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Table 2.3.
Bronchial length ratios. Method A; values were computed from the
ratio of short/long conjugate tube lengths for each bifurcation for which
measurements were available in bronchial generations one through seven. The
overall mean and variance were calculated for these seven generations using the
number of ratios indicated. Method B; the sums of all short and all long branches
in bronchial generations one to seven were obtained. The ratio of these sums
was then calculated for each generation. The overall mean and variance were
calculated for these seven generations [
22
]. Reproduced with permission.
Lung cast
Method A
Method B
Human
0.65
±
0
.
02
(
n
=
226
)
0.62
±
0
.
03
0.67
±
0
.
02
(
n
=
214
)
0.67
±
0
.
02
Dog
±
.
(
n
=
)
±
.
Rat
0.63
0
03
100
0.55
0
05
Hamster
0.62
±
0
.
02
(
n
=
107
)
0.59
±
0
.
07
complex and captures the modulation with period ln
N
[
72
]. In the next section we
argue that this result is representative of many physiologic phenomena and should not
be restricted to bronchial airways.
It is worthwhile to point out that in addition to the classical and fractal (renormal-
ization) scaling discussed above other kinds of scaling appear in physiologic webs. For
example, the Fibonacci scaling introduced in the problem set has been observed in a
number of biological and anatomic relations. The historical observation that the ratio of
the total height to the vertical height of the navel in humans approximates the golden
mean has been verified in systematic studies [
15
]. We can also apply this reasoning
to the asymmetric branching within the mammalian lung. We assume that the daugh-
ter branches in the bronchial tree have the lengths
and hypothesize that these
lengths satisfy Fibonacci scaling. To test this hypothesis Goldberger
et al.
[
22
]used
a data set of detailed morphometric measurements of silicone-rubber bronchial casts
from four mammalian species. The results of analysis of these data for human, dog, rat
and hamster lungs are shown in Table
2.3
. In this table we see a consistency with the
hypothesis that the Fibonacci ratio is a universal scaling ratio in the bronchial tree.
Results such as these suggest a regularity underlying the variability found in natural
phenomena. We often lose sight of this variability due to our focus on the regularity and
controllability that modern society imposes on our thinking and our lives. Consequently,
to understand natural phenomena we must examine the variability together with the
regularity. The phrase “thinking outside the box” attempts to prompt thinking outside
the averaged boundaries in order to capture the variability.
α
and
β
2.2
Physiology in fractal dimensions
Physical science contains a number of assumptions that are so basic to the phenom-
ena being examined that their significance is often lost and it is not obvious what the
assumptions have to do with what is being measured. The nature of measurement and
