Biomedical Engineering Reference
In-Depth Information
in which the left-hand side term denotes the macroscopic convection term,
while the four terms on the right-hand side correspond to the macroscopic
conduction, thermal dispersion, interfacial convective heat transfer, and blood
perfusion, respectively.
For the solid tissue phase:
(1
+
a
f
h
f
s
s
s
ε
)
ρ
s
c
s
∂
T
∂
∂x
j
ε
)
k
s
∂
T
f
(1
−
=
−
T
−
T
∂t
∂x
j
+
ρ
f
c
pf
ω
s
+(1
f
T
−
T
−
ε
)
S
m
(1.26)
in which the left-hand side term denotes the thermal inertia term, while the
four terms on the right-hand side correspond to the macroscopic conduction,
interfacial convective heat transfer, blood perfusion heat source, and metabolic
heat source, respectively.
The resulting equations (1.25) and (1.26) appear to be a correct form for
the case of thermal nonequilibrium and are expected to clear up possible con-
fusions associated with the blood perfusion term. The continuity equation
(1.13), Darcy's law (1.17), and the two-energy equations (1.25) and (1.26)
form a closed set of the macroscopic governing equations. The present model
in a multidimensional and anisotropic form is quite general and can be applied
to find both velocity and temperature fields, as we prescribe the spatial distri-
butions of permeability tensor, porosity, interfacial heat transfer coecient,
metabolic reaction rate, and perfusion rate. It is interesting to note that,
when the velocity field, porosity, and metabolic reaction are prescribed, we
only need to know the local value of the lumped convection-perfusion param-
eter, namely,
a
f
h
f
+
ρ
f
c
p
f
ω
(in addition to appropriate thermal boundary
conditions) to solve the two-energy equations (1.25) and (1.26) for the blood
and tissue temperatures,
f
and
s
.
T
T
1.4.3 Pennes Model
It should be noted that most existing bioheat transfer models already reside
in the present model based on the theory of porous media. We shall revisit
some of the existing models and try to generate them from the present general
model, starting with the Pennes model (1948), which in our notation runs as
(1
s
s
ε
)
ρ
s
c
s
∂
T
∂
∂x
j
ε
)
k
s
∂
T
(1
−
=
−
∂t
∂x
j
s
)+(1
+
ρ
f
c
p
f
ω
Pennes
(
T
a
0
−
T
−
ε
)
S
m
(1.27)
where
ω
Pennes
is the mean blood perfusion rate, while
T
a
0
is the mean
brachial artery temperature. We compare the Pennes model against the energy
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