## Abstract

In response to exercise performed before or after altitude decompression, physiological changes are suspected to affect the formation and growth of decompression bubbles. We hypothesized that the work to change the size of a bubble is done by gas pressure gradients in a macro- and microsystem of thermodynamic forces and that the number of bubbles formed through time follows a Poisson process. We modeled the influence of tissue O_{2} consumption on bubble dynamics in the O_{2}transport system in series against resistances, from the alveolus to the microsystem containing the bubble and its surrounding tissue shell. Realistic simulations of experimental decompression procedures typical of actual extravehicular activities were obtained. Results suggest that exercise-induced elevation of O_{2} consumption at altitude leads to bubble persistence in tissues. At the same time, exercise-enhanced perfusion leads to an overall suppression of bubble growth. The total volume of bubbles would be reduced unless increased tissue motion simultaneously raises the rate of bubble formation through cavitation processes, thus maintaining or increasing total bubble volume, despite the exercise.

- macro- and microsystem
- O
_{2}serial transport - O
_{2}tissue uptake - O
_{2}window - N
_{2}supersaturation - Poisson process

astronauts performing extravehicular activities (EVAs), when exposed to a reduced absolute pressure in their space suits, may experience decompression illness (DCI) (6-8). The primary cause of DCI is the formation and growth of gas bubbles within tissues evolving from excess dissolved gases (7, 8). It has been suggested that metabolic gases make up significant fractions of the gas in bubbles during altitude decompression (11, 41). In tissues supersaturated with an inert gas, typically N_{2}, in the presence of O_{2} (30, 45), CO_{2}, and water vapor, de novo bubbles are generated from primordial gaseous entities, “gas micronuclei.” The initial explosive bubble growth involves the surrounding tissue (11, 40) and may recruit all dissolved gases. To reduce bubble formation and growth, a denitrogenation or N_{2} “washout” procedure consisting of prebreathing a hyperoxic mixture is performed before ascent to a constant working altitude pressure. We refer to the overall sequence of O_{2} prebreathe, ascent, and time at altitude as a “decompression profile.” When referring to the actual process of pressure reduction, however, we use the simpler term “decompression.” Bubbles may form, grow, and decay during the sojourn at altitude, usually disappearing on recompression to sea level.

It has become increasingly apparent that skeletal muscle exercise, regardless of when it is performed, influences the onset of DCI (20, 48). A possible explanation is that exercise may create gas micronuclei (44). In particular, high-intensity exercise before decompression may create gas micronuclei (19), which increase the risk of DCI. Also, mechanical movement of body structures may cause cavitation (19) and increase the production of bubbles after decompression (20). Although exercise may accelerate N_{2}elimination, it does not invariably precipitate bubble formation (20) and, therefore, may even induce a protection against DCI. Experimental results from Webb et al. (48) indicated that moderate exercise performed during the O_{2}-prebreathe period enhanced the tissue N_{2} washout and reduced the incidence of DCI. Here, we develop a bubble formation-and-growth model (FGM) to answer the following questions. First, how do exercise-induced mechanisms impact formation and growth of gas bubbles? Second, are these mechanisms competing, and if so how? In the accompanying study (11a), the FGM will be validated in a survival analysis to predict the incidence of DCI in the National Aeronautics and Space Administration Altitude Experimental Data Set.

A bubble is defined as a volume of gas in a tissue that follows the phenomenological laws of ideal gases, diffusion and surface tension (40). Bubble growth is controlled by the classical laws of motion: the pressure of the gas provides the driving force to expand the bubble, while the inertia and elastic recoil of the tissue, together with the interfacial tension of the bubble wall, provide resistance to expansion (40). The actual work to change the bubble volume is accomplished solely by pressure gradients of gases across the interface between the bubble and surrounding tissue (21). During breathing of pure O_{2} after decompression, the N_{2} pressure gradient is directed from the tissue to the alveolus. Although this pressure gradient creates a flux that tends to remove N_{2} from the vicinity of the tissue bubble, N_{2} still diffuses into the bubble, which enlarges (40). In our hypotheses, the relevant system in which thermodynamic forces act consists of two distinct spatial and functional subsystems. First, there is a microsystem consisting of tissue volume containing the bubble, its boundary layer, and a tissue shell. This microsystem then interacts with a macrosystem as the alveolus-arterial blood-tissue shell-venous blood serial cascade of structural or functional barriers. We then developed the FGM to explain how bubble growth is influenced by exercise-induced changes in the O_{2} physiological resistances in series in both systems.

Because bona fide bubbles appear to form randomly (47,50), we hypothesized that their spatial and temporal distributions in small units of tissue volume follow a Poisson process. Informally, the Poisson process asserts that the event of bubble formation occurs independently through time in any of a large number of small units of tissue volume, but with a small probability in any given unit at a given time (32, 38). This process is characterized by parameters that may depend on the type, intensity, duration, and chronology of exercise. At working altitude pressure, the total volume of all bubbles in tissue is propagated in time through the growth-and-decay mechanism, which applies independently for each bubble relative to its time of formation.

We demonstrated the potential mechanisms of exercise by applying the FGM in simulations to calculate total bubble volume for several variations of decompression procedures typical of actual EVAs. These variations were primarily characterized by differences in O_{2} consumption, blood flow, and bubble formation rates (Poisson process). The results of our simulations suggested that exercise-induced elevation of O_{2} consumption at altitude facilitated the persistence of bubbles in tissues, whereas exercise-enhanced perfusion tended to suppress bubble growth. The total volume of bubbles would be reduced unless increased tissue motion simultaneously raises the rate of bubble formation through cavitation processes, thus maintaining or increasing total bubble volume, despite the exercise.

