Electric field depth–focality tradeoff in transcranial magnetic stimulation: simulation comparison of 50 coil designs

Abstract

Background

Various transcranial magnetic stimulation (TMS) coil designs are available or have been proposed. However, key coil characteristics such as electric field focality and attenuation in depth have not been adequately compared. Knowledge of the coil focality and depth characteristics can help TMS researchers and clinicians with coil selection and interpretation of TMS studies.

Objective

To quantify the electric field focality and depth of penetration of various TMS coils.

Methods

The electric field distributions induced by 50 TMS coils were simulated in a spherical human head model using the finite element method. For each coil design, we quantified the electric field penetration by the half-value depth, d_1/2, and focality by the tangential spread, S_1/2, defined as the half-value volume (V_1/2) divided by the half-value depth, S_1/2 = V_1/2/d_1/2.

Results

The 50 TMS coils exhibit a wide range of electric field focality and depth, but all followed a depth–focality tradeoff: coils with larger half-value depth cannot be as focal as more superficial coils. The ranges of achievable d_1/2 are similar between coils producing circular and figure-8 electric field patterns, ranging 1.0–3.5 cm and 0.9–3.4 cm, respectively. However, figure-8 field coils are more focal, having S_1/2 as low as 5 cm² compared to 34 cm² for circular field coils.

Conclusions

For any coil design, the ability to directly stimulate deeper brain structures is obtained at the expense of inducing wider electrical field spread. Novel coil designs should be benchmarked against comparison coils with consistent metrics such as d_1/2 and S_1/2.

Introduction

Transcranial magnetic stimulation (TMS) is a non-invasive technique that is used or investigated for numerous research and therapeutic applications, including the study of normal and pathological brain function and the treatment of neurological and psychiatric disorders.^1,2 TMS uses brief, intense pulses of electric current delivered to a coil placed on the subject’s head to generate an electric field in the brain via electromagnetic induction. The induced electric field modulates the neural transmembrane potentials and, thereby, neural activity. The locus of activation in the brain is approximately in the area where the induced electrical field is maximal^3,4; this location, in turn, depends on the stimulating coil geometry and placement. For the purpose of determining a map of direct neural activation by TMS, the induced electric field distributions generated by different coil types have been characterized by theoretical calculations,^5–13 numerical simulation models,^14–21 and measurements of the electric currents induced in phantoms^22,23 or in vivo.^24,25 However, the analytic studies use idealized circular and/or figure-8 coil geometries, and only a handful of commercial coils have been modeled in computational studies.^21,26–28 Thus, electric field distribution data for many commercial or experimental TMS coils are still lacking. Knowledge of the electric field spatial distribution of specific coils and how it compares to other coils is valuable in the design and interpretation of basic research and clinical studies, as well as in the development of novel coils.

Two electric field spatial features of particular interest are depth of penetration and focality. Proposed or implemented coil designs have often been developed with the objective of improving one or both of these field characteristics. For example, in applications where the stimulation target is small (e.g., hand muscle representations in the primary motor cortex), localization of the induced electric field is important in order to minimize the stimulation of non-target regions. There has also been substantial interest in direct, non-invasive stimulation of brain regions deeper than the superficial cortex. For example, the subcallosal cingulate cortex, which lies approximately 6 cm from the head surface, is a putative target for the treatment of depression.^29,30 However, the design of TMS coils to stimulate such deep brain targets is limited by the rapid attenuation of the electric field in depth. It has been mathematically proven that at the quasi-static frequencies used in TMS the electric field is always stronger on the surface than inside of a spherically symmetric volume conductor.³¹ Further, the electric field in the center of a uniformly conducting sphere is always zero.^10,13,32,33 Therefore, direct stimulation with TMS of regions near the center of the head appears impossible.

It is known that coils with larger dimensions generate an electric field that penetrates deeper but is less focal than that of smaller coils.^3,33–35 However, this electric field depth–focality tradeoff is not always acknowledged when novel TMS coil designs are proposed and has not been characterized with a uniform set of metrics. This is a serious limitation to clinical and basic neuroscience applications because stimulation of non-target brain regions may affect clinical outcomes, and certainly affects the degree to which any observed changes in behavior can be attributed to stimulation of the deep target alone rather than the neighboring regions that are also affected by the broader field. Furthermore, the electric field depth and focality may relate to risk of accidental seizure and other adverse side effects. Consequently, the safety guidelines for the use of repetitive TMS which are based on studies with ~70 mm figure-8 coils³⁶ should be applied with caution to coil designs generating significantly different electric field characteristics.

