Numerical Simulation And Analysis For Electrical Impedance Tomograph

Electrical impedance tomography(EIT) is a developing biomedical imaging technology. By means of injecting safe electric currents into a person through electrodes attached to the skin, it measures the resulting electric voltages on the surface, and sends data to the computer. By computing, the impedance or resistivity distribution is obtained and revealed as a gray or color image. It is the one's picture of the interior, because different tissues have different impedances or resistances. EIT has caused a wide attention for its safety, low cost and excellent application prospect in geophysical exploration, dam detecting, tracing of contaminants, and the search for underwater objects, etc., see [5,10, 11,12, 13]. The research covers the subjects of mathematics, physics, electronics and biomedical engineering.EIT is an inverse problem of elliptic partial differential equation. Different from the traditional problems of mathematical physics(direct problems) which are well-posed, inverse problems is to solve something unknown starting from only part of information and are usually ill-posed. It is the difficulty of inverse problems. On account of the promotion of practical problems, in 1960s, the former Soviet Union mathematician Tikhonov([2]) proposed the famous regularization method for ill-posed problems and led the research on ill-posed and inverse problems to a new stage. In China the research of inverse problems began from 80s of twenty century on the initiative of mathematician Feng Kang.Because of nonlinearity and severe ill-posedness, it is very complicated and difficult to solve EIT numerically. Algorithms known to me can be categorized as1. noniterative algorithms based on global linearization,2. direct methods,3. iterative solvers tackling the nonlinear problem.Noniterative algorithms based on global linearization can be built by stopping any iterative algorithm after the first step, a prominent example is the NOSER algorithm. The class of direct methods splits into two subclasses: factorization methods use special singular testfunctions to characterize inclusions in a homogeneous background medium and direct methods that implement a constructive existence and uniqueness proof. As far as I know both direct methods are not able to deal with finite electrode models but need to apply currents and measure the voltages along the whole boundary of the object(in mathematical terms: they need to observe the Neumann-to-Dirichlet mapping). Their use for a realistic setting is therefore limited. Iterative algorithms are appropriate methods for EIT, but the convergence proof has not been completed so far, and in addition that stability of numerical solution is bad for the ill-posedness of inverse problem, realtime reconstruction is difficult for the high computational cost. Cheney([6]) proposed the famous NOSER algorithm to reduce the cost, which takes homogeneous distribution as the initial guess, saving the Jacobian in advance and iterates only one step. But it loses the accuracy and is only appropriate for the medium approximating to be homogeneous.Under the supervision of Prof. Yuan Yirang, the author use Levenberg-Marquardt iterative algorithm combined with various numerical method for partial differential equations to simulate EIT problem. L-M method is a trust region method, also a regularization algorithm. It can be used to obtain stable numerical solution for nonlinear inverse problem([18,19]). Two models are considered: continuum model and electrode model(shunt model). Finite element methods, finite volume element methods and cell-centered finite volume methods are used to solve EIT. It is the first time that finite volume element methods and cell-centered finite volume methods are applied in EIT . We verify the correctness of the continuum model, reliability and feasibility of the iterative algorithm by numerical simulation with exact solution of inverse problem and apply the methods to the electrode model. A class of current patterns are proposed to simplify completely the computation of Jacobian matrix and reduce the number of direct problems solved per iteration to the least.The thesis consists of three chapters.In Chapter 1, the numerical simulation and analysis based on finite element method for EIT is addressed. The method and technique of numerical simulation on two- and three-dimensional domains are studied. At first, continuum model of EIT is considered. The computational technique for an inverse boundary value problem is proposed based on invariance of pseudo element stiffness matrix. This offers a significant reduction in computation of numerical integration arising from finite element methods. The results of numerical simulation on a 3-dimensional domain show that numerical solutions approach those exact solutions gradually with space step length becoming smaller, which verifies the correctness of its continuum model, the reliability and feasibility of the algorithm. A method to reduce the number of direct problems to be solved at each iteration step is proposed as well. These methods have been applied practically in simulation of electrical impedance tomography.In Chapter 2, the numerical simulation and analysis based on finite volume element method for EIT is addressed, numerical simulations and analysis for it on two- and three-dimensional domains are presented. In this chapter a modified symmetric finite volume element method is proposed, semi-positive definiteness and existence of solution for this scheme are proved; element geometry matrix is introduced, which is helpful for simplifying the calculation of coefficient matrix; patch approximation for electrical resistivity is present to lower the scale of this inverse problem; the computational formula of Jacobian matrix of error functional is obtained, a class of electrical current patterns is proposed, under which the number of direct problems to solve in each iteration can be reduced to the least. A series of numerical experiments verify the correctness of the continuum model, the reliability and feasibility of the algorithm. These methods have been applied successfully in practical simulation of electrical impedance tomography.In Chapter 3, the numerical simulation and analysis based on cell-centered finite volume element method for EIT is addressed. Numerical simulations and analysis on two- and three-dimensional domains are presented in this paper. Cell-centered finite volume scheme for Neumann boundary value problem is proposed and proved to be semi-positive definite; the formula of Jacobian matrix of error functional is derived, fast algorithm of which is presented. The results of numerical experiments verify the correctness of continuum model, the reliability and feasibility of this algorithm. These methods in this chapter have been successful in numerical simulation of three-dimensional practical imaging.