The D-bar method for electrical impedance tomography-demystified

TSS-ConvNet for electrical impedance tomography image reconstruction

Objective.The objective of this study was to propose a novel data-driven method for solving ill-posed inverse problems, particularly in certain conditions such as time-difference electrical impedance tomography for detecting the location and size of bubbles inside a pipe.Approach.We introduced a new layer architecture composed of three paths: spatial, spectral, and truncated spectral paths. The spatial path processes information locally, whereas the spectral and truncated spectral paths provide the network with a global receptive field. This unique architecture helps eliminate the ill-posedness and nonlinearity inherent in the inverse problem. The three paths were designed to be interconnected, allowing for an exchange of information on different receptive fields with varied learning abilities. Our network has a bottleneck architecture that enables it to recover signal information from noisy redundant measurements. We named our proposed model truncated spatial-spectral convolutional neural network (TSS-ConvNet).Main results.Our model demonstrated superior accuracy with relatively high resolution on both simulation and experimental data. This indicates that our approach offers significant potential for addressing ill-posed inverse problems in complex conditions effectively and accurately.Significance.The TSS-ConvNet overcomes the receptive field limitation found in most existing models that only utilize local information in Euclidean space. We trained the network on a large dataset covering various configurations with random parameters to ensure generalization over the training samples.

ACT5 Electrical Impedance Tomography System

OBJECTIVE

This article introduces the Adaptive Current Tomograph 5 (ACT5) Electrical Impedance Tomography (EIT) system. ACT5 is a 32 electrode applied-current multiple-source EIT system that can display real-time images of conductivity and susceptivity at 27 frames per second. The adaptive current sources in ACT5 can apply fully programmable current patterns with frequencies varying from 5 kHz to 500 kHz. The system also displays real-time ECG readings during the EIT imaging process.

METHODS

The hardware and software design and specifications are presented, including the current source design, FPGA hardware, safety features, calibration, and shunt impedance measurement.

RESULTS

Images of conductivity and susceptivity are presented from ACT5 data collected on tank phantoms and a human subject illustrating the system's ability to provide real-time images of pulsatile perfusion and ECG traces.

SIGNIFICANCE

The portability, high signal-to-noise ratio, and flexibility of applied currents over a wide range of frequencies enable this instrument to be used to obtain useful human subject data with relative clinical ease.

Fast absolute 3D CGO-based electrical impedance tomography on experimental tank data

Objective.To present the first 3D CGO-based absolute EIT reconstructions from experimental tank data.Approach.CGO-based methods for absolute EIT imaging are compared to traditional TV regularized non-linear least squares reconstruction methods. Additional robustness testing is performed by considering incorrect modeling of domain shape.Main Results.The CGO-based methods are fast, and show strong robustness to incorrect domain modeling comparable to classic difference EIT imaging and fewer boundary artefacts than the TV regularized non-linear least squares reference reconstructions.Significance.This work is the first to demonstrate fully 3D CGO-based absolute EIT reconstruction on experimental data and also compares to TV-regularized absolute reconstruction. The speed (1-5 s) and quality of the reconstructions is encouraging for future work in absolute EIT.

Applications of Electrical Impedance Tomography in Neurology

Introduction

Electrical impedance tomography (EIT) is a non-invasive technique utilized in various medical applications, including brain imaging and other neurological diseases. Recognizing the physiological and anatomical characteristics of organs based on their electrical properties is one of the main applications of EIT, as each variety of tissue structure has its own electrical characteristics. The high potential of brain EIT is established in real-time supervision and early recognition of cerebral brain infarction, hemorrhage, and other diseases. In this paper, we review the studies on the neurological applications of EIT.

Methods

EIT calculates the internal electrical conductivity distribution of an organ by measuring its surface impedance. A series of electrodes are placed on the surface of the target tissue, and small alternating currents are injected. The related voltages are then observed and analyzed. The electrical permittivity and conductivity distributions inside the tissue are reconstructed by measuring the electrode voltages.

Results

The electrical characteristic of biological tissues is remarkably dependent on their structures. Some tissues are better electrical conductors than the others since they have more ions that can carry the electrical charges. This difference is attributed to changes in cellular water content, membrane properties, and destruction of tight junctions within cell membranes.

Conclusion

EIT is an extremely practical device for brain imaging, capturing fast electrical activities in the brain, imaging epileptic seizures, detecting intracranial bleeding, detecting cerebral edema, and diagnosing stroke.

