Biomedical Engineering Reference
In-Depth Information
the limb. Using his experimental data, Pennes proposed what is known today
as the Pennes bioheat equation. In his model, he assumed that the net heat
transferred from the blood to tissue is proportional to the temperature dif-
ference between the arterial blood entering the tissue and the venous blood
leaving from the tissue, and introduced the Pennes perfusion heat source.
The Pennes bioheat equation has been used for various bioheat transfer
problems and found satisfactory for roughly describing the effect of blood flow
on the tissue temperature. However, a number of researchers including Wulff
(1974) and Klinger (1978) pointed out serious shortcomings in his model due
to its inherent simplicity, namely, assuming uniform perfusion rate without
accounting for blood flow direction, neglecting the important anatomical fea-
tures of the circulatory network system such as countercurrent arrangement
of the system, and choosing only the venous blood stream as the fluid stream
equilibrated with the tissue.
Possible modifications have been proposed by some researchers, so as to
remedy these shortcomings. Wulff (1974) and Klinger (1978) considered the
local blood mass flux to account the blood flow direction, whereas Chen and
Holmes (1980) examined the effect of thermal equilibration length on the blood
temperature and added the dispersion and microcirculatory perfusion terms
to the Klinger equation.
In this section, we exploit VAT described in the foregoing sections to obtain
a complete set of the volume averaged governing equations for bioheat transfer
and blood flow. Most shortcomings in existing models can be overcome.
1.4.2 Two-Energy Equation Model Based on VAT
Before actually integrating the energy equation (1.10), it may be quite instruc-
tive to focus our attention on the volume average of the convection term. Using
equations (1.5) and (1.6), it is straightforward to show
ε
∂
∂x
j
ρ
f
c
p
f
u
j
T
f
∂x
j
ερ
f
c
p
f
u
j
T
f
∂
∂x
j
ρ
f
c
p
f
∂
f
+
=
u
j
T
1
V
+
ρ
f
c
p
f
u
j
Tn
j
dA
(1.18)
A
int
where the first term on the right-hand side describes the macroscopic convec-
tion, while the second term on the right-hand side takes account of the thermal
dispersion (Nakayama et al. 2004). It is the last term on the right-hand side
that corresponds to the blood “perfusion” heat source. Thus, the blood perfu-
sion heat source term is identified as an extra surface integral term resulting
from changing the sequence of integration and derivation, as we obtain the
macroscopic energy equation by integrating the microscopic convection term
over a local control volume.
Having expanded the integrated convection term, we may readily trans-
form both the energy equation (1.10) for the blood flow and the conduction
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