Page 33 - Computational Modeling in Biomedical Engineering and Medical Physics

Physical, mathematical, and numerical modeling 19

Consequently the dynamics of the unsteady diffusion advection energy Eq. (1.23)
is led by either the diffusion time scale, τ diffusion Eq. (1.19), or by the velocity time
scale, τ transport . As only one time-scale may prevail in numerical simulations of the cou-
pled convection heat transfer problem, either the largest (diffusion usually) or the
smallest (transport usually) prevails. If the transport time scale is dominant (smallest)
then τ transport is selected, and the energy equation has then to be rescaled accordingly.
As an example, in modeling the heat transfer in localized hyperthermia it is com-
mon to use a homogenization technique in lieu of the general heat transfer model, for
example, the bioheat equation (Pennes, 1948) (discussed later in this chapter, and in :
Magnetic Stimulation). This approach assumes that the ROI is a homogeneous, con-
tinuum medium with distributed heat sink/source that accounts for the hemodynamic
heat transfer

@T
ρc 1 ρ C b ω T 2 T a Þ 5 r krTð Þ 1 p Joule ; ð1:26Þ
ð
b
@t
which is a particular form of the energy equation. Here T a is the arterial blood tem-
3

perature (37 C), ω [1/s] is the blood perfusion rate, ρ and ρ b [kg/m ] are mass densi-
ties of tissue and blood, respectively, C and C b [J/(kg K)] are specific heat capacities
3
of tissue and blood, respectively, and p Joule [W/m ] is the heat source, for example, the
Joule effect. The scaling of Eq. (1.26) yields the order of magnitude relation
L 2 ρC
τB ; ; ð1:27Þ
α ρ C b ω
|{z} b
|fflffl{zfflffl}
τ diffusion τ transport

which shows off two concurring time scales: a diffusion time scale and a transport
time scale. If the ROI is a spherical volume of liver tissue of diameter 1 cm, using
3 3
ρ 5 1000 kg/m , k 5 0.512 W/(m K), C 5 3600 J/(kg K), ρ b 5 1000 kg/m ,
C b 5 4180 J/(kg K), ω 5 6.4 3 10 23 21 Morega (EHB), the scaling relation
s
Eq. (1.27) yields τ diffusion B 800 s and τ transport B 140 s. Therefore hyperthermia
procedure may be successful provided that the heat source is intense enough to
raise the tissue temperature at the required hyperthermia level, and to compensate
for the heat loss through hemodynamic flow for the duration of the procedure.
If the general heat transfer model, Eqs. (1.23), (1.24) is recognized for larger vessels (arter-
ies and veins) then, in general, the velocity time constant prevails. Furthermore the pace of
the pulsating arterial blood flow (e.g., 60 80 bpm) may be much smaller than the diffusion
time constant. To avoid a lengthy and cumbersome numerical simulation, it may be more
convenient to replace the pulsatile flow with an equivalent stationary flow, with an average
flow rate (r.m.s.). The stationary velocity field is then used in the transient heat transfer