Vol. 87, Issue 4, 1521-1531, October 1999
MODELING IN PHYSIOLOGY
A model of extravascular bubble evolution: effect of changes in
breathing gas composition
J. F.
Himm and
L. D.
Homer
Naval Medical Research Institute, Bethesda, Maryland 20889-5607
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ABSTRACT |
Observations of bubble evolution in rats after
decompression from air dives (O. Hyldegaard and J. Madsen.
Undersea Biomed. Res. 16:
185-193, 1989; O. Hyldegaard and J. Madsen.
Undersea Hyperbaric Med. 21:
413-424, 1994; O. Hyldegaard, M. Moller, and J. Madsen.
Undersea Biomed. Res. 18:
361-371, 1991) suggest that bubbles may resolve more safely when
the breathing gas is a heliox mixture than when it is pure
O2. This is due to a transient
period of bubble growth seen during switches to
O2 breathing. In an attempt to
understand these experimental results, we have developed a multigas-multipressure mathematical model of bubble evolution, which
consists of a bubble in a well-stirred liquid. The liquid exchanges gas
with the bubble via diffusion, and the exchange between liquid and
blood is described by a single-exponential time constant for each inert
gas. The model indicates that bubbles resolve most rapidly in spinal
tissue, in adipose tissue, and in aqueous tissues when the breathing
gas is switched to O2 after surfacing. In addition, the model suggests that switching to heliox breathing may prolong the existence of the bubble relative to breathing
air for bubbles in spinal and adipose tissues. Some possible
explanations for the discrepancy between model and experiment are discussed.
gas bubble evolution; breathing gas change; inert gas; countercurrent exchange; oxygen
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INTRODUCTION |
THE LEADING CANDIDATE in the etiology of decompression
sickness (DCS) is the evolution of gas bubbles in the tissues and
circulatory system of the diver, at least in cases where symptoms
appear shortly after surfacing (3). Numerous investigators have
developed models of bubble evolution in tissue (2, 6, 22, 32), and some
work has been done in modeling bubble evolution in blood vessels (12a).
In addition, bubble-related models have been important in the
development of decompression procedures at the University of
Pennsylvania (9), the Naval Medical Research Institute (25a, 26, 36),
and the Defence and Civilian Institute of Environmental Medicine (30).
However, few of the models have been validated experimentally (34). The
recent observations by Hyldegaard et al. (16, 17, 19) of the growth
and dissolution of bubbles in rats after decompression from an air dive
offer an opportunity to validate some of these models.
The first series of experiments performed by Hyldegaard and Madsen (16)
looked at the effect of breathing gas switches on the evolution of gas
bubbles in adipose tissue after a dive to ~75 feet seawater (fsw) for
4 h. The bubbles observed in the fat tissue developed naturally as a
consequence of the dive and were not introduced into the tissue by the
investigators. Observations of the bubbles began ~30 min after
decompression, and switches in the breathing gas occurred during the
following 30 min. Bubbles grew in rats that continued to breathe air
throughout the observation period. When the breathing gas was switched
to a heliox mixture, bubbles consistently diminished in size; when the
breathing gas was switched to pure
O2, bubbles resolved in about the
same time frame as during heliox breathing. The initial response to the O2 breathing was a stabilization
or growth of the bubble, followed by a more rapid resolution of the
bubble than was seen during the heliox breathing. One bubble continued
to grow throughout the O2
breathing. If the breathing mixture was switched a second time from
heliox to air or
N2O-O2
while the bubble was still of an appreciable size, the bubble would
resume growing.
The second series of experiments (19) involved bubbles injected into
the white matter of the spine after a 4-h dive to 70 fsw. Results for
this series of experiments were similar to those in which bubbles were
injected into the adipose tissue. Most bubbles grew throughout the
observation period, whereas heliox or
O2 breathing led to a resolution
of the bubble. Again, O2 breathing
caused an initial growth of the bubble, followed by a more rapid
dissolution than was seen during heliox breathing. A second breathing
gas switch to
N2O-O2
while the bubble was still appreciable in size led to resumed growth of
the bubble.
The third series of experiments (17) involved bubbles injected into
muscle and tendon after a 1-h dive to 80 fsw. The effect of the
breathing gas switches on the injected bubbles was similar in muscle
and tendon. Bubbles resolved rapidly after the switch. Continued
breathing of air led to a period of growth for bubbles in muscle
tissue, whereas bubbles diminished steadily after injection into tendon.
