Page 34 - Computational Modeling in Biomedical Engineering and Medical Physics

20 Computational Modeling in Biomedical Engineering and Medical Physics

problem solution to avoid solving for a coupled pulsating hemodynamic flow and the associ-
ated transientheattransferproblem simultaneously [Chapter 8: Hyperthermia and Ablation
(Thermotherapy Methods)]. Finer details (e.g., the oscillations in the arterial blood tempera-
ture duetothe pulsatingflow) maybelostbut thedynamicsofthe tissue temperatureraise
is well characterized [Chapter 7: Magnetic Stimulation and Chapter 8: Hyperthermia and
Ablation (Thermotherapy Methods)].
Propagation time scales may occur in EMF radiation problems. For a system without
internal EMF sources, placed in an external EMF, e.g., produced by a radiant EMF
external power source, the time scales for EMF interactions may be those of the exter-
nal excitation. Assuming linear media, Maxwell’s laws for immobile media yield the
general diffusion propagation PDE for the EMF presented in H
2
@H @ H
ΔH 5 μσ 1 με ; ð1:28Þ
@t @t 2
which may lead to the following order of magnitude relation
1 1 1
Bμσ ; με : ð1:29Þ
L 2 τ diffusion τ 2 propagation

2
Twotime-scales areseentoemerge: an EMFdiffusion time scale, τ diffusion 5 L =μσ,
p ffiffiffiffiffiffi
and an EMF propagation time scale, τ propagation 5 L= με. Depending of the material prop-
erties and the length scale, the smallest of two is usually observed to follow in detail the
dynamics of the process.
The scaling relation Eq. (1.29) conveys also valuable information on the penetration
depth for the two EMF transmission mechanisms. For instance, for a harmonic EMF power
source operating at ω 5 2πf, the time constant is τ source 5 1/f.Relation Eq. (1.29) indicates
1
two penetration depth scales: a diffusion penetration depth, δ diffusion B p ffiffiffiffi,and a propagation
ω με
1
penetration depth, δ propagation B p ffiffiffiffi.
ω με
Correlating the two power sources, Ohmic and dielectric, pointed out by Eq. (1.13)
for contributing to the first law energy balance with the penetration depths outlined by
Eq. (1.29), it may be inferred that the hot spots inside the system that correspond to the
two power sources may have different localizations, and their cumulative effect may actu-
ally show off a different thermal image to be the object of interest. This aspect will be ana-
lyzed in Chapter 8: Hyperthermia and Ablation (Thermotherapy Methods).

1.9 Properties of anatomic media

Electrical properties
Numerical modeling of living tissue is not possible without proper knowledge of their
physical properties, since any interaction phenomenon (electromagnetic, thermal, or