Michael Pidcock

Electrode modelling in electrical impedance tomography

In electrical impedance tomography (EIT), measurements of an applied electrical current and the corresponding electrical potential are made on a finite number of electrodes placed on the boundary of an object. These measurements are then used to reconstruct the electrical conductivity distribution in the interior of the object. Iterative solutions of this inverse problem involve frequent solution of the forward problem, and it is therefore important to be able to model the current flow through the electrodes. This paper discusses one such model and describes how the related boundary value problem can be solved using semi-analytical and numerical techniques. Some conclusions regarding the proportion of the boundary that should be covered by electrodes are also drawn.

The boundary inverse problem for the Laplace equation in two dimensions

We have studied the problem of determining part of the boundary of a domain where a potential satisfies the Laplace equation. The potential and its normal derivative have prescribed values on the known part of the boundary that encloses while its normal derivative must vanish on the remaining part. We establish a sufficient condition for the potential to be monotonic along the unknown boundary. This allows us to use the potential to parametrize the boundary. Two methods are presented that solve the problem under this assumption. The first one solves the problem in a closed form and it can be used to define a parameter that will describe the ill-posedness of the problem. The effect of this parameter on the second method presented has been determined for a particular numerical example.

Optimal experiments in electrical impedance tomography

Electrical impedance tomography (EIT) is a noninvasive imaging technique which aims to image the impedance within a body from electrical measurements made on the surface. The reconstruction of impedance images is a ill-posed problem which is both extremely sensitive to noise and highly computationally intensive. The authors define an experimental measurement in EIT and calculate optimal experiments which maximize the distinguishability between the region to be imaged and a best-estimate conductivity distribution. These optimal experiments can be derived from measurements made on the boundary. The analysis clarifies the properties of different voltage measurement schemes. A reconstruction algorithm based on the use of optimal experiments is derived. It is shown to be many times faster than standard Newton-based reconstruction algorithms, and results from synthetic data indicate that the images that it produces are comparable.

Tikhonov regularization for electrical impedance tomography on unbounded domains

The mathematical analysis of geoelectric applications leads to the inverse problem of electric impedance tomography on unbounded domains. We introduce appropriate function spaces for this setting and discuss the analytic properties of the related forward operator on unbounded domains with Lipschitz boundaries. For the numerical approximation we consider Tikhonov regularization for a finite number of measurements. The main theorem states that this yields an approximation process which converges with an optimal rate to a minimum norm solution. Finally, numerical results in two and three dimensions, which are obtained from simulated, noisy data, confirm the theoretical findings.

Analytic and semi-analytic solutions in electrical impedance tomography. I. Two-dimensional problems

The authors give analytic and semi-analytic solutions to a number of problems which are related to the image reconstruction problem of electrical impedance tomography (EIT) in two dimensions.

An image reconstruction algorithm for three-dimensional electrical impedance tomography

Electrical impedance tomography (EIT) has been studied by many authors and in most of this work it has been considered to be a two-dimensional problem. Many groups are now turning their attention to the full three-dimensional case in which the computational demands become much greater. It is interesting to look for ways to reduce this demand and here the authors describe an implementation of an algorithm that is able to achieve this by precomputing many of the quantities needed in the image reconstruction. The algorithm is based on a method called NOSER introduced some years ago, by Cheney et al. (1990). Here, the authors have significantly extended the method by introducing a more realistic electrode model into the analysis. They have given explicit formulae for the quantities involved so that the reader can reproduce their results.

Function spaces and optimal currents in impedance tomography

Abstract The main objective of this paper is to compare – analytically as well as numerically – different approaches for obtaining optimal input currents in impedance tomography. Following the approaches described in, e.g. [Cheney and Isaacson, IEEE Trans. Biomed. Eng. 39: 852–860, 1992, Isaacson, IEEE Trans. Med. Imag. 5: 91–95, 1986, Ito and Kunisch, SIAM J. Contr. Optim. 33: 643–666., 1995, Knowles, An optimal current functional for electrical impedance tomography, 2004], we aim at constructing input currents j, which contain the most information about the difference between the unknown physical conductivity σ* and a given approximation σ 0. The differences can be measured by different discrepancy functionals and the optimal input currents which maximize these functionals depend on the function spaces chosen for defining j and on the norm for measuring the discrepancy. Moreover, the definition of the appropriately weighted Sobolev spaces depends on σ and this subsequently influences the iteration for maximizing the functionals. Numerical experiments illustrate features of the optimal input currents obtained for several combinations of function spaces. The reconstructions with these optimal currents are compared with those with standard input currents (sinusoid and dipole). The differences between the optimal currents obtained by different function space settings are significant. Two newly developed optimal currents can yield qualitatively better reconstructions.

An adaptive current tomograph using voltage sources

An adaptive electric current tomography system that contains a novel front-end analog architecture was developed. Programmable voltage sources were used to deliver currents into the study object and to avoid the difficulties of obtaining high-quality current sources. Through inverting an admittance matrix, the system is capable of achieving a desired current drive pattern by applying a computed voltage pattern. The tomograph, operating at 9.6 kHz, comprises 32 driving electrodes and 32 voltage measurement electrodes. The study of system noise performance shows high SNR in the data acquisition which is enhanced by a digital demodulation scheme. In vitro reconstruction images have been obtained with the data collected by the tomograph.<<ETX>>

POMPUS: an optimized EIT reconstruction algorithm

Electrical impedance tomography (EIT) is a non-invasive imaging technique which aims to image the impedance of material within a test volume from electrical measurements made on the surface. The reconstruction of impedance images is an ill-posed problem which is both extremely sensitive to noise and highly computationally intensive. This paper defines an experimental measurement in EIT and calculates optimal experiments which maximize the distinguishability between the region to be imaged and a best estimate conductivity distribution. These optimal experiments can be derived from measurements made on the boundary. We describe a reconstruction algorithm, known as POMPUS, which is based on the use of optimal experiments. We have shown that, given some mild constraints, if POMPUS converges, it converges to a stationary point of our objective function. It is demonstrated to be many times faster than standard, Newton based, reconstruction algorithms. Results using synthetic data indicate that the images produced by POMPUS are comparable to those produced by these standard algorithms.

A probe for organ impedance measurement

In this paper, we describe the theory and practical implementation of an electrical impedance probe for making in vivo measurements of the electrical admittance of living tissue. The probe uses concentric annular electrodes and is shown to sample a more localized, yet greater, volume of tissue than the standard four-electrode probe. We have developed a mathematical model for the conduction of current between the probe electrodes assuming that we are investigating a uniform, isotropic, semi-infinite region and taking into account the contact impedance between the electrodes and the organ. The electric fields produced by the probe have been calculated by solving a weakly singular Fredholm integral equation of the second kind. The size and position of the probe electrodes have been optimized to maximize both the accuracy in the admittance measurement and insensitivity to contact impedance. A probe and driving hardware have been constructed and experimental results are provided showing the accuracy of admittance measurements at 50 and 640 KHz.