INTRODUCTION

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REDUCTION IN AMBIENT PRESSURE is experienced by divers, aviators, and astronauts. The Shuttle and the Russian Space Station Mir atmospheres are at a pressure of 101.3 kPa, but astronauts or cosmonauts are exposed to a reduced absolute pressure in the space suit when they are performing extravehicular activity (EVA). Decompression may lead to the formation and growth of gas bubbles within tissues (3, 4, 16), with a resultant risk of decompression illness (DCI). In humans, the composition of gas bubbles in tissues is not amenable to direct experimental verification. Furthermore, analysis of intravascular bubbles does not provide information about the fractions of dissolved gas in tissues participating in bubble formation (8). O₂ and CO₂ rapidly permeate in and out of the bubbles, compared with N₂ (13). After sufficient time, the gas in bubbles presumably equilibrates with metabolic levels of O₂ and CO₂ in mixed venous blood or tissue, and with the body saturation level for water vapor pressure (8, 13). In contrast to hyperbaric decompressions, metabolic gases form a large fraction of the gas in bubbles during hypobaric decompressions (15). Moreover, theoretical simulations utilizing a system of mathematical equations (13) suggested a significant role for metabolic gases in bubbles during hypobaric decompressions.

Our hypothesis is that metabolic gases are involved in the initial growth phase of bubbles. An analysis is made by using experimental results from human exposures conducted in altitude chambers that simulate EVA procedures. Oftentimes, venous gas emboli (VGE) can be detected in the venous blood flow (7, 9) when a Doppler ultrasound bubble detector is used. Bubbles spawned in capillaries of tissues were mobilized into the venous return by flexing the limb during the period of bubble monitoring. Models that account for the tensions of dissolved gases in tissues and predict VGE incidence were statistically fitted to the data. We evaluated whether the incorporation of an additional mechanistic parameter into a model produced a better fit to the observed response. The model with the best fit to the data is assumed to support the more realistic hypothesis.

	METHODS

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Glossary

FIO2	Fraction of inspired O2, dimensionless
FIN2	Fraction of inspired N2, dimensionless
i	Subscript for the ith record (426 records)
L	Likelihood function, dimensionless
LL	Natural logarithm of the likelihood function, dimensionless
PACO2	Alveolar CO2 partial pressure of 5.33 kPa (40 mmHg)
PAH2O	Alveolar H2O pressure of 6.27 kPa (47 mmHg)
PAN2	Alveolar N2 partial pressure, kPa
PaN2	Arterial N2 tension, kPa
PAO2	Alveolar O2 partial pressure, kPa
PB	Total absolute pressure of the breathing medium; pressure at altitude, kPa
Pother	Additional dissolved gas tissue tension, kPa
P	Probability of occurrence of VGE, dimensionless
PtiN2(0)	Initial N2 tissue tension just before the procedure of interest, kPa
PtiN2(t)	Dissolved N2 tissue tension at a time t; estimated dissolved N2 tissue tension at the end of denitrogenation, kPa
R	Respiratory exchange ratio, CO2/O2, dimensionless
t1/2	Tissue half time for washin and washout of N2, min
t	Time of interest, usually the end of the O2 prebreathing, min
CO2	Amount of CO2 eliminated, l/min
O2	Rate of O2 uptake, l/min
yi	VGE outcome, 1 if VGE occurred, or 0 if none, dimensionless

Estimation of Dissolved N₂ Tension in Tissues

(Eq. 1)

Assuming a perfusion-limited system (2-4), an approximation of the N₂ partial tension of dissolved inert gas in the tissue during any N₂ partial pressure change in the breathing medium is provided by the classic exponential model

(Eq. 2)

where ln (2) is the natural logarithm of 2 and t is the elapsed O₂ prebreathe time. The dissolved N₂ tissue tension at the end of each denitrogenation stage was estimated by using iterations with Eq. 2 for each National Aeronautics and Space Administration (NASA) pre-EVA denitrogenation. MATHEMATICA software, version 2.2 (24), was employed for this purpose. The ascent time for each pressure change was included as part of the denitrogenation time.

NASA Hypobaric Data Set

Subjects. There were 164 volunteers (37 women and 127 men), who participated in 426 hypobaric exposures at the Johnson Space Center between 1982 and 1990. The average age was 31.38 ± 7.2 yr. Their individual characteristics were representative of the astronaut population. Women were included during the latter part of these studies. All were required to pass the United States Air Force Class III Flight Physical examination. The subjects signed an informed consent and were free to withdraw from the tests at any time.

