Improved forward EEG calculations using local mesh refinement of realistic head geometries

Structural connectivity to reconstruct brain activation and effective connectivity between brain regions

OBJECTIVE

Understanding how brain regions interact to perform a specific task is very challenging. EEG and MEG are two non-invasive imaging modalities that allow the measurement of brain activation with high temporal resolution. Several works in EEG/MEG source reconstruction show that estimating brain activation can be improved by considering spatio-temporal constraints but only few of them use structural information to do so.

APPROACH

In this work, we present a source reconstruction algorithm that uses brain structural connectivity, estimated from diffusion MRI (dMRI), to constrain the EEG/MEG source reconstruction. Contrarily to most source reconstruction methods which reconstruct activation for each time instant, the proposed method estimates an initial reconstruction for the first time instants and a multivariate autoregressive model that explains the data in further time instants. This autoregressive model can be thought as an estimation of the effective connectivity between brain regions. We called this algorithm iterative Source and Dynamics reconstruction (iSDR).

MAIN RESULTS

This paper presents the overall iSDR approach and how the proposed model is optimized to obtain both brain activation and brain region interactions. The accuracy of our method is demonstrated using synthetic data in which it shows a good capability to reconstruct both activation and connectivity. iSDR is also tested with real data obtained from (Wakeman D and Henson R 2015 A multi-subject, multi-modal human neuroimaging dataset Scientific Data 2 15001) (face recognition task). The results are in phase with other works published with the same data and others that used different imaging modalities with the same task showing that the choice of using an autoregressive model gives relevant results.

SIGNIFICANCE

This work shows that complex EEG/MEG datasets can be explained by an initial state and a MAR model for effective connectivity. This is a compact way to describe brain dynamics and offers a direct access to effective connectivity.

Unsupervised Segmentation of Head Tissues from Multi-modal MR Images for EEG Source Localization

In this paper, we present and evaluate an automatic unsupervised segmentation method, hierarchical segmentation approach (HSA)-Bayesian-based adaptive mean shift (BAMS), for use in the construction of a patient-specific head conductivity model for electroencephalography (EEG) source localization. It is based on a HSA and BAMS for segmenting the tissues from multi-modal magnetic resonance (MR) head images. The evaluation of the proposed method was done both directly in terms of segmentation accuracy and indirectly in terms of source localization accuracy. The direct evaluation was performed relative to a commonly used reference method brain extraction tool (BET)-FMRIB's automated segmentation tool (FAST) and four variants of the HSA using both synthetic data and real data from ten subjects. The synthetic data includes multiple realizations of four different noise levels and several realizations of typical noise with a 20% bias field level. The Dice index and Hausdorff distance were used to measure the segmentation accuracy. The indirect evaluation was performed relative to the reference method BET-FAST using synthetic two-dimensional (2D) multimodal magnetic resonance (MR) data with 3% noise and synthetic EEG (generated for a prescribed source). The source localization accuracy was determined in terms of localization error and relative error of potential. The experimental results demonstrate the efficacy of HSA-BAMS, its robustness to noise and the bias field, and that it provides better segmentation accuracy than the reference method and variants of the HSA. They also show that it leads to a more accurate localization accuracy than the commonly used reference method and suggest that it has potential as a surrogate for expert manual segmentation for the EEG source localization problem.

Improved method for retinotopy constrained source estimation of visual-evoked responses

Retinotopy constrained source estimation (RCSE) is a method for noninvasively measuring the time courses of activation in early visual areas using magnetoencephalography (MEG) or electroencephalography (EEG). Unlike conventional equivalent current dipole or distributed source models, the use of multiple, retinotopically mapped stimulus locations to simultaneously constrain the solutions allows for the estimation of independent waveforms for visual areas V1, V2, and V3, despite their close proximity to each other. We describe modifications that improve the reliability and efficiency of this method. First, we find that increasing the number and size of visual stimuli results in source estimates that are less susceptible to noise. Second, to create a more accurate forward solution, we have explicitly modeled the cortical point spread of individual visual stimuli. Dipoles are represented as extended patches on the cortical surface, which take into account the estimated receptive field size at each location in V1, V2, and V3 as well as the contributions from contralateral, ipsilateral, dorsal, and ventral portions of the visual areas. Third, we implemented a map fitting procedure to deform a template to match individual subject retinotopic maps derived from functional magnetic resonance imaging (fMRI). This improves the efficiency of the overall method by allowing automated dipole selection, and it makes the results less sensitive to physiological noise in fMRI retinotopy data. Finally, the iteratively reweighted least squares (IRLS) method was used to reduce the contribution from stimulus locations with high residual error for robust estimation of visual evoked responses.

