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 a least squares problem and solved by a projected version of the Levenberg–Marquardt method. Moreover, we analyze the well-posedness 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 (J. Math. Biol. 79, 2033–2068, 2019). We are able to reconstruct the activation instants as well as the locations of the activations with high accuracy relative to the noise level.
Highlights
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 a least squares problem and solved by a projected version of the Levenberg–Marquardt method
In the numerical examples we 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
Summary
This work is concerned with an inverse problem in cardiac electrophysiology. In particular, the activation instants of the excitation wave in the myocardium are estimated from the arrival times of the wave at the epicardium. To briefly explain the problem we recall that the electro-physiologic activity of the heart is often modeled using the bidomain equations, whose numerical solution is very expensive. As noted above it has become a standard procedure to rely for the mathematical description of the excitation process in the myocardium on various forms of Eikonal equations, see for instance [10, 17, 19] These are reduced forms of the bidomain equations, a reaction diffusion system which describes the electrical activity of the heart. We calculate the gradient of the least squares cost functional with respect to these activation instants It can be expressed in terms of the normal derivative of the solution to the adjoint state equation on the surface of activation sites. The numerical examples illustrate the feasibility of the approach and are carried out on the 2D unit square with artificial data
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