Abstract
We reported changes in alveolararterial Po _{2} gradient, ventilationperfusion heterogeneity, and arterialalveolar Pco _{2} gradient during partial liquid ventilation (PLV) in healthy piglets (E. A. Mates, P. TarczyHornoch, J. Hildebrandt, J. C. Jackson, and M. P. Hlastala. In: Oxygen Transport to Tissue XVII, edited by C. Ince. New York: Plenum, 1996, vol. 388, p. 585–597). Here we develop two mathematical models to predict transient and steadystate (SS) gas exchange conditions during PLV and to estimate the contribution of diffusion limitation to SS arterialalveolar differences. In the simplest model, perfluorocarbon is represented as a uniform flat stirred layer and, in a more complex model, as an unstirred spherical layer in a ventilated terminal alveolar sac. Timedependent solutions of both models show that SS is established for various inert and respiratory gases within 5–150 s. In fluidfilled unventilated terminal units, all times to SS increased sometimes by hours, e.g., SF_{6} exceeded 4 h. SS solutions for the ventilated spherical model predicted minor endcapillary disequilibrium of inert gases and significant disequilibrium of respiratory gases, which could explain a large portion of the arterialalveolar Pco _{2} gradient measured during PLV (14). We conclude that, during PLV, diffusion gradients for gases are generally small, except for CO_{2}.
 liquid breathing
 perfluorocarbon liquids
 mathematical model
 gas exchange
partial liquid ventilation (PLV) is a technique of ventilatory support in which the air spaces of the lung are partially replaced with liquid perfluorocarbon (PFC) and then periodically insufflated with O_{2}enriched gas with use of a conventional mechanical ventilator. PLV was first described by Fuhrman et al. (3) and has been shown to improve oxygenation and lung mechanics in animal models and in humans with acute respiratory distress syndrome (2, 4, 8, 10, 11). We previously showed that PLV in healthy piglets causes mild increases in arterialalveolar Po _{2}and Pco _{2} gradients [(aa)Do _{2} and (aa)Dco _{2}] (1214). Compared with conventional gas ventilation with 100% O_{2}, there was a 50% increase in ventilationperfusion (V˙a/Q˙) heterogeneity and a 50% increase in O_{2} shunt, both of which can contribute to the alveolararterial difference. We hypothesized, but were unable to verify experimentally, that a diffusion barrier exists across the PFC in the lung periphery and that it is responsible for a significant portion of measured increases in alveolararterial differences in healthy animals during PLV. To test the feasibility of this hypothesis, we developed two mathematical models of gaseous diffusion in partially PFCfilled lung subunits.
In our experimental studies we used the multiple inert gas elimination technique (MIGET) to measureV˙a/Q˙heterogeneity in healthy piglets during PLV (13, 14, 20). The use of this method raised the question of whether inert and respiratory gas exchange reaches steady state during PLV within a time frame similar to conventional gas ventilation. Steady state refers to the condition in which, given a constant source of a gas infused into mixed venous blood, the ratio of input to output partial pressures across the lung (i.e., P_{a}/
In recent publications, PLV has been shown to improve gas exchange in humans with acute lung injury (4, 8, 11). We have focused our efforts on studying the effects of PLV in healthy animals to shed light on the fundamental differences in gas exchange between gas and liquidfilled lungs. Many of the equations in traditional gas exchange theory are based on the assumptions that steadystate mass flux exists and that there is a negligible diffusion barrier in the alveolus (e.g., Berggren shunt and Bohr dead space). These assumptions need to be critically evaluated in the novel situation of a fluidfilled lung. Despite mild increases in (aa)Do _{2} and (aa)Dco _{2} during PLV in healthy animals, oxygenation and ventilation can be achieved surprisingly well through a liquidfilled lung. The success of PLV in a clinical setting may depend on altering our thinking about shunt and dead space when we add a highsolubility fluid with diffusion resistance to the air space of the lung. The models described here have been helpful in exploring these ideas.
