Incorporating a Spatial Prior into Nonlinear D-Bar EIT Imaging for Complex Admittivities

Evaluation of adjacent and opposite current injection patterns for a wearable chest electrical impedance tomography system

Objective.Wearable electrical impedance tomography (EIT) can be used to monitor regional lung ventilation and perfusion at the bedside. Due to its special system architecture, the amplitude of the injected current is usually limited compared to stationary EIT system. This study aims to evaluate the performance of current injection patterns with various low-amplitude currents in healthy volunteers.Approach.A total of 96 test sets of EIT measurement was recorded in 12 healthy subjects by employing adjacent and opposite current injection patterns with four amplitudes of small current (i.e. 1 mA, 500 uA, 250 uA and 125 uA). The performance of the two injection patterns with various currents was evaluated in terms of signal-to-noise ratio (SNR) of thorax impedance, EIT image metrics and EIT-based clinical parameters.Main results.Compared with adjacent injection, opposite injection had higher SNR (p< 0.01), less inverse artifacts (p< 0.01), and less boundary artifacts (p< 0.01) with the same current amplitude. In addition, opposite injection exhibited more stable EIT-based clinical parameters (p< 0.01) across the current range. For adjacent injection, significant differences were found for three EIT image metrics (p< 0.05) and four EIT-based clinical parameters (p< 0.01) between the group of 125 uA and the other groups.Significance.For better performance of wearable pulmonary EIT, currents greater than 250 uA should be used in opposite injection, 500 uA in adjacent one, to ensure a high level of SNR, a high quality of reconstructed image as well as a high reliability of clinical parameters.

Image Reconstruction Using Supervised Learning in Wearable Electrical Impedance Tomography of the Thorax

Electrical impedance tomography (EIT) is a non-invasive technique for visualizing the internal structure of a human body. Capacitively coupled electrical impedance tomography (CCEIT) is a new contactless EIT technique that can potentially be used as a wearable device. Recent studies have shown that a machine learning-based approach is very promising for EIT image reconstruction. Most of the studies concern models containing up to 22 electrodes and focus on using different artificial neural network models, from simple shallow networks to complex convolutional networks. However, the use of convolutional networks in image reconstruction with a higher number of electrodes requires further investigation. In this work, two different architectures of artificial networks were used for CCEIT image reconstruction: a fully connected deep neural network and a conditional generative adversarial network (cGAN). The training dataset was generated by the numerical simulation of a thorax phantom with healthy and illness-affected lungs. Three kinds of illnesses, pneumothorax, pleural effusion, and hydropneumothorax, were modeled using the electrical properties of the tissues. The thorax phantom included the heart, aorta, spine, and lungs. The sensor with 32 area electrodes was used in the numerical model. The ECTsim custom-designed toolbox for Matlab was used to solve the forward problem and measurement simulation. Two artificial neural networks were trained with supervision for image reconstruction. Reconstruction quality was compared between those networks and one-step algebraic reconstruction methods such as linear back projection and pseudoinverse with Tikhonov regularization. This evaluation was based on pixel-to-pixel metrics such as root-mean-square error, structural similarity index, 2D correlation coefficient, and peak signal-to-noise ratio. Additionally, the diagnostic value measured by the ROC AUC metric was used to assess the image quality. The results showed that obtaining information about regional lung function (regions affected by pneumothorax or pleural effusion) is possible using image reconstruction based on supervised learning and deep neural networks in EIT. The results obtained using cGAN are strongly better than those obtained using a fully connected network, especially in the case of noisy measurement data. However, diagnostic value estimation showed that even algebraic methods allow us to obtain satisfactory results.

Semi-Siamese U-Net for separation of lung and heart bioimpedance images: A simulation study of thorax EIT

Electrical impedance tomography (EIT) is widely used for bedside monitoring of lung ventilation status. Its goal is to reflect the internal conductivity changes and estimate the electrical properties of the tissues in the thorax. However, poor spatial resolution affects EIT image reconstruction to the extent that the heart and lung-related impedance images are barely distinguishable. Several studies have attempted to tackle this problem, and approaches based on decomposition of EIT images using linear transformations have been developed, and recently, U-Net has become a prominent architecture for semantic segmentation. In this paper, we propose a novel semi-Siamese U-Net specifically tailored for EIT application. It is based on the state-of-the-art U-Net, whose structure is modified and extended, forming shared encoder with parallel decoders and has multi-task weighted losses added to adapt to the individual separation tasks. The trained semi-Siamese U-Net model was evaluated with a test dataset, and the results were compared with those of the classical U-Net in terms of Dice similarity coefficient and mean absolute error. Results showed that compared with the classical U-Net, semi-Siamese U-Net exhibited performance improvements of 11.37% and 3.2% in Dice similarity coefficient, and 3.16% and 5.54% in mean absolute error, in terms of heart and lung-impedance image separation, respectively.

