## Abstract

For altitude decompressions, we hypothesized that reported onset times of limb decompression illness (DCI) pain symptoms follow a probability distribution related to total bubble volume [V_{b·}(*t*)] as a function of time. Furthermore, we hypothesized that the probability of ever experiencing DCI during a decompression is associated with the cumulative volume of bubbles formed. To test these hypotheses, we first used our previously developed formation-and-growth model (*Am J Physiol Regulatory Integrative Comp Physiol* 279: R2304–R2316, 2000) to simulate Vb·(t) for 20 decompression profiles in which 334 human subjects performed moderate repetitive skeletal muscle exercise (827 kJ/h) in an altitude chamber. Using survival analysis, we determined that, for a controlled condition of exercise, the fraction of the subject population susceptible to DCI can be approximately expressed as a power function of the formation-and-growth model-predicted cumulative volume of bubbles throughout the altitude exposure. Furthermore, for this fraction, the probability density distribution of DCI onset times is approximately equal to the ratio of the time course of formation-and growth-modeled total bubble volume to the predicted cumulative volume.

- time-varying volume of bubbles
- cumulative volume of bubbles
- rate of bubble formation
- instantaneous probability of pain
- susceptibility to decompression illness
- Cox-Snell residuals

altitude decompression illness (DCI) may occur in response to acute reduction in ambient atmospheric pressure, for example, as experienced by astronauts during extravehicular activities (EVAs). The actual manifestation of DCI is characterized by its extreme variability to individual response (1, 26, 33). To account for this uncertainty, survival analysis models have been used to provide estimates of DCI risk as a function of exposure time (7). Typically, these models have attempted to differentiate between decompression profiles in terms of a constant, the tissue ratio of the dissolved N_{2} tissue tension to the ambient pressure at altitude. However, it is generally agreed that gas bubbles are either the primary cause or the precipitating factor of DCI pain symptomatology (29). The underlying mechanism that causes DCI may be related to the onset (19), growth (13, 19, 25, 28), and/or density of bubbles in tissues (29). Indeed, the series of events including the onset and growth of tissue bubbles produces a volume of bubbles in the tissue, which in turn may be the stimulus or cause for pain. It is difficult to evaluate, however, whether the onset, the growth, or an increased density of tissue bubbles actually causes DCI pain symptoms.

Because bubbles in tissues are not amenable to direct experimental observation, it is necessary to characterize the processes of bubble growth and decay by mathematical models (16). Risk models have been used to express overall DCI risk as a function of non-time-dependent mechanistic variables (28), such as maximum bubble size (25). However, to specify the time dependence of the hazard or probability density function (pdf) for time to DCI in survival analysis, time-varying physiological variables must be used.

In our accompanying study (11a), we used the formation-and-growth model (FGM) for various simulated conditions of exercise and decompression profiles to calculate the expected total volume of bubbles from a region of tissue at a given time [V_{b·}(*t*)]. In the present study, we hypothesized that reported onset times of limb DCI pain symptoms follow a probability distribution with kernel V_{b·}(*t*). Roughly speaking, this means that the instantaneous probability of symptom onset at *time t* is proportional to V_{b·}(*t*). Furthermore, we hypothesized that the probability of ever experiencing DCI during a decompression is directly related to the cumulative volume of bubbles formed. We tested these hypotheses in a survival analysis using data from 20 actual decompression profiles taken from the National Aeronautics and Space Administration (NASA) Experimental Altitude Data Set of human exposures between 1982 and 1998.

It has been shown in experimentation with animals that skeletal muscle exercise at altitude pressure of ∼30 kPa is associated with bubble formation, whereas in the absence of activity, bubbles form only at a higher-altitude pressure of ∼5.53 kPa (2). Furthermore, a definite association between intensity of exercise and rate of bubble formation in limb tissues was observed (2, 17, 18). Because the NASA experiments were designed to simulate a limited range of exercise workloads typical of EVAs during NASA Space Shuttle flights, there was not enough variability in the data set to allow estimation of bubble formation rate as a function of workload. Therefore, in our survival analysis, we assumed a constant rate of bubble formation. Nevertheless, it has been shown (31) that the bubble growth rate depends on the severity of the decompression. Consequently, we expected a rich variety of bubble growth rates due to the different levels of denitrogenation associated with the profiles.

In this study, we found that, for a controlled condition of exercise, there was a definite association between the model-predicted time course of total bubble volume in tissues and the probability distribution of the time of onset of DCI pain. We also found that overall propensity for DCI is correlated with the predicted cumulative volume of bubbles through the altitude exposure.

