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Functional magnetic stimulation: theory and coil optimization
Bioelectrochemistry and Bioenergetics 47 Ž1998. 213–219
Functional magnetic stimulation: theory and coil optimization Jarmo Ruohonen
a,b,)
, Paolo Ravazzani c , Ferdinando Grandori
c
a
c
BioMag Laboratory, Medical Engineering Centre, Helsinki UniÕersity Hospital, Finland b Bioengineering Centre, Pro JuÕentute Don Gnocchi Foundation, Milan, Italy CNR Centre of Biomedical Engineering, Piazza Leonardo da Vinci 32, I-20133 Milan, Italy Received 9 January 1998; revised 2 September 1998; accepted 25 September 1998
Abstract Excitable tissue can be stimulated non-invasively by means of externally applied changing magnetic fields. This paper considers theoretically magnetic stimulation to functionally stimulate the tissue. We optimize the stimulating coil to accomplish functional magnetic stimulation ŽFMS. with the minimum cost expressed in terms of driving energy needed to provoke excitation. Magnetic stimulation is less discomfortable than functional electrical stimulation ŽFES. and therefore, the use of FMS opens up completely new views to the restoration of movement and therapy in patients with neuromuscular system impairment. Advances in coil design can make FMS a promising application field for magnetic stimulation. q 1998 Elsevier Science S.A. All rights reserved. Keywords: Coil design; Coil shape; Peripheral nerves; Neuromuscular rehabilitation; Functional stimulation; FES
1. Introduction Excitable neuromuscular cells can be stimulated non-invasively by strong magnetic field pulses that induce an electric field and a flow of current in the tissue, thereby leading to local depolarization of the cell membrane w1,2x. The applications of magnetic stimulation are apparently immense since tissues as diverse as the brain, peripheral nerves and the heart can be stimulated. However, further progress is limited by the unsatisfactory capability of focusing and controlling the excitation and by the large amount of energy required to induce a strong enough electric field in the tissue. Therapeutic use of magnetic stimulation appears an exciting and promising application in the near future, being particularly true for functional andror rehabilitative stimulation. Presently, there is a great interest in functional electrical stimulation ŽFES., a generic term for the use of electricity to augmenting or restoring function in patients suffering from movement impairment w3,4x. FES includes techniques that go beyond simple muscle stimulation; the terms FES, FNS Žfunctional neuromuscular stimulation., and EMS Želectric muscle stimulation. are often used
) Corresponding author. Tel: q358-9-471-5541; fax: q3589 -4 7 1 -5 7 8 1 ; e-m ail: jarm o @ b io m ag .h elsin k i.fi; In tern et: www.biomag.helsinki.firtms
interchangeably. In FES, an electric current is fed through implanted or surface electrodes and may generate or improve movement in patients with several pathologies, with beneficial effects that occasionally persist even after the period of therapy w5,6x. It has been recently realized that functional magnetic stimulation ŽFMS. may provide a fascinating option or additional tool to the conventional FES w7–10x. The often painful sensation experienced by the subject in electrical stimulation is largely reduced or even abolished in magnetic stimulation, which promotes the use of FMS, e.g., in muscular atrophy prevention, muscle control and force reinforcement. However, to accomplish comfortable, safe and widespread use of FMS, it is necessary that one can overcome many limitations of magnetic stimulation, including the poor reproducibility and high power demands as well as the large dimensions of the available devices. Moreover, the present magnetic stimulators have been designed exclusively for brain stimulation and will not operate optimally in FMS, mainly because the dominant activating mechanisms are different. In this study, after briefly reviewing the necessary mathematical tools, we describe a method to optimize coils for use in functional magnetic stimulation. Optimization is accomplished by studying how the coil should be wound and which coil type Žsingle, 8-shaped or 4-leaf. should be used to achieve effective stimulation. We measure the efficacy of the coil as the peak magnetic energy that is
0302-4598r98r$ - see front matter q 1998 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 2 - 4 5 9 8 Ž 9 8 . 0 0 1 9 1 - 3
J. Ruohonen et al.r Bioelectrochemistry and Bioenergetics 47 (1998) 213–219
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required to induce a given electric field. Finally, we present optimal coils that can be used with commercially available stimulator electronics units.
2. Theory 2.1. Basic principles of magnetic stimulation In magnetic stimulation, a current pulse I Ž t . driven through a stimulating coil sets up a strong magnetic field pulse, which induces an electric field E in the underlying tissue. The E sets free charges in the tissue into coherent motion both in the intra- and extracellular spaces, depolarizing or hyperpolarizing the cell membranes that interrupt the free motion of charges. The chain of events in electrical stimulation is similar, although the driving force is the electrical current injected in the tissue through electrodes. The high magnetic field strength of ; 1 T required in magnetic stimulation is achieved by discharging a capacitor charged to some kilovolts through the coil. The resulting current pulse is usually one decaying sine wave that lasts 200 to 300 ms; the peak current in the coil will be some kiloamperes. The strength of E is linearly proportional to d I Ž t .rdt. The induced E decreases quickly with distance from the coil and it is hence favorable, but not necessary, to place the coil against the skin. Strongest responses are normally obtained with the coil parallel to the skin. The stimulating coil has one or more tightly wound circular or nearly circular wings. The active region to be positioned above the target is roughly the edge of the wing.
