Abstract
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 (tworegion model) or surrounded by a boundary layer within a wellstirred tissue (threeregion model). The contrasting premises regarding the bubbletissue 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 tworegion 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. Tworegion models that are incorrectly applied in such cases yield results that may be reinterpreted in terms of their threeregion model equivalents but only if the parameters in the tworegion model transform into consistent values in the threeregion model. When such transforms yield inconsistent parameter values for the threeregion 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
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 wellstirred tissue compartment but is immediately surrounded by a welldefined boundary layer through which diffusionlimited exchange of gas between bubble and tissue occurs. The model developed by Gernhardt (4) is a typical example of these “threeregion” 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 “tworegion” 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 bubbletissue 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 × 10^{6} dyn/cm^{2})
α_{b}
Solubility of gas in blood (ml gas/ml tissueATA)
α_{t}
Solubility of gas in tissue (ml gas/ml tissueATA)
α_{t,m}
Solubility of mth diffusible gas in tissue (ml gas/ml tissueATA)
λ
Coefficient associated with the sink term of the diffusion equation (μm^{−1}) (1 μm = 10^{−6} m)
ς
Surface tension (dyn/cm)
τ
Tissue time constant (min)
A _{i}
Bubble surface area (μm^{2})
c
Tissue gas concentration (mol/ml)
c_{i}
Gas concentration at the inner surface of boundary layer (mol/ml)
c_{o}
Gas concentration at the outer surface of boundary layer (mol/ml)
D _{b}
Diffusion constant of gas associated with boundary layer (cm^{2}/min)
D _{t}
Diffusion constant of gas in tissue (cm^{2}/min)
D _{x,m}
Diffusion constant of mth diffusible gas, in tissue or associated with boundary layer (cm^{2}/min)
h
Boundary layer thickness (μm)
h _{e}
“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/cm^{2}/μm^{3})
N _{i,D}
Molar quantity of all diffusible gases in bubble at the end of each integration step
N _{i,m}
Molar quantity of mth diffusible gas in bubble at the end of each integration step
P
Local tissue gas tension (ATA)
P_{m}
Local tissue gas tension for mth diffusible gas (ATA)
P_{a}
Arterial gas tension (ATA)
P_{v}
Venous gas tension (ATA)
P_{amb}
Ambient pressure (ATA)
P_{i}
Gas pressure in bubble (ATA)
P_{t}
Tissue gas tension far away from bubble (ATA)
P_{t,m}
Tissue tension of mth diffusible gas far away from bubble (ATA)
P_{i,D}
Sum of diffusible gas partial pressures in bubble (ATA)
P_{i,m}
Partial pressure of mth diffusible gas in bubble (ATA)
P_{i,n}
Partial pressure of nth infinitely diffusible gas in bubble (ATA)
Q˙
Blood flow per unit volume of tissue (min^{−1})
r
Radial distance from the center of bubble (μm)
r _{i}
Inner radius of boundary layer (μm)
r _{o}
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 tissueATA)
s
Dummy variable of integration in time (min)
t
Time (min)
Δt
Integration step size (min)
V_{i}
Bubble volume (μm^{3})
V_{t}
Tissue volume (μm^{3})
X
Product of P_{i}V_{i} (ATAμm^{3})
Initial value of P_{i,m}V_{i} for mth diffusible gas at each integration step (ATAμm^{3})
Δ(P_{i,m}V_{i})
Change in P_{i,m}V_{i} for mth diffusible gas at each integration step (ATAμm^{3})
BACKGROUND
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 .
The Diffusion Equation
Neglecting convection due to bubble movement, gas diffusion through the tissue without sources or sinks is described by
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
Effects of surface tension at the gasliquid interface of the bubble are incorporated through use of the Laplace equation, which, neglecting tissue viscoelastic effects, is
The Mass Balance Equation
The rate of change of the dissolved gas tension P_{t} 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
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 threeregion and tworegion models to derive ODEs for describing gas bubble dynamics.
Model Descriptions
Threeregion model.
