Time-harmonic impedance tomography using the T-matrix method

A time-harmonic formulation for the electrical impedance tomography (EIT) inverse problem accounting for electrodynamic effects is derived. This work abandons the standard electrostatic impedance model for a full-wave T-matrix model. The advantage of this method is an accurate physical model that includes finite frequency effects, such as diffusion phenomena, and electrode contact impedance effects. This model offers the potential for increased resolution and larger invertible contrast objects than other methods when used on experimental data, because it may represent a more realistic physical model. Also, an accurate gradient matrix is used in the Newton iterative method so the image reconstruction converges in a few iterations. These advantages are realized with no increase in the computational complexity of this algorithm, compared to the static finite element model. A calibration technique is suggested for measurement systems, to test the validity of a theoretical model that includes electrode contact impedance effects.     

Electrode models for electric current computed tomography

This paper develops a mathematical model for the physical properties of electrodes suitable for use in electric current computed tomography (ECCT). 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, 29.5 omega.cm were studied. Values of "effective" contact impedance zeta used in the numerical approximations were 58.0, 35.0, 15.0, and 7.5 omega.cm2, 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.     

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     

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.     

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     

Magnetic resonance driven electrical impedance tomography: a simulation study

Magnetic resonance electrical impedance tomography (MREIT) is a method for reconstructing a three-dimensional image of the conductivity distribution in a target volume using magnetic resonance (MR). In MREIT, currents are applied to the volume through surface electrodes and their effects on the MR induced magnetic fields are analyzed to produce the conductance image. However, current injection through surface electrodes poses technical problems such as the limitation on the safely applicable currents. In this paper, we present a new method called magnetic resonance driven electrical impedance tomography (MRDEIT), where the magnetic resonance in each voxel is used as the applied magnetic field source, and the resultant electromagnetic field is measured through surface electrodes or radio-frequency (RF) detectors placed near the surface. Because the applied magnetic field is at the RF frequency and eddy currents are the integral components in the method, a vector wave equation for the electric field is used as the basis of the analysis instead of a quasi-static approximation. Using computer simulations, it is shown that complex permittivity images can be reconstructed using MRDEIT, but that improvements in signal detection are necessary for detecting moderate complex permittivity changes.     

Rapid assessment of electrode characteristics for impedance imaging

Electrical impedance imaging is the technique for producing images of the resistivity of internal body structures based on measurements of voltage and current from electrodes applied to the body's surface. When a multiplicity of electrodes are applied in one or more rows around a body structure such as the thorax or limb, it is useful to be able to rapidly assess the general status of the electrode-body interface to determine if the skin has been suitably prepared, and that electrode and skin impedance are suitably low. In addition, assessment of the impedance of individual electrodes should precede acquisition of data for image formation. This communication presents techniques for assessing the overall skin and electrode impedances relative to the impedance of the body interior, and for assessing the integrity of each electrode's contact impedance.     

Finite-element method for the impedance analysis of traveling-wavemodulators

A low-frequency finite-element method (FEM) developed to calculate the impedance of multi-strip-line electrodes on traveling-wave modulators is discussed. In this method, the divergence theorem is used to evaluate the electrode capacitance from the node potential values of discrete elements. This method is used to calculate the impedance of electrodes on anisotropic-inhomogeneous dielectric media. The effect of nonzero electrode thickness and a groove excavated at the electrode gap is analyzed     

Three-dimensional electrical impedance tomography based on thecomplete electrode model

In electrical impedance tomography an approximation for the internal resistivity distribution is computed based on the knowledge of the injected currents and measured voltages on the surface of the body. It is often assumed that the injected currents are confined to the two-dimensional (2-D) electrode plane and the reconstruction is based on 2-D assumptions. However, the currents spread out in three dimensions and, therefore, off-plane structures have significant effect on the reconstructed images. In this paper we propose a finite element-based method for the reconstruction of three-dimensional resistivity distributions. The proposed method is based on the so-called complete electrode model that takes into account the presence of the electrodes and the contact impedances. Both the forward and the inverse problems are discussed and results from static and dynamic (difference) reconstructions with real measurement data are given. It is shown that in phantom experiments with accurate finite element computations it is possible to obtain static images that are comparable with difference images that are reconstructed from the same object with the empty (saline filled) tank as a reference.     

