Journal of Applied Physiology
Vol. 82, No. 6,
pp. 1963-1971,
June 1997
GAS EXCHANGE, MECHANICS, AND AIRWAYS
Evaluation of estimates of alveolar gas exchange by using a
tidally ventilated nonhomogenous lung model
Thierry
Busso and
Peter A.
Robbins
University Laboratory of Physiology, Oxford OX1 3PT, United
Kingdom
ABSTRACT
- INTRODUCTION
- METHODS
- RESULTS AND DISCUSSION
- ACKNOWLEDGEMENTS
- FOOTNOTES
- REFERENCES
ABSTRACT 
Busso, Thierry, and Peter A. Robbins. Evaluation of
estimates of alveolar gas exchange by using a tidally ventilated nonhomogenous lung model. J. Appl.
Physiol. 82(6): 1963-1971, 1997.
The purpose
of this study was to evaluate algorithms for estimating
O2 and
CO2 transfer at the
pulmonary capillaries by use of a nine-compartment tidally
ventilated lung model that incorporated inhomogeneities in
ventilation-to-volume and ventilation-to-perfusion ratios.
Breath-to-breath O2 and
CO2 exchange at the capillary level and at the mouth were simulated by using realistic cyclical breathing patterns to drive the model, derived from 40-min recordings in six resting subjects. The SD of the breath-by-breath gas exchange at
the mouth around the value at the pulmonary capillaries was 59.7 ± 25.5% for O2 and 22.3 ± 10.4% for CO2. Algorithms
including corrections for changes in alveolar volume and for changes in alveolar gas composition improved the estimates of pulmonary exchange, reducing the SD to 20.8 ± 10.4% for
O2 and 15.2 ± 5.8% for
CO2. The remaining imprecision of
the estimates arose almost entirely from using end-tidal measurements
to estimate the breath-to-breath changes in end-expiratory alveolar gas
concentration. The results led us to suggest an alternative method that
does not use changes in end-tidal partial pressures as explicit
estimates of the changes in alveolar gas concentration. The proposed
method yielded significant improvements in estimation for the model
data of this study.
breath-to-breath variability; oxygen uptake; carbon dioxide
elimination; ventilation-to-volume ratios; ventilation-to-perfusion
ratios
INTRODUCTION
VARIOUS METHODS have been proposed to estimate
breath-to-breath gas transfer at the pulmonary capillaries from
measurements made at the mouth combined with compensation for changes
in pulmonary gas stores (2, 3, 7, 11, 13). Estimation of the change in
pulmonary gas stores includes allowing for both changes in alveolar gas
concentrations and for changes in alveolar volume. A single-compartment
model of the lung represented by a nominal volume at the end of each
expiration is used in this process. Whereas end-tidal measurements are
used to estimate the composition of alveolar gas, the methods proposed
to estimate alveolar gas exchange (2, 3, 7, 11, 13) differ in the
assessment of the end-expiratory nominal lung volume used in the
computations.
Because no direct measurement of gas transfer at the pulmonary
capillaries is available, the different estimates cannot be compared
with a standard reference. The various proposed methods yield estimates
of alveolar gas exchange with differences in breath-to-breath variability (5). The technique proposed by Swanson (11) yields a
relatively smooth estimate of breath-to-breath alveolar gas exchange
(5, 10). Nevertheless, it is difficult to assess whether the
interbreath variability in alveolar gas exchange arises from artifacts
in the computation methods or is a physiological phenomenon (5).
The purpose of this study was to evaluate the methods for estimating
alveolar gas exchange proposed by Auchincloss et al. (2), Wessel et al.
(13), and Swanson (11) by using a mathematical model of a tidally
ventilated and steadily perfused lung that includes within its
structure unevenness in ventilation-to-volume and
ventilation-to-perfusion ratios. Alveolar gas transfer has been
simulated with the model by using realistic breathing patterns derived
from experimental data. The simulated gas exchange at the pulmonary
capillaries has been compared with the estimates obtained with the use
of alternative algorithms applied to the simulated gas exchange and
end-tidal values at the mouth. The following questions have been
addressed: 1) How much variability is there in gas exchange at the level of the pulmonary
capillaries that is induced by the normal variability in
breathing pattern? 2) How do the
three different algorithms compare in estimating the gas exchange at
the level of pulmonary capillaries?
3) How much error occurs through
using the estimates of lung volume or change in lung volume obtained
via the algorithm compared with using
true1
(model) values for lung volume? 4)
How much error occurs through using the end-tidal values as estimates
of alveolar gas composition compared with using true (model) values for
alveolar gas composition? The findings led us to develop a new method
for estimating breath-to-breath alveolar gas transfer not using
explicit estimation of the change in alveolar composition between two
subsequent breaths. Finally, the variability in the estimates of
alveolar gas exchange from the different algorithms has been compared
by using the real experimental data associated with the breathing
patterns that were used to drive the model.
METHODS
Glossary
| FAgas |
Alveolar end-expiratory gas fraction
|
FA gas |
Alveolar end-expiratory gas fraction from previous breath
|
 gas |
Mixed-expired gas fraction
|
| FETgas |
End-tidal gas fraction
|
| FETgas |
End-tidal gas fraction from previous breath
|
 gas |
Mixed-inspired gas fraction
|
| VI |
Inspired volume
|
| VE |
Expired volume
|
| VL |
End-expiratory nominal lung volume
|
| VL |
End-expiratory nominal lung volume from previous breath
|
| VAgas |
Volume of gas exchanged for breath at the alveolar level
|
 Agas |
Rate of gas exchange for breath at the alveolar level
|
| V0 gas, 0 gas |
Estimate of VA gas and
A gas for method
of Wessel et al. (13)
|
| VFRCgas, FRCgas |
Estimate of VA gas and
A gas for method
of Auchincloss et al. (2)
|
| Vmgas |
Volume of gas exchanged for breath at the mouth
|
 FAgas |
Change in alveolar end-expiratory gas fraction, i.e.,
FAgas  FAgas
|
| FETgas |
Change in end-tidal gas fraction,
i.e., FETgas FETgas
|
| Sgas |
Change in pulmonary gas store
|
| VL |
Change in end-expiratory nominal lung volume, i.e., VL VL
|
Model simulations.
