A mathematical model is presented that predicts the electric field induced in the arm during magnetic stimulation of a peripheral nerve. The arm is represented as a homogeneous, cylindrical volume conductor. The electric field arises from two sources: the time-varying magnetic field and the accumulation of charge on the arm surface. In magnetic stimulation both of these contributions are significant. The magnitude of the electric field is greatest near the surface of the arm, and is well localized. Various coil orientations are examined; the smallest electric fields are induced when the coil is perpendicular to the arm surface, the largest when the coil is parallel. These results are consistent with many experimental observations in the literature, and aid in the basic understanding of magnetic stimulation of the peripheral nervous system. Key words: magnetic stimulation charge accumulation electric field MUSCLE & NERVE 13:734-741 1990

A THEORETICAL CALCULATION OF THE ELECTRIC FIELD INDUCED BY MAGNETIC STIMULATION OF A PERIPHERAL NERVE BRADLEY J. ROTH, PhD, LEONARD0 G. COHEN, MD, MARK HALLETT, MD, WALTER FRIAUF, and PETER J. BASSER, PhD

I n the last decade magnetic stimulation has emerged as a useful method for stimulating nerves. 5,18 A capacitor is discharged through a coil, and the resulting current pulse produces a magnetic field. Since the magnetic field is changing with time, an electric field is also induced. This techni ue has been used to map the motor cortex, to study central motor conduction in patients with multiple s c l e r ~ s i s ~and ” ~ degenerative ataxic disorder^,^ and to stimulate peri heral nerves in routine nerve conduction studies. Magnetic stimulation is noninvasive and less painful than stimulation using electrodes on the skin. In clinical applications of magnetic stimulation, the nerve lies within the human body, a conductor, while the coil is held in air, an insulator. T h e magnetic field is not changed by the presence of

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From the Biomedical Engineering and Instrumentation Branch. Division of Research Services (Drs. Roth and Basser, and Mr. Friauf). and Human Cortical Physiology Unit, Human Motor Control Section, Medical Neurology Branch, National Institute of Neurological Disorders and Stroke, (Drs. Cohen and Hallett) National Institutes of Health, Bethesda, Maryland. Acknowledgments: We thank Dr Ronald Levin for writing the software to make the contour plots. Address reprint requests to Dr. Roth, Building 13, Room 3W13, National Institutes of Health, Bethesda. MD 20892. Accepted for publication October 4, 1989 CCC 01 48-639X/90/080734-08 $04.00 Not subject to copyright within the United States. Published by John Wiley & Sons, Inc.

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tissue, but the electric field is changed. In this paper we calculate the electric field in a cylindrical volume conductor, taking into account the tissueair boundary. Our goal is to predict the magnitude and spatial distribution of the electric field induced in the arm during magnetic stimulation of a peripheral nerve. The electric field induced in the human body has been studied during investigations of the biophysical effects of electromagnetic fields,’ 1323s25 such as those occurring near high voltage power lines, l 4 during hyperthermia therapy for cancer, 10,13 or from pulsed magnetic field therapy for bone healing. l 5 S ~ i e g e and l ~ ~ Elliott et a18 calculated the electric field induced in a conducting sphere and cylinder by a uniform time-dependent magnetic field. In simulations of power deposition during radio frequency hyperthermia therapy, 2dimensional models of the body coupled to a uniform magnetic field were considered. During magnetic stimulation of a peripheral nerve, the fundamental interaction between the electromagnetic field and the tissue is identical to that discussed in these previous studies. However, the tissue and coil geometries are different, and in general the magnetic and electric fields produced during magnetic stimulation are quite sensitive to the eoil geometry and its position and orientation relative to the arm.