Source estimates for MEG/EEG visual evoked responses constrained by multiple, retinotopically-mapped stimulus locations

Studying the human visual system with high temporal resolution is a significant challenge due to the limitations of the available, noninvasive measurement tools. MEG and EEG provide the millisecond temporal resolution necessary for answering questions about intracortical communication involved in visual processing, but source estimation is ill-posed and unreliable when multiple; simultaneously active areas are located close together. To address this problem, we have developed a retinotopy-constrained source estimation method to calculate the time courses of activation in multiple visual areas. Source estimation was disambiguated by: (1) fixing MEG/EEG generator locations and orientations based on fMRI retinotopy and surface tessellations constructed from high-resolution MRI images; and (2) solving for many visual field locations simultaneously in MEG/EEG responses, assuming source current amplitudes to be constant or varying smoothly across the visual field. Because of these constraints on the solutions, estimated source waveforms become less sensitive to sensor noise or random errors in the specification of the retinotopic dipole models. We demonstrate the feasibility of this method and discuss future applications such as studying the timing of attentional modulation in individual visual areas.

Development of volume conductor and source models to localize epileptic foci

SUMMARY

There is increasing interest in mapping and source reconstruction from electrocorticoencephalographic (ECoG) grid data and comparison to surface EEG evaluations of epileptic patients. ECoG mapping onto three-dimensional renderings of the individual cortical anatomy derived from magnetic resonance images and computed tomography (CT) is performed after coregistration of anatomical and functional coordinate systems. Source reconstructions from ECoG and EEG are compared using different source models and realistically shaped volume conductor models. Realistically shaped volume conductor models for EEG source reconstruction are a prerequisite for improved localization accuracy. Individual boundary element method (BEM) models derived from MRI represent the "gold standard" and can approximate isotropic homogeneous head compartments and thus give an improved description of the head shape as compared with classical oversimplifying spherical shell models. Anisotropic volume conduction properties of the bone layer or the white matter fibers can be described by the finite element method (FEM); unfortunately, these models require a huge computational effort and are thus not used in daily applications. To avoid this computational effort, head models derived from an averaged MRI dataset can be used. Highly refined models with a large number of nodes and thus better numerical accuracy can be used by this approach, because the setup is performed only once and the decomposed models or precomputed leadfield matrices are saved for later application. Individual image data are not at all needed, if an overlay of the reconstruction results with the anatomy is not desired. With precomputed leadfield matrices and linear interpolation techniques, at least standardized BEM and FEM volume conductor models derived from averaged MRI datasets can achieve the same computational speed as analytical spherical models. The smoothed cortical envelope is used as a realistically shaped single-shell volume conductor model for ECoG source reconstruction, whereas three-compartment BEM-models are required for EEG. The authors describe how to localize ECoG-grid electrode positions and how to segment the cortical surface from coregistered magnetic resonance and CT images. Landmark-based coregistration is performed using common fiducials in both image modalities. Another more promising automatic approach is based on mutual three-dimensional volume gray-level information. The ECoG electrode positions can be retrieved from three-dimensional CT slices manually using cursors in thresholded images with depth information. Alternatively, the smoothed envelope of the cortical surface segmented from the MRI is used to semiautomatically determine the grid electrode positions by marking the four corners and measuring distances along the smoothed surface. With extended source patches for cortically constrained scans and current density reconstructions, results from ECoG and surface EEG data were compared. Single equivalent dipoles were used to explain the EEG far fields, and results were compared with the original current density distributions explaining the ECoG data. The authors studied the performance of spherical and realistically shaped BEM volume conductor models for EEG and ECoG source reconstruction in spherical and nonspherical parts of the head with simulations and measured epileptic spike data. Only small differences between spherical and realistically shaped models were found in the spherical parts of the head, whereas realistically shaped models are superior to spherical approximations in both single-shell ECoG and three-shell EEG cases in the nonspherical parts, such as the temporal lobe areas. The ECoG near field is more complicated to interpret than the surface EEG far field and cannot be explained in general by simple equivalent dipoles. However, from simulations with realistically shaped volume conductor models and cortically constrained source models, the authors studied how the bone and skin layer act as spatial low pass filters that smooth and simplify the surface EEG maps generated by much more complicated-looking source configurations derived from measured ECoG data.

