Electroencephalography and clinical Neurophysiology, 1991, 81:47-56 ~" 1991 Elsevier Scientific Publishers Ireland, Ltd. 0924-980X/91/$03.50 ADONIS 0924980X9100057D
47
ELMOCO 89174
A theoretical calculation of the electric field induced in the cortex during magnetic stimulation Bradley J. Roth a, Joshua M. Saypol a, Mark Hallett b and Leonardo G. Cohen b " Biomedical Engineering and Instrumentation Branch, Division of Research Services, and ~ Human Cortical Physiology Unit, Human Motor Control Section, Medical NeuroloD" Branch, National Institute of Neurological Disorders and Stroke. National Institutes of Health, Bethe;da, MD 20892 (U.S.A.) (Accepted for publication: 9 February 1990)
Summary We present a mathematical model for calculating the electric field induced in the head during magnetic stimulation of the cortex. The electric field arises from 2 sources: (1) the changing magnetic field creates an electric field in the tissue by electromagnetic induction, and (2) a charge distribution arises on the surface of the head and produces its own electrostatic field. A 3-sphere model is used to represent the brain, skull and scalp. The electric field as a function of the coil position, shape and orientation is computed numerically. The charge distribution partially shields the brain from the stimulus. The electric field is insensitive to the skull conductivity, in contrast with electrical stimulation using surface electrodes. Different coil shapes and orientations are considered, and a figure-of-eight coil is shown to deliver the largest and most focal stimulus. Key words: Magnetic stimulation; Electric field; Cortex
X
Magnetic stimulation is a new technique for activating neurons in the cortex (Barker et al. 1985). Stimulation occurs when a brief current pulse is passed through a coil near the head, producing an electric field in the brain by electromagnetic induction. This technique is used to map the motor cortex (Cohen et al. 1988) and in central motor conduction studies (Barker et al. 1987; Mills et al. 1987; Claus et al. 1988). Possible clinical applications of magnetic stimulation include the diagnosis of multiple sclerosis, amyotrophic lateral sclerosis and other degenerative ataxic disorders, and intraoperarive monitoring of the corticospinal tract during spinal cord surgery (for review, see Hallett and Cohen 1989). The technique is non-invasive and is less painful than stimulating the brain electrically through scalp surface electrodes. Although the use of magnetic stimulation is growing rapidly, the technique has been applied clinically without a complete theoretical understanding of the induced electric field distribution. In this paper we present a mathematical calculation of the electric field produced in the brain during magnetic stimulation, with the goal of providing some theoretical insight to guide future applications of this technique. We use the 3-sphere
Correspondence to: Bradley J. Roth, Building 13, Room 3W13, National Institutes of Health, Bethesda, MD 20892 (U.S.A.).
y~
!z
6)
Z
Y S Cortex Fig. 1. Schematic drawing of the 3-sphere model of the head. Onefourth of the sphere is cut away to show the 3 layers; the outermost layer represents the scalp, the middle layer the skull, and the inner sphere the cortex. A spherical coordinate system (r, 0 and ~) and its relationship to the Cartesian system (x, y, z) is also shown. The relationship between the x, y, and z coordinates and the anatomical features on the head is indicated.
48
B.J. ROTH ET AL.
model introduced by Rush and Driscoll (1968, 1969) to represent the head as 3 concentric layers: the scalp, skull, and cortex (Fig. 1). This model has been used to interpret electroencephalographic and magnetoencephalographic data (Nunez 1981) and to describe electrical stimulation of the motor cortex using electrodes on the scalp surface (Grandori and Rossini 1988).
Y
Methods
The electric field induced in the brain during magnetic stimulation is produced by 2 sources: (1) the current in the stimulating coil produces an electric field by electromagnetic induction, E A, and (2) the charge that accumulates on the surface of the head produces an electric field within the tissue, E~. The total electric field in the cortex, E, is the sum of E A and E~. The electric field produced by induction, EA, is discussed frequently in the literature (Barker et al. 1987; Maccabee et al. 1988; Reuter et al. 1988). The magnitude of this electric field depends on the rate of change of the current in the coil, and the spatial distribution of the field can be calculated from the coil geometry (Cohen et al. 1990). The electric field produced by charge, E~, is rarely considered in discussions of magnetic stimulation, although often a charge distribution must exist on the surface of the head in order to satisfy the fundamental laws governing electric fields (Roth et al. 1990). To understand why a charge distribution arises, consider what happens if E A is not everywhere directed parallel to the surface of the head. Charge flows in the direction of E a until it reaches the head surface. Since air is an insulator, the charge cannot leave the surface and therefore accumulates, producing its own electric field E~. Charge continues to accumulate until the components of E , and E A perpendicular to the surface of the head cancel. This process is analogous to the charging of a capacitor, and it occurs very quickly compared with the rise time of current pulses used in magnetic stimulation. Charge accumulation not only occurs on the head surface but also at the interface between any two tissues with different conductivities (see Appendix). Consider the boundary between the skull and scalp. The charge accumulation at the interface between the two tissues causes the electric field lines to change directions, flowing more perpendicular to the boundary in the low-conductivity skull and more parallel to the boundary in the high-conductivity scalp. We calculate the total electric field in the head during magnetic stimulation, taking into account the electric fields produced by both induction and charge. The details of the numerical computation are described in the appendix. Briefly, the electric field due to induction, E A, is obtained from an integral expression derived
Z
10 rnrn
Fig. 2. Magnitude of the electric field 3 m m below the surface of the cortex produced by a circular coil (bold circle) with its center l0 m m above the vertex (left inset). The curves represent a contour m a p of the magnitude of the electric field projected into the y-z plane (right inset). The contour lines are labeled in units of V / m . The peak field is 89 V / m and is located approximately midway between the two 80 V / m curves. The arrows indicate the direction of the electric field.
from electromagnetic theory (Cohen et al. 1990). The electric potential is determined from Laplace's equation and the boundary conditions at the interfaces between tissues; then E~ is calculated by taking the negative gradient of the potential (Roth et al. 1990). Laplace's equation is solved using a finite difference approximation and a systematic overrelaxation iteration (Rice 1983).
Results
Fig. 2 shows the electric field produced during magnetic stimulation by a circular coil oriented tangentially to the head with its center 10 mm above the vertex (top of the head), an orientation often used clinically (Barker et al. 1987; Hess et al. 1987; Mills et al. 1987; Claus et al. 1988) 1. The coil has a radius of 50 mm, is wound with 8 turns, and carries a current increasing at a rate of
1The 10 m m spacing between the coil and the scalp is somewhat larger than used clinically. We chose this spacing to account for the thickness of the coil, which is neglected in our model.
ELECTRIC FIELD INDUCED DURING MAGNETIC STIMULATION TABLE I Parameters in the 3-sphere model. Tissue
Outer radius ( m m )
Conductivity (S/m)
Cortex Skull Scalp
80.0 85.0 92.0
0.450 0.00563 0.450
100 A//zsec. The radius of the head, the thickness of the scalp and skull, a n d the conductivities of the tissues are the same as used by Rush and Driscoll (1969) a n d are
49 listed in T a b l e I. The calculated electric field is a 3 - d i m e n s i o n a l vector field. To plot this field graphically, we use a c o n t o u r m a p of the m a g n i t u d e of the electric field over a spherical surface 3 m m below the surface of the cortex. F o r this coil o r i e n t a t i o n the electric field E A is everywhere parallel to the head surface, so n o charge distrib u t i o n is induced. The electric field is directed in concentric loops, as indicated by the arrows in Fig. 2. The coil c u r r e n t is in the counterclockwise direction, as viewed from above, a n d is increasing with time, so that
a)
I------4
10ram
Fig. 3. Magnitude of the electric field produced by (a) induction, (b) charge, and (c) the total electric field, 3 mm below the surface of the brain, produced by a circular coil (bold circle) with one edge 10 mm above the vertex. The contour lines are labeled in units of V/m; the peak field in c is 89 V/m at the vertex. The arrows indicate the direction of the electric field.
