# Resistivity Tensor Imaging via Network Discretization of Faraday’s Law

Natural Science > Mathematical Sciences

The muscle or nerve system has a fascinating fiber structure. This formation was intended to help the brain communicate with the constituent parts of the body and make movement possible. Considering these structures from a directional point of view, it is important to have knowledge about the direction of fibers so to understand the connection of components of the brain or the mechanics of a body. Information about such physiology comes from imaging. Diffusion tensor imaging is a well-established method to find the direction of fibers using the property that water diffuses to the fiber direction more easily. To make determinations about the direction of fibers, one should find a 3 by 3 symmetric matrix (or tensor) and the fiber direction is given by the integral curve of the first eigenvector field. The paper summarized herein by KAIST researchers is to develop a different approach by constructing resistivity tensor images using the fact that the electrical current also flows in the direction of fibers more easily. Faraday’s law of currents is discretized using a network method.

A resistivity (or conductivity) body with cell membrane, muscle fiber, or nerve fiber structure must be investigated using an anisotropic tensor. The purpose of this research is to develop a network based numerical method for resistivity tensor imaging. The introduced algorithm requires local computations only and hence the computation complexity is same as the order of space dimensions. This simplicity is important in achieving a fast reconstruction method for three spatial dimensions.

Classically, the conductivity of a body is represented using Maxwell’s equation,

Here, voltage and is the conductivity are represente. However, if the current data is given, one should use Faraday’s law. In the paper, the resistivity tensor image is obtained using three sets of current data and then solving accordingly,

In the algorithm, three sets of current data are used and the equation is discretized by a network method. By improving the network structure, better images are then obtained. The algorithm is for the two space dimensions. However, it can be extended to three space dimensions.

The goal of this research was to obtain directional information of fibers. The fiber direction was obtained from the integral curves of the first eigenvector field.

Kindly see the research group website for more information

DOI: http://dx.doi.org/10.1137/16M1074643

* Lab information

http://amath.kaist.ac.kr/pde_lab/