The understanding of electrical injury pathophysiology has benefited greatly from models of electrical current propagation in tissue.1 However, the accuracy of such models could potentially be improved by incorporating quantitative values for tissue. While it is not possible with the present technology to measure the conductivity of deep tissue noninvasively, it has recently been proposed that the conductivity tensor can be approximated from the diffusion tensor as measured by diffusion MRI.2–4 Here, we show how the conductivity tensor can be derived from the water diffusion tensor by a differential effective medium approximation.2,3 Electrical conductivity maps of human brain white matter are also presented. The basis for a general relation between conductivity and diffusion in porous media, such as biological tissue, stems from the mutual restriction of both the ionic and water mobility by the geometry of the medium. The effective restricted mobilities can then be related through a parameterization of the geometric boundary conditions. The claim is not, of course, that there is a fundamental relationship between the free mobilities, but rather that the restricted mobilities are related through the geometry. If we assume, in the spirit of the Neumann principle, that the conductivity and water self-diffusion tensors share eigenvectors, then the tensors can be related by a similarity transformation, = R (D)RT, where is the conductivity tensor, R is the column matrix of the water diffusion tensor eigenvectors, and (D) is the diagonalized conductivity tensor as a function of the diagonalized water diffusion tensor D. We can measure the water diffusion tensor with diffusion MRI, and we wish to find the function (D) that relates the conductivity v and diffusion dv eigenvalues. The relationship between the general transport eigenvalues v, representing either v or dv, can be related to the geometry of the medium by the Sen-Scala-Cohen effective medium relation:
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