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1 Wyle Laboratories, Houston, Texas 77058; 2 Department of Anesthesiology, Duke University Medical Center, Durham, North Carolina 27710; and 3 Environmental Physiology Laboratory, NASA Johnson Space Center, Houston, Texas 77058
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ABSTRACT |
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 Top
 Abstract
 Introduction
Background
Results
Discussion
References
Appendix A
Appendix B
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Mathematical models of bubble evolution in tissue have recently been incorporated into risk functions for predicting the incidence of decompression sickness (DCS) in human subjects after diving and/or flying exposures. Bubble dynamics models suitable for these applications assume the bubble to be either contained in an unstirred tissue (two-region model) or surrounded by a boundary layer within a well-stirred tissue (three-region model). The contrasting premises regarding the bubble-tissue system lead to different expressions for bubble dynamics described in terms of ordinary differential equations. However, the expressions are shown to be structurally similar with differences only in the definitions of certain parameters that can be transformed to make the models equivalent at large tissue volumes. It is also shown that the two-region model is applicable only to bubble evolution in tissues of infinite extent and cannot be readily applied to bubble evolution in finite tissue volumes to simulate how such evolution is influenced by interactions among multiple bubbles in a given tissue. Two-region models that are incorrectly applied in such cases yield results that may be reinterpreted in terms of their three-region model equivalents but only if the parameters in the two-region model transform into consistent values in the three-region model. When such transforms yield inconsistent parameter values for the three-region model, results may be qualitatively correct but are in substantial quantitative error. Obviation of these errors through appropriate use of the different models may improve performance of probabilistic models of DCS occurrence that express DCS risk in terms of simulated in vivo gas and bubble dynamics.
decompression sickness; perfusion; boundary layer
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INTRODUCTION |
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Top
Abstract
Introduction
Background
Results
Discussion
References
Appendix A
Appendix B
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DIFFUSION AND PERFUSION processes thought to govern extravascular gas bubble growth and resolution in tissues have been modeled in terms of ordinary differential equations (ODEs) for various studies of gas bubble behavior in animals and humans. Such models have recently been used in probabilistic treatments of the occurrence of decompression sickness (DCS) in humans (5, 6, 14). These applications entail rigorous determination of model parameter values, such as gas solubilities and diffusivities, that force model behavior into closest possible conformance to observed DCS incidences and times of DCS occurrence in large and heterogeneous "training" data sets. The procedure involved is highly computation intensive and can be undertaken only with bubble dynamics models that can be applied with a minimum of computational overhead to assess DCS risk accumulation during complex pressure and breathing gas profiles.
Bubble dynamics models suitable for these applications fall into one of two classes on the basis of different conceptualizations of the tissue surrounding the gas bubble. In the first model class, the bubble is immersed in a well-stirred tissue compartment but is immediately surrounded by a well-defined boundary layer through which diffusion-limited exchange of gas between bubble and tissue occurs. The model developed by Gernhardt (4) is a typical example of these "three-region" models, which consist of bubble, boundary layer, and tissue regions. In the other model class, the bubble is immersed in an unstirred tissue compartment, and gas exchange between bubble and tissue is limited by bulk diffusion through the tissue. The model developed originally by Van Liew and Hlastala (18) and later elaborated by Hlastala and Van Liew (7), Van Liew (16), and Van Liew and Burkard (2, 17) is typical of these "two-region" models, which consist of only bubble and tissue regions.
Models in either class accommodate the influences of diffusion and surface tension on bubble growth in a physiologically perfused medium and can be readily extended to include effects of tissue elasticity. Their contrasting conceptualizations of the bubble-tissue system yield different equations for the rate of change of bubble radius as a function of time. Although these equations are structurally similar, with differences only in the definitions of certain parameters, the different assumptions under which they are derived impose important limitations on application of the models. Improper application of the models violating these limitations can lead to quantitative model behavior that is inappropriate for the values of the model parameters used. The purpose of this paper is to illuminate these issues and their implications in applying the two classes of models to physiological decompression problems.
