A depth-averaged heart model for the inverse problem of cardiac electrophysiology

The inverse problem of cardiac electrophysiology is notoriously ill-posed, but is nonetheless extremely useful. In particular, it is difficult to reconstruct the transmembrane voltage in the volume of the heart, since an infinite dimensional space of transmembrane voltages can produce the same observation on the body surface. A widely used alternative is to consider only the outer surface of the heart, the epicardium, and solve a Cauchy problem for the Laplace equation in the torso domain. However, this approach only allows reconstruction of the extracellular potential on the epicardium, thus missing the information contained in the volume of the heart and in the transmembrane voltage. We propose a new methodology for reconstructing activation maps from torso surface data, which incorporates information from the myocardial volume, while solving a surface problem on the heart. We formulate a static forward model, derived from the bidomain model, by averaging equations in the heart. The averaged equations are coupled with the usual Laplace equations in the surrounding domains. For solving the inverse problem, this << depth-averaged >> forward model is used as a constraint in an optimal control problem that allows to recover depth-averaged transmembrane voltage and extracellular potential in the heart, corresponding to observations on the body surface. Activation maps are computed by post-processing the recovered signals in the heart. This method retains the ability to include the interactions between the heart, torso and blood cavities, while maintaining the simplicity of the usual inverse procedure. The inverse problem is solved with a simple linear system, using the Lagrangian formalism, and thus in a single iteration. We emphasize on the post-processing techniques for recovering activation maps. We observe that using a threshold on the transmembrane voltage allows to recover smoother and more accurate activation maps than with a maximal deflection method on the extracellular potential.