Vol. 279, Issue 6, R2317-R2328, December 2000
Predicting time to decompression illness during exercise at
altitude, based on formation and growth of bubbles
Philip P.
Foster1,
Alan H.
Feiveson2, and
Aladin M.
Boriek1
1 Department of Medicine, Baylor College of Medicine,
Houston 77030; and 2 Medical Sciences Division, National
Aeronautics and Space Administration Johnson Space Center, Houston,
Texas 77058
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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 [Vb·(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
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INTRODUCTION |
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 N2 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 [Vb·(t)]. In the present study, we hypothesized that reported onset times of limb
DCI pain symptoms follow a probability distribution with kernel
Vb·(t). Roughly speaking, this means that the
instantaneous probability of symptom onset at time t is
proportional to Vb·(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
 |
Parameter of the Poisson process, dimensionless
|
 |
Parameter of the Poisson process, dimensionless
|
 i |
Area under the curve
Vb · i(t), mm2
|
| FIO2 |
Fraction of O2 in the inspired medium, dimensionless
|
| fi(t,) |
Probability density function,
Vb · i(t)/i,
approximated by lognormal probability density function
|
| fA(t,) |
Probability density function for profile A
|
| fA*(t,) |
Perturbed probability density function for profile A;
lognormal scale parameters ( i) reduced by
25%
|
| Fi(t,) |
Cumulative distribution function
|
| FGM |
Formation-and-growth model
|
 |
Parameter estimated from experimental data, dimensionless
|
| µi |
Lognormal location parameter, dimensionless
|
P O2 |
Tension drop of O2 dissolved in tissue due to
O2 consumption, Pa.
|
 i |
Probability that a subject is susceptible to DCI in the ith
profile
|
 ti(t) |
Blood flow in the tissue shell, m3/min
|
| R |
Respiratory exchange ratio
|
| sb,i |
Solubility of the ith gas in the blood,
ml · ml 1 · 100 Pa1
|
| sti,i |
Solubility of the ith gas in the tissue,
ml · ml1 · 100 Pa1
|
| Si(t,) |
Survivorship function
|
| i |
Lognormal scale parameter, dimensionless
|
| Ti |
Total time of exposure to altitude, min
|
| tij |
Time to onset of DCI, min
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| t1/2,N2 |
Half time for tissue washin and washout of N2, 360 (rest)
and 300 (exercise), min
|
| t1/2,O2 |
Half time for tissue washin and washout of O2, 313.2 (rest)
and 261 (exercise), min
|
 |
Parameter estimated from experimental data, dimensionless
|
| Vb·(t) |
Volume of bubbles in the tissue region, mm3
|
| Vb · i(t) |
Volume of bubbles in the tissue region for ith profile,
mm3
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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.
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Fig. 1.
Flow chart of methodology. FGM, formation-and-growth
model; DCI, decompression illness; see Glossary for
definition of other abbreviations.
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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 O2 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 Table
1. 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% O2 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 O2
with a reduced N2 partial pressure
(FIO2 = 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 O2-N2 mixture
(FIO2 = 0.60) at an altitude of 41.4 kPa (6.0 psia); in the other profiles, subjects breathed pure
O2 at 30 kPa (4.3 psia,
FIO2 = 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 O2 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% O2-balance N2
mixture. Still at 70.3 kPa, the gas was switched to 100%
O2 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% O2 and
then 100% O2 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 Vb·(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 Vb·(t) separately for each profile
and 2) using the collection [Vb · i(t)] as explanatory
variables in a survival analysis. [Here,
Vb · i(t) denotes Vb·(t) for the ith profile.] As an
illustration, Fig. 2A shows
Vb · i(t) for profiles
A-D. Use of the FGM obviates the need for direct measurement
of bubbles in tissue.
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Fig. 2.
A: calculated bubble volume in the tissue region. The
first 4 historical profiles of the data set (A-D) are
shown for an estimated intensity of bubble formation corresponding to
= 0.017. The maximum volume of bubbles is reduced in long
O2 prebreathe procedures. However, in such procedures,
Vb·(t) reaches a plateau and slowly decreases
as time increases. B: transformation of total volume of
tissue bubbles [Vb · i(t)] into the
product of a true lognormal probability density function (pdf),
fi(t,), by the calculated cumulative volume
of bubbles (i). Expected
Vb · i(t) for profile A. A
close fit was obtained by the product of
fi(t,) and i (solid line).
