22.1 INTRODUCTION

In Chapter 12 it was pointed out that the origin of the biomagnetic fiel= d is the electric activity of biological tissue. This bioelectric activi= ty produces an electric current in the volume conductor which induces th= e biomagnetic field. This correlation between the bioelectric and biomag= netic phenomena is, of course, not limited to the generation of the bioe= lectric and biomagnetic fields by the same bioelectric sources. This cor= relation also arises in the stimulation of biological tissue.

Magnetic stimulation is a = method for stimulating excitable tissue with an electric current induced= by an external time-varying magnetic field. It is important to note= here that, as in the electric and magnetic detection of the bioelectric= activity of excitable tissues, both the electric and the magnetic stimu= lation methods excite the membrane with electric current. The for= mer does that directly, but the latter does it with the electric current= which is induced within the volume conductor by the time-varying applie= d magnetic field. =

The reason for using a time-v= arying magnetic field to induce the stimulating current is, on the one h= and, the different distribution of stimulating current and, on the other= hand, the fact that the magnetic field penetrates unattenuated through = such regions as the electrically insulating skull. This makes it possibl= e to avoid a high density of stimulating current at the scalp in stimula= ting the central nervous system and thus avoid pain sensation. Also, no = physical contact of the stimulating coil and the target tissue is requir= ed, unlike with electric stimulation.

The first documents on magnet= ic stimulation described the stimulation of the retina by Jacques d'Arso= nval (1896) and Silvanus P. Thompson (1910). The retina is known to be v= ery sensitive to stimulation by induced currents, and field strengths as= low as 10 mT rms at 20 Hz will cause a stimulation (L=F6vsund, =D6berg,= and Nilsson, 1980).

From the pioneering works of = d'Arsonval and Thompson it took some time before the magnetic method was= applied to neuromuscular stimulation. Bickford and Fremming (1965) used= a damped 500 Hz sinusoidal magnetic field and demonstrated muscular sti= mulation in animals and humans. Magnetic stimulation of nerve tissue was= also demonstrated by =D6berg (1973). The first successful magnetic stim= ulation of superficial nerves was reported by Polson et al. in 1982 (Pol= son, Barker, and Freeston, 1982).

Transcranial stimulation of t= he motor cortex is the most interesting application of magnetic stimulat= ion because the magnetic field (unlike the electric current) penetrates = through the skull without attenuation. The first transcranial stimulatio= n of the central nervous system was achieved in 1985 (Barker and Freesto= n, 1985; Barker, Freeston, Jalinous, Merton, and Morton, 1985; Barker, J= alinous, and Freeston, 1985). A more complete history of magnetic stimul= ation may be found from a review article of Geddes (1991).

22.2 THE DESIGN OF STIMULATOR COILS

A magnetic stimulator includes a coil that is placed on the surface of t= he skin. To induce a current into the underlying tissue, a strong and ra= pidly changing magnetic field must be generated by the coil. In practice= , this is generated by first charging a large capacitor to a high voltag= e and then discharging it with a thyristor switch through a coil. The pr= inciple of a magnetic stimulator is illustrated in Figure 22.1.

The Faraday-Henry law states = that if an electric conductor, which forms a closed circuit, is linked b= y a time-varying magnetic flux F, a current is observed in the ci= rcuit. This current is due to the electromotive force (emf) induced by t= he time-varying flux. The magnitude of emf depends on the rate of change= of the magnetic flux dF/dt. The direction of emf is such that th= e time-varying magnetic field that results from it is always opposite to= that of dF/dt; therefore,

(22.1)

where =3D electromotive force (emf) [V]

F =3D magnetic flux [Wb =3D Vs]

t =3D time [s]

Corresponding to a magnetic field = the flux , linking the circuit is= given by , = where the integral is taken over any surface whose periphery is the circ= uit loop.
If the flux is due to a coil'= s own current I, the flux is defined as: F =3D LI, where <= i>L is the inductance of the coil and the emf can be written

(22.2)

where L =3D inductance= of the coil [H =3DWb/A =3D Vs/A]

