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1st International Conference on Advancements of Medicine and Health Care through Technology, MediTech2007, 27-29th September, 2007, Cluj-Napoca, ROMANIA Energy Efficient Coils for Magnetic Stimulation of Peripheral Nerves Laura Cret, Mihaela Plesa, Radu V. Ciupa, Dan D. Micu and Lucian Man Abstract — The preoccupation for improving the quality of life, for persons with different handicaps, led to extended research in the area of functional stimulation. Due to its advantages compared to electrical stimulation, the technique of magnetic stimulation of nerve fibers represents a new direction of research in modern medicine. This study starts from a major limitation of the coils used for magnetic stimulation: their inability to specifically stimulate the target tissue, without activating the surrounding areas. The first goal of this study was to determine the optimal configuration of coils for some specific applications and evaluate the distribution of the electric field induced. Once the coil configuration established, we address other issues that need to be solved: achieving smaller coils, reducing power consumption (the low efficiency of power transfer from the coil to the tissue is a major drawback) and reducing coil heating. Keywords: biomedical engineering, biotechnology. 1. INTRODUCTION where V is the transmembrane voltage, Ex the axial component of the induced electric field, λ the space constant of the cable and τ the time constant. The term on the right of the equation represents the activation function, equal to the spatial derivative of the electric field induced along the nerve fiber. Never the less, for finite cables, two additional difference equations are required to describe the transmembrane potential at the boundaries of the cable. This is because there is no current flows through the high impedance of the membrane termination. Therefore, if L is the length of the nerve fiber, d the fiber diameter, ρa the axoplasmatic resistivity, cm the membrane capacitance per unit area and Iionic the membrane current due to other (than capacitive) channels, the following equation has to be considered, for both x=0 and x=L boundary conditions (Δx → 0) [1]: I dV d dV d + ionic + = E x (2) dt c m πdΔx 4ρ a c m Δx dx 4ρ a c m Δx Thus, for the ends of the nerve fiber, the activation function is given by the magnitude of the axial component of the electric field and not the magnitude of the gradient of this field as in (1). This electric field can be produced by electrical or magnetic means. The second method is presented in this paper. The principle of magnetic nerve stimulation is emphasized in figure 1. The preoccupation for improving the quality of life, for persons with different handicaps, led to extended research in the area of functional stimulation. Due to its advantages compared to electrical stimulation, magnetic stimulation of the human nervous system is now a common technique in modern medicine. A disadvantage consists, however, in the fact that the need of focal stimulation can not always be fulfilled. This is why the design of coils with special geometries can help achieving this goal. Another drawback consists of the low efficiency of power transfer from the coil to the tissue. The present paper starts by emphasizing the theoretical background of magnetic stimulation. Then, a computer model with all its characteristics is presented. Finally, the total electric field and energy transfer parameters are computed for coils with special design and important conclusions are drawn. 2. THEORETICAL BACKGROUND Neuronal structures can be modeled in the form of a cable and the membrane response can be computed by solving the equations describing the transmembrane potential across the membrane of the cable in the presence of induced electric fields. The relation between the transmembrane potential along an infinitely long nerve fiber in the presence of induced electric fields is given by [1]: ∂E x ∂V ∂ 2V + V − λ2 = − λ2 τ (1) 2 ∂t ∂3 x ∂x 1424 M a g n e ti c f i e l d M a g n e t ic c o il N erv e fib re Eddy c u rre n ts = f ( x) Figure 1. An alternating current flowing through the magnetic coil causes a time varying magnetic field, which induces an electric field. L. Cret, M. Plesa, R.V. Ciupa, D.D. Micu and L. Man are with the Technical University of Cluj-Napoca, Romania, phone: +40-264-401462; fax: +40-264-592-903; e-mail: [email protected]. 