Incorporating a priori information into the Sheffield filtered backprojection algorithm

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.

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.

Calderón's Method with a Spatial Prior for 2-D EIT Imaging of Ventilation and Perfusion

Bedside imaging of ventilation and perfusion is a leading application of 2-D medical electrical impedance tomography (EIT), in which dynamic cross-sectional images of the torso are created by numerically solving the inverse problem of computing the conductivity from voltage measurements arising on electrodes due to currents applied on electrodes on the surface. Methods of reconstruction may be direct or iterative. Calderón’s method is a direct reconstruction method based on complex geometrical optics solutions to Laplace’s equation capable of providing real-time reconstructions in a region of interest. In this paper, the importance of accurate modeling of the electrode location on the body is demonstrated on simulated and experimental data, and a method of including a priori spatial information in dynamic human subject data is presented. The results of accurate electrode modeling and a spatial prior are shown to improve detection of inhomogeneities not included in the prior and to improve the resolution of ventilation and perfusion images in a human subject.

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.

A second order Calderón's method with a correction term and a priori information

Calderón's method is a direct linearized reconstruction method for the inverse conductivity problem with the attribute that it can provide absolute images of both conductivity and permittivity with no need for forward modeling. In this work, an explicit relationship between Calderón's method and the D-bar method is provided, facilitating a "higher-order" Calderón's method in which a correction term is included, derived from the relationship to the D-bar method. Furthermore, a method of including a spatial prior is provided. These advances are demonstrated on simulated data and on tank data collected with the ACE1 EIT system.

A Review on Electrical Impedance Tomography Spectroscopy

Electrical Impedance Tomography Spectroscopy (EITS) enables the reconstruction of material distributions inside an object based on the frequency-dependent characteristics of different substances. In this paper, we present a review of EITS focusing on physical principles of the technology, sensor geometries, existing measurement systems, reconstruction algorithms, and image representation methods. In addition, a novel imaging method is proposed which could fill some of the gaps found in the literature. As an example of an application, EITS of ice and water mixtures is used.

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.

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.

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.

Effects of incompatible boundary information in EIT on the convergence behavior of an iterative algorithm

In electrical impedance tomography, currents are applied to the body through electrodes that are attached to the surface and the corresponding surface voltages are measured. Based on these boundary measurements, the internal admittivity distribution of the body can be reconstructed. In order to improve the image quality it is necessary and useful to apply physiologically meaningful prior information into the image reconstruction. Such prior information usually can be obtained from other sources. For example, information on the object's boundary shape and internal structure can be obtained from computed tomography and magnetic resonance imaging scan. However, this type of prior information may change from time to time and from person to person. As these changes are limited anatomically and physiologically, the prior information including the possible changes can be presented in a number of variational forms. The aim of this paper is to find which form of prior information is more compatible for a specific imaged object at the time of imaging. This paper proposes a new method for selecting the most appropriate form of prior information, through the procedure of iterative image reconstruction by using the information obtained from boundary measurements. The method is based on the principle that incompatible prior information causes errors which are able to affect the image reconstruction's convergence behavior. In this method, according to the various forms of prior information available, several image reconstruction configurations are designed. Then, through monitoring the convergence behavior in an iterative image reconstruction, the configuration with compatible prior information can be found among those different configurations. As an example, the prior information regarding the imaged object's boundary shape and internal structure was studied by computer simulation. Results were shown and discussed.

Two-dimensional finite element modelling of the neonatal head

Electrical impedance tomography (EIT) could allow the early diagnosis of infant brain injury following birth asphyxia. The purpose of this work was to determine the effect of variations in skull, scalp or cerebrospinal fluid (CSF) resistivity, as these vary in clinical conditions and could degrade image quality. These factors were investigated using finite element models of the adult and neonatal head. The results suggest that there is a wide range over which the resistivity of the neonatal skull has little effect on the sensitivity to a central impedance change. The scalp and CSF appear to shunt current away from the brain; when their resistivity was decreased from normal values, this shunting effect increased and caused a decrease in sensitivity to a central resistance change. The resistivity of neonatal skull has not, to our knowledge, been directly measured and will anyway vary within and between individuals; this work suggests that EIT will be relatively insensitive to variations in neonatal skull impedance.

Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography

Image reconstruction in electrical impedance tomography using the sensitivity theorem is generally based on the assumption that the initial conductivity distribution of the body being imaged is uniform. The technique of image reconstruction using this method is described and reconstructed images are presented. Improvements in image quality and accuracy are demonstrated when accurate a priori 'anatomical' information, in the form of a model of the distribution of conductivity within the region to be imaged, is used to construct the sensitivity matrix. In practice correct a priori information is not available, for example the conductivity values within the various anatomical regions will not be known. An iterative algorithm is presented which allows the conductivity parameters of the a priori model to be determined during reconstruction.

Three-dimensional image reconstruction for electrical impedance tomography

Very little work has been conducted on three-dimensional aspects of electrical impedance tomography (EIT), partly due to the increased computational complexity over the two-dimensional aspects of EIT. Nevertheless, extending EIT to three-dimensional data acquisition and image reconstruction may afford significant advantages such as an increase in the size of the independent data set and improved spatial resolution. However, considerable challenges are associated with the software aspects of three-dimensional EIT systems due to the requirement for accurate three-dimensional forward problem modelling and the derivation of three-dimensional image reconstruction algorithms. This paper outlines the work performed to date to derive a three-dimensional image reconstruction algorithm for EIT based on the inversion of the sensitivity matrix approach for a finite right circular cylinder. A comparison in terms of the singular-value spectra and the singular vectors between the sensitivity matrices for a three-dimensional cylinder and a two-dimensional disc has been performed. This comparison shows that the three-dimensional image reconstruction algorithm recruits more central information at lower condition numbers than the two-dimensional image reconstruction algorithm.