Integral equations and boundary-element solution for static potential in a general piece-wise homogeneous volume conductor

A survey on integral equations for bioelectric modeling

Bioelectric modeling problems, such as electroencephalography, magnetoencephalography, transcranial electrical stimulation, deep brain stimulation, and transcranial magnetic stimulation, among others, can be approached through the formulation and resolution of integral equations of theboundary element method(BEM). Recently, it has been realized that thecharge-based formulationof the BEM is naturally well-suited for the application of thefast multipole method(FMM). The FMM is a powerful algorithm for the computation of many-body interactions and is widely applied in electromagnetic modeling problems. With the introduction of BEM-FMM in the context of bioelectromagnetism, the BEM can now compete with thefinite element method(FEM) in a number of application cases. This survey has two goals: first, to give a modern account of the main BEM formulations in the literature and their integration with FMM, directed to general researchers involved in development of BEM software for bioelectromagnetic applications. Second, to survey different techniques and available software, and to contrast different BEM and FEM approaches. As a new contribution, we showcase that the charge-based formulation is dual to the more common surface potential formulation.

Estimating the influence of stroke lesions on MEG source reconstruction

Source reconstruction of magnetoencephalography (MEG) has been used to assess brain reorganization after brain damage, such as stroke. Lesions result in parts of the brain having an electrical conductivity that differs from the normal values. The effect this has on the forward solutions (i.e., the propagation of electric currents and magnetic fields generated by cortical activity) is well predictable. However, their influence on source localization results is not well characterized and understood. This is specifically a concern for patient studies with asymmetric (i.e., within one hemisphere) lesions focusing on asymmetric and lateralized brain activity, such as language. In particular, it is good practice to consider the level of geometrical detail that is necessary to compute and interpret reliable source reconstruction results. To understand the effect of lesions on source estimates and propose recommendations to researchers working with clinical data, in this study we consider the trade off between improved accuracy and the additional effort to compute more realistic head models, with the aim to answer the question whether the additional effort is worth it. We simulated and analyzed the effects of a stroke lesion (i.e., an asymmetrically distributed CSF-filled cavity) in the head model with three different sizes and locations when performing MEG source reconstruction using a finite element method (FEM). We compared the effect of the lesion with a homogeneous head model that neglects the lesion. We computed displacement and attenuation/amplification maps to quantify the localization errors and signal magnitude modulation. We conclude that brain lesions leading to asymmetrically distributed CSF-filled cavities should be modeled when performing MEG source reconstruction, especially when investigating deep sources or post-stroke hemispheric lateralization of functions. The strongest effects are not only visible in perilesional areas, but can extend up to 20 mm from the lesion. Bigger lesions lead to stronger effects impacting larger areas, independently from the lesion location. Lastly, we conclude that more priority should be given to usability and accessibility of the required computational tools, to allow researchers with less technical expertise to use the improved methods that are available but currently not widely adopted yet.

Magnetoencephalography Signal Processing, Forward Modeling, Magnetoencephalography Inverse Source Imaging, and Coherence Analysis

Magnetoencephalography (MEG) is a noninvasive functional imaging technique for the brain. MEG directly measures the magnetic signal due to neuronal activation in gray matter with high spatial localization accuracy. The first part of this article covers the overall concepts of MEG and the forward and inverse modeling techniques. It is followed by examples of analyzing evoked and resting-state MEG signals using a high-resolution MEG source imaging technique. Next, different techniques for connectivity and network analysis are reviewed with examples showing connectivity estimates from resting-state and epileptic activity.

Boundary Element Fast Multipole Method for Enhanced Modeling of Neurophysiological Recordings

OBJECTIVE

A new numerical modeling approach is proposed which provides forward-problem solutions for both noninvasive recordings (EEG/MEG) and higher-resolution intracranial recordings (iEEG).

METHODS

The algorithm is our recently developed boundary element fast multipole method or BEM-FMM. It is based on the integration of the boundary element formulation in terms of surface charge density and the fast multipole method originating from its inventors. The algorithm still possesses the major advantage of the conventional BEM - high speed - but is simultaneously capable of processing a very large number of surface-based unknowns. As a result, an unprecedented spatial resolution could be achieved, which enables multiscale modeling.

RESULTS

For non-invasive EEG/MEG, we are able to accurately solve the forward problem with approximately 1 mm anatomical resolution in the cortex within 1-2 min given several thousand cortical dipoles. Targeting high-resolution iEEG, we are able to compute, for the first time, an integrated electromagnetic response for an ensemble (2,450) of tightly packed realistic pyramidal neocortical neurons in a full-head model with 0.6 mm anatomical cortical resolution. The neuronal arbor is comprised of 5.9 M elementary 1.2 μm long dipoles. On a standard server, the computations require about 5 min.

