Electrical impedance tomography using the extended Kalman filter

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.

Review on Electrical Impedance Tomography: Artificial Intelligence Methods and its Applications

Electrical impedance tomography (EIT) has been a hot topic among researchers for the last 30 years. It is a new imaging method and has evolved over the last few decades. By injecting a small amount of current, the electrical properties of tissues are determined and measurements of the resulting voltages are taken. By using a reconstructing algorithm these voltages then transformed into a tomographic image. EIT contains no identified threats and as compared to magnetic resonance imaging (MRI) and computed tomography (CT) scans (imaging techniques), it is cheaper in cost as well. In this paper, a comprehensive review of efforts and advancements undertaken and achieved in recent work to improve this technology and the role of artificial intelligence to solve this non-linear, ill-posed problem are presented. In addition, a review of EIT clinical based applications has also been presented.

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.

A dynamic oppositional biogeography-based optimization approach for time-varying electrical impedance tomography

Dynamic electrical impedance tomography-based image reconstruction using conventional algorithms such as the extended Kalman filter often exhibits inferior performance due to the presence of measurement noise, the inherent ill-posed nature of the problem and its critical dependence on the selection of the initial guess as well as the state evolution model. Moreover, many of these conventional algorithms require the calculation of a Jacobian matrix. This paper proposes a dynamic oppositional biogeography-based optimization (OBBO) technique to estimate the shape, size and location of the non-stationary region boundaries, expressed as coefficients of truncated Fourier series, inside an object domain using electrical impedance tomography. The conductivity of the object domain is assumed to be known a priori. Dynamic OBBO is a novel addition to the family of dynamic evolutionary algorithms. Moreover, it is the first such study on the application of dynamic evolutionary algorithms for dynamic electrical impedance tomography-based image reconstruction. The performance of the algorithm is tested through numerical simulations and experimental study and is compared with state-of-the-art gradient-based extended Kalman filter. The dynamic OBBO is shown to be far superior compared to the extended Kalman filter. It is found to be robust to measurement noise as well as the initial guess, and does not rely on a priori knowledge of the state evolution model.

A comparison framework for temporal image reconstructions in electrical impedance tomography

Electrical impedance tomography (EIT) provides low-resolution images of internal conductivity distributions, but is able to achieve relatively high temporal resolutions. Most EIT image reconstruction algorithms do not explicitly account for the temporal constraints on the measurements or physiological processes under investigation. Instead, algorithms typically assume both that the conductivity distribution does not change during the acquisition of each EIT data frame, and that frames can be reconstructed independently, without consideration of the correlation between images. A failure to account for these temporal effects will result in aliasing-related artefacts in images. Several methods have been proposed to compensate for these effects, including interpolation of raw data, and reconstruction algorithms using Kalman and temporal filtering. However, no systematic work has been performed to understand the severity of the temporal artefacts nor the extent to which algorithms can account for them. We seek to address this need by developing a temporal comparison framework and figures of merit to assess the ability of reconstruction algorithms to account for temporal effects. Using this approach, we compare combinations of three reconstruction algorithms using three EIT data frame types: perfect, realistic and interpolated. The results show that, without accounting for temporal effects, artefacts are present in images for dynamic conductivity contrasts at frequencies 10-20 times slower than the frame rate. The proposed methods show some improvements in reducing these artefacts.

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.

Electrical impedance tomography reconstruction through simulated annealing with total least square error as objective function

The EIT reconstruction problem can be solved as an optimization problem where the divergence between a simulated impedance domain and the observed one is minimized. This optimization problem can be solved by a combination of Simulated Annealing (SA) for optimization and Finite Element Method (FEM) for simulation of the impedance domain. This combination has usually a very high computational cost, since SA requires an elevated number of objective function evaluations and those, obtained through FEM, are often expansive enough to make the whole process inviable. In here it is presented a new approach for EIT image reconstructions using SA and partial evaluations of objective functions based on overdetermined linear systems. This new reconstruction approach is evaluated with experimental data and compared with previous approaches.

