Journal of Applied Physiology
Vol. 81, No. 5,
pp. 2287-2296,
November 1996
CONTROL OF BREATHING, CIRCULATION, AND TEMPERATURE
modeling in physiology
Breath-to-breath relationships between respiratory cycle variables in
humans at fixed end-tidal PCO2 and
PO2
Thierry
Busso,
Pei-Ji
Liang, and
Peter A.
Robbins
University Laboratory of Physiology, Oxford OX1 3PT, United
Kingdom
ABSTRACT
- INTRODUCTION
- METHODS
- RESULTS
- DISCUSSION
- ACKNOWLEDGEMENTS
- FOOTNOTES
- REFERENCES
ABSTRACT 
Busso, Thierry, Pei-Ji Liang, and Peter A. Robbins.
Breath-to-breath relationships between respiratory cycle variables in humans at fixed end-tidal PCO2 and
PO2. J. Appl. Physiol. 81(5): 2287-2296, 1996.
This study examined the statistical properties of breath-to-breath variations in the inspiratory and expiratory volumes and times during rest and light exercise. Sixty data sets were
analyzed. Initial data and residuals after fitting time-series models
were examined for 1) sustained periodicities with use of spectral analysis, 2) temporal changes in signal power with
use of evolutionary spectral analysis, and 3) auto- and cross
correlations with use of a portmanteau test. The major findings were as
follows: 1) no sustained periodic components were detected;
2) temporal changes in signal power were normally present,
but these did not affect significantly the results from time-series
modeling; 3) for all variables, a simple autoregressive
moving average (ARMA) AR1MA1 model generally
described the autocorrelation; 4) considerable cross
correlation remained between residuals from the
AR1MA1 model; 5) relationships
between variables could be described by using a multivariate
time-series model; 6) residual fluctuations in end-tidal
PCO2 had little influence; and 7)
responses were broadly similar between rest and exercise, although some
quantitative differences were found. The multivariate model provides a
description of the structure of the interrelationships between cycle
variables in a quantitative and a qualitative form.
control of breathing; chemoreflex feedback loop; correlated
variability; time-series models; constant power spectrum
INTRODUCTION
AS DEMONSTRATED BY Priban (26), the breath-by-breath
fluctuations in respiratory cycle variables are not purely random. Correlations between successive breaths were observed for tidal volume
(VT) and cycle duration (TT). This result was
extended by Bolton and Marsh (5) to include inspiratory and expiratory volumes (VTI and VTE)
and times (TI and TE).
Time-series models, or autoregressive moving average (ARMA), models
have been used to analyze the autoregressive structure of these data
sequences. The application of a first-order model to steady-state
breathing in humans showed that the values for VT and
TT in a given breath were dependent on the values from the
preceding breath (3, 4). The interpretation of these results, however,
has been complicated by the influence of the feedback loops involving
blood gas tensions. Periodic oscillations arising from chemical
feedback have been observed for VT and TT (7, 18, 21). It thus becomes unclear whether the autocorrelation is
due to these feedback loops or to other intrinsic factors.
To try to resolve this issue, animal preparations have been employed in
which the respiratory controller has been isolated from the effects of
chemical feedback and where the properties of the respiratory
controller can be studied independently of any instability induced by
chemical feedback. The results from such experiments on the respiratory
controller in an open-loop state have not been clear-cut. On the one
hand, breath-to-breath correlations in TI, TE,
and phrenic activity have been observed in an "isolated respiratory
centre" preparation by Benchetrit and Bertrand (2) in the
anesthetized cat. On the other hand, Khatib et al. (20) did not observe
such correlation in phrenic activity in anesthetized, vagotomized, and
artificially ventilated rats. These authors suggested that the
discrepancy between their findings and those of Benchetrit and Bertrand
could be due to failure of Benchetrit and Bertrand to check their data
sequences for nonstationarity, which could have corrupted the analysis
of the breath-to-breath correlation.
In humans, the respiratory controller can be investigated in a
quasi-open-loop state by use of the technique of dynamic end-tidal forcing, where a computer-controlled gas-mixing system is used to
adjust the inspiratory gas composition on a breath-by-breath basis to
maintain the subject's end-tidal O2 and CO2
pressures (PETO2 and
PETCO2, respectively) close to
target values, despite variations in ventilation. In essence,
this system attempts to break the physiological feedback loop at the
point where ventilation influences
PETCO2 and
PETO2. Liang et al. (22)
demonstrated the presence of breath-by-breath correlation of minute
ventilation (
E) in data that had been
obtained using this technique. In most cases, there was also a slow
change over time in the variance of the data that was detected by using
evolutionary spectral analysis (28). However, these slow variations
over time in variance did not affect the observed breath-to-breath
correlations for ventilation.
The purpose of the present study was to extend the work of Liang et al.
(22) to the respiratory cycle variables, VTI,
VTE, TI, and TE. The
first aim was to determine whether these respiratory variables have
statistical properties similar to or different from breath-by-breath
ventilation and whether their statistical properties fit with the
findings of Benchetrit and Bertrand (2) or Kathib et al. (20) for the
isolated respiratory center preparation. The second aim was to study
the cross correlations between these variables within and between
breaths. As with the study by Liang et al. (22), all data analyzed had
been gathered by using the end-tidal forcing technique to keep the
chemical stimulation at as constant a level as possible.
