Estimating respiratory mechanical parame

1 Introduction

POST-OPERATIVE analysis of respiratory mechanics in
mechanically ventilated patients is useful for evaluating
patient status and assessing the effect of therapy in intensive
care units (ICUs). It is important to have knowledge of two
main quantities which characterise breathing mechanical
properties; total compliance, often measured under static
conditions as an indicator of lung and chest-wall elasticity
(Ross1 et al., 1985; GILLARD et al., 1990), and total flow
resistance, which reflects properties of both the tissue and
the peripheral airways (GoTrERIED et al., 1985; BATESet al.,
1986).
Several studies characterising the main aspects of
breathing mechanics have been published in recent years.
These have used different lumped-parameter models,
ranging from the simple two-element resistance-compliance
linear model to more sophisticated physiological models
which include tissue viscoelasticity, the inertial effects of the
airways and branching networks (LUTCHENand COSTA,
First received 3 April and in final form 29 December 1992

@ IFMBE: 1994

Medical & Biological Engineering & Computing

1990), to non-linear models (BEN-HAIM et aL, 1988).
However, working with non-linear models precludes the use
of many powerful concepts usually adopted in the clinical
investigation of respiratory mechanics (for example, the use
of frequency-domain analysis. Bode diagrams and input
impedance). For this reason, we preferred to the limit the
present analysis to linear models.
For identification purposes, the estimation of parameters
in the linear models has usually been achieved starting from
relatively controllable measurement set-ups, either in
experiments performed on animals (BATESet aL, 1986; 1989)
or in clinical trials carried out on conscious volunteers
(MICHAELSON et aL, 1975; EVLES et al., 1982). In such
experiments, the operating conditions can be carefully
checked before each trial and the accuracy of the
measurements is guaranteed.

The situation, however, is rather different when
respiratory mechanics have to be analysed in routine
post-operative ICUs, because the safety and well-being of
the patients impose pressing constraints on measurements.
For example, in order to avoid the proliferation of
signal-acquisition devices at a patient's bed, flow is not
measured exactly at the airway opening but is assumed to

March 1994

153

be equal to the flow supplied by the ventilator, which is
generally directly available at the ventilator output panel.
Moreover, the frequency and shape of flow signals are
generally fixed or can be varied only to a minor extent.
Despite these intrinsic limitations, however, achieving an
accurate estimation of the main respiratory parameters in
routine ICUs may be of the greatest potential clinical value
for the diagnosis of and therapy for post-operative patients.

In fact, knowledge of parameters such as respiratory
resistance and compliance can provide useful suggestions
for optimising ventilation, for example in the choice of the
level of positive end-expiratory pressure (PEEP) ( S U T E R et
al., 1975; Rossl et al., 1985; SYDOW et al., 1991) or
differential lung ventilation (DLV) (ZANDSTRAet al., 1989).
In this study, we analysed the possibility of achieving
accurate estimations of respiratory parameters in mechanically ventilated patients, using pressure and flow data taken
under the operating conditions typical of a routine ICU.
Different models of respiratory input impedance were used
and their ability to reproduce real data compared. The
results obtained may contribute to the choice of the model
most suited for describing breathing mechanics in
mechanically ventilated patients. This might be of potential
aid for improving medical decisions in intensive care
monitoring.

]O 1 _

I

|

I

I I IIII

Ira"

I

I

I IIIII

t

I

1

I I lilt

I

I

I

I I II~

I

1

I

I I I

-(3

2

-~

O)

10o

4.

4,

E

10-1

I

t

I

I I I111

104

10~
f r e q u e n c y , Hz

10]

102

Fig. 1 Magnitude o f the catheter transfer function plotted against
frequency on a logarithmic scale; asterisks denote
experimental points; the continuous curve represents the
best fitting obtained using eqn. 2; the optimal values of
parameters are 6 = 0.3424, co, = 57.9673

where co, and 6 are the catheter natural pulsation and
damping factor, respectively, to be estimated on the basis
of experimental data.
If we write the transfer function magnitude in terms of
frequency, we obtain

