Investigating Transmembrane Current Source Formulation for Solving the ECG Inverse Problem

The ElectroCardioGraphy (ECG) inverse problem has gained a rising interest over the past decades as a tool to non-invasively retrieve cardiac electrical information from body surface measurements. However it is still very challenging due to its ill-posed nature and needs to be regularized. Many formulations of this problem are well established in the cardiac electrophysiology community: equivalent dipole, potential, and activation time-based formulations. Surprisingly, the ECG inverse problem has never been investigated in terms of transmembrane current source model. The latter are derived from the bidomain theory in terms of transmembrane potentials or local activation time, but not in terms of transmembrane current sources. In this paper, we propose a volumetric transmembrane current based formulation of the ECG inverse problem and compare it with the epicardial potential based approach. This was done by studying the performance of two L1 based and two L2 based regularizations for solving the inverse problem. In [1], the Alternating Direction Method of Multipliers (ADMM) framework combined with the discrepancy principle is successfully used to update the regularization parameter in sparse regularization. We adopt the same strategy and adapt this scheme for Tikhonov regularization. Performance of each formulation is assessed using realistic data generated by a 3D coupled heart-torso propagation model in an idealized geometry. Results in terms of localization error and electrical mappings visualization are presented. It is shown on the one hand that the transmembrane current formulation gives promising results and deserves further investigation. On the other hand, the discrepancy principle applied to a split variable problem was successfully applied to Tikhonov regularization.