Computational aspects of finite element modeling in EEG source localization

Semi-analytic three-shell forward calculation for magnetoencephalography

In previous studies, the magnetic lead field theorem in the quasi-static approximation was derived and used for the development of a method for the forward problem of MEG. It was applied and tested on a single-shell model of the human head and the question whether one shell is adequate enough for the calculation of the magnetic field is the main reason for this study. This forward method is based on the fundamental concept that one can calculate the lead field for MEG by decomposing it into two parts: the lead field of an arbitrary volume conductor that is already known and the gradient of basis functions that have to be harmonic, here derived from spherical harmonics. The problem then is reduced to evaluating the coefficients found in the basis functions. In this research we aim to improve the accuracy of the forward model, hence improving the localization accuracy in inverse methods by introducing a more detailed realistic head model. We here generalize the algorithm developed for a single-shell volume conductor to a three-shell volume conductor representing the brain, the skull and the skin with homogenous and isotropic conductivities in realistic ratios. The expansion to three shells could be tested as the three-shell algorithm is approaching the single-shell with high accuracy in special cases where three-shell solutions can also be calculated using a single-shell solution, especially for higher levels of expansion. The deviation in the calculation of the lead field is also evaluated when using three shells with realistic conductivities. The magnetic field turned out to differ to an important measurable extend in particular for deeper sources, making the three-shell algorithm substantially more accurate for these dipole locations.

A hybrid boundary element-finite element approach for solving the EEG forward problem in brain modeling

This article introduces a hybrid BE-FE method for solving the EEG forward problem, leveraging the strengths of both the Boundary Element Method (BEM) and Finite Element Method (FEM). FEM accurately models complex and anisotropic tissue properties for realistic head geometries, while BEM excels in handling isotropic tissue regions and dipolar sources efficiently. The proposed hybrid method divides regions into homogeneous boundary element (BE) regions that include sources and heterogeneous anisotropic finite element (FE) regions. So, BEM models the brain, including dipole sources, and FEM models other head layers. Validation includes inhomogeneous isotropic/anisotropic three- and four-layer spherical head models, and a four-layer MRI-based realistic head model. Results for six dipole eccentricities and two orientations are computed using BEM, FEM, and hybrid BE-FE method. Statistical analysis, comparing error criteria of RDM and MAG, reveals notable improvements using the hybrid FE-BE method. In the spherical head model, the hybrid BE-FE method compared with FEM demonstrates enhancements of at least 1.05 and 38.31% in RDM and MAG criteria, respectively. Notably, in the anisotropic four-layer head model, improvements reach a maximum of 88.3% for RDM and 93.27% for MAG over FEM. Moreover, in the anisotropic four-layer realistic head model, the proposed hybrid method exhibits 55.4% improvement in RDM and 89.3% improvement in MAG compared to FEM. These findings underscore the proposed method is a promising approach for solving the realistic EEG forward problems, advancing neuroimaging techniques and enhancing understanding of brain function.

New Strategy for Finite Element Mesh Generation for Accurate Solutions of Electroencephalography Forward Problems

The finite element method (FEM) is a numerical method that is often used for solving electroencephalography (EEG) forward problems involving realistic head models. In this study, FEM solutions obtained using three different mesh structures, namely coarse, densely refined, and adaptively refined meshes, are compared. The simulation results showed that the accuracy of FEM solutions could be significantly enhanced by adding a small number of elements around regions with large estimated errors. Moreover, it was demonstrated that the adaptively refined regions were always near the current dipole sources, suggesting that selectively generating additional elements around the cortical surface might be a new promising strategy for more efficient FEM-based EEG forward analysis.

