INTRODUCTION

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PROBABILISTIC MODELS of the risk of human decompression sickness (DCS) have been successful in describing the occurrence and even the time of occurrence of DCS (9, 13, 15, 17, 18). With rare exceptions (14, 19), only inert gases have been considered in such decompression modeling, on the assumption that the role of inert gases in the development of DCS is of overwhelming importance. In nearly all decompression models, inspired O₂ is treated as a "free" quantity and is not linked to the risk of DCS. O₂ is less available as a dissolved gas when it is bound to hemoglobin and when it is converted to the very soluble gas CO₂. That view is substantiated by measurements of tissue O₂ levels of only a few Torr under normoxic conditions (2).

The most successful probabilistic model has not performed well in predicting DCS risk in dives that use a high fraction (~100%) of O₂ in the breathing gas during decompression (9, 13), underpredicting the occurrence of DCS in these O₂ decompression dives by ~60%. In a subsequent prospective trial of O₂ decompression procedures, severe underprediction again occurred (11). These results contradicted the expectation of no O₂ effect found in a moderately large study of dives (19) with direct ascent after breathing mixtures with a PO₂ range of 0.2-1.3 atmospheres absolute (ata). The emphasis of the present study is to develop modifications to the previous model to identify a specific O₂ influence on the accumulation of DCS risk. The ideal modification would improve, or not disturb, the model's success with N₂-O₂ data while better describing the DCS outcomes observed in the O₂ decompression data. Such an improved model could then be applied to the practical optimization of the use of O₂ to accelerate decompression.

The O₂ effects explored here are of two very different forms, both based on observed physiology. In our first model a PO₂-dependent alteration of the N₂ washin-washout kinetics acknowledges the pharmacological ability of PO₂ to alter central and peripheral circulation. Anderson et al. (1) demonstrated a progressive and significant reduction in cumulative N₂ excretion with increasing inspired O₂, although the difficult experimental procedure did not allow quantitative estimates of actual N₂ kinetic parameters. In our second model, some of the inspired O₂ is treated as an inert gas, adding to the tissue level of N₂ in leading to DCS risk. Hyperoxia is known to greatly increase PO₂ in tissues (2, 5), and some prior decompression studies concluded that O₂ was approaching N₂ in its DCS risk potency (3, 4, 8, 10). Tikuisis and Nishi (14) explored a bubble-based DCS risk model that included an explicit O₂ contribution, but they did not apply it to data as extensive as those used here, nor did they use it to predict time of DCS occurrence, which is the focus of the present study.

All our models are based on survival functions and are intended to predict the risk of occurrence of an undesirable outcome due to a risk-generating event, in this case the occurrence of DCS after a hyperbaric exposure. We construct a mathematical model that relates a small number of measured variables (time, pressure, gas mix) to a binary outcome (DCS: yes/no). Although we borrow from the terminology of physiology when we use a label such as "partial pressure of gas in tissue," we have made no direct physiological measurements. Gas terminology is used to aid visualization of a risk function. The success or failure of such a model rests strictly on its ability to predict the probability of occurrence of the outcome.

	DATA

The data sets used in fitting models in this report were taken from carefully controlled and well-documented experimental dives conducted in the United States, Canada, and Great Britain, described in detail elsewhere (data sources are described in Ref. 16 with additional sources in Refs. 6 and 11). The basic data set (group A in Table 1) used in earlier model development (6, 9) contains 3,322 dives. The data set with ~100% O₂ breathed during decompression (group B in Table 1) contains 1,013 dives.