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METHODS Physical Description of the Effect of an Air-Tissue Surface on the Electric Field. There are two

sources of electric fields: a charge distribution and a time-varying magnetic field. Clearly, during magnetic stimulation a time-dependent magnetic field induces an electric field by electromagnetic induction (Faraday’s However, the role a charge distribution plays in magnetic stimulation is not obvious. To understand intuitively how a charge distribution arises, consider a circular coil in air oriented perpendicular to a flat tissue surface. The electric field lines produced by the changing magnetic field are circles pointing in the opposite direction to the current in the coil when the current is increasing (Fig. la). Note that these field lines cross the air-tissue surface. If this field is imposed on the tissue, then charge flows along the electric field lines until it reaches the tissue surface. Since the charge cannot flow into the air, which is an insulator, it accumulates on the tissueair boundary; positive charge to the right of the coil, negative charge to the left. This surface charge distribution creates its own electric field (Fig. lb). Charge accumulates until the component of the electric field perpendicular to the surface (the normal component) produced by charge

is equal and opposite to the normal component of the electric field produced by the time-changing magnetic field. The total electric field is the sum of the electric fields due to charge and electromagnetic induction. The accumulation of charge on the tissue surface is similar to the charging of a capacitor, with a time constant that depends on the dielectric properties of the tissue. For most tissues this time constant is small relative to the rise time of the coil current (typically 100 Fsec”). We assume that charge accumulation is so rapid that it occurs instantaneously. Under this quasistatic approximation” the lines of current form closed loops within the tissue (these current loops produce their own magnetic field, but this field is negligible compared to the magnetic field produced by the current in the coil). If we rotate the coil so that it lies parallel to the tissue- air boundary, then the electric field induced by the magnetic field never crosses the tissue surface, and no charge accumulation occurs. A small circular roil located near and parallel to the surface of the arm has approximately this geometry. However, if the orientation of the coil is changed, or if the tissue surface is not flat, then charge on the surface may become important. Mathematical Calculation of the Effect of an AirTissue Surface on the Electric Fieid. T o calculate

the electric field in the tissue, we first review some electromagnetic theory. The electric field vector, E , and the magnetic field vector, B , can be determined from a scalar potential, @, and a vector potential. A l g

b)

0

The symbols V and V x are the gradient and the curl operators, respectively, and represent two types of differentiation with respect to position that arise when working with vectors.22 T h e symbol dlat indicates the partial derivative with respect to time. The scalar potential, @, is the same as the voltage, V , familiar to electrophysiologists, and the source of the scalar potential is charge. The vector potential, A , is less familiar. However it is often encountered in, and is essential to, electromagnetic theory.Ig The source of the vector potential is current. It is not surprising that the magnetic field can be obtained from the vector potential and the electric field is due, in part, to the scalar

.r Tissue

FIGURE 1. The electric field produced by a coil near tissue. This figure shows the qualitative behavior of the electric field lines, but is not intended to be quantitatively correct. (a) The electric field produced by the coil due to electromagnetic induction, €*, and (b) the electric field produced by charge on the tissue surface, 4.

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potential; it is well known that current produces a magnetic field while charge produces an electric field. The contribution of the vector potential to the electric field represents electromagnetic induction. The vector potential only contributes to the electric field if A, and therefore the current and magnetic field, is changing with time. Under most situations of biological interest the contribution of the vector potential to the electric field is negligible, so that E is equal to the voltage gradient. However when stimulating nerves with a timedependent current through a coil, in which a kiloAmp of current is produced in 100 psec,” the vector potential term must be considered. Under these conditions, Ohm’s law is valid in the form J = uE, where u is the conductivity of the tissue and J is the current density. However, Ohm’s law written as V = IR does not correctly describe current flow in biological tissue. T h e electric field is not simply the gradient of the voltage; both the scalar and vector potentials contribute to E . When there is little or no charge accumulation on the tissue surface (eg, for a small circular coil lying near and parallel to the arm surface), the concept of a voltage induced in the tissue is of no use, and speaking in terms of currents in the tissue as being due to voltage drops is confusing and incorrect.*l Consider a coil with N turns carrying a current, Z(t), which is varying with time, t. The vector potential, A(r,t), is given by’9

~ V sec/A m), r is where p(Jis a constant ( 4 lo-’ the position where the vector potential is calculated (the field point), r‘ is the position of a small segment of the coil (the source point), and dl’ is a vector pointing tangent to the coil (Fig. 2a). We sum the contributions from every segment of the coil using the rules of vector addition, with each segment having a weight that depends on the distance between the field and source points, lr-r‘l. The scalar potential, @(r,t),can be related to the surface char e density, s(r’,t), by an equation similar to Eq. 2 18

(3)

b)

