A direct reconstruction algorithm for electrical impedance tomography

Hardware for cell culture electrical impedance tomography: A critical review

Human cell cultures are powerful laboratory tools for biological models of diseases, drug development, and tissue engineering. However, the success of biological experiments often depends on real-time monitoring of the culture state. Conventional culture evaluation methods consist of end-point laborious techniques, not capable of real-time operation and not suitable for three-dimensional cultures. Electrical Impedance Tomography (EIT) is a non-invasive imaging technique with high potential to be used in cell culture monitoring due to its biocompatibility, non-invasiveness, high temporal resolution, compact hardware, automatic operation, and high throughput. This review approaches the different hardware strategies for cell culture EIT that are presented in the literature, discussing the main components of the measurement system: excitation circuit, voltage/current sensing, switching stage, signal specifications, electrode configurations, measurement protocols, and calibration strategies. The different approaches are qualitatively discussed and compared, and design guidelines are proposed.

Supervised Descent Learning for Thoracic Electrical Impedance Tomography

OBJECTIVE

The absolute image reconstruction problem of electrical impedance tomography (EIT) is ill-posed. Traditional methods usually solve a nonlinear least squares problem with some kind of regularization. These methods suffer from low accuracy, poor anti-noise performance, and long computation time. Besides, the integration of a priori information is not very flexible. This work tries to solve EIT inverse problem using a machine learning algorithm for the application of thorax imaging.

METHODS

We developed the supervised descent learning EIT (SDL-EIT) inversion algorithm based on the idea of supervised descent method (SDM). The algorithm approximates the mapping from measured data to the conductivity image by a series of descent directions learned from training samples. We designed a training data set in which the thorax contour, and some general structure of lungs, and heart are embedded. The algorithm is implemented in both two-, and three-dimensional cases, and is evaluated using synthetic, and measured thoracic data. Results, and conclusion: For synthetic data, SDL-EIT shows better accuracy, and anti-noise performance compared with traditional Gauss-Newton inversion (GNI) method. For measured data, the result of SDL-EIT is reasonable compared with computed tomography (CT) scan image.

SIGNIFICANCE

Using SDL-EIT, prior information can be easily integrated through the specifically designed training data set, and the image reconstruction process can be accelerated. The algorithm is effective in inverting measured thoracic data. It is a potential algorithm for human thorax imaging.

B-Spline Level Set Method for Shape Reconstruction in Electrical Impedance Tomography

A B-spline level set (BLS) based method is proposed for shape reconstruction in electrical impedance tomography (EIT). We assume that the conductivity distribution to be reconstructed is piecewise constant, transforming the image reconstruction problem into a shape reconstruction problem. The shape/interface of inclusions is implicitly represented by a level set function (LSF), which is modeled as a continuous parametric function expressed using B-spline functions. Starting from modeling the conductivity distribution with the B-spline based LSF, we show that the shape modeling allows us to compute the solution by restricting the minimization problem to the space spanned by the B-splines. As a consequence, the solution to the minimization problem is obtained in terms of the B-spline coefficients. We illustrate the behavior of this method using simulated as well as water tank data. In addition, robustness studies considering varying initial guesses, differing numbers of control points, and modeling errors caused by inhomogeneity are performed. Both simulation and experimental results show that the BLS-based approach offers clear improvements in preserving the sharp features of the inclusions in comparison to the recently published parametric level set method.

Beltrami-net: domain-independent deep D-bar learning for absolute imaging with electrical impedance tomography (a-EIT)

OBJECTIVE

To develop, and demonstrate the feasibility of, a novel image reconstruction method for absolute electrical impedance tomography (a-EIT) that pairs deep learning techniques with real-time robust D-bar methods and examine the influence of prior information on the reconstruction.

APPROACH

A D-bar method is paired with a trained convolutional neural network (CNN) as a post-processing step. Training data is simulated for the network using no knowledge of the boundary shape by using an associated nonphysical Beltrami equation rather than simulating the traditional current and voltage data specific to a given domain. This allows the training data to be boundary shape independent. The method is tested on experimental data from two EIT systems (ACT4 and KIT4) with separate training sets of varying prior information.

MAIN RESULTS

Post-processing the D-bar images with a CNN produces significant improvements in image quality measured by structural SIMilarity indices (SSIMs) as well as relative [Formula: see text] and [Formula: see text] image errors.