A finite-element study of the effects of electrode position on themeasured impedance change in impedance cardiography

Traditional impedance cardiography (ICG) technique uses band electrodes both for delivering current to and measuring impedance change in the thorax. The use of spot electrodes increases the ease of electrode placement and comfort level for patients. Research has shown that changes in thoracic impedance can have multiple causes. In this study, we used finite element modeling to investigate the sources of impedance change for both band-electrode and spot-electrode ICG, and focused on how differences in electrode location affect the contribution of different sources to changes in impedance. The ultimate purpose is to identify the optimal electrode type and placement for the sensing of stroke volume (SV). Our models were built on sets of end-diastolic and end-systolic magnetic resonance images of a healthy human subject. The results showed that the effect of ventricular contraction is opposite to that of the other changes in systole: the expansion of major vessels, decrease in blood resistivity due to increased blood flow velocity, and decrease in lung resistivity due to increased blood perfusion. Ventricular contraction, the only factor that tends to increase systolic impedance, has a larger effect than any of the other factors. When spot electrodes are placed on the anterior chest wall near the heart, ventricular contraction is so dominant that the measured impedance increases from end-diastole to end-systole, and the change represents 82% of the contribution from ventricular contraction. When using the common band-electrode configuration, the change in measured impedance is a more balanced combination of the four effects, and ventricular contraction is overcome by the other three factors so that the impedance decreases. These results suggest that the belief that ICG can be used to directly measure SV based on the change in the whole thoracic impedance may be invalid, and that spot electrodes may be more useful for understanding local physiological events such as ventricular volume change. These findings are supported by previously reported experimental observations.     

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.     

Influence of the electrode radius on the impedance spectra of cell-covered disc electrode

Non-destructive methods for cell characterization are required, for example, for the evaluation of stem cell differentiations or for the long-term monitoring of (nano) materials in vitro. Microelectrode-based chips and electrical impedance spectroscopy can be used to monitor the growth of human mesenchymal stem cells on electrodes or to investigate effects on cell barrier function and on cell adhesion without any chemical marker. In this article, it was investigated how the area of planar disc electrode affects the impedance of cell-covered electrodes. The sensitivity of impedance measurement with cells (defined as the ratio of the impedance magnitude of electrode with cells to the value without cells) was examined with respect to the electrode radius. Under given conditions, the impedance magnitude of cell-covered electrode was most distinguished from the value of uncovered electrode when the electrode radius was 200 μm.     

A reconstruction algorithm for electrical impedance tomography data collected on rectangular electrode arrays

A three-dimensional reconstruction algorithm in electrical impedance imaging is presented for determining the conductivity distribution beneath the surface of a medium, given surface voltage data measured on a rectangular array of electrodes. Such an electrode configuration may be desirable for using electrical impedence tomography to detect tumors in the human breast. The algorithm is based on linearizing the conductivity about a constant value. Here, we describe a simple implementation of the algorithm on a four-electrode--by-four-electrode array and the reconstructions obtained from numerical and experimental tank data. The results demonstrate significantly better spatial resolution in the plane of the electrodes than with respect to depth.     

Electrical impedance tomography using induced currents

The mathematical basis of a new imaging modality, induced current electrical impedance tomography (EIT), is investigated, The ultimate aim of this technique is the reconstruction of conductivity distribution of the human body, from voltage measurements made between electrodes placed on the surface, when currents are induced inside the body by applied time varying magnetic fields. In this study the two-dimensional problem is analyzed. A specific 9-coil system for generating nine different exciting magnetic fields (50 kHz) and 16 measurement electrodes around the object are assumed, The partial differential equation for the scaler potential function in the conductive medium is derived and finite element method (FEM) is used for its solution. Sensitivity matrix, which relates the perturbation in measurements to the conductivity perturbations, is calculated. Singular value decomposition of the sensitivity matrix shows that there are 135 independent measurements. It is found that measurements are less sensitive to changes in conductivity of the object's interior. While in this respect induced current EIT is slightly inferior to the technique of injected current EIT (using Sheffield protocol), its sensitivity matrix is better conditioned. The images obtained are found to be comparable to injected current EIT images In resolution. Design of a coil system for which parameters such as sensitivity to inner regions and condition number of the sensitivity matrix are optimum, remains to be made.     

Methods for Compensating for Variable Electrode Contact in EIT

Electrical impedance tomography (EIT) is an imaging modality that currently shows promise for the detection and characterization of breast cancer. A very significant problem in EIT imaging is the proper modeling of the interface between the body and the electrodes. We have found empirically that it is very difficult, in a clinical setting, to assure that all electrodes make satisfactory contact with the body. In addition, we have observed a capacitive effect at the skin/electrode boundary that is spatially heterogeneous. To compensate for these problems, we have developed a hybrid nonlinear–linear reconstruction algorithm using the complete electrode model in which we first estimate electrode surface impedances, by means of a Levenberg–Marquardt iterative optimization procedure with an analytically computed Jacobian matrix. We, subsequently, use a linearized algorithm to perform a 3-D reconstruction of perturbations in both contact impedances, and in the spatial distributions of conductivity and permittivity. Results show that, with this procedure, artifacts due to electrodes making poor contact can be greatly reduced. If the experimental apparatus physically applies voltages and measures currents, we show that it is preferable to compute the reconstruction with respect to the Dirichlet-to-Neumann map rather than the Neumann-to-Dirichlet map if there is a significant possibility that electrodes will be fully disconnected. Finally, we test our electrode compensation algorithms for a set of clinical data, showing that we can significantly improve the fit of our model to the measurements by allowing the electrode surface impedances to vary.      