The model of the lung was composed of nine separate compartments and
was similar to the model of Arieli and Fahri (1) in the steadily
perfused case. Each compartment was modeled as a single alveolar space
ventilated through a constant-volume dead space and steadily perfused
by a set of capillaries. The relative values for end-expiratory
alveolar volume, ventilation, and perfusion for each unit were taken
from the data of West (14). The values for the sum of all nine
compartments were set to 2,500 ml for functional residual capacity
(FRC), 5.1 l/min BTPS for alveolar ventilation, and 6 l/min for pulmonary blood flow. Each alveolar unit
was connected to a portion of the anatomical dead space. The total
anatomical dead space was given a value of 150 ml, and the fraction
associated with each lung unit was set to be the same as the fractional
ventilation. All nine portions of anatomical dead space were joined
onto a single apparatus dead space set to 100 ml.
The variables of the model were first initialized by using the mass
balance equations for a steadily ventilated homogenous (i.e., ideal)
lung. Initial gas pressures within the alveoli were set to be the same
in all compartments and were estimated from the alveolar gas equations
by using a total alveolar ventilation of 5.1 l/min
BTPS,
O2 uptake
(O2) set to 300 ml/min
STPD and CO2 uptake
(CO2) to 240 ml/min
STPD. End-capillary
PO2 (PaO2) and
PCO2
(PaCO2), respectively, were taken
to be 4 Torr lower and 1 Torr higher than the alveolar values,
respectively, based on the data of West (14). End-capillary contents of
O2 and
CO2 were calculated
from PaO2 and
PaCO2. The
O2 dissociation curve was taken
from Lodbell (9), and the CO2
content was calculated from the equation of Douglas et al. (6). Mixed
venous contents of O2 and
CO2 were then calculated to give
the predetermined O2 (300 ml/min) and CO2 (240 ml/min) by using the Fick principle. This gave values for the mixed
venous contents of 14.6 and 52.6 ml/100 ml, for
O2 and
CO2, respectively. These mixed
venous compositions were used throughout the model simulations for all
the nine lung compartments.
Once the venous contents for O2
and CO2 have been set, the model
can be set in motion by using some cyclical breathing pattern. The
ventilation of each lung unit is calculated every 20 ms from the
breathing pattern and the relevant fraction of total ventilation ascribed to each lung unit. The model equations were applied for all of
the nine lung units through the respiratory cycle, with an incremental
time of 20 ms. The capillary gas exchange for each unit for the 20-ms
period is calculated (assuming no diffusion limitation) from the
alveolar PO2 and
PCO2
(PAO2 and
PACO2,
respectively), the mixed venous
PO2 and
PCO2, and the fractional blood flow
ascribed to the unit. From the ventilation and gas exchange over the
20-ms period, the PO2 and
PCO2 for the alveolar space and dead
space of each unit may be updated. This cycle is repeated every 20 ms.
No gradient for diffusive and convective mixing was assumed in the
computations.
The second stage of the initialization process is to determine
appropriate values for PO2 and
PCO2 in each of the nine compartments
of the model. This was achieved by using a typical breathing pattern,
as illustrated in Fig. 1. The ventilatory profile used for this initialization procedure was adapted for each
experimental data set to match the inspiratory and expiratory durations
to the mean values for the data set. The alveolar ventilation was
maintained at 5.1 l/min. In each lung unit, there will be small changes
in volume in the model occurring over each respiratory cycle because
CO2 output from the pulmonary
capillaries will not in general equal
O2 uptake by the capillaries. To
prevent this from causing progressive decreases or increases in
alveolar volume over time, an adjustment was made to expiratory flow
for each lung unit in the current breath based on the difference
between O2 and
CO2 capillary gas exchange that
had occurred in the preceding breath. The respiratory cycle was
repeated for successive breaths, until
PAO2 and
PACO2 at end expiration did
not differ between two subsequent breaths by >0.01 Torr in any lung compartment (this took between 40 and 57 breaths). Figure
2 gives an example of intrabreath variation
in PAO2 and
PACO2 and gas fluxes for a
tidal volume (VT) of 590 ml,
an inspiratory time of 1,500 ms, and an expiratory time of 2,500 ms.
Fig. 1.
Profile of total respiratory flow (in l/s) used to initialize model
simulations for a tidal volume of 590 ml, inspiratory time of 1,500 ms,
and expiratory time of 2,500 ms.
[View Larger Version of this Image (11K GIF file)]
Fig. 2.
Intrabreath variations in alveolar (alv.) gas pressures and gas
exchange at the capillary level for breathing pattern given in Fig. 1.
A and B: alveolar PO2 and
PCO2 in 3 lung compartments (units),
respectively. C: mean alveolar gas
pressures (volume-weighted) averaged across 9 lung compartments.
D: total gas exchange at pulmonary
capillaries summed across 9 lung compartments.
[View Larger Version of this Image (21K GIF file)]
Once the model had been fully initialized, it could be used to simulate
breath-to-breath gas exchange in real breathing sequences. Data were
collected over 40 min from six seated subjects during air breathing at
rest. The volumes and flows were measured by a
turbine volume-measuring device and pneumotachograph connected in
series (8). PO2 and
PCO2 at the mouth were measured
continuously by using a mass spectrometer. Table 1 gives the characteristics of the
respiratory data collected. The flow data from the pneumotachograph
were calibrated every half cycle by using the values for inspired
and expired tidal volume recorded by the turbine
volume-measuring device. The flows were then corrected to
BTPS and integrated to yield a
continuous record of volume. The volume sequence was then smoothed to
remove any long-term trends in FRC and scaled so that the mean alveolar ventilation was equal to the value designated in the model.
|
Table 1.
