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In-Depth Information
A
1
0
B
2.58
2.4
-1
2.0
1.6
1.2
-2
0.8
0.4
-3
0
-0.4
-0.8
-4
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2
0
246810121416182022242628
GENERATION (Z)
LOG GENERATION (LOG Z)
Figure 2.12.
The human-lung-cast data of Weibel and Gomez for twenty-three generations are indicated by
the circles and the prediction using classical scaling for the average diameter is given by the
straight-line segment on the left. The fit is quite good until
z
=
10, after which there is a
systematic deviation of the anatomic data from the theoretical curve [
91
].Thesamedataason
the left are plotted on log-log graph paper on the right and fit to an inverse power law. Adapted
from [
93
].
Recall that the classical scaling argument neglects the variability in the linear scales
at each generation and uses only average values for the lengths and diameters. This dis-
tribution of linear scales at each generation accounts for the deviation in the average
diameter from a simple exponential form. The problem is that the seemingly obvi-
ous classical scaling sets a characteristic scale size. This will clearly fail for complex
webs where no characteristic scale exists. We find that the fluctuations in the lin-
ear sizes are inconsistent with simple scaling, but are compatible with more general
scaling theory. This is essentially the renormalization-group argument that was given
earlier.
At each generation
z
of the bronchial tree the diameter of the bronchial tube takes
on a distribution of values. The average diameter
d
of the tube as a function of the
generation number is given by the solution to the renormalization-group relation (
2.16
)
with a complex power-law index (
2.17
). The best fit to the bronchial-tube data is given in
Figure
2.12
for both classical scaling (on the left) and fractal (renormalization) scaling
(on the right). We see that the inverse power law with
n
(
z
)
0 in the index (
2.17
) captures
the general behavior of the scaling over all generations. The apparent modulation of the
general scaling can be fit by using the
n
=
=
1in(
2.17
) so that the scaling index becomes
