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

CutFEM forward modeling for EEG source analysis

Introduction

Source analysis of Electroencephalography (EEG) data requires the computation of the scalp potential induced by current sources in the brain. This so-called EEG forward problem is based on an accurate estimation of the volume conduction effects in the human head, represented by a partial differential equation which can be solved using the finite element method (FEM). FEM offers flexibility when modeling anisotropic tissue conductivities but requires a volumetric discretization, a mesh, of the head domain. Structured hexahedral meshes are easy to create in an automatic fashion, while tetrahedral meshes are better suited to model curved geometries. Tetrahedral meshes, thus, offer better accuracy but are more difficult to create.

Methods

We introduce CutFEM for EEG forward simulations to integrate the strengths of hexahedra and tetrahedra. It belongs to the family of unfitted finite element methods, decoupling mesh and geometry representation. Following a description of the method, we will employ CutFEM in both controlled spherical scenarios and the reconstruction of somatosensory-evoked potentials.

Results

CutFEM outperforms competing FEM approaches with regard to numerical accuracy, memory consumption, and computational speed while being able to mesh arbitrarily touching compartments.

Discussion

CutFEM balances numerical accuracy, computational efficiency, and a smooth approximation of complex geometries that has previously not been available in FEM-based EEG forward modeling.

Neural stochastic differential equations network as uncertainty quantification method for EEG source localization

EEG source localization remains a challenging problem given the uncertain conductivity values of the volume conductor models (VCMs). As uncertain conductivities vary across people, they may considerably impact the forward and inverse solutions of the EEG, leading to an increase in localization mistakes and misdiagnoses of brain disorders. Calibration of conductivity values using uncertainty quantification (UQ) techniques is a promising approach to reduce localization errors. The widely-known UQ methods involve Bayesian approaches, which utilize prior conductivity values to derive their posterior inference and estimate their optimal calibration. However, these approaches have two significant drawbacks: solving for posterior inference is intractable, and choosing inappropriate priors may lead to increased localization mistakes. This study used the Neural Stochastic Differential equations Network (SDE-Net), a combination of dynamical systems and deep learning techniques that utilizes the Wiener process to minimize conductivity uncertainties in the VCM and improve the inverse problem. Results revealed that SDE-Net generated a lower localization error rate in the inverse problem compared to Bayesian techniques. Future studies may employ new stochastic dynamical systems-based techniques as a UQ technique to address further uncertainties in the EEG Source Localization problem. Our code can be found here:https://github.com/rrwabina/SDENet-UQ-ESL.

High-Resolution EEG Source Localization in Segmentation-Free Head Models Based on Finite-Difference Method and Matching Pursuit Algorithm

Electroencephalogram (EEG) is a method to monitor electrophysiological activity on the scalp, which represents the macroscopic activity of the brain. However, it is challenging to identify EEG source regions inside the brain based on data measured by a scalp-attached network of electrodes. The accuracy of EEG source localization significantly depends on the type of head modeling and inverse problem solver. In this study, we adopted different models with a resolution of 0.5 mm to account for thin tissues/fluids, such as the cerebrospinal fluid (CSF) and dura. In particular, a spatially dependent conductivity (segmentation-free) model created using deep learning was developed and used for more realist representation of electrical conductivity. We then adopted a multi-grid-based finite-difference method (FDM) for forward problem analysis and a sparse-based algorithm to solve the inverse problem. This enabled us to perform efficient source localization using high-resolution model with a reasonable computational cost. Results indicated that the abrupt spatial change in conductivity, inherent in conventional segmentation-based head models, may trigger source localization error accumulation. The accurate modeling of the CSF, whose conductivity is the highest in the head, was an important factor affecting localization accuracy. Moreover, computational experiments with different noise levels and electrode setups demonstrate the robustness of the proposed method with segmentation-free head model.

Multimodal Neuroimaging Using Concurrent EEG/fNIRS for Poststroke Recovery Assessment: An Exploratory Study

Background

Persistent motor deficits are very common in poststroke survivors and often lead to disability. Current clinical measures for profiling motor impairment and assessing poststroke recovery are largely subjective and lack precision.

