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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.

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    To simplify our inference algorithm, for certain dynamic models such as the FitzHugh–Nagumo, we can use the method of variation of parameters in order to have the evolution function dependence on \({\user2{w}}_{0:n}\) brought down to depend only on w 0 and \({\user2{v}}_{0:n}\) (see Supplemental materials Eq. (4)).

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    This parameter can also be viewed, in a game theoretic framework, as quantifying the amount of cooperation or “trust” the players (tensors) have in one another.

  3. 3.

    Minimization is performed using a standard Matlab implementation of the sequential quadratic programming method described in Ref. 28.


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MEA data set recorded at Italian Institute of Technology by using the high-resolution 4096-channel MEA platform of 3Brain GmbH, Switzerland. This work was supported in part by the McDonnell International Scholars Academy Fellowship, and also in part by a National Science found CCF-0963742. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the NSF.

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Correspondence to Elad Gilboa.

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Associate Editor Zahra Moussavi oversaw the review of this article.

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Gilboa, E., La Rosa, P.S. & Nehorai, A. Estimating Electrical Conductivity Tensors of Biological Tissues Using Microelectrode Arrays. Ann Biomed Eng 40, 2140–2155 (2012).

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  • Inverse solution
  • Electrical conductivity
  • Bidomain model
  • Tensor field
  • Parallel optimization
  • Alternating optimization
  • Microelectrode array
  • Biological tissues