Model Of Electrical Impedance Tomography Research Articles

A complete electrode model for plasma impedance probes

Plasma impedance probes measure the impedance spectrum of an antenna immersed in a plasma. The 1964 work of Balmain remains the standard method to interpret these data, using the peak in the magnitude at the upper-hybrid frequency to infer plasma electron density. The primary limitations of Balmain's model are the assumption of a homogenous plasma and a cylindrical dipole. This work presents a numerical model applicable to inhomogeneous plasma and arbitrary antenna geometry based on the cold, fluid approximation given by Balmain. This model solves Poisson's equation using the finite element method and accounts for the effects of the dipole using the plasma complete electrode model (PCEM). The PCEM is developed in this article and accounts for the voltage shunting effects of the dipole elements, the discrete current to the dipole, and the plasma sheath surrounding the dipole. The sheath is incorporated as a contact impedance between the dipole and the plasma in a manner analogous to the complete electrode model of electrical impedance tomography. The first portion of this paper presents the mathematical framework of the PCEM, starting from Maxwell's equations. The second part of the paper compares the output of this numerical method to Balmain's work and to data collected by an impedance probe in the Space Physics Simulation Chamber at the U.S. Naval Research Laboratory. The PCEM results agree with both the observed data and the prior modeling done by Balmain. An additional consequence of the numerical study is the observation that some second-order resonances not predicted by Balmain's model can be attributed to the presence of the plasma sheath.

Damage detection of CFRP laminates based on Adamax-INN model for EIT

Carbon fiber reinforced polymer (CFRP) is electrically conductive, making it possible to detect structural damage of CFRP laminates with electrical impedance tomography (EIT). However, the inverse problem of EIT is severely non-linear, ill-posed, and underdetermined, thus limiting the resolution and accuracy of images reconstructed with EIT. This paper solves the inverse problem of EIT based on invertible neural networks (INN) and optimizes INN with Adamax gradient descent algorithm. Simulation and experimental results demonstrated that, compared with traditional EIT image reconstruction algorithms and radial basis function (RBF) neural network algorithm, the Adamax-INN model largely reduced the artifacts in reconstructed images, improved the damage recognition accuracy and edge clarity, and displayed good noise immunity.

Applications of the duality theory of convex analysis to the complete electrode model of electrical impedance tomography

The duality theory of convex analysis is applied to the complete electrode model (CEM), which is a standard model in electrical impedance tomography (EIT). This results in a dual formulation of the CEM and a general error estimate. This new formulation of the CEM is written in terms of current fields and is shown to have a unique solution. Using this formulation, the general error estimate is proved, from which two a posteriori error estimates and a well known asymptotic result on CEM solutions are obtained. The first a posteriori error estimate assesses the accuracy of solutions to approximate problems, and the second one assesses the accuracy of approximate solutions. Numerical tests to apply this second estimate are performed, employing the finite element method to obtain approximate solutions.

Bridging and Improving Theoretical and Computational Electrical Impedance Tomography via Data Completion

In computational PDE-based inverse problems, a finite amount of data is collected to infer unknown parameters in the PDE. In order to obtain accurate inferences, the collected data must be informative about the unknown parameters. How to decide which data is most informative and how to efficiently sample it is the notoriously challenging task of optimal experimental design (OED). In this context, the best, and often infeasible, scenario is when the full input-to-output (ItO) map, i.e., an infinite amount of data, is available: This is the typical setting in many theoretical inverse problems, which is used to guarantee the unique parameter reconstruction. These two different settings have created a gap between computational and theoretical inverse problems, where finite and infinite amounts of data are used, respectively. In this article we aim to bridge this gap while circumventing the OED task. This is achieved by exploiting the structures of the ItO data from the underlying inverse problem, using the electrical impedance tomography (EIT) problem as an example. To accomplish our goal, we leverage the rank structure of the EIT model and formulate the ItO matrix\textemdash the discretized ItO map\textemdash as an $\mathcal{H}$-matrix whose off-diagonal blocks are low rank. This suggests that, when equipped with the matrix completion technique, one can recover the full ItO matrix, with high probability, from a subset of its entries sampled following the rank structure: The data in the diagonal blocks is informative and should be fully sampled, while data in the off-diagonal blocks can be subsampled. This recovered ItO matrix is then utilized to present the full ItO map up to a discretization error, paving the way to connect with the problem in the theoretical setting where the unique reconstruction of parameters is guaranteed. This strategy achieves two goals: (I) it bridges the gap between the finite- and infinite-dimensional settings for numerical and theoretical inverse problems and (II) it improves the quality of computational inverse solutions. We detail the theory for the EIT model and provide numerical verification to both EIT and optical tomography problems.

