardial activation can be simulated over the meshfree particle representation. Fiber orientations and other material properties of myocardium are then attached to sample nodes according to their geometrical locations, and over the meshfree particle representation spatial variation of these properties is approximated using the shape function of EFGM. In our framework the geometry of myocardium is first defined by a meshfree particle representation that is, a sufficient number of sample nodes without explicit connectivities are placed in and inside the surface of myocardium. An element-free Galerkin method (EFGM) is proposed to simulate the propagation of myocardial electrical activation without explicit mesh constraints using a monodomain model.
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