Biomedical Engineering Reference
In-Depth Information
equation (1.26) for the solid tissue phase and find the following relationship:
s )= a f h f
s + ρ f c p f ω
s
f
f
ρ f c pf ω Pennes ( T a 0
T
T
T
T
T
(1.28)
Perhaps, Pennes considered that the blood perfusion is the predominant
heat source for the tissue, and did not bother to describe the interfacial convec-
tive heat transfer between the blood and tissue via the vascular wall. Instead,
he introduced T a 0 to adjust the total heat transfer, which takes place as the
blood enters and leaves the tissue. We may assume T a 0
f for small vessels,
T
and find
a f h f
ρ f c pf
ω Pennes = ω +
(1.29)
Thus, Pennes' perfusion rate may be regarded as an effective one that
includes interfacial convective heat transfer as well. Pennes assumed that
blood enters the smallest vessels of the microcirculation at T a 0 , where all
heat transfer between the blood and tissue takes place. The assumption of the
complete thermal equilibration with the surrounding tissue is valid only when
Peclet number is suciently small.
1.4.4 Wulff Model and Klinger Model
Wulff (1974) criticized the Pennes model, pointing out that the moving blood
through a tissue convects heat in any direction, not just in the direction of
the local tissue temperature gradient. He assumed that the blood temperature
f is equivalent to the tissue temperature within a tissue control volume and
proposed a new bioheat transfer equation. The equation later generalized by
Klinger (1978) runs in our notation as
T
(1
s
s
s
ε ) ρ s c s
T
∂x j
ε ) k s
T
ρ f c pf
u j
T
(1
=
+(1
ε ) S m
∂t
∂x j
∂x j
(1.30)
We can obtain a similar equation by combining equations (1.25) and (1.26)
setting
f =
s as follows:
T
T
s
ερ f c p f +(1
ε ) ρ s c s
T
∂x j
s
+ ρ f c p f
u j
T
∂t
( εk f +(1
+(1
s
s
∂x j
ε ) k s )
T
T
=
+ εk dis jk
ε ) S m
(1.31)
∂x j
∂x k
 
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