Modeling and Analysis for Risk Assessment of Decompression Sickness

Raj S. Chhikara, Ph.D., Professor; Floyd M. Spears, Ph.D., Associate Professor; and Thomas T. English, graduate student, UHCL

Aviators and astronauts may experience altitude decompression sickness (DCS) as a result of reduced environment pressure. When astronauts have to perform extra-vehicular activities or when there is a damage or loss of the cabin or space suit pressure, they may be exposed to acute environment pressure reduction and thus run the risk of DCS. Scientists at the NASA Johnson Space Center have conducted, over several years, experimental tests using hypobaric chambers and simulated extra-vehicular activities to determine the DCS incidence and its onset time for human subjects.

In testing, each subject is monitored for Doppler detectable bubbles, and the test is terminated either upon incidence of a DCS onset or when the test period is over. The test duration is recorded in each case, and an observation is either the DCS onset time or the test termination (censored) time for a test subject.

Empirical studies of decompression sickness have been conducted by NASA using experimental data collected from the hypobaric chamber tests. Conkin et al.¹ modeled the probability of DCS occurrence by relating it to decompression stress as measured by the relative change in atmosphere that the human body can sustain without incurring DCS. This decompression stress measure is called the tissue ratio (TR) index. Gerth and Vann² and Tikuisis et al.³ based their developments on certain bubble dynamic models to characterize the DCS occurrence rate. Analyses reported in these and other more recent studies undertaken by the researchers from NASA and the US Air Force focus on the selection of a probabilistic model that statistically provides the best possible fit for experimental DCS data.

Statistical analyses have mostly been limited to parameter estimation for a model and do not provide the significance level or reliability measure for the data-fitted model.

The maximum likelihood method is invariably used in estimating the model parameters and the probability density or distribution function, whereas the Kaplan-Meier method is used for the empirical estimate of the underlying distribution function. The goodness-of-fit is judged either by comparing the graphs of the two estimates of the distribution function, or by the chi square test based on the maximum likelihood value resulting from a model fit. Both the log normal and the log logistic probability models provide a reasonable model fit to the NASA experimental DCS data.

Besides the DCS onset time observations, data are available on a number of physiological and related variables.⁴ Previous data analysis studies conducted by NASA and the US Air Force have made use of these variables to explain the variability in DCS onset time. In modeling the DCS onset time response, the important variables are ambient pressure (P2) at test altitude, nitrogen (N2) pressure, particularly that determined in the theoretical 360-minute half-time compartment (PN2360), exercise, and preoxygenation, among others. A log logistic model involving some of these explanatory variables has been fitted to the DCS data. Conkin, et al.¹ and Kannan, et al.,⁵ among others, have empirically shown that a log logistic model provides a reasonable model-fit and that several of these variables contribute significantly to the occurrence of DCS.

In the present study, we apply the Cox proportional hazard model to analyze DCS data obtained from NASA. These data consist of 1321 test duration times of which 1154 are censored and 167 are DCS onset time observations. The following covariates are used in this modeling: P2, PN2360, ALTTIME (altitude time), TR360, and EXER (exercise). The choice of these variables is partly based on the fact that their measurements are also available for the test subjects for which DCS data are analyzed.

We first discuss the application of the Cox proportional hazard model and then determine the significance of each of these variables and interactions among them. The following results show the significant variables including interaction terms:

The model with interaction has a likelihood ratio test statistic of 281, while the model without interaction has a likelihood ratio test statistics of 279. Results show no significant advantage to the interaction term model over the non-interaction term model.

The covariate EXER is an exercise indicator variable; hence, it takes value as either 0 or 1. This presented computational difficulty in obtaining residuals associated with EXER as a covariate in the model. As such EXER is used as a stratified variable and not as a covaraite in the model. This resolution, in turn, presents no computational difficulty in determining the residuals using the S-Plus routine resid().

Wei⁶ proposed an omnibus test of proportional hazard between two groups using data containing censored observations. We consider here an application of Wei’s test to verify the assumption of proportional hazards between the two groups stratified by EXER. The Wei test statistics is a measure of goodness-of-fit with a computed value of 1.44. Using the non-exercise group, we compute a truncation value of 27/263 = 0.1 (number observed/number in sample) needed to look up the p-value from the applicable table of cumulative probabilities.⁷ This yields a p-value less than 0.001, which allows us to reject the hypothesis of proportional hazards between the exercise and non-exercise groups and thus provides a justification for the use of above stratification.

Finally, we assessed the assumption of proportional hazards for the stratified model by examining the rescaled Schoenfeld residuals and found that proportional hazards is a reasonable assumption. The estimated baseline survival curves for the stratified model with the average covariate values as input are given in Fig. 1.

Figure 1. Survival for No Exercise and Exercise Groups with Average Covariates

References
¹J. Conkin, K. V. Kumar, M. R. Powell, P. P. Foster, and J. M. Waligora, "A Probabilistic Model of Hypobaric Decompression Sickness Based on 66 Chamber Tests," Aviation, Space, and Environmental Medicine 67.2 (1996): 176-83.
²W. A. Gerth and R. D. Vann, "Statistical Bubble Dynamics Algorithms for Assessment of Altitude Decompression Sickness Incidence," Technical Report, Southeastern Center for Electrical Engineering Education, Duke University, North Carolina (1994).
³P. Tikuisis, K. A. Gault, and R. Y. Nishi, "Prediction of Decompression Illness Using Bubble Models," Undersea & Hyperbaric Medicine 21 (1994): 129-43.
⁴J. Conkin, S. R. Bedhl, and H. D. Van Liew, "A Computerized Databank of Decompression Sickness Incidence in Altitude Chambers," Aviation, Space, and Environmental Medicine 63 (1992): 819-24.
⁵N. Kannan, A. Raychandhuri, and A. A. Pilmanis, "A Loglogistic Model for Altitude Decompression Sickness," Aviation, Space, and Environmental Medicine 69 (1998).
⁶L. J. Wei, "Testing Goodness of Fit for Proportional Hazards Model With Censored Observations," JASA 79 (1984): 649-52.
⁷J. A. Koziol and D. P. Byar, "Percentage Points of the Asymptotic Distributions of One and Two Sample K-S Statistics for Truncated or Censored Data," Technometrics 17.4 (1975): 507-10
⁸R. S. Chhikara, F. M. Spears, and T. T. English, "Cox Proportional Hazard Model for Altitude Decompression Sickness." Technical Report, University of Houston-Clear Lake (Aug. 1999).

Publications
Chhikara, R. S., F. M. Spears, and T. T. English. "Cox Proportional Hazard Model for Altitude Decompression Sickness," Proceedings of the 15th International Workshop on Statistical Modeling, Bilbao, Spain, July 17-21, 2000. 400-03.

Presentations
Chhikara, R. S. "Statistical Modeling in Environmental Physiology." Invited Presentation, Conf. of Texas Statisticians, Sam Houston State University, Hunstville, TX, March 31, 2000.
English, T. T. "Cox Proportional Hazard Model for Altitude Decompression Sickness," Student Research Conf., University of Houston-Clear Lake, April 26, 2000.
Chhikara, R. S. "Cox Proportional Hazard Model for Altitude Decompression Sickness," 15th International Workshop of Statistical Modeling, Bilbao, Spain, July 17-21, 2000.
English, T. T. "Goodness-of-Fit Test for the Cox Proportional Hazard Model," Master Thesis Presentation, University of Houston-Clear Lake, Aug. 10, 2000.