Formulation of the Transfer Matrix of the Invers Electrophysiology of the Heart Activation Inside Homogeneous and Inhomogeneous Volume Conductor

This work addresses the mathematical modeling of the Inverse Problem in cardiac electrophysiology. The Forward Problem aims to compute the potential field on the body surface generated by electrical sources within the heart, while the Inverse Problem seeks to reconstruct these internal sources based on body surface measurements. This research focuses on solving the Inverse Problem by formulating it in terms of transmembrance potentials (TMPs) rather than current density vectors, which significantly reduces the solution domain and computational cost. Three modeling scenarios are explored: an isotropic volume source in an infinite homogeneous volume conductor, an anisotropic source in a homogeneous medium, and an anisotropic source in an inhomogeneous medium. Each scenario leads to a different lead field transfer matrix formulation, ultimately enabling the reconstruction of cardiac TMPs from body surface potential maps (BSPMs). The linearity of the resulting system validates the application of advanced linear regularization methods.