### Glossary

- (a-a)Po
_{2}(*t*) - Alveolar-arterial Po
_{2}difference, Pa - A
_{b}(*t*) - Surface area of the bubble at
*time t*, m^{2} - α
- Parameter of the Poisson process, dimensionless
- (a-
)Po
_{2}(*t*) - Arterial-venous Po
_{2}difference, 8,000 Pa (∼60 mmHg) at rest - β
- Parameter of the Poisson process, dimensionless
- D
_{i} - Diffusivity of the
*i*th gas species in the tissue, m^{2}/min - ε
- Thickness of the diffusion barrier (protein layer), 2 × 10
^{−6}m - Fi
_{O2} - Fraction of O
_{2}in the inspired medium, dimensionless - h
- Constant of proportionality = 2, dimensionless
- J
_{N} - Net flux of all gas species across the boundary layer, mol · m
^{−2}· min^{−1} - J
_{i} - Molar flux of the
*i*th gas species across the boundary layer, mol · m^{−2}· min^{−1} - k
_{1} - Tissue gas exchange rate constant for washin and washout of N
_{2}, min^{−1} - k
_{2} - Tissue gas exchange rate constant for washin and washout of O
_{2}, min^{−1} - M
_{b}(*t*) - Number of gas moles in the bubble, mol
- m
_{CO2}(*t*) - Number of dissolved moles of CO
_{2}in the tissue, mol - m
_{H2}_{O}(*t*) - Number of moles of water vapor from the tissue shell, mol
- m(
*t*) - Mean of the Poisson process, dimensionless
- M
_{ti}(*t*) - Number of gas moles in the tissue shell, mol
- ν
- Tissue elastic recoil from Ref. 14, 3.7 × 10
^{3}Pa (= 3.7 × 10^{4}dyn/cm^{2}) - N(
*t*) - Total number of bubbles in the
*n*tissue units, dimensionless - N
_{i}(*t*) - Number of bubbles formed in the
*i*th unit at*time t*, dimensionless - Ω
- Constant of proportionality estimated from Table 2, 0.87, dimensionless
- P
- Ambient pressure, Pa
- Pa
_{d,O2}(*t*) - Partial tension of O
_{2}fraction dissolved in the arterial blood, Pa - Pa
_{i}(*t*) - Alveolar partial pressure of gas
*i*, Pa - Pa
_{i}(*t*) - Arterial tension of gas
*i*, Pa - Pa
_{Hb O2}(*t*) - Partial tension of O
_{2}fraction bound to Hb, Pa - P
_{b,i} - Partial pressure of the
*i*th gas in the bubble, Pa - P
_{b,mg}(*t*) - Pressure of metabolic gases in the bubble, Pa
- Pi
_{i}(*t*) - Partial pressure of gas
*i*in inspired breathing medium, Pa - Pti
_{i} - Tissue tension of the
*i*th gas, Pa - P
_{i}(*t*) - Tension of gas
*i*in the mixed venous blood, Pa - Pv˙
_{O2} - Tension drop of dissolved tissue O
_{2}due to O_{2}consumption, Pa - P
_{O2} - Mixed venous Po
_{2}, Pa - P
_{w}(*t*) - O
_{2}window, Pa - Ψ
_{O2}(*t*) - Arterial partial tension of dissolved O
_{2}that is not utilized in tissue metabolism, Pa - Φ
_{O2}(*t*) - Overall O
_{2}pressure gradient in the macro- and microsystem, Pa - ϕ
_{1}(*t*) - Sum of pressures due to surface tension and tissue elastic recoil, Pa
- Q˙ti(
*t*) - Blood flow in the tissue shell, m
^{3}/min *R*- Universal gas constant, N · m · mol
^{−1}· K^{−1} - R
- Respiratory exchange ratio: 0.7–1.12, 0.82 at rest, dimensionless
- R
_{b}(*t*) - Radius of a bubble at
*time t*, m -
_{b}(*t*) - Mean radius of bubbles from the entire region at
*time t*, m -
_{b,max} - Maximum mean radius of bubbles from
*n*units at*time t*, m - R
_{q} - Circulatory convective resistance, Pa · l
^{−1}· min - s
_{b,i} - Solubility of the
*i*th gas in the blood, ml · ml^{−1}· 100 Pa^{−1} - s
_{ti,i} - Solubility of the
*i*th gas in the tissue, ml · ml^{−1}· 100 Pa^{−1} - τ
- Surface tension of the tissue from Refs. 13 and 14, 10
^{−2}N/m (= 10 dyn/cm) *T*- Temperature, Kelvin
- T
- Total time of exposure to altitude, min
*t*- Time of interest measured from first pressure change (prebreathe), min
*t*_{alt}- Time of exposure to altitude immediately after decompression, min
*t*_{b ij}- Time of onset of the
*j*th generated bubble in the*i*th unit of tissue volume measured from t_{alt}(t_{alt}= 0), min - t
_{1/2,N2} - Half time for tissue washin and washout of N
_{2}, min - t
_{1/2,O2} - Half time for tissue washin and washout of O
_{2}, min - V
_{b}(*t*) - Volume of the bubble at
*time t*, m^{3} - V
_{b·}(*t*) - Volume of bubbles in the tissue region, m
^{3} - V
_{b · max} - Maximum volume of bubbles in the tissue region, m
^{3} - v(t)
- Intensity of the Poisson process, bubbles formed/min
- V˙ti
_{O2}(*t*) - O
_{2}uptake in the tissue shell, m^{3}/min - Vti(
*t*) - Volume of the tissue shell at
*time t*, m^{3} - V
_{tot}(*t*) - Volume of the tissue element at
*time t*, m^{3} - V
_{tu} - Volume of the tissue unit (realized as a cube), m
^{3} - x
_{i} - Molar fraction of the
*i*th gas species, dimensionless

## METHODS

### Growth-and-Decay Model for a Single Bubble

#### Macro- and microsystems of gas exchange.

We define a tissue element to be a small spherical unit of tissue containing the bubble (Fig.1
*A*), where significant gas exchanges take place (4, 40). We assumed that every tissue element contains a single bubble and that the ratio of the volume of the tissue element (V_{tot}) to the bubble volume (V_{b}) is constant. The part of the tissue element that does not include the bubble per se will be referred to as the homogeneous tissue shell. We proceed to derive a differential equation relating the bubble radius (R_{b}), and hence V_{b}, to physical and physiological parameters obtainable from the characteristics of the decompression profile.

