Design of Coils for Magnetic Neural Stimulation. Efficiency Criteria and Technical Solutions [627650]

TABEMED2000 – 1 DESIGN OF COILS FOR MAGNETIC NEURAL STIMULATION. EFFICIENCY CRITERIA AND TECHNICAL SOLUTIONS Mihaela Morega POLITEHNICA University of Bucharest, e-mail:[anonimizat] ABSTRACT: Magnetic stimulation is a relatively new technique, developed in the last three decades for neuro-muscular rehabilitation, therapy and stimulation tests. The method is based on the electric field produced through electromagnetic induction in the area of the target cellular tissue (neural or muscular fibers). The variable magnetic field is produced at the skin surface, by a current-carrying stimulating coil. The equivalence between the electric current applied through electrodes in the electric stimulation method and the so called “activating function” (AF) created through electromagnetic induction is considered in this paper. The activating function is the spatial derivative of the electric field component, tangential to the cellular fiber (axon or muscular long cylindrical cell). Several efficiency criteria are established in order to analyze the magnitude and spatial distribution of AF and different forms of the stimulating coils are compared. The mathematical model considers the analytical solution to an electric field problem applied to a simplified model of anatomical tissue: a straight nerve bundle embedded in a homogeneous conductive material. The literature provides sufficient arguments that support this model. 1. INTRODUCTION The so called neural magnetic stimulation procedure is a non-invazive, non-contact and painless alternative to the electric stimulation (through implanted electrodes) of the peripherial nervous system and spinal cord. This procedure is currently applied in phisio-therapy and rehabilitation treatments and consists of inducing electric field in relatively good conducting tissues. The magnetic field source is a coil, placed above the surface of the skin in the stimulated (target) area and fed by pulsed current. The current is produced by discharging a capacitor on the R-L equivalent electric circuit of the induction coil. The waveform of the current, the waveform of the magnetic flux density and of the induced electric field are similar. Magnetic stimulation was first performed in 1965, by Bickford and Fremming and reported in their studies on the frog sciatic nerve excitation, extended later to humans [7]. The technique evolved, and magnetic stimulation is used today with a multitude of commercially available stimulators, both in rehabilitation and in diagnosis. Several problems are associated with the improvement of the method and this paper presents a number of solutions concerning the geometry of the stimulating coil, in order to gain better results: (1) stimulus peak value is maximized versus other values and is focused on the target area; (2) the induced electric field strength is maximized at the target area depth and minimized at other depth levels in tissues; (3) the inductance of the stimulating coil depends on its geometry and it also influences the waveform of the stimulating current and, consequently, the efficiency of the procedure; the computation of the inductance, for a given geometry, would allow for an efficient adjustment of the other elements in the electric circuit, that are responsible for the waveform of the stimulating current. The literature presents several attempts to create stimulating coils with convenient forms, easy to manipulate, but also with good stimulating parameters [1], [2], [5], [6], [7], [8]. However, none of these models takes into account more than one optimization criterium, that leads to solutions which solve one certain procedure problem, yet ignoring the negative effects.

Design of Coils for Magnetic Neural Stimulation. Efficiency Criteria and Technical Solutions
TABEMED2000 – 2 2. DEFINITION AND EVALUATION OF THE ACTIVATING FUNCTION In the magnetic stimulation of nerves, based on the cable theory of cylindrical cells [4], it is assumed that the so called “activating function” (AF) is the spatial derivative of the induced electric field component, tangential to the nerve direction [6]. We adopted an idealized model for the stimulated tissue, in order to compare the efficiency of different geometric designs of the stimulating coils. The model is introduced by Esselle and Stuchly [1] and consists of a semi-infinite space (fig. 1), namely the (xOy, z < 0) semispace in a Cartesian coordinate system; this could be an anatomical structure (for example, the torso, an arm or a leg), with the skin surface on the (xOy, z = 0) plane, and a nerve trunk embedded at a certain depth, and, for example, parallel to the (Ox) direction. The stimulating coil is situated above the (xOy, z = 0) surface and the electric field analytical solution cited earlier gives the electric field strength and the activating function produced by an element (idl) of the current carrying winding. The authors [1] assume that the semi-infinite space is homogeneous and isotropic, although the electric field solution does not depend on the structure in the (Oz) direction.
Fig. 1 The geometry of the computational model The model also assumes the quasistady regime; the frequency of the excitation current (f < 10kHz) and the usual tissues conductivity (σ ≅ 1 S / m ) l e a d t o a p e n e t r a t i o n d e p t h m u c h l a r g e r t h a n t h e characteristic dimensions of anatomical domains, which in turn are larger than the dimensions of the coils currently used in magnetic stimulation. In association to fig. 1, Esselle and Stuchly [1] derived the solution of the electric field problem, the components dEx and dEy of the electric field strength (they found dEz = 0): , (1) , (2) and the expression of the elementary AF, , (3)

