Journal of Applied Physiology
Vol. 82, No. 2, pp. 667-677, February 1997
CONTROL OF BREATHING, CIRCULATION, AND TEMPERATURE

Extended models of the ventilatory response to sustained isocapnic hypoxia in humans

Pei-Ji Liang, Daphne A. Bascom, and Peter A. Robbins

University Laboratory of Physiology, Parks Road, Oxford OX1 3PT, United Kingdom

ABSTRACT

INTRODUCTION

METHODS

RESULTS

DISCUSSION

ACKNOWLEDGEMENTS

FOOTNOTES

REFERENCES

ABSTRACT

Liang, Pei-Ji, Daphne A. Bascom, and Peter A. Robbins. Extended models of the ventilatory response to sustained isocapnic hypoxia in humans. J. Appl. Physiol. 82(2): 667-677, 1997.The purpose of this study was to examine extensions of a model of hypoxic ventilatory decline (HVD) in humans. In the original model (model I) devised by R. Painter, S. Khamnei, and P. Robbins (J. Appl. Physiol. 74: 2007-2015, 1993 [Medline] ), HVD is modeled entirely by a modulation of peripheral chemoreflex sensitivity. In the first extension (model II), a more complicated dynamic is used for the change in peripheral chemoreflex sensitivity. In the second extension (model III), HVD is modeled as a combination of both the mechanism of Painter et al. and a component that is independent of peripheral chemoreflex sensitivity. In all cases, a parallel noise structure was incorporated to describe the stochastic properties of the ventilatory behavior to remove the correlation of the residuals. Data came from six subjects from a study by D. A. Bascom, J. J. Pandit, I. D. Clement, and P. A. Robbins (Respir. Physiol. 88: 299-312, 1992). For model II, there was a significant improvement in fit for two out of six subjects. The reasons for this were not entirely clear. For model III, the fit was again significantly improved in two subjects, but in this case the subjects were those who had the most marked undershoot and recovery of ventilation at the relief of hypoxia. In these two subjects, the chemoreflex-independent component contributed ~50% to total HVD. In the other four subjects, the chemoreflex-independent component contributed ~10% to total HVD. It is concluded that in some subjects, but not in others, there may be a component of HVD that is independent of peripheral chemoreflex sensitivity.

sustained isocapnic hypoxia; hypoxic ventilatory decline; peripheral chemoreflex sensitivity

INTRODUCTION

THE VENTILATORY RESPONSE to sustained hypoxia is biphasic: there is an initial rapid increase in ventilation, which is then followed by a slow decline (7, 18). This slow decline is known variously as hypoxic ventilatory decline or depression (HVD) or hypoxic ventilatory "roll-off."

In anesthetized animals, HVD may be independent of the peripheral chemoreflex sensitivity, since 1) in anesthetized cats, the peripheral chemoreflex sensitivity to CO₂ is unchanged during HVD induced by central hypoxia (16); 2) in anesthetized cats, the rapid ventilatory response at the onset and relief of hypoxia is almost symmetrical (4); and 3) respiratory depression during hypoxia has been a consistent finding in anesthetized animals that have undergone peripheral chemodenervation (12).

In conscious humans, however, HVD may arise predominantly through an alteration of peripheral chemoreflex sensitivity. The evidence for this is as follows: 1) the rapid ventilatory response at the relief of sustained hypoxia is much smaller than the rapid response at the onset of the period of hypoxia (6, 7, 9); 2) the reintroduction of hypoxia after a brief relief from sustained hypoxia will cause a ventilatory response with a smaller magnitude compared with the first exposure to hypoxia (3, 8); and 3) brief exposures to additional hypoxia during the development of HVD elicit progressively smaller ventilatory responses as the HVD develops (2).

On the basis of the above findings, Painter et al. (13) developed a mathematical model to describe HVD during sustained hypoxia in humans in which HVD arises as the result of a progressive decline in peripheral chemoreflex sensitivity. The ventilatory response to sustained hypoxia and the asymmetrical magnitudes of the ventilatory on- and off-transients at the onset and relief of hypoxia are both described well by this model. However, as noted by Painter et al., this model fails to describe very well the ventilatory behavior after the step out of hypoxia in cases in which there may be some undershoot and subsequent recovery of the ventilatory response. One possible cause is that the assumptions about the dynamics are too simple. Another possible cause of the undershoot and recovery sometimes observed at the relief of hypoxia might be the existence of an additional component of HVD in conscious humans, which is independent of peripheral chemoreflex sensitivity, as has been described in the studies relating to anesthetized animals.

