Rapid computation of TMS-induced E-fields using a dipole-based magnetic stimulation profile approach

A Closed Formalism for Anatomy-Independent Projection and Optimization of Magnetic Stimulation Coils on Arbitrarily Shaped Surfaces

INTRODUCTION

Transcranial magnetic stimulation (TMS) is a popular method for the noninvasive stimulation of neurons in the brain. It has become a standard instrument in experimental brain research and has been approved for a range of diagnostic and therapeutic applications. These applications require appropriately shaped coils. Various applications have been established or approved for specific coil designs with their corresponding spatial electric field distributions. However, the specific coil implementation may no longer be appropriate from the perspective of available material and manufacturing opportunities or considering the latest understanding of how to achieve induced electric fields in the head most efficiently. Furthermore, in some cases, field measurements of coils with unknown winding or a user-defined field are available and require an actual implementation. Similar applications exist for magnetic resonance imaging coils.

OBJECTIVE

This work aims at introducing a complete formalism free from heuristics, iterative optimization, and ad-hoc or manual steps to form practical stimulation coils with individual turns to either equivalently match an existing coil or produce a given field. The target coil can reside on practically any sufficiently large or closed surface adjacent to or around the head.

METHODS

The method derives an equivalent field through vector projection exploiting the well-known Huygens' and Love's equivalence principle. In contrast to other coil design or optimization approaches recently presented, the procedure is an explicit forward Hilbert-space vector projection or basis change. For demonstration, we map a commercial figure-of-eight coil as one of the most widely used devices and a more intricate coil recently approved clinically for addiction treatment (H4) onto a bent surface close to the head for highest efficiency and lowest field energy.

RESULTS

The resulting projections are within ≤4% of the target field and reduce the necessary pulse energy by more than 40%.

A review of algorithms and software for real-time electric field modeling techniques for transcranial magnetic stimulation

Transcranial magnetic stimulation (TMS) is a device-based neuromodulation technique increasingly used to treat brain diseases. Electric field (E-field) modeling is an important technique in several TMS clinical applications, including the precision stimulation of brain targets with accurate stimulation density for the treatment of mental disorders and the localization of brain function areas for neurosurgical planning. Classical methods for E-field modeling usually take a long computation time. Fast algorithms are usually developed with significantly lower spatial resolutions that reduce the prediction accuracy and limit their usage in real-time or near real-time TMS applications. This review paper discusses several modern algorithms for real-time or near real-time TMS E-field modeling and their advantages and limitations. The reviewed methods include techniques such as basis representation techniques and deep neural-network-based methods. This paper also provides a review of software tools that can integrate E-field modeling with navigated TMS, including a recent software for real-time navigated E-field mapping based on deep neural-network models.

Electric Field Modeling in Personalizing Transcranial Magnetic Stimulation Interventions

The modeling of transcranial magnetic stimulation (TMS)-induced electric fields (E-fields) is a versatile technique for evaluating and refining brain targeting and dosing strategies, while also providing insights into dose-response relationships in the brain. This review outlines the methodologies employed to derive E-field estimations, covering TMS physics, modeling assumptions, and aspects of subject-specific head tissue and coil modeling. We also summarize various numerical methods for solving the E-field and their suitability for various applications. Modeling methodologies have been optimized to efficiently execute numerous TMS simulations across diverse scalp coil configurations, facilitating the identification of optimal setups or rapid cortical E-field visualization for specific brain targets. These brain targets are extrapolated from neurophysiological measurements and neuroimaging, enabling precise and individualized E-field dosing in experimental and clinical applications. This necessitates the quantification of E-field estimates using metrics that enable the comparison of brain target engagement, functional localization, and TMS intensity adjustments across subjects. The integration of E-field modeling with empirical data has the potential to uncover pivotal insights into the aspects of E-fields responsible for stimulating and modulating brain function and states, enhancing behavioral task performance, and impacting the clinical outcomes of personalized TMS interventions.

