Biomedical Engineering Reference
In-Depth Information
depends on the ratios between the ladder network parameters, since its value varies
from one scenario to the other.
For the
simulated nominal case
observed in Fig.
5.12
, with averaged ratios
from (
5.42
), the calculated fractional-order value is
n
=
0
.
59, corresponding in the
53
◦
. This value does not corre-
spond to our simulation results, due to the fact that the values for
λ
are sub-unitary,
therefore not fulfilling the condition for convergence. Since the formula for
n
is
valid only for positive values of the logarithm, for
λ<
1 the formula (
5.25
) will be
inaccurate.
In the
simulated pathologic case
, one may observe in Fig.
5.12
that in the fre-
quency interval
ω
impedance plots to a phase constancy of about
−
10
−
4
,
10
−
1
(rad/s) the constant-phase effect is visible. The
magnitude decreases with 45 dB over four decades, which results in a change of
about
∈[
]
11 dB/dec. The phase exhibits a phase-locking within this frequency range,
around the value of
−
48
◦
.From
−
−
n
·
20 dB/dec
=−
11 we have
n
≈
0
.
55, and
90
◦
=−
48
◦
, it follows that
n
from
0
.
53. If one calculates the fractional-
order value from (
5.25
) with averaged ratio values from (
5.43
), one comes up with
n
=
−
n
·
≈
0
.
59 for the admittance in (
5.26
), which corresponds closely to the value ob-
served in Fig.
5.12
. This result proves that the formula (
5.25
) for calculating
n
is
valid in the limit if and only if all convergence conditions are fulfilled.
In the
simulated extended case
,Fig.
5.13
shows the effect of increasing the num-
ber of cells, while keeping the same ratios as in (
5.43
). Increasing the number of
cells will help convergence in the limit; recall here that the formula (
5.25
) was de-
rived from (
5.17
) assuming
N
. Increasing the number of cells in the ladder
network leads indeed to a constant-phase behavior corresponding to a similar phase
value as in the pathologic case, but its effect will be visible over a broader frequency
band. It is worth to notice that the frequency band is linearly dependent with the in-
creasing in the number of cells.
Our findings justify the use of a FO parametric model to characterize the res-
piratory input impedance [
69
]. Hence, we established that the origins of the FO
behavior are not only the viscoelastic properties of the lung tissue, typically visible
at low frequencies, but also the fractal structure of the respiratory tree. It is inter-
esting to note that both viscoelasticity and diffusion appear at low frequencies; the
diffusion is not tackled in this topic. The proposed model allows variations in the
parameters by altering the elastic modulus
E
and cartilage fraction
κ
,aswellas
variations in the airway geometry by altering the airway radius
R
, length
, and
thickness
h
. Although preserving its fractal structure, these alterations can be cor-
related to airway remodeling in pathology, leading to different values in the ratios,
e.g. those given in (
5.43
). The results depicted in Figs.
5.12
and
5.13
indicate that
viscoelastic and diffusion phenomena are not the only origin of the phase constancy
(non-integer order) models for the lungs, but also the intrinsic recurrent geometry
has a similar contribution. Although simple, the nominal case of this model proves
to be reasonably close to the data measured from the healthy subjects, showing that
it is able to capture the intrinsic properties of the respiratory tree.
→∞