The threeregion model considers the gas bubble to be covered by an unstirred boundary layer of constant and uniform thickness within a wellstirred 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 quasistatic approximation2 (9); that is, by ignoring the timedependent term ∂c /∂t. Setting ∂c/∂t = 0, the diffusion Eq. 2
becomes
To relate concentrations c_{i} and c_{o} to pressures, we assume 1) gas concentrations in the boundary layer follow Henry’s law with instantaneous equilibration of gas tensions at r
_{i} and r
_{o} and2) same solubility of gas in the boundary layer as in the tissue. Then, we have c_{i} = α_{t}P_{i}and c_{o} = α_{t}P_{t}. 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
), substituting for ∂c/∂r on the right side using Eq. 8, expressing V_{i} andA
_{i} in terms of r
_{i}, and converting concentrations into partial pressures. We obtain
With large r
_{i}, the factor (1/h + 1/r
_{i}) in the numerator ofEq. 9
reduces to (1/h), yielding the expression for dr
_{i}/dt derived by Gernhardt (4) under the assumption that r
_{i} ≅ r
_{o}. Also, under this assumption, the expressions for the concentration gradients in both Eqs. 7a
and
7b
reduce to (c_{o} − c_{i})/h. Equation 8
reduces to a similar expression, (c_{o} − c_{i})/h
_{e}, without the assumption r
_{i} ≅ r
_{o} if we define an “effective” boundary layer thicknessh
_{e}, which varies as a function ofr
_{i}
The threeregion model given by Eq. 9 is a simplification of a more complete solution of Eq. 2, including the ∂c/∂t term, developed by Tikuisis (13). As would be expected, their expression for the concentration gradient derived without the quasistatic approximation is very complex, involving bothr 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.
Tworegion model.
The bubble in the tworegion 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 P_{t} 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 P_{t} that would exist in the absence of the bubble (Fig. 1). The difference is largest at the bubbletissue interface, and approaches zero far away from the bubble. With the added sink term, Eq. 2
becomes
Equation 11b
is solved for the following boundary conditions: P = P_{i} at r = r
_{i} and ∂P/∂r = 0 as r → ∞. The first condition is the same as in the threeregion model. Under the second condition, P_{t} 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 quasistatic approximation, i.e., neglecting the timedependent 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)
The pressure gradient at the bubble surface is obtained by differentiating Eq. 12
As before, we expand the differential on the left side of Eq. 3
by using Eq. 4, express V_{i} andA
_{i} in terms of r
_{i}, and substitute c = α_{t}P and Eq. 13
into the result to obtain the following ODE for the rate of change of bubble radius
Equation 14 differs from the expression originally derived by Van Liew and Hlastala (18) for the tworegion model. These workers wrote the diffusion equation with P_{a} in place of P_{t} in the sink term and, neglecting the effects of surface tension and changing hydrostatic pressure, obtained a solution under the boundary condition P = P_{a} as r → ∞. As a result, their solution differs from our Eq. 14, with P_{a} substituted for P_{t}, the denominator replaced by P_{i} (and hence larger by 2ς/3r _{i}) and no term involving dP_{amb}/dt in the numerator. Recent applications of the tworegion model have retained the larger denominator from the original derivation and, hence, do not fully incorporate the effects of surface tension on dr _{i}/dt (2, 5, 6, 14, 16, 17).
RESULTS
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 threeregion model and Eqs. 14 and 5b for the tworegion model. The altitude profile consisted of 30 min of 100% oxygen prebreathe at sealevel 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 constantsD _{b} and D _{t} were assumed to be the same. The parameters Q˙ and λ were defined to yield a tissue halftime of 360 min.
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 tworegion model because of the much larger value of 1/λ (250 μm) relative toh (3 μm). Differences between the two models in the times to attain maximum radius decrease with increasing initial bubble size.
Figure 2 shows the effect of decreasing tissue volume on bubble growth and gas tension P_{t} in the wellstirred region of the threeregion model. For these calculations, bubbles grew from nuclei of 10μm 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 × 10^{4}. With larger tissue volumes, P_{t} can be computed by using Eq. 5b for mass balance with infinite tissue volume rather than the exact Eq. 5a for the finite tissue.