METHODS
The experimental data were taken from a study described in a previous
report (25). Most of the statistical methods employed in this study
have been fully described and discussed by Liang et al. (22) in
relation to ventilation.
Recording and treatment of the data.
Ventilatory data were available from five subjects over periods of 43 min, with the subject seated at rest or performing exercise at 70 W on
a cycle ergometer. Each protocol was repeated six times by each
subject. During each experiment,
PETO2 was held constant at
100 Torr and PETCO2 at 2-3
Torr (4-5 Torr in subject 807) above the
subject's natural value during rest or exercise. Breath-by-breath VTI, VTE,
TI, and TE were recorded continuously during
each 43-min experiment. Details of the experimental methods have been
described elsewhere (25).
The first 50 breaths in the rest experiments and the first 300 breaths
in the exercise experiments were discarded to remove any initial
transient responses. The mean values and the coefficients of variation
of each variable for the remainder of the data sequences are listed for
each subject in Table 1.
Table 1.
Range in numbers of breaths per data sequence and means and CVs across
data sequences for variables of breathing pattern
| Subj. No. |
No. of Breaths, Range
|
VTI, liters
|
VTE, liters
|
TI, s
|
TE, s
|
| Mean ± SD |
CV, % |
Mean ± SD |
CV, % |
Mean ± SD |
CV, % |
Mean ± SD |
CV, % |
|
| Rest |
| 797-R
|
595-701 |
0.713 ± 0.096 |
11.12 ± 1.59 |
0.756 ± 0.093 |
10.23 ± 1.47 |
1.278 ± 0.051 |
15.08 ± 6.82 |
2.112 ± 0.146 |
10.73 ± 0.89 |
| 799-R
|
550-701 |
0.561 ± 0.040 |
17.54 ± 2.92 |
0.553 ± 0.055 |
19.51 ± 5.94 |
1.438 ± 0.094 |
21.19 ± 8.77 |
2.413 ± 0.166 |
14.81 ± 4.96 |
| 800-R
|
346-643 |
0.850 ± 0.040 |
11.14 ± 1.82 |
0.882 ± 0.047 |
11.81 ± 2.10 |
1.517 ± 0.101 |
13.36 ± 2.29 |
2.346 ± 0.219 |
15.05 ± 2.32 |
| 807-R
|
840-946 |
1.317 ± 0.052 |
15.63 ± 3.37 |
1.322 ± 0.047 |
17.08 ± 3.05 |
1.035 ± 0.049 |
17.36 ± 2.92 |
1.371 ± 0.044 |
17.74 ± 1.92 |
| 811-R |
586-694 |
0.692 ± 0.084 |
10.99 ± 2.70 |
0.712 ± 0.087 |
10.93 ± 2.00 |
1.417 ± 0.110 |
12.85 ± 0.76 |
2.020 ± 0.152 |
14.74 ± 1.47 |
| Exercise |
| 797-E
|
631-741 |
1.447 ± 0.087 |
10.43 ± 1.62 |
1.510 ± 0.080 |
9.89 ± 1.23 |
0.988 ± 0.021 |
14.08 ± 3.41 |
1.269 ± 0.060 |
9.46 ± 0.91 |
| 799-E
|
419-640 |
1.673 ± 0.115 |
11.97 ± 0.82 |
1.626 ± 0.109 |
12.42 ± 0.94 |
1.293 ± 0.164 |
17.17 ± 1.65 |
1.835 ± 0.340 |
14.61 ± 3.10 |
| 800-E
|
709-985 |
1.683 ± 0.059 |
10.57 ± 2.07 |
1.693 ± 0.049 |
11.08 ± 2.08 |
1.036 ± 0.074 |
14.67 ± 2.87 |
1.138 ± 0.113 |
14.30 ± 2.57 |
| 807-E
|
714-821 |
1.976 ± 0.131 |
10.83 ± 0.85 |
2.010 ± 0.133 |
11.87 ± 1.00 |
0.949 ± 0.029 |
14.33 ± 2.99 |
1.183 ± 0.066 |
13.76 ± 1.66 |
| 811-E
|
703-916 |
2.017 ± 0.171 |
4.99 ± 0.59 |
1.988 ± 0.163 |
5.93 ± 0.71 |
1.023 ± 0.053 |
7.14 ± 1.20 |
1.271 ± 0.111 |
9.30 ± 1.69 |
|
|
VTI and VTE,
inspiratory and expiratory volume; TI and TE,
inspiratory and expiratory time; CV, coefficient of variation.
|
|
Before any further analysis, the mean value, together with any linear
trend, was removed from each respiratory cycle variable for each data
sequence. The resulting deviations were analyzed before and after the
application of the demodulation process proposed by Liang et al. (22),
which aims to remove any slow variations in variance of the respiratory
variables. Demodulation of the original sequence consisted of dividing
the ith observation of each respiratory cycle variable,
xi, by the corresponding value of the
modulating function (an autoregressive estimate of the variance),
ci, computed as follows
|
(1)
|
c0 was computed from the variance of
the first 20 values of the corresponding variable. The mean time
constant for the smoothing of the variance was 66 s for the data
obtained at rest and 47 s for the data obtained during exercise.
Specific periodicities.