Test procedures for simulated EVAs. The chamber tests were not all the same, since 1) denitrogenation periods varied; 2) the breathing gas during the denitrogenation was 100% O₂ at 101.3 kPa [14.7 lb./in.² absolute (psia)]; 3) however, some tests used a staged decompression, a prolonged stay at an atmosphere of 70.3 kPa (10.2 psia) enriched with O₂ and with a reduced N₂ partial pressure (26% O₂-74% N₂), as part of the denitrogenation process; 4) at altitude, after the final decompression, the breathing gas was 100% O₂ or O₂-N₂ mixtures; 5) the pressure at altitude ranged from 29.64 kPa (4.3 psia) to 44.80 kPa (6.5 psia); and 6) the time at altitude varied from 3 to 6 h. In all these tests, no exercise was performed during the O₂ prebreathe, and subjects were reclined or seated. Each subject was exposed to a particular denitrogenation and decompression profile; several of the same profiles constituted a test. Twenty tests were conducted. When test groups were separated with respect to gender, the total number of groups was 23. A low level of exercise during the simulated EVA involving upper limbs with an average metabolic rate of 837 kJ/h (200 kcal/h) was performed.

Dependent variable. VGE in the pulmonary artery were detected with a Doppler ultrasound bubble detector in the precordial position at ~15-min intervals throughout the exposure. The bubbles were mobilized by flexing the joints and straining the muscle groups of a limb (1) at the time of bubble monitoring. It is assumed that we measure the "limb bubble-formation tendency" before the time of monitoring. We code the presence of VGE in the pulmonary artery as 1, or 0 if no bubbles were detected.

Independent variable. We define a dose for the decompression (2, 14) as tissue ratio (TR), which is the ratio of the calculated dissolved N₂ tissue tension for a given tissue to the ambient pressure

(Eq. 3)

where Pti_N2(t) is the dissolved N₂ tissue tension at the end of the denitrogenation period, as obtained from Eq. 2, and PB is the ambient pressure at altitude. In Eq. 3, TR is expressed by using only the calculated dissolved N₂ tissue tension. A second determination of dose, i.e., TR', is considered to be

(Eq. 4)

where P_other is the magnitude of dissolved gas tissue tensions other than N₂. The addition of P_other tests whether dissolved gas tissue tension other than N₂ plays a significant role in bubble formation during hypobaric exposures.

Statistical Analysis

Logistic regression model. The probability of VGE occurrence is modeled as a function of the decompression dose, TR or TR'. An appropriate statistical model for the probability of occurrence of an event is often assumed to be logistic (6). In terms of a function of dose, the probability P of occurrence of VGE is defined by the logistic regression equation, namely

(Eq. 5)

where ₀ and ₁ are unknown parameters. The use of a similar probability function for the VGE dose-response relationship, the Hill equation, is documented elsewhere (2, 17, 20). We rewrote Eq. 5 into a form comparable to the Hill equation

(Eq. 6)

where b₀ = ₁ and b₁ = exp (₀/₁) are two statistical parameters. Moreover, unknown physiological parameters, t_1/2 and P_other, also must be optimally determined. The three-parameter model with P_other = 0 thus becomes a four-parameter model when P_other is taken to be a nonzero quantity, and this must be determined so that it maximizes the likelihood function.

Maximum-likelihood method. Maximum-likelihood estimation is the preferred technique (3, 4, 11, 14, 20, 22) to estimate unknown parameters in a model so that the probability function is maximized. Given a set of data, the likelihood function L is defined as the product of the probability densities for the subjects' outcomes

(Eq. 7)

where P_i is the probability and y_i denotes the VGE outcome (1 or 0) for the ith record; total number of records was 426. It is customary to use the natural logarithm of the likelihood function

(Eq. 8)

The log likelihood function, LL(b₀, b₁), is always negative, since ln (P_i) and ln (1  P_i) are negative as P_i lies between 0 and 1. The maximum likelihood estimates of b₀ and b₁ that maximize the LL function can be obtained by solving the following equation

(Eq. 9)

for the 20 different denitrogenation procedures. We used the NONLIN module of SYSTAT version 5.03 (23) to solve Eq. 9 and to estimate unknown parameters in the models with the sum of absolute values, |LL|, minimized by using the quasi-Newton algorithm. In what follows, the absolute value of the LL function will be noted LL. For simplicity, the lowest LL value will be designated as the "best" LL value.