Hybrid method for improving Tikhonov-based reconstruction quality in electrical impedance tomography

Purpose

Electrical impedance tomography (EIT) has shown its potential in the field of medical imaging. Physiological or pathological variation would cause the change of conductivity. EIT is favorable in reconstructing conductivity distribution inside the detected area. However, due to its ill-posed and nonlinear characteristics, reconstructed images suffer from low spatial resolution.

Approach

Tikhonov regularization method is a popular and effective approach for image reconstruction in EIT. Nevertheless, excessive smoothness is observed when reconstruction is conducted based on Tikhonov method. To improve Tikhonov-based reconstruction quality in EIT, an innovative hybrid iterative optimization method is proposed. An efficient alternating minimization algorithm is introduced to solve the optimization problem.

Results

To verify image reconstruction performance and anti-noise robustness of the proposed method, a series of simulation work and phantom experiments is carried out. Meanwhile, comparison is made with reconstruction results based on Landweber, Newton-Raphson, and Tikhonov methods. The reconstruction performance is also verified by quantitative comparison of blur radius and structural similarity values which further demonstrates the excellent performance of the proposed method.

Conclusions

In contrast to Landweber, Newton-Raphson, and Tikhonov methods, it is found that images reconstructed by the proposed method are more accurate. Even under the impact of noise, the proposed method outperforms comparison methods.

Improved resolution of D-bar images of ventilation using a Schur complement property and an anatomical atlas

BACKGROUND

Electrical impedance tomography (EIT) is a nonionizing imaging technique for real-time imaging of ventilation of patients with respiratory distress. Cross-sectional dynamic images are formed by reconstructing the conductivity distribution from measured voltage data arising from applied alternating currents on electrodes placed circumferentially around the chest. Since the conductivity of lung tissue depends on air content, blood flow, and the presence of pathology, the dynamic images provide regional information about ventilation, pulsatile perfusion, and abnormalities. However, due to the ill-posedness of the inverse conductivity problem, EIT images have a coarse spatial resolution. One method of improving the resolution is to include prior information in the reconstruction.

PURPOSE

In this work, we propose a technique in which a statistical prior built from an anatomical atlas is used to postprocess EIT reconstructions of human chest data. The effectiveness of the method is demonstrated on data from two patients with cystic fibrosis.

METHODS

A direct reconstruction algorithm known as the D-bar method was used to compute a two-dimensional reconstruction of the conductivity distribution in the plane of the electrodes. Reconstructions using one step in an iterative (regularized) Newton's method were also computed for comparison. An anatomical atlas consisting of 1589 synthetic EIT images computed from X-ray computed tomography (CT) scans of 74 adult male subjects was computed for use as a statistical prior. The resolution of the D-bar images was then improved by maximizing the conditional probability density function of an image that is consistent with the a priori information and the statistical model. A new method to evaluate the accuracy of the EIT images using CT scans of the imaged patient as ground truth is presented. The novel approach is tested on data from two patients with cystic fibrosis.

RESULTS AND CONCLUSIONS

The D-bar images resulted in better structural similarity index measures (SSIM) and multiscale (MS) SSIM measures for both subjects using the mask or amplitude evaluation approach than the one-step (regularized) Newton's method. Further improvement was achieved using the Schur complement (SC) approach, with MS-SSIM values of 0.718 and 0.682 using SC evaluated with the mask and amplitude approach, respectively, for Patient 1, and MS-SSIM values of 0.726 and 0.692 using SC evaluated with the mask and amplitude approach, respectively, for Patient 2. The results from applying an anatomical atlas and statistical prior to EIT data from two patients with cystic fibrosis suggest that the spatial resolution of the EIT image can be improved to reveal pathology that may be difficult to discern in the original EIT image. The novel metric of evaluation is consistent with the appearance of improved spatial resolution and provides a new way to evaluate the accuracy of EIT reconstructions when a CT scan is available.

