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

Standardized hierarchical adaptive Lp regression for noise robust focal epilepsy source reconstructions

OBJECTIVE

To investigate the ability of standardization to reduce source localization errors and measurement noise uncertainties for hierarchical Bayesian algorithms with L1- and L2-norms as priors in electroencephalography and magnetoencephalography of focal epilepsy.

METHODS

Description of the standardization methodology relying on the Hierarchical Bayesian framework, referred to as the Standardized Hierarchical Adaptive Lp-norm Regularization (SHALpR). The performance was tested using real data from two focal epilepsy patients. Simulated data that resembled the available real data was constructed for further localization and noise robustness investigation.

RESULTS

The proposed algorithms were compared to their non-standardized counterparts, Standardized low-resolution brain electromagnetic tomography, Standardized Shrinking LORETA-FOCUSS, and Dynamic statistical parametric maps. Based on the simulations, the standardized Hierarchical adaptive algorithm using L2-norm was noise robust for 10 dB signal-to-noise ratio (SNR), whereas the L1-norm prior worked robustly also with 5 dB SNR. The accuracy of the standardized L1-normed methodology to localize focal activity was under 1 cm for both patients.

CONCLUSIONS

Numerical results of the proposed methodology display improved localization and noise robustness. The proposed methodology also outperformed the compared methods when dealing with real data.

SIGNIFICANCE

The proposed standardized methodology, especially when employing the L1-norm, could serve as a valuable assessment tool in surgical decision-making.

Lattice layout and optimizer effect analysis for generating optimal transcranial electrical stimulation (tES) montages through the metaheuristic L1L1 method

Introduction

This study focuses on broadening the applicability of the metaheuristic L1-norm fitted and penalized (L1L1) optimization method in finding a current pattern for multichannel transcranial electrical stimulation (tES). The metaheuristic L1L1 optimization framework defines the tES montage via linear programming by maximizing or minimizing an objective function with respect to a pair of hyperparameters.

Methods

In this study, we explore the computational performance and reliability of different optimization packages, algorithms, and search methods in combination with the L1L1 method. The solvers from Matlab R2020b, MOSEK 9.0, Gurobi Optimizer, CVX's SeDuMi 1.3.5, and SDPT3 4.0 were employed to produce feasible results through different linear programming techniques, including Interior-Point (IP), Primal-Simplex (PS), and Dual-Simplex (DS) methods. To solve the metaheuristic optimization task of L1L1, we implement an exhaustive and recursive search along with a well-known heuristic direct search as a reference algorithm.

Results

Based on our results, and the given optimization task, Gurobi's IP was, overall, the preferable choice among Interior-Point while MOSEK's PS and DS packages were in the case of Simplex methods. These methods provided substantial computational time efficiency for solving the L1L1 method regardless of the applied search method.

Discussion

While the best-performing solvers show that the L1L1 method is suitable for maximizing either focality and intensity, a few of these solvers could not find a bipolar configuration. Part of the discrepancies between these methods can be explained by a different sensitivity with respect to parameter variation or the resolution of the lattice provided.

Multi-compartment head modeling in EEG: Unstructured boundary-fitted tetra meshing with subcortical structures

This paper introduces an automated approach for generating a finite element (FE) discretization of a multi-compartment human head model for electroencephalographic (EEG) source localization. We aim to provide an adaptable FE mesh generation tool for EEG studies. Our technique relies on recursive solid angle labeling of a surface segmentation coupled with smoothing, refinement, inflation, and optimization procedures to enhance the mesh quality. In this study, we performed numerical meshing experiments with the three-layer Ary sphere and a magnetic resonance imaging (MRI)-based multi-compartment head segmentation which incorporates a comprehensive set of subcortical brain structures. These experiments are motivated, on one hand, by the sensitivity of non-invasive subcortical source localization to modeling errors and, on the other hand, by the present lack of open EEG software pipelines to discretize all these structures. Our approach was found to successfully produce an unstructured and boundary-fitted tetrahedral mesh with a sub-one-millimeter fitting error, providing the desired accuracy for the three-dimensional anatomical details, EEG lead field matrix, and source localization. The mesh generator applied in this study has been implemented in the open MATLAB-based Zeffiro Interface toolbox for forward and inverse processing in EEG and it allows for graphics processing unit acceleration.