DISCUSSION
In their original derivation of the tworegion model, Van Liew and Hlastala (18) solved the diffusion equation with P_{a} in place of P_{t} in the sink term and under the boundary condition P = P_{a} as r → ∞. As pointed out by Ball et al. (1), P_{a} rapidly equilibrates with the inspired gas and is, hence, practically always less than the bubble pressure P_{i}. The sink term with P_{a} can thus serve as a physiological source of gas for bubble growth only under extreme and very shortlived conditions. To overcome this problem, Van Liew and Hlastala (18) noted without derivation that P_{t} could be substituted for P_{a} in their original solution for dr _{i}/dt. Contrary to the remarks of Ball et al. (1), we have shown here how this form of the tworegion 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 = P_{a} asr → ∞ in deriving complete solutions to the partial differential equation model including the ∂P/∂t term inEq. 11b. Their solutions are valid under the same constraint required here [P_{a} (or P_{t}) must be independent of r] but also require that P_{a} be independent of t. Such solutions are, therefore, of limited utility in practical physiological decompression problems.
Unfortunately, neither P_{a} nor P_{t} 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 tworegion model compared with that of the threeregion model. The tworegion model encompasses a very large volume compared with intercapillary dimensions, whereas the threeregion model encompasses only the volume between nearestneighbor capillaries. Choice of either P_{t} or P_{a} in the definition of the sink term in the tworegion model is, therefore, arbitrary but only use of P_{t} allows consideration of bubble growth with gas washout after decompression.
The tworegion 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 ≥ r
_{i}) is determined by the blood flow and the local arterialvenous (AV) gas tension difference and is given by α_{b}Q˙(P_{a} − P_{v}), where P_{v} is the inert gas partial pressure in venous blood. Under the assumption that gas exchange between tissue and blood is perfusion limited, P_{v} is equal to the prevailing local tissue tension P. Therefore, the local rate of gas transport due to perfusion is equal to α_{b}Q˙(P_{a} − P), which can be expanded as
Substituting for P from Eq. 12, we obtain the following expression for the r dependence of this transport rate
Equation 16 shows how gas transport by perfusion varies in the tissue when it is not well stirred. The AV difference is dependent onr. Tissueblood gas exchange at all points throughout the tissue includes a positionindependent component given by the first term on the right side of Eq. 16, which is the same as inEq. 5b for a tissue of infinite extent. As shown in Fig. 1, tissueblood gas exchange is modified by a diffusionlimited contribution from bubbletissue 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(P_{i}V_{i})/dt term in the ancillary massbalance equation for P_{t}. Consequently, Eq. 5a cannot be used to incorporate the influence of bubble growth on the tissue dissolved gas tension P_{t} in the tworegion model, while it is appropriately used for this purpose in the threeregion model. This analytic property of the threeregion 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 tworegion 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 tworegion model with given λ and D _{t} is equivalent to a threeregion model with D _{b} =D _{t} and h = 1/λ, provided the tissue volume V_{t} in the threeregion model is sufficiently high to render the d(P_{i}V_{i})/dt term in Eq.5a negligible. Recall λ^{2} = 1/τD _{t} (see Eq. 11b ). Because of the reciprocal relationship between h and λ, h increases with both tissue halftime (= 0.693τ) and diffusion constantD _{t} (= D _{b}) in equivalent two and threeregion models (Fig. 3). With tissue halftime ranging from 0.36 to 360 min andD _{t} ranging from 1.3 × 10^{−8} to 1.3 × 10^{−3} cm^{2}/min in a tworegion model, the boundary layer thickness in the equivalent threeregion model ranges from a fraction of a micrometer to several millimeters.
