Novel TMS coils designed using an inverse boundary element method

In the following, some of the most important properties that assess the efficiency and performance of a TMS coil are listed.

2.1.1. Stored magnetic energy

2.1.2. Power dissipation

2.1.3. Mechanical forces

The high currents flowing in the TMS coil induce significant mechanical forces between coil elements, which are responsible for the audible click when a magnetic stimulator fires. So in designing TMS coils, it is important to consider the force and torque experienced to avoid damage or changes in the coil structure due to excessive mechanical stress.

Moreover, this issue is specially crucial in the concurrent application of TMS and functional MRI, where the operation of the coil within a large external field produces intense forces that can lead to mechanical failure of the TMS system.

In this particular case, the torque, m experienced by a surface current, J in the presence of the axial main magnetic field, ${{\mathbf{B}}_{0}}$, is given by

$$\begin{eqnarray}&&\mathbf{m}={\int}_{S}\mathbf{r}\times \left(\mathbf{J}\times {{\mathbf{B}}_{0}}\right)~{{\text{d}}^{3}}r,\end{eqnarray} \tag{ 1 }$$

where the integration domain, S, is the coil surface, and $\mathbf{r}$ the vector pointing from the center of the coil to the corresponding current element.

2.1.4. Induced electric field

2.1.5. Penetration

2.1.6. Focality

There are several definitions of focality in the literature; in general, more focal stimulation means a smaller stimulation area with the maximum field. Here, we have employed the focality defined through the effective surface area (Deng et al 2013)

$$\begin{eqnarray}&&{{S}_{1/2}}=\frac{{{V}_{1/2}}}{{{d}_{1/2}}}\end{eqnarray} \tag{ 2 }$$

where V_1/2 is the volume inside the brain in which the stimulus is over $50 \%$ of the maximum. This metric takes into account that it is harder to have a focal stimulus deeper in the head.

In this work, it is assumed that the range of frequencies of the employed TMS pulses is below 50 MHz, at these frequencies the electromagnetic properties of the body allow us to use a quasi-static approximation (Miranda et al 2003). Additionally, it is also considered that the currents flowing in the TMS coil are likewise in the quasi-static regime.

Under these conditions, and by using a boundary element method (BEM), the current under search can be modelled in terms of the nodal values of the stream function and elements of the local geometry (Cobos Sanchez et al 2010b).

Let us first assume that the conducting surface, S, on which we want to find the optimal current, is divided into Q triangular elements with N nodes, which are lying at each vertex of the element. The current density at a given point at the surface can be then written as

$$\begin{eqnarray}&&\mathbf{J}\left(\mathbf{r}\right)\approx \underset{n=1}{\overset{N}{\sum}}{{\psi}_{n}}~\boldsymbol{\jmath}^{{n}}\left(\mathbf{r}\right),\quad \quad \mathbf{r}\in S,\end{eqnarray} \tag{ 3 }$$

where $\psi ={{\left[{{\psi}_{1}},{{\psi}_{2}},\ldots,{{\psi}_{N}}\right]}^{T}}$ is the vector containing the nodal values of the stream function, and $\boldsymbol{\jmath}^{{n}}$, are functions related to the curl of the shape functions (Cobos Sanchez et al 2010b). The stream function nodal values $\psi \in {{\mathbb{R}}^{N}}$ are the optimization variables in this work.

By using this current density model, equation (3), the discretized expressions for the required physical magnitudes involved in the inverse problem can be obtained, and so matrix equations that transform ψ to the various coil properties (Poole et al 2010) can be then constructed