Resistor mesh model of a spherical head: Part 2: A review of applications to cortical mapping

A resistor mesh model (RMM) has been validated with reference to the analytical model by consideration of a set of four dipoles close to the cortex. The application of the RMM to scalp potential interpolation was detailed in Part 1. Using the RMM and the same four dipoles, the different methods of cortical mapping were compared and have shown the potentiality of this RMM for obtaining current and potential cortical distributions. The lead-field matrices are well-adapted tools, but the use of a square matrix of high dimension does not permit the inverse solution to be improved in the presence of noise, as a regularisation technique is necessary with noisy data. With the RMM, the transfer matrix and the cortical imaging technique proved to be easy to implement. Further development of the RMM will include application to more realistic head models with more accurate conductivities.

Resistor mesh model of a spherical head: Part 1: Applications to scalp potential interpolation

A resistor mesh model (RMM) has been implemented to describe the electrical properties of the head and the configuration of the intracerebral current sources by simulation of forward and inverse problems in electroencephalogram/event related potential (EEG/ERP) studies. For this study, the RMM representing the three basic tissues of the human head (brain, skull and scalp) was superimposed on a spherical volume mimicking the head volume: it included 43 102 resistances and 14 123 nodes. The validation was performed with reference to the analytical model by consideration of a set of four dipoles close to the cortex. Using the RMM and the chosen dipoles, four distinct families of interpolation technique (nearest neighbour, polynomial, splines and lead fields) were tested and compared so that the scalp potentials could be recovered from the electrode potentials. The 3D spline interpolation and the inverse forward technique (IFT) gave the best results. The IFT is very easy to use when the lead-field matrix between scalp electrodes and cortex nodes has been calculated. By simple application of the Moore-Penrose pseudo inverse matrix to the electrode cap potentials, a set of current sources on the cortex is obtained. Then, the forward problem using these cortex sources renders all the scalp potentials.

A computationally efficient method for accurately solving the EEG forward problem in a finely discretized head model

OBJECTIVE

Solution of the forward problem using realistic head models is necessary for accurate EEG source analysis. Realistic models are usually derived from volumetric magnetic resonance images that provide a voxel resolution of about 1 mm3. Electrical models could, therefore contain, for a normal adult head, over 4 million elements. Solution of the forward problem using models of this magnitude has so far been impractical due to issues of computation time and memory.

METHODS

A preconditioner is proposed for the conjugate-gradient method that enables the forward problem to be solved using head models of this magnitude. It is applied to the system matrix constructed from the head anatomy using finite differences. The preconditioner is not computed explicitly and so is very efficient in terms of memory utilization.

RESULTS

Using a spherical head model discretized into over 4 million volumes, we have been able to obtain accurate forward solutions in about 60 min on a 1 GHz Pentium III. L2 accuracy of the solutions was better than 2%.

CONCLUSIONS

Accurate solution of the forward problem in EEG in a finely discretized head model is practical in terms of computation time and memory.

SIGNIFICANCE

The results represent an important step in head modeling for EEG source analysis.

An advanced boundary element method (BEM) implementation for the forward problem of electromagnetic source imaging

The forward problem of electromagnetic source imaging has two components: a numerical model to solve the related integral equations and a model of the head geometry. This study is on the boundary element method (BEM) implementation for numerical solutions and realistic head modelling. The use of second-order (quadratic) isoparametric elements and the recursive integration technique increase the accuracy in the solutions. Two new formulations are developed for the calculation of the transfer matrices to obtain the potential and magnetic field patterns using realistic head models. The formulations incorporate the use of the isolated problem approach for increased accuracy in solutions. If a personal computer is used for computations, each transfer matrix is calculated in 2.2 h. After this pre-computation period, solutions for arbitrary source configurations can be obtained in milliseconds for a realistic head model. A hybrid algorithm that uses snakes, morphological operations, region growing and thresholding is used for segmentation. The scalp, skull, grey matter, white matter and eyes are segmented from the multimodal magnetic resonance images and meshes for the corresponding surfaces are created. A mesh generation algorithm is developed for modelling the intersecting tissue compartments, such as eyes. To obtain more accurate results quadratic elements are used in the realistic meshes. The resultant BEM implementation provides more accurate forward problem solutions and more efficient calculations. Thus it can be the firm basis of the future inverse problem solutions.