50 the electric field induced in the head is oriented in the clockwise direction. Because of the curvature of the head, the maximum of the electric field (89 V / m ) does not occur directly below the coil winding. Suppose this coil is moved 50 mm toward the back of the head (in the negative y direction), so that the front edge of the coil is 10 mm above the vertex. The electric field due to induction, E A, is maximum (141 V / m ) at the vertex and zero under the center of the coil (Fig. 3a) 2. For this coil orientation, E a is no longer parallel to the head surface and a charge distribution develops. The electric field due to the charge, E~, is maximum (58 V / m ) slightly behind the vertex and is smaller than EA by about a factor of 2.3 (Fig. 3b). Although the contour lines indicate only the magnitudes of the fields, the arrows show that near the v e r t e x E A is in the opposite direction to E~. Thus the total electric field, shown in Fig. 3c, has a smaller maximum (89 V / m ) than the field due only to induction. The charge accumulation reduces the peak electric field produced by the coil. The electric field falls off with the depth into the brain, which may be important in determining what types of neuron are stimulated. Fig. 4 shows the magnitude of the electric field at the vertex as a function of depth below the surface of the head, using the coil geometry and orientation presented in Fig. 3. At the surface of the cortex the electric field is 104 V / m , but l0 mm below the surface it has fallen by 38% to 65 V/re. During magnetic stimulation, the electric field induced in the scalp is about twice as large as that in the brain. In electric stimulation this ratio is much larger, with the scalp field often more than 10 times that in the cortex (Rush and Driscoll 1969). This difference provides one explanation why magnetic stimulation is less painful than electric stimulation; the pain receptors in the skin are not stimulated by the relatively small electric fields induced in the scalp. Not only the magnitude but also the direction of the electric field is important in the stimulation of cortical neurons (Amassian et al. 1988; Coburn 1989). Fig. 5 shows the 3 components of the total electric field 3 mm below the surface of the cortex. These components are associated with a spherical coordinate system (r, 0, q~), as shown in Fig. 1: the r direction is radially outward from the center of the head, the ~ direction is directed toward the right ear (at the vertex the 8 direction is along a coronal axis), and the q5 direction is at right angles to the r and 8 directions (along a sagittal axis) 3.
2 The contour lines of E A would be circular if plotted on a flat surface parallel to the coil instead of on the surface of the head. 3 In an analogy to a globe, the north and south poles (8 = 0 and 180) are located at the ears, 8 measures latitude, and rp measures longitude. The equator (8 = 90) passes through the nose and the vertex.
B.J. ROTH ET AL. E (V/m)
250 200 150
~
100
- Scalp - - Skull
50 0 0
10
20
30
40
50
60
70 80
90
depth (mm) Fig. 4. Magnitude of the electric field at the vertex as a function of depth below the head surface, for the coil orientation shown in Fig. 3.
The smallest component of the electric field is in the radial direction (Fig. 5a); the 8 and ~ components are a factor of 100 larger than the r component (Fig. 5b and c) 4. The 8 component is maximum under the edge of the coil, at the vertex; there is a smaller maximum approximately below the center of the coil. The component is maximum under the edge of the coil, about 45 ° removed from the center of the head in either direction. Amassian et al. (1988) showed experimentally that the stimulation threshold changes with coil orientation. Fig. 6 shows the magnitude of the electric field at the vertex, 3 mm below the surface of the cortex, as a function of the angle, ~b, between the coil and a plane tangent to the head at the vertex (~b = 0 corresponds to a tangential orientation). The magnitude of the electric field decreases as ~b increases from 0 to 90, with the greatest change at small angles. The tangential orientation induces the largest electric field. However, the spatial distribution of the electric field is more focal with the coil perpendicular to the head. For the tangential orientation with one edge above the vertex (Fig. 3c), the width of the region in which the electric field is greater than 90% of its maximum value is 14 mm along the sagittal axis and 48 mm along a coronal axis. With the coil perpendicular to the head surface and aligned with the coronal axis (~b = 90), these widths are 10 and 24 mm (Fig. 7). Along the coronal axis the focality of the electric field increases by a factor of two by rotating the coil 90 °. These predictions are consistent with the experimental observations by Amassian et al. (1988) and Hallett et al. (1989).
4 Cohen (1989) has suggested that the electric field induced in a sphere during magnetic stimulation has no radial component. If correct, his suggestion implies that the radial component calculated here reflects the residual error in our calculation (less than 1%).
ELECTRIC FIELD INDUCED DURING MAGNETIC STIMULATION
51
) b 10 mm
Fig. 5. The (a) r, (b) 0, and (c) ,~ components of the electric field 3 mm below the surface of the brain, produced by a circular coil (bold circle) 50 mm in diameter, with its front edge 10 mm above the vertex. The contour lines are labeled in units of V/re. In b and c the arrows indicate the 0 and q~directions.
I n electric s t i m u l a t i o n using surface electrodes, the skull conductivity is extremely i m p o r t a n t in d e t e r m i n i n g the m a g n i t u d e of the electric field in the cortex ( N u n e z 1981). In magnetic stimulation, however, the i n d u c e d electric field is insensitive to the skull conductivity. A t 3 m m below the surface of the cortex the m a x i m u m electric field decreased b y 0.05% for a skull c o n d u c t i v i t y double that of Rush a n d Driscoll's, a n d increased b y
0.03% for a skull c o n d u c t i v i t y half of their value 5. In fact, to a very good a p p r o x i m a t i o n (0.06%), both the skull a n d scalp can be ignored a n d the cortex treated as
5 The observation that the electric field is nearly independent of the skull conductivity is consistent with Cohen's (1989) suggestion that there is no radial component of the electric field induced in a sphere during magnetic stimulation.
52
B.J. R O T H ET AL.
E (V/m) 100 8O 60 4O 200 0
f
I
I
I
I
I
I
I
r
10
20
30
40
50
60
70
80
90
(degrees) Fig. 6. Magnitude of the electric field at the vertex, 3 m m below the surface of the cortex, for different coil orientations, ~k.
if it were exposed directly to the air. The skull resistance plays a very m i n o r role in magnetic stimulation, which is in stark contrast to the major role it plays in electric stimulation.
b-----4
10mm Fig. 7. Magnitude of the electric field 3 mm below the surface of the cortex produced by a circular coil (bold line) oriented perpendicular to the head surface, with its nearest edge 10 mm above the vertex. The contour lines are labeled in units of V/m.
Fig. 8. X-ray photograph of the 6 coils. Clockwise from upper left: coil a, coil f, coil e, coil d, coil b, coil a.
ELECTRIC F I E L D I N D U C E D D U R I N G M A G N E T I C S T I M U L A T I O N
53
a
b
10 mm
c
e
Fig. 9. a-f: magnitude of the electric field 3 mm below Ihe surface of the brain for the 6 coils shown in Fig. 8.
54
B.J. ROTH ET AL.
