Biomedical Engineering Reference
In-Depth Information
n)ω
n
cos
nπ
2
L
r
ω
2
T(jω)
=
A(
1
−
−
+
B
j
A(
1
n)ω
n
sin
nπ
2
+
−
(1.5)
This function describes the behavior of the balloon in a plethysmograph, while un-
dergoing sinusoidal forced oscillations. One year later, in 1970, he published the
results obtained by identifying such a model on excised cat lungs [
62
]. He then sug-
gests to do the PV approximation with a transfer function which has an imaginary
part independent on frequency. This special property gives a phase angle which de-
creases slightly with frequency (quasi-constant). Playing with these models on the
data for the PV curves, he discusses the viscoelastic properties of the rubber balloon
versus the excised cat lungs. In doing so, he combines several idealized mechanical
elements to express viscoelasticity in a mechanical context. Some fragile steps are
then directed towards concepts of stress relaxation and dynamic hysteresis of the
lungs.
Two decades later, Hantos and co-workers in 1992 revised the work of Hilde-
brandt and introduced the impedance as the ratio of pressure and flow, in a model
structure containing a resistance
R
r
, inertance
L
r
and compliance
C
r
element, as in
(
3.9
)[
57
]. This model proved to have significant success at low frequencies and has
been used ever since by researchers to characterize the respiratory impedance.
In the same context of characterizing viscoelasticity, Suki provided an overview
of the work done by Salazar, Hildebrandt and Hantos, establishing possible scenar-
ios for the origin of viscoelastic behavior in the lung parenchyma [
143
]. The authors
acknowledge the validity of the models from (
1.1
) and the FO impedance from [
57
]:
1
C
r
s
β
r
Z
r
(s)
=
(1.6)
in which the real part denotes elastance and the imaginary part the viscance of the
tissue. This model was then referred to as
the constant-phase model
because the
phase is independent of frequency, implying a frequency-independent mechanical
efficiency. Five classes of systems admitting power-law relaxation or constant-phase
impedance are acknowledged [
143
].
•
Class 1
:
systems with nonlinear constitutive equations
; a nonlinear differential
equation may have a
At
−
n
solution to a step input. Indeed, lung tissue behaves
nonlinearly, but this is not the primary mechanism for having constant-phase be-
havior, since the forced oscillations are applied with small amplitude to the mouth
of the patient to ensure linearity. Moreover, the input to the system is not a step,
but rather a multisine.
•
Class 2
:
systems in which the coefficients of the constitutive differential equations
are time-varying
; the linear dependence of the pressure-volume curves in loga-
rithmic time scale does not support this assumption. However, on a larger time
interval, the lungs present time-varying properties.
•
Class 3
:
systems in which there is a continuous distribution of time constants that
are solutions to integral equations
. By aid of Kelvin bodies and an appropriate