FIGURE 2. (a) The geometry for calculating the vector potential from the current in Eq. 2; r is a vector to the point where the vector potential is calculated, r‘ is a vector to a position on the coil, and dl’ is a vector tangent to the coil. (b) The geometry for calculating the scalar potential from the surface charge in Eq. 3; r is a vector to the position where the scalar potential is calculated, r’ is a vector to a position on the air-tissue surface, n is a unit vector perpendicularto the tissue surface, and da‘ is an element of area.

small element of area on the tissue-air surface (Fig. 2b). In an inhomogeneous body, charge can build up on any surface between tissues with different conductivities (eg, bone and muscle). In this article, we consider only homogeneous isotropic tissues and limit our investigation to the effect of the tissue surface on the electric field. Using Equations 1, 2, and 3, (and following Reitz et allg to compute the gradient of l/lr-r’l), we write the total electric field in the tissue, E(r,t), as the sum of two terms: EA due to the current integrated over the coil, and E , due to the charge integrated over the tissue surface, where

and

where E , ~is a constant (8.854 x C/V m), K is the dielectric constant of the tissue, and du‘ is a

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We discuss two examples in which different approaches are used to calculate the electric field. First, we consider a coil near a tissue with a flat surface, which would correspond to magnetic stimulation of a peripheral nerve using a coil that has a radius much smaller than the radius of the arm. In this case we relate the surface charge to the normal component of the electric field due to induction, E A , and then use Eq. 5 to determine E,. Second, we consider a coil near a cylindrical volume conductor representing the arm, corresponding to stimulation of a peripheral nerve using a large circular coil. In this case there is no simple way to determine the charge distribution so we cannot use Eq. 5 to calculate E,. Instead we use a boundary value method, described below. The simplest example of magnetic stimulation is a circular coil near a flat tissue surface. This example can model magnetic stimulation of any tissue when the radius of the coil is small compared to the curvature of the body; for instance using a small coil for magnetic stimulation of a peripheral nerve or the cortex. The normal component of the electric field produced by a surface charge, s(r',t), over a plane surl9 Recall that the charge accumuface is s(T',~)/~KE,. lated on the air-tissue surface is just enough that the normal component of the electric field due to charge is equal and opposite to the normal component of the electric field due to the timedependent magnetic field. If n is a unit vector perpendicular to the surface pointing outward from the tissue (Fig. 2b), then the surface charge is A Flat, Planar Surface.

indicates the dot product of where the symbol the vectors (the projection of EA onto the direction of n). We can calculate the total electric field by first calculating EAfrom current in the coil (Eq. 4), then use E A to determine the charge distribution (Eq. 6), and finally compute E, from the charge (Eq. 5). The dielectric constant, K, in Eqs. 5 and 6 cancels, and the conductivity of the tissue never enters our equations, so the electric field is independent of the tissue properties and only depends on the geometry of the tissue surface and the coil. This calculation was implemented on a computer using the technique described by Cohen et a1.6 In brief, Eq. 4 was solved for a linear segment of wire, the coil was approximated by a polygon, and the integral in Eq. 4 was replaced by a sum ''*I'

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over the sides of the polygon. We used a 50-mm diameter circular coil, which we approximated as a 64-sided polygon. We assumed that the coil consisted of 10 turns of wire, and in each turn the current was changing at a rate of 100 A/psec. The surface charge density was calculated at 4225 points 2 mm apart on a square grid over the tissue surface. The integral in Eq. 5 was approximated by a summation over point charges at each node in this grid, with the charge $ven as the charge density times an area of 4 mm . A common application of ma netic stimulation is to excite periphera1 nerve^.^ p16,18We model the arm as a cylindrical volume conductor of radius a. For this case, we cannot use Eq. 5 to solve for E, because we do not know the surface charge distribution, To solve for the electric field, we recall that at the tissue surface the normal components of the electric fields due to charge and electromagnetic induction are equal and opposite. The electric field due to charge is the negative gradient of the scalar potential, @, so at the surface of the arm we have the boundary condition A Cylindrical Volume Conductor.

5

(7) where alan denotes the spatial derivative in the direction perpendicular to the surface (a@/& = V

A theoretical calculation of the electric field induced by magnetic stimulation of a peripheral nerve.

A mathematical model is presented that predicts the electric field induced in the arm during magnetic stimulation of a peripheral nerve. The arm is re...
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