This paper presents an explicit reconstruction formula for magnetic resonance electrical property tomography (MREPT). We derive a Dbar problem from the time-harmonic Maxwell equations under the assumptions that , , and , where the z-axis is parallel to the body axis, is the measured magnetic field, and . Then, by using the generalized Cauchy formula, the electrical conductivity and permittivity are explicitly expressed in terms of their boundary values and the measured magnetic field. We also propose an iterative algorithm based on the explicit reconstruction formula without the assumption that . Numerical simulations show that the proposed methods can reconstruct the electrical properties even with noisy data.
: 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.
Magnetic resonance electrical properties tomography (MREPT) noninvasively reconstructs highresolution electrical property (EP) maps using MRI scanners and is useful for diagnosing cancerous tissues. However, conventional MREPT methods have limitations: sensitivity to noise in the numerical Laplacian operation, difficulty in reconstructing three-dimensional (3D) EPs and no guarantee of convergence in the iterative process. We propose a novel, iterative 3D reconstruction MREPT method without a numerical Laplacian operation. We derive an integral representation of the electric field using its Helmholtz decomposition with Maxwell's equations, under the assumption that the EPs are known on the boundary of the region of interest with the approximation that the unmeasurable magnetic field components are zero. Then, we solve the simultaneous equations composed of the integral representation and Ampere's law using a convex projection algorithm whose convergence is theoretically guaranteed. The efficacy of the proposed method was validated through numerical simulations and a phantom experiment. The results showed that this method is effective in reconstructing 3D EPs and is robust to noise. It was also shown that our proposed method with the unmeasurable component H − enhances the accuracy of the EPs in a background and that with all the components of the magnetic field reduces the artifacts at the center of the slices except when all the components of the electric field are close to zero.
Recently, magnetic-resonance-based electrical properties tomography, by which the electrical properties (EPs), namely conductivity and permittivity, of biological tissues are reconstructed, has been an active area of study. We previously proposed an explicit reconstruction method based on the Dbar equation and its explicit solution given by the generalized Cauchy formula. In this method, as in some other conventional methods, the values of EPs on the boundary of the region of interest must be specified by the Dirichlet boundary condition of the partial differential equation. However, it is difficult to know the precise values in practical situations. In this paper, we propose a novel method that reconstructs EPs without the prior information of boundary EP values by deriving a new representation formula of the solution of the Dbar equation with the complex-derivative boundary condition. Numerical simulations and phantom experiments show that the proposed method can reconstruct EPs without knowledge of the boundary EP values. Therefore, the proposed method greatly enhances the applicability of the current EPT methods to practical situations.
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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