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Magnetic stimulation of peripheral nerves: computation of the induced electric field in a cylinder-like structure
Advances in Engineering Sofhvare 22 (1995) 29-35
0 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 0965~9978/95/$09SO
0965-9978(94)00056-S
ELSEVIER
Magnetic stimulation of peripheral nerves: computation of the induced electric field in a cylinder-like structure P. Ravazzani,* § J. Ruohonenj & F. Grandori* * Systems Theory Centre-CNR and Department of Biomedical Engineering, Polytechnic of Milan, Via Ponzio 34/5,20133 Milan, Italy i Medical Engineering Centre, Helsinki University Central Hospital, Stenbickinkatu 9, 00290 Helsinki, Finland (Received 11 August 1994; accepted 7 November 1994) The uncertain site of excitation is the main factor that prevents magnetic stimulation of peripheral nerves from becoming a routine clinical tool. Long and straight fibres are thought to be excited by the first spatial derivative of the electric field component parallel to the fibre and by the electric field component perpendicular to the fibre. The authors have developed an analytical model to compute both of these field features inside pseudo-cylindrical structures. Here they analyse the computation of the electric fields induced with different stimulator arrangements: they examine the effects of changes in stimulator coil location, orientation, size, and geometry and of changes in the dimensions of the limbs and in the position of the nerve inside the limb. Key words: magnetic stimulation, peripheral nerves, computer modelling, electromagnetic induction.
for unbounded and semi-infinite media. Only recently models have been developed for bounded media, like the homogeneous sphere to model the human headG9 and cylindrical structures to model the
INTRODUCTION A suitable electric field E can excite or inhibit neurones. In non-invasive electromagnetic stimulation, E can be produced in the nervous tissue by two different methods: (1) in electrical stimulation, E is produced by driving current
through
limbs.‘e”
An analytical model based on approximating cylinderlike limbs by the stretched homogeneousprolate spheroid is analysed here in detail. The model is employed to reconstruct some practical arrangements and to study the influence of various geometrical factors on the final results. Modelling of E is particularly important since the site of excitation is not well known in magnetic stimulation of peripheral nerves. Accurate prediction of E gives additional information about the feature of E responsible for activation. Moreover, modelling allows the design of more efficient coil shapes and placements with
surface electrodes; (2) in magnetic
stimulation, E is induced by large time-varying currents flowing in a coil which is located above the tissue. Here, is discussed the latter technique, and in particular the computation of E and its gradient inside homogeneous cylinder-like structures that model a human limb. Magnetic stimulation provides a fascinating means for assessing noninvasively the intact nervous system and several nervous disorders in man. It is typically used to determine the conduction velocities of motor and sensory nerves: when a stimulus is given, for example, at the elbow, a muscle twitch in the hand can be observed
respect to the target structures.
Computation of E with the proposed model reduces
after a time delay, which depends on the stimulus location and the conduction velocity.
to: (1) computation of the external (to the bounding tissue) magnetic induction B due to an internal point-
The technique was introduced some 10 years ago,’ but the first theoretical predictions of the induced E were presented only in the beginning of 1990~,~-~
like current dipole Q; (2) computation of the magnetic flux through a surface bounded by the excitation coil. Analytical solutions to the first problem are available only in some simplified geometries, and numerical methods are required in more realistically-shaped
3 To whom correspondence should be addressed. 29
30
P. Ravazzani. J. Ruohonen, F. Grandori
geometries. Magnetic flux can be usually computed by discretizing the flux integral into a finite sum. THE MODEL In this section some theories of biomagnetism and magnetobiology are linked together; biomagnetism is the study of magnetic fields generated by organisms; magnetobiology is the study of biological effects from magnetic fields, magnetic stimulation being a special branch of it. The so-called reciprocity theorem16 implies that at a point it4 inside a medium, the electric field E induced by an external current loop C at a point N can be computed from the magnetic induction B produced at N by a current dipole Q at M. Hence, on practical grounds, E can be computed using virtually the same equations as in biomagnetism when computing the magnetic flux + through the loop C caused by a known source current density. The latter is a well-studied problem and it is therefore sensible to employ here the already existing solutions (for review, seeHamllainen and colleagues17). The quasi-static approximation is valid in magnetic stimulation of the nervous system, and then the reciprocity theorem gives:7’8Y11 E(r) = - Tk
ek jc B(Q . ek,r’) . n(r’)dS’
(1)
k=l
where Z(t) is the current in the coil, el, e2 and e3 are orthogonal unit vectors, and n(r’) is a unit normal vector of the area bounded by the coil. The surface integral is recognized to be the magnetic flux through a surface bounded by the coil. In practice, it is computed by dividing the surface into a number of subsurfaces small enough so that B can be considered constant on them. Typically, it is enough to divide the coil surface into 12 subareas whose weights are optimally chosen.‘* While the coil shape is accounted for in the surface integral in eqn (l), the shape of the medium modelling the tissue contributes to the magnetic induction B. Finding B outside a medium due to a current source inside is the so-called biomagnetic forward problem. Geselowitz” has modified the Ampere-Laplace law to give B due to a general current density; for a dipolar source Q at r’ the result is:
x dS(r’)
(2)
where the electrical potential I’ is (from Horacek2’):
(3)
satisfying Laplace’s equation with appropriate boundary conditions. Here, g(r) is the conductivity of the tissue and Au is the conductivity difference over the boundary S of the conductor in the direction of dS(r’). The vector surface element dS(r’) = n(r’)dS is oriented along the outward unit normal n. In a general geometry, one usually solves iteratively the surface potential after which B can be obtained by numerical integration. On the other hand, only an analytical model allows an easy analysis of changes in the parameters. So far, analytical solutions for B are available only in simplified homogeneous geometries; an analytical solution of B in the stretched prolate spheroid, which was originally given by Cut& and Cohen2’ is adapted here. This can be considered to be a good approximation of a limb when the spheroid major axis is much longer than the minor axis, i.e. when the spheroid looks like a cylinder. The expressions for B and of the resulting induced E are given in the Appendix. The aim of magnetic stimulation is to depolarize the cell membrane in such a way that the membrane potential exceeds a threshold value and the neuron generates a propagating depolarization front, or an action potential. Among others, Roth and Basser22have predicted that a z-axial axon is depolarized where the zaxial gradient El of the z-component of E is most negative. Actually, this view is only in partial agreement with experiments; recently, strong evidence was found of excitation of a peripheral nerve with such coil locations and orientations that do not produce Ei within the nerve.23 It was concluded that the electric field component perpendicular to the nerve also produces excitation. However, here discussion is limited to only such practically interesting situations, in which the mechanism of excitation is thought to depend solely on the magnitude, sign and time-course of E:. The Appendix gives Ei for nerves parallel to the major axis of the prolate spheroid. COMPUTER SIMULATIONS. VARIOUS PARAMETERS
INFLUENCE OF
Various geometrical conditions were simulated, including different coil shapes, radii, locations, orientations, various sizes of the spheroid, and various depths of the target nerve. A 5-cm radius coil in contact with and perpendicular to the surface of a 100~cmlong and 8-cm thick prolate spheroid was considered as the reference arrangement, which was then modified. E was computed 1Omm below the surface and 20mm below the coil, which roughly corresponds to a realistic situation when stimulating the median nerve at the elbow. The induced electric field strength scales linearly with the rate of change of the coil current which was of lOOA/ps in nearly all simulations.