Therefore, there is a fundamental difference between magnetic stimulation of the brain and peripheral nervous system in that whereas stimulation of the straight peripheral nerves requires a high gradient of E, bent and short cortical neurons are dominantly activated at the site of the maximum E w14–16x. With magnetic stimulation it is possible to achieve activation also with the electric field induced transverse to the length of the nerve w17x. Here, we limit the attention to the gradient activation mechanism, which is believed to be the dominant mechanism also in electrical stimulation of the nerve fibers w13x. 2.3. The induced electric field The electric field induced in the body is calculated using the so-called reciprocity model described in Ref. w18x. Explicit formula for d E xrd x induced in various homogeneous conductor models are given in Ref. w19x. Here, we assume that the underlying tissue is an unbounded homogeneous medium, which is considered to be a reasonably accurate and simple model to approximate the limbs in coil optimization w20,21x. In the calculation of the induced field, each loop of the coil winding is approximated by a filament in the center of the wire. The estimated threshold values of d E xrd x for provoking motor-evoked potentials range from y5 to y15 mVrmm2 w13,15,17,22x. We require here that the peak value is d E xrd x s y10 mVrmm2 . We omit the possible influence of the shape of E on the threshold value w22x. The electric field computation is confined to a 10-mm-deep plane below the coil’s bottom surface, corresponding to the typical distance of superficial distal nerves from the skin.
2.2. The cellular response to magnetic stimulation
2.4. Stimulator’s figure of merit: the peak magnetic energy
The response of long axons to magnetic stimulation is well-understood w11–13x. The activation is supposed to occur most effectively at the site of greatest variation of the electric field along the nerve. A simplified description of the transmembrane potential change V of an axon is given by the cable equation w13x:
Figures of merit are required to evaluate and compare different stimulators and coils; important figures of merit include power consumption, coil heating and peak magnetic energy in the coil w23x. Our objective is to device a small, cheap and effective stimulator suitable for FMS. We assess the goodness with the peak magnetic energy, because it represents the energy that must be transmitted by or stored in the stimulator’s components Žcoil, capacitor, thyristor switch and power source.. The price, volume and weight of the components, and hence also of the stimulator, are proportional to W. The energy is:
Vyl
2
E 2V E x2
qt
EV Et
s yl
2
E 2 Ex E x2
Ž 1.
where l s Žp rm dr4R i .1r2 and t s rm c m are the axon’s space and time constants, x is the distance along the distal axon and E x is the x-component of the electric field. The membrane resistance and capacitance per unit length are rm and c m . The intracellular resistivity is R i and the axon diameter is d. The right-hand term, d E xrd x, is the socalled activating function and the strongest depolarization is achieved where it is most negative. Eq. Ž1. holds also for long bent axons. At bends, a high effective yd E xrd x is achieved in a homogeneous E.
Ws
1 2
2 LImax
Ž 2.
where L is the coil’s inductance and Imax the peak current in the coil. An effective coil should have a small L and induce a high E with low Imax . It is not possible to realize both, since whereas the required Imax decreases with the number of turns and with coil dimensions and winding
J. Ruohonen et al.r Bioelectrochemistry and Bioenergetics 47 (1998) 213–219
density, the inductance L grows. The current I Ž t . in the coil is w23x: IŽ t. s
Uo Lv
ey a t sin Ž v t .
Ž 3. 2
.y1
2
where a s Rr2 L and v s Ž LC y a and Uo is the discharge capacitor’s initial voltage. The capacitance is C and the resistance in the circuit is R. The peak current is computed from Eq. Ž3. requiring that the initial rate of change d Irdt s UorL Ž t s 0. is sufficient to induce the given d E xrd x and that I Ž t . peaks at t s t , with t as the rise time from zero to peak. The cell membrane behaves as a leaky integrator, so that the efficacy of stimulation depends on the duration of the stimulus. We fix t s 100 ms, which is obtained by changing the capacitance C, and suffices because our interest is to compare the efficacy of different coils. In some calculations that follow we will assume that the current rises linearly, so that Imax s t d Irdt. The inductance L is a sum of self inductances of individual loops and of mutual inductances between all loops. The mutual inductance between loops i and j is obtained from Neumann’s equation: Mi j s
mo
d li P d l j
HH 4p i j < r y r < i
Ž 4.
j
Above, the integration is performed along the wire; in this study, the loops are approximated by 300 filamentary line segments. The self-inductance of individual loops of the coil is estimated as the mutual inductance computed from Eq. Ž4. between filamentary loops in the center and in the surface of the wire w24x. 2.5. The definition of coil geometry Three coil types useful for peripheral magnetic stimulation can be distinguished according to the number of their wings: single, 8-shaped w25x and 4-leaf coils w26,27x. We cast the coil geometry in terms of six parameters Žsee Fig. 1.: Ži. w—outer width of the wing; Žii. h—outer height of the wing; Žiii. s—length of a straight edge in the wing Ž s F h.; Živ. N—total number of turns in the coil; Žv.
215
K—number of layers, i.e., concentric turns in one level; and Žvi. coil type. The shape of a loop centered at the origin of the xy-plane is given by:
°< y < F 0.5s, x s y0.5w, ~ y s " Ž h y s . (0.25 y x rw 2
¢y s "h(0.25 y x rw 2
2
2
q 0.5s,y 0.5w - x - 0,
,0 F x F 0.5w.
Ž 4a . Depending on the parameters, the wings can be circular, D-shaped or elliptic. The coils are wound as tight as possible so that the space between adjacent loops is equal to the space required by the insulation.