### Glossary

- α
- Parameter of the Poisson process, dimensionless
- β
- Parameter of the Poisson process, dimensionless
- δ
_{i} - Area under the curve V
_{b · i}(*t*), mm^{2} - Fi
_{O2} - Fraction of O
_{2}in the inspired medium, dimensionless - f
_{i}(*t,*β) - Probability density function, V
_{b · i}(*t*)/δ_{i}, approximated by lognormal probability density function - f
_{A}(*t*,β) - Probability density function for
*profile A* - f
_{A}^{*}(*t*,β) - Perturbed probability density function for
*profile A*; lognormal scale parameters (ς_{i}) reduced by 25% - F
_{i}(*t*,β) - Cumulative distribution function
- FGM
- Formation-and-growth model
- λ
- Parameter estimated from experimental data, dimensionless
- μ
_{i} - Lognormal location parameter, dimensionless
- Pv˙
_{O2} - Tension drop of O
_{2}dissolved in tissue due to O_{2}consumption, Pa. - π
_{i} - Probability that a subject is susceptible to DCI in the
*i*th profile - Q˙ti(
*t*) - Blood flow in the tissue shell, m
^{3}/min - R
- Respiratory exchange ratio
- s
_{b,i} - Solubility of the
*i*th gas in the blood, ml · ml^{−1}· 100 Pa^{−1} - s
_{ti,i} - Solubility of the
*i*th gas in the tissue, ml · ml^{−1}· 100 Pa^{−1} - S
_{i}(*t*,β) - Survivorship function
- ς
_{i} - Lognormal scale parameter, dimensionless
- T
_{i} - Total time of exposure to altitude, min
*t*_{ij}- Time to onset of DCI, min
*t*_{1/2,N2}- Half time for tissue washin and washout of N
_{2}, 360 (rest) and 300 (exercise), min *t*_{1/2,O2}- Half time for tissue washin and washout of O
_{2}, 313.2 (rest) and 261 (exercise), min - θ
- Parameter estimated from experimental data, dimensionless
- V
_{b·}(*t*) - Volume of bubbles in the tissue region, mm
^{3} - V
_{b · i}(*t*) - Volume of bubbles in the tissue region for
*i*th profile, mm^{3}

## METHODS

A flow chart of the overall methodology for estimating the probability of experiencing DCI at altitude as a function of time is shown in Fig. 1.

#### Historical experiments.

The data used in our analysis consisted of unpublished test results on 334 exposures obtained by NASA investigators in the course of estimating DCI risk for EVA operations. Subjects were healthy and well-motivated women (*n* = 35) and men (*n* = 299) of about average fitness. The subjects were representative of the astronaut population and participated in various studies between 1982 and 1998. The age, height, and weight (means ± SD) were as follows: 31.4 ± 7 yr, 175.7 ± 7.9 cm, and 74.4 ± 10.9 kg. Voluntary and informed consent forms were obtained from subjects before they entered each study. The studies were approved by the Institutional Review Board at the NASA Johnson Space Center. All individuals were required to pass the US Air Force Class III flight physical examination. Subjects were free to withdraw from the study at any stage.

#### Test procedure for altitude exposures.

In the NASA test program, subjects were tested under various predefined denitrogenation and decompression profiles. To permit calibration of the FGM, we initially considered only those 20 profiles that had similar expected propensities for bubble formation (but not necessarily bubble growth). In so doing, we assumed that this constraint would be satisfied only by profiles with similar exercise regimens. These altitude exposures involved moderate repetitive skeletal muscle exercise of the upper or lower limbs with an average metabolic rate of 827 kJ/h (200 kcal/h). Subjects were exercising for 16 min and rested for 4 min and repeated this cycle throughout each altitude exposure. In addition, subjects were required to walk the few steps between simulated EVA workstations. During the O_{2} prebreathe period, no exercise was performed and subjects were reclined or seated.

Despite the uniformity restriction on exercise, and presumably bubble formation*,* the 20 profiles involved different prebreathe procedures so that the expected time course of bubble growth would be expected to vary considerably. On the basis of our hypothesis, this variation should be reflected in different DCI risks among the profiles. A comprehensive list of the profiles is shown in Table1. These profiles are listed in chronological order, *profile A*, tested in 1982, being the first. Profiles involving a single type of altitude exposure are identified by a capital letter only; those requiring a series of repetitive prebreathe protocols and exposures are designated by a letter with an additional identifying number.

Specific differences between the profile types were as follows:*1*) Denitrogenation periods ranged from 3.5 to 26.16 h.*2*) The number of repetitive altitude exposures varied between two and six; for example, *profiles I1–I6*involved a cumulative implementation of six repetitive altitude exposures with return to ambient atmospheric pressure after 67.5 h. *3*) In 6 profiles, the breathing gas during denitrogenation was 100% O_{2} at 101.3 kPa (14.7 psia); in the other 14 profiles, a staged denitrogenation was used with a prolonged stay at 70.3 kPa (10.2 psia) enriched with O_{2}with a reduced N_{2} partial pressure (Fi
_{O2} = 0.28 or 0.265). *4*) Times in the final ascent to altitude ranged from 4 to 30 min (mean 17.36 ± 10.5 min). *5*) In *profile L*, subjects breathed an O_{2}-N_{2} mixture (Fi
_{O2} = 0.60) at an altitude of 41.4 kPa (6.0 psia); in the other profiles, subjects breathed pure O_{2} at 30 kPa (4.3 psia, Fi
_{O2} = 1). *6*) The time at altitude varied from 3 to 6 h.