Objective

A multimodal neuroimaging approach was developed based on concurrent functional near-infrared spectroscopy (fNIRS) and electroencephalography (EEG) to identify biomarkers associated with motor function recovery and document the poststroke cortical reorganization.

Methods

EEG and fNIRS data were simultaneously recorded from 9 healthy controls and 18 stroke patients during a hand-clenching task. A novel fNIRS-informed EEG source imaging approach was developed to estimate cortical activity and functional connectivity. Subsequently, graph theory analysis was performed to identify network features for monitoring and predicting motor function recovery during a 4-week intervention.

Results

The task-evoked strength at ipsilesional primary somatosensory cortex was significantly lower in stroke patients compared with healthy controls ( P < .001). In addition, across the 4-week rehabilitation intervention, the strength at ipsilesional premotor cortex (PMC) ( R = 0.895, P = .006) and the connectivity between bilateral primary motor cortices (M1) ( R = 0.9, P = .007) increased in parallel with the improvement of motor function. Furthermore, a higher baseline strength at ipsilesional PMC was associated with a better motor function recovery ( R = 0.768, P = .007), while a higher baseline connectivity between ipsilesional supplementary motor cortex (SMA)–M1 implied a worse motor function recovery ( R = −0.745, P = .009).

Conclusion

The proposed multimodal EEG/fNIRS technique demonstrates a preliminary potential for monitoring and predicting poststroke motor recovery. We expect such findings can be further validated in future study.

Three-Dimensional Innervation Zone Imaging from Multi-Channel Surface EMG Recordings

There is an unmet need to accurately identify the locations of innervation zones (IZs) of spastic muscles, so as to guide botulinum toxin (BTX) injections for the best clinical outcome. A novel 3D IZ imaging (3DIZI) approach was developed by combining the bioelectrical source imaging and surface electromyogram (EMG) decomposition methods to image the 3D distribution of IZs in the target muscles. Surface IZ locations of motor units (MUs), identified from the bipolar map of their MU action potentials (MUAPs) were employed as a prior knowledge in the 3DIZI approach to improve its imaging accuracy. The performance of the 3DIZI approach was first optimized and evaluated via a series of designed computer simulations, and then validated with the intramuscular EMG data, together with simultaneously recorded 128-channel surface EMG data from the biceps of two subjects. Both simulation and experimental validation results demonstrate the high performance of the 3DIZI approach in accurately reconstructing the distributions of IZs and the dynamic propagation of internal muscle activities in the biceps from high-density surface EMG recordings.

Noninvasive Brain-Computer Interfaces Based on Sensorimotor Rhythms

Brain-computer interfaces (BCIs) have been explored in the field of neuroengineering to investigate how the brain can use these systems to control external devices. We review the principles and approaches we have taken to develop a sensorimotor rhythm EEG based brain-computer interface (BCI). The methods include developing BCI systems incorporating the control of physical devices to increase user engagement, improving BCI systems by inversely mapping scalp-recorded EEG signals to the cortical source domain, integrating BCI with noninvasive neuromodulation strategies to improve learning, and incorporating mind-body awareness training to enhance BCI learning and performance. The challenges and merits of these strategies are discussed, together with recent findings. Our work indicates that the sensorimotor-rhythm-based noninvasive BCI has the potential to provide communication and control capabilities as an alternative to physiological motor pathways.

Noninvasive imaging of internal muscle activities from multi-channel surface EMG recordings

Surface Electromyogram (sEMG) technology provides a non-invasive way for rapid monitoring muscle activities, but its poor spatial resolution and specificity limit its application in clinic. To overcome these limitations, a noninvasive muscle activity imaging (MAI) approach has been developed and used to reconstruct internal muscle activities from multi-channel sEMG recordings. A realistic geometric hand model is developed from high-resolution MR images and a distributed bioelectric dipole source model is employed to describe the internal muscle activity space of the muscles. The finite element method and weighted minimum norm method are utilized solve the forward and inverse problems respectively involved in the proposed MAI technique. A series of computer simulations was conducted to test the performance of the proposed MAI approach. Results show that reconstruction results achieved by the MAI technique indeed provide us more detailed and dynamic information of internal muscle activities, which enhance our understanding of the mechanisms underlying the surface EMG recordings.

