Biomedical Engineering Reference
In-Depth Information
Differentiating (4.46) with respect to
r
,
dP
−
4*8
LQ
2
μ
T
=
+
2
απ
rL
=
0
b
5
dr
π
r
Rearranging,
πα
μ
26
r
2
Q
=
b
(4.38)
16
π
,
α
b
,
and
μ
are all constants. Hence
3
Qr
∝
α
b
is relatively difficult. However, some of the experimental data
suggest that the exponent on the radius could vary from 2.7 to 3.2.
Using (4.16) in conjunction with Murray's law forms the basis for modeling
the microvascular branching geometry as a means to providing insight into blood
flow distribution, blood pressure distribution, X-ray based indicator for dilution
curves, and to get more insight into the impact of changes in the vessel dimensions,
capillary recruitment, and functional changes of vessel diameters.
Obtaining
QQQ
r
=+
=+
0
1
2
3
3
3
r
r
0
1
2
−
1/3
−
1/3
⎡
3
⎤
⎡
3
⎤
r
⎛⎞
r
r
r
⎛⎞
r
1
=+
⎢
1
2
⎥
and
2
=
2
⎢
1
+
2
⎥
Rearranging
⎜⎟
⎜⎟
r
⎝⎠
r
r
r
⎝⎠
r
⎢
⎥
⎢
⎥
⎣
⎦
⎣
⎦
0
1
0
1
1
EXAMPLE 4.11
Calculate the volumetric flow rate in a 5-mm vascular graft of a length of 5 cm. Assume
the viscosity if 3 cP and
α
b
is 780 erg/mL.s [6]. Then calculate the work required to drive
the flow.
Solution: From (4.38),
26
2
6
−−
3 1
πα
r
π
(0.5[cm]) * 780[erg.cm
s
]
Q
2
b
2,500 cm
⎡
62
s
⎤
50 cm
3
s
=
=
=
=
⎣
⎦
16
16 * 3 * 10
−
3
[gm.cm
−
1
s
−
1
]
μ
The branching and the predictions of Murray's law agree well with the branch-
ing in the lungs. However, it does not predict the angle of bifurcation and the
