Well-posed formulation of the inverse problem of electrocardiography

It has long been speculated that incorporation of available time constraints into the inverse electrocardiography problem could improve the accuracy of maps of epicardial potential or activation reconstructed from body surface potential measurements. However, all prior formulations of this problem have remained ill-posed, and the best way to utilize these constraints has been unclear. By making proper use of the timing information, we show that the inverse electrocardiography problem (for calculation of ventricular surface activation isochrones) is formally well-posed under anisotropic bidomain conditions and the assumption that ventricular muscle action potential phase 0 is a step discontinuity. In practical terms, this implies that non-regularized stable activation map solutions are possible if correlates of derived body surface potential derivative discontinuity times can be identified from the noisy analog signals, and only a small number of ventricular surface activation function extrema occur during a unit of time resolution defined by phase zero duration over the spatial extent of a bidomain point. We include a quasi-realistic numerical example illustrating the ease with which the extrema of the endocardial and epicardial activation maps are computed via Jump Maps derived from body surface potentials (this being the crucial step in rendering images of ventricular surface activation in this approach). The efficient signal processing algorithm used to accomplish this task is well suited to the setting of multiple extrema occurring during overlapping phase zero time intervals.