2 Methods

Pressure at the airway opening and ventilatory flow were
measured in eight post-cardiac surgery patients mechanically ventilated in the ICU. Pressure was measured with a
flexible catheter (180 cm long, 1 mm diameter), connected
at one end to a rigid endotracheal tube (35 cm long, 9 mm
diameter) and at the other to a Hewlett Packard quartz
pressure transducer model 1290A OPT 006. The flow data,
as originally generated, were collected directly from a
Siemens Elema Servo-Ventilator model 900 C provided
with digital outputs. Special care was taken to have
negligible leaks in the ventilator-patient circuit. In fact,
leaks evaluated by computing the average air flow over ten
cycles were less than 1 per cent. The acquisition system

consisted of an IBM PC/AT computer with an A/D
convertor model PCL812PG, whose sampling frequency
was fixed at 128 Hz.
The frequency response of the catheter was evaluated by
imposing periodic pressure signals of different frequencies
at one end and measuring pressure at both ends with
identical transducers. The experimental values of the
transfer function magnitude thus obtained are shown in
Fig. 1.
According to the transmission-line theory, the transfer
function linking input and output pressure signals in a
catheter should be described using an infinite number of
complex conjugate poles and zeros, which alternate in the
complex plane. However, because the spectrum of pressure
signals used in this study does not exceed 10 Hz, we deemed
it sufficient to reproduce the catheter transfer function using
only two complex conjugate poles. As we see later, this
choice allows the catheter response to be compensated for
over the entire frequency range of practical interest.
According to the above statements, we can write the

following theoretical expression for the catheter transfer
function in the Laplace Transform domain:

co.
Gca,(s) = s2 + 2&o,s + ~o,
154

(1)

I Gr

l-

(2)
-

+ 4(o. 6 co

where f = co/2~z is the frequency.
The two unknown parameters 6 and co, in eqn. 2 were

obtained by an identification procedure, i.e. minimising the
following least-squares criterion function:
N

Fc,,(6, co,)= ~

[IGca,(L, co.,g)l-G.pe(L)] 2

(3)

k=l

where N is the number of frequencies examined in the
catheter experiment, fk =cok/2~ is the kth of these
frequencies and Gspe(fk) are the corresponding experimental values of the transfer function modulus (Fig. 1).
In Fig. 1, we show the theoretical pattern of [Gc,t[ versus
frequency, computed by substituting optimal values of 6
and co, in eqn. 2. As is eVident from Fig. 1, eqn. 1 provides
a satisfactory reproduction of the catheter transfer function
over the entire frequency range of practical interest
( f < 10 Hz). However, for frequencies above 10 Hz, the real
transfer function significantly differs from the theoretical
one owing to the effect of additional complex conjugate
poles and zeros.
Before analysing the pattern of respiratory input
impedance in mechanically ventilated patients, both flow
and pressure signals were filtered to eliminate highfrequency noise. For this purpose, we implemented a
numerical FIR low-pass filter with a cut-off frequency of
13"5 Hz. This choice enabled us to filter all frequencies with
a poor signal-to-noise ratio (SNR) and for which the
catheter compensation was inadequate (Fig. 1). The FIR
low-pass filter was designed by the Hamming window
technique with 129 coefficients. As is well known, this kind
of filter does not appreciably alter signal magnitude or
phase in the passband, and so does not introduce significant
errors into the computation of input impedance.
Finally, the pressure signals, after low-pass filtering, were
sent to a correcting unit with transfer function 1/Gc,t(co).
This unit ensured a fairly accurate compensation of the

Medical & Biological Engineering & Computing

March 1994

R

C

~

T

Fig. 2 Electrical analogue of the traditional two-element model
catheter effect in the overall frequency range of practical
interest.
The design of all filtering and compensation units was
performed on an IBM PC using the MATLAB software
package for linear systems analysis (MOLER et al., 1987).
The frequency spectra were obtained by the FFT algorithm.
To this end, all signals were resampled to have exactly 256
samples per respiratory period. Resampling was achieved
using linear interpolation of the original experimental data.
As described below, the filtered and compensated
pressure and flow signals were analysed, both in the time
and frequency domains, to derive a theoretical expression
and parameter estimates for respiratory input impedance.
Minimisation of least-squares criterion functions for
estimation of input impedance and catheter transfer
function parameters were carried out on a Digital
MicroVAX II computer, using the Powell algorithm
(PowELL, 1964).
3 Modelling analysis