Origin of changes in the epigastric impedance signal as determinedby a three-dimensional model

The impedance technique can be used for the noninvasive monitoring of the stomach, with the epigastric impedance signal being a function of the position and configuration of the electrodes. The changes observed could be due to meal resistivity, meal volume and contractions of the gastric smooth muscles. The effect of these various factors on the epigastric impedance signal was determined for several electrode configurations using a three-dimensional model of the abdomen. When the voltage sensing electrodes are located close to the current electrodes, the signal varies linearly with meal volume. The relationship between meal resistivity and the epigastric impedance signal is nonlinear. The impedance signal varies linearly with the contractions of the circular smooth muscles of the stomach.     

Electrical impedance tomography problem with inaccurately known boundary and contact impedances

In electrical impedance tomography (EIT) electric currents are injected into a body with unknown electromagnetic properties through a set of contact electrodes at the boundary of the body. The resulting voltages are measured on the same electrodes and the objective is to reconstruct the unknown conductivity function inside the body based on these data. All the traditional approaches to the reconstruction problem assume that the boundary of the body and the electrode-skin contact impedances are known a priori. However, in clinical experiments one usually lacks the exact knowledge of the boundary and contact impedances, and therefore, approximate model domain and contact impedances have to be used in the image reconstruction. However, it has been noticed that even small errors in the shape of the computation domain or contact impedances can cause large systematic artefacts in the reconstructed images, leading to loss of diagnostically relevant information. In a recent paper (Kolehmainen , 2006), we showed how in the 2-D case the errors induced by the inaccurately known boundary can be eliminated as part of the image reconstruction and introduced a novel method for finding a deformed image of the original isotropic conductivity using the theory of TeichmÜller mappings. In this paper, the theory and reconstruction method are extended to include the estimation of unknown contact impedances. The method is implemented numerically and tested with experimental EIT data. The results show that the systematic errors caused by inaccurately known boundary and contact impedances can efficiently be eliminated by the reconstruction method.      

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.     

Identification of the impedance model of an implanted cochlear prosthesis from intracochlear potential measurements

Those suffering from a severe to profound sensorineural hearing loss can obtain substantial benefit from a cochlear implant prosthesis. An electrode array implanted in the inner ear stimulates auditory nerve fibers by direct injection of electrical current. A major limitation of today's technology is the imprecise control of intracochlear current flow, particularly the relatively wide spread of neural excitation. A better understanding of the intracochlear electrical fields is, therefore, required. This paper analyzes the structure of intracochlear potential measurements in relation to both the subject's anatomy and to the properties of the electrode array. An electrically equivalent network is proposed, composed of small lumped circuits for the interface impedance and for the cochlear tissues. The numerical methods required to estimate the model parameters from high-quality electrical potential recordings are developed. Finally, some models are presented for subjects wearing a Clarion CII device with a HiFocus electrode and discussed in terms of model reliability.     

A reconstruction algorithm for breast cancer imaging with electrical impedance tomography in mammography geometry

The conductivity and permittivity of breast tumors are known to differ significantly from those of normal breast tissues, and electrical impedance tomography (EIT) is being studied as a modality for breast cancer imaging to exploit these differences. At present, X-ray mammography is the primary standard imaging modality used for breast cancer screening in clinical practice, so it is desirable to study EIT in the geometry of mammography. This paper presents a forward model of a simplified mammography geometry and a reconstruction algorithm for breast tumor imaging using EIT techniques. The mammography geometry is modeled as a rectangular box with electrode arrays on the top and bottom planes. A forward model for the electrical impedance imaging problem is derived for a homogeneous conductivity distribution and is validated by experiment using a phantom tank. A reconstruction algorithm for breast tumor imaging based on a linearization approach and the proposed forward model is presented. It is found that the proposed reconstruction algorithm performs well in the phantom experiment, and that the locations of a 5-mm-cube metal target and a 6-mm-cube agar target could be recovered at a target depth of 15 mm using a 32 electrode system.