The purpose of the present study was to explore extensions to the model of Painter et al. First, we wished to explore whether the dynamics of the model were too simple. Second, we wished to explore whether there is an additional component for HVD that is not related to the peripheral chemoreflex sensitivity. In particular, we wished to determine 1) whether either extension would improve the fit of the model to the data at the relief of hypoxia and 2) in the case of the model with the additional component for HVD, what proportion of HVD would be attributed to the component that is independent of the peripheral chemoreflex sensitivity.

In the original model of Painter et al., the parameter estimation was performed with a simple least squares fitting technique (this corresponds to a measurement-only noise model). With respiratory data, use of this technique generally results in residuals that are correlated, and this can make statistical comparisons between different models difficult. To overcome this, a parallel noise structure was used in conjunction with the deterministic models to remove this correlation. The noise model was fitted by using a Kalman filter algorithm.

METHODS

Data

Both the strength of the stimulus (PET_O2 = 50 Torr) and background PET_CO2 were the same as for the data used by Painter et al. (13). However, the data differ in the sense that the protocol used to generate the data modeled in the present study was prolonged to allow for the reintroduction of hypoxia to provide a second on-transient. In the data modeled by Painter et al., there was no second on-transient. This second on-transient may help to discriminate between a component of HVD that arises through modulation of peripheral chemoreflex sensitivity and a component that does not act via such a mechanism.

Models

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

Fitting Technique

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

Sensitivity Analysis

Model data were generated for the input function associated with the protocol. The more complex extended model (model III) was used with a set of physiologically reasonable parameter values (actually, the parameter values determined for subject 758) and a breath duration of 3 s. The model parameters were then estimated repeatedly for the data set, with the exception that one of the parameters would be fixed, in turn, to the value used to generate the data and then to values ranging from 80 to 120% of this value (except for k_p, where a range of 0.0-0.02 was used, since its value is close to zero). The sum of squared residuals could then be plotted against the value of the fixed parameter. If the parameter is not identifiable, then the plot should be flat, whereas if the value of the parameter affects the fit, then the plot should have a definite minimum (zero) at the parameter value used to generate the data.

RESULTS

Sensitivity Analysis

Fitting Technique

Average parameter values from both the simple least squares fitting technique and the Kalman filter algorithm are shown for the model by Painter et al. (13) in Table 1. In general, the parameter estimates from the two techniques appear quite similar. However, for the time constants associated with the rapid on-transients (T_p1, T_p3), the Kalman filter algorithm consistently yielded lower values than did the simple least squares fitting procedure. For the time constant for the hypoxic ventilatory decline T_h, the reverse was true, with the values from the Kalman filter algorithm being consistently higher than those from the simple least squares fitting procedure. Consistent effects across all the six subjects are significant at the level P < 0.05 (sign test) and suggest some element of bias in estimating these values with a simple least squares fitting routine.

Table 1. Parameter values estimated by using least squares fitting technique and by using a Kalman filter for each subject for model I Subject No.  c, l/min g100, l/min gh, l/min kp Tp1, s Tp2, s Tp3, s Th, s dp, s Least squares 743 12.7 139 295 0.06 48.1 5.3 33.9 205 0 758 8.8 79 272 0.05 51.2 3.2 56.8 296 0 766 6.6 92 433 0.01 24.7 4.7 11.5 759 0 796 14.7 59 244 0.06 17.9 1.6 3.9 299 1 818 18.7 257 1,436 0.02 8.9 6.7 33.4 344 0 824 15.0 222 804 0.02 27.1 30.9 19.5 312 0 Kalman filter 743 14.4 136 273 0.04 31.2 4.8 26.7 273 0 758 10.2 84 335 0.03 46.8 3.9 56.3 315 0 766 6.5 88 407 0.01 15.6 5.7 5.8 832 0 796 12.7 50 165 0.11 2.5 1.2 0.8 380 3 818 18.3 249 1,006 0.03 6.3 4.1 18.7 392 2 824 16.0 210 729 0.01 12.3 16.3 12.1 359 0 See text for symbol definitions.