Sensitivity analysis of the complete electrode model for electrical impedance tomography

Electrical impedance tomography (EIT) is an imaging technique that reconstructs the conductivity distribution in the interior of an object using electrical measurements from the electrodes that are attached around the boundary. The Complete Electrode Model (CEM) accurately incorporates the electrode size, shape, and effective contact impedance into the forward problem for EIT. In this work, the effect of the conductivity distribution and the electrode contact impedance on the solution of the forward problem is addressed. In particular, the sensitivity of the electric potential with respect to a small-amplitude perturbation in the conductivity, and with respect to some defective electrodes is studied. The Gâteaux derivative is introduced as a tool for the sensitivity analysis and the Gâteaux differentiability of the electric potential with respect to the conductivity and to the contact impedance of the electrodes is proved. The derivative is then expressed as the unique solution to a variational problem and the discretization is performed with Finite Elements of type P1. Numerical simulations for different 2D and 3D configurations are presented. This study illustrates the impact of the presence of perturbations in the parameters of CEM on EIT measurements. Finally, the 2D inverse conductivity problem for EIT is numerically solved for some configurations and the results confirm the conclusions of the numerical sensitivity analysis.

Integral formulation of the complete electrode model of electrical impedance tomography

We model electrical impedance tomography (EIT) based on the minimum energy principle. It results in a constrained minimization problem in terms of current density. The new formulation is proved to have a unique solution within appropriate function spaces. By characterizing its solution with the Lagrange multiplier method, we relate the new formulation to the so-called shunt model and the complete electrode model (CEM) of EIT. Based on the new formulation, we also propose a new numerical method to solve the forward problem of EIT. The new solver is formulated in terms of current. It was shown to give similar results to that of the traditional finite element method, with simulations on a 2D EIT model.

Depth Analysis of Planar Array for 3D Electrical Impedance Tomography

Electrical impedance tomography (EIT) imaging modality has great potential on industrial applications with the advantages of being high temporal resolution. It is especially useful in cases, such as geophysical detection, landmine detection, and detections on non-transparent region, where the measurement data are only available from single surface, for data acquisition. Instead of the circular EIT model that uses the traditional circular electrode model, in this paper, planar array EIT is implemented, aiming to visualize a pipeline transporting a two-phase flow. The planar array can explore spatial information within its detectable region by producing 3D images, which have a higher spatial resolution in the axis direction than a traditional EIT with a dual-plane electrode sensor. However, in solving the inverse problem of a 3D subsurface EIT using a planar array, the images may be degraded, especially in cases where the location of the target is relatively deep. The total variation (TV) algorithm as block prior assumption-based regularization method has the potential to improve the image quality, and some works have shown that TV reconstructs sharper images, which provides an advantage when representing spatial information. In this paper, the performance of subsurface EIT using the TV algorithm for 3D visualization is presented based on simulations and experiments, and the results of quantitative measurement of depth are discussed.

