Contrast enhancement in EIT imaging of the brain.

We consider electrical impedance tomography (EIT) imaging of the brain. The brain is surrounded by the poorly conducting skull which has low conductivity compared to the brain. The skull layer causes a partial shielding effect which leads to weak sensitivity for the imaging of the brain tissue. In this paper we propose an approach based on the Bayesian approximation error approach, to enhance the contrast in brain imaging. With this approach, both the (uninteresting) geometry and the conductivity of the skull are embedded in the approximation error statistics, which leads to a computationally efficient algorithm that is able to detect features such as internal haemorrhage with significantly increased sensitivity and specificity. We evaluate the approach with simulations and phantom data.

A new computational approach for cortical imaging.

Estimation of current or potential distribution on the cortex is used to obtain information about neural sources from the scalp recorded electroencephalogram. If the active sources in the brain are superficial, the estimated field distribution on the cortex also yields information about the active source configuration. In these cases, these methods can be used as source localization methods. In this study, we concentrate on finite-element-based cortex potential estimation. Usually these methods require surface interpolation of the recorded voltages at the electrodes onto the entire scalp surface. We propose a new computational approach which does not require the use of surface interpolation but does it implicitly and uses only the recorded data at the electrodes. We refer to this method as the systematic approach (SA). We compare the SA with the surface interpolation approach (IA) and show that the SA is able to produce somewhat better accuracy than the IA. However, the main asset is that the sensitivity of the cortical potential maps to the regularization parameter is significantly lower than with the IA.

Tikhonov regularization and prior information in electrical impedance tomography.

The solution of impedance distribution in electrical impedance tomography is a nonlinear inverse problem that requires the use of a regularization method. The generalized Tikhonov regularization methods have been popular in the solution of many inverse problems. The regularization matrices that are usually used with the Tikhonov method are more or less ad hoc and the implicit prior assumptions are, thus, in many cases inappropriate. In this paper, we propose an approach to the construction of the regularization matrix that conforms to the prior assumptions on the impedance distribution. The approach is based on the construction of an approximating subspace for the expected impedance distributions. It is shown by simulations that the reconstructions obtained with the proposed method are better than with two other schemes of the same type when the prior is compatible with the true object. On the other hand, when the prior is incompatible with the true object, the method will still give reasonable estimates.

The inverse problem of depth dose curve estimation.

The inverse problem of the depth dose curve is formulated and a proposition for its solution is presented. The solution is based on the approximation of the observation equation with a numerical quadrature operator and the regularization of this inverse problem with a smoothness side constraint. The problem formulation is applicable for both the electron and the photon depth dose curve estimation. Moreover, the method is equivalent for, for example, all energies, field sizes and source-to-phantom distances. Simulations show that the estimation error is smaller with the proposed method than with direct linear interpolation. The main result of the paper, however, is the formulation of the problem that allows feasible extensions and modifications for different measurement situations.

Simultaneous reconstruction of internal tissue region boundaries and coefficients in optical diffusion tomography.

In this paper we propose a new numerical method to the inverse problem in optical diffusion tomography. We consider the reconstruction of the diffusion and absorption coefficients (kappa, mu(a)) within a domain omega which is known to consist of a set of disjoint regions of distinct tissue types. The assumption is that the regions of different tissues are bounded by smooth boundary curves and have constant absorption and diffusion coefficients. The goal in the proposed method is to reconstruct simultaneously the boundaries of the tissue regions together with the absorption and diffusion coefficients within these regions. The solution of the problem is based on the finite element method and subdivision of the elements. The performance of the proposed method is evaluated by simulations in which the optical parameters (kappa, mu(a)) are relevant in medical applications of optical tomography. It is shown that the proposed method is able to recover both the boundaries and the coefficients with good accuracy.

Statistical inversion for medical x-ray tomography with few radiographs: I. General theory.

In x-ray tomography, the structure of a three-dimensional body is reconstructed from a collection of projection images of the body. Medical CT imaging does this using an extensive set of projections from all around the body. However, in many practical imaging situations only a small number of truncated projections are available from a limited angle of view. Three-dimensional imaging using such data is complicated for two reasons: (i) typically, sparse projection data do not contain sufficient information to completely describe the 3D body, and (ii) traditional CT reconstruction algorithms, such as filtered backprojection, do not work well when applied to few irregularly spaced projections. Concerning (i), existing results about the information content of sparse projection data are reviewed and discussed. Concerning (ii), it is shown how Bayesian inversion methods can be used to incorporate a priori information into the reconstruction method, leading to improved image quality over traditional methods. Based on the discussion, a low-dose three-dimensional x-ray imaging modality is described.