The first TMS system used a circular coil (see, e.g., Figure 1(2, 4, 5)) due to its simple geometry and, hence, ease of construction.^37,38 However, the circular coil induces a nonfocal ring-shaped electric field maximum potentially stimulating a swath of brain regions under the coil perimeter. Attempts have been made to focalize the stimulation site with circular coils by introducing an angulated extension in the winding³⁹ or by modifying the winding density^40,41 or concavity.⁴² These approaches, however, only marginally enhance the circular coil’s ability to produce a focal field.⁴³ Thus far, the most significant improvement of TMS focality has been the introduction of the figure-8 coil (see, e.g., Figure 1(30, 31)), which has generally been credited to Ueno and colleagues,⁴⁴ although this coil configuration had been proposed earlier for the purpose of more localized sensing of magnetoencephalographic sources,⁴⁵ which is the inverse problem of focal TMS. The figure-8 configuration consists of a pair of adjacent circular loops with current flow in opposite directions, producing a relatively focal electric field maximum under the center of the coil where the two loops meet.⁴⁴ Cohen and Cuffin studied the focality of the figure-8 coil and found improvements with decreasing loop diameter down to 2.5 cm, at which point heating and stress limitations would necessitate sophisticated coil fabrication techniques.^32,46 Nevertheless, the search for even more focal TMS coils continues. The “cloverleaf” coil design consists of four sets of nearly circular windings (Figure 1(45)) and has been shown to be more efficient in stimulating long fibers compared to the figure-8 coil.^34,47–51 The “slinky” coil design consists of multiple circular or rectangular loops joined together at one edge and fanned out to form a half toroid (Figure 1(23, 24)).^12,52–55 Knäulein and Weyh introduced a figure-8 coil with eccentric windings producing higher density of winding turns toward the center of the coil than in the periphery (Figure 1(22)),⁵⁶ which has been shown to have better focality compared to the regular figure-8 and slinky coils of the same outer loop diameter.^57–59 The 3-D differential coil (Figure 1(41)) consists of a small figure-8 coil with a third loop positioned perpendicular to the center of the figure-8 coil and flanked by two additional loops to restrict the area of stimulation.⁶⁰ This 3-D differential coil has been shown to provide more focal stimulation compared to the figure-8 and slinky coils.^60,61 Other efforts to control the TMS electric field focality include the use of litz wire to construct small diameter figure-8 coils (Figure 1(21)),^62,63 conductive shielding plates (Figure 1(32)),^64–68 active shields (Figure 1(33)),^68,69 and ferromagnetic cores (Figure 1(3, 10, 11, 34, 40)).^28,70

Figure 1.

Simulation models of 50 TMS coil configurations: (1) Animal mini-coil,¹⁴⁵ (2) Magstim 50 mm circular coil (P/N 9999),¹⁴⁶ (3) 50 mm circular coil with iron core,⁷⁰ (4) Magstim 70 mm circular coil (P/N 3192),¹⁴⁶ (5) Magstim 90 mm circular coil (P/N 3192),¹⁴⁶ (6) Magstim animal MST coil, (7) Magstim human MST coil (S/N MP39), (8) Brainsway H coil,¹⁴⁷ (9) Brainsway H1 coil,^{76,80–83,147} (10) H1 coil with frontal iron core,⁷⁹ (11) H1 coil with lateral iron core,⁷⁹ (12) Brainsway H1L coil,⁸⁰ (13) Brainsway H2 coil,^76,80,81,147 (14) Brainsway HADD coil,¹¹⁶ (15) Magstim cap coil, (16) crown coil,³ (17) crown coil with back-splayed winding, (18) crown coil with back-spaced winding, (19) supraorbital halo coil,^98,99 (20) MRI z-gradient coil in parallel-current (Helmholz) mode,¹⁰⁰ (21) 3-layer double coil,⁶² (22) double butterfly,^56,58,59 (23) circular slinky-7 coil,^52,53 (24) rectangular slinky-7 coil,^12,53,54 (25) Magstim 25 mm figure-8 (P/N 1165),^21,146 (26) Cadwell Corticoil,¹⁴⁸ (27) Cadwell B-shaped coil,²¹ (28) 50 mm V-coil,¹³ (29) MagVenture C-B65 butterfly coil,¹⁴⁹ (30) MagVenture MC-B70 butterfly coil,^26,27,149 (31) Magstim 70 mm figure-8 coil (P/N 9925, 3190),¹⁴⁶ (32) 70 mm figure-8 with shielding plate,⁶⁵ (33) 70 mm figure-8 with active shield (5 turns),⁶⁹ (34) Neuronetics iron-core figure-8 coil (CRS 2100),^28,68 (35) MagVenture D-B80 butterfly coil,^27,149 (36) MagVenture MST twin coil, (37) Magstim double cone coil (P/N 9902),¹⁴⁶ (38) eccentric double cone coil with center-dense windings, (39) eccentric double cone coil with center-sparse windings, (40) stretched C-core coil,^3,68 (41) 3-D differential coil,⁶⁰ (42) 3-D coil array #1,¹¹⁴ (43) 3-D coil array #2,¹¹⁴ (44) 3-D coil array #3,¹¹⁴ (45) Cadwell cloverleaf coil,^48,49 (46) circular coil array,^110,150 (47) Magstim 70 mm figure-8 coil in reversed-current mode,¹⁵¹ (48) 70 mm figure-8 with active shield (7 turns),⁶⁹ (49) MRI z-gradient coil opposing-current (Maxwell) mode,¹⁰⁰ and (50) MRI x- (or y-) gradient (Golay) coil.¹⁰² The last configuration, labeled “0”, is the ideal “flux ball” coil whose windings are parallel to the circles of latitude of the spherical model and cover the whole head.¹²⁹