Means and coefficients of variation of respiratory variables
|
| Subject No. |
No. of Breaths
|
VTI, liters
|
VTE, liters
|
TI, s
|
TE, s
|
E, l/min
|
| Mean
|
CV, % |
Mean |
CV, % |
Mean |
CV, % |
Mean |
CV, % |
Mean |
CV, % |
|
| 952 |
630 |
0.62 |
18.6 |
0.60 |
20.0
|
1.46 |
18.0 |
2.32 |
26.2 |
9.7 |
17.6 |
| 971 |
367
|
0.93 |
25.8 |
0.93 |
27.3 |
2.96 |
51.9 |
3.31 |
24.2
|
9.5 |
36.3 |
| 973 |
664 |
0.63 |
39.1 |
0.68 |
40.0
|
1.57 |
24.0 |
1.78 |
19.9 |
12.0 |
30.4 |
| 997 |
504
|
0.51 |
8.7 |
0.52 |
8.9 |
1.93 |
18.7 |
2.80 |
12.6
|
6.6 |
11.7 |
| 998 |
534 |
0.69 |
29.6 |
0.71 |
28.0
|
1.98 |
32.3 |
2.76 |
24.2 |
9.0 |
21.5 |
| 999 |
418
|
0.79 |
14.0 |
0.79 |
14.8 |
2.18 |
21.4 |
3.60 |
17.5
|
8.4 |
17.4 |
|
|
VTI, inspiratory tidal volume;
VTE, expiratory tidal volume; TI,
inspiration time; TE, expiration time;
E, expired ventilation; CV, coefficient
of variation.
|
|
The respiratory volume sequences were then used to drive the model to
simulate breath-to-breath variations in pulmonary gas stores and
pulmonary gas exchange. The breath-to-breath values for alveolar gas
exchange were summed across all lung compartments through the
respiratory cycle to provide true (model) values for breath-to-breath
gas exchange at the pulmonary capillaries. The profiles of the gas
tensions at the frontier between anatomic and apparatus dead spaces
were used to compute the breath-to-breath gas exchange at the mouth and
the end-tidal composition. These variables were used to calculate the
estimates of breath-to-breath alveolar gas exchange by using the
algorithms described below.
Algorithms for computing breath-to-breath gas exchange at pulmonary
capillaries.
Auchincloss et al. (2) proposed the basic algorithm for estimating
breath-to-breath gas exchange at the pulmonary capillaries with the use
of a correction for changes in pulmonary gas stores. The
breath-to-breath gas transfer at the alveolar level is given for each
gas by the measurement at the mouth minus the change in pulmonary gas
stores
(Sgas)
|
(1)
|
Note
that for CO2 elimination, the
signs in Eq. 1 should be inverted. The
Sgas is computed as follows
|
(2)
|
VL
is estimated by assuming that the net transfer of
N2 at the alveolar level is equal
to zero
|
(3)
|
Auchincloss
et al. (2) and other groups (3, 7) assumed that the
VL term could be treated
as a constant and equal to the subject's FRC measured at rest with
other techniques. If FAgas and
FAgas
are approximated by
FETgas and
FETgas
and VL is set to equal
FRC, Eqs. 1, 2, and
3 describe the method proposed by
Auchincloss et al. (2).
Wessel et al. (13) proposed to omit in Eq. 1 the term containing
VL (thus assumed to be 0)
because the differences in alveolar fractions between two successive
breaths could be considered small. With
FAgas or
VL set equal to 0, Eqs. 1, 2, and 3 yield
|
(4)
|
which,
when the alveolar fractions are approximated by using end-tidal
fractions, is the estimate of alveolar gas exchange proposed by Wessel
et al. (13).
Swanson (11) adopted a different approach to assign a value to
VL. Instead of setting
VL either to 0 or to FRC,
he computed an effective lung volume (ELV) that was taking part in gas
exchange by determining the value of
VL that minimized
breath-to-breath variations in gas exchange at the pulmonary
capillaries.
All three algorithms were applied to the model data for end-tidal
values together with values for gas exchange at the mouth. The gas
exchange at the pulmonary capillary calculated by using each algorithm
could then be compared with the actual gas exchange at the pulmonary
capillaries. Additionally, to evaluate the error caused by the various
assumptions relating to VL in
the three algorithms, alveolar gas exchange was estimated by using
Eqs. 1, 2, and 3 with the true breath-to-breath
alveolar volumes of the model. Moreover, to evaluate the error caused
by using end-tidal values as estimates of alveolar fractions in the
algorithms, the computations were repeated by using the true
FAgas,
averaged according to the volume of gas within each compartment of the
lung.
Finally, the algorithms were applied to the real (nonmodel)
breath-to-breath gas exchange and end-tidal values associated with the
respiratory data used for driving the ventilation of the model lung.
This enables a comparison between the variations in gas exchange
computed by using real data with those computed from the model for the
three algorithms by using a similar breathing pattern. FRC for the
original data sets was estimated from anthropometric data (4).
RESULTS AND DISCUSSION
Precision of the estimates at the pulmonary capillaries.
Figure 3 illustrates successive values for
gas exchange over 50 breaths for one of the model simulations. First,
it can be seen that, under conditions of breathing air at rest, there
is little variation in true O2
exchange at the pulmonary capillaries. In contrast, variations in
breathing pattern induce considerable variability into true
CO2 exchange at the pulmonary
capillaries. It can also be seen that all three algorithms reduce
considerably the breath-by-breath variations in gas exchange when
compared with gas exchange values at the mouth. There is the suggestion that the algorithms may differ in the accuracy with which they estimate
the exchange at the pulmonary capillaries, and this is explored further
in Table 2.
Fig. 3.
