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

Magnetic resonance electrical property tomography 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 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=$128 MHz for a 3 tesla MRI machine) from the positive circularly polarized component of the magnetic field ${\bf H}=(H_x,H_y,H_z)$. Previous methods are usually based on an assumption of local homogeneity ($\nablaγ\approx 0$) 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 magnetic resonance electrical property tomography 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 $H^+$, 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.

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