The data were examined for the presence of any specific periodicity by
use of spectral analysis. A smoothed periodogram was calculated for
each data set with use of a combination of three Daniell windows with
lengths of 3, 5, and 7. Then the six individual spectra calculated for
each subject and protocol were averaged. The 95% confidence interval
for the mean spectral density was calculated on the basis that the
estimates of spectral density follow a 
2 distribution
(27). The procedure is described in more detail by Liang et al. (22).
Evolutionary spectral analysis.
The data sequences were examined for the presence of variations in
power over time by use of evolutionary spectral analysis (28). The
principle of this technique is to apply a double window in time and
frequency domains. The width of the window used in the time domain was
100 breaths, and the bandwidth used in the frequency domain was 3/40.
Values for evolutionary spectral density were then calculated for the
corresponding times and frequencies. A two-way analysis of the variance
was applied on the logarithmic values of spectral density with use of
factors of time and frequency to test for differences in spectral
density with time. The 2 tests were used to examine
the significance of the factors of "time" and "interaction + residual error." When the interaction and the time terms were not
significant, it was concluded that the power of the data sequence was
constant. When the time term but not the interaction term was
significant, the data sequence was considered to be uniformly
modulated. A significant interaction term led to the conclusion that
the data sequence was not uniformly modulated. On a similar set of
simulated respiratory data, evolutionary spectral analysis was found to
yield a test statistic that was somewhat oversensitive, and for this
reason a level of significance of P < 0.0025 was employed for
the 2 test (22).
Simple ARMA model fitting.
The data were fitted with ARMA models, which may be written in the form
|
(2)
|
where xn,
xn 
1, ... ,
xn p are the deviations
from the mean of the respiratory variables
(VTI, VTE,
TI, or TE) for the breaths n to
n 
p, 
n,
n 1, ... , n q are samples of
uncorrelated noise, and p and q are the orders of the
ARMA model. In this study, three low-order models were fit to the data
sequences: AR1 (a1 coefficient),
AR2 (a1 and a2
coefficients), and AR1MA1
(a1 and c1 coefficients). The
model parameters were estimated for each data set by maximum
likelihood.
Multivariate ARMA model fitting.
The data were fitted with multivariate models of the form
|
(3)
|
where xn is column vector of the
respiratory variables, A, B, C, and D
are matrices, and 
n is a column vector of
samples of uncorrelated noise. The choice of the precise form of the
models was dependent on the results from simple ARMA model fitting and
is described in more detail in RESULTS.
The parameters of the different multivariate models were estimated by
maximum likelihood.
Portmanteau test.
A modified portmanteau test (6) was used to examine the whiteness of
each variable and the independence of any pair of variables within each
data set. The test is based on the statistic Q, defined as
follows
|
(4)
|
where n is the number of observations used to
compute the likelihood; rk is the estimated
autocorrelation value or the cross-correlation value with lag of
k (depending on whether whiteness or independence,
respectively, is being tested) for the sequences of the original
deviations or residuals after the fitting of the time-series models;
and K is an integer, the value of which was chosen to be 10 + m, where m is the number of parameters in the model.
Q was not always computed from a lag of 1 because of the cross
correlations. First, expiratory variables can depend on inspiratory
variables within the same breath, and therefore Q was computed
from a lag of 0 in these cases. Second, the effect of
PETCO2 on other respirtory
variables is lagged because of the pure delay of the chemoreflex
feedback, and therefore the dependence of the respiratory variables on
PETCO2 was tested using Q
computed from a lag of 2. In all cases, the portmanteau statistic
Q was compared with the 2 distribution with
degrees of freedom equal to 10 to test whether the sum of auto- or
cross correlations was significantly different from zero. Rejection of
the null hypothesis led to the conclusion that the data series could
not be accepted as uncorrelated (white) if computed from
autocorrelation or as independent of the other series if computed from
cross-correlations.
Comparison of the model coefficients among protocols, models, or data
treatments was undertaken using analysis of variance. All the
statistical calculations were done using S-plus statistical software
(Statistical Sciences, Oxford, UK).
RESULTS
Specific periodicities.
Smoothed periodograms were estimated for each data set for
VTI, VTE,
TI, and TE. No obvious peaks in the power
spectrum were apparent. Examples of smoothed periodograms for
VTI, VTE,
TI, and TE averaged for the six data sets for
subject 797 at rest are shown in Fig.
1. The power is concentrated in the lower
frequencies, and a smooth decline in the power can be traced within the
95% confidence intervals, with no peaks occurring at any particular frequency. These results generally indicate that there were no marked
specific periodic components within the data for any of the respiratory
variables examined.
Fig. 1.
Averaged smoothed periodograms and associated 95% confidence intervals
for inspiratory and expiratory volume (VTI and
VTE) and inspiratory and expiratory time
(TI and TE) for subject 797 at rest.
[View Larger Version of this Image (39K GIF file)]
Variations in power spectrum over time.