Initial bubble diameters were highly variable; spinal bubbles were
roughly 300 µm; and adipose, muscle, and tendon bubbles were ~600
µm. Bubbles that resolved typically did so within ~3 h in spinal
and adipose tissues and within ~4 h in muscle and tendon.
Thus the experiments of Hyldegaard et al. (16, 17, 19) indicate that
changes in breathing gas composition can affect the time course of
bubble evolution. Bubbles resolve more rapidly when the breathing gas
is switched to 80% He-20% O2 or
100% O2 than with air breathing
after surfacing. Bubbles resolved more slowly, or not at all, when air
was the breathing gas. Bubbles in adipose tissue, spinal white matter,
and tendon could undergo a transient period of growth after the
breathing gas was switched to pure
O2. Such growth was not observed
after the breathing gas was switched to heliox. On the basis of these
results (and on the standard assumption that DCS is alleviated by
eliminating bubbles), the Israeli Navy has modified its DCS treatment
procedure by recompressing patients on
He-O2 instead of pure
O2 in cases where neurological
symptoms are manifested (23; A. Shupak, personal communication).
To determine the relative importance of various physiological factors
in the evolution of a bubble in tissue, we have developed a detailed
model of bubble evolution in a liquid. The specific aim is to test
whether one should expect the breathing of certain gases to be
inherently advantageous in helping resolve extravascular bubbles or
whether certain sequences of "gas switches" are advantageous. When dives similar to those performed by Hyldegaard et al. (16, 17, 19)
are simulated, several discrepancies between the bubble model and the
experimental results arise with the assumption of normal physiological
parameters for the rat tissues. Possible sources of the discrepancies,
including altered tissue perfusion, altered
O2 consumption in the tissue, and
countercurrent exchange (CCE) of the inert breathing gases, are
explored with use of the model.
| |
DESCRIPTION OF THE MODEL |
Here we discuss the assumptions on which the model is based. The
translation of those assumptions into equations is shown in the
APPENDIX.
The model describes a spherical bubble in a homogeneous liquid of
finite volume. The bubble exchanges gas with the surrounding liquid at
a finite rate. The rate is computed using the steady-state mass
transfer coefficient for a sphere in a stagnant liquid, although the
system is unsteady state. This approximation is adequate only when the
transport rate and the rate of change of the bubble radius are
sufficiently slow. Therefore, the model is expected to give a poor
description of bubble dynamics during rapid changes of hydrostatic
pressure or when the bubble is "small" (i.e.,
/R > 25,000 dyn/cm2, where is surface
tension and R is bubble radius).
The dissolved gases are uniformly distributed throughout the liquid;
i.e., it behaves as a well-mixed compartment. The liquid is
equilibrated with a third phase that flows past it. For our purposes,
the liquid can be thought of as a volume of tissue being perfused by
blood. The assumption that the tissue and blood are equilibrated means
that when Henry's law is obeyed in tissue and blood, the rate of gas
exchange between them is governed by the familiar single-exponential
kinetic model. Single-exponential kinetics, therefore, are used for the
inert gases N2 and He and for
CO2, but not for
O2, the solubility of which in
blood does not follow ideal solution behavior because of binding to Hb.
Although CO2 also binds to Hb, its
high effective solubility in tissue (see
APPENDIX) leads to a virtually
constant tissue partial pressure of 0.0461 ATA under all conditions
explored here, so the simpler single-exponential kinetics were used.
The effects of CCE are taken into account by requiring that the
concentration of an inert gas in the arterial blood supplied to the
tissue be between the concentration in the arterial blood in the lung
and the concentration in the tissue. The CCE process has a greater
effect on inert gas exchange for the more diffusible gases.
Because the process of bubble nucleation is not understood, a bubble
that does not exchange gas with the surrounding tissue is assumed to be
present at the start of the dive. During decompression the bubble
begins to exchange gas with the surrounding tissue when the sum of
partial pressures of the gases in the tissue exceeds the sum of the
partial pressures of these same gases in the bubble. If the rate of
decompression is slow enough, this partial pressure condition will not
be met, and the bubble will not evolve.