The Discontinuous Galerkin Finite Element Method for Solving the MEG and the Combined MEG/EEG Forward Problem

In Electro- (EEG) and Magnetoencephalography (MEG), one important requirement of source reconstruction is the forward model. The continuous Galerkin finite element method (CG-FEM) has become one of the dominant approaches for solving the forward problem over the last decades. Recently, a discontinuous Galerkin FEM (DG-FEM) EEG forward approach has been proposed as an alternative to CG-FEM (Engwer et al., 2017). It was shown that DG-FEM preserves the property of conservation of charge and that it can, in certain situations such as the so-called skull leakages, be superior to the standard CG-FEM approach. In this paper, we developed, implemented, and evaluated two DG-FEM approaches for the MEG forward problem, namely a conservative and a non-conservative one. The subtraction approach was used as source model. The validation and evaluation work was done in statistical investigations in multi-layer homogeneous sphere models, where an analytic solution exists, and in a six-compartment realistically shaped head volume conductor model. In agreement with the theory, the conservative DG-FEM approach was found to be superior to the non-conservative DG-FEM implementation. This approach also showed convergence with increasing resolution of the hexahedral meshes. While in the EEG case, in presence of skull leakages, DG-FEM outperformed CG-FEM, in MEG, DG-FEM achieved similar numerical errors as the CG-FEM approach, i.e., skull leakages do not play a role for the MEG modality. In particular, for the finest mesh resolution of 1 mm sources with a distance of 1.59 mm from the brain-CSF surface, DG-FEM yielded mean topographical errors (relative difference measure, RDM%) of 1.5% and mean magnitude errors (MAG%) of 0.1% for the magnetic field. However, if the goal is a combined source analysis of EEG and MEG data, then it is highly desirable to employ the same forward model for both EEG and MEG data. Based on these results, we conclude that the newly presented conservative DG-FEM can at least complement and in some scenarios even outperform the established CG-FEM approaches in EEG or combined MEG/EEG source analysis scenarios, which motivates a further evaluation of DG-FEM for applications in bioelectromagnetism.

Forward and inverse effects of the complete electrode model in neonatal EEG

The effect of the complete electrode model on electroencephalography forward and inverse computations is explored. A realistic neonatal head model, including a skull structure with fontanels and sutures, is used. The electrode and skull modeling differences are analyzed and compared with each other. The results suggest that the complete electrode model can be considered as an integral part of the outer head model. To achieve optimal source localization results, accurate electrode modeling might be necessary.

This paper investigates finite element method-based modeling in the context of neonatal electroencephalography (EEG). In particular, the focus lies on electrode boundary conditions. We compare the complete electrode model (CEM) with the point electrode model (PEM), which is the current standard in EEG. In the CEM, the voltage experienced by an electrode is modeled more realistically as the integral average of the potential distribution over its contact surface, whereas the PEM relies on a point value. Consequently, the CEM takes into account the subelectrode shunting currents, which are absent in the PEM. In this study, we aim to find out how the electrode voltage predicted by these two models differ, if standard size electrodes are attached to a head of a neonate. Additionally, we study voltages and voltage variation on electrode surfaces with two source locations: 1) next to the C6 electrode and 2) directly under the Fz electrode and the frontal fontanel. A realistic model of a neonatal head, including a skull with fontanels and sutures, is used. Based on the results, the forward simulation differences between CEM and PEM are in general small, but significant outliers can occur in the vicinity of the electrodes. The CEM can be considered as an integral part of the outer head model. The outcome of this study helps understanding volume conduction of neonatal EEG, since it enlightens the role of advanced skull and electrode modeling in forward and inverse computations.

NEW & NOTEWORTHY The effect of the complete electrode model on electroencephalography forward and inverse computations is explored. A realistic neonatal head model, including a skull structure with fontanels and sutures, is used. The electrode and skull modeling differences are analyzed and compared with each other. The results suggest that the complete electrode model can be considered as an integral part of the outer head model. To achieve optimal source localization results, accurate electrode modeling might be necessary.

Electroencephalography (EEG) forward modeling via H(div) finite element sources with focal interpolation

The goal of this study is to develop focal, accurate and robust finite element method (FEM) based approaches which can predict the electric potential on the surface of the computational domain given its structure and internal primary source current distribution. While conducting an EEG evaluation, the placement of source currents to the geometrically complex grey matter compartment is a challenging but necessary task to avoid forward errors attributable to tissue conductivity jumps. Here, this task is approached via a mathematically rigorous formulation, in which the current field is modeled via divergence conforming H(div) basis functions. Both linear and quadratic functions are used while the potential field is discretized via the standard linear Lagrangian (nodal) basis. The resulting model includes dipolar sources which are interpolated into a random set of positions and orientations utilizing two alternative approaches: the position based optimization (PBO) and the mean position/orientation (MPO) method. These results demonstrate that the present dipolar approach can reach or even surpass, at least in some respects, the accuracy of two classical reference methods, the partial integration (PI) and St. Venant (SV) approach which utilize monopolar loads instead of dipolar currents.