31
Magnetic stimulation of peripheral nerves
The computation was done with a personal computer (80486 at 25 MHz). Depending on the required resolution, it usually took some hours to run the model equations. By computing some of the geometrical factors in advance, the rest of the computation took just minutes.
induced by the edge-tangential single coil; additionally, the figure-of-eight coil induces a more focused field, having a field pattern similar to that with the erect coil in Fig. l(a).
Effect of coil orientation
Changes in E and E: due to variations in coil radius were computed when the coil was erect above and touched the spheroid surface so that its windings were parallel to the nerve. It was found that the magnitude of E decreases with decreasing coil radius (Fig. 2(a)) but the maximum amplitude of E: is about constant when the coil radius is between 2 and 5cm (Fig. 2(b)). In peripheral nerve stimulation, the locations of the virtual cathode and the virtual anode are of great importance. The virtual cathode (anode) occurs where the nerve is depolarized (hyperpolarized), i.e. where EL is the most negative (positive). Their locations change when the coil radius is varied (Fig. 2(b)): the anode-cathode distance increases from 1.5 to 4.4cm when the radius increases from 1 to 5cm.
Maps of E and E: were computed for various orientations and placements of the coil to reconstruct typical clinical experiments. In Fig. l(a)-(c) are illustrated E and EL for a single coil, which is 1Omm above and initially erect over the spheroid surface, then tilted by 45”, and finally with its windings edgetangential to the spheroid (see inserts in the figure). Likewise, Fig. l(d) illustrates the figure-of-eight coil, which is a pair of adjacent edge-tangential coils with opposite current directions. In these simulations the rate of change of the current was set to 1A/s. The most important consequence of the results illustrated in Fig. 1 is that the magnitudes of the induced E and EL are about doubled when the single coil is tilted from the erect to the edge-tangential orientation. On the other hand, both E and Et are notably more localized when the coil is erect. The figure-of-eight coil is commonly used in magnetic stimulation since it has been found to activate nerves more readily than the single coil. Furthermore, the stimulation has been found to be more focused. This behaviour is explained by Fig. l(d) where the figure-ofeight coil is seento induce E and .,$ that are twice those
Effect of coil radius
Effect of spheroid length The spheroid length was varied between 30 and 1OOcm. Although the magnitude of E remains gpproximately unchanged (Fig. 3(a)), the gradient E, of the field along the axon varies moderately (Fig. 3(b)). The peak value of E: decreaseswhen the spheroid is shorter, and the tails of the curves extend farther away from Ez’
Electricfield E (a) Erectcoil =pggq
hdb
max.53 nV/m
m
4cm
(b) 45-tiltedcoil
(c) Edge-tangential coil 73-
cd=
(d) Figure-of-eightcoil u.
max.l80nV/m
,
Fig. 1. Computed maps of the induced E and its spatial gradient Ei in a spheroid due to stimulation with: (a) an erect coil perpendicular to the surface; (b) 45”-tilted coil; (c) edge-tangential coil; (d) figure-of-eight coil. The coil radii were 3 cm; the spheroid major axes were 8 and 100cm; the results were obtained for coils 10mm above the spheroid and on a plane 2 mm below the coil. The rate of change of the current was assumed to be 1A/s.
32
P. Ravazzani, J. Ruohonen, F. Grandori IJ.0
3 1.0 J.0
4.0
-4.0
-2.0
0.0
2.0
4.0
6.0
ZlnJ
4.0
4.0
-2.0
0.0 1 WI
2.0
4.0
6.0
0.6
-2.J
I
6.6
4.0
: 4.0
1 L i 0.0
-2.0
2.0
4.0
.! -! 6.0
= 14
-6.0
Fig. 2. The magnitude of the induced field E (a) and E: (b) plotted along the courseof a z-axial lo-mm deepaxon when the radius of an erect coil is varied from 1 to Scm (step of 1cm). The coil was in contact with the surface.The rate of changeof the current wasassumedto be 100Alps.
-4.0
-2.0
0.0 rlcnl
2.0
4.0
6.0
Fig. 4. The magnitudeof the induced field E (a) and E: (b)
plotted along the courseof a z-axial axon when the spheroid radius is varied from 2 to 4cm. Other details as in Fig. 2.
the centre of the coil. The sites of maximal depolarization and hyperpolarization remain approximately unchanged. Effect of spheroid radius
! : 0.0 4.0
40
-2.0
0.0
20
4.0
6.0
1 I4 0.6
b j
0.4 ’
:
’
‘.
~
With decreasing spheroid radius the magnitude of E decreases,although the distance between the nerve and the coil is kept constant (Fig. 4(a)). This is because the computation points are closer to the z-axis, where E: has to vanish as one could observe from the model equations (seethe Appendix). Interestingly, variations in the spheroid radius between 2 and 4cm have only negligible influence on the induced E: (Fig. 4(b)). Therefore, one can conclude that variations in the radius of limbs (the spheroid radius) have only negligible effects on the magnetic stimulation of peripheral nerves. This is an important observation, since limb radii vary greatly between individuals.
-3ocm -40
Effect of nerve depth
-80
-0.6
i -60
“’ -40
-IO0
‘.’ -2.0
0.0
20
4.0
60
z[cml
Fig. 3. The magnitude of the induced field E (a) and E: (b)
plotted along the courseof a z-axial axon when the spheroid length is varied from 30 to 100cm.Other details as in Fig. 2.