In parallel with the attempts to optimize the focality of magnetic stimulation, there have been efforts to increase the depth of stimulation. With conventional circular or figure-8 coils, the TMS-induced electric field is restricted to superficial cortical targets due to its rapid attenuation in depth. The double cone coil, formed by two large adjacent circular windings fixed at an angle (Figure 1(37)), has been shown to induce a more deeply penetrating and less focal electric field compared to a planar 70-mm figure-8 coil.²⁷ The double cone coil has been used for activation of the pelvic floor and lower limb motor representation at the interhemispheric fissure,⁷¹ as well as transsynaptic activation of the anterior cingulate cortex via stimulation of the medial frontal cortex.⁷² This coil is also highly efficient at seizure induction,⁷³ which is an advantage in the context of magnetic seizure therapy, but a potential source of risk in subconvulsive applications.

A family of coil designs called Hesed (H) coils has been proposed to achieve effective stimulation of deep brain structures.^74–77 The H coils have complex winding patterns and larger dimensions compared to conventional TMS coils, and consequently can be expected to have slower electric field attenuation with depth, at the expense of reduced focality. Measurements of the electric field induced by some H coils in a saline-filled phantom brain reveal circular field patterns that are similar to those produced by large circular coils.⁷⁶ It has been proposed to use high-permeability ferromagnetic cores to improve the electric efficiency, field penetration, and focality of H coils.^78,79 The magnetic field associated with a particular coil design can be influenced by placing a ferromagnetic core within the field. However, the reported improvements from placing ferromagnetic cores in H coils were minor.^78,79 H coils are currently being evaluated for the treatment of a variety of psychiatric and neurological disorders, including major depression,^80–87 schizophrenia,^88,89 dystonia,⁹⁰ autism,⁹¹ pain relief,⁹² and chronic migraine.⁹³ However, the comparative advantage of H coils over conventional 70-mm figure-8 coils with regard to depth of stimulation has been disputed.^94–96 In addition, a randomized, sham-controlled, observer-blinded study in patients with benign essential blepharospasm has shown that the H coil used in that study has similar clinical effects to a 90-mm circular coil.⁹⁰

Other designs for deep brain TMS coils have been proposed, such as the stretched C-shaped ferromagnetic core coil,^3,68,97 circular crown coil,³ and large halo coil.⁹⁸ The halo coil has been shown to elicit lower limb and trunk muscle contractions in a cynomolgus monkey, suggesting that this coil design is capable of directly stimulating relatively deep brain regions.⁹⁹ The effects on the brain of low-field magnetic stimulation using MRI gradient coils have also been investigated.^100–102 These deep TMS and MRI coils have larger dimensions than conventional coils and H coils, and are expected to provide slower decay rate of the electric field with distance, at the expense of reduced focality.

Another line of work on improving magnetic stimulation targeting concerns the use of independently-controlled multi-channel coil arrays.^103–115 The major advantage of such a system is the flexibility of controlling the spatial electric field distribution and the ability to stimulate several brain areas simultaneously.¹¹¹ However, multi-channel arrays do not circumvent the fundamental limitations of electric field targeting.³¹ Furthermore, high resolution targeting with multi-channel TMS requires small coil windings. Unfortunately, decreasing coil size is associated with higher electric losses, mechanical stress, and driving current.⁹ The need for sophisticated control electronics further complicates the design. Another putative advantage of the independently-controlled multi-channel system is temporal summation of spatially distinct electric field pulses. Specifically, it has been proposed that the neuronal activation threshold in depth can be reduced relative to the superficial neural threshold by applying several consecutive pulses from a set of coils at different spatial locations.^77,116 However, our previous study showed theoretically that synchronous firing of all TMS coil channels is more effective for stimulating deep neurons than sequential firing.³

Understanding the electric field depth–focality tradeoff could help TMS clinicians and researchers with coil selection and interpretation of TMS studies, and can inform the selection of coil designs for magnetic seizure therapy. In the present paper we systematically compare the electric field penetration and focality for 50 commercial and experimental TMS coil designs in a spherical head model. Even though more anatomically accurate head models have been developed to reveal local detail of the TMS electric field and current density distribution,^{17,19,117–121} the spherical model remains a useful reduction for parametric and comparative studies of global TMS coil characteristics such as electric field penetration and focality.^{3,9,13,15,32–34,122} The results obtained with the spherical model are in reasonable agreement with a variety of quantitative and qualitative observations of TMS,^{121,123–126} and are not limited to a particular subject’s head anatomy or coil placement, which is an advantage for comparative studies of global TMS coil characteristics. Furthermore, the spherical model provides a standardized measure that can be easily replicated by various researchers evaluating coil designs.

