Adaptive mesh refinement techniques for electrical impedance tomography

Adaptive techniques in electrical impedance tomography reconstruction

We present an adaptive algorithm for solving the inverse problem in electrical impedance tomography. To strike a balance between the accuracy of the reconstructed images and the computational efficiency of the forward and inverse solvers, we propose to combine an adaptive mesh refinement technique with the adaptive Kaczmarz method. The iterative algorithm adaptively generates the optimal current patterns and a locally-refined mesh given the conductivity estimate and solves for the unknown conductivity distribution with the block Kaczmarz update step. Simulation and experimental results with numerical analysis demonstrate the accuracy and the efficiency of the proposed algorithm.

Reconstruction convergence and speed enhancement in electrical impedance tomography for domains with known internal boundaries

An improved approach for electrical impedance tomography (EIT) image reconstruction, based on modifying the forward and inverse solutions, is proposed. In this approach, the EIT forward problem is solved via the finite element method (FEM) using two types of elements. The inverse problem is solved by the modified Newton-Raphson method, whereas the condition number of the Hessian matrix is being monitored. At the early stage of the reconstruction, first-order elements are used, and if the condition number exceeds the allowable limit, the algorithm restarts. Otherwise, if the reconstruction error becomes lower than a predefined threshold, second-order elements are employed in the forward solution in order to preserve the precision of the final results. The latter stage converges in very few iterations. Since the solution speed with the first-order FEM is considerably higher than the second-order FEM, the reconstruction speed improves considerably by this approach, whereas the accuracy of the results is guaranteed by the well-conditioned Hessian matrix. Numerical simulations and experiments are followed by comparisons with other reconstruction methods which demonstrate the reliability and high solution speed of this approach. According to the results, the convergence of the proposed method is significantly improved, and its speed is 2-200 times higher than the previously developed methods with the same level of precision.

Adaptive mesh refinement techniques for 3-D skin electrode modeling

In this paper, we develop a 3-D adaptive mesh refinement technique. The algorithm is constructed with an electric impedance tomography forward problem and the finite-element method in mind, but is applicable to a much wider class of problems. We use the method to evaluate the distribution of currents injected into a model of a human body through skin contact electrodes. We demonstrate that the technique leads to a significantly improved solution, particularly near the electrodes. We discuss error estimation, efficiency, and quality of the refinement algorithm and methods that allow for preserving mesh attributes in the refinement process.

Discretization error analysis and adaptive meshing algorithms for fluorescence diffuse optical tomography: part II

In the first part of this work, we analyze the effect of discretization on the accuracy of fluorescence diffuse optical tomography (FDOT). Our error analysis provides two new error estimates which present a direct relationship between the error in the reconstructed fluorophore concentration and the discretization of the forward and inverse problems. In this paper, based on these error estimates, we develop two new adaptive mesh generation algorithms for the numerical solutions of the forward and inverse problems in FDOT, with the objective of error reduction in the reconstructed optical images due to discretization while keeping the size of the discretized forward and inverse problems within the allowable limits. We present three-dimensional numerical simulations to demonstrate the improvements in accuracy, resolution and detectability of small heterogeneities in reconstructed images provided by the use of the new adaptive mesh generation algorithms. Finally, we compare our algorithms both analytically and numerically with the existing conventional adaptive mesh generation algorithms.

Fast intersections on nested tetrahedrons (FINT): An algorithm for adaptive finite element based distributed parameter estimation

A variety of biomedical imaging techniques such as optical and fluorescence tomography, electrical impedance tomography, and ultrasound imaging can be cast as inverse problems, wherein image reconstruction involves the estimation of spatially distributed parameter(s) of the PDE system describing the physics of the imaging process. Finite element discretization of imaged domain with tetrahedral elements is a popular way of solving the forward and inverse imaging problems on complicated geometries. A dual-adaptive mesh-based approach wherein, one mesh is used for solving the forward imaging problem and the other mesh used for iteratively estimating the unknown distributed parameter, can result in high resolution image reconstruction at minimum computation effort, if both the meshes are allowed to adapt independently. Till date, no efficient method has been reported to identify and resolve intersection between tetrahedrons in independently refined or coarsened dual meshes. Herein, we report a fast and robust algorithm to identify and resolve intersection of tetrahedrons within nested dual meshes generated by 8-similar subtetrahedron subdivision scheme. The algorithm exploits finite element weight functions and gives rise to a set of weight functions on each vertex of disjoint tetrahedron pieces that completely cover up the intersection region of two tetrahedrons. The procedure enables fully adaptive tetrahedral finite elements by supporting independent refinement and coarsening of each individual mesh while preserving fast identification and resolution of intersection. The computational efficiency of the algorithm is demonstrated by diffuse photon density wave solutions obtained from a single- and a dual-mesh, and by reconstructing a fluorescent inclusion in simulated phantom from boundary frequency domain fluorescence measurements.

Numerical modelling errors in electrical impedance tomography

Electrical impedance tomography (EIT) is a non-invasive technique that aims to reconstruct images of internal impedance values of a volume of interest, based on measurements taken on the external boundary. Since most reconstruction algorithms rely on model-based approximations, it is important to ensure numerical accuracy for the model being used. This work demonstrates and highlights the importance of accurate modelling in terms of model discretization (meshing) and shows that although the predicted boundary data from a forward model may be within an accepted error, the calculated internal field, which is often used for image reconstruction, may contain errors, based on the mesh quality that will result in image artefacts.

