Vascular
Branching
167
4
=
R,Q’
where the Poiseuille resistance,
R,,
to steady flow
Q,
is
(5.4.1)
(5.4.2)
where
I
is the length of the vessel along which blood flows. Thus, a
larger vessel radius is more advantageous. This is because the flowing
blood encounters a smaller resistance.
However, a greater volume of
blood is required for perfusion through a vessel with a larger radius,
hence a greater demand on metabolic energy:
v
=
m21
(5.4.3)
where
V
is the blood vessel volume.
lumen radius, r2, assuming the vessel
is
cylindrical:
The amount of volume flow,
Q,
is
proportional to the square
of
the
Q
=
m2v
(5.4.4)
where v is linear blood flow velocity.
The optimal radius is therefore the one that can minimize the
resistance
to
blood flow, as well as the power of expenditure. This can
be formulated as:
1
2
P,
=
k,
+
k,r
r
where
k,
and
kz
are constants.
Differentiate
Po
with respect to r, we have:
dP,
-4k,
-=-
+2k,r
=
0
dr
r5
(5.4.5)
(5.4.6)