Introduction of Sample Based Prior into the D-Bar Method Through a Schur Complement Property

Electrical impedance tomography (EIT) is a non-invasive medical imaging technique in which images of the conductivity in a region of interest in the body are computed from measurements of voltages on electrodes arising from low-frequency, low-amplitude applied currents. Mathematically, the inverse conductivity problem is nonlinear and ill-posed, and the reconstructions have characteristically low spatial resolution. One approach to improve the spatial resolution of EIT images is to include anatomically and physiologically-based prior information in the reconstruction algorithm. Statistical inversion theory provides a means of including prior information from a representative sample population. In this paper, a method is proposed to introduce statistical prior information into the D-bar method based on Schur complement properties. The method presents an improvement of the image obtained by the D-bar method by maximizing the conditional probability density function of an image that is consistent with a prior information and the model, given a D-bar image computed from the voltage measurements. Experimental phantoms show an improved spatial resolution by the use of the proposed method for the D-bar image reconstructions.

Shape-Driven Difference Electrical Impedance Tomography

This work proposes a novel shape-driven reconstruction approach for difference electrical impedance tomography (EIT). In the proposed approach, the reconstruction problem is formulated as a shape reconstruction problem and solved via an explicit and geometrical methodology, where the geometry of the embedded inclusions is represented by a shape and topology description function (STDF). To incorporate more geometry and prior information directly into the reconstruction and to provide better flexibility in the solution process, the concept of a moving morphable component (MMC) is applied here implying that MMC is treated as the basic building block of the embedded inclusions. Simulations, phantom studies, and in vivo pig data are used to test the proposed approach for the most popular biomedical application of EIT - lung imaging - and the performance is compared with the conventional linear approach. In addition, the modality's robustness is studied in cases where (i) modeling errors are caused by inhomogeneity in the background conductivity, and (ii) uncertainties in the contact impedances and reference state are present. The results of this work indicate that the proposed approach is tolerant to modeling errors and is fairly robust to typical EIT uncertainties, producing greatly improved image quality compared to the conventional linear approach.

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.

The D-bar method for electrical impedance tomography-demystified

Electrical impedance tomography (EIT) is an imaging modality where a patient or object is probed using harmless electric currents. The currents are fed through electrodes placed on the surface of the target, and the data consists of voltages measured at the electrodes resulting from a linearly independent set of current injection patterns. EIT aims to recover the internal distribution of electrical conductivity inside the target. The inverse problem underlying the EIT image formation task is nonlinear and severely ill-posed, and hence sensitive to modeling errors and measurement noise. Therefore, the inversion process needs to be regularized. However, traditional variational regularization methods, based on optimization, often suffer from local minima because of nonlinearity. This is what makes regularized direct (non-iterative) methods attractive for EIT. The most developed direct EIT algorithm is the D-bar method, based on Complex Geometric Optics solutions and a nonlinear Fourier transform. Variants and recent developments of D-bar methods are reviewed, and their practical numerical implementation is explained.

Reconstruction of Organ Boundaries With Deep Learning in the D-Bar Method for Electrical Impedance Tomography

OBJECTIVE

Medical electrical impedance tomography is a non-ionizing imaging modality in which low-amplitude, low-frequency currents are applied on electrodes on the body, the resulting voltages are measured, and an inverse problem is solved to determine the conductivity distribution in the region of interest. Due the ill-posedness of the inverse problem, the boundaries of internal organs are typically blurred in the reconstructed image.