The assumption of uniformity within the tissue volume might be the most
unsatisfactory one that we are forced to make. On the one hand, the
bubble is treated as being in a quiescent fluid for the purpose of
calculating gas flux across the interface. This is consistent with the
properties of a tissue, where the fluid content of the tissue is bound
within cells. On the other hand, the tissue is assumed to be well
stirred because of the convective flow of the blood. In addition, not
only has it been shown experimentally (25) and via simulation (14) that
inert gas kinetics in muscle are not simply described, this assumption also precludes the inclusion of potentially important geometry-specific information, e.g., the distance from the bubble to the nearest blood
vessels. A more complete treatment of the tissue might be attempted,
but there are too many unknowns regarding the interaction of a bubble
with the tissue. Direct effects of the bubble include possible changes
in the blood flow in the smaller vessels near the evolving bubble, the
accumulation of surfactants at the tissue-bubble interface interfering
with the transport of gases into and out of the bubble, and the
contribution of tissue elasticity to the internal bubble pressure.
Indirect effects may include the triggering of the immune system,
leading to changes in perfusion and
O2 consumption, as well as
swelling of the tissue.
Thermal, viscous, and compressibility effects were ignored, which
invalidates the model only at very small diameters (a few micrometers),
when implosion is rapid and internal pressure is high. The partial
pressure of water in the bubble is taken to be the vapor pressure of
water at ambient temperature. Its dependency on hydrostatic pressure
and temperature is ignored, but these effects are significant only for
tiny bubbles.
Similar assumptions are used in the Burkhard-Van Liew model (6),
wherein the tissue-blood exchange is assumed to follow single-exponential kinetics, and gas concentration in the tissue is
determined by assuming that the gas is uniformly distributed throughout
the volume. However, the gas exchange between the bubble and tissue is
determined from a steady-state solution to the diffusion equation with
a source/sink term, but with no convective term. Their model does not
incorporate CCE and uses an indirect method to calculate the partial
pressure of CO2 in the tissue.
Our model does incorporate the presence of the metabolic gases
O2,
CO2, and
H2O within the tissue and the
bubble; the effect of a nonzero surface tension on the total pressure
within the bubble; the consumption of
O2 and simultaneous generation of
CO2 within the tissue; the
presence of dissolved CO2 in the
form of carbonic acid, bicarbonate, and carbonate; the effect of
changes in hydrostatic pressure on bubble volume; the effects of CCE on inert gas uptake and elimination; and the complicated solubility curve
for O2 in whole blood. It is the
most complete simulation of an extravascular bubble exposed to multiple
gases of which we are aware.
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RESULTS |
For bubbles in adipose and spinal tissues, the simulated dive is to
75.9 and 69.3 fsw, respectively, for 4 h, with decompression stops at
66 fsw for 3 min, 33 fsw for 6 min, and 16.5 fsw for 11 min. Ascent
rate was 39.4 fsw/min, and ascent time between stops is included in the
time at a given stop (O. Hyldegaard, personal communication). The
breathing gas was changed 30 min after surfacing to approximate the
actual experimental conditions, and the initial bubble radius is taken
to be ~3 µm for both tissues. These bubbles evolved to a maximum
radius comparable to that observed experimentally. For bubbles in
muscle and tendon, the simulated dive is to 82.5 fsw for 1 h, with
2-min decompression stops at 66, 33, and 16.5 fsw, with travel times of
15, 30, 15, and 30 s between stops, respectively. Gas switches for the
muscle and tendon simulations occurred ~60 min after surfacing.
Initial bubble radius for the muscle and tendon simulations was taken
to be 290 µm. These bubbles grew, after decompression, to a radius of
~300 µm, corresponding to the size of the bubbles in the rat experiments.
Because bubbles in the experiments were generally isolated from other
bubbles, we assumed a maximum bubble density of 1,000 bubbles/cm3, corresponding to a
spherical tissue volume of radius 0.06 cm. Simulations with larger and
smaller volumes were also done, with the qualitative effects of gas
switching being unaffected by the tissue volume. The diffusive time
scale (length2/distance) for this
distance for all gases discussed here is on the order of 1-10 min.
Distances to blood vessels (except in tendon) are more typically on the
order of 
100 µm, leading to diffusive time scales of at most a few
tens of seconds. Table 1 shows the parameter values used for the simulations.
Figure 1 shows simulation results for a
bubble evolving in spinal white matter. For the top
curves, matching the experimental conditions of
Hyldegaard et al. (19), the initial breathing gas is air. For the
bottom curves, the breathing gas is
80% He-20% O2. The simulation
results indicate that for air and heliox breathing during the dive,
switching the breathing gas to 100%
O2 is most effective at speeding
dissolution of a bubble, and switching it to a heliox mixture is least
effective. Delaying the gas switch did not change the qualitative
effects of the switch.
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Fig. 1.
