Three-dimensional electrical impedance tomography based on the complete electrode model

Compressed sampling for boundary measurements in three-dimensional electrical impedance tomography

Electrical impedance tomography (EIT) utilizes electrodes on a medium's surface to produce measured data from which the conductivity distribution inside the medium is estimated. For the cases that relocation of electrodes is impractical or no a priori assumptions can be made to optimize the electrodes placement, a large number of electrodes may be needed to cover all possible imaging volume. This may occur in dynamically varying conductivity distribution in 3D EIT. Three-dimensional EIT then requires inverting very large linear systems to calculate the conductivity field, which causes significant problems regarding storage space and reconstruction time in addition to that data acquisition for a large number of electrodes will reduce the achievable frame rate, which is considered as major advantage of EIT imaging. This study proposes an idea to reduce the reconstruction complexity based on the well-known compressed sampling theory. By applying the so-called model-based CoSaMP algorithm to large size data collected by a 256 channel system, the size of forward operator and data acquisition time is reduced to those of a 32 channel system, while accuracy of reconstruction is significantly improved. The results demonstrate great capability of compressed sampling for overriding the challenges arising in 3D EIT.

Reducing computational costs in large scale 3D EIT by using a sparse Jacobian matrix with block-wise CGLS reconstruction

Electrical impedance tomography (EIT) is a fast and cost-effective technique to provide a tomographic conductivity image of a subject from boundary current-voltage data. This paper proposes a time and memory efficient method for solving a large scale 3D EIT inverse problem using a parallel conjugate gradient (CG) algorithm. The 3D EIT system with a large number of measurement data can produce a large size of Jacobian matrix; this could cause difficulties in computer storage and the inversion process. One of challenges in 3D EIT is to decrease the reconstruction time and memory usage, at the same time retaining the image quality. Firstly, a sparse matrix reduction technique is proposed using thresholding to set very small values of the Jacobian matrix to zero. By adjusting the Jacobian matrix into a sparse format, the element with zeros would be eliminated, which results in a saving of memory requirement. Secondly, a block-wise CG method for parallel reconstruction has been developed. The proposed method has been tested using simulated data as well as experimental test samples. Sparse Jacobian with a block-wise CG enables the large scale EIT problem to be solved efficiently. Image quality measures are presented to quantify the effect of sparse matrix reduction in reconstruction results.

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.

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.

Methods for compensating for variable electrode contact in EIT

Electrical impedance tomography (EIT) is an imaging modality that currently shows promise for the detection and characterization of breast cancer. A very significant problem in EIT imaging is the proper modeling of the interface between the body and the electrodes. We have found empirically that it is very difficult, in a clinical setting, to assure that all electrodes make satisfactory contact with the body. In addition, we have observed a capacitive effect at the skin/electrode boundary that is spatially heterogeneous. To compensate for these problems, we have developed a hybrid nonlinear-linear reconstruction algorithm using the complete electrode model in which we first estimate electrode surface impedances, by means of a Levenberg-Marquardt iterative optimization procedure with an analytically computed Jacobian matrix. We, subsequently, use a linearized algorithm to perform a 3-D reconstruction of perturbations in both contact impedances, and in the spatial distributions of conductivity and permittivity. Results show that, with this procedure, artifacts due to electrodes making poor contact can be greatly reduced. If the experimental apparatus physically applies voltages and measures currents, we show that it is preferable to compute the reconstruction with respect to the Dirichlet-to-Neumann map rather than the Neumann-to-Dirichlet map if there is a significant possibility that electrodes will be fully disconnected. Finally, we test our electrode compensation algorithms for a set of clinical data, showing that we can significantly improve the fit of our model to the measurements by allowing the electrode surface impedances to vary.

Computational modelling of blood-flow-induced changes in blood electrical conductivity and its contribution to the impedance cardiogram

Studies have shown that blood-flow-induced change in electrical conductivity is of equal importance in assessment of the impedance cardiogram (ICG) as are volumetric changes attributed to the motion of heart, lungs and blood vessels. To better understand the sole effect of time-varying blood conductivity on the spatiotemporal distribution of trans-thoracic electric fields (i.e. ICG), this paper presents a segmented high-resolution (1 mm(3)) thoracic cardiovascular system, in which the time-varying pressures, flows and electrical conductivities of blood in different vessels are evaluated using a set of coupled nonlinear differential equations, red blood cell orientation and cardiac cycle functions. Electric field and voltage simulations are performed using two and four electrode configurations delivering a small alternating electric current to an anatomically realistic and electrically accurate model of modelled human torso. The simulations provide a three-dimensional electric field distribution and show that the time-varying blood conductivity alters the voltage potential difference between the electrodes by a maximum of 0.28% for a cardiac output of about 5 L min(-1). As part of a larger study, it is hoped that this initial model will be useful in providing improved insights into blood-flow-related spatiotemporal electric field variations and assist in the optimal placement of electrodes in impedance cardiography experiments.

