Page 234 - Computational Modeling in Biomedical Engineering and Medical Physics

Magnetic stimulation and therapy 223

1=ðg m r i 1 r e ÞÞ 5
and time constants of the membrane λ 5 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r m =ðr i 1 r e Þ and
ð
τ 5 c m =g m 5 r m c m are also identified in the equation; the parameters λ and τ depend
on cell dimension (cylinder radius) and dielectric properties.
The right term in Eq. (7.3b) evidence the rise of the transmembrane voltage, from
2
rest to the depolarization threshold, Δu m 5 r e λ i s , where AF 5 r e i s is evidenced as the
AF for electrical stimulation.
Experiments performed by Sudhansu (1990) proved that Δu m 10 mV for a suc-
cessful depolarization of axons. Further processing of AF and in connection with the
cable model (Fig. 7.1) one could reach the specific expression of AF for MS.

1 r e Δx
AF 5 r e i s 5 i s Δxð Þ r e ΔxÞ 5 ½ i e x 1 ΔxÞ 2 i e xðÞ 2 i m ΔxŠ; ð7:4aÞ
ð
ð
2 2
ð ΔxÞ ð ΔxÞ
where i m Δx ,, i s Δx looks like a reasonable approximation for the membrane under the
depolarization threshold (Morega, 1999); under this assumption i m Δx is neglected, yielding
! !
2 3
Δx Δx
V e x 1 2 V e x 1 Δxð Þ V e xðÞ 2 V e x 1
r e Δx r e Δx 6 2 2 7
6
7
AF 5 2 ½ i e x 1 ΔxÞ 2 i e xðފ 5 6 2 7
ð
ð ΔxÞ ð 2 6 r e Δx r e Δx 7
ΔxÞ 4
5
2 2
ð7:4bÞ
!
Δx
V e x 1 ΔxÞ 2 2V e x 1 1 V e xðÞ
ð
2 @ !
52 2 5 2 2 @V e 5 2 @E x ;
ð ΔxÞ 2 @x @x @x
AF 5 2ð@E x =@xÞ shows that in the MS of a cable-like fiber, the active physical quantity is
the spatial derivative of the induced electric field along the fiber (in fact, the tangential com-
ponent of the electric field strength relative to the fiber direction is derived concerning the
same space coordinate as its direction). In these circumstances, a computational model for
magnetic stimulation should focus not only on the assessment of the induced electric field
but also especially on the derivative of its strength along the fiber.
A computational model for the induced electric field and the
activating function
Finding the distribution of the electric field inside a semiinfinite dispersive space (which
could be assimilated to an anatomical structure), as the result of electromagnetic induction
from an external current-carrying coil is the objective of this subsection. The computa-
tional model is based on analytical methods applied in electromagnetism and its impor-
tance lies in highlighting some physical aspects that are fundamental for medical
procedures based on the generation of the electric field by electromagnetic induction. It
was introduced and discussed in Esselle and Stuchly (1992) for applications in PMS and
further used for the assessment of the electric field and AF distribution and optimization of