Statistical inversion for medical x-ray tomography with few radiographs: II. Application to dental radiology.

Diagnostic and operational tasks in dental radiology often require three-dimensional information that is difficult or impossible to see in a projection image. A CT-scan provides the dentist with comprehensive three-dimensional data. However, often CT-scan is impractical and, instead, only a few projection radiographs with sparsely distributed projection directions are available. Statistical (Bayesian) inversion is well-suited approach for reconstruction from such incomplete data. In statistical inversion, a priori information is used to compensate for the incomplete information of the data. The inverse problem is recast in the form of statistical inference from the posterior probability distribution that is based on statistical models of the projection data and the a priori information of the tissue. In this paper, a statistical model for three-dimensional imaging of dentomaxillofacial structures is proposed. Optimization and MCMC algorithms are implemented for the computation of posterior statistics. Results are given with in vitro projection data that were taken with a commercial intraoral x-ray sensor. Examples include limited-angle tomography and full-angle tomography with sparse projection data. Reconstructions with traditional tomographic reconstruction methods are given as reference for the assessment of the estimates that are based on the statistical model.

Estimation of event-related synchronization changes by a new TVAR method.

The modeling of nonstationary electroencephalogram (EEG) with time-varying autoregressive (TVAR) models is discussed. The classical least squares TVAR approach is modified so that prior assumptions about the signal can be taken into account in an optimal way. The method is then applied to the estimation of event-related synchronization changes in the EEG. The results show that the new approach enables effective estimation of the parameter evolution of the time-varying EEG with better time resolution compared to previous methods. The new method also allows single-trial analysis of the event-related synchronization.

A Kalman filter approach to track fast impedance changes in electrical impedance tomography.

In electrical impedance tomography (EIT), an estimate for the cross-sectional impedance distribution is obtained from the body by using current and voltage measurements made from the boundary. All well-known reconstruction algorithms use a full set of independent current patterns for each reconstruction. In some applications, the impedance changes may be so fast that information on the time evolution of the impedance distribution is either lost or severely blurred. In this paper, we propose an algorithm for EIT reconstruction that is able to track fast changes in the impedance distribution. The method is based on the formulation of EIT as a state-estimation problem and the recursive estimation of the state with the aid of the Kalman filter. The performance of the proposed method is evaluated with a simulation of human thorax in a situation in which the impedances of the ventricles change rapidly. We show that with optimal current patterns and proper parameterization, the proposed approach yields significant enhancement of the temporal resolution over the conventional reconstruction strategy.

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

In electrical impedance tomography 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. It is often assumed that the injected currents are confined to the two-dimensional (2-D) electrode plane and the reconstruction is based on 2-D assumptions. However, the currents spread out in three dimensions and, therefore, off-plane structures have significant effect on the reconstructed images. In this paper we propose a finite element-based method for the reconstruction of three-dimensional resistivity distributions. The proposed method is based on the so-called complete electrode model that takes into account the presence of the electrodes and the contact impedances. Both the forward and the inverse problems are discussed and results from static and dynamic (difference) reconstructions with real measurement data are given. It is shown that in phantom experiments with accurate finite element computations it is possible to obtain static images that are comparable with difference images that are reconstructed from the same object with the empty (saline filled) tank as a reference.

Subspace regularization method for the single-trial estimation of evoked potentials.

A method for the single-trial estimation of the evoked potentials is proposed. The method is based on the so-called subspace regularization approach in which the second-order statistics of the set of the measurements is used to form a prior information model for the evoked potentials. The method is closely related to the Bayesian estimation. The performance of the proposed method is evaluated using realistic simulations. As a specific application the method is applied to the estimation of the target responses in the P300 test.

Single-trial estimation of multichannel evoked-potential measurements.

A method for single-trial estimation of multichannel evoked potentials is presented. The proposed method is based on the regularized least squares scheme. The spatial correlation between the channels is used as additional information in the estimation procedure. Amplitude estimates obtained with the proposed method is compared with the estimates calculated without using the spatial information. The performance of the method is evaluated using simulated and real data of P300 responses measured using auditory stimuli. The multichannel approach is shown to give realistic and comparable information about the amplitude differences of the P300 peak between different channels.

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.

Assessment of errors in static electrical impedance tomography with adjacent and trigonometric current patterns.