Fast and accurate computational E-field dosimetry for group-level transcranial magnetic stimulation targeting

Transcranial magnetic stimulation (TMS) is used to study brain function and treat mental health disorders. During TMS, a coil placed on the scalp induces an E-field in the brain that modulates its activity. TMS is known to stimulate regions that are exposed to a large E-field. Clinical TMS protocols prescribe a coil placement based on scalp landmarks. There are inter-individual variations in brain anatomy that result in variations in the TMS-induced E-field at the targeted region and its outcome. These variations across individuals could in principle be minimized by developing a large database of head subjects and determining scalp landmarks that maximize E-field at the targeted brain region while minimizing its variation using computational methods. However, this approach requires repeated execution of a computational method to determine the E-field induced in the brain for a large number of subjects and coil placements. We developed a probabilistic matrix decomposition-based approach for rapidly evaluating the E-field induced during TMS for a large number of coil placements due to a pre-defined coil model. Our approach can determine the E-field induced in over 1 Million coil placements in 9.5 h, in contrast, to over 5 years using a brute-force approach. After the initial set-up stage, the E-field can be predicted over the whole brain within 2-3 ms and to 2% accuracy. We tested our approach in over 200 subjects and achieved an error of <2% in most and <3.5% in all subjects. We will present several examples of bench-marking analysis for our tool in terms of accuracy and speed. Furthermore, we will show the methods' applicability for group-level optimization of coil placement for illustration purposes only. The software implementation link is provided in the appendix.

In-vivoverified anatomically aware deep learning for real-time electric field simulation

Objective.Transcranial magnetic stimulation (TMS) has emerged as a prominent non-invasive technique for modulating brain function and treating mental disorders. By generating a high-precision magnetically evoked electric field (E-field) using a TMS coil, it enables targeted stimulation of specific brain regions. However, current computational methods employed for E-field simulations necessitate extensive preprocessing and simulation time, limiting their fast applications in the determining the optimal coil placement.Approach.We present an attentional deep learning network to simulate E-fields. This network takes individual magnetic resonance images and coil configurations as inputs, firstly transforming the images into explicit brain tissues and subsequently generating the local E-field distribution near the target brain region. Main results. Relative to the previous deep-learning simulation method, the presented method reduced the mean relative error in simulated E-field strength of gray matter by 21.1%, and increased the correlation between regional E-field strengths and corresponding electrophysiological responses by 35.0% when applied into another dataset.In-vivoTMS experiments further revealed that the optimal coil placements derived from presented method exhibit comparable stimulation performance on motor evoked potentials to those obtained using computational methods. The simplified preprocessing and increased simulation efficiency result in a significant reduction in the overall time cost of traditional TMS coil placement optimization, from several hours to mere minutes.Significance.The precision and efficiency of presented simulation method hold promise for its application in determining individualized coil placements in clinical practice, paving the way for personalized TMS treatments.

A fast direct solver for surface-based whole-head modeling of transcranial magnetic stimulation

When modeling transcranial magnetic stimulation (TMS) in the brain, a fast and accurate electric field solver can support interactive neuronavigation tasks as well as comprehensive biophysical modeling. We formulate, test, and disseminate a direct (i.e., non-iterative) TMS solver that can accurately determine global TMS fields for any coil type everywhere in a high-resolution MRI-based surface model with ~ 200,000 or more arbitrarily selected observation points within approximately 5 s, with the solution time itself of 3 s. The solver is based on the boundary element fast multipole method (BEM-FMM), which incorporates the latest mathematical advancement in the theory of fast multipole methods-an FMM-based LU decomposition. This decomposition is specific to the head model and needs to be computed only once per subject. Moreover, the solver offers unlimited spatial numerical resolution. Despite the fast execution times, the present direct solution is numerically accurate for the default model resolution. In contrast, the widely used brain modeling software SimNIBS employs a first-order finite element method that necessitates additional mesh refinement, resulting in increased computational cost. However, excellent agreement between the two methods is observed for various practical test cases following mesh refinement, including a biophysical modeling task. The method can be readily applied to a wide range of TMS analyses involving multiple coil positions and orientations, including image-guided neuronavigation. It can even accommodate continuous variations in coil geometry, such as flexible H-type TMS coils. The FMM-LU direct solver is freely available to academic users.