Finite element modeling of human head from MRI

The FEM geometry modeling of realistic head is a key issue for the research on FEM-based EEG/MEG. In this paper, a methodology is developed to construct this kind of model. By using this method, a five-layer realistic head FEM model is obtained, and with its application in FEM-based EEG, a satisfying result shows the reliability of the model.

Computed basis functions for finite element analysis based on tomographic data

In bioelectromagnetics, the structures in which the electromagnetic field is to be computed are sometimes defined by a fine grid of voxels (3-D cells) whose tissue types are obtained by tomography. A novel finite element method is proposed for such cases. A simple, regular mesh of cube elements is constructed, each containing the same, integer number of voxels. There may be several different tissues present within an element, but this is accommodated by computing element basis functions that approximately respect the interface conditions between different tissues. Results are presented for a test model of 128 (3) voxels, consisting of nested dielectric cubes, driven by specified charges. The electrostatic potential computed with the new method agrees well with that of a conventional finite element code: the rms difference along the sample line is 1.5% of the highest voltage. Results are also presented for the potential due to a current dipole placed in a brain model of 181 × 217 × 181 voxels, derived from MRI data. The new method gives potentials that are different to those obtained by treating each voxel as an element by 1% of the peak voltage, yet the global finite element matrix has a dimension which is more than 50 times smaller.

Noninvasive estimation of global activation sequence using the extended Kalman filter

A new algorithm for 3-D imaging of the activation sequence from noninvasive body surface potentials is proposed. After formulating the nonlinear relationship between the 3-D activation sequence and the body surface recordings during activation, the extended Kalman filter (EKF) is utilized to estimate the activation sequence in a recursive way. The state vector containing the activation sequence is optimized during iteration by updating the error variance/covariance matrix. A new regularization scheme is incorporated into the "predict" procedure of EKF to tackle the ill-posedness of the inverse problem. The EKF-based algorithm shows good performance in simulation under single-site pacing. Between the estimated activation sequences and true values, the average correlation coefficient (CC) is 0.95, and the relative error (RE) is 0.13. The average localization error (LE) when localizing the pacing site is 3.0 mm. Good results are also obtained under dual-site pacing (CC = 0.93, RE = 0.16, and LE = 4.3 mm). Furthermore, the algorithm shows robustness to noise. The present promising results demonstrate that the proposed EKF-based inverse approach can noninvasively estimate the 3-D activation sequence with good accuracy and the new algorithm shows good features due to the application of EKF.

Multimodal functional neuroimaging: integrating functional MRI and EEG/MEG

Noninvasive functional neuroimaging, as an important tool for basic neuroscience research and clinical diagnosis, continues to face the need of improving the spatial and temporal resolution. While existing neuroimaging modalities might approach their limits in imaging capability mostly due to fundamental as well as technical reasons, it becomes increasingly attractive to integrate multiple complementary modalities in an attempt to significantly enhance the spatiotemporal resolution that cannot be achieved by any modality individually. Electrophysiological and hemodynamic/metabolic signals reflect distinct but closely coupled aspects of the underlying neural activity. Combining fMRI and EEG/MEG data allows us to study brain function from different perspectives. In this review, we start with an overview of the physiological origins of EEG/MEG and fMRI, as well as their fundamental biophysics and imaging principles, we proceed with a review of the major advances in the understanding and modeling of neurovascular coupling and in the methodologies for the fMRI-EEG/MEG simultaneous recording. Finally, we summarize important remaining issues and perspectives concerning multimodal functional neuroimaging, including brain connectivity imaging.

Three-dimensional brain current source reconstruction from intra-cranial ECoG recordings

We have investigated 3-dimensional brain current density reconstruction (CDR) from intracranial electrocorticogram (ECoG) recordings by means of finite element method (FEM). The brain electrical sources are modeled by a current density distribution and estimated from the ECoG signals with the aid of a weighted minimum norm estimation algorithm. A series of computer simulations were conducted to evaluate the performance of ECoG-CDR by comparing with the scalp EEG based CDR results. The present computer simulation results indicate that the ECoG-CDR provides enhanced performance in localizing single dipole sources which are located in regions underneath the implanted subdural ECoG grids, and in distinguishing and imaging multiple separate dipole sources, in comparison to the CDR results as obtained from the scalp EEG under the same conditions. We have also demonstrated the applicability of the present ECoG-CDR method to estimate 3-dimensional current density distribution from the subdural ECoG recordings in a human epilepsy patient. Eleven interictal epileptiform spikes (seven from the frontal region and four from parietal region) in an epilepsy patient undergoing surgical evaluation were analyzed. The present promising results indicate the feasibility and applicability of the developed ECoG-CDR method of estimating brain sources from intracranial electrical recordings, with detailed forward modeling using FEM.