To illustrate the notation in Table 1 describing repetitive exposures, consider the following example: In *profile I1*, subjects breathed pure O_{2} at ambient pressure (101.3 kPa) for 60 min and then ascended for 15 min to 70.3 kPa and stayed at this pressure for 705 min breathing a 26.5% O_{2}-balance N_{2}mixture. Still at 70.3 kPa, the gas was switched to 100% O_{2} for 40 min. Finally, the ascent to the working altitude pressure of 30 kPa took 25 min. This working altitude pressure in*profile I1* lasted 180 min and was the first of the six repetitive altitude exposures (*I1–I6*). The same subjects started *profile I2* by a 6-min ascent to 70.3 kPa, where they sojourned for 80 min breathing 26.5% O_{2} and then 100% O_{2} for 40 min. The final ascent time to 30 kPa was 25 min. Subjects were then exposed to the subsequent *profiles I3–I6*.

#### Response variable.

Subjects were asked to report the first incidence of a musculoskeletal DCI pain symptom. The primary response variable was the elapsed time from the beginning of the altitude exposure to the first report of DCI; otherwise the total test time was recorded. Concurrent tabulation of a dichotomous indicator of whether DCI occurred was also recorded.

#### Explanatory variables.

In our hypothesis, we assumed that V_{b·}(*t*), the trace of total bubble volume with time, is associated with the likelihood of DCI occurring and the time of its occurrence. Following this hypothesis, we proceeded to explain the DCI incidence times and frequency in the NASA data set by *1*) using the FGM (11a) to calculate V_{b·}(*t*) separately for each profile and *2*) using the collection [V_{b · i}(*t*)] as explanatory variables in a survival analysis. [Here, V_{b · i}(*t*) denotes V_{b·}(*t*) for the *i*th profile.] As an illustration, Fig. 2
*A* shows V_{b · i}(*t*) for *profiles A–D*. Use of the FGM obviates the need for direct measurement of bubbles in tissue.

The FGM utilizes exercise parameters of two main types: *1*) physiological and *2*) those defining the Poisson process for bubble formation. Values of physiological parameters (Pv˙_{O2}, R, *t*
_{1/2,N2}, and *t*
_{1/2,O2}) correspond to those for a typical human subject and are shown in Table2. Here, Pv˙_{O2} is the tension drop of O_{2} dissolved in tissue due to O_{2} consumption, R is the respiratory exchange ratio, and*t*
_{1/2,N2} and*t*
_{1/2,O2} are the tissue half times for washin and washout of N_{2} and O_{2}, respectively. The correct assessment of tissue half times is essential to operation of the FGM. For rest periods in single-exposure profiles, we selected the constant working value *t*
_{1/2,N2} = 360 min, which has been well documented (6-8). We then selected a corresponding exercise value of 300 min to be in the low range of values estimated during moderate exercise (7). To obtain *t*
_{1/2,O2}, we used the relationship *t*
_{1/2,O2} = 0.87*t*
_{1/2,N2} (11a). In reality, these half times are also affected by the lengths of prebreathe periods. In particular, we believed that, for repetitive exposures with long prebreathe periods (*profiles G2, H2*, and*I2–I6*), the assumption of near-constant tissue half times is not suitable; hence, these profiles were not used in the subsequent survival analysis. We did, however, compare the overall DCI incidence in these profiles with that predicted by the model under the assumption of constant half times (see results and discussion
*).* Although the FGM could accommodate variable half times if they were known, we were unable to do so here, because a correct assessment of such half times is not available. Other relevant parameters used in the FGM are listed in Tables3 and 4.

The parameters β and α characterize the Poisson process, which emulates bubble formation in the FGM. In particular, β affects the rate of bubble formation, whereas α is proportional to tissue unit volume. To effectively predict DCI, total bubble volume must be normalized to tissue volume. We therefore calculated V_{b·}(*t*) for a fixed hypothetical tissue region comprised of, e.g., 50 units of tissue volume (11a). The expected number of bubbles formed per tissue unit over the course of altitude exposure, α/β, was normalized to an arbitrary number, in this case six. Although the size of a tissue unit is not explicitly used in running the FGM, the assumption is that, with an average of only six bubbles per exposure, tissue units are small enough to reflect the essential gas exchange characteristics of the model. The mean total number of bubbles simulated in a run of the FGM was equal to 300 (= 50 units × 6 bubbles); however, because of the random property of the Poisson process, the actual number of simulated bubbles ranged from 269 to 343 (302 ± 19). In our FGM, the rate of bubble formation in a tissue unit depends on the ratio α/β. However, with β assumed constant over profiles, fixing α/β serves also to fix tissue volume. This requirement is equivalent to specifying that the parameter β of the Poisson process is the same for all profiles (seeresults and discussion).