3. Simulation results 3.1. Single Õs. 8-shaped Õs. 4-leaf coils We first examine the effects of the coil type and dimensions on the peak magnetic energy W in the coil. Fig. 2 compares the single, 8-shaped and 4-leaf coils with varying wing diameter. Other parameters were held constant. The coils were wound of N s 20 circular turns in K s 5 layers. The wire diameter was 2 mm with 0.1 mm insulation around the wire. The data were calculated requiring that the peak value d E xrd x s y10 mVrmm2 in a plane 10 mm from and parallel to the coil. The coil current was linearly rising with t s 100 ms. Fig. 2 shows that the 4-leaf coil is more effective than the single or 8-shaped coil. While the single coil requires at minimum 1100 J and the 8-shaped 550 J, the 4-leaf coil requires only 280 J of energy to induce y10 mVrmm2 . Other advantages of the 4-leaf coil are discussed in w27x. The minimum energy W is obtained when the wing size is ; 25–40 mm, depending on the coil type. For large coil size W grows monotonically because I levels off to a constant value while L grows monotonically. When the 4-leaf coil becomes very large, the regions of high yd E xrd x induced by each wing do not sum up
Fig. 1. Definition of the coil geometry. A four-leaf coil over a straight nerve Ždotted line. is shown. Ža. Three variables define the loop shape: width w, height h and segment length s. Žb. The profile of the wings is defined by the number of turns and the number of layers K in one level. The wing in the figure has 24 loops in 7 layers. In the definition of the geometry we use the total number of turns in all wings, N. The direction of the current in the wings is depicted by the arrows.
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J. Ruohonen et al.r Bioelectrochemistry and Bioenergetics 47 (1998) 213–219
since the 4 wings begin to behave as two separate 8-shaped coils. For the same reason, for large wing size Žexceeding the range of Fig. 2. the 8-shaped coil will behave as two separate single coils and thus be less effective than one single coil.
3.2. Wing shape, size and winding profile Fig. 2. Comparison of single, 8-shaped and 4-leaf coils: the peak magnetic energy W required to induce y10 mVrmm2 is displayed as a function of the wing diameter. For all coils, N s 20 and K s 5. The wire diameter is 2 mm and there is 0.1 mm of insulation around the wire; hence, the smallest possible diameter is ; 23 mm. The wings are circular Ž hs w and ss 0.. The nerve is 10 mm below the coil. The current rises linearly with t s100 ms.
forming a single zone of high yd E xrd x in the center of the coil. In Fig. 2, when the wing size exceeds 80 mm, the 8-shaped coil becomes more effective than the 4-leaf coil,
Four-leaf coils made of 40-mm-diameter wings of different shape, size and winding profile are compared in Fig. 3. In Fig. 3Ža. the peak magnetic energy in the coil W is plotted as a function of the winding layers K for different number of turns N in the coil. More energy is needed when the number of layers K is small Ža solenoidal coil. than when there are many layers Žflat coil.. A similar W can be reached with several coil profiles. The effects of changes in the wing shape are analyzed in Fig. 3Žb. and Žc.. First, the segment length s is varied in Fig. 3Žb. while holding the winding profile and the wing’s
Fig. 3. The effect of wing shape and winding profile on the peak magnetic energy W for a 4-leaf coil. Ža. W as a function of the number of layers K for N s 20, 40 and 60 total turns. Žb. W as a function of the segment length s for wings with h s w s 40 mm. When s s 0, the wings are circular; when s s 40 mm, the wings are D-shaped. Žc. Contours of W and inductance L for varying wing width w and height h. The basic coil in all figures has N s 20 total turns in K s 5 layers; the wing diameter is w s h s 40 mm and s s 0 mm. Other details as in Fig. 2.
J. Ruohonen et al.r Bioelectrochemistry and Bioenergetics 47 (1998) 213–219
outer width and height constant Ž N s 20, K s 5, h s w s 40 mm.. The energy W decreases from 390 J to 250 J when the loop shape goes from circular Ž s s 0 mm. to D-shaped Ž s s 40 mm.. Second, in Fig. 3Žc., the loop width w and height h are varied Žwith s s 0., while the coil profile is held constant Ž N s 40, K s 5.. The circular wing shape is clearly preferable over elliptic shapes; the minimum energy of 278 J is achieved with circular wings with the outer diameter of 25 mm. Although in Fig. 3Žc. the minimum W is achieved with the smallest possible wing diameter, this does not happen with all N and K. Also, the electronics pose practical limits to the minimum possible inductance L Žsee, w23x.. For instance, requiring that L G 9 mH in Fig. 3Žc., the minimum W s 820 J is achieved with w s h s 65 mm. 3.3. Optimal coils for FMS The search method of Rosenbrock w28x was employed to find the parameter values that yield the minimum peak magnetic energy W in the coil that is required to induce d E xrd x s y10 mVrmm2 in a plane 10 mm below the coil. We limited the search to coils that can be directly used with the today’s commercial stimulators without modifications to the stimulator’s electrical circuitry. Therefore, we required that the coil inductance must be L s 20– 30 mH. Also, we assumed that the pulse shape is realistic,
217
i.e., a damped sine pulse given by Eq. Ž3.. We held constant the rise time from zero to peak t s 100 ms and the resistance in the circuit R s 50 m V, which are typical values in today’s stimulators. The capacitance C was calculated requiring that I Ž t . peaks at t s t . In addition, it was required that each wing had the same shape and profile and an equal number of turns. Otherwise, the coil shape parameters could vary freely within physically feasible limits Že.g., coils size ) 0 mm.. Although the results in Figs. 2 and 3 showed that the 4-leaf coil outperforms the other coil types, we repeated the optimization also for the 8-shaped coil, since it is the most frequently used coil geometry. Fig. 4 depicts the optimum coils and the resulting d E xrd x in a plane below the coils. Two optimal 4-leaf coil are shown in Ža. and Žb. that have a different wire diameter Ž2 and 3 mm.. Interestingly, with both wire gauges the optimum wing size and shape are the same, h s w s s s 42 mm. The coil wound of 2-mm wire is more effective with W s 106 J than 3-mm wire with W s 134 J. The optimal 8-shaped coil in Fig. 4Žc., wound of 2-mm wire, is far less effective, demanding W s 239 J. Greater wire size allows the use of hollow wire for continuous water cooling w23x. The optimum coils had L s 20 mH, so that to have t s 100 ms, the capacitance must be C s 225 mF. With the 4-leaf coils of Fig. 4Ža. and Žb. d E xrd x s y10 mVrmm2 is achieved with d Irdt s 55 and 62 Arms, respectively,
Fig. 4. Results of minimizing the peak magnetic energy W in the coil requiring that the inductance L ranges from 20 to 30 mH. Optimal coils: Ža. 4-leaf coil wound of 2-mm wire Ž w s h s s s 42 mm, N s 56, K s 9.. Žb. 4-leaf coil wound of 3-mm wire Ž w s h s s s 42 mm, N s 60, K s 6.. Žc. 8-shaped coil wound of 2-mm wire Ž w s h s s s 50 mm, N s 42, K s 11.. The top figures show the coil geometry from above and list L and W. The optimal location of the nerve is indicated by the dotted line. The bold arrows show the current direction in each wing. The bottom maps show contours of d E xrd x induced in a plane 10 mm below the bottom surface of the coils assuming that d Irdt s 100 Arms. The zero contour is bold, the positive contours dashed and the negative contours solid. Contour step is 2 mVrmm2 . Peak values of yd E xrd x are listed.