Breath-to-breath gas exchange at mouth, together with estimates
at level of pulmonary capillaries, using various algorithms plotted
around the true (model) value, over 50 successive breaths for model simulations. Respiratory flow data from
subject 952. O2,
O2 consumption;
CO2,
CO2 output;
VL, end-expiratory lung volume from previous breath; FRC, functional residual capacity; ELV,
effective lung volume. Bottom:
estimates of alveolar
O2 and
CO2
obtained by minimizing their respective breath-to-breath variations (Eq. 7) subject for each
breath to the constraints (Eqs. 5 and 6) that
O2
and
CO2
lie between the estimates from Wessel et al. (Ref. 13;
VL = 0) and
from Auchincloss et al. (Ref. 2;
VL = FRC).
[View Larger Version of this Image (34K GIF file)]
|
Table 2.
Standard deviations for breath-to-breath differences between estimates
of gas exchange and true (model) values, expressed as % of mean level
of gas exchange
|
| Subject No. |
Mouth
|
Estimates of Gas Exchange for Model Data
|
Eq. 7 |
| VL = 0 FET FET
|
VL = FRC FET
FET |
VL = ELV
FET FET
|
VL = VA FET
FET |
VL = FRC
FA FA
|
VL = FRC FET
FA
|
|
| O2,
ml/min STPD |
| 952 |
45.7 |
19.9 |
16.1
|
13.1 |
14.2 |
2.9 |
3.0 |
8.6 |
| 971 |
104.9 |
35.8
|
53.5 |
32.1 |
55.3 |
5.9 |
8.0 |
18.8 |
| 973 |
60.7
|
35.0 |
41.9 |
30.1 |
42.0 |
4.0 |
3.9 |
19.9 |
| 997
|
36.4 |
17.2 |
12.1 |
11.4 |
11.3 |
1.7 |
1.6 |
9.9
|
| 998 |
70.2 |
27.8 |
34.3 |
22.9 |
29.8 |
4.7 |
5.0
|
14.8 |
| 999 |
40.5 |
18.7 |
17.0 |
15.3
|
16.4 |
3.1 |
3.2 |
9.9
|
| Mean ± SD
|
59.7 ± 25.5 |
25.7 ± 8.3 |
29.2 ± 16.7 |
20.8 ± 8.9 |
28.2 ± 17.6 |
3.7 ± 1.5 |
4.1 ± 2.2 |
13.6 ± 4.9 |
| CO2,
ml/min STPD |
| 952 |
17.6 |
13.3 |
14.8
|
11.1 |
14.0 |
1.6 |
1.8 |
7.1 |
| 971 |
40.5 |
20.7
|
46.7 |
20.3 |
53.4 |
3.7 |
5.6 |
13.4 |
| 973 |
27.6
|
24.6 |
37.2 |
24.1 |
39.5 |
2.7 |
2.3 |
15.7 |
| 997
|
12.5 |
11.7 |
9.4 |
9.7 |
9.5 |
0.9 |
1.0 |
7.3
|
| 998 |
21.4 |
16.2 |
30.3 |
14.6 |
28.1 |
3.0 |
3.7
|
11.2 |
| 999 |
14.1 |
12.5 |
14.2 |
11.3
|
14.9 |
1.5 |
1.5 |
7.2
|
| Mean ± SD |
22.3 ± 10.4 |
16.5 ± 5.2 |
25.4 ± 14.9 |
15.2 ± 5.8 |
26.6 ± 17.2 |
2.2 ± 1.1 |
2.7 ± 1.7 |
10.3 ± 3.7 |
|
|
O2, O2
consumption; CO2,
CO2 output; VL, end-expiratory nominal lung
volume from previous breath; VA, alveolar volume from
previous breath; FRC, functional residual capacity; ELV, effective lung
capacity; FET, end-tidal gas fraction; FA,
alveolar end-expiratory gas fraction.
|
|
Table 2 shows the standard deviation (SD) for the breath-by-breath
estimates of gas exchange around their true values, expressed as a
percentage of the mean level of gas exchange. Results are shown for
each breathing pattern and for each algorithm. The algorithm of Swanson
(11) gave better estimates at the pulmonary capillaries when compared with the other two algorithms. Generally, assuming VL = 0 [algorithm of Wessel et al. (13)] gave better results than
assuming VL = FRC
[algorithm of Auchincloss et al. (2)], although this was
not always the case. The ELV values were 1,310 ± 550 ml
BTPS for
O2 and 1,180 ± 820 ml
BTPS for
CO2 (mean ± SD for the 6 data
sets). Higher values for
VL like FRC induced a degradation in the estimates of alveolar gas exchange.
The effect of replacing the constant values for
VL (i.e., 0, FRC, or ELV)
with the real time varying values is also shown in Table 2. This
generally degraded the estimates of the gas transfer at the capillary
level, in particular when compared with the estimates from Swanson's
algorithm (11). This finding clearly suggests that improving the volume
estimates in these algorithms is unlikely to improve the
breath-to-breath estimate of pulmonary gas exchange. The effect of
replacing the estimates of
FAgas and FAgas
(i.e., the end-tidal values) with the true alveolar compositions is
shown in Table 2. This yielded a much more precise estimation of the
gas transfer at the capillary level, even though a single constant
value (FRC) was used for
VL. Finally, the effect
of replacing only the estimates of
FAgas
with the true changes in alveolar composition and using
FETgas to
estimate
FAgas is
also shown in Table 2. This did not degrade the estimation of gas transfer at the capillary level when compared with the estimates using
the true values for both
FAgas and
FAgas. A
comparison of the effects of using end-tidal vs. true alveolar values
in the algorithms is shown for O2
in Fig. 4 and for
CO2 in Fig. 5. In
these figures, the precision of the estimates for
O2 and CO2 at the pulmonary
capillaries is plotted as a function of VL. The errors associated
with the use of end-tidal fractions were close to those for alveolar
fractions at low values of
VL. They diverged as the
value of VL increased.
