Page 244 - Computational Modeling in Biomedical Engineering and Medical Physics

Magnetic stimulation and therapy 233

2
The applicator coils have 20 turns each, with 4 mm overall cross-sections areas.
The calculated inductance of such a coil is 10.89 μH, consistent with values reported
in the literature (e.g., Darabant et al., 2013). They are powered by ac voltage sources,
at 5 V (effective value), which keeps the electric current density for the coil conductor
(Copper) within safe limits.
The PDE Eq. (7.11) was integrated numerically using FEM, Comsol. Linear elements
for the magnetic field (vector potential) were used. The algebraic system was solved using a
Flexible Generalized Minimal Residual solver Morikuni et al. (2012),without
preconditioning.

Numerical simulation results
Fig. 7.7 singles out the lumbar spine and shows a slice distribution of the magnetic flux
density(left). When asinglecoilisused, B max B 3.5 mT, whereas for two, focused coils,
B max B 4.7 mT. As the mathematical model Eq. (7.11) is linear and the electric and mag-
netic properties are constant, the stimulation levels for other excitations are readily avail-
able, by scaling the solution with the scaling factor of the field source (ampere-turns, or
“total” current of the applicator). The cut-line in Fig. 7.7, middle is used to approximately
draw the magnetic (and electric) field along the nerve principal direction (its real trajectory
is much more complex but traceable though).
The magnetic flux density (Fig. 7.7, right) shows off a bell-shaped profile, with
maxima consistent with reported results in similar cases (e.g., Darabant et al., 2013).
The electric field strength (its effective value) is rendered in Fig. 7.8 using slices
selected to showcase the distribution in a vertical plane that passes through the nerve’s
central branch, and in horizontal planes elected to intercept the horizontal prolonga-
tions (xOy plane) of the nerve.

Figure 7.7 A slice distribution of the magnetic flux density through the spine (left) and the mag-
netic flux density (right) along a line traced through the spine (middle).