Model Comparison

Figure 4 shows average data and model fits for each subject for model I, Fig. 5 shows these data for model II, and Fig. 6 shows these data for model III. These were calculated from the individual data sets and model fits by interpolating the data every 3 s and then averaging across all the individual responses for each subject. The average of the residuals sequences for each subject was calculated in the same way, and the 95% confidence intervals (calculated as ±1.96 SE) were plotted along with the averaged ventilatory output. For much of the response, the outputs of all three models are very similar. However, for the two subjects (subjects 758, 796) who show the greatest undershoot and recovery at the relief of hypoxia, model III appears to describe this process more accurately than do the other two models.

When model II is compared with model I, F-ratio tests on the sum of squared prediction errors revealed that there was no significant improvement in fit for four out of the six subjects but there was an improvement in fit for the other two (subjects 766, 824). Careful inspection of Figs. 4 and 5 suggests that there may have been some improvement in the fit during the on-transients, but no improvement was observed during the recovery from hypoxia, where visual inspection suggests the fit is least good.

When model III is compared with model I, F-ratio tests revealed that there was, again, no significant improvement in fit for four out of the six subjects but that for the two other subjects (subjects 758, 796) model III fitted significantly better than model I. Visual inspection of Figs. 4 and 6 suggests that for these two subjects the undershoot and recovery in ventilation were most marked and that model III did produce an apparent improvement in fit to this portion of the response.

A whiteness test (portmanteau test) on the prediction errors showed that 26 out of 31 sequences could be accepted as white for model I, 27 out of 31 sequences for model II, and 28 out of 31 sequences for model III. Thus fitting a stochastic component to the data makes the use of an F-ratio test possible without the complications of trying to correct the degrees of freedom for correlation present within the residuals.

Overall parameter values for each subject for each model are shown in Table 2. For all three models, the time constants for the rapid ventilatory response to the onset and the relief of hypoxia (T_p1, T_p2, and T_p3) are faster than the time constant for the slow change in the peripheral chemoreflex sensitivity T_h. In most cases, the time constant for the rapid ventilatory response at the relief of hypoxia (T_p2) appears faster than those for the rapid ventilatory response at the onset of hypoxia (T_p1 and T_p3). However, this effect does not reach statistical significance overall.

Table 2. Parameter values estimated by using a Kalman filter for each subject for each model Subject No.  c, l/min g100, l/min gh, l/min kp Tp1, s Tp2, s Tp3, s Th, s gd, l/min Td, s TI, s dp, s f Rv/Rw P Value (F Ratio) Model I 743 14.4 136 273 0.04 31.2 4.8 26.7 273 0 0.74 2.39 758 10.2 84 335 0.03 46.8 3.9 56.3 315 0 0.65 3.01 766 6.5 88 407 0.01 15.6 5.7 5.8 832 0 0.81 0.59 796 12.7 50 165 0.11 2.5 1.2 0.8 380 3 0.81 1.02 818 18.3 249 1,006 0.03 6.3 4.1 18.7 392 2 0.67 1.78 824 16.0 210 729 0.01 12.3 16.3 12.1 359 0 0.87 0.53 Model II 743 18.8 121 189 0.01 29.3 3.9 19.4 141 141 0 0.75 3.49 NS 758 10.9 68 238 0.02 40.4 2.9 39.6 101 239 0 0.65 3.00 NS 766 6.8 70 226 0.0 15.5 5.2 4.5 287 287 0 0.77 0.65 <0.05 796 13.0 48 148 0.09 5.7 3.9 1.2 293 0.2 1 0.81 1.03 NS 818 19.6 207 727 0.02 5.1 3.2 13.3 279 88 2 0.68 2.48 NS 824 17.1 184 549 0.01 12.3 14.5 10.3 168 168 0 0.84 0.47 <0.05 Model III 743 20.2 136 281 0.0 27.8 4.1 23.5 328  12.4 300 0 0.77 1.94 NS 758 12.5 135 352 0.0 46.1 15.0 53.3 324  55.4 52 0 0.60 4.12 <0.05 766 7.2 91 431 0.0 20.4 5.4 6.9 869  4.5 232 0 0.72 0.74 NS 796 16.7 102 252 0.0 11.5 1.9 5.4 499  53.1 22 3 0.76 1.25 <0.05 818 18.5 248 1,009 0.03 6.2 2.8 18.7 395 0.0 2 0.67 1.80 NS 824 19.0 213 723 0.0 16.1 32.6 12.9 319  16.9 738 0 0.81 0.59 NS NS, not significant. See text for other definitions.