Variability over time of the power spectrum was tested for each
respiratory variable by evolutionary spectral analysis. The number of
data sets for which the null hypothesis of having a constant power
spectrum over time could be accepted, according to the respiratory
variable, was 2-6 of 30 data sets for rest and 4-9 of 30 data
sets for exercise (Table 2). The number of uniformly modulated data sets, where the overall shape of the power
spectrum can be accepted as constant but where the total power may
vary, was 17-28 for rest and 24-27 for exercise. "Demodulation" of the data sets by use of the autoregressive estimate of the variance
altered these results considerably. After demodulation, 16-28 data
sets for rest and 22-27 for exercise could be accepted as having a
constant power spectrum. The number of uniformly modulated data sets
rose to 24-29 for rest and 28-29 for exercise. These results
generally indicate that, for each variable tested, the absolute
variance associated with the variable changes over time but that the
relative distribution of power within the power spectrum remains
constant over time.
Table 2.
Results of evolutionary spectral analysis
| Subj. No. |
No. of Data Sets
|
VTI
|
VTE
|
TI
|
TE
|
| Constant Power |
Uniformly Modulated |
Constant
Power |
Uniformly Modulated |
Constant Power
|
Uniformly Modulated |
Constant Power |
Uniformly
Modulated |
|
| Rest |
| 797-R
|
6 |
0 (5) |
4 (6) |
0 (4) |
5 (6)
|
0 (3) |
3 (4) |
0 (6) |
5 (6) |
| 799-R |
6 |
0 (5)
|
6 (6) |
0 (4) |
5 (6) |
0 (2) |
2 (5) |
2 (5)
|
5 (5) |
| 800-R |
6 |
1 (5) |
4 (5) |
1 (5) |
4 (5)
|
1 (6) |
5 (6) |
4 (6) |
6 (6) |
| 807-R |
6 |
0 (5)
|
5 (5) |
0 (6) |
6 (6) |
0 (1) |
2 (4) |
0 (5)
|
6 (6) |
| 811-R |
6 |
1 (5) |
6 (6) |
1 (5)
|
5 (6) |
2 (4) |
5 (5) |
0 (6) |
6 (6)
|
| Total |
30 |
2 (25) |
25 (28) |
2 (24)
|
25 (29) |
3 (16) |
17 (24) |
6 (28) |
28 (29)
|
| Exercise |
| 797-E |
6
|
0 (5) |
5 (6) |
0 (6) |
5 (6) |
1 (5) |
5 (6)
|
1 (6) |
4 (6) |
| 799-E |
6 |
4 (5) |
6 (6) |
1 (6)
|
6 (6) |
1 (5) |
6 (6) |
4 (6) |
6 (6) |
| 800-E |
6
|
3 (5) |
5 (5) |
3 (5) |
6 (5) |
1 (5) |
5 (6)
|
0 (3) |
6 (5) |
| 807-E |
6 |
1 (4) |
3 (5) |
1 (4)
|
5 (5) |
0 (3) |
5 (6) |
0 (5) |
5 (6)
|
| 811-E |
6 |
1 (6) |
5 (6) |
2 (6) |
5 (6)
|
1 (4) |
5 (5) |
1 (5) |
6 (6) |
| Total |
30
|
9 (25) |
24 (28) |
7 (27) |
27 (28) |
4 (22)
|
26 (29) |
6 (25) |
27 (29) |
|
|
No. of data sets that could be accepted as constant power over
time or uniformly modulated over time for original data sets and for
data sets after demodulation (nos. in parentheses).
|
|
Correlations within data before modeling.
Table 3 illustrates the degree of auto- and
cross correlation for the individual respiratory cycle variables in the
original data before and after demodulation. The results generally
indicate that 1) few sequences could be accepted as white
and/or independent of the other sequences, 2) the
results were similar for the data sequences before and after
demodulation, and 3) in approximately one-half of the data
sequences, no dependence on
PETCO2 was detected.
Table 3.
Results of portmanteau tests on data before time-series modeling
|
VTI |
VTE
|
TI |
TE
|
PETCO2
|
|
| Rest
|
| VTI |
0 (0) |
0 (0) |
9 (7)
|
9 (6) |
8 (10) |
| VTE |
0 (0)
|
4 (5) |
5 (6) |
13 (13) |
11 (13) |
| TI
|
7 (6) |
9 (7) |
0 (0) |
2 (1) |
11 (14)
|
| TE |
3 (4) |
13 (14) |
1 (3)
|
1 (1) |
13 (17)
|
| Exercise
|
| VTI |
1 (0) |
0 (0) |
7 (8)
|
4 (7) |
16 (18) |
| VTE |
0 (0)
|
1 (1) |
1 (5) |
3 (7) |
18 (18) |
| TI
|
3 (5) |
7 (7) |
0 (0) |
3 (0) |
13 (17)
|
| TE |
1 (1) |
7 (8) |
0 (1) |
0 (0)
|
12 (14) |
|
|
Values represent no. of data sets (total = 30) for each
variable (row) that could be accepted as independent of selected
variables (columns): results obtained with data before demodulation and results obtained with data after demodulation (in parentheses). PETCO2, end-tidal
PCO2.
|
|
Simple ARMA models.
For each data set, VTI,
VTE, TI, and TE were
fitted with AR1, AR2, and
AR1MA1 models. The whiteness of the
corresponding residuals was used to assess the goodness of fit. The
number of data sets accepted as having white residuals is given for
each variable and protocol in Table 4. The
results clearly indicate that the AR1MA1 model
accounted best for the autocorrelation in the data sets before and
after the demodulation process.
Table 4.
