Transcranial magnetic stimulation (TMS) is a non-invasive method for brain stimulation widely employed in neuroscience to investigate and probe brain function and connectivity. Furthermore, TMS has received FDA approval for treating depression [1] as well as various other psychiatric and neurological disorders [2, 3]. During a TMS session, a coil placed on the scalp and driven by brief strong current pulses induces an electric field (E-field) in the brain. Enhancing the focality and depth of this induced brain E-field provides greater flexibility and selectivity in targeting deep brain regions. Consequently, prior studies have attempted to design coils with improved focus and penetration depth [4–19]. We previously developed a computational method for designing focal-deep (fdTMS) coils that achieve optimal trade-offs between focality, depth, and energy. Numerical studies demonstrated that coils designed using this framework can outperform the state-of-the-art figure-8 coils [20]. In this paper, we extend the fdTMS framework beyond its original numerical formulations by explicitly addressing practical implementation constraints. We present an optimization pipeline that turns theoretical windings into manufacturable coil geometries with adequate clearance for wire cross-sections and hybrid-layer builds, improving energy efficiency. We then benchmark the resulting designs against commercial figure-8 coils, showing superior depth–focality trade-offs. E-field simulations and bench experiments confirm that the optimized coils deliver significantly greater focality while remaining feasible for high-current TMS operation.

Figure-8 type coils consist of two circular coils placed side-by-side [21]. Until recently, these coils were considered to provide an optimal depth–focality trade-off [21]. Computational coil design methods employing the stream function approach [22] have recently been used to design coils that improve energy efficiency [11, 19, 23–32], reduce sound artifact [33], and produce more focused and deeply penetrating E-fields [11, 20]. These studies have also revealed that to achieve a more focal stimulation, increased coil energy is required [11]. Additionally, coil supports that better conform to the head will better achieve energy trade-offs than non-conformal ones [11, 23]. However, perfectly conformed coils cannot be used across individual head shapes or different cortical targets, leading to ‘hat’ shaped coils [23]. This study describes an optimized hat-shaped coil support that outperforms flat supports in energy efficiency while providing ergonomic adaptability across head sizes, cranial morphologies, and scalp positions.

The half-maximum E-field threshold—defined relative to the peak and therefore scale-independent—is commonly used as a figure of merit for benchmarking coil focality and stimulation depth [21]. We tested the sensitivity of this choice by analyzing alternative thresholds and by designing coils to optimize depth–focality trade-offs under these thresholds. Designs based on the half-maximum criterion were nearly indistinguishable from those based on other thresholds, indicating that the half-maximum region is a robust figure of merit for ranking coils by E-field distribution shape despite reasonable variations in activation threshold.

The optimized fdTMS coils concentrate smaller windings near the center and adopt more intricate patterns than conventional figure-8 designs. A typical fdTMS coil comprises a compact figure-8 core augmented by four ‘cancellation’ loops and two larger lateral ‘biasing’ loops. Because the figure-8 loops are small and the wire cross-section is constrained for energy efficiency, meeting driver-compatible inductance would otherwise require multilayer windings that increase energy demand. We developed a winding-synthesis method that applies multiple layers only in critical regions and single layers elsewhere, limiting the energy penalty while achieving the target inductance. We also created a semi-automatic workflow to generate a 3D coil former with integrated grooves for the windings and features that enhance subject safety and comfort. Performance was quantified and benchmarked against commercial coils using E-field simulations, bench measurements, and human motor mapping.

In summary, this paper contributes (i) a manufacturability-aware pipeline that converts Pareto-optimal surface currents into realizable multilayer windings under explicit constraints on wire cross-section, winding spacing, and inductance; (ii) a hybrid multilayer discretization strategy that preserves the targeted intracranial E-field while satisfying manufacturing constraints; (iii) a ‘hat’ support that improves placement flexibility over practical cortical targets while remaining partially conformal; and (iv) experimental validation including E-field mapping and human motor mapping, which has not been previously demonstrated for focal-deep designs derived from optimal-current theory.