MATHEMATICAL MODELS
In prior publications we presented two different models of gas exchange during PLV: 1) a twocompartment wellmixed model used to estimate times to steady state (13) and 2) a spherical gas exchange model used to estimate steady state arterialalveolar differences across a PFC diffusion barrier (14). Here we expand on both models, adding a gas compartment to the onedimensional wellmixed model, providing time and spacedependent numerical solutions to the spherical model, and providing a full discussion of the underlying assumptions and model behavior. We explore solutions to the time rate of change of partial pressures of O_{2}, CO_{2}, and six MIGET gases in PFC after a step change in input partial pressures.
A comparison of two separate model configurations is particularly enlightening, since the in vivo PFCfilled alveolus probably includes some features of both. The wellstirred compartment model reflects a PFC layer with complete convective mixing and no diffusion limitation within the gas exchange unit, whereas the spherical shell model imitates a perfectly still diffusion barrier interposed between gas and blood. The true nature of gas exchange in PFC lies somewhere between these models. With each breath, PFC probably moves in and out of some alveoli and small airways and exists as small stagnant puddles in others.
Glossary
Solubility of a tracer gas in a solvent (ml gas ⋅ 100 ml solvent^{−1} ⋅ Torr^{−1})
C
Concentration of a tracer gas in a solvent (ml gas/ml solvent)
D
Molecular diffusion coefficient (cm^{2}/s)
M
Mass of tracer gas in a solvent (ml gas)
MIGET
Multiple inert gas elimination technique
n
Number of gas exchange units in a piglet lung
P
Partial pressure of a tracer gas (Torr)
PFC
Perfluorochemical
PLV
Partial liquid ventilation
Q˙
Blood flow (ml/s)
RR
Respiratory rate (min^{−1})
r
Radial distance from center of gas compartment (cm)
r _{c}
Radius of gas exchange unit at the capillary boundary (cm)
r _{g}
Radius of gas compartment (cm)
t
Time (s)
τ
Time constant (s)
T
Temperature (°C or K)
T _{98}
Time to 98% of steady state (s)
Vd
Dead space (ml/breath)
Vt
Tidal volume (ml/breath)
a
Arterial
a
Alveolar
b
Blood
c
Capillary
g
Alveolar gas
gi
Inspired gas
pfc
Perfluorocarbon
Mixed venous
Model assumptions.
In both models we assumed that the blood and gas compartments on either side of the PFC are well mixed. The models also assume that diffusion barriers at the capillary membrane and the PFCgas interface are negligible. Because the presence of the PFC in the alveolar space does not affect gas exchange properties of the alveolarcapillary membrane, the assumption of complete equilibrium across the membrane is as valid as in the gasfilled lung. Blood flow and ventilation are assumed continuous and nonpulsatile (i.e., Q˙ andV˙a are constant).
Timedependent gas exchange in a wellstirred threecompartment model.
Figure 1 schematically describes this model, in which blood is delivered to the capillary compartment at a flow rate Q˙ (ml/s) and ventilation through the gas compartment occurs at a rate V˙a (ml/s). A tracer gas may enter the gas exchange unit dissolved in blood at partial pressure
The rate at which P_{pfc} approaches steadystate equilibrium is determined by τ, the time for the exponential term to decrease by 63%. At 4τ, steadystate equilibrium is >98% complete. The standard MIGET theory assumes that P_{c} = P_{g} = constant; i.e., after a change in the infusate, the time at which gas exchange measurements are taken is much longer than τ, so the exponential term in Eq. 3 becomes negligible.
Equation 3
shows that when PFC is present in the alveolus and β_{pfc} > β_{b}, τ is prolonged, especially if β_{pfc} is greater than both β_{b} and β_{g}. For gases in which this holds true, larger volumes of PFC result in longer times to equilibrium. For O_{2}, τ is actually prolonged in the absence of PFC, because β_{g} > β_{pfc}. It is also prolonged asV˙a approaches zero (i.e., shunt), because PFC must equilibrate to a higher final value, i.e., input partial pressures
Timedependent gas exchange in a spherical shell with radial diffusion.
