A new method for spatially selective, non-invasive activation of neurons: concept and computer simulation

Abstract

Currently available non-invasive neurostimulation devices, using skin electrodes or externally applied magnetic coils, are not capable of producing a local stimulation maximum deep inside a homogeneous conductor, because of a fundamental limitation inherent to the Laplace equation. In this paper, a new neurostimulation method (the DeepFocus method) is presented, which avoids this limitation by using an indirect method of producing electric currents inside tissues: First, cylinder-shaped ferromagnetic rotating disks of non-permanent magnetic material are placed near the skin and magnetized by a non-rotating magnetic coil. Each of the disks rotates at high speed around its own axis of symmetry, thus producing a purely electric Lorentz force field having a non-zero divergence outside the disk, and therefore giving rise to charge accumulations inside the tissues. Subsequently, the magnetic field is switched off suddenly, causing a re-distribution of charge, and hence short-lived electrical currents, which can be used to activate neurons. Two magnet configurations are presented in this paper, and analyzed by computer simulation, showing that the DeepFocus method produces a maximum current density (the ‘focus’) deep inside the conducting body. The field strength thus created in the focus (7.9 V/m) is strong enough to activate thick myelinated fibers, but can be kept below the threshold for C-fibers, which makes the new method a possible tool for pain mitigation by targeted neurostimulation.

Appendices

Appendix A-calculation of the divergence of the ${{\bf F}_{{\mathcal{M}}}}$ field from a rotating magnet

Following the formula for ${{\bf F}_{{\mathcal{M}}}}$ due to a rotating magnet according to Landau [8] (Eq. 9), the divergence of the ${\breve{\bf{F}}_{{\mathcal{M}}}}$ field equals (for v ≪ c):

$$ \nabla \cdot \breve{\bf{F}}_{{\mathcal{M}}} = - \nabla \cdot ( ({\varvec{\Upomega}} \times {\bf r}) \times {\bf B}) $$

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in which ${{\varvec{\Upomega}},}$ r, and B are all defined in an inertial, non-rotating frame of reference ${{\mathcal{F}}_{\rm inertial},}$ of which the origin coincides with the center point of the magnet, located on its axis of rotation. The vector r is a 3D spatial position vector in the non-rotating frame (with r = 0 referring to the origin of ${\mathcal{F}_{\rm inertial}}$), B is the magnetic field, and ${{\varvec{\Upomega}}}$ is the rotation vector.

Using the general equality ∇·(P ×  Q) =  Q·(∇ ×  P) − P·(∇ ×  Q) which is valid for any P, Q, we have:

$$ \nabla \cdot (({\varvec{\Upomega}} \times {\bf r}) \times {\bf B}) = {\bf B} \cdot (\nabla \times ({\varvec{\Upomega}} \times {\bf r})) - ({\varvec{\Upomega}} \times {\bf r}) \cdot \underbrace{(\nabla \times {\bf B})}_{{\rm equals \ 0 }} $$

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As indicated by the underbrace, ∇ ×  B = 0 for ${{\bf x} \in {\mathcal{R}},}$ because there are no (primary) current sources inside ${{\mathcal{R}}.}$ To calculate the other term, we use Stokes’ theorem to calculate ${\nabla \times ({\varvec{\Upomega}} \times {\bf r}):}$

$$ \begin{aligned} \nabla \times ({\varvec{\Upomega}} \times {\bf r}) &= \lim_{\buildrel{r(\delta r)(\delta\varphi)=\delta{\mathcal{A}}} \over {\delta{\mathcal{A}}\rightarrow 0}} \frac{1}{r(\delta r)(\delta\varphi)} \iint_{\delta{\mathcal{A}}} r\,{\rm d}r\,{\rm d}\varphi \nabla \times ({\varvec{\Upomega}} \times {\bf r}) \\ & \buildrel{\rm Stokes} \over {=} \lim_{\buildrel{r(\delta r)(\delta\varphi)=\delta{\mathcal{A}}}\over{\delta{\mathcal{A}}\rightarrow 0}} {\frac{\hat{\bf e}_{\Upomega}}{r\delta r\delta\varphi}} \oint\limits_{{\rm contour \ of\ {\delta{\mathcal{A}}}}} {\rm d}S \Upomega r \\ &= \lim_{\buildrel{r(\delta r)(\delta\varphi)=\delta{\mathcal{A}}} \over {\delta\mathcal{A}\rightarrow 0}} {\frac{\hat{\bf e}_{\Upomega}}{r(\delta r)(\delta\varphi)}}\left((r+\delta r)\delta\varphi \Upomega (r + \delta r) - (r\delta\varphi)\Upomega r \right) \\ & = 2{\varvec{\Upomega}} \\ \end{aligned} $$

