Boundary Value-Free Magnetic Resonance Electrical Properties Tomography Based on the Generalized Cauchy Formula With the Complex-Derivative Boundary Condition

Abstract

Recently, magnetic-resonance-based electrical properties tomography, by which the electrical properties (EPs), namely conductivity and permittivity, of biological tissues are reconstructed, has been an active area of study. We previously proposed an explicit reconstruction method based on the Dbar equation and its explicit solution given by the generalized Cauchy formula. In this method, as in some other conventional methods, the values of EPs on the boundary of the region of interest must be specified by the Dirichlet boundary condition of the partial differential equation. However, it is difficult to know the precise values in practical situations. In this paper, we propose a novel method that reconstructs EPs without the prior information of boundary EP values by deriving a new representation formula of the solution of the Dbar equation with the complex-derivative boundary condition. Numerical simulations and phantom experiments show that the proposed method can reconstruct EPs without knowledge of the boundary EP values. Therefore, the proposed method greatly enhances the applicability of the current EPT methods to practical situations.

Electrical Properties Tomography: A Methodological Review

“…Gradient-based EPT (G-EPT) continues from Equation (58) and writes it in terms of absolute and relative transmit phases with respect to a reference element, as presented in Equation (17) [44][45][46][47][48]. We then obtain…”

“…Note that the formulation is presented as a three-dimensional problem, but, in the case of an E-polarized field structure, it can be simplified to a two-dimensional setting. Cauchy-based EPT shares a lot of similarities with foIC-EPT, but the electric field strength is derived via a Cauchy integral which allows for the computation of the EPs in a direct manner through complex analysis [57][58][59][60].…”