This study has been presented in part in abstract form.¹²⁷

Methods

Electric field simulation

The TMS coil and human head models, and the electric field solution were implemented with the finite element method package MagNet (Infolytica, Inc., Canada). The human head was modeled by a homogeneous sphere with 8.5 cm radius and isotropic conductivity of 0.33 S m⁻¹. The cortical surface was assumed to be at a depth of 1.5 cm from the surface of the head.¹²⁸ The distinct head tissue layers (scalp, skull, corticospinal fluid, and brain) were not differentiated, since magnetically induced electric field in a sphere is insensitive to radial variations of conductivity.¹³

We modeled 50 TMS coil configurations shown in Figure 1. In addition to modeling commercial coils from Brainsway (Jerusalem, Israel), Cadwell (Kennewick, WA, USA), Magstim (Whitland, Wales, UK), MagVenture (Farum, Denmark), and Neuronetics (Malvern, PA, USA), we have also included various coil designs proposed for enhancing the field focality or penetration. The MagVenture twin coil (Figure 1(36) and Figure 5) was modeled for various opening angles between the centers of the two loops to examine the effect of this parameter on the electric field depth–focality profile. The coils were modeled based on published data, manufacturer’s specifications, coil X-rays, and inductance measurements. Detailed descriptions of the various coils are provided in Table S1 in the Supplementary Material. The minimum spacing between the coil windings and the surface of the head model was 5 mm to account for the thickness of the coil insulation. Ferromagnetic cores were modeled with a linear, homogeneous, isotropic material, with relative permeability of 1000 and electrical conductivity of 1 S m^−1.79 Finally, we simulated the “flux ball” coil whose windings are parallel to the circles of latitude of the spherical model and cover the whole head (Figure 1, coil #0).¹²⁹ While this coil configuration is not physically realizable for brain stimulation, it illustrates the limiting case for large coils where the magnetic field intensity within the sphere becomes uniform, resulting in linear—the slowest possible—electric field attenuation in depth.^3,129

Figure 5.

Electric field S_1/2 –d_1/2 locus for the MagVenture MST twin coil (#36) for inter-loop opening angles ranging from 90° to 180°. The field characteristics of one of the twin coil windings is also plotted for comparison.

Some clinical studies using H coils have not provided details of the coil geometry^90–93 or were published after the initial submission of this paper.¹³⁰ Since geometric details of these coils were not available to us for this study, these H coils were not modeled.

The electric field was computed with the 3-D electromagnetic time-harmonic solver of MagNet. MagNet first solves for the magnetic field H via the edge-element version of the T-Ω method.^131,132 Fields are assumed to be time-harmonic with angular frequency ω =2π×5 kHz. MagNet solves the equation

$$\nabla \times \left[{(\sigma +\text{j}\omega \epsilon )}^{-1}\nabla \times \mathbf{H}\right]+j\omega \mu \mathbf{H}=0$$ (1)

where σ is the electrical conductivity, and ε and μ are electric permittivity and magnetic permeability of the medium, respectively. The tissue permeability was set to that of free space since head tissues are non-magnetic. The tissue permittivity was also set to that of free space since the quasistatic approximation is applicable to TMS¹²¹ and since permittivity will not affect the normalized electric field depth and focality metrics used in this study, as discussed in the next section. The electric field was subsequently computed using Ampère’s and Ohm’s Laws

$$\mathbf{E}={\sigma }^{-1}\nabla \times \mathbf{H}.$$ (2)

Electric field characterization

We compared the overall electric field distribution features of the various coils by plotting the field magnitude and direction on the brain surface (Figure 2).

Figure 2.

Induced electric field distribution on the brain surface by the 50 TMS coils from Figure 1. The electric field magnitude is plotted with a color map normalized to the field maximum in the brain, E_max, for each coil. The arrows indicate the electric field direction.

We quantified the electric field penetration by the half-value depth, d_1/2, defined as the radial distance from the cortical surface to the deepest point where the electric field strength E is half of its maximum value on the cortical surface, E_max.¹³³ The d_1/2 metric has units of length and quantifies the extent of the field along the radial brain dimension.

Focality is traditionally defined as the area where the electric field strength exceeds a certain value relative to the maximum at a given depth (e.g., the cortical surface). For example, the half-value area, A_1/2, is defined as the area of the cortex where the electric field exceeds half of the maximum electric field strength. Thus, focality quantifies the electric field spread in the two dimensions tangential to the brain surface. However, focality metrics such as A_1/2 are sensitive to the chosen surface depth where the area is calculated. Therefore, we adopted a more robust quantification of the electric field tangential spread, defined as

$$S_{1/2}=\frac{V_{1/2}}{d_{1/2}}$$ (3)

where V_1/2 is the half-value volume—the volume of the brain region that is exposed to an electric field as strong as or stronger than half of the maximum electric field.^123,133 Figure 3 illustrates the definitions of d_1/2 and V_1/2 for three different coils. Like A_1/2, the spread metric S_1/2 has units of area. The lower S_1/2, the more focal the electric field is. Collectively, d_1/2 and S_1/2 characterize the 3-D spatial extent of the TMS electric field.

Figure 3.