Simulated effect of a breathing gas switch on growth of gas bubbles in
rat spinal white matter. Top curves
assume air breathing during dive; bottom
curves assume heliox breathing during dive. A switch to
O2 breathing is most effective at
reducing bubbles. Vertical dashed line, time of gas switch (30 min
after surfacing).
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Figure 2 shows similar results for a bubble
evolving in adipose tissue. Again, switching the breathing gas to 100%
O2 is the most effective way to
speed dissolution, and switching it to heliox is the least effective.
For the air dives, the switch to heliox breathing initially speeds
dissolution relative to air breathing but has the effect of slowing
dissolution at later times. The initially beneficial effects of heliox
breathing occur at 2-3 h after decompression, corresponding to the
observation period of the rat experiments. In addition, the model
predicts that bubbles in adipose tissue will persist for many hours
longer than was observed experimentally.
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Fig. 2.
Simulated effect of a breathing gas switch on growth of gas bubbles in
rat adipose tissue. Top curves assume
air breathing during dive; bottom
curves assume heliox breathing during dive. A switch to
O2 breathing is most effective at
reducing bubbles. Note transient period of growth of bubble after
switch to O2 breathing from
breathing air; switch to heliox breathing is initially more effective
at reducing bubbles than is continued breathing of air.
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Figures 3 and
4 show the simulation results for bubble
evolution in muscle and tendon, respectively. Once again, the switch to
O2 is most effective in reducing
the bubbles, and switches to heliox or air are less effective. In
contrast to the situation in adipose tissue or spinal white matter, a
switch to heliox breathing is slightly more effective at resolving
bubbles than a switch to air breathing. These results are consistent,
whether the diving gas is air or heliox.
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Fig. 3.
Simulated effect of a breathing gas switch on growth of gas bubbles in
rat muscle tissue after an air dive. A switch to
O2 breathing is most effective at
reducing bubbles, and heliox breathing is slightly more effective at
reducing bubbles than continued breathing of air. Curves for a heliox
dive are very similar for bubble size and evolution.
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Fig. 4.
Simulated effect of a breathing gas switch on growth of gas bubbles in
rat tendon after an air dive. Curves for a heliox dive are very similar
for bubble size and evolution. A switch to
O2 breathing is most effective at
reducing bubbles, and heliox breathing is slightly more effective at
reducing bubbles than is breathing of air.
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DISCUSSION |
Our simulations indicate that the heliox mixture may be the least
effective of the three breathing gases modeled in resolving gas bubbles
in spinal and adipose tissue and that
O2 breathing is much more
effective in reducing bubbles than air or heliox breathing,
irrespective of the tissue. What is the source of the discrepancy
between the experimental results of Hyldegaard et al. (16, 17, 19) and
the simulation results? Two possibilities are as follows:
1) the physiological factors
affecting bubble evolution in the tissue are affected by the dive,
whether during the dive or after, and
2) the transport of gases to the
tissue is not described, as mentioned above, by the single-exponential function that can be determined from the tissue perfusion rate.
In possibility 1 we include changes in
O2 consumption and tissue
perfusion. During a dive the possibility exists that
O2 consumption and tissue
perfusion are lower than on the surface, which leads to an effective
increase in these factors on surfacing. After the dive, damage to the
tissue or bubble growth in blood vessels may lead to a reduction in
these factors. In possibility 2 we include the process of CCE, wherein gas diffuses between parallel blood
vessels, leading to an effective lengthening of the tissue time
constants for the gases. The amount of gas exchanged between the
vessels will be greater for the more diffusible gases.
Figures 5 and 6
show simulation results for bubbles evolving in spinal white matter and
in adipose tissue with the assumption of a 90% reduction in
O2 consumption rate that occurs at
the same time as the gas switch. For the switch to pure
O2, a transient increase in bubble
radius is observed, similar to that seen experimentally. Although lower
O2 consumption reproduces some
aspects of bubble evolution seen experimentally, the qualitative
aspects of bubble evolution (i.e., that switching the breathing gas to
heliox is perhaps the least effective of the breathing gas switches in
speeding bubble dissolution in fatty tissue) seem unaffected.
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Fig. 5.
Simulated effect of a breathing gas switch and a 90% reduction in
O2 consumption on evolution of gas
bubbles. In rat spinal white matter,
O2 breathing is still most
effective at resolving bubbles, but reduced
O2 consumption leads to a
transient period of growth in bubble after switch in breathing gases.
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Fig. 6.
In adipose tissue, reduction in O2
consumption delays bubble resolution significantly, even though
O2 switch ultimately leads to most
rapid resolution of bubble.
