PDE-induced connection of moving frames for the Atlas of the cardiac electric propagation on 2D atrium
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As another critical implementation of moving frames for partial differential equations, this paper proposes a novel numerical scheme by aligning one of three orthogonal unit vectors at each grid point along the direction of a wave propagation to construct an organized set of frames, called a connection. This connection characterizes the geometry of wave propagation depending on (1) the initial point, (2) type of wave, and (3) shape of the domain with conduction properties. The constructed connection is differentiated again to derive the Riemann curvature tensor of orthonormal bases corresponding to important physical and biological meanings in wave propagation. As a practical application, the proposed scheme is applied to diffusion-reaction equations to obtain the Atlas, or a geometric map with connections, of an atrium with cardiac fibers, for the quantitative and qualitative analysis of cardiac action potential propagation, which could contribute to the clinical and surgical planning of atrial fibrillation.
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