Number of residual data sets accepted as white after fitting of
ARMA models
| Variable |
Rest
|
Exercise
|
| AR1 |
AR2
|
AR1MA1 |
AR1
|
AR2 |
AR1MA1
|
|
| VTI |
12 (15) |
20 (20) |
26 (23)
|
9 (14) |
15 (20) |
28 (22) |
| VTE
|
11 (9) |
16 (17) |
22 (19) |
7 (10) |
18 (21)
|
25 (25) |
| TI |
6 (7) |
11 (15) |
18 (24)
|
10 (7) |
16 (14) |
22 (24) |
| TE |
7 (11)
|
17 (19) |
26 (24) |
7 (10) |
12 (18) |
23 (28) |
|
|
Values represent results obtained with data before demodulation;
values in parentheses represent results obtained with data after
demodulation (total = 30). ARMA, autoregressive moving average; AR1, AR2, AR1MA1,
ARMA models.
|
|
The model parameter estimates averaged for each protocol are given in
Table 5. The major findings were that
1) the coefficients for VTE and
VTI were significantly greater
(P < 0.01) for the exercise data than for the data obtained
at rest and 2) no statistical difference was observed between
the parameters estimated from the original data and the demodulated
data.
Table 5.
Coefficients for AR1, AR2, and
AR1MA1 models fitted to original data sets and
for AR1MA1 model fitted to demodulated data sets
| Variable |
AR1
|
AR2
|
AR1MA1
|
AR1MA1 on Demodulated Data
|
| a1 |
a1
|
a2 |
a1
|
c1 |
a1
|
c1
|
|
| Rest
|
| VTI |
0.341 ± 0.095 |
0.306 ± 0.073 |
0.073 ± 0.086 |
0.522 ± 0.192 |
0.248 ± 0.239 |
0.516 ± 0.214 |
0.227 ± 0.245 |
| VTE |
0.308 ± 0.106 |
0.264 ± 0.076 |
0.088 ± 0.086 |
0.555 ± 0.266 |
0.348 ± 0.329 |
0.496 ± 0.292 |
0.285 ± 0.326 |
| TI
|
0.299 ± 0.095 |
0.257 ± 0.081 |
0.132 ± 0.065 |
0.727 ± 0.171 |
0.491 ± 0.230 |
0.702 ± 0.227 |
0.465 ± 0.283 |
| TE
|
0.330 ± 0.102 |
0.292 ± 0.083 |
0.101 ± 0.068 |
0.695 ± 0.188 |
0.438 ± 0.255 |
0.596 ± 0.325 |
0.362 ± 0.351 |
| Exercise
|
| VTI |
0.326 ± 0.107 |
0.287 ± 0.087 |
0.098 ± 0.057 |
0.683 ± 0.212 |
0.442 ± 0.244 |
0.589 ± 0.249 |
0.334 ± 0.271 |
| VTE |
0.295 ± 0.094 |
0.258 ± 0.080 |
0.102 ± 0.053 |
0.648 ± 0.251 |
0.436 ± 0.277 |
0.625 ± 0.245 |
0.399 ± 0.279 |
| TI
|
0.306 ± 0.089 |
0.260 ± 0.080 |
0.125 ± 0.051 |
0.728 ± 0.159 |
0.488 ± 0.193 |
0.595 ± 0.242 |
0.355 ± 0.257 |
| TE |
0.362 ± 0.061 |
0.307 ± 0.052 |
0.142 ± 0.055 |
0.735 ± 0.135 |
0.452 ± 0.173 |
0.710 ± 0.128 |
0.439 ± 0.169 |
|
|
Values are means ± SD averaged across subjects.
|
|
Having modeled the autocorrelative structure of each respiratory
variable, the next step is to determine whether this also accounts
adequately for the cross correlation observed between respiratory
variables. This has been done by examining the degree of cross
correlation in the residuals after fitting of the
AR1MA1 model. The results are shown in Table
6. The conclusions that can be drawn are as
follows: 1) the results are similar for rest and exercise
protocols, 2) the residuals for VTE
and TE generally could not be accepted as independent of
the residuals for VTI and TI in
most of the data sets, and 3) VTI and
TI were commonly dependent on VTE
and TE, respectively. Thus the cross correlations observed
between respiratory variables cannot be due entirely to the
autocorrelation within respiratory variables.
Table 6.
Results of portmanteau tests on residuals after fitting of
AR1MA1 model
|
VTI |
VTE
|
TI |
TE
|
|
| Rest
|
| VTI |
26 |
4 |
14 |
17
|
| VTE |
0 |
22 |
7 |
20
|
| TI |
11 |
13 |
18 |
5
|
| TE |
6 |
21 |
9 |
26
|
| Exercise
|
| VTI |
28 |
9 |
22 |
11
|
| VTE |
0 |
25 |
5 |
14
|
| TI |
19 |
25 |
22 |
4 |
| TE |
1
|
18 |
2 |
23 |
|
|
Values represent no. of data sets (total = 30) for each
variable (row) that could be accepted as independent of selected
variables (columns).
|
|
Multivariate ARMA models.