When a nerve is deeper, it is exposed to lower E and EL (Fig. 5). In a very superficial 5-mm deep nerve, an increase of 3mm in the depth of the nerve causes an almost 100% decrease in the induced E and a 40% decreasein E:. The change in the fields is less dramatic for nerves deeper than 1Omm. Hence, the depth of the
Magnetic stimulation of peripheral nerves
,
-5llXll -8rm -1llWll -I,llllll -17mm
1.2
-1.2
1
4.0
-4.0
-2.0
0.0 z l-4
2.0
4.0
60
Fig. 5. The magnitudeof the induced field E (a) and Ei (b) plotted along the course of a z-axial axon when the nervedepth is varied from 5 to 17mm. Other details as in Fig. 2. nerve is of crucial importance in determining the activation point or stimulation thresholds. In addition, the deeper the nerve, the sharper are the peaks of E and E:. Therefore, for deep nerves, the membrane potential of a longer segment of the nerve may be driven above the threshold, which probably causes some of the variation frequently seen in experimental determination of the latencies of the motor responseselicited by magnetic stimulation.
DISCUSSION So far, the theoretical knowledge of the technique of the magnetic stimulation of peripheral nerves is based on modelling the induced electric fields either with simplified analytical solutions that neglect the boundaries of the tissue or with complicated numerical methods. Direct solution of E from Faraday’s law is possible only when the boundaries are either neglected or the tissue is substituted by a semi-infinite medium.2-5 Unfortunately, there is no easy way to take into account the boundaries. A good example of a clever, but an extremely complicated analytical solution to the problem of an infinite-length cylinder has been recently presented by Esselle and Stuchly,” who employ several-fold numerical integration. Due to the intrinsic complexity, this kind of a model has few future extensions to account for
33
more complicated geometries. The same is true for numerical methods that take into account realistic geometries of the media:‘4124they are complicated, very time consuming, and poor in terms of flexibility. Such methods are advisable when a well-known, defined, and detailed structure has to be modelled and when there is no need to analyse the effectsfrom changes in the parameters. A simple analytical model was developed to compute the electric field induced in a finite-length cylinder-like structure (prolate spheroid). This analytical model allows easy changes in all the parameters involved in the stimulation, such as coil size, location, and shape, size of the limbs, and location of the nerves. In its analytical formulation, the model cannot account for bones and other changes in conductivity characteristics. To examine these effects, it is necessary to compute numerically the surface integrals of eqns (2) and (3). It is expected that, for example, the wrist bones have a great effect on E, but one should note that to analyse these effects, it is not enough to have a good model, but one should also know well the conductivity characteristics of the wrist. In the future, numerical methods such as finite element and boundary element methods may demonstrate their value, but their correctness must always be checked, at least in some simplified cases,with analytical expressions. Till now, the activation of a peripheral nerve by magnetic stimulation was considered substantially due to E:, that is the gradient of the component of E along the course of the nerve.= Actually, this view has been only partially justified in experiments, in that excitation of a peripheral nerve can also be obtained with a coil location and orientation producing no E: along the nerve.23 In that sense, probably also the magnitude of the electric field E and the component of E perpendicular to the nerve play a role in nervous activation.23 The model showed that E and EL induced by various stimulating coils show great differences in terms of field intensities. Especially, possibilities of focusing E and E: vary greatly. A future analysis includes new coil shapes that focus the fields better. An analysis of geometrical factors, when the coil was erect above the limb so that its windings were parallel to the nerve, showed the locations of the excitation sites are sensitive to changes in coil radius. On the other hand, the size of the medium, i.e. spheroid radius and length , have no influence on the sites of excitation or on the magnitude of E:. Instead, the spheroid length has notable effects on the magnitude of E. As one would expect, the depth of the nerve inside the limb is an utterly important factor in determining the activation threshold while it has no influence on the site of activation. Simulations with other coil arrangements and orientations yielded similar results. A last note regards the choice of an optimal stimulation coil. The magnitude of E decreases with
34
P. Ravazzani. J. Ruohonen, F. Grandori
decreasing coil radius, but a small coil induces a more focused E than a large coil (Fig. 2). Thus Et is proportionally higher with smaller coils. To date, the radius of commercially available coils is typically between 4 and 7 cm, but a smaller coil radius of some2 cm is desirable for several reasons: (1) the intrinsic forces are smaller; (2) a lower inductance is obtained implying a greater rate of change of current and a greater EL; (3) exposure to magnetic and electric fields is reduced; (4) handling and positioning the coil becomes easier.