We consider two gas transport systems: a macrosystem, in which gases move from the alveolus to the tissue element and vice versa, and a microsystem for gas exchanges across the diffusion barrier inside the tissue element. In the macrosystem, CO_{2} moves outward from the tissue element to the alveolus, and this flux is considered in the positive direction. Similarly, after decompression during breathing of N_{2}-O_{2} mixtures, N_{2} moves in the positive direction, from the supersaturated tissue element to the alveolus. In contrast, O_{2} moves from the alveolus toward the tissue element, and the flux has a negative direction. Finally, water in the tissue fluid also tends to move toward the tissue element (negative direction) as it vaporizes to fill the empty space created by the forming bubble. In the microsystem (bubble-diffusion barrier-tissue shell), we establish signs for the gradient and flux of a gas across the diffusion barrier to be “positive” if it has same direction as the gas flux in the macrosystem (21, 28). During the initial explosive bubble growth phase, all gases in this system diffuse into the bubble (11). Thus CO_{2} and N_{2} have “negative” fluxes in this system, whereas O_{2} has a positive flux. Water vapor diffuses into the de novo bubble; hence, it too has a positive flux in this system (Fig. 1
*B*).

#### Diffusion of gases across the diffusion barrier in the microsystem.

The volumes of gases are expressed under standard body conditions of temperature, ambient pressure, and saturated with water vapor (btps). In a three-dimensional coordinate system, the molar flux of the *i*th permeating ideal gas species (J_{i}) obeys Fick's first phenomenological law of diffusion (4, 14, 17, 18, 33, 39, 43) and can then be estimated by J_{i} = −(P/*RT*)D_{i}∇x_{i}, where P is the ambient pressure, *R* is the universal gas constant,*T* is the temperature in degrees Kelvin, D_{i}is the diffusion coefficient, and ∇x_{i} is the molar fraction gradient of the *i*th gas in the macrosystem. For air breathing, gas exchange dynamics involve four relevant species: CO_{2}, N_{2}, O_{2}, and H_{2}O (*i* = 1, … ,4). In the macrosystem, ∇x_{i} is defined along a direct path to the center of the bubble, where the inward direction is negative and outward is positive. Resulting fluxes have signs in accordance with the “macrosystem” rule (inward = negative; outward = positive). In the microsystem, we establish that a flux is positive if it has the same direction as in the macrosystem (opposite direction = negative). Therefore, the CO_{2} and N_{2} fluxes are negative (Fig. 1
*B*). The net flux (28) (J_{N}) into or out of the bubble is expressed as follows
Equation 1Applying Henry's law to the dissolved tissue gases surrounding the bubble and using the ideal gas equations for gas pressures inside the bubble (14), J_{N}(*t*) can be approximately expressed in terms of partial pressures of the*i*th gas as a function of time. As reported previously (14, 28), the net flux is thus expressed as follows
Equation 2where s_{ti,i} is the solubility of the *i*th gas species in the tissue element, ε is the thickness of the diffusion barrier, and Pti_{i}(*t*) and P_{b,i}(*t*) are the partial tissue tension and pressure within the bubble of the *i*th gas, respectively. When referring to a specific gas species, we use the convention of replacing the subscript *i* by the gas name (e.g., s_{ti,N2} instead of s_{ti,i}). Time is expressed in minutes and measured from the start of the prebreathe period to the end of the exposure to altitude. The prebreathe period begins at *time t* = *t*
_{0} = 0, when partial pressures of gases in the breathing medium start to change from the equilibrium of standard atmospheric conditions. Arrival at working altitude pressure (end of depressurization), occurs at*time t* = *t*
_{alt}.

#### Moles of gas within the tissue shell and in the bubble.

To evaluate the net flux (*Eq. 2
*), we next calculate the number of moles of each gas crossing the diffusion barrier. Suppose a particular bubble forms at*t*
_{b} > *t*
_{alt}. For*t* > *t*
_{b}, let M_{ti}(*t*) be the total number of moles of gas in the tissue shell. From Henry's law (14), we have
Equation 3
where m_{CO2}(*t*) and m_{H2}
_{O}(*t*) are the number of moles of dissolved CO_{2} and water vapor, respectively, Vti(*t*) is the tissue shell volume pertaining to the bubble at *time t* [which has been in existence for (*t* − *t*
_{b}) min], and Φ_{O2}(*t*) is the O_{2} overall pressure difference between the macro- and the microsystem. Using the equation of state of an ideal gas (14), we estimate the number of moles in the bubble
Equation 4where V_{b}(*t*) is the volume of the bubble at *time t*.

#### Estimation of the bubble radius.

From the law of conservation of mass (21, 28), the number of moles diffusing into and out of the bubble per minute is
Equation 5for *t* > *t*
_{b}, where*A*
_{b}(*t*) is the surface area of the bubble and M˙_{ti}(*t*) andM˙_{b}(*t*) are gas mole uptake into and out of the tissue shell (microsystem) and into and out of the bubble, respectively. (We use the convention that the overdot denotes differentiation with respect to time.) By definition of the microsystem, we assume Vti to be proportional to V_{b}(*h* = Vti/V_{b}≥1) and h to be sufficiently small so that any gases entering or leaving the tissue shell are being involved in exchanges across the diffusion barrier of the microsystem. Finally, for *t* > *t*
_{b},* Eq.5
* can be rewritten in terms of the bubble radius R_{b}(*t*)
Equation 6where L(*t*) and K(*t*) are quantities derived in the
. For a given decompression profile, values of K(*t*), L(*t*), and J_{N}(*t*) may be obtained as a function of time through measurements of inspired pressure and fraction of gases. Details of the calculation of K˙(*t*) andL˙(*t*) are given in the
.*Equation 6 *has no analytic solution for R_{b}(*t*) and must be solved numerically.

#### Gas transport in the macrosystem: estimation of pressures and/or tensions.

To obtain L(*t*) and K(*t*) in *Eq. 6
*, it is first necessary to estimate the pressure gradients in the macrosystem. These may be calculated from values of the absolute pressure and inspired fractions of N_{2} and O_{2}and expired CO_{2} in the breathing medium for each phase of our decompression profiles. In addition, within the macrosystem, we consider partial pressures of each gas: inside the alveolus [Pa(*t*)], inside the pulmonary capillary [Pa(*t*)], and in the mixed venous blood [P
(*t*)]. For all gases, we assume P
(*t*) = Pti(*t*).