TABLE 1I
Discussion
Parameters describing the coils. Coil
Number of turns
d I / d t (A//~sec)
Manufacturer
a b c d e f
14 14 20 19 20 15
150 150 150 80 160 150
Cadwell a Cadwell Cadwell Novametrix b Novametrix Cadwell
a Cadwell Lab., Inc., Kennewick, WA. h Novametrix Medical Systems Inc., Wallingford, CT.
Cohen et al. (1990) calculated the electric field produced in air by the 6 coils shown in Fig. 8. We have calculated the electric field in the brain for the same 6 coils. The number of turns and rate of change of current for each coil is supplied by the manufacturers (Table II) 6. A contour plot of the magnitude of the electric field 3 mm below the brain surface produced by each coil is shown in Fig. 9. The first coil (Fig. 9a) is circular with a radius of 39 mm. Its field is very similar to the one shown in Fig. 3c. Fig. 9b and 9c show circular coils with pointed ends. Although similar to a circular coil, a coil of this variety produces a somewhat different field near the point. Clearly the size of the coil also plays a role in determining both the magnitude and the spatial distribution of the two fields. Fig. 9d and 9e show the electric field produced by spiral coils. The fields they produce are broader and sensitive to the radius of the spiral. Fig. 9f shows a figure-of-eight coil, consisting of 2 circular coils placed side by side. Its peak field is greater than any other coil's and is also more localized, as demonstrated experimentally by Cohen et al. (1988). One measure of locality is the area of tissue with an electric field strength of half the peak field. If we assume stimulation to occur where the magnitude of the electric field is above a threshold value, this area corresponds to the area of stimulation at a stimulus strength twice threshold. Coil D has the largest area, approximately 14 × 103 mm 2, followed by coil A (10 × 103), B (9 × 103), E (9 × 103), C (7 × 103) and F (2 × 103). The relative strength and focality of the electric fields produced by these coils vary with depth, and only a depth of 3 mm below the cortex surface has been considered here.
6 The quoted rates of change of the current in Table 1I correspond to the 100% setting for the particular manufacturer and coil. Another way to compare coils is to assume the same rate of change of current for each coil. In that case, the electric field from coil D would be almost twice as large as shown in Fig. 9, and therefore its stimulus strength would be comparable to the figure-of-eight coil, coil F.
Our results show that for most coil geometries and orientations it is essential to account for the charge distribution when calculating the electric field produced in the brain during magnetic stimulation. The effect of the charge is to decrease the peak electric field, by 36% for the coil geometry shown in Fig. 3, thereby shielding the brain from the stimulus. Another consequence of the charge distribution is to force the electric field to be directed almost parallel to the surface of the head, with only a small radial component. The electric field in the cortex is quite insensitive of the conductivity of the skull and scalp; to a good approximation the cortex can be considered in isolation. Our model predicts which coils produce the largest and most focal electric fields. A focal stimulus is important when only one particular region of the cortex is to be stimulated, for instance in cortical mapping studies. If a circular coil is rotated from a tangential orientation to a perpendicular one, the magnitude of the electric field decreases but becomes more focal. Smaller coils produce weak and focal electric fields. However, the figure-of-eight coil delivers a larger and more focal stimulation than other coil geometries, consistent with the experimental observations by Cohen et al. (1988, 1989). One major assumption in our model is that the cortex can be modeled as an isotropic, homogeneous volume conductor. Another is that the brain is spherical. Given the extreme complexity of the brain, these assumptions may not be valid. More detailed models might be developed using finite element routines to follow the exact shape of the brain surface (Rusinko et al. 1984). The 3-sphere model provides much insight into the spatial distribution of the electric field in the head during magnetic stimulation. However, knowledge of the electric field distribution does not immediately indicate which neurons in the brain are stimulated. There is an additional question regarding how the electric field interacts with the neuron. Often the location of stimulation is assumed to be where the electric field is maximum (Barker et al. 1987). However, theoretical studies indicate that for a long axon stimulation occurs not where the electric field is largest but instead where its spatial derivative along the axon is maximum (Ferguson et al. 1989; Reilly 1989; Roth and Basser 1990). It is not known which prediction is valid for stimulation of a short fiber, cell body, or dendritic tree. It is clear that the relative orientation of the nerve and the electric field is important (Amassian et al. 1988). Evidence also exists that the polarity of the current in the coil influences the stimulation threshold (Barker et al. 1987; Mills et al. 1987). There have been several studies of how electric fields stimulate neurons in the central
ELECTRIC FIELD I N D U C E D D U R I N G M A G N E T I C STIMULATION
nervous system (Chan and Nicholson 1986; Tranchina and Nicholson 1986), including 2 exhaustive reviews (Ranck 1975; Coburn 1989). However, none of these studies define a definitive relationship between the electric field distribution in the motor cortex and the resulting transmembrane potential induced in the cortical neurons. Because of the complexity of the neuron-electric field interaction, the problem of modeling the interaction of the electric field with the neurons may prove to be much more difficult than determining the electric field distribution in the brain.
55
where the residual, R i, depends on the potential at the node r, 0 and 0 and its nearest neighbors W=
1
2
, rA0 ]
r sin0 Aq~
X
1 + -~-/Oi(r +Ar) +
+
~-~-0 ] + 2 r 2 tan0
Ar ~2
We would like to thank Dr. Ronald Levin for writing the software to make the contour plots, Drs. Peter Basser and Eduardo DuclaSoares for their careful reading of the manuscript, and B.J. Hessie for editorial assistance.
Ar 2
Oi(0 + A0) +
~r2 )Oi(0 _ a 0 ) 2 r 2 tan0 +
rsinOA 0
[oi(~+AO)+Oi(O-A~)]
Numerical computation of the electric field In electromagnetic theory, the electrical field arises from the scalar and vector potentials, • and A (Jackson 1975) 0A Ot
For each iteration, Eqn. (A3) is evaluated at each node in the tissue until the largest residual is smaller than a user-selected tolerance. Convergence is accelerated by the addition of a factor X, called the relaxation parameter, which lies between 1 and 2 (Rice 1983) O i+l -= o i +
vO
(A1)
The first term, E A = - 0 A / a t, is due to electromagnetic induction and has been analyzed by Cohen et al. (1990). The second term, E , = -V'O, is produced by a charge distribution and is the term traditionally encountered in electrophysiology. Within the brain, scalp and skull the potential v obeys Laplace's equation (Jackson 1975) ~720 = 0
(A2)
Since we are using spherical geometry, we express Laplace's equation in a spherical coordinate System (Jackson 1975), where r is the distance from the center of the sphere, 0 is the angle to the z-axis (latitude), and 0 is the polar angle in the x-y plane (longitude) (Fig. 1). We approximate Laplace's equation with a finite difference equation and then solve the resulting set of linear equations using systematic overrelaxafion iteration (Rice 1983). The potential is determined on a grid of nodes separated by a distance A r in the radial direction, and by angles A0 and Aq, in the 0 and q, directions. The (i + 1)th iteration of the potential, • i+1, at point r, 0 and 0 can be obtained from the ith iteration, • ~, by O i+l = o i +
-0 i (A4)
Appendix
E-
r zl0 ]
Ri
(A3)
v Henceforth • is referred to as simply the potential, while A retains the name vector potential.