• The magnetic field at a series of H points, $\left\{{{\mathbf{r}}_{1}},{{\mathbf{r}}_{2}},\ldots,{{\mathbf{r}}_{H}}\right\}$
  $$\begin{eqnarray}&&{{b}_{{{x}_{i}}}}={{B}_{{{x}_{i}}}}\psi,\quad \quad {{b}_{{{x}_{i}}}}\in {{\mathbb{R}}^{H}},\quad \quad {{B}_{{{x}_{i}}}}\in {{\mathbb{R}}^{H\times N}},\,{{x}_{i}}=x,\,y,\,z.\end{eqnarray} \tag{ 4 }$$
  The coefficient ${{B}_{{{x}_{i}}}}(h,n)$, is the x_i  −  component of the magnetic induction produced by the current element associated to the nth-node in the prescribed hth-point.
• The magnetic stored energy in the coil
  $$\begin{eqnarray}&&W={{\psi}^{T}}L\psi,\quad \quad L\in {{\mathbb{R}}^{N\times N}},\end{eqnarray} \tag{ 5 }$$
  where L is the inductance matrix, a full symmetric positive definite matrix (Cobos Sanchez et al 2012).
• The resistive power dissipation of the coil
  $$\begin{eqnarray}&&P={{\psi}^{T}}R\psi,\quad \quad R\in {{\mathbb{R}}^{N\times N}},\end{eqnarray} \tag{ 6 }$$
  where R is the resistance matrix, a full symmetric positive definite matrix (Cobos Sanchez et al 2012).
• The electric field induced in a series of M points inside of the conducting system (Cobos Sanchez et al 2010b), $\left\{{{\mathbf{r}}_{1}},{{\mathbf{r}}_{2}},\ldots,{{\mathbf{r}}_{M}}\right\}$
  $$\begin{eqnarray}&&{{e}_{{{x}_{i}}}}={{E}_{{{x}_{i}}}}\psi,\quad \quad {{e}_{{{x}_{i}}}}\in {{\mathbb{R}}^{M}},\quad \quad {{E}_{{{x}_{i}}}}\in {{\mathbb{R}}^{M\times N}},~{{x}_{i}}=x,y,z.\end{eqnarray} \tag{ 7 }$$
• The net torque experienced by the surface current in an external static magnetic field, ${{\mathbf{B}}_{0}}$
  $$\begin{eqnarray}&&\mathbf{m}=\mathbf{M}\psi,\quad \quad \mathbf{M}\in {{\mathbb{R}}^{3\times N}},\end{eqnarray}$$
  where the nth-column of the matrix $\mathbf{M}$ represents the torque vector produced by ${{\mathbf{B}}_{0}}$ in the current associate to the nth-node.
  $$\begin{eqnarray}&&\mathbf{M}(n)={{{\int}^{}}_{S}}\mathbf{r}\times \left[\,\boldsymbol{J}^{{n}}\left(\mathbf{r}\right)\times {{\mathbf{B}}_{0}}\right]\,{{\text{d}}^{3}}r.\end{eqnarray}$$

Having found the wire pattern, the value of the coil inductance is evaluated by using FastHenry © (Kamon et al 1994), a multipole impedance extraction tool, and assuming the coil wires have a 1 mm diameter.

Full details of the induced E-field of each TMS coil solution are acquired using an existing direct BEM (Cobos Sanchez et al 2009), which allows calculation of the electric field induced by the coil windings in conducting systems.

Moreover, sinusoidal variation of the electric and magnetic fields (f  =  5 kHz) has been adopted, and unless it is stated, an arbitrary coil current with peak value of 1 kA has been considered.

Additionally, depth and focality metrics (section 2.1.6) were employed to describe the spatial characteristics of the stimulation; where the human head has been modelled by a homogeneous sphere of 8.5 cm radius and isotropic conductivity. The cortical surface was assumed to be at a depth of 1.5 cm from the surface of the head (Deng et al 2013), so the cortex is described by a sphere of 7.0 cm radius.

Finally, in order to evaluate the mechanical behaviour when performing TMS and functional MRI concurrently, the net torque of each stimulator is calculated in three different scenarios, in which the main magnetic field, ${{\mathbf{B}}_{0}}$, is considered to be parallel to each Cartesian axis and homogeneous; that is, ${{\mathbf{B}}_{0}}={{B}_{0}}{{\mathbf{\hat{x}}}_{i}},~{{\mathbf{\hat{x}}}_{i}}=~\mathbf{\hat{x}},\mathbf{\hat{y}},{{\mathbf{\hat{z}}}_{i}}$.