REFERENCES 1. Barker, A. T., Jalinous, R. & Freeston, I., Non-invasive magnetic stimulation of the human motor cortex. Lancet, 1, (1985) 1106-7. 2. Grandori, F. & Ravazzani, P., Magnetic stimulation of the motor cortex. Theoretical considerations. IEEE Trans. Biomed. Engng., 38 (1991) 180-91. 3. Esselle, K. P. & Stuchly, M. A., Neural stimulation with magnetic fields: analysis of induced electric fields. IEEE Trans. Biomed. Engng., 39 (1992) 693-700. 4. Tofts, P. S., The distribution of induced currents in magnetic stimulation of the nervous system. Phys. Med. Biol., 8 (1990) 1119-28. 5. Nagarajan, S. S., Durand, D. M. & Warman, E. N., Effects of induced electric fields on finite neuronal structures: a simulation study. IEEE Trans. Biomed. Engng., 40 (1993) 1175588. 6. Eaton, H., Electric field induced in a spherical volume conductor from arbitrary coils: applications to magnetic stimulation and MEG. Med. Biolog. Engng. Comput., 30 (1992) 433-40. 7. Heller, L. & van Hulsteyn, D. B., Brain stimulation using electromagnetic sources:theoretical aspects,Biophys. J., 63 (1992) 129-38. 8. Ravazzani, P., Grandori, F.: Piavani, A. & Vardanega, M. G., Intracranial electric fields produced by magnetic stimulation in a spherical model. In Proc. 14th Ann. Znt. Conf. IEEE EMBS, Paris, France, 29 Ott-1 Nov, 4, 1992, pp. 1411-13. 9. Ravazzani, P., Ruohonen, J. Jz Grandori, F., Electric fields induced in magnetic stimulation of the nervous system. Influence of volume conductor boundaries. In Proc. 16th Ann. Znt. Co& IEEE EMBS, Baltimore, USA, 3-6 November, 1994 pp. 325-6. 10. Ruohonen, J., Magnetic stimulation of the human nervous system. MSc thesis, Helsinki University of Technology, Department of Technical Physics, Espoo, Finland, September, 1993. 11. Ruohonen, J., Ravazzani, P. & Grandori, F., Magnetic stimulation of peripheral nerves: computation of the induced electric fields. In Proc. 13th Southern Biomed. Engng Conf., Washington, DC, 16-17 April, 1994, pp. 1065-8. 12. Ruohonen, J., Ravazzani, P. & Grandori, F., An analytical model to predict the electric field and excitation zones due to magnetic stimulation of peripheral nerves. IEEE Trans. Biomed. Engng., 42 (1995) 158-61. 13. Ruohonen, J., Ravazzani, P., Nilsson, J., Panizza, M. & Grandori, F. A volume-conduction analysis of magnetic stimulation of peripheral nerves. IEEE Trans. Biomed. Engng (submitted). 14. Roth, B. J., Cohen, L. G., Hallet, M., Friauf, W. & Basser, P. J.. A theoretical calculation of the electric field induced
by magneticstimulation of a peripheral nerve. Muscle & Nerve, 13 (1990) 734-41. 15. Esselle,K. P. Jr Stuchly, M. A., Quasi-static electric field in a cylindrical volume conductor inducedby external coils. IEEE Trans. Biomed. Engng., 41 (1994) 151-8. 16. Corson, D. R. & Lorrain, P., In Introduction to Electromagnetic Fields and Waves. W. H. Freeman, San Francisco, CA, 1962, pp. 481-5. 17. Hamlllinen, M., Hari, R., Ihnoniemi, R. J., Knuutila, J. & Lounasmaa, 0. V., Magnetoencephalography-theory, instrumentation, and applications to noninvasive studies of the working human brain. Rev. Mod. Phys., 65 (1993) 413-97. 18. Roth, B. J. & Sato, S., Accurate and effective formulas for averaging the magnetic field over a circular coil. In Biomagnetism: Clinical Aspects, Proc. 8th Znt. Conf. Biomagnetism, Miinster, Aug. 1991, ed. M. Hoke, S. N. Erni, Y. C. Okada & G. L. Romani, Elsevier, Amsterdam, 1992, pp. 797-800. 19. Geselowitz, D., On the magnetic field generated outside an inhomogeneous volume conductor by internal sources. IEEE Trans. Magn., 6 (1970) 346-7. 20. Horacek, B., Digital model for studies in magnetocardiography. IEEE Trans. Magn., 9 (1973) 440-4. 21. Cuffin, B. N. & Cohen, D., Magnetic fields of a dipole in special volume conductor shapes. IEEE Trans. Biomed. Engng., 24 (1977) 372-81. 22. Roth, B. J. & Basser,P. J., A model of the stimulation of a nerve fiber by electromagnetic induction. IEEE Trans. Biomed. Engng., 37 (1990) 588-97. 23. Ruohonen, J., Panizza, M., Nilsson, J., Ravazzani, P. & Grandori, F., Transverse-field activation mechanism in magnetic stimulation of peripheral nerves. Electroencephalogr. clin. Neurophysiol. (submitted). 24. De Leo, R., Cerri, G., Balducci, D., Moglie, F., Scarpino, 0. & Guidi, M., Computer modeling of brain cortex excitation by magnetic field pulses. J. Med. Engng. Technol., 16 (1992) 149-56.
APPENDIX Equation (1) shows that to obtain E one has to find B due to a general Q. For the cylinder this problem is complicated and was not found in the literature. Hence,
a classical method of stretching out a prolate spheroid is used; the prolate spheroid can assume the shape of a sphere or approximate a thin finite-length circular cylinder. The prolate spheroidal co-ordinates (7: [, ‘p) are illustrated in Fig. A.1. For a spheroid whose major axis is parallel to the z-axis, the relationships between
the spheroidal and Cartesian co-ordinates are: x=c (T/2- l)(l-(2)cos cp $I= y = c (q2 -. l)( 1 - <‘) sin cp { z = C7g
(A.11
It is useful to divide the total B into two parts: B = hip + %I, where Bdir is due to the dipole in an infinite medium being equal to the first term in eqn (2). The contribution of the volume boundaries Bvol is given
35
Magnetic stimulation of peripheral nerves
Fig. A.l. The prolate spheroidalco-ordinatesystem.The point P(n,<,‘p) is definedby the intersectionof a spheroid,a hyperboloid,
and a plane. Focal distanceis 2c.
by Cuffin and Cohen.” Simplified expressions of the Cartesian components are:
--
B
2vaAQ9
“’
h9Jlr;-1
F
~m2a+,,cos(m~t)
n=l
m=l
(A.2f) where (A.2a)
&,9 =
(
H
.)
2Pn,m($)
sin(m(p’)[(m - l)(n + m)
- Cm+ l)L,m+l(11, E,dl BY>< =-
a,,, = (2n + l)(-l)m
7
(A.2b) P,,,, Q,,m = associated Legendre functions of the first and the second kind, Pd,,, = derivative of P,,, with respect to its argument. The metric coefficients are:
~~a,,~sin(m$)[(n+m) (
n=l
m=l
x (n- m+ l)~,,,-l (rl,& rla)+ ~,,,+1(77,5, dl (A.2c)
x (n + m)(n - m + lPL,,-l(rl,C, 77,)
)
AQlp m 6rid +-ch9,,=l%,04l,l(rl,
(A.31
( h, = c;/(q2 - l)(l - t2) In the above, the primed variables apply to coil elements; the unprimed apply to field points. The surface of the spheroid is denoted by va and c is the semi-focal distance. The Bvol due to a dipole Q, is obtained by replacing n with <, and vice versa, in the expressions for dipole Qt. Dipole moments QE= Q, = Q, = 1 A m can be assumed without loss of generality. The z-axial derivative of E, is:
a,,, cos(mcp’)[Cm- 1)
+ Cm+ l&,+1 (~6 dl
h,=cJ ($- E2)l(l - t2) h,=q/m
(A.2d)
(A.2e)
‘%iip(r, r’: Q,) dz
where the differentiations of Bdipand Bvol with respect to i- can be easily obtained.
