13.4 Scaling Laws in Life Systems
573
Let the volume flow through one of the vessels at level k be called Q k and u k be
the average blood speed at level k (averaged over the cross-sectional area). Then
Q k = A k u k
so that the total flow at level k will be
N k Q k = N k A k u k .
Assumption 13.2 The metabolic rate and blood volume flow are proportional to
body size. (As we noted in Sect. 9.7 describing the Basal Metabolism Rate, this
direct proportionality is called isometric scaling. If the body size in a scaling
relationship has an exponent different than one, then the scaling is allometric.) A
linear dependence should be expected. Individual animal cell sizes do not vary
much. Each cell requires a certain supply of nutrients and oxygen, and a system
to expel waste and carbon dioxide. The body mass is proportional to the number of
cells.
The basal metabolic rate B is usually calculated from the rate of oxygen
consumption. This rate is proportional to the total volume flow:
B = f 0 N 1 Q 1 .
where f 0 is independent of body size.
Assumption 13.3 The blood is incompressible, noting that water has a very large
bulk modulus and that blood is made principally of water. Conservation of mass
then gives
N k Q k = N k+1 Q k+1
We will have
B = f C N C A C u C
= f C A C u C
V b
πr 2
C l C S 2
3/4
.
Assumption 13.4 The blood volume in an animal using blood oxygenation is
proportional to the body size (mass M). Since all cells require a nearby blood flow,
this proportionality is expected. The proportionality has been checked by empirical
studies of mammals and birds. We have
V v = V b0 M .
573
Let the volume flow through one of the vessels at level k be called Q k and u k be
the average blood speed at level k (averaged over the cross-sectional area). Then
Q k = A k u k
so that the total flow at level k will be
N k Q k = N k A k u k .
Assumption 13.2 The metabolic rate and blood volume flow are proportional to
body size. (As we noted in Sect. 9.7 describing the Basal Metabolism Rate, this
direct proportionality is called isometric scaling. If the body size in a scaling
relationship has an exponent different than one, then the scaling is allometric.) A
linear dependence should be expected. Individual animal cell sizes do not vary
much. Each cell requires a certain supply of nutrients and oxygen, and a system
to expel waste and carbon dioxide. The body mass is proportional to the number of
cells.
The basal metabolic rate B is usually calculated from the rate of oxygen
consumption. This rate is proportional to the total volume flow:
B = f 0 N 1 Q 1 .
where f 0 is independent of body size.
Assumption 13.3 The blood is incompressible, noting that water has a very large
bulk modulus and that blood is made principally of water. Conservation of mass
then gives
N k Q k = N k+1 Q k+1
We will have
B = f C N C A C u C
= f C A C u C
V b
πr 2
C l C S 2
3/4
.
Assumption 13.4 The blood volume in an animal using blood oxygenation is
proportional to the body size (mass M). Since all cells require a nearby blood flow,
this proportionality is expected. The proportionality has been checked by empirical
studies of mammals and birds. We have
V v = V b0 M .