The precision of the estimates using the true alveolar composition
reached a minimum for a
VL close to FRC.
Fig. 4.
SD of breath-by-breath differences between estimated and true (model)
O2 at capillary level, using
end-tidal
(FETO2) and true (model) alveolar fractions
(FAO2)
for subject 952.
[View Larger Version of this Image (12K GIF file)]
Fig. 5.
SD of the breath-by-breath differences between estimated and true
(model) CO2 at capillary
level, using
FETCO2 and true (model)
FACO2
for subject 952.
[View Larger Version of this Image (12K GIF file)]
The results above show clearly that the limitations of the algorithms
in compensating for Sgas arise
more from the inadequacy of using breath-to-breath differences in the
end-tidal measurements as an assessment of breath-to-breath differences
in alveolar composition than from the assessment of
VL. Furthermore, the results
show that the changes in end-tidal composition generally overestimate the real changes in alveolar composition. This can be seen from 1) the fact that when end-tidal
values are used the effective VL
appears systematically less than FRC;
2) the SD for the breath-to-breath differences in end-tidal values are 2.10 ± 1.16 Torr (mean ± SD for the 6 data sets) for O2 and
1.33 ± 0.65 Torr for CO2
compared with the SD for the breath-to-breath differences in true mean alveolar values (1.87 ± 0.75 Torr for
O2 and 1.04 ± 0.31 Torr for CO2); and
3) the tendency for variations in
alveolar composition to be greatest where the
ventilation-to-volume ratio is greatest, and these will have a
greater influence on end-tidal values that are flow-weighted means than
on alveolar values that are volume-weighted means.
Development of an alternative algorithm.
The above results lead us to suggest an alternative algorithm that does
not use the end-tidal values to calculate the change in alveolar
fraction directly. We use FRC as
VL,
FETgas as
FAgas, and
assume that the true
FAgas
lies between 0 and
FETgas.
From this, it follows that true
Agas
lies between
0gas
[algorithm of Wessel et al. (13)] and
FRCgas
[algorithm of Auchincloss et al. (2)].
AO2
and
ACO2
are then estimated by minimizing their respective breath-to-breath
variation, subject to the constraints that for each breath
i
|
(5)
|
where
Min is a function that returns the minimum value in the list and Max is
a function that returns the maximum value in the list, and
|
(6)
|
The minimization functions may be written
as
|
(7)
|
and solved as an iterative process. The iterative solution
was undertaken as follows. First, the starting values for the
AO2 and
ACO2
of each breath were set midway between the estimates from the Wessel
and Auchincloss algorithms (Refs. 13 and 2, respectively)
|
(8)
|
where
the superscript 0 indicates that they are the starting values for the
procedure. Then, for each iteration j,
the values for
and
were calculated sequen- tially, starting at
i = 1, so as to minimize the
breath-by-breath variation in the gas-exchange estimates, subject to
the bounds of Eqs. 5 and 6. The procedure is as follows. First,
for i = 1(9)
Then, for i = 2 through
n 1 (in increasing value for
i, so that
is available before
is calculated)
|
(10)
|
and(11)
Finally, for i = n(12)
This procedure was repeated (next increment in
j) until the mean change in gas
exchange from one iteration ( j) to
the next (j + 1) was <0.1 ml/min
|
(13)
|
This was achieved for our data sets at between 6 and 13 iterations. An example of the solution with the bounds provided by the
Wessel and Auchincloss algorithms (Refs. 13 and 2) is shown in Fig.
6.
Fig. 6.
Breath-to-breath estimates of gas exchange at pulmonary capillaries
over 20 successive breaths for model simulations. Respiratory flow data
are from subject 952:
0CO2
[algorithm of Wessel et al. (13)] and
FRCCO2
[algorithm of Auchincloss et al. (2)]. Line represents
estimates of alveolar
CO2 that minimize
breath-to-breath variations (Eq. 7)
with respect to the constraint that alveolar CO2 lies between
0CO2
and
FRCCO2
(Eq. 6).
[View Larger Version of this Image (14K GIF file)]
To check that the starting point for the iterations has no
effect on the final solution, the iterative process was also begun with different starting points. Both
= 
and = 
were tried, and in all data analyzed, the final solutions were the
same.
A sample of the results obtained with this algorithm is shown in Fig.
3, and this suggests some reduction in the error around the simulated
gas exchange at the pulmonary capillaries compared with the other
algorithms. The reduction in the error is confirmed by the results in
Table 2, which show in every case, for both CO2 and
O2, an improvement in the estimate
over the estimates obtained via the other algorithms.
The assumed values of
FAO2
and
FACO2
are not explicitly calculated in the above algorithm. However, some idea of their likely values can be obtained from the ratio
|
(14)
|
Because
of the constraints imposed on
and
, the ratios for
O2
(RO2) and for
CO2 (RCO2)
lie between 0 and 1. A value close to 0 for
Rgas implies
Agas
corresponds closely to the estimate from Wessel's algorithm where
FAgas = 0, and a value close to 1 corresponds closely to the estimate from
Auchincloss's algorithm where
FAgas = FETgas.
The mean breath-to-breath value of
RO2 was 0.55 ± 0.44 (mean ± SD averaged across the subjects), which was similar to
RCO2 at 0.57 ± 0.46 (mean ± SD averaged across the subjects).
By assuming RCO2 is equal to
RO2, it is possible to calculate
the breath-to-breath
ACO2
from the minimization of breath-to-breath
or vice versa. If breathto-breath
ACO2
is calculated from the minimization of JO2,
then the overall SD of estimate around the true value is
10.1 ± 3.1% of the mean
CO2, very close to the value of 10.3 ± 3.7% for the minimization of
JCO2.