Particle Swarm Optimization Applied to EEG Source Localization of Somatosensory Evoked Potentials

One of the most important steps in presurgical diagnosis of medically intractable epilepsy is to find the precise location of the epileptogenic foci. Electroencephalography (EEG) is a noninvasive tool commonly used at epilepsy surgery centers for presurgical diagnosis. In this paper, a modified particle swarm optimization (MPSO) method is used to solve the EEG source localization problem. The method is applied to noninvasive EEG recording of somatosensory evoked potentials (SEPs) for a healthy subject. A 1 mm hexahedra finite element volume conductor model of the subject's head was generated using T1-weighted magnetic resonance imaging data. Special consideration was made to accurately model the skull and cerebrospinal fluid. An exhaustive search pattern and the MPSO method were then applied to the peak of the averaged SEP data and both identified the same region of the somatosensory cortex as the location of the SEP source. A clinical expert independently identified the expected source location, further corroborating the source analysis methods. The MPSO converged to the global minima with significantly lower computational complexity compared to the exhaustive search method that required almost 3700 times more evaluations.

Effects of sutures and fontanels on MEG and EEG source analysis in a realistic infant head model

In infants, the fontanels and sutures as well as conductivity of the skull influence the volume currents accompanying primary currents generated by active neurons and thus the associated electroencephalography (EEG) and magnetoencephalography (MEG) signals. We used a finite element method (FEM) to construct a realistic model of the head of an infant based on MRI images. Using this model, we investigated the effects of the fontanels, sutures and skull conductivity on forward and inverse EEG and MEG source analysis. Simulation results show that MEG is better suited than EEG to study early brain development because it is much less sensitive than EEG to distortions of the volume current caused by the fontanels and sutures and to inaccurate estimates of skull conductivity. Best results will be achieved when MEG and EEG are used in combination.

Cortical sources of ERP in prosaccade and antisaccade eye movements using realistic source models

The cortical sources of event-related-potentials (ERP) using realistic source models were examined in a prosaccade and antisaccade procedure. College-age participants were presented with a preparatory interval and a target that indicated the direction of the eye movement that was to be made. In some blocks a cue was given in the peripheral location where the target was to be presented and in other blocks no cue was given. In Experiment 1 the prosaccade and antisaccade trials were presented randomly within a block; in Experiment 2 procedures were compared in which either prosaccade and antisaccade trials were mixed in the same block, or trials were presented in separate blocks with only one type of eye movement. There was a central negative slow wave occurring prior to the target, a slow positive wave over the parietal scalp prior to the saccade, and a parietal spike potential immediately prior to saccade onset. Cortical source analysis of these ERP components showed a common set of sources in the ventral anterior cingulate and orbital frontal gyrus for the presaccadic positive slow wave and the spike potential. In Experiment 2 the same cued- and non-cued blocks were used, but prosaccade and antisaccade trials were presented in separate blocks. This resulted in a smaller difference in reaction time between prosaccade and antisaccade trials. Unlike the first experiment, the central negative slow wave was larger on antisaccade than on prosaccade trials, and this effect on the ERP component had its cortical source primarily in the parietal and mid-central cortical areas contralateral to the direction of the eye movement. These results suggest that blocked prosaccade and antisaccade trials results in preparatory or set effects that decreases reaction time, eliminates some cueing effects, and is based on contralateral parietal-central brain areas.

