Biomedical Engineering Reference
In-Depth Information
Fig. 6.6
Parameter evolution in singular tubes (
left
) and in the entire level (
right
), for levels 16-24
may observe that the evolution in a single tube, in consecutive levels is quasi-linear
for both parameters (Fig.
6.6
-left). However, since the total parameter values from
Fig.
6.6
-right depend on the total number of tubes within each level, they change
as an exponential decaying function. When represented on a logarithmic scale, one
can observe a quasi-linear behavior, as in Fig.
6.6
-right.
In a similar manner as the electrical impedance, one may obtain
H(s)
, which
defines the relation from velocity (input) to force (output)
F(s)/v(s)
, with
s
the
Laplace operator. The transfer function of a cell in the ladder network consisting of
one damper and one spring, is
K
s
H(s)
=
B
+
(6.12)
10
−
5
,
10
5
which can be evaluated over a range of frequencies, e.g.
ω
, with the
result depicted in Fig.
6.7
. In this figure '24' denotes that the
H(s)
is calculated at
level 24; '23' denotes that
H(s)
is calculated at level 23, etc.
Due to the fact that the network is dichotomous and symmetric, we can obtain
the total mechanical impedance using the network structure as in Fig.
6.5
, with
B
m
and
K
m
calculated with (
6.11
). Since the Kelvin-Voigt elements corresponding to
one level are in parallel, their transfer function
H
m
will be in series with the spring
in the level
m
∈[
]
1. The next corresponding transfer function is in parallel with the
damper in the level
m
−
−
1, as depicted schematically by Fig.
6.9
. In this manner, the
total transfer function
H(s)
can be determined, starting at level 24 [
26
].
The lung parenchyma consists of interwoven collagen (infinitely stiff) and elastin
(elastic) fibers. Each level in the respiratory tree has a specific balance between these
two components. In our model we take this balance into account in (
4.14
), in func-
tion of the cartilage percent (Table
2.1
). Following this reasoning, a similar repre-
sentation of the mechanical model is given in Fig.
6.8
. Here, the cylinders represent
the collagen fibers within one level, which are interconnected with elastin fibers,
represented by inextensible unstressed strings. This representation varies from that
of Bates in that it represents the total collagen-elastin distribution in a level and not
