Frequency Domain: Parametric Model Selection and Evaluation - The Human Respiratory System - page 98

The last model (FO4) proposed for evaluation in this chapter is based on FO3, but
takes into account the ability of fractional-order terms to model constant elements
in the gain of the term. This results in a simplification of the FO3 model to

1
C r s β r

L r s α r Z FO4 (s) = + (7.5)

which does not contain the resistance term R r . Indeed, the theory of fractional-order
appearance in ladder networks shows that the effects of R r are indirectly captured in
the values of the FO terms and FO coefficients [ 76 , 118 ]. Hence, if it turns out that
the R r term in FO3 will not give significant values, then FO4 will have less model
parameters to be interpreted by the clinicians.
In order to illustrate the above rationale, two groups of respiratory impedance
data will be employed in an example. The remainder of this section presents the
results of these models and makes a discussion on the model parameter values.

7.2.1 Relation Between Model Parameters and Physiology

Recalling here the identification procedure described in Sect. 3.1 using ( 3.8 ), one
obtains the complex impedance by means of its real and imaginary parts as a func-
tion of frequency. From the real and imaginary parts of the complex impedance,
the model parameters of ( 7.5 ) were identified. The modeling errors were calculated
with ( 3.12 ).
From the identified model parameters one can derive the tissue damping G r and
elastance H r , defined as [ 57 , 58 ]

cos β r

1
C r ω β r

π
2

G r =

sin β r

(7.6)

1
C r ω β r

π
2

H r =

both in (l/kPa). The hysteresivity coefficient η r (dimensionless) is defined as [ 42 ]

G r
H r

η r = (7.7)

This parameter characterizes the heterogeneity of the lung tissue and has been
shown to vary significantly with pathology. Since all these parameters from ( 7.6 )
and ( 7.7 ) are frequency dependent, the lumped identified values will in fact repre-
sent an averaged value over the 4-48 Hz frequency range.
Apart from the identified model parameters, some additional parameters are in-
troduced in this analysis. The real part of the complex impedance at 6 Hz ( R 6) can
be used to characterize the total resistance at this frequency, a parameter often en-
countered in clinical studies. The resonant frequency ( Frez ) could also be used as a
classifying parameter, since it has been shown that the balance between elastic and
inertial properties change with pathology.