Real-Time Computation of Brain E-Field for Enhanced Transcranial Magnetic Stimulation Neuronavigation and Optimization

Transcranial Magnetic Stimulation (TMS) coil placement and pulse waveform current are often chosen to achieve a specified E-field dose on targeted brain regions. TMS neuronavigation could be improved by including real-time accurate distributions of the E-field dose on the cortex. We introduce a method and develop software for computing brain E-field distributions in real-time enabling easy integration into neuronavigation and with the same accuracy as 1st -order finite element method (FEM) solvers. Initially, a spanning basis set (< 400) of E-fields generated by white noise magnetic currents on a surface separating the head and permissible coil placements are orthogonalized to generate the modes. Subsequently, Reciprocity and Huygens' principles are utilized to compute fields induced by the modes on a surface separating the head and coil by FEM, which are used in conjunction with online (real-time) computed primary fields on the separating surface to evaluate the mode expansion. We conducted a comparative analysis of E-fields computed by FEM and in real-time for eight subjects, utilizing two head model types (SimNIBS's 'headreco' and 'mri2mesh' pipeline), three coil types (circular, double-cone, and Figure-8), and 1000 coil placements (48,000 simulations). The real-time computation for any coil placement is within 4 milliseconds (ms), for 400 modes, and requires less than 4 GB of memory on a GPU. Our solver is capable of computing E-fields within 4 ms, making it a practical approach for integrating E-field information into the neuronavigation systems without imposing a significant overhead on frame generation (20 and 50 frames per second within 50 and 20 ms, respectively).

A fast direct solver for surface-based whole-head modeling of transcranial magnetic stimulation

Background

When modeling transcranial magnetic stimulation (TMS) in the brain, a fast and accurate electric field solver can support interactive neuronavigation tasks as well as comprehensive biophysical modeling.

Objective

We formulate, test, and disseminate a direct (i.e., non-iterative) TMS solver that can accurately determine global TMS fields for any coil type everywhere in a high-resolution MRI-based surface model with ~200,000 or more arbitrarily selected observation points within approximately 5 sec, with the solution time itself of 3 sec.

Method

The solver is based on the boundary element fast multipole method (BEM-FMM), which incorporates the latest mathematical advancement in the theory of fast multipole methods - an FMM-based LU decomposition. This decomposition is specific to the head model and needs to be computed only once per subject. Moreover, the solver offers unlimited spatial numerical resolution.

Results

Despite the fast execution times, the present direct solution is numerically accurate for the default model resolution. In contrast, the widely used brain modeling software SimNIBS employs a first-order finite element method that necessitates additional mesh refinement, resulting in increased computational cost. However, excellent agreement between the two methods is observed for various practical test cases following mesh refinement, including a biophysical modeling task.

Conclusion

The method can be readily applied to a wide range of TMS analyses involving multiple coil positions and orientations, including image-guided neuronavigation. It can even accommodate continuous variations in coil geometry, such as flexible H-type TMS coils. The FMM-LU direct solver is freely available to academic users.