339 1st International Conference on Advancements of Medicine and Health Care through Technology, MediTech2007, 27-29th September, 2007, Cluj-Napoca, ROMANIA (3)) with dI / dt t =0 = U 0 / L . Assuming that the activation of the nerve fiber occurs for a preset value of the electric field E, we obtain U0, the necessary initial voltage on the capacitor that would lead to activation. The energy dissipated in the circuit during one pulse of duration Δt is [4]: According to the electromagnetic field theory, the electric filed inside the tissue can be computed by means of the scalar electric potential and the vector magnetic potential [2]: ∂A (3) E=− − gradV 424 3 ∂3 t1 12 EV Δt ∫ EA WJ = R I 2 (t )dt The first term of the electric field is called “primary electric field”, and it is due directly to the electromagnetic induction phenomenon, while the second term represents the “secondary electric field”, due to charge accumulation on the tissue-air boundary. According to formula (3), the computation of the electric filed due to electromagnetic induction is done by means of the magnetic vector potential [2]: μ ⋅ N ⋅ I( t ) dl (4) A( r , t ) = 0 ∫ r 4π (6) 0 The peak magnetic energy in the coil WB required to induce a given electric field is [4]: WB = 1 2 LI peak 2 (7) The temperature rise in the coil after one pulse of duration Δt is (assuming there is no cooling) [4]: η Δt 2 (8) ΔT = I (t )dt cσA2 ∫0 where η is the resistivity, σ the density, c the specific heat and A the cross-sectional area of the copper wire of the coil. Those three quantities are evaluated to establish the parameters of energy transfer from the coil to the target tissue. coil where the vector dl represents the differential element of the coil, r is the distance from the coil element to the field point, and N is the number of turns of the coil. For coils of different shapes, one can compute A using the following technique: The contour of the coil is divided into a certain number of equal segments, and the magnetic vector potential in the calculus point is obtained by adding the contribution of each segment to the final value. The secondary electric field depends on the geometry of the tissue-air interface, considered a cylindrical surface. This term is computed knowing that on the surface, the 3. RESULTS Magnetic coil design is one of the most important aspect of the technique of magnetically stimulating the nervous system. Recent advances in this area have improved significantly the focality. The planar “butterfly” coil was the first major improvement over the conventional planar circular coil. But more recently, 3-D design has shown even greater promise. This paper presents the electric field generated by a family of “slinky” coils, and some studies regarding the electromagnetic field penetration in the tissue for different parameters of these coils (radius, configuration, number of turns per loop). Considering a coil with N turns, the “slinky-k” coils are generated by spatially locating these turns at successive angles of i x 180 / (k − 1) degrees, were i = 0, 1, …, k-1 [5]. If the current passing through this coil is I, then the central leg carries the total current N x I. With this definition, the circular coil is considered a “slinky-1” coil, and the figure of eight is a “slinky-2” coil. The geometry of the problem is given in figure 2. boundary condition to be fulfilled is: n ⋅ E A = −n ⋅ E V (continuity of the normal component of the current density vector). The electric potential inside this domain, V, is numerically evaluated by solving Laplace equation (ΔV = 0) with