METHODS

A deep learning approach is introduced in the D-bar method for reconstructing a 2-D slice of the thorax to recover the boundaries of organs. This is accomplished by training a deep neural network on labeled pairs of scattering transforms and the boundaries of the organs in the data from which the transforms were computed. This allows the network to "learn" the nonlinear mapping between them by minimizing the error between the output of the network and known actual boundaries. Further, a "sparse" reconstruction is computed by fusing the results of the standard D-bar reconstruction with reconstructed organ boundaries from the neural network.

RESULTS

Results are shown on simulated and experimental data collected on a saline-filled tank with agar targets simulating the conductivity of the heart and lungs.

CONCLUSIONS AND SIGNIFICANCE

The results demonstrate that deep neural networks can successfully learn the mapping between scattering transforms and the internal boundaries of structures.

Induced-Current Learning Method for Nonlinear Reconstructions in Electrical Impedance Tomography

Electrical impedance tomography (EIT) is an attractive technique that aims to reconstruct the unknown electrical property in a domain from the surface electrical measurements. In this work, the induced-current learning method (ICLM) is proposed to solve nonlinear electrical impedance tomography (EIT) problems. Specifically, the cascaded end-to-end convolutional neural network (CEE-CNN) architecture is designed to implement the ICLM. The CEE-CNN greatly decreases the nonlinearities in EIT problems by designing a combined objective function and introducing multiple labels. A noticeable characteristic of the proposed CNN scheme is that the input parameters are chosen as both induced contrast current (ICC) and the updated electrical field from a spectral analysis and the output is chosen as ICC, which is fundamentally different from prevailing CNN schemes. Further, several skip connections are introduced to focus on learning only the unknown part of ICC. ICLM is verified with both numerical and experimental tests on typical EIT problems, and it is found that ICLM is able to solve typical EIT problems in less than 1 second with high image qualities. More importantly, it is also highly robust to measurement noises and modeling errors, such as inaccurate boundary data.

DYNAMIC OPTIMIZED PRIORS FOR D-BAR RECONSTRUCTIONS OF HUMAN VENTILATION USING ELECTRICAL IMPEDANCE TOMOGRAPHY

A method of including dynamic spatial priors in the 2-D D-bar reconstruction algorithm is presented for use on time-difference reconstructions of human subject thoracic data. The conductivity values for the prior are updated at each frame in the reconstruction using an optimization method applied to the scattering transform. The updates of the dynamic spatial priors are guided by a principle component analysis of the data to determine the timepoint in the ventilatory (or cardiac) cycle. The effectiveness of the method is demonstrated on human subject ventilatory data.

A Moving Morphable Components Based Shape Reconstruction Framework for Electrical Impedance Tomography

This paper presents a new computational framework in electrical impedance tomography (EIT) for shape reconstruction based on the concept of moving morphable components (MMC). In the proposed framework, the shape reconstruction problem is solved in an explicit and geometrical way. Compared with the traditional pixel or shape-based solution framework, the proposed framework can incorporate more geometry and prior information into shape and topology optimization directly and therefore render the solution process more flexibility. It also has the afford potential to substantially reduce the computational burden associated with shape and topology optimization. The effectiveness of the proposed approach is tested with noisy synthetic data and experimental data, which demonstrates the most popular biomedical application of EIT: lung imaging. In addition, robustness studies of the proposed approach considering modeling errors caused by non-homogeneous background, varying initial guesses, differing numbers of candidate shape components, and differing exponent in the shape and topology description function are performed. The simulation and experimental results show that the proposed approach is tolerant to modeling errors and is fairly robust to these parameter choices, offering significant improvements in image quality in comparison to the conventional absolute reconstructions using smoothness prior regularization and total variation regularization.

B-Spline-Based Sharp Feature Preserving Shape Reconstruction Approach for Electrical Impedance Tomography

This paper presents a B-spline-based shape reconstruction approach for electrical impedance tomography (EIT). In the proposed approach, the conductivity distribution to be reconstructed is assumed to be piecewise constant. The geometry of the inclusions is parameterized using B-spline curves, and the EIT forward solver is modified as a set of control points representing the inclusions' boundary to the data on the domain boundary. The low-order representation decreases the computational demand and reduces the ill-posedness of the EIT reconstruction problem. The performance of the proposed B-spline-based approach is tested with simulations that demonstrate the most popular biomedical application of EIT: lung imaging. The approach is experimentally validated using water tank data. In addition, robustness studies of the proposed approach considering varying initial guesses, inaccurately known contact impedances, differing numbers of control points, and degree of B-spline are performed. The simulation and experimental results show that the B-spline-based approach offers improvements in image quality in comparison to the traditional Fourier series-based reconstruction approach, as measured by quantitative metrics such as relative size coverage ratio and relative contrast. Inasmuch, the proposed approach is demonstrated to offer clear improvement in the ability to preserve the sharp properties of the inclusions to be imaged.