In the following, the proposed approach is demonstrated with the design of seven TMS coil examples, which have been chosen to elucidate the method and to illustrate some of the many different requirements that can be prototyped with this technique to design TMS coils of almost any geometry.

The geometry of the first coil example is that of a rectangular flat form of $14~\text{cm}\times 7.5$ cm located at the xy  −  plane, and it is designed to have minimum stored-energy and maximise B_y in a prescribed region of interest (ROI), which is made up of 400 points inside a spherical region of radius 2 cm that is centred on the z-axis, and 4 cm below the conducting surface as shown in figure 1.

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The relevant functional describing this problem is a particular case of equation (8), which can be written as

$$\begin{eqnarray}&&\mathcal{F}\left(\psi \right)=\,\parallel {{B}_{y}}\psi -b_{y}^{t}{{\parallel}_{2}}\,+\,\gamma W\left(\psi \right).\end{eqnarray} \tag{ 11 }$$

where $b_{y}^{t}$ has been set to have a constant value, in order to achieve an optimised stimulation in the ROI due to the y-component of the magnetic field.

By using the same rectangular planar surface as in section 3.2, a slightly more complex coil design problem can be posed by maximizing the magnitude of the B-field in the same the prescribed spherical volume of interest (ROI), figure 1, while having minimum power dissipation (or equivalently minimum resistance). The functional describing this particular problem can be obtained by imposing $\beta =\gamma =0$ in equation (8); where $b_{x}^{t},b_{y}^{t}$ and $b_{z}^{t}$ are chosen to have a constant value to optimise the modulus of the magnetic field induced in the ROI.

By using the same rectangular planar surface as in sections 3.2 and 3.3 we can explore the design of TMS coil with minimum power dissipation and an optimized magnitude of the E-field in the same the prescribed spherical volume of interest (ROI), figure 1(b). The functional describing this particular problem can be obtained by imposing $\alpha =\gamma =0$ in equation (8). The target fields $e_{x}^{t},e_{y}^{t}$ and $e_{z}^{t}$ are chosen to have a constant value to optimise the modulus of electric field induced in the ROI.

Koponen et al (2014) have shown the design of minimum-energy TMS coils wound on a spherical surface, this represents an interesting approach as this particular geometry is well matched to the head.

Here we investigate the design of a minimum inductance (or equivalently minimum stored energy) spherical TMS coil of radius 9 cm, which is designed to produce an optimised E-field in a spherical ROI of 2 cm radius and centred 5 cm above the centre of the conducting sphere, figure 2(a).

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The functional describing this particular problem can be obtained by imposing $\alpha =\delta =0$ in equation (8); where $e_{x}^{t},e_{y}^{t}$ and $e_{z}^{t}$ are chosen to have a constant value to optimise the modulus of electric field induced in the ROI.

The undesired stimulation in non-target cortex regions can be reduced by introducing a second region of interest (ROI2), in which the E-field produced by the stimulator is minimized. This strategy is evaluated by designing a minimum stored-energy spherical TMS coil of radius 9 cm, constructed to produce an E-field which is both a maximum in a spherical ROI and minimum in a second ROI2. Both volumes of interest are of 2 cm radius and formed by 400 points, where ROI2 is concentric to the conducting sphere and ROI is shifted by 5 cm in the positive z-direction, figure 2(a). The functional of Coil 5 design problem is

$$\begin{eqnarray}&&\mathcal{F}=\beta e\left(\psi \right)+{{\beta}^{\prime}}{{e}^{\prime}}\left(\psi \right)+\delta P\left(\psi \right).\end{eqnarray} \tag{ 12 }$$

where the terms e and ${{e}^{\prime}}$ are sum-of-squares of the error in electric field at ROI and ROI2 respectively, whereas β and ${{\beta}^{\prime}}$ the corresponding weighting parameters.