Neumann boundary conditions inside the tissue. The electric current required to induce the electric field ( A is proportional to I – see (4)) is delivered by a magnetic stimulator (LRC circuit). The current waveform through the discharging of a capacitor, with an initial voltage U0, to the coil is [3,4]: (5) I = U 0 ωL ⋅ sin (ωt ) exp(− αt ) where α = R /( 2 L ) , ω = 1 / LC − α 2 , C is the capacitance, and R and L are the resistance and inductance of the coil, respectively. We focus on stimulators with a fixed rise time of the current I(t) from 0 to peak, which is sufficient for comparing relative figures of merit of the stimulators. The inductance is evaluated by taking the line integral of the vector potential around the coil, for unit current [2]: Slinky_5 coil 10 [mm] Cylinder radius: 25 [mm] 6,25 [mm] nerve L = ∫ A ⋅ dl . This technique permits the computation of tissue inductances of the special coils we designed to improve focality. Given the values of L and R, capacitance C is obtained from requiring that the rise time of the current is fixed. Because of the same requirement, we may substitute dI/dt (the derivative of the current, that comes up in equation Figure 2. Slinky coil over a cylindrical surface modeling the hand First, we considered a set of 5 coils with the same number of turns (8) and the same radius of the loop (25[mm]). 340 1st International Conference on Advancements of Medicine and Health Care through Technology, MediTech2007, 27-29th September, 2007, Cluj-Napoca, ROMANIA The number of turns per loop is 8, for “slinky-1”, 4-4 for “slinky-2”, 2-4-2 for “slinky-3”; 2-2-2-2 for “slinky-4” and 1-2-2-2-1 for “slinky-5”. For each one, we computed the total electric field along the nerve fiber (figure 2). The results obtained are given in figure 3. Another issue investigated is the impact that the loops radius may have on the electromagnetic field distribution along the nerve fiber and its penetration inside the tissue. For a point situated exactly under the symmetry center of a Slinky_3 coil and different values of the radius raza, the total electric field is plotted in figure 4. 90 Slinky1 Slinky2 Slinky3 Slinky4 Slinky5 80 70 70 50 60 40 30 20 50 40 30 10 0 raza=25mm raza=15mm raza=38.1mm 80 E-total [V/m] E-total [V/m] 60 90 20 0 50 100 150 200 250 z [mm] 300 350 400 450 10 500 Figure 3. Electric field strength induced by a set of 8 turns (25 mm radius) Slinky coils along the nerve fiber – the underlined configurations in table 1 0 0 50 100 150 200 250 z [mm] 300 350 400 450 500 a) 140 The focality criterion is given by the ratio between the primary and secondary peak of the total electric field. Therefore, focalization requires the maximization of E in the center of the coil and minimization of the same quantity along the periphery. Table 1 gives the value of this ratio for some coils of different configurations and radii of the turns. 120 E-total [V/m] 100 Table1. Evaluation of focality ratio for a set of Slinky coils over a cylindrical surface Coil type Number of Turn Inductivity Focality turns Radius [μH] ratio (Configuration) [mm] Slinky_1 8 25 3,9 1 Slinky_2 8 (4-4) 15 25 38,1 25 25 15 25 38,1 25 25 15 25 38,1 25 15 25 38,1 25 25 1,4 2,8 4,9 0,87 5,4 1,2 2,5 4,4 2,4 2,8 1,2 2,2 3,8 2,3 1,1 2,2 3,7 7,4 3,8 2,512 2,19 1,99 2,19 2,19 3,21 2,83 3 2,47 3,57 3,28 2,85 2,76 3,55 4,04 3,5 3,71 3,5 3,71 Slinky_3 Slinky_4 4 (2-2) 12 (6-6) 8 (2-4-2) 8 (3-2-3) 8 (1-6-1) 8 (2-2-2-2) 8 (1-3-3-1) Slinky_5 8 (1-2-2-2-1) 16 (2-4-4-4-2) 11 (1-3-3-3-1) raza=25mm raza=15mm raza=38.1mm 80 60 40 20 0 0 5 10 15 Depth [mm] 20 25 b) Figure 4. The electric field induced by a Slinky_3 coil, for different values of the turn radius a) Along the nerve fiber and b) For different depths inside the tissue For the same depth, one can see that a lower value of the loop radius would lead to a larger electric field induced in the tissue. On the other hand, the decreasing slope of the field is more significant for lower values of the radius. Finally, we changed the configuration of the initial Slinky_3 coil by modifying the number of turns per loop: configuration 1 is the initial one, with 2-4-2 turns/loop, configuration 2 has 3-2-3 turns/loop and configuration 3 has 1-6-1 turns/loop. Figure 5 shows the impact of these new configurations on the field penetration inside the tissue. 