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

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.

Pressure-Poisson equation in numerical simulation of cerebral arterial circulation and its effect on the electrical conductivity of the brain

BACKGROUND AND OBJECTIVE

This study considers dynamic modeling of the cerebral arterial circulation and reconstructing an atlas for the electrical conductivity of the brain. Electrical conductivity is a governing parameter in several electrophysiological modalities applied in neuroscience, such as electroencephalography (EEG), transcranial electrical stimulation (tES), and electrical impedance tomography (EIT). While high-resolution 7-Tesla (T) Magnetic Resonance Imaging (MRI) data allow for reconstructing the cerebral arteries with a cross-sectional diameter larger than the voxel size, electrical conductivity cannot be directly inferred from MRI data. Brain models of electrophysiology typically associate each brain tissue compartment with a constant electrical conductivity, omitting any dynamic effects of cerebral blood circulation. Incorporating those effects poses the challenge of solving a system of incompressible Navier-Stokes equations (NSEs) in a realistic multi-compartment head model. However, using a simplified circulation model is well-motivated since, on the one hand, the complete system does not always have a numerically stable solution and, on the other hand, the full set of arteries cannot be perfectly reconstructed from the MRI data, meaning that any solution will be approximative.

METHODS

We postulate that circulation in the distinguishable arteries can be estimated via the pressure-Poisson equation (PPE), which is coupled with Fick's law of diffusion for microcirculation. To establish a fluid exchange model between arteries and microarteries, a boundary condition derived from the Hagen-Poisseuille model is applied. The relationship between the estimated volumetric blood concentration and the electrical conductivity of the brain tissue is approximated through Archie's law for fluid flow in porous media.

RESULTS

Through the formulation of the PPE and a set of boundary conditions (BCs) based on the Hagen-Poisseuille model, we obtained an equivalent formulation of the incompressible Stokes equation (SE). Thus, allowing effective blood pressure estimation in cerebral arteries segmented from open 7T MRI data.

CONCLUSIONS

As a result of this research, we developed and built a useful modeling framework that accounts for the effects of dynamic blood flow on a novel MRI-based electrical conductivity atlas. The electrical conductivity perturbation obtained in numerical experiments has an appropriate overall match with previous studies on this subject. Further research to validate these results will be necessary.

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.

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.

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.

Reconstructing subcortical and cortical somatosensory activity via the RAMUS inverse source analysis technique using median nerve SEP data

This study concerns reconstructing brain activity at various depths based on non-invasive EEG (electroencephalography) scalp measurements. We aimed at demonstrating the potential of the RAMUS (randomized multiresolution scanning) technique in localizing weakly distinguishable far-field sources in combination with coinciding cortical activity. As we have shown earlier theoretically and through simulations, RAMUS is a novel mathematical method that by employing the multigrid concept, allows marginalizing noise and depth bias effects and thus enables the recovery of both cortical and subcortical brain activity. To show this capability with experimental data, we examined the 14-30 ms post-stimulus somatosensory evoked potential (SEP) responses of human median nerve stimulation in three healthy adult subjects. We aim at reconstructing the different response components by evaluating a RAMUS-based estimate for the primary current density in the nervous tissue. We present source reconstructions obtained with RAMUS and compare them with the literature knowledge of the SEP components and the outcome of the unit-noise gain beamformer (UGNB) and standardized low-resolution brain electromagnetic tomography (sLORETA). We also analyzed the effect of the iterative alternating sequential technique, the optimization technique of RAMUS, compared to the classical minimum norm estimation (MNE) technique. Matching with our previous numerical studies, the current results suggest that RAMUS could have the potential to enhance the detection of simultaneous deep and cortical components and the distinction between the evoked sulcal and gyral activity.

Sensitivity Analysis Highlights the Importance of Accurate Head Models for Electrical Impedance Tomography Monitoring of Intracerebral Hemorrhagic Stroke

OBJECTIVE

Electrical impedance tomography (EIT) has been proposed as a novel tool for diagnosing stroke. However, so far, the clinical feasibility is unresolved. In this study, we aim to investigate the need for accurate head modeling in EIT and how the inhomogeneities of the head contribute to the EIT measurement and affect its feasibility in monitoring the progression of a hemorrhagic stroke.

METHODS

We compared anatomically detailed six- and three-layer finite element models of a human head and computed the resulting scalp electrode potentials and the lead fields of selected electrode configurations. We visualized the resulting EIT measurement sensitivity distributions, computed the scalp electrode potentials, and examined the inverse imaging with selected cases. The effect of accurate tissue geometry and conductivity values on the EIT measurement is assessed with multiple different hemorrhagic perturbation locations and sizes.

RESULTS

Our results show that accurate tissue geometries and conductivity values inside the cranial cavity, especially the highly conductive cerebral spinal fluid, significantly affect EIT measurement sensitivity distribution and measured potentials.

CONCLUSIONS

We can conclude that the three-layer head models commonly used in EIT literature cannot depict the current paths correctly in the head. Thus, our study highlights the need to consider the detailed geometry of the cerebrospinal fluid (CSF) in EIT.

SIGNIFICANCE

The results clearly show that the CSF should be considered in the head EIT calculations.

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.

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.