To model the cross correlation between variables, a multivariate ARMA
model structure was employed. The design of the model, in terms of
which coefficients should be nonzero, was based on the following:
1) the AR1MA1 model could be taken as
a good model of the autoregressive structure, and 2) the
coefficients for cross correlation should be based on the marked
dependencies that remained between the residuals after the fit of the
AR1MA1 model. The particular model employed was
as follows
|
(5)
|
The estimates for the coefficients for the above model are given in
Table 7. The coefficient of
VTE(n 1) in fitting
VTE(n) was not statistically different
from zero for the rest and exercise protocols. The coefficient of the moving average term in fitting VTE(n)
and TEn was not statistically different
from zero for the exercise protocol. The coefficient of
VTI(n) in fitting
VTE(n) those of
VTI(n 1) in fitting
VTE(n) and those of
TEn 1 in fitting
TIn were statistically greater for the
exercise data than for the data obtained at rest. Conversely, the
coefficient of TIn 1 in
fitting TIn and those of
TEn 1 in fitting
TEn were statistically lower for the
exercise data.
To determine the degree of success or otherwise of the multivariate
model in describing the correlations between the respiratory variables,
the independence of the residuals after fitting of the multivariate
model was again examined using the portmanteau test. The results are
shown in Table 8. Only the residuals for TE during exercise failed to be independent of the other
variables in a reasonable number of cases.
Table 8.
Results of portmanteau tests on residuals after fitting of multivariate
model
|
VTI |
VTE
|
TI |
TE
|
|
| Rest
|
| VTI |
25 |
24 |
11 |
17
|
| VTE |
15 |
11 |
15 |
12
|
| TI |
13 |
14 |
19 |
20
|
| TE |
16 |
14 |
21 |
17
|
| Exercise
|
| VTI |
17 |
19 |
18 |
13
|
| VTE |
13 |
15 |
16 |
12
|
| TI |
15 |
17 |
17 |
15 |
| TE |
8
|
6 |
13 |
6 |
|
|
Values represent no. of data sets (total = 30) for each
variable (row) that could be accepted as independent of selected
variables (columns).
|
|
Influence of PETCO2 on model
coefficients.
The influence of residual fluctuations of
PETCO2 not controlled by the
end-tidal forcing system on the above results was examined by fitting
the data sets with a multivariate model that matched the one above, but
where PETCO2 and
E had been included as additional variates. For each respiratory cycle variable (apart from
PETCO2), a further
coefficient for PETCO2 at breath
n 2 was added to the model. The lag of n 2 was
used to allow for the pure delay associated with the transport delay of
the respiratory gases from the lung to the chemoreceptors (9).
The structure for modeling PETCO2 was as described by Liang et
al. (22). The overall model structure was as
follows
|
(6)
|
The coefficients of the multivariate model including
PETCO2 were then compared with
those from the model excluding
PETCO2. Only the coefficient of
VTEn 
1 in the fitting of VTIn was
significantly reduced by the inclusion of
PETCO2 (0.381 vs. 0.374, P < 0.05). The independence or otherwise of the
residuals for VTI,
VTE, TI, and TE
on the residuals for PETCO2 was
tested by the portmanteau test. The residuals for
VTI, VTE,
TI, and TE could be accepted as independent of
the residuals for PETCO2 in
42-47 of 60 data sets for each variable for rest and exercise
protocols. Thus it appears that residual fluctuations in
PETCO2 not controlled by
the end-tidal forcing system had little influence on the
results obtained.
DISCUSSION
The findings of this study concern the statistical properties of the
breathing pattern associated with the human respiratory controller when
dynamic end-tidal forcing has been used in an attempt to open the
feedback loop between ventilation and blood gas tensions.
VTI, VTE,
TI, and TE were shown to have properties similar to the composite variable of ventilation, as
previously described for the same data set by Liang et al.
(22). The breath-to-breath fluctuations of each respiratory variable
could be fitted with a simple AR1MA1 model, the
coefficients of which were not affected by the slow variations in
variance over time that were observed in the majority of data sets.
However, there remained considerable cross correlations between
different variables for the residuals after fitting of the
AR1MA1 model. The use of a multivariate
statistical analysis enabled these correlations between the respiratory
variables within and between breaths to be modeled in many of the data
sequences. The significant correlations, as determined by the
multivariate approach, are summarized in Fig.
2. Before the physiological implications of
these outcomes are discussed, possible confounding factors need to be
considered. These factors include 1) the influence of any
instability in chemical stimulation arising through imperfections in
the technique of end-tidal forcing and 2) the nonstationarity and/or nonlinearity of many of the data sequences examined.
Fig. 2.
Summary diagram of breath-to-breath relationships between respiratory
cycle variables as determined from fitting multivariate autoregressive
moving-average model. Ex, effect of exercise on correlation.
[View Larger Version of this Image (13K GIF file)]
Instability arising from chemical feedback.
Systems with intact feedback loops commonly display specific
periodicities. However, no such sustained specific periodicities were
detected within our data with use of spectral analysis. This result is
in keeping with the result for ventilation for the same data (22) and
is similar to that from one other study of respiratory cycle variables
in conscious humans where no end-tidal forcing had been employed (19).