C: pdf, fi(t,) = Vb · i(t)/i, for
profiles A-D. Long O2 prebreathe procedures
( 810 min = 2 + 718 + 90 min) of profiles C
and D resulted in a significantly lower maximum volume of
bubbles than profiles A and B, whereas
fi(t,) curves of profiles B-D
are almost superimposed.
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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
(PO2, R, t1/2,N2, and t1/2,O2) correspond to those for a
typical human subject and are shown in Table
2. Here, PO2 is
the tension drop of O2 dissolved in tissue due to
O2 consumption, R is the respiratory exchange ratio, and
t1/2,N2 and
t1/2,O2 are the tissue half times for
washin and washout of N2 and O2, 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 t1/2,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 t1/2,O2, we
used the relationship t1/2,O2 = 0.87t1/2,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 Tables
3 and 4.
The parameters and characterize the Poisson process, which
emulates bubble formation in the FGM. In particular, affects the
rate of bubble formation, whereas is proportional to tissue unit
volume. To effectively predict DCI, total bubble volume must be
normalized to tissue volume. We therefore calculated
Vb·(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, /, 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 /. However, with assumed constant over profiles, fixing / serves also to fix
tissue volume. This requirement is equivalent to specifying that the
parameter of the Poisson process is the same for all profiles (see
RESULTS AND DISCUSSION).
Lognormal approximation.
For each profile, Vb · 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 2A shows
Vb · i(t) for profiles
A-D. Because intensive computation was required to
numerically solve the differential equations to obtain
Vb · i(t), we approximated each Vb · i(t) by a kernel of a lognormal
probability density function; i.e.
|
(1)
|
where the parameters i, µi, and
i were estimated by the method of least squares using
SYSTAT software (34). The fit was excellent for the all
decompression profiles (R2 0.985). Figure 2B
illustrates a typical fit (R2 = 0.99) for
profile A. The filled circles represent the pointwise values
of Vb · i(t) obtained using the FGM, and
the solid line is the function given by Eq. 1. The lognormal
parameters (i, µi, and i)
serve to summarize the essential characteristics that distinguish the
profiles as determined by the FGM. In particular, i and
µi are the respective scale and location parameters of fi(t,), the lognormal pdf (21),
proportional to Vb · i(t), and
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 N2 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,
i, that a subject is susceptible to DCI varies among profiles and is modeled by the relationship
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(2)
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where and are parameters to be estimated. Thus for
> 0, test profiles with relatively large time-integrated
bubble volume (i) would be those most likely to induce
DCI. The probability of being resistant or cured is simply 1 
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 fi(t,); i.e.
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(3)
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This pdf is approximately proportional to
Vb · i(t), calculated by the FGM for a
particular value of the Poisson process parameter . Values of the
parameters µi and i were obtained directly
from Vb · i(t) before fitting the DCI
response data. In other words, fi(t,)
summarizes the dynamic process of expected bubble formation and growth
during application of the ith 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 , , and 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) = i
F(t,), where
Fi(t,) = 
0tfi(u,)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 iF(Ti,), where
Ti is the total exposure time for the ith
profile. For the jth exposure of the ith profile,
let tij be the time to onset of DCI, in cases
where the latter occurs. Under our survival analysis with known,
the log likelihood (10) of the data as an explicit
function of and is given by
![log L<IT>=</IT><LIM><OP>∑</OP><LL>i</LL></LIM> <LIM><OP>∑</OP><LL>j</LL></LIM> a<SUB>ij</SUB>[log<IT> &thgr;+&lgr; </IT>log<IT> &dgr;</IT><SUB>i</SUB><IT>+</IT>log f<SUB>i</SUB>(<IT>t</IT><SUB>ij</SUB><IT>, &bgr;</IT>)]](/content/vol279/issue6/fulltext/R2317/img009.gif)
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(4)
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where aij = 1 if DCI was observed on the
jth exposure and ith profile; otherwise
aij = 0. The first term in Eq. 4 is the contribution from cases where DCI occurred; the second term comes from
cases without DCI. When 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
i > 0 (Table 1).
