Regional regularization of the electrocardiographic inverse problem: a model study using spherical geometry

This study examines the use of a new regularization scheme, called regional regularization, for solving the electrocardiographic inverse problem. Previous work has shown that different time frames in the cardiac cycle require varying degrees of regularization. This reflects differences in potential magnitudes, gradients, signal-to-noise ratio (SNR), and locations of electrical activity. One might expect, therefore, that a single regularization parameter and a uniform level of regularization may also be insufficient for a single potential map of a single time frame because in one map there are regions of high and low potentials and potential gradients. Regional regularization is a class of methods that subdivides a given potential map into functional "regions" based on the spatial characteristics of the potential ("spatial frequencies"). These individual regions are regularized separately and recombined into a complete map. This paper examines the hypothesis that such regionally regularized maps are more accurate than if all regions were taken together and solved with an averaged level of regularization. In a homogeneous concentric spheres model, Legendre polynomials are used to decompose a torso potential map into a set of submaps, each with a different degree of spatial variation. The original torso map is contaminated with data noise, or geometrical error or both, and regional regularization improves the epicardial potential reconstruction by up to 25% [relative error (RE)]. Regional regularization also improves the reconstructed location of peaks. A practical goal is to extend the application of this method to the realistic torso geometry, but because Legendre decomposition is limited to geometries with spherical symmetry, other methods of map decomposition must be found. Singular value decomposition (SVD) is used to decompose the maps into component parts. Its individual submaps also have different levels of spatial variation; moreover, it is generalizable to any vector, does not require spherical symmetry, and is extremely efficient numerically. Using SVD decomposition for regional regularization, significant improvement was achieved in the map quality in the presence of data noise.