EEG source imaging

OBJECTIVE

Electroencephalography (EEG) is an important tool for studying the temporal dynamics of the human brain's large-scale neuronal circuits. However, most EEG applications fail to capitalize on all of the data's available information, particularly that concerning the location of active sources in the brain. Localizing the sources of a given scalp measurement is only achieved by solving the so-called inverse problem. By introducing reasonable a priori constraints, the inverse problem can be solved and the most probable sources in the brain at every moment in time can be accurately localized.

METHODS AND RESULTS

Here, we review the different EEG source localization procedures applied during the last two decades. Additionally, we detail the importance of those procedures preceding and following source estimation that are intimately linked to a successful, reliable result. We discuss (1) the number and positioning of electrodes, (2) the varieties of inverse solution models and algorithms, (3) the integration of EEG source estimations with MRI data, (4) the integration of time and frequency in source imaging, and (5) the statistical analysis of inverse solution results.

CONCLUSIONS AND SIGNIFICANCE

We show that modern EEG source imaging simultaneously details the temporal and spatial dimensions of brain activity, making it an important and affordable tool to study the properties of cerebral, neural networks in cognitive and clinical neurosciences.

Effects of skull thickness, anisotropy, and inhomogeneity on forward EEG/ERP computations using a spherical three-dimensional resistor mesh model

Bone thickness, anisotropy, and inhomogeneity have been reported to induce important variations in electroencephalogram (EEG) scalp potentials. To study this effect, we used an original three-dimensional (3-D) resistor mesh model described in spherical coordinates, consisting of 67,464 elements and 22,105 nodes arranged in 36 different concentric layers. After validation of the model by comparison with the analytic solution, potential variations induced by geometric and electrical skull modifications were investigated at the surface in the dipole plane and along the dipole axis, for several eccentricities and bone thicknesses. The resistor mesh permits one to obtain various configurations, as local modifications are introduced very easily. This has allowed several head models to be designed to study the effects of skull properties (thickness, anisotropy, and heterogeneity) on scalp surface potentials. Results show a decrease of potentials in bone, depending on bone thickness, and a very small decrease through the scalp layer. Nevertheless, similar scalp potentials can be obtained using either a thick scalp layer and a thin skull layer, and vice versa. It is thus important to take into account skull and scalp thicknesses, because the drop of potential in bone depends on both. The use of three different layers for skull instead of one leads to small differences in potential values and patterns. In contrast, the introduction of a hole in the skull highly increases the maximum potential value (by a factor of 11.5 in our case), because of the absence of potential drop in the corresponding volume. The inverse solution without any a priori knowledge indicates that the model with the hole gives the largest errors in both position and dipolar moment. Our results indicate that the resistor mesh model can be used as a robust and user-friendly simulation tool in EEG or event-related potentials. It makes it possible to build up real head models directly from anatomic magnetic resonance imaging without tessellation, and is able to take into account head heterogeneities very simply by changing volume elements conductivity. Hum. Brain Mapping 21:84-95, 2004.

Construction of patient-specific surface models from MR images: application to bioelectromagnetism

Patient-specific geometric models are needed in many engineering problems. This work reports a novel software tool developed to construct individualized triangulated surface models from MR images. The program consists of three main parts: segmentation, triangulation and registration. The software tool was developed under the UNIX operating system. The application area demonstrated in this work is bioelectromagnetism but the program can be used as well in other engineering problems. The tool has been successfully applied in numerous cases, both for the thorax and the head.

Realistic human head model for EEG from both the geometry and conductivity aspects

This paper describes modelling and simulation of a had model which incorporates both the geometries and conductivities of the human head. It focuses on the inclusion of tissue conductivity inhomogeneity in a realistically-shaped head model, and investigates the impact of this inclusion on the potential distribution within the model. The result show that the impact, which has been neglected in realistic head models so far, is significant.

