574
13 Modeling Biological Systems
Assumption 13.5 Capillary radii and length do not depend on body size. This is an
important, but not obvious, property. In essence, the size of a capillary is determined
by the need for diffusion of nutrients and oxygen to nearby cells. Diffusion would
be impeded if the capillary were larger in radius. If it were longer, the concentration
of nutrients and oxygen would be further diminished at its far end while it supplies
near-end cells. These constraints are strictly local, depending on a small region of
cells near the capillary, and not on the total mass of all the cells.
Assumption 13.6
(a) The network preserves area:
N k πr
2
k = N k+1 πr
2
k+1
i.e.
α k = 1
This is an impedance-matching assumption. If the total area of the arteries
before a node does not match the total area after that node, the blood flow would
show pressure reflection at the node, and more heart power would be needed to
pump the blood against the increased resistance.
(b) The network is ‘space-filling’, in the sense that the volume around all the arteries
before a given node matches the volume after that node. This reads
4
3
πN k l
3
k =
4
3
πN k+1 l
3
k+1
i.e.
ϕ k = 1 .
The argument for this assumption is one of economy in the utilization of the
available volume. Alternative models now exist which involve the thickness of
the arterial walls, the pulsation character of the blood in the initial few nodes,
and the metabolic requirements needed for sudden increases in activity. 13
(c) The total number of levels, C, is much greater than one. This makes the quantity
S 3 asymptotically reach a constant value not dependent on C, and so not
dependent on the body size.
For example, the asymptotic value of S 3 when ν k = n, so N k = N C /n C−k ,
becomes
13 M.E.F. Apol et al., Revisiting the evolutionary origin of allometric metabolic scaling in biology,
Functional Ecology 22, 1070–1080 (2008).
13 Modeling Biological Systems
Assumption 13.5 Capillary radii and length do not depend on body size. This is an
important, but not obvious, property. In essence, the size of a capillary is determined
by the need for diffusion of nutrients and oxygen to nearby cells. Diffusion would
be impeded if the capillary were larger in radius. If it were longer, the concentration
of nutrients and oxygen would be further diminished at its far end while it supplies
near-end cells. These constraints are strictly local, depending on a small region of
cells near the capillary, and not on the total mass of all the cells.
Assumption 13.6
(a) The network preserves area:
N k πr
2
k = N k+1 πr
2
k+1
i.e.
α k = 1
This is an impedance-matching assumption. If the total area of the arteries
before a node does not match the total area after that node, the blood flow would
show pressure reflection at the node, and more heart power would be needed to
pump the blood against the increased resistance.
(b) The network is ‘space-filling’, in the sense that the volume around all the arteries
before a given node matches the volume after that node. This reads
4
3
πN k l
3
k =
4
3
πN k+1 l
3
k+1
i.e.
ϕ k = 1 .
The argument for this assumption is one of economy in the utilization of the
available volume. Alternative models now exist which involve the thickness of
the arterial walls, the pulsation character of the blood in the initial few nodes,
and the metabolic requirements needed for sudden increases in activity. 13
(c) The total number of levels, C, is much greater than one. This makes the quantity
S 3 asymptotically reach a constant value not dependent on C, and so not
dependent on the body size.
For example, the asymptotic value of S 3 when ν k = n, so N k = N C /n C−k ,
becomes
13 M.E.F. Apol et al., Revisiting the evolutionary origin of allometric metabolic scaling in biology,
Functional Ecology 22, 1070–1080 (2008).