#### N_{2} tissue tension.

On the downstream side of the macrosystem flow, we estimated Pti_{N2} for *t* >*t*
_{alt} using the classical exponential equation (8, 11, 14). Using this method, we computed Pti_{N2} by increments at fixed times*t*
_{1},*t*
_{2}, … ,*t _{n}
*, where

*t*

_{n−1}=

*t*

_{alt}and

*t*=

_{n}*t*. The incremental expression for Pti

_{N2}is given by Equation 7 where Pa

_{N2}is the N

_{2}arterial tension derived from the alveolar gas equations and partial pressures (11, 42) and k

_{1}is the tissue gas exchange rate constant for N

_{2}.

#### O_{2} transport.

On the upstream side of the macrosystem flow portrayed in Fig.1
*B*, O_{2} is driven through a series of interfaces (12) into the tissue element, where it is dissolved. To correctly model the flow, it is necessary to estimate pressure differences across various encountered interfaces as follows (Fig.1
*C*). First, because of the alveolar membrane, there is an alveolar-arterial pressure difference [(a-a)Po
_{2}(*t*)], which tends to increase with Pi
_{O2}(*t*) (5, 15, 37). On the basis of our own measured values of Pi
_{O2}(*t*), we estimated (a-a)Po
_{2}(*t*) with an algorithm that uses values given by Clark and Lambertsen (5). We then obtained Pa_{O2}(*t*) by subtraction from Pa
_{O2}(*t*). Second, the O_{2} transfer to the capillaries occurs when O_{2} is transported in physical solution or bound to Hb. Above the threshold Pa_{O2}(*t*) of 13.33 kPa (100 Torr, 1 kPa = 7.50062 Torr), the O_{2} saturation is assumed to be 100% and the unbound O_{2} remains in physical solution. In terms of tension, the arterial O_{2} tension is made up of two components, the O_{2} dissolved fraction [Pa_{d,O2}(*t*)] and the Hb-bound O_{2} [Pa_{Hb O2}(*t*)]. We used an algorithm derived from the O_{2} dissociation curve (29,35, 36) to estimate Pa_{Hb O2}(*t*) as a function of Pa_{O2}(*t*). Third, the O_{2} supply to the mitochondria results in the withdrawal of O_{2} from further utilization in the O_{2} transport process across the bubble boundary layer. Di Prampero and Ferretti (9, 10) derived a relationship between the O_{2} tissue consumption [V˙ti_{O2}(*t*)] and the arteriovenous O_{2} difference [(a-
)Po
_{2}(*t*)]. The relationship is V˙ti_{O2}(*t*) = [(a-
)Po
_{2}(*t*)]/R_{q}= Pv˙_{O2}(*t*)/R_{q}, where R_{q} is the circulatory convective resistance and Pv˙_{O2} is the tension drop of dissolved tissue O_{2} due to O_{2} consumption. Finally, O_{2} supplied to the tissue element can potentially participate in bubble gas exchanges of the microsystem. Thus the tension of the dissolved O_{2} that is applied to the microsystem at *time t*[Ψ_{O2}(*t*)] can be written in the form
Equation 8Dissolution of O_{2} in the tissue element follows Henry's law and is described by an exponential relation similar to*Eq. 7
*. As was the case for N_{2}, Φ_{O2}(*t*), the overall O_{2}pressure gradient in the macro- and microsystem that applies from the alveolus to the tissue element can be expressed incrementally at fixed times *t*
_{1},*t*
_{2},… ,*t _{n}
* ≡

*t*, so that Equation 9 where k

_{2}is the tissue gas exchange rate constant for O

_{2}. Indeed, metabolism lowers the O

_{2}tension in the tissue element below Pa

_{O2}, creating a phenomenon known as the “O

_{2}window” or “inherent unsaturation” (11,16, 23, 34, 42). The sum of partial tensions of the dissolved gases in the tissues is usually less than atmospheric pressure. The O

_{2}window [P

_{w}(

*t*)] is calculated by subtracting the sum of pressures of all dissolved gases to the ambient pressure and can be written using Φ

_{O2}(

*t*) so that P

_{w}(

*t*) = Φ

_{O2}(

*t*) + {P(1 − Fi

_{O2}) − [Pti

_{N2}(

*t*) + Pti

_{CO2}(

*t*) + Pti

_{H2}

_{O}(

*t*)]}.

*Equations7*and

*9*are then used in the to calculate L(

*t*) and K(

*t*).

#### Number and onset times of bubbles in tissue.

Yount (50) proposed a stochastic model for accretion and deletion of skin molecules that leads to an exponential distribution for the number of micronuclei. Here, we consider a stochastic model of bubble formation in which micronuclei evolve into bubbles at random times after decompression following a Poisson process. Consider a tissue region divided into n units of tissue that are identical in makeup and perfusion (40). Each tissue unit is independent, and there is no diffusion from one unit to another. All tissue units are the same size, and units are evenly distributed in space. In contrast, bubbles in one unit are of different age and size. Schematically, the units are illustrated as cubes of constant volume V_{tu} in Fig. 1
*D*. For a total time at altitude of T minutes, N_{i}(*t*) (i = 1, … , n), the number of bubbles formed in the *i*th unit of tissue volume up to *time t* (in minutes, 0 < *t* ≤ T) is assumed to follow a nonhomogeneous Poisson process (32,38) with intensity v(*t*), v(*t*) = α*e*
^{−βt}. The parameters α and β are driven by the decompression procedure and the level of exercise or rest. For a given procedure, α also serves as a scaling factor, being proportional to V_{tu.} The decreasing exponential form of v(*t*) reflects the rapidly decreasing propensity to form bubbles as exposure time increases (6,22).

To be a Poisson process, N_{i}(0) = 0, N_{i}(*t*) must have independent increments; i.e., the number of bubbles formed in nonoverlapping intervals of time must be statistically independent, and two or more bubbles cannot form simultaneously. Physically, the latter two requirements correspond to temporal and spatial independence of bubble formation. It can be shown under these assumptions (32) that N_{i}(*t*) has a Poisson distribution with mean
Equation 10In other words, the probability p to obtain k bubbles up to*time t* is p[N_{i}(*t*) = k] = exp[−m(*t*)][m(*t*)]^{k}/k! (k = 0, 1, …).

