A Convex Optimized Estimator of the Laplacian Operator for Bioelectric Simulations

Simulations of bioelectric potentials in the direct problem usually require numerical integration in the spatial dimensions to obtain both the transmembrane current diffusion and the extracellular potentials in homogeneous conductivity conditions. Given that the Laplacian of a potential field in said conditions is a spatially linear operator, we propose its implementation with non-uniformly spaced point clouds using a static-matrix formulation which avoids the integration in the spatial domain. We also analyzed the effect of severe irregular sampling in these calculations. Matrix estimators in the 3-dimensional space were built for the Laplacian operator in point clouds defining lines and surfaces. An optimized algorithm was proposed, in which the spatial convolution of the Laplacian impulse response is locally and globally estimated using convex programming techniques, sparse matrix representations, and basic concepts of Graph Theory. We benchmarked the behavior of the estimated synthetic transmembrane and extracellular potentials with simple geometrical substrates. Our proposal paves the way towards simplifying spatial evolution in computer simulations and its use in more clinically realistic environments, such as non-homogeneous conductivity in volume conductor problems and patient-based arrhythmia simulations from point clouds in Electrophysiology Laboratory.