An electrical impedance tomography-based approach to monitor in vitro sodium chloride dissolution from pharmaceutical tablets

An approach to monitor in vitro dissolution process from pharmaceutical tablets utilizing electrical impedance tomography (EIT) is introduced. In the demonstration, a tablet containing sodium chloride (NaCl) was dissolution tested using tap water as a dissolution medium within an apparatus similar to the United States Pharmacopoeia dissolution apparatus II. During the process, the three-dimensional sodium chloride concentration distribution was monitored with EIT measurements as a function of time. For EIT measurements, an array of electrodes was attached on the boundary of the dissolution vessel, a set of alternating electric currents was injected through the electrodes, and the resulting voltages were measured. With these data and by applying mathematical algorithms, an approximation for the spatial/temporal concentration distribution inside the vessel was computed. It was found that the computed distributions were relatively homogeneous. A NaCl release curve was computed by integrating the concentration distribution over the vessel volume, and the final value of the curve matched well with the reference point based on the weight loss of the tablet. Finally, EIT monitoring is suggested to be used for research and product development purposes.

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

The D-bar algorithm based on Nachman's 2-D global uniqueness proof for the inverse conductivity problem (Nachman, 1996) is implemented on a chest-shaped domain. The scattering transform is computed on this chest-shaped domain using trigonometric and adjacent current patterns and the complete electrode model for the forward problem is computed with the finite element method in order to obtain simulated voltage measurements. The robustness and effectiveness of the method is demonstrated on a simulated chest with errors in input currents, output voltages, electrode placement, and domain modeling.

An implementation of CalderOn's method for 3-D limited-view EIT

Mathematical interest in electrical impedance tomography has been strong since the publication of CalderOn's foundational paper. This paper introduced the idea of applying external voltage patterns to a medium such that, assuming that the medium is sufficiently close to a constant admittivity, the reconstruction can be accomplished directly by inverse Fourier transform. Motivated by CalderOn's method, we have developed a variant of the algorithm which is applicable to the case of measurement on only a part of the boundary and on discrete electrodes. Here we determine voltage or current patterns to apply to the electrodes which optimally approximate CalderOn's special functions in the interior. Furthermore, in three dimensions and higher, CalderOn's method allows each point in Fourier space to be computed in a multiplicity of ways. We show that by making use of the inherent redundancy in our measurements, we can significantly improve the quality of the static images produced by our algorithm.

A broadband high-frequency electrical impedance tomography system for breast imaging

Bio-electric impedance signatures arise primarily from differences in cellular morphologies within an organ and can be used to differentiate benign and malignant pathologies, specifically in the breast. Electrical impedance tomography (EIT) is an imaging modality that determines the impedance distribution within tissue and has been used in prior work to map the electrical properties of breast at signal frequencies ranging from a few kHz to 1 MHz. It has been suggested that by extending the frequency range, additional information of clinical significance may be obtained. We have, therefore, developed a new EIT system for breast imaging which covers the frequency range from 10 kHz to 10 MHz. The instrument developed here is a distributed processor tomograph with 64 channels, capable of generating and measuring voltages and currents. Electrical benchmarking has shown the system to have a SNR greater than 94 dB up to 2 MHz, 90 dB up to 7 MHz, and 65 dB at 10 MHz. In addition, the system measures impedances to an accuracy of 99.7 % and has channel-to-channel variations of less than 0.05 %. Phantom imaging has demonstrated the ability to image across the entire frequency range in both single- and multiplane configurations. Further, 96 women have participated safely in breast exams with the system and the associated conductivity spectra obtained from 3-D image reconstructions range from 0.0237 S/m at 10 kHz to 0.2174 S/m at 10 MHz. These findings are consistent with impedance values reported in the literature.

