Electric field modeling in personalizing TMS interventions

Abstract

The modeling of TMS-induced electric fields (E-fields) serves as 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-field responsible for stimulating and modulating brain function and states, enhancing behavioral task performance, and impacting the clinical outcomes of personalized TMS interventions.

1. Introduction

Transcranial magnetic stimulation (TMS) is a widely used noninvasive brain stimulation technique with applications in both neuroscience research and therapeutic interventions for neurological and psychiatric disorders (1). By delivering brief current pulses through a transducing coil, TMS generates a magnetic field that induces a time-varying electric field (E-field) in the brain. The spatial features of the induced E-field are critical for determining the region and depth of brain tissue stimulated. The spatial properties of the E-field are influenced by several key determinants: coil positioning, individual head anatomy as well as tissue electrical properties (conductivities), and relative orientation of cortical neurons with respect to the E-field. Achieving precise and focal E-field delivery is crucial for target engagement and avoiding undesired effects on off-target regions. The ability to modulate neural activity in specific brain networks with high spatial resolution is key to improving the efficacy and selectivity of TMS interventions.

Computational modeling has emerged as a valuable tool for understanding the properties and effects of the TMS induced E-field. These models enable quantifying the delivered “E-field dose” in terms of strength, focality, and spatial distribution, accounting for subject specific anatomy. They also enable the determination of optimal TMS parameters that maximize the E-field intensity in the region of interest (ROI), and hence target engagement, potentially enhancing the efficiency of TMS interventions (2,3).

In this review, we present the state-of-the-art of using E-field models in personalized TMS interventions. We will discuss computational methods used to optimize coil placement, explore experimental applications of E-field modeling incorporating physiological measures, and highlight the potential for future developments in this field. By organizing the existing knowledge on E-field modeling in TMS, we aim to contribute to a deeper understanding of its implications for improving the efficacy and precision of personalized TMS interventions.

2. Case for individualization

TMS dosing conventionally relies on defining the treatment intensity as the maximum stimulator output (%MSO), whether through a fixed intensity or an individually titrated motor threshold (MT), even for non-motor cortex stimulation. At a fixed %MSO, E-field modeling shows large inter-individual variability in stimulation intensity and spread (4,5). This variability primarily arises from varying coil-to-cortex distances and cortical gyrification across individuals, influenced by factors like age-related brain atrophy (6), sex (7), and cognitive impairment (8). Coil-to-cortex distances significantly impact the E-field, as the E-field strength attenuates with increasing distance (9), and distance-adjusted MT (9,10) have been proposed.

E-field modeling has informed TMS depression treatment by guiding coil targeting and exploring the dose–response relationship between E-field and clinical outcomes. For example, E-field models have been used in patients with alcohol use disorders (AUD) receiving TMS targeting the dorsolateral prefrontal cortex (DLPFC), medial prefrontal cortex (MPFC), and motor cortex (MC) (11). The E-field in MPFC was significantly lower than that in the DLPFC and MC. Patients with AUD, who exhibit decreased gray matter volume compared to healthy controls, may require higher E-field doses to achieve the desired therapeutic effect (11). In another study on depressed adolescents receiving MRI-guided rTMS treatment, E-field modeling was used to compare DLPFC targeting approaches (3). The conventional 5-cm rule targeting resulted in the lowest DLPFC E-field compared to the F3 and MRI-guided targeting, which could partially explain negative clinical trial results utilizing the 5-cm rule in adolescents (12). Furthermore, the F3 method exhibited more variability in the induced E-field across individuals compared to the 5-cm and MRI-guided targets. Higher induced E-field strength in the left DLPFC correlated with greater reductions in Children’s Depression Rating Scale–Revised (CDRS) scores (3). Similarly, adult studies have also reported that higher induced E-field in the left and right DLPFC correlated with greater reduction in Inventory of Depressive Symptomatology (IDS) scores (13,14).