Glossary
Atmospheres absolute (1 ATA = 1,013 kPa = 1.013 × 106 dyn/cm2)
b
Solubility of gas in blood (ml gas/ml tissue-ATA)
t
Solubility of gas in tissue (ml gas/ml tissue-ATA)
t,m
Solubility of mth diffusible gas in tissue (ml gas/ml tissue-ATA)
Coefficient associated with the sink term of the diffusion equation
(µm
1) (1 µm = 106 m)
Surface tension (dyn/cm)
Tissue time constant (min)
Ai
Bubble surface area (µm2)
c
Tissue gas concentration (mol/ml)
ci
Gas concentration at the inner surface of boundary layer (mol/ml)
co
Gas concentration at the outer surface of boundary layer (mol/ml)
Db
Diffusion constant of gas associated with boundary layer (cm2/min)
Dt
Diffusion constant of gas in tissue (cm2/min)
Dx,m
Diffusion constant of mth diffusible gas, in tissue or associated with boundary layer (cm2/min)
h
Boundary layer thickness (µm)
he
"Effective" boundary layer thickness (µm)
j
Number of diffusible gases in tissue
k
Total number of all gases in tissue, including solvent vapor
m, n
Dummy summation indexes
M
Tissue modulus of elasticity (dyn/cm2/µm3)
Ni,D
Molar quantity of all diffusible gases in bubble at the end of each integration step
Ni,m
Molar quantity of mth diffusible gas in bubble at the end of each integration step
P
Local tissue gas tension (ATA)
Pm
Local tissue gas tension for mth diffusible gas (ATA)
Pa
Arterial gas tension (ATA)
Pv
Venous gas tension (ATA)
Pamb
Ambient pressure (ATA)
Pi
Gas pressure in bubble (ATA)
Pt
Tissue gas tension far away from bubble (ATA)
Pt,m
Tissue tension of mth diffusible gas far away from bubble (ATA)
Pi,D
Sum of diffusible gas partial pressures in bubble (ATA)
Pi,m
Partial pressure of mth diffusible gas in bubble (ATA)
Pi,n
Partial pressure of nth infinitely diffusible gas in bubble (ATA)
Blood flow per unit volume of tissue (min1)
r
Radial distance from the center of bubble (µm)
ri
Inner radius of boundary layer (µm)
ro
Outer radius of boundary layer (µm)
r
Outer radius of tissue (µm)
RT
Product of gas constant and temperature (in units to express solubility in ml gas/ml tissue-ATA)
s
Dummy variable of integration in time (min)
t
Time (min)
t
Integration step size (min)
Vi
Bubble volume (µm3)
Vt
Tissue volume (µm3)
X
Product of PiVi (ATA-µm3)
P0i,mV0i
Initial value of Pi,mVi for mth diffusible gas at each integration step (ATA-µm3)
(Pi,mVi)
Change in Pi,mVi for mth diffusible gas at each integration step (ATA-µm3)
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BACKGROUND |
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Top
Abstract
Introduction
Background
Results
Discussion
References
Appendix A
Appendix B
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The basic equations for either model class are the diffusion equation, which describes diffusion of gas through tissue; the Fick equation, which allows calculation of the gas flux through the bubble surface; and the mass balance equation, which determines tissue gas tension. Gases are considered to be ideal. For simplicity, we will neglect solvent vapor pressure and consider cases involving only a single diffusible gas. Generalization of the results to more complex cases is presented in APPENDIX A.
The Diffusion Equation
Neglecting convection due to bubble movement, gas diffusion through the tissue without sources or sinks is described by
|
(1) |
|
(2) |
The Fick Equation
The rate of change of molar concentration of gas in the bubble equals the molar flux of gas through the bubble surface. Thus
|
(3) |
/3)r3i and
Ai = 4r2i are the volume and
surface area of the bubble, respectively.
Effects of surface tension at the gas-liquid interface of the bubble are incorporated through use of the Laplace equation, which, neglecting tissue viscoelastic effects, is
|
(4) |
is the
gas-liquid surface tension.
The Mass Balance Equation
The rate of change of the dissolved gas tension Pt in the tissue at large distances from the bubble is derived from mass balance considerations, assuming equilibration of tissue gas with venous blood gas. The rate of gas uptake by the tissue is the amount carried by the blood per unit time less the flux into the gas bubble. Thus
|
(5a) |
is blood flow per unit tissue volume,
Pa is gas partial pressure in arterial blood,
Vt is the tissue volume, and t and
b are the gas solubilities in tissue and blood,
respectively (in moles per unit volume per unit pressure). For a tissue
of infinite volume, division by Vt reduces Eq. 5a
to
|
(5b) |
t/(b) = is
the time constant associated with blood-tissue gas exchange. The tissue
half-time is (ln 2) or 0.693.
Equations 2, 3, and 5 are solved with appropriate boundary conditions to obtain expressions for bubble growth or resolution in tissues according to the presence or absence of gas supersaturation. The equations are coupled and can be solved only numerically. Although numerical solutions are feasible for a given set of parameter values, excessive computational requirements preclude their use in application to DCS studies involving parameter optimization about large training data sets. Therefore, simplifying assumptions are made to obtain expressions of bubble growth that are easier to handle computationally. We examine below simplifications made in the three-region and two-region models to derive ODEs for describing gas bubble dynamics.
Model Descriptions
Three-region model.