Modeling of the human skull in EEG source analysis

We used computer simulations to investigate finite element models of the layered structure of the human skull in EEG source analysis. Local models, where each skull location was modeled differently, and global models, where the skull was assumed to be homogeneous, were compared to a reference model, in which spongy and compact bone were explicitly accounted for. In both cases, isotropic and anisotropic conductivity assumptions were taken into account. We considered sources in the entire brain and determined errors both in the forward calculation and the reconstructed dipole position. Our results show that accounting for the local variations over the skull surface is important, whereas assuming isotropic or anisotropic skull conductivity has little influence. Moreover, we showed that, if using an isotropic and homogeneous skull model, the ratio between skin/brain and skull conductivities should be considerably lower than the commonly used 80:1. For skull modeling, we recommend (1) Local models: if compact and spongy bone can be identified with sufficient accuracy (e.g., from MRI) and their conductivities can be assumed to be known (e.g., from measurements), one should model these explicitly by assigning each voxel to one of the two conductivities, (2) Global models: if the conditions of (1) are not met, one should model the skull as either homogeneous and isotropic, but with considerably higher skull conductivity than the usual 0.0042 S/m, or as homogeneous and anisotropic, but with higher radial skull conductivity than the usual 0.0042 S/m and a considerably lower radial:tangential conductivity anisotropy than the usual 1:10.

Accuracy and run-time comparison for different potential approaches and iterative solvers in finite element method based EEG source analysis

Accuracy and run-time play an important role in medical diagnostics and research as well as in the field of neuroscience. In Electroencephalography (EEG) source reconstruction, a current distribution in the human brain is reconstructed noninvasively from measured potentials at the head surface (the EEG inverse problem). Numerical modeling techniques are used to simulate head surface potentials for dipolar current sources in the human cortex, the so-called EEG forward problem.In this paper, the efficiency of algebraic multigrid (AMG), incomplete Cholesky (IC) and Jacobi preconditioners for the conjugate gradient (CG) method are compared for iteratively solving the finite element (FE) method based EEG forward problem. The interplay of the three solvers with a full subtraction approach and two direct potential approaches, the Venant and the partial integration method for the treatment of the dipole singularity is examined. The examination is performed in a four-compartment sphere model with anisotropic skull layer, where quasi-analytical solutions allow for an exact quantification of computational speed versus numerical error. Specifically-tuned constrained Delaunay tetrahedralization (CDT) FE meshes lead to high accuracies for both the full subtraction and the direct potential approaches. Best accuracies are achieved by the full subtraction approach if the homogeneity condition is fulfilled. It is shown that the AMG-CG achieves an order of magnitude higher computational speed than the CG with the standard preconditioners with an increasing gain factor when decreasing mesh size. Our results should broaden the application of accurate and fast high-resolution FE volume conductor modeling in source analysis routine.

A full subtraction approach for finite element method based source analysis using constrained Delaunay tetrahedralisation

A mathematical dipole is widely used as a model for the primary current source in electroencephalography (EEG) source analysis. In the governing Poisson-type differential equation, the dipole leads to a singularity on the right-hand side, which has to be treated specifically. In this paper, we will present a full subtraction approach where the total potential is divided into a singularity and a correction potential. The singularity potential is due to a dipole in an infinite region of homogeneous conductivity. The correction potential is computed using the finite element (FE) method. Special care is taken in order to evaluate the right-hand side integral appropriately with the objective of achieving highest possible convergence order for linear basis functions. Our new approach allows the construction of transfer matrices for fast computation of the inverse problem for anisotropic volume conductors. A constrained Delaunay tetrahedralisation (CDT) approach is used for the generation of high-quality FE meshes. We validate the new approach in a four-layer sphere model with a highly conductive cerebrospinal fluid (CSF) and an anisotropic skull compartment. For radial and tangential sources with eccentricities up to 1 mm below the CSF compartment, we achieve a maximal relative error of 0.71% in a CDT-FE model with 360 k nodes which is not locally refined around the source singularity and therefore useful for arbitrary dipole locations. The combination of the full subtraction approach with the high quality CDT meshes leads to accuracies that, to the best of the author's knowledge, have not yet been presented before.

Cortical source localization of infant cognition

Neuroimaging techniques such as positron emission topography (PET) and functional magnetic resonance imaging (fMRI) have been utilized with older children and adults to identify cortical sources of perceptual and cognitive processes. However, due to practical and ethical concerns, these techniques cannot be routinely applied to infant participants. An alternative to such neuroimaging techniques appropriate for use with infant participants is high-density electroencephalogram (EEG) recording and cortical source localization techniques. The current article provides an overview of a method developed for such analyses. The method consists of four steps: (1) recording high-density (e.g., 128-channel) EEG. (2) Analysis of individual participant raw segmented data with independent component analysis (ICA). (3) Estimation of equivalent current dipoles (ECDs) that represent cortical sources for the observed ICA component clusters. (4) Calculation of component activations in relation to experimental factors. We discuss an example of research applying this technique to investigate the development of visual attention and recognition memory. We also describe the application of "realistic head modeling" to address some of the current limitations of infant cortical source localization.