SIGNIFICANCE

This work demonstrates that more general networks can be trained without being specific about boundary shape, a key challenge in EIT image reconstruction. The work is promising for future studies involving databases of anatomical atlases.

A Parametric Level Set-Based Approach to Difference Imaging in Electrical Impedance Tomography

This paper presents a novel difference imaging approach based on the recently developed parametric level set (PLS) method for estimating the change in a target conductivity from electrical impedance tomography measurements. As in conventional difference imaging, the reconstruction of conductivity change is based on data sets measured from the surface of a body before and after the change. The key feature of the proposed approach is that the conductivity change to be reconstructed is assumed to be piecewise constant, while the geometry of the anomaly is represented by a shape-based PLS function employing Gaussian radial basis functions (GRBFs). The representation of the PLS function by using GRBF provides flexibility in describing a large class of shapes with fewer unknowns. This feature is advantageous, as it may significantly reduce the overall number of unknowns, improve the condition number of the inverse problem, and enhance the computational efficiency of the technique. To evaluate the proposed PLS-based difference imaging approach, results obtained via simulation, phantom study, and in vivo pig data are studied. We find that the proposed approach tolerates more modeling errors and leads to a significant improvement in image quality compared with the conventional linear approach.

A Fidelity-Embedded Regularization Method for Robust Electrical Impedance Tomography

Electrical impedance tomography (EIT) provides functional images of an electrical conductivity distribution inside the human body. Since the 1980s, many potential clinical applications have arisen using inexpensive portable EIT devices. EIT acquires multiple trans-impedance measurements across the body from an array of surface electrodes around a chosen imaging slice. The conductivity image reconstruction from the measured data is a fundamentally ill-posed inverse problem notoriously vulnerable to measurement noise and artifacts. Most available methods invert the ill-conditioned sensitivity or the Jacobian matrix using a regularized least-squares data-fitting technique. Their performances rely on the regularization parameter, which controls the trade-off between fidelity and robustness. For clinical applications of EIT, it would be desirable to develop a method achieving consistent performance over various uncertain data, regardless of the choice of the regularization parameter. Based on the analysis of the structure of the Jacobian matrix, we propose a fidelity-embedded regularization (FER) method and a motion artifact reduction filter. Incorporating the Jacobian matrix in the regularization process, the new FER method with the motion artifact reduction filter offers stable reconstructions of high-fidelity images from noisy data by taking a very large regularization parameter value. The proposed method showed practical merits in experimental studies of chest EIT imaging.

Translational-circular scanning for magneto-acoustic tomography with current injection

Background

Magneto-acoustic tomography with current injection involves using electrical impedance imaging technology. To explore the potential applications in imaging biological tissue and enhance image quality, a new scan mode for the transducer is proposed that is based on translational and circular scanning to record acoustic signals from sources.

Methods

An imaging algorithm to analyze these signals is developed in respect to this alternative scanning scheme. Numerical simulations and physical experiments were conducted to evaluate the effectiveness of this scheme. An experiment using a graphite sheet as a tissue-mimicking phantom medium was conducted to verify simulation results. A pulsed voltage signal was applied across the sample, and acoustic signals were recorded as the transducer performed stepped translational or circular scans. The imaging algorithm was used to obtain an acoustic-source image based on the signals.

Results

In simulations, the acoustic-source image is correlated with the conductivity at the sample boundaries of the sample, but image results change depending on distance and angular aspect of the transducer. In general, as angle and distance decreases, the image quality improves. Moreover, experimental data confirmed the correlation.

Conclusion

The acoustic-source images resulting from the alternative scanning mode has yielded the outline of a phantom medium. This scan mode enables improvements to be made in the sensitivity of the detecting unit and a change to a transducer array that would improve the efficiency and accuracy of acoustic-source images.

Direct EIT reconstructions of complex admittivities on a chest-shaped domain in 2-D

Electrical impedance tomography (EIT) is a medical imaging technique in which current is applied on electrodes on the surface of the body, the resulting voltage is measured, and an inverse problem is solved to recover the conductivity and/or permittivity in the interior. Images are then formed from the reconstructed conductivity and permittivity distributions. In the 2-D geometry, EIT is clinically useful for chest imaging. In this work, an implementation of a D-bar method for complex admittivities on a general 2-D domain is presented. In particular, reconstructions are computed on a chest-shaped domain for several realistic phantoms including a simulated pneumothorax, hyperinflation, and pleural effusion. The method demonstrates robustness in the presence of noise. Reconstructions from trigonometric and pairwise current injection patterns are included.