In electrical impedance tomography (EIT), difference imaging is often preferred over static imaging. This is because of the many unknowns in the forward modelling which make it difficult to obtain reliable absolute resistivity estimates. However, static imaging and absolute resistivity values are needed in some potential applications of EIT. In this paper we demonstrate by simulation the effects of different error components that are included in the reconstruction of static EIT images. All simulations are carried out in two dimensions with the so-called complete electrode model. Errors that are considered are the modelling error in the boundary shape of an object, errors in the electrode sizes and localizations and errors in the contact impedances under the electrodes. Results using both adjacent and trigonometric current patterns are given.

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.

Estimation of the dynamics of event-related desynchronisation changes in electroencephalograms.

A method for the estimation of medium rate transitions of non-stationary electroencephalograms (EEG) is proposed. The method is applicable to such EEG dynamics that are between (a) fast transitions for which segmentation procedures are used and (b) slow transitions for which adaptive filters work properly. The estimation of the transition dynamics is based on a novel time-varying autoregressive model. This model belongs to the class of deterministic regression time-varying autoregressive models and its parametrisation allows only simultaneous transitions in all coefficient evolutions. Data from 22 patients was analysed. The performance of the method is first evaluated with realistic simulations of known transition dynamics and it is shown to be able to track medium-rate transitions. The method is then applied to the estimation of the dynamics of event related desynchronisation. It is shown that the proposed method is able to estimate the transitions which are less apparent, such as from a multi-infarct patient.

Effects of electrode properties on EEG measurements and a related inverse problem.

A trend in EEG measurements is to increase the number of measurement electrodes in order to improve the spatial resolution of the recorded voltage distribution at the scalp. It is assumed that this would implicate better accuracy in the EEG inverse estimates. However, this does not necessarily hold. The reason for this is that the electrodes create a well conducting shunting "layer" on the scalp which affects the voltage distribution. This may decrease the information obtained and may therefore worsen the inverse estimates. Electrodes in EEG inverse problems are commonly modeled as point electrodes. This model cannot take into account the possible shunting effect of the electrodes. In this study the measurement electrodes are modeled using the so-called complete electrode model which takes into account the actual size of the electrode, the contact impedance between the skin and the electrode and also the shunting effect of the electrodes. In this paper the effects of the electrode size and the contact impedance on the voltage distribution are studied by simulations. It is shown that, depending on the size and the contact impedance of the electrodes, increasing the number of electrodes does not necessarily improve the accuracy of the inverse estimates. We also conclude that the use of the point electrode model is quite adequate in normal EEG studies. The use of a complete electrode model is necessary if electrodes cover more than 50% of the surface area.

Effects of local skull inhomogeneities on EEG source estimation.

The accuracy of the head model affects the solutions of the EEG inverse problems. If a simple three-sphere model and standard conductivity values for brain, skull and scalp regions are used, significant errors may occur in the dipole localisation. One of the most sensitive head model parameters is the conductivity of the skull. A realistic three-dimensional finite-element model provides a method to study the effect of inhomogeneities of the skull on the solutions of EEG inverse problems. In this paper the effect of a local skull conductivity inhomogeneity on source estimation accuracy is analyzed by computer simulations for different numbers of electrodes. It is shown that if the inhomogeneity of the skull conductivity is not taken into account, localisation errors of approximately 1 cm can be encountered in the equivalent current dipole estimation. This modelling error introduces a bias to the solution which cannot be compensated by increasing the number of electrodes.

Approximation errors and model reduction in optical tomography.

Model reduction is often required in optical diffusion tomography (ODT), typically due to limited available computation time or computer memory. In practice, this often means that we are bound to use sparse meshes in the model for the forward problem. Conversely, if we are given more and more accurate measurements, we have to employ increasingly accurate forward problem solvers in order to exploit the information in the measurements. In this paper we apply the approximation error theory to ODT. We show that if the approximation errors are estimated and employed, it is possible to use mesh densities that would be unacceptable with a conventional measurement model.

Simulation of nonstationary EEG.

In this paper we present a systematic method for generating simulations of nonstationary EEG. Such simulations are needed, for example, in the evaluation of tracking algorithms. First a state evolution process is simulated. The states are initially represented as segments of stationary autoregressive processes which are described with the corresponding predictor coefficients and prediction error variances. These parameters are then concatenated to give a piecewise time-invariant parameter evolution. The evolution is projected onto an appropriately selected set of smoothly time-varying functions. This projection is used to generate the final EEG simulation. As an example we use this method to simulate the EEG of a drowsy rat. This EEG can be described as toggling between two states that differ in the degree of synchronization of the activity-inducing neuron clusters.