Image reconstruction using interval simulated annealing in electrical impedance tomography

Electrical impedance tomography (EIT) is an imaging technique that attempts to reconstruct the impedance distribution inside an object from the impedance between electrodes placed on the object surface. The EIT reconstruction problem can be approached as a nonlinear nonconvex optimization problem in which one tries to maximize the matching between a simulated impedance problem and the observed data. This nonlinear optimization problem is often ill-posed, and not very suited to methods that evaluate derivatives of the objective function. It may be approached by simulated annealing (SA), but at a large computational cost due to the expensive evaluation process of the objective function, which involves a full simulation of the impedance problem at each iteration. A variation of SA is proposed in which the objective function is evaluated only partially, while ensuring boundaries on the behavior of the modified algorithm.

Electrical impedance tomography reconstruction through Simulated Annealing with incomplete evaluation of the objective function

The EIT reconstruction problem is approached as an optimization problem where the difference between a simulated impedance domain and the observed one is minimized. This optimization problem is often solved by Simulated Annealing (SA), but at a large computational cost due to the expensive evaluation process of the objective function. We propose here, a variation of SA applied to EIT where the objective function is evaluated only partially, while ensuring upper boundaries on the deviation on the behavior of the modified SA. The reconstruction method is evaluated with simulated and experimental data.

Frequency-division multiplexing for electrical impedance tomography in biomedical applications

Electrical impedance tomography (EIT) produces an image of the electrical impedance distribution of tissues in the body, using electrodes that are placed on the periphery of the imaged area. These electrodes inject currents and measure voltages and from these data, the impedance can be computed. Traditional EIT systems usually inject current patterns in a serial manner which means that the impedance is computed from data collected at slightly different times. It is usually also a time-consuming process. In this paper, we propose a method for collecting data concurrently from all of the current patterns in biomedical applications of EIT. This is achieved by injecting current through all of the current injecting electrodes simultaneously, and measuring all of the resulting voltages at once. The signals from various current injecting electrodes are separated by injecting different frequencies through each electrode. This is called frequency-division multiplexing (FDM). At the voltage measurement electrodes, the voltage related to each current injecting electrode is isolated by using Fourier decomposition. In biomedical applications, using different frequencies has important implications due to dispersions as the tissue's electrical properties change with frequency. Another significant issue arises when we are recording data in a dynamic environment where the properties change very fast. This method allows simultaneous measurements of all the current patterns, which may be important in applications where the tissue changes occur in the same time scale as the measurement. We discuss the FDM EIT method from the biomedical point of view and show results obtained with a simple experimental system.

A fisher information matrix interpretation of the NOSER algorithm in electrical impedance tomography

In this paper, we employ the concept of the Fisher information matrix (FIM) to reformulate and improve on the "Newton's One-Step Error Reconstructor" (NOSER) algorithm. FIM is a systematic approach for incorporating statistical properties of noise, modeling errors and multi-frequency data. The method is discussed in a maximum likelihood estimator (MLE) setting. The ill-posedness of the inverse problem is mitigated by means of a nonlinear regularization strategy. It is shown that the overall approach reduces to the maximum a posteriori estimator (MAP) with the prior (conductivity vector) described by a multivariate normal distribution. The covariance matrix of the prior is a diagonal matrix and is computed directly from the Fisher information matrix. An eigenvalue analysis is presented, revealing the advantages of using this prior to a Gaussian smoothness prior (Laplace). Reconstructions are shown using measured data obtained from a shallow breathing of an adult human subject. The reconstructions show that the FIM approach clearly improves on the original NOSER algorithm.