Using the GRID to improve the computation speed of electrical impedance tomography (EIT) reconstruction algorithms

In our group at University College London, we have been developing electrical impedance tomography (EIT) of brain function. We have attempted to improve image quality by the use of realistic anatomical meshes and, more recently, non-linear reconstruction methods. Reconstruction with linear methods, with pre-processing, may take up to a few minutes per image for even detailed meshes. However, iterative non-linear reconstruction methods require much more computational resources, and reconstruction with detailed meshes was taking far too long for clinical use. We present a solution to this timing bottleneck, using the resources of the GRID, the development of coordinated computing resources over the internet that are not subject to centralized control using standard, open, general-purpose protocols and are transparent to the user. Optimization was performed by splitting reconstruction of image series into individual jobs of one image each; no parallelization was attempted. Using the GRID middleware 'Condor' and a cluster of 920 nodes, reconstruction of EIT images of the human head with a non-linear algorithm was speeded up by 25-40 times compared to serial processing of each image. This distributed method is of direct practical value in applications such as EIT of epileptic seizures where hundreds of images are collected over the few minutes of a seizure and will be of value to clinical data collection with similar requirements. In the future, the same resources could be employed for the more ambitious task of parallelized code.

Imbalances in Regional Lung Ventilation

Imbalances in regional lung ventilation, with gravity-dependent collapse and overdistention of nondependent zones, are likely associated to ventilator-induced lung injury. Electric impedance tomography is a new imaging technique that is potentially capable of monitoring those imbalances. The aim of this study was to validate electrical impedance tomography measurements of ventilation distribution, by comparison with dynamic computerized tomography in a heterogeneous population of critically ill patients under mechanical ventilation. Multiple scans with both devices were collected during slow-inflation breaths. Six repeated breaths were monitored by impedance tomography, showing acceptable reproducibility. We observed acceptable agreement between both technologies in detecting right-left ventilation imbalances (bias = 0% and limits of agreement = -10 to +10%). Relative distribution of ventilation into regions or layers representing one-fourth of the thoracic section could also be assessed with good precision. Depending on electrode positioning, impedance tomography slightly overestimated ventilation imbalances along gravitational axis. Ventilation was gravitationally dependent in all patients, with some transient blockages in dependent regions synchronously detected by both scanning techniques. Among variables derived from computerized tomography, changes in absolute air content best explained the integral of impedance changes inside regions of interest (r(2) > or = 0.92). Impedance tomography can reliably assess ventilation distribution during mechanical ventilation.

Electrical impedance tomography of human brain function using reconstruction algorithms based on the finite element method

Electrical impedance tomography (EIT) is a recently developed technique which enables the internal conductivity of an object to be imaged using rings of external electrodes. In a recent study, EIT during cortical evoked responses showed encouraging changes in the raw impedance measurements, but reconstructed images were noisy. A simplified reconstruction algorithm was used which modelled the head as a homogeneous sphere. In the current study, the development and validation of an improved reconstruction algorithm are described in which realistic geometry and conductivity distributions have been incorporated using the finite element method. Data from computer simulations and spherical or head-shaped saline-filled tank phantoms, in which the skull was represented by a concentric shell of plaster of Paris or a real human skull, have been reconstructed into images. There were significant improvements in image quality as a result of the incorporation of accurate geometry and extracerebral layers in the reconstruction algorithm. Image quality, assessed by blinded subjective expert observers, also improved significantly when data from the previous evoked response study were reanalysed with the new algorithm. In preliminary images collected during epileptic seizures, the new algorithm generated EIT conductivity changes which were consistent with the electrographic ictal activity. Incorporation of realistic geometry and conductivity into the reconstruction algorithm significantly improves the quality of EIT images and lends encouragement to the belief that EIT may provide a low-cost, portable functional neuroimaging system in the foreseeable future.

Comparison of algorithms for non-linear inverse 3D electrical tomography reconstruction

Non-linear electrical impedance tomography reconstruction algorithms usually employ the Newton-Raphson iteration scheme to image the conductivity distribution inside the body. For complex 3D problems, the application of this method is not feasible any more due to the large matrices involved and their high storage requirements. In this paper we demonstrate the suitability of an alternative conjugate gradient reconstruction algorithm for 3D tomographic imaging incorporating adaptive mesh refinement and requiring less storage space than the Newton-Raphson scheme. We compare the reconstruction efficiency of both algorithms for a simple 3D head model. The results show that an increase in speed of about 30% is achievable with the conjugate gradient-based method without loss of accuracy.

Optimal imaging with adaptive mesh refinement in electrical impedance tomography

In non-linear electrical impedance tomography the goodness of fit of the trial images is assessed by the well-established statistical chi2 criterion applied to the measured and predicted datasets. Further selection from the range of images that fit the data is effected by imposing an explicit constraint on the form of the image, such as the minimization of the image gradients. In particular, the logarithm of the image gradients is chosen so that conductive and resistive deviations are treated in the same way. In this paper we introduce the idea of adaptive mesh refinement to the 2D problem so that the local scale of the mesh is always matched to the scale of the image structures. This improves the reconstruction resolution so that the image constraint adopted dominates and is not perturbed by the mesh discretization. The avoidance of unnecessary mesh elements optimizes the speed of reconstruction without degrading the resulting images. Starting with a mesh scale length of the order of the electrode separation it is shown that, for data obtained at presently achievable signal-to-noise ratios of 60 to 80 dB, one or two refinement stages are sufficient to generate high quality images.