The finite element method for a realistic head model of electrical brain activities: preliminary results

Electrical source localization by LORETA in patients with epilepsy: Confirmation by postoperative MRI

BACKGROUND

Few studies have been conducted that have compared electrical source localization (ESL) results obtained by analyzing ictal patterns in scalp electroencephalogram (EEG) with the brain areas that are found to be responsible for seizures using other brain imaging techniques. Additionally, adequate studies have not been performed to confirm the accuracy of ESL methods.

MATERIALS AND METHODS

In this study, ESL was conducted using LORETA (Low Resolution Brain Electromagnetic Tomography) in 9 patients with lesions apparent on magnetic resonance imaging (MRI) and in 6 patients who did not exhibit lesions on their MRIs. EEGs of patients who underwent surgery for epilepsy and had follow-ups for at least 1 year after operations were analyzed for ictal spike, rhythmic, paroxysmal fast, and obscured EEG activities. Epileptogenic zones identified in postoperative MRIs were then compared with localizations obtained by LORETA model we employed.

RESULTS

We found that brain areas determined via ESL were in concordance with resected brain areas for 13 of the 15 patients evaluated, and those 13 patients were post-operatively determined as being seizure-free.

CONCLUSION

ESL, which is a noninvasive technique, may contribute to the correct delineation of epileptogenic zones in patients who will eventually undergo surgery to treat epilepsy, (regardless of neuroimaging status). Moreover, ESL may aid in deciding on the number and localization of intracranial electrodes to be used in patients who are candidates for invasive recording.

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.

Optimal spacing of surface electrode arrays for brain-machine interface applications

Brain-machine interfaces (BMIs) use signals recorded directly from the brain to control an external device, such as a computer cursor or a prosthetic limb. These control signals have been recorded from different levels of the brain, from field potentials at the scalp or cortical surface to single neuron action potentials. At present, the more invasive recordings have better signal quality, but also lower stability over time. Recently, subdural field potentials have been proposed as a stable, good quality source of control signals, with the potential for higher spatial and temporal bandwidth than EEG. Here we used finite element modeling in rats and humans and spatial spectral analysis in rats to compare the spatial resolution of signals recorded epidurally (outside the dura), with those recorded from subdural and scalp locations. Resolution of epidural and subdural signals was very similar in rats and somewhat less so in human models. Both were substantially better than signals recorded at the scalp. Resolution of epidural and subdural signals in humans was much more similar when the cerebrospinal fluid layer thickness was reduced. This suggests that the less invasive epidural recordings may yield signals of similar quality to subdural recordings, and hence may be more attractive as a source of control signals for BMIs.

Influence of anisotropic conductivity on EEG source reconstruction: investigations in a rabbit model

The aim of our work was to quantify the influence of white matter anisotropic conductivity information on electroencephalography (EEG) source reconstruction. We performed this quantification in a rabbit head using both simulations and source localization based on invasive measurements. In vivo anisotropic (tensorial) conductivity information was obtained from magnetic resonance diffusion tensor imaging and included into a high-resolution finite-element model. When neglecting anisotropy in the simulations, we found a shift in source location of up to 1.3 mm with a mean value of 0.3 mm. The averaged orientational deviation was 10 degree and the mean magnitude error of the dipole was 29%. Source localization of the first cortical components after median and tibial nerve stimulation resulted in anatomically verified dipole positions with no significant anisotropy effect. Our results indicate that the expected average source localization error due to anisotropic white matter conductivity is within the principal accuracy limits of current inverse procedures. However, larger localization errors might occur in certain cases. In contrast, dipole orientation and dipole strength are influenced significantly by the anisotropy. We conclude that the inclusion of tissue anisotropy information improves source estimation procedures.

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.

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.

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.

The impact of inhomogeneous tissue anisotropy on potential distribution within head model

This paper investigates the influence of inhomogeneous head tissue conductivity on EEGs. A head model with inhomogeneity of radial and tangential conductivity is built up. Based on this model the EEG signals are calculated using the finite elements method. The results show that the inhomogeneous radial and tangential conductivity of skull tissue, which has been ignored so far, can cause about 5% difference compared with commonly used homogeneous conductivity head models.

