An asymptotic formula for boundary potential perturbations in a semilinear elliptic equation related to cardiac electrophysiology

In this paper, we provide a representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction in a simplified monodomain model describing the electrical activity of the heart. We derive such a result in the case of a nonlinear problem. Our long-term goal is the solution of the inverse problem related to the detection of regions affected by heart ischemic disease, whose position and size are unknown. We model the presence of ischemic regions in the form of small inhomogeneities. This leads to the study of a boundary value problem for a semilinear elliptic equation. We first analyze the well-posedness of the problem establishing some key energy estimates. These allow us to derive rigorously an asymptotic formula of the boundary potential perturbation due to the presence of the inhomogeneities, following an approach similar to the one introduced by Capdeboscq and Vogelius in [A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, Math. Model. Numer. Anal. 37 (2003) 159-173] in the case of the linear conductivity equation. Finally, we propose some ideas of the reconstruction procedure that might be used to detect the inhomogeneities.