Examples of electric field characterization for (a) double-cone (#37), (b) 90-mm circular (#5), and (c) MRI x- (or y-) gradient (#50) coils. The left column shows the respective coil and the spherical model representation of the brain. The middle column shows the electric field strength contour and color maps on the quarter-sphere segment of the brain outlined in black on the left. The right column shows the location of the maximum induced electric field on the brain surface, E_max (green circle), and the location of the deepest point where the electric field strength is E_max/2 (yellow circle). Note that the deepest point where E =E_max/2 does not necessarily fall on the same radial line as E_max. The yellow arrow represents the half-value depth, d_1/2, which is the radial distance from the cortical surface to the deepest point where the electric field strength is half of its maximum value on the cortical surface. The red portions of the quarter-sphere indicate the regions of the brain exposed to electric field as strong as or stronger than E_max/2; the total volume of these regions is V_1/2. The half-value spread, S_1/2, is defined as S_1/2 = V_1/2/d_1/2.

The fraction of E_max relative to which the various depth and focality metrics are defined, typically $1/\sqrt{2}$,^{60,61,65,66,79,134} 1/2,^33,122 or 1–1/ e,¹³⁵ is arbitrary and can be set to any value between 0 and 1 depending on the objectives of the evaluation. Furthermore, since the depth and focality metrics are normalized to the peak electric field, E_max, they describe purely the relative spatial characteristics of the electric field and depend only on the coil geometry and placement. The peak electric field strength, as well as the pulse shape and width affect the focality of neural activation with TMS, but these parameters can be controlled by the pulse generation circuit of the TMS device and are therefore excluded from this characterization of the intrinsic spatial features of the coil electric field.¹³⁶

Finally, the direction of the induced electric field in the brain is another important determinant of the effective stimulation strength and focality.^137–139 However, most TMS coils can be flexibly positioned and oriented on the head; therefore, the field direction relative to the brain is not an intrinsic property of the coil, but rather depends on the specific coil positioning. Therefore, we did not characterize the field direction as part of the intrinsic coil focality properties.

Results

Figure 2 shows the induced electric field distribution on the brain surface. Symmetric circular type coils (#1–7, 15, 16, 19, and 20) induce a single loop of eddy current with axial symmetry around the coil central axis. The H coils (#8–14), asymmetric crown coils (#17, 18), circular coil array (#46), and reverse current figure-8 coil (#47) produce electric field distributions similar to the single loop pattern of the circular coils. Symmetric figure-8 type coils (#21–26, 28, 29, 31, 32, 34–40) induce two loops of eddy current with reflection symmetry with respect to the x-z and y-z planes. The asymmetric figure-8 coils (#27, 30), coils arrays #42–44, and active shield figure-8 coil (#33) produce electric field distributions similar to those of the symmetric figure-8 coils. The 3-D differential coil (#41), cloverleaf coil (#45), active shield figure-8 coil (#48), MRI z-gradient coil in opposing-current mode (#49), and MRI x- (or y-) gradient coil (#50) produce more complex electric field distributions with multiple eddy current loops. Finally, the flux ball coil (#0) produces a broad, circular, symmetric electric field distribution.

Figure 4 shows the TMS electric field half-value spread S_1/2 as a function of the half-value depthd_1/2 for the 50 simulated TMS coils. The solid and dashed lines are curves of best fit of the points corresponding to the symmetric circular (#1–7, 15, 16, 19, and 20) and figure-8 (#21–26, 28, 29, 31, 32, 34–40) type coils, respectively. Coils #34* and #40*—are the air-core counterparts (core relative permeability set to 1) to ferromagnetic-core coils #34 and 40, respectively. The TMS coils exhibit a wide range of S_1/2 and d_1/2, but, as expected, are all subject to a depth–focality tradeoff. The range of d_1/2 is similar between the simulated circular and figure-8 type coils, 1.0–3.5 cm and 0.9–3.4 cm, respectively. However, figure-8 type coils are more focal compared to circular type coils, with S_1/2 range of 5–261 cm² and 34–273 cm², respectively. For either circualr or figure-8 type coils with d_1/2 < 2 cm, S_1/2 and d_1/2 follow approximately a power law demonstrated by their near linear relationship on the logarithmic axes plot in Figure 4. Coils with deeper electric field penetration cannot be as focal as more superficial coils. Figure S1 in the Supplementary Material shows analogous plots of V_1/2 and A_1/2 versus d_1/2, demonstrating consistent depth–focality tradeoff for these alternative focality metrics as well.

Figure 4.

Electric field focality quantified by the half-value spread, S_1/2, as a function of the half-value depth, d_1/2, for the TMS coils in Figure 1. The definitions of metrics d_1/2 and S_1/2 are illustrated in Figure 2. The solid and dashed lines are curves of best fit of the points corresponding to symmetric circular (#1–7, 15, 16, 19, and 20) and figure-8 (#21–26, 28, 29, 31, 32, 34–40) type coils, respectively. Coils #34* and #40* are the air-core counterparts to ferromagnetic-core coils #34 and 40, respectively. For large coil sizes the depth–focality tradeoff trends for both circular and figure-8 type coils converge to the flux ball (coil #0) which corresponds to uniform magnetic field in the head and has S_1/2 = 308 cm² and d_1/2 = 3.5 cm. The dotted line is the S_1/2 –d_1/2 locus for the MagVenture MST twin coil (#36) for a range of inter-loop opening angles, replicated from Figure 4.