#### Lognormal approximation.

For each profile, V_{b · i}(*t*) was calculated pointwise at equal intervals of time (10 min) throughout the altitude exposure by numerical solution of the differential equation (11a, 35). Figure 2
*A* shows V_{b · i}(*t*) for *profiles A–D*. Because intensive computation was required to numerically solve the differential equations to obtain V_{b · i}(*t*), we approximated each V_{b · i}(*t*) by a kernel of a lognormal probability density function; i.e.
Equation 1where the parameters δ_{i}, μ_{i}, and ς_{i} were estimated by the method of least squares using SYSTAT software (34). The fit was excellent for the all decompression profiles (R^{2} ≥ 0.985). Figure 2
*B*illustrates a typical fit (R^{2} = 0.99) for*profile A*. The filled circles represent the pointwise values of V_{b · i}(*t*) obtained using the FGM, and the solid line is the function given by *Eq. 1
*. The lognormal parameters (δ_{i}, μ_{i}, and ς_{i}) serve to summarize the essential characteristics that distinguish the profiles as determined by the FGM. In particular, ς_{i} and μ_{i} are the respective scale and location parameters of f_{i}(*t,*β), the lognormal pdf (21), proportional to V_{b · i}(*t*), and δ_{i} is approximately equal to the total cumulative bubble volume; i.e., the area under the probability density function.

#### Survival analysis.

Using other altitude test data, Conkin et al. (7) used survival analysis to fit the pdf of time to DCI with a family of log-logistic distributions whose means were expressed as a linear combination of profile-specific tissue ratios. Here, we have expanded on this idea in two ways. First, standard survival analysis methodology assumes that the defining event (in this case, reported DCI) is a certainty if testing is carried out for a sufficiently long time. In our application, this may not be realistic, because N_{2}supersaturation drops as the test continues; thus a subject who does not experience DCI by the end of a test might never experience DCI, even if the test were extended indefinitely. It has also been observed (23) that there are individuals who are highly resistant to DCI, even under relatively stressful experimental conditions. Any such individuals in the population of test subjects would probably not have experienced DCI, even if the tests had been extended well beyond their actual termination times. Although we do not attempt to identify these subjects in our analysis, we do account for their presence at a population level by allowing for a certain proportion (the “cured” fraction) of the population to be “DCI resistant.” The remaining subjects are considered “susceptible” to DCI. For this group of subjects, we assumed that DCI would eventually occur if testing were indefinitely continued. The actual probability, π_{i}, that a subject is susceptible to DCI varies among profiles and is modeled by the relationship
Equation 2where θ and λ are parameters to be estimated. Thus for λ > 0, test profiles with relatively large time-integrated bubble volume (δ_{i}) would be those most likely to induce DCI. The probability of being resistant or cured is simply 1 − π_{i}. Examples of survival analysis with cured fractions can be found elsewhere (20, 22).

A second major innovation is that, for susceptible subjects, we made use of both of our hypotheses to assume that the pdf of time to DCI is the lognormal pdf *f _{i}
*(

*t*,β); i.e. Equation 3This pdf is approximately proportional to V

_{b · i}(

*t*), calculated by the FGM for a particular value of the Poisson process parameter β. Values of the parameters μ

_{i}and ς

_{i}were obtained directly from V

_{b · i}(

*t*) before fitting the DCI response data. In other words, f

_{i}(

*t,*β) summarizes the dynamic process of expected bubble formation and growth during application of the

*i*th profile. Subsequent verification that

*Eq. 3*can indeed be used a priori to define the conditional probability distribution of time to DCI for susceptible subjects is a powerful statement in support of our hypothesis.

#### Estimation of parameters in the survival analysis.

We used the method of maximum likelihood (11, 12, 21, 26, 27,33) using the optimization package of STATA (24) to estimate the unknown parameters β, θ, and λ for the 20 profiles in the study. The probability that a test exposure results in DCI before *time t* is a product of two terms: *1*) the probability that the subject tested in the exposure was susceptible (*Eq. 2
*) and *2*) the conditional probability of DCI occurring before *time t* given that the subject was indeed susceptible. We thus have *P*(DCI before *time t*) = θδ_{i}
^{λ}F(*t,*β), where F_{i}(*t*,β) = ∫_{0}
^{t}f_{i}(u,β)du is the conditional cumulative distribution function of time to DCI for susceptible subjects. Conversely, the probability that a test ends without DCI occurring is 1 − θδ_{i}
^{λ}F(T_{i,}β), where T_{i} is the total exposure time for the *i*th profile. For the *j*th exposure of the *i*th profile, let *t*
_{ij} be the time to onset of DCI, in cases where the latter occurs. Under our survival analysis with β known, the log likelihood (10) of the data as an explicit function of θ and λ is given by
Equation 4
where a_{ij} = 1 if DCI was observed on the*j*th exposure and *i*th profile; otherwise a_{ij} = 0. The first term in *Eq. 4
* is the contribution from cases where DCI occurred; the second term comes from cases without DCI. When δ_{i} = 0, DCI cannot occur according to the model (*Eq. 2
*), and the likelihood function (*Eq. 4
*) is therefore undefined. The summation in *Eq.4
* is only over those 13 single-exposure profiles with δ_{i} > 0 (Table 1).