#### Mean bubble radius and total volume of bubbles.

The mean bubble radius for the given region of tissue at *time t* can be written as
Equation 11where N(*t*) = Σ_{i=1}
^{I}N_{i}(*t*) and R_{b,ij}(*t*) is the radius of the *j*th bubble in the *i*th tissue unit. By convention, we take R_{b,ij}(*t*) = 0 if the bubble has not formed by *time t*. Under the additional assumption that bubbles form and grow independently over the *n* tissue units, N(*t*) is also a Poisson process (32, 38). If the region is homogeneous in the sense that all the Poisson processes N_{i}(*t*) have identical values of α and β, then the mean of N(*t*) is simply (nα/β)(1 − e^{−βt}). With the assumption of spherical bubbles, the total volume of bubbles in the region of tissue at*time t* is
Equation 12

### Experimental Design

We applied the FGM to four types of decompression profiles (*A–D*) typical of chamber tests and actual EVAs (Table1). These profiles shared the following properties: *1*) the duration of prebreathe was 210 min (P = 101.13 kPa, Fi
_{O2} = 1);*2*) ascent time was 6 min; and *3*) Fi
_{O2} = 1. The altitude pressure was 30 kPa for *profiles A–C* compared with 60 kPa for*profile D*. In *profiles A* and *D*, physiological parameters Pv˙_{O2}, R,*t*
_{1/2,N2}, and*t*
_{1/2,O2} were set to known values of 8 kPa, 0.82, 360 min (7), and 313.2 min, respectively. These values are consistent with the case of no exercise. In addition, *t*
_{1/2,O2} was calculated using*Eq. EA13
*. Using experimental data, we found in another study (11a) that the Poisson process parameter β = 0.017 for profiles without prebreathe exercise but with mild exercise (817 kJ) at altitude. We assumed that, for a control case of no exercise at any time, β would be reduced by 20% (to ∼0.014), reflecting the decreased propensity to form bubbles. In contrast, *profile B*incorporated mild exercise at altitude to emulate the moderate workloads performed by astronauts during ordinary EVA. For this case, we assumed Pv˙_{O2} = 10 kPa, R = 0.95,*t*
_{1/2,N2} = 200 min, and*t*
_{1/2,O2} = 174 min. *Profile C* also simulates exercise, but with two phases (10 min of heavy exercise followed by 25 min of light exercise) during prebreathe. Physiological parameters were set at Pv˙_{O2} = 12 kPa, R = 1.12, *t*
_{1/2,N2} = 60 min, and *t*
_{1/2,O2} = 52.2 min (phase 1) and then at Pv˙_{O2} = 10 kPa, R = 0.95,*t*
_{1/2,N2} = 80 min, and*t*
_{1/2,O2} = 69.5 min.

For each profile type, various simulations of bubble formation and growth were analyzed to examine the effect of exercise on
_{b}(*t*) and V_{b·}(*t*). For *simulations A9–A11* (Table 1), we modified values of Pv˙_{O2} while keeping all other parameters fixed to isolate the effect of V˙ti_{O2}. These higher values are representative of heavier exercise workloads required to perform EVA tasks during some orbital missions. For *simulations A7*and *A8*, β was increased by a factor of 3 to emulate the effect of an increased rate of bubble formation, but with physiological parameters remaining at no-exercise levels. This would be the case if mechanical motion takes place without significant additional O_{2} consumption. It has been conjectured that mechanical motion of tissues may cause cavitation (19) and increase the bubble formation rate (20). For simulations involving*profiles A, C*, and *D*, the parameter α was chosen to make the expected number of bubbles per tissue unit (≈ α/β) equal to 6.0 over 50 tissue units.

For the case of exercise at altitude (*simulations B1–B14*), we investigated the effect of bubble formation rate on maximum bubble volume. This rate was again controlled by varying β. During this exercise, more units of tissue would be recruited for bubble formation; therefore, we increased *n*, the number of tissue units, to 100. Also, the density of bubbles per tissue unit would be expected to be greater than at rest; hence, we changed α so that the mean number of bubbles per unit was 25.

### Simulation Process

An overview of the simulation is illustrated in Fig.2. Working values of solubilities of gases (N_{2}, O_{2}, and CO_{2}) in the tissue element were chosen to lie in the range of similar values for blood (Table 2). Diffusivities of gases were chosen as ∼75% of corresponding values for water (Table2) and 200% of the values for lipids. Values of other physical constants are listed in the *Glossary*.

From the Poisson distribution with mean m(*t*) (*Eq.*
10), we generated N_{i}(T), the total number of bubbles formed over a decompression period of T minutes. Values of α and β defining m(*t*) are given in Table 1. Methodology for generating random numbers from the Poisson distribution is well known (32). Next, we used a property of Poisson processes (32) that relates the conditional distribution of event times to the mean, when the total number of events is known. This enabled us to generate random bubble creation times*t*
_{b,ij} [j = 1, … , N_{i}(T); see
].

For each creation time, *Eq. 6
* was constructed using*Eqs. 1-5
* and *
EA1-EA13,* where*t*
_{b} in the
is replaced by*t*
_{b,ij}, i.e., onset time of the *j*th bubble in the *i*th unit of tissue volume. Numerical solutions R_{b,ij}(*t*) for R_{b}(*t*) in *Eq.* 6 were then found for each simulated bubble radius at 10-min intervals using Mathematica software version 3.0.1 (49). Solution of *Eq. 6
*proved difficult. In general, a combination of nonstiff Adams or stiff Gear, Fehlberg order 4–5, or Runge Kutta methods for nonstiff equations was required to achieve convergence. The Gel'fand-Lokutsiyevski chasing method was used for solving the boundary value problem. Processing of 300 bubbles took ∼2 min to run on a 300-mHz personal computer with 64 MB of random access memory. Finally, the R_{b,ij}(*t*) was used in*Eqs. 11
* and *
12
* to calculate
_{b}(*t*) and V_{b·}(*t*).