Effects of dipole position, orientation and noise on the accuracy of EEG source localization

BACKGROUND

The electroencephalogram (EEG) reflects the electrical activity in the brain on the surface of scalp. A major challenge in this field is the localization of sources in the brain responsible for eliciting the EEG signal measured at the scalp. In order to estimate the location of these sources, one must correctly model the sources, i.e., dipoles, as well as the volume conductor in which the resulting currents flow. In this study, we investigate the effects of dipole depth and orientation on source localization with varying sets of simulated random noise in 4 realistic head models.

METHODS

Dipole simulations were performed using realistic head models and using the boundary element method (BEM). In all, 92 dipole locations placed in temporal and parietal regions of the head with varying depth and orientation were investigated along with 6 different levels of simulated random noise. Localization errors due to dipole depth, orientation and noise were investigated.

RESULTS

The results indicate that there are no significant differences in localization error due tangential and radial dipoles. With high levels of simulated Gaussian noise, localization errors are depth-dependent. For low levels of added noise, errors are similar for both deep and superficial sources.

CONCLUSION

It was found that if the signal-to-noise ratio is above a certain threshold, localization errors in realistic head models are, on average the same for deep and superficial sources. As the noise increases, localization errors increase, particularly for deep sources.

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.

Evaluation of inverse methods and head models for EEG source localization using a human skull phantom

We used a real-skull phantom head to investigate the performances of representative methods for EEG source localization when considering various head models. We describe several experiments using a montage with current sources located at multiple positions and orientations inside a human skull filled with a conductive medium. The robustness of selected methods based on distributed source models is evaluated as various solutions to the forward problem (from the sphere to the finite element method) are considered. Experimental results indicate that inverse methods using appropriate cortex-based source models are almost always able to locate the active source with excellent precision, with little or no spurious activity in close or distant regions, even when two sources are simultaneously active. Superior regularization schemes for solving the inverse problem can dramatically help the estimation of sparse and focal active zones, despite significant approximation of the head geometry and the conductivity properties of the head tissues. Realistic head models are necessary, though, to fit the data with a reasonable level of residual variance.

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.

Factors affecting the accuracy of the boundary element method in the forward problem--I: Calculating surface potentials

A comprehensive review of factors affecting the accuracy of the boundary element method (BEM) for calculating surface potentials is presented. A relative-error statistic is developed which is only sensitive to calculation errors that could affect the inverse solution for source position, and insensitive to errors that only affect the solution for source strength. The factors considered in this paper are: numerical approximations intrinsic to the BEM, such as constant-potential versus linear-potential basis functions and sharp-edged versus smooth-surfaced volumes; aspects of the volume conductor including the volume shape, density of surface elements, and element shape; source position and orientation; and effects of "refinements" in the numerical methods. The effects of these factors are considered in both smooth-shaped (spheres and spheroids) and sharp-edged (cubes) volume conductors. This represents the first attempt to assess the effects of many of these factors pertaining to the numerical methods commonly used in fields such as electrocardiography (ECG) and electroencephalography (EEG). Strategies for obtaining the most accurate solutions are presented.

The resolution-field concept

The concept of a resolution field provides a means to compare arbitrary estimators of a brain activity of interest (AOI), represented, for example, by the amplitude of a dipole at a certain location of interest with well-defined and known direction. Like the lead field, it represents a vector field with the property that a measure of the impact of a hypothetical dipole at an arbitrary point in the brain is obtained by calculating the scalar product with the respective dipole moment. While in the case of the lead field this measure of impact quantifies the contribution of a hypothetical dipole to the data recorded in a specific measurement channel, in the case of the resolution field it quantifies the contribution of a hypothetical dipole to the estimate of the AOI. The resolution-field concept, which uses elements of the Backus-Gilbert theory and is closely related to the concept of a resolution matrix, is illustrated with examples based on a simulated measurement with a 148-channel magnetometer system.

