PDE-induced connection of moving frames for the Atlas of the cardiac electric propagation on 2D atrium

TL;DR

This paper introduces a novel numerical scheme that constructs a geometric connection to analyze wave propagation in cardiac tissue, aiding in understanding atrial fibrillation.

Contribution

It proposes a new method for aligning moving frames along wave directions to create a connection and curvature map of cardiac electric activity.

Findings

Successfully applied to cardiac diffusion-reaction equations

Generated an Atlas of atrial wave propagation

Provides insights for clinical planning of atrial fibrillation

Abstract

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.