The cost of performing E-field modeling for TMS targeting and dosing involves expenses related to imaging, data processing, computational resources, system equipment, and personnel training. Constructing a subject-specific head model for accurate E-field simulations require at least a high-resolution structural MRI scan (T1-weighted; ideally also a T2-weighted scan, and optionally a diffusion-weighted scan). When acquiring fMRI to guide TMS targeting, the cost of additional structural imaging may be marginal. It may also be feasible to approximate the field strength from a template head model in the absence of individual neuroimaging data (15), though this may result in trade-off with accuracy. The cost of software for E-field modeling can vary. Certain software packages are open-source, easily accessible, and user-friendly, while others are commercial, demanding licensing fees and necessitating a higher degree of specialized expertise to operate. Additional computational resources, such as high-performance computing services may be needed to run optimization algorithms that require many E-field simulations. Finally, to extract the greatest benefit from the simulations and optimization, neuronavigation and coil holder systems are required for precise TMS coil placement and tracking during the intervention.

3. Computational approach

In the following sections, we describe E-field modeling methodology (refer to Figure 1), including TMS physics, coil and head model complexity and various numerical techniques.

Figure 1.

Pipeline of E-field modeling methods: head imaging using magnetic resonance imaging (MRI), tissue segmentation, diffusion weighted/tensor imaging (DWI/DTI), surface or volumetric meshing of the segmentation, coil model selection and intensity specifications, coil placement on scalp, (numerical) E-field solver, E-field extraction/visualization/statistics, and optional repeated utilization to find the most optimal coil scalp placement, and usage in neuronavigation.

3.1. TMS physics and what quantities are being solved

The physics underlying TMS is the phenomenon of electromagnetic induction, a changing magnetic field induces an E-field within a conductor. The magnetic field pulses have relatively low frequencies (< 10 kHz), thus can be analyzed within the quasi-static regime (16). In this regime, conduction currents within the head are assumed to have negligible inductive coupling effects. As a result, the magnetic field can be completely described by the magnetic vector potential (A) induced by the TMS coil within free space. The overall induced E-field (E) is composed of two distinct contributions. The primary component, ${\mathit{E}}_{\text{p}}$, is linked to the coil’s changing magnetic field: ${\mathit{E}}_{\text{p}}=-\frac{\partial }{\partial t}\mathit{A}$. To calculate ${\mathit{E}}_{\text{p}}$, the Biot-Savart law can be employed (17). The secondary E-field component, ${\mathit{E}}_s$, originates from electric charge accumulation at tissue interfaces with varying conductivities, generating an additional effect (${\mathit{E}}_{\text{s}}=-\nabla \phi$; where φ is the electric potential). The determination of ${\mathit{E}}_{\text{s}}$ involves solving the relationship:

$$\nabla \cdot \left(\sigma \left(-\nabla \phi -\frac{\partial }{\partial t}\mathit{A}\right)\right)=\nabla \cdot \left(\sigma \left(-{\mathit{E}}_{\text{s}}-{\mathit{E}}_{\text{p}}\right)\right)=0,$$ (Equation 1)

where σ is the tissue conductivity. This equation is subject to the Neumann boundary condition, which ensures the continuity of the normal component of the current density through any boundary (no charge accumulation in time).

3.2. Head modeling

The TMS-induced E-field is relatively insensitive to nominal changes in tissue (18). Due to the low skull conductivity, ${\mathit{E}}_{\text{s}}$ generated by charges at the scalp–air and scalp–skull boundaries are largely blocked from reaching the brain. Consequently, the E-fields within the brain predominantly affected by the boundary shapes of intracranial compartments, particularly skull–cerebrospinal fluid (CSF) and CSF–gray matter (GM) interfaces (19,20).

To represent the intricate composition of head tissues, modeling approaches range from approximations using spherical models to more precise mesh-based representations incorporating polyhedra, to account for realistic tissue volume and surface. Concentric sphere and multisphere (21) head model have been used due to its simplified geometry that allows for analytical solutions to be computed (22). However, spherical models fall short in accurately representing head shape or complex geometric details, such as cortical gyrification (23) that substantially influences the local E-field (19). In contrast, anatomically realistic head models derived from individual neuroimaging modalities, such as structural magnetic resonance imaging (MRI), necessitate numerical solutions but provide significantly enhanced E-field predictions compared to spherical approximations. Different tissues (scalp, skull, CSF, GM, white matter (WM)) are first segmented from the structural images of the head, then converted into a mesh of interconnected elements. The accurate representation of the discretized tissues (computational mesh) is generally important to avoid anomalies during the subsequent numerical solving step. Quality of the mesh can be improved by using adaptive mesh refinement for thin tissues (24), as well as preventing intersecting defects, self-intersections, and ensuring an adequate number of elements (25).