The dependence of L on β appears implicitly through the effect of the latter on V_{b · i}(*t*), which in turn determines f_{i}(*t*
_{ij},β) and F(T_{i},β). Although β is unknown, we minimized −log L in*Eq. 4
* (equivalent to maximizing log L) with respect to θ and λ for fixed trial values of β ranging from 0.005 to 0.07. For each of these trial values, the FGM had to be rerun to recompute f_{i}(*t*
_{ij,}β), F(*t*
_{ij,}β), and the resulting minimum value of −log L, e.g., Q(β) (Fig. 1). Ideally, the final estimate of β should be the one minimizing Q(β) (Fig.3). However, because the output of the FGM is affected by random bubble formation, assumed to follow a Poisson process, the values of Q(β) show a slight random scatter (Fig. 3). Therefore, our final estimate of β was actually obtained by minimizing a quadratic function of β fitted to the values of Q(β) (Fig. 3). For each trial value of β and for our final estimate, we always set α equal to 6β so that V_{b · i}(*t*) corresponded to a volume of tissue with a mean of 300 bubbles formed during the altitude exposure (see *Explanatory variables*).

## RESULTS AND DISCUSSION

#### Skeletal muscle exercise and bubble formation.

A plot of Q(β) vs. β with the quadratic approximation is shown for values of β ranging from 0.005 to 0.070 in Fig. 3. The maximum value of the quadratic function occurred at β̂ =˙ 0.017. Note the small random scatter induced by the finite expected sample size of 300 bubbles produced by the Poisson process. The value β = 0.017, which provides the best fit to the data, is presumed to produce the most realistic simulation of the intensity of bubble formation in tissues.

In this study, our hypotheses were based on the assumption that the number and onset times of bubble formation in limb tissues, from gas micronuclei through nucleation processes, are caused by skeletal muscle exercise at altitude. It has been shown for fluids that spontaneous nucleation can only occur above strikingly high supersaturation thresholds of ∼200–380 atm (14, 15). In contrast, for human subjects performing skeletal muscle exercise, bubbles are nucleated spontaneously at much lower supersaturation (15). Bubble formation is caused by the relative motion of internal structures and increases with exercise (31). Once bubbles are present, their rate of growth depends on the severity of the exercise and decompression (11a, 31). However, without exercise at altitude, it is questionable whether bubbles would form (2) for the intensity of decompressions modeled in this work. In particular, moderate N_{2} supersaturation at altitude affects bubble growth but does not in itself provide a mechanism for bubble formation. Furthermore, although nucleation sites or gas micronuclei in tissues can be generated by hydrostatic compression, as in diving, this is not the case for altitude decompressions (30). The same exercise regimen of ∼827 kJ/h was performed at altitude across profiles, and no exercise was performed during the O_{2} prebreathe. Therefore, we presumed the intensity of bubble formation (characterized by β = 0.017) to be identical across all profiles.

#### Relationship between total volume of bubbles and DCI pain symptoms.

After we found that the bubble formation rate was best represented by β = β̂ = 0.017, we ran the FGM (Fig. 1) for this value of β and obtained final estimates of the parameters θ and λ, which characterize the proportion of susceptible subjects. The resulting values were θ̂ = 0.285 ± 0.036 andλ̂ = 0.150 ± 0.078. Therefore, the estimated probability of a subject being susceptible to DCI for the*i*th profile is equal to π_{i} = 0.285 δ_{i}
^{0.150} (*Eq. 2
*). The small value forλ̂ reflects the property that the π, the probability of susceptibility as a function of δ, the cumulative volume of bubbles, vs. time curve is approximately constant except for δ < 0.3. This result is not surprising, since the profiles, although different, were designed to meet operational standards bounding the anticipated risk of DCI. Despite the relatively large standard error of 0.078 for λ = 0.150, it would not be realistic to take λ = 0, because π would be equal to θ for all values of δ (including zero). Under such a condition, even when no bubbles are present, a subject would still have a probability of θ = 0.285 of being susceptible to DCI.