(A5)
)k' R i
At the interfaces between tissues the potential must obey two boundary conditions. First, the potential is continuous across the interface between regions 1 and 2 O1 = 02
(A6)
Second, the current density, J, perpendicular to the interface (in our case, in the radial direction) is continuous. By Ohm's law, the current density is equal to the conductivity, o, times the electric field: a = aE
(A7)
The electric field perpendicular to the tissue interface is the sum of the radial component of the electric field produced by the vector potential, E~., and the negative derivative of the potential in the radial direction, - a O / a t . Using Eqn. (A7), we find that between regions 1 and 2 the potential obeys the condition o, (E~,
00~ (E~, Or ) =02
002
Or )
(A8)
We approximate the derivatives in Eq. (A8) with finite differences, using a 3-point approximation to preserve second-order accuracy in our algorithm (Rice 1983). The discrete approximations of the boundary conditions in Eqns. (A6) and (A8) are enforced at each iteration. Once we obtain the potential at every node, the electric field is calculated using a central difference approximation to take the negative gradient of the potential. Our calculations were performed with a relaxation parameter of 1.7. a residual tolerance of 0.01 /~V, and a
56
grid spacing of A0 = 11.25 °, Aq~= 11.25 °, and Ar = 1.17 mm in the scalp, 1.25 m m in the skull, and 2.0 m m in the brain (where r < 60 mm, we switched to a larger value of ,Ar = 5.0 mm to reduce the number of nodes in the computation). The total number of nodes was 16,384. The algorithm iterated approximately 5500 times before convergence. We experimented with a stricter tolerance of 0.001 ~V and a greater number of nodes, but found that this altered our maximum electric field by only 1.6%. The calculations were done on a VAX-11 750 computer using programs written in Fortran. A typical computation of the potential took 1 h and 45 min of CPU time.
References Amassian, V.E., Cracco, R.Q. and Maccabee, P.J. Basic mechanisms of magnetic coil excitation of nervous system in humans and monkeys and their applications. In: IEEE Special Symposium on Maturing Technologies and Emerging Horizons. IEEE, New York, 1988: 10-17. Barker, A.T., Jalinous, R. and Freeston, I.L. Non-invasive magnetic stimulation of human motor cortex. Lancet, 1985, i: 1101-1107. Barker, A.T., Freeston, 1.L., Jalinous, R. and Jarratt, J.A. Magnetic stimulation of the human brain and peripheral nervous system: an introduction and the results of an initial clinical evaluation. Neurosurgery, 1987, 20: 100-109. Chan, C.Y. and Nicholson, C. Modulation by applied electric fields of Purkinje and stellate cell activity in the isolated turtle cerebellum. J. Physiol. (Lond.), 1986, 371: 89-114. Claus, D., Harding, A.E., Hess, C.W., Mills, K.R., Murray, N.M.F. and Thomas, P.K. Central motor conduction in degenerative ataxic disorders: a magnetic stimulation study. J. Neurol. Neurosurg. Psychiat., 1988, 51: 790-795. Coburn, B. Neural modeling in electrical stimulation. CRC Crit. Rev. Biomed. Eng., 1989, 17: 133-178. Cohen, D. Some principles of focal magnetic stimulation. 1989 International Motor Evoked Potential Symposium, Chicago, IL, Aug. 18-20, 1989. Cohen. L.G., Bandinelli, S., Lelli, S. and Hallett, M. Noninvasive mapping of hand motor somatotopic area using magnetic stimulation. J. Clin. Neurophysiol., 1988, 5: 371-372. Cohen, L.G, Sato, S., Rose, D., Bandinelli, S., Kufta, C. and Hallett, M. Correlation of transcranial magnetic stimulation (TCMS) direct cortical stimulation (DCS) and somatosensory evoked potentials (SEP) for mapping of hand motor representation area (HMRA). Neurology, 1989, 39 (Suppl. 1): 375. Cohen, L.G, Roth, B.J., Nilsson, J., Dang, N., Panizza, M., Bandinelli, S., Friauf, W. and Hallett, M. Effects of coil design on delivery of focal magnetic stimulation. Technical considerations. Electroenceph. clin. Neurophysiol., 1990, 75: 350-357. Ferguson, A.S., Durand, D. and Dalbasti, T. Optimization of coil
B.J. ROTH ET AL. design for neuronal excitation by magnetic stimulation. In: IEEE Engineering in Medicine and Biology Soc., l lth Annual Conf. IEEE, New York, 1989: 1254-1255. Grandori, F. and Rossini, P. Electrical stimulation of the motor cortex: theoretical considerations. Ann. Biomed. Eng.. 1988, 16: 639-652. Hallett, M. and Cohen, L.G. Magnetism: a new method for stimulation of nerve and brain. JAMA, 1989, 262: 538-541. Hallett, M., Cohen, L.G., Nilsson, J. and Panizza, M. Differences between electrical and magnetic stimulation of human peripheral nerve and motor cortex. In: S. Chokroverty (Ed.), Magnetic Stimulation in Clinical Neurophysiology. Butterworth, Stoneham~ MA, 1989: 275-287. Hess, C.W., Mills, K.R. and Murray, N.M.F. Responses in small hand muscles from magnetic stimulation of the human brain. J. Physiol. (Lond.), 1987, 388: 397-419. Jackson, J.D. Classical Electrodynamics. Wiley, New York, 1975. Maccabee, P.J., Amassian, V.E., Cracco, R.Q and Cadwell, J.A. An analysis of peripheral motor nerve stimulations in humans using the magnetic coil. Electroenceph. olin. Neurophysiol., 1988, 70: 524-533. Mills, K.R., Murray, N.M.F. and Hess. C.W. Magnetic and electrical transcranial brain stimulation: physiological mechanisms and clinical applications. Neurosurgery, 1987, 20: 164-168. Nunez, P.L. Electric Fields of the Brain. Oxford University Press, New York, 1981. Ranck, Jr., J.B, Which elements are excited in electrical stimulation of mammalian central nervous system: a review. Brain Res., 1975, 98: 417-440. Reilly, J.P. Peripheral nerve stimulation by induced electric currents: exposure to time-varying magnetic fields. Med. Biol. Eng. Cornput., 1989, 27: 101-110. Reuter, C., Battocletti, J.H., Myklebust, J. and Maiman, D. Magnetic stimulation in peripheral nerves. In: IEEE Engineering in Medicine and Biology Society 10th Annual Int. Conf. IEEE, New York, 1988: 928-929. Rice, J.R. Numerical Methods, Software, and Analysis. McGraw-Hill, New York, 1983. Roth, B.J. and Basset, P.J. A model of the stimulation of a nerve fiber by electromagnetic induction. IEEE Trans. Biomed. Eng., 1990, 37: 588-597. Roth, B.J., Cohen, L.G., Hallett, M., Friauf, W. and Basser, P.J. A theoretical calculation of the electric field induced by magnetic stimulation of a peripheral nerve. Muscle Nerve, 1990, 13: 734-741. Rush, S. and Driscoll, D.A. Current distribution in the brain from surface electrodes. Anesth. Anal&, 1968, 47: 717-723. Rush, S. and DriscolL D.A. EEG electrode sensitivity. An application of reciprocity. IEEE Trans. Biomed. Eng., 1969, 16: 15-22. Rusinko, J.B., Sepulveda, N.G. and Walker, C.F. Three-dimensional numerical solution of field distribution. In: Proc. 37th Annual Conf. on Engineering in Medicine and Biology. Alliance for Engineering in Medicine and Biology, Washington, DC, 1984. Tranchina, D. and Nicholson, C. A model for the polarization of neurons by extrinsically applied electric fields. Biophys. J., 1986, 50: 1139-1156.
