Electrical impedance tomography and D-Bar equation

Abstract

The D-Bar method is based on solving for the complex geometrical optics (CGO) solutions to the Schrödinger equation, which leads to solving a series of DBar equations numerically. Previous numerical efforts to solve the D-Bar equation include an integral equation solver and a direct PDE solver that uses finite difference method. This thesis introduces and analyzes some alternative solving strategies under the finite element method (FEM) framework.

The computation efficiency aspect of the solver is then discussed. A specially designed FEM mesh is proposed and the performance of the reconstruction routine is roughly compared with that of the finite difference solver.

Finally, a detailed ventilation-perfusion V / Q index analysis is carried out and V /Q index is calculated using reconstructed conductivities.

We argue that the original instability of the classical FEM formulation comes from the inappropriate treatment of the boundary condition. One chapter is therefore specifically dedicated to analyzing this issue, where three alternative boundary conditions are considered. Some numerical simulations are provided to support our argument.

In electrical impedance tomography (EIT), body conductivity is recovered from voltage and current measurements made on the boundary of the body. The major advantage of EIT over other imaging modalities such as CT and MRI is that it is inexpensive, non-invasive and fast. Mainstream reconstruction methods can be categorized into linearized and non-linear, indirect and direct methods. This thesis is focused on the numerical implementations of one of the non-linear direct methods: the D-Bar method.

Two different FEM formulations of the D-Bar Equation will be discussed in detail. The first one is the classical Galerkin method, which proves to be unstable in general. We also notice that it shares a similar stability issue on a uniform grid with the finite difference solver developed in [13]. The stability issue is resolved by introducing the second formulation, the Least Square Finite Element Method (LSFEM). The solver is proven to have a unique solution and the corresponding FEM problem is second order convergent. Reconstruction results using LSFEM D-Bar solver are demonstrated.