In susceptible subjects, V_{b·}(*t*) rises soon after decompression, quickly attains a maximum, and then gradually declines throughout the rest of the exposure (Fig. 2
*A*). This property is a result of the following sequence of events: *1*) immediately after decompression, the level of dissolved N_{2}in tissues is maximal, *2*) bubble growth is terminated by the O_{2} window (11a), *3*) the breathing of enriched O_{2} mixtures creates a positive N_{2} gradient from tissue to alveolus, thus facilitating the removal of N_{2}from tissue before it diffuses into bubbles and simultaneously N_{2} diffuses out of bubbles, *4*) O_{2}permeates rapidly into bubbles driving out N_{2}, and*5*) because of its high permeation coefficient, O_{2} quickly exits, thus causing more bubble decay (16,30).

The question has been raised whether the risk of DCI (28) or venous gas emboli (6, 8, 12) can be adequately explained by the rise and decay of bubble volume in tissue. The role of bubbles may be only to initiate cascades of physiological or chemical reactions that eventually give rise to pain symptoms (28). It has also been suggested that symptoms relate to total volume of evolved gas, size of individual bubbles (27), or bubble density (29). This postulate was the prime motivation in an earlier study (25) for testing certain bubble models against data that measured the amount of venous gas emboli by ultrasound Doppler monitoring. However, the size and number of tissue bubbles cannot be measured by Doppler bubble monitoring; thus we are not able to verify the relationship by direct observation. Instead, in the present study, we use V_{b·}(*t*) as a surrogate for direct measurements in tissues to predict the time to DCI pain symptoms. A good fit to the data would then corroborate our mechanistic hypotheses as being realistic (see *Model prediction and goodness of fit*).

#### Model prediction and goodness of fit.

The Cox-Snell residual plot (4) illustrates the consistency of the observed DCI onset times with the FGM prediction for β = 0.017. The logarithm of the *i*th Cox-Snell residual, Z_{i}, should be approximately equal to K_{i}, the double log transform of the corresponding Kaplan-Meier survivor function estimate applied to the residuals (4). Profiles with repetitive exposures separated by short time intervals (*I3–I6*) are not conducive to DCI, because they produce intense N_{2} washout. A similar degree of washout could be expected with *profile K*, which had a long prebreathe (480 min) with pure O_{2.} After applying the FGM to *profiles I3–I6* and *K*, we predicted no bubble growth whatsoever (δ_{i} = 0). For these cases, there is no pdf, and Cox-Snell residuals are therefore not defined. Consequently, the goodness-of-fit analysis was restricted to the remaining profiles with δ_{i} > 0.

In Fig. 4
*A*, we plotted K_{i} against Z_{i} for all observations for which DCI occurred (*n* = 49). In general, cases of early onset of DCI appear as points near the lower left corner of the data in Fig.4
*A*. A Cox-Snell residual plot commonly exhibits characteristics as seen in Fig. 4
*A*. A perfect match would have all points lying on the solid line, but random variation of the time to DCI would necessarily manifest itself as deviations from the line. Unlike residual plots after linear regression, points in Cox-Snell plots do not appear to have “random” scatter, even if the statistical model is correct, because they contain accumulated error and are thus highly dependent on each other. No direct goodness of fit test is available with Cox-Snell residuals. However, we used a parametric bootstrap (9) to obtain the sampling distribution of the root-mean-square (RMS) discrepancy between Z_{i} and K_{i} under the null hypothesis of a correct model. For our data, we obtained RMS = 0.287. This was exceeded 63 times in 100 bootstrap iterations; therefore, we concluded that our Cox-Snell residuals were consistent with an excellent model (*P* > 0.63). (*P *< 0.05 would indicate that data are not in agreement with the model.) Clearly, the expected V_{b · i}(*t*) is a good predictor of times to onset of DCI.

We made the assumption that DCI incidence can be predicted from characteristics of bubbles and that symptoms may be secondary to bubbles (28). A possible relationship between the amount of bubbles in tissues and the intensity of the pain stimulus has been suggested (5). Indeed, multiple processes may occur between bubble formation and pain DCI symptoms. Damaging tissue bubbles may not induce or be related to the amount of detectable intravascular bubbles. Furthermore, muscle tissue may not be the site of DCI symptoms. We are uncertain where damaging bubbles are located in the body and in which critical tissue they arise. Nevertheless, this historical NASA data set showed that the expected V_{b · i}(*t*) in the selected critical tissue was associated with the onset of DCI pain symptoms.