Test of statistical hypotheses. The likelihood ratio test statistic (14, 20) is used to assess whether additional parameters added to a model improve the goodness of fit. The degrees of freedom of a test statistic are equal to the number of parameters estimated in a full model minus the number of parameters in a hypothesized model. The likelihood ratio test statistic is transformed, from a ratio to a difference, by taking twice the difference between the two corresponding LL values. The transformed statistic follows a ² distribution. Finally, a ² table is entered with the calculated likelihood ratio and the difference in degrees of freedom between the two models to determine the appropriate P value.

	RESULTS

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Table 1 lists the family of models tested, the fitted parameters in each model, and the LL value obtained in each case. The three-parameter model with only N₂ dissolved tissue tension in the expression of dose defined by Eq. 3 (P_other = 0), resulted in an LL value of 255.01. The ability to describe the response variable improved by accounting for an additional dissolved gas tissue tension other than N₂ in the dose expression of Eq. 4. The four-parameter model (P_other  0) returned an LL of 251.70, the best LL value that we encountered, indicating that it has the best fit to the data. A decrease of 3.31 LL units by the addition of P_other in dose is statistically significant at P = 0.012 under the likelihood ratio test. We also define an upper and lower boundary of the LL value (14, 20) to further evaluate the goodness of fit of our best model. The hypothesis that dose of decompression is not useful in predicting VGE defines the "null" model; it is considered as an upper boundary. This is a constant-probability model with one scaling parameter that is estimated by the odds ratio of cases with VGE vs. no VGE. This model returned a value of 290.46 for LL, which significantly differs from all the other models tested. The LL value of 240.08 corresponded to the low-boundary case of the "discontinuous" model, which consisted of separate null models for the 23 groups of data. This discontinuous model, of course, is too data specific to be effective for prediction beyond the range of observations utilized. The discontinuous model and the four-parameter model are, however, not statistically different, since the likelihood ratio test based on the 19 degrees of freedom for the ² statistics yielded a P value >0.50.

Table 2 lists information for the best fit four-parameter model shown in Table 1. The estimated value for P_other was 19.1 kPa (143 mmHg). The asymptotic SE for P_other was 4.64, which is small, relative to the parameter estimate. The asymptotic correlation matrix in Table 3 indicates that P_other was poorly correlated with b₀; therefore, it merits retaining P_other in the model (12). However, P_other was significantly correlated with b₁, implying that a change in P_other would have caused large changes in b₁. The estimates of b₀ and b₁ were several times larger than their SE values, thus indicating their significance in the model. The t_1/2 had a value of 329 min.

Because t_1/2 was estimated from trial-and-error modification of potential models, the study of these models allows further insight into relations between t_1/2 and P_other. Close inspection of Fig. 1 shows that the distribution of isopressure isopleths (LL vs. t_1/2, at six different values of P_other) follows a specific pattern. The parabolic shape of these isopleths is a property of the maximum likelihood optimization. As P_other increases in value, the isopleth becomes steeper. The minimum on the isopleth corresponds to the best fit of this model with the data. The intercept of the 19.1 isopleth with the 329-min t_1/2 corresponds to the best fit model. On the other hand, the t_1/2 estimate for the three-parameter model located at the minimum of the 0 kPa isopleth has a value of 420 min. Figure 1 also shows that our best fit model is robust, since slight variations of P_other, e.g., within 2 kPa of 19.1 kPa, hardly affect the LL value. Furthermore, variations of P_other within the range of the SE (4.64 kPa) do not significantly affect the estimated model.

View larger version (24K): [in this window] [in a new window]   Fig. 1.   Plot of six isopressure isopleths on natural logarithm of likelihood function (LL) vs. half time t1/2 corresponding to six additional dissolved gas tissue tension (Pother) values of 0, 0.5, 17.7, 19.1, 21.0, or 30.0 kPa. Three isopleths, 17.7, 19.1, and 21.0 kPa, are almost superimposed; their LL minima are nearly equal, occurring at the same t1/2. Because there is no significant difference among the three cases, the 19.1-kPa isopleth is taken to be the best fit model with the lower LL number. The 17.7-kPa isopleth corresponds to the estimates of mixed venous blood (or tissue) tensions of O2, CO2, and water vapor pressure.