However, in other studies of the human respiratory system with intact
chemical feedback loops, periodic components of respiratory variability
have often been observed (18, 21, 23). The fairly tight control
exercised on the end-tidal gases by the end-tidal forcing technique in
our study would be expected to reduce instability arising from chemical feedback. However, spectral analysis cannot totally exclude brief periods of periodicity or instability arising through small
fluctuations in alveolar gas tensions. Variations in
PETO2 and
PETCO2 can potentially influence breathing pattern. For the data of the current study, PETO2 was maintained at ~100
Torr. This mean value is associated with a very low level of
sensitivity of the respiratory controller to small changes in
PO2 (11). Thus it is most unlikely that any
fluctuations in PETO2 affect the
correlational structure of the breathing pattern. For
CO2, this is not the case, and for about one-half of the
original data sequences some dependence of the respiratory variables on
PETCO2 was detected. Consequently, the influence of fluctuations in
PETCO2 on the observed correlations between respiratory cycle variables has to be examined, despite the
fact that the end-tidal forcing system was employed to minimize these. For this purpose, the influence of
PETCO2 was examined within
the multivariate ARMA model. Only one coefficient (relating VTI to the preceding
VTE) was significantly depressed by the
inclusion of PETCO2 in the model,
and this effect was small (2% change). Moreover, the residuals for the
respiratory cycle parameters from the multivariate model were in most
cases independent of the residuals for
PETCO2. Therefore, although the
end-tidal forcing technique did not eliminate all the fluctuations in
the chemical stimuli, the residual fluctuations do not seem to affect
the correlational structure of the breathing pattern to any very great
extent.
Slowly changing variance.
Evolutionary spectral analysis was used to assess whether the power
spectrum of the data sequences was constant over time. For most of the
sequences of respiratory variables, the associated power spectrum was
not constant, although it was often uniformly modulated, as observed
for ventilation (22). Using a likelihood ratio test, Ackerson et al.
(1) suggested that respiratory data sequences were often nonstationary.
Unfortunately, they did not give any results that could be compared
with ours.
In keeping with previous results for ventilation (22), a modulating
function built from an autoregressive estimate of the variance could
often be used to reduce the original data to sequences that could be
accepted as having a constant power spectrum. Thus the observed
variations in power spectrum can be treated as arising from slow
variations in variance over time of the respiratory sequences.
The application of ARMA models requires the data sequence to be
stationary, and therefore the application of ARMA models to our data is
potentially inappropriate. However, because our data sequences could
often be reduced to those that could be accepted as stationary by
demodulation, the AR1MA1 model could be applied to original and demodulated data to test the influence of the observed
slowly changing variance of the data on the estimates of the model
coefficients. No statistical difference was observed between the
results from the two sets of data. Thus we conclude that the variance
of the data changes too slowly to affect the short-term
breath-to-breath correlations in our time series.
Correlations between successive breaths.
Autocorrelative structure within variables of the respiratory cycle has
been observed previously in humans with intact chemical feedback loops
and described using AR1 and AR2 models
(3, 4, 19, 23). During sleep, a first-order autoregressive structure
has been observed for VT and TI, although
variability in TE appeared mostly to be due to periodic
oscillations without underlying autoregressive structure (23). In the
current study where dynamic end-tidal forcing has been used to open the
chemical feedback loops, an AR1MA1 model
described the fluctuations of all the respiratory cycle variables more
satisfactorily than AR1 and AR2 models.
However, inasmuch as we are not aware of any other study of an
AR1MA1 model in humans with intact chemical
feedback loops, it is not possible to conclude that there are any
differences in autoregressive structure between the different states.
In animal studies of the respiratory controller where the chemical
feedback loops have been opened, the observations have been somewhat
contradictory. The anesthetized cat showed breath-to-breath correlations for TI and TE and phrenic activity
in an isolated respiratory center preparation (2). On the other hand,
although they observed correlations between successive breaths for
VT in spontaneously breathing bivagotomized rats, Khatib et
al. (20) failed to find such correlations for phrenic activity in
paralyzed and artificially ventilated bivagotomized rats. These authors suggested that the autoregressive structure of VT resulted
from the presence of chemical feedback. Khatib et al. suggested that the discrepancy between their results and those of Benchetrit and
Bertrand (2) might be due to differences between the species studied or
to nonstationarity within the data of Benchetrit and Bertrand.
Our results in conscious humans, where we have used dynamic end-tidal
forcing to open the chemical feedback loops in a functional sense, are
in much closer agreement with the findings of Benchetrit and Bertrand
(2) than with Khatib et al. (20). Although the original data sequences
could not be accepted as stationary in almost all cases, they could be
transformed into sequences that could be accepted as stationary by
demodulation, and this process did not modify significantly the
coefficients of the ARMA models fitted to the data. Thus we can be
reasonably confident that the difference between our result and that of
Khatib et al. is not due to nonstationarity within our data sets.
Interrelationships between respiratory variables.
In the original data sets, almost every pair of variables describing
the respiratory cycle was correlated. To make progress, the
autocorrelation within the data was first modeled using the AR1MA1 model. Once this autocorrelative
structure had been removed by eliminating those pairs of variables for
which cross correlation may have been induced by intrinsic
autocorrelation, the residuals from the model could then be examined
for any remaining cross correlation. This reduced the pairs of
variables for which cross correlation was detected. After this
procedure, however, there remained a considerable dependence of
VTE and TE on preceding VTI and TI and a considerable
dependence of VTI on preceding
VTE and TI on preceding
TE. These results led us to fit a multivariate ARMA model
that included the above dependencies together with the
AR1MA1 structure for the autocorrelative
components.