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 tworegion model can be reconciled by reinterpretation in terms of their corresponding correct threeregion model equivalents. For example, a tworegion 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 tworegion model are incorrect but can be viewed as inadvertant applications of the threeregion model withD _{b} = D _{t}, h = 1/λ, and finite tissue volumes. However, for this reinterpretation to be valid, the parameter values in the tworegion model must transform into consistent values of the parameters in the equivalent threeregion model. This does not hold for parameters used in the cited work, where bubble growth was considered in tissues with halftimes of 5, 40, and 360 min and an assumed D _{t} = 0.00132 cm^{2}/min (17). Respective values of h in the equivalent threeregion model are 0.098, 0.276, and 0.828 cm. These correspond to boundary layer volumes of 0.004, 0.088, and 2.379 cm^{3} around bubbles of 2μm 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 initialsize bubble in the 360min halftime tissue is too large to consider a bubble density as high as 1 bubble/cm^{3} liquid. With the assumed value of D _{t} = 0.00132 cm^{2}/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 threeregion model by altering only the value of the diffusion constant, D _{b}. At larger _{i}, dr _{i}/dt is proportional to D _{b}/h in the threeregion model (1/h + 1/r _{i} ≅ 1/h) and to λD _{t} in the tworegion model (λ + 1/r _{i} ≅ λ). Under these conditions, a tworegion model with given λD _{t} is equivalent to a threeregion model with D _{b}/h = λD _{t}, and V_{t} ≫ maximum V_{i}. Thus lower values of h might be used while retaining near equivalence of the two models if D _{b}is decreased with respect to D _{t}(D _{b}/D _{t} < 1). WithD _{b} = 1.32 × 10^{−6}cm^{2}/min (D _{b}/D _{t} = 10^{−3}), values of h in the 5, 40, and 360min halftime tissues of the equivalent threeregion model would be reduced to 0.98, 2.76 and 8.28 μm, respectively. These correspond to respective boundary layer volumes of 7.69 × 10^{−11}, 4.18 × 10^{−10}and 4.5 × 10^{−9} cm^{3} around bubbles of 2μm radius. Even the largest of these volumes would be small enough to consider the influence of as many as 10^{5}bubbles/cm^{3} of host liquid. However, as illustrated in Fig.4, the 1/r _{i} term in the expressions for dr _{i}/dt is not negligible with the conditions and parameter values considered here. Figure 4shows bubble volume vs. time in a 360min halftime tissue during the same pressure/respired gas schedule used to generate Fig. 2, as determined by a tworegion model and its threeregion model equivalent, assuming that 1/r _{i} is negligible andD _{b} = D _{t} × 10^{−3}. The tissue volume of 10 cm^{3} assumed for the threeregion model was sufficiently large to render the d(P_{i}V_{i})/dt term in Eq. 5a negligible. The volumetime profiles for the two models would be identical if the 1/r _{i} 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 tworegion model is more than three orders of magnitude higher than that achieved in the threeregion model. This large difference arises from a strong dependence of the solution of the nonlinear differential equation for dr _{i}/dt on the initial bubble radius when this radius is comparable to h or 1/λ. Thus, when the growth or resolution of the relatively small bubbles that are thought to cause DCS is considered, the equivalence of two and threeregion models at large tissue volumes holds only ifD _{b} = D _{t}. If h assumes inconsistent values under this condition, results obtained from incorrect implementations of the tworegion model cannot be quantitatively reconciled with those from properly applied threeregion model equivalents.
As illustrated in Fig. 3, two and threeregion models can be formulated that retain equivalence at V_{t} ≫ maximum V_{i} and with lower values of h across the physiological range of tissue halftimes by assuming lower values ofD _{b} = D _{t}. The threeregion models of such equivalent pairs can then be exercised by using successively smaller values of V_{t} to correctly examine the influences of competition between bubbles and blood for dissolved gas. Viewed in this fashion, results from the threeregion model for large V_{t} in Fig. 2 could be modeled by using a tworegion model with all parameters but the diffusivity D _{t}unchanged. Model equivalence in this case requiresD _{t} = 1.73 × 10^{−10}cm^{2}/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 V_{t} in the threeregion model would yield results considerably different from those illustrated. The combination of more reasonable values for D _{b} and h used to obtain the illustrated results are possible because, unlike in the tworegion model, these parameters vary independently of the tissue halftime in the threeregion 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 abovecited work but at values of the parameters much different from those seemingly allowed with incorrect application of the tworegion model.
In summary, the two and threeregion models are structurally similar and can be made equivalent at large tissue volumes through appropriate transformation of certain parameters. Whereas the threeregion model can be applied to bubble growth in small tissue volumes, the tworegion model is readily applied only to bubble growth in tissues of infinite extent. Tworegion models that are incorrectly applied to problems involving bubble growth in finite tissue volumes yield results that may be reinterpreted in terms of their threeregion model equivalents, if the parameters in the tworegion model transform into consistent values in the threeregion model. When such transforms yield inconsistent values for the threeregion model, the tworegion 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).
Acknowledgments
The authors gratefully acknowledge Dr. Edward D. Thalmann and Dr. Richard D. Vann for critical reviews of this manuscript.
