Page 243 - Computational Modeling in Biomedical Engineering and Medical Physics

232 Computational Modeling in Biomedical Engineering and Medical Physics

where J 5 σE 1 jωD, the magnetic flux law, B 5 r 3 A (it introduces the magnetic
vector field, A), Faraday’s law, E 52 jωA, and the constitutive (material) laws for the
electric field, D 5 εE, and the magnetic field, B 5 μH, Mocanu (1982) yields the
Helmholtz PDE for the magnetic field strength, H (underscore denotes complex
quantity)
σ
2
2
2
ΔH 5 γ H; γ 5 ω με ; ε 5 ε 1 : ð7:11Þ

jω
p ffiffiffiffiffiffiffiffi
Here j 5 2 1, ω 5 2πf is the angular velocity, ε is the complex permittivity, ε is

the electric permittivity, σ is the electrical conductivity, μ is the magnetic permeability
(free space), and B is the magnetic flux density. The electric conduction, σE,and displace-
ment, jωD, currents are the sources of the magnetic field. Magnetic insulation boundary
condition closes the model (nUB 5 0, where n is the outward normal to the boundary).
The electromagnetic field produced by the coils diffuses through the body and
causes the excitation of neurons (Ugawa et al., 1989; Maccabee et al., 1991). The
induced electric field depends on the coil type (shape), its position, and, of course, on
the electric properties of the tissues. The working frequency is f 5 100 Hz, and
Table 7.1 lists the properties that are used (Gabriel, 1996; Andreuccetti et al., 2020).
Duetothe limitations in thehardwareresources availableatthe time (Baerov et al.,
2019), the thorax is presented as an ensemble of equivalent, homogenized subdomain. Its
electrical permittivity and conductivity have average, volume-weighted values over the ana-
tomic subdomains. The spinal cord and spinal nerves (Liu et al., 2015) are homogenized
too, their properties have average, volume-weighted values of the nerve, white matter, gray
matter, and cerebrospinal fluid, respectively (Table 7.2).

Table 7.1 Volume-weighted average electrical properties for different anatomical regions at
f 5 100 Hz.

Tissue Relative permittivity ( ) Electrical conductivity (S/m)
Vertebrae 5.85 3 10 3 2.01 3 10 22
Intervertebral disc 6.1 3 10 1 8.30 3 10 21
6 21
Spinal cord and spinal nerves 1.51 3 10 5.43 3 10
Thorax, average 4 3 10 3 3.3 3 10 21

Table 7.2 Electrical properties for tissues used in numerical simulations.
Tissue Relative permittivity ( ) Relative permittivity ( )
Nerve 4.66 3 10 5 2.8 3 10 22
White matter 1.67 3 10 6 5.81 3 10 22
Gray matter 3.91 3 10 6 8.9 3 10 22
2
Cerebrospinal fluid 1.09 3 10 2