IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 39, NO. 7, JULY 1992

693

Neural Stimulation with Magnetic Fields: Analysis of Induced Electric Fields Karu P. Esselle, Member, IEEE, and Maria A. Stuchly, Fellow, IEEE

Abstract-Spatial distributions of the derivative of the electric field induced in a planar semi-infinite tissue model by various current-carrying coils and their utility in neural stimulation are evaluated. Analytical expressions are obtained for the electric field and its spatial derivatives produced by an infinitely short current element. Fields and their derivatives for an arbitrarily shaped coil are then obtained by numerical summation of contributions from all the elements forming the coil. The simplicity of the solution and a very short computation time make this method particularly attractive for gaining a physical insight into the spatial behavior of the stimulating parameter and for the optimization of coils. Such analysis is useful as the first step before undertaking a more complex numerical analysis of a model more closely representing the tissue geometry and heterogeneity.

INTRODUCTION INCE mid-1980’s high magnitude magnetic fields produced by coils external to the body have been used for stimulation of motor neurons in the brain cortex and peripheral nerves. Magnetic stimulation offers some advantages compared to electrical stimulation, such as its noncontact, noninvasive nature, and a lack of or minimal discomfort to the patient [ 11, [2]. This technique has been successfully used to diagnose a number of disorders of the neuro-muscular system [3]-[5] and to map the motor-cortex [6]-[8]. Some of the limitations of the technique have been attributed to the poor focusing of the interacting field 191. Little attention has been paid to developing quantitative models until recently when Basser and Roth established such models [lo], [11]. These models indicate that in magnetic stimulation, the magnitude, sign and time-course of the spatial derivative of the induced electric field along the axis of the axon (nerve fiber) determine whether stimulation occurs and where along the axon it occurs. Depending on the sign of the field derivative the axon is depolarized (negative derivative) or hyperpolarized. All neurons within the volume, where the electric field derivative is negative and above a threshold value for the given neurons, are stimulated [ 113. In clinical practice, it is im-

S

Manuscript received May 20, 1991; revised October 15, 1991. K. P. Esselle is with the Bureau of Radiation and Medical Devices, Health and Welfare Canada, Ottawa, Ont. KIA OL2, Canada. M. A. Stuchly was with the Bureau of Radiation and Medical Devices, Health and Welfare Canada, Ottawa, Ont. KIA OL2, Canada. She is now with the Department of Electrical and Computer Engineering, University of Victoria, Victoria, B.C. V8W 3P6, Canada. IEEE Log Number 9200697.

portant to control the location and size of this volume. Furthermore, it is important to control the location of the volume where hyperpolarization occurs, as it may block the propagation of the action potential. The importance of evaluation of the induced electric field and its derivative in tissue due to current in the stimulating coil is apparent. In some investigations published earlier, the air-tissue interface was neglected, and the induced electric field was evaluated using analytical or semianalytical techniques [ 121. Although this is acceptable as a first approximation, an accurate analysis should take into account the presence of surface charges at the interface unless the interface is everywhere parallel to the coil. Various numerical techniques have been developed to account for the interface but they all require considerable amount of computations. For example, numerical techniques have been used to calculate the induced fields from a circular coil above a semi-infinite tissue space [lo], [ 131, a homogeneous tissue cylinder [ 141, and a multilayered sphere [ 131, [ 151. Fields produced by a double-circular coil (forming a figure eight), a spiral coil and a pointed circular coil (used in one of the commercially available stimulators) have also been calculated for a multilayered spherical tissue model [ 131. The main limitation of these methods is the long computational time and therefore the difficulty in exploring various options. Numerical solutions present inherent difficulties in gaining a physical insight into the dependence of the field spatial derivative of the electric field on the coil geometry and the neuron location. In this contribution, analytical expressions are derived for the induced electric field and its spatial derivatives in a semi-infinite tissue half-space due to an infinitesimal element of the current-carrying coil. The electric field and derivatives induced by an arbitrarily-shaped coil are then obtained by numerically combining the contributions from many small elements forming the coil. Since the only numerical step in this method is a one-dimensional integration, the method is significantly faster than the previous method of analyzing tissue half-space problems in which an additional two-dimensional numerical integration of surface charge is required [ 131, [ 141. We have analyzed circular, square, double circular and square, and quadruple square coils. The spatial distributions of the electric field derivatives are illustrated. Normalized curves are provided for the peak electric field spatial derivative, the

0018-9294/92$03.00

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 39, NO. 7. JULY 1992

694

locations of the peaks (stimulation points) and the stimulation-region size where the electric field spatial derivative is 90% of its maximum value. These data illustrate the dependence of the essential stimulation parameter on the coil size and the distance from the coil to the nerve. In the analysis, the nerves are assumed to be long, approximately straight and parallel to the interface.

THEORY The geometry of the problem is illustrated in Fig. 1. In the absence of tissue, the electric field produced by a N-turn cog can be calculated using the magnetic vector potential A and electric szalar potential 9.This primary electric field denoted by EP is given by [16]

I

Tissue-Air

YY

Homogeneous Tissue Half-space Fig. 1 . A stimulating coil of an arbitrary shape above a tissue half-space. The coil has N turns, each carrying current 1.