Footnotes

Address for reprint requests: R. Srini Srinivasan, Wyle Laboratories, 1290 Hercules Dr., Suite 120, Houston, TX 770582769.

This work was supported in part by National Aeronautics and Space Administration Cooperative Agreement NCC 942 and US Navy contract N0463A97M0126 (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.
 Copyright © 1999 the American Physiological Society
Appendix
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 ink > 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, P_{i,D}, is given by elaborating Eq.4
The total flux of diffusible gases across the bubble surface is given as before by the Fick equation
By expanding the differential on the left side of Eq. EA2
and by using Eq. EA1, with the assumption that the infinitely diffusible gases remain at constant partial pressure in the bubble, we obtain
By equating the right sides of Eqs. EA2
and
EA3,substituting V
_{i} = (4π/3)
The change in bubble radius with time is obtained by integratingEq. EA4 numerically by using the expression for the partial pressure gradient for each gas at the bubble surface appropriate to the model. For the threeregion model, the appropriate expression is given by Eq. 8 with c, c_{i}, and c_{o} replaced by P_{m}, P_{i,m}, and P_{t,m}, respectively, where P_{t,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 tworegion model, the appropriate expression is given by Eq.13, with P_{i} and P_{t} replaced by P_{i,m} and P_{t,m}, respectively. In this case, P_{t,m} is given by Eq. 5b, as discussed in the text (see discussion).
The partial pressure of each diffusible gas in the bubble, P_{i,m}, at the end of each integration step is computed as follows by using the Dalton’s law of partial pressures
Equation EA4 provides a comprehensive description of bubble dynamics for either a tworegion or a threeregion model, including the effects of bubbletissue gas fluxes, surface tension, tissue elasticity, and changes in ambient hydrostatic pressure (Boyle’s law effects). The tworegion model version of this equation differs from that described by Burkard and Van Liew (2), because they used an expression for dr _{i}/dt based on the original singlegas 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 V_{i} and P_{i,m} in each time step (2) is not required with the present formulation.
Appendix
Mass Balance in the ThreeRegion 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
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
Also, P(r, t) = P_{t}(t), independent of r, and therefore
Equating the right sides of Eqs. EB2
and
EB4
to the right side of Eq. EB5
(according to Eq. EB1
) yields the following mass balance equation for the threeregion model
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. EB2.This term is the same as the last term on the right side of Eq.5a, except for the solubility factor α_{t} (including the factor RT), which is canceled out by division. The second term within brackets involving dr _{i}/dt arises by differentiating (V_{i} − V_{o}) and substituting dr _{o}/dt = dr _{i}/dt (see Eq. EB4 ). 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 10^{−6}cm^{3}) and can, therefore, be ignored. Note that as V_{t} → ∞, the terms in brackets vanish, and Eq.EB6 reduces to Eq. 5b.
For a given decompression profile, the time course of changes in bubble radius is determined by solving Eqs. 9
and
EB6
simultaneously. We derive below another form of Eq. EB6
that requires slightly less computations in each integration step by solving the mass balance equation with some approximations. Ignoring the small difference (V_{i} − V_{o}), (V_{t} + V_{i} − V_{o}) ≈ V_{t}. Then, with use of Eq. EB3, Eq. EB1
becomes
Multiplying both sides of Eq. EB7
by e^{t/τ}(integrating factor) and integrating between t and t + Δt, we obtain
The integral in Eq. EB8
above can be evaluated by approximatingX(s) by a straight line in the interval [t, t + Δt], i.e., by lettingX(s) = X(t) + [X(t + Δt) − X(t)/Δt](s − t) for t ≤ s ≤ t + Δt. Simplifying the result, we get the following expression for tissue tension at time (t + Δt)
Note that as V_{t} → ∞, 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. EB9, we first solve Eq. 9 for r _{i} assuming P_{t}to be constant in the interval [t, t + Δt], in accord with the quasistatic approximation. We then use Eq. EB9 to update P_{t}. Thus Eq. EB9 needs to be used only once during each integration step. Also, it should be noted that expressions similar to Eq. EB9 for P_{t}(t + Δt) can be derived using other approximations for X(s) in the interval (t, t + Δt) (e.g., exponential).