Design of Coils for Magnetic Neural Stimulation. Efficiency Criteria and Technical Solutions
TABEMED2000 – 3 produced at a specific location P(x,y,z), by the element (idl) of the current carrying winding, which is situated at (x0,y0,z0); i s t h e d i s t a n c e b e t w e e n ( x0,y0,z0) and (x,y,z) and is its projection on the (xOy, z = 0) plane, µ0 is the magnetic permeability of air and is the time derivative of the stimulating current. 3. FIELD SOLUTION FOR CIRCULAR TURNS We analyzed different circular and polygonal coils [2], [3] and the best results (focalization and maximization of peak values of AF) were found for quadruple coils, with symmetric repartition of circular or squared turns, in a flower–like form. The circular coils will be discussed further. In order to compute the contribution of the total amperturns to the electric field solution, one has to integrate the equations (1) and (2) along the entire stimulating winding. A similar procedure should be applied to equation (3) for the calculation of AF. A change of variables, conveniant for coils with circular turns is applied here. According to fig. 2., the elementary turn is circular, of radius r, and it can rotate by an angle α, maintaining the fix point A in the (xOy, z = 0) plane.
α = 0, the turn is in the (xOy, z = 0) plane the turn can rotate by 0 ≤ α ≤ π/2, with A as a fix point Fig. 2 The elementary turn , (4) , (5) , (6) . (7) With the new variables, the elementary activating function (3) becomes

Design of Coils for Magnetic Neural Stimulation. Efficiency Criteria and Technical Solutions
TABEMED2000 – 4 (8) and it allows for the evaluation of AF produced by horizontal turns (α = 0), as well as by inclined turns (α ≠ 0) relatively to the skin surface. The reason of using inclined turns will be discussed later. 4. EFFICIENCY CRITERIA AND TECHNICAL SOLUTIONS In order to better compare the performance of coils of different forms, the elementary stimulating coil has four circular turns, of radius r = 0.02 m. The time derivative of the current is , a customary value for this application [1]. The stimulated fiber is considered embedded in the tissue, at z = -0.01 m; it has the (Ox) direction, at y = 0. By the cable theory [4], the negative peaks of AF produce depolarization (activation) of the exciting fiber, while the positive peaks produce hyperpolarization (inhibition) of the fiber. One can choose the polarity of the stimulating current that favor the negative peaks. Three configurations were compared: (a) the concentrated coil, (b) the double coil, (c) the quadruple coil. Figure 3 presents the geometry of the coils and the corresponding distribution of AF: the 2D distribution of AF in the (xOy, z = -0.01 m) plane and respectively, the distribution along the fiber (on the (Ox) direction, at y = 0, where the highest negative peak occurs, and at z = -0.01 m depth). The superiority of the quadruple coil is evident, for the following criteria: * the best focalisation (F) or concentration of AF, estimated here by the projection of the negative peak on (xOy, z = -0.01 m) plane, considered as the target (stimulated) area; F should be minimized; * the efficiency ratio (ER), defined by the absolute value of the ratio between the negative and the positive peaks, is the highest; the positive peaks of AF should be minimized, in order to avoid hyperpolarization of the surrounding regions; * the negative peak of AF is located at the cross-section of its symmetry axes (x = 0, y = 0 in the considered coordinate system), that makes the application of the stimulus in the targeted area more accurately than in the other cases. It is expected that quadruple coils with eccentric position of turns could better concentrate AF at the intersection of their axes of symmetry, as Zhang and Edrich [8] illustrate with eccentric, yet concentrated coils. They also suggest a different problem that occurs in magnetic stimulation: the uneven distribution of AF in the depth of the tissue ((Oz) direction). For horizontal turns, the peak values of AF are maximum at the skin surface and decrease (aprox. exponentially) in depth; that could cause unpleasant, even painful reaction at the skin surface, a region that is rich in sensitive nerves. A possibility to reduce AF values at the skin surface and increase them in depth is the use of combined coils, made of horizontal and inclined turns. This is the reason for considering, in our model, the possibility to rotate the turns (fig. 2). Figure 4 presents the distribution of AF in depth of the tissue domain ((Oz) direction) in three cases: (a) the quadruple coil introduced earlier (it has 8 turns, 2 for each section), (b) a combination of a similar quadruple coil with 3 turns for each section and a quadruple coil with 1 turn for each section, with the turns inclined (α = π/3) and with opposite current polarity, (c) the same combination as (b), but with double magnitude for the radius of turns. As fig. 4 shows, AF at z = 0 rated by AF at z = -0.01 m (the considered target area) is aprox. 2.5 in case (a), 1.8 in case (b) and 1.15 in case (c). The inclination and radius of turns could provide a good control for the localization of the AF values in depth of the tissue.