In order to derive the most suitable theoretical model for
analysing respiratory data, we started our study by
examining the frequency pattern of input impedance Z(e))
(i.e. the ratio of pressure and flow Fast Fourier transforms)
using Bode diagrams. As is well known from automatic
control theory, examination of Bode diagrams allows the
number and the approximate position of poles and zeros in
a transfer function to be easily evaluated, and hence the
most suitable structure for the model to be chosen.
As an example, let us consider the traditional two-element
model shown in Fig. 2, where R denotes the respiratory
resistance, including the resistance of the endotracheal tube,
and C is the compliance of the total respiratory system,
which includes the contributions of lungs and chest wall. In
this model, the input impedance magnitude settles at a
constant value as frequency increases, and the phase goes
from -re/2 to zero.
However, all Bode diagrams observed in mechanically
ventilated patients are basically different from those
102

magnitude
of Z m m

I

~"

I

~I'l

I I 11

I

I

t

I

I

predicted by the simple model of Fig. 2. A typical example
of one of these diagrams, obtained in patient PG, is shown
in Fig. 3. All the other patient diagrams share its main
features, with only minor changes from one case to another.
As is evident from Fig. 3a, the input impedance
magnitude, plotted against frequency on a logarithmic scale,
does not tend to a constant value below 10 Hz, but rather
exhibits a linear decrease with increasing frequency. Such
behaviour is indicative of the presence of a second real pole,
at a frequency below 10 Hz. This statement is supported by
further examination of the input impedance phase plotted
against frequency on a semi-logarithmic scale. It is clear
from Fig. 3b that the phase does not tend to zero, as
predicted by the classic two-element model, but settles at
about - zr/2.
Bode diagrams suggested the use of the following
expression for the input impedance in the frequency
domain:
Z(o~) = G

(4)

jco(l + 2o)%)

where G is a gain factor, j = x~--1, and T= and % are the
time constants of a real zero and of a real pole, respectively.
Eqn. 4 can be realised with two different electrical
analogues using only capacitances and resistances (Fig. 4).
In the present study, we chose the electrical analogue of
Fig. 4a because of its greater physical meaning. In fact, this
model can be obtained from the two-element model of Fig.
2 by adding a parallel compliance Cp. As pointed out in
Section 5, this additional parameter may account for both
the elastic characteristics of the upper airways and the
ventilator-patient circuit compliance.
Parameter estimation was performed in the frequency
and time domains to compare and validate the estimates.
The influence of experimental conditions on the accuracy
of parameter estimates was also evaluated. The three-model
parameters were estimated by minimising least-squares
criterion functions, i.e. the sums of squared differences
between the experimental data and the corresponding
model predictions.
3.1 Time domain
In the time domain, we defined the following criterion
function:
N

V,(0) = }-" [p(kr)

pm(kr, 0)3 2

-

(5)

k = l

where O is the parameter vector [RCCp]', N is the number
of data points, p(kT) the kth experimental pressure at the

I I

phase o f
Z rod

101

Hg slq

i

E

iw5

A

I

I

iiii

~

'

i

i

i

i

i~tl

i

l

i

i

) i i i

-l
-2

10o
10-I

Fig.3

1 + jco'c:

I

I

I

I

I lltl

i

I

i

I0-~
f, Hz
a

i

i

I0.

-3
10-I

I

( I i i i

I0-~

I0.

f, Hz
b

Examples of Bode diagrams obtained for patient PG." (a) input impedance versus frequency on a logarithmic scale.," (b) input
impedance phase versus frequency on a semilogarithmie scale, these patterns are indicative of the presence of a second real pole
below 10 Hz

Medical & Biological Engineering & Computing

March 1994

155

R

R

C

O

T

T

o

?

O
Fig. 5

R

1

Electrical analogue of the Mead four-element model

The output equation coincides with Xcp(t), and so

1t-Fig. 4

•
T

C

b
Two alternative electrical analogues of the input impedance
expression of eqn. 4; the circuit of (a) was used throughout
the present work to assign model parameters and
characterise the input impedance frequency pattern

airway opening, T the resampling period and pro(t, O) the
pressure predicted by the model.
If we take pressures at the end of the two compliances,
Xcp(t) and Xc(t), as state variables, the model differential
equations can be written as follows in matrix form:
(6)

k(t) = ax(t) + Bq(t)

where x(t) = [Xce(t) Xc(t)]' is the state vector, the dot denotes
the time derivatwe, q(t) is ventilatory flow, and matrices A
and B are

A=

RCp
1

R

P

B=

(8)

pro(t, O) = Xcp(t)

(7)

RC.

