Vascular Branching
169
Where the first term is simply the rate of pressure-volume work and the
second term is dependent on the volume of the vessel, assuming
cylindrical in shape, with radius
r
and length,
I
(Fig. 5.4.1). Steady flow,
Q,
and pressure drop, Ap are considered here.
Power
is
the rate of energy use, i.e.
where p(t) and Q(t) are the pulsatile pressure and flow respectively and
the instantaneous power is then
(5.4.11)
Equation (5.4.1) can be rewritten, employing Poiseuille’s formula, as
P,
=
(7)Q’
8771
+
k(m’1)
m
(5.4.12)
The optimal vascular system in this concept is plausible only when
individual vessel segments are optimized.
Minimum rate of work
is
obtained, by differentiating Po with respect to r,
giving
(5.4.13)
(5.4.14)
which is identical to eqn. (5.4.7), specifying the cube-law (eqn. (5.4.8)).
For this radius, the corresponding minimum rate of energy is
3k
2
P,=-ml
2
(5.4.15)
