Biomedical Engineering Reference
In-Depth Information
Fig. 5.4
Impedance by means of complex (
left
) and Bode-plot (
right
) representation, for the
R
-
C
(
continuous line
)andthe
R
-
L
-
C
(
dashed line
) model structures
Depending on the number of cells in the ladder (
N
), the constant-phase behavior
will emerge over a wider range of frequencies. This result is applicable to any kind
of ladder network (airways, arteries, etc.). However, the fractional-order value and
coefficients will change according to the properties (morphology, geometry) of the
system.
In practice, the respiratory tract can be simulated as follows. The relations de-
rived in Chap.
4
for resistance (
4.62
), inertance (
4.63
), and compliance (
4.64
)are
used to calculate the total level values as in (
5.1
), (
5.2
), and (
5.3
). Notice that the
values in the trachea
R
e
1
,
L
e
1
, and
C
e
1
need to take into account the flow and
pressure effects in the upper airways (mouth, nose, larynx, pharynx). Since we
do not model the upper airways, we need to take the values from literature [
121
]:
R
UA
=
0
.
002 kPa/(l/s
2
), and
C
UA
=
0
.
25 l/kPa. To find the
level values, one can make use of the ratios from Table
5.1
. The last compartment
needs to model the gas compression effect; hence, from literature, we introduce the
series impedance consisting of [
54
]
R
GC
=
0
.
2 kPa/(l/s),
L
UA
=
0
.
06 kPa/(l/s
2
),
0
.
05 kPa/(l/s),
L
GC
=
and
C
GC
=
6 l/kPa. This last impedance is closing the ladder network, being in par-
allel with the cell
N
24. The total admittance from (
5.16
) is then calculated. The
equivalent total input impedance (including the upper airways and the gas compres-
sion compartment) is depicted by means of its real-imaginary parts in Fig.
5.4
-left,
respectively, by its equivalent Bode-plot representation in Fig.
5.4
-right. Notice that,
in these figures, we show the impedance in two cases: when the airway tube is mod-
eled by an
R
-
C
element, and an
R
-
L
-
C
element, respectively. This comparison
allows capturing the effect of the inertance element, while the frequency is increas-
ing.
In the Bode plot, a variation of the phase between 0
◦
=
26
◦
and
−
can be observed
10
−
4
,
10
2
∈[
]
in the frequency interval
ω
(rad/s). However, this is not the constant-
phase effect as expected from theory, because the fractional-order value
n
would
have to be zero, or 0
.
3, respectively. If we use the analytical form of (
5.25
) to calcu-
late this fractional-order value, we obtain
n
0
.
59. This mismatch between simula-
tion and theory is due to the fact that the condition of
λ>
1 is not fulfilled in (
5.19
).
We shall discuss this aspect later on.
=
