The boundary element method in the forward and inverse problem of electrical impedance tomography

In this paper, a new formulation of the reconstruction problem of electrical impedance tomography (EIT) is proposed. Instead of reconstructing a complete two-dimensional picture, a parameter representation of the gross anatomy is formulated, of which the optimal parameters are determined by minimizing a cost function. The two great advantages of this method are that the number of unknown parameters of the inverse problem is drastically reduced and that quantitative information of interest (e.g., lung volume) is estimated directly from the data, without image segmentation steps. The forward problem of EIT is to compute the potentials at the voltage measuring electrodes, for a given set of current injection electrodes and a given conductivity geometry. In this paper, it is proposed to use an improved boundary element method (BEM) technique to solve the forward problem, in which flat boundary elements are replaced by polygonal ones. From a comparison with the analytical solution of the concentric circle model, it appears that the use of polygonal elements greatly improves the accuracy of the BEM, without increasing the computation time. In this formulation, the inverse problem is a nonlinear parameter estimation problem with a limited number of parameters. Variants of Powell's and the simplex method are used to minimize the cost function. The applicability of this solution of the EIT problem was tested in a series of simulation studies. In these studies, EIT data were simulated using a standard conductor geometry and it was attempted to find back this geometry from random starting values. In the inverse algorithm, different current injection and voltage measurement schemes and different cost functions were compared. In a simulation study, it was demonstrated that a systematic error in the assumed lung conductivity results in a proportional error in the lung cross sectional area. It appears that our parametric formulation of the inverse problem leads to a stable minimization problem, with a high reliability, provided that the signal-to-noise ratio is about ten or higher.     

An efficient 3-D eddy-current solver using an independent impedance method for transcranial magnetic stimulation

In many important bioelectromagnetic problem settings, eddy-current simulations are required. Examples are the reduction of eddy-current artifacts in magnetic resonance imaging and techniques, whereby the eddy currents interact with the biological system, like the alteration of the neurophysiology due to transcranial magnetic stimulation (TMS). TMS has become an important tool for the diagnosis and treatment of neurological diseases and psychiatric disorders. A widely applied method for simulating the eddy currents is the impedance method (IM). However, this method has to contend with an ill conditioned problem and consequently a long convergence time. When dealing with optimal design problems and sensitivity control, the convergence rate becomes even more crucial since the eddy-current solver needs to be evaluated in an iterative loop. Therefore, we introduce an independent IM (IIM), which improves the conditionality and speeds up the numerical convergence. This paper shows how IIM is based on IM and what are the advantages. Moreover, the method is applied to the efficient simulation of TMS. The proposed IIM achieves superior convergence properties with high time efficiency, compared to the traditional IM and is therefore a useful tool for accurate and fast TMS simulations.     

An experimental study in electrical impedance tomography usingbackprojection reconstruction

This paper reports on experiments designed to evaluate the performance of the equipotentials backprojection method under conditions modeling those of proposed applications of electrical impedance tomography. Small spherical targets were placed inside a saline-filled tank with dimensions similar to a human torso. Data were acquired with a computer-based instrument that applies current to pairs of electrodes located on two horizontal planes and records potential differences between electrodes of a third plane. The relative contrast produced by nonconducting spheres in a uniform saline background was measured on the reconstructed images and used to determine system sensitivity to target volume and to the radial and vertical positions of single spheres. Results show that for radial positions within a critical radius sensitivity is always maximum when the spheres center is on the recording plane and decreases gradually when the target is moved outside this plane. Localization of simple targets in 3-D, with data acquired from multiple recording planes, appears feasible. The results provide guidelines for the interpretation of images with complex 3-D conductivity distributions.     

Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography

The problem this paper addresses is how to use the two-dimensional D-bar method for electrical impedance tomography with experimental data collected on finitely many electrodes covering a portion of the boundary of a body. This requires an approximation of the Dirichlet-to-Neumann, or voltage-to-current density map, defined on the entire boundary of the region, from a finite number of matrix elements of the current-to-voltage map. Reconstructions from experimental data collected on a saline filled tank containing agar heart and lung phantoms are presented, and the results are compared to reconstructions by the NOSER algorithm on the same data.     

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 in this paper we 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. [3]. In this paper we have significantly extended the method by introducing a more realistic electrode model into the analysis. We have given explicit formulae for the quantities involved so that the reader can reproduce our results.     

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.     

Finite element implementation of Maxwell's equations for image reconstruction in electrical impedance tomography

Traditionally, image reconstruction in electrical impedance tomography (EIT) has been based on Laplace's equation. However, at high frequencies the coupling between electric and magnetic fields requires solution of the full Maxwell equations. In this paper, a formulation is presented in terms of the Maxwell equations expressed in scalar and vector potentials. The approach leads to boundary conditions that naturally align with the quantities measured by EIT instrumentation. A two-dimensional implementation for image reconstruction from EIT data is realized. The effect of frequency on the field distribution is illustrated using the high-frequency model and is compared with Laplace solutions. Numerical simulations and experimental results are also presented to illustrate image reconstruction over a range of frequencies using the new implementation. The results show that scalar/vector potential reconstruction produces images which are essentially indistinguishable from a Laplace algorithm for frequencies below 1 MHz but superior at frequencies reaching 10 MHz.     