A physiologically plausible spatio-temporal model for EEG signals recorded with intracerebral electrodes in human partial epilepsy

Stereoelectroencephalography (depth-EEG signals) is a presurgical investigation technique of drug-resistant partial epilepsy, in which multiple sensor intracerebral electrodes are used to directly record brain electrical activity. In order to interpret depth-EEG signals, we developed an extended source model which connects two levels of representation: (1) a distributed current dipole model which describes the spatial distribution of neuronal sources; (2) a model of coupled neuronal populations which describes their temporal dynamics. From this extended source model, depth-EEG signals were simulated from the forward solution at each electrode sensor located inside the brain. Results showed that realistic transient epileptiform activities (spikes) are obtained under specific conditions in the model in terms of degree of coupling between neuronal populations and spatial extent of the source. In particular, the cortical area involved in the generation of epileptic spikes was estimated to vary from 18 to 25 cm2, for brain conductivity values ranging from 30 to 35 x 10(-5) S/mm, for high coupling degree between neuronal populations and for a volume conductor model that accounts for the three main tissues of the head (brain, skull, and scalp). This study provides insight into the relationship between spatio-temporal properties of cortical neuronal sources and depth-EEG signals.

Geometry-adapted hexahedral meshes improve accuracy of finite-element-method-based EEG source analysis

Mesh generation in finite-element- (FE) method-based electroencephalography (EEG) source analysis generally influences greatly the accuracy of the results. It is thus important to determine a meshing strategy well adopted to achieve both acceptable accuracy for potential distributions and reasonable computation times and memory usage. In this paper, we propose to achieve this goal by smoothing regular hexahedral finite elements at material interfaces using a node-shift approach. We first present the underlying theory for two different techniques for modeling a current dipole in FE volume conductors, a subtraction and a direct potential method. We then evaluate regular and smoothed elements in a four-layer sphere model for both potential approaches and compare their accuracy. We finally compute and visualize potential distributions for a tangentially and a radially oriented source in the somatosensory cortex in regular and geometry-adapted three-compartment hexahedra FE volume conductor models of the human head using both the subtraction and the direct potential method. On the average, node-shifting reduces both topography and magnitude errors by more than a factor of 2 for tangential and 1.5 for radial sources for both potential approaches. Nevertheless, node-shifting has to be carried out with caution for sources located within or close to irregular hexahedra, because especially for the subtraction method extreme deformations might lead to larger overall errors. With regard to realistic volume conductor modeling, node-shifted hexahedra should thus be used for the skin and skull compartments while we would not recommend deforming elements at the grey and white matter surfaces.

An EEG forward solution using second-order anisotropic FEM

A 2(nd) order finite element method (FEM) algorithm has been developed to solve the anisotropic electroencephalogram (EEG) forward problem and has been evaluated by comparing with analytic solutions in a multispheres concentric head model. The present simulation study indicates that the 2(nd) order FEM provides substantially enhanced numerical accuracy and computational efficiency, as compared with the 1(st) order FEM for comparable numbers of tetrahedron elements.

Computational aspects of the EEG forward problem solution for real head model using finite element method

The real head model has been used for the accurate scalp potential modeling. The realistic shapes of head tissues were derived from a set of 2-D magnetic resonance images (MRI) by extracting surface boundaries for the major tissues such as the scalp, the skull, the cerebrospinal fluid (CSF), the white matter, and the gray matter. From boundary data a 3-D volume generic head model has been constructed and a mesh for an arbitrary complexity head shape has been generated for finite-element method (FEM) modeling. This paper first addresses the use of this realistic finite elements head model to solve the EEG forward problem. The accuracy and computational time of the potential modeling are then examined.

