Department of Electrical and Electronics Engineering, Bilkent University, Ankara, Turkey.
Phys Med Biol. 2012 Aug 21;57(16):5113-40. doi: 10.1088/0031-9155/57/16/5113. Epub 2012 Jul 27.
Most algorithms for magnetic resonance electrical impedance tomography (MREIT) concentrate on reconstructing the internal conductivity distribution of a conductive object from the Laplacian of only one component of the magnetic flux density (∇²B(z)) generated by the internal current distribution. In this study, a new algorithm is proposed to solve this ∇²B(z)-based MREIT problem which is mathematically formulated as the steady-state scalar pure convection equation. Numerical methods developed for the solution of the more general convection-diffusion equation are utilized. It is known that the solution of the pure convection equation is numerically unstable if sharp variations of the field variable (in this case conductivity) exist or if there are inconsistent boundary conditions. Various stabilization techniques, based on introducing artificial diffusion, are developed to handle such cases and in this study the streamline upwind Petrov-Galerkin (SUPG) stabilization method is incorporated into the Galerkin weighted residual finite element method (FEM) to numerically solve the MREIT problem. The proposed algorithm is tested with simulated and also experimental data from phantoms. Successful conductivity reconstructions are obtained by solving the related convection equation using the Galerkin weighted residual FEM when there are no sharp variations in the actual conductivity distribution. However, when there is noise in the magnetic flux density data or when there are sharp variations in conductivity, it is found that SUPG stabilization is beneficial.
大多数磁共振电阻抗断层成像（MREIT）算法都专注于从内部电流分布产生的磁通密度（∇²B（z））的仅一个分量的拉普拉斯重建导电物体的内部电导率分布。在这项研究中，提出了一种新的算法来解决基于∇²B（z）的 MREIT 问题，该问题在数学上被表述为稳态标量纯对流方程。用于解决更一般的对流扩散方程的数值方法被利用。已知，如果场变量（在这种情况下是电导率）存在急剧变化或存在不一致的边界条件，则纯对流方程的解在数值上是不稳定的。基于引入人工扩散的各种稳定技术被开发出来处理这种情况，在本研究中，流线迎风 Petrov-Galerkin（SUPG）稳定化方法被合并到 Galerkin 加权残值有限元法（FEM）中，以数值求解 MREIT 问题。当实际电导率分布没有急剧变化时，使用 Galerkin 加权残值 FEM 求解相关的对流方程，可以成功地重建电导率。然而，当磁通密度数据存在噪声或电导率存在急剧变化时，发现 SUPG 稳定化是有益的。