Dynamic imaging in electrical impedance tomography of the human chest with online transition matrix identification

One of the electrical impedance tomography objectives is to estimate the electrical resistivity distribution in a domain based only on electrical potential measurements at its boundary generated by an imposed electrical current distribution into the boundary. One of the methods used in dynamic estimation is the Kalman filter. In biomedical applications, the random walk model is frequently used as evolution model and, under this conditions, poor tracking ability of the extended Kalman filter (EKF) is achieved. An analytically developed evolution model is not feasible at this moment. The paper investigates the identification of the evolution model in parallel to the EKF and updating the evolution model with certain periodicity. The evolution model transition matrix is identified using the history of the estimated resistivity distribution obtained by a sensitivity matrix based algorithm and a Newton-Raphson algorithm. To numerically identify the linear evolution model, the Ibrahim time-domain method is used. The investigation is performed by numerical simulations of a domain with time-varying resistivity and by experimental data collected from the boundary of a human chest during normal breathing. The obtained dynamic resistivity values lie within the expected values for the tissues of a human chest. The EKF results suggest that the tracking ability is significantly improved with this approach.

Direct reconstruction of pharmacokinetic-rate images of optical fluorophores from NIR measurements

In this paper, we present a new method to form pharmacokinetic-rate images of optical fluorophores directly from near infra-red (NIR) boundary measurements. We first derive a mapping from spatially resolved pharmacokinetic rates to NIR boundary measurements by combining compartmental modeling with a diffusion based NIR photon propagation model. We express this mapping as a state-space equation. Next, we introduce a spatio-temporal prior model for the pharmacokinetic-rate images and combine it with the state-space equation. We address the image formation problem using the extended Kalman filtering framework. We analyze the computational complexity of the resulting algorithms and evaluate their performance in numerical simulations. An important feature of our approach is that the reconstruction of fluorescence concentrations and compartmental modeling are combined into a single step 1) to take advantage of the inherent temporal correlations in dynamic NIR measurements, and 2) to incorporate spatio-temporal a priori information on pharmacokinetic-rate images. Simulation results show that the resulting algorithms are more robust and lead to higher signal-to-noise ratio as compared to existing approaches where the reconstruction of concentrations and compartmental modeling are treated separately. Additionally, we reconstructed pharmacokinetic-rate images using in vivo data obtained from three patients with breast tumors. The reconstruction results show that the pharmacokinetic rates of indocyanine green are higher inside the tumor region as compared to the surrounding tissue.

Reconstructions of conductive and insulating targets using the D-bar method on an elliptical domain

The D-bar algorithm based on A Nachman's 2D global uniqueness proof for the inverse conductivity problem (Nachman 1996 Ann. Math. 143 71-96) is implemented on an elliptical domain. The scattering transform is computed on an ellipse and the complete electrode model (CEM) for the forward problem is computed with the finite element method (FEM) in order to obtain static conductivity reconstructions of conductive and insulating targets in a saline-filled tank. It is demonstrated that the spatial artifacts in the image are significantly reduced when the domain is properly modeled in the reconstruction, as opposed to being modeled as a disk.

Temporal image reconstruction in electrical impedance tomography

Electrical impedance tomography (EIT) calculates images of the body from body impedance measurements. While the spatial resolution of these images is relatively low, the temporal resolution of EIT data can be high. Most EIT reconstruction algorithms solve each data frame independently, although Kalman filter algorithms track the image changes across frames. This paper proposes a new approach which directly accounts for correlations between images in successive data frames. Image reconstruction is posed in terms of an augmented image x and measurement vector y, which concatenate the values from the d previous and future frames. Image reconstruction is then based on an augmented regularization matrix R, which accounts for a model of both the spatial and temporal correlations between image elements. Results are compared for reconstruction algorithms based on independent frames, Kalman filters and the proposed approach. For low values of the regularization hyperparameter, the proposed approach performs similarly to independent frames, but for higher hyperparameter values, it uses adjacent frame data to reduce reconstructed image noise.