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Fig. 5. The peak values of yd E x rd x induced at different distances from the coil’s bottom surface for the coils of Fig. 4 Žthe letters a, b and c denote the corresponding items of Fig. 4.. d Ird t s100 Arms.
corresponding to an initial capacitor voltage Uo of 1100 and 1235 V. From Eq. Ž3., the resulting peak currents will be 3250 A and 3650 A. Fig. 5 plots the peak d E xrd x for the coils of Fig. 4 as a function of the distance from the coil’s bottom surface when d Irdt s 100 Arms. It is clearly important that the coil is placed as close to the nerve as possible, since even a small increase in the distance will cause a notable decrease in the field strength. For instance, at a depth of 12 mm the field will be 80% of the field at 10 mm. Hence, to have the same d E xrd x at 12 mm as at 10 mm, the required energy will be about 1.5-fold. The 4-leaf coil has so far been used only in a paper by Roth et al. w27x. These authors constructed a 35-mH 4-leaf coil, which induced a peak d E xrd x f y2.7 mVrmm2 in a plane 12 mm below the bottom surface of the coil when the coil was driven with d Ird t s 25.7 Arms. Making the same assumptions as above in Fig. 4 Ž R s 50 m V and t s 100 ms, obtained with C s 124 mF. and requiring that d Irdt s 95.2 Arms so that the peak d E xrd x s y10 mVrmm2 , the resulting Imax s 5835 A and W s 596 J. The energy required by the coils in Fig. 4 for nerve depth of 12 mm would be 163 J, 201 J and 341 J, for Ža., Žb. and Žc., respectively.
4. Conclusions There are some important differences between electrical and magnetic stimulation. The body tissue does not resist magnetic fields, whereas it resists considerably the currents in electrical stimulation; hence, the effects of the electrical and geometrical properties of the tissue and bones may differ greatly. Second, whereas the pulse shape in magnetic stimulation is always a damped sinusoid Žor with great damping, monophasic. and its duration cannot be easily changed, electrical stimulation allows adjustment of the pulse shape and duration for effective stimulation. A drawback of magnetic stimulation is that it may be unable of directly exciting muscle fibers; in an experiment with
cats magnetic stimulation could excite muscles via stimulation of motor nerve axons only w29x. Several FMS experiments have already been described in literature. Lin et al. assessed the possibility of FMS to help bladder emptying and training in individuals with spinal cord injury w9x. Craggs et al. discussed the possibility to use FMS of the phrenic nerve for respiratory muscle function w7x. Sheriff et al. reported that FMS over the sacrum could profoundly suppress detrusor hyper-reflexia resulting from spinal cord injury w10x. However, to enable the use of magnetic stimulation as an alternative of electrical stimulation for functional or rehabilitation purposes, it is crucial to optimize the available stimulating devices, and the key to optimization is the coil design w23x. Few papers in literature address coil optimization. In one study, 8-shaped and 4-leaf coils with various angles between the wings, as to better fit the shape of the tissue surface, were analyzed along with square and circular loop shapes w21x. The effect of the size of the circular coil was analyzed in, e.g., Ref. w19x and the degree to which the field can be concentrated in Ref. w30x. These studies considered the relationship between the amplitude of the driving current and the magnitude of the induced field, which, however, is not a measure of the efficacy of the coil or the stimulator. In context of repetitive brain stimulation, the analysis has been extended to minimize the stimulator’s power consumption, coil heating and the peak magnetic energy w23x. These last results do not apply directly for FMS, since the activating mechanisms in brain and peripheral stimulation are thought to be different. For the stimulation of curved axons or near terminations like in nerve root stimulation, coils similar to those used in brain stimulation may be useful. Coil design depends greatly on the application, there being no globally optimal solution w23x. In this paper, we analyzed where the copper loops inside the coil should be placed to achieve the required electric field gradient along the nerve with the minimum driving energy in the coil. For this purpose, we cast the coil geometry in terms of six parameters that define the size, shape, type and winding profile of the coil. One prominent result of analyzing how changes in the geometry factors affect the driving energy was that single and 8-shaped coils are much less effective than 4-leaf coils. Another conclusion was that D-shaped wings are better over elliptic or circular wings and that the winding profile should preferably be flat with many concentric layers. The optimum outer wing diameter was found to be 40–50 mm. Although smaller coils have been suggested on the basis of field modeling, our results indicate that the reduced inductance of small coils is not enough to compensate for the greater driving current. Although this preliminary study limited the discussion to the devices for functional andror rehabilitative magnetic stimulation, the results from coil optimization are useful in all applications of magnetic stimulation, including the whole neurological and neurophysiological realm.
