Reconstruction algorithms for electrical impedance mammography

Abstract

Electrical impedance tomography (EIT) is a medical imaging method which aims to reconstruct internal electric properties, such as conductivity and permittivity, of an object from electrical measurements conducted on its boundary. Within this dissertation, we investigate the application of EIT for breast cancer detection. Initially, the focus of the work is to introduce the reader to the mathematical formulation of EIT and what makes its associated inverse problem, namely the inverse conductivity problem, both nonlinear and ill-posed. After the introductory chapters, an extension of a direct algorithm dependent on a linearized integral equation approach to the inverse conductivity problem in two dimensions is derived and tested numerically. The proposed algorithm is capable of simultaneously approximating both electrical conductivity and permittivity distributions either from single-time, time difference (tdEIT) and multiple times voltage measurements at different angular frequencies. Then, the design of a novel three-dimensional sensing head with a hemispherical geometry based on a brassiere is described. The proposed design is modelled in EIDORS and we present three-dimensional conductivity reconstructions of the interior of the hemispherical domain when it contains one or two inclusions with conductivities three times (or more) greater than the background, thereby simulating tumours in a human breast. We present novel expressions for the associated potential function related to Laplace’s equation in three dimensions, specifically, for a hemispherical domain. Two expressions for the potential function subject to two different idealized Neumann boundary conditions are obtained, first by a method which uses the Neumann Green’s function of the hemisphere and then by separation of variables. Both derived expressions are infinite series containing associated Legendre functions. A convergence analysis of the derived expressions is presented. One of the expressions is determined to be convergent in the interior of the domain but divergent on the hemispherical boundary whereas the other expression is determined to be convergent both within and on the domain. Finally, we derive a system of integral equations which, when solved, would provide an alternative approach of obtaining the three-dimensional potential function which solves the Laplace’s equation in a spherical domain.