Electrical Impedance Tomography Problem Research Articles

Approximation of continuous EIT data from electrode measurements with Bayesian methods

The electrical impedance tomography (EIT) in its classical formulation seeks to estimate the electric conductivity distribution inside the body from the knowledge of the Dirichlet-to-Neumann (DtN) map of the conductivity equation at the boundary. Numerical methods for the solution of the EIT problem have been developed based on this formulation, most notably the d-bar method and the layer stripping algorithm. In practice, however, the EIT data (electrode data), collected by using a fixed number of contact electrodes, is tantamount to knowledge of the resistance matrix, a mapping between given current configuration and the corresponding vector of measured electrode voltages. Forward models corresponding to the DtN data and the electrode data differ in terms of the boundary values and no direct connection between them has been established. In this article, we analyze the relation between the two boundary data types, and propose to approximate the DtN data from the measured resistance matrix for solving the EIT inverse problem within the Bayesian framework, leveraging a sample based prior and a principal component model reduction.

Uniqueness and Lipschitz stability in electrical impedance tomography with finitely many electrodes

For the linearized reconstruction problem in electrical impedance tomography with the complete electrode model, Lechleiter and Rieder (2008 Inverse Problems 24 065009) have shown that a piecewise polynomial conductivity on a fixed partition is uniquely determined if enough electrodes are being used. We extend their result to the full non-linear case and show that measurements on a sufficiently high number of electrodes uniquely determine a conductivity in any finite-dimensional subset of piecewise-analytic functions. We also prove Lipschitz stability, and derive analogue results for the continuum model, where finitely many measurements determine a finite-dimensional Galerkin projection of the Neumann-to-Dirichlet operator on a boundary part.

Shape optimizationviaa levelset and a Gauss-Newton method

In the context of shape optimization via level-set methods, we propose a general framework for a Gauss-Newton method to optimize quadratic functionals. Our approach provides a natural extension of the shape derivative as a vector field defined in the whole working domain. We implement and discuss this method in two cases: first a least-square error minimization reminiscent of the Electrical Impedance Tomography problem, and second the compliance problem with volume constraints.

Three-dimensional Pulmonary Monitoring Using Focused Electrical Impedance Measurements.

Lung pathologies such as edema, atelectasis or pneumonia are potentially life threatening conditions. Especially in critically ill and mechanically ventilated patients, an early diagnosis and treatment is crucial to prevent an Acute Respiratory Distress Syndrome [1]. Thus, continuous monitoring tool for the lung condition available at the bedside would be highly appreciated. One concept for this is Electrical Impedance Tomography (EIT). In EIT, an electrode belt of typically 16 or 32 electrodes is attached at the body surface and multiple impedance measurements are performed. From this, the conductivity change inside the body is reconstructed in a two-dimensional image. In various studies, EIT proved to be a useful tool for quantifying recruitment maneuvers, the assessment of the ventilation homogeneity, the detection of lung edema or perfusion monitoring [2, 3, 4, 5]. Nevertheless, the main problem of EIT is the low spatial resolution (compared to CT) and the limitation to two dimensional images. In this paper, we try to address the latter issue: Instead of projecting conductivity changes onto a two-dimensional image, we adjust electrode positions to focus single tetrapolar measurements to specific, three-dimensional regions of interest. In earlier work, we defined guidelines to achieve this focusing [6, 7]. In this paper, we demonstrate in simulations and in a water tank experiment that applying these guidelines can help to detect pathologies in specific lung regions.

Convergence of finite element approximation for electrical impedance tomography with the complete electrode model

This paper discusses the convergence of the finite element approximation for the forward problem of electrical impedance tomography with the complete electrode model. When the domain is polygonal and conforming finite element method is employed with linear triangle or tetrahedron elements, we prove that the estimated voltages on the electrodes converge to the true ones up to an additive constant, as the sizes of elements in the underlying mesh all tend to zero.

Application of meshless local Petrov Galerkin method (MLPG5) for EIT forward problem