J. Ruohonen et al.r Bioelectrochemistry and Bioenergetics 47 (1998) 213–219
Acknowledgements This work has been partially coordinated within the framework of the program ‘Magnetic Stimulation’ of the Centro di Ingegneria Biomedica CNR and the Istituto Nazionale di Riposo e Cura per Anziani ŽINRCA.. The support of the Finnish Technology Development Centre ŽTEKES. is also acknowledged.
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Functional magnetic stimulation: theory and coil optimization Jarmo Ruohonen
a,b,)
, Paolo Ravazzani c , Ferdinando Grandori
c
a
c
BioMag Laboratory, Medical Engineering Centre, Helsinki UniÕersity Hospital, Finland b Bioengineering Centre, Pro JuÕentute Don Gnocchi Foundation, Milan, Italy CNR Centre of Biomedical Engineering, Piazza Leonardo da Vinci 32, I-20133 Milan, Italy Received 9 January 1998; revised 2 September 1998; accepted 25 September 1998
Abstract Excitable tissue can be stimulated non-invasively by means of externally applied changing magnetic fields. This paper considers theoretically magnetic stimulation to functionally stimulate the tissue. We optimize the stimulating coil to accomplish functional magnetic stimulation ŽFMS. with the minimum cost expressed in terms of driving energy needed to provoke excitation. Magnetic stimulation is less discomfortable than functional electrical stimulation ŽFES. and therefore, the use of FMS opens up completely new views to the restoration of movement and therapy in patients with neuromuscular system impairment. Advances in coil design can make FMS a promising application field for magnetic stimulation. q 1998 Elsevier Science S.A. All rights reserved. Keywords: Coil design; Coil shape; Peripheral nerves; Neuromuscular rehabilitation; Functional stimulation; FES
1. Introduction Excitable neuromuscular cells can be stimulated non-invasively by strong magnetic field pulses that induce an electric field and a flow of current in the tissue, thereby leading to local depolarization of the cell membrane w1,2x. The applications of magnetic stimulation are apparently immense since tissues as diverse as the brain, peripheral nerves and the heart can be stimulated. However, further progress is limited by the unsatisfactory capability of focusing and controlling the excitation and by the large amount of energy required to induce a strong enough electric field in the tissue. Therapeutic use of magnetic stimulation appears an exciting and promising application in the near future, being particularly true for functional andror rehabilitative stimulation. Presently, there is a great interest in functional electrical stimulation ŽFES., a generic term for the use of electricity to augmenting or restoring function in patients suffering from movement impairment w3,4x. FES includes techniques that go beyond simple muscle stimulation; the terms FES, FNS Žfunctional neuromuscular stimulation., and EMS Želectric muscle stimulation. are often used
) Corresponding author. Tel: q358-9-471-5541; fax: q3589 -4 7 1 -5 7 8 1 ; e-m ail: jarm o @ b io m ag .h elsin k i.fi; In tern et: www.biomag.helsinki.firtms
interchangeably. In FES, an electric current is fed through implanted or surface electrodes and may generate or improve movement in patients with several pathologies, with beneficial effects that occasionally persist even after the period of therapy w5,6x. It has been recently realized that functional magnetic stimulation ŽFMS. may provide a fascinating option or additional tool to the conventional FES w7–10x. The often painful sensation experienced by the subject in electrical stimulation is largely reduced or even abolished in magnetic stimulation, which promotes the use of FMS, e.g., in muscular atrophy prevention, muscle control and force reinforcement. However, to accomplish comfortable, safe and widespread use of FMS, it is necessary that one can overcome many limitations of magnetic stimulation, including the poor reproducibility and high power demands as well as the large dimensions of the available devices. Moreover, the present magnetic stimulators have been designed exclusively for brain stimulation and will not operate optimally in FMS, mainly because the dominant activating mechanisms are different. In this study, after briefly reviewing the necessary mathematical tools, we describe a method to optimize coils for use in functional magnetic stimulation. Optimization is accomplished by studying how the coil should be wound and which coil type Žsingle, 8-shaped or 4-leaf. should be used to achieve effective stimulation. We measure the efficacy of the coil as the peak magnetic energy that is
0302-4598r98r$ - see front matter q 1998 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 2 - 4 5 9 8 Ž 9 8 . 0 0 1 9 1 - 3
J. Ruohonen et al.r Bioelectrochemistry and Bioenergetics 47 (1998) 213–219
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required to induce a given electric field. Finally, we present optimal coils that can be used with commercially available stimulator electronics units.
2. Theory 2.1. Basic principles of magnetic stimulation In magnetic stimulation, a current pulse I Ž t . driven through a stimulating coil sets up a strong magnetic field pulse, which induces an electric field E in the underlying tissue. The E sets free charges in the tissue into coherent motion both in the intra- and extracellular spaces, depolarizing or hyperpolarizing the cell membranes that interrupt the free motion of charges. The chain of events in electrical stimulation is similar, although the driving force is the electrical current injected in the tissue through electrodes. The high magnetic field strength of ; 1 T required in magnetic stimulation is achieved by discharging a capacitor charged to some kilovolts through the coil. The resulting current pulse is usually one decaying sine wave that lasts 200 to 300 ms; the peak current in the coil will be some kiloamperes. The strength of E is linearly proportional to d I Ž t .rdt. The induced E decreases quickly with distance from the coil and it is hence favorable, but not necessary, to place the coil against the skin. Strongest responses are normally obtained with the coil parallel to the skin. The stimulating coil has one or more tightly wound circular or nearly circular wings. The active region to be positioned above the target is roughly the edge of the wing.