0 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 0965~9978/95/$09SO
0965-9978(94)00056-S
ELSEVIER
Magnetic stimulation of peripheral nerves: computation of the induced electric field in a cylinder-like structure P. Ravazzani,* § J. Ruohonenj & F. Grandori* * Systems Theory Centre-CNR and Department of Biomedical Engineering, Polytechnic of Milan, Via Ponzio 34/5,20133 Milan, Italy i Medical Engineering Centre, Helsinki University Central Hospital, Stenbickinkatu 9, 00290 Helsinki, Finland (Received 11 August 1994; accepted 7 November 1994) The uncertain site of excitation is the main factor that prevents magnetic stimulation of peripheral nerves from becoming a routine clinical tool. Long and straight fibres are thought to be excited by the first spatial derivative of the electric field component parallel to the fibre and by the electric field component perpendicular to the fibre. The authors have developed an analytical model to compute both of these field features inside pseudo-cylindrical structures. Here they analyse the computation of the electric fields induced with different stimulator arrangements: they examine the effects of changes in stimulator coil location, orientation, size, and geometry and of changes in the dimensions of the limbs and in the position of the nerve inside the limb. Key words: magnetic stimulation, peripheral nerves, computer modelling, electromagnetic induction.
for unbounded and semi-infinite media. Only recently models have been developed for bounded media, like the homogeneous sphere to model the human headG9 and cylindrical structures to model the
INTRODUCTION A suitable electric field E can excite or inhibit neurones. In non-invasive electromagnetic stimulation, E can be produced in the nervous tissue by two different methods: (1) in electrical stimulation, E is produced by driving current
through
limbs.‘e”
An analytical model based on approximating cylinderlike limbs by the stretched homogeneousprolate spheroid is analysed here in detail. The model is employed to reconstruct some practical arrangements and to study the influence of various geometrical factors on the final results. Modelling of E is particularly important since the site of excitation is not well known in magnetic stimulation of peripheral nerves. Accurate prediction of E gives additional information about the feature of E responsible for activation. Moreover, modelling allows the design of more efficient coil shapes and placements with
surface electrodes; (2) in magnetic
stimulation, E is induced by large time-varying currents flowing in a coil which is located above the tissue. Here, is discussed the latter technique, and in particular the computation of E and its gradient inside homogeneous cylinder-like structures that model a human limb. Magnetic stimulation provides a fascinating means for assessing noninvasively the intact nervous system and several nervous disorders in man. It is typically used to determine the conduction velocities of motor and sensory nerves: when a stimulus is given, for example, at the elbow, a muscle twitch in the hand can be observed
respect to the target structures.
Computation of E with the proposed model reduces
after a time delay, which depends on the stimulus location and the conduction velocity.
to: (1) computation of the external (to the bounding tissue) magnetic induction B due to an internal point-
The technique was introduced some 10 years ago,’ but the first theoretical predictions of the induced E were presented only in the beginning of 1990~,~-~
like current dipole Q; (2) computation of the magnetic flux through a surface bounded by the excitation coil. Analytical solutions to the first problem are available only in some simplified geometries, and numerical methods are required in more realistically-shaped
3 To whom correspondence should be addressed. 29
30
P. Ravazzani. J. Ruohonen, F. Grandori
geometries. Magnetic flux can be usually computed by discretizing the flux integral into a finite sum. THE MODEL In this section some theories of biomagnetism and magnetobiology are linked together; biomagnetism is the study of magnetic fields generated by organisms; magnetobiology is the study of biological effects from magnetic fields, magnetic stimulation being a special branch of it. The so-called reciprocity theorem16 implies that at a point it4 inside a medium, the electric field E induced by an external current loop C at a point N can be computed from the magnetic induction B produced at N by a current dipole Q at M. Hence, on practical grounds, E can be computed using virtually the same equations as in biomagnetism when computing the magnetic flux + through the loop C caused by a known source current density. The latter is a well-studied problem and it is therefore sensible to employ here the already existing solutions (for review, seeHamllainen and colleagues17). The quasi-static approximation is valid in magnetic stimulation of the nervous system, and then the reciprocity theorem gives:7’8Y11 E(r) = - Tk
ek jc B(Q . ek,r’) . n(r’)dS’
(1)
k=l
where Z(t) is the current in the coil, el, e2 and e3 are orthogonal unit vectors, and n(r’) is a unit normal vector of the area bounded by the coil. The surface integral is recognized to be the magnetic flux through a surface bounded by the coil. In practice, it is computed by dividing the surface into a number of subsurfaces small enough so that B can be considered constant on them. Typically, it is enough to divide the coil surface into 12 subareas whose weights are optimally chosen.‘* While the coil shape is accounted for in the surface integral in eqn (l), the shape of the medium modelling the tissue contributes to the magnetic induction B. Finding B outside a medium due to a current source inside is the so-called biomagnetic forward problem. Geselowitz” has modified the Ampere-Laplace law to give B due to a general current density; for a dipolar source Q at r’ the result is:
x dS(r’)
(2)
where the electrical potential I’ is (from Horacek2’):
(3)
satisfying Laplace’s equation with appropriate boundary conditions. Here, g(r) is the conductivity of the tissue and Au is the conductivity difference over the boundary S of the conductor in the direction of dS(r’). The vector surface element dS(r’) = n(r’)dS is oriented along the outward unit normal n. In a general geometry, one usually solves iteratively the surface potential after which B can be obtained by numerical integration. On the other hand, only an analytical model allows an easy analysis of changes in the parameters. So far, analytical solutions for B are available only in simplified homogeneous geometries; an analytical solution of B in the stretched prolate spheroid, which was originally given by Cut& and Cohen2’ is adapted here. This can be considered to be a good approximation of a limb when the spheroid major axis is much longer than the minor axis, i.e. when the spheroid looks like a cylinder. The expressions for B and of the resulting induced E are given in the Appendix. The aim of magnetic stimulation is to depolarize the cell membrane in such a way that the membrane potential exceeds a threshold value and the neuron generates a propagating depolarization front, or an action potential. Among others, Roth and Basser22have predicted that a z-axial axon is depolarized where the zaxial gradient El of the z-component of E is most negative. Actually, this view is only in partial agreement with experiments; recently, strong evidence was found of excitation of a peripheral nerve with such coil locations and orientations that do not produce Ei within the nerve.23 It was concluded that the electric field component perpendicular to the nerve also produces excitation. However, here discussion is limited to only such practically interesting situations, in which the mechanism of excitation is thought to depend solely on the magnitude, sign and time-course of E:. The Appendix gives Ei for nerves parallel to the major axis of the prolate spheroid. COMPUTER SIMULATIONS. VARIOUS PARAMETERS
INFLUENCE OF
Various geometrical conditions were simulated, including different coil shapes, radii, locations, orientations, various sizes of the spheroid, and various depths of the target nerve. A 5-cm radius coil in contact with and perpendicular to the surface of a 100~cmlong and 8-cm thick prolate spheroid was considered as the reference arrangement, which was then modified. E was computed 1Omm below the surface and 20mm below the coil, which roughly corresponds to a realistic situation when stimulating the median nerve at the elbow. The induced electric field strength scales linearly with the rate of change of the coil current which was of lOOA/ps in nearly all simulations.