Fast computational E-field dosimetry for transcranial magnetic stimulation using adaptive cross approximation and auxiliary dipole method (ACA-ADM)

Transcranial Magnetic Stimulation (TMS) is a non-invasive brain stimulation technique that uses a coil to induce an electric field (E-field) in the brain and modulate its activity. Many applications of TMS call for the repeated execution of E-field solvers to determine the E-field induced in the brain for different coil placements. However, the usage of solvers for these applications remains impractical because each coil placement requires the solution of a large linear system of equations. We develop a fast E-field solver that enables the rapid evaluation of the E-field distribution for a brain region of interest (ROI) for a large number of coil placements, which is achieved in two stages. First, during the pre-processing stage, the mapping between coil placement and brain ROI E-field distribution is approximated from E-field results for a few coil placements. Specifically, we discretize the mapping into a matrix with each column having the ROI E-field samples for a fixed coil placement. This matrix is approximated from a few of its rows and columns using adaptive cross approximation (ACA). The accuracy, efficiency, and applicability of the new ACA approach are determined by comparing its E-field predictions with analytical and standard solvers in spherical and MRI-derived head models. During the second stage, the E-field distribution in the brain ROI from a specific coil placement is determined by the obtained rows and columns in milliseconds. For many applications, only the E-field distribution for a comparatively small ROI is required. For example, the solver can complete the pre-processing stage in approximately 4 hours and determine the ROI E-field in approximately 40 ms for a 100 mm diameter ROI with less than 2% error enabling its use for neuro-navigation and other applications. Highlight: We developed a fast solver for TMS computational E-field dosimetry, which can determine the ROI E-field in approximately 40 ms for a 100 mm diameter ROI with less than 2% error.

Fast And Accurate Population Level Transcranial Magnetic Stimulation via Low-Rank Probabilistic Matrix Decomposition (PMD)

Transcranial magnetic stimulation (TMS) is used to study brain function and treat mental health disorders. During TMS, a coil placed on the scalp induces an E-field in the brain that modulates its activity. TMS is known to stimulate regions that are exposed to a large E-field. Clinical TMS protocols prescribe a coil placement based on scalp landmarks. There are inter-individual variations in brain anatomy that result in variations in the TMS-induced E-field at the targeted region and its outcome. These variations across individuals could in principle be minimized by developing a large database of head subjects and determining scalp landmarks that maximize E-field at the targeted brain region while minimizing its variation using computational methods. However, this approach requires repeated execution of a computational method to determine the E-field induced in the brain for a large number of subjects and coil placements. We developed a probabilistic matrix decomposition-based approach for rapidly evaluating the E-field induced during TMS for a large number of coil placements. Our approach can determine the E-field induced in over 1 Million coil placements in 9.5 hours, in contrast, to over 5 years using a bruteforce approach. After the initial set-up stage, the E-field can be predicted over the whole brain within 2-3 milliseconds and to 2% accuracy. We tested our approach in over 200 subjects and achieved an error of < 2% in most and < 3.5% in all subjects. We will present several examples of bench-marking analysis for our tool in terms of accuracy and speed across and its applicability for population level optimization of coil placement.

Highlights

A method for practical E-field informed population-level TMS coil placement strategies is developed.This algorithm enables the determination of E-field informed optimal coil placement in seconds enabling it’s use for close-loop and on-the-fly reconfiguration of TMS.After the initial set-up stage of less than 10 hours, the E-field can be predicted for any coil placement across the whole brain within in 2-3 milliseconds.

