Reconstruction algorithms for two and three-dimensional electrical impedance tomography (EIT) are developed. EIT is a relatively new medical imaging technique in which electrodes are placed on the surface of the body, current is applied on the electrodes, and the resulting voltage is measured on the electrodes. Since the various organs and tissues in the body frequently have different conductive properties, an image is formed from the reconstructed conductivity distribution. In this research the D- bar reconstruction algorithm for 2-D EIT based on the paper by Nachman [Ann. Of Math. 143 (1996)] will be further developed through a study of the algorithm applied to discontinuous conductivities, a study of electrode effects on the algorithm, incorporating boundary reconstruction in the numerical algorithm, more accurate computations of the scattering transform, and adapting the algorithm to non-circular geometries. This work is expected to improve the accuracy of the method and bring it to a clinically useful stage. In addition, a 3-D D-bar algorithm will be developed for EIT data collected on planar electrode arrays. The particular clinical application for the 3-D algorithm is breast cancer detection.
This work will provide further insight into the D-bar method in two and three dimensions. The terminology "D-bar method" comes from the techniques in inverse scattering which result in a D-bar equation, which is a certain type of partial differential equation in complex variables, to be solved in the reconstruction process. The 2-D D-bar algorithm is the only direct (non-iterative) method for EIT solving the full nonlinear conductivity problem to have been successfully implemented on experimental EIT data. At present, no direct numerically implemented method exists for EIT in three dimensions. The 2-D EIT problem has an important application as a method for monitoring patients with acute respiratory distress syndrome (ARDS). ARDS is a life-threatening condition in which inflammation of the lungs and fluid in the air sacs leads to low blood oxygen levels. The treatment is mechanical ventilation, which has potential side-effects to the lungs. By placing electrodes around the patient's thorax, EIT can be used to continually monitor ARDS patients, providing information about their condition and aiding in determining the proper ventilator settings to minimize side-effects. A 3-D D-bar reconstruction algorithm with the main application of breast cancer detection has the potential to reconstruct the conductivity values with more accuracy than existing algorithms and thus may improve the specificity of EIT as a method of breast cancer detection. EIT as a tool for breast cancer detection is in the early stages, and the development and study of a 3-D D-bar algorithm will provide a deeper understanding of the effectiveness of EIT for this application.