The target fields are chosen to have a constant value in the ROI and null in the ROI2, so as to produce a strong stimulation in the desired region, while reducing the undesired stimulation in the non target volume.

In this experiment we have studied the reduction of the size of the ROI, in order to improve the focality of the TMS coil. To this end, a minimum inductance spherical TMS coil of radius 9 cm, capable of stimulating with an optimised B-field a spherical ROI of 0.5 cm radius and 7 cm above the centre of the conducting sphere, figure 2(b), has been evaluated. This particular problem can be formulated by imposing $\beta =\delta =0$ in equation (8). The target fields $b_{x}^{t},b_{y}^{t}$ and $b_{z}^{t}$ are chosen to have a constant value to optimise the modulus of the magnetic field induced in the ROI.

Closed geometries, such as spherical shapes, suffer the significant disadvantage of offering no natural aperture for patient access. This known problem has already been addressed in Koponen et al (2014) work, where it is suggested to constraint the current distribution to have a negligible value at the region opposite to the target.

Here we propose the use of open geometries as an efficient solution to overcome this type of problem. More precisely, we have studied the design of a TMS coil with hemispherical shape with a radius of 9 cm, figure 2(c), which is also well matched to the head and allows a favourable patient access. It is also designed to produced a minimum inductance and an optimized E-field in a prescribed spherical volume of interest (ROI) of radius 2 cm and shifted by 5 cm in the positive z-direction, figure 2(c).

The functional describing this particular problem can be obtained by imposing $\alpha =\gamma =0$ in equation (8). The target fields $e_{x}^{t},e_{y}^{t}$ and $e_{z}^{t}$ are chosen to have a constant value to optimise the modulus of electric field induced in the ROI.

In the construction of the functional equation (8) we must find the optimal choice of the regularisation parameters α, β, γ and δ. For the designs tackled here, they have been chosen so that the self-inductance and the resistance of the TMS coil are less than a prescribed value ($L\leqslant 10\mu$ H and $R\leqslant 100$ m$\Omega$ for flat rectangular TMS coils, and $L\leqslant 18\mu$ H and $R\leqslant 200$ m$\Omega$ for spherical and hemispherical coils).

It is worth noting, that this election of maximum values of R and L are just illustrative; for each particular problem, the election of the regularisation parameters can be studied so as to yield a solution with a concrete desired coil feature (Cobos Sanchez et al 2012).

A standard figure-of-eight coil has been chosen for sake of comparison, where each circular lobe has a diameter of 70 mm, containing 8 loops of 1 mm-thick wire, as can be seen in figure 3(a). We refer to this stimulator as Coil 0, and its performance parameters are found in table 1.

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Table 1. TMS coil performance parameters: L is the inductance, R the resistance, S_1/2 the effective surface area (focality), D_1/2 the penetration or depth and N_contours is the number of levels in which the stream-function is contoured to produce the wire paths. Simulated values of L and R were obtained using FastHenry © (Kamon et al 1994) using 1 mm diameter circular cross-section wire.

	L (μH)	R ($\text{m} \Omega$)	D1/2 (cm)	S1/2 (cm2)	Ncontours
Coil 0	13.3	127	1.2	16.1	16
Coil 1	9.3	70	1.2	13.9	16
Coil 2	6.9	56	1.3	15.9	16
Coil 3	7.1	57	1.3	16.2	16
Coil 4	17.4	174	2.5	48.0	18
Coil 5	10.1	155	1.6	31.7	18
Coil 6	5.0	73	1.4	14.3	18
Coil 7	14.7	125	2.3	41.4	18

The wire-paths of the solution to Coil 1, Coil 2 and Coil 3 problems are shown in figures 3(b)–(d) respectively. All of them are two lobed TMS coils (red wires indicate reversed current flow with respect to blue), where it can be seen that the winding density is more concentrated over the region of stimulation for all cases.