In silicostudy of the effects of cerebral circulation on source localization using a dynamical anatomical atlas of the human head

Objective.This study focuses on the effects of dynamical vascular modeling on source localization errors in electroencephalography (EEG). Our aim of thisin silicostudy is to (a) find out the effects of cerebral circulation on the accuracy of EEG source localization estimates, and (b) evaluate its relevance with respect to measurement noise and interpatient variation.Approach.We employ a four-dimensional (3D + T) statistical atlas of the electrical properties of the human head with a cerebral circulation model to generate virtual patients with different cerebral circulatory conditions for EEG source localization analysis. As source reconstruction techniques, we use the linearly constraint minimum variance (LCMV) beamformer, standardized low-resolution brain electromagnetic tomography (sLORETA), and the dipole scan (DS).Main results.Results indicate that arterial blood flow affects source localization at different depths and with varying significance. The average flow rate plays an important role in source localization performance, while the pulsatility effects are very small. In cases where a personalized model of the head is available, blood circulation mismodeling causes localization errors, especially in the deep structures of the brain where the main cerebral arteries are located. When interpatient variations are considered, the results show differences up to 15 mm for sLORETA and LCMV beamformer and 10 mm for DS in the brainstem and entorhinal cortices regions. In regions far from the main arteries vessels, the discrepancies are smaller than 3 mm. When measurement noise is added and interpatient differences are considered in a deep dipolar source, the results indicate that the effects of conductivity mismatch are detectable even for moderate measurement noise. The signal-to-noise ratio limit for sLORETA and LCMV beamformer is 15 dB, while the limit is under 30 dB for DS.Significance.Localization of the brain activity via EEG constitutes an ill-posed inverse problem, where any modeling uncertainty, e.g. a slight amount of noise in the data or material parameter discrepancies, can lead to a significant deviation of the estimated activity, especially in the deep structures of the brain. Proper modeling of the conductivity distribution is necessary in order to obtain an appropriate source localization. In this study, we show that the conductivity of the deep brain structures is particularly impacted by blood flow-induced changes in conductivity because large arteries and veins access the brain through that region.

Brainstorm-DUNEuro: An integrated and user-friendly Finite Element Method for modeling electromagnetic brain activity

Human brain activity generates scalp potentials (electroencephalography - EEG), intracranial potentials (iEEG), and external magnetic fields (magnetoencephalography - MEG). These electrophysiology (e-phys) signals can often be measured simultaneously for research and clinical applications. The forward problem involves modeling these signals at their sensors for a given equivalent current dipole configuration within the brain. While earlier researchers modeled the head as a simple set of isotropic spheres, today's magnetic resonance imaging (MRI) data allow for a detailed anatomic description of brain structures and anisotropic characterization of tissue conductivities. We present a complete pipeline, integrated into the Brainstorm software, that allows users to automatically generate an individual and accurate head model based on the subject's MRI and calculate the electromagnetic forward solution using the finite element method (FEM). The head model generation is performed by integrating the latest tools for MRI segmentation and FEM mesh generation. The final head model comprises the five main compartments: white-matter, gray-matter, CSF, skull, and scalp. The anisotropic brain conductivity model is based on the effective medium approach (EMA), which estimates anisotropic conductivity tensors from diffusion-weighted imaging (DWI) data. The FEM electromagnetic forward solution is obtained through the DUNEuro library, integrated into Brainstorm, and accessible with either a user-friendly graphical interface or scripting. With tutorials and example data sets available in an open-source format on the Brainstorm website, this integrated pipeline provides access to advanced FEM tools for electromagnetic modeling to a broader neuroscience community.