I =3D current in the coil [A]

and other variables are as in Equation 22.1. =
The magnitude of induced emf = is proportional to the rate of change of current, dI/dt. The coef= ficient of proportionality is the inductance L. The term dI/dt= depends on the speed with which the capacitors are discharged; the = latter is increased by use of a fast solid-state switch (i.e., fast thyr= istor) and minimal wiring length. Inductance L is determined by t= he geometry and constitutive property of the medium. The principal facto= rs for the coil system are the shape of the coil, the number of turns on= the coil, and the permeability of the core. For typical coils used in p= hysiological magnetic stimulation, the inductance may be calculated from= the following equations:

Fig. 22.1

Multiple-Layer Cylinder Coil

The inductance of a multiple-layer cylinder coil (Figure 22.2A) is:

(22.3)

where L =3D inductance= of the coil [H]

=B5 =3D permeability of the coil core [V= s/Am]

N =3D number of turns on the coil=

r =3D coil radius [m]

l =3D coil length [m]

s =3D coil width [m]

The following example is give= n of the electric parameters of a multiple-layer cylinder coil (Rossi et= al., 1987): A coil having 19 turns of 2.5 mm=B2 copper wound in three l= ayers has physical dimensions of r =3D 18 mm, l =3D 22 mm,= and s =3D 6 mm. The resistance and the inductance of the coil we= re measured to be 14 mΩ and 169 =B5H, respectively.

Flat Multiple-Layer Disk Coil

The inductance of a flat multiple-layer disk coil (Figure 22.2B) is

(22.4)

where N, r, and s are the same as in the equation above.

A coil having 10 turns of 2.5= mm=B2 copper wire in one layer has physical dimensions of r =3D 14 ... = 36 mm. The resistance and the inductance of the coil had the measured va= lues of 10 mΩ and 9.67 =B5H, respectively.

Long Single-Layer Cylinder Coil

The inductance of a long single-layer cylinder coil (Figure 22.2C) is

(22.5)

where N, r, and l are again the same as in the equation ab= ove.

Fig. 22.2

22.3 CURRENT DISTRIBUTION IN MAGNETIC STIMULATION

The magnetic permeability of biological tissue is approximately that of = a vacuum. Therefore the tissue does not have any noticeable effect on th= e magnetic field itself. The rapidly changing field of the magnetic impu= lse induces electric current in the tissue, which produces the stimulati= on.

Owing to the reciprocity t= heorem, the current density distribution of a magnetic stimulator is= the same as the sensitivity distribution of such a magnetic detector ha= ving a similar construction. (Similarly, this is, of course, true for el= ectric stimulators and detectors as well (Malmivuo, 1992a,b).) Note that= in the lead field theory, the reciprocal energization equals the applic= ation of stimulating energy. The distribution of the current density in = magnetic stimulation may be calculated using the method introduced by Ma= lmivuo (1976) and later applied for the MEG (Malmivuo, 1980). As mention= ed in Section 14.3, there are also other methods for calculating the sen= sitivity distribution of MEG detectors. They give accurate results in si= tuations having less symmetry and are therefore more complicated and, un= fortunately, less illustrative (Durand, Ferguson, and Dalbasti, 1992; Ea= ton, 1992; Esselle and Stuchly, 1992).

Single Coil

The current distribution of a single coil, producing a dipolar field, wa= s presented earlier in this book in Sections 12.11 and 14.2. The stimula= tion energy distribution can be readily seen in the form of vector field= s from Figure 14.2 and is not repeated here. Figure 22.3 illustrates the= iso-intensity lines and half-intensity volume for a coil with a 50 mm r= adius. The concepts of iso-intensity line and half-intensity volume are = reciprocal to the isosensitivity line and half-sensitivity volume, discu= ssed in Section 11.6.1. As discussed in Section 12.3.3, because of cylin= drical symmetry the iso-intensity lines coincide with the magnetic field= lines. The reader may again compare the effect of the coil radius on th= e distribution of the stimulus current by comparing Figures 22.3 and 14.= 3..