The focality advantage of figure-8 over circular type coils diminishes as the d_1/2 increases. The depth–focality tradeoff curves for both coil types converge to the flux ball (coil #0) characteristics which correspond to linear electric field decay from the circles of latitude to the sphere axis where the field is zero.³ Therefore, the maximum achievable d_1/2 is half the brain radius, 3.5 cm in our model (see Figure 4). The maximum S_1/2 is 308 cm², representing the most unfocal coil configuration possible. The most focal circular type coil is the mini-coil (#1) designed for stimulation in non-human primates, with d_1/2 and S_1/2 of 1.0 cm and 34 cm², respectively. The most focal figure-8 type coil is the three-layered double coil (#21), with d_1/2 and S_1/2 of 0.9 cm and 5 cm², respectively.

Developed for deep brain TMS, the H coils (#8–14) have relatively nonfocal, near circular electric field distribution. The H coils have deeper electric field penetration (d_1/2 = 1.7–2.4 cm) compared to conventional circular coils (e.g., #4–7, 15; d_1/2 = 1.4–1.9 cm). Larger diameter circular coils (e.g., #16–19, d_1/2 = 2.5–2.7 cm), in turn, produce more deeply penetrating electric field than H coils, at the expense of reduced focality. Large double-cone type coils (#36, 39) provide deeper stimulation (d_1/2 = 2.5–3.1 cm) than H coils with comparable or better focality.

The insertion of a ferromagnetic core in a coil can alter the focality and depth of the electric field, but is not able to circumvent the depth–focality tradeoff limits observed for air core coils. The effects of inserting a cylindrical ferromagnetic core (#3) through the center of a 50-mm circular coil (#2) are minor, reducing d_1/2 and S_1/2 by 3% and 5%, respectively. Placing a ferromagnetic core lateral to a circular coil can increase the field focality without sacrificing half-value depth. For example, adding a partial cylindrical core frontal (#10) or lateral (#11) to the H1 coil (#9) reduces S_1/2 by 6% and 0.8% while increasing d_1/2 by 5% and 1.4%, respectively. However, the resultant depth and focality are still inferior to the performance of comparable figure-8 type coils. When a C-shaped ferromagnetic core (#34) is added to a figure-8 coil (#34*), both d_1/2 and S_1/2 are increased by 13% and 34%, respectively. On the other hand, adding a stretched C-core (#40) to a partial toroidal coil (#40*) reduces both d_1/2 and S_1/2 by 11% and 46%, respectively.

Finally, Figure 5 shows the electric field S_1/2 and d_1/2 profile for the twin coil for opening angles ranging from 90° to 180°. The field characteristics for a circular coil identical to one of the twin coil windings are plotted for comparison. The spread S_1/2 of the twin coil increases monotonically as the inter-loop opening angle widens. However, d_1/2 increases for opening angles from 90° to 110°, and decreases for opening angles from 110° to 180°. The focality locus in Figure 5 is replicated with a dotted line in Figure 4 for comparison with the other coils.

Discussion

Electric field depth–focality tradeoff

Among the TMS coil designs, there is a tradeoff between electric field depth of penetration and focality, as illustrated in Figure 4 and Figure S1 in the Supplementary Material. In general, coils with larger dimensions produce electric field with greater d_1/2 and S_1/2. In contrast, smaller coils produce electric field that is more localized and superficial. For conventional and smaller coil sizes (d_1/2 < 2 cm), S_1/2 and d_1/2 are related approximately by a power law for either circular or figure-8 type coils. It is noteworthy that none of the coil designs was able to overcome the depth–focality tradeoff set by the figure-8 type coils—no TMS coil performance lies significantly to the right of the dashed curve in Figure 4. This finding supports the claim that the figure-8 coil focality can only be modified somewhat but not improved substantially.³² Hence, no TMS coil can achieve deep and focal stimulation simultaneously, consistent with previous findings.^31,33

In this work we used a fixed size spherical model with radius of 8.5 cm. Variation in head anatomy could affect the electric field depth of penetration and focality. We have previously investigated the sensitivity of the induced electric field to anatomical variability (variation in head diameter, scalp and skull thickness, brain volume, and tissue electrical properties) in the context of magnetic seizure therapy.¹⁴⁰ The metrics d_1/2 and S_1/2 are most sensitive to the head diameter, but considered as percentage of their maximum values, which are head diameter dependent, they both scale similarly with head size. Furthermore, variability in head size is equivalent to variation in coil size for fixed head size, which is already illustrated in Figure 4. Therefore, variation in head anatomy would not change qualitatively the relative performance of the different coils.

This study focused on a comparison of the geometric characteristics of the electric field induced by various TMS coils. It did not characterize the effect of coil current waveform and amplitude, coil placement relative to anatomical head landmarks, and neural activation thresholds. Clearly these aspects of stimulation have to be considered in a more complete analysis of the extent of neural activation by TMS. In previous modeling studies of TMS we have considered the coil current waveform and amplitude and the neural response threshold.^123,140 Anatomically realistic TMS models^120,121 can be particularly useful in studying the effect of the complex tissue structure of the head on the electric field for various coil placements.