Accuracy and precision of navigated transcranial magnetic stimulation

Objective.Transcranial magnetic stimulation (TMS) induces an electric field (E-field) in the cortex. To facilitate stimulation targeting, image-guided neuronavigation systems have been introduced. Such systems track the placement of the coil with respect to the head and visualize the estimated cortical stimulation location on an anatomical brain image in real time. The accuracy and precision of the neuronavigation is affected by multiple factors. Our aim was to analyze how different factors in TMS neuronavigation affect the accuracy and precision of the coil-head coregistration and the estimated E-field.Approach.By performing simulations, we estimated navigation errors due to distortions in magnetic resonance images (MRIs), head-to-MRI registration (landmark- and surface-based registrations), localization and movement of the head tracker, and localization of the coil tracker. We analyzed the effect of these errors on coil and head coregistration and on the induced E-field as determined with simplistic and realistic head models.Main results.Average total coregistration accuracies were in the range of 2.2-3.6 mm and 1°; precision values were about half of the accuracy values. The coregistration errors were mainly due to head-to-MRI registration with average accuracies 1.5-1.9 mm/0.2-0.4° and precisions 0.5-0.8 mm/0.1-0.2° better with surface-based registration. The other major source of error was the movement of the head tracker with average accuracy of 1.5 mm and precision of 1.1 mm. When assessed within an E-field method, the average accuracies of the peak E-field location, orientation, and magnitude ranged between 1.5 and 5.0 mm, 0.9 and 4.8°, and 4.4 and 8.5% across the E-field models studied. The largest errors were obtained with the landmark-based registration. When computing another accuracy measure with the most realistic E-field model as a reference, the accuracies tended to improve from about 10 mm/15°/25% to about 2 mm/2°/5% when increasing realism of the E-field model.Significance.The results of this comprehensive analysis help TMS operators to recognize the main sources of error in TMS navigation and that the coregistration errors and their effect in the E-field estimation depend on the methods applied. To ensure reliable TMS navigation, we recommend surface-based head-to-MRI registration and realistic models for E-field computations.

Computation of transcranial magnetic stimulation electric fields using self-supervised deep learning

Electric fields (E-fields) induced by transcranial magnetic stimulation (TMS) can be modeled using partial differential equations (PDEs). Using state-of-the-art finite-element methods (FEM), it often takes tens of seconds to solve the PDEs for computing a high-resolution E-field, hampering the wide application of the E-field modeling in practice and research. To improve the E-field modeling's computational efficiency, we developed a self-supervised deep learning (DL) method to compute precise TMS E-fields. Given a head model and the primary E-field generated by TMS coils, a DL model was built to generate a E-field by minimizing a loss function that measures how well the generated E-field fits the governing PDE. The DL model was trained in a self-supervised manner, which does not require any external supervision. We evaluated the DL model using both a simulated sphere head model and realistic head models of 125 individuals and compared the accuracy and computational speed of the DL model with a state-of-the-art FEM. In realistic head models, the DL model obtained accurate E-fields that were significantly correlated with the FEM solutions. The DL model could obtain precise E-fields within seconds for whole head models at a high spatial resolution, faster than the FEM. The DL model built for the simulated sphere head model also obtained an accurate E-field whose average difference from the analytical E-fields was 0.0054, comparable to the FEM solution. These results demonstrated that the self-supervised DL method could obtain precise E-fields comparable to the FEM solutions with improved computational speed.

The effect of meninges on the electric fields in TES and TMS. Numerical modeling with adaptive mesh refinement

BACKGROUND

When modeling transcranial electrical stimulation (TES) and transcranial magnetic stimulation (TMS) in the brain, the meninges - dura, arachnoid, and pia mater - are often neglected due to high computational costs.

OBJECTIVE

We investigate the impact of the meningeal layers on the cortical electric field in TES and TMS while considering the headreco segmentation as the base model.

METHOD

We use T1/T2 MRI data from 16 subjects and apply the boundary element fast multipole method with adaptive mesh refinement, which enables us to accurately solve this problem and establish method convergence at reasonable computational cost. We compare electric fields in the presence and absence of various meninges for two brain areas (M1_HAND and DLPFC) and for several distinct TES and TMS setups.

RESULTS

Maximum electric fields in the cortex for focal TES consistently increase by approximately 30% on average when the meninges are present in the CSF volume. Their effect on the maximum field can be emulated by reducing the CSF conductivity from 1.65 S/m to approximately 0.85 S/m. In stark contrast to that, the TMS electric fields in the cortex are only weakly affected by the meningeal layers and slightly (∼6%) decrease on average when the meninges are included.

CONCLUSION

Our results quantify the influence of the meninges on the cortical TES and TMS electric fields. Both focal TES and TMS results are very consistent. The focal TES results are also in a good agreement with a prior relevant study. The solver and the mesh generator for the meningeal layers (compatible with SimNIBS) are available online.