Increasing image resolution in electrical impedance tomography

An effective approach to increase the image resolution in static electrical impedance tomography is proposed, in which the image with local high resolution is reconstructed by fine meshing only the impedance abnormal element in the finite element model based on a genetic algorithm. Experimental results from a laboratory phantom are presented     

Using compound electrodes in electrical impedance tomography

A compound electrode composed of a large outer electrode to inject current and a small inner electrode to sense voltage was developed and used to measure voltages from a physical phantom. The measured voltages were smaller in amplitude than those from conventional electrodes, demonstrating that the compound electrode can minimize contact impedance voltage drop from the measured data. A finite-element model was used for the compound electrode and incorporated into the regularized Newton-Raphson reconstruction algorithm. A sensitivity study showed that the reconstructed resistivity distributions are less dependent on the unknown contact resistance values for a compound electrode than a conventional electrode and that the use of a compound electrode results in improved images for the reconstruction algorithm     

Measuring lung resistivity using electrical impedance tomography

We propose the use of electrical impedance tomography (EIT) imaging techniques in the measurement of lung resistivity for detection and monitoring of apnea and edema. In EIT, we inject currents into a subject using multiple electrodes and measure boundary voltages to reconstruct a cross-sectional image of internal resistivity distribution. We found that a simplified, therefore fast, version of the impedance imaging method can be used for detection and monitoring of apnea and edema. We have showed the feasibility of this method through computer simulations and human experiments. We speculate that the EIT imaging technique will be more reliable than the current impedance apnea monitoring method, since we are monitoring the change of internal lung resistivity. However, more study is required to verify that this method performs better in the presence of motion artifact than the conventional two-electrode impedance apnea monitoring method. Future work should include experiments which carefully simulate different kinds of motion artifacts.     

Weighted least-squares criteria for electrical impedance tomography

Methods are developed for the design of electrical impedance tomographic reconstruction algorithms with specified properties. Assuming a starting model with constant conductivity or some other specified background distribution, an algorithm with the following properties is found. (1) The optimum constant for the starting model is determined automatically. (2) The weighted least-squares error between the predicted and measured power dissipation data is as small as possible. (3) The variance of the reconstructed conductivity from the starting model is minimized. (4) Potential distributions with the largest volume integral of gradient squared have the least influence on the reconstructed conductivity, and therefore distributions most likely to be corrupted by contact impedance effects are deemphasized. (5) Cells that dissipate the most power during the current injection tests tend to deviate least from the background value. For a starting model with nonconstant conductivity, the reconstruction algorithm has analogous properties.     

Finite element modeling of electrode-skin contact impedance inelectrical impedance tomography

In electrical impedance tomography (EIT), the measured voltages are sensitive to electrode-skin contact impedance because the contact impedance and the current density through it are both high. Large electrodes were used to provide a more uniform current distribution and reduce the contact impedance. A large electrode differs from a point electrode in that it has shunting and edge effects that cannot be modeled by a single resistor. The finite-element method (FEM) was used to study the electric field distributions underneath an electrode, and three models were developed: a FEM model, a simplified FEM model, and a weighted load model. The FEM models considered both shunting and edge effects and closely matched the experimental measurements. It is concluded that FEM models of electrodes can be used to improve the performance of an electrical impedance tomography reconstruction algorithm     

A quantitative method based on total relative change for dynamic electrical impedance tomography

We proposed a new method based on total relative change (TRC) from measured boundary voltages to quantify the volume changes of fluid during electrical impedance tomography (EIT) monitoring. The results showed that TRC linearly correlated with the volume of infused saline solution into a phantom, and the slope of TRC changes was approximately linear with the infusion speed. A inserted copper tube at different positions did not affect TRC significantly. The linear relationship between TRC and volume change indicates that TRC could be a good quantitative index for dynamic EIT.     

Statistical estimation of resistance/conductance by electrical impedance tomography measurements

This paper is built upon the assumption that in electrical impedance tomography, vectors of voltages and currents are linearly dependent through a resistance matrix. This linear relationship was confirmed experimentally and may be derived analytically under certain assumptions regarding electrodes (Isaacson, 1991). Given measurement data consisting of voltages and currents, we treat this relationship as a linear statistical model. Thus, our goal is not to reconstruct the image but directly estimate its electromagnetic properties reflected in the resistance and/or conductance matrix using electrical impedance tomography (EIT) measurements of voltages and currents on the periphery of the body. Since no inverse problem is involved the algorithm for estimation merely reduces to one matrix inversion. We estimate the impedance resistance matrix using well established statistical inference techniques for linear regression models. We provide a comprehensive treatment for a two-dimensional homogeneous body of a circular shape, by which many concepts of electrical impedance tomography, such as width of electrodes, the difference between voltage-current and current-voltage systems are illustrated. Our theory may be applied to various tests including EIT hardware calibration and whether the medium is homogeneous. These tests are illustrated on phantom agar data.     