The three-region model considers the gas bubble to be covered by an
unstirred boundary layer of constant and uniform thickness within a
well-stirred tissue mass. The uniform gas tension in the tissue outside
the boundary layer is determined from Eq. 5a, which allows the
gas partial pressure in afferent arterial blood to vary with changes in
ambient pressure or breathing gas. The concentration gradient across
the boundary layer is obtained from Eq. 2 with the quasi-static
approximation2 (9); that is, by
ignoring the time-dependent term 
c /t. Setting
c/t = 0, the diffusion Eq. 2 becomes
|
(6) |
c/r
|
(7a) |
|
(7b) |
ri) is the
thickness of the boundary layer. Notice that, when multiplied by the
appropriate surface areas, these expressions give the same flux through
the boundary layer at ri and
ro. Substituting ro = ri + h into Eq. 7a yields
|
(8) |
tPi
and co = tPt. The desired ODE for the rate of change of bubble radius is obtained from Eq. 3 by expanding the differential on the left side by using Eq. 4 (see APPENDIX A), substituting for c/r on the
right side using Eq. 8, expressing Vi and
Ai in terms of ri, and
converting concentrations into partial pressures. We obtain
|
(9) |
, allowing diffusion through
the entire tissue mass, and with constant Pamb, Eq. 9 reduces to the quasi-static form of the solution given by Epstein
and Plesset (3) for bubble evolution in isobaric, unperfused media.
Also, with no gas flux through the boundary layer
(Db = 0), Eq. 9 reduces to the
differential equation for Boyle's law effects on a spherical bubble
with surface tension.
With large ri, the factor
(1/h + 1/ri) in the numerator of
Eq. 9 reduces to (1/h), yielding the expression for
dri/dt derived by Gernhardt (4) under the
assumption that ri 
ro.
Also, under this assumption, the expressions for the concentration
gradients in both Eqs. 7a and 7b reduce to
(co ci)/h. Equation 8
reduces to a similar expression,
(co ci)/he, without
the assumption ri ro if we
define an "effective" boundary layer thickness
he, which varies as a function of
ri
|
(10) |
c/t term, developed by Tikuisis (13). As would be
expected, their expression for the concentration gradient derived
without the quasi-static approximation is very complex, involving both
r and t as independent variables and infinite sums. A
simpler formulation similar to that outlined here has also been used by
Kunkle (10) to study bubble growth in fluids and by Kunkle and Beckman
(11) to calculate bubble dissolution times after recompression.
Two-region model. The bubble in the two-region model lacks a boundary layer and is immersed in an unstirred but uniformly perfused tissue to permit absorption or release of gas by the blood at every point in the tissue. Because the tissue is unstirred, the tissue inert gas tension P is nonuniform and depends on the radial distance r. The effects of perfusion are accommodated by adding a sink (or source) term to Eq. 2. In defining this term, the bubble and the tissue, with its embedded sink, must be recognized as the only two regions in the model. Capillaries, with associated arterial and venous gas tensions, are included in the tissue per se and do not explicitly exist. In the absence of bubbles, there is no gas diffusion through the tissue and the tissue gas tension Pt is determined by Eq. 5b. The presence of a bubble alters this tension in the vicinity of the bubble, leading to a gas loss or gain that is proportional to the difference between the prevailing local gas tension P and the tissue gas tension Pt that would exist in the absence of the bubble (Fig. 1). The difference is largest at the bubble-tissue interface, and approaches zero far away from the bubble. With the added sink term, Eq. 2 becomes
|
(11a) |
tP. Dividing by tDt,
we get
|
(11b) |
2 = b/tDt = 1/Dt. Note that the term
2(P Pt) acts as a sink if P > Pt and as a source if P < Pt.
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P/r = 0 as r . The first condition is the
same as in the three-region model. Under the second condition, Pt is defined as the tissue tension too far from the bubble
to be influenced by the bubble. Thus it is governed only by perfusion as described by Eq. 5b. The solution to Eq. 11b is
obtained under the quasi-static approximation, i.e., neglecting the
time-dependent term P/t. The general solution contains
both positive and negative exponential terms. However, the positive
exponential term vanishes due to the boundary condition at , leading
to the following solution at any time, t (18)
|
(12) |
|
(13) |
tP and Eq. 13 into the result to
obtain the following ODE for the rate of change of bubble radius
|
(14) |
t is expressed in
volume units to absorb the constant RT. Note that Eq. 14, like
Eq. 9 for the three-region model, includes the effects of
surface tension and changing hydrostatic pressure (Boyle's law
effects) on bubble evolution. With constant Pamb and
= 0 (and hence = 0), Eq. 14 also
reduces to the quasi-static form of the expression given by Epstein and
Plesset (3) for bubble evolution in isobaric, unperfused media.
Equation 14 differs from the expression originally derived by
Van Liew and Hlastala (18) for the two-region model. These workers
wrote the diffusion equation with Pa in place of
Pt in the sink term and, neglecting the effects of surface
tension and changing hydrostatic pressure, obtained a solution under
the boundary condition P = Pa as r . As a
result, their solution differs from our Eq. 14, with
Pa substituted for Pt, the denominator replaced by Pi (and hence larger by 2/3ri)
and no term involving dPamb/dt in the numerator.