In this section, we describe our proposed approach for designing and fabricating fdTMS coils. Firstly, we introduce the coil design parameters and figures of merit used to assess coil performance. Detailed methodologies for optimizing these parameters were previously presented [20]. Secondly, we describe the methods used to modify the optimal design, making it compatible with our coil fabrication procedures. In the third part, we outline the procedures for coil fabrication and provide details about the validation measurements of the coil’s E-field. Lastly, we provide additional details about figures of merit used for computational comparisons with conventional coils.

2.1. Coil design parameters

The coil windings were assumed to reside on a surface that is approximated by a triangle mesh consisting of nodes and flat triangle cells. The winding paths were chosen such that when driven by a TMS coil driver they will approximate the E-field generated by a surface current on (this process is given in the next section). To determine the optimal current, it was assumed that it is in the span of the seed currents , where , denotes Cartesian position, and are nodal finite elements on the triangle mesh [22]. In other words, surface current distributions are defined on a coil surface (figure 1) as

Figure 1. Coil current density parameters and E-field figures of merit definition. (A) Each seed current corresponding to internal nodes of the coil support triangle mesh is linearly combined to generate surface current distributions. (B) The E-field generated in a spherical head model by each coil is computed, and stimulation depth and volume figures of merit are extracted.

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where is a vector of weights, each (where ) is a real number, and and . Note that was assumed to be time-harmonic to simplify the exposition. However, because of the relatively low-frequency content of TMS pulses, the results apply to other current waveforms as well.

The above seed currents are known to span all piecewise linear currents with zero divergence on the triangle mesh [34], which thereby forms an adequate basis for approximately including all admissible E-fields generated by non-dissipative surface current distributions on .

2.2. Coil performance figures of merit

The fdTMS coil optimization takes as input a triangle mesh (or parametrization) of the surface and a set of seed current distributions , where . Then, it finds Pareto optimal currents that achieve optimal trade-offs with respect to stimulation energy, depth, and volume. The coil figures of merit are defined as follows:

(i)  

Minimum stimulation volume: the stimulated volume was defined as where denotes peak E-field at location induced by the surface current , denotes vector magnitude, is a unit step function, the integration is over the brain region (denoted ), and is the E-field at the targeted depth location. The value of is one if is above the stimulation threshold and zero otherwise. As a result, equation (2) measures the volume of the region above threshold.

(ii)  

Maximum depth of stimulation: the depth of stimulation was defined along a line chosen as a line that intersects at and is perpendicular to the center of the surface current support, i.e. , is the head unit normal pointing inwards at the point nearest to the coil center. In accordance, the stimulation depth is where denotes the point on the cortex closest to . For example, figure 1(B) depicts the coil placed centered about and oriented perpendicular to the z-axis. In this case, the line in blue denotes = . Furthermore, markers were included at and the lowest point with E-field above threshold . The value of is the distance between these two points and the choice of pointing toward the brain results in being the deepest point stimulated.

(iii)  

Minimum energy: TMS pulses have relatively low-frequency temporal variation and their induced magnetic field is negligibly affected by the presence of the head. The magnetic energy stored in the current distribution can be computed using the Biot–Savart law [35] as

where is the permeability of free space.

In addition to the aforementioned figures of merit, we combined and to define spread ( as the average transverse surface area of the stimulated region, calculated as [21, 36].8 Reducing either or is equivalent to an increase in focality. Moreover, safety considerations impose limits on the maximum E-field strength. For a given , we assumed that E-field strength exceeding in the brain is unacceptable. Therefore, currents in the span of the modes that result in an E-field that exceeds in the brain were excluded from the admissible designs. Throughout, we used thresholded spread metrics, , where . In prior research, a common selection was . In this scenario, , , and are equal to the figures of merit , , and , respectively [20, 21, 36, 37]. Consequently, this led to representing the sub-volume of the brain where the E-field equals or exceeds half of its peak value, and representing the greatest depth where the E-field equals or exceeds ½ of its peak value.