47
ELMOCO 89174
A theoretical calculation of the electric field induced in the cortex during magnetic stimulation Bradley J. Roth a, Joshua M. Saypol a, Mark Hallett b and Leonardo G. Cohen b " Biomedical Engineering and Instrumentation Branch, Division of Research Services, and ~ Human Cortical Physiology Unit, Human Motor Control Section, Medical NeuroloD" Branch, National Institute of Neurological Disorders and Stroke. National Institutes of Health, Bethe;da, MD 20892 (U.S.A.) (Accepted for publication: 9 February 1990)
Summary We present a mathematical model for calculating the electric field induced in the head during magnetic stimulation of the cortex. The electric field arises from 2 sources: (1) the changing magnetic field creates an electric field in the tissue by electromagnetic induction, and (2) a charge distribution arises on the surface of the head and produces its own electrostatic field. A 3-sphere model is used to represent the brain, skull and scalp. The electric field as a function of the coil position, shape and orientation is computed numerically. The charge distribution partially shields the brain from the stimulus. The electric field is insensitive to the skull conductivity, in contrast with electrical stimulation using surface electrodes. Different coil shapes and orientations are considered, and a figure-of-eight coil is shown to deliver the largest and most focal stimulus. Key words: Magnetic stimulation; Electric field; Cortex
X
Magnetic stimulation is a new technique for activating neurons in the cortex (Barker et al. 1985). Stimulation occurs when a brief current pulse is passed through a coil near the head, producing an electric field in the brain by electromagnetic induction. This technique is used to map the motor cortex (Cohen et al. 1988) and in central motor conduction studies (Barker et al. 1987; Mills et al. 1987; Claus et al. 1988). Possible clinical applications of magnetic stimulation include the diagnosis of multiple sclerosis, amyotrophic lateral sclerosis and other degenerative ataxic disorders, and intraoperarive monitoring of the corticospinal tract during spinal cord surgery (for review, see Hallett and Cohen 1989). The technique is non-invasive and is less painful than stimulating the brain electrically through scalp surface electrodes. Although the use of magnetic stimulation is growing rapidly, the technique has been applied clinically without a complete theoretical understanding of the induced electric field distribution. In this paper we present a mathematical calculation of the electric field produced in the brain during magnetic stimulation, with the goal of providing some theoretical insight to guide future applications of this technique. We use the 3-sphere
Correspondence to: Bradley J. Roth, Building 13, Room 3W13, National Institutes of Health, Bethesda, MD 20892 (U.S.A.).
y~
!z
6)
Z
Y S Cortex Fig. 1. Schematic drawing of the 3-sphere model of the head. Onefourth of the sphere is cut away to show the 3 layers; the outermost layer represents the scalp, the middle layer the skull, and the inner sphere the cortex. A spherical coordinate system (r, 0 and ~) and its relationship to the Cartesian system (x, y, z) is also shown. The relationship between the x, y, and z coordinates and the anatomical features on the head is indicated.
48
B.J. ROTH ET AL.
model introduced by Rush and Driscoll (1968, 1969) to represent the head as 3 concentric layers: the scalp, skull, and cortex (Fig. 1). This model has been used to interpret electroencephalographic and magnetoencephalographic data (Nunez 1981) and to describe electrical stimulation of the motor cortex using electrodes on the scalp surface (Grandori and Rossini 1988).
Y
Methods
The electric field induced in the brain during magnetic stimulation is produced by 2 sources: (1) the current in the stimulating coil produces an electric field by electromagnetic induction, E A, and (2) the charge that accumulates on the surface of the head produces an electric field within the tissue, E~. The total electric field in the cortex, E, is the sum of E A and E~. The electric field produced by induction, EA, is discussed frequently in the literature (Barker et al. 1987; Maccabee et al. 1988; Reuter et al. 1988). The magnitude of this electric field depends on the rate of change of the current in the coil, and the spatial distribution of the field can be calculated from the coil geometry (Cohen et al. 1990). The electric field produced by charge, E~, is rarely considered in discussions of magnetic stimulation, although often a charge distribution must exist on the surface of the head in order to satisfy the fundamental laws governing electric fields (Roth et al. 1990). To understand why a charge distribution arises, consider what happens if E A is not everywhere directed parallel to the surface of the head. Charge flows in the direction of E a until it reaches the head surface. Since air is an insulator, the charge cannot leave the surface and therefore accumulates, producing its own electric field E~. Charge continues to accumulate until the components of E , and E A perpendicular to the surface of the head cancel. This process is analogous to the charging of a capacitor, and it occurs very quickly compared with the rise time of current pulses used in magnetic stimulation. Charge accumulation not only occurs on the head surface but also at the interface between any two tissues with different conductivities (see Appendix). Consider the boundary between the skull and scalp. The charge accumulation at the interface between the two tissues causes the electric field lines to change directions, flowing more perpendicular to the boundary in the low-conductivity skull and more parallel to the boundary in the high-conductivity scalp. We calculate the total electric field in the head during magnetic stimulation, taking into account the electric fields produced by both induction and charge. The details of the numerical computation are described in the appendix. Briefly, the electric field due to induction, E A, is obtained from an integral expression derived
Z
10 rnrn
Fig. 2. Magnitude of the electric field 3 m m below the surface of the cortex produced by a circular coil (bold circle) with its center l0 m m above the vertex (left inset). The curves represent a contour m a p of the magnitude of the electric field projected into the y-z plane (right inset). The contour lines are labeled in units of V / m . The peak field is 89 V / m and is located approximately midway between the two 80 V / m curves. The arrows indicate the direction of the electric field.
from electromagnetic theory (Cohen et al. 1990). The electric potential is determined from Laplace's equation and the boundary conditions at the interfaces between tissues; then E~ is calculated by taking the negative gradient of the potential (Roth et al. 1990). Laplace's equation is solved using a finite difference approximation and a systematic overrelaxation iteration (Rice 1983).
Results
Fig. 2 shows the electric field produced during magnetic stimulation by a circular coil oriented tangentially to the head with its center 10 mm above the vertex (top of the head), an orientation often used clinically (Barker et al. 1987; Hess et al. 1987; Mills et al. 1987; Claus et al. 1988) 1. The coil has a radius of 50 mm, is wound with 8 turns, and carries a current increasing at a rate of
1The 10 m m spacing between the coil and the scalp is somewhat larger than used clinically. We chose this spacing to account for the thickness of the coil, which is neglected in our model.
ELECTRIC FIELD INDUCED DURING MAGNETIC STIMULATION TABLE I Parameters in the 3-sphere model. Tissue
Outer radius ( m m )
Conductivity (S/m)
Cortex Skull Scalp
80.0 85.0 92.0
0.450 0.00563 0.450
100 A//zsec. The radius of the head, the thickness of the scalp and skull, a n d the conductivities of the tissues are the same as used by Rush and Driscoll (1969) a n d are
49 listed in T a b l e I. The calculated electric field is a 3 - d i m e n s i o n a l vector field. To plot this field graphically, we use a c o n t o u r m a p of the m a g n i t u d e of the electric field over a spherical surface 3 m m below the surface of the cortex. F o r this coil o r i e n t a t i o n the electric field E A is everywhere parallel to the head surface, so n o charge distrib u t i o n is induced. The electric field is directed in concentric loops, as indicated by the arrows in Fig. 2. The coil c u r r e n t is in the counterclockwise direction, as viewed from above, a n d is increasing with time, so that
a)
I------4
10ram
Fig. 3. Magnitude of the electric field produced by (a) induction, (b) charge, and (c) the total electric field, 3 mm below the surface of the brain, produced by a circular coil (bold circle) with one edge 10 mm above the vertex. The contour lines are labeled in units of V/m; the peak field in c is 89 V/m at the vertex. The arrows indicate the direction of the electric field.
