An inverse problem involving a viscous Eikonal equation with applications in electrophysiology

TL;DR

This paper presents a method to reconstruct cardiac activation times and locations from boundary data using a viscous Eikonal equation, employing a Levenberg Marquardt approach and shape gradient techniques for high-accuracy results.

Contribution

It introduces a novel inverse problem formulation for cardiac activation reconstruction using a viscous Eikonal model and develops an efficient numerical solution with joint location and timing estimation.

Findings

High-accuracy reconstruction of activation times

Successful joint localization of activation sites

Robustness against noise in data

Abstract

In this work we discuss the reconstruction of cardiac activation instants based on a viscous Eikonal equation from boundary observations. The problem is formulated as an least squares problem and solved by a projected version of the Levenberg Marquardt method. Moreover, we analyze the wellposeness of the state equation and derive the gradient of the least squares functional with respect to the activation instants. In the numerical examples we also conduct an experiment in which the location of the activation sites and the activation instants are reconstructed jointly based on an adapted version of the shape gradient method from https://link.springer.com/article/10.1007/s00285-019-01419-3. We are able to reconstruct the activation instants as well as the locations of the activations with high accuracy relative to the noise level.