Methods and evaluations of MRI content-adaptive finite element mesh generation for bioelectromagnetic problems

In studying bioelectromagnetic problems, finite element analysis (FEA) offers several advantages over conventional methods such as the boundary element method. It allows truly volumetric analysis and incorporation of material properties such as anisotropic conductivity. For FEA, mesh generation is the first critical requirement and there exist many different approaches. However, conventional approaches offered by commercial packages and various algorithms do not generate content-adaptive meshes (cMeshes), resulting in numerous nodes and elements in modelling the conducting domain, and thereby increasing computational load and demand. In this work, we present efficient content-adaptive mesh generation schemes for complex biological volumes of MR images. The presented methodology is fully automatic and generates FE meshes that are adaptive to the geometrical contents of MR images, allowing optimal representation of conducting domain for FEA. We have also evaluated the effect of cMeshes on FEA in three dimensions by comparing the forward solutions from various cMesh head models to the solutions from the reference FE head model in which fine and equidistant FEs constitute the model. The results show that there is a significant gain in computation time with minor loss in numerical accuracy. We believe that cMeshes should be useful in the FEA of bioelectromagnetic problems.

EEG human head modelling based on heterogeneous tissue conductivity

This paper studies the effect of heterogeneous tissue conductivity in a human head model for the EEG forward problem. Firstly, the tissue heterogeneity in conductivity was characterised from measured data in the literature. Then a method was developed to include this feature in modelling. Finally, the effect of tissue heterogeneity on EEG signals was studied. Based on these studies the paper concludes that the inclusion of tissue heterogeneity is significant in accurate head modelling for the EEG problem.

Generic head models for atlas-based EEG source analysis

We describe a method for using a generic head model, in the form of an anatomical atlas, to produce EEG source localizations. The atlas is fitted to the subject by a nonrigid warp using a set of surface landmarks. The warped atlas is used to compute a finite element model (FEM) of the forward mapping or lead-fields between neural current generators and the EEG electrodes. These lead-fields are used to localize current sources from the subject's EEG data and the sources are then mapped back to the anatomical atlas. This approach provides a mechanism for comparing source localizations across subjects in an atlas-based coordinate system, which can be used in the large fraction of EEG studies in which MR images are not available. The Montreal brain atlas was used as the reference anatomical atlas and 10 individual MR volumes were used to evaluate the method. The atlas was fitted to each subject's head by a thin-plate-spline (TPS) warp. The spatial locations of a generic 155-electrode configuration were used to constrain the warp. For the purposes of evaluation, dipolar sources were placed on the inner cortical surface in the atlas geometry and transferred to each subject's brain space using a polynomial warp. The parameters of the warp were computed using an intensity-based matching of the atlas and subject brains, thus ensuring that the sources were placed at approximately the same anatomical location in each case. Data were simulated in the subject geometry and a dipole fit was performed on these data using an FEM of the TPS warped atlas. The source positions found in the warped atlas were transferred back to the original atlas and compared to the original position. Sources were simulated at 972 locations evenly distributed over the inner cortical surface of the atlas. The mean error over all 10 subjects was 8.1 mm in the subject space and 15.2 mm in the atlas space. In comparison, using an affine transformation of the electrodes into atlas space and an FEM model generated from the atlas produced mean errors of 22.3 mm in subject space and 19.6 mm in atlas space. With a standard three-shell spherical model the errors were 27.2 mm in the subject space and 34.7 mm when mapped to atlas space.

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.

Representation of bioelectric current sources using Whitney elements in the finite element method

Bioelectric current sources of magneto- and electroencephalograms (MEG, EEG) are usually modelled with discrete delta-function type current dipoles, despite the fact that the currents in the brain are naturally continuous throughout the neuronal tissue. In this study, we represent bioelectric current sources in terms of Whitney-type elements in the finite element method (FEM) using a tetrahedral mesh. The aim is to study how well the Whitney elements can reproduce the potential and magnetic field patterns generated by a point current dipole in a homogeneous conducting sphere. The electric potential is solved for a unit sphere model with isotropic conductivity and magnetic fields are calculated for points located on a cap outside the sphere. The computed potential and magnetic field are compared with analytical solutions for a current dipole. Relative difference measures between the FEM and analytical solutions are less than 1%, suggesting that Whitney elements as bioelectric current sources are able to produce the same potential and magnetic field patterns as the point dipole sources.