Another approach to making comparison between models is the graphic illustration of goodness of fit as shown in Figs. 2 and 3. Each circle represents a group of subjects, and the size of a circle is proportional to the corresponding group size; there were 23 groups. For a given group, the value on the y-axis is the observed incidence of VGE in this group. The sigmoidal curves show the predicted probability of VGE by two different models: the best fit four-parameter model (Fig. 2) and a three-parameter model (Fig. 3). The models were fitted using a NASA data set consisting of 426 individual observations, as described earlier. The position of all the circles around the curve provides a visual impression of the fit; circles close to the sigmoidal curve indicate a better fit of the model to the data. However, this goodness of fit criterion based on the examination of group incidence is limited because of the lack of reliability of the visual interpretation of observed incidences. Because the model fit takes into account the size of groups of data as weight for each group of subjects, the model represents the larger groups of subjects better than it does the smaller groups. Overall, the best fit model (Fig. 2) does not seem to over- or underpredict the incidence of VGE, except for a few small groups of subjects. In contrast, Fig. 3 depicts a poor fit for the three-parameter model with a 240-min t_1/2, since it did over- and underestimate the VGE incidence even in larger groups of subjects, and circles are dispersed away from the model curve. Although the model fit is weighted according to the size of subject groups, Fig. 3 shows that a model with an inappropriate t_1/2 and absence of P_other fails to accurately predict the observed incidence, even in the case of larger groups of subjects.

View larger version (15K): [in this window] [in a new window]   Fig. 2.   Display of goodness of fit between our best model and observed incidence in groups of tests composing the data set. Dose is defined as tissue ratio (TR') from Eq. 4; parameter estimates were t1/2 = 329 min, and Pother = 19.1 kPa. Area of a circle is proportional to no. of subjects in a group; smallest circle has 3 subjects, and largest circle has 59. Estimated probability curve intercepts centers of larger circles (59, 35, and 28 subjects). Three subjects in topmost smallest circle have a venous gas emboli (VGE) incidence of 1; therefore, model underestimates the outcome for this group.

View larger version (16K): [in this window] [in a new window]   Fig. 3.   Display of goodness of fit of a 3-parameter model with t1/2 of 240 min; a model with a poor fit to the data (LL = 268.35). Dose is defined as TR from Eq. 3. Model over- and underpredicts incidence of VGE for all groups of data.

Because PA_N2 (or Pa_N2) was estimated by using Eq. 1, it was appropriate to determine whether variations in the respiratory exchange ratio R affect the predictability of the model. We added R as an additional parameter to the four-parameter model and evaluated the model for values of R between 0.70 and 1.0. The ability to describe the response variable is not affected by including R as a parameter. The model is robust, since variations of R between 0.70 and 1.00 do not influence the LL value (251.70). Clearly, variations in R and, therefore, small changes in dissolved N₂ tissue fraction have no influence on the model.

	DISCUSSION

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Presence of Bubbles

Mechanistic Hypotheses

The initial explosive-growth phase precipitates a series of events. Bubble size is inversely proportional to surface tension pressure, and the driving force for diffusion of gas into a bubble increases as surface tension pressure diminishes (12). Nitrogen tension in the tissue becomes a driving force for diffusion, causing N₂ to diffuse from tissue to nascent bubble. Once molecules of N₂ are captured inside the newly generated bubble, they are involved in the N₂ partial pressure of the bubble as an outward radial pressure. In contrast, pressure due to surface tension is an inward radial pressure that tends to reduce the bubble volume. Thus, in modeling bubble growth, the surface tension pressure should be subtracted from N₂ dissolved tissue tension. Similarly, tissue elastic recoil is also an inward radial pressure as well as O₂ ambient pressure; both quantities should then be subtracted from the N₂ dissolved tissue tension.

Underlying mechanisms of bubble growth suggest that surface tension pressure and tissue elastic recoil are not involved in P_other. If P_other is not due to inward radial elastic forces, it should then be caused by gas(es) in physical solution in the tissue before the initial explosive-growth phase. The positive sign of P_other unmasks the contribution of an outward radial pressure due to this (these) dissolved gas(es).

Assuming that P_other is due to gas(es) previously dissolved in tissues, it is questionable whether the tissue gas(es) tension could follow various distributions with time. Indeed, an alternative to the single-exponential tissue-gas exchange for N₂ and to the constant second-term P_other of Eq. 4 has to be examined. The number of plausible tissue types considered in the analysis may, in fact, be more than one (18). It has been shown (in dogs) that tissue isobaric gas exchange for ¹³³Xe (21) is better described by two or three exponential processes in series. The exponential-series analysis (18) assigns one exponential term or t_1/2 for inert-gas washout for each tissue present in the expression of dose. These models were applied to predict probability of DCI in diving (11, 22), and it was found that the prediction of DCI incidence was similar for both the series and parallel arrangement of tissues (11). It is questionable how a detailed description of the tissue gas exchange for N₂ including another exponential term in lieu of P_other in Eq. 4 would predict the VGE outcome in the NASA hypobaric exposures. Therefore, we tested (although analysis is not shown) an expression of dose derived from a double-exponential gas exchange in a single tissue or a monoexponential gas exchange in a parallel arrangement of two tissues, each tissue with a different exponential term (22). This dose was then used in the logistic model. The addition of more parameters in the model even further reduced the goodness of fit. In contrast to some DCI data from hyperbaric exposures (11, 22), a monoexponential gas exchange in a single tissue for N₂, as described in our analysis, better fitted the data of hypobaric exposures than a complex gas-exchange kinetics with two exponential terms. The rationale behind these observations is that P_other is not an additional pressure due to N₂, which would be disregarded by the Pti_N2(t) term of Eq. 4, and that noninert gas(es) would assist N₂ during the initial explosive bubble-growth phase.