To illustrate the importance of the role of V_{b · i}(*t*) in the onset of DCI in critical tissue, we reduced the lognormal scale parameters (ς_{i}) by 25%, so that the resulting conditional density functions f_{i}(*t*,β) were no longer proportional to V_{b · i}(*t*). However, μ_{l}, the lognormal mean, and δ_{i}, the area under V_{b · i}(*t*), were not modified. Figure4
*C* shows a plot of the original f_{A}(*t*,β) and the perturbed function, f_{A}
^{*}(*t*,β), corresponding to*profile A*. Attempting the survival analysis with f_{A}
^{*}(*t*,β) led to a poor fit, even with reestimation of the parameters θ and λ, producing Cox-Snell residuals (Fig. 4
*B*) with an inflated RMS of ∼0.85. In particular, the actual DCI onset times tended to occur much earlier than predicted by the incorrect conditional density, resulting in a preponderance of data points distributed under the equality line. In this case, the hypothesis of a correct model would be strongly rejected (*P* ≈ 0.01). This example lends support to the conclusion that the excellent match in Fig. 4
*A* using the (correct) proportional f_{i}(*t*,β) was due to the influence of the FGM and not simply to data fitting. In other words, a correct representation of V_{b · i}(*t*) is necessary to predict the time of onset of DCI pain symptoms. This suggests a relationship between onset/growth of bubbles and onset time of pain.

#### Incidence of DCI pain symptoms.

Recall that profiles with δ_{i} = 0 and/or profiles with repetitive exposures could not be directly included in the survival analysis. However, ignoring times of onset, we were able to compare predicted with observed DCI incidence rates for all 20 profiles in Table 1. In so doing, profiles with δ_{i} = 0 were predicted to have no DCI cases, since π_{i}, the susceptible fraction of subjects, would then be equal to zero (*Eq. 2
*). In Fig.5, the proportion of observed DCI cases (number of DCI cases divided by number of exposures) is plotted against the predicted incidence θδ_{i}
^{λ}F(T_{i},β) (see *Estimation of parameters in the survival analysis*). Each profile is represented by an open circle with diameter proportional to the number of exposures. The solid line represents perfect agreement. Points corresponding to δ_{i} = 0 (*profiles I3–I6* and *K*) were slightly perturbed to make them distinguishable; in agreement with the FGM, no cases of DCI were observed on any of these profiles.

DCI incidence was substantially underpredicted for *profiles G2* and *H2* (Fig. 5). Both of these profiles involved two exposures to altitude, with each exposure following a short prebreathe with 100% O_{2}. Before the O_{2} prebreathe for the second exposure, however, a long additional prebreathe period with an O_{2}-N_{2} mixture was inserted (Table 1). For all profiles, we ran the FGM with a constant resting*t*
_{1/2,N2} of 360 min based on an approximation in the literature (6-8). However, in the case of *profiles G2* and *H2*, using*t*
_{1/2,N2} = 360 min does not adequately account for the degree of tissue resaturation with N_{2}during the long prebreathe after the initial exposure. As a consequence, actual DCI incidence was considerably higher (21.4 vs. 2.8% for *G2* and 16.7 vs. 1.8% for *H2*).

Nevertheless, observed and predicted DCI incidence were zero on*profiles I3–I6*, which also involved repetitive exposures. *Profiles I3* and *I4* followed two successive initial exposures (*profiles I1* and *I2*) and a long prebreathe with N_{2}-O_{2.} After completion of *profile I4*, another long prebreathe was inserted followed in quick succession by *profiles I5* and*I6*. We hypothesized that for *profiles I3–I6*the incorrect value of *t*
_{1/2,N2} was immaterial because of the virtually complete N_{2} washout after the second of the two initial exposures. In the case of*profiles G2* and *H2*, enough N_{2} was left after the single initial exposure so that, by the end of the long prebreathe, tissues were again highly saturated with N_{2}. A further complication is that any error in*t*
_{1/2,N2} affects*t*
_{1/2,O2}, which in turn causes error in the O_{2} pressure gradient and the O_{2} window, both critical to calculations in the FGM. Clearly, a dynamic reassessment of tissue half times is necessary for complex repetitive exposures such as*profiles G2* and *H2*. If true dynamic half-time values were available in such cases, they could be used in the FGM to obtain more realistic predictions of DCI incidence.

The statistical fit takes into account the group size of subjects for each profile as weight for each profile. Therefore, larger groups of data are better represented than smaller groups. Because of the relatively small sample sizes in each profile, one would not expect all points to be close to the line, even if the FGM were the perfect model. For example, *point D* appears to be a gross underprediction; however, this case corresponded to *profile D*, which had three exposures, two of which resulted in DCI.

#### Physiological and physical characteristics of subjects.

In designing a prebreathe or exercise protocol for operational use, the predicted overall risk of DCI is an important controlling criterion. It is important to note that the term “overall” means that the risk is meant to apply a priori to a randomly chosen subject from a population, without prior knowledge of the subject's particular propensity to incurring DCI. We expected, however, that the propensity to DCI would vary considerably among individuals in accordance with key FGM physiological parameters. The blood flow in tissue,Q˙ti(*t*), regulated by several mechanisms, including neural control and hormonal control systems, varies considerably from subject to subject and also in time, even at rest. Exercise-induced metabolic changes involved in modulating blood flow may also differ locally. The tissue half times for O_{2} and N_{2}washin and washout are defined by*t*
_{1/2,i} = log(2)/[*s*
_{b,i}Q˙ti(*t*)/*s*
_{ti,i}Vti(*t*)], where *s*
_{b,i} is the relatively stable blood solubility, s_{ti,i} is the tissue solubility varying with the tissue composition (32), e.g., level of hydration and biochemical factors, and Vti(*t*) is the volume of the tissue involved in gas exchange. Also, physical parameters, e.g., diffusivity of gases, elastic recoil influenced by density and texture of the tissue, surface tension, and thickness of the boundary layer, were selected from the literature (3, 13, 30, 32) and may vary in the same subject from time to time and across subjects. We were unable to adjust the values of these parameters in the FGM for each subject or according to subject variations of the internal milieu. Therefore, this accumulation of approximations, especially for repetitive exposures, might lead to inexact predictions of DCI occurrence in some individuals.

