NIH Public Access
`Author Manuscript
`Brain Stimul. Author manuscript; available in PMC 2013 February 09.
`Published in final edited form as:
`Brain Stimul. 2013 January ; 6(1): 1–13. doi:10.1016/j.brs.2012.02.005.
`
`Electric field depth–focality tradeoff in transcranial magnetic
`stimulation: simulation comparison of 50 coil designs
`
`Zhi-De Deng1,2, Sarah H. Lisanby1,3, and Angel V. Peterchev1,4,*
`Zhi-De Deng: zd2119@columbia.edu; Sarah H. Lisanby: sarah.lisanby@duke.edu; Angel V. Peterchev:
`angel.peterchev@duke.edu
`1Department of Psychiatry and Behavioral Sciences, Duke University, Durham, NC, USA
`2Department of Electrical Engineering, Columbia University, New York, NY, USA
`3Department of Psychology and Neuroscience, Duke University, Durham, NC, USA
`4Department of Biomedical Engineering and Department of Electrical and Computer Engineering,
`Duke University, Durham, NC, USA
`
`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, d1/2, and focality by the tangential spread, S1/2,
`defined as the half-value volume (V1/2) divided by the half-value depth, S1/2 = V1/2/d1/2.
`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 d1/2 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 S1/2 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 d1/2 and S1/2.
`
`*Corresponding author. Department of Psychiatry and Behavioral Sciences, Duke University, Box 3950 DUMC, Durham, NC 27710,
`tel. +1 919 684 0383, fax +1 919 681 9962, angel.peterchev@duke.edu.
`Disclosure
`Mr. Deng is inventor on patent applications and invention disclosures on TMS technology. Dr. Lisanby has served as Principal
`Investigator on industry-sponsored research grants to Columbia/RFMH or Duke (Brainsway, ANS/St. Jude Medical, NeoSync);
`equipment loans to Columbia or Duke (Magstim, MagVenture); is co-inventor on a patent application on TMS technology; is
`supported by grants from NIH (R01MH091083-01, 5U01MH084241-02, 5R01MH060884-09), Stanley Medical Research Institute,
`and National Alliance for Research on Schizophrenia and Depression; and has no consultancies, speakers bureau memberships, board
`affiliations, or equity holdings in related industries. Dr. Peterchev is inventor on patents and patent applications on TMS technology,
`including TMS technology licensed to Rogue Research; is Principal Investigator on a research grant to Duke from Rogue Research
`and equipment donations to Columbia (Magstim, ANS/St. Jude Medical); has received patent royalties from Rogue Research through
`Columbia; and is supported by NIH grant R01MH091083 and Wallace H. Coulter Foundation Translational Partners grant.
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`Keywords
`transcranial magnetic stimulation; electric field; depth; focality; simulation
`
`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 maximal3,4; 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 phantoms22,23 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
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`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 density40,41 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
`
`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
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`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.
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`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
`
`Some clinical studies using H coils have not provided details of the coil geometry90–93 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
`
`V x [ (a-+jwtT 1V x u] +jwµH=O
`
`(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
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`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).
`
`(2)
`
`We quantified the electric field penetration by the half-value depth, d1/2, 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, Emax.133 The d1/2 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, A1/2, 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 A1/2
`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
`
`(3)
`
`where V1/2 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 d1/2 and V1/2 for three different coils. Like A1/2, the spread
`metric S1/2 has units of area. The lower S1/2, the more focal the electric field is. Collectively,
`d1/2 and S1/2 characterize the 3-D spatial extent of the TMS electric field.
`
`The fraction of Emax relative to which the various depth and focality metrics are defined,
`1/ Y2
`,60,61,65,66,79,134 1/2,33,122 or 1–1/ e,135 is arbitrary and can be set to any
`typically
`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, Emax, 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.
`
`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,
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`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 S1/2 as a function of the half-value
`depthd1/2 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 S1/2 and d1/2, but, as expected, are
`all subject to a depth–focality tradeoff. The range of d1/2 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 S1/2 range of 5–261 cm2 and
`34–273 cm2, respectively. For either circualr or figure-8 type coils with d1/2 < 2 cm, S1/2 and
`d1/2 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 V1/2 and A1/2 versus d1/2, demonstrating consistent depth–focality tradeoff for these
`alternative focality metrics as well.
`
`The focality advantage of figure-8 over circular type coils diminishes as the d1/2 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 d1/2 is half the
`brain radius, 3.5 cm in our model (see Figure 4). The maximum S1/2 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 d1/2 and S1/2 of
`1.0 cm and 34 cm2, respectively. The most focal figure-8 type coil is the three-layered
`double coil (#21), with d1/2 and S1/2 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 (d1/2 = 1.7–2.4
`cm) compared to conventional circular coils (e.g., #4–7, 15; d1/2 = 1.4–1.9 cm). Larger
`diameter circular coils (e.g., #16–19, d1/2 = 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 (d1/2 = 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 d1/2 and S1/2 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 S1/2 by 6% and 0.8% while increasing d1/2 by 5%
`and 1.4%, respectively. However, the resultant depth and focality are still inferior to the
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`performance of comparable figure-8 type coils. When a C-shaped ferromagnetic core (#34)
`is added to a figure-8 coil (#34*), both d1/2 and S1/2 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 d1/2 and S1/2 by 11% and 46%, respectively.
`
`Finally, Figure 5 shows the electric field S1/2 and d1/2 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 S1/2 of the twin coil
`increases monotonically as the inter-loop opening angle widens. However, d1/2 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 d1/2 and S1/2. In
`contrast, smaller coils produce electric field that is more localized and superficial. For
`conventional and smaller coil sizes (d1/2 < 2 cm), S1/2 and d1/2 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 d1/2 and S1/2 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 models120,121 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 d1/2. The figure-8 type coil maintains its focality advantage over the circular
`
`Brain Stimul. Author manuscript; available in PMC 2013 February 09.
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`NIH-PA Author Manuscript
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`NIH-PA Author Manuscript
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`NIH-PA Author Manuscript
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`LUMENIS EX1045
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`Deng et