On an inverse problem with applications in cardiac electrophysiology

In this paper, we study the monodomain model of cardiac electrophysiology, which is widely used to describe the propagation of electrical signals in cardiac tissue. The forward problem, described by a reaction-diffusion equation coupled with an ordinary differential equation in a domain containing a perfectly insulating region, is first analysed to establish its well-posedness under standard assumptions on the conductivity and ionic current terms. We then investigate the inverse problem of identifying perfectly insulating regions within the cardiac tissue, which serve as mathematical representations of ischemic areas. These regions are characterised by a complete lack of electrical conductivity, impacting the propagation of electrical signals. We prove that the geometry and location of these insulating regions can be uniquely determined using only partial boundary measurements of the transmembrane potential. Our approach combines tools from elliptic and parabolic PDE theory, Carleman estimates, and the analysis of unique continuation properties. These results contribute to the theoretical understanding of diagnostic methods in cardiology.