For model II, in five out of six subjects, the parameter values for T_I and T_h are of the same order of magnitude (for three subjects, the values for T_I and T_h are the same). In only one subject (subject 796) are the values radically different. In this one subject, the very low value for T_I compared with T_h means that the dynamic for g_p is very close to first order, but for the other five subjects this seems not to be true. This alteration in dynamic has had consistent (albeit often small) effects on the other parameter values of the model. In particular, when model II is compared with model I, _c is consistently increased and g₁₀₀ and g_h consistently decreased across all the subjects.

Turning to the parameters for model III, some effect was attributed to the extra component of the model by the fitting process in five out of six subjects. The values estimated for both g_d and T_d were quite variable among the different subjects. Table 3 shows the percentage contributions to HVD of the component acting via modulation of the peripheral chemoreflex sensitivity _p and the component that is independent of the peripheral chemoreflex (_d) for each subject. These figures were obtained by calculating the steady-state values for _p and _d under conditions of a sustained steady PET_O2 of 100 Torr and under conditions of a sustained steady PET_O2 of 50 Torr. For _p, this was done by setting the derivatives of Eqs. 2 and 3 to zero and obtaining an explicit expression for _p, and, for _d, by setting the derivative of Eq. 18 to zero and obtaining an explicit expression for _d. The averaged contributions calculated across all the six subjects were 73.8% for the component acting via the peripheral chemoreflex and 26.2% for the peripheral chemoreflex-insensitive component.

Table 3. Ventilations in euoxia and percentage of HVD attributed to each component for each model Subject No. Ventilation, l/min HVD, %   c  p  d  E  p  d Model I 743 14.4 7.0 21.4 100 0 758 10.2 3.4 13.6 100 0 766 6.5 1.9 8.4 100 0 796 12.7 5.9 18.6 100 0 818 18.3 10.1 28.4 100 0 824 16.0 4.6 20.6 100 0 Model II 743 18.8 2.7 21.5 100 0 758 10.9 2.1 13.0 100 0 766 6.8 0.9 7.7 100 0 796 13.0 4.7 17.7 100 0 818 19.6 6.5 26.1 100 0 824 17.1 4.1 21.2 100 0 Model III 743 20.2 1.7  0.2 21.8 76 24 758 12.5 1.7  0.7 13.5 48 52 766 7.2 1.1  0.1 8.3 93 7 796 16.7 1.3  0.7 17.3 40 60 818 18.5 10.1 0.0 28.6 100 0 824 19.0 2.6  0.2 21.4 86 14 HVD, hypoxic ventilatory decline; E, minute ventilation. See text for other definitions.

When we compared parameter values across all models, one striking feature we noticed was the consistent nature of the differences in value for k_p. For five out of six subjects, the value for k_p for model II was smaller than for model I, and for five out of six subjects, the value for k_p was smaller for model III compared with model II. For model I, a positive value for k_p was obtained for all six subjects; for model II, a positive value was obtained for five out of six subjects; but for model III, a positive value was only obtained for one out of the six subjects. Physiologically, k_p represents the component of peripheral chemoreflex drive that is present in high oxygen, and this suggests that model I attributes a higher proportion of the total ventilation to peripheral chemoreflex drive in euoxia than does model II, and that model II attributes a higher proportion than does model III. This is confirmed by inspection of Table 3, where the contributions of each component to ventilation have been calculated for each of the models in euoxia. The mean contributions to ventilation by the peripheral chemoreflex drive are significantly different among models, being highest with model I and lowest with model III (analysis of variance, P < 0.05). In the case of model III, the reduction of total ventilation by _d is small in euoxia, as would be expected by the fact that the saturation is close to 1.

DISCUSSION

Fitting Technique

It is of some interest to determine whether the values for the stochastic component of the model ( f, R_v/R_w) were similar to those obtained from a previous study under conditions of steady chemical stimulation (10). From this previous study, the mean ± SE values for f and R_v/R_w were 0.69 ± 0.06 and 0.95 ± 0.13, respectively, for the resting data and 0.81 ± 0.03 and 0.44 ± 0.11, respectively, for the data during exercise. The mean values for f and R_v/R_w in the present study were 0.76 ± 0.04 and 1.55 ± 0.41, respectively, for model I; 0.75 ± 0.03 and 1.85 ± 0.53, respectively, for model II; and 0.72 ± 0.03 and 1.74 ± 0.53, respectively, for model III. Thus the mean values for f in the present study lay between the values for rest and exercise from the previous study, whereas the values for R_v/R_w appeared to be a little higher. When the values obtained from the fits for the three different models of the present study were compared, analysis of variance showed that there was no significant difference in the estimated value of R_v/R_w between the three models (P > 0.05), but there was a significant difference between the estimated values of f for the different models (P < 0.05). One interpretation of this is that when the model structure varies, the residual correlation that has to be explained by the stochastic process ( f ) varies too, but not necessarily the estimate for R_v/R_w.

