Effect of domain shape modeling and measurement errors on the 2-D D-bar method for EIT

Rapid patient-specific FEM meshes from 3D smart-phone based scans

Objective.The objective of this study was to describe and evaluate a smart-phone based method to rapidly generate subject-specific finite element method (FEM) meshes. More accurate FEM meshes should lead to more accurate thoracic electrical impedance tomography (EIT) images.Approach.The method was evaluated on an iPhone®that utilized an app called Heges, to obtain 3D scans (colored, surface triangulations), a custom belt, and custom open-source software developed to produce the subject-specific meshes. The approach was quantitatively validated via mannequin and volunteer tests using an infrared tracker as the gold standard, and qualitatively assessed in a series of tidal-breathing EIT images recorded from 9 subjects.Main results.The subject-specific meshes can be generated in as little as 6.3 min, which requires on average 3.4 min of user interaction. The mannequin tests yielded high levels of precision and accuracy at 3.2 ± 0.4 mm and 4.0 ± 0.3 mm root mean square error (RMSE), respectively. Errors on volunteers were only slightly larger (5.2 ± 2.1 mm RMSE precision and 7.7 ± 2.9 mm RMSE accuracy), illustrating the practical RMSE of the method.Significance.Easy-to-generate, subject-specific meshes could be utilized in the thoracic EIT community, potentially reducing geometric-based artifacts and improving the clinical utility of EIT.

Signal-to-noise ratio variance impact on the image reconstruction of electrical resistance tomography in solutions with high background conductivity

Electrical Resistance Tomography (ERT) has the potentialities of non-intrusive techniques and high temporal resolution which are essential characteristics for multiphase flow measurements. However, high background conductivities, such as saline water in oil extraction, impose a limitation in ERT image reconstruction. Focusing on the operational limits of an ERT tomography system operating in different conductivity backgrounds from 0.010 to 4.584 S/m, the impact on the image reconstruction was assessed via signal-to-noise variance. The signal-to-noise ratio (SNR) variance had a strong correlation (p-value = 5.40 × 10^-15) with the image reconstruction quality at the threshold of 30 dB, reaching a correlation value of r = -0.92 in the range of 0.010-0.246 S/m. Regarding the position error of the phantom, p-value = 1.30 × 10^-5 and r = -0.66 were attained. The global results revealed that the correlation of the mean of the SNR (p-value = 5 × 10^-4 and r = 0.55) was kept unaltered through the whole conductivity range, showing that such a statistical index can induce bias in establishing the operational limits of the hardware.

Complementary use of priors for pulmonary imaging with electrical impedance and ultrasound computed tomography

For medical professionals caring for patients undergoing mechanical ventilation due to respiratory failure, the ability to quickly and safely obtain images of pulmonary function at the patient's bedside would be highly desirable. Such images could be used to provide early warnings of developing pulmonary pathologies in real time, thereby reducing the incidence of complications and improving patient outcomes. Electrical impedance tomography (EIT) and low-frequency ultrasound computed tomography (USCT) are two imaging techniques with the potential to provide real-time non-ionizing pulmonary monitoring in the ICU setting, and each method has its own unique advantages as well as drawbacks. In this work, we describe a new algorithm for a system in which the strengths of the two modalities are combined in a complementary fashion. Specifically, preliminary reconstructions from each modality are used as priors to stabilize subsequent reconstructions, providing improved spatial resolution, sharper organ boundaries, and enhanced appearance of pathologies and other features. Results are validated using three numerically simulated thoracic phantoms representing pulmonary pathologies.

Measurement-Based Domain Parameter Optimization in Electrical Impedance Tomography Imaging

This paper discusses the optimization of domain parameters in electrical impedance tomography-based imaging. Precise image reconstruction requires accurate, well-correlated physical and numerical finite element method (FEM) models; thus, we employed the Nelder–Mead algorithm and a complete electrode model to evaluate the individual parameters, including the initial conductivity, electrode misplacement, and shape deformation. The optimization process was designed to calculate the parameters of the numerical model before the image reconstruction. The models were verified via simulation and experimental measurement with single source current patterns. The impact of the optimization on the above parameters was reflected in the applied image reconstruction process, where the conductivity error dropped by 6.16% and 11.58% in adjacent and opposite driving, respectively. In the shape deformation, the inhomogeneity area ratio increased by 11.0% and 48.9%; the imprecise placement of the 6th electrode was successfully optimized with adjacent driving; the conductivity error dropped by 12.69%; and the inhomogeneity localization exhibited a rise of 66.7%. The opposite driving option produces undesired duality resulting from the measurement pattern. The designed optimization process proved to be suitable for correlating the numerical and the physical models, and it also enabled us to eliminate imaging uncertainties and artifacts.