The model was solved numerically using an Euler
integration scheme with an integration step equal to the
resampling period T.
The accuracy in parameter estimation was assessed by
the percentage standard derivation of each estimate,
provided by the Powell algorithm. Furthermore, model
capacity to reproduce the experimental data was evaluated
by computing the root-mean-square error per cent (rmse%)
defined as follows:
/ F,(O)
100
rinse% =/,~](N - n) •
Ppeak

(9)

where n is the number of p a r a m e t e r s and Ppeak is the peak
value of pressure at the airway opening.
In addition, we found it interesting to estimate the
parameter values, the percentage root-mean-square error in
the time domain using the two-element model of Fig. 2 and
the Mead four-element model shown in Fig. 5 . These
estimates were acquired t h r o u g h eqns. 5 and 9. The
corresponding model predicted pressures p,,(t) were
computed using equations similar to eqns. 6-8, solved with
an Euler integration scheme.
3.2 Frequency domain
The most c o m m o n way of fitting experimental and model
data in the frequency domain is to use real and imaginary
parts of input impedance (Lu'rCHEN, 1990). Thus, we can

Table 1 Parameter estimates and their percentage standard deviations (in parentheses) obtained
from eight patients using the three-element model of Fig. 4a in the time (upper values) and
frequency (lower values) domains. The resistances and compliances are in mm Hg s litre- 1 and
in ml mm Hg- 1, respectively

156

Patient

R

(SD, per cent)

C

(SD, per cent)

Cp

(SD, per cent)

SC

7.80
7-88

(0.5)
(5.8)

45.8
47.8

(0-3)
(8-5)

4-75
4-69

(1-7)
(6-4)

MG

6-73
7.04

(1.4)
(29.6)

29-5
25-9

(0.5)
(16.1)

15.38
13.29

(1.4)
(28.0)

PG

7-33
7-39

(0.4)
(6.6)

32.4
30.7

(0.2)
(5.6)

5.49
5.44

(1.2)
(8.0)

GE

11.95
11.19

(0.4)
(4.8)

37.6
45-6

(0.3)
(21.2)

2.49
3.27

(2.7)
(1.5)

BA

9-93
10.62

(0-5)
(3.6)

40-6
55-8

(0-3)
(4.8)

4.98
4.56

(1-7)
(4.4)

BF

6-12
6-99

(0-7)
(8.1)

51.4
73.5

(0-2)
(11.4)

20-10
15-26

(1.3)
(7.4)

AE

10-49
10.40

(1.0)
(8.9)

30.8
33.6

(0-3)
(6-0)

15-70
14-67

(1.3)
(7.5)

SM

6-40
7-85

(1.3)
(8.6)

37-7
54.9

(0-5)
(9.3)

14-06
12.27

(2.1)
(8.2)

Medical & Biological Engineering & Computing

March 1994

i
iO)

t,q

'%.:"" !,'

"g"v" v""'[

N

E
0

1

2

3

4
f, Hz

5

6

7

-20'

8

0

1

2

3

4
f, Hz

a

6

5

7

8

b

Fig. 6 Frequency pattern of the (a) real and (b) imaginary parts of respiratory input impedance measured in patient PG (dotted line),
simulated with the three-element model of Fig. 4a (continuous line) ; the simulation was performed using parameter values estimated
in the frequency domain (see Table 1)

write the following criterion function:
M

FI(0 ) = ~

[Re (Z(fk)) -- Re (Z,,(fk, O))]2

k=[

+ Jim (Z(fk)) -- Im (Z,,(fk, 0))] z }

(10)

where Re (Z(fk)) and Im (Z(fk)) are the real and imaginary
parts of the experiment input impedance at the kth
frequency fk, Re(Z,,(fk, 0)) and Im(Z,,(fk, 0)) are the
corresponding real and imaginary parts of input impedance
predicted by the model, and M is the number of frequency
data points. In computing the criterion function, the
frequency was chosen to ensure a sufficient signal-to-noise
ratio. The value of this frequency was subject-dependent,
but never exceeded 8-10 Hz.
The real and imaginary parts of input impedance
predicted by the model were computed through the
following expressions, which can be deduced from the
electrical analogue of Fig. 4a:
RC 2

Re (Zm(fk, 0)) =

(11)

D

lm (am(fk, 0)) = --

C + C v + (RC)ZCpoo 2

(12)

OOkD

where D = (C + Cp)2+ (RCCpoJ) 2.
We also computed the percentage standard deviation of
estimates in the frequency domain as an index of parameter
accuracy, and the root-mean-square error using an equation
similar to eqn. 9.