For the estimates of breath-to-breath
AO2
based on the minimization of
JCO2,
the overall SD of the estimate around the true value is 15.9 ± 6.2%, slightly higher than the value of 13.6 ± 4.9% based
on the minimization of
JO2,
but still substantially below the values obtained with the other
algorithms. These findings suggest that the relationships between the
breath-to-breath changes in mean alveolar compositions and end-tidal
compositions are affected by the breathing pattern in a broadly similar
way for O2 as for CO2.
Variability in gas exchange in both simulated data and real data.
As a final study, the various algorithms were applied to the original
experimental data (gas exchange at the mouth and end-tidal values) from
which the breathing patterns driving the model were obtained. The
segment which matches that in Fig. 3 is shown for the real data in
Fig.7. Similar trends can be observed for
the variability in breath-to-breath gas exchange in the real data and
in the model data. Furthermore, the differences in breath-to-breath variability for the different algorithms observed in the model data
appear to be reflected in the results for the real data. A more
detailed examination of this can be made from Table
3, which gives the coefficient of variation
for each algorithm for both the model data and the real data. For each
algorithm, the variations are similar for the model and the real data,
except in the case of the algorithm of Auchincloss et al. (2), where using FRC as an estimate of VL
caused the estimates substantially to be more variable for the real
data compared with the model data. The method developed in the present
study for estimating gas exchange at the pulmonary capillaries yielded
less variability in alveolar gas exchange compared with the other
algorithms when applied to the real (nonmodel) data.
Fig. 7.
Breath-to-breath gas exchange at mouth, together with estimates at
level of pulmonary capillaries. Various algorithms for real (nonmodel)
data match those in Fig. 3.
[View Larger Version of this Image (34K GIF file)]
|
Table 3.
Coefficients of variation for gas exchange at mouth and of estimates at
capillary level for both model and actual data
|
| Subject No. |
Gas Exchange for
Model Data
|
Gas Exchange for Actual Data
|
| Capillary |
Mouth |
Estimates
at capillary level with VL =
|
Eq. 7
|
Mouth |
Estimates at capillary level with
VL =
|
Eq. 7 |
| 0 |
FRC |
ELV |
0
|
FRC |
ELV
|
|
| O2,
ml/min STPD |
| 952 |
1.5 |
45.4 |
20.2
|
16.1 |
13.3 |
8.9 |
50.5 |
19.6 |
64.1 |
19.2 |
12.1
|
| 971 |
4.8 |
102.1 |
35.5 |
53.3 |
31.6 |
19.4 |
129.9
|
40.4 |
66.5 |
40.2 |
33.5 |
| 973 |
3.6 |
61.2 |
36.0
|
42.7 |
32.5 |
20.8 |
64.7 |
27.6 |
126.1 |
27.6
|
20.2 |
| 997 |
1.3 |
35.7 |
17.2 |
12.1 |
12.0 |
10.0
|
34.8 |
19.0 |
39.7 |
18.9 |
16.4 |
| 998 |
3.2 |
71.0
|
28.9 |
35.1 |
23.9 |
15.7 |
64.9 |
28.4 |
82.4 |
28.0
|
18.1 |
| 999 |
1.8 |
40.1 |
19.0 |
17.2
|
15.1 |
10.3 |
38.3 |
19.6 |
114.5 |
19.6 |
16.1
|
| Mean ± SD
|
2.7 ± 1.4 |
59.3 ± 24.9 |
26.1 ± 8.5 |
29.4 ± 16.8 |
21.4 ± 9.2 |
14.2 ± 5.2 |
63.9 ± 34.8 |
25.8 ± 8.3 |
82.2 ± 32.7 |
25.6 ± 8.3 |
19.4 ± 7.4 |
| CO2,
ml/min STPD |
| 952 |
14.6 |
22.2 |
21.1
|
20.3 |
18.4 |
15.5 |
22.0 |
20.8 |
63.1 |
20.7 |
14.6
|
| 971 |
29.5 |
44.7 |
38.4 |
54.6 |
37.4 |
30.5 |
39.7
|
32.8 |
60.7 |
32.8 |
27.1 |
| 973 |
29.8 |
42.2 |
42.4
|
49.3 |
41.6 |
33.9 |
36.7 |
35.6 |
117.4 |
35.5
|
30.5 |
| 997 |
11.3 |
16.9 |
18.3 |
15.4 |
15.4 |
14.0
|
19.1 |
19.1 |
28.5 |
19.1 |
16.9 |
| 998 |
22.2 |
27.9
|
30.6 |
39.7 |
29.9 |
24.5 |
30.5 |
29.9 |
82.2 |
29.9
|
23.8 |
| 999 |
14.4 |
20.3 |
21.4 |
21.2
|
19.9 |
16.7 |
18.9 |
18.8 |
107.4 |
18.7 |
16.1
|
| Mean ± SD |
20.3 ± 8.1 |
29.0 ± 11.7 |
28.7 ± 10.0 |
33.4 ± 16.7 |
27.1 ± 10.8 |
22.5 ± 8.4 |
27.8 ± 9.1
|
26.2 ± 7.5 |
76.6 ± 32.8 |
26.1 ± 7.5 |
21.5 ± 6.6 |
|
|
|
The model of the lung used in this study included
1) inhomogeneities in the
ventilation-to-volume ratio,
2) inhomogeneities in
ventilation-to-perfusion ratio, and
3) unevenness in the cyclical nature
of the ventilation. In the present study, the variability in the true
O2 transfer at the alveolar level
was six- to tenfold lower than for
CO2. The physiological reason for
this is the difference in shape of the blood
O2 and
CO2 dissociation curves. The model used did not consider certain other possible sources of inhomogeneity. These include incomplete gas mixing in alveoli and airways, cyclical variations in the perfusion of the lung of both cardiac and respiratory origin, and intra- and interbreath variations in ventilation-perfusion distribution. Interbreath variations could also arise from fluctuations in the pulmonary blood flow and the mixed venous composition. Despite
the much greater complexity of the real lung when compared with the
model, it is noteworthy that the variability in
O2 and
CO2, both measured at the
mouth and estimated at the alveolar level for the simulated data, were
comparable with the variability for the original experimental data
(Table 3). The degree of variability is also similar to earlier
observations made at rest (5, 7). Moreover, the patterns of
O2 and
CO2 showed some
remarkable similarities between model and experimental data (Figs. 3
and 7). These findings indicate that variations in the cyclic nature of
the ventilation and the ventilation-volume-perfusion inequalities underlie a major part of the variability in gas exchange.