CONCLUSION

Our results indicate that the BEM-FMM approach may be well suited to support numerical multiscale modeling pertinent to modern high-resolution and submillimeter iEEG.

SIGNIFICANCE

Based on the speed and ease of implementation, this new algorithm represents a method that will greatly facilitate simulations at multi-scale across a variety of applications.

Individual head models for estimating the TMS-induced electric field in rat brain

In transcranial magnetic stimulation (TMS), the initial cortical activation due to stimulation is determined by the state of the brain and the magnitude, waveform, and direction of the induced electric field (E-field) in the cortex. The E-field distribution depends on the conductivity geometry of the head. The effects of deviations from a spherically symmetric conductivity profile have been studied in detail in humans. In small mammals, such as rats, these effects are more pronounced due to their less spherical head, proportionally much thicker neck region, and overall much smaller size compared to the TMS coils. In this study, we describe a simple method for building individual realistically shaped head models for rats from high-resolution X-ray tomography images. We computed the TMS-induced E-field with the boundary element method and assessed the effect of head-model simplifications on the estimated E-field. The deviations from spherical symmetry have large, non-trivial effects on the E-field distribution: for some coil orientations, the strongest stimulation is in the brainstem even when the coil is over the motor cortex. With modelling prior to an experiment, such problematic coil orientations can be avoided for more accurate targeting.

Real-time computation of the TMS-induced electric field in a realistic head model

Transcranial magnetic stimulation (TMS) is often targeted using a model of TMS-induced electric field (E). In such navigated TMS, the E-field models have been based on spherical approximation of the head. Such models omit the effects of cerebrospinal fluid (CSF) and gyral folding, leading to potentially large errors in the computed E-field. So far, realistic models have been too slow for interactive TMS navigation. We present computational methods that enable real-time solving of the E-field in a realistic five-compartment (5-C) head model that contains isotropic white matter, gray matter, CSF, skull and scalp. Using reciprocity and Geselowitz integral equation, we separate the computations to coil-dependent and -independent parts. For the Geselowitz integrals, we present a fast numerical quadrature. Further, we present a moment-matching approach for optimizing dipole-based coil models. We verified and benchmarked the new methods using simulations with over 100 coil locations. The new quadrature introduced a relative error (RE) of 0.3-0.6%. For a coil model with 42 dipoles, the total RE of the quadrature and coil model was 0.44-0.72%. Taking also other model errors into account, the contribution of the new approximations to the RE was 0.1%. For comparison, the RE due to omitting the separation of white and gray matter was >11%, and the RE due to omitting also the CSF was >23%. After the coil-independent part of the model has been built, E-fields can be computed very quickly: Using a standard PC and basic GPU, our solver computed the full E-field in a 5-C model in 9000 points on the cortex in 27 coil positions per second (cps). When the separation of white and gray matter was omitted, the speed was 43-65 cps. Solving only one component of the E-field tripled the speed. The presented methods enable real-time solving of the TMS-induced E-field in a realistic head model that contains the CSF and gyral folding. The new methodology allows more accurate targeting and precise adjustment of stimulation intensity during experimental or clinical TMS mapping.

Comparative performance of the finite element method and the boundary element fast multipole method for problems mimicking transcranial magnetic stimulation (TMS)

OBJECTIVE

A study pertinent to the numerical modeling of cortical neurostimulation is conducted in an effort to compare the performance of the finite element method (FEM) and an original formulation of the boundary element fast multipole method (BEM-FMM) at matched computational performance metrics.

APPROACH

We consider two problems: (i) a canonic multi-sphere geometry and an external magnetic-dipole excitation where the analytical solution is available and; (ii) a problem with realistic head models excited by a realistic coil geometry. In the first case, the FEM algorithm tested is a fast open-source getDP solver running within the SimNIBS 2.1.1 environment. In the second case, a high-end commercial FEM software package ANSYS Maxwell 3D is used. The BEM-FMM method runs in the MATLAB^® 2018a environment.

MAIN RESULTS

In the first case, we observe that the BEM-FMM algorithm gives a smaller solution error for all mesh resolutions and runs significantly faster for high-resolution meshes when the number of triangular facets exceeds approximately 0.25 M. We present other relevant simulation results such as volumetric mesh generation times for the FEM, time necessary to compute the potential integrals for the BEM-FMM, and solution performance metrics for different hardware/operating system combinations. In the second case, we observe an excellent agreement for electric field distribution across different cranium compartments and, at the same time, a speed improvement of three orders of magnitude when the BEM-FMM algorithm used.

SIGNIFICANCE

This study may provide a justification for anticipated use of the BEM-FMM algorithm for high-resolution realistic transcranial magnetic stimulation scenarios.