Electrical conductivity values are assigned to different tissues based on empirical data or literature. These conductivity values are typically assumed to be constant within the mesh element and tissues. Conductivity anisotropy can be modeled in the WM using tensors derived from diffusion-weighted MRI and has been shown to increase E-field magnitude perpendicular to the gyrus (23). When considering WM conductivity anisotropy, a notable enhancement of up to +7% in WM E-field magnitude is anticipated for strictly perpendicular TMS coil placements, rising to approximately +40% otherwise. In WM regions with large principal fiber bundle directions connecting different lobes of the brain (e.g., corpus callosum, arcuate fasciculus), the anisotropy-caused E-field increase is most pronounced.

3.3. Coil modeling

After the head model generation, the TMS coil geometry and characteristics need to be incorporated. This involves representing the coil’s winding geometry, size, orientation, and current flow. The coil′s field output can be modeled using varying levels of abstraction to strike a balance between computational efficiency and accuracy. These approaches encompass treating the coil’s wire windings as continuous, segmented entities, or even as electric and magnetic dipoles. Gomez et al. (25) recommended to model coil winding with rectangular cross-sections and discrete turn or use over 3000 distributed current dipoles or >200 distributed magnetic dipoles to reduce the numerical error below 2% (25). Recently, calibrated magnetic dipole coil models for the 25 most widely used coil types have been made available (26). Analyzing the E-field characteristics of distinct TMS coils can guide the selection.

3.4. E-field solver

Various approaches (Table S1) are available to solve for the E-field distribution. For spherical head models, analytical expressions can be obtained for the overall E-field (17,27). Numerical techniques primarily adopt a fundamental approach of approximating the unknown field through a summation of basis functions.

Classical numerical methods employed for TMS simulations encompass the finite difference method (FDM), finite element method (FEM), and boundary element method (BEM). FDM starts from a head voxel approximation (e.g., tissue segmentation; (28)) and minimizes the residual error of nodal potentials and gradients in the voxel mesh (29). In contrast, FEM integrates a testing function multiplied with the residual approximating the same solution. FEM holds a distinct advantage over FDM due to its capacity to represent smooth head tissue surfaces through adaptable polyhedral-type elements. BEM formulates the problem as equivalent integral equations assuming constant conductivities between the tissue surfaces (30–34).

To speed up computations, classical numerical techniques can be optimized through modifications, including the adoption of different strategies for direct, using precalculated matrix factorization (35) and inversion (36), or accelerated iterative (37) solvers using multi-grid methods (28). For specific applications, if the E-field is only needed in limited cortical region under the coil (not whole brain), electromagnetic equivalency principles (reciprocity (2,38) or Huygen’s principle (39)) can be exploited.

Novel approaches using machine learning techniques offer possibilities to determine subject-specific E-fields without the time-consuming anatomical head modeling efforts ((40), Table S1). In such approaches, the mapping from individual structural MRI to E-fields (computed by conventional numerical methods) is learned through a supervised learning framework (41). Alternatively, given a head model and ${\mathit{E}}_{\text{p}}$, a self-supervised deep learning model can be trained to solve for the total E-field (42).

3.5. Software packages

Open-source and commercial software can be utilized to run parts or an entire TMS E-field modeling pipeline, including coil placement optimization. For example, SimNIBS (35) can take MRI (single T1, optional: T2/DTI) data as inputs, generate head and coil model, and solve TMS simulations numerically. SCIRun (43) is another open-source package that simulate TMS E-fields with an input head model. Other commercial tools such as Sim4Life (44) and other multiphysics packages such as ANSYS Maxwell (45) or COMSOL (46) can also be utilized to perform E-field simulations. Some of these multiphysics tools may require additional software (e.g., SimpleWare ScanIP (47)) to generate the head model from the imaging data, but are more flexible in incorporating non-standard coil models or more complex geometries.