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Simulation results for bubbles evolving in spinal white matter and
adipose tissue with the assumption of an increase in
O2 consumption on surfacing
(intended to simulate a possible suppression of
O2 consumption during the dive)
indicate that bubble dissolution is speeded slightly, but the
qualitative effects of the different gas switches are unchanged.
Another physiological factor that may be affected by
decompression is the blood supply to the tissue. Changes in
perfusion rate may also affect bubble evolution. Lowering the perfusion rate by 50% at the time of the gas switch slows the rate of
dissolution slightly, and a 50% increase in perfusion speeds the rate
of dissolution slightly, but the qualitative effects of the gas
switches are again unchanged. A combination of increased
O2 consumption and increased
perfusion speeds dissolution over that seen for either increase alone.
In possibility 2, where gas kinetics
are not simply related to the tissue perfusion rate, we will assume
that these effects are due to CCE. [There is evidence from
experimental work and simulation that gas kinetics in tissue are not
single exponential in nature. The experimental work of
Novotny et al. (25) showed that xenon kinetics were not explained by a
model of parallel compartments and that other factors, such as CCE or
mixing between compartments, are important. The simulation work of Himm
et al. (14) proposed that the relative dispersion in tissue transit times in rat muscle may be explained by assuming that there is mixing
of inert gas between compartments. However, because data on the
kinetics of gas washout in all these tissues are not available, we will
use the simpler, single-exponential model.] To simulate the
effects of CCE, the concentration of inert gas seen by the tissue is
taken to be a weighted average of the concentration in the lung and the
concentration in the tissue, as described in Eq. 1. O2 is not
subject to the CCE (15)
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(1)
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where
Pli is concentration of
gas i in the lung,
Ptii is concentration of
gas i in the tissue,
Pai is concentration of
gas i in the blood supplying the
tissue, ceff is
efficiency of the CCE process,
Di is diffusion
coefficient of gas i, and
D0 is reference diffusion rate.
CCE of inert gases has the effect of delaying the washin and washout of
inert gas in the tissue, thereby slowing the bubble evolution. The more
diffusible a gas, the greater is the effective time constant for
washout of the gas. The effect of CCE is illustrated in Figs.
7-10.
Note the effect of the CCE in adipose tissue in Fig. 7. Because air is
used during the dive, CCE slows the washout of the
N2 from the tissue, so the bubble
lifetime is ~50% longer. On switching the breathing gas to
O2, bubbles resolve most rapidly. However, the CCE of inert gases leads to a transient period of growth
of the bubble. The switch to heliox leads to an initially faster rate
of dissolution without CCE. With CCE, heliox breathing leads to a rapid
decrease in size of the bubble initially and then a more gradual rate
at later times, leading to complete resolution of the bubble in a
shorter period of time than required during air breathing. In contrast,
continued air breathing without CCE causes a more rapid resolution of
the bubble.
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Fig. 7.
Simulated effect of highly efficient (0.8) countercurrent exchange
(CCE) on evolution of gas bubbles in adipose tissue. With this high
level of CCE, heliox is initially more effective than
O2 at reducing bubbles and is more
effective than continued breathing of air. Note more pronounced
transient growth of bubble after switch to
O2.
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Fig. 8.
Simulated effect of moderately efficient (0.5) CCE on evolution of gas
bubbles in spinal white matter. With this level of CCE, heliox
breathing is now more effective than air breathing at resolving air
bubbles.
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Fig. 9.
Simulated effect of highly efficient (0.8) CCE on evolution of gas
bubbles in muscle tissue. With this level of CCE, heliox breathing is
initially as effective as O2 at
reducing bubbles.
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Fig. 10.
Simulated effect of highly efficient (0.8) CCE on evolution of gas
bubbles in tendon. Similar to bubble evolution in muscle at this level
of CCE, heliox is initially as effective as
O2 and better than continued
breathing of air.
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Figure 8 shows the simulated effect of gas switching on the evolution
of gas bubbles in spinal white matter. The dive simulated is similar to
that for adipose tissue, 4 h at ~70 fsw, with a 20-min decompression.
The breathing gas switch occurs 30 min after surfacing. The
bottom curves show the effects of gas
switching without CCE, and the top
curves, with associated symbols, assume a
ceff = 0.5, which
is a moderate level of CCE. With this moderate level of CCE, heliox
breathing, although not as effective as
O2 breathing, is now more
effective at resolving bubbles than continued air breathing. Again,
this is in contrast to the simulated results without CCE.