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in which the projection of the 3D vector r on the plane ${{\mathcal{Y}}}$ is described by the 2D polar coordinates (r,φ), and the plane ${{\mathcal{Y}}}$ is perpendicular to the rotation vector ${{\varvec{\Upomega}},}$ and the small surface area ${\delta{\mathcal{A}}}$ is located entirely within the plane ${{\mathcal{Y}}.}$ Furthermore, ${{\varvec{\Upomega}} = \Upomega\hat{{\bf e}}_{\Upomega}}$ and r =  |r|. Combining these results, we now have

$$ - \nabla \cdot (({\varvec{\Upomega}} \times {\bf r}) \times {\bf B}) = - {\bf B} \cdot (\nabla \times ({\varvec{\Upomega}} \times {\bf r})) = - 2{\bf B} \cdot {\varvec{\Upomega}} $$

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Appendix B- Design and mechanical stability of the rotating ferromagnetic disks

Given the fact that the magnetized disks described in this paper rotate at high angular velocities (up to 60,000 rpm), the mechanical stability of the rotating disks has to be considered with extra care. In order to ensure that each individual disk can withstand the centripetal accelerations associated with these high rotational velocities, the soft ferromagnetic material of each disk (i.e. the ‘iron powder’ core, which is a non-permanent magnet, consisting of 50%Ni-50%Fe alloy powder) is placed inside a kevlar ring (see Fig. 11).

Fig. 11

a Single rotating disk (diameter = 17.8 mm), consisting of a soft-ferromagnetic ‘iron powder’ (50%Ni-50%Fe) core MD, and a very strong kevlar ring K that fits tightly around the core MD. The core MD is a non-permanent magnet, i.e. it is magnetic only as long as an external magnetic field (from e.g. a coil) is applied to it. The kevlar ring is strong enough to withstand the pressures due to the high rotational velocity (up to 60,000 rpm), even in the ‘worst case scenario’ that the core MD would disintegrate suddenly and completely. b Here, the extra non-rotating components are shown that have been omitted in Figs 6, 7, 8, 9. Only the left end and the right end of one single array of disks are depicted; the middle portion of the array is left out for reasons of visibility. The non-rotating magnetic coils NRC are connected to the current source (cf Fig. 4b) and serve to produce the magnetic field that magnetizes the soft-ferromagnetic cores (MD) of the rotating disks. Each of the rotating disks (K, with core MD) is identical to the disk shown above in (a). Non-rotating soft-ferromagnetic disks (NRSF) are located between the rotating disks, in order to guide the magnetic field (B) through the array of disks efficiently and hence achieve maximum magnetization of the cores MD of the rotating disks. All non-rotating components are rigidly connected to a stationary non-metal support structure NMSS. The rotating disks are all rigidly connected to a rotating axis (cf RA in Fig. 4b), which is not shown here

In the “worst case scenario" that the magnetic core of a disk, rotating at 60,000 rpm, would disintegrate completely and instantaneously, the maximum momentary pressure on the inner surface of the kevlar ring would be approximately 850 MPa, resulting in a hoop stress (i.e. circumferential stress) inside the kevlar ring of approximately 2,060 MPa. This would still be well below the typical critical tensile stress value for kevlar: for example, the tensile stress value of the DuPont Kevlar 49 Aramid Fiber is 3,620 MPa. Newer materials can resist even higher stresses: the tensile stress of the Toyobo Zylon PBO-AS Poly(p-phenylene-2,6-benzobisoxazole) Fiber (Toyobo Co. Ltd., Osaka, Japan) equals 5,800 MPa.

Appendix C- Alternating pattern for B and Ω to avoid induction effects

As has been mentioned in Sect. 3, it is important to avoid direct induction effects (i.e., ${{\frac{\partial}{\partial t}}{\bf B}}$ effects) inside the tissues during a ‘quenching event’. Let the electric field that is caused by induction be denoted as E _induction. Then, according to e.g. Jackson [7], the electric field due to non-zero ${{\frac{\partial}{\partial t}}{\bf B}}$ equals:

$$ {\bf E}_{\rm induction}({\bf x}) = - {\frac{\partial}{\partial t}} {\bf A}({\bf x}) $$