Reconstruction of Layered Biological Tissues via Electrical Impedance Tomography

The reconstruction of layered biological tissues, via electrical impedance tomography, is carried out by utilizing two alternative linear transformations, namely, Legendre polynomial expansion and Fourier Bessel transform. Both transformations are performed on a recently proposed image series expansion scheme in conjunction with the WKB approximation. The resultant spectrum posses a crucially important locality feature, assigning analytically to each spectral image term a local impedance associated with a unique layer, and thereby leading to efficient and accurate novel reconstruction procedures. The quality of the reconstruction procedures is verified through numerical simulations, revealing small reconstruction errors even in the presence of noisy data.

Estimation of Respiratory Gas Exchange: A Comparative Study of Linear and Nonlinear Model-Based Estimation Techniques

Monitoring of respiratory gas exchange in humans is an important task in order to establish the physical condition of the patient and to control important physiological indices. In a previous work, we proposed an approximated linear dynamical model of gas exchange within a controlled volume, to be used as a basis for Kalman filtering technique in order to improve the accuracy of the estimates. In this paper, we propose an alternative nonlinear dynamical model of the same phenomenon and suggest the use of a nonlinear estimation technique. A simulation study demonstrates that operative conditions exist where the latter results are more accurate than the estimates based on the linear model. A set of controlled experiments are also designed in order to create situations in which the above difference is significant. In the paper it is evidenced that, in different operative conditions, the analysis both of simulated and experimental data, give systematically the same indications about the choice of the filtering method. The conclusive result of this paper is that a nonlinear model, and the corresponding nonlinear estimation technique, turn out to be convenient when the operative volume and the accuracy of the instrumentation of the experimental set up are both low (operative volumes of about ten liters and flows measurements errors with variances not less than 1 [liter/min]2). It should be also remarked that the proposed model-based estimation techniques, both linear and nonlinear, are highly superior to conventional methods used in medical practice. The present study provides insights and guidelines that can also be useful when dealing with similar gas exchange estimation problems in many other different application areas.

Dynamic electrical impedance imaging of a chest phantom using the Kalman filter

A dynamic complex impedance imaging technique is developed with the aid of the linearized Kalman filter (LKF) for real-time reconstruction of the human chest. The forward problem is solved by an analytical method based on the separation of variables and Fourier series. The inverse problem is treated as a state estimation problem. The nonlinear measurement equation is linearized about the best homogeneous impedivity value as an initial guess, and the impedivity distribution is estimated with the aid of the Kalman estimator. The Kalman gain matrix is pre-computed and stored off-line to minimize the on-line computational time. Simulation and phantom experiment are reported to illustrate the reconstruction performances in the sense of spatio-temporal resolution in a simplified geometry of the human chest.

Electrical impedance tomography method for reconstruction of biological tissues with continuous plane-stratification

A novel electrical impedance tomography method is introduced for reconstruction of layered biological tissues with continuous plane-stratification. The algorithm implements the recently proposed reconstruction scheme for piecewise constant conductivity profiles, based on an improved Prony method in conjunction with Legendre polynomial expansion (LPE). It is shown that the proposed algorithm is capable of successfully reconstructing continuous conductivity profiles with moderate (WKB) slop. Features of the presented reconstruction scheme include, an inherent linearity, achieved by the linear LPE transform, a locality feature, assigning analytically to each spectral component a local electrical impedance associated with a unique location, and effective performance even in the presence of noisy measurements.

Dynamic electrical impedance imaging with the interacting multiple model scheme

In this paper, an effective dynamical EIT imaging scheme is presented for on-line monitoring of the abruptly changing resistivity distribution inside the object, based on the interacting multiple model (IMM) algorithm. The inverse problem is treated as a stochastic nonlinear state estimation problem with the time-varying resistivity (state) being estimated on-line with the aid of the IMM algorithm. In the design of the IMM algorithm multiple models with different process noise covariance are incorporated to reduce the modeling uncertainty. Simulations and phantom experiments are provided to illustrate the proposed algorithm.