50 the electric field induced in the head is oriented in the clockwise direction. Because of the curvature of the head, the maximum of the electric field (89 V / m ) does not occur directly below the coil winding. Suppose this coil is moved 50 mm toward the back of the head (in the negative y direction), so that the front edge of the coil is 10 mm above the vertex. The electric field due to induction, E A, is maximum (141 V / m ) at the vertex and zero under the center of the coil (Fig. 3a) 2. For this coil orientation, E a is no longer parallel to the head surface and a charge distribution develops. The electric field due to the charge, E~, is maximum (58 V / m ) slightly behind the vertex and is smaller than EA by about a factor of 2.3 (Fig. 3b). Although the contour lines indicate only the magnitudes of the fields, the arrows show that near the v e r t e x E A is in the opposite direction to E~. Thus the total electric field, shown in Fig. 3c, has a smaller maximum (89 V / m ) than the field due only to induction. The charge accumulation reduces the peak electric field produced by the coil. The electric field falls off with the depth into the brain, which may be important in determining what types of neuron are stimulated. Fig. 4 shows the magnitude of the electric field at the vertex as a function of depth below the surface of the head, using the coil geometry and orientation presented in Fig. 3. At the surface of the cortex the electric field is 104 V / m , but l0 mm below the surface it has fallen by 38% to 65 V/re. During magnetic stimulation, the electric field induced in the scalp is about twice as large as that in the brain. In electric stimulation this ratio is much larger, with the scalp field often more than 10 times that in the cortex (Rush and Driscoll 1969). This difference provides one explanation why magnetic stimulation is less painful than electric stimulation; the pain receptors in the skin are not stimulated by the relatively small electric fields induced in the scalp. Not only the magnitude but also the direction of the electric field is important in the stimulation of cortical neurons (Amassian et al. 1988; Coburn 1989). Fig. 5 shows the 3 components of the total electric field 3 mm below the surface of the cortex. These components are associated with a spherical coordinate system (r, 0, q~), as shown in Fig. 1: the r direction is radially outward from the center of the head, the ~ direction is directed toward the right ear (at the vertex the 8 direction is along a coronal axis), and the q5 direction is at right angles to the r and 8 directions (along a sagittal axis) 3.
2 The contour lines of E A would be circular if plotted on a flat surface parallel to the coil instead of on the surface of the head. 3 In an analogy to a globe, the north and south poles (8 = 0 and 180) are located at the ears, 8 measures latitude, and rp measures longitude. The equator (8 = 90) passes through the nose and the vertex.
B.J. ROTH ET AL. E (V/m)
250 200 150
~
100
- Scalp - - Skull
50 0 0
10
20
30
40
50
60
70 80
90
depth (mm) Fig. 4. Magnitude of the electric field at the vertex as a function of depth below the head surface, for the coil orientation shown in Fig. 3.
The smallest component of the electric field is in the radial direction (Fig. 5a); the 8 and ~ components are a factor of 100 larger than the r component (Fig. 5b and c) 4. The 8 component is maximum under the edge of the coil, at the vertex; there is a smaller maximum approximately below the center of the coil. The component is maximum under the edge of the coil, about 45 ° removed from the center of the head in either direction. Amassian et al. (1988) showed experimentally that the stimulation threshold changes with coil orientation. Fig. 6 shows the magnitude of the electric field at the vertex, 3 mm below the surface of the cortex, as a function of the angle, ~b, between the coil and a plane tangent to the head at the vertex (~b = 0 corresponds to a tangential orientation). The magnitude of the electric field decreases as ~b increases from 0 to 90, with the greatest change at small angles. The tangential orientation induces the largest electric field. However, the spatial distribution of the electric field is more focal with the coil perpendicular to the head. For the tangential orientation with one edge above the vertex (Fig. 3c), the width of the region in which the electric field is greater than 90% of its maximum value is 14 mm along the sagittal axis and 48 mm along a coronal axis. With the coil perpendicular to the head surface and aligned with the coronal axis (~b = 90), these widths are 10 and 24 mm (Fig. 7). Along the coronal axis the focality of the electric field increases by a factor of two by rotating the coil 90 °. These predictions are consistent with the experimental observations by Amassian et al. (1988) and Hallett et al. (1989).
4 Cohen (1989) has suggested that the electric field induced in a sphere during magnetic stimulation has no radial component. If correct, his suggestion implies that the radial component calculated here reflects the residual error in our calculation (less than 1%).
ELECTRIC FIELD INDUCED DURING MAGNETIC STIMULATION
51
) b 10 mm
Fig. 5. The (a) r, (b) 0, and (c) ,~ components of the electric field 3 mm below the surface of the brain, produced by a circular coil (bold circle) 50 mm in diameter, with its front edge 10 mm above the vertex. The contour lines are labeled in units of V/re. In b and c the arrows indicate the 0 and q~directions.
I n electric s t i m u l a t i o n using surface electrodes, the skull conductivity is extremely i m p o r t a n t in d e t e r m i n i n g the m a g n i t u d e of the electric field in the cortex ( N u n e z 1981). In magnetic stimulation, however, the i n d u c e d electric field is insensitive to the skull conductivity. A t 3 m m below the surface of the cortex the m a x i m u m electric field decreased b y 0.05% for a skull c o n d u c t i v i t y double that of Rush a n d Driscoll's, a n d increased b y
0.03% for a skull c o n d u c t i v i t y half of their value 5. In fact, to a very good a p p r o x i m a t i o n (0.06%), both the skull a n d scalp can be ignored a n d the cortex treated as
5 The observation that the electric field is nearly independent of the skull conductivity is consistent with Cohen's (1989) suggestion that there is no radial component of the electric field induced in a sphere during magnetic stimulation.
52
B.J. R O T H ET AL.
E (V/m) 100 8O 60 4O 200 0
f
I
I
I
I
I
I
I
r
10
20
30
40
50
60
70
80
90
(degrees) Fig. 6. Magnitude of the electric field at the vertex, 3 m m below the surface of the cortex, for different coil orientations, ~k.
if it were exposed directly to the air. The skull resistance plays a very m i n o r role in magnetic stimulation, which is in stark contrast to the major role it plays in electric stimulation.
b-----4
10mm Fig. 7. Magnitude of the electric field 3 mm below the surface of the cortex produced by a circular coil (bold line) oriented perpendicular to the head surface, with its nearest edge 10 mm above the vertex. The contour lines are labeled in units of V/m.
Fig. 8. X-ray photograph of the 6 coils. Clockwise from upper left: coil a, coil f, coil e, coil d, coil b, coil a.
ELECTRIC F I E L D I N D U C E D D U R I N G M A G N E T I C S T I M U L A T I O N
53
a
b
10 mm
c
e
Fig. 9. a-f: magnitude of the electric field 3 mm below Ihe surface of the brain for the 6 coils shown in Fig. 8.
54
B.J. ROTH ET AL.