31
Magnetic stimulation of peripheral nerves
The computation was done with a personal computer (80486 at 25 MHz). Depending on the required resolution, it usually took some hours to run the model equations. By computing some of the geometrical factors in advance, the rest of the computation took just minutes.
induced by the edge-tangential single coil; additionally, the figure-of-eight coil induces a more focused field, having a field pattern similar to that with the erect coil in Fig. l(a).
Effect of coil orientation
Changes in E and E: due to variations in coil radius were computed when the coil was erect above and touched the spheroid surface so that its windings were parallel to the nerve. It was found that the magnitude of E decreases with decreasing coil radius (Fig. 2(a)) but the maximum amplitude of E: is about constant when the coil radius is between 2 and 5cm (Fig. 2(b)). In peripheral nerve stimulation, the locations of the virtual cathode and the virtual anode are of great importance. The virtual cathode (anode) occurs where the nerve is depolarized (hyperpolarized), i.e. where EL is the most negative (positive). Their locations change when the coil radius is varied (Fig. 2(b)): the anode-cathode distance increases from 1.5 to 4.4cm when the radius increases from 1 to 5cm.
Maps of E and E: were computed for various orientations and placements of the coil to reconstruct typical clinical experiments. In Fig. l(a)-(c) are illustrated E and EL for a single coil, which is 1Omm above and initially erect over the spheroid surface, then tilted by 45”, and finally with its windings edgetangential to the spheroid (see inserts in the figure). Likewise, Fig. l(d) illustrates the figure-of-eight coil, which is a pair of adjacent edge-tangential coils with opposite current directions. In these simulations the rate of change of the current was set to 1A/s. The most important consequence of the results illustrated in Fig. 1 is that the magnitudes of the induced E and EL are about doubled when the single coil is tilted from the erect to the edge-tangential orientation. On the other hand, both E and Et are notably more localized when the coil is erect. The figure-of-eight coil is commonly used in magnetic stimulation since it has been found to activate nerves more readily than the single coil. Furthermore, the stimulation has been found to be more focused. This behaviour is explained by Fig. l(d) where the figure-ofeight coil is seento induce E and .,$ that are twice those
Effect of coil radius
Effect of spheroid length The spheroid length was varied between 30 and 1OOcm. Although the magnitude of E remains gpproximately unchanged (Fig. 3(a)), the gradient E, of the field along the axon varies moderately (Fig. 3(b)). The peak value of E: decreaseswhen the spheroid is shorter, and the tails of the curves extend farther away from Ez’
Electricfield E (a) Erectcoil =pggq
hdb
max.53 nV/m
m
4cm
(b) 45-tiltedcoil
(c) Edge-tangential coil 73-
cd=
(d) Figure-of-eightcoil u.
max.l80nV/m
,
Fig. 1. Computed maps of the induced E and its spatial gradient Ei in a spheroid due to stimulation with: (a) an erect coil perpendicular to the surface; (b) 45”-tilted coil; (c) edge-tangential coil; (d) figure-of-eight coil. The coil radii were 3 cm; the spheroid major axes were 8 and 100cm; the results were obtained for coils 10mm above the spheroid and on a plane 2 mm below the coil. The rate of change of the current was assumed to be 1A/s.
32
P. Ravazzani, J. Ruohonen, F. Grandori IJ.0
3 1.0 J.0
4.0
-4.0
-2.0
0.0
2.0
4.0
6.0
ZlnJ
4.0
4.0
-2.0
0.0 1 WI
2.0
4.0
6.0
0.6
-2.J
I
6.6
4.0
: 4.0
1 L i 0.0
-2.0
2.0
4.0
.! -! 6.0
= 14
-6.0
Fig. 2. The magnitude of the induced field E (a) and E: (b) plotted along the courseof a z-axial lo-mm deepaxon when the radius of an erect coil is varied from 1 to Scm (step of 1cm). The coil was in contact with the surface.The rate of changeof the current wasassumedto be 100Alps.
-4.0
-2.0
0.0 rlcnl
2.0
4.0
6.0
Fig. 4. The magnitudeof the induced field E (a) and E: (b)
plotted along the courseof a z-axial axon when the spheroid radius is varied from 2 to 4cm. Other details as in Fig. 2.
the centre of the coil. The sites of maximal depolarization and hyperpolarization remain approximately unchanged. Effect of spheroid radius
! : 0.0 4.0
40
-2.0
0.0
20
4.0
6.0
1 I4 0.6
b j
0.4 ’
:
’
‘.
~
With decreasing spheroid radius the magnitude of E decreases,although the distance between the nerve and the coil is kept constant (Fig. 4(a)). This is because the computation points are closer to the z-axis, where E: has to vanish as one could observe from the model equations (seethe Appendix). Interestingly, variations in the spheroid radius between 2 and 4cm have only negligible influence on the induced E: (Fig. 4(b)). Therefore, one can conclude that variations in the radius of limbs (the spheroid radius) have only negligible effects on the magnetic stimulation of peripheral nerves. This is an important observation, since limb radii vary greatly between individuals.
-3ocm -40
Effect of nerve depth
-80
-0.6
i -60
“’ -40
-IO0
‘.’ -2.0
0.0
20
4.0
60
z[cml
Fig. 3. The magnitude of the induced field E (a) and E: (b)
plotted along the courseof a z-axial axon when the spheroid length is varied from 30 to 100cm.Other details as in Fig. 2.
