Mathematical framework for a new microscopic electrical impedance tomography system

This work presents a mathematical framework of a new microscopic electrical impedance tomography (micro-EIT) system which aims to produce cross-sectional conductivity images of a biological tissue sample or cells inside a small hexahedral container. Unlike conventional micro-EIT systems which have much in common with a standard EIT system, the proposed micro-EIT system has a unique electrode configuration and associated data collection method. The first pair of driving electrodes are located on the left and right sides of the container facing each other. They fully cover the sides so that the induced current density is uniform and parallel when the container contains no anomaly. The second pair of driving electrodes are thin and long and located at the middle of the front and back sides. There are many miniature electrodes on the front, bottom and back sides for voltage measurements. The top of the container is open for sample manipulations. This electrode configuration provides a large number of voltage measurements from the three surfaces subject to two current injections. In this paper, we provide a mathematical framework of this micro-EIT system for the development of image reconstruction algorithms. We construct an inversion formula of the conductivity from the acquired boundary voltage data. Numerical simulations show that the proposed algorithm successfully reconstructs conductivity images of multiple anomalies. In terms of the image quality, the new micro-EIT system is advantageous over a conventional EIT method adopting multiple current injection patterns.