The complete electrode model for imaging and electrode contact compensation in electrical impedance tomography

Electrical Impedance Tomography (EIT) is an imaging modality which currently shows promise for the detection and characterization of breast cancer. A very significant problem in EIT imaging is the proper modeling of the interface between the body and the electrodes. We have found empirically that it is very difficult, in a clinical setting, to assure that all electrodes make satisfactory contact with the body. In addition, we have observed a capacitive effect at skin/electrode boundary that is spatially heterogeneous. To compensate for these problems, we have developed a hybrid nonlinear-linear reconstruction algorithm in which we first estimate electrode surface impedances, using a Newton-type iterative optimization procedure with an analytically computed Jacobian matrix. We subsequently make use of a linearized algorithm to perform a three-dimensional reconstruction of perturbations in both contact impedances and in the spatial distributions of conductivity and permittivity. Results show that, using this procedure, artifacts due to electrodes making poor contact can be greatly reduced.

The complete electrode model for EIT in a mammography geometry

We have developed an EIT system for simultaneous use in a mammography examination, allowing for highly accurate co-registration between the two modalities. In this pre-clinical study, we investigate the importance of properly modeling the interface between the electrodes and the medium being imaged. We have implemented the complete electrode model for a parallel-plane mammography geometry, in which currents are injected into the medium through two planar sets of electrodes above and below the medium. We make use of the ACT4 device to conduct saline-tank experiments showing the improvement of the complete model over an ave-gap model, which ignores both the conductivity of the electrodes and the surface impedance. The experimental results show an improvement in both forward modeling accuracy and in the quality of the resulting reconstructed images using the complete electrode model, as compared to the ave-gap model.

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.

Accounting for hardware imperfections in EIT image reconstruction algorithms

Electrical impedance tomography (EIT) is a non-invasive technique for imaging the conductivity distribution of a body section. Different types of EIT images can be reconstructed: absolute, time difference and frequency difference. Reconstruction algorithms are sensitive to many errors which translate into image artefacts. These errors generally result from incorrect modelling or inaccurate measurements. Every reconstruction algorithm incorporates a model of the physical set-up which must be as accurate as possible since any discrepancy with the actual set-up will cause image artefacts. Several methods have been proposed in the literature to improve the model realism, such as creating anatomical-shaped meshes, adding a complete electrode model and tracking changes in electrode contact impedances and positions. Absolute and frequency difference reconstruction algorithms are particularly sensitive to measurement errors and generally assume that measurements are made with an ideal EIT system. Real EIT systems have hardware imperfections that cause measurement errors. These errors translate into image artefacts since the reconstruction algorithm cannot properly discriminate genuine measurement variations produced by the medium under study from those caused by hardware imperfections. We therefore propose a method for eliminating these artefacts by integrating a model of the system hardware imperfections into the reconstruction algorithms. The effectiveness of the method has been evaluated by reconstructing absolute, time difference and frequency difference images with and without the hardware model from data acquired on a resistor mesh phantom. Results have shown that artefacts are smaller for images reconstructed with the model, especially for frequency difference imaging.

Experimental justification for using 3D conductivity reconstructions in electrical impedance tomography

Conductivity imaging of the breast using electrical impedance tomography (EIT) is a three-dimensional (3D) problem since the induced currents are free to travel through the entire tissue volume. It is therefore necessary to determine the effect this 3D current flow has on the image reconstruction problem and to ascertain how much benefit is gained by using a more appropriate 3D model to estimate the conductivity distribution. In addition, it is important to consider how much is gained if measurements are collected from multiple circular arrays of electrodes positioned around the breast as opposed to just a single plane of electrodes. We used a 64 electrode EIT system to collect data from a series of high contrast saline phantoms to determine the benefits gained by using a 3D model and the incorporation of out-of-plane measurements. We found that it is preferable to use a 3D mesh even when looking only at a single plane through the object of interest and that this 3D mesh should extend in the axial direction at least one radius away from the plane of interest. Further, out-of-plane measurements enhance axial information and improve the quantification of reconstructed inclusions by a factor of 2.2 in the particular case presented here. These findings should ultimately be incorporated to clinical imaging with EIT when circular electrode arrays are employed.

Dogleg trust-region application in electrical impedance tomography

This paper presents a trust-region implementation for image reconstruction of conductivity changes in electrical impedance tomography. A dogleg trust-region algorithm is applied in different cases to detect abnormalities. The dogleg algorithm approximates a Levenberg-Marquardt step within the trust region of the model function with a quadratic model. The comparison of Levenberg-Marquardt and dogleg algorithms is presented using the reconstructed images. This comparison of two techniques suggests the implementation of the dogleg method could result in the reduction of the execution time to less than 50% of that of the Levenberg-Marquardt algorithm without any quantifiable loss of quality of reconstructed images.