4. Using the E-field: metrics, optimal brain targeting

4.1. E-field metrics

Given a brain ROI, the voxel-wise E-field magnitude can be summarized using standard descriptive statistics such as maximum, mean, median, or percentile. In choosing a suitable summary metric, it is important for the metric to be resilient against computational inaccuracies. Since computational methods adopt a variational framework to reduce the error, pointwise estimates (e.g., the E-field maximum at tissue boundaries) could be inaccurate, even if the solution is globally accurate. In addition, the E-field maximum may not be located within the targeted region due to local gyrification. In an example illustrated in Figure 2A, the TMS coil was placed to maximize the E-field strength at the motor cortex target (black dot), however, since the adjacent gyrus has smaller coil-to-cortex distance, the peak E-field is induced outside of the target (within an arbitrarily chosen ROI of 15mm radius) in the post-central gyrus (red dot). The average E-field within the ROI is also not a reliable metric in many cases, since cortical E-field distribution is skewed to smaller values (Figure 2B). Other metrics such as median and percentiles also depend heavily on the size of the ROI (see Figure 2B). Alternatively, certain metrics aim to determine an E-field value that guarantees a specified volume with the ROI exposed to an E-field exceeding that strength. For instance, the notation “E₁₀₀” denotes the E-field strength value that at least 100 ROI voxels are exposed to (48,49). The E₁₀₀ (and its variations such as E₅₀, E₂₀, etc.) is invariant to ROI size (if there are an adequate number of voxels) as illustrated in Figure 2.

Figure 2.

(A) Simulated TMS-induced cortical E-field (Ernie model generated from SimNIBS’s mri2mesh pipeline using MagVenture Cool-B65 coil model at stimulation intensity of 50%MSO). The TMS coil was placed to maximize the E-field strength at the motor cortex target (black dot in precentral gyrus), however, since the adjacent gyrus has smaller coil-to-cortex distance, the peak E-field is induced outside of the target in the postcentral gyrus (red dot). (B) E-field values extracted from a large ROI (whole-brain gray matter) and a smaller ROI (turquoise, 15-mm radius sphere outlined in (A). Different E-field metrics (mean, median, 95%tile, 99%tile, E₁₀₀, and maximum) are indicated. E₁₀₀ denotes the E-field strength that at least 100 ROI voxels are exposed to. Note that some E-field metrics, such as the mean, median, and percentiles, depend heavily on the size of the ROI, whereas E₁₀₀ is invariant to ROI size.

4.2. TMS coil placement optimization for maximal ROI E-field

E-field dosimetry tools offer the capability to select the most favorable coil placement from an array of potential setups, aiming to maximize the E-field delivered to a specific target ROI. These ROIs are commonly derived either from structural or functional neuroimaging data. In this section, we briefly review methods for target identification and strategies for coil optimization.

4.2.1. Target identification

Various approaches are utilized to identify ROIs for TMS targeting, employing both structural and functional neuroimaging methods. Structural-based strategies rely on anatomical (50) or scalp (3) landmarks to pinpoint TMS targets. For example, DTI-guided TMS specifically targets fiber bundles near cortical surfaces anatomically connected to deeper brain structures (51–54). Alternatively, fMRI-guided TMS often employs seed-based functional activation or connectivity during task or at rest (55–57). Individualized fMRI-based targeting has been shown to be more effective than group-based targeting using TMS (58,59). An alternative functional-based method involves positron emission tomography (PET) imaging, which quantifies regional cerebral blood flow within a predefined ROI (60,61). Despite its potential to enhance TMS targeting accuracy, PET-guided rTMS has been found to be equally effective as the 5-cm rule in a depression study (62). These approaches have limitations as they focus solely on the seed region and may not account for activations beyond it (63–65). Graph-based network analysis of fMRI provides a holistic view of whole-brain connectivity to identify ROIs based on connectivity parameters (66–68). Recent studies have integrated fMRI analysis with computational techniques to optimize coil placement and orientation within predefined ROIs (66,69–71).