A Review of Electrical Impedance Tomography in Lung Applications: Theory and Algorithms for Absolute Images

Electrical Impedance Tomography (EIT) is under fast development, the present paper is a review of some procedures that are contributing to improve spatial resolution and material properties accuracy, admitivitty or impeditivity accuracy. A review of EIT medical applications is presented and they were classified into three broad categories: ARDS patients, obstructive lung diseases and perioperative patients. The use of absolute EIT image may enable the assessment of absolute lung volume, which may significantly improve the clinical acceptance of EIT. The Control Theory, the State Observers more specifically, have a developed theory that can be used for the design and operation of EIT devices. Electrode placement, current injection strategy and electrode electric potential measurements strategy should maximize the number of observable and controllable directions of the state vector space. A non-linear stochastic state observer, the Unscented Kalman Filter, is used directly for the reconstruction of absolute EIT images. Historically, difference images were explored first since they are more stable in the presence of modelling errors. Absolute images require more detailed models of contact impedance, stray capacitance and properly refined finite element mesh where the electric potential gradient is high. Parallelization of the forward program computation is necessary since the solution of the inverse problem often requires frequent solutions of the forward problem. Several reconstruction algorithms benefit by the Bayesian inverse problem approach and the concept of prior information. Anatomic and physiologic information are used to form the prior information. An already tested methodology is presented to build the prior probability density function using an ensemble of CT scans and in vivo impedance measurements. Eight absolute EIT image algorithms are presented.

Dominant-Current Deep Learning Scheme for Electrical Impedance Tomography

OBJECTIVE

Deep learning has recently been applied to electrical impedance tomography (EIT) imaging. Nevertheless, there are still many challenges that this approach has to face, e.g., targets with sharp corners or edges cannot be well recovered when using circular inclusion training data. This paper proposes an iterative-based inversion method and a convolutional neural network (CNN) based inversion method to recover some challenging inclusions such as triangular, rectangular, or lung shapes, where the CNN-based method uses only random circle or ellipse training data.

METHODS

First, the iterative method, i.e., bases-expansion subspace optimization method (BE-SOM), is proposed based on a concept of induced contrast current (ICC) with total variation regularization. Second, the theoretical analysis of BE-SOM and the physical concepts introduced there motivate us to propose a dominant-current deep learning scheme for EIT imaging, in which dominant parts of ICC are utilized to generate multi-channel inputs of CNN.

RESULTS

The proposed methods are tested with both numerical and experimental data, where several realistic phantoms including simulated pneumothorax and pleural effusion pathologies are also considered.

CONCLUSIONS AND SIGNIFICANCE

Significant performance improvements of the proposed methods are shown in reconstructing targets with sharp corners or edges. It is also demonstrated that the proposed methods are capable of fast, stable, and high-quality EIT imaging, which is promising in providing quantitative images for potential clinical applications.

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.

EIT Imaging of admittivities with a D-bar method and spatial prior: experimental results for absolute and difference imaging

Electrical impedance tomography (EIT) is an emerging imaging modality that uses harmless electrical measurements taken on electrodes at a body's surface to recover information about the internal electrical conductivity and or permittivity. The image reconstruction task of EIT is a highly nonlinear inverse problem that is sensitive to noise and modeling errors making the image reconstruction task challenging. D-bar methods solve the nonlinear problem directly, bypassing the need for detailed and time-intensive forward models, to provide absolute (static) as well as time-difference EIT images. Coupling the D-bar methodology with the inclusion of high confidence a priori data results in a noise-robust regularized image reconstruction method. In this work, the a priori D-bar method for complex admittivities is demonstrated effective on experimental tank data for absolute imaging for the first time. Additionally, the method is adjusted for, and tested on, time-difference imaging scenarios. The ability of the method to be used for conductivity, permittivity, absolute as well as time-difference imaging provides the user with great flexibility without a high computational cost.