Biomedical Engineering Reference
In-Depth Information
Chapter 5
Ladder Network Models as Origin
of Fractional-Order Models
5.1 Fractal Structure and Ladder Network Models
5.1.1 An Elastic Airway Wall
In this section, we make use of the formulas (
4.62
), (
4.63
), and (
4.64
), which are cal-
culated with the morphologic values from Table
2.1
. With the resistance, inertance,
and capacitance values at hand, one is able to build an electrical network. Suppose
we have the electrical network as depicted in Fig.
5.1
, which preserves the geome-
try of the respiratory tree. In this network,
Zl
m
denotes the longitudinal impedance,
whereas
Zt
m
denotes the transversal impedance of the airway tubes and
m
denotes
the level in the respiratory tree (
m
1
,...,N
).
Assuming that the flow
Q
is symmetric with respect to each bifurcation (divides
equally trough the branches) one can define the equivalent level impedances and
admittances as a function of powers 2. Hence, the total resistance per level is given
by [
108
]:
=
R
em
=
R
em
/
2
m
−
1
(5.1)
with
R
em
the resistance in a single branch. Similarly, the total inertance per level is
given by
L
em
/
2
m
−
1
L
em
=
(5.2)
with
L
em
the inertance in a single branch; finally, the total capacitance in a level is
given by
2
m
−
1
(5.3)
with
C
em
the capacitance in a single branch. Using these relations, one can simplify
the electrical network from Fig.
5.1
to an equivalent ladder network, schematically
depicted in Fig.
5.2
. In this ladder network,
Zl
m
, with
m
C
em
·
C
em
=
=
1
,...,N
denoting the
longitudinal impedance, which is defined as
Zl
m
(s)
L
em
s
. Since both re-
sistance and inertance in each level are divided by 2
m
−
1
, we can use the equivalent
R
em
+
=
