A Fast Solver for the Complex Eikonal Equation to Initiate Cardiac Arrhythmias

Tomás Banduc, Simone Pezzuto, Francisco Sahli Costabal

Abstract: Atrial fibrillation (AF) is the most common type of arrhythmia and a main cause of cardio-embolic stroke. The requirement for less invasive and optimal strategies for the targeted treatment of AF, such as radio-frequency ablation, has brought in silico models at the forefront of research in the patient-specific setting. Typically, simulations of sustained episodes of arrhythmia are performed with the monodomain model of cardiac electrophysiology by triggering spiraling fronts with specific pacing protocols. Eikonal-based solvers have been created to initiate these patterns of reentry. However, they rely on iterative algorithms that can become computationally expensive and do not manage synchronized patterns of activation, thus failing to broaden simulation capabilities to achieve more structurally complex events of arrhythmia. In this work, we propose a non-iterative method to interpolate initial conditions for the initiation of stable AF episodes. Our method is based on the domain parametrization given by the logarithmic map on Riemannian manifolds, which we use to construct a closed-form solution for the complex eikonal equation. Our method is simple, fast to compute and comparable in terms of stability and maintenance of spirals to traditional solvers. An example code for the proposed solver is available at https://github.com/tbanduc/logmap-phase-demo.git.