Ladder Network Models as Origin of Fractional-Order Models - The Human Respiratory System - page 63

Fig. 5.6 Asymmetric representation for the first four generations, in its electrical equivalent

Fig. 5.7 Number of branches
for each generation, in the
asymmetric ( top )and
symmetric ( bottom )
generation. Notice that the
Y-axis is logarithmic

It has been demonstrated by a systematic analysis that the airway tree in differ-
ent species shows a common fractal structure, in spite of some gross differences in
airway morphology [ 164 ]. Nevertheless, let us investigate the case of asymmetric
branching in the human lungs. The Horsfield representation will be used, as from
[ 65 ], with the values listed in Table 2.2 . In this scenario, an airway of level m bifur-
cates into two daughters: one of order m

Δ , with Δ the
asymmetry index. As a result of the asymmetry, the electrical network becomes as
in Fig. 5.6 . Figure 5.7 shows the number of branches that are in one generation, for
the symmetric and asymmetric lung structure. Notice the different slope which char-
acterizes the space-filling distribution; the top figure shows that the slope is lower
in the asymmetric tree section than in the symmetric tree section.
Since the symmetry is lost, one cannot simplify the electrical network to its lad-
der network equivalence as in Fig. 5.2 . Therefore, one must calculate explicitly the
impedance from level 36 to level 1. To avoid complex numerical formulations, the
impedance along the longest path was calculated, as in [ 54 ]. One should notice that
from level 26 onward, the asymmetry index is zero, therefore symmetric bifurca-
tion occurs (recall here Table 2.2 ). The effect of this change in the asymmetry index
is visible in Fig. 5.7 , i.e. a change in the slope. The initial values in the trachea
are imposed similarly as in the symmetric case [ 121 ]. Figure 5.8 shows the total
impedance by means of its complex representation (left) and its Bode plot (right),

+ 1 and one of order m + 1 +