A Boundary-Value-Free Reconstruction Method for Magnetic Resonance Electrical Properties Tomography Based on the Neumann-Type Integral Formula over a Circular Region

Abstract

: The electrical properties (EPs) of biological tissue, consisting of conductivity and permittivity, provide useful information for the diagnosis of malignant tissues and the evaluation of heat absorption rates. Recently, magnetic resonance electrical properties tomography (MREPT), by which EPs are reconstructed from internal magnetic field data measured by using magnetic resonance imaging (MRI), has been actively studied. We previously proposed an explicit pointwise reconstruction method for MREPT based on a complex partial differential equation (PDE), known as the D-bar equation, of the electric field and its explicit solution given by an integral formula. In this method, as in some other conventional methods, EP values on the boundary of the region of interest must be given as a Dirichlet boundary condition of the PDE. However, it is difficult to know these values precisely in practical situations. Therefore, in this paper, we propose a novel method for reconstructing EPs in a circular region without any knowledge of boundary EP values. Starting from the integral solution to solve the D-bar equation in a circular region with the Neumann boundary condition, we show that the contour integral term of the integral formula is eliminated by using Faraday's law and solve the PDE based only on magnetic field data measured by using MRI. Numerical simulations show that the proposed method yields a good reconstruction results without any knowledge of boundary EP values.

Use of dielectric padding to eliminate low convective field artifact in cr‐MREPT conductivity images

“…Because γ ≠ 0, it follows that if the convective field is 0, E z is also 0. In Fushimi and Nara 14 and Nara et al, 15 a 2-step algorithm for MREPT has been proposed and implemented, where they first estimate E z from the knowledge of B 1 + and then obtain admittivity by γ = 2F x iμ 0 E Z , which is undefined (illdefined) in the LCF region. They have recognized the fact that in the low electric field region there is a spot-like artifact, and with additive noise it is more prominent and widened.…”

“…In Ammari et al, the forward problem of MREPT is also modeled as a PDE and an inverse problem approach is used to iteratively find the EPs by minimizing the norm of the difference between the measured and calculated . More recently, in Fushimi and Nara and Nara et al, a 2‐stage algorithm is developed whereby the axial electric field is also estimated together with the EPs.…”

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

“…(13) solves the Dbar equation with the complex-derivative BC. In our previous method [16], we used the Neumann derivative ∂ n f , where ∂ n represents the normal derivative on C. Thus, the previous representation formula was only valid for a circular region. Eq.…”

“…We previously proposed a Dbar-equation-based reconstruction method [15,16]. In this method, with the two-dimensional (2D) plane approximation H z = 0 and ∂ z H + = 0, the following complex form of Maxwell's equations was derived:∂…”

“…To this end, we solve the Dbar equation with the boundary value of the complex-derivative of the electric field instead of giving the Dirichlet boundary values of EPs. This is a generalization of our previous method [16] that solves the Neumann problem without the Dirichlet boundary value, although that method is valid only for the special case that the ROI is a circular domain.…”