Iterative sinc-convolution method for solving planar D-bar equation with application to EIT

The numerical solution of D-bar integral equations is the key in inverse scattering solution of many complex problems in science and engineering including conductivity imaging. Recently, a couple of methodologies were considered for the numerical solution of D-bar integral equation, namely product integrals and multigrid. The first one involves high computational complexity and other one has low convergence rate disadvantages. In this paper, a new and efficient sinc-convolution algorithm is introduced to solve the two-dimensional D-bar integral equation to overcome both of these disadvantages and to resolve the singularity problem not tackled before effectively. The method of sinc-convolution is based on using collocation to replace multidimensional convolution-form integrals- including the two-dimensional D-bar integral equations - by a system of algebraic equations. Separation of variables in the proposed method allows elimination of the formulation of the huge full matrices and therefore reduces the computational complexity drastically. In addition, the sinc-convolution method converges exponentially with a convergence rate of O(e-cN). Simulation results on solving a test electrical impedance tomography problem confirm the efficiency of the proposed sinc-convolution-based algorithm.

A direct D-bar reconstruction algorithm for recovering a complex conductivity in 2-D

A direct reconstruction algorithm for complex conductivities in W^2,∞ (Ω), where Ω is a bounded, simply connected Lipschitz domain in ℝ², is presented. The framework is based on the uniqueness proof by Francini [Inverse Problems 20 2000], but equations relating the Dirichlet-to-Neumann to the scattering transform and the exponentially growing solutions are not present in that work, and are derived here. The algorithm constitutes the first D-bar method for the reconstruction of conductivities and permittivities in two dimensions. Reconstructions of numerically simulated chest phantoms with discontinuities at the organ boundaries are included.

Precise two-dimensional D-bar reconstructions of human chest and phantom tank via sinc-convolution algorithm

BACKGROUND

Electrical Impedance Tomography (EIT) is used as a fast clinical imaging technique for monitoring the health of the human organs such as lungs, heart, brain and breast. Each practical EIT reconstruction algorithm should be efficient enough in terms of convergence rate, and accuracy. The main objective of this study is to investigate the feasibility of precise empirical conductivity imaging using a sinc-convolution algorithm in D-bar framework.

METHODS

At the first step, synthetic and experimental data were used to compute an intermediate object named scattering transform. Next, this object was used in a two-dimensional integral equation which was precisely and rapidly solved via sinc-convolution algorithm to find the square root of the conductivity for each pixel of image. For the purpose of comparison, multigrid and NOSER algorithms were implemented under a similar setting. Quality of reconstructions of synthetic models was tested against GREIT approved quality measures. To validate the simulation results, reconstructions of a phantom chest and a human lung were used.

RESULTS

Evaluation of synthetic reconstructions shows that the quality of sinc-convolution reconstructions is considerably better than that of each of its competitors in terms of amplitude response, position error, ringing, resolution and shape-deformation. In addition, the results confirm near-exponential and linear convergence rates for sinc-convolution and multigrid, respectively. Moreover, the least degree of relative errors and the most degree of truth were found in sinc-convolution reconstructions from experimental phantom data. Reconstructions of clinical lung data show that the related physiological effect is well recovered by sinc-convolution algorithm.

CONCLUSIONS

Parametric evaluation demonstrates the efficiency of sinc-convolution to reconstruct accurate conductivity images from experimental data. Excellent results in phantom and clinical reconstructions using sinc-convolution support parametric assessment results and suggest the sinc-convolution to be used for precise clinical EIT applications.

[Electrical impedance tomography in acute lung injury]

Electrical impedance tomography has been described as a new method of monitoring critically ill patients on mechanical ventilation. It has recently gained special interest because of its applicability for monitoring ventilation and pulmonary perfusion. Its bedside and continuous implementation, and the fact that it is a non-ionizing and non-invasive technique, makes it an extremely attractive measurement tool. Likewise, given its ability to assess the regional characteristics of lung structure, it could be considered an ideal monitoring tool in the heterogeneous lung with acute lung injury. This review explains the physical concept of bioimpedance and its clinical application, and summarizes the scientific evidence published to date with regard to the implementation of electrical impedance tomography as a method for monitoring ventilation and perfusion, mainly in the patient with acute lung injury, and other possible applications of the technique in the critically ill patient. The review also summarizes the limitations of the technique and its potential areas of future development.