341 1st International Conference on Advancements of Medicine and Health Care through Technology, MediTech2007, 27-29th September, 2007, Cluj-Napoca, ROMANIA 90 increase for the case of the cylindrical surface (more than 10 times for Slinky_3, 4 and 5 coils)! 2-4-2 3-2-3 1-6-1 80 70 Table 2. Energetic parameters of a set of Slinky coils (turn radius 25 mm) E-total [V/m] 60 Coil Slinky1 Slinky2 L(μH) 3.9 2.8 WJ (J) 181 162.55 WB (J) 105.04 70.51 3.07 ΔT (°C) 3.42 Ipeak (A) 7339.7 7096.9 C (mF) 0.51 0.71 50 40 30 20 10 0 0 50 100 150 200 250 300 z [mm] 350 400 450 500 a) 2-4-2 3-2-3 1-6-1 E-total [V/m] 80 60 40 20 0 0 5 10 15 Depth [mm] 20 4. CONCLUSIONS The main conclusions arising from the work presented in this paper are: • The focality of a slinky coil improves when the turns are distributed over a larger number of directions (the Slinky_5 coil generates the largest ration between the primary and secondary peaks of the induced electric field); • The total electric field decreases as it penetrates inside the tissue. This phenomenon is affected by the radius of the loops for a slinky coil, and there would also be an optimum configuration of the coil to minimize this effect; • From the point of view of energy transfer from the coil to the target tissue, the figure of 8 coil is the most efficient of the set. One can conclude that the right choice of the turns dimensions and geometrical disposition has a strong influence on the magnetic stimulation efficiency, even if it’s the application that finally establishes the relative importance of the parameters. 120 100 Slinky3 Slinky4 Slinky5 2.5 2.2 2.2 284.47 261.03 331.4 111.99 92.27 117.14 5.38 4.93 6.27 9465.3 9158.9 10320 0.8 0.9 0.9 25 b) Figure 5. Electric field induced by a Slinky_3 coil with different configurations of the turns a) Along the nerve fiber and b) For different depths inside the tissue 5. REFERENCES The second configuration seems to show the most rapid drop of the induced electric field along with the distance – the worst case, while the initial configuration represents the best case. Finally, we compare the energetic efficiency of the set of 8 turns Slinky coils. The coil’s wire has a circular crosssectional area, with 1[mm] radius; the insulation gap between turns is 0,2[mm] and the target point is situated 16,25[mm] below the center of the coil (there are 10[mm] between the coil and the tissue, and the target point is 6,25[mm] inside the tissue – figure 2). We require that the value of the induced electric field in the target point is equal to 100[V/m]. Table 2 gives the energetic parameters for these coils of 25[mm] radius. One can observe that the Slinky_2 coil is the most efficient one in the set! If one would compare the necessary amount of energy required to produce the same field value in the target point (100[V/m]) for a flat tissue-air interface [3], one can see a very significant [1] Nagarajan S., Durand D., Warman E., Effects of Induced Electric Fields on Finite Neuronal Structures: A stimulation Study, IEEE Transactions on Biomedical Engineering, vol. 40, nr. 11, November, 1993; [2] Roth B.J., Basser P.J., A Model of the Stimulation of a Nerve Fiber by Electromagnetic Induction, IEEE Transactions on Biomedical Engineering, vol. 37, nr. 6, June, 1990; [3] Cret L., Ciupa R., Remarks on the Optimal Design of Coils for Magnetic Stimulation, ISEM, Bad Gastein, Austria, 2005. [4] Ruohonen J., Virtanen J., Ilmoniemi R., Coil Optimisation for Magnetic Brain Stimulation, Annals of Biomedical Engineering, Vol 25, 1997; [5] Lin V., Hsiao I., Dhaka V., Magnetic Coil Design Considerations for Functional Magnetic Stimulation, IEEE Transactions on Biomedical Engineering, vol. 47, nr. 5, May, 2000; 342