Mapping human brain function with MEG and EEG: methods and validation

We survey the field of magnetoencephalography (MEG) and electroencephalography (EEG) source estimation. These modalities offer the potential for functional brain mapping with temporal resolution in the millisecond range. However, the limited number of spatial measurements and the ill-posedness of the inverse problem present significant limits to our ability to produce accurate spatial maps from these data without imposing major restrictions on the form of the inverse solution. Here we describe approaches to solving the forward problem of computing the mapping from putative inverse solutions into the data space. We then describe the inverse problem in terms of low dimensional solutions, based on the equivalent current dipole (ECD), and high dimensional solutions, in which images of neural activation are constrained to the cerebral cortex. We also address the issue of objective assessment of the relative performance of inverse procedures by the free-response receiver operating characteristic (FROC) curve. We conclude with a discussion of methods for assessing statistical significance of experimental results through use of the bootstrap for determining confidence regions in dipole-fitting methods, and random field (RF) and permutation methods for detecting significant activation in cortically constrained imaging studies.

A second-order finite element algorithm for solving the three-dimensional EEG forward problem

A finite element algorithm has been developed to solve the electroencephalogram (EEG) forward problem. A new computationally efficient approach to calculate the stiffness matrix of second-order tetrahedral elements has been developed for second-order tetrahedral finite element models. The present algorithm has been evaluated by means of computer simulations, by comparing with analytic solutions in a multi-spheres concentric head model. The developed finite element method (FEM) algorithm has also been applied to address questions of interest in the EEG forward problem. The present simulation study indicates that the second-order FEM provides substantially enhanced numerical accuracy and computational efficiency, as compared with the first-order FEM for comparable numbers of tetrahedral elements. The anisotropic conductivity distribution of the head tissue can be taken into account in the present FEM algorithm. The effects of dipole eccentricity, size of finite elements and local mesh refinement on solution accuracy are also addressed in the present simulation study.

The magnetic lead field theorem in the quasi-static approximation and its use for magnetoencephalography forward calculation in realistic volume conductors

The equation for the magnetic lead field for a given magnetoencephalography (MEG) channel is well known for arbitrary frequencies omega but is not directly applicable to MEG in the quasi-static approximation. In this paper we derive an equation for omega = 0 starting from the very definition of the lead field instead of using Helmholtz's reciprocity theorems. The results are (a) the transpose of the conductivity times the lead field is divergence-free, and (b) the lead field differs from the one in any other volume conductor by a gradient of a scalar function. Consequently, for a piecewise homogeneous and isotropic volume conductor, the lead field is always tangential at the outermost surface. Based on this theoretical result, we formulated a simple and fast method for the MEG forward calculation for one shell of arbitrary shape: we correct the corresponding lead field for a spherical volume conductor by a superposition of basis functions, gradients of harmonic functions constructed here from spherical harmonics, with coefficients fitted to the boundary conditions. The algorithm was tested for a prolate spheroid of realistic shape for which the analytical solution is known. For high order in the expansion, we found the solutions to be essentially exact and for reasonable accuracies much fewer multiplications are needed than in typical implementations of the boundary element methods. The generalization to more shells is straightforward.

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.

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

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

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.

Influence of skull anisotropy for the forward and inverse problem in EEG: simulation studies using FEM on realistic head models

For the sake of realism in the description of conduction from primary neural currents to scalp potentials, we investigated the influence of skull anisotropy on the forward and inverse problems in brain functional imaging with EEG. At present, all methods available for cortical imaging assume a spherical geometry, or when using realistic head shapes do not consider the anisotropy of head tissues. However, to our knowledge, no study relates the implication of this simplifying hypothesis on the spatial resolution of EEG for source imaging. In this paper, a method using finite elements in a realistic head geometry is implemented and validated. The influence of erroneous conductivity values for the head tissues is presented, and results show that the conductivities of the brain and the skull in the radial orientation are the most critical ones. In the inverse problem, this influence has been evaluated with simulations using a distributed source model with a comparison of two regularization techniques, with the isotropic model working on data sets produced by a nonisotropic model. Regularization with minimum norm priors produces source images with spurious activity, meaning that the errors in the head model totally annihilate any localization ability. But nonlinear regularization allows the accurate recovery of simultaneous spots of activity, while the restoration of very close active regions is profoundly disabled by errors in the head model. We conclude that for robust cortical source imaging with EEG, a realistic head model taking anisotropy of tissues into account should be used.

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.