Moreover, in order to produce a high stimulus in a given region, use of maximum $|B|$ and $|E|$ conditions can be used, which in general, lead to different solutions. Nonetheless, in the case of the flat designs tackled here, it has been found a similitude between the wire arrangements of Coil 2 (figures 3(c)) and Coil 3 (figure 3(d)). This fact is also confirmed when evaluating the characteristics of each TMS coil in table 1, where it can be seen that Coil 1, Coil 2 and Coil 3 provide a more efficient performance than Coil 0, which has remarkably high values of L and R compared to those obtained for the TMS coils design with the stream function IBEM.

The wire arrangements of the four rectangular TMS coils can also be found in figure 4 along with the colour map of the modulus of the E-field induced in the spherical cortex model (Coil 0 in figure 4(a), Coil 1 in figure 4(b), Coil 2 in figure 4(c) and Coil 3 in figure 4(d)).

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As it is expected from their wire paths, there is a similarity between the electric field distribution induced by Coil 2 and Coil 3. Nonetheless it is worth noting that the maximum E-field induced by Coil 3 (about 53.0 V m⁻¹) is significantly higher than that one found for Coil 2 (49.5 V m⁻¹). The highest electric field induced at the cortex surface is produced by Coil 1, which is about 60.2 V m⁻¹.

Similar facts can also be appreciated when evaluating the E-field induced inside the cortex by Coil 0, Coil 1, Coil 2 and Coil 3 when energized, as can be seen in figures 5(a)–(d) respectively, where it is depicted the colour-coded $|E|$ maps on a xy-plane with z  =  4 cm and the grey line delineates the region that we want to stimulate (ROI). It is clear that these planar TMS coils generate an electric field that satisfies the initial requirements of stimulating the prescribed ROI.

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The wire-paths of the solution to Coil 4, Coil 5 and Coil 6 problems are shown in figures 6(a)–(c) respectively, where red wires indicate reversed current flow with respect to blue. As expected, there is a higher density of winding turns over the region of stimulation. The performance parameters of each spherical TMS coil can be found in table 1.

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These wire arrangements can also be found in figure 7 along with the colour map of the modulus of the E-field distribution produced in the spherical cortex model by Coil 4 (figure 7(a)), Coil 5 (figure 7(b)) and Coil 6 (figure 7(c)). For all cases, the E-field induced pattern is a kind of a mirror image of the coil current distribution, forming two loops of eddy currents with reflection symmetry respect the x  =  y plane.

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It can be seen that all the spherical TMS coils presented here satisfy the initial requirements of stimulating the prescribed ROI, where the highest values of the electric field are found.

Additionally, Coil 5 has been designed to mitigated the undesired stimulation in the prescribed non target cortex region (ROI2), in order to evaluate this fact, the electric field induced in the ROI2 by Coil 5 is depicted in figure 8(a), which can be compared to that one generated by its counterpart Coil 4 in figure 8(b). It can be seen how the values obtained for Coil 5 are lower than those produced in the case of Coil 4, being the reduction of the electric field induced greater than $70 \%$ at many points in the ROI2. The price we have to pay for the mitigation of the undesired stimulation is a significant reduction of the electric field induced in the target region ROI; and even the increase of the E-field induced in other head parts.

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Coil 6 solution induced a significant lower E-field in the cortex compared to those produced by the other spherical coils, nonetheless it fulfil the initial requirements of producing a more focal and superficial stimulation in the cortex, as the values of d_1/2 and S_1/2 show in table 1.

Figure 6(d) shows the wire path of the Coil 7, which is again concentrated over the region of stimulation. It is formed by two lobes of nine turns, which have been produce by contouring the optimal stream function with the same number of levels. The relevant parameters for Coil 7 are detailed in table 1, which indicate that the hemispherical Coil 7 have slightly better performance that than its spherical counterpart Coil 4.

The electric field induced in the cortex by this hemispherical TMS stimulator and the corresponding one produced by Coil 4 present a somewhat similar pattern, although the latter has a higher magnitude over a wider cortex surface, as it can be seen in figures 7(a) and (d).