In the group A dives, there are 190 DCS and 110 marginal cases, giving an overall DCS incidence of 6.1%. (The Appendix lists the data by file names in the primary database of the Naval Medical Research Institute, which is available from the authors.) Marginal cases are mild events considered to be related to the hyperbaric exposure but not severe enough to warrant recompression treatment. These events are given a value of 0.1 DCS case on the basis of the experience of senior diving medical officers (9). Although the majority of dives in group A used compressed air (21% O₂), a large number of dives were performed with moderately enriched O₂ atmospheres. In most of these nonair dives a constant PO₂ of 0.7 ata was breathed, either throughout the dive or with interspersed periods of air breathing. Other nonair dives used a range of constant fraction of O₂ throughout the dive from 10 to 40%, resulting in PO₂ of 0.21-1.4 atmospheres absolute (ata) (19). None of the nonair dives used a significantly higher PO₂ during decompression than during the dive itself. The high PO₂ values (up to 4.0 ata) in the single-air category come from 58 short-duration (<3 min) dives from a submarine escape experiment, in which high pressures were present for <1 min. Without these 58 profiles, the upper limit of the PO₂ range for single-air dives would be 1.5 ata. Only two of the DCS cases in group A come from these escape dives.

Group B contains 33 DCS and 17 marginal cases, for an incidence of 3.4%. The dives in group B are of two types: 1) air dives that use ~100% O₂ during decompression and 2) air dives followed by ~100% O₂ during surface decompression procedures. Surface decompression involves omitting much of the usual decompression requirement, traveling quickly to the surface, and then recompression in a dry hyperbaric chamber, usually to a fixed pressure, after a brief interval at the surface. To allow for incomplete delivery of O₂ to the diver, we assume that immersed divers breathed 99.5% O₂ and dry divers 98% O₂. The consequences of choosing these particular values are discussed later. PO₂ within group B is 0.21-2.8 ata, with the majority of the O₂ exposures at 1.9 or 2.2 ata, corresponding to decompression stop depths of 30 and 40 feet of seawater.

The data include time of occurrence for all DCS cases and for many of the marginal cases. The time of symptom occurrence is represented in the data as an interval (T₁  T₂) over which symptoms appeared, where T₁ is the latest time the diver was known to be entirely free of symptoms and T₂ is the time at which definite symptoms were first reported. The methods and rules of establishing T₁  T₂ for most reported dives are described in detail elsewhere (16).

	MODELS

The best-fitting model from our most recent N₂-O₂ modeling effort (9, 13) was used as the base model for this study (model 0). This model allows for exponential washin and a mixed exponential-linear washout of inert gas partial pressure (9, 12, 13). Risk accumulation for this model is characterized by an instantaneous risk (r) proportional to the sum of the risks of each of its three parallel compartments

(Eq. 1)

where A_i is a scale factor and Pti_i is the inert (N₂) gas burden for the ith compartment. The inert gas burden represents all inert gas pressure in the compartment, including that in any bubbles present, as though it had remained in solution. P_amb is the ambient pressure, Thr_i is an estimated threshold parameter (9) for the ith compartment, and P_met is a small constant contribution of metabolic gases (venous PO₂ and PCO₂ and water vapor pressure), with a numerical value of 0.19 atmospheres. Pti_i is a function of the arterial inert gas partial pressure (Pa_N2); a time constant (_i), which conceptually represents blood perfusion to the tissue; and an estimated linear-exponential kinetic crossover parameter (PXO_i)

(Eq. 2)

where Ps_N2 is the partial pressure of dissolved N₂ in the tissue.

If Pti_i  (PXO_i + P_amb  P_met), only dissolved gas is present and Pti_i equals Ps_N2 and gas exchange is simply exponential. If Pti_i > (PXO_i + P_amb  P_met), then a bubble is deemed to be present and excess gas comes out of solution, such that the Ps_N2 remains constant at a level of (PXO_i + P_amb  P_met). Thus, when depth and Pa_N2 are constant, exchange becomes linear with time. The parameters A_i, Thr_i, PXO_i, and _i are estimated by fitting to the observed data.