Recent applications of the two-region model have retained the larger
denominator from the original derivation and, hence, do not fully
incorporate the effects of surface tension on
dri/dt (2, 5, 6, 14, 16, 17).
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RESULTS |
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Top
Abstract
Introduction
Background
Results
Discussion
References
Appendix A
Appendix B
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The dynamics of bubble growth after a decompression from sea level to
altitude were computed by using each of the models with the parameter
values shown in Table 1. Because comparison
of model results is meaningful only for large tissue volumes, Eq. 5b was used in both cases, i.e., by using Eqs. 9 and 5b for the three-region model and Eqs. 14 and 5b for the two-region model. The altitude profile consisted of
30 min of 100% oxygen prebreathe at sea-level pressure (1 ATA)
followed by ascent at 5,000 ft/min to an indefinitely long residence at
25,000 ft breathing pure oxygen. The diffusion constants
Db and Dt were assumed to be
the same. The parameters and were defined to yield a
tissue half-time of 360 min.
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Table 2 shows the maximum radius and the time to maximum
radius of the bubble computed by using the two models for different values of the initial radius. Both the minimum radius for bubble growth
and the maximum radius reached are smaller for the two-region model
because of the much larger value of 1/ (250 µm) relative to
h (3 µm). Differences between the two models in the times to attain maximum radius decrease with increasing initial bubble size.
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Figure 2 shows the effect of decreasing tissue volume on
bubble growth and gas tension Pt in the well-stirred region
of the three-region model. For these calculations, bubbles grew from nuclei of 10-µm radius that were assumed to be ever present in the
tissue. Bubble growth begins to exert a significant effect on the
tissue inert gas tension if the tissue volume does not exceed the
maximum bubble volume by more than a factor of ~5 × 104. With larger tissue volumes, Pt can be
computed by using Eq. 5b for mass balance with infinite tissue
volume rather than the exact Eq. 5a for the finite
tissue.
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DISCUSSION |
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Top
Abstract
Introduction
Background
Results
Discussion
References
Appendix A
Appendix B
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In their original derivation of the two-region model, Van Liew and
Hlastala (18) solved the diffusion equation with Pa in place of Pt in the sink term and under the boundary
condition P = Pa as r . As pointed out by
Ball et al. (1), Pa rapidly equilibrates with the inspired
gas and is, hence, practically always less than the bubble pressure
Pi. The sink term with Pa can thus serve as a
physiological source of gas for bubble growth only under extreme and
very short-lived conditions. To overcome this problem, Van Liew and
Hlastala (18) noted without derivation that Pt could be
substituted for Pa in their original solution for
dri/dt. Contrary to the remarks of Ball et
al. (1), we have shown here how this form of the two-region model can
be derived directly from the diffusion equation. Our solution also
includes the previously neglected effects of surface tension and
changing hydrostatic pressure. Hlastala and Van Liew (7) and Meisel et
al. (12) also used the boundary condition P = Pa as
r in deriving complete solutions to the partial
differential equation model including the P/t term in
Eq. 11b. Their solutions are valid under the same constraint
required here [Pa (or Pt) must be
independent of r] but also require that Pa be
independent of t. Such solutions are, therefore, of limited
utility in practical physiological decompression problems.
Unfortunately, neither Pa nor Pt in the definition of the sink term in Eq. 11a corresponds to a readily conceptualized physical model. This is because the model implies that a gas tension equal to the chosen value is present everywhere in the tissue. This implication underscores the much larger scale involved in the two-region model compared with that of the three-region model. The two-region model encompasses a very large volume compared with intercapillary dimensions, whereas the three-region model encompasses only the volume between nearest-neighbor capillaries. Choice of either Pt or Pa in the definition of the sink term in the two-region model is, therefore, arbitrary but only use of Pt allows consideration of bubble growth with gas washout after decompression.
The two-region model as formulated here is seriously limited by its
applicability only to a bubble in a tissue of infinite extent. This
limitation and its implications become clear if we examine the nature
of the sink for gas diffusing out of the bubble and the maintenance of
mass balance in the system. The rate of gas uptake or release by the
perfusate at any point in the tissue (r 
ri) is determined by the blood
flow and the local arterial-venous (A-V) gas tension difference and is
given by
b(Pa Pv),
where Pv is the inert gas partial pressure in venous blood.