Therefore, there is a fundamental difference between magnetic stimulation of the brain and peripheral nervous system in that whereas stimulation of the straight peripheral nerves requires a high gradient of E, bent and short cortical neurons are dominantly activated at the site of the maximum E w14–16x. With magnetic stimulation it is possible to achieve activation also with the electric field induced transverse to the length of the nerve w17x. Here, we limit the attention to the gradient activation mechanism, which is believed to be the dominant mechanism also in electrical stimulation of the nerve fibers w13x. 2.3. The induced electric field The electric field induced in the body is calculated using the so-called reciprocity model described in Ref. w18x. Explicit formula for d E xrd x induced in various homogeneous conductor models are given in Ref. w19x. Here, we assume that the underlying tissue is an unbounded homogeneous medium, which is considered to be a reasonably accurate and simple model to approximate the limbs in coil optimization w20,21x. In the calculation of the induced field, each loop of the coil winding is approximated by a filament in the center of the wire. The estimated threshold values of d E xrd x for provoking motor-evoked potentials range from y5 to y15 mVrmm2 w13,15,17,22x. We require here that the peak value is d E xrd x s y10 mVrmm2 . We omit the possible influence of the shape of E on the threshold value w22x. The electric field computation is confined to a 10-mm-deep plane below the coil’s bottom surface, corresponding to the typical distance of superficial distal nerves from the skin.
2.2. The cellular response to magnetic stimulation
2.4. Stimulator’s figure of merit: the peak magnetic energy
The response of long axons to magnetic stimulation is well-understood w11–13x. The activation is supposed to occur most effectively at the site of greatest variation of the electric field along the nerve. A simplified description of the transmembrane potential change V of an axon is given by the cable equation w13x:
Figures of merit are required to evaluate and compare different stimulators and coils; important figures of merit include power consumption, coil heating and peak magnetic energy in the coil w23x. Our objective is to device a small, cheap and effective stimulator suitable for FMS. We assess the goodness with the peak magnetic energy, because it represents the energy that must be transmitted by or stored in the stimulator’s components Žcoil, capacitor, thyristor switch and power source.. The price, volume and weight of the components, and hence also of the stimulator, are proportional to W. The energy is:
Vyl
2
E 2V E x2
qt
EV Et
s yl
2
E 2 Ex E x2
Ž 1.
where l s Žp rm dr4R i .1r2 and t s rm c m are the axon’s space and time constants, x is the distance along the distal axon and E x is the x-component of the electric field. The membrane resistance and capacitance per unit length are rm and c m . The intracellular resistivity is R i and the axon diameter is d. The right-hand term, d E xrd x, is the socalled activating function and the strongest depolarization is achieved where it is most negative. Eq. Ž1. holds also for long bent axons. At bends, a high effective yd E xrd x is achieved in a homogeneous E.
Ws
1 2
2 LImax
Ž 2.
where L is the coil’s inductance and Imax the peak current in the coil. An effective coil should have a small L and induce a high E with low Imax . It is not possible to realize both, since whereas the required Imax decreases with the number of turns and with coil dimensions and winding
J. Ruohonen et al.r Bioelectrochemistry and Bioenergetics 47 (1998) 213–219
density, the inductance L grows. The current I Ž t . in the coil is w23x: IŽ t. s
Uo Lv
ey a t sin Ž v t .
Ž 3. 2
.y1
2
where a s Rr2 L and v s Ž LC y a and Uo is the discharge capacitor’s initial voltage. The capacitance is C and the resistance in the circuit is R. The peak current is computed from Eq. Ž3. requiring that the initial rate of change d Irdt s UorL Ž t s 0. is sufficient to induce the given d E xrd x and that I Ž t . peaks at t s t , with t as the rise time from zero to peak. The cell membrane behaves as a leaky integrator, so that the efficacy of stimulation depends on the duration of the stimulus. We fix t s 100 ms, which is obtained by changing the capacitance C, and suffices because our interest is to compare the efficacy of different coils. In some calculations that follow we will assume that the current rises linearly, so that Imax s t d Irdt. The inductance L is a sum of self inductances of individual loops and of mutual inductances between all loops. The mutual inductance between loops i and j is obtained from Neumann’s equation: Mi j s
mo
d li P d l j
HH 4p i j < r y r < i
Ž 4.
j
Above, the integration is performed along the wire; in this study, the loops are approximated by 300 filamentary line segments. The self-inductance of individual loops of the coil is estimated as the mutual inductance computed from Eq. Ž4. between filamentary loops in the center and in the surface of the wire w24x. 2.5. The definition of coil geometry Three coil types useful for peripheral magnetic stimulation can be distinguished according to the number of their wings: single, 8-shaped w25x and 4-leaf coils w26,27x. We cast the coil geometry in terms of six parameters Žsee Fig. 1.: Ži. w—outer width of the wing; Žii. h—outer height of the wing; Žiii. s—length of a straight edge in the wing Ž s F h.; Živ. N—total number of turns in the coil; Žv.
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K—number of layers, i.e., concentric turns in one level; and Žvi. coil type. The shape of a loop centered at the origin of the xy-plane is given by:
°< y < F 0.5s, x s y0.5w, ~ y s " Ž h y s . (0.25 y x rw 2
¢y s "h(0.25 y x rw 2
2
2
q 0.5s,y 0.5w - x - 0,
,0 F x F 0.5w.
Ž 4a . Depending on the parameters, the wings can be circular, D-shaped or elliptic. The coils are wound as tight as possible so that the space between adjacent loops is equal to the space required by the insulation.