DUNEuro-A software toolbox for forward modeling in bioelectromagnetism

Accurate and efficient source analysis in electro- and magnetoencephalography using sophisticated realistic head geometries requires advanced numerical approaches. This paper presents DUNEuro, a free and open-source C++ software toolbox for the numerical computation of forward solutions in bioelectromagnetism. Building upon the DUNE framework, it provides implementations of modern fitted and unfitted finite element methods to efficiently solve the forward problems of electro- and magnetoencephalography. The user can choose between a variety of different source models that are implemented. The software's aim is to provide interfaces that are extendable and easy-to-use. In order to enable a closer integration into existing analysis pipelines, interfaces to Python and MATLAB are provided. The practical use is demonstrated by a source analysis example of somatosensory evoked potentials using a realistic six-compartment head model. Detailed installation instructions and example scripts using spherical and realistic head models are appended.

A comprehensive study on electroencephalography and magnetoencephalography sensitivity to cortical and subcortical sources

Signal-to-noise ratio (SNR) maps are a good way to visualize electroencephalography (EEG) and magnetoencephalography (MEG) sensitivity. SNR maps extend the knowledge about the modulation of EEG and MEG signals by source locations and orientations and can therefore help to better understand and interpret measured signals as well as source reconstruction results thereof. Our work has two main objectives. First, we investigated the accuracy and reliability of EEG and MEG finite element method (FEM)-based sensitivity maps for three different head models, namely an isotropic three and four-compartment and an anisotropic six-compartment head model. As a result, we found that ignoring the cerebrospinal fluid leads to an overestimation of EEG SNR values. Second, we examined and compared EEG and MEG SNR mappings for both cortical and subcortical sources and their modulation by source location and orientation. Our results for cortical sources show that EEG sensitivity is higher for radial and deep sources and MEG for tangential ones, which are the majority of sources. As to the subcortical sources, we found that deep sources with sufficient tangential source orientation are recordable by the MEG. Our work, which represents the first comprehensive study where cortical and subcortical sources are considered in highly detailed FEM-based EEG and MEG SNR mappings, sheds a new light on the sensitivity of EEG and MEG and might influence the decision of brain researchers or clinicians in their choice of the best modality for their experiment or diagnostics, respectively.

Parametrizing the Conditionally Gaussian Prior Model for Source Localization with Reference to the P20/N20 Component of Median Nerve SEP/SEF

In this article, we focused on developing the conditionally Gaussian hierarchical Bayesian model (CG-HBM), which forms a superclass of several inversion methods for source localization of brain activity using somatosensory evoked potential (SEP) and field (SEF) measurements. The goal of this proof-of-concept study was to improve the applicability of the CG-HBM as a superclass by proposing a robust approach for the parametrization of focal source scenarios. We aimed at a parametrization that is invariant with respect to altering the noise level and the source space size. The posterior difference between the gamma and inverse gamma hyperprior was minimized by optimizing the shape parameter, while a suitable range for the scale parameter can be obtained via the prior-over-measurement signal-to-noise ratio, which we introduce as a new concept in this study. In the source localization experiments, the primary generator of the P20/N20 component was detected in the Brodmann area 3b using the CG-HBM approach and a parameter range derived from the existing knowledge of the Tikhonov-regularized minimum norm estimate, i.e., the classical Gaussian prior model. Moreover, it seems that the detection of deep thalamic activity simultaneously with the P20/N20 component with the gamma hyperprior can be enhanced while using a close-to-optimal shape parameter value.

Zeffiro User Interface for Electromagnetic Brain Imaging: a GPU Accelerated FEM Tool for Forward and Inverse Computations in Matlab

This article introduces the Zeffiro interface (ZI) version 2.2 for brain imaging. ZI aims to provide a simple, accessible and multimodal open source platform for finite element method (FEM) based and graphics processing unit (GPU) accelerated forward and inverse computations in the Matlab environment. It allows one to (1) generate a given multi-compartment head model, (2) to evaluate a lead field matrix as well as (3) to invert and analyze a given set of measurements. GPU acceleration is applied in each of the processing stages (1)-(3). In its current configuration, ZI includes forward solvers for electro-/magnetoencephalography (EEG) and linearized electrical impedance tomography (EIT) as well as a set of inverse solvers based on the hierarchical Bayesian model (HBM). We report the results of EEG and EIT inversion tests performed with real and synthetic data, respectively, and demonstrate numerically how the inversion parameters affect the EEG inversion outcome in HBM. The GPU acceleration was found to be essential in the generation of the FE mesh and the LF matrix in order to achieve a reasonable computing time. The code package can be extended in the future based on the directions given in this article.

