follows
from Eq.3.6.4
that
the
required condition of minimum power occurs when
(3.6.5)
which
is
usually referred
to
as
the
"cube law," or as "Murray's law" after
its first author.
The
cube law has been used for many years and has shown considerable
presence in the cardiovascular system [5-14].
It
is important
to
recall
that
it
rests on two important assumptions:
(i)
that
the
flow
under consideration
is
steady fully developed Poiseuille
flow,
and
(ii)
that
the
optimality criterion
being used
is
that
of minimizing
the
total
rate
of energy expenditure for
dynamic and metabolic purposes.
Other laws have been considered by a number of authors to address
observed departures or scatter away from the cube law [15-17].
It
has been
found, for example,
that
in the
aorta
and its first generation of major
(l
•
)
radius
=a
pumping power
=
Hs
t
SAME
FLOW
RATE
l
radius
=0/2
(J
)
•
pumping power
=
16
Hs
FIGURE
3.6.1.
Pumping
power required
to
maintain
steady
fully developed
Poiseuille flow.
If
the
radius
of
a
tube
is halved,
the
pumping
power required
to
drive
the
same
flow
rate
through
it
increases by a factor
of
16 (1600%). Con-
versely, if
the
radius of a
tube
is doubled, only
about
6%
of
the
power is required
to
drive
the
same
flow
rate
through
it.

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3.7
Arterial Bifurcation
51
branches, a "square law" in which
the
flow
rate
is proportional
to
a
2
may
be more appropriate
than
the
cube law.
At
more peripheral regions of
the
arterial tree, however, measured
data
suggest
that
the
cube law eventually
prevails, even if with considerable scatter.
Because in Poiseuille
flow
the
shear stress on
the
tube
wall is proportional
to
the
ratio
qja
3
(Eq.3.4.7),
the
cube law is consistent with constant shear
rate
in
the
vascular system.
That
is,
at
higher levels of
the
arterial tree, as
the
diameters of vessel segments become smaller,
flow
rates within
them
also become smaller
but
in accordance with
the
cube law, leaving
the
shear
stress on vessel walls unchanged. Because
the
range of diameters in
the
arterial tree extends over three
to
four orders of magnitude, any departure
from this precarious design would lead
to
a range of shear stress of many
orders of magnitude.
If
the
flow
rate
varies in accordance with a square
law
(q
ex:
a
2
)
or a quartic law
(q
ex:
a
4
),
for example,
the
shear stress
on
the
vessel walls would
be
proportional
to
1ja
or
to
a,
respectively.
Thus as
a
varies by three
to
four orders of magnitude, so will
the
shear
stress,
and
in
the
case of
the
square law
the
higher shear will occur in
the
smaller vessels. Because
the
lumen of vessels of all sizes are lined with
essentially
the
same
type
of endothelial tissue, these scenarios seem unlikely
on theoretical grounds. Thus
the
form of
the
shear stress in Poiseuille
flow
provides a strong theoretical
support
for
the
cube law, on grounds
that
are
quite different from those on which
the
law was originally based.