Biomedical Engineering Reference
In-Depth Information
5.2 Effects of Structural Asymmetry
In his recent publication, Weibel discusses the reduction of diameter and length by a
constant factor for both blood vessels and airways [
164
]. He recognizes the theoret-
ical contributions of Murray [
107
], i.e. that the dissipation of energy due to flow of
blood or air in a branched tube system can be minimized if the diameter of the two
daughter-branches are related to the diameter of the parent as in
d
parent
=
d
2
.
In the context of fractal geometry, the reduction factor depends on the fractal dimen-
sion
FD
of the branching tree such that the correct formula is
d
1
=
d
1
+
2
−
1
/FD
.
d
parent
·
In the case of Hess-Murray law,
FD
3 because the tree is considered to be space-
filling [
34
]. In his investigations, Weibel found that the slope of the conducting
airway diameters against the generations was given by
d(m)
=
d
0
·
=
2
−
m/
3
, with
d
0
the tracheal diameter and
m
the airway generation. He then concludes that the con-
ducting airways of the human lung are designed as a self-similar and space-filling
fractal tree, with a homothety factor of 2
−
1
/
3
0
.
79 (similitude ratio). However, as
discussed in Chap.
3
and in the beginning of Chap.
4
, this average has a significant
variance in the first generations. Hence, the average value changes in the diffusion
zone (airways from 16th generation onward). These observations and the fact that
Weibel himself discusses that a small change in the homothety factor results in a
dramatic increase in peripheral bronchiolar resistance imply that the lung must be
capable to adjust itself to the optimality conditions. Indeed, a closer analysis reveals
that the homothety factor is about 0.79 in the sixth generation, but it increases slowly
to about 0.9 in the 16th generation, with an average of 0.85 for the small airways
[
164
]. The physiological implications of this observation are:
=
•
the flow resistance decreases in the small airways and
•
a small reduction in the homothety factor does not affect significantly the lung
function.
In the context of the above observations, one may explore the possibility of the
respiratory system as a multi-fractal structure. A self-similar multi-fractal spatial
distribution forms the basis for breaking the symmetry of bifurcation design within a
tree. In [
167
], the author discusses the implications of self-affine scaling. It turns out
that the fractal dimension changes when calculated from different reference points.
Therefore, the slope determining the homothety factor changes when viewed at a
fine or coarse grained diameter scale. This latter observation is of interest in the
context of this topic, since it supports the idea of a multi-fractal structure. For ex-
ample, the average of the radius ratio changes from 2
−
0
.
17
0
.
88 to 0
.
89 when only
the first 16 generations are taken into account, respectively, to 0
.
87 for the alveoli
(generations 17-24). This implies that the homothety factor changes, depending on
the spatial location within the tree. On the other hand, if we analyze the radius ratio
from generations 1 to 24 in steps of 4, we obtain an average of 0
.
85, whereas if we
use steps of 2, we obtain an average homothety factor of 0
.
86. These changes might
not seem significant, but one should recall that they originate by the symmetric ge-
ometry of the respiratory tree. However, when asymmetry is considered, one deals
with several homothety factors, i.e. as schematically drawn in Fig.
5.6
.
=