The dependence of L on appears implicitly through the effect of the
latter on Vb · i(t), which in turn
determines fi(tij,) and
F(Ti,). Although is unknown, we minimized log L in
Eq. 4 (equivalent to maximizing log L) with respect to and for fixed trial values of ranging from 0.005 to 0.07. For
each of these trial values, the FGM had to be rerun to recompute fi(tij,),
F(tij,), and the resulting minimum value
of log L, e.g., Q() (Fig. 1). Ideally, the final estimate of should be the one minimizing Q() (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() show a slight random scatter (Fig. 3).
Therefore, our final estimate of was actually obtained by
minimizing a quadratic function of fitted to the values of Q()
(Fig. 3). For each trial value of and for our final estimate, we
always set equal to 6 so that
Vb · i(t) corresponded to a volume of
tissue with a mean of 300 bubbles formed during the altitude exposure
(see Explanatory variables).
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Fig. 3.
Model calibration for the intensity of bubble formation
in this controlled exercise regimen (817 kJ) using the experimental
data. We plotted the log-likelihood value against the Poisson parameter
. A logarithmic scale is used for . Because of random generation
of bubbles and onset times through the Poisson process,
Vb · i(t) may not vary in a monotonic
way; thus the log-likelihood values are scattered. To estimate the
minimum of the log-likelihood function, we performed a regression using
a quadratic function. The expected minimal value was obtained for
= 0.017 (log L = 386.21) and is indicated by
an arrow.
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RESULTS AND DISCUSSION |
Skeletal muscle exercise and bubble formation.
A plot of Q() vs. with the quadratic approximation is shown for
values of ranging from 0.005 to 0.070 in Fig. 3. The maximum value
of the quadratic function occurred at 
0.017. Note the
small random scatter induced by the finite expected sample size of 300 bubbles produced by the Poisson process. The value = 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 N2 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 O2 prebreathe. Therefore, we presumed
the intensity of bubble formation (characterized by = 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
= = 0.017, we ran the FGM (Fig. 1) for this
value of and obtained final estimates of the parameters and
, which characterize the proportion of susceptible subjects. The
resulting values were 
= 0.285 ± 0.036 and
= 0.150 ± 0.078. Therefore, the estimated
probability of a subject being susceptible to DCI for the
ith profile is equal to i = 0.285 i0.150 (Eq. 2). The small value for
reflects the property that the , the probability of
susceptibility as a function of , the cumulative volume of bubbles,
vs. time curve is approximately constant except for < 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
= 0.150, it would not be realistic to take = 0, because would be equal to for all values of (including
zero). Under such a condition, even when no bubbles are present, a
subject would still have a probability of = 0.285 of being
susceptible to DCI.
In susceptible subjects, Vb·(t) rises soon
after decompression, quickly attains a maximum, and then gradually
declines throughout the rest of the exposure (Fig. 2A). This
property is a result of the following sequence of events: 1)
immediately after decompression, the level of dissolved N2
in tissues is maximal, 2) bubble growth is terminated by the
O2 window (11a), 3) the breathing of enriched
O2 mixtures creates a positive N2 gradient from
tissue to alveolus, thus facilitating the removal of N2
from tissue before it diffuses into bubbles and simultaneously
N2 diffuses out of bubbles, 4) O2
permeates rapidly into bubbles driving out N2, and
5) because of its high permeation coefficient,
O2 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 Vb·(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 = 0.017. The logarithm of the ith Cox-Snell
residual, Zi, should be approximately equal to
Ki, 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 N2 washout. A similar degree
of washout could be expected with profile K, which had a
long prebreathe (480 min) with pure O2. After applying the
FGM to profiles I3-I6 and K, we predicted no
bubble growth whatsoever (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 i > 0.
In Fig. 4A, we plotted
Ki against Zi 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.
4A. A Cox-Snell residual plot commonly exhibits
characteristics as seen in Fig. 4A. 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
Zi and Ki 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 Vb · i(t) is a good predictor of times
to onset of DCI.
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Fig. 4.
Predicted onset times of pain DCI, goodness of fit,
and importance of Vb · i(t).
