基于磁共振的Larmor频率下电导率和介电常数分布重构方法
Magnetic resonance-based reconstruction method of conductivity and permittivity distributions at the Larmor frequency
摘要：
磁共振电特性断层扫描是一种最近的医学成像模式，用于使用射频磁场可视化人体的电组织特性。它利用了在磁共振成像系统中由射频磁场感应的涡电流通过麦克斯韦方程反映组织内的电导率（$σ$）和介电常数（$ε$）分布的事实。相应的逆问题包括从磁场的正圆极化分量${\bf H}=(H_x,H_y,H_z)$重建在拉莫尔频率（对于3特斯拉MRI机器，$ω/2π=$ 128 MHz）下的导纳分布（$γ=σ+iωε$）。以前的方法通常是基于局部均匀性（$\nablaγ\approx 0$）的假设，简化了控制方程。然而，包括同质性假设的先前方法易于在$γ$变化的区域中产生伪影。因此，最近的工作已经寻求不假设局部均匀性的重建方法。本文提出了一种新的磁共振电特性层析成像重建方法，该方法不需要对$γ$作任何局部均匀性假设。我们发现，$γ$是一个半椭圆型偏微分方程的解决方案，其系数只取决于测量数据$H^+$，这使我们能够计算一个模糊版本的$γ$。为了提高重建图像的分辨率，我们开发了一种新的优化算法，最大限度地减少数据和模型数据之间的不匹配作为一个高度非线性函数的$γ$。数值模拟，以说明所提出的重建方法的潜力。
英文摘要：
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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