Biomedical Engineering Reference
In-Depth Information
2.1. Body Size Dependence
So, naturalists observe, a flea has smaller fleas that on him prey; and
these have smaller still to bite 'em, And so proceed ad infinitum.
Thus, every poet in his kind is bit by him that comes behind.
—Jonathan Swift
We posit that the effect of body size on all physiological variables is deter-
mined by the scaling properties of hierarchical resource-distribution networks,
such as the cardiovascular and respiratory systems, which deliver essential nu-
trients and metabolites to cells. There are three main assumptions, all presumed
to be derivative from the processes of evolution and natural selection, that define
the theory and are postulated to characterize the resource-distribution networks
(12,16): (i) they are space filling in order to service all cells through the supply
of nutrients and the removal of wastes, (ii) the energy to deliver resources is
minimized, and (iii) their terminal units (e.g., capillaries) are invariant. We now
review how these three assumptions are used to derive the scaling of metabolic
rate with body size.
In order to describe the network we need to determine how the radii,
r
k
, and
lengths,
l
k
, of the tubes change throughout the network;
k
denotes the level of the
branching, beginning with the aorta at
k
= 0 and terminating at the capillaries
where
k
=
N
. The number of branches per node (the branching ratio),
n
, is as-
sumed to be constant throughout the network. To characterize the branching we
introduce scale factors via the ratios C
k
=
r
k
+1
/
r
k
and H
k
=
l
k
+1
/
l
k
. Since capillaries
are invariant units, these scale factors completely determine the network except
for the number of levels, which is a function of body size.
The first assumption (i), that networks are space-filling (31), ensures that all
tissues are supplied by capillaries. The organism is composed of many groups of
cells, referred to here as "service volumes,"
v
N
, which are supplied by a single
capillary. The total volume to be filled, or serviced, is given by
V
=
N
N
v
N
, where
N
N
is the number of capillaries. For a network with many levels,
N
, complete
space-filling implies that this same volume,
V
, is filled at all scales by an analo-
gous volume,
v
k
, defined by branches at each level
k
. Since
r
k
<<
l
k
,
v
k
l
k
3
, so
space-filling constrains only branch lengths,
l
k
. Thus,
V
N
k
v
k
N
k
l
k
3
, and since
V
is independent of
k
, we have H
n
-1/3
. We assume this relation is valid
throughout the network, although it becomes less realistic for small values of
k
.
A more explicit statement of assumption (ii) is that the continuous feedback
implicit in evolutionary adaptation has lead to resource-distribution networks
that, on average, minimize the energy required to support flow through the sys-
tem. There are two independent contributions to energy loss: energy dissipated
by viscous forces, which is only important in smaller vessels, and energy re-
flected at branch points, which is entirely eliminated by impedance matching. In
large vessels (e.g., arteries) pulse waves suffer little attenuation or dissipation,
