Non-invasive focalized stimulation in deep brain using the spatially symmetric array

As an emerging neuromodulation method, transcranial magnetic stimulation (TMS) is widely used in the clinical diagnosis and scientific research of mental diseases.^1–4 Achieving effective stimulation of the deep brain is critical to the treatment of psychiatric diseases and the exploration of the causes of psychiatric diseases.^5–8 As the most widely commercially used TMS coil, the figure of eight coil (the FOE coil) can generate a focalized induced electrical field (E-field) 1.50–2 cm below the human scalp and enable modulation of cortical excitability. However, most of the important deep brain regions are located at depths of more than 4 cm below the human scalp, and they are beyond the effective stimulation depth of the FOE coil.^9,10 For example, as the key neural basis for the study of major depression, the ventromedial prefrontal cortex (VMPFC) region is located 7 cm below the scalp.

In order to improve the stimulation depth and focalization in non-invasive brain stimulation, many efforts have been made on the design of TMS coils.^11–19 The double cone coil is the earliest formed deep brain stimulation coil, and its stimulation depth is about 3–4 cm. The focusing area of the double cone coil reaches 94.40 cm², which has the risk of inducing epilepsy.^11–13 The H-coil is a representative deep brain stimulation coil, which can effectively stimulate the tissues 3–6 cm below the scalp. However, the stimulation depth improvement of the H-coil is at the cost of weakening focalization.^14–17 Studies have shown that under the same excitation conditions, the volume of brain tissue stimulated by the H-coil is six times that stimulated by the FOE coil.¹⁸ At a depth of 10 cm, a multi-coil configuration called the Triple Halo Coil (THC) can generate a magnetic field seven times stronger than that generated by the FOE coil. The design of the THC coil aimed to maximize the depth of stimulation without concern for stimulation focalization.¹⁹ 

In this paper, the spatially symmetric array based on the curved θ-type coil (the θ-SSA) is introduced as a new TMS array with a special geometrical structure. The finite element method (FEM) is adopted to obtain the relationships between the geometric parameters of the proposed array and the spatial distribution characteristics of the intracranial induced E-field. The influence of the stimulation current parameters on the spatial coordinates of the stimulation target point is analyzed with the mathematical method. Results show that the θ-SSA has great advantages over the traditional TMS coil in deep brain stimulation performance. The θ-SSA makes it possible to achieve focalized stimulation in the deep brain, and fewer non-targeted tissues will be exposed to strong stimulation. In addition, the location of the stimulation target point generated by the array can be flexibly and continuously adjusted.

Until now, the Figure of Eight coil (FOE coil) has been the most commonly used TMS coil because of its good focalization. The traditional FOE coil pair consists of two circular coil units, and the current directions of the two coil units are identical at the tangent point. The targeted point is right below the tangent point of the FOE coil pair. There is only one tangent point in the FOE coil, which is advantageous to narrowing down the stimulating area; however, it limits the enhancement of stimulation intensity, and the focusing area is restricted to the superficial cortical targets.

According to the superposition principle of the electromagnetic field, if the number of central tangent points of the magnetic coil pair is increased to form two central segments with the identical current direction, the amplitude of the induced E-field below the coil could be further increased and would be helpful in increasing the stimulation depth. In addition, considering the ergonomic structure of the human head, the curved stimulation coil can physically limit the induced E-field in a smaller domain, which could be beneficial in improving the stimulation focalization. When multiple curved coils are designed in a specific location to form a stimulation array, the superposition effect of the intracranial induced E-field can be further improved.

Based on these considerations, the θ-SSA is designed and proposed in this paper. The geometry of the θ-SSA is shown in Fig. 1. Different from the traditional planar TMS arrays, which are composed of multiple flat coils, the θ-SSA consists of four curved θ-type coils with identical geometric parameters. D (cm) describes the vertical distance between the upper θ-type coil and the lower θ-type coil on the same side of the head. The parameter D (cm) is marked in Fig. 1.

Each curved θ-type coil contains two units. The stimulation units in the θ-SSA are located in eight quadrants of the 3D coordinate system (as marked in Fig. 1), and they are symmetrical relative to the YZ plane and XZ plane. The 3D geometry and the planar winding structure of the θ-type coil are shown in Fig. 2. The red lines with arrows between copper conductors indicate the flowing direction of the stimulation current. It should be noticed that in practice, the θ-type coil is tightly wound into multi-turn and multi-layer structures and there is no large gap between copper wires. The red transparent ellipse indicates the area where the induced E-field is strengthened below the θ-type coil. The θ-type coil can be obtained by winding the copper wire into a common planar semi-ellipse coil pair and then bending the initial coil pair along its central axis.