4.2.2. Coil placement optimization

Identifying optimal coil placements involves evaluating numerous candidate coil configurations to select the one that maximizes the chosen metric of the ROI E-field component or magnitude. Generally, minor variations in coil location and orientation result in gradual spatial changes of the E-field (e.g., (2)). The precision in coil placement that can be achieved by manual or robotic coil holding is on the order of 1 mm and 1° orientation (49,72). The number of candidate coil configurations should be chosen to balance computational effort and practicality of the physical coil placement.

The search through candidate coil configurations represents an optimization problem, which to date, has been solved for TMS using (1) a genetic algorithm (73) and (2) a discrete search approach (e.g., (2,35)). Inspired from natural selection in evolution, genetic algorithms starts with a random set (‘population’) of coil placements and iteratively produce new instances (‘generations’) that, on average, improve the previous ones, eventually converging to the optimal placement (73). The discrete search is another approach, which is implemented in the prominent open-source software package SimNIBS (35). This method uses a search procedure on a discrete grid of candidates to find the optimal coil placement with maximal E-field averaged across the ROI mesh elements. SimNIBS serves also as the computational backbone for three TMS targeting pipelines: TArgeted functional Network Stimulation (TANS, (66)), the TMS targeting pipeline by Balderston et al. (74), and Targeting and Analysis Pipeline (TAP, (49)). In additional to the head model generated by SimNIBS, the labeled target ROI (*.nii NIFTI file format) is provided as input. The TMS targeting pipeline and TANS both aim to determine the best coil placement using fMRI data. The former does so by defining a spherical region around task-evoked peak activity (within a group mask), while the latter maximizes the cortical overlap between resting-state activity and the simulated E-field. Both pipelines have similar computational complexities, and the overall run-time has been reported to be 24–48h (74), including head model generation (up to 15h), solve and output file writing time (20–30minutes per coil position, for several hundred candidate positions). Since each simulation does not depend on another, all simulations could be run in parallel on a high-performance compute cluster. Both pipelines focus on maximizing the ROI E-field magnitude only. Additionally, TANS also allows for minimizing non-ROI E-fields.

SimNIBS offers a more efficient way to perform coil placement optimization, in which only a single target coordinate together with radius specifies all gray matter elements of the ROI. With the Pardiso direct solver a single TMS simulation takes only a few seconds computation time (see Table 1). Alternatively, the auxiliary dipole method (ADM) is implemented in SimNIBS, which offers an even faster technique to find the optimal coil placement. ADM exploits a specific electromagnetic equivalency principle of reciprocity which allows to drastically reduce computation for dense coil placement optimization (i.e., up to millions of candidate positions) time on a regular computer to a few minutes.

TAP uses ADM and allows for additional functionality such as optimizing the coil placement based on an E-field component (i.e., perpendicular to cortex derived from normal direction of closest sulci wall). This optimization constraint is supported by SimNIBS and may restrict the coil orientation with the coil handle also being perpendicular to the sulci wall of a TMS coil. A coil orientation perpendicular to the sulci wall has been reported in studies of the motor cortex (75–77) and is commonly applied in TMS interventions. TAP also offers possibilities to scale the TMS coil current intensity based on %MSO and to deliver a particular ROI-induced E-field value for a chosen metric.

5. Clinical applications

Present E-field modeling aims to develop better methods for achieving consistent TMS dosing across individuals to reduce variability in interventional outcomes. Within the context of TMS motor mapping, hotspots and input–output functions are assessed. The sigmoidal input–output curve estimates the amount of neuronal recruitment as a function of stimulus intensity, represented as either %MSO or simulated motor E-fields (78,79). E-field modeling has been instrumental in quantifying the requisite intensity and pinpointing the representation of muscle groups within the primary motor cortex (79,80). Cortical locations and thresholds for muscle activation can be color-coded to visualize MEPs, statistical measures, and combined with simulated E-field estimates (79–82). The approach provides an opportunity to systematically understand the dose–response relationships of different muscle groups, including their interindividual variability.