Strategies for controlling electric field focality

Several strategies have been deployed for making the electric field more focal. As illustrated in Figure 4 and Figure S1, a figure-8 type coil is always more focal than a circular type coil with the same d_1/2. The figure-8 type coil maintains its focality advantage over the circular type coil of the same diameter as long as the two windings of the figure-8 remain relatively close.³⁴ For example, as illustrated in Figure 5, S_1/2 of the twin coil (#36) increases as the inter-loop opening angle increases, exceeding that of a single loop for angles above 103°.

Focality can be enhanced by reducing the diameter of the coil windings; smaller coils are more focal because the magnetic flux is more confined. The most focal of the circular and figure-8 type coils are the animal mini-coil (#1) and the 3-layer double coil (#21), respectively, both having a mean loop diameter of approximately 2 cm, approaching the smallest practical diameter.⁴⁶ It should be noted, however, that some of the smallest simulated coils were not designed for human cortical stimulation; coils #1, 6, and 25 were specifically designed for animal TMS where the electric field focality diminishes due to the smaller head size of animals.¹⁴¹ Very small coils may be impractical for human TMS since they require excessively high currents to achieve threshold-level brain stimulation, because of the rapid attenuation of the electric field in depth.⁹ Finally, different effective diameter of the coil windings contributes to differences between commercial coils with otherwise comparable specifications. For example, comparing two common “70 mm” figure-8 coils—the Magstim 70-mm figure-8 (#31) and the MagVenture B70 butterfly (#30)—show that S_1/2, A_1/2, and d_1/2 of the MagVenture coil are 6%, 8%, and 4% smaller compared to the Magstim coil which has a larger average winding diameter, consistent with the findings of previous comparisons of these two coils.²⁶

Another approach to improve field focality is to concentrate the coil windings over the target area while distributing the return current paths away from the target. For example, in the case of the slinky coil (#23 and 24) the windings are fanned out while maintaining tangency of all the windings along one edge. Similarly, the focality of the crown coil (#16) can be improved by spreading (#17) or shifting (#18) the return paths away from the target. The eccentric figure-8 coils (#22 and #38) achieve improved focality in a similar fashion.

Various shielding strategies have been proposed for the improvement of TMS coil focality. Conductive plates (passive shields) or ancillary powered windings (active shields) could be used to create secondary fields that block the main coil field outside the region of interest. These techniques are widely applied in the design of MRI scanners.¹⁴² By attaching a passive shield (#32) or an active shield with an appropriate number of turns (#33) to the 70-mm figure-8 coil (#31), d_1/2 decreases by 16% and 24% and S_1/2 decreases by 35% and 41%, respectively. Although similar reductions in S_1/2 and d_1/2 can be achieved by using smaller figure-8 coils (e.g., #25, 26), the shields provide more flexibility since a range of field characteristics can be obtained by tuning the conductive plate geometry or the current amplitude in the shielding coil without the need to modify the stimulation coil itself.⁶⁹ Shielding the coil could, however, increase losses due to the added conductive paths and reduce inductance which may affect the stimulator operation. Furthermore, some active shield configurations could reduce the coil focality—compare coil #48 with coils #31 and 33.

The insertion of a ferromagnetic core can increase or decrease to varying degrees the focality and depth of the electric field. For example, inserting a cylindrical ferromagnetic core through the center of a small circular coil (coil #3 versus #2) marginally increases focality but reduces stimulation depth. Placing a ferromagnetic core over one side of the winding in an H-coil with a circular type electric field pattern (coils #10 and 11 versus #9) enhances the magnetic flux and electric field under the core, similarly to the field concentration under the center of figure-8 type coils, consequently decreasing S_1/2 and increasing d_1/2 up to about 6%. Inserting ferromagnetic cores in figure-8 coils can have different effects on the depth–focality profile depending on the coil geometry. A C-shaped ferromagnetic core increases both d_1/2 and S_1/2 in coil #34 versus coil #34*, but reduces both d_1/2 and S_1/2 in coil #40 versus #40*. Importantly, there does not appear to be a way of using ferromagnetic cores to improve the focality of figure-8 type coils beyond the depth–focality tradeoff curve associated with air-core figure-8 coils.