Under the assumption that gas exchange between tissue and blood is
perfusion limited, Pv is equal to the prevailing local
tissue tension P. Therefore, the local rate of gas transport due to
perfusion is equal to
b(Pa P), which can be
expanded as
|
(15) |
Substituting for P from Eq. 12, we obtain the following expression for the r dependence of this transport rate
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(16) |
Equation 16 shows how gas transport by perfusion varies in the
tissue when it is not well stirred. The A-V difference is dependent on
r. Tissue-blood gas exchange at all points throughout the
tissue includes a position-independent component given by the first
term on the right side of Eq. 16, which is the same as in
Eq. 5b for a tissue of infinite extent. As shown in Fig. 1,
tissue-blood gas exchange is modified by a diffusion-limited
contribution from bubble-tissue gas exchange, which is largest close to
the bubble and vanishingly small as r . This
contribution is given by the second term on the right side of Eq. 16, which is seen by comparison to Eq. 15 to equal the sink
term in Eq. 11a. All gas losses or gains by the bubble,
therefore, occur entirely through the sink term in the diffusion
equation. It is erroneous to include an additional accounting for these
losses or gains via a d(PiVi)/dt term
in the ancillary mass-balance equation for Pt.
Consequently, Eq. 5a cannot be used to incorporate the
influence of bubble growth on the tissue dissolved gas tension
Pt in the two-region model, while it is appropriately used
for this purpose in the three-region model. This analytic property of
the three-region model makes it particularly well suited to simulate
how dissolved gas depletion by bubble growth influences the evolution
of one or more bubbles in a given tissue. Modeling of such interactions
in a two-region model is also possible but only by considering the
anisotropy of the diffusion field and explicitly solving the gradient
equation in all directions around each bubble (8), a process that is too tedious for application in probabilistic models of DCS occurrence.
It is clear by comparing Eqs. 9 and 14 that a
two-region model with given and Dt is
equivalent to a three-region model with Db = Dt and h = 1/, provided the tissue
volume Vt in the three-region model is sufficiently high to
render the d(PiVi)/dt term in Eq. 5a negligible. Recall 2 = 1/Dt (see Eq. 11b). Because of the
reciprocal relationship between h and , h increases
with both tissue half-time (= 0.693) and diffusion constant
Dt (= Db) in equivalent two-
and three-region models (Fig. 3). With
tissue half-time ranging from 0.36 to 360 min and
Dt ranging from 1.3 × 108 to 1.3 × 103 cm2/min in a two-region model, the
boundary layer thickness in the equivalent three-region model ranges
from a fraction of a micrometer to several millimeters.
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The high boundary layer volumes corresponding to high values of the
boundary layer thickness limit the extent to which results obtained by
using certain incorrect implementations of the two-region model can be
reconciled by reinterpretation in terms of their corresponding correct
three-region model equivalents. For example, a two-region model was
recently used with Eq. 5a to examine how increasing numbers of
growing bubbles affect the dissolved gas tensions and maximum bubble
volumes in a given volume of supersaturated liquid (17). Such
implementations of the two-region model are incorrect but can be viewed
as inadvertant applications of the three-region model with
Db = Dt, h = 1/, and
finite tissue volumes. However, for this reinterpretation to be valid,
the parameter values in the two-region model must transform into
consistent values of the parameters in the equivalent three-region
model. This does not hold for parameters used in the cited work, where bubble growth was considered in tissues with half-times of 5, 40, and
360 min and an assumed Dt = 0.00132 cm2/min (17). Respective values of h in the
equivalent three-region model are 0.098, 0.276, and 0.828 cm. These
correspond to boundary layer volumes of 0.004, 0.088, and 2.379 cm3 around bubbles of 2-µm radius, the assumed initial
size of the bubbles (17). Because these volumes increase as the bubbles grow from initial size and can never exceed the total liquid volume of
the tissue, they are too large to allow consideration of bubble number
densities of several hundred or more bubbles per cubic centimeter of
liquid, as was attempted. In fact, the boundary layer volume for the
initial-size bubble in the 360-min half-time tissue is too large to
consider a bubble density as high as 1 bubble/cm3 liquid.
With the assumed value of Dt = 0.00132 cm2/min, dissolved gas depletion by growing bubbles plays
only a minimal role at even the highest bubble number densities that can be reasonably considered.
The question remains whether it is possible to obtain acceptable values
of h in the equivalent three-region model by altering only the
value of the diffusion constant, Db. At large
ri, dri/dt is
proportional to Db/h in the three-region
model (1/h + 1/ri 1/h) and
to Dt in the two-region model
( + 1/ri ). Under these conditions, a
two-region model with given Dt is equivalent to
a three-region model with Db/h = Dt, and Vt 
maximum
Vi. Thus lower values of h might be used while
retaining near equivalence of the two models if Db
is decreased with respect to Dt
(Db/Dt < 1). With
Db = 1.32 × 106
cm2/min
(Db/Dt = 103),
values of h in the 5-, 40-, and 360-min half-time tissues of the equivalent three-region model would be reduced to 0.98, 2.76 and
8.28 µm, respectively. These correspond to respective boundary layer
volumes of 7.69 × 1011, 4.18 × 1010
and 4.5 × 109 cm3 around bubbles of 2-µm
radius. Even the largest of these volumes would be small enough to
consider the influence of as many as 105
bubbles/cm3 of host liquid. However, as illustrated in Fig.