Fig. 22.3 Iso-intensity lines (dashed black), induced stimulation= currrent lines (solid blue) and half-intensity volume (green) for a sti= mulation coil with 50 mm radius. The distance of the coil plane from the= scalp is 10 mm.

Quadrupolar Coil Configuration

The coils can be equipped with cores of highly permeable material. One a= dvantage of this arrangement is that the magnetic field that is produced= is better focused in the desired location. Constructing the permeable c= ore in the form of the letter U results in the establishment of a= quadrupolar magnetic field source. With a quadrupolar magnetic field th= e stimulating electric current field in the tissue has a linear instead = of circular form. In some applications the result is more effective stim= ulation. On the other hand, a quadrupolar field decreases as a function = of distance faster than that of a dipolar coil. Therefore, the dipolar c= oil is more effective in stimulating objects that are located deeper wit= hin the tissue.

The first experiments with th= e quadrupolar magnetic field were made by Rossi et al. (1987). The distr= ibution of the stimulating electric current field of a figure of eight c= oil system was calculated by Malmivuo (1987). This method has subsequent= ly been applied to magnetic stimulation by many scientists (Ueno, Tashir= o, and Harada, 1988).

The sensitivity distributions= of dipolar and quadrupolar magnetometer coils were discussed in detail = in Section 14.2. The sensitivity distributions shown in Figures 14.4 and= 14.5 are similarly applicable to magnetic stimulation as well and are t= herefore not reproduced here.

22.4 STIMULUS PULSE

The experimental stimulator examined by Irwin et al. (1970) had a multic= apacitor construction equaling a capacitance of 4760 =B5F. This was char= ged to 90-260 V and then discharged with a bank of eight thyristors thro= ugh the stimulating coil. The result was a magnetic field pulse of 0.1-0= .2 T, 5 mm away from the coil. The length of the magnetic field pulse wa= s of the order of 150-300 =B5s. Today's commercial magnetic stimulators = generate magnetic energies of some 500 J and use typically 3 ... 5 kV to= drive the coil. Peak fields are typically 2 T, risetimes of order 100 =B5= s, and peak values of dB/dt =3D 5=D710⁴ T/s.

The energy required to stimul= ate tissue is proportional to the square of the corresponding magnetic f= ield. According to Faraday's induction law, this magnetic field is in tu= rn approximately proportional to the product of the electric field magni= tude and the pulse duration (Irwin et al., 1970):

(22.6)

Thus

(22.7)

<= /tr>

where W =3D energy req= uired to stimulate tissue

B =3D magnetic flux density
E =3D electric field

t =3D pulse duration

The effectiveness of the stim= ulator with respect to energy transfer is proportional to the square roo= t of the magnetic energy stored in the coil when the current in the coil= reaches its maximum value. A simple model of a nerve fiber is to regard= each node as a leaky capacitor that has to be charged. Measurements wit= h electrical stimulation indicate that the time constant of this leaky c= apacitor is of the order of 150-300 =B5s. Therefore, for effective stimu= lation the current pulse into the node should be shorter than this (Hess= , Mills, and Murray, 1987). For a short pulse in the coil less energy is= required, but obviously there is a lower limit too. =

22.5 ACTIVATION OF EXCITABLE TISSUE BY TIME-VARYING MAGNETIC FIELDS<= /h1> The actual stimulation of excitable tissue by a time-varying magnetic fi= eld results from the flow of induced current across membranes. Without s= uch flow a depolarization is not produced and excitation cannot result. = Unfortunately, one cannot examine this question in a general sense but r= ather must look at specific geometries and structures. To date this has = been done only for a single nerve fiber in a uniform conducting medium w= ith a stimulating coil whose plane is parallel to the fiber (Roth and Ba= sser, 1990).
In the model examined by Roth= and Basser, the nerve is assumed to be unmyelinated, infinite in extent= and lying in a uniform unbounded conducting medium, the membrane is des= cribed by Hodgkin-Huxley equations. The transmembrane voltage Vm is shown to satisfy the equation

(22.8)

where V_m =3D= transmembrane voltage

λ/td> =3D membrane space constant
τ =3D membrane time constant
x =3D orientation of the fiber

E_x =3D x compon= ent of the magnetically induced electric field (proportional to the x= component of induced current density).

