Discrete mathematical models for electrical impedance tomography

Electrical Impedance Tomography (EIT) is a non-invasive, portable and low-cost medical imaging technique. Different current patterns are injected to the surface of a conductive body and the corresponding voltages are measured also on the boundary. These mea-surements are the data used to infer the interior conductivity distribution of the object. However, it is well known that the reconstruction process is extremely ill-posed due to the low sensitivity of the boundary voltages to changes in the interior conductivity distribution. The reconstructed images also suffer from poor spatial resolution. In tomographic systems, the spatial resolution is related to the number of applied current patterns and to the number and positions of electrodes which are placed at the surface of the object under examination. Two mammographic sensors were recently developed at the University of Mainz in collaboration with Oxford Brookes University. These prototypes consist of a planar sensing head of circular geometry with twelve large outer (active) electrodes arranged on a ring of radius 4.4cm where the external currents are injected and a set of, respectively thirty six and fifty four point-like high-impedance inner (passive) electrodes arranged in a hexagonal pattern where the induced voltages are measured. Two 2D reconstruction methods were proposed for these devices, one based on resistor network models and another one which uses an integral equation formulation. The novelty of the device and hence of these imaging techniques consists exactly in the distinct use of active and passive electrodes. The 2D images of the conductivity distribution of the interior tissue of the breast provide only information about the existence and location of the tumour. In this thesis different circular designs for the sensing head of this EIT device were analysed. The 2D resistor network approach was adapted to the different data collection geometries and the sensitivity of the reconstructions with respect to errors in the simulate data were investigated before any modifications to the original design were made. A novel 3D reconstruction algorithm was also developed for a simpler geometry of the sensing head which consisted of a rectangular array of thirty six electrodes (twenty active+ sixteen passive). This electrode configuration as well as the proposed imaging technique are intended to be used for breast cancer detection. The algorithm is based on linearizing the conductivity about a constant value and allows real-time reconstructions. The perfor-mance of the algorithm was tested on numerically simulated data and small inclusions with conductivities three or four times the background lying beneath the data collection surface were successfully detected. The results were fairly stable with respect to the noise level in the data and displayed very good spatial resolution in the plane of electrodes