This paper deals with the application of Meshless Local Petrov Galerkin method(MLPG) for solving the forward problem associated with Electrical Impedance Tomography(EIT). Out of six variants of MLPG, here we have used MLPG5 to solve the forward problem. Use of Heaviside function as test function eliminates the domain integral term from the weak formulation. Radial point interpolation method(RPIM) is used and the resulting shape functions possess Kronecker delta function property. Hence no modification is required in the weak form to impose essential boundary conditions. Aforementioned advantages have motivated us to apply this meshless method for EIT forward problem. Numerical accuracy of this method is determined by finding the relative error between numerical solution and exact solution. There are several factors that can affect the accuracy such as local sub domain size(αsub), influence domain size(αinf), nodal density, type and shape of radial basis functions used. A detailed parametric study is carried out in this paper. Radial basis functions(RBF) such as Multiquadrics(MQ), Inverse Multiquadrics(INVMQ) and Gaussian are used for function interpolation along with linear polynomial. Shape parameter of RBFs is also varied to study the effect on accuracy. The solution obtained from the meshless method and analytical method are compared with solutions from Finite Element Method using linear triangular elements. Relative error is found to be minimum when αsub lies in the range of 0.6–0.7 for MQ and INVMQ and αinf > 5 since the boundary voltage profile should be the same when current is injected into the circular medium from any angle under homogeneous conditions. For Gaussian function, minimum error is observed for αsub in the interval 0.5–0.6 and αinf > 5. Gaussian has shown the lowest error compared to MQ and INVMQ. Forward problem in EIT usually deals with nonhomogeneous medium and therefore we need to extend this study to nested domains. Since it is very difficult to get exact solution for nonhomogeneous medium, same conductivity value is given for both background(outer domain) and the inhomogeneity(inner domain) with some additional coupling conditions to be satisfied at the interface. Resulting boundary potentials are matched with the exact solution of homogeneous medium. MQ is used for MLPG5 implementation and error is calculated for different values of αsub and αinf. Minimum error is obtained at αsub = 0.6 for all values of αinf. It is better to select αinf > 5 to ensure the boundary profiles to overlap in all current injection cycles.

A new regularization algorithm based on the neighborhood method for electrical impedance tomography

Regularization can be used to address the ill-posed inverse problem of electrical impedance tomography (EIT). However, the most commonly used regularization methods have their own inherent issues, such as the over-smoothness of reconstructed edges and unstable solution due to measurement noise. In this paper, a new regularization method is proposed, which utilizes the total differences of neighboring pixels (TDN) as the penalty term by utilizing spatial constraints from four or even more directions. With the use of more effective spatial constraints, the solution can be kept more stable. To obtain the sharp edges between different materials in the reconstructed image, the L1 norm was utilized as the form of the penalty term. Moreover, targeting the reconstruction of square objects, an improved image reconstruction approach based on this method was investigated. Two new criteria for the evaluation of reconstruction quality were proposed accordingly. The simulation and experimental results demonstrated that the TDN regularization performed robustly against the boundary measurement noise as well as possessing the ability to preserve the relatively sharp edges of interesting objects.

Study on the inverse problem of electrical impedance tomography based on self-diagnosis regularization

The inverse problem of electrical impedance tomography (EIT) is seriously ill-posed, which restricts the clinical application of EIT. Regularization is an important numerical method to improve the stability of the EIT inverse problem as well as the resolution of the imaging. This paper proposes a self-diagnosis regularization method based on Tikhonov regularization and diagonal weight regularization method (DWRM). Firstly, the ill-posedness of the inverse problem is analyzed by sensitivity. Then, the performance of the self-diagnosis regularization is analyzed through the singular value theory. Finally, some simulated experiments including simulations and flume experiment are carried out and verify that the self-diagnosis regularization has better image quality and anti-noise ability than those of traditional regularization methods. The self-diagnosis regularization method weakens the ill-posedness of inverse problem of EIT and can prompt the practical application of EIT.

An Automatic Detection of the ROI Using Otsu Thresholding in Nonlinear Difference EIT Imaging

Inverse problem of electrical impedance tomography is highly ill-posed therefore often prior information is used to have a satisfactory and stable solution. Recently, introduced non-linear differential imaging method estimates the initial and difference in conductivities simultaneously and is efficient in handling modeling errors. The non-linear parameterization of conductivity enables to use different regularization schemes for background and region of interest (ROI). Identifying the ROI without any prior information can be beneficial in improving the reconstruction performance. Therefore, in this paper, automatic detection of ROI is introduced using Otsu thresholding method and is then used with non-linear differential imaging. The proposed non-linear differential imaging with Otsu method (NDIWO) considers different regularization methods, i.e., total variational approach with ROI and smoothness prior with background regions during reconstruction. Numerical and experimental studies are performed to test NDIWO method for two-phase flow and thorax imaging and the performance is compared with absolute and linear difference imaging. The results indicate that the proposed NDIWO method has improved reconstruction performance compared with conventional absolute and linear difference imaging.

Deep D-Bar: Real-Time Electrical Impedance Tomography Imaging With Deep Neural Networks.

