A new method for regularization parameter determination in the inverse problem of electrocardiography

Computing the potentials on the heart's epicardial surface from the body surface potentials constitutes one form of the inverse problem of electrocardiography. An often-used approach to overcoming the ill-posed nature of the inverse problem and stabilizing the solution is via zero-order Tikhonov regularization, where the squared norms of both the surface potential residual and the solution are minimized, with a relative weight determined by a so-called regularization parameter. This paper looks at the composite residual and smoothing operator (CRESO) and L-curve methods currently used to determine a suitable value for this regularization parameter, t, and proposes a third method that works just as well and is much simpler to compute. This new zero-crossing method selects t such that the squared norm of the surface potential residual is equal to t times the squared norm of the solution. Its performance was compared with those of the other two methods, using three stimulation protocols of increasing complexity. The first of these protocols involved a concentric spheres model for the heart and torso and three current dipoles placed inside the inner sphere as the source distribution. The second replaced the spheres with realistic epicardial and torso geometries, while keeping the three-dipole source configuration. The final simulation kept the realistic epicardial and torso geometries, but used epicardial potential distributions corresponding to both normal and ectopic activation of the heart as the source model. Inverse solutions were computed in the presence of both geometry noise, involving assumed erroneous shifts in the heart position, and of Gaussian measurement noise added to the torso surface potentials. It was verified that in an idealistic situation, in which correlated geometry noise dominated the uncorrelated Gaussian measurement noise, only the CRESO approach arrived at a value for t. Both L-curve and zero-crossing approaches did not work. Once measurement noise dominated geometry noise, all three approaches resulted in comparable t values. It was also shown, however, that often under low measurement noise conditions none of the three resulted in an optimum solution.