22.1 INTRODUCTION
In Chapter 12 it was pointed out that the origin of the biomagnetic field is the electric activity of biological tissue. This bioelectric activity produces an electric current in the volume conductor which induces the biomagnetic field. This correlation between the bioelectric and biomagnetic phenomena is, of course, not limited to the generation of the bioelectric and biomagnetic fields by the same bioelectric sources. This correlation also arises in the stimulation of biological tissue.
Corresponding to a magnetic field
and other variables are as in Equation 22.1.
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 stimulation methods excite the membrane with electric current. The former does that directly, but the latter does it with the electric current which is induced within the volume conductor by the time-varying applied magnetic field.
The reason for using a time-varying magnetic field to induce the stimulating current is, on the one hand, 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 possible to avoid a high density of stimulating current at the scalp in stimulating the central nervous system and thus avoid pain sensation. Also, no physical contact of the stimulating coil and the target tissue is required, unlike with electric stimulation.
The first documents on magnetic stimulation described the stimulation of the retina by Jacques d'Arsonval (1896) and Silvanus P. Thompson (1910). The retina is known to be very sensitive to stimulation by induced currents, and field strengths as low as 10 mT rms at 20 Hz will cause a stimulation (Lövsund, Öberg, 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 stimulation in animals and humans. Magnetic stimulation of nerve tissue was also demonstrated by Öberg (1973). The first successful magnetic stimulation of superficial nerves was reported by Polson et al. in 1982 (Polson, Barker, and Freeston, 1982).
Transcranial stimulation of the motor cortex is the most interesting application of magnetic stimulation because the magnetic field (unlike the electric current) penetrates through the skull without attenuation. The first transcranial stimulation of the central nervous system was achieved in 1985 (Barker and Freeston, 1985; Barker, Freeston, Jalinous, Merton, and Morton, 1985; Barker, Jalinous, and Freeston, 1985). A more complete history of magnetic stimulation may be found from a review article of Geddes (1991).
(22.1)
where 
= electromotive force (emf) [V] F = magnetic flux [Wb = Vs] t = time [s] 
the flux 
, linking the circuit is given by 
, where the integral is taken over any surface whose periphery is the circuit loop.
If the flux is due to a coil's own current I, the flux is defined as: F = LI, where L is the inductance of the coil and the emf can be written
(22.2)
where L = inductance of the coil [H =Wb/A = Vs/A] I = current in the coil [A] The magnitude of induced emf is proportional to the rate of change of current, dI/dt. The coefficient 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 thyristor) and minimal wiring length. Inductance L is determined by the geometry and constitutive property of the medium. The principal factors 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 physiological magnetic stimulation, the inductance may be calculated from the following equations:
Multiple-Layer Cylinder Coil
The inductance of a multiple-layer cylinder coil (Figure 22.2A) is:
 | (22.3) |
| where | L | = inductance of the coil [H] |
| µ | = permeability of the coil core [Vs/Am] | |
| N | = number of turns on the coil | |
| r | = coil radius [m] | |
| l | = coil length [m] | |
| s | = coil width [m] |
Flat Multiple-Layer Disk Coil
The inductance of a flat multiple-layer disk coil (Figure 22.2B) is
where N, r, and s are the same as in the equation above.
Long Single-Layer Cylinder Coil
The inductance of a long single-layer cylinder coil (Figure 22.2C) is
where N, r, and l are again the same as in the equation above.
The following example is given of the electric parameters of a multiple-layer cylinder coil (Rossi et al., 1987): A coil having 19 turns of 2.5 mm² copper wound in three layers has physical dimensions of r = 18 mm, l = 22 mm, and s = 6 mm. The resistance and the inductance of the coil were measured to be 14 mW and 169 µH, respectively.
(22.4) A coil having 10 turns of 2.5 mm² copper wire in one layer has physical dimensions of r = 14 ... 36 mm. The resistance and the inductance of the coil had the measured values of 10 mW and 9.67 µH, respectively.
(22.5)
A) Multiple-layer cylinder coil.
B) Flat multiple-layer disk coil.
C) Long single-layer cylinder coil.
 | (22.6) |
Thus
 | (22.7) |
| where | W | = energy required to stimulate tissue |
| B | = magnetic flux density | |
| E | = electric field | |
| t | = pulse duration |
The effectiveness of the stimulator with respect to energy transfer is proportional to the square root 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 with electrical stimulation indicate that the time constant of this leaky capacitor is of the order of 150-300 µs. Therefore, for effective stimulation 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.
 | (22.8) |
| where | Vm | = transmembrane voltage |
| l | = membrane space constant | |
| t | = membrane time constant | |
| x | = orientation of the fiber | |
| Ex | = x component 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 maximum 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 = 0, y = 0, while the fiber is parallel to the x axis and at y = 2.5 cm and z = 1.0 cm. They consider a coil with radius of 2.5 cm wound from 30 turns of wire of 1.0 mm radius. The coil, located at a distance of 1.0 cm from the fiber, is a constituent of an RLC circuit; and the time variation is that resulting from a voltage step input. Assuming C = 200 µF and R = 3.0W, an overdamped current waveform results. From the resulting stimulation it is found that excitation results at x = 2.0 cm (or -2.0 cm, depending on the direction of the magnetic field) which corresponds to the position of maximum 
Ex /x. The threshold applied voltage for excitation is determined to be 30 V. (This results in a peak coil current of around 10 A.) These design 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 µs) need only half the stored energy of those with longer risetimes (> 180 µs). The use of a variable field risetime also enables membrane time constant to be measured and this may contain useful diagnostic information.