Validating the boundary element method for forward and inverse EEG computations in the presence of a hole in the skull

Holes in the skull may have a large influence on the EEG and ERP. Inverse source modeling techniques such as dipole fitting require an accurate volume conductor model. This model should incorporate holes if present, especially when either a neuronal generator or the electrodes are close to the hole, e.g., in case of a trephine hole in the upper part of the skull. The boundary element method (BEM) is at present the preferred method for inverse computations using a realistic head model, because of its efficiency and availability. Using a simulation approach, we have studied the accuracy of the BEM by comparing it to the analytical solution for a volume conductor without a hole, and to the finite difference method (FDM) for one with a hole. Furthermore, we have evaluated the influence of holes on the results of forward and inverse computations using the BEM. Without a hole and compared to the analytical model, a three-sphere BEM model was accurate up to 5-10%, while the corresponding FDM model had an error <0.5%. In the presence of a hole, the difference between the BEM and the FDM was, on average, 4% (1.3-11.4%). The FDM turned out to be very accurate if no hole is present. We believe that the difference between the BEM and the FDM represents the inaccuracy of the BEM. This inaccuracy in the BEM is very small compared to the effect that holes can have on the scalp potential (up to 450%). In regard to the large influence of holes on forward and inverse computations, we conclude that holes in the skull can be treated reliably by means of the BEM and should be incorporated in forward and inverse modeling.

A standardized boundary element method volume conductor model

OBJECTIVES

We used a 3-compartment boundary element method (BEM) model from an averaged magnetic resonance image (MRI) data set (Montreal Neurological Institute) in order to provide simple access to realistically shaped volume conductor models for source reconstruction, as compared to individually derived models. The electrode positions were transformed into the model's coordinate system, and the best fit dipole results were transformed back to the original coordinate system. The localization accuracy of the new approach was tested in a comparison with simulated data and with individual BEM models of epileptic spike data from several patients.

METHODS

The standard BEM model consisted of a total of 4770 nodes, which describe the smoothed cortical envelope, the outside of the skull, and the outside of the skin. The electrode positions were transformed to the model coordinate system by using 3-5 fiducials (nasion, left and right preauricular points, vertex, and inion). The transformation consisted of an averaged scaling factor and a rigid transformation (translation and rotation). The potential values at the transformed electrode positions were calculated by linear interpolation from the stored transfer matrix of the outer BEM compartment triangle net. After source reconstruction the best fit dipole results were transformed back into the original coordinate system by applying the inverse of the first transformation matrix.

RESULTS

Test-dipoles at random locations and with random orientations inside of a highly refined reference BEM model were used to simulate noise-free data. Source reconstruction results using a spherical and the standardized BEM volume conductor model were compared to the known dipole positions. Spherical head models resulted in mislocation errors at the base of the brain. The standardized BEM model was applied to averaged and unaveraged epileptic spike data from 7 patients. Source reconstruction results were compared to those achieved by 3 spherical shell models and individual BEM models derived from the individual MRI data sets. Similar errors to that evident with simulations were noted with spherical head models. Standardized and individualized BEM models were comparable.

CONCLUSIONS

This new approach to head modeling performed significantly better than a simple spherical shell approximation, especially in basal brain areas, including the temporal lobe. By using a standardized head for the BEM setup, it offered an easier and faster access to realistically shaped volume conductor models as compared to deriving specific models from individual 3-dimensional MRI data.

Dissociating top-down attentional control from selective perception and action

Research into the neural mechanisms of attention has revealed a complex network of brain regions that are involved in the execution of attention-demanding tasks. Recent advances in human neuroimaging now permit investigation of the elementary processes of attention being subserved by specific components of the brain's attention system. Here we describe recent studies of spatial selective attention that made use of positron emission tomography (PET), functional magnetic resonance imaging (fMRI), and event-related brain potentials (ERPs) to investigate the spatio-temporal dynamics of the attention-related neural activity. We first review the results from an event-related fMRI study that examined the neural mechanisms underlying top-down attentional control versus selective sensory perception. These results defined a fronto-temporal-parietal network involved in the control of spatial attention. Activity in these areas biased the neural activity in sensory brain structures coding the spatial locations of upcoming target stimuli, preceding a modulation of subsequent target processing in visual cortex. We then present preliminary evidence from a fast-rate event-related fMRI study of spatial attention that demonstrates how to disentangle the potentially overlapping hemodynamic responses elicited by temporally adjacent stimuli in studies of attentional control. Finally, we present new analyses from combined neuroimaging (PET) and event-related brain potential (ERP) studies that together reveal the timecourse of activation of brain regions implicated in attentional control and selective perception.