3. Simulation results 3.1. Single Õs. 8-shaped Õs. 4-leaf coils We first examine the effects of the coil type and dimensions on the peak magnetic energy W in the coil. Fig. 2 compares the single, 8-shaped and 4-leaf coils with varying wing diameter. Other parameters were held constant. The coils were wound of N s 20 circular turns in K s 5 layers. The wire diameter was 2 mm with 0.1 mm insulation around the wire. The data were calculated requiring that the peak value d E xrd x s y10 mVrmm2 in a plane 10 mm from and parallel to the coil. The coil current was linearly rising with t s 100 ms. Fig. 2 shows that the 4-leaf coil is more effective than the single or 8-shaped coil. While the single coil requires at minimum 1100 J and the 8-shaped 550 J, the 4-leaf coil requires only 280 J of energy to induce y10 mVrmm2 . Other advantages of the 4-leaf coil are discussed in w27x. The minimum energy W is obtained when the wing size is ; 25–40 mm, depending on the coil type. For large coil size W grows monotonically because I levels off to a constant value while L grows monotonically. When the 4-leaf coil becomes very large, the regions of high yd E xrd x induced by each wing do not sum up
Fig. 1. Definition of the coil geometry. A four-leaf coil over a straight nerve Ždotted line. is shown. Ža. Three variables define the loop shape: width w, height h and segment length s. Žb. The profile of the wings is defined by the number of turns and the number of layers K in one level. The wing in the figure has 24 loops in 7 layers. In the definition of the geometry we use the total number of turns in all wings, N. The direction of the current in the wings is depicted by the arrows.
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since the 4 wings begin to behave as two separate 8-shaped coils. For the same reason, for large wing size Žexceeding the range of Fig. 2. the 8-shaped coil will behave as two separate single coils and thus be less effective than one single coil.
3.2. Wing shape, size and winding profile Fig. 2. Comparison of single, 8-shaped and 4-leaf coils: the peak magnetic energy W required to induce y10 mVrmm2 is displayed as a function of the wing diameter. For all coils, N s 20 and K s 5. The wire diameter is 2 mm and there is 0.1 mm of insulation around the wire; hence, the smallest possible diameter is ; 23 mm. The wings are circular Ž hs w and ss 0.. The nerve is 10 mm below the coil. The current rises linearly with t s100 ms.
forming a single zone of high yd E xrd x in the center of the coil. In Fig. 2, when the wing size exceeds 80 mm, the 8-shaped coil becomes more effective than the 4-leaf coil,
Four-leaf coils made of 40-mm-diameter wings of different shape, size and winding profile are compared in Fig. 3. In Fig. 3Ža. the peak magnetic energy in the coil W is plotted as a function of the winding layers K for different number of turns N in the coil. More energy is needed when the number of layers K is small Ža solenoidal coil. than when there are many layers Žflat coil.. A similar W can be reached with several coil profiles. The effects of changes in the wing shape are analyzed in Fig. 3Žb. and Žc.. First, the segment length s is varied in Fig. 3Žb. while holding the winding profile and the wing’s
Fig. 3. The effect of wing shape and winding profile on the peak magnetic energy W for a 4-leaf coil. Ža. W as a function of the number of layers K for N s 20, 40 and 60 total turns. Žb. W as a function of the segment length s for wings with h s w s 40 mm. When s s 0, the wings are circular; when s s 40 mm, the wings are D-shaped. Žc. Contours of W and inductance L for varying wing width w and height h. The basic coil in all figures has N s 20 total turns in K s 5 layers; the wing diameter is w s h s 40 mm and s s 0 mm. Other details as in Fig. 2.
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outer width and height constant Ž N s 20, K s 5, h s w s 40 mm.. The energy W decreases from 390 J to 250 J when the loop shape goes from circular Ž s s 0 mm. to D-shaped Ž s s 40 mm.. Second, in Fig. 3Žc., the loop width w and height h are varied Žwith s s 0., while the coil profile is held constant Ž N s 40, K s 5.. The circular wing shape is clearly preferable over elliptic shapes; the minimum energy of 278 J is achieved with circular wings with the outer diameter of 25 mm. Although in Fig. 3Žc. the minimum W is achieved with the smallest possible wing diameter, this does not happen with all N and K. Also, the electronics pose practical limits to the minimum possible inductance L Žsee, w23x.. For instance, requiring that L G 9 mH in Fig. 3Žc., the minimum W s 820 J is achieved with w s h s 65 mm. 3.3. Optimal coils for FMS The search method of Rosenbrock w28x was employed to find the parameter values that yield the minimum peak magnetic energy W in the coil that is required to induce d E xrd x s y10 mVrmm2 in a plane 10 mm below the coil. We limited the search to coils that can be directly used with the today’s commercial stimulators without modifications to the stimulator’s electrical circuitry. Therefore, we required that the coil inductance must be L s 20– 30 mH. Also, we assumed that the pulse shape is realistic,
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i.e., a damped sine pulse given by Eq. Ž3.. We held constant the rise time from zero to peak t s 100 ms and the resistance in the circuit R s 50 m V, which are typical values in today’s stimulators. The capacitance C was calculated requiring that I Ž t . peaks at t s t . In addition, it was required that each wing had the same shape and profile and an equal number of turns. Otherwise, the coil shape parameters could vary freely within physically feasible limits Že.g., coils size ) 0 mm.. Although the results in Figs. 2 and 3 showed that the 4-leaf coil outperforms the other coil types, we repeated the optimization also for the 8-shaped coil, since it is the most frequently used coil geometry. Fig. 4 depicts the optimum coils and the resulting d E xrd x in a plane below the coils. Two optimal 4-leaf coil are shown in Ža. and Žb. that have a different wire diameter Ž2 and 3 mm.. Interestingly, with both wire gauges the optimum wing size and shape are the same, h s w s s s 42 mm. The coil wound of 2-mm wire is more effective with W s 106 J than 3-mm wire with W s 134 J. The optimal 8-shaped coil in Fig. 4Žc., wound of 2-mm wire, is far less effective, demanding W s 239 J. Greater wire size allows the use of hollow wire for continuous water cooling w23x. The optimum coils had L s 20 mH, so that to have t s 100 ms, the capacitance must be C s 225 mF. With the 4-leaf coils of Fig. 4Ža. and Žb. d E xrd x s y10 mVrmm2 is achieved with d Irdt s 55 and 62 Arms, respectively,
Fig. 4. Results of minimizing the peak magnetic energy W in the coil requiring that the inductance L ranges from 20 to 30 mH. Optimal coils: Ža. 4-leaf coil wound of 2-mm wire Ž w s h s s s 42 mm, N s 56, K s 9.. Žb. 4-leaf coil wound of 3-mm wire Ž w s h s s s 42 mm, N s 60, K s 6.. Žc. 8-shaped coil wound of 2-mm wire Ž w s h s s s 50 mm, N s 42, K s 11.. The top figures show the coil geometry from above and list L and W. The optimal location of the nerve is indicated by the dotted line. The bold arrows show the current direction in each wing. The bottom maps show contours of d E xrd x induced in a plane 10 mm below the bottom surface of the coils assuming that d Irdt s 100 Arms. The zero contour is bold, the positive contours dashed and the negative contours solid. Contour step is 2 mVrmm2 . Peak values of yd E xrd x are listed.