Clearly, two types of tissue gas exchange appear in the expression for dose, TR', in the best fit four-parameter model. Our best predictor for the total driving tension of the tissue ratio is made up of two pressure terms, each with a different relation to time. The two types of tissue gas exchange were 1) an elimination of dissolved N₂ exponentially related with time, as described by others (5, 18); and 2) a gas tension in terms of P_other, which remains constant, presumably due to metabolic gases. Among all dissolved gases in tissues, this exponential elimination appears to be a characteristic of N₂ tissue gas exchange.

The aforementioned mechanistic premises allow further insight into properties of P_other in bubble formation. However, no direct measurement could verify any postulate about P_other. Our statistical analysis shows a correlation between P_other and the model parameter b₁, and thus a decrease in the LL value may be due to an improvement in the model caused by the addition of P_other in determining dose.

It is also unclear whether the magnitude for P_other obtained in this analysis is a coincidence. The value of 143 mmHg (19.1 kPa) is approximately the tension (BTPS) of metabolic gases in tissue or in mixed venous blood. This finding suggests that nearly all the dissolved gas, other than the N₂ that remains in physical solution in the tissue, has also been utilized to separate the gas phase.

The dissolved tissue tension of all gases involved in the bubble growth, or driving tension, warrants a brief description. Figure 4 shows two dose-response curves of the best fit four-parameter model corresponding to the two altitude pressures of 30 and 45 kPa, respectively. As a result of a complete denitrogenation [Pti_N2(t) = 0], the driving tension still available is ~19.1 kPa; P_other, presumably because of metabolic gases, remains constant, whereas the dissolved N₂ tissue tension in the expression of dose depends on the denitrogenation procedure. In contrast, without preliminary denitrogenation, and according to our analysis, the total driving tension that can potentially generate bubbles in tissues is 93.3 kPa; it is calculated by subtracting the arteriovenous O₂ difference (8 kPa) from the standard pressure (101.3 kPa). The pressure difference of 8 kPa is due to a phenomenon known as the "oxygen window" (15) because metabolism lowers partial O₂ tension in tissues below the value in arterial blood. The 95% confidence intervals were computed based on the propagation of error formula (14). It is seen that the confidence intervals are narrower in the case of 30 kPa than in the case of 45 kPa. The confidence interval provides a range for the parametric values, but it does not establish the accuracy of the estimate. Furthermore, the larger the sample size, the narrower the confidence intervals (3).

View larger version (42K): [in this window] [in a new window]   Fig. 4.   Probability of VGE predicted by best model as function of dissolved tissue tension of gases generating bubbles. Two isopressure isopleths are displayed for 2 altitude pressures of 30 kPa (top curve) and 45 kPa (bottom curve). Nonzero probability of VGE starts to rise for a driving pressure of ~20 kPa. At altitude pressure of 30 kPa, without denitrogenation, model predicts occurrence of VGE with probability 1 when a dissolved gas tissue tension is 100 kPa. The 95% confidence interval is shown by shaded area.

Fractions of Gases

The mechanistic role of metabolic gases in bubble formation appears to be inversely proportional to the excess of dissolved N₂ in tissue; comparison of hypobaric and hyperbaric exposures indicates significant difference. The P_other value derived from these NASA hypobaric decompressions is higher than values from direct measurements in diving experiments with guinea pigs breathing air or gas mixtures (8); metabolic gases fraction in bubbles was ~10%, with the balance of 90% due to inert gas. In human dives, there is a slight chance that O₂ could be 40% as effective as N₂ in producing a risk of DCI (19). In hyperbaric conditions, dissolved N₂ tissue tensions and fractions are large, whereas metabolic tissue tensions and fractions remain constant. Therefore, the fraction of inert gas that is likely to participate in the bubble-formation process should be greater in hyperbaric decompression than in hypobaric decompression. Moreover, a detailed model of tissue gas exchange considered in terms of either a series or parallel arrangement of tissues may provide a better fit to the data from diving exposures.

Conclusions