Figures 9 and 10 show simulation results of the effect of breathing gas
switches on bubble evolution in muscle and tendon, respectively. For
these curves,
ceff = 0.8, which
is a high degree of CCE. At this high level, a switch to heliox
breathing is initially as effective as a switch to
O2 breathing, which leads to a
very rapid resolution of the bubble. However, at longer times, the rate
of bubble dissolution slows. Continued breathing of air is the least
effective at resolving the bubble. These results are in contrast to the
results without CCE, where the switch to heliox breathing was only
slightly more effective than continued breathing of air.
Conclusions.
Because our model treats each tissue as a single well-mixed
compartment, gas exchange between tissue and blood is probably more
rapid in our model than in an actual tissue. We would expect bubble
evolution in an actual tissue to proceed more slowly than modeled here,
with the qualitative aspects being unaffected by the assumption of
single-exponential kinetics for tissue-blood exchange. Our model
reproduces qualitatively some aspects of the experimental results of
Hyldegaard et al. (16, 17, 19) under the assumption of reduced
O2 consumption or CCE of inert
gases but differs from the experimental results in some important ways, including which gas mixtures are best at resolving bubbles most rapidly. Possible sources of the discrepancies are mechanisms that may
have effects similar to CCE, such as gas exchange with other tissue
through which the arterial blood passes before it reaches the tissue of
interest, or other tissue properties not accounted for here, such as
tissue elasticity. An elastic response of the tissue, as in
Gernhardt's model (9), would tend to increase the internal bubble
pressure as the bubble grows, leading to more rapid dissolution of the
bubble. Further experimental work, perhaps with heliox as the breathing
gas during the dive, might help in determining the source of the discrepancies.
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APPENDIX |
For a single spherical bubble the mole flux of a gas
j away from its surface is
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(2)
|
where
Jj is mole flux
(mol · cm
2 · s1),
kj is external
mass transfer coefficient (cm/s),
Csurfj is concentration of
dissolved gas at the outer bubble surface
(mol/cm3), and
C
j is concentration of
dissolved gas far from the surface
(mol/cm3).
We make the following assumptions:
1) There is equilibrium between
phases across the phase interface.
2) There is no resistance to mass
transport in the gas phase; therefore, the partial pressure of each gas
at the inner bubble surface
(Psurf) equals its bulk value
(Pbub) in the bubble.
3) Henry's law applies to each
dissolved gas; i.e., the concentration of each dissolved gas
(Cdis) is proportional to the
partial pressure of gas with which it would be in equilibrium (Pdis), with the proportionality
factor being a concentration-independent solubility or partition
coefficient (K)
Equation 1 now becomes
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(3)
|
where
Kj is the
tissue-gas partition coefficient
(mol · cm3 · atm1).
Mechanical equilibrium at the phase boundary requires that when the
hydrostatic pressure within the bubble equals the sum of the partial
pressures of all gases in the bubble
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(4)
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where
is ambient hydrostatic pressure
(dyn/cm2) and is surface
tension (dyn/cm).
The above summation is over j, and
partial pressure of H2O
(PH2O) is taken
as constant. The mass balance on the gas inside the bubble (with the
assumption of sphericity) gives
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(5)
|
where
R is bubble radius (cm) and
nbubj is number of
moles of gas j inside the bubble.
Assuming ideal gases, we have
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(5a)
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(5b)
|
which
can be rewritten using Eq. 3 as
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(5c)
|
where
RG is gas law
constant (82.057 atm · cm3 · mol1 · °K1)
and T is absolute temperature
(°K).
