Coil design theory

Before designing the coil, it is important to calculate the required parameters such as coil inductance and H field intensity so we can use the minimum turns and optimal design to reach the required values. The inductance and magnetic field intensity were calculated mathematically. The HaroldWheeler formula was used to calculate the inductance13,14. This formula is applied at "low" frequencies (<30MHz) using enameled copper wire. The inductance (L) can be calculated as follows:

$$L = frac{{N^{2} A^{2} }}{{30A - 11D_{i} }}$$

(1)

where Di denotes the inner diameter of the coil set to 20mm and N and A represent the number of turns and cross-sectional area of each turn of the coil, respectively (Fig.1). Further, the area of the coil can be calculated as

$$A = frac{{D_{i} + N(W + S)}}{2}$$

(2)

where W denotes the width of the wire (considered 3 to maintain a smaller size), and S denotes the spacing between the coil's turns, which is set as 0.2mm. As we required a smaller coil, the turn of the coil (N) was kept at 7 to enable the focus most of the H field on the murine brains. Solving Eqs.(1) and (2), the calculated inductance of the coil was 2.04 H. Furthermore, the H field intensity (B) was calculated as

$$B = frac{{mu_{0} NI}}{2R}$$

(3)

where I is the input current of the coil, which is set to 1000A, and R is the total radius of the coil. Therefore, the calculated H field intensity was 0.43T.

Physical dimensions of the HaroldWheeler formula.

After calculating its parameters, we designed and analyzed the coil using an FEM simulation tool (Ansys Maxwell). The frequency of stimulation was conducted with 20Hz due to its beneficial biological effects on the Alzheimers disease model brain3. And, for thermal profile analysis, Femtet (Murata Software Co., Ltd., Tokyo, Japan) was used. The designed coil is shown in Fig.2.

Coil design and parameters.

The design parameters are summarized in Table 1.

The designed coil was simulated in Ansys Maxwell, and the inductance of the coil was analyzed (Table 2). The calculated result is almost identical to the previously calculated inductance.

Furthermore, the magnetic field intensity on the coil's surface was analyzed and matched with the calculated results shown in Fig.3. The magnetic field intensity was simulated using Finite element simulation tool with finite element method11. Therefore, the magnetic field in this study is the vector sum of the magnetic field intensity. The coil's maximum magnetic field intensity is generated at its center (0.44T), which matched our calculated value.

Magnetic field intensity of the designed coil.

To design a new rTMS coil, we optimized its design by simulating the thermal stability and focusing degree. The circular coil was adopted in this study because it demonstrates superior fine focusing ability and generates less heat compared to figure 8-shaped coil.To confirm this, we performed several simulations. In the initial stage of this study, we designed a figure 8-shaped coil with similar parameters and kept a reference plane 5mm apart to compare the magnetic field pattern with circular coil shown in Fig.4.

Simulation setup of coils for simulation of magnetic field intensity: (a) designed circular coil; (b) figure 8-shaped coil.

After running the magnetic field analysis simulations, we have observed that figure 8-shaped coil generates two focusing magnetic field patterns when placed close to the subject as shown in Fig.5b, whereas the circular coil is still producing a single focusing magnetic field pattern when placed close to the subject as shown in Fig.5a. Also, the magnetic field pattern of figure 8-shaped coil is much wider than circular coil as we are trying to focus on a smaller subject and the extra field generated by figure 8-shaped coil will be not of any use.

Comparison of magnetic field intensity on observation plane: (a) designed circular coil; (b) figure 8-shaped coil.

To examine the focus area and quantitative values of focality of two different coils, we performed 3D plot analysis of magnetic field intensity on an observing plane kept 5mm apart from coil as shown in Fig.6.

3D plot of magnetic field intensity of (a and b) Circular coil, (c and d) Figure 8-shaped coil with respect to distance.

Figure6a, b shows the 3D magnetic field distribution of circular coil across the observing plane. It is observed that the high intensity field is focused (which is displayed with red in the plot). Whereas Fig.6c, d shows 3D magnetic field distribution of figure-8 shaped coil across the observing plane where the intensity is not concentrated on a single area, rather it has two different peaks of intensity.

Further, the magnetic field distribution of two different coils is plotted on a graph for better comparison of focality as shown in Fig.7. Therefore, it is concluded that circular coil has more concentrated focusing ability than figure 8-shaped coil.

Graphical comparison of magnetic field intensity and pattern of circular coil and figure-8shaped coil.

