Understanding the Coefficient of a PDE in the context of Electrical Tissue Property Imaging
In this blog, I discuss how the coefficient of the elliptic partial differential equation \(\sum_{i=1}^3 \partial_i \big( a_{ij}\,\partial_j u \big) = 0\) in a body arises in the context of electrical tissue property imaging, where \(u\) denotes the electrical potential. In brief, bioimpedance is directly linked to this coefficient, and several companies , such as InBody and Sciospec , are actively developing bioimpedance-based devices. This blog is based on the book *Electromagnetic Tissue Properties MRI* (Imperial College Press) written by Jin Keun Seo, Eung Je Woo, Ulrich Katscher, and Yi Wang. The mathematical model for electrical tissue property imaging is derived from an appropriate reduction of Maxwell’s equations. In the time-harmonic regime, the electric field \( \mathbf{E} \), current density \( \mathbf{J} \), magnetic field \( \mathbf{H} \), and magnetic flux density \( \mathbf{B} \) satisfy the following relations: Gauss's law: \( \nab...