A model of electrical impedance tomography implemented in nerve-cuff for neural-prosthetics control

Objective: In neural interfaces for peripheral nerve a trade-off exists between the level of invasiveness and the selectivity of neural recordings. In this study, we implement electrical impedance tomography (EIT) in a nerve cuff with the aim to investigate the achievable level of selectivity. Approach: Established modelling approaches in neural-EIT are expanded on to be used, for the first time, on myelinated fibres which are abundant in mammalian peripheral nerves and transmit motor commands. The model is then used to evaluate the viability of using EIT with a nerve cuff to record neural activity in peripheral nerves. Main results: Fibre impedance models indicate activity in unmyelinated fibres can be screened out from activity in myelinated fibres using operating frequencies above 100 Hz. At 1 kHz the transverse impedance magnitude, which is perpendicular to the fibre length axis, of inactive intra-fascicle tissue and the fraction change during neural activity are estimated to be 1142 Ω cm and −8.8 × 10−4, respectively. At 1 kHz and 10 mm spacing between the impedance measurement electrode pair, the longitudinal impedance magnitude, which is parallel to the fibre length axis, and the fraction change during neural activity are estimated to be 328 Ω cm and −0.30, respectively. We show that a novel EIT drive and measurement electrode pattern which utilises longitudinal current and longitudinal differential boundary voltage measurements could distinguish activity in different fascicles, as well as simultaneous activity in multiple fascicles, of a three-fascicle mammalian nerve using simulated data. Significance: The results of this study provide an estimate of the transient change in impedance of intra-fascicle tissue during neural activity in mammalian nerve, and present a viable EIT electrode pattern, both of which are critical steps towards implementing EIT in a nerve cuff for a recording neural interface.

Smoothened Complete Electrode Model

This work reformulates the complete electrode model of electrical impedance tomography in order to enable more efficient numerical solution. The model traditionally assumes constant contact conductances on all electrodes, which leads to a discontinuous Robin boundary condition since the gaps between the electrodes can be described by vanishing conductance. As a consequence, the regularity of the electromagnetic potential is limited to less than two square-integrable weak derivatives, which negatively affects the convergence of, e.g., the finite element method. In this paper, a smoothened model for the boundary conductance is proposed, and the unique solvability and improved regularity of the ensuing boundary value problem are proven. Numerical experiments demonstrate that the proposed model is both computationally feasible and compatible with real-world measurements. In particular, the new model allows faster convergence of the finite element method.

The MUSIC algorithm for impedance tomography of small inclusions from discrete data

We consider a point-electrode model for electrical impedance tomography and show that current-to-voltage measurements from finitely many electrodes are sufficient to characterize the positions of a finite number of point-like inclusions. More precisely, we consider an asymptotic expansion with respect to the size of the small inclusions of the relative Neumann-to-Dirichlet operator in the framework of the point electrode model. This operator is naturally finite-dimensional and models difference measurements by finitely many small electrodes of the electric potential with and without the small inclusions. Moreover, its leading-order term explicitly characterizes the centers of the small inclusions if the (finite) number of point electrodes is large enough. This characterization is based on finite-dimensional test vectors and leads naturally to a MUSIC algorithm for imaging the inclusion centers. We show both the feasibility and limitations of this imaging technique via two-dimensional numerical experiments, considering in particular the influence of the number of point electrodes on the algorithm’s images.

Wearable sensors for patient-specific boundary shape estimation to improve the forward model for electrical impedance tomography (EIT) of neonatal lung function

Electrical impedance tomography (EIT) could be significantly advantageous to continuous monitoring of lung development in newborn and, in particular, preterm infants as it is non-invasive and safe to use within the intensive care unit. It has been demonstrated that accurate boundary form of the forward model is important to minimize artefacts in reconstructed electrical impedance images. This paper presents the outcomes of initial investigations for acquiring patient-specific thorax boundary information using a network of flexible sensors that imposes no restrictions on the patient's normal breathing and movements. The investigations include: (1) description of the basis of the reconstruction algorithms, (2) tests to determine a minimum number of bend sensors, (3) validation of two approaches to reconstruction and (4) an example of a commercially available bend sensor and its performance. Simulation results using ideal sensors show that, in the worst case, a total shape error of less than 6% with respect to its total perimeter can be achieved.

