The inverse problem of electrocardiography: a solution in terms of single- and double-layer sources of the epicardial surface

Abstract

An approach to the inverse problem of electrocardiography that involves an estimation of the electric potentials (double-layer equivalent sources) on the heart's epicardial surface from the electrocardiographic potentials that are measurable on the body surface has received considerable attention. This report deals with a heretofore unexplored extension of this approach, one that yields, in addition to the electric potentials on the epicardial surface, the normal components of their gradients (single-layer equivalent sources). We show that this formulation has at least three advantages over the formulation in term of epicardial potentials alone: (1) single-layer equivalent sources, which reflect the flow of current across the epicardial surface, are well suited for the imaging of regional ischemia and infarction; (2) the transfer matrix linking the epicardial and body-surface potentials for this formulation is less ill conditioned than that for the formulation in terms of potentials alone; (3) the input vector for inverse calculations consists of spatially filtered (rather that directly measured and therefore noise) body-surface potentials. To establish the feasibility of this new formulation of the inverse problem and to compare it with the formulation in terms of potentials alone, we used a realistically shaped boundary-element model of human torso. By calculating singular values less ill conditioned. We then directly calculated epicardial and body-surface potentials for a single dipole located centrally and for three simultaneously active dipoles located eccentrically in the torso's heart region and used these results to test three methods that are prerequisites of a successful inverse solution: Tikhonov regularization, linearly constrained least squares, and an L-curve method. The feasibility of the new formulation was demonstrated by the fact that the method based on the linearly constrained least squares improved on overregularized Tikhonov solutions over a wide range of regularization parameters, and it yielded solutions that were more accurate than the best-possible Tikhonov solutions. Moreover, the L-curve solution procedure, which requires no a priori information about the solution, yielded slightly underregularized, but accurate, estimates for the optimal regularization parameter and the corresponding best-possible Tikhonov solution. Our results also showed that replacing--in the interest computational economy--quadrature formulas for the planar triangles with various approximate formulas for the nodes of the model reduces the accuracy of the inverse solution.