For the algorithm of Swanson, the ELV values for the experimental data
were 125 ± 98 ml BTPS for alveolar
exchange of O2 and 42 ± 94 ml
BTPS for
CO2, which were lower than the
values obtained with the simulated data from the model (Table
4). For the algorithm developed in the
present study (Eq. 7), the mean
value for RO2 was 0.38 ± 0.43 (mean ± SD averaged across the subjects), and the mean value for
RCO2 was 0.47 ± 0.46 (mean ± SD averaged across the subjects). The ratios for both
O2 and
CO2 were systematically lower for
the experimental data than for the simulated data. These findings
suggest that the overestimation of the breath-to-breath changes in mean
alveolar fractions by using end-tidal measurement is greater for the
actual data than for the simulated data from the lung model. This could
arise from the dependence of the end-tidal values in life on other
factors that were not included in the model and also from the
possibility that the data used to generate the model did not account
for the full degree of inhomogeneity within the lung. Additionally,
RCO2 was systematically greater than RO2 for the experimental data.
This difference did not appear with the data simulated by using the
model. The explanation for this is not entirely clear.
|
Table 4.
ELV and FRC values for data simulated by using model and for actual
experimental data
|
| Subject No. |
ELV for
O2, ml
|
ELV for
CO2, ml
|
FRC, ml
|
| Model data |
Actual data
|
Model data |
Actual data |
Model data
|
Actual data* |
|
| 952 |
1,570 |
206 |
1,399 |
102
|
2,500 |
3,270 |
| 971 |
729 |
223 |
497 |
61 |
2,500
|
2,790 |
| 973 |
904 |
27 |
621 |
46 |
2,500 |
3,330
|
| 997 |
2,213 |
139 |
2,642 |
184 |
2,500 |
2,820 |
| 998
|
981 |
169 |
549 |
12 |
2,500 |
2,880 |
| 999 |
1,478
|
16 |
1,343 |
63 |
2,500 |
3,300 |
|
|
*
Predicted for individual subjects by using anthropometric
data (Ref. 4).
|
|
In conclusion, it appears from the above results that many of the
features of breath-by-breath gas exchange can be simulated by using an
irregularly ventilated inhomogenous model of the lung. The new method
proposed in the present study for the estimation of the
breath-to-breath gas exchange at the pulmonary capillaries appears
promising, although the usefulness of the algorithm would clearly have
to be evaluated further under conditions of hypoxia and during
transients in pulmonary blood flow and/or in inspiratory gas
composition.
ACKNOWLEDGEMENTS
The authors thank Dr. J. Sato for helpful collaboration.
FOOTNOTES
The Laboratoire de Physiologie and Groupement d'Intérêt
Public Exercice, Saint-Etienne (France) provided financial support for
T. Busso.
Present address of T. Busso: Laboratoire de Physiologie, CHU de
Saint-Etienne, Hôpital de Saint-Jean-Bonnefonds Pavillon 12, 42055 Saint-Etienne cedex 2, France.
1
Here and elsewhere, the term "true" is
used in the statistical sense to distinguish correct values from their
estimates obtained via the various algorithms.
Address for reprint requests: P. A. Robbins, Univ. Laboratory of
Physiology, Parks Rd., Oxford OX1 3PT, UK (E-mail:
peter.robbins{at}physiol.ox.ac.uk).
Received 11 April 1996; accepted in final form 17 December 1996.
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0161-7567/97 $5.00
Copyright © 1997 the American Physiological Society