Application of stochastic Galerkin FEM to the complete electrode model of electrical impedance tomography

The aim of electrical impedance tomography is to determine the internal conductivity distribution of some physical body from boundary measurements of current and voltage. The most accurate forward model for impedance tomography is the complete electrode model, which consists of the conductivity equation coupled with boundary conditions that take into account the electrode shapes and the contact resistances at the corresponding interfaces. If the reconstruction task of impedance tomography is recast as a Bayesian inference problem, it is essential to be able to solve the complete electrode model forward problem with the conductivity and the contact resistances treated as a random field and random variables, respectively. In this work, we apply a stochastic Galerkin finite element method to the ensuing elliptic stochastic boundary value problem and compare the results with Monte Carlo simulations.

Reconstruction of electrical impedance tomography (EIT) images based on the expectation maximum (EM) method

Electrical impedance tomography (EIT) calculates the internal conductivity distribution within a body using electrical contact measurements. The image reconstruction for EIT is an inverse problem, which is both non-linear and ill-posed. The traditional regularization method cannot avoid introducing negative values in the solution. The negativity of the solution produces artifacts in reconstructed images in presence of noise. A statistical method, namely, the expectation maximization (EM) method, is used to solve the inverse problem for EIT in this paper. The mathematical model of EIT is transformed to the non-negatively constrained likelihood minimization problem. The solution is obtained by the gradient projection-reduced Newton (GPRN) iteration method. This paper also discusses the strategies of choosing parameters. Simulation and experimental results indicate that the reconstructed images with higher quality can be obtained by the EM method, compared with the traditional Tikhonov and conjugate gradient (CG) methods, even with non-negative processing.

Use of anisotropic modelling in electrical impedance tomography; Description of method and preliminary assessment of utility in imaging brain function in the adult human head

Electrical Impedance Tomography (EIT) is an imaging method which enables a volume conductivity map of a subject to be produced from multiple impedance measurements. It has the potential to become a portable non-invasive imaging technique of particular use in imaging brain function. Accurate numerical forward models may be used to improve image reconstruction but, until now, have employed an assumption of isotropic tissue conductivity. This may be expected to introduce inaccuracy, as body tissues, especially those such as white matter and the skull in head imaging, are highly anisotropic. The purpose of this study was, for the first time, to develop a method for incorporating anisotropy in a forward numerical model for EIT of the head and assess the resulting improvement in image quality in the case of linear reconstruction of one example of the human head. A realistic Finite Element Model (FEM) of an adult human head with segments for the scalp, skull, CSF, and brain was produced from a structural MRI. Anisotropy of the brain was estimated from a diffusion tensor-MRI of the same subject and anisotropy of the skull was approximated from the structural information. A method for incorporation of anisotropy in the forward model and its use in image reconstruction was produced. The improvement in reconstructed image quality was assessed in computer simulation by producing forward data, and then linear reconstruction using a sensitivity matrix approach. The mean boundary data difference between anisotropic and isotropic forward models for a reference conductivity was 50%. Use of the correct anisotropic FEM in image reconstruction, as opposed to an isotropic one, corrected an error of 24 mm in imaging a 10% conductivity decrease located in the hippocampus, improved localisation for conductivity changes deep in the brain and due to epilepsy by 4–17 mm, and, overall, led to a substantial improvement on image quality. This suggests that incorporation of anisotropy in numerical models used for image reconstruction is likely to improve EIT image quality.

State estimation in process tomography—Three‐dimensional impedance imaging of moving fluids

AbstractIn this paper, we consider three‐dimensional impedance imaging of rapidly varying objects. We especially concentrate on the case where the target is a moving fluid and the objective is to track the concentration distribution of a substance dissolved in the fluid. The observations are made as in ordinary electrical impedance tomography (EIT), but in the reconstruction we employ the convection–diffusion model to yield information on the temporal behavior of the object. The observation model of EIT together with the evolution model constitute the state‐space representation of the system, and the reconstruction problem of EIT can be described as a state estimation problem. The state estimation problem is solved using the Kalman filter and the fixed‐interval smoother algorithms. The performance of the state estimation approach is evaluated in a simulation study. Copyright © 2007 John Wiley & Sons, Ltd.