TABLE 1I
Discussion
Parameters describing the coils. Coil
Number of turns
d I / d t (A//~sec)
Manufacturer
a b c d e f
14 14 20 19 20 15
150 150 150 80 160 150
Cadwell a Cadwell Cadwell Novametrix b Novametrix Cadwell
a Cadwell Lab., Inc., Kennewick, WA. h Novametrix Medical Systems Inc., Wallingford, CT.
Cohen et al. (1990) calculated the electric field produced in air by the 6 coils shown in Fig. 8. We have calculated the electric field in the brain for the same 6 coils. The number of turns and rate of change of current for each coil is supplied by the manufacturers (Table II) 6. A contour plot of the magnitude of the electric field 3 mm below the brain surface produced by each coil is shown in Fig. 9. The first coil (Fig. 9a) is circular with a radius of 39 mm. Its field is very similar to the one shown in Fig. 3c. Fig. 9b and 9c show circular coils with pointed ends. Although similar to a circular coil, a coil of this variety produces a somewhat different field near the point. Clearly the size of the coil also plays a role in determining both the magnitude and the spatial distribution of the two fields. Fig. 9d and 9e show the electric field produced by spiral coils. The fields they produce are broader and sensitive to the radius of the spiral. Fig. 9f shows a figure-of-eight coil, consisting of 2 circular coils placed side by side. Its peak field is greater than any other coil's and is also more localized, as demonstrated experimentally by Cohen et al. (1988). One measure of locality is the area of tissue with an electric field strength of half the peak field. If we assume stimulation to occur where the magnitude of the electric field is above a threshold value, this area corresponds to the area of stimulation at a stimulus strength twice threshold. Coil D has the largest area, approximately 14 × 103 mm 2, followed by coil A (10 × 103), B (9 × 103), E (9 × 103), C (7 × 103) and F (2 × 103). The relative strength and focality of the electric fields produced by these coils vary with depth, and only a depth of 3 mm below the cortex surface has been considered here.
6 The quoted rates of change of the current in Table 1I correspond to the 100% setting for the particular manufacturer and coil. Another way to compare coils is to assume the same rate of change of current for each coil. In that case, the electric field from coil D would be almost twice as large as shown in Fig. 9, and therefore its stimulus strength would be comparable to the figure-of-eight coil, coil F.
Our results show that for most coil geometries and orientations it is essential to account for the charge distribution when calculating the electric field produced in the brain during magnetic stimulation. The effect of the charge is to decrease the peak electric field, by 36% for the coil geometry shown in Fig. 3, thereby shielding the brain from the stimulus. Another consequence of the charge distribution is to force the electric field to be directed almost parallel to the surface of the head, with only a small radial component. The electric field in the cortex is quite insensitive of the conductivity of the skull and scalp; to a good approximation the cortex can be considered in isolation. Our model predicts which coils produce the largest and most focal electric fields. A focal stimulus is important when only one particular region of the cortex is to be stimulated, for instance in cortical mapping studies. If a circular coil is rotated from a tangential orientation to a perpendicular one, the magnitude of the electric field decreases but becomes more focal. Smaller coils produce weak and focal electric fields. However, the figure-of-eight coil delivers a larger and more focal stimulation than other coil geometries, consistent with the experimental observations by Cohen et al. (1988, 1989). One major assumption in our model is that the cortex can be modeled as an isotropic, homogeneous volume conductor. Another is that the brain is spherical. Given the extreme complexity of the brain, these assumptions may not be valid. More detailed models might be developed using finite element routines to follow the exact shape of the brain surface (Rusinko et al. 1984). The 3-sphere model provides much insight into the spatial distribution of the electric field in the head during magnetic stimulation. However, knowledge of the electric field distribution does not immediately indicate which neurons in the brain are stimulated. There is an additional question regarding how the electric field interacts with the neuron. Often the location of stimulation is assumed to be where the electric field is maximum (Barker et al. 1987). However, theoretical studies indicate that for a long axon stimulation occurs not where the electric field is largest but instead where its spatial derivative along the axon is maximum (Ferguson et al. 1989; Reilly 1989; Roth and Basser 1990). It is not known which prediction is valid for stimulation of a short fiber, cell body, or dendritic tree. It is clear that the relative orientation of the nerve and the electric field is important (Amassian et al. 1988). Evidence also exists that the polarity of the current in the coil influences the stimulation threshold (Barker et al. 1987; Mills et al. 1987). There have been several studies of how electric fields stimulate neurons in the central
ELECTRIC FIELD I N D U C E D D U R I N G M A G N E T I C STIMULATION
nervous system (Chan and Nicholson 1986; Tranchina and Nicholson 1986), including 2 exhaustive reviews (Ranck 1975; Coburn 1989). However, none of these studies define a definitive relationship between the electric field distribution in the motor cortex and the resulting transmembrane potential induced in the cortical neurons. Because of the complexity of the neuron-electric field interaction, the problem of modeling the interaction of the electric field with the neurons may prove to be much more difficult than determining the electric field distribution in the brain.
55
where the residual, R i, depends on the potential at the node r, 0 and 0 and its nearest neighbors W=
1
2
, rA0 ]
r sin0 Aq~
X
1 + -~-/Oi(r +Ar) +
+
~-~-0 ] + 2 r 2 tan0
Ar ~2
We would like to thank Dr. Ronald Levin for writing the software to make the contour plots, Drs. Peter Basser and Eduardo DuclaSoares for their careful reading of the manuscript, and B.J. Hessie for editorial assistance.
Ar 2
Oi(0 + A0) +
~r2 )Oi(0 _ a 0 ) 2 r 2 tan0 +
rsinOA 0
[oi(~+AO)+Oi(O-A~)]
Numerical computation of the electric field In electromagnetic theory, the electrical field arises from the scalar and vector potentials, • and A (Jackson 1975) 0A Ot
For each iteration, Eqn. (A3) is evaluated at each node in the tissue until the largest residual is smaller than a user-selected tolerance. Convergence is accelerated by the addition of a factor X, called the relaxation parameter, which lies between 1 and 2 (Rice 1983) O i+l -= o i +
vO
(A1)
The first term, E A = - 0 A / a t, is due to electromagnetic induction and has been analyzed by Cohen et al. (1990). The second term, E , = -V'O, is produced by a charge distribution and is the term traditionally encountered in electrophysiology. Within the brain, scalp and skull the potential v obeys Laplace's equation (Jackson 1975) ~720 = 0
(A2)
Since we are using spherical geometry, we express Laplace's equation in a spherical coordinate System (Jackson 1975), where r is the distance from the center of the sphere, 0 is the angle to the z-axis (latitude), and 0 is the polar angle in the x-y plane (longitude) (Fig. 1). We approximate Laplace's equation with a finite difference equation and then solve the resulting set of linear equations using systematic overrelaxafion iteration (Rice 1983). The potential is determined on a grid of nodes separated by a distance A r in the radial direction, and by angles A0 and Aq, in the 0 and q, directions. The (i + 1)th iteration of the potential, • i+1, at point r, 0 and 0 can be obtained from the ith iteration, • ~, by O i+l = o i +
-0 i (A4)
Appendix
E-
r zl0 ]
Ri
(A3)
v Henceforth • is referred to as simply the potential, while A retains the name vector potential.