## RESULTS AND DISCUSSION

### Effects of O_{2} Consumption on Bubble Dynamics

*Simulations A1* and *A9–A11* were run with different levels of O_{2} consumption at altitude (Pv˙_{O2} = 8, 12, 18, and 20 kPa) with bubble formation parameters (α and β) and blood flow fixed at resting control values (Table 1). Bubble growth to
_{b,max} was essentially the same for all four levels of O_{2} consumption. These results are consistent with those of Van Liew et al. (42), who noted that a large O_{2} window, especially during O_{2} breathing, reduced the bubble enlargement. They also reported no significant change in the O_{2} window values at arteriovenous pressure differences <100 kPa (Fi
_{O2} = 1). Similarly, we observed that increasing O_{2} extraction (Pv˙_{O2} > 9 kPa) had little or no effect on the O_{2} window, and therefore maximal bubble growth was not affected. However, the average bubble size decreased slowly in time when V˙ti_{O2} increased, whereas low restingV˙ti_{O2} values facilitated faster decay (Fig.3
*A*). Because O_{2}has a greater permeation coefficient than N_{2}, short transients of O_{2} permeate rapidly into the bubble at rest [Pv˙_{O2}(*t*) = 8 kPa]; simultaneously, N_{2} exits the bubble to the surrounding tissue (40). Then, O_{2} rapidly permeates out of the bubble, resulting finally in a rapid bubble decay. In contrast, tissue O_{2} extraction is enhanced during exercise; thus a relatively small amount of O_{2} diffuses into the bubble and is exchanged for N_{2}. Therefore, N_{2} builds up in the bubble, which in turn reduces the bubble decay rate.

### Exercise Decreases the Mean Bubble Radius

Aerobic exercise-enhanced blood flow, before and/or after decompression, generates a cascade of events in the macro- and microsystem as follows. Augmentation of blood flow [Q˙ti(*t*); *Eq. EA12
*] causes a decrease in the O_{2} and N_{2} tissue washin and washout half times, *t*
_{1/2,i} = (ln 2)/k_{i}, where k_{i} is the tissue gas exchange rate constant. As a result, excess N_{2} in the tissue element is carried away before it can diffuse into postdecompression bubbles. The fast N_{2} removal by blood precludes bubble enlargement (40). Also, little or no N_{2} is carried to the tissue when breathing enriched O_{2} mixtures. In contrast, a greater amount of O_{2} physically dissolves in the tissue element, which in turn moderately reduces the O_{2} window.

Hills and LeMessurier (16) reported that, after 15 min of exposure of rabbits to an hyperoxic medium, the O_{2} window was large. Here, we agree that only a small amount of O_{2}would dissolve in the tissue during this time interval. However, during several hours of hyperoxic O_{2} prebreathe/altitude exposure, greater amounts of O_{2} would dissolve in tissues, thereby eventually reducing the O_{2} window. According to Lambertsen et al. (23), other mechanisms such as hypercapnic vasodilatation may also facilitate the O_{2} transfer to the tissue, which in turn accelerates the window reduction. Lambertsen et al. also showed in humans, that arterial hypercapnia, a highly potent vasodilator of cerebral blood vessels, in conjunction with breathing for 15 min at 3.5 atm (fraction of CO_{2} in the inspired medium = 0.02, Fi
_{O2} = 0.98) induced a significant elevation of P
_{O2} (146 kPa) in internal jugular venous blood samples, as compared with breathing without hypercapnia (Fi
_{O2} = 1, P
_{O2} = 10.13 kPa). In both cases Pa_{O2} = 266 kPa; however, hypercapnia via cerebral vasodilatation reduced the O_{2}window in very limited time.

Observations in pigs breathing hyperoxic mixtures showed that the bubble incidence in the pulmonary artery was reduced as Pa_{O2} increased (34). Here, for pure O_{2} breathing, we illustrate how P = Pi
_{O2} affects
_{b}(*t*) through change in Pa_{O2}. Figure 3
*B* shows
_{b}(*t*) for the case of no exercise at two different altitudes: P = 30 kPa (*simulation A1*) and P = 60 kPa (*simulation D1*). At 60 kPa, the larger value of Pa_{O2} induces a significant reduction in
_{b,max}. Therefore, we postulate that the O_{2} window inhibits bubble enlargement at the beginning of the altitude exposure. Later, because of increased perfusion, the higher O_{2} flow to tissues augments the amount of dissolved O_{2} in the unit of tissue volume. This, in turn, decreases the O_{2} window, thus tending to slow the rate of bubble decay. However, the greater amounts of physically dissolved O_{2} facilitate increased O_{2} exchange within the microsystem. According to Van Liew and Burkard (41), many short O_{2} transients would permeate rapidly in the bubble, resulting finally in a marked decay rate of bubble radii.

Figure 4 compares
_{b,max} for decompression profiles whose physiological parameters (Pv˙_{O2}, R,*t*
_{1/2,N2} and*t*
_{1/2,O2}) have the characteristics of no exercise, moderate exercise at altitude, and heavy exercise during prebreathe, but where β is held fixed. First, we compared the effect of exercise when β is set to the resting value of 0.014 (*simulations A1, B1*, and *C1*). Then a similar comparison was made for β = 0.017 (*simulations A2, B2*, and *C2*). For both values of β, we observed that
_{b,max} dropped by ∼64%, (from 34.1 to 12.4 μm for β = 0.014 and from 44.5 to 16.2 μm for β = 0.017). When the effect of heavy exercise during prebreathe was compared with the case where there is no exercise, the drop was more dramatic, i.e., ∼83% for both values of β. Results of these simulations illustrate how an augmentation of blood flow, which decreases *t*
_{1/2,N2} and*t*
_{1/2,O2}, is paralleled by a reduction of bubble radius regardless of the intensity of the Poisson process and the level of V˙ti_{O2}.

Also, with an acceleration of the nucleation process (β increased from 0.014 to 0.017), the increase in
_{b,max} was much more pronounced for the no-exercise case than for the case of exercise at altitude. However, the relative increase was about the same (30%), and for the prebreathe case the relative change appeared less (∼20%). This is probably because, regardless of the nucleation rate, only small amounts of dissolved N_{2} remain to be available for bubble growth after the start of decompression.

