Combining initialization and solution inverse methods for inverse electrocardiography

Abstract

Many methods have been developed over the last 25 years for the inverse problem of electrocardiography. The 2 main classes of such methods that have been of recent interest use 2 alternative equivalent source formulations. Activation-based methods were use as a source model a dipole layer on the heart surface with identical known transmembrane potential waveforms at each node. Potential-based methods use as a source model potential values at each time instant at each node on a closed surface surrounding the sources. The former is highly constrained, nonlinear, and nonconvex, whereas the latter is loosely constrained but linear. Recent interest in activation-based iterative solutions has focused on initialization. Much recent interest in potential-based solutions has focused on devising constraints, which impose physiologically meaningful waveform shapes. Here we describe work using one method to initialize a second for both formulations. For activation-based solutions, we initialize by embedding the solution set for the nonconvex optimization problem, which results from fixing waveforms and allowing only activation sample times to vary, into a less constrained solution set, thus formulating a convex problem. To do so, we replace the hard constraints on waveform shape by softer constraints. We solve this convex optimization problem combining constraints on the residual with the waveform shape constraints and imposing smoothness of the activation surface. The resulting optimal solution will not be a feasible set of activation waveforms, but we use it to obtain a good initialization for subsequent Newton-type iterations for the nonlinear, nonconvex problem. For the potential-based problem, we use a recent method from our group, Wavefront-Based Potential Reconstruction. Wavefront-Based Potential Reconstruction uses a constraint based on segmenting the reconstructed potentials from a previous iteration or time step into 3 spatial regions: activated, not yet activated, and transition. The first 2 are modeled as spatially constant, the third by a specified transition function. This constraint is used in a Tikhonov-like regularization, producing wavefronts without the typical regularized smoothing. Wavefront-Based Potential Reconstruction requires an initialization to remove reference drift to create the 3-region segmentation. We use a different potential-based method, the “isotropy method” from Huiskamp and Greensite, to estimate the required initialization parameters. In both cases, the combination of these 2 inversemethods shows, in simulations based on measured canine heart surface data and data from the ECGSim software package, improved results compared with standard approaches.