Model coefficients corresponding to the cross correlations were
statistically different from zero, but the autocorrelation of
VTE became nonsignificant in rest and exercise
protocols with the inclusion of VTI in the
multivariate model. Thus the autoregressive structure for
VTE in the simple ARMA model would appear to
arise because of its dependence on the preceding
VTI, rather than through a true
breath-to-breath dependence independent of
VTI. This may be explained by the relatively
passive nature of expiration (14) and its mechanical dependence on the
volume of the previous inspiration. It appeared with the exercise data
that the cross correlation between VTE and the
previous VTI was increased and that the moving average term for VTE was not statistically
different from zero. It is possible that a greater VT would
induce a greater mechanical dependence between
VTE and the previous
VTI for the exercise protocol than for the rest
protocol.
The cross correlations between VTI and
VTE will be affected in rest and exercise
protocols by the fact that over time the total volume inspired will be
approximately equal to the total volume expired. The fact that the
dependence of VTE on the previous VTI is much greater than the dependence of
VTI on VTE reflects the
greater variability of end-inspiratory volumes than end-expiratory volumes, as can be observed on a standard spirometer record.
The significance of the weak, but statistically significant, negative
relationship between VTE and preceding
TI is not clear. We are not aware of any results comparable
to ours concerning this negative cross correlation.
The link between, on the one hand, TE and, on the other
hand, preceding TI and VTI has been
previously documented in conscious humans. Correlations between
TE and preceding TI were shown during steady
breathing at a constant inspiratory gas composition in resting humans
(10, 24). TE has also been shown to be dependent on
preceding TI in some subjects with use of a multivariate
time-series model (19). However, the true dependence of TE
on VTI is not supported by all observations.
Rafferty et al. (29) investigated the separate effects on
TE of changes in TI and
VTI by using auditory feedback to enable the
subjects to fix some of the variables of the breathing pattern. This
study showed that TE changed in parallel with
TI when VTI was maintained constant
but that TE did not change with VTI
when TI was maintained constant. In addition, the study by
Benchetrit and Bertrand (2) of the anesthetized cat with open chemical
feedback loops has shown in most cases a dependence of TE
on preceding TI but no dependence on the value of
integrated phrenic activity (2).
The link that we observed between TI and preceding
TE in our study is less well established. Such a link has
been observed in some subjects in a previous study using a multivariate
time-series model in conscious humans (19). However, in this study, the stationarity of the data was not tested, and the influence of the
chemical feedback loops on the respiratory instability was not clearly
identified. Although not demonstrated in the majority of studies,
TI has been shown to vary with changes in TE
mediated by electrical activation of vagal afferents and mechanical
activation of the receptors in animals (12, 15, 30). Using a
multivariate ARMA model, Benchetrit and Bertrand (2) observed in the
isolated respiratory center preparation a correlation between
TI and preceding TE. As in our study, the link
between TE and subsequent TI was generally
weaker in these studies than the converse relationship between
TI and subsequent TE.
The multivariate ARMA model yielded broadly similar results between the
rest and exercise protocols. Nevertheless, some quantitative differences were found. These included a decrease in the
autocorrelation for TI and for TE and an
increase in the cross correlation between TI and preceding
TE for exercise compared with rest. Additionally, the
moving-average term for TE was not statistically different from zero for the exercise protocol. These findings may suggest that
the next-neighboring events have a greater importance for the timing
variables during exercise than during rest.
From the results above, it appears that the breath-to-breath
relationships between the respiratory cycle variables observed in this
study in conscious humans are generally in good agreement with the
results of Benchetrit and Bertrand (2) in an isolated respiratory
center preparation in the anesthetized cat. These authors concluded
that the dependence of a given breath on preceding breaths in the
absence of chemical and vagal feedback loops was the result of a
central mechanism acting as a short-term memory. Since then,
memory-like mechanisms in the brain stem contributing to the smoothing
of the respiratory output have received additional support (13).
In the current study, the chemical respiratory drive was maintained as
constant as possible to study the respiratory controller isolated from
the effects of the chemoreflex feedback loop. Furthermore, there is
also evidence that the vagal feedback mediated by the pulmonary stretch
receptors does not affect breathing pattern in humans, provided that
the overall level of ventilation is not too high (16, 17). It is thus
likely that the relationships between respiratory cycle variables
observed for the data in resting subjects arise essentially from
central mechanisms within the respiratory controller. During exercise,
the increase in ventilation, and thus in VT, may well
enhance the role of the mechanical feedback loop in determining
breathing pattern (8, 14). Despite this possibility, only the
magnitudes of some auto- and cross correlations fitted using the
multivariate model were modified during exercise compared with rest.
ACKNOWLEDGEMENTS
We thank Dr. J. J. Pandit for the material that made this study
possible. T. Busso is grateful to the Laboratoire de Physiologie and
the Groupement d'Intérêt Public Exercise,
Saint-Etienne (France) for financial support.
FOOTNOTES
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.
Address for reprint requests: P. A. Robbins, University Laboratory of
Physiology, Parks Rd., Oxford OX1 3PT, UK.
Received 28 November 1995; accepted in final form 14 June 1996.
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