Brain electrical impedance at various frequencies: the effect of hypoxia

Non-invasive multi-frequency measurements of transcephalic impedance, both reactance and resistance, can efficiently detect cell swelling of brain tissue and can be used for early detection of threatening brain damage. We have performed experiments on piglets to monitor transcephalic impedance during hypoxia. The obtained results have confirmed the hypothesis that changes in the size of cells modify the tissue impedance. During tissue inflammation after induced hypoxia, cerebral tissue exhibits changes in both reactance and resistance. Those changes are remarkably high, up to 71% over the baseline, and easy to measure especially at certain frequencies. A better understanding of the electrical behaviour of cerebral tissue during cell swelling would lead us to develop effective non-invasive clinical tools and methods for early diagnosis of cerebral edema and brain damage prevention.

Optimum design of electrode structure and parameters in electrical impedance tomography

In electrical impedance tomography (EIT) the electrode structure and parameters significantly influence measurement sensitivity and image quality, so how to optimize the electrode structure and parameters is one of the key problems in research today. This paper presents a method to optimize the EIT electrode structure and parameters based on coercive equipotential node models. The coercive equipotential mode of the compound electrode has been established based on that of the line electrode. A simulation study for the line electrode and the compound electrode of EIT has been made on a simulation software platform. The influences of different electrode structures and parameters on measurement sensitivity and the image reconstruction quality are studied. For line electrode simulation studies, two important conclusions are drawn. First, a narrower electrode is helpful in improving the imaging quality. Second, although it is known that a wider electrode is beneficial in decreasing the contact impedance, using a too wide electrode causes the measurement sensitivity to decrease. Furthermore the electrode width that leads to the best measurement sensitivity is different for different measurement depths. The compound electrode has four parameters: the excitation electrode width, the measurement electrode width, the space between the excitation electrode and the measurement electrode, and the distance between two adjacent compound electrodes. These parameters have mutual restrictions and complex influences on each other. It is unwise to optimize the design of a compound electrode by only using the overlay rate of electrodes. A simulation study of EIT electrode structure and parameter influences can be carried out according to this paper to determine the optimum design of the electrode structure and its parameters.

Front-tracking image reconstruction algorithm for EIT-monitored cryosurgery using the boundary element method

The effectiveness of cryosurgery, treatment of tumors by freezing, is highly dependent on knowledge of transient freezing extent, and therefore relies heavily on real-time imaging techniques for monitoring. Electrical impedance tomography (EIT) holds much promise for this application. In cryosurgery there is a three order of magnitude change in impedance across the freezing boundary and there is a priori knowledge of the freezing origin. Furthermore, an EIT image of the tissue can be done prior to the cryosurgery. In this study, we have developed an EIT front tracking reconstruction algorithm which takes advantage of these particular attributes of cryosurgery. The method tracks the freezing interface rather than the impedance distribution in the freezing tissue. In addition to drastically reducing the number of parameters needed to define the image, the computational complexity is further reduced by using the more appropriate boundary element method (BEM) for solution to the forward problem. The front-tracking method was found to converge rapidly and accurately to a variety of simulated phantom images.

Improving the forward solver for the complete electrode model in EIT using algebraic multigrid

Image reconstruction in electrical impedance tomography is an ill-posed nonlinear inverse problem. Linearization techniques are widely used and require the repeated solution of a linear forward problem. To account correctly for the presence of electrodes and contact impedances, the so-called complete electrode model is applied. Implementing a standard finite element method for this particular forward problem yields a linear system that is symmetric and positive definite and solvable via the conjugate gradient method. However, preconditioners are essential for efficient convergence. Preconditioners based on incomplete factorization methods are commonly used but their performance depends on user-tuned parameters. To avoid this deficiency, we apply black-box algebraic multigrid, using standard commercial and freely available software. The suggested solution scheme dramatically reduces the time cost of solving the forward problem. Numerical results are presented using an anatomically detailed model of the human head.