Improving the finite element forward model of the human head by warping using elastic deformation

As the use of realistic geometry in the forward model of electrical impedance tomography (EIT) of brain function appears to improve image reconstruction, the generation of patient-specific finite element meshes has been the subject of much recent work. This paper presents a more rapid method of generating more geometrically accurate finite element meshes of the human head by warping existing meshes such that the surface boundary beneath the electrodes closely matches that of the subject with minimal degradation to the quality of the mesh. Pre-existing meshes of spheres and adult head models incorporating key internal anatomical features are warped, using elastic deformation, to match a phantom latex tank incorporating a real skull. The algorithm is described and tests are carried out to optimize the key parameters to ensure minimal degradation of mesh quality and distortion of internal features. Results show that the algorithm operating with the optimum parameters produces meshes of sound quality and could represent an important step in the timely and productive creation of forward models in clinical applications.

A boundary-element approach for the complete-electrode model of EIT illustrated using simulated and real data

Electrical impedance tomography (EIT) allows non-invasive monitoring and control of dynamic industrial processes. Although reconstruction of tomographic images is useful for process visualization, for industrial process control, estimation of key process parameters is frequently more appropriate than visualization. Often such parameters comprise geometric information. The proposed direct problem solution method naturally encodes geometric information. The complete-electrode model for the direct problem of electrical tomography is solved with a boundary-element approach assuming a piecewise constant conductivity distribution with unknown conductivity and unknown interface location. The boundary-element discretization of the complete-electrode model is presented. In practical applications it is essential to provide an assessment of reliability of the estimates. To achieve this aim, confidence intervals are constructed in a manner which allows sufficient flexibility for the summary of non-Gaussian distributions. Hypothesis testing is proposed to detect changes in electrical impedance on the electrode surface which can be used to identify malfunctioning or poorly contacting electrodes. The proposed data analysis is applied to both computationally simulated and real experimental data.

A piecewise-constant binary model for electrical impedance tomography

In this paper, we consider the electrical impedance tomography problem in a computational approach. This inverse problem is the recovery of the electrical conductivity $\sigma$ in a domain from boundary measurements, given in the form of the Neumann-to-Dirichlet map. We formulate the inverse problem as a variational one, with a fitting term and a regularization term. We restrict the minimization with respect to the unknown $\sigma$ to piecewise-constant functions defined on rectangular domains in two dimensions. We borrow image segmentation techniques to solve the minimization problem. Several experimental results of conductivity reconstruction from synthetic data are shown, with and without noise, that validate the proposed method.

Numerical implementation of the factorization method within the complete electrode model of electrical impedance tomography

In electrical impedance tomography, one tries to recover the spatial conductivitydistribution inside a body from boundary measurements of current and voltage.In many practically important situations, the object has known backgroundconductivity but it is contaminated by inhomogeneities.The factorization method of Andreas Kirsch provides a tool for locatingsuch inclusions. In earlier work, it has been shown, both theoretically andnumerically, that the inhomogeneities can be characterized by the factorization techniqueif the input current can be controlled and the potential can be measured everywhere onthe object boundary. However, in real-world electrode applications, one can onlycontrol the net currents through certain surface patches and measure thecorresponding potentials on the electrodes. In this work,the factorization method is translated to the framework of the complete electrode modelof electrical impedance tomography and its functionality is demonstrated through two-dimensionalnumerical experiments. Special attention is paid to the efficient implementation of thealgorithm in polygonal domains.

Excitation patterns in three-dimensional electrical impedance tomography

Electrical impedance tomography (EIT) is a non-invasive technique that aims to reconstruct images of internal electrical properties of a domain, based on electrical measurements on the periphery. Improvements in instrumentation and numerical modeling have led to three-dimensional (3D) imaging. The availability of 3D modeling and imaging raises the question of identifying the best possible excitation patterns that will yield to data, which can be used to produce the best image reconstruction of internal properties. In this work, we describe our 3D finite element model of EIT. Through singular value decomposition as well as examples of reconstructed images, we show that for a homogenous female breast model with four layers of electrodes, a driving pattern where each excitation plane is a sinusoidal pattern out-of-phase with its neighboring plane produces better qualitative images. However, in terms of quantitative imaging an excitation pattern where all electrode layers are in phase produces better results.