Majorization–Minimization Total Variation Solution Methods for Electrical Impedance Tomography

Inverse problems arise in many areas of science and engineering, such as geophysics, biology, and medical imaging. One of the main imaging modalities that have seen a huge increase in recent years is the noninvasive, nonionizing, and radiation-free imaging technique of electrical impedance tomography (EIT). Other advantages of such a technique are the low cost and ubiquitousness. An imaging technique is used to recover the internal conductivity of a body using measurements from electrodes from the body’s surface. The standard procedure is to obtain measurements by placing electrodes in the body and measuring conductivity inside the object. A current with low frequency is applied on the electrodes below a threshold, rendering the technique harmless for the body, especially when applied to living organisms. As with many inverse problems, EIT suffers from ill-posedness, i.e., the reconstruction of internal conductivity is a severely ill-posed inverse problem and typically yields a poor-quality solution. Moreover, the desired solution has step changes in the electrical properties that are typically challenging to be reconstructed by traditional smoothing regularization methods. To counter this difficulty, one solves a regularized problem that is better conditioned than the original problem by imposing constraints on the regularization term. The main contribution of this work is to develop a general ℓp regularized method with total variation to solve the nonlinear EIT problem through a iteratively reweighted majorization–minimization strategy combined with the Gauss–Newton approach. The main idea is to majorize the linearized EIT problem at each iteration and minimize through a quadratic tangent majorant. Simulated numerical examples from complete electrode model illustrate the effectiveness of our approach.

Robust imaging using electrical impedance tomography: review of current tools

Electrical impedance tomography (EIT) is a medical imaging technique with many advantages and great potential for development in the coming years. Currently, some limitations of EIT are related to the ill-posed nature of the problem. These limitations are translated on a practical level by a lack of genericity of the developed tools. In this paper, the main robust data acquisition and processing tools for EIT proposed in the scientific literature are presented. Their relevance and potential to improve the robustness of EIT are analysed, in order to conclude on the feasibility of a robust EIT tool capable of providing resistivity or difference of resistivity mapping in a wide range of applications. In particular, it is shown that certain measurement acquisition tools and algorithms, such as faulty electrode detection algorithm or particular electrode designs, can ensure the quality of the acquisition in many circumstances. Many algorithms, aiming at processing acquired data, are also described and allow to overcome certain difficulties such as an error in the knowledge of the position of the boundaries or the poor conditioning of the inverse problem. They have a strong potential to faithfully reconstruct a quality image in the presence of disturbances such as noise or boundary modelling error.

Graph Convolutional Networks for Model-Based Learning in Nonlinear Inverse Problems

The majority of model-based learned image reconstruction methods in medical imaging have been limited to uniform domains, such as pixelated images. If the underlying model is solved on nonuniform meshes, arising from a finite element method typical for nonlinear inverse problems, interpolation and embeddings are needed. To overcome this, we present a flexible framework to extend model-based learning directly to nonuniform meshes, by interpreting the mesh as a graph and formulating our network architectures using graph convolutional neural networks. This gives rise to the proposed iterative Graph Convolutional Newton-type Method (GCNM), which includes the forward model in the solution of the inverse problem, while all updates are directly computed by the network on the problem specific mesh. We present results for Electrical Impedance Tomography, a severely ill-posed nonlinear inverse problem that is frequently solved via optimization-based methods, where the forward problem is solved by finite element methods. Results for absolute EIT imaging are compared to standard iterative methods as well as a graph residual network. We show that the GCNM has good generalizability to different domain shapes and meshes, out of distribution data as well as experimental data, from purely simulated training data and without transfer training.

A Point-Matching Method of Moment with Sparse Bayesian Learning Applied and Evaluated in Dynamic Lung Electrical Impedance Tomography

Dynamic lung imaging is a major application of Electrical Impedance Tomography (EIT) due to EIT's exceptional temporal resolution, low cost and absence of radiation. EIT however lacks in spatial resolution and the image reconstruction is very sensitive to mismatches between the actual object's and the reconstruction domain's geometries, as well as to the signal noise. The non-linear nature of the reconstruction problem may also be a concern, since the lungs' significant conductivity changes due to inhalation and exhalation. In this paper, a recently introduced method of moment is combined with a sparse Bayesian learning approach to address the non-linearity issue, provide robustness to the reconstruction problem and reduce image artefacts. To evaluate the proposed methodology, we construct three CT-based time-variant 3D thoracic structures including the basic thoracic tissues and considering 5 different breath states from end-expiration to end-inspiration. The Graz consensus reconstruction algorithm for EIT (GREIT), the correlation coefficient (CC), the root mean square error (RMSE) and the full-reference (FR) metrics are applied for the image quality assessment. Qualitative and quantitative comparison with traditional and more advanced reconstruction techniques reveals that the proposed method shows improved performance in the majority of cases and metrics. Finally, the approach is applied to single-breath online in-vivo data to qualitatively verify its applicability.