Boundary element method volume conductor models for EEG source reconstruction

OBJECTIVES

The boundary element method (BEM) approximates the different compartments of volume conductor models by closed triangle meshes with a limited number of nodes. The shielding effect of the weakly conducting skull layer of the human head leads to decreasing potential gradients from the inside to the outside. Thus, there may be an optimum distribution of nodes to the compartments for a given number of nodes corresponding to a fixed computational effort, resulting in improved accuracy as compared to standard uniform distributions.

METHODS

Spherical and realistically shaped surfaces are approximated by 500, 1000, 2000, and 3000 nodes, each leading to BEM models with 1500-9000 nodes in total. Electrodes are placed on extended 10/20-system positions. Potential distributions of test-dipoles at 4000 random positions within the innermost compartment are calculated. Dipoles are then fitted using 192 different models to find the optimum node distribution.

RESULTS

Fitted dipole positions for all BEM models are evaluated to show the dependency of the averaged and maximum localization errors on their node distributions. Dipoles close to the innermost boundary exhibit the largest localization errors, which mainly depend on the refinement of this compartment's triangle mesh.

CONCLUSIONS

More than 500 nodes per compartment are needed for reliable BEM models. For a state-of-the-art model consisting of 6000 nodes overall, the best model consists of 3000, 2000, and 1000 nodes from the inside to the outside.

Fast realistic modeling in bioelectromagnetism using lead-field interpolation

The practical use of realistic models in bioelectromagnetism is limited by the time-consuming amount of numerical calculations. We propose a method leading to much higher speed than currently available, and compatible with any kind of numerical methods (boundary elements (BEM), finite elements, finite differences). Illustrated with the BEM for EEG and MEG, it applies to ECG and MCG as well. The principle is two-fold. First, a Lead-Field matrix is calculated (once for all) for a grid of dipoles covering the brain volume. Second, any forward solution is interpolated from the pre-calculated Lead-Fields corresponding to grid dipoles near the source. Extrapolation is used for shallow sources falling outside the grid. Three interpolation techniques were tested: trilinear, second-order Bézier (Bernstein polynomials), and 3D spline. The trilinear interpolation yielded the highest speed gain, with factors better than x10,000 for a 9,000-triangle BEM model. More accurate results could be obtained with the Bézier interpolation (speed gain approximately 1,000), which, combined with a 8-mm step grid, lead to intrinsic localization and orientation errors of only 0.2 mm and 0.2 degrees. Further improvements in MEG could be obtained by interpolating only the contribution of secondary currents. Cropping grids by removing shallow points lead to a much better estimation of the dipole orientation in EEG than when solving the forward problem classically, providing an efficient alternative to locally refined models. This method would show special usefulness when combining realistic models with stochastic inverse procedures (simulated annealing, genetic algorithms) requiring many forward calculations.

Comparison study of different head model structures with homogeneous/inhomogeneous conductivity

Most of the human head models used in dipole localisation research, which have been reported in the literature to date, assume a simplified cranial structure wherein the head is modelled as a set of distinct homogenous tissue compartments. The inherent inhomogeneity of the tissues has so far been ignored in these models due to the difficulties involved in obtaining the conductivity characteristics with sufficiently high enough spatial resolution throughout the head. A technique for developing an inhomogeneous head model based on the generation of pseudo-conductivity values from the existing but sparse conductivity values is proposed in this paper. Comparative studies are conducted on different model structures and different mechanisms for generating the pseudo conductivities. An evaluation of the results of these studies as reported in this paper, shows that contrary to current simplifying assumptions, tissue inhomogeneity has a major influence on the computation of electrical potential distributions in the head.

Improving the accuracy of the boundary element method by the use of second-order interpolation functions

The boundary element method (BEM) is a widely used method to solve biomedical electromagnetic volume conduction problems. The commonly used formulation of this method uses constant interpolation functions for the potential and flat triangular surface elements. Linear interpolation for the potential on a flat triangular mesh turned out to yield a better accuracy. In this paper, we introduce quadratic interpolation functions for the potential and quadratically curved surface elements, resulting from second-order spatial interpolation. Theoretically, this results in an accuracy that is inversely proportional to the third power of element size. The method is tested on a four concentric sphere geometry, representative for electroencephalogram modeling, and compared to previous solutions of this problem in literature. In addition, a cylindrical test configuration is used. We conclude that the use of quadratic interpolation functions for the potential and of quadratically curved surface elements in BEM results in a significant increase in accuracy and in some cases even a reduction of the computation time with the same number of nodes involved in the calculations.

