On the Correctness of the Transmembrane Potential Based Inverse Problem of ECG

Solving the inverse problem of electrocardiography in terms of transmembrane potentials (TMPs) is considered to be a promising approach in noninvasive imaging of cardiac electrical activity. However, the correctness of the statement of this problem (i.e. existence, uniqueness and stability of the solution) was not explored with sufficient mathematical rigor. In this work, we aim to eliminate this gap and provide some mathematical background for the facts that are well adopted in the engineering community. We consider the inverse problem of TMP reconstruction from known body surface potentials using two models: First, a 'microscopic' model assuming knowledge of geometrical cellular contours. Furthermore, we analyze a conventional isotropic steady-state version of the bidomain model under the assumption that the body and myocardial domains are enclosed by an infinitely smooth boundary. The extracellular potentials, TMP and the body potentials are supposed to be functions belonging to a wide class of Sobolev-Slobodetckij functional spaces. The main conclusions of this work can be formulated as follows: For the cellular model, TMPs can be uniquely found on the surface of the intracellular domain. For the more practical, bidomain model, the null-space of the inverse problem includes a constant TMP on the myocardial surface and an infinite set of TMP distributions in the myocardial domain. Therefore, the TMPs inside the myocardium are not uniquely defined, while the TMPs on the myocardial surface can be reconstructed up to an arbitrary additive constant. The obtained results can be used as a sound basis for creating numerical methods for noninvasive mapping of the heart.