Three-Dimensional Cardiac Electrical Imaging From Intracavity Recordings

A novel approach is proposed to image 3-D cardiac electrical activity from intracavity electrical recordings with the aid of a catheter. The feasibility and performance were evaluated by computer simulation studies, where a 3-D cellular-automaton heart model and a finite-element thorax volume conductor model were utilized. The finite-element method (FEM) was used to simulate the intracavity recordings induced by a single-site and dual-site pacing protocol. The 3-D ventricular activation sequences as well as the locations of the initial activation sites were inversely estimated by minimizing the dissimilarity between the intracavity potential "measurements" and the model-generated intracavity potentials. Under single-site pacing, the relative error (RE) between the true and estimated activation sequences was 0.03 +/- 0.01 and the localization error (LE) (of the initiation site) was 1.88 +/- 0.92 mm, as averaged over 12 pacing trials when considering 25 microV additive measurement noise using 64 catheter electrodes. Under dual-site pacing, the RE was 0.04 +/- 0.01 over 12 pacing trials and the LE over 24 initial pacing sites was 2.28 +/- 1.15 mm, when considering 25 microV additive measurement noise using 64 catheter electrodes. The proposed 3-D cardiac electrical imaging approach using intracavity electrical recordings was also tested under various simulated conditions and robust inverse solutions obtained. The present promising simulation results suggest the feasibility of obtaining 3-D information of cardiac electrical activity from intracavity recordings. The application of this inverse method has the potential of enhancing electrocardiographic mapping by catheters in electrophysiology laboratories, aiding cardiac resynchronization therapy, and other clinical applications.

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.

Solving the ECG forward problem by means of a meshless finite element method

The conventional numerical computational techniques such as the finite element method (FEM) and the boundary element method (BEM) require laborious and time-consuming model meshing. The new meshless FEM only uses the boundary description and the node distribution and no meshing of the model is required. This paper presents the fundamentals and implementation of meshless FEM and the meshless FEM method is adapted to solve the electrocardiography (ECG) forward problem. The method is evaluated on a single-layer torso model, in which the analytical solution exists, and tested in a realistic geometry homogeneous torso model, with satisfactory results being obtained. The present results suggest that the meshless FEM may provide an alternative for ECG forward solutions.

Estimation of in vivo brain-to-skull conductivity ratio in humans

The electrical conductivity value of the human skull is important for biophysics research of the brain. In the present study, the human brain-to-skull conductivity ratio was estimated through in vivo experiments utilizing intra-cranial electrical stimulation in two epilepsy patients. A realistic geometry inhomogeneous head model including the implanted silastic grids was constructed with the aid of the finite element method, and used to estimate the conductivity ratio. Averaging over 49 sets of measurements, the mean value and standard deviation of the brain-to-skull conductivity ratio were found to be 18.7 and 2.1, respectively.

Effects of holes on EEG forward solutions using a realistic geometry head model

Holes in the skull and the scalp are associated with intracranial monitoring procedures. The purpose of the present study is to evaluate the effects of holes on extracranial electroencephalogram (EEG) and intracranial electrocorticogram (ECoG) recordings. The finite difference method (FDM) was used to model the head volume conductor with a hole of varying size. A current dipole was used to simulate the brain electrical activity with varying locations within the brain. The effects of the holes were assessed by comparing the forward potential distributions with and without a hole. The present computer simulation results indicate that the effect of a hole on the scalp EEG and ECoG recordings strongly depends on the dipole location and orientation. For a superficial radial dipole located under a hole of radius ranging from 5 mm to 40 mm, the relative error (RE) varies from 0.99% to 93.07% for the EEG and from 0.025% to 16.72% for the ECoG. The correlation coefficient (CC) varies from 99.99% to 21.1% and from 100% to 99.75% for the EEG and EcoG, respectively. For radial dipoles, the strongest effect on the EEG and ECoG occurs when the dipole is located below the center of the hole, while for tangential dipoles, the strongest effect occurs when the dipole is located below the border of the hole. The effect of a hole on the EEG is much larger than upon the ECoG.