(A5)
)k' R i
At the interfaces between tissues the potential must obey two boundary conditions. First, the potential is continuous across the interface between regions 1 and 2 O1 = 02
(A6)
Second, the current density, J, perpendicular to the interface (in our case, in the radial direction) is continuous. By Ohm's law, the current density is equal to the conductivity, o, times the electric field: a = aE
(A7)
The electric field perpendicular to the tissue interface is the sum of the radial component of the electric field produced by the vector potential, E~., and the negative derivative of the potential in the radial direction, - a O / a t . Using Eqn. (A7), we find that between regions 1 and 2 the potential obeys the condition o, (E~,
00~ (E~, Or ) =02
002
Or )
(A8)
We approximate the derivatives in Eq. (A8) with finite differences, using a 3-point approximation to preserve second-order accuracy in our algorithm (Rice 1983). The discrete approximations of the boundary conditions in Eqns. (A6) and (A8) are enforced at each iteration. Once we obtain the potential at every node, the electric field is calculated using a central difference approximation to take the negative gradient of the potential. Our calculations were performed with a relaxation parameter of 1.7. a residual tolerance of 0.01 /~V, and a
56
grid spacing of A0 = 11.25 °, Aq~= 11.25 °, and Ar = 1.17 mm in the scalp, 1.25 m m in the skull, and 2.0 m m in the brain (where r < 60 mm, we switched to a larger value of ,Ar = 5.0 mm to reduce the number of nodes in the computation). The total number of nodes was 16,384. The algorithm iterated approximately 5500 times before convergence. We experimented with a stricter tolerance of 0.001 ~V and a greater number of nodes, but found that this altered our maximum electric field by only 1.6%. The calculations were done on a VAX-11 750 computer using programs written in Fortran. A typical computation of the potential took 1 h and 45 min of CPU time.
References Amassian, V.E., Cracco, R.Q. and Maccabee, P.J. Basic mechanisms of magnetic coil excitation of nervous system in humans and monkeys and their applications. In: IEEE Special Symposium on Maturing Technologies and Emerging Horizons. IEEE, New York, 1988: 10-17. Barker, A.T., Jalinous, R. and Freeston, I.L. Non-invasive magnetic stimulation of human motor cortex. Lancet, 1985, i: 1101-1107. Barker, A.T., Freeston, 1.L., Jalinous, R. and Jarratt, J.A. Magnetic stimulation of the human brain and peripheral nervous system: an introduction and the results of an initial clinical evaluation. Neurosurgery, 1987, 20: 100-109. Chan, C.Y. and Nicholson, C. Modulation by applied electric fields of Purkinje and stellate cell activity in the isolated turtle cerebellum. J. Physiol. (Lond.), 1986, 371: 89-114. Claus, D., Harding, A.E., Hess, C.W., Mills, K.R., Murray, N.M.F. and Thomas, P.K. Central motor conduction in degenerative ataxic disorders: a magnetic stimulation study. J. Neurol. Neurosurg. Psychiat., 1988, 51: 790-795. Coburn, B. Neural modeling in electrical stimulation. CRC Crit. Rev. Biomed. Eng., 1989, 17: 133-178. Cohen, D. Some principles of focal magnetic stimulation. 1989 International Motor Evoked Potential Symposium, Chicago, IL, Aug. 18-20, 1989. Cohen. L.G., Bandinelli, S., Lelli, S. and Hallett, M. Noninvasive mapping of hand motor somatotopic area using magnetic stimulation. J. Clin. Neurophysiol., 1988, 5: 371-372. Cohen, L.G, Sato, S., Rose, D., Bandinelli, S., Kufta, C. and Hallett, M. Correlation of transcranial magnetic stimulation (TCMS) direct cortical stimulation (DCS) and somatosensory evoked potentials (SEP) for mapping of hand motor representation area (HMRA). Neurology, 1989, 39 (Suppl. 1): 375. Cohen, L.G, Roth, B.J., Nilsson, J., Dang, N., Panizza, M., Bandinelli, S., Friauf, W. and Hallett, M. Effects of coil design on delivery of focal magnetic stimulation. Technical considerations. Electroenceph. clin. Neurophysiol., 1990, 75: 350-357. Ferguson, A.S., Durand, D. and Dalbasti, T. Optimization of coil
B.J. ROTH ET AL. design for neuronal excitation by magnetic stimulation. In: IEEE Engineering in Medicine and Biology Soc., l lth Annual Conf. IEEE, New York, 1989: 1254-1255. Grandori, F. and Rossini, P. Electrical stimulation of the motor cortex: theoretical considerations. Ann. Biomed. Eng.. 1988, 16: 639-652. Hallett, M. and Cohen, L.G. Magnetism: a new method for stimulation of nerve and brain. JAMA, 1989, 262: 538-541. Hallett, M., Cohen, L.G., Nilsson, J. and Panizza, M. Differences between electrical and magnetic stimulation of human peripheral nerve and motor cortex. In: S. Chokroverty (Ed.), Magnetic Stimulation in Clinical Neurophysiology. Butterworth, Stoneham~ MA, 1989: 275-287. Hess, C.W., Mills, K.R. and Murray, N.M.F. Responses in small hand muscles from magnetic stimulation of the human brain. J. Physiol. (Lond.), 1987, 388: 397-419. Jackson, J.D. Classical Electrodynamics. Wiley, New York, 1975. Maccabee, P.J., Amassian, V.E., Cracco, R.Q and Cadwell, J.A. An analysis of peripheral motor nerve stimulations in humans using the magnetic coil. Electroenceph. olin. Neurophysiol., 1988, 70: 524-533. Mills, K.R., Murray, N.M.F. and Hess. C.W. Magnetic and electrical transcranial brain stimulation: physiological mechanisms and clinical applications. Neurosurgery, 1987, 20: 164-168. Nunez, P.L. Electric Fields of the Brain. Oxford University Press, New York, 1981. Ranck, Jr., J.B, Which elements are excited in electrical stimulation of mammalian central nervous system: a review. Brain Res., 1975, 98: 417-440. Reilly, J.P. Peripheral nerve stimulation by induced electric currents: exposure to time-varying magnetic fields. Med. Biol. Eng. Cornput., 1989, 27: 101-110. Reuter, C., Battocletti, J.H., Myklebust, J. and Maiman, D. Magnetic stimulation in peripheral nerves. In: IEEE Engineering in Medicine and Biology Society 10th Annual Int. Conf. IEEE, New York, 1988: 928-929. Rice, J.R. Numerical Methods, Software, and Analysis. McGraw-Hill, New York, 1983. Roth, B.J. and Basset, P.J. A model of the stimulation of a nerve fiber by electromagnetic induction. IEEE Trans. Biomed. Eng., 1990, 37: 588-597. Roth, B.J., Cohen, L.G., Hallett, M., Friauf, W. and Basser, P.J. A theoretical calculation of the electric field induced by magnetic stimulation of a peripheral nerve. Muscle Nerve, 1990, 13: 734-741. Rush, S. and Driscoll, D.A. Current distribution in the brain from surface electrodes. Anesth. Anal&, 1968, 47: 717-723. Rush, S. and DriscolL D.A. EEG electrode sensitivity. An application of reciprocity. IEEE Trans. Biomed. Eng., 1969, 16: 15-22. Rusinko, J.B., Sepulveda, N.G. and Walker, C.F. Three-dimensional numerical solution of field distribution. In: Proc. 37th Annual Conf. on Engineering in Medicine and Biology. Alliance for Engineering in Medicine and Biology, Washington, DC, 1984. Tranchina, D. and Nicholson, C. A model for the polarization of neurons by extrinsically applied electric fields. Biophys. J., 1986, 50: 1139-1156.