Model Comparison

In statistical terms, for model II, two subjects (subjects 766, 824) showed a significant reduction in the sum of squared residuals, as compared with model I. These subjects do not appear to have any particularly distinguishing features in terms of their response to the hypoxic protocol. In addition, a direct visual comparison of the model fit between model II and model I did not suggest much by way of difference between them. Overall, it is difficult to be certain exactly why these two subjects show a significant improvement in fit, but we feel that it is most likely to be related to differences during and immediately after the onset of hypoxia. The model does not appear to improve the fit to the period immediately following the relief from hypoxia, which was one of the specific purposes of the study.

For model III, again only two subjects (subjects 758, 796) showed a significant reduction in the total sum of squared residuals when compared with model I. These two subjects consistently showed an apparent undershoot and then recovery of ventilation at the relief of hypoxia in all the individual experimental repeats of the protocol. In other subjects, an undershoot followed by recovery was either not observed or appeared inconsistently in some, but not all, of the repeats. Inspection of the plots of the model fit and ensemble-averaged residuals over time shows that it is during the period following the relief from hypoxia that the models are most distinct. We conclude that the use of model III may significantly improve the fit in some subjects, who show a reasonable degree of undershoot and recovery of ventilation after the relief of hypoxia, but that this is not the case for all subjects.

The two subjects in whom the fit of model III was a significant improvement over the original model of Painter et al. (13) had the two largest magnitudes for the gain term g_d of the additional chemoreflex-independent component. In these two subjects, slightly over 50% of HVD (mean 56%) was attributed to the peripheral chemoreflex-independent component. In the other four subjects, the value was very much smaller, with a mean value of 11%. Again, this emphasizes the importance of intersubject differences.

For the two subjects with a substantial contribution to ventilation from the peripheral chemoreflex-independent compartment, the time constants associated with this compartment were rather fast (22 and 50 s) and are consistent with the comments of Bascom et al. (3) that the baseline ventilation appears to recover more quickly than does the peripheral chemoreflex sensitivity. These values do not really correlate well with the sorts of values previously obtained for the development of HVD in anesthetized animals [onset of HVD 149.7 ± 8.5 (SE) s; offset of HVD 105.5 ± 10.1 s (17)]. Thus, on the basis of data from these two subjects, it would appear that this peripheral chemoreflex-insensitive component cannot really be equated simply with a chemoreflex-insensitive form of HVD observed in anesthetized animals, as postulated in the introductory part of this paper. However, for the other three subjects in whom some component of the ventilatory response was attributed to the peripheral chemoreflex-independent term, the time constants were all somewhat longer than those associated with HVD in anesthetized animals. Overall, we feel that it would be unwarranted to draw any very firm conclusions from these values.

One notable feature that differs among the models is the parameter estimates determined for k_p. Physiologically, k_p is a parameter relating to the contribution of the peripheral chemoreflex to ventilation in hyperoxia when the arterial blood is fully saturated. We consider that the positive values for k_p in the original model of Painter et al. (13) may be necessary to fit the slightly lower level of euoxic ventilation seen after the relief of hypoxia, when compared with the original baseline ventilation. In model III, this feature of the data can be modeled via the additional term _d. Which of these approaches in a physiological sense is correct is difficult to determine. Although there is some evidence to suggest that the peripheral chemoreflex is inactivated by hyperoxia in humans (11), in which case model III might represent the physiology better, it is also possible that positive values of k_p might simply represent some nonlinearity of the response of the peripheral chemoreflex to saturation in this range.

ACKNOWLEDGEMENTS

This work was supported by The Wellcome Trust. P.-J. Liang was supported by a Run Run Shaw studentship.

FOOTNOTES

Address for reprint requests: P. A. Robbins, University Laboratory of Physiology, Parks Road, Oxford OX1 3PT, UK.

Received 11 October 1995; accepted in final form 24 September 1996.

REFERENCES