A fast method to derive realistic FEM models based on BEM models

A fast method for constructing a FEM head model based on the relevant BEM head model is presented. The method has been evaluated and shown to provide an alternative means of deriving FEM head models. The availability of such fast method would facilitate the realistic head modeling for EEG/MEG research.

Integration of EEG/MEG with MRI and fMRI

EEG and MEG are important functional neuroimaging modalities for studying the temporal dynamics of neural activities and interactions, but the accurate localization of neural activities still remains a challenging problem. Combining EEG/MEG with MRI or/and functional MRI (fMRI) holds promise to significantly increase the spatial resolution of electromagnetic source imaging, and at the same time, allows tracing the rapid neural processes and information pathways within the brain, which cannot be achieved using these modalities in isolation. In this paper, we review some recent progresses in multimodal neuroimaging, with special emphasis on the integration of EEG, MEG with MRI and fMRI. Some examples are shown to illustrate the importance of the combined source analysis in clinical and experimental studies.

A cortical potential imaging study from simultaneous extra- and intracranial electrical recordings by means of the finite element method

In the present study, we have validated the cortical potential imaging (CPI) technique for estimating cortical potentials from scalp EEG using simultaneously recorded electrocorticogram (ECoG) in the presence of strong local inhomogeneity, i.e., Silastic ECoG grid(s). The finite element method (FEM) was used to model the realistic postoperative head volume conductor, which includes the scalp, skull, cerebrospinal fluid (CSF) and brain, as well as the Silastic ECoG grid(s) implanted during the surgical evaluation in epilepsy patients, from the co-registered magnetic resonance (MR) and computer tomography (CT) images. A series of computer simulations were conducted to evaluate the present FEM-based CPI technique and to assess the effect of the Silastic ECoG grid on the scalp EEG forward solutions. The present simulation results show that the Silastic ECoG grid has substantial influence on the scalp potential forward solution due to the distortion of current pathways in the presence of the extremely low conductive materials. On the other hand, its influence on the estimated cortical potential distribution is much less than that on the scalp potential distribution. With appropriate numerical modeling and inverse estimation techniques, we have demonstrated the feasibility of estimating the cortical potentials from the scalp EEG with the implanted Silastic ECoG gird(s), in both computer simulations and in human experimentation. In an epilepsy patient undergoing surgical evaluation, the cortical potentials were reconstructed from the simultaneously recorded scalp EEG, in which main features of spatial patterns during interictal spike were preserved and over 0.75 correlation coefficient value was obtained between the recorded and estimated cortical potentials. The FEM-based CPI technique provides a means of connecting the simultaneous recorded ECoG and the scalp EEG and promises to become an effective tool to evaluate and validate CPI techniques using clinic data.

A finite difference method with reciprocity used to incorporate anisotropy in electroencephalogram dipole source localization

Many implementations of electroencephalogram (EEG) dipole source localization neglect the anisotropical conductivities inherent to brain tissues, such as the skull and white matter anisotropy. An examination of dipole localization errors is made in EEG source analysis, due to not incorporating the anisotropic properties of the conductivity of the skull and white matter. First, simulations were performed in a 5 shell spherical head model using the analytical formula. Test dipoles were placed in three orthogonal planes in the spherical head model. Neglecting the skull anisotropy results in a dipole localization error of, on average, 13.73 mm with a maximum of 24.51 mm. For white matter anisotropy these values are 11.21 mm and 26.3 mm, respectively. Next, a finite difference method (FDM), presented by Saleheen and Kwong (1997 IEEE Trans. Biomed. Eng. 44 800-9), is used to incorporate the anisotropy of the skull and white matter. The FDM method has been validated for EEG dipole source localization in head models with all compartments isotropic as well as in a head model with white matter anisotropy. In a head model with skull anisotropy the numerical method could only be validated if the 3D lattice was chosen very fine (grid size < or = 2 mm).