Effect of conductivity uncertainties and modeling errors on EEG source localization using a 2-D model

This paper presents a sensitivity study of electroencephalography-based source localization due to errors in the head-tissue conductivities and to errors in modeling the conductivity variation inside the brain and scalp. The study is conducted using a two-dimensional (2-D) finite element model obtained from a magnetic resonance imaging (MRI) scan of a head cross section. The effect of uncertainty in the following tissues is studied: white matter, gray matter, cerebrospinal fluid (CSF), skull, and fat. The distribution of source location errors, assuming a single-dipole source model, is examined in detail for different dipole locations over the entire brain region. We also present a detailed analysis of the effect of conductivity on source localization for a four-layer cylinder model and a four-layer sphere model. These two simple models provide insight into how the effect of conductivity on boundary potential translates into source location errors, and also how errors in a 2-D model compare to errors in a three-dimensional model. Results presented in this paper clearly point to the following conclusion: unless the conductivities of the head tissues and the distribution of these tissues throughout the head are modeled accurately, the goal of achieving localization accuracy to within a few millimeters is unattainable.

Computational aspects of finite element modeling in EEG source localization

A comparison is made of two different implementations of the finite element method (FEM) for calculating the potential due to dipole sources in electroencephalography (EEG). In one formulation (the direct method) the total potential is the unknown that is solved for and the dipole source is directly incorporated into the model. In the second formulation (the subtraction method) the unknown is the difference between the total potential and the potential due to the same dipole in an infinite region of homogeneous conductivity, corresponding to the region where the dipole is located. Both methods have the same FEM system matrix. However, the subtraction method requires an additional calculation of flux integrations along the edges of the elements in the computation of the right-hand side (RHS) vector. It is shown that the subtraction method is usually more accurate in the forward modeling, provided the flux integrations are computed accurately. Errors in calculating the flux integrations may result in large errors in the forward solution due to the ill-conditioned nature of the FEM system matrix caused by the Neumann boundary condition. To minimize the errors, closed-form expressions for the flux integrations are used for both linear and quadratic triangular elements. It is also found that FEM forward modeling errors may cause false extrema in the least-square objective function obtained from the boundary potential, near boundaries between media of differing conductivity. Multiple initial guesses help eliminate the possibility of the solution getting trapped in these false extrema.

A systematic evaluation of the spherical model accuracy in EEG dipole localization

This paper presents a study of the intrinsic localization error bias due to the use of a spherical geometry model on EEG simulated data obtained from realistically shaped models. About 2000 dipoles were randomly chosen on the segmented cortex surface of a particular subject. Forward calculations were performed using a uniformly meshed model for each dipole located at a depth greater than 20 mm below the brain surface, and locally refined models were used for shallower dipoles. Inverse calculations were performed using four different spherical models and another uniformly meshed model. It was found that the best spherical model lead to localization errors of 5-6 mm in the upper part of the head, and of 15-25 mm in the lower part. The influence of the number of electrodes upon this intrinsic bias was also studied. It was found that using 32 electrodes instead of 19 improves the localization by 2.7 mm on average, while using 63 instead of 32 electrodes lead to improvements of less than 1 mm. Finally, simulations involving two simultaneously active dipoles (one in the vicinity of each auditory cortex) show localization errors increasing by about 2-3 mm.

Improved dipole localization using local mesh refinement of realistic head geometries: an EEG simulation study

A systematic evaluation of dipole localization accuracy using the boundary element method is presented. EEG simulations are carried out with dipoles located in the right parietal and temporal regions of the head. Uniformly meshed and locally refined head models are considered in both spherical and realistic geometries. An initial study determines the influence upon the localization accuracy of the dipole depth below the brain surface, of its orientation (radial and tangential), and of the global and local mesh densities. Simulated potential data are computed analytically in the spherical case, and numerically using a very fine (locally refined) model in the realistic case. Results in both geometries show that in order to get localization errors of about 2-4 mm, uniformly meshed models may be used for dipoles located at depths greater than 20 mm, whereas locally refined models should be used for shallower dipoles. Two other studies show how the localization accuracy depends upon the size of the local refinement area and upon the number of electrodes (19, 32, 63). Results show that a large number of electrodes brings significant improvements, especially for shallow and tangential dipoles.