The geometry of the θ-type coil can be described as a × b (mm²) with θ (°). a and b represent the length and width of the θ-type coil, respectively, and θ stands for the bending angle. The parameters a (mm), b (mm), and θ (°) are marked in Fig. 2.

Each θ-type coil of the θ-SSA is placed tangent to the human scalp and bent away from the human head to reduce the non-longitudinal component accumulation of the induced E-field and improve the stimulation focalization. The spatially symmetrical structure of the array enhances the superposition effect of the induced E-field in the deep brain, and the double layer design of the array is beneficial in further improving the stimulation depth.

1. The θ-SSA: The coils in this paper are wound by copper wire with a conductivity of 6.00 × 10⁷ S/m. The size of copper is 3 × 4 mm². The coils have 16 turns and 4 layers. The cross sections of the stimulation current are in a fixed rectangle of 12 × 16 mm². The total turns of the coils are 4 × 4. The frequency of the stimulation current applied in the array is set at 5 kHz. The θ-type coil was first modeled on the SolidWorks platform and then imported to the COMSOL Multiphysics platform.

2. The human head model: The anatomically realistic human head model comes from the Population Head Model (PHM) Repository. The PHM Repository was developed by Lee et al. using the SimNIBS pipeline, which was utilized to segment anatomical regions from Human Connectome Project MRI images.²⁰ The biological conductivity of the gray matter is set at 1.07 × 10⁻¹ S/m.²¹ 

   To analyze the longitudinal distribution characteristics of the intracranial induced E-field, an elliptical area with the half major axis equal to 10 cm and half minor axis equal to 4.50 cm is taken on the intracranial YOZ plane as the target plane. The target plane is shown by the blue dotted line in Fig. 3.

3. Meshes: The human head model and the proposed array model were imported to the COMSOL Multiphysics platform and the finite-element (FE) method was adopted to calculate the spatial distribution of the intracranial E-field generated by the array. To disregard mesh differences when comparing the stimulation performance of arrays with different geometric parameters, all Finite Element (FE) models have the same mesh size and are computed twice.

The details about the mesh size in this paper are as follows: the maximum element size is 6.60 × 10⁻³ m, the minimum element size is 2 × 10⁻² m, the maximum element growth is 1.40, and the curvature factor is 0.4. The level of the grid is fine enough, and the computation converges smoothly.

To explore the three-dimensional distribution of the induced E-field generated by the θ-SSA within the brain, three intracranial test lines [test line X (Y = 0, Z = 1.40 cm), test line Y (X = 0, Z = 1.40 cm), and test line Z (X = 0, Y = 0)] and a longitudinal target plane are set as shown in Fig. 3.

1. Stimulation intensity: Under identical current excitation, the higher amplitude of the induced E-field means stronger stimulation, and it is more possible to change the nerve cell membrane potential.

To meet the depth requirement of deep brain stimulation, in this paper, the stimulation depth is set 11 cm below the scalp. The center of the human head coincides with the origin of the coordinate axis, and the scalp vertex is located at Z = 12.40 cm. Accordingly, E_z.max has been adopted on test line X and test line Y at Z = 1.40 cm to characterize the stimulation intensity in the deep brain.²² 

1. Focalization: S₇₀ and V₇₀ are adopted to evaluate the stimulation focalization from 2D and 3D levels, respectively. S₇₀ represents the focusing area on the target plane where the amplitude of the induced E-field exceeds E_z.max/√2. V₇₀ represents the focusing volume of the gray matter within which the induced E-field exceeds E_z.max/√2.^23–25 

2. Longitudinal attenuation ratio: The attenuation feature of the induced E-field is evaluated with the ratio δ = E_GM.max/E_z.max, and E_GM.max is the maximum induced E-field on the surface of the gray matter. A larger ratio (δ) means when the intensity of the induced E-field on the surface of the gray matter remains unchanged, a stronger induced E-field can be obtained within the deep brain area, and the attenuated performance of the intracranial induced E-field is better.^25,26

In this part, the influence of the geometric parameters a (mm), b (mm), θ (°), and D (cm) of the proposed array on the distribution of the induced E-field in the deep brain is studied in detail. The geometric parameters of the four θ-type coils are the same, and the stimulation currents applied to the coils are with an amplitude of 1 A and a frequency of 5 kHz.^26,27

When the parameters of the stimulation currents in the four coils of the array are identical, the array will generate a focusing E-field in the deep brain, and the stimulation target point is P (X = 0, Y = 0, Z = 1.40 cm).