4 Results

In Table 1, we present the estimated values of model
parameters and percentage standard deviations obtained in
the eight patients using the model of Fig. 4a. Two values
are shown for each parameter, with reference to the
estimations performed in the time and frequency domains.
The two different estimations of the same parameter
agreed fairly well. In no case was a parameter estimated in
the time domain with a percentage standard deviation
greater than 2-7 per cent. The error was greater when
T

15

-~

g 10
T 5
E
E 0

.

-10

N

0

I

2

3

4

5

6

7

8

E

fHz

Fig. 7

0

,

,. .~. ,* :'.-" ~ .' v" . . . . . .

N
rv

estimations were performed in the frequency domain, but
even in this case, we had a mean percentage standard
deviation of about 10 per cent.
Using the values of model parameters estimated in the
frequency domain, we then tried to reproduce the variations
of the resistive and reactive parts of the input impedance
versus frequency. Two examples are shown in Figs. 6 and
7 with reference to patients PG and BA. It is evident from
examination of these Figures that the real part of the input
impedance tends to zero more rapidly than the imaginary
part. This behaviour is in accordance with that predicted
by eqns. 11 and 12, but cannot be understood with the
traditional two-element model of Fig. 2.
An example of the results obtained in the time domain
is shown in Fig. 8. Here we show the experimental time
pattern of the ventilatory flow in patient PG (Fig. 8a), and
the corresponding experimental and model time patterns of
pressure at the airway opening (Fig. 8b). As is clear from
this Figure, model prediction of pressure at the airway
opening is almost indistinguishable from the real trace.
Finally, as specified above, we estimated the input
impedance parameters in the time domain using not only
the three-element model of Fig. 4a, but also the two-element
model (Fig. 2) and the four-element model proposed by
MEAD (1969) (Fig. 5).
The results of these estimations, shown in Table 2, can
be briefly summarised as follows. The two-element model
provides a poor fit of the experimental data; the real part
of the input impedance in particular cannot be satisfactorily
reproduced with this model.
Estimations of the parameters R, C, and Cp in the Mead
model are not significantly different from those obtained
using the three-element model of Fig. 4a. The additional
parameter R,, which represents possible resistance of the
upper airways in the Mead model, was estimated in only
three subjects (Table 2). In the remaining five cases, R,
assumes a very low value, practically indistinguishable from
zero.
A final comparison between the performances of the three
models is presented in Table 3, with reference to the
estimations achieved in the time domain. In this Table, we
present the estimated root-mean-square error of each model
computed using eqn. 9. It is clear from this Table that the

-20

II

0

.

I

'i

;'

.

.

2

_.

......

.

3

.

4

-

.. ,.,.--~--~ '.~/', ,", .... :v-- "-"" . . . . .

.

"

.

5

6

7

fHz
b

Frequency pattern of the (a) real and (b) imaginary parts of respirator)' input impedance measured in patient BA (dotted line),
simulated with the three-element model of Fig. 4a (continuous line) ; the simulation was performed usin9 parameter t'al~s est#nated
in the frequency domain (see Table 1)

Medical & Biological Engineering & Computing

March 1994

157

Table 2 Comparison of estimates of model parameters and their percentage standard detJiations (in
parentheses) obtained in the time domain by the models of Figs. 2, 4a and 5_ The resistances and the
compliances are in mm Hy s litre- 1 and in ml/mm H9, respectively

two elements

three elements

Patient

R

C

R

SC

6'23
(0-7)

48"4
(0.4)

7"80
(0"5)

MG

2"41
(1.8)

42.3
(0.7)

PG

5-23
(0.6)

GE

C

Cp

R

45"8
(0"3)

4.75
(1"7)

6"88
(1-1)

43"4
(0.4)

7-15
(2"7)

1"37
(4"6)

6"73
(1.4)

29-5
(0,5)

15"38
(1.4)

