Page 31 - Computational Modeling in Biomedical Engineering and Medical Physics

Physical, mathematical, and numerical modeling 17

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where ρ [kg/m ] is the mass density, c [J/(kg K)] is the specific heat at constant pres-
sure, k [W/(m K)] is the thermal conductivity, and T [K] is the temperature.
Assuming that the thermal size of the system, that is, the size (length) of the diffusion
path is L [m], then Eq. (1.17) indicates the following order of magnitude balance
between its constitutive terms
ΔT ΔT
ρc Bk 2 ; qw; ð1:18Þ
0
τ diffusion L
where ΔT is the order of magnitude of the temperature temporal and spatial “excur-
sion,” qw is the order of magnitude of the heat source. The scaling relation Eq. (1.18)
0
shows off the diffusion time constant, τ diffusion [s], which is a measure of the time needed
for the thermal transient to vanish hence for the system to reach a final steady state. As
the material properties and the space scales are known in direct problems then
Eq. (1.12) yields

ρc 1 L 2
τ diffusion 5 5 ; ð1:19Þ
k L 2 α
2
where α [m /s] is called diffusivity, here thermal diffusivity—a quantity found in the
material properties data sheets.
If the heat source is dynamic, then τ diffusion has to be compared with the source
(excitation) time scale to decide the adequate form of the mathematical model to be
used. For instance, if the heat source is the Joule power produced by a harmonic con-
duction current [e.g., the electroablation, Chapter 8: Hyperthermia and Ablation
2
(Thermotherapy Methods)], qw5 ρ J , where ρ el [Ω m] is the electrical conductivity
el 0
0
2
and J 0 [A/m ] is the order of magnitude of the electrical current density, then the
time scale of the excitation is 20 ms for an electrical current source operating at 50 Hz.
Assuming that the ROI is a spherical volume of liver tissue with the radius B1 cm,
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and its properties are k 5 0.502 W/m K, ρ 5 1060 kg/m , c 5 3600 J/(kg K)
(Valvano, 2010), then the relation Eq. (1.19) yields τ diffusion B760 s, which predicts
roughly the time to reach some desired steady state temperature.
However, because τ diffusion ,, 20 ms it will be reasonable to consider that the Joule
2
power isproducedbyanequivalent(r.m.s.)DCcurrent density, J DC [A/m ], whose distri-
bution is governed by an equivalent potential (Laplace) problem, which yields

ΔT ΔT 2
ρc p Bk ; ρ J : ð1:20Þ
el DC
Δt L 2
This approach avoids integrating, in the same time, a quasistationary EMF problem
and the associated heat transfer problem—a considerably more laborious and, overall,
less relevant path to follow.