In many instances the peak E-field on the cortex was less than twice . Furthermore, TMS pulse current could be increased or decreased, thereby allowing for the coil to stimulate deeper or shallower regions. As such, the energy and spread of a fixed coil is not a constant but a function of targeted depth. To account for this in the coil benchmarks we additionally considered the stimulation volume, spread, and energy for each coil as a function of targeted depth, i.e. , , and , respectively, where . and , and were computed by driving the coil with a current that results in an E-field of at . Furthermore, targeted depths that result in a peak cortical E-field above were considered unreachable and excluded from the evaluation of , , and .

Finally, we considered the design of coils with the alternative choice of used in some publications [11]. We found that , , and for designs with either or achieve similar trade-offs for shallow depths, and the ones designed with achieved superior performance for larger depths. As such, we used and recommend to design fdTMS coils.

2.3. Coil winding generation

Here we discuss how we converted the current density into coil windings. Our approach started by adopting the standard coil winding generation approach that defines the stream function of the current , where . The contour lines of the stream function point in the direction of the surface current and trace out regions of constant current. Furthermore, since the current is dependent on the gradient of the stream function, an approximation of the surface current was made by placing wires on contour lines that are an equal elevation distance apart (i.e. equispaced contour intervals). For example, if we choose equispaced contour intervals, then, the wires are placed at contour elevation levels of

where These wires were then typically connected serially to generate a coil design. This approach has been shown to yield coil designs that match the surface current distribution and by proxy E-fields. For a fixed capacitor capacitance and voltage, the peak current and resonant frequency are inversely proportional to the square root of the coil inductance. The value of is typically chosen to result in a coil design that matches a desired inductance that results in a specified TMS pulse width and peak current.

For fdTMS coil designs, the contour lines typically concentrate near the coil center, and it is difficult to attain the TMS driver required inductance () without using multiple layers of wire. Adding multiple winding layers results in a design that is less energy efficient than a single layer design. To achieve the desired inductance while accommodating enough contour intervals to achieve a desired inductance we adopted two strategies.

One evaluated strategy was to add constraints to the optimization to ensure that the stream function would allow enough concentric loops to fit the required number of turns of the windings. For a given choice of , the closest any two windings can theoretically be is . Following established practice [38], we set the additional constraints to our fdTMS design framework to guarantee that , where is the wire width. This was done by enforcing , i.e. by constraining the maximum value of the L-infinity norm and as such ensuring fit for wires of width . Note that this approach differs from previous optimizations that used the L-infinity norm as the cost function, which have been shown to yield designs with more spread-out concentric windings [24], or from alternative winding-spacing optimization approaches [39, 40] since our results only needed to satisfy a fixed constraint to fit a prescribed wire size. Correspondingly, outside this region, the optimization is unconstrained and may generate currents where it best supports deep focal stimulation.

We iteratively implemented this constraint by first running the fdTMS framework and adding constraints where the condition was violated. The value of was determined using the stream function of the previous design iteration. The constraint of was approximated by 16 linear constraints by using the same approach that was used for the E-field constraint in [11]. This procedure converged after 2–5 iterations for all cases tested here. For any given , could only be reduced marginally while maintaining the same performance. As a result, these additional constraints still required the use of multiple layers.

The second evaluated strategy adopted a hybrid approach where the coil has multiple layers only on critical regions where the concentric windings are densest. This strategy exploited the fact that the induced E-field is relatively insensitive to small perturbations of the winding layout. Instead of using uniformly spaced stream-function contours, we selected non-uniform contour levels as winding locations and chose them to minimize the mismatch between the ideal fdTMS E-field and the E-field produced by the hybrid-layer coil implementation. Specifically, the majority of fdTMS coils consisted of three distinct types of sub-coils: figure-8, biasing, and cancellation windings. The biasing and cancellation windings allowed enough space to be implemented using one layer, while the figure-8 winding required three layers. We first partitioned the coil surface into the three subregions containing the figure-8, biasing, and cancellation loops, respectively. For each part, we specified the number of concentric loops in our implementation and number of layers. Each concentric loop consisted of wire placed at a contour line. As a result, the design was specified by the contour line elevation levels and layers for each coil sub-region. The contour line height levels were determined by running an optimization to minimiz