Magnetic resonance-based reconstruction method of conductivity and permittivity distributions at the Larmor frequency

Magnetic resonance electric properties tomography (MREPT) is a recent medical imaging modality for visualizing the electrical tissue properties of the human body using radio-frequency magnetic fields. It uses the fact that in magnetic resonance imaging (MRI) systems the eddy currents induced by the radio-frequency magnetic fields reflect the conductivity (σ) and permittivity (ε) distributions inside the tissues through Maxwell's equations. The corresponding inverse problem consists of reconstructing the admittivity distribution (γ = σ + iωε)at the Larmor frequency (ω/2π = MHz for a 3 Tesla MRI machine) from the positive circularly polarized component of the magnetic field H =(H_x, H_y, H_z. Previous methods are usually based on an assumption of local homogeneity (∀_γ ≈0)which simplifies the governing equation. However, previous methods that include the assumption of homogeneity are prone to artifacts in the region where γ varies. Hence, recent work has sought a reconstruction method that does not assume local-homogeneity. This paper presents a new MREPT reconstruction method which does not require any local homogeneity assumption on γ. We find that γ is a solution of a semi-elliptic partial differential equation with its coefficients depending only on the measured data H⁺:= (H_x+iH_y)/2, which enable us to compute a blurred version of γ. To improve the resolution of the reconstructed image, we developed a new optimization algorithm that minimizes the mismatch between the data and the model data as a highly nonlinear function of γ. Numerical simulations are presented to illustrate the potential of the proposed reconstruction method.