6"84
(1.4)

29-3
(0,5)

15"67
(1.5)

0"00
(--)

36.5
(0.3)

7,33
(0.4)

32-4
(0.2)

5.49
(1.2)

6.88
(1.5)

30-1
(1.5)

7,79
(5.4)

1.04
(12.3)

~0-19
(0.7)

37-5
(0.8)

11-95
(0.4)

37-6
(0-3)

2.49
(2.7)

I0.01
(1.2)

36.I
(0-6)

3-8t
(5-8)

1-37
(5-7)

BA

7.30
(1-1)

44.4
(0.6)

9,93
(0-5)

40.6
(0.3)

4.98
(1.7)

9.94
(0-5)

40,6
(0-3)

500
(1,7)

0.00
(--)

BF

2.62
(1-9)

59-3
(0-8)

6.I2
(0-7)

51.4
(0.2)

20.10
(1-3)

6.14
(1-6)

51-4
(0.5)

20.21
(2.0)

0.00
(--)

AE

3"96
(1.3)

41"5
(0.5)

10-49
(1.0)

30"8
(0.3)

15-70
(1.3)

10-49
(1.6)

30-8
(0.6)

15"70
(1.5)

0.00
(--)

SM

2.94
(2-2)

45.5
(0.6)

6-40
(1.3)

37.7
(0-3)

14.06
(2.1)

6-10
(2.8)

37-9
(1.0)

13.05
(3-2)

0.00
(--)

root-mean-square error significantly decreases from the
two-element to the three-element model. However, it does
not appreciably change when the fourth parameter R, is
taken into account.
5 Discussion

Graphic analysis of the respiratory impedance frequency
pattern in mechanically ventilated patients (Figs. 3, 6 and
7) demonstrates that, although flow and pressure signals
become quite noisy after the first harmonics, useful
information can be extracted up to 6-8 Ha. This is
supported by comparison of parameter values estimated in
the time and frequency domains (Table t). In the second
case, more accurate values for the parameters can be
obtained if frequencies up to 6-8 Hz are included in the
estimation procedure.
1
14

0.8

"70 3

0.6

12

0.4

10

0.2
0

~5 6

-0,2
r"

>o

-O.d
-0.6

2

-0.8

0

-1
0

6

time,s
a

Fig. 8

158

four elements

0

2

4

6

time,s
b

Time pattern of (a) ventilator flow in patient PG and (b)
correspondin9 pressure at the airway openin9 ; the
continuous line represents the pressure curve measured in
the patient. the dotted line is the simulation curve obtained
with the three-element model of Fig. 4a; this simulation was
performed usin9 parameter values estimated in the time
domain (see Table 1)

C

Cp

R,

The most important new aspect arising from examination
of the input impedance frequency pattern between 0"2 and
8 Hz is that the magnitude decreases linearly with frequency
when plotted on a logarithmic scale (Fig. 3a). At the same
time, the phase does not return t o zero but settles at about
--~/2. The real part of the input impedance exhibits an
evident decrease in the low-frequency range. It is well
known that this behaviour is indicative of the existence of
a second real pole in the input impedance expression, in
addition to the unique pole at the origin predicted by the
classic two-element R C model (Fig. 2).
This suggests that the two-element model is not adequate
for describing respiratory data measured in conditions
typical of a post-operative I C U . In support of this
hypothesis, we also noted that all the estimations performed
with the two-element model were characterised by
excessively high values of the root-mean-square error (Table
3), and that the real part of input impedance could not be
acceptably reproduced.
Several authors have observed that the real part of
respiratory impedance decreases with frequency, even in
healthy subjects, in the range 0.1-4 Hz (HANTOS et al., 1986;
LtSTCHEN et al., 1988) and that the traditional-two-element
model is not adequate to describe respiratory mechanics in
these patients. Various four-element models, such as the
Mead model (MEAD, 1969), the Otis model (OTIs et al., 1956)
and the viscoelastic model by Bates (BATES et at., 1986),
have been proposed.
However, examination of Bode diagram in our patients
(Fig. 3) and comparison of p a r a m e t e r values estimated in
the time and frequency domains (Tables 2 and 3) clearly
demonstrate that introduction of a fourth resistive
parameter is not necessary to properly describe respiratory
data in post-operative mechanically ventilated cardiac
patients. In particular, all the p r o p o s e d four-element models
exhibit an input impedance with two real zeros and two
real poles. This means that the phase should approach zero
as frequency increases. Such behaviour, however, has never
been observed in patient data. We maintain that it is
impossible to identify a second real zero in pressure and
flow signals monitored in routine post-operative ICUs.
The most suitable model for analysing respiratory data