The mathematical problem for electrical impedance tomography (EIT) is a highly nonlinear ill-posed inverse problem requiring carefully designed reconstruction procedures to ensure reliable image generation. D-bar methods are based on a rigorous mathematical analysis and provide robust direct reconstructions by using a low-pass filtering of the associated nonlinear Fourier data. Similarly to low-pass filtering of linear Fourier data, only using low frequencies in the image recovery process results in blurred images lacking sharp features, such as clear organ boundaries. Convolutional neural networks provide a powerful framework for post-processing such convolved direct reconstructions. In this paper, we demonstrate that these CNN techniques lead to sharp and reliable reconstructions even for the highly nonlinear inverse problem of EIT. The network is trained on data sets of simulated examples and then applied to experimental data without the need to perform an additional transfer training. Results for absolute EIT images are presented using experimental EIT data from the ACT4 and KIT4 EIT systems.

Electrical Impedance Tomography With a Lab‐on‐Chip for Imaging Cells in Culture

Electrical Impedance Tomography (EIT) is a non‐invasive, non‐ionizing, and inexpensive imaging modality that is used to image the conductivity distribution inside the subject under test. EIT is an emerging imaging technique that has the potential to be used in a variety of (bio)medical applications. A technology that is easy to integrate into a small portable device and also easy to setup. In this work a custom made impedance analyzer is used as a measurement device. The working principle is based on the different conductivity distributions of the material under test, this due to inhomogeneous bioelectrical properties. However there is one major downside of this technique, the reconstruction problem of EIT is severely ill‐posed. This means that the definition of a correct model is essential. Because of this ill‐posed condition, a comparison of different models is done. In this work, an in depth study is performed to achieve the most optimal way of solving the inverse problem, which leads to noise suppression and reproducible results. This technology, integrated in a lab‐on‐chip for monitoring cellular growth, is based on a spatial reconstructed imaging technique using electrical impedance tomography.

Generalized linearization techniques in electrical impedance tomography

Electrical impedance tomography aims at reconstructing the interior electrical conductivity from surface measurements of currents and voltages. As the current–voltage pairs depend nonlinearly on the conductivity, impedance tomography leads to a nonlinear inverse problem. Often, the forward problem is linearized with respect to the conductivity and the resulting linear inverse problem is regarded as a subproblem in an iterative algorithm or as a simple reconstruction method as such. In this paper, we compare this basic linearization approach to linearizations with respect to the resistivity or the logarithm of the conductivity. It is numerically demonstrated that the conductivity linearization often results in compromised accuracy in both forward and inverse computations. Inspired by these observations, we present and analyze a new linearization technique which is based on the logarithm of the Neumann-to-Dirichlet operator. The method is directly applicable to discrete settings, including the complete electrode model. We also consider Frechet derivatives of the logarithmic operators. Numerical examples indicate that the proposed method is an accurate way of linearizing the problem of electrical impedance tomography.

Solving the electrical impedance tomography inverse problem for logarithmic conductivity

PurposeIn the context of electrical impedance tomography (EIT), this paper aims to evaluate limitations of estimating conductivity or resistivity, as well as the improvements achieved with the use of an alternate description of the solution space, the logarithmic conductivity.Design/methodology/approachA quantitative analysis is performed, solving the inverse EIT problem by using the Gauss–Newton and non-linear conjugate gradient methods for a numerical phantom of 15 elements. A property of symmetry is studied for the direct EIT problem for a phantom of 385,601 elements.FindingsSolving the inverse EIT problem in logarithmic conductivity is more robust to the initial guess, as solutions are kept within physical bounds (conductivity positiveness). Also, convergence is faster and less dependent on the final values of the estimates.Research limitations/implicationsLogarithmic conductivity provides an advantageous description of the solution space for the EIT inverse problem. Similar estimation problems might be subject to analogous conclusions.Originality/valueThis study provides a novel analysis, quantitatively comparing the effect of different variables to solve the inverse EIT problem.

A New Divide-and-Conquer Method for 3-D Electrical Impedance Tomography

Electrical impedance tomography (EIT) is a patient-safe approach for imaging applications and is a promising technology for biomedical imaging. By applying a small electrical current to living tissue and measuring the electrical potential at different points of the tissue’s boundary, it is possible to solve the inverse problem and to generate a map of the tissue’s conductivities. Although it is a promising method, the reconstruction process remains a complicated task, and real-time nonlinear image reconstruction algorithms have not yet been widely adopted. Recent studies have proposed the use of artificial neural networks (ANN) to solve the EIT inverse problem. ANNs can give a nonlinear conductivity distribution. Although several studies have examined 2-D finite-element models, very little work has been done on 3-D problems. In this paper, a solution based on the divide-and-conquer method and ANNs is used to solve the nonlinear inverse problem without any linearization. The solution presented here reduces the difficulty of training large ANNs, which are commonly required to solve a 3-D EIT problem with artificial intelligence algorithms.