To ensure uniform dosing across participants, a study using rTMS to enhance working memory in a group of older adults used E-field models to deliver a predetermined E-field dose to an individual peak fMRI activation hotspot in the left lateral parietal cortex (48). The E-field dose was set at 56 V/m, which represents the participant-averaged E₁₀₀ value from pilot findings (48). The E-field model also considered hair thickness to further refine the coil-to-cortex distance. Utilizing this E-field model-inform dosing strategy, a site-specific effect of rTMS were observed, resulting in the downregulation of working memory. Another approach to achieving uniform dosing is the “A Personalized E-field X” (APEX) method (83), which integrates MT measurements and E-field modeling to reduce over- and under-dosing in the prefrontal cortex stimulation. The E-field at the motor hotspot that corresponded to 100% rMT stimulation intensity was modeled. Stimulation intensity at the prefrontal cortex was set to achieve the equivalent E-field strength as 100% rMT stimulation over the motor cortex. However, this method has not been used in any clinical study prospectively.

E-field-based coil placement optimization pipelines have been proposed to enhance treatment outcomes in patients with psychiatric disorders. For instance, the TAP was used in a group of depressed adolescents; it was shown that the model-derived coil placement produced a higher E-field in the DLPFC target compared to the 5-cm, F3, and MRI-guided targets (3). Similarly, the TANS approach was used in a sample of healthy controls and depressed patients to maximize induced E-field within a targeted functional network and therefore improving the stimulation specificity (66). TANS achieved significantly more selective stimulation of the frontoparietal network in the patient group, and fMRI-based targeting of the somatomotor network in the healthy controls compared to ADM. Overall, TANS increased the on-target value by approximately 51% compared to using the ADM method. This method produces higher intervariability compared to the ADM method but could be beneficial for subtypes of depression. However, TANS has not yet been integrated into a prospectively treatment study for TMS target selection.

Some of these E-field optimization techniques are being integrated into multiple ongoing clinical trials. For example, the TAP approach is presently being used in the depression treatment study (Concurrent fMRI-guided rTMS and Cognitive Therapy for Treatment of Major Depressive Episodes study, NCT03289923). TAP informs the optimal TMS coil placement to deliver stimulation to an individually targeted region of the DLPFC identified via task fMRI. Additionally, the Adaptive Trial for the Treatment of Depressive Symptoms associated with Concussion using Repetitive Transcranial Magnetic Stimulation Protocols study (NCT05426967) is conducting in-between comparisons between scalp targeting, structural targeting, and individualized connectome targeting and E-field modeling to deliver maximum E-field magnitude to the target region. These studies aim to improve and optimize therapeutic effects by using E-field modeling to account for individual differences and maximize dose delivery and target engagement.

6. Conclusions

There are several pivotal avenues where the integration of E-field modeling can profoundly impact the practice of TMS. First, E-field modeling holds promise for enhancing TMS dosing precision. Subject-specific, anatomically accurate computational models can provide detailed insights into the distribution and magnitude of induced E-fields within the brain. This capability can enhance the precision of TMS dosing, allowing for individualized stimulation strategies that account for individual variability in head anatomy and tissue conductivity. The emergence of rapid E-field solvers presents opportunities for real-time applications within neuronavigation and closed-loop paradigms. By combining E-field modeling with systems such as electroencephalography, TMS coil placement and intensity can be dynamically updated to optimize stimulation based on real-time brain activity patterns. The combination of TMS with neuroimaging methods opens doors to systematically mapping dose–response relationships for both motor and non-motor regions of interest. Finally, a compelling direction involves synergizing E-field simulations with neuronal and plasticity models to gain deeper insights into the mechanisms underlying TMS effects. This integration offers the potential to unravel how TMS modulates neural circuits and plasticity processes, thereby illuminating the intricate workings of TMS-induced neurophysiological changes. Finally, it is essential to acknowledge that despite the promising insights provided by E-field modeling, there may be limitations and challenges associated with its application. To fully harness the potential benefits, experimental verification of strategies informed by E-field modeling is necessary.