Design of Coils for Magnetic Neural Stimulation. Efficiency Criteria and Technical Solutions
TABEMED2000 – 5 (a) 1 × 4 turns F ≅ 12.5⋅10-6 m2 ER = 1 (b) 2 × 2 turns F ≅ 6.25⋅10-6 m2 ER = 1 (c) 4 × 1 turns F ≅ 6.25⋅10-6 m2 ER = 2.08 geometry of the coils repartition of AF in the (xOy, z = -0.01m) plane repartition of AF along the fiber (on the (Ox) direction, at y = 0 and z = -0.01 m) Fig. 3 Performance of three geometric configurations of stimulating coils 5. CONCLUSIONS This work analyzes several forms of stimulating coils used in magnetic stimulation of the excitable tissue (nerve and muscle long fibers), in order to obtain a better distribution of the activating function (AF), the spatial derivative of the induced electric field component, tangential to the fiber direction. The quadruple coils produce the best concentration of AF at the target stimulating area, minimizing the inhibition effects on surrounding regions. The combination of quadruple coils with horizontal and inclined turns could result into the control of the magnitude of AF in depth of stimulated tissue and could concentrate the peak values at the fiber depth level, rather than at the skin surface.

Design of Coils for Magnetic Neural Stimulation. Efficiency Criteria and Technical Solutions
TABEMED2000 – 6 case (a) – quadruple horizontal coil AF1 case (b) – combination of a quadruple horizontal coil and a quadruple coil with inclined turns AF2 case (c) – combination of a quadruple horizontal coil and a quadruple coil with inclined turns; the turns have double radius compared with case (a) and (b) AF3 Fig. 4 Distribution of AF in depth of tissue for three forms of the stimulating coils It is posible that the forms of the coils could be refined, considering quadruple coils with eccentric turns. This geometry requires a slight modification of the equations (5) – (7), for circular turns that are not tangential to the coordinate axes. A further study of stimulating coils will take into account the computation of the inductance for each form considered for the coils, because the spatial distribution of turns affects the inductance and its value influences the waveform of the stimulating current. 6. REFERENCES [1] Esselle K., Stuchly Maria, Neural Stimulation with Magnetic Fields: Analysis of Induced Electric Fields, IEEE Trans. on BME, vol. 39, no. 7, 1992, p. 693-700. [2] Morega Mihaela, Morega Al., M., Procedeu tehnic si model matematic pentru stimularea pe cale magnetica a sistemului nervos, Rev. EEA Electrotehnica, vol. 44/1996, nr. 9-10, p. 14-20. [3] Morega Mihaela, Bioelectromagnetism, Ed. Matrix Rom, Bucuresti, 1999. [4] Plonsey R., Barr R.C., Bioelectricity. A quantitative approach, Plenum Press, New York, 1988. [5] Ren C., et. al., A novel electric design for electromagnetic stimulation – t h e s l i n k y c oil, IEEE Trans. On BME, vol. 42, no. 9, 1995, p. 918-925. [6] Stuchly Maria, Esselle K., Factors Affecting Neural Stimulation with Magnetic Fields, Bioelectromagnetics Supplement 1, 1992, p. 191-204. [7] Sudhansu Chokroverty, Editor, Magnetic Stimulation in Clinical Neurophysiology, Butterworths Publishers, Reed Publishing Inc., 1990. [8] Zhang T., Edrich J., Eccentric Coils for Focused Neuromagnetic Stimulation and Biomagnetic Detection, Proc. of the18th Annual Int. Conf. IEEE-EMBS, Oct. 31 – Nov. 3, 1996, Amsterdam