Differentiation of Eq. 5c with respect
to time yields
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(6)
|
Equations
2, 4, and 6 are
combined to yield
![<FR><NU>d<IT>R</IT></NU><DE>d<IT>t</IT></DE></FR> = − <FR><NU><IT>R</IT><SUB>G</SUB><IT>T</IT> &Sgr;[<IT>k<SUB>j</SUB> K</IT><SUP>ti-gas</SUP><SUB><IT>j</IT></SUB> (P<SUP>bub</SUP><SUB><IT>j</IT></SUB> − P<SUP>∞</SUP><SUB><IT>j</IT></SUB>)] + <FR><NU><IT>R</IT></NU><DE>3</DE></FR> <FR><NU>d&rgr;</NU><DE>d<IT>t</IT></DE></FR></NU><DE><FENCE>&rgr; − P<SUB>H<SUB>2</SUB>O</SUB> + <FR><NU>4&sfgr;</NU><DE>3<IT>R</IT></DE></FR></FENCE></DE></FR>](/content/vol87/issue4/fulltext/1521/img012.gif)
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(7)
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For
the mass transfer coefficient, we use the result that for a sphere in a
quiescent liquid the dimensionless Sherwood number (Sh) is 2.0 (31), so
for any of the gases j
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(8)
|
or
where
D is the diffusivity
(cm2/s). Substitution into
Eq. 7 results in the final form of the
equation for the rate of change of the bubble radius
![<FR><NU>d<IT>R</IT></NU><DE>d<IT>t</IT></DE></FR> = − <FR><NU><IT>R</IT><SUB>G</SUB><IT>T</IT>&Sgr;[<IT>D<SUB>j</SUB>K</IT><SUB><IT>j</IT></SUB>(P<SUP>bub</SUP><SUB><IT>j</IT></SUB> − P<SUP>∞</SUP><SUB><IT>j</IT></SUB>)] + <FR><NU><IT>R</IT><SUP>2</SUP></NU><DE>3</DE></FR> <FR><NU>d&rgr;</NU><DE>d<IT>t</IT></DE></FR></NU><DE><IT>R</IT><FENCE>&rgr; − P<SUB>H<SUB>2</SUB>O</SUB> + <FR><NU>4&sfgr;</NU><DE>3<IT>R</IT></DE></FR></FENCE></DE></FR>](/content/vol87/issue4/fulltext/1521/img015.gif)
|
(9)
|
An expression for
dPj/dt
is also necessary. To obtain it, first note that a mass balance around the finite closed system shows that
|
(10)
|
where
Vl is the volume of the condensed
phase (cm3). An expression for
dnbubj/dt
is acquired by differentiating Eq. 5a
|
(11)
|
This is substituted into Eq. 10 to yield
|
(12)
|
If the system is open to mass transport, then dissolved gas is
exchanged with the surroundings of the volume element, which in this
case is arterial blood. If the tissue is treated as a well-mixed
compartment with a constant tissue-blood partition coefficient, then
the kinetics of this exchange are single exponential and
Eq. 12 becomes
|
(13)
|
Here,
Partj is the tension of
gas j in arterial blood and
j is the time constant for
exchange of gas j between tissue and
blood, given by
|
(14)
|
where
Kti-bloodj is
tissue-blood partition coefficient for gas
j
{[mol · (cm3
tissue)1] · [mol · (cm3
blood)1]1},
equal to the ratio of the solubility in tissue to the solubility in
blood
(Kti-gasj/Kblood-gasj) and 
is specific rate of perfusion
(cm3
blood · cm3
tissue1 · s1).
For all gases except O2, the
arterial gas tensions in the lung are computed from the breathing gas
composition by applying the usual correction for the presence of water
vapor and CO2. The effects of CCE
are simulated by shifting the arterial gas tensions from the value in
the lung toward the gas tension in the tissue. The magnitude of the
shift is larger for the more diffusible gases and is given by
|
(15)
|
where
0 < 
< 1 is a factor describing the efficiency of the CCE
process for a reference gas with diffusion coefficient
Dref. No CCE is
described by = 0, whereas perfect exchange is given by = 1. The
arterial O2 tension is computed
after West (37). For O2 the
tissue-blood exchange is a bit different, because its solubility in
blood is not concentration independent, and hence the tissue-blood
partition coefficient is not constant. The Hill equation
[P50 = 36 Torr,
n = 2.6 (8), where
P50 is
O2 half-saturation pressure of Hb
and n is Hill's coefficient]
was used to determine the O2
concentration in blood as a function of its partial pressure and was
used to obtain the partition coefficient as a function of
Pbloodox
|
(16)
|
Also,
for O2 there is an additional term
on the right-hand side of Eq. 13 that
accounts for metabolic consumption. Incorporating these refinements
into Eq. 13 as written for
O2 results in
|
(13`)
|
Here, Rox is the rate
of O2 consumption per unit volume
(mol · cm3 · s1)
and is believed to be independent of
O2 tension in adipose and spinal
nerve tissue. Similarly, for CO2
there is a term accounting for metabolic generation
|
(13``)
|
where
RQ is the respiratory quotient (mol
CO2/mol
O2).
To describe the system we still require an independent expression for
dPbubj/dt.
To obtain it, the rate-related Eqs. 2,
4, and 8 are combined
with the mass balance of Eq. 11 to yield the expression
|
(17)
|
Equations
9, 13, and 17 are
independent and define the system. For
n diffusing gases, there are
(2n + 1) equations.