Figure 1 illustrates the handling of inert gas partial pressure in model 0 for a dive with O₂ decompression. In the hypothetical dive shown, two possible washout curves are plotted: one for a diver who breathes air (solid curve) throughout the decompression and another for a diver who breathes 100% O₂ (dashed curve) during a portion of the decompression. The duration of the O₂ period is indicated by the drop in Pa_N2 below that for breathing air. During the O₂ breathing period, N₂ washout accelerates because Pa_N2, the asymptote (or forcing function) for the model's calculated N₂ partial pressure (Pti_N2), is then essentially zero. Because model 0 considers DCS risk to be proportional only to the area between the Pti_N2 curve and P_amb, risk is reduced, both in magnitude and duration, because of the O₂ breathing period. This risk reduction agrees qualitatively with the idea that breathing O₂ during decompression reduces the risk of DCS, but comparison of predictions with observed DCS incidence indicates that the reduction is too large (9, 13).

View larger version (17K): [in this window] [in a new window]   Fig. 1.   Model 0: preferential washout of N2 partial pressure in tissue (PtiN2) during O2 breathing. Dashed line (PaN2 with air), arterial pressure of N2 during air breathing; thin solid curve (PtiN2 with air), model's washin-washout response; thick dashed line at middle bottom (PaN2 during O2 interval), drop in N2 pressure during 100% O2 breathing; thick dashed curve (PtiN2 during O2 interval), model's response.

O₂-induced kinetic modifications. The first class of modification (model 1) changes the inert gas kinetic time constants for each compartment as a function of inspired PO₂. This type of modification is based on experimental results in which a reduction of whole body N₂ washout was observed with exposure to increasing PO₂ (1). This reduced N₂ washout is attributed to simultaneously observed reductions in cardiovascular parameters, including heart rate and blood flow. These combined effects can be modeled as O₂-induced reduction of perfusion rate, resulting in increased kinetic time constants. In model 1 the modified time constant for each compartment is defined as

(Eq. 3)

where _0,i is the unmodified inert gas time constant for the ith compartment (to be estimated by fitting to data), PO₂ is the inspired O₂ pressure, and P_set and k are parameters to be estimated from the data. P_set is a pressure threshold above which pressures of O₂ begin to cause kinetic slowing and k is simply a scale factor necessary to modulate the effect. There is no effect if PO₂ is less than P_set.

View larger version (13K): [in this window] [in a new window]   Fig. 2.   Model 1: possible kinetic slowing effects as a function of PO2. Depending on values of estimated parameters Pset and k, model 1 may produce a wide range of slowing factors. Solid line, slowing function resulting from best-fitting estimated parameter values.

O₂ as an inert gas. In this model, O₂, at sufficiently high partial pressures, can contribute to bubble formation or growth (3-5, 8, 14). Model 2 introduces the "O₂ effect" as a direct additive term in the supersaturation part of the risk function. Thus for the inert gas term in Eq. 1

(Eq. 4)

These burdens of N₂ and O₂ are governed by their own kinetic time constants. As in model 0, exponential and linear kinetics are possible. If Pti_i  (P_amb + PXO_i  P_met), then washout is exponential with independent N₂ and O₂ kinetics. However, if Pti_i > (P_amb + PXO_i  P_met), then the N₂ and O₂ washouts become linked, such that the sum of partial pressures of dissolved N₂ and O₂ remains constant at the level of (P_amb + PXO_i  P_met).

(Eq. 5)

Model evaluation. The risk functions, each model's set of equations leading to Eq. 1, were cast in standard risk (or hazard) function form to predict the probability of each observed dive in the data set and then into a likelihood (or log likelihood, LL) function. Details, especially those required to properly account for time of DCS onset, have been presented previously (17). Parameter estimation, propagation of errors, and formulation of likelihood ratio (LR) tests used standard methods, as in prior work (9, 15, 17, 18).

Results of fitting. Ideally, the O₂ effect parameters of any model would describe the data from group B in Table 1 and allow the basic parameters (those relating to Eq. 1) to better describe the data in group A. Table 2 lists the best-fit parameters and SEs estimated for each model.