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in which A is the vector potential associated with the magnetic field, with B = ∇ × A. If for future systems the E _induction would become too large (because it would approach the critical limit of 6.2 V/m), there is a simple remedy to avoid these ${{\frac{\partial}{\partial t}}{\bf B}}$ induction effects: as can be seen in Eq. 21, the quenching current due to one single rotating magnetic disk depends on the dot product ${{\bf B} \cdot {\varvec{\Upomega}}}.$ As a result, the pattern of the quenching currents does not change if both B and ${{\varvec{\Upomega}}}$ would change sign simultaneously (i.e., if both B and ${{\varvec{\Upomega}}}$ would be multiplied with −1 simultaneously). See Fig. 12a. Therefore, for example, instead of letting the B and ${{\varvec{\Upomega}}}$ point in the same direction for all disks of the 150 disks (10 rows of 15 disks each) near the bottom of the conducting body R in Fig. 6a (and the diagram on the left side in Fig. 12b), one could choose to multiply the B and ${{\varvec{\Upomega}}}$ of only row #2, row #4, row #6, row #8 and row #10 with −1, thus creating an alternating pattern of B-field and Ω directions (see the diagram on the right side in Fig. 12b), which reduces the ${{\frac{\partial}{\partial t}}{\bf B}}$ induction effects considerably (because the B-field in the body ${{\mathcal{R}}}$ is now the sum of the counteracting contributions from the various rows), but leaves the ${\breve{\bf{F}}_{{\mathcal{M}}}}$ and ρ_comp and J _quench distributions completely invariant. In Fig. 12c, the absolute value |A(x)| for all points along the line segment λ, situated on the bottom surface of the conducting body ${{\mathcal{R}}}$ (see thick line in diagram on the right in Fig. 12b), is rendered. These |A| values have been calculated using the software package Amperes 6.1 from Integrated Engineering Software (IES), Winnipeg, Canada. As can be seen in the graph in Fig. 12c, the maximum value of the |A(x)| in the graph is about 1.45 milliWeber/m. For τ =  1 ms and |A(x)| =  1.45 milliWeber/m, Eq. 29 yields |E _induction| =  1.45 V/m, which is well below the critical limit of 6.2 V/m.

Fig. 12

Invariance of ${\breve{\bf F}_{{\mathcal{M}}}({\bf x})}$ and ρ_comp(x) and J _quench(x) if both the magnetic field B and the rotation vector ${{\varvec{\Upomega}}}$ are reversed (i.e., multiplied by − 1), and the application of this invariance to avoid induction ${({\frac{\partial}{\partial t}}{\bf B})}$ effects during ‘quenching events’. a Two times the same situation (the left and right image in a, respectively) consisting of a single rotating soft-ferromagnetic disk. With respect to the left image, the B-vector and the ${{\varvec{\Upomega}}}$ -vector in the right image are multiplied with − 1, which leaves ${\breve{\bf F}_{{\mathcal{M}}}({\bf x})}$ and ρ_comp(x) and J _quench(x) invariant for any point P, because ${{({\varvec{\Upomega}} \times {\bf r}({\bf x})) \times {\bf B}({\bf x}) = ((- {\varvec{\Upomega}}) \times {\bf r}({\bf x})) \times (- {\bf B}({\bf x}))}}$ and ${{\rho_{\rm comp}({\bf x}) = 2 \varepsilon_r\varepsilon_0 {\bf B}({\bf x}) \cdot {\varvec{\Upomega}} = 2 \varepsilon_r\varepsilon_0 (- {\bf B}({\bf x})) \cdot (- {\varvec{\Upomega}})}.}$ b Application of this invariance to the bottom layer of magnets in the ‘sandwich’ configuration. At the left: bottom layer of magnets in which the B and ${{\varvec{\Upomega}}}$ vectors point in the same direction for all 150 rotating magnets. At the right: same situation, but now with B and ${{\varvec{\Upomega}}}$ reversed (multiplied with − 1) for row #2, #4, #6, #8, and #10. This leaves the ${\breve{\bf F}_{{\mathcal{M}}}({\bf x})}$ and ρ_comp(x) and J _quench(x) invariant, i.e.: the ${{\breve{\bf F}}_{{\mathcal{M}}}({\bf x})}$ and ρ_comp(x) and J _quench(x) distributions inside the ${{\mathcal{R}}}$ at the left are the same as those inside the ${{\mathcal{R}}}$ at the right. c Graph of the absolute value of the vector potential |A| as function of the position (Y coordinate) along the line segment λ, which is situated on the bottom surface of the conducting body ${{\mathcal{R}}}$ parallel to the Y-axis, and passes through the center of the bottom surface (see image at the right in b). The maximum value of the |A(x)| in the graph is about 1.45 milliWeber/m. For τ =  1 ms and |A(x)| =  1.45 milliWeber/m, Eq. 29 yields |E _induction| =  1.45 V/m, which is well below the critical limit of 6.2 V/m