Magnetic Resonance Electrical Impedance Tomography (MREIT): Convergence and Reduced Basis Approach

This article considers the inverse problem of magnetic resonance electrical impedance tomography (MREIT) in two dimensions. A rigorous mathematical framework for this inverse problem as well as the existing Harmonic $B_z$ Algorithm as a solution algorithm are presented. The convergence theory of this algorithm is extended, such that the usage of an approximative forward solution of the underlying partial differential equation (PDE) in the algorithm is sufficient for convergence. Motivated by this result, a novel algorithm is developed, the aim of which is to speed up the existing Harmonic $B_z$ Algorithm. This is achieved by combining it with an adaptive variant of the reduced basis method, a model order reduction technique. In a numerical experiment, a high-resolution image of the Shepp--Logan phantom is reconstructed and both algorithms are compared.

Detecting stochastic inclusions in electrical impedance tomography

This work considers the inclusion detection problem of electrical impedance tomography with stochastic conductivities. It is shown that a conductivity anomaly with a random conductivity can be identified by applying the factorization method or the monotonicity method to the mean value of the corresponding Neumann-to-Dirichlet map provided that the anomaly has high enough contrast in the sense of expectation. The theoretical results are complemented by numerical examples in two spatial dimensions.

Reconstruction of a piecewise constant conductivity on a polygonal partition via shape optimization in EIT

In this paper, we develop a shape optimization-based algorithm for the electrical impedance tomography (EIT) problem of determining a piecewise constant conductivity on a polygonal partition from boundary measurements. The key tool is to use a distributed shape derivative of a suitable cost functional with respect to movements of the partition. Numerical simulations showing the robustness and accuracy of the method are presented for simulated test cases in two dimensions.

An Equilibrated Fluxes Approach to the Certified Descent Algorithm for Shape Optimization Using Conforming Finite Element and Discontinuous Galerkin Discretizations

The certified descent algorithm (CDA) is a gradient-based method for shape optimization which certifies that the direction computed using the shape gradient is a genuine descent direction for the objective functional under analysis. It relies on the computation of an upper bound of the error introduced by the finite element approximation of the shape gradient. In this paper, we present a goal-oriented error estimator which depends solely on local quantities and is fully-computable. By means of the equilibrated fluxes approach, we construct a unified strategy valid for both conforming finite element approximations and discontinuous Galerkin discretizations. The new variant of the CDA is tested on the inverse identification problem of electrical impedance tomography: both its ability to identify a genuine descent direction at each iteration and its reliable stopping criterion are confirmed.

Image reconstruction using simulated annealing in electrical impedance tomography: a new approach

Image reconstruction in electrical impedance tomography (EIT) deals with an ill-posed and non-linear inverse problem. It has the objective of minimizing the difference between simulated (virtual) object data and electric voltage measurements performed on a non-simulated (real) object. In this paper, a new approach to the simulated annealing method applied to the reconstruction of EIT images is described. The main advantage of this approach is that all conductivity parameters are updated simultaneously. Most methods that employ simulated annealing to the problem of EIT usually evaluate each conductivity parameter individually resulting in high computational cost. The algorithm was tested both with computationally generated data and with measurements performed on a physical tank. In both cases, the method was able to make data inversion, determining the position, the dimensions and the conductivity of materials in an opaque object plane.

The effect of error and regularization norms on strain and damage identification via electrical impedance tomography in piezoresistive nanocomposites

Electrical impedance tomography (EIT) has recently emerged as a promising tool for the structural health monitoring (SHM) and non-destructive evaluation (NDE) of piezoresistive nanofiller-modified composites. With few exceptions in SHM and NDE, the vast majority of studies have formulated the EIT problem such that it endeavors to minimize a voltage error vector in the least-squares sense while simultaneously using an additional least-squares term for regularization. Unfortunately, the use of least-squares minimization and regularization has important limitations especially in the context of SHM and NDE. For example, the least-squares error minimization is extremely sensitive to outlier data. Such outlier data is a very real possibility in SHM applications due to damaged electrodes. Additionally, utilizing a least-squares term for regularization renders EIT incapable of resolving discontinuous conductivity changes such as would be induced by fracture events. More sophisticated minimization and regularization techniques that obviate these limitations are well known to medical and mathematical practitioners of EIT; however, these methods have not been explored in SHM and NDE applications. This is an important oversight because the needs of SHM and NDE differ considerably from the needs of medical imaging. Therefore, we herein explore the use of different minimization and regularization norms on EIT for experimental strain identification, detection of artificially induced through-hole damage, and detection of impact damage.