Fast Individualized High-resolution Electric Field Modeling for Computational TMS Neuronavigation

Transcranial Magnetic Stimulation (TMS) is a non-invasive method for safe and painless activation of cortical neurons. On-line visualization of the induced Electric field (E-field) has the potential to improve quantitative targeting and dosing of stimulation, however present commercially available systems are limited by simplified approximations of the anatomy. Here, we developed a near real-time method to accurately approximate the induced E-field of a freely moving TMS coil with an individualized high-resolution head model. We use a set of magnetic dipoles around the head to approximate the total E-field of a moving TMS coil. First, we match the incident field of the dipole basis set with the incident E-field of the moving coil. Then, based on the principle of superposition and uniqueness of the solutions, we apply same basis coefficients to the total E-field of the basis set. The computed E-fields results show high similarity with an established TMS solver both in terms of the amplitude and the spatial distribution patterns. The proposed method enables rapid visualization of the E-field with ~100 ms of computation time enabling interactive planning, targeting, dosing and coil positioning tasks for TMS neuronavigation.

Rapid whole-brain electric field mapping in transcranial magnetic stimulation using deep learning

Transcranial magnetic stimulation (TMS) is a non-invasive neurostimulation technique that is increasingly used in the treatment of neuropsychiatric disorders and neuroscience research. Due to the complex structure of the brain and the electrical conductivity variation across subjects, identification of subject-specific brain regions for TMS is important to improve the treatment efficacy and understand the mechanism of treatment response. Numerical computations have been used to estimate the stimulated electric field (E-field) by TMS in brain tissue. But the relative long computation time limits the application of this approach. In this paper, we propose a deep-neural-network based approach to expedite the estimation of whole-brain E-field by using a neural network architecture, named 3D-MSResUnet and multimodal imaging data. The 3D-MSResUnet network integrates the 3D U-net architecture, residual modules and a mechanism to combine multi-scale feature maps. It is trained using a large dataset with finite element method (FEM) based E-field and diffusion magnetic resonance imaging (MRI) based anisotropic volume conductivity or anatomical images. The performance of 3D-MSResUnet is evaluated using several evaluation metrics and different combinations of imaging modalities and coils. The experimental results show that the output E-field of 3D-MSResUnet provides reliable estimation of the E-field estimated by the state-of-the-art FEM method with significant reduction in prediction time to about 0.24 second. Thus, this study demonstrates that neural networks are potentially useful tools to accelerate the prediction of E-field for TMS targeting.

Degree of improving TMS focality through a geometrically stable solution of an inverse TMS problem

The Transcranial Magnetic Stimulation (TMS) inverse problem (TMS-IP) investigated in this study aims to focus the TMS induced electric field close to a specified target point defined on the gray matter interface in the M1_HAND area while otherwise minimizing it. The goal of the study is to numerically evaluate the degree of improvement of the TMS-IP solutions relative to the well-known sulcus-aligned mapping (a projection approach with the 90^∘ local sulcal angle). In total, 1536 individual TMS-IP solutions have been analyzed for multiple target points and multiple subjects using the boundary element fast multipole method (BEM-FMM) as the forward solver. Our results show that the optimal TMS inverse-problem solutions improve the focality - reduce the size of the field "hot spot" and its deviation from the target - by approximately 21-33% on average for all considered subjects, all observation points, two distinct coil types, two segmentation types, two intracortical observation surfaces under study, and three tested values of the field threshold. The inverse-problem solutions with the maximized focality simultaneously improve the TMS mapping resolution (differentiation between neighbor targets separated by approximately 10 mm) although this improvement is quite modest. Coil position/orientation and conductivity uncertainties have been included into consideration as the corresponding de-focalization factors. The present results will change when the levels of uncertainties change. Our results also indicate that the accuracy of the head segmentation critically influences the expected TMS-IP performance.