Abstract
Probabilistic models of human decompression sickness (DCS) have been successful in describing DCS risk observed across a wide variety of N_{2}O_{2}dives but have failed to account for the observed DCS incidence in dives with high
 oxygen effects
 gasexchange kinetics
 risk function
 hazard function
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 O_{2}is treated as a “free” quantity and is not linked to the risk of DCS. O_{2} is less available as a dissolved gas when it is bound to hemoglobin and when it is converted to the very soluble gas CO_{2}. That view is substantiated by measurements of tissue O_{2} 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 O_{2} in the breathing gas during decompression (9, 13), underpredicting the occurrence of DCS in these O_{2} decompression dives by ∼60%. In a subsequent prospective trial of O_{2} decompression procedures, severe underprediction again occurred (11). These results contradicted the expectation of no O_{2} effect found in a moderately large study of dives (19) with direct ascent after breathing mixtures with a
The O_{2} effects explored here are of two very different forms, both based on observed physiology. In our first model a
All our models are based on survival functions and are intended to predict the risk of occurrence of an undesirable outcome due to a riskgenerating 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 welldocumented 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% O_{2} 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
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 ingroup A used compressed air (21% O_{2}), a large number of dives were performed with moderately enriched O_{2} atmospheres. In most of these nonair dives a constant
Group B contains 33 DCS and 17 marginal cases, for an incidence of 3.4%. The dives ingroup B are of two types:1) air dives that use ∼100% O_{2} during decompression and2) air dives followed by ∼100% O_{2} 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 O_{2} to the diver, we assume that immersed divers breathed 99.5% O_{2}and dry divers 98% O_{2}. The consequences of choosing these particular values are discussed later.
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 (T _{1} −T _{2}) over which symptoms appeared, whereT _{1} is the latest time the diver was known to be entirely free of symptoms andT _{2} is the time at which definite symptoms were first reported. The methods and rules of establishing T _{1}− T _{2} for most reported dives are described in detail elsewhere (16).
MODELS
The bestfitting model from our most recent N_{2}O_{2}modeling effort (9, 13) was used as the base model for this study (model 0). This model allows for exponential washin and a mixed exponentiallinear 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
If Pti_{i} ≤ (PXO_{i} + P_{amb} − P_{met}), only dissolved gas is present and Pti_{i} equals
Figure 1 illustrates the handling of inert gas partial pressure in model 0 for a dive with O_{2} 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% O_{2} (dashed curve) during a portion of the decompression. The duration of the O_{2} period is indicated by the drop in
O_{2}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
Figure 2 shows a range of effects for several values of P_{set} andk that model 1 might have on an N_{2} kinetic time constant over the
O_{2} as an inert gas.
In this model, O_{2}, at sufficiently high partial pressures, can contribute to bubble formation or growth (35, 8, 14). Model 2 introduces the “O_{2} effect” as a direct additive term in the supersaturation part of the risk function. Thus for the inert gas term in Eq. 1
Not all the O_{2} pressure will be considered to be available to contribute to DCS risk. We limit the contribution of O_{2} to pressures above a certain level,
Model evaluation.
The risk functions, each model’s set of equations leading toEq. 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).
Each of the O_{2} effect models is a modification of, and can be simplified to, model 0; therefore, an LR test (7, 18) is used to test for the significance of the added parameters contained in each modification. A proposed model will have a significantly improved fit to the data (at P = 0.5) if its LL exceeds the model 0 LL (smaller negative number) by at least 1.92 for one added parameter and 2.98 for two added parameters, out to 6.30 for six added parameters (7). Each model was fitted to the combined data set (A +B). Models 1 and 2 allow for up to six new parameters (2 per kinetic compartment) to be estimated, in addition to the kinetic time constants, scale factors, thresholds, and linearexponential crossover parameters, which are common to all. Some or all of the added parameters may not add significantly to the improvement of the fit, as judged by the LR test. Final results for each model were chosen among many parameter estimation runs to include only those parameters the existence of which was justified atP < 0.05.
Results of fitting.
Ideally, the O_{2} 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. Table2 lists the bestfit parameters and SEs estimated for each model.
The best fit of model 1 improved LL by 11.1 units with only two additional estimated parameters, applied tocompartment 2. The improvement is significant at P < 0.01. Inmodel 1, estimated O_{2} effect parameters result in no alteration of the N_{2}based kinetics for <1.7 ata inspired