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Fig. 5. The peak values of yd E x rd x induced at different distances from the coil’s bottom surface for the coils of Fig. 4 Žthe letters a, b and c denote the corresponding items of Fig. 4.. d Ird t s100 Arms.
corresponding to an initial capacitor voltage Uo of 1100 and 1235 V. From Eq. Ž3., the resulting peak currents will be 3250 A and 3650 A. Fig. 5 plots the peak d E xrd x for the coils of Fig. 4 as a function of the distance from the coil’s bottom surface when d Irdt s 100 Arms. It is clearly important that the coil is placed as close to the nerve as possible, since even a small increase in the distance will cause a notable decrease in the field strength. For instance, at a depth of 12 mm the field will be 80% of the field at 10 mm. Hence, to have the same d E xrd x at 12 mm as at 10 mm, the required energy will be about 1.5-fold. The 4-leaf coil has so far been used only in a paper by Roth et al. w27x. These authors constructed a 35-mH 4-leaf coil, which induced a peak d E xrd x f y2.7 mVrmm2 in a plane 12 mm below the bottom surface of the coil when the coil was driven with d Ird t s 25.7 Arms. Making the same assumptions as above in Fig. 4 Ž R s 50 m V and t s 100 ms, obtained with C s 124 mF. and requiring that d Irdt s 95.2 Arms so that the peak d E xrd x s y10 mVrmm2 , the resulting Imax s 5835 A and W s 596 J. The energy required by the coils in Fig. 4 for nerve depth of 12 mm would be 163 J, 201 J and 341 J, for Ža., Žb. and Žc., respectively.
4. Conclusions There are some important differences between electrical and magnetic stimulation. The body tissue does not resist magnetic fields, whereas it resists considerably the currents in electrical stimulation; hence, the effects of the electrical and geometrical properties of the tissue and bones may differ greatly. Second, whereas the pulse shape in magnetic stimulation is always a damped sinusoid Žor with great damping, monophasic. and its duration cannot be easily changed, electrical stimulation allows adjustment of the pulse shape and duration for effective stimulation. A drawback of magnetic stimulation is that it may be unable of directly exciting muscle fibers; in an experiment with
cats magnetic stimulation could excite muscles via stimulation of motor nerve axons only w29x. Several FMS experiments have already been described in literature. Lin et al. assessed the possibility of FMS to help bladder emptying and training in individuals with spinal cord injury w9x. Craggs et al. discussed the possibility to use FMS of the phrenic nerve for respiratory muscle function w7x. Sheriff et al. reported that FMS over the sacrum could profoundly suppress detrusor hyper-reflexia resulting from spinal cord injury w10x. However, to enable the use of magnetic stimulation as an alternative of electrical stimulation for functional or rehabilitation purposes, it is crucial to optimize the available stimulating devices, and the key to optimization is the coil design w23x. Few papers in literature address coil optimization. In one study, 8-shaped and 4-leaf coils with various angles between the wings, as to better fit the shape of the tissue surface, were analyzed along with square and circular loop shapes w21x. The effect of the size of the circular coil was analyzed in, e.g., Ref. w19x and the degree to which the field can be concentrated in Ref. w30x. These studies considered the relationship between the amplitude of the driving current and the magnitude of the induced field, which, however, is not a measure of the efficacy of the coil or the stimulator. In context of repetitive brain stimulation, the analysis has been extended to minimize the stimulator’s power consumption, coil heating and the peak magnetic energy w23x. These last results do not apply directly for FMS, since the activating mechanisms in brain and peripheral stimulation are thought to be different. For the stimulation of curved axons or near terminations like in nerve root stimulation, coils similar to those used in brain stimulation may be useful. Coil design depends greatly on the application, there being no globally optimal solution w23x. In this paper, we analyzed where the copper loops inside the coil should be placed to achieve the required electric field gradient along the nerve with the minimum driving energy in the coil. For this purpose, we cast the coil geometry in terms of six parameters that define the size, shape, type and winding profile of the coil. One prominent result of analyzing how changes in the geometry factors affect the driving energy was that single and 8-shaped coils are much less effective than 4-leaf coils. Another conclusion was that D-shaped wings are better over elliptic or circular wings and that the winding profile should preferably be flat with many concentric layers. The optimum outer wing diameter was found to be 40–50 mm. Although smaller coils have been suggested on the basis of field modeling, our results indicate that the reduced inductance of small coils is not enough to compensate for the greater driving current. Although this preliminary study limited the discussion to the devices for functional andror rehabilitative magnetic stimulation, the results from coil optimization are useful in all applications of magnetic stimulation, including the whole neurological and neurophysiological realm.
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Acknowledgements This work has been partially coordinated within the framework of the program ‘Magnetic Stimulation’ of the Centro di Ingegneria Biomedica CNR and the Istituto Nazionale di Riposo e Cura per Anziani ŽINRCA.. The support of the Finnish Technology Development Centre ŽTEKES. is also acknowledged.
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