4, the 1/ri term in the
expressions for dri/dt is not negligible
with the conditions and parameter values considered here. Figure 4
shows bubble volume vs. time in a 360-min half-time tissue during the
same pressure/respired gas schedule used to generate Fig. 2, as
determined by a two-region model and its three-region model equivalent,
assuming that 1/ri is negligible and
Db = Dt × 103.
The tissue volume of 10 cm3 assumed for the three-region
model was sufficiently large to render the
d(PiVi)/dt term in Eq. 5a
negligible. The volume-time profiles for the two models would be
identical if the 1/ri term contributed negligibly
to the time course of bubble evolution in the two models, but this is
clearly not the case. The maximum bubble volume achieved in the
two-region model is more than three orders of magnitude higher than
that achieved in the three-region model. This large difference arises
from a strong dependence of the solution of the nonlinear differential
equation for dri/dt on the initial bubble
radius when this radius is comparable to h or 1/. Thus, when
the growth or resolution of the relatively small bubbles that are
thought to cause DCS is considered, the equivalence of two- and
three-region models at large tissue volumes holds only if
Db = Dt. If h assumes
inconsistent values under this condition, results obtained from
incorrect implementations of the two-region model cannot be
quantitatively reconciled with those from properly applied three-region
model equivalents.
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As illustrated in Fig. 3, two- and three-region models can be
formulated that retain equivalence at Vt maximum
Vi and with lower values of h across the
physiological range of tissue half-times by assuming lower values of
Db = Dt. The three-region
models of such equivalent pairs can then be exercised by using
successively smaller values of Vt to correctly examine the
influences of competition between bubbles and blood for dissolved gas.
Viewed in this fashion, results from the three-region model for large
Vt in Fig. 2 could be modeled by using a two-region model
with all parameters but the diffusivity Dt
unchanged. Model equivalence in this case requires Dt = 1.73 × 1010
cm2/min, or a value more than six orders of magnitude
smaller than used in earlier work (17). Because this diffusivity is
more than three orders of magnitude smaller than the diffusivity used to obtain the illustrated results, further examination of the influence
of decreasing Vt in the three-region model would yield results considerably different from those illustrated. The combination of more reasonable values for Db and h used
to obtain the illustrated results are possible because, unlike in the
two-region model, these parameters vary independently of the tissue
half-time in the three-region model. Thus, as shown in Fig. 2,
essential qualitative features of the interactions between increasing
numbers of bubbles in a given tissue volume are evident as described in
the above-cited work but at values of the parameters much different
from those seemingly allowed with incorrect application of the
two-region model.
In summary, the two- and three-region models are structurally similar and can be made equivalent at large tissue volumes through appropriate transformation of certain parameters. Whereas the three-region model can be applied to bubble growth in small tissue volumes, the two-region model is readily applied only to bubble growth in tissues of infinite extent. Two-region models that are incorrectly applied to problems involving bubble growth in finite tissue volumes yield results that may be reinterpreted in terms of their three-region model equivalents, if the parameters in the two-region model transform into consistent values in the three-region model. When such transforms yield inconsistent values for the three-region model, the two-region model results may be qualitatively correct but are in substantial quantitative error. Obviation of these errors through appropriate use of the different models may improve performance of probabilistic models of DCS occurrence that express DCS risk in terms of simulated in vivo gas and bubble dynamics (5, 6).
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APPENDIX A |
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Top
Abstract
Introduction
Background
Results
Discussion
References
Appendix A
Appendix B
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Generalization to Multiple Diffusible Gases
The simplified models presented here are readily generalized to accommodate multiple diffusible gases, tissue viscoelastic effects, and the influences of gases such as water vapor that can be presumed to be always in equilibrium between bubble and surroundings (5). In the following, the term "diffusible gas" will denote a gas with finite diffusivity in the liquid, to distinguish such gases from infinitely diffusible gases that are always in equilibrium between bubble and surroundings. In a mixture of j diffusible gases in k > j gases, with gases j + 1, j + 2, ... , k always in equilibrium between bubble and surroundings, the sum of the diffusible gas partial pressures in the bubble, Pi,D, is given by elaborating Eq. 4
|
(A1) |
The total flux of diffusible gases across the bubble surface is given as before by the Fick equation
|
(A2) |
t,m is the solubility of
diffusible gas m in tissue;
Dx,m is the diffusivity of diffusible
gas m in either the tissue or boundary layer, depending on the
model; and
(dPm/dr)r=ri
is the partial pressure gradient of diffusible gas m
at the bubble surface.