For *profile A*, we investigated the effect of increasing β from its resting value of 0.014 to large values reflecting intense bubble formation with physiological parameters remaining fixed (Pv˙_{O2} = 8 kPa, R = 0.82, *t*
_{1/2,N2} = 360 min and*t*
_{1/2,O2} = 313.2 min). We observed that in *simulations A1, A2, A7,* and*A8*,
_{b,max} increased from 34.1 to 49.8 μm as β increased from 0.014 to 0.051 (Fig. 4). As β increases, early formation of bubbles before N_{2} supersaturation drops, leading to larger radii than the resting case (*simulation A1*). The pattern of increase was nonlinear, tending toward a plateau for large values of β. This is because, no matter how early the bubble is formed, for a fixed N_{2} pressure gradient, the amount of N_{2} available for diffusion into a bubble is limited.

### Does Exercise at Altitude Increase the Volume of Tissue Bubbles?

In general, it has been observed that there is an increased incidence of DCI symptoms for subjects exercising at altitude after decompression (20). However, a diving experiment showed that exercise during decompression actually reduced Doppler-detectable venous gas emboli (19). Therefore, it is unclear how exercise affects the incidence of tissue bubbles. Using the FGM, we evaluated the effect of exercise in terms of how much the bubble formation process would have to be accelerated to achieve a value of V_{b · max} equal to that in a no-exercise case. For example, *simulations A5* (no exercise) and *B5*(exercise at altitude) produced about the same value of V_{b · max} (0.049 mm^{3}). In the latter case, β had to be increased by a factor of ∼2.5 (from 0.014 to 0.38) to achieve the same V_{b · max}. In other words, ∼9.5 times as many bubbles would have to be generated with exercise at altitude to achieve the same maximum volume as at rest. Despite similar values of V_{b · max}, V_{b·}(*t*) differed considerably. For the exercise case (*simulation B5*), a relatively large number of smaller bubbles formed earlier, whereas in the no-exercise case (*simulation A5*), larger bubbles were formed, but mostly at later times (Fig.5
*A*).

As demonstrated in Fig. 4, exercise-enhanced blood flow reduces
_{b,max} and, therefore, V_{b · max}for a fixed number of bubbles formed. In this case, the only way V_{b · max} could be increased is through a more intense generation of bubbles. In *simulations B3–B14*, we calculated V_{b · max} for *simulation B*(exercise at altitude), with β ranging from 0.036 to 1.5. In Fig.5
*B*, V_{b · max} increases rapidly as a function of β when β is small and reaches a plateau of ∼0.1 mm^{3} for β > 0.3. The reason is that, similar to the no-exercise case, the N_{2} pressure gradient limits the supply of N_{2} available for bubble growth in the tissue region.

### Stability

We found that varying the initial minimal value of the bubble radius, within a range of 10^{−8}–10^{−4} m, did not affect the bubble growth dynamics; thus it appears that the differential *Eq. 6 *is stable, and therefore, our model is robust with respect to the choice of the size of micronuclei. Also, the variability (<2.3%) inherent in the Poisson process did not affect the reproducibility of the simulation results for various realizations of the same profile. *Simulations A1* and*A3* represent two realizations of *profile A* (no exercise). As shown in Fig. 3
*B*, there is very little difference in
_{b}(*t*) between these two examples. As a further illustration of variability, we found that
_{b,max} varied with a standard deviation of ∼1.6% over 13 similar realizations. Also, the time of maximum bubble radius varied with a standard deviation of ∼3%. However, this small amount of variation does not explain the different decompression outcomes observed over a population of subjects. Significant variability may be associated with age, body mass index, time of the day, seasonal variation, body temperature, body chemistry, and previous injuries (2).

### Review of Assumptions

Physical parameters, e.g., s_{ti,i}, D_{i} , ν, ,τ, ε, and h, were selected only for a hypothetical tissue derived as a mixture of known values (3, 4, 14, 24, 41, 46) for blood and lipids. In reality, values of these parameters would be expected to vary for a given subject over time and also between subjects. In addition, the site of formation of critical tissue bubbles is unclear. Not only is there a lack of knowledge about where damaging bubbles are located in the body and where they arise (42), but there are also uncertainties inherent in our calculations. Several simplifications were made as follows. First, the tissue was assumed to be perfused by an infinite number of infinitesimally small capillaries (17,18), and the tissue element was assumed to be well stirred. The overall O_{2} pressure gradient in the macro- and microsystem [Φ_{O2}(*t*)] was considered homogeneous in the simulated large population of tissue elements/bubbles. Second, we assumed a resting value of *t*
_{1/2,N2} of 360 min, which has been well documented for “standard” NASA and US Air Force altitude exposures (6-8). We then obtained a corresponding value of *t*
_{1/2,O2} (313.2 min), through a derived relationship to *t*
_{1/2,N2}(see *Eq. EA13
*). Third, blood flow in tissues and exercising skeletal muscles has not yet been measured during altitude. We therefore approximated*t*
_{1/2,N2} and*t*
_{1/2,O2} using a rough linear relationship between O_{2} consumption and corresponding hypothetical blood flow (1, 25, 26) for an average subject performing mild, moderate, and heavy exercise. Furthermore, physical properties of the tissue such as the overall solubility are modified by exercise-induced tissue hyperemia. Therefore, *Eq. EA13
* should be modified with adjustments of blood flow with exercise along with the solubility of the tissue. However, we were unable to assess the change of tissue solubility with exercise. Fourth, we neglected the possible effects of the expected local temperature rise (1–2°C) in critical tissues during the simulated submaximal exercises. From the equation of state of an ideal gas, this minor change of ≤0.64% on the Kelvin scale would have negligible effect on the bubble volume. With this small temperature rise, physical constants such as solubilities and diffusivities of gases in tissues would also be expected to remain nearly constant (24). Fifth, even though the metabolic production of CO_{2} may increase at the onset of our simulated submaximal aerobic exercise, we assumed that the dissolved CO_{2} tissue tension (Pti_{CO2}) remains approximately constant during the ensuing steady-state phase due to the concurrent decrease in arterial CO_{2} tension (Pa_{CO2}) (27). Sixth, the effects of gas diffusion and coalescence between bubbles within a unit of tissue volume were neglected. Seventh, an increase of intramuscular interstitial pressure during skeletal muscle contractions, which may affect the tissue elastic recoil (ν) and bubble growth, was not considered in our study.