Reconstruction convergence and speed enhancement in electrical impedance tomography for domains with known internal boundaries

An improved approach for electrical impedance tomography (EIT) image reconstruction, based on modifying the forward and inverse solutions, is proposed. In this approach, the EIT forward problem is solved via the finite element method (FEM) using two types of elements. The inverse problem is solved by the modified Newton-Raphson method, whereas the condition number of the Hessian matrix is being monitored. At the early stage of the reconstruction, first-order elements are used, and if the condition number exceeds the allowable limit, the algorithm restarts. Otherwise, if the reconstruction error becomes lower than a predefined threshold, second-order elements are employed in the forward solution in order to preserve the precision of the final results. The latter stage converges in very few iterations. Since the solution speed with the first-order FEM is considerably higher than the second-order FEM, the reconstruction speed improves considerably by this approach, whereas the accuracy of the results is guaranteed by the well-conditioned Hessian matrix. Numerical simulations and experiments are followed by comparisons with other reconstruction methods which demonstrate the reliability and high solution speed of this approach. According to the results, the convergence of the proposed method is significantly improved, and its speed is 2-200 times higher than the previously developed methods with the same level of precision.

An implementation of CalderOn's method for 3-D limited-view EIT

Mathematical interest in electrical impedance tomography has been strong since the publication of CalderOn's foundational paper. This paper introduced the idea of applying external voltage patterns to a medium such that, assuming that the medium is sufficiently close to a constant admittivity, the reconstruction can be accomplished directly by inverse Fourier transform. Motivated by CalderOn's method, we have developed a variant of the algorithm which is applicable to the case of measurement on only a part of the boundary and on discrete electrodes. Here we determine voltage or current patterns to apply to the electrodes which optimally approximate CalderOn's special functions in the interior. Furthermore, in three dimensions and higher, CalderOn's method allows each point in Fourier space to be computed in a multiplicity of ways. We show that by making use of the inherent redundancy in our measurements, we can significantly improve the quality of the static images produced by our algorithm.

Electrical impedance tomography guided ventilation therapy

PURPOSE OF REVIEW

Computed tomography (CT) in patients with acute respiratory distress syndrome has shown that intrapulmonary gas is not homogeneously distributed. Although regional ventilation can be studied by isotope and magnetic resonance techniques while aeration of the lungs can be imaged using CT, these techniques are not available at the bedside. Recently, electrical impedance tomography has been introduced as a true bedside technique which provides information on regional ventilation distribution.

RECENT FINDINGS

Electrical impedance tomography can reliably determine regional ventilation in healthy lungs and various models of induced lung injury when compared with CT, electron beam CT, and single photon emission CT. In healthy volunteers and patients with acute lung injury, relative impedance changes on the electrical impedance tomography image demonstrate an excellent correlation with regional changes in lung air content detected by CT. In a limited number of patients with respiratory dysfunction, gas exchange was found to improve when electrical impedance tomography was used to adjust ventilator settings, improving regional ventilation and avoiding tidal alveolar collapse.

SUMMARY

In view of recently published data, it can be concluded that, in critically ill patients, electrical impedance tomography determines reliable regional ventilation. Therefore, this technique has the potential to become a valuable bedside tool.

Noncontact scanning electrical impedance imaging

We are interested in applying electrical impedance imaging to a single cell because it has potential to reveal both cell anatomy and cell function. Unfortunately, classic impedance imaging techniques are not applicable to this small scale measurement due to their low resolution. In this paper, a different method of impedance imaging is developed based on a noncontact scanning system. In this system, the imaging sample is immersed in an aqueous solution allowing for the use of various probe designs. Among those designs, we discuss a novel shield-probe design that has the advantage of better signal-to-noise ratio with higher resolution compared to other probes. Images showing the magnitude of current for each scanned point were obtained using this configuration. A low-frequency linear physical model helps to relate the current to the conductivity at each point. Line-scan data of high impedance contrast structures can be shown to be a good fit to this model. The first two-dimensional impedance image of biological tissues generated by this technique is shown with resolution on the order of 100 mum. The image reveals details not present in the optical image.