The secondary electric field dEs due to the coil-element dl is equal to Since the current in the stimulating coil changes very slowly with time (i.e., quasi-static conditions prevail), the current distribution is assumed to be uniform along the coil. Consequently, both the charge on the coil and the electric potential 9 are zero, and EP is equal to

where i ,9, 2 are unit vectors in x,, y , z directions, respectively. The total electric field dE due to the coil-element is then

dE = dEP + dES Since the time-retardation of electromagnetic fields can be neglect5d under quasi-static conditions, the magnetic potential A can be evaluated using the following line integral along the coil:

- _ -

47rR

-

v*

Under quasi-static conditions, the boundary conditions at the air-tissue interface require the current normal to the interface to be zero [14],i.e.,

(3) where 21 is an element of the coil, p, is the permeability H /m), N is the number of turns, of free space (47r X I is the current per turn, and R is the distance between the element and the point where the electric field is calculated. Assuming that the origin of the coordinate system is located on the air-tissue interface right below the coilelement (as shown in Fig. l),the coordinates of the coilelement are (0, 0, z,), and the primary electric field produced by the coil element at an arbitrary point with coordinates (x, y, z ) is given by

p,N(dZ/dt) 21

adE, = 0 a t z = 0

(9)

where U is $e tissue conductivity an: dE, is the z-component of dE. Expanding the vector d l into x, y, z components :

21 = 2 dl,

+ 9 dly + 2 dl,

(10)

and using (7)-(9) the following boundary condition is obtained:

dE, = -

poN(dZ/dt)dl, - a* = 0 at

az

4aR

z

= 0.

(11)

Since \E is a rotationally symmetric solution of Laplace's equation, it is expanded as a Bessel integral with(4) out a loss of generality [ 171 :

* io U(X)eE"J,(Xp)d h (12) m,J , is the Bessel function of first 03

=

where

R

=

Jx2

+ 9 + (Z - z,)~.

(5)

In the presence of tissue, surface charges are induced at the air-tissue interface, which in turn create a secondary electric field. As in previous investigations [13]-[151, we assume that this secondary electric field can be derived from another scalar potential function \k, which is a solution of Laplace's equation:

v2\E= 0.

(6)

where p = kind, 0-order and U(A) is an unknown function. In order to determine U @ ) , we substitute (12)into (ll),and also expand the 1/R term in (11) as follows [17]:

_1 -R

Jp2

+ (Z for z

1

03

1

= -

z0l2

< z,.

0

e-x(z"-z)J,(hp)dX

ESSELLE AND STUCHLY: NEURAL STIMULATION WITH MAGNETIC FIELDS

By equating the coefficients of J , ( h p ) , it can be shown that U(X) = -

p,N(dl/dt) dlZe-"" 4wx

(14)

Substitution of (14) into (12) leads to the following analytical solution for 'k: p , = - pON(dz/dt)

47r

dl,

so x m

e-A(,,-~)

J , ( x ~ )d),.

(15)

695

where inside the tissue half-space. The same property was previously shown by others [ 191, [20]. A corollary of this conclusion is that the electric field is independent of any tissue heterogeneities along z-axis. Therefore, all the expressions derived here are equally valid for heterogeneous tissue half-space provided that the tissue conductivity vanes only in the direction perpendicular to the airtissue interface (either continuously or in steps). The electric field produced by the coil element at (0, 0, z,) is obtained by combining (19), (20) and (23):

The electric field produced by the coil-element is obtained by substitutgg (15) into (8). According to (8), the x-component of d E is equal to

Differentiating (15) with respect to x and using the Bessel function property d [ J,(x)] /dr = - J l ( x ) [ 181, where J 1 is the Bessel function of first kind and first order, one obtains

This result can be further generalized by assuming that the coil element is located at_(x,, yo, z,) rather than at (0, 0, 2,). The electric field dE is then given by dE

= -

47r

. (1

-y)Ii

Following the approach taken by Wait [17], this expression is further reduced to the much simpler form:

_ a'k _ - p,N(dZ/dt) dl, ax 4Ir

);(

(1 -

y). (18)

Substituting this in (16), the x-component of dE is obtained as

where p = J ( x - x,)2

R

Similarly, the y-component of d z is found as

(20) From (15), the z-component of lows:

& is determined as fol-

Since the integral in the above expression is equal to 1/ R , as shown in (13), a'k az

p,N(dZ/dt) dl, 41rR '

=

J(x - x,)'

+ ( y - yo)' + ( y - yJ2 + (Z

(26) -

zJ2.