### Perspectives

Our simulations suggest that exercise-induced elevation of O_{2} consumption at altitude leads to bubble persistence in tissues. At the same time, however, exercise-enhanced perfusion leads to an overall suppression of bubble growth. The total volume of bubbles would be reduced unless increased tissue motion simultaneously raises the rate of bubble formation (larger values of the Poisson process parameter β) through cavitation processes, thus maintaining or increasing total bubble volume, despite the exercise. Whether the rate of bubble formation and the incidence of DCI are associated with specific types of mechanical movement of body structures (19) remains to be investigated. Furthermore, measurements of cardiac output and local blood flows in skeletal muscles of a limb, presently under way, will provide further insight into the effects of exercise-induced circulatory changes, particularly during the O_{2} prebreathe, on the incidence of altitude bubbles and DCI.

## Acknowledgments

The authors acknowledge Drs. Bruce D. Butler, Joseph R. Rodarte, and Michael B. Reid for critically reading the manuscript. The authors thank Dr. Johnny Conkin for useful advice and Dr. Michael L. Gernhardt for the many discussions.

## Appendix

The introduces the intermediate steps necessary before calculation of R_{b}(t) using *Eq. 6
*. After a partial O_{2} pressure change, mixing of inspired O_{2}-enriched mixtures with lung resident gas is assumed to be approximately complete within 1 min. Pa
_{O2} and Pa
_{N2}are estimated from the alveolar gas equations (11, 42).

#### Calculation of L(t) and L˙(t).

Various homeostatic processes maintain m_{CO2}(*t*) and m_{H2}
_{O}(*t*) relatively constant; hence, these two terms can be neglected in the differentiation of M_{ti}(*t*) and M_{b}(*t*) in *Eq*. 5. When *Eqs.2-5
* are combined, the quantity
Equation A1and its time derivative L˙(*t*) occur explicitly in the expression for*R*
_{b}(*t*) given by *Eq. 6
*. To obtain L˙(*t*), we used *Eqs. 7
* and *
9
* to calculate
Equation A2
Equation A3

#### Calculation of K(t) and K˙(t).

The expression
Equation A4also occurs in the derivation of *Eq. 6
*. Dissolved gases surrounding the bubble add to the surface tension pressure and to the elastic recoil of the tissue, which resists expansion. With the use of the Laplace law (4, 13, 14) and application to all dissolved gases, P_{b,N2}(*t*), part of the expression of K(*t*), may be expressed as follows
Equation A5
In *Eq. EA5
*, ϕ_{1}(*t*) is the sum of pressures due to surface tension and tissue elastic recoil. With the assumption of perfect spherical bubble, ϕ_{1}(*t*) can be expressed in terms of the radius R_{b}(*t*)
Equation A6where τ is the surface tension at the bubble-tissue interface and ν is the tissue elastic recoil. The last term in *Eq.EA5
*, total pressure in the bubble due to metabolic gases, is expressed in terms of its components
Equation A7By use of *Eqs. EA5-EA7,* it follows that
Equation A8where ϕ_{1}(*t*) is readily obtained from*Eq. EA6
*. Note that Pti_{CO2} and Pti_{H2}
_{O}, being constant, do not contribute to K˙(*t*).

#### Approximation of the O_{2} pressure in the bubble.

To obtain the net flux across the diffusion barrier with *Eq.3
*, we need P_{b,O2}(t). According to Fick's first law (33), the flux of O_{2} passing through the surface of the boundary layer at *time t*,*Ĵ*
_{O2}, is approximately given by
Equation A9From the law of conservation of mass (14, 33), we have
Equation A10It has been shown that, for N_{2}, linear gas exchange kinetics in tissues are invoked for long half times for which the tension of dissolved gases exceeds ambient pressure (30,31). With the assumption of similar kinetics for O_{2}, M_{ti,O2}(*t*) is approximately linear in*t* for *t* > *t*
_{b}. Therefore, we readily obtainM˙_{ti,O2}(*t*) ≅ M_{ti,O2}(*t*)/(*t* −*t*
_{b}); M_{ti,O2}(*t*
_{b}) = 0. Also, according to Henry's Law, we can write
Equation A11From *Eqs. EA9-EA11
*, it follows that
Equation A12(Recall that h = Vti/V_{b} is assumed constant.)

#### Relationship between O_{2} and N_{2} tissue half times.

Let Q˙ti(*t*) be the blood flow in the homogeneous tissue with volume Vti(*t*) at *time t*. Also let*s*
_{b,N2} and*s*
_{ti,N2} be the solubilities of N_{2} in blood and tissue, respectively, and let*s*
_{b,O2} and*s*
_{ti,O2} be similar characteristics for O_{2} (Table 2). The half times k_{1} and k_{2} for N_{2} and O_{2} (4,14) are then given by
Equation A13where *i* = 1 or 2. Therefore, given k_{1}, we may obtain k_{2} = Ωk_{1}without knowledge of blood flow and tissue shell volume, where Ω is a constant [*t*
_{1/2,O2} = (1/Ω)*t*
_{1/2,N2} with 1/Ω = 1/(0.0227/0.02) × (0.0150/0.146) ≈ 0.87].

#### Generation of simulated bubble creation times.

In terms of our application, Parzen's result can be stated as follows: “Given N_{i}(T), the bubble creation times*t*
_{b,ij} are distributed as order statistics corresponding to N_{i}(T) independent random variables with common cumulative distribution function F(*t*) = m(*t*)/m(T) (*t*
_{alt} ≤ *t* ≤ T).” In accordance with this result, we generated the unordered times *t*
_{b,ij}
^{′} = F^{−1}(U_{ij}) = −(1/β)log[1 − U_{ij}(1 − e^{−βT})], where the U_{ij} were independent uniform (0,1) random variates [j = 1,…,N_{i}(T)]. The*t*
_{b,ij}
^{′} were then sorted in ascending order to obtain the *t*
_{b,ij}.

## Footnotes

This study was supported by National Aeronautics and Space Administration Cooperative Agreement NCC9-58.

Address for reprint requests and other correspondence: P. P. Foster, Pulmonary and Critical Care Section, Dept. of Medicine, Baylor College of Medicine, 6550 Fannin St., Smith Tower, Suite 1225, Houston, TX 77030 (E-mail: philipf{at}bcm.tmc.edu).

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

*advertisement*” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

- Copyright © 2000 the American Physiological Society