By expanding the differential on the left side of Eq. A2 and by using Eq. A1, with the assumption that the infinitely diffusible gases remain at constant partial pressure in the bubble, we obtain
|
|
(A3) |
By equating the right sides of Eqs. A2 and A3,
substituting Vi = (4/3)r3i and
Ai = 4r2i, and using Eq. A1 to replace Pi,D, we get the following expression for dri/dt
|
(A4) |
The change in bubble radius with time is obtained by integrating Eq. A4 numerically by using the expression for the partial pressure gradient for each gas at the bubble surface appropriate to the model. For the three-region model, the appropriate expression is given by Eq. 8 with c, ci, and co replaced by Pm, Pi,m, and Pt,m, respectively, where Pt,m is the tissue tension for the diffusible gas denoted by m as given by Eq. 5a or Eq. 5b for that gas. For the two-region model, the appropriate expression is given by Eq. 13, with Pi and Pt replaced by Pi,m and Pt,m, respectively. In this case, Pt,m is given by Eq. 5b, as discussed in the text (see DISCUSSION).
The partial pressure of each diffusible gas in the bubble, Pi,m, at the end of each integration step is computed as follows by using the Dalton's law of partial pressures
![P<SUB>i,<IT>m</IT></SUB> = P<SUB>i,<IT>D</IT></SUB> <FENCE><FR><NU><IT>N</IT><SUB>i,<IT>m</IT></SUB></NU><DE><IT>N</IT><SUB>i,<IT>D</IT></SUB></DE></FR></FENCE> = P<SUB>i,<IT>D</IT></SUB> <FENCE><FR><NU>P<SUP>0</SUP><SUB>i,<IT>m</IT></SUB>V<SUP>0</SUP><SUB>i</SUB> + &Dgr;(P<SUB>i,<IT>m</IT></SUB>V<SUB>i</SUB>)</NU><DE><LIM><OP>∑</OP><LL><IT>m</IT>=1</LL><UL><IT>j</IT></UL></LIM> [P<SUP>0</SUP><SUB>i,<IT>m</IT></SUB>V<SUP>0</SUP><SUB>i</SUB> + &Dgr;(P<SUB>i,<IT>m</IT></SUB>V<SUB>i</SUB>)]</DE></FR></FENCE>](/content/vol86/issue2/fulltext/732/img027.gif)
|
(A5) |
(Pi,mVi) is the change in
Pi,mVi product over the integration interval, obtained by integrating Eq. A2.
Equation A4 provides a comprehensive description of bubble dynamics for either a two-region or a three-region model, including the effects of bubble-tissue gas fluxes, surface tension, tissue elasticity, and changes in ambient hydrostatic pressure (Boyle's law effects). The two-region model version of this equation differs from that described by Burkard and Van Liew (2), because they used an expression for dri/dt based on the original single-gas derivation (18) in which effects of surface tension, tissue elasticity, and changing hydrostatic pressure were neglected. Their separate equation for Boyle's law effects with iterative approximation to find Vi and Pi,m in each time step (2) is not required with the present formulation.
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APPENDIX B |
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Top
Abstract
Introduction
Background
Results
Discussion
References
Appendix A
Appendix B
|
|---|
Mass Balance in the Three-Region Model With a Finite Tissue Volume
We assume that there is no gas exchange or blood flow in the boundary layer region and that the tissue volume including the boundary layer remains constant. Expressing the rate of change of gas content in moles per unit time, we have
|
|
is the outer radius of a
spherically shaped tissue, and
|
![= <LIM><OP>∫</OP><LL><IT>r</IT><SUB>o</SUB></LL><UL><IT>r</IT><SUB>∞</SUB></UL></LIM> &agr;<SUB>b</SUB><A><AC>Q</AC><AC>˙</AC></A>[P<SUB>a</SUB> − P(<IT>r</IT>, <IT>t</IT>)]4&pgr;<IT>r</IT><SUP> 2</SUP>d<IT>r</IT>](/content/vol86/issue2/fulltext/732/img031.gif)
|
Mass balance implies that, in any given time interval, the sum of changes in bubble and tissue gas contents equals the amout of gas transported by perfusion. Therefore
|
![= <LIM><OP>∫</OP><LL><IT>r</IT><SUB>o</SUB></LL><UL><IT>r</IT><SUB>∞</SUB></UL></LIM> &agr;<SUB>b</SUB><A><AC>Q</AC><AC>˙</AC></A>[P<SUB>a</SUB> − P(<IT>r</IT>, <IT>t</IT>)]4&pgr;<IT>r</IT><SUP>2</SUP>d<IT>r</IT>](/content/vol86/issue2/fulltext/732/img033.gif)
|
(B1) |
t and
b are expressed in appropriate units to include the
factor RT. With t thus expressed, we use Vi = (4/3)r3i and Eqs.