Calculation of electrical potentials on the surface of a realistic head model by finite differences

We present a method for the calculation of electrical potentials at the surface of realistic head models from a point dipole generator based on a 3D finite-difference algorithm. The model was validated by comparing calculated values with those obtained algebraically for a three-shell spherical model. For a 1.25 mm cubic grid size, the mean error was 4.9% for a superficial dipole (3.75 mm from the inner surface of the skull) pointing in the radial direction. The effect of generator discretization and node spacing on the accuracy of the model was studied. Three values of the node spacing were considered: 1, 1.25 and 1.5 mm. The mean relative errors were 4.2, 6.3 and 9.3% respectively. The quality of the approximation of a point dipole by an array of nodes in a spherical neighbourhood did not depend significantly on the number of nodes used. The application of the method to a conduction model derived from MRI data is demonstrated.

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.

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

A method for semi-automatically constructing realistic surface meshes of 3 head structures--scalp, skull and brain--from a stack of MR images is described. Then an evaluation is given for both spherical and realistic dipolar models, using the boundary element method (BEM). In both cases, locally refined models were considered. Two characteristic mesh parameters were defined: the global and the local mesh densities (in triangles per cm2). In spherical geometries, numerical and analytical solutions were compared, and in the realistic case, all models were compared to a highly refined one, considered as a reference. Both geometries gave comparable results. It was found that for "deep dipoles" located at more than 20-30 mm under the brain surface, meshes with a global density of 0.5 tri/cm2 gave "acceptable" results, whereas for more superficial dipoles (2-3 mm < depth < 20-30 mm), it was necessary to locally refine meshes near the source location up to a local density of about 5-8 tri/cm2, to get comparable results.

Validation of a detailed computer model for the electric fields in the brain

A computer model has been designed for the calculation of the electrical fields in the head, based on the finite difference method. This method has not previously been applied for head modelling. The model was validated by using three concentric spheres and comparing it with an analytic model. Three levels of accuracy were tested. The forward solutions show that the finite difference algorithm works correctly and, by selecting the size of the volume elements properly, accurate results are obtained. The model will be applied to accurate and realistic geometries of the human head obtained from magnetic resonance images.

Identification of the hemisphere activated by hemifield visual stimulation using a single equivalent dipole model

Topographic amplitude distribution of the hemifield pattern visual evoked potential (PVEP) shows substantial intersubject variability. Many subjects have larger P100 amplitudes paradoxically over the hemisphere ipsilateral to the stimulated field, whereas others show larger responses over the stimulated hemisphere. The present study was designed to determine whether a single equivalent dipole model could correctly identify the field of stimulation, and therefore the hemisphere activated, under conditions in which the surface distribution is variable. Under conditions used in the present study, visual examination of the surface amplitude distribution of the P100 peak could not be reliably used to identify the hemifield that was stimulated. Of 28 hemifield PVEPs, obtained from 14 normal subjects, only 13 showed higher amplitude ipsilateral to the field of stimulation. Thus, neither examination of EP wave forms nor topographic maps provided an accurate means for determination of which hemifield was stimulated or which hemisphere was activated. The single equivalent dipole model correctly identified the stimulated hemisphere for 25/28 hemifield PVEPs. Orientation of the equivalent dipole accounted for much of the variability in surface amplitude distribution, with tangential orientations obtained in subjects with ipsilaterally predominant P100 surface topography. Although the dipole model improved identification of the stimulated hemisphere, dipoles were located anterior or inferior to the occipital lobe in some subjects. Results suggest that dipole modeling can provide useful information regarding the source of surface recorded potentials.

Method to reduce blur distortion from EEG's using a realistic head model

A mathematical procedure, which we call "Deblurring," was developed to reduce spatial blur distortion of scalp-recorded brain potentials due to transmission through the skull and other tissues. Deblurring estimates potentials at the superficial cerebral cortical surface from EEG's recorded at the scalp using a finite element model of each subject's scalp, skull and cortical surface constructed from their magnetic resonance images (MRI's). Simulations indicate that Deblurring is numerically stable, while a comparison of deblurred data with a direct cortical recording from a neurosurgery patient suggests that the procedure is valid. Application of Deblurring to somatosensory evoked potential data recorded at 124 scalp sites suggests that the method produces a dramatic improvement in spatial detail, and merits further development.