Given the vast number of possible multi-winding coil configurations, in this study we only examined a handful of them (coils #41–50). Some of the 3-D coil arrays (#42–44) display electric field characteristics similar to symmetric figure-8 coils with focality and depth comparable to medium/small size coils such as the MagVenture B65 (#29) or the Cadwell Corticoil (#26). The reverse-current figure-8 (#47), intended for sham TMS, exhibits circular electric field characteristics, since the opposing winding current directions at the coil center cancel the electric field there. The increased d_1/2 and S_1/2 indicate potential disadvantages of the use of current reversal in figure-8 coils for sham TMS.¹⁴³ The 3-D differential coil (#41), designed for very focal stimulation,⁶⁰ is indeed the most superficial of the simulated coils but the S_1/2 is inferior to that of small figure-8 coil. However, by the half-power area focality metric, $A_{1/\sqrt{2}}$, the 3-D differential coil has greater focality compared to the figure-8 and slinky coils,⁶⁰ demonstrating that the relative advantages of coil designs are in some cases dependent on the focality metric definitions. Similar observations apply to the cloverleaf coil (#45). Finally, other coils, such as the MRI gradient coils (#49 and 50) produce even more nonfocal stimulation than circular coils, which may have therapeutic applications.¹⁰²

Strategies for controlling electric field depth

In this study we modeled all H coil designs used in clinical studies and published before the initial submission of this paper. H coil designs used in some recent studies^90–93,130 have winding geometries that can be classified as figure-8 type (Zangen A 2012, personal communication, Feb. 23, 2012) and may therefore have different depth–focality characteristics compared to the H coils modeled in this study.

Our results indicate that the modeled H coils (#8, 9, 12–14) have depth–focality characteristics consistent with large circular coils (see Figure 4). Therefore, d_1/2 of the H coil is smaller compared to figure-8 type coils with similar focality. Conversely, for a given d_1/2, the H coils induce a more diffuse field than figure-8 type coils. Some studies have indicated that H coils have significantly deeper field penetration than a conventional figure-8 coil. Specifically, at a depth of 4.5 cm, the electric field strength of the H1 and H2 coils is reported to be 66% and 81% of the maximal field induced at the surface of the skull, respectively, compared to 12% for the 70-mm figure-8 coil.^76,82 Furthermore, at 150% of motor threshold, the H2 coil reportedly enables effective activation of the medial prefrontal and subcortical frontal brain regions down to depth of 7 to 8 cm below the skull, compared to 2.5 cm for the figure-8 coil.⁷⁶ These results are based on electric field calculations along a lateral superior–inferior line in the brain and peak electric field measurements on transverse slices in a saline head phantom, but not on radial distance from the brain surface that is used in our definition of depth. Characterization of the field penetration by a distance that is not purely radial results in the electric field tangential spread contributing to the apparent depth of the field, thus confounding coil comparisons. Indeed, our results indicate a more moderate d_1/2 advantage of 22%–72% of the H coils (#8, 9, 12–14) over the 70-mm Magstim figure-8 coil (#31), at the expense of S_1/2 increase of 406%–811%. Our results may help interpret studies that claim no significant depth advantage of an H coil (#8) over a 70-mm figure-8 coil (#31).^94–96

The large S_1/2 of the H coil electric field compared to conventional figure-8 coils could be contributing to increased risk of seizures, consistent with reports of accidental seizure induction with H coils.^84–86,88 On the other hand, the large S_1/2 of H coils could also contribute to therapeutic effectiveness, with reported remission rate in depression⁸⁰ nearly twice that reported with the Neuronetics ferromagnetic-core figure-8 coil (#34),¹⁴⁴ though stimulation parameters besides the coils differed in the two studies and a head-to-head comparison of the efficacy of these two coils has not been conducted.

Large diameter circular coils (e.g., #16–19) produce more deeply penetrating electric field than the H coils, at the expense of further reduced focality. Large double-cone type coils (#36 and 39) can provide deeper stimulation than the modeled H coils with comparable focality and may be better suited for spatially confined deep TMS. The half-value depth of large double-cone type coils can be increased by widening the opening angle between the two coil wings. As illustrated in Figure 5, the twin coil d_1/2 increases for opening angles up to 110°, at which point the reverse trend occurs.

Conclusions

We have presented the most comprehensive comparison of TMS coil electric field penetration and focality to date. We showed that the various TMS coils differ markedly in their electric field characteristics, but they all are subject to a consistent depth–focality tradeoff. Specifically, deeper electric field penetration is achieved at the expense of reduced focality. Consequently, coil designs for deep brain TMS are unfocal and may be associated with increased risk of seizures and other side effects. The ease of seizure induction is an advantage in the context of magnetic seizure therapy, but would be considered a disadvantage for subconvulsive applications of TMS. Figure-8 type coils are fundamentally more focal than circular type coils. None of the simulated coils exhibits depth–focality tradeoff better than that of the figure-8 type coils. MRI gradient coils induce more diffuse electric field compared to TMS coils with comparable depth of field penetration. The addition of a ferromagnetic core can change a coil’s depth–focality profile depending on the coil geometry and core placement, but does not overcome the depth–focality tradeoff limit of figure-8 type air-core coils.

Understanding the depth–focality tradeoff can help TMS researchers and clinicians to appropriately select coils and interpret TMS studies. Evaluation of novel coil designs with consistent penetration and focality metrics such as half-value depth d_1/2 and spread S_1/2 can provide objective comparison among coil designs. Such comparison, potentially coupled with more detailed simulations in realistic head models, would be a logical precursor to clinical investigations with novel coils. Comparison of the d_1/2 and S_1/2 metrics could also be useful in interpreting the differential clinical efficacy and side effects reported with different coils. This study illustrates the utility of computational modeling to inform coil design for various applications of TMS and magnetic seizure therapy.