Page 236 - Computational Modeling in Biomedical Engineering and Medical Physics

Magnetic stimulation and therapy 225

where A 1 is the magnetic vector potential of the incident magnetic field and V 2 is the
electric potential produced by charge separation at the air tissue interface; the indices
are further removed for simplification and Eq. (7.5) becomes

μ di
dE 5 dE x i 1 dE y j 1 dE z k 52 0 dl 2 d gradVÞ: ð7:6Þ
ð
4πR dt
The scalar electric potential could be derived as the solution of the Laplace equation in the
conductive half-space by the separation of variables, as presented in Esselle and Stuchly (1992);
the complete solution is expressed with Bessel functions. The term d gradVÞ needed for the
ð
complete computation of Eq. (7.6) is fully derived in Morega (1999) and the result is as follows

μ di dl x ð x x 0 Þdl z z z 0 dl y ð y y 0 Þdl z z z 0
dE 52 0 1 1 1 i 1 1 1 1 j : ð7:7Þ
4π dt R ρ 2 R R ρ 2 R
As one could observe, the electric field strength has a null component along z-axis
direction, normal to the skin surface. In Fig. 7.2, the long cylindrical fiber is shown in
the direction of the x-axis, which implies that only the x-component in Eq. (7.7) is
important for the evaluation of the AF, that is, the expression of the field derivative
@E x =@x is only needed and AF results by the integration of its expression along the
H
coil contour, AF 5 d @E x =@x , where
Γ
! ( " ! # )
2 2 2
ð
ð
@E x μ di ð x x 0 Þ ð x x 0 Þ 2 y2y 0 Þ z z 0 ð x x 0 Þ z z 0 Þ
0
d 5 dl x 1 1 1 1 dl z : ð7:8Þ
2
@x 4π dt R 3 ρ 4 R ρ R 3
Equation (7.8) is built as the differential of the electric field component, derived
from Eq. (7.7) on the same x-direction.
The activating function produced by circular coils
Coils commonly used in PMS are wound with circular turns, either concentrated or distrib-
uted in various modes, such as double coil (in the shape of figure eight) or quadruple coil
(like a flower) with the turns in a parallel or inclined positions relative to the body surface,
like a butterfly or o slinky coil (e.g., Ren et al., 1995). To compute AF as shown here,
Eq. (7.8) must be integrated on the contour of the stimulation coil (all ampere-turns) and
that respective contour must be described by a suitable geometry capable of providing a
convenient expression for the analytical calculation, or the integration must be approached
numerically. In Morega (2000) change of variables is applied for different geometries and
positions of coils with circular turns. According to Fig. 7.3, the elementary turn is circular,
of radius r; its basic position, shown in Fig. 7.3A,iscoplanarwiththe referenceplane (xOy,
z 5 0), tangential to the axes. The turn can be rotated at an angle α, keeping point A
fixed, as shown in Fig. 7.3B; “d” is a reference axis attached to the moving turn.