As shown in Fig. 4(a), as the length of the θ-type coil increases from 35 to 55 mm, the stimulation intensity increases by 54.10%, and the intracranial longitudinal attenuation ratio δ increases by 13.40%. The focusing area and the volume increase by 1.20 cm² and 5.20 cm³, respectively. In addition, for parameter b, when b increases from 20 to 40 mm, the stimulation intensity increases by 90%, and intracranial longitudinal attenuation ratio δ increases by 20.50%. The focusing area and the volume increase by 1.90 cm² and 2.90 cm³, respectively.

The effects of parameter a and parameter b on the spatial distribution characteristics of the intracranial induced E-field are similar, but the influence of parameter b on the spatial distribution characteristics of the intracranial induced E-field is more obvious than that of parameter a.

The increase in a and b is beneficial in enhancing the stimulation intensity. The reason is that according to the Maxwell equations, the stimulation intensity of the induced magnetic field is related to the length and shape of the stimulation current’s integral path. As the circumference of the coil increases, the amplitude of the induced magnetic field increases, and the induced E-field is enhanced as well. It shows that with the increase in the coil size, the longitudinal attenuation performance of the intracranial induced E-field is improved, which agrees with the results of previous studies that TMS coils with larger dimensions can provide sufficient field strength and penetrate deeper.^9,27 It also proves that the stimulation focalization will be weakened with the increase in a or b. Previous studies have indicated that TMS coils with larger dimensions weaken the focalization, which is similar to the trend of the θ-SSA.^27,28

The influence of θ on the spatial distribution characteristics of the intracranial induced E-field is shown in Fig. 5(a). As we can see, with the increase in θ, the stimulation intensity of the intracranial target increased slightly, and the variation range was about 4.90%. This is because the direction of the stimulating current in the conductors away from the centre of the coil is opposite to that of the central conductors of the coil. When the induced E-field generated by the central conductors is in the direction of +Z, the induced E-field generated by the conductors away from the centre is in the direction of −Z. With the increase in the bending angle, the distance between the conductors away from the centre of the coil and the head increases, and the induced E-field in the −Z direction is also reduced; hence, the induced E-field in the +Z direction in the intracranial target area will increase slightly.

When θ was increased from 50° to 90°, the longitudinal attenuation increased by 9.20%, the focusing area on the intracranial target plane increased by 1.40 cm², and the focusing volume in the gray matter decreased by 9.30 cm³.

The above-mentioned results show that the bending characteristics of the θ-type coil have a large impact on the spatial distribution of the intracranial induced electric field and with the increase in the bending angle, the intensity of deep brain stimulation, the longitudinal attenuation, and 3D focalization can be improved. Considering the stress borne by the coil skeleton and copper wire in the actual processing process, the maximum bending angle selected in this paper is 90°.

When the positions of the stimulation units at the lower layer of the array (unit V to unit VIII) are kept unchanged and the stimulation units at the upper layer (unit I to unit IV) move up, the vertical distance D increases from 6 to 11 cm. When a = 40 mm, b = 30 mm, and θ = 80°, the spatial distribution characteristics of the intracranial induced E-field under different vertical distances D are shown in Fig. 5(b).

With the increase in D, the stimulation intensity in the deep brain decreases by 10.40%, the intracranial longitudinal attenuation ratio δ decreases by 15.50%, S₇₀ increases by 5.70 cm², and V₇₀ increases by 111.20 cm³. Results prove that the increase in the vertical distance between the upper and lower coils in the array will significantly weaken the focalization in the deep brain. The closer the coils of the array are placed to each other, the more conducive it is to improve the spatial distribution characteristics of the intracranial induced E-field.

The stimulation target point P is at the position with the strongest induced electric field on the target plane in the deep brain. In the analysis of the previous part, the stimulation current in each stimulation unit of the array is identical, and the stimulation target point P (0, 0, and 1.40 cm) generated by the array is located at the center of the head.

When the configuration of stimulation current parameters in each stimulation unit changes, the spatial superposition effect of the intracranial induced E-field will be affected to realize the flexible adjustment of spatial coordinates of the stimulation target point P.