![= <FENCE><AR><R><C><A><AC>V</AC><AC>˙</AC></A><SUP>2,<IT>j</IT> − 1</SUP><SUB><SC>a</SC><SUB>gas</SUB></SUB> if <A><AC>V</AC><AC>˙</AC></A><SUP>2,<IT>j</IT> − 1</SUP><SUB><SC>a</SC><SUB>gas</SUB></SUB> ∈ [Min (<A><AC>V</AC><AC>˙</AC></A><SUP>1</SUP><SUB>0<SUB>gas</SUB></SUB>, <A><AC>V</AC><AC>˙</AC></A><SUP>1</SUP><SUB>FRC<SUB>gas</SUB></SUB>), Max (<A><AC>V</AC><AC>˙</AC></A><SUP>1</SUP><SUB>0<SUB>gas</SUB></SUB>, <A><AC>V</AC><AC>˙</AC></A><SUP>1</SUP><SUB>FRC<SUB>gas</SUB></SUB>)]</C></R><R><C>Min (<A><AC>V</AC><AC>˙</AC></A><SUP>1</SUP><SUB>0<SUB>gas</SUB></SUB>, <A><AC>V</AC><AC>˙</AC></A><SUP>1</SUP><SUB>FRC<SUB>gas</SUB></SUB>) if <A><AC>V</AC><AC>˙</AC></A><SUP>2,<IT>j</IT> − 1</SUP><SUB><SC>a</SC><SUB>gas</SUB></SUB> < Min (<A><AC>V</AC><AC>˙</AC></A><SUP>1</SUP><SUB>0<SUB>gas</SUB></SUB>, <A><AC>V</AC><AC>˙</AC></A><SUP>1</SUP><SUB>FRC<SUB>gas</SUB></SUB>)</C></R><R><C>Max (<A><AC>V</AC><AC>˙</AC></A><SUP>1</SUP><SUB>0<SUB>gas</SUB></SUB>, <A><AC>V</AC><AC>˙</AC></A><SUP>1</SUP><SUB>FRC<SUB>gas</SUB></SUB>) if <A><AC>V</AC><AC>˙</AC></A><SUP>2,<IT>j</IT> − 1</SUP><SUB><SC>a</SC><SUB>gas</SUB></SUB> > Max (<A><AC>V</AC><AC>˙</AC></A><SUP>1</SUP><SUB>0<SUB>gas</SUB></SUB>, <A><AC>V</AC><AC>˙</AC></A><SUP>1</SUP><SUB>FRC<SUB>gas</SUB></SUB>)</C></R></AR></FENCE>](/content/vol82/issue6/fulltext/1963/img013.gif)
![<A><AC>V</AC><AC>˙</AC></A><SUP><IT>i</IT>,<IT>j</IT></SUP><SUB><SC>a</SC><SUB>gas</SUB></SUB> = <FENCE><AR><R><C><IT>y</IT> if <IT>y</IT> ∈ [Min (<A><AC>V</AC><AC>˙</AC></A><SUP><IT>i</IT></SUP><SUB>0<SUB>gas</SUB></SUB>, <A><AC>V</AC><AC>˙</AC></A><SUP><IT>i</IT></SUP><SUB>FRC<SUB>gas</SUB></SUB>), Max (<A><AC>V</AC><AC>˙</AC></A><SUP><IT>i</IT></SUP><SUB>0<SUB>gas</SUB></SUB>, <A><AC>V</AC><AC>˙</AC></A><SUP><IT>i</IT></SUP><SUB>FRC<SUB>gas</SUB></SUB>)]</C></R><R><C>Min (<A><AC>V</AC><AC>˙</AC></A><SUP><IT>i</IT></SUP><SUB>0<SUB>gas</SUB></SUB>, <A><AC>V</AC><AC>˙</AC></A><SUP><IT>i</IT></SUP><SUB>FRC<SUB>gas</SUB></SUB>) if <IT>y</IT> < Min (<A><AC>V</AC><AC>˙</AC></A><SUP><IT>i</IT></SUP><SUB>0<SUB>gas</SUB></SUB>, <A><AC>V</AC><AC>˙</AC></A><SUP><IT>i</IT></SUP><SUB>FRC<SUB>gas</SUB></SUB>)</C></R><R><C>Max (<A><AC>V</AC><AC>˙</AC></A><SUP><IT>i</IT></SUP><SUB>0<SUB>gas</SUB></SUB>, <A><AC>V</AC><AC>˙</AC></A><SUP><IT>i</IT></SUP><SUB>FRC<SUB>gas</SUB></SUB>) if <IT>y</IT> > Max (<A><AC>V</AC><AC>˙</AC></A><SUP><IT>i</IT></SUP><SUB>0<SUB>gas</SUB></SUB>, <A><AC>V</AC><AC>˙</AC></A><SUP><IT>i</IT></SUP><SUB>FRC<SUB>gas</SUB></SUB>)</C></R></AR></FENCE>](/content/vol82/issue6/fulltext/1963/img017.gif)
![= <FENCE><AR><R><C><A><AC>V</AC><AC>˙</AC></A><SUP><IT>n</IT> − 1, <IT>j</IT></SUP><SUB><SC>a</SC><SUB>gas</SUB></SUB> if <A><AC>V</AC><AC>˙</AC></A><SUP><IT>n</IT> − 1,<IT>j</IT></SUP><SUB><SC>a</SC><SUB>gas</SUB></SUB> ∈ [Min (<A><AC>V</AC><AC>˙</AC></A><SUP><IT>n</IT></SUP><SUB>0<SUB>gas</SUB></SUB>, <A><AC>V</AC><AC>˙</AC></A><SUP><IT>n</IT></SUP><SUB>FRC<SUB>gas</SUB></SUB>), Max (<A><AC>V</AC><AC>˙</AC></A><SUP><IT>n</IT></SUP><SUB>0<SUB>gas</SUB></SUB>, <A><AC>V</AC><AC>˙</AC></A><SUP><IT>n</IT></SUP><SUB>FRC<SUB>gas</SUB></SUB>)]</C></R><R><C>Min (<A><AC>V</AC><AC>˙</AC></A><SUP><IT>n</IT></SUP><SUB>0<SUB>gas</SUB></SUB>, <A><AC>V</AC><AC>˙</AC></A><SUP><IT>n</IT></SUP><SUB>FRC<SUB>gas</SUB></SUB>) if <A><AC>V</AC><AC>˙</AC></A><SUP><IT>n</IT> − 1,<IT>j</IT></SUP><SUB><SC>a</SC><SUB>gas</SUB></SUB> < Min (<A><AC>V</AC><AC>˙</AC></A><SUP><IT>n</IT></SUP><SUB>0<SUB>gas</SUB></SUB>, <A><AC>V</AC><AC>˙</AC></A><SUP><IT>n</IT></SUP><SUB>FRC<SUB>gas</SUB></SUB>)</C></R><R><C>Max (<A><AC>V</AC><AC>˙</AC></A><SUP><IT>n</IT></SUP><SUB>0<SUB>gas</SUB></SUB>, <A><AC>V</AC><AC>˙</AC></A><SUP><IT>n</IT></SUP><SUB>FRC<SUB>gas</SUB></SUB>) if <A><AC>V</AC><AC>˙</AC></A><SUP><IT>n</IT> − 1,<IT>j</IT></SUP><SUB><SC>a</SC><SUB>gas</SUB></SUB> > Max (<A><AC>V</AC><AC>˙</AC></A><SUP><IT>n</IT></SUP><SUB>0<SUB>gas</SUB></SUB>, <A><AC>V</AC><AC>˙</AC></A><SUP><IT>n</IT></SUP><SUB>FRC<SUB>gas</SUB></SUB>)</C></R></AR></FENCE>](/content/vol82/issue6/fulltext/1963/img019.gif)
