Magnetic resonance electrical properties tomography has attracted attentions as an imaging modality for reconstructing the electrical properties (EPs), namely conductivity and permittivity, of biological tissues. Current reconstruction algorithms assume that EPs are locally homogeneous, which results in the so-called tissue transition-region artifact. We previously proposed a reconstruction algorithm based on a Dbar equation that governed electric fields. The representation formula of its solution was given by the generalized Cauchy formula. Although this method gives an explicit reconstruction formula of EPs when two-dimensional approximation holds, an iterative procedure is required to deal with three-dimensional problems, and the convergence of this method is not guaranteed. In this paper, we extend our previous method to derive an explicit reconstruction formula of EPs that is effective even when the magnetic field and EPs vary along the body axis. The proposed method solves a linear system of equation derived from the generalized Cauchy formula using the conjugate gradient method with fast Fourier transform algorithm instead of directly performing a forward calculation, as was done in our previous method. Numerical simulations with cylinder and human-head models and phantom experiments show that the proposed method can reconstruct EPs precisely without iteration even in the three-dimensional case.
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