The XOY plane is used to divide the θ-SSA into upper parts (unit I to unit IV) and lower parts (unit V to unit VIII), and the amplitude ratio of the stimulation current in the upper units to that in the lower units is set to α.

When a = 40 mm, b = 30 mm, θ = 80°, and D = 11 cm, keeping the current amplitude of the lower stimulation units unchanged and increasing the stimulation current amplitude in the upper units to increase α from 1.0 to 3.0, the results show that the stimulation target point gradually moves upward toward the +Z direction.

The comparison of the induced electric field distribution on the intracranial target plane at α = 1.5 and α = 2.5 is taken as an example, shown in Fig. 6. Figures 6(a) and 6(b) show the distribution of the intracranial induced E-field on the target plane at α = 1.5 and α = 2.5, respectively. With the increase in α, the intracranial focused field [the dark red area in Figs. 6(a) and 6(b)] moves upward obviously. When α increased from 1.5 to 2.3, the Z-ordinate of the stimulation target point P increased from 2.40 to 6.10 cm. Figure 6(c) shows the distribution of the induced E-field along test line Z at α = 1.5 and α = 2.5, and it also agrees with the result that the stimulation target point moves upward.

The reason is that when α = 1, the currents in the upper and lower stimulation units are equal, the induced E-fields in the upper and lower half of the brain are symmetrical, and the stimulation target point P is located at the center of the brain. With the increase in the current amplitude in the upper stimulation units, the induced E-field in the upper half of the head is stronger than that in the lower half, resulting in the upward movement of point P. When α < 2, the Z coordinate of the stimulation target point increases quickly, and when α > 2, the increase in the Z ordinate of the stimulation target point is relatively slowed down.

With the use of the polynomial numerical fitting method, the mathematical relationships between the Z ordinate of the stimulation target point and α can be obtained as

The correlation coefficient R²_α = 0.98, which reaches a significant level, and the fitting effect is good.

Similarly, the XOZ plane is taken to divide the array into front parts (units I, IV, V, and VIII) and rear parts (units II, III, VI, and VII). The front stimulation units are located in the −Y direction, and the rear stimulation units are located in the +Y direction. The current amplitude ratio of the front units to the rear stimulation units is β.

When a = 40 mm, b = 30 mm, θ = 80°, and D = 11 cm, keeping the current amplitude of the rear stimulation units unchanged and increasing the stimulation current amplitude in the front stimulation units to increase β from 1.0 to 3.0, it is found that the stimulation target point P gradually moves toward the +Y direction, that is, in the direction with a smaller amplitude of current.

The comparison of the induced electric field distribution on the intracranial target plane at β = 2 and β = 4 is taken as an example, shown in Fig. 7. Figures 7(a) and 7(b) show the distribution of the intracranial induced E-field on the target plane at β = 2 and β = 4, respectively. With the increase in β, the intracranial focused field [the dark red area in Figs. 7(a) and 7(b)] moves toward the +Y direction. When β increased from 2 to 4, the Y-ordinate of the stimulation target point P increased from 1.4 to 2.5 cm. Figure 7(c) shows the distribution of the induced E-field along test line Y at β = 2 and β = 4.

In the relationship between the ratio α and the Z-coordinate of the stimulation target point P, as the ratio α increases, the Z-coordinate moves toward the stimulation with the larger coil. However, in the relationship between the ratio β and the Y-coordinate of the stimulation target point P, the Y-coordinate moves toward the stimulation units with smaller stimulation currents as the ratio β increases. The reason is that under this circumstance, in the Y direction, the superposition effect of the induced E-field generated by two stimulation units in a coil pair is the main factor affecting the formation of the stimulation target point P. When the amplitudes of the induced E-field vectors generated by the two stimulation units are close and the included vector angle is small, it is beneficial to form a higher synthetic E-field. If the stimulation current applied to a certain stimulation unit in the coil is large, the induced E-field generated by it needs to travel further to attenuate to be close to the amplitude of the induced E-field generated by another stimulation unit, so the stimulation target point P will be farther away from the stimulation unit with larger current and closer to the stimulation unit with smaller current. This rule is consistent with the previous research results that took the FOE coil as an example.²⁹ 

Using the polynomial numerical fitting method, the mathematical relationships between the Y-coordinate of the stimulation target point and β can be obtained as

The correlation coefficient R²_β = 0.99. The fitting curves of α, β, and the coordinates of simulation target point P are shown by the black dotted line in Fig. 8. In addition, the red solid lines with points represent the actual data to be fitted.

