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

An approach to the inverse problem of electrocardiography that involves an estimation of the electric potentials (double-layer equivalent sources) on the heartʹs 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 terms 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 than directly measured and therefore noisy) 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 the human torso. By calculating singular values of the transfer matrices for this model, we found that one for the new formulation is 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 torsoʹs 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 of 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.