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.

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.

Spatially resolved electrical impedance methods for cell and particle characterization

Electrical impedance is an established technique used for cell and particle characterization. The temporal and spectral resolution of electrical impedance have been used to resolve basic cell characteristics like size and type, as well as to determine cell viability and activity. Such electrical impedance measurements are typically performed across the entire sample volume and can only provide an overall indication concerning the properties and state of that sample. For the study of heterogeneous structures such as cell layers, biological tissue, or polydisperse particle mixtures, an overall measured impedance value can only provide limited information and can lead to data misinterpretation. For the investigation of localized sample properties in complex heterogeneous structures/mixtures, the addition of spatial resolution to impedance measurements is necessary. Several spatially resolved impedance measurement techniques have been developed and applied to cell and particle research, including electrical impedance tomography, scanning electrochemical microscopy, and microelectrode arrays. This review provides an overview of spatially resolved impedance measurement methods and assesses their applicability for cell and particle characterization.

Noise-robust bioimpedance approach for cardiac output measurement

OBJECTIVE

Congestive heart failure is a problem affecting millions of Americans. A continuous, non-invasive, telemonitoring device that can accurately monitor cardiac metrics could greatly help this population, reducing unnecessary hospitalizations and cost.

APPROACH

Machine learning (ML) algorithms trained on electrical-impedance tomography (EIT) data are presented for portable cardiac monitoring. The approach was validated on a simulated thorax and a measured tank experiment. A highly detailed 4D chest model (finite element method mesh and conductivity profiles) was developed utilizing the 4D XCAT phantom to provide realistic data. The ML algorithms were trained using databases that assumed the presence of poorly contacting electrodes without any assumptions of knowing which electrodes would be bad in the experiment. The trained ML algorithms were compared to EIT evaluated with and without removing bad electrodes.

MAIN RESULTS

A regression support vector machine and a deep neural network (DNN) were found to be the most accurate and robust to poorly contacting electrodes while not needing to know which electrodes were in poor contact in the simulated and measured experiments, respectively.

SIGNIFICANCE

Although the ML algorithms are not always better than EIT (with bad electrodes removed), the comparable results without needing a priori knowledge of which electrodes are bad is seen as a very promising feature. An evaluation of computational costs showed that the DNN required comparable computational power to the other methods while requiring less memory, which could make the DNNs an attractive algorithm for a low-power, portable system. This work represents an important validation of the method using measured data, and model development, which is needed to apply this method on real clinical data. Additionally, the developed 4D simulated thorax model could be an important tool within the EIT community.

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 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.

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.

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.

Robust computation in 2D absolute EIT (a-EIT) using D-bar methods with the 'exp' approximation

OBJECTIVE

Absolute images have important applications in medical electrical impedance tomography (EIT) imaging, but the traditional minimization and statistical based computations are very sensitive to modeling errors and noise. In this paper, it is demonstrated that D-bar reconstruction methods for absolute EIT are robust to such errors.

APPROACH

The effects of errors in domain shape and electrode placement on absolute images computed with 2D D-bar reconstruction algorithms are studied on experimental data.

MAIN RESULTS

It is demonstrated with tank data from several EIT systems that these methods are quite robust to such modeling errors, and furthermore the artefacts arising from such modeling errors are similar to those occurring in classic time-difference EIT imaging.

SIGNIFICANCE

This study is promising for clinical applications where absolute EIT images are desirable but previously thought impossible.

