Estimating electrical conductivity tensors of biological tissues using microelectrode arrays

Finding the electrical conductivity of tissue is highly important for understanding the tissue's structure and functioning. However, the inverse problem of inferring spatial conductivity from data is highly ill-posed and computationally intensive. In this paper, we propose a novel method to solve the inverse problem of inferring tissue conductivity from a set of transmembrane potential and stimuli measurements made by microelectrode arrays (MEA). We first formalize the discrete forward model of transmembrane potential propagation, based on a reaction-diffusion model with an anisotropic inhomogeneous electrical conductivity-tensor field. Then, we propose a novel parallel optimization algorithm for solving the complex inverse problem of estimating the electrical conductivity-tensor field. Specifically, we propose a single-step approximation with a parallel block-relaxation optimization routine that simplifies the joint tensor field estimation problem into a set of computationally tractable subproblems, allowing the use of efficient standard optimization tools. Finally, using numerical examples of several electrical conductivity field topologies and noise levels, we analyze the performance of our algorithm, and discuss its application to real measurements obtained from smooth-muscle cardiac tissue, using data collected with a high-resolution MEA system.