4 and 9 to obtain
|
(B2) |
Also, P(r, t) = Pt(t), independent of r, and therefore
|
|
(B3) |
/3)r3o is the volume
of the bubble including the boundary layer. Vt is the total
volume of tissue including the boundary layer region. Note that the
tissue need not be spherical in shape; in other words, the outer radius r is irrelevant. Using Eq. B3, we
obtain
|
|
(B4) |
![<LIM><OP>∫</OP><LL><IT>r</IT><SUB>o</SUB></LL><UL><IT>r</IT><SUB>∞</SUB></UL></LIM> &agr;<SUB>b</SUB><A><AC>Q</AC><AC>˙</AC></A>[P<SUB>a</SUB> − P(<IT>r</IT>, <IT>t</IT>)]4&pgr;<IT>r</IT><SUP>2</SUP>d<IT>r</IT>](/content/vol86/issue2/fulltext/732/img039.gif)
|
|
(B5) |
Equating the right sides of Eqs. B2 and B4 to the right side of Eq. B5 (according to Eq. B1) yields the following mass balance equation for the three-region model
|
|
(B6) |
The first term within brackets on the right side of the above equation
is due to gas flux through the bubble surface given by Eq. B2.
This term is the same as the last term on the right side of Eq. 5a, except for the solubility factor t (including the factor RT), which is canceled out by division. The second term
within brackets involving dri/dt arises by
differentiating (Vi Vo) and
substituting dro/dt = dri/dt (see Eq. B4). Our
simulation results indicate that this term does not significantly
change the maximum bubble radius (<0.33% by using the parameter
values shown in Table 1 and a tissue volume of 106
cm3) and can, therefore, be ignored. Note that as
Vt , the terms in brackets vanish, and Eq. B6 reduces to Eq. 5b.
For a given decompression profile, the time course of changes in bubble
radius is determined by solving Eqs. 9 and B6
simultaneously. We derive below another form of Eq. B6 that
requires slightly less computations in each integration step by solving
the mass balance equation with some approximations. Ignoring the small
difference (Vi Vo),
(Vt + Vi Vo) 
Vt. Then, with use of Eq. B3, Eq. B1 becomes
|
(B7) |
Multiplying both sides of Eq. B7 by et/
(integrating factor) and integrating between t and t + t, we obtain
![P<SUB>t</SUB>(<IT>t</IT> + &Dgr;<IT>t</IT>)<IT>e</IT><SUP>(<IT>t</IT>+&Dgr;<IT>t</IT>)/&tgr;</SUP> − P<SUB>t</SUB>(<IT>t</IT>)<IT>e</IT><SUP><IT>t</IT>/&tgr;</SUP> = P<SUB>a</SUB>[<IT>e</IT><SUP>(<IT>t</IT>+&Dgr;<IT>t</IT>)/&tgr;</SUP> − <IT>e</IT><SUP><IT>t</IT>/&tgr;</SUP>]](/content/vol86/issue2/fulltext/732/img044.gif)
|
|
(B8) |
t is the integration step size and
s is the dummy variable of integration.
The integral in Eq. B8 above can be evaluated by approximating
X(s) by a straight line in the interval
[t, t + t], i.e., by letting
X(s) = X(t) + [X(t + t) X(t)/t](s t)
for t 
s t + t.
Simplifying the result, we get the following expression for tissue
tension at time (t + t)
|
|
(B9) |
Note that as Vt , the last term within brackets on
the right side of the above equation becomes 0, and the solution
reduces to that of Eq. 5b. In applying Eq. B9, we first
solve Eq. 9 for ri assuming Pt
to be constant in the interval
[t, t + t], in accord with the
quasi-static approximation. We then use Eq. B9 to update
Pt. Thus Eq. B9 needs to be used only once during
each integration step. Also, it should be noted that expressions
similar to Eq. B9 for
Pt(t + t) can be derived using
other approximations for X(s) in the interval
(t, t + t) (e.g., exponential).
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ACKNOWLEDGEMENTS |
|---|
The authors gratefully acknowledge Dr. Edward D. Thalmann and Dr. Richard D. Vann for critical reviews of this manuscript.
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FOOTNOTES |
|---|
This work was supported in part by National Aeronautics and Space Administration Cooperative Agreement NCC 9-42 and US Navy contract N0463A-97-M-0126 (to W. A. Gerth).
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. §1734 solely to indicate this fact.
1 Subscript i denotes inner surface and quantities inside the bubble. Subscript o refers to the outer surface of the bubble, when a boundary layer is present.
2 This approximation may not hold during very rapid changes in ambient pressure or breathing gas. In such cases, changes in bubble radius may have to be determined by using the complete diffusion equation.
Address for reprint requests: R. Srini Srinivasan, Wyle Laboratories, 1290 Hercules Dr., Suite 120, Houston, TX 77058-2769.
Received 22 January; accepted in final form 23 September 1998.
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REFERENCES |
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Top
Abstract
Introduction
Background
Results
Discussion
References
Appendix A
Appendix B
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