Real-Time Implementation of Calderón's Method on Subject-Specific Domains

A real-time implementation of Calderón's method for the reconstruction of a 2-D conductivity from electrical impedance tomography data is presented, in which domain-specific modeling is taken into account. This is the first implementation of Calderón's method that accounts for correct modeling of non-symmetric domain boundaries in image reconstruction. The domain-specific Calderón's method is derived and reconstructions from experimental tank data are presented, quantifying the distortion when correct modeling is not included in the reconstruction algorithm. Reconstructions from human subject volunteers are presented, demonstrating the method's effectiveness for imaging changes due to ventilation and perfusion in the human thorax.

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

Electrical Impedance Tomography (EIT) aims to recover the internal conductivity and permittivity distributions of a body from electrical measurements taken on electrodes on the surface of the body. The reconstruction task is a severely ill-posed nonlinear inverse problem that is highly sensitive to measurement noise and modeling errors. Regularized D-bar methods have shown great promise in producing noise-robust algorithms by employing a low-pass filtering of nonlinear (nonphysical) Fourier transform data specific to the EIT problem. Including prior data with the approximate locations of major organ boundaries in the scattering transform provides a means of extending the radius of the low-pass filter to include higher frequency components in the reconstruction, in particular, features that are known with high confidence. This information is additionally included in the system of D-bar equations with an independent regularization parameter from that of the extended scattering transform. In this paper, this approach is used in the 2-D D-bar method for admittivity (conductivity as well as permittivity) EIT imaging. Noise-robust reconstructions are presented for simulated EIT data on chest-shaped phantoms with a simulated pneumothorax and pleural effusion. No assumption of the pathology is used in the construction of the prior, yet the method still produces significant enhancements of the underlying pathology (pneumothorax or pleural effusion) even in the presence of strong noise.

Direct 2-D reconstructions of conductivity and permittivity from EIT data on a human chest

A novel direct D-bar reconstruction algorithm is presented for reconstructing a complex conductivity distribution from 2-D EIT data. The method is applied to simulated data and archival human chest data. Permittivity reconstructions with the aforementioned method and conductivity reconstructions with the previously existing nonlinear D-bar method for real-valued conductivities depicting ventilation and perfusion in the human chest are presented. This constitutes the first fully nonlinear D-bar reconstructions of human chest data and the first D-bar permittivity reconstructions of experimental data. The results of the human chest data reconstructions are compared on a circular domain versus a chest-shaped domain.

A Real-time D-bar Algorithm for 2-D Electrical Impedance Tomography Data

The aim of this paper is to show the feasibility of the D-bar method for real-time 2-D EIT reconstructions. A fast implementation of the D-bar method for reconstructing conductivity changes on a 2-D chest-shaped domain is described. Cross-sectional difference images from the chest of a healthy human subject are presented, demonstrating what can be achieved in real time. The images constitute the first D-bar images from EIT data on a human subject collected on a pairwise current injection system.

Calderón's method on an elliptical domain

One possible application for electrical impedance tomography is in medical imaging where lung and heart function may be monitored. One drawback of current algorithms is that they are implemented for use in a circular domain, but a human thorax is more elliptical than circular. In this paper, a reconstruction algorithm based on the work of Calderón (1980 Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro) pp 65-75) on the inverse conductivity problem is derived for an elliptical domain. It is explained how this reconstruction algorithm uses a transformed Dirichlet-to-Neumann map. Experimental results from an elliptical tank are given to show how correct domain modelling reduces the artefacts produced by this version of Calderón's reconstruction algorithm.

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.

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.

Compensation of modelling errors due to unknown domain boundary in electrical impedance tomography

Electrical impedance tomography is a highly unstable problem with respect to measurement and modeling errors. This instability is especially severe when absolute imaging is considered. With clinical measurements, accurate knowledge about the body shape is usually not available, and therefore an approximate model domain has to be used in the computational model. It has earlier been shown that large reconstruction artefacts result if the geometry of the model domain is incorrect. In this paper, we adapt the so-called approximation error approach to compensate for the modeling errors caused by inaccurately known body shape. This approach has previously been shown to be applicable to a variety of modeling errors, such as coarse discretization in the numerical approximation of the forward model and domain truncation. We evaluate the approach with a simulated example of thorax imaging, and also with experimental data from a laboratory setting, with absolute imaging considered in both cases. We show that the related modeling errors can be efficiently compensated for by the approximation error approach. We also show that recovery from simultaneous discretization related errors is feasible, allowing the use of computationally efficient reduced order models.