One additional detail will be noted regarding the partitioning of
CO2. The
CO2 gas is equilibrated with the
dissolved CO2 across the phase
interface according to
|
(18)
|
where
CCO2(aq) is the
concentration of unhydrated dissolved
CO2. The unhydrated
CO2 exchanges with carbonic acid,
bicarbonate, and carbonate
|
(19)
|
The equilibrium coefficients are defined as follows
![&agr;<SUB>2</SUB> = <FR><NU>[H<SUB>2</SUB>CO<SUB>3</SUB>]</NU><DE>[CO<SUB>2</SUB>(aq)][H<SUB>2</SUB>O]</DE></FR>](/content/vol87/issue4/fulltext/1521/img032.gif)
|
(20a)
|
![&agr;<SUB>3</SUB> = <FR><NU>[HCO<SUP>−</SUP><SUB>3</SUB>][H<SUP>+</SUP>]</NU><DE>[H<SUB>2</SUB>CO<SUB>3</SUB>]</DE></FR>](/content/vol87/issue4/fulltext/1521/img033.gif)
|
(20b)
|
![&agr;<SUB>4</SUB> = <FR><NU>[CO<SUP>2−</SUP><SUB>3</SUB>][H<SUP>+</SUP>]</NU><DE>[HCO<SUP>−</SUP><SUB>3</SUB>]</DE></FR>](/content/vol87/issue4/fulltext/1521/img034.gif)
|
(20c)
|
where
the square brackets denote molar concentrations (mol/l). The fraction
(frac) of the total dissolved CO2
that is undissociated at equilibrium is found to be
![frac = <FR><NU>1</NU><DE>1 + &agr;<SUB>2</SUB>[H<SUB>2</SUB>O] + &agr;<SUB>2</SUB>&agr;<SUB>3</SUB> <FR><NU>[H<SUB>2</SUB>O]</NU><DE>[H<SUP>+</SUP>]</DE></FR> + &agr;<SUB>2</SUB>&agr;<SUB>3</SUB>&agr;<SUB>4</SUB> <FR><NU>[H<SUB>2</SUB>O]</NU><DE>[H<SUP>+</SUP>]<SUP>2</SUP></DE></FR></DE></FR>](/content/vol87/issue4/fulltext/1521/img035.gif)
|
(21)
|
In the calculations for CO2 it is
assumed that the unhydrated form is equilibrated with the other forms,
and the total dissolved CO2 is
quantified by using "effective" values of the tissue-gas and
blood-gas partition coefficients to account for the fact that only a
fraction of the dissolved CO2
shows up as CO2(aq), as follows
|
(22)
|
The
molar concentration of protons at physiological pH is of course
At pH 7.4 it is calculated that frac has a value of only
0.0016. Consequently, the effective tissue-gas partition coefficient is
high enough to make the tissue an efficient sink/source of CO2, and it is found that the
tension of dissolved CO2 is
essentially a constant 0.0461 ATA.
| |
ACKNOWLEDGEMENTS |
The authors are grateful to Dr. G. Albin for invaluable scientific
contribution to this paper.
| |
FOOTNOTES |
This work was supported by Naval Medical Research and Development
Command Work Unit 62233N MM33P30.004-1509.
The opinions and assertions contained herein are the private ones of
the authors and are not to be construed as official or reflecting the
views of the US Navy or the naval service at large.
Address for reprint requests and other correspondence: J. F. Himm, Code
31, Diving and Environmental Physiology Dept., Naval Medical Research
Institute, 8901 Wisconsin Ave., Bethesda, MD 20889-5607 (E-mail: himmj{at}nmripo.nmri.navy.mil).
Received 19 February 1997; accepted in final form 14 May 1999.
| |
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TOP
ABSTRACT
INTRODUCTION
![+ <FR><NU>ϝ</NU><DE><IT>K</IT><SUP>ti-gas</SUP><SUB>ox</SUB></DE></FR> [<IT>K</IT><SUP>blood-gas</SUP><SUB>ox</SUB>(P<SUP>art</SUP><SUB>ox</SUB>)P<SUP>art</SUP><SUB>ox</SUB> − <IT>K</IT><SUP>blood-gas</SUP><SUB>ox</SUB>(P<SUP>∞</SUP><SUB>ox</SUB>)P<SUP>∞</SUP><SUB>ox</SUB>]](/content/vol87/issue4/fulltext/1521/img025.gif)
![[H<SUP>+</SUP>] = 10<SUP>−7.4</SUP>](/content/vol87/issue4/fulltext/1521/img037.gif)