The above-mentioned results prove that the coordinates of the deep brain stimulation target point P can be flexibly adjusted by changing the amplitude ratio of stimulation currents in the array.

Since the focusing area of the θ-SSA is located at the center of the human head with X = 0, with the mathematical relationships between the Y-coordinate, Z-coordinate, and current amplitude ratio of the array, the location of the stimulation target point P can be determined. Compared with the traditional planar array, one advantage of this array is that the array can steer the intracranial stimulated area, the location of the stimulation target point is not limited to the intersection of the array,^30,31 and continuous adjustment of the position of the intracranial stimulation target point can be realized.

To enable meaningful comparison, a traditional FOE coil with an inner diameter R₀ = 28 mm, an outer diameter R₁ = 40 mm, and a height of 16 mm is taken as the reference coil in this paper. The resistance of the FOE coil is equal to the resistance of the θ-type coil (a = 40 mm, b = 30 mm, and θ = 80°), and the resistance is about 4 mΩ. Under the same excitation conditions, the three-dimensional distribution of induced E-field generated by the FOE coil and the θ-SSA (a = 40 mm, b = 30 mm, θ = 80°, and D = 11 cm) is shown in Fig. 9.

As we can see from Fig. 9, the induced E-field generated by the FOE coil in the brain is concentrated in the superficial tissues, while the θ-SSA can produce an obvious focusing area in the deep brain. The intracranial longitudinal attenuation ratio δ of the FOE coil is 5.25 × 10⁻², which is 77.30% lower than that of the θ-SSA. The focusing area of the θ-SSA is 34.30 cm² at a stimulation depth of 11 cm, and the focusing area of the FOE coil is 34 cm² at a stimulation depth of 6.6 cm. It means under the same stimulation conditions, when an approximately equal focusing area is generated, the stimulation depth of the array is 1.67 times that of the traditional FOE coil.

After satisfying the therapeutic demands, the practical possibility of realizing the θ-SSA is considered and discussed based on our previous experience in developing coils and the advanced coil manufacturing technology in Wuhan National High Magnetic Field Center (WHMFC).^32–34 The θ-type coil in this paper can be wound by copper wires with a rectangular cross section of 3 × 4 mm² and conductivity of 6.00 × 10⁷ S/m. It can be supported by a θ-shaped framework processed with an epoxy, featuring high strength and high corrosion-resistance properties. The outermost layer of the coil can also be reinforced with epoxy. Then the θ-SSA can be built by placing the four coils at certain positions around the human head with the help of fixing brackets.

This paper aims to propose and introduce a new stimulation coil array. Different from the traditional planar TMS array, the array is composed of multiple symmetrically placed curved stimulation coils. This special spatial structure enables the array to realize focalized stimulation in the deep brain. The relationships between array geometric parameters and spatial distribution characteristics of the intracranial induced E-field are discussed with the finite element numerical analysis method, and the relationships between array stimulation current parameters and spatial coordinates of the stimulation target point in the deep brain are obtained by the mathematical method. Results show that the array can produce an obvious focusing area 11 cm below the scalp. Under the same stimulation conditions, the array can improve the intracranial longitudinal attenuation characteristics by 77% compared with the traditional FOE coil. When producing an equal focusing area, the stimulation depth of the array is 1.67 times that of the traditional FOE coil. This design provides a flexible way to continuously adjust the spatial coordinates of the stimulation point in the deep brain and makes it possible to generate an induced E-field in the deep brain with fewer non-targeted tissues exposed to strong stimulation.

This work was supported by the National Natural Science Foundation of China (Grant No. 51407015) and by the Beijing Municipal Science and Technology Commission (Grant Nos. Z151100003215011 and Z161100001016010).

The authors have no conflicts to disclose.

Xiao Fang: Conceptualization (lead); Formal analysis (lead); Writing – original draft (lead); Writing – review & editing (equal). Chen Yun: Formal analysis (equal); Writing – original draft (equal). Chaoxu Zeng: Data curation (equal); Writing – review & editing (lead). Hongfa Ding: Funding acquisition (equal); Supervision (equal). Yongheng Huang: Formal analysis (equal); Validation (equal); Writing – original draft (equal). Wei Liu: Methodology (equal); Validation (equal); Writing – original draft (equal). Yaoyao Luo: Validation (equal); Visualization (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.