	PREDICTION OF DCS

Table 3 lists the DCS occurrence predicted by each of the candidate models for the data used in fitting, along with the 95% confidence limits of each prediction obtained from propagation of errors. The last column in Table 3 gives predictions from model 0 fit to group A only (model 0A). As expected, model 0 predicts DCS in the combined data better than model 0A just by calibration to the combined A + B data. For example, the total DCS predicted by model 0 increased to 238, from 216 predicted by model 0A, compared with 236 observed cases. This improvement is accomplished by increased prediction of DCS for all data types except saturation dives. However, model 0 continues to underpredict DCS incidence in group B (by 25.1%) and fails to include the observed value within the 95% confidence limits of its prediction in group B, either as a whole or in its subsets.

It is clear from Table 3 that models 1 and 2 have most of the desired predictive ability: prediction of DCS occurrence in group A dives centered nearly on the observed value and prediction of DCS occurrence in group B, which includes the observed value within its confidence limits. Also, models 1 and 2 have maintained the quality of prediction of model 0A for dives in group A.

Tables similar to Table 3 can be used in ² tests of "goodness of fit," where large values of the test statistic are taken as "failure" of the model to describe the distribution of the data. We can test each model's ability to predict DCS within each data group by separately considering the five categories of group A and two of group B from Table 3. The resulting model 0, 1, and 2 test statistics are 6.6, 2.9, and 3.3 for group A [4 degrees of freedom (df)] and 2.9, 1.0, and 1.3 for group B (1 df), respectively. None of these models "fails" to fit: all these ² values yield P > 0.05. Similarly, we can break the 26 categories in the Appendix into the 21 belonging to group A and the 5 belonging to group B. The resulting model 0, 1, and 2 test statistics are 23.3, 19.7, and 20.0 for group A and 11.8, 6.6, and 6.7 for group B. All group A tests yield P > 0.05 for 20 df. For group B, model 0 has P < 0.05 and models 1 and 2 have P > 0.05 for 4 df. This data categorization provides an indication that model 0 does not predict DCS occurrence in the dives of group B as well as models 1 and 2. However, the outcomes of such ² tests are clearly dependent on the choice of categorization. From results such as these and from many other instances where arbitrary but "reasonable" recategorization of data leads to "large" ² statistics, we believe that such tests are only useful as a rough guide to identify problem areas. These areas can be identified more readily using line-by-line comparisons of observed and predicted results.

The inclusion of time of occurrence in our data allowed us to compare the predictive performance of the candidate models with the observed time distribution of DCS incidence. Figure 3 shows the observed and predicted DCS cases in each 1-h interval after surfacing for the dives in group B. Negative times indicate relatively rare events occurring during decompression before the divers reach the surface. Model 0A clearly underpredicts occurrence as a function of time throughout. Model 0 shows substantial improvement over model 0A, with increased prediction for at least 8 h after the divers surface. Models 1 and 2 have nearly identical predictions of occurrence in all time intervals but tend to overpredict in the 2- to 5-h range. Because almost one-third of the DCS cases are observed within the 1st h after surfacing, a good prediction here is particularly important. Here, the prediction of model 2 (8.9 cases/h) comes closest to matching this value observed in the 1st h (10.6) but differs only slightly from that of model 1 (8.7).

View larger version (19K): [in this window] [in a new window]   Fig. 3.   Time of occurrence of decompression sickness (DCS). Predicted and observed DCS cases are shown for each hour after diver surfaces.

	DISCUSSION

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Both models of an O₂ contribution to DCS successfully described the expanded data set. Are the fully parameterized models plausible in light of the supposed underlying physiology? Because model 1 was intended to incorporate the experimental observations of Anderson et al. (1), we compared the behavior of this model with those observations. They reported 9 and 17% reductions in the volume of whole body N₂ elimination compared with normoxic levels over 2 h of washout at 2.0 and 2.5 ata PO₂, respectively. By use of the best-fit parameters shown in Table 2, the time constant for N₂ elimination in the second of three compartments in model 1 was increased by factors of 2.55 and 6.67 at 2.0 and 2.5 ata PO₂, respectively. A decrease in blood flow of >80% is large but not inconceivable. Over a 2-h washout period, these increased calculated time constants would result in 60 and 85% reductions, respectively, of the N₂ elimination expected from the unmodified time constant of 57.6 min, taking into account the asymmetric washout due to the mixed linear-exponential kinetics. It is reasonable to ignore the very fast and very slow compartments of the model compared with the experiment of Anderson et al. If compartment 2 represents ~15-20% of the total N₂ gas volume, then the calculated reductions in N₂ elimination would translate approximately into the reported 9 and 17% whole body reductions.

