Biomedical Engineering Reference
In-Depth Information
1.0
0.8
n
=
0
a
ε a
u
u 0
0.6
n = 2
n
=
5
0.4
ω
a L
E
=
=
0.5
u 0
0.2
0
0
0.2
0.4
0.6
0.8
1.0
x / L
FIGURE 1.6
Effect of the exponent n on perfusion rate.
along the blood vessel of length L, where ω a is t he average perfusion rate such
that the total amount of perfusion is given by ω a L , irrespective of the value
of n . The exponent n may take any value equal to zero (i.e., Chato's case)
or greater than zero, such that we can compare the results against Chato's
and elucidate the effect of blood pressure on the bioheat transfer for fixed
total amount of perfusion. As we substitute the foregoing equation into the
continuity equations (1.37) and (1.40), we readily obtain
ω a L x
1+ n
a = u 0
ε a
u
(1.68)
L
u 0 + ω a L x
L
1+ n
v =
ε v
u
(1.69)
where u 0 is the apparent blood velocity at x = 0. As illustrated in Figure 1.6,
the exponent n controls the distribution of the perfusion rate. For a large
exponent n , the perfusion bleed-off takes place rather suddenly toward the
end of the vessel, indicating poor blood circulation.
Upon substituting these velocity distributions into the momentum equa-
tions (1.38) and (1.41), we obtain
ε a
K a
u 0 L 1
x =0
x = L =( ε a
v )
ε v
K v
E
2+ n
0
L =
a + ε v
p
p
p
p
µ
(1.70)
where
E = ω a L
u 0
(1.71)
is the dimensionless perfusion bleed-off rate, while µ is the viscosity.
Thus, the pressure difference within the body may never be large since
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