Generalized inverse problem in electrocardiography: theoretical, computational and experimental results

The electrical behavior within the heart gives rise to electrostatic potentials on the body surface. These potentials are related to potentials on the epicardial surface and are dependent on the geometry and resistive properties of the passive volume conductor between the heart and body surface. The determination of detailed information about the heart from noninvasive electrical measurements taken on the body surface is defined as the inverse problem in electrocardiography. In this study the generalized inverse problem in electrocardiography is solved for an anisotropic inhomogeneous volume conductor utilizing epicardial and torso potentials. The strategy includes a multicomponent computer model which consists of 1) a finite element program to solve the electrocardiographic field equation by utilizing the Ritz technique to reformulate the differential equation into a global integral equation; 2) a penalty method algorithm that is applied to the Dirichlet condition to assure accuracy at the boundaries; and 3) a local Tikhonov regularizing algorithm, used to constrain the solution by restoring continuous dependence of the solution on the data. This is achieved by utilizing a general discrepancy principle that makes use of measurement errors of torso potentials and discretization errors to optimize the choice of the regularization parameter. Objectives included theoretical, computational, and experimental studies of the effectiveness of the homogeneous assumption using a realistic geometry as well as optimization of the a priori regularization parameter. The studies were carried out using a concentric spheres model and a realistic torso model, which is the computational equivalent of an electrolytic tank. Forward and inverse calculations were performed using both models. It was shown that the multi-component computer model was effective for solving forward and inverse problems in anisotropic, inhomogeneous media exhibiting realistic geometry. It was also shown that the homogeneous assumption is not valid for recovering detailed electrical information on the epicardial surface through an anisotropic, inhomogeneous torso. The difficulty lies, not in the new ideas, but in escaping the old ones, which ramify, for those brought up as most of us have been, into every corner of our minds.