It is interesting that it is the axial derivative of this field that is = the driving force for an induced voltage. For a uniform system in which = end effects can be ignored, excitation will arise near the site of maxim= um changing current and not maximum current itself.
In the example considered by = Roth and Basser the coil lies in the xy plane with its center at = x =3D 0, y =3D 0, while the fiber is parallel to the x<= /i> axis and at y =3D 2.5 cm and z =3D 1.0 cm. They consid= er a coil with radius of 2.5 cm wound from 30 turns of wire of 1.0 mm ra= dius. The coil, located at a distance of 1.0 cm from the fiber, is a con= stituent of an RLC circuit; and the time variation is that result= ing from a voltage step input. Assuming C =3D 200 =B5F and R =3D 3.0Ω, an overdamped current waveform results. From the resul= ting stimulation it is found that excitation results at x =3D 2.0= cm (or -2.0 cm, depending on the direction of the magnetic field) which= corresponds to the position of maximum E_x/x. The threshold applied voltage for excitation is determined t= o be 30 V. (This results in a peak coil current of around 10 A.) These d= esign conditions could be readily realized.
The effect of field risetime = on efficiency of stimulation has been quantified (Barker, Freeston, and = Garnham, 1990; Barker, Garnham, Freeston, 1991). Stimulators with short = risetimes (< 60 =B5s) need only half the stored energy of those with= longer risetimes (> 180 =B5s). The use of a variable field ris= etime also enables membrane time constant to be measured and this may co= ntain useful diagnostic information.

22.6 APPLICATION AREAS OF MAGNETIC STIMULATION OF NEURAL TISSUE

= Magnetic stimulation can be applied to nervous stimulation either centra= lly or peripherally.
The main benefit of magnetic = stimulation is that the stimulating current density is not concentrated = at the skin, as in electric stimulation, but is more equally distributed= within the tissue. This is true especially in transcranial magnetic sti= mulation of the brain, where the high electric resistivity of the skull = does not have any effect on the distribution of the stimulating current.= Therefore, magnetic stimulation does not produce painful sensations at = the skin, unlike stimulation of the motor cortex with electrodes on the = scalp (Mills, Murray, and Hess,1986; 1988; Rimpil=E4inen et al., 1990, 1= 991).
Another benefit of the magnet= ic stimulation method is that the stimulator does not have direct skin c= ontact. This is a benefit in the sterile operation theater environment.
As mentioned at the beginning= of this chapter, the first papers introducing the clinical application = of magnetic stimulation were published in 1985. Now magnetic stimulators= for clinical applications are produced by several manufacturers. It may= be predicted that the magnetic stimulation will be applied particularly= to the stimulation of cortical areas, because in electric stimulation i= t is difficult to produce concentrated stimulating current density distr= ibutions in the cortical region and to avoid high current densities on t= he scalp.
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where		=3D electromotive force (emf) [V]
	F	=3D magnetic flux [Wb =3D Vs]
	t	=3D time [s]

where	L	=3D inductance= of the coil [H =3DWb/A =3D Vs/A]
	I	=3D current in the coil [A]

where	L	=3D inductance= of the coil [H]
	=B5	=3D permeability of the coil core [V= s/Am]
	N	=3D number of turns on the coil=
	r	=3D coil radius [m]
	l	=3D coil length [m]
	s	=3D coil width [m]

where	W	=3D energy req= uired to stimulate tissue
	B	=3D magnetic flux density
	E	=3D electric field
	t	=3D pulse duration

where	V_m	=3D= transmembrane voltage
	λ/td>	=3D membrane space constant
	τ	=3D membrane time constant
	x	=3D orientation of the fiber
	E_x	=3D x compon= ent of the magnetically induced electric field (proportional to the x= component of induced current density).