Excitation patterns in three-dimensional electrical impedance tomography

Electrical impedance tomography (EIT) is a non-invasive technique that aims to reconstruct images of internal electrical properties of a domain, based on electrical measurements on the periphery. Improvements in instrumentation and numerical modeling have led to three-dimensional (3D) imaging. The availability of 3D modeling and imaging raises the question of identifying the best possible excitation patterns that will yield to data, which can be used to produce the best image reconstruction of internal properties. In this work, we describe our 3D finite element model of EIT. Through singular value decomposition as well as examples of reconstructed images, we show that for a homogenous female breast model with four layers of electrodes, a driving pattern where each excitation plane is a sinusoidal pattern out-of-phase with its neighboring plane produces better qualitative images. However, in terms of quantitative imaging an excitation pattern where all electrode layers are in phase produces better results.

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.

Distributed network imaging and electrical impedance tomography of minimally invasive surgery

Minimally invasive surgery has become highly dependent on imaging. For instance, the effectiveness of cryosurgery in treating cancer is dependent on knowledge of freezing extent, and relies on real-time imaging techniques for monitoring. However, medical imaging is often very expensive and therefore not available to most of the world population. Here we propose the concept of distributed network imaging (DNI) which could make medical imaging and minimally invasive surgery available to all who need these advanced medical modalities. We demonstrate the concept through electrical impedance tomography (EIT) of cryosurgery. The central idea is to develop an inexpensive measurend (data collection hardware) at a remote site and then to connect the measurend apparatus to an advanced image reconstruction server, which can serve a large number of distributed measurends at remote sites, using existing communication conduits (Ethernet, telephone, satellite, etc.). These conduits transfer the raw data from the measurend to the server and the reconstructed image from the server to the measurend. Electrical impedance tomography (EIT) is an imaging modality which utilizes tissue impedance variation to construct an image. The EIT measurend which consists of electrodes, a power supply, and means to measure voltage is inexpensive, and therefore suitable for DNI. EIT is also very well-suited to imaging cryosurgery since frozen tissue impedance is much higher than that of unfrozen tissue. In this study, we first develop numerical models to illustrate the theoretical ability of EIT to image cryosurgery. We begin with a simplified two dimensional model, and then extend the study to the more appropriate three dimensional model. Our simulated finite element phantoms and pixel-based Newton-Raphson reconstruction algorithms were able to produce easily identifiable images of frozen regions within tissue. Then, we demonstrate the feasibility of the DNI concept though a case study using EIT to image an in vitro liver cryosurgery procedure through a modem. We find that the acquired raw data packets are less than 5KB per image and the images, using compression, do not exceed 50KB per image.

Three-dimensional forward solver and its performance analysis for magnetic resonance electrical impedance tomography (MREIT) using recessed electrodes

In magnetic resonance electrical impedance tomography (MREIT), we try to reconstruct a cross-sectional resistivity (or conductivity) image of a subject. When we inject a current through surface electrodes, it generates a magnetic field. Using a magnetic resonance imaging (MRI) scanner, we can obtain the induced magnetic flux density from MR phase images of the subject. We use recessed electrodes to avoid undesirable artefacts near electrodes in measuring magnetic flux densities. An MREIT image reconstruction algorithm produces cross-sectional resistivity images utilizing the measured internal magnetic flux density in addition to boundary voltage data. In order to develop such an image reconstruction algorithm, we need a three-dimensional forward solver. Given injection currents as boundary conditions, the forward solver described in this paper computes voltage and current density distributions using the finite element method (FEM). Then, it calculates the magnetic flux density within the subject using the Biot-Savart law and FEM. The performance of the forward solver is analysed and found to be enough for use in MREIT for resistivity image reconstructions and also experimental designs and validations. The forward solver may find other applications where one needs to compute voltage, current density and magnetic flux density distributions all within a volume conductor.

The application of the generalized vector sample pattern matching method for EIT image reconstruction

This paper presents a new application of a generalized vector sample pattern matching (GVSPM) method for image reconstruction of conductivity changes in electrical impedance tomography. GVSPM is an iterative method for linear inverse problems. The key concept of the GVSPM is that the objective function is defined in terms of an angular component between the inner product of the known vector and solution of a system of equations. Comparisons are presented between images of simulated and experimental data, reconstructed using truncated singular value decomposition and GVSPM. In both cases, a normalized sensitivity matrix is constructed using the finite volume method to solve the forward problem.