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March 1994

Table 3 Comparison of percentage root-mean- square errors
obtained in the time domain using the models of Figs. 2, 4a and 5

Patient

two elements

three elements

four elements

SB
MG
PG
GE
BA
BF
AE
SM

3.60
6.07
2.71
5'91
6"58
7.41
4-88
6.73

2.25
1-55
1.74
3"92
1"76
1-91
2-06
2"86

2"23
1-55
1"73
3"89
1.64
1-91
2"05
2"86

in ICUs therefore seems to be a three-element model with
two compliances but only one resistance (Fig. 4). This is
supported by the fact that the root-mean-square error
significantly improves from the two-element to the
three-element model (Table 3). At the same time, inclusion
of a parallel compliance Cp significantly alters the estimates
of the other parameters (Table 2). In other words, it is
necessary to consider compliance in order to correctly
evaluate the elastic and resistive properties of respiratory
mechanics.
In particular, when using the modified three-element
model we obtained values of resistance R significantly
higher than those derived from the two-element model. This
can be understood if we consider that the main effect of the
additional parameter Cp is to cause a sharp low-frequency
decrease in the real part of the input impedance. Cp has less
effect on the estimation of the other capacitance, C,
provided the ratio of the two capacitances C/Cp is
sufficiently high.
When comparing the Mead four-element model and the
three-element model of Fig. 4a, we observed no significant
change in the parameter values in five cases out of eight
(Table 2). In the remaining three cases, the addition of upper
airway resistance Ra only slightly modified the other
parameter estimates. Moreover, and most importantly, in
no case did the root-mean-square error appreciably
improve as a consequence of the introduction of a fourth
resistive parameter (Table 3). We can thus conclude that a
fourth parameter is not identifiable in the ICU data at our
disposal.
At this point, it is of interest to consider the physical
significance of the additional parameter Cp included in the
three-element model. A parallel compliance might be
introduced by the ventilator-patient circuit. In fact, the
ventilatory systems usually adopted in cardiac ICUs have
a pneumatic connection to the patient, the compressible
volume of which is not negligible (EPSTEINand EPSTEIN,
1979; GUILLAUMEand BORRELLO,1991). Of course, in the
operating conditions typical of post-operative ICUs, flow
is not measured directly at the airway opening of the
patient, but is assumed equal to the flow imposed by the
ventilator. Owing to the compliance of the ventilator, the
amount of flow entering the airways may be rather different
from that assumed in the computations, thus altering the
value of the input impedance.
In particular, Epstein and Epstein (EPSTEINand EPSTEIN,
1979) report that, in practical ventilators designed for
adults, the magnitude of compliance generally lies in the
range 2.0-4-5 ml mm Hg- 1, a value which may account for
almost 100 per cent of the parameter Cp in patients SC, GE
and BA, but only 22 per cent of it in patient BF. However,
it should be noted that compliance of the ventilator-patient
circuit may vary widely due to causes not easily controllable
in routine ICUs (such as the type of tubing or the level of
water in the humidifier). It is therefore not easy to provide
specific experimental values for this parameter a priori.
Medical & Biological Engineering & Computing