A direct reconstruction algorithm for electrical impedance tomography

A direct (noniterative) reconstruction algorithm for electrical impedance tomography in the two-dimensional (2-D), cross-sectional geometry is reviewed. New results of a reconstruction of a numerically simulated phantom chest are presented. The algorithm is based on the mathematical uniqueness proof by A. I. Nachman [1996] for the 2-D inverse conductivity problem. In this geometry, several of the clinical applications include monitoring heart and lung function, diagnosis of pulmonary embolus, diagnosis of pulmonary edema, monitoring for internal bleeding, and the early detection of breast cancer.     

On optimal current patterns for electrical impedance tomography

We develop a statistical criterion for optimal patterns in planar circular electrical impedance tomography. These patterns minimize the total variance of the estimation for the resistance or conductance matrix. It is shown that trigonometric patterns (Isaacson, 1986), originally derived from the concept of distinguishability, are a special case of our optimal statistical patterns. New optimal random patterns are introduced. Recovering the electrical properties of the measured body is greatly simplified when optimal patterns are used. The Neumann-to-Dirichlet map and the optimal patterns are derived for a homogeneous medium with an arbitrary distribution of the electrodes on the periphery. As a special case, optimal patterns are developed for a practical EIT system with a finite number of electrodes. For a general nonhomogeneous medium, with no a priori restriction, the optimal patterns for the resistance and conductance matrix are the same. However, for a homogeneous medium, the best current pattern is the worst voltage pattern and vice versa. We study the effect of the number and the width of the electrodes on the estimate of resistivity and conductivity in a homogeneous medium. We confirm experimentally that the optimal patterns produce minimum conductivity variance in a homogeneous medium. Our statistical model is able to discriminate between a homogenous agar phantom and one with a 2 mm air hole with error probability (p-value) 1/1000.     

Reconstruction of three-dimensional data for electrical impedance tomography

A two-dimensional reconstruction algorithm based on a modified version of the method of sensitivity regions is used to reconstruct data obtained from a three-dimensional finite element model. By using data obtained from off-drive-plane measurements an improved image of changes in resistivity on the drive plane is obtained.<>     

Anatomically constrained electrical impedance tomography forthree-dimensional anisotropic bodies

As shown previously for two-dimensional geometries, anisotropy effects should not be ignored in electrical impedance tomography (EIT) and structural information is important for the reconstruction of anisotropic conductivities. Here, we describe the static reconstruction of an anisotropic conductivity distribution for the more realistic three-dimensional (3-D) case. Boundaries between different conductivity regions are anatomically constrained using magnetic resonance imaging (MRI) data. The values of the conductivities are then determined using gradient-type-algorithms in a nonlinear-indirect approach. At each iteration, the forward problem is solved by the finite element method. The approach is used to reconstruct the 3-D conductivity profile of a canine torso. Both computational performance and simulated reconstruction results are presented together with a detailed study on the sensitivity of the prediction error with respect to different parameters. In particular, the use of an intracavity catheter to better extract interior conductivities is demonstrated     

Tissue spectroscopy with electrical impedance tomography: computersimulations

A method is proposed by which bioelectrical spectroscopy could be combined with electrical impedance tomography (EIT) to provide noninvasive characterization of tissue. Multifrequency (2-200 kHz) EIT measurements were simulated with a numerical model for a volume of porcine liver immersed in an electrolytic tank. From the reconstructed EIT images the tissue characterization method was then applied enabling a plot of complex resistivity to be drawn for any selected pixel in the image. Simulations were performed for a small volume of degraded tissue embedded in the normal tissue to examine its effect on the derived spectroscopic parameters. The method could have an application in transplant surgery for screening organs for tissue degradation     

Regularized reconstruction in electrical impedance tomography usinga variance uniformization constraint

This paper describes a new approach to reconstruction of the conductivity field in electrical impedance tomography. Our goal is to improve the tradeoff between the quality of the images and the numerical complexity of the reconstruction method. In order to reduce the computational load, we adopt a linearized approximation to the forward problem that describes the relationship between the unknown conductivity and the measurements. In this framework, we focus on finding a proper way to cope with the ill-posed nature of the problem, mainly caused by strong attenuation phenomena; this is done by devising regularization techniques well suited to this particular problem. First, we propose a solution which is based on Tikhonov regularization of the problem. Second, we introduce an original regularized reconstruction method in which the regularization matrix is determined by space-uniformization of the variance of the reconstructed conductivities. Both methods are nonsupervised, i.e., all tuning parameters are automatically determined from the measured data. Tests performed on simulated and real data indicate that Tikhonov regularization provides results similar to those obtained with iterative methods, but with a much smaller amount of computations. Regularization using a variance uniformization constraint yields further improvements, particularly in the central region of the unknown object where attenuation is most severe. We anticipate that the variance uniformization approach could be adapted to iterative methods that preserve the nonlinearity of the forward problem. More generally, it appears as a useful tool for solving other severely ill-posed reconstruction problems such as eddy current tomography