The circular coil had a better focus than the figure 8-shaped coil. Moreover, the intensity of the circular coil was higher than that of the figure 8-shaped coil. The major drawback of the figure 8-shaped coil is that it focuses on two different places that reduce the magnetic field intensity. Therefore, a circular coil was selected for the experiment. Moreover, it is important to maintain the thermal stability of the rTMS coil. So we further studied the thermal profile of two different coils as our aim was to avoid excess heating produced by the coil. After running the temperature analysis simulations, we found out that figure 8-shaped coil is generating more heat than circular coil because of higher amount of turns as shown in Fig.6. Thermal analysis of the coil was simulated using Femtet (Fig.8). A remarkable difference in temperature was noted between the two coils. Accordingly, we concluded that circular coil is more efficient in terms of magnetic field focusing and thermal stability when the subject is small and close to the coil.

Comparison of thermal analysis of the designed: (a) circular coil; and (b) figure 8-shaped coil.

Before using the fabricated rTMS coil, it was necessary to calculate the optimal magnetic field discharge conditions. A head model was required to interpret the simulation results. There are many reports on human models but few on rodents, especially murine head models. Therefore, brain imaging images of C57BL6 mice were obtained using 7.0T MRI (Fig.9a). After the depths of the various layers constituting the head were measured, a real murine brain model was simulated (Fig.9b).

Murine head structure analysis and simulation model construction: (a) brain MRI images of C57BL/6 mice; (b) analytical model of different layers (scalp, skull, dura matter, arachnoid matter, subarachnoid space, pia matter, brain) of the murine head.

Furthermore, a circular coil was simulated using a murine brain model. The objective was to study the change in magnetic field intensity of the coil induced to the murine brain with change in distance, so the coil position was varied (0, 2, 5, 8, and 10mm away from the model) and the readings were recorded. The magnetic field simulation results are shown in Fig.10 and Table 3.

Magnetic field intensity simulation using the mice brain model. The distance between the coil and brain model varied: (a) 0mm apart; (b) 2mm apart; (c) 5mm apart; (d) 8mm apart; and (e) 10mm apart.

With increasing distance, the magnetic field induced in the brain decreased. The maximum H field was obtained when there was no gap between the coil and the mouse head. The best reading was obtained 2mm apart, and the intensity of the H field was sufficient to stimulate the brain cells.

Since the conductivity of different tissues is different, according to the MRI data, electromagnetic properties and thickness were fed in different tissues of murine brain model in the simulation tool. The thickness and electromagnetic properties of different tissue layers are summarized in the Table 4. The murine head model was modeled using 6 different tissue layers (scalp, skull, dura matter, arachnoid matter, and brain). According to measurement of the electric field strength on murine brain model, it was obtained as 136.14V/m (Fig.11).

Simulation result of electric field intensity of murine brain model.

All animal experiments were approved by the Institutional Animal Care and Use Committee of CHA University (IACUC210116). Isoflurane was administered via a VEVO COMPACT ANESTHESIA SYSTEM, and anesthesia induction was performed by positioning the nose of each mouse into a small nose cone delivering 3% isoflurane in pure medical oxygen. Anesthetized animals were fixed stereotaxically and rTMS stimulation was applied. After shaving the mouse's neck and making a minimal incision, a Tesla meter (FW Bell's model 8010) probe was inserted into the skull to measure the Tesla under the skull18. The mice were housed in four cages and maintained on a daily 12:12h lightdark cycle in a temperature-controlled room. The animals were provided standard rodent food and water ad libitum. The mice were allowed to acclimatize to the new environment inside the cage for 7days prior to the start of the study, and the ears were punctured 3days prior to confirmation.

MRI was performed using a 7.0T small animal scanner (Biospin 70/20 USR; Bruker, Fllanden, Switzerland). A quadrature birdcage coil (inner diameter, 72mm) was used for excitation, and an actively decoupled 4-channel phased array surface coil was used to receive the signal. T2-weighted images were acquired from C57BL/6N mice under isoflurane anesthesia (5% for induction, 1.5% for maintenance) using a turbo rapid acquisition with refocusing echoes (Turbo RARE) sequence with the following parameters: repetition time (TR)/echo time (TE)=3000/60ms; number of averages=4; field of view=3030mm2; image matrix=192192; and in-plane resolution=0.1560.1560.75 mm319.

Additionally, to confirm difference in output of rTMS coil for mouse form it of conventionally used for human application, the electric field intensities from those were compared by the same stimulation conditions at a place in air. As for human rTMS, a conventional clinical device, Brain-Stim-A of Remed Co. which acquired approval of Korean government was used.

Data are presented as meanstandard error. Statistical comparison between each group was performed on values calculated through simulation and magnetic field applied values by distance using one-way ANOVA using SPSS version 21.0 (IBM, Chicago, IL, USA). A value of p<0.05 was considered statistically significant as different.

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