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the next generation, 2
V
n
+
1
. Unlike in the more general case considered by da Vinci, the
value in each daughter branch was assumed by him to be the same. Rohrer argued that,
since the bronchial-tube diameters decrease by the same proportion from one genera-
tion to the next, the diameter should decrease exponentially with generation number as
determined in the previous section. If we assume that the tube length scales in the same
way as the tube diameter, that is,
l
n
+
1
=
λ
l
n
, this argument yields for the volume
d
n
l
n
=
V
n
=
π
2
V
n
+
1
(2.18)
and
3
V
n
.
V
n
+
1
=
λ
(2.19)
Consequently, if volume is conserved from one generation to the next the scaling
parameter must satisfy the relation
3
2
λ
=
1
(2.20)
and the optimal scaling parameter is given by
2
−
1
/
3
λ
=
.
(2.21)
Over half a century after Rohrer [
66
] formulated his principle, the same exponen-
tial scaling relationship was reported by Weibel and Gomez in their classic paper on
bronchial airways [
91
]. We refer to the exponential reduction in the diameter of the air-
way with generation number as classical scaling and find in the following section that it
applies to the bronchial tree only in a limited domain.
The same scaling parameter was obtained by Wilson [
102
], who explained the pro-
posed exponential decrease in the average diameter of a bronchial tube with generation
number by showing that this is the functional form for which a gas of given composition
can be provided to the alveoli with minimum metabolism or entropy production in the
respiratory musculature. He assumed that minimum entropy production is the design
principle for biological systems to carry out a given function. It is also noteworthy that
if we set the daughter diameters in da Vinci's equation (
2.1
) equal to one another we
obtain
2
1
/α
d
1
.
d
0
=
(2.22)
Using the value of
given by the energy minimization argument, we determine that the
size of the diameter is reduced by 2
−
1
/
3
in each successive bifurcation. Consequently
we obtain exactly the same scaling result for the diameter of the bronchial airway from
da Vinci as from scientists nearly half a millennium later.
In classical scaling the index is
α
3 so that the branching structure fills the avail-
able space. A less space-filling value of the scaling index is obtained for the arterial
system, where it has empirically been determined that
α
=
7. For general non-integer
branching index, the scaling relation (
2.22
) defines a fractal tree. Such trees have no
characteristic scale length and were first organized and discussed as a class by Man-
delbrot [
44
], the father of fractals. The classical approach relied on the assumption that
biological processes, like their physical counterparts, are continuous, homogeneous and
α
=
2
.
