David Isaacson

Electrical Impedance Tomography

Electrical impedance tomography (EIT) is an imaging modality that estimates the electrical properties at the interior of an object from measurements made on its surface. Typically, currents are injected into the object through electrodes placed on its surface, and the resulting electrode voltages are measured. An appropriate set of current patterns, with each pattern specifying the value of the current for each electrode, is applied to the object, and a reconstruction algorithm uses knowledge of the applied current patterns and the measured electrode voltages to solve the inverse problem, computing the electrical conductivity and permittivity distributions in the object. This article focuses on the type of EIT called adaptive current tomography (ACT) in which currents are applied simultaneously to all the electrodes. A number of current patterns are applied, where each pattern defines the current for each electrode, and the subsequent electrode voltages are measured to generate the data required for image reconstruction. A ring of electrodes may be placed in a single plane around the object, to define a two-dimensional problem, or in several layers of such rings, to define a three-dimensional problem. The reconstruction problem is described and two algorithms are discussed, a one-step, two-dimensional (2-D) Newton-Raphson algorithm and a one-step, full three-dimensional (3-D) reconstructor. Results from experimental data are presented which illustrate the performance of the algorithms.

Existence and uniqueness for electrode models for electric current computed tomography

The following experiment is considered. To a body of given conductivity, a number of electrodes are attached, through which current is sent. On the same electrodes, the resulting voltages are measured. This experiment can be described by a number of mathematical models [K.-S. Cheng et al., IEEE Transactions on Biomedical Engineering, 36 (1989), pp. 918–924]. These models are discussed and their predictions compared with experiment. In particular, a model is exhibited that is capable of predicting the experimentally measured voltages to within 0.1 percent. For this model, existence and uniqueness of the associated electrical potential is proved.

Electrode models for electric current computed tomography

A mathematical model for the physical properties of electrodes suitable for use in electric current computed tomography is discussed. The model includes the effects of discretization, shunt, and contact impedance. The complete model was validated by experiment. Bath resistivities of 284.0, 139.7, 62.3, and 29.5 Omega -cm were studied. Values of effective contact impedance used in the numerical approximations were 58.0, 35.0, 15.0, and 7.5 Omega -cm/sup 2/, respectively. Agreement between the calculated and experimentally measured values was excellent throughout the range of bath conductivities studied. It is desirable in electrical impedance imaging systems to model the observed voltages to the same precision as they are measured in order to be able to make the highest-resolution reconstructions of the internal conductivity that the measurement precision allows. The complete electrode model, which includes the effects of discretization of the current pattern, the shunt effect due to the highly conductive electrode material, and the effect of an effective contact impedance, allows calculation of the voltages due to any current pattern applied to a homogeneous resistivity field.<<ETX>>

NOSER: An algorithm for solving the inverse conductivity problem

The inverse conductivity problem is the mathematical problem that must be solved in order for electrical impedance tomography systems to be able to make images. Here we show how this inverse conductivity problem is related to a number of other inverse problem. We then explain the workings of an algorithm that we have used to make images from electrical impedance data measured on the boundary of a circle in two dimensions. This algorithm is based on the method of least squares. It takes one step of a Newtons method, using a constant conductivity as an initial guess. Most of the calculations can therefore be done analytically. The resulting code is named NOSER, for Newtons One‐Step Error Reconstructor. It provides a reconstruction with 496 degrees of freedom. The code does not reproduce the conductivity accurately (unless it differs very little from a constant), but it yields useful images. This is illustrated by images reconstructed from numerical and experimental data, including data from a human chest.

Distinguishability of Conductivities by Electric Current Computed Tomography

We give criteria for the distinguishability of two different conductivity distributions inside a body by electric current computed tomography (ECCT) systems with a specified precision. It is shown in a special case how these criteria can be used to determine the measurement precision needed to distinguish between two different conductivity distributions. It is also shown how to select the patterns of current to apply to the body in order to best distinguish given conductivity distributions with an ECCT system of finite precision.

ACT3: a high-speed, high-precision electrical impedance tomograph

Presents the design, implementation, and performance of Rensselaers third-generation adaptive current tomograph, ACT3. This system uses 32 current sources and 32 phase-sensitive voltmeters to make a 32-electrode system that is capable of applying arbitrary spatial patterns of current. The instrumentation provides 16 b precision on both the current values and the real and reactive voltage readings and can collect the data for a single image in 133 ms. Additionally, the instrument is able to automatically calibrate its voltmeters and current sources and adjust the current source output impedance under computer control. The major system components are discussed in detail and performance results are given. Images obtained using stationary agar targets and a moving pendulum in a phantom as well as in vivo resistivity profiles showing human respiration are shown.<<ETX>>

An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem

The 2D inverse conductivity problem requires one to determine the unknown electrical conductivity distribution inside a bounded domain ⊂ R from knowledge of the Dirichletto-Neumann map. The problem has geophysical, industrial, and medical imaging (electrical impedance tomography) applications. In 1996 A Nachman proved that the Dirichlet-to-Neumann map uniquely determines C2 conductivities. The proof, which is constructive, outlines a direct method for reconstructing the conductivity. In this paper we present an implementation of the algorithm in Nachman’s proof. The paper includes numerical results obtained by applying the general algorithms described to two radially symmetric cases of small and large contrast. (Some figures in this article are in colour only in the electronic version; see www.iop.org)

An electric current tomograph

A description is given of an instrument designed to acquire data for the construction of images of internal body structures based on measurements of electrical impedance made from a set of electrodes applied around the periphery of the body. The instrument applies currents at 15 kHz in any desired pattern to 32 electrodes and measures the resulting voltage at each electrode. The construction of a test phantom is also described and the results of initial studies showing the distinguishability of targets of differing sizes and conductivities placed in the phantom are reported. The system is able to distinguish the presence of 9-mm-diameter insulators or conductors placed in the center of a 30-cm-diameter circular tank of salt water. This system is capable of implementing an adaptive process of produce the best currents to distinguish the unknown conductivity from a homogeneous conductivity.<<ETX>>

Current source design for electrical impedance tomography

Questions regarding the feasibility of using electrical impedance tomography (EIT) to detect breast cancer may be answered by building a sufficiently precise multiple frequency EIT instrument. Current sources are desirable for this application, yet no current source designs have been reported that have the required precision at the multiple frequencies needed. We have designed an EIT current source using an enhanced Howland topology in parallel with a generalized impedance converter (GIC). This combination allows for nearly independent adjustment of output resistance and output capacitance, resulting in simulated output impedances in excess of 2 Gohms between 100 Hz and 1 MHz. In this paper, the theoretical operation of this current source is explained, and experimental results demonstrate the feasibility of creating a high precision, multiple frequency, capacitance compensated current source for EIT applications.

Electric current computed tomography eigenvalues

In electric current computed tomography, patterns of currents or voltages are applied to electrodes on the surface of a body and the resulting voltages or currents are measured. A reconstruction and display of an approximation to the impedance inside the body is then made based on these external measurements.It is shown how the problems of choosing current patterns, electrode size and number can be studied in terms of the spectral properties of certain pseudodifferential operators.The proofs given here are simpler and more general than those in [ IEEE Trans. Medical Imaging, 5 (1986), pp. 91–95] and [Clin. Phys. Physiol. Meas. Suppl. A, (1987), pp. 39–46].