A: goodness of fit for the calibrated FGM ( = 0.017). For DCI cases, we plotted the double log of the Kaplan-Meier
survivorship function estimates against the log of the Cox-Snell
residuals obtained from FGM. Overall,  lie close to the
solid equality line (perfect match). B: in contrast, an
inexact Vb · i(t) led to a poor fit. A
25% reduction of the scale parameter for the 13 log-normal pdf
produced a prediction of late onsets of DCI ( are over
the equality line). We are unable to show the goodness of fit of the
model for the onset times in the case of no occurrence of DCI. In
A and B, x-axis is log of Cox-Snell
residuals (Zi) obtained from FGM and y-axis is
double log of Kaplan-Meier survival estimate (Ki).
C: an inexact Vb · i(t) with
an example: the case of profile A. Comparison of the
original pdf, fA(t,), obtained with the
calibrated FGM and used in the calculation of data in A,
with the perturbed pdf,
fA*(t,), used in B
is shown.
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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
Vb · i(t) in the selected critical
tissue was associated with the onset of DCI pain symptoms.
To illustrate the importance of the role of
Vb · i(t) in the onset of DCI in
critical tissue, we reduced the lognormal scale parameters
(i) by 25%, so that the resulting conditional density
functions fi(t,) were no longer proportional
to Vb · i(t). However, µl,
the lognormal mean, and i, the area under
Vb · i(t), were not modified. Figure
4C shows a plot of the original
fA(t,) and the perturbed function,
fA*(t,), corresponding to
profile A. Attempting the survival analysis with
fA*(t,) led to a poor fit, even
with reestimation of the parameters and , producing Cox-Snell
residuals (Fig. 4B) 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. 4A using the (correct)
proportional fi(t,) was due to the influence of the FGM and not simply to data fitting. In other words, a correct representation of Vb · 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 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 i = 0 were predicted to have no DCI cases, since 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 iF(Ti,)
(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 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.
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Fig. 5.
Predicted occurrence of pain DCI. Display of goodness of
fit between the predicted (FGM) and the observed incidence. Area of a
is proportional to number of subjects in the
decompression profile. Illustration of the goodness of fit for profiles
with no DCI is possible with the study of DCI occurrence only. Solid
equality line represents the perfect agreement. Equality line
intercepts centers of 5 profiles (I3-I6 and
K) with no DCI (a slight perturbation made them
distinguishable). However, in one such profile (I2), the FGM
overpredicted the incidence of DCI. In profile D (3 subjects), DCI occurrence (2 subjects) was underpredicted. Because key
FGM physiological parameters have been selected for a typical subject,
they may not apply to all individuals. This could lead to a biased
estimation of the overall risk of DCI over an exposure period,
particularly in the small sample size profiles of our data set.
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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% O2. Before the O2 prebreathe for the
second exposure, however, a long additional prebreathe period with an
O2-N2 mixture was inserted (Table 1). For all
profiles, we ran the FGM with a constant resting
t1/2,N2 of 360 min based on an
approximation in the literature (6-8). However, in
the case of profiles G2 and H2, using
t1/2,N2 = 360 min does not adequately
account for the degree of tissue resaturation with N2
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 N2-O2. 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 t1/2,N2 was
immaterial because of the virtually complete N2 washout
after the second of the two initial exposures. In the case of
profiles G2 and H2, enough N2 was
left after the single initial exposure so that, by the end of the long
prebreathe, tissues were again highly saturated with N2. A
further complication is that any error in
t1/2,N2 affects t1/2,O2, which in turn causes error in the
O2 pressure gradient and the O2 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,
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 O2 and N2
washin and washout are defined by
t1/2,i = log(2)/[sb,iti(t)/sti,iVti(t)], where sb,i is the relatively stable blood
solubility, sti,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 = 0, which would in turn make = 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 (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 i, the
area under the Vb · i(t) curves in Fig. 2, A and B. Subjects could then be classified DCI
resistant for the ith profile if i were below
some threshold value, e.g., 0.1 (Table 1). Under this ideal scenario,
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
i is directly related to i for
a typical subject having FGM parameters that are about the mean values.
We also used the approximation i 
i
, where and 
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 , is estimated for the moderate repetitive altitude exercise typical of
actual EVAs. Given , 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.
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ACKNOWLEDGEMENTS |
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.
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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).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 20 April 2000; accepted in final form 3 August 2000.
| |
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ABSTRACT
INTRODUCTION
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