When a nerve is deeper, it is exposed to lower E and EL (Fig. 5). In a very superficial 5-mm deep nerve, an increase of 3mm in the depth of the nerve causes an almost 100% decrease in the induced E and a 40% decreasein E:. The change in the fields is less dramatic for nerves deeper than 1Omm. Hence, the depth of the
Magnetic stimulation of peripheral nerves
,
-5llXll -8rm -1llWll -I,llllll -17mm
1.2
-1.2
1
4.0
-4.0
-2.0
0.0 z l-4
2.0
4.0
60
Fig. 5. The magnitudeof the induced field E (a) and Ei (b) plotted along the course of a z-axial axon when the nervedepth is varied from 5 to 17mm. Other details as in Fig. 2. nerve is of crucial importance in determining the activation point or stimulation thresholds. In addition, the deeper the nerve, the sharper are the peaks of E and E:. Therefore, for deep nerves, the membrane potential of a longer segment of the nerve may be driven above the threshold, which probably causes some of the variation frequently seen in experimental determination of the latencies of the motor responseselicited by magnetic stimulation.
DISCUSSION So far, the theoretical knowledge of the technique of the magnetic stimulation of peripheral nerves is based on modelling the induced electric fields either with simplified analytical solutions that neglect the boundaries of the tissue or with complicated numerical methods. Direct solution of E from Faraday’s law is possible only when the boundaries are either neglected or the tissue is substituted by a semi-infinite medium.2-5 Unfortunately, there is no easy way to take into account the boundaries. A good example of a clever, but an extremely complicated analytical solution to the problem of an infinite-length cylinder has been recently presented by Esselle and Stuchly,” who employ several-fold numerical integration. Due to the intrinsic complexity, this kind of a model has few future extensions to account for
33
more complicated geometries. The same is true for numerical methods that take into account realistic geometries of the media:‘4124they are complicated, very time consuming, and poor in terms of flexibility. Such methods are advisable when a well-known, defined, and detailed structure has to be modelled and when there is no need to analyse the effectsfrom changes in the parameters. A simple analytical model was developed to compute the electric field induced in a finite-length cylinder-like structure (prolate spheroid). This analytical model allows easy changes in all the parameters involved in the stimulation, such as coil size, location, and shape, size of the limbs, and location of the nerves. In its analytical formulation, the model cannot account for bones and other changes in conductivity characteristics. To examine these effects, it is necessary to compute numerically the surface integrals of eqns (2) and (3). It is expected that, for example, the wrist bones have a great effect on E, but one should note that to analyse these effects, it is not enough to have a good model, but one should also know well the conductivity characteristics of the wrist. In the future, numerical methods such as finite element and boundary element methods may demonstrate their value, but their correctness must always be checked, at least in some simplified cases,with analytical expressions. Till now, the activation of a peripheral nerve by magnetic stimulation was considered substantially due to E:, that is the gradient of the component of E along the course of the nerve.= Actually, this view has been only partially justified in experiments, in that excitation of a peripheral nerve can also be obtained with a coil location and orientation producing no E: along the nerve.23 In that sense, probably also the magnitude of the electric field E and the component of E perpendicular to the nerve play a role in nervous activation.23 The model showed that E and EL induced by various stimulating coils show great differences in terms of field intensities. Especially, possibilities of focusing E and E: vary greatly. A future analysis includes new coil shapes that focus the fields better. An analysis of geometrical factors, when the coil was erect above the limb so that its windings were parallel to the nerve, showed the locations of the excitation sites are sensitive to changes in coil radius. On the other hand, the size of the medium, i.e. spheroid radius and length , have no influence on the sites of excitation or on the magnitude of E:. Instead, the spheroid length has notable effects on the magnitude of E. As one would expect, the depth of the nerve inside the limb is an utterly important factor in determining the activation threshold while it has no influence on the site of activation. Simulations with other coil arrangements and orientations yielded similar results. A last note regards the choice of an optimal stimulation coil. The magnitude of E decreases with
34
P. Ravazzani. J. Ruohonen, F. Grandori
decreasing coil radius, but a small coil induces a more focused E than a large coil (Fig. 2). Thus Et is proportionally higher with smaller coils. To date, the radius of commercially available coils is typically between 4 and 7 cm, but a smaller coil radius of some2 cm is desirable for several reasons: (1) the intrinsic forces are smaller; (2) a lower inductance is obtained implying a greater rate of change of current and a greater EL; (3) exposure to magnetic and electric fields is reduced; (4) handling and positioning the coil becomes easier.