Another human decompression study attempted to analyze N₂ exchange retardation from high O₂ pressures (19). Over the experimental range of 0.2-1.3 ata PO₂, the single N₂ time constant did not appear to change, but parameter uncertainty allows the ~90-min time constant to slow to as much as ~130 min, which would represent an 18% reduction in N₂ elimination over a 2-h washout period.

Model 2 represents an approach fundamentally different from model 1, in that O₂, when present in pressures greater than P_seti (Table 2), contributes directly to the risk generating overpressure, as defined in Eqs. 1 and 4. The estimated value of 1.03 ata for P_set2 requires that no O₂ effect on DCS risk be seen at pressures lower than this. This is a plausible threshold, in that O₂ levels in the tissue can be kept low until the hemoglobin dissociation curve is fully saturated above ~1.0 ata. A P_set of 1.03 ata allows for a contribution to DCS risk accumulation of 25-60% of the O₂ present during decompression in the dives of group B. This result is in general agreement with some animal studies (3, 4, 8, 10), which called for a 25-33% contribution from O₂. A prior human study (19) did not require an O₂ effect on risk but placed an upper bound of 40% contribution up to 1.3 ata PO₂ and thus is consistent with the present result. We note that combinations of models 1 and 2, incorporating a kinetic slowing and a direct contribution effect, were not successful in improving the fit relative to model 1 or model 2 as fit separately.

Our O₂ effect modifications were intended to remedy the failure of model 0 to account for the DCS incidence observed in the O₂ data. Because our data coding of the inspired O₂ level in this data set is critical in all models, we should ask whether our data misrepresented the diver's actual gas exposure. In particular, we have explored the possibility that the coding of dry chamber O₂ decompressions, which form the bulk of group B, at 98% O₂ is incorrect because of imperfect delivery of the gas. Estimates from experienced investigators suggest that the minimum O₂ fraction likely to be present in the face mask in dry exposures is ~85-95% (R. Y. Nishi, personal communication). If the actual O₂ exposures were much less than our indicated 98%, model 0, without a specific O₂ contribution to DCS risk, might be able to account for the DCS incidence observation in group B. To explore this, model 0 was calibrated to a series of altered data sets, with these dry O₂ exposures in group B modified to 60-90%. Only at 70% O₂ was model 0 able to accurately predict the DCS outcome in groups A and B. With the data coded at 80%, the model's predictions were minimally changed from those shown for model 0 in Table 3 (first "predicted" column). Thus our coding of the data at 98% does not directly "create" the need for an O₂ effect; even at a conservative value of 85%, model 0 fails to describe the O₂ data. Similarly, inward skin flux of ambient N₂ from the air-filled chamber would increase the total body N₂ content but is unlikely to correspond to 20-30% of air breathing.

A third O₂ effect model added a fourth parallel risk compartment to Eq. 1, in which risk accumulation was based solely on PO₂ rather than on PN₂. The best fit of this model improved the LL by only 3.8 (LL = 1196.3) with two additional estimated parameters: a time constant and a scale factor. Although this was a statistically significant improvement, it was not as impressive as those of models 1 and 2. This model's prediction of DCS incidence in group A was similar to that of model 0, and its prediction of DCS in group B (29.5 ± 6.9), although an improvement, was again less impressive than that of model 1 or model 2. Its relatively poor fit and its problematic tie to plausible physiology led us to abandon the model.

The present results suggest that use of O₂ much over 1 ata has drawbacks that warrant consideration in optimizing decompression. This does not mean that O₂ is not useful during decompression, only that O₂ is not totally free of concern for causing DCS. Either of the two new models can be used for O₂ decompression optimization.