The total electric field produced by the coil can be obtained by numerically integrating the (25) along the length of the coil (with respect to dl ). This method of evaluating the electric field is computationally much more efficient than methods used previously. The parameter important in nerve stimulation is the spatial derivative of the electric field component tangential to the nerve [lo], [1I]. For example, for a nerve (assumed straight and long) parallel to the x-axis, the stimulation parameter (activating function) is aEx/ax. The electric field spatial derivatives in the x- and y-directions, produced by a coil-element at (x,, yo, z,) can be obtained by differentiating dE, and dE, with respect to x and y, respectively. The results are

(22)

Combining (8) and (22), one obtains

This leads to an interesting conclusion that the electric field is always parallel to the air-tissue interface every-

(27)

-

[

( Y - Y,Y P4 - (x - xJ2

1

~

696

lEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 7, JULY 1992

h = l cm

a=5 cm

N=10

*[I + y ] d l z

where p and R are given by (26) and (27), respectively. As previously, the total gradients aE,/ax and aE,/ay can be obtained by numerically integrating the above expressions along the length of the coil. It is interesting to notice from (25), (28), and (29) that if both z and z, are changed by the same amount (so that z - zo remains contant), both the electric field and its derivatives remain unchanged. In other words, the electric field and its derivatives are independent of the location of the air-tissue interface. RESULTS Stimulating coils with square, circular, double-square, double-circular, and quadruple-square shapes were analyzed. Coils oriented parallel, perpendicular, and at an angle to the air-tissue interface were considered. It was found that, for all coil shapes and orientations analyzed, the spatial derivatives of the electric field components in two orthogonal directions are equal in magnitude, but opposite in sign, i.e., aEX/ax = -aE,/ay forz e 0. (This -+ results from V E = 0.) Because of this property, we present graphs of only one spatial derivative (aE,/ax) in this paper. It should be noted that aE,/ax is the stimulation parameter for nerves parallel to x-axis, and aE,/ay is the stimulation parameter for nerves parallel to y-axis. For convenience and to facilitate scaling, all the spatial derivatives are multiplied by a characteristic dimension of the coil a (length of a square, diameter of a circle, etc.). It can be shown that if all the dimensions are scaled by the same factor, the value of a(dE,/ax) remains unchanged. For example, the value of a(dE,/ax) at the point (2, 3 , - 1) due to a 5-cm diameter coil placed 1 cm above the interface is equal to the value of a(dE,/dx) at the point (4,6 , -2) due to a 10-cm diameter coil placed 2 cm above the interface, provided that both coils carry the same current pulse. All the results presented in this paper are for coils placed symmetrically about the x-axis. The spatial derivative is skew-symmetric (symmetric with a change of sign), except for quadruple coils where it is symmetric about the x-axis, and therefore results are presented only for the y > 0 region. Fig. 2 shows the spatial derivative aEx/ax produced by a square coil parallel to the interface. Results for a parallel circular coil are shown in Fig. 3. It can be seen, that these single coils produce two positive peaks and two negative peaks of the spatial derivative. Nerves parallel to x-axis can be stimulated at the negative peaks. Since

T O 230 x

0 0

-10.0

-5.0

0.0

5.0

.o

x (cm) Fig. 2. 3-D and contour plots of a ( a E , / a x ) on a plane t-cm below a 5 X 5 cm square coil parallel to the interface. The 10-turn coil cames a current rising at 100 A/ps. I a(aE,/ax) I has four peaks equal to 374 V / m , at ( + 2 . 5 cm, + 2 . 5 cm). Contour interval is equal to 10%of this peak value. h=l cm a=5 cm

N=10

To L'

fl

>

0 0

-10.0

-5.0

0.0

5.0

).O

x (cm) Fig. 3. 3-D and contour plots of u(aE,/ax) on a plane 1-cm below a 5-cm diameter circular coil parallel to the interface. The 10-turn coil carries a current rising at 100 A/ps. 1 a(aE,/ax) I has four peaks equal to 334 V / m , at ( k 2 . 1 5 cm, k2.15 cm). Contour interval is equal to 10% of the peak value.

ESSELLE AND STUCHLY: NEURAL STIMULATION WITH MAGNETIC FIELDS

691

0.30

-

0 Circular C o i l

0.25 A Double-Square Coil

c

U

lU

z

0.20

s ?+ v

z

0.1 5

-

< X

0.1 0

w

cg -

- 0.05 0

0.00 0.0

0.2

0.4

0.6

0.8

1 .O

h/a Fig. 4. The normalized maximum spatial derivative versus h / u where h is the height of the coil above the nerve, for square, circular and doublesquare coils parallel to the interface. For single coils, N is the number of turns. Double-square coil is composed of two square coils placed side by side, each having N / 2 turns.

Fig. 5. The location ( k X m , fY,) of the peaks of 1 a(aE,/ax) I as a function of h / a . All coils are parallel to the interface.

0.6

aEy/ay = -aEx/ax, nerves parallel to y-axis can be stimulated at the positive peaks of aE,/ax. The locations of positive and negative peaks are interchanged if the current direction is reversed, or if the current decreases with time (i.e., d / d t < 0). Comparing data in Figs. 2 and 3, one may conclude that a square coil produces a smaller stimulation area than a circular coil of comparable size. The peak value of the spatial derivative magnitude I dEx/ax Imax decreases with the depth in tissue h, as shown in Fig. 4. This normalized graph can be used to obtain data for parallel square, circular and double-square coils of any size. When h

Neural stimulation with magnetic fields: analysis of induced electric fields.

Spatial distributions of the derivative of the electric field induced in a planar semi-infinite tissue model by various current-carrying coils and the...
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