Conductivity and permittivity image reconstruction at the larmor frequency using MRI

This paper presents a new method of providing both conductivity (θ) and permittivity (-) images at the MR Larmor frequency in magnetic resonance electrical property tomography (MREPT), a relatively new MR-based electrical tissue property imaging modality. In MREPT, the RF coil of the MR scanner is used to feed a sinusoidal current at the Larmor frequency, ε/2π (approximately 128 MHz for a 3 Tesla MRI machine), to an imaging object within the MR scanner. Inside the object, this injection current induces a time-harmonic magnetic field, H = (H_x,H_y,H_z), that is influenced by the body's admittivity distribution, δ = θ + iε, via Maxwell's equations. Currently, the positive rotating field, H ₊ = (H_x + iH_y)/2, is the only measurable quantity which can be obtained from B1 mapping techniques. The inverse problem of MREPT is to reconstruct δ from the H₊ data. Existing MREPT reconstruction methods use the local homogeneity assumption (δ = 0) to simplify the relation between δ and H₊, so that the admittivity is proportional to the ratio of the Laplacian ₂H₊ to H₊. However, the conventional reconstruction method based on this local homogeneity assumption gives rise to serious reconstruction errors at large |δ||δ|. Hence, the major issue in MREPT is to find a reconstruction formula without assuming the local homogeneity condition. The proposed method removes the assumption of ( θ θx δ, θ θy δ) = 0 (but still requires θ θz δ ? 0). We have found an exact relation between δ and H+, so that δ is a solution of a semilinear elliptic PDE with coefficients depending only upon H+. Numerical simulations show that the proposed method successfully reconstructs discontinuous admittivities.