The boundary element method in the forward and inverse problem of electrical impedance tomography

In this paper, a new formulation of the reconstruction problem of electrica l impedance tomography (EIT) is proposed. Instead of reconstructing a compl ete two-dimensional picture, a parameter representation of the gross anatom y is formulated, of which the optimal parameters are determined by minimizi ng a cost function, The two great advantages of this method are that the nu mber of unknown parameters of the inverse problem is drastically reduced an d that quantitative information of interest (e.g., lung volume) is estimate d directly from the data, without image segmentation steps. The forward problem of EIT is to compute the potentials at the voltage meas uring electrodes, for a given set of current injection electrodes and a giv en conductivity geometry. In this paper, it is proposed to use an improved boumdary element method (BEM) technique to solve the forward problem, in wh ich flat boundary elements are replaced by polygonal ones. From a compariso n with the analytical solution of the concentric circle model, it appears t hat the use of polygonal elements greatly improves the accuracy of the BEM, without increasing the computation time. In this formulation, the inverse problem is a nonlinear parameter estimatio n problem with a limited number of parameters. Variants of Powell's and the simplex method are used to minimize the cost function. The applicability o f this solution of the EIT problem was tested in a series of simulation stu dies. In these studies, EIT data were simulated using a standard conductor geometry and it was attempted to find back this geometry from random starti ng values. In the inverse algorithm, different current injection and voltag e measurement schemes and different cost functions were compared. In a simu lation study, it was demonstrated that a systematic error in the assumed lu ng conductivity results in a proportional error in the lung cross sectional area. It appears that our parametric formulation of the inverse problem leads to a stable minimization problem, with a high reliability, provided that the s ignal-to-noise ratio is about ten or higher.