The Research Progress of Electrical Impedance Tomography for Lung Monitoring

Medical imaging can intuitively show people the internal structure, morphological information, and organ functions of the organism, which is one of the most important inspection methods in clinical medical diagnosis. Currently used medical imaging methods can only be applied to some diagnostic occasions after qualitative lesions have been generated, and the general imaging technology is usually accompanied by radiation and other conditions. However, electrical impedance tomography has the advantages of being noninvasive and non-radiative. EIT (Electrical Impedance Tomography) is also widely used in the early diagnosis and treatment of some diseases because of these advantages. At present, EIT is relatively mature and more and more image reconstruction algorithms are used to improve imaging resolution. Hardware technology is also developing rapidly, and the accuracy of data collection and processing is continuously improving. In terms of clinical application, EIT has also been used for pathological treatment of lungs, the brain, and the bladder. In the future, EIT has a good application prospect in the medical field, which can meet the needs of real-time, long-term monitoring and early diagnosis. Aiming at the application of EIT in the treatment of lung pathology, this article reviews the research progress of EIT, image reconstruction algorithms, hardware system design, and clinical applications used in the treatment of lung diseases. Through the research and introduction of several core components of EIT technology, it clarifies the characteristics of EIT system complexity and its solutions, provides research ideas for subsequent research, and once again verifies the broad development prospects of EIT technology in the future.

An efficient Point-Matching Method-of-Moments for 2D and 3D Electrical Impedance Tomography Using Radial Basis functions

AbstractObjective: The inverse problem of computing conductivity distributions in 2D and 3D objects interrogated by low frequency electrical signals, which is called Electrical Impedance Tomography (EIT), is treated using a Method-of-Moment technique.

METHODS

A Point-Matching-Method-of-Moment technique is used to formulate a global integral equation solver. Radial Basis Functions are adopted to express the conductivity distribution. Single-step quadratic-norm (L2) and iterative total variation (L1) regularization techniques are exploited to solve the inverse problem.

RESULTS

Simulation and experimental tests on a circular reconstruction domain show satisfactory performance in deriving conductivity distribution, achieving a Correlation Coefficient (CC) up to 0:863 for 70 dB voltage SNR and 0:842 for 40 dB voltage SNR. The proposed methodology with L2-norm regularization provided better results than traditional iterative Gauss-Newtons approach, whereas with L1-norm regularization it showed promising performance. Moreover, 3D reconstructions on a cylindrical cavity demonstrated superior results near the electrodes planes compared to those of the conventional linearized approach. Finally, application to EIT medical data for dynamic lung imaging successfully revealed the breath-cycle conductivity changes.

CONCLUSION

The results show that the proposed method can be effective for both 2D and 3D EIT and applicable to many applications.

SIGNIFICANCE

Strong conductivity variations are successfully tackled with a very good Correlation Coefficient. In contrast to conventional EIT solutions based on weak-form and linearization on small conductivity changes, the proposed method requires only one step to converge with L2-norm regularization. The proposed method with L1-norm regularization also achieves good reconstruction quality with a low number of iterations.

Fusing electrical and elasticity imaging

Electrical and elasticity imaging are promising modalities for a suite of different applications, including medical tomography, non-destructive testing and structural health monitoring. These emerging modalities are capable of providing remote, non-invasive and low-cost opportunities. Unfortunately, both modalities are severely ill-posed nonlinear inverse problems, susceptive to noise and modelling errors. Nevertheless, the ability to incorporate complimentary datasets obtained simultaneously offers mutually beneficial information. By fusing electrical and elastic modalities as a joint problem, we are afforded the possibility to stabilize the inversion process via the utilization of auxiliary information from both modalities as well as joint structural operators. In this study, we will discuss a possible approach to combine electrical and elasticity imaging in a joint reconstruction problem giving rise to novel multi-modality applications for use in both medical and structural engineering. This article is part of the theme issue 'Synergistic tomographic image reconstruction: part 1'.

A DIRECT RECONSTRUCTION ALGORITHM FOR THE ANISOTROPIC INVERSE CONDUCTIVITY PROBLEM BASED ON CALDERÓN'S METHOD IN THE PLANE

A direct reconstruction algorithm based on Calderón's linearization method for the reconstruction of isotropic conductivities is proposed for anisotropic conductivities in two-dimensions. To overcome the non-uniqueness of the anisotropic inverse conductivity problem, the entries of the unperturbed anisotropic tensors are assumed known a priori, and it remains to reconstruct the multiplicative scalar field. The quasi-conformal map in the plane facilitates the Calderón-based approach for anisotropic conductivities. The method is demonstrated on discontinuous radially symmetric conductivities of high and low contrast.