Krylov subspace iterative techniques: on the detection of brain activity with electrical impedance tomography

In this paper, we review some numerical techniques based on the linear Krylov subspace iteration that can be used for the efficient calculation of the forward and the inverse electrical impedance tomography problems. Exploring their computational advantages in solving large-scale systems of equations, we specifically address their implementation in reconstructing localized impedance changes occurring within the human brain. If the conductivity of the head tissues is assumed to be real, the pre-conditioned conjugate gradients (PCGs) algorithm can be used to calculate efficiently the approximate forward solution to a given error tolerance. The performance and the regularizing properties of the PCG iteration for solving ill-conditioned systems of equations (PCGNs) is then explored, and a suitable preconditioning matrix is suggested in order to enhance its convergence rate. For image reconstruction, the nonlinear inverse problem is considered. Based on the Gauss-Newton method for solving nonlinear problems we have developed two algorithms that implement the PCGN iteration to calculate the linear step solution. Using an anatomically detailed model of the human head and a specific scalp electrode arrangement, images of a simulated impedance change inside brain's white matter have been reconstructed.

Weighted regularization in electrical impedance tomography with applications to acute cerebral stroke

We apply electrical impedance tomography to detect and localize brain impedance changes associated with stroke. Forward solutions are computed using the finite-element method in two dimensions. We assume that baseline conductivity values are known for the major head tissues, and focus on changes in the brain compartment only. We use singular-value decomposition (SVD) to show that different impedance measurement patterns, which are theoretically equivalent by the reciprocity theorem, have different sensitivities to the brain compartment in the presence of measurement noise. The inverse problem is solved in part by standard means, using iterated SVD, and regularizing by truncation. To improve regularization we introduce a weighting scheme which normalizes the sensitivity matrix for voxels at different depths. This increases the number of linearly independent components which contribute to the solution, and forces the different measurement patterns to have similar sensitivity. When applied to stroke, this weighted regularization improves image quality overall.

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.

The number of electrodes and basis functions in EIT image reconstruction

In electrical impedance tomography, many factors affect the image reconstruction results. Among them are the number of electrodes (NOE) and the number of conductivity basis functions (NOCBF) for image reconstruction. The NOCBF generally reflects the density of the mesh with which images are reconstructed. How and to what extent do these factors affect the image reconstruction and corresponding images? In this area detailed analysis is still lacking. This study aims to address the above question. In this study, image reconstruction and its ill-posed condition were analysed by singular value decomposition (SVD) and spectral expansion theory with different configurations of NOE and NOCBF. The results in this study indicate that for a circular 2D plane object with electrodes evenly located around the boundary: (1) Under certain conditions, increasing the NOE enables us to improve the ill-posed condition in image reconstruction and hence improve the image quality. Generally more improvement is expected near the image periphery than in the image centre. (2) Increasing the NOCBF generally worsens the ill-posed condition. But it enables the solution to be sought in a finer subspace and may be able to improve the image quality on the periphery, while generally the result in image centre depends more on the prior information incorporated in the regularization.

A MATLAB package for the EIDORS project to reconstruct two-dimensional EIT images

The EIDORS (electrical impedance and diffuse optical reconstruction software) project aims to produce a software system for reconstructing images from electrical or diffuse optical data. MATLAB is a software that is used in the EIDORS project for rapid prototyping, graphical user interface construction and image display. We have written a MATLAB package (http://venda.uku.fi/ vauhkon/) which can be used for two-dimensional mesh generation, solving the forward problem and reconstructing and displaying the reconstructed images (resistivity or admittivity). In this paper we briefly describe the mathematical theory on which the codes are based on and also give some examples of the capabilities of the package.

Errors due to the truncation of the computational domain in static three-dimensional electrical impedance tomography

In electrical impedance tomography (EIT), an approximation for the internal resistivity distribution is computed based on the knowledge of the injected currents and measured voltages on the surface of the body. The currents spread out in three dimensions and therefore off-plane structures have a significant effect on the reconstructed images. A question arises: how far from the current carrying electrodes should the discretized model of the object be extended? If the model is truncated too near the electrodes, errors are produced in the reconstructed images. On the other hand if the model is extended very far from the electrodes the computational time may become too long in practice. In this paper the model truncation problem is studied with the extended finite element method. Forward solutions obtained using so-called infinite elements, long finite elements and separable long finite elements are compared to the correct solution. The effects of the truncation of the computational domain on the reconstructed images are also discussed and results from the three-dimensional (3D) sensitivity analysis are given. We show that if the finite element method with ordinary elements is used in static 3D EIT, the dimension of the problem can become fairly large if the errors associated with the domain truncation are to be avoided.