Randomized Multiresolution Scanning in Focal and Fast E/MEG Sensing of Brain Activity with a Variable Depth

We focus on electro-/magnetoencephalography imaging of the neural activity and, in particular, finding a robust estimate for the primary current distribution via the hierarchical Bayesian model (HBM). Our aim is to develop a reasonably fast maximum a posteriori (MAP) estimation technique which would be applicable for both superficial and deep areas without specific a priori knowledge of the number or location of the activity. To enable source distinguishability for any depth, we introduce a randomized multiresolution scanning (RAMUS) approach in which the MAP estimate of the brain activity is varied during the reconstruction process. RAMUS aims to provide a robust and accurate imaging outcome for the whole brain, while maintaining the computational cost on an appropriate level. The inverse gamma (IG) distribution is applied as the primary hyperprior in order to achieve an optimal performance for the deep part of the brain. In this proof-of-the-concept study, we consider the detection of simultaneous thalamic and somatosensory activity via numerically simulated data modeling the 14-20 ms post-stimulus somatosensory evoked potential and field response to electrical wrist stimulation. Both a spherical and realistic model are utilized to analyze the source reconstruction discrepancies. In the numerically examined case, RAMUS was observed to enhance the visibility of deep components and also marginalizing the random effects of the discretization and optimization without a remarkable computation cost. A robust and accurate MAP estimate for the primary current density was obtained in both superficial and deep parts of the brain.

A hybrid analytical–numerical algorithm for determining the neuronal current via electroencephalography

Specific mental processes are associated with brain activation of a unique form, which are, in turn, expressed via the generation of specific neuronal electric currents. Electroencephalography (EEG) is based on measurements on the scalp of the electric potential generated by the neuronal current flowing in the cortex. This specific form of EEG data has been employed for a plethora of medical applications, from sleep studies to diagnosing focal epilepsy. In recent years, there have been efforts to use EEG data for a more ambitious purpose, namely to determine the underlying neuronal current. Although it has been known since 1853, from the studies by Helmholtz, that the knowledge of the electric potential of the external surface of a conductor is insufficient for the determination of the electric current that gave rise to this potential, the important question of which part of the current can actually be determined from the knowledge of this potential remained open until work published in 1997, when it was shown that EEG provides information only about the irrotational part of the current, which will be denoted by Ψ ; moreover, an explicit formula was derived in the above work relating this part of the current, the measured electric potential, and a certain auxiliary function, v s , that depends on the geometry of the various compartments of the brain–head system and their conductivities. In the present paper: (i) Motivated by recent results which show that, in the case of ellipsoidal geometry, the assumption of the L 2 minimization of the current yields a unique solution, we derive an analogous analytic formula characterizing this minimization for arbitrary geometry. (ii) We show that the above auxiliary function can be computed numerically via a line integral from the values of a related function v s computed via OpenMEEG; moreover, we propose an alternative approach to computing the auxiliary function v s based on the construction of a certain surrogate model. (iii) By expanding Ψ in terms of an inverse multiquadric radial basis we implement the relevant formulae numerically. The above algorithm performs well for synthetic data; its implementation with real data only requires the knowledge of the coordinates of the positions where the given EEG data are obtained.

The multipole approach for EEG forward modeling using the finite element method

For accurate EEG forward solutions, it is necessary to apply numerical methods that allow to take into account the realistic geometry of the subject's head. A commonly used method to solve this task is the finite element method (FEM). Different approaches have been developed to obtain EEG forward solutions for dipolar sources with the FEM. The St. Venant approach is frequently applied, since its high numerical accuracy and stability as well as its computational efficiency was demonstrated in multiple comparison studies. In this manuscript, we propose a variation of the St. Venant approach, the multipole approach, to improve the numerical accuracy of the St. Venant approach even further and to allow for the simulation of additional source scenarios, such as quadrupolar sources. Exploiting the multipole expansion of electric fields, we demonstrate that the newly proposed multipole approach achieves even higher numerical accuracies than the St. Venant approach in both multi-layer sphere and realistic head models. Additionally, we exemplarily show that the multipole approach allows to not only simulate dipolar but also quadrupolar sources.