The best fit of model 2 improved the LL fit by 10.5 with two additional estimated parameters applied tocompartment 2. This improvement is also significant at P < 0.01. The estimated N_{2} time constant is substantially longer (slower) for model 2 in compartment 2than for model 0. Although this slower time constant will result in less uptake of inert gas, it will also slow washout, thus allowing for longer risk accumulation for many dives. The specific O_{2} effect parameters for this model apply a direct risk addition tocompartment 2 through the “combined” Pti_{i}(Eq. 4
). This O_{2}based contribution replaces overpressure “lost” due to the slower N_{2} washin and applies this added risk specifically to the high
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 togroup A only (model 0A). As expected, model 0 predicts DCS in the combined data better thanmodel 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 1and 2 have most of the desired predictive ability: prediction of DCS occurrence ingroup A dives centered nearly on the observed value and prediction of DCS occurrence ingroup 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 χ^{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 Bfrom 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 χ^{2} values yieldP > 0.05. Similarly, we can break the 26 categories in the into the 21 belonging to group A and the 5 belonging to group B. The resultingmodel 0, 1, and2 test statistics are 23.3, 19.7, and 20.0 for group A and 11.8, 6.6, and 6.7 for group B. Allgroup A tests yieldP > 0.05 for 20 df. Forgroup B, model 0 hasP < 0.05 and models 1 and 2 haveP > 0.05 for 4 df. This data categorization provides an indication that model 0 does not predict DCS occurrence in the dives ofgroup B as well asmodels 1 and2. However, the outcomes of such χ^{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” χ^{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 linebyline 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. Figure3 shows the observed and predicted DCS cases in each 1h interval after surfacing for the dives ingroup 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 and2 have nearly identical predictions of occurrence in all time intervals but tend to overpredict in the 2 to 5h range. Because almost onethird 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 ofmodel 1 (8.7).
DISCUSSION
Both models of an O_{2} 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 N_{2} elimination compared with normoxic levels over 2 h of washout at 2.0 and 2.5 ata
Another human decompression study attempted to analyze N_{2} exchange retardation from high O_{2} pressures (19). Over the experimental range of 0.2–1.3 ata
Model 2 represents an approach fundamentally different from model 1, in that O_{2}, when present in pressures greater than
Our O_{2} effect modifications were intended to remedy the failure of model 0 to account for the DCS incidence observed in the O_{2} data. Because our data coding of the inspired O_{2} 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 O_{2} decompressions, which form the bulk of group B, at 98% O_{2} is incorrect because of imperfect delivery of the gas. Estimates from experienced investigators suggest that the minimum O_{2}fraction likely to be present in the face mask in dry exposures is ∼85–95% (R. Y. Nishi, personal communication). If the actual O_{2} exposures were much less than our indicated 98%, model 0, without a specific O_{2} contribution to DCS risk, might be able to account for the DCS incidence observation ingroup B. To explore this,model 0 was calibrated to a series of altered data sets, with these dry O_{2} exposures ingroup B modified to 60–90%. Only at ≤70% O_{2} wasmodel 0 able to accurately predict the DCS outcome in groups A andB. With the data coded at ≥80%, the model’s predictions were minimally changed from those shown formodel 0 in Table 3 (first “predicted” column). Thus our coding of the data at 98% does not directly “create” the need for an O_{2} effect; even at a conservative value of 85%, model 0 fails to describe the O_{2} data. Similarly, inward skin flux of ambient N_{2}from the airfilled chamber would increase the total body N_{2} content but is unlikely to correspond to 20–30% of air breathing.
A third O_{2} effect model added a fourth parallel risk compartment to Eq.1
, in which risk accumulation was based solely on
The present results suggest that use of O_{2} much over 1 ata has drawbacks that warrant consideration in optimizing decompression. This does not mean that O_{2} is not useful during decompression, only that O_{2} is not totally free of concern for causing DCS. Either of the two new models can be used for O_{2} decompression optimization.
Acknowledgments
We are indebted to several colleagues for their advice and guidance: R. Y. Nishi for consultation on modeling and data collection; P. Tikuisis, E. D. Thalmann, and L. D. Homer for insights on physiology and modeling; A. L. Harabin for several critical reviews through which the manuscript was much improved; and S. Mannix for valuable editorial assistance.
Footnotes

Address for reprint requests: E. C. Parker, Albert R. Behnke Diving Medicine Research Center, Naval Medical Research Institute, 8901 Wisconsin Ave., Bethesda, MD 208895607.

This work was supported by Naval Medical Research and Development Command Work Unit 0603713N M0099.01A1510.

The opinions and assertions contained herein are the private ones of the authors and are not to be construed as official or reflecting the views of the Navy Department or the naval service at large.
 Copyright © 1998 the American Physiological Society