Noniterative analytical formula for inverse scattering of multiply scattering point targets

This paper derives, in the exact framework of multiple scattering theory for point targets, a noniterative analytical formula for the nonlinear inversion of the target scattering strengths from the scattering or response matrix that can be applied after the target positions have been estimated in a previous step via, e.g., time-reversal multiple signal classification or another approach. The new formula provides a noniterative analytical alternative to the iterative numerical solution approach for the same problem presented in a recent paper [A. J. Devaney, E. A. Marengo, and F. K. Gruber, "Time-reversal-based imaging and inverse scattering of multiply scattering point targets," J. Acoust. Soc. Am. 118, 3129-3138 (2005)]. The two methods (noniterative versus iterative) are comparatively investigated with two numerical examples.

Recent advances in diffuse optical imaging

We review the current state-of-the-art of diffuse optical imaging, which is an emerging technique for functional imaging of biological tissue. It involves generating images using measurements of visible or near-infrared light scattered across large (greater than several centimetres) thicknesses of tissue. We discuss recent advances in experimental methods and instrumentation, and examine new theoretical techniques applied to modelling and image reconstruction. We review recent work on in vivo applications including imaging the breast and brain, and examine future challenges.

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.

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.

Imbalances in Regional Lung Ventilation

Imbalances in regional lung ventilation, with gravity-dependent collapse and overdistention of nondependent zones, are likely associated to ventilator-induced lung injury. Electric impedance tomography is a new imaging technique that is potentially capable of monitoring those imbalances. The aim of this study was to validate electrical impedance tomography measurements of ventilation distribution, by comparison with dynamic computerized tomography in a heterogeneous population of critically ill patients under mechanical ventilation. Multiple scans with both devices were collected during slow-inflation breaths. Six repeated breaths were monitored by impedance tomography, showing acceptable reproducibility. We observed acceptable agreement between both technologies in detecting right-left ventilation imbalances (bias = 0% and limits of agreement = -10 to +10%). Relative distribution of ventilation into regions or layers representing one-fourth of the thoracic section could also be assessed with good precision. Depending on electrode positioning, impedance tomography slightly overestimated ventilation imbalances along gravitational axis. Ventilation was gravitationally dependent in all patients, with some transient blockages in dependent regions synchronously detected by both scanning techniques. Among variables derived from computerized tomography, changes in absolute air content best explained the integral of impedance changes inside regions of interest (r(2) > or = 0.92). Impedance tomography can reliably assess ventilation distribution during mechanical ventilation.

Markov chain Monte Carlo techniques and spatial–temporal modelling for medical EIT

Many imaging problems such as imaging with electrical impedance tomography (EIT) can be shown to be inverse problems: that is either there is no unique solution or the solution does not depend continuously on the data. As a consequence solution of inverse problems based on measured data alone is unstable, particularly if the mapping between the solution distribution and the measurements is also nonlinear as in EIT. To deliver a practical stable solution, it is necessary to make considerable use of prior information or regularization techniques. The role of a Bayesian approach is therefore of fundamental importance, especially when coupled with Markov chain Monte Carlo (MCMC) sampling to provide information about solution behaviour. Spatial smoothing is a commonly used approach to regularization. In the human thorax EIT example considered here nonlinearity increases the difficulty of imaging, using only boundary data, leading to reconstructions which are often rather too smooth. In particular, in medical imaging the resistivity distribution usually contains substantial jumps at the boundaries of different anatomical regions. With spatial smoothing these boundaries can be masked by blurring. This paper focuses on the medical application of EIT to monitor lung and cardiac function and uses explicit geometric information regarding anatomical structure and incorporates temporal correlation. Some simple properties are assumed known, or at least reliably estimated from separate studies, whereas others are estimated from the voltage measurements. This structural formulation will also allow direct estimation of clinically important quantities, such as ejection fraction and residual capacity, along with assessment of precision.

Induced-current electrical impedance tomography: a 2-D theoretical simulation

A reconstruction algorithm, based on the modified Newton-Raphson algorithm, was developed for induced-current electrical impedance tomography and studied in theoretical two-dimensional geometry representing a human thorax. The finite-volume method was applied for the discretization of the physical domain, resulting in a symbolic representation of the Jacobian matrix, which is accurate and fast to construct. Several system configurations, differing in the number of excitation coils and electrodes, were simulated, and the performance in thoracic imaging was studied. It was found that a six-coil system shows a significant 40% improvement of conductivity values reconstruction over the three-coil system (an error of 2.06 omega(-1) compared with 3.44 omega(-1)). A number of 32 electrodes was found to be sufficient, being the smallest number of electrodes to still provide a reasonable performance (only 4.2% degradation in average conductivity error compared with the maximum possible 106-electrode system).