A second factor capable of explaining the presence of a
parallel compliance Cp might be shunting of the upper
airways. Several authors (PESLIN et el., 1985; CAt:BERtHS
and VAN DE WOESTIJNE, 1989) report that when forced
oscillations are applied to the mouth to measure respiratory
impedance, part of the flow is lost by motion of the upper
airway walls, especially of the cheeks.
A decrease in the real part of respiratory input impedance
above 4 Hz, similar to that reported in our study, has also
been observed by various authors in patients with chronic
obstructive pulmonary diseases (MICHAELSONe t eL, 1975;
EYLES and PIM~EL, 1981; EYLES et eL, 1982; LUTCHEN et
eL, 1990). However, they failed to observe a significant
dependence of Re (Z) on frequency in normal adult subjects.
This observation is in accordance with the findings of other
authors who used the interrupter technique (CHENG et el.,
1959; FRANKet eL, 1971). In these studies, airway resistance
was underestimated if measured in patients with airway
obstruction, in contrast to findings in normal subjects.
The abnormalities in the input impedance pattern
observed in chronic patients are believed to be primarily a
consequence of an increase in resistance, produced by
diffuse obstruction, which causes the extrathoracic and
upper airways to become a more important contributor to
shunting flow (LUTCHENe t eL, 1990). This would result in
a greater negative phase angle as frequency increases
(MICHAELSON et eL, 1975).
In mechanically ventilated patients, however, the
contribution of the cheeks, pharynx and extrathoracic
trachea to airway compliance is absent, because the patients
are intubated. Moreover, according to data obtained by
FOUKE et aL (1989), in healthy subjects, the compliance p.u.
length of the human intrathoracic trachea is probably no
greater than 0-03-0.05 cm 2 mm Hg- t. Using this information, and considering the typical length of the human
intrathoracic pathway, we argue that the overall compliance
of the airways in mechanically ventilated patients only
contributes to a minor portion of the estimated values of
Cp. In support of this hypothesis, MCRCIANO et aL (1982)
and GoTrFRIED et el. (1985) observed that the interrupter
technique works properly even in patients with severe
obstructive diseases, provided that the upper airways are
bypassed, as by tracheostomy or endotracheal intubation.
A third hypothesis is that the parameter Cp may not be
imputable to a real capacitance which shunts the flow
imposed by the ventilator, but may rather be affected by
the status of the lungs and chest wall. Parallel pathway
inhomogeneities, as proposed by OTIS et el. (1956), or
viscoelasticity of the lungs and chest tissues, as suggested
by BATESet el. (1986), might be responsible for the observed
low-frequency decrease in the real part of input impedance.
Indeed, in this case, the three-element model would
simply be a simplification of the Otis or Bates four-element
models. This simplification would be justified by the fact
that the fourth parameter is not identifiable in the pressure
and flow data available in routine cardiac ICUs. However,
the decrease in the real part of respiratory impedance,
attributed to pathway inhomogeneities in the Otis model,
occurs at frequencies much lower than those reported in the
present study. For example, LUTCHEN et al. (1988)
demonstrated that frequencies below 0.25 Hz should be
included to correctly estimate pulmonary inhomogeneities
in healthy subjects by the Otis model. Even in patients with
respiratory failure, the Otis model can be identified only
including frequencies below 1.0Hz (AvANZOLINI and
BARBINI,1984; AVANZOLINIet el., 1986).
In this regard, it is worth noting that all the patients
analysed were classified as respiratorily normal before
surgery on the basis of spirometric data (forced vital

March 1994

159

capacity, forced expiratory volume in 1 s, m i d - m a x i m u m
expiratory flow rate and flow rates with 50 and 25 per cent
of vital capacity remaining). Moreover, the number of
low-frequency d a t a points at our disposal ( f < 1.0 Hz) was
very modest. This explains why the Otis model is not
a p p r o p r i a t e for reproducing the patterns of input
impedance observed in this study.
In conclusion, the most i m p o r t a n t new result arising from
the present study is that a three-element model should be
used to analyse pressure and flow respiratory d a t a in
routine post-operative ICUs. W h a t e v e r is physical meaning,
inclusion of a parallel compliance is absolutely necessary to
achieve better characterisation of elastic and resistive
respiratory parameters in mechanically ventilated patients.

Acknowledgments--This work was supported by the Italian
Ministry for University, Scientific & Technological Research
(MURST) and by the US Research Foundation under NSF grant
BCS-9011168 and NIH grant HL-31248.

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Author's biography
Paolo Barbini was born in Monteroni d'Arbia,
Siena, Italy, in 1952. He received the Dr. Ing.
degree in Electronic Engineering from the
University of Firenze, Italy, in 1975, and the
Diploma in Biomedical Engineering from the
University of Bologna, in 1978. From 1975 to
1976 he was with the School of Engineering,
University of Firenze, and in 1976 he joined the
School of Medicine, University of Siena, where
he is currently Full Professor of Biomedical Engineering and Head
of the Clinical Engineering Unit. His main research interests
include measurement, modelling and identification of the
cardiorespiratory system.

Medical & Biological Engineering & Computing

March 1994