REFERENCES 1. Barker, A. T., Jalinous, R. & Freeston, I., Non-invasive magnetic stimulation of the human motor cortex. Lancet, 1, (1985) 1106-7. 2. Grandori, F. & Ravazzani, P., Magnetic stimulation of the motor cortex. Theoretical considerations. IEEE Trans. Biomed. Engng., 38 (1991) 180-91. 3. Esselle, K. P. & Stuchly, M. A., Neural stimulation with magnetic fields: analysis of induced electric fields. IEEE Trans. Biomed. Engng., 39 (1992) 693-700. 4. Tofts, P. S., The distribution of induced currents in magnetic stimulation of the nervous system. Phys. Med. Biol., 8 (1990) 1119-28. 5. Nagarajan, S. S., Durand, D. M. & Warman, E. N., Effects of induced electric fields on finite neuronal structures: a simulation study. IEEE Trans. Biomed. Engng., 40 (1993) 1175588. 6. Eaton, H., Electric field induced in a spherical volume conductor from arbitrary coils: applications to magnetic stimulation and MEG. Med. Biolog. Engng. Comput., 30 (1992) 433-40. 7. Heller, L. & van Hulsteyn, D. B., Brain stimulation using electromagnetic sources:theoretical aspects,Biophys. J., 63 (1992) 129-38. 8. Ravazzani, P., Grandori, F.: Piavani, A. & Vardanega, M. G., Intracranial electric fields produced by magnetic stimulation in a spherical model. In Proc. 14th Ann. Znt. Conf. IEEE EMBS, Paris, France, 29 Ott-1 Nov, 4, 1992, pp. 1411-13. 9. Ravazzani, P., Ruohonen, J. Jz Grandori, F., Electric fields induced in magnetic stimulation of the nervous system. Influence of volume conductor boundaries. In Proc. 16th Ann. Znt. Co& IEEE EMBS, Baltimore, USA, 3-6 November, 1994 pp. 325-6. 10. Ruohonen, J., Magnetic stimulation of the human nervous system. MSc thesis, Helsinki University of Technology, Department of Technical Physics, Espoo, Finland, September, 1993. 11. Ruohonen, J., Ravazzani, P. & Grandori, F., Magnetic stimulation of peripheral nerves: computation of the induced electric fields. In Proc. 13th Southern Biomed. Engng Conf., Washington, DC, 16-17 April, 1994, pp. 1065-8. 12. Ruohonen, J., Ravazzani, P. & Grandori, F., An analytical model to predict the electric field and excitation zones due to magnetic stimulation of peripheral nerves. IEEE Trans. Biomed. Engng., 42 (1995) 158-61. 13. Ruohonen, J., Ravazzani, P., Nilsson, J., Panizza, M. & Grandori, F. A volume-conduction analysis of magnetic stimulation of peripheral nerves. IEEE Trans. Biomed. Engng (submitted). 14. Roth, B. J., Cohen, L. G., Hallet, M., Friauf, W. & Basser, P. J.. A theoretical calculation of the electric field induced
by magneticstimulation of a peripheral nerve. Muscle & Nerve, 13 (1990) 734-41. 15. Esselle,K. P. Jr Stuchly, M. A., Quasi-static electric field in a cylindrical volume conductor inducedby external coils. IEEE Trans. Biomed. Engng., 41 (1994) 151-8. 16. Corson, D. R. & Lorrain, P., In Introduction to Electromagnetic Fields and Waves. W. H. Freeman, San Francisco, CA, 1962, pp. 481-5. 17. Hamlllinen, M., Hari, R., Ihnoniemi, R. J., Knuutila, J. & Lounasmaa, 0. V., Magnetoencephalography-theory, instrumentation, and applications to noninvasive studies of the working human brain. Rev. Mod. Phys., 65 (1993) 413-97. 18. Roth, B. J. & Sato, S., Accurate and effective formulas for averaging the magnetic field over a circular coil. In Biomagnetism: Clinical Aspects, Proc. 8th Znt. Conf. Biomagnetism, Miinster, Aug. 1991, ed. M. Hoke, S. N. Erni, Y. C. Okada & G. L. Romani, Elsevier, Amsterdam, 1992, pp. 797-800. 19. Geselowitz, D., On the magnetic field generated outside an inhomogeneous volume conductor by internal sources. IEEE Trans. Magn., 6 (1970) 346-7. 20. Horacek, B., Digital model for studies in magnetocardiography. IEEE Trans. Magn., 9 (1973) 440-4. 21. Cuffin, B. N. & Cohen, D., Magnetic fields of a dipole in special volume conductor shapes. IEEE Trans. Biomed. Engng., 24 (1977) 372-81. 22. Roth, B. J. & Basser,P. J., A model of the stimulation of a nerve fiber by electromagnetic induction. IEEE Trans. Biomed. Engng., 37 (1990) 588-97. 23. Ruohonen, J., Panizza, M., Nilsson, J., Ravazzani, P. & Grandori, F., Transverse-field activation mechanism in magnetic stimulation of peripheral nerves. Electroencephalogr. clin. Neurophysiol. (submitted). 24. De Leo, R., Cerri, G., Balducci, D., Moglie, F., Scarpino, 0. & Guidi, M., Computer modeling of brain cortex excitation by magnetic field pulses. J. Med. Engng. Technol., 16 (1992) 149-56.
APPENDIX Equation (1) shows that to obtain E one has to find B due to a general Q. For the cylinder this problem is complicated and was not found in the literature. Hence,
a classical method of stretching out a prolate spheroid is used; the prolate spheroid can assume the shape of a sphere or approximate a thin finite-length circular cylinder. The prolate spheroidal co-ordinates (7: [, ‘p) are illustrated in Fig. A.1. For a spheroid whose major axis is parallel to the z-axis, the relationships between
the spheroidal and Cartesian co-ordinates are: x=c (T/2- l)(l-(2)cos cp $I= y = c (q2 -. l)( 1 - <‘) sin cp { z = C7g
(A.11
It is useful to divide the total B into two parts: B = hip + %I, where Bdir is due to the dipole in an infinite medium being equal to the first term in eqn (2). The contribution of the volume boundaries Bvol is given
35
Magnetic stimulation of peripheral nerves
Fig. A.l. The prolate spheroidalco-ordinatesystem.The point P(n,<,‘p) is definedby the intersectionof a spheroid,a hyperboloid,
and a plane. Focal distanceis 2c.
by Cuffin and Cohen.” Simplified expressions of the Cartesian components are:
--
B
2vaAQ9
“’
h9Jlr;-1
F
~m2a+,,cos(m~t)
n=l
m=l
(A.2f) where (A.2a)
&,9 =
(
H
.)
2Pn,m($)
sin(m(p’)[(m - l)(n + m)
- Cm+ l)L,m+l(11, E,dl BY>< =-
a,,, = (2n + l)(-l)m
7
(A.2b) P,,,, Q,,m = associated Legendre functions of the first and the second kind, Pd,,, = derivative of P,,, with respect to its argument. The metric coefficients are:
~~a,,~sin(m$)[(n+m) (
n=l
m=l
x (n- m+ l)~,,,-l (rl,& rla)+ ~,,,+1(77,5, dl (A.2c)
x (n + m)(n - m + lPL,,-l(rl,C, 77,)
)
AQlp m 6rid +-ch9,,=l%,04l,l(rl,
(A.31
( h, = c;/(q2 - l)(l - t2) In the above, the primed variables apply to coil elements; the unprimed apply to field points. The surface of the spheroid is denoted by va and c is the semi-focal distance. The Bvol due to a dipole Q, is obtained by replacing n with <, and vice versa, in the expressions for dipole Qt. Dipole moments QE= Q, = Q, = 1 A m can be assumed without loss of generality. The z-axial derivative of E, is:
a,,, cos(mcp’)[Cm- 1)
+ Cm+ l&,+1 (~6 dl
h,=cJ ($- E2)l(l - t2) h,=q/m
(A.2d)
(A.2e)
‘%iip(r, r’: Q,) dz
where the differentiations of Bdipand Bvol with respect to i- can be easily obtained.