The application of the FGM with average parameters to each profile could lead to biased estimation of the overall risk of DCI over a given exposure period. For example, the FGM parameters pertaining to a typical subject might be such that bubbles do not grow; hence, we would have δ = 0, which would in turn make π = 0. Suppose, however, that there is a hypothetical fraction (e.g., 25%) of subjects with FGM parameters in a range where tissue bubbles would grow and DCI could occur. In this case, the probability of a randomly chosen subject acquiring DCI would not be even close to zero. We think that the above is the most likely explanation for the large discrepancy between the very small predicted incidences of DCI on *profiles G2* and*H2* (∼2% for both) and the observed incidences (17 and 21%, respectively). This bias could be mitigated by applying the FGM to more than one set of FGM parameters per profile, reflecting the individual variation. That would require knowledge of the distribution for FGM parameters in a population of subjects. This information, while theoretically obtainable, is not amenable to experimental measurement.

Because of the inherent variability in propensity for DCI in the population of subjects, a more likely explanation is that the subject was intrinsically resistant to DCI; i.e., values of the FGM parameters were such that bubble growth in that individual was stifled. In our statistical analysis, we allowed for both possibilities by treating a certain proportion of subjects as DCI resistant. The probability distribution of time to DCI applies only for the nonresistant or susceptible fraction of the population (π_{i}). Under our general hypothesis relating DCI incidence to bubble growth, knowledge of the FGM parameters for each subject tested would permit us to use the FGM to calculate subject-specific values of δ_{i}, the area under the V_{b · i}(*t*) curves in Fig.2, *A* and *B*. Subjects could then be classified DCI resistant for the *i*th profile if δ_{i} were below some threshold value, e.g., 0.1 (Table 1). Under this ideal scenario, π_{i} would be directly estimated by simply counting the ratio of susceptible subjects to total number of subjects. However, most of the FGM parameters are not measurable. The practical recourse is not to attempt identification of susceptibility for each subject. Instead, to account for individual variation, we assume that π_{i} is directly related to δ_{i} for a typical subject having FGM parameters that are about the mean values. We also used the approximation π_{i} ≅ θδ_{i}
^{γ}, where θ and γ are constants estimated from the NASA data set.

In summary, we found that, for a controlled condition of exercise, the fraction of the subject population susceptible to DCI can be approximately expressed in terms of a power function of the predicted cumulative volume of bubbles through the altitude exposure. Furthermore, for this fraction, the probability density distribution of DCI onset times is approximately equal to the ratio of the time course of total bubble volume to the predicted cumulative volume.

### Perspectives

Studies in the literature have implicated a role for mechanical movement of body structures in the formation of decompression bubbles (17, 18). It has long been known that skeletal muscle exercise is associated with the rate of bubble formation during decompression (2, 31). In the present study, an index of the rate of bubble formation (11a), the Poisson parameter β, is estimated for the moderate repetitive altitude exercise typical of actual EVAs. Given β, the volume of bubbles in the tissue region for a typical subject, expressed as a function of time by our mechanistic model (FGM), serves as a kernel for the probability distribution of the onset times of limb DCI pain symptoms. Furthermore, the proportion of subjects susceptible to DCI is expressed in terms of the cumulative volume of bubbles, calculated from the FGM. Improvements could be made in the FGM by incorporating time-varying and subject-specific values of physiological parameters, e.g., tissue blood flow, elastic recoil, and surface tension.

## Acknowledgments

The authors acknowledge Drs. Bruce D. Butler, Joseph R. Rodarte, and Michael B. Reid for critically reading the manuscript. The authors thank Dr. Johnny Conkin for useful advice and help in editing the data and Dr. Michael L. Gernhardt for the many discussions.

## Footnotes

This study was supported by National Aeronautics and Space Administration Cooperative Agreement NCC9-58.

Address for reprint requests and other correspondence: P. P. Foster, Pulmonary and Critical Care Section, Dept. of Medicine, Baylor College of Medicine, 6550 Fannin St., Smith Tower, Suite 1225, Houston, TX 77030 (E-mail: philipf{at}bcm.tmc.edu).

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- Copyright © 2000 the American Physiological Society