A realistic, accurate and fast source modeling approach for the EEG forward problem

The aim of this paper is to advance electroencephalography (EEG) source analysis using finite element method (FEM) head volume conductor models that go beyond the standard three compartment (skin, skull, brain) approach and take brain tissue inhomogeneity (gray and white matter and cerebrospinal fluid) into account. The new approach should enable accurate EEG forward modeling in the thin human cortical structures and, more specifically, in the especially thin cortices in children brain research or in pathological applications. The source model should thus be focal enough to be usable in the thin cortices, but should on the other side be more realistic than the current standard mathematical point dipole. Furthermore, it should be numerically accurate and computationally fast. We propose to achieve the best balance between these demands with a current preserving (divergence conforming) dipolar source model. We develop and investigate a varying number of current preserving source basis elements n (n=1,…,n=5). For validation, we conducted numerical experiments within a multi-layered spherical domain, where an analytical solution exists. We show that the accuracy increases along with the number of basis elements, while focality decreases. The results suggest that the best balance between accuracy and focality in thin cortices is achieved with n=4 (or in extreme cases even n=3) basis functions, while in thicker cortices n=5 is recommended to obtain the highest accuracy. We also compare the current preserving approach to two further FEM source modeling techniques, namely partial integration and St. Venant, and show that the best current preserving source model outperforms the competing methods with regard to overall balance. For all tested approaches, FEM transfer matrices enable high computational speed. We implemented the new EEG forward modeling approaches into the open source duneuro library for forward modeling in bioelectromagnetism to enable its broader use by the brain research community. This library is build upon the DUNE framework for parallel finite elements simulations and integrates with high-level toolboxes like FieldTrip. Additionally, an inversion test has been implemented using the realistic head model to demonstrate and compare the differences between the aforementioned source models.

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 FieldTrip-SimBio pipeline for EEG forward solutions

Background

Accurately solving the electroencephalography (EEG) forward problem is crucial for precise EEG source analysis. Previous studies have shown that the use of multicompartment head models in combination with the finite element method (FEM) can yield high accuracies both numerically and with regard to the geometrical approximation of the human head. However, the workload for the generation of multicompartment head models has often been too high and the use of publicly available FEM implementations too complicated for a wider application of FEM in research studies. In this paper, we present a MATLAB-based pipeline that aims to resolve this lack of easy-to-use integrated software solutions. The presented pipeline allows for the easy application of five-compartment head models with the FEM within the FieldTrip toolbox for EEG source analysis.

Methods

The FEM from the SimBio toolbox, more specifically the St. Venant approach, was integrated into the FieldTrip toolbox. We give a short sketch of the implementation and its application, and we perform a source localization of somatosensory evoked potentials (SEPs) using this pipeline. We then evaluate the accuracy that can be achieved using the automatically generated five-compartment hexahedral head model [skin, skull, cerebrospinal fluid (CSF), gray matter, white matter] in comparison to a highly accurate tetrahedral head model that was generated on the basis of a semiautomatic segmentation with very careful and time-consuming manual corrections.

Results

The source analysis of the SEP data correctly localizes the P20 component and achieves a high goodness of fit. The subsequent comparison to the highly detailed tetrahedral head model shows that the automatically generated five-compartment head model performs about as well as a highly detailed four-compartment head model (skin, skull, CSF, brain). This is a significant improvement in comparison to a three-compartment head model, which is frequently used in praxis, since the importance of modeling the CSF compartment has been shown in a variety of studies.

Conclusion

The presented pipeline facilitates the use of five-compartment head models with the FEM for EEG source analysis. The accuracy with which the EEG forward problem can thereby be solved is increased compared to the commonly used three-compartment head models, and more reliable EEG source reconstruction results can be obtained.

Electronic supplementary material

The online version of this article (10.1186/s12938-018-0463-y) contains supplementary material, which is available to authorized users.

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.