When is the Inverse Problem of Electrocardiography Well-Posed

Abstract

The inverse problem of electrocardiography [1] can be formulated as a reconstruction of maps of epicardial potentials and/or activation times from observed body-surface potential distributions. The problem is formulated as a Fredholm equation of the first kind, where the kernel has a finite L²-norm. Therefore, the body-surface potential data—being the “convolution” of the source with a well-behaved transfer function—is a smoothed version of the desired source and the least-squares solution is unstable in the presence of measurement and modelling noise. Thus, when the inverse problem of electrocardiography is formulated in this manner it is ill-posed, i.e. the solution does not depend continuously on the data, and regularization methods produce severely filtered solution candidates [2, 3, 4, 5]. Almost since the inverse problem of electrocardiography was first stated, mechanisms for incorporating inherent time constraints have been proposed, but none has led to a well-posed formulation. We will demonstrate in this chapter that the ill-posedness of the inverse problem of electrocardiography resolves when one supplements the usual bidomain assumptions with the assumption that phase 0 of the ventricular action potential is a step discontinuity. This extends work found in [6, 7, 8].