To simulate gas exchange in a functional subunit of lung (Fig.2), we chose a spherically shaped structure with an outer layer of capillary blood surrounding a layer of PFC that, in turn, surrounds a gasfilled center. The branching, spacefilling nature of lung architecture is too complex for smallscale mathematical modeling. We chose to model gas exchange at the level of the terminal alveolar duct and represented them as smooth spheres. If the anatomic subunit is larger than this, the surface area of a smooth sphere would greatly underestimate the surfacetovolume ratio. On the other hand, representing a structure as small as an alveolus by a closed sphere would overestimate the ratio, since alveoli are roughly hexagonal cups. We therefore compromised on a structure the size of a single terminal alveolar sac to be portrayed by a sphere with dimensions derived accordingly.
We assumed that the capillary and alveolar gas compartments were individually well mixed and that uniform radial diffusion occurred in the PFC. Mass exchange between the compartments is dependent on the interfacial area bounding two adjacent regions. The area of the capillaryPFC interface is fixed at 4π
We use three coupled differential equations to describe mass flux between blood, PFC, and gas. Equation 4
represents the rate of change of mass (β ⋅ V ⋅ P) of a dissolved gas in the capillary blood compartment. It is equal to the rate of gas delivery to the capillary space via blood flow, the rate of gas removal via blood flowing out of the capillary, and the rate of diffusive gas flux across the alveolar capillary membrane into the PFC. Equation5
describes radial diffusion in the PFC shell, which has spherical symmetry (1). Equation 6
represents the rate of change of mass in the central air space determined by addition of gas via inspiration, subtraction of gas removed by expiration, and subtraction of gas diffusing across the airliquid interface from the PFC layer adjacent to the compartment
The system of three partial differential equations was solved numerically to determine the partial pressure profiles in the PFC layer from the capillaryPFC interface to the PFCgas interface. Spatial derivatives were determined by finite difference, and time derivatives were solved using LSODE, a timeintegrating algorithm developed by Hindmarsh (7). The executable program was submitted as a batch job in which each simulation was solved numerically using an IBM model RS6000 computer running Unix version 4.2. P_{c} and P_{g}are equal to P_{pfc}(r) at the r _{c}and r _{g} boundaries. The time to steadystate equilibrium (T _{98}) was defined as the time for the numerical solutions to converge to 98% of the analytically determined P_{c} and P_{g} for a steadystate gas diffusion in a spherical shell, as defined by Crank (1) (see Eqs. 711 ). The two calculated mass flow rates across the capillaryPFC and PFCgas boundaries were nearly equal at “steady state” by use of these criteria.
Steadystate gas exchange in a spherical shell with radial diffusion.
Under steadystate conditions, the time rate of change of compartmental partial pressures is zero and mass flow is equal across all boundaries. We used Crank’s (1) steadystate solution to Eq. 5
describing the concentration profile as a function of radial position [C(r)] in a spherical shell to simplify the above system of equations and to analytically calculate bloodgas partial pressure differences
PARAMETER ESTIMATES
Parameter values were chosen to correspond to the dimensions of lung structure and function of healthy piglets weighing 2–4 kg. Piglets this size typically have a functional residual capacity of 30 ml/kg and respiratory rate (RR) of 20 breaths/min. For calculation purposes, an average weight of 2.5 kg was used. As discussed above, our gas exchange unit represents a terminal sac in the lung of a piglet. HaefeliBleuer and Weibel (5) measured the outer diameter of human terminal sacs (an alveolar duct plus 2 alveoli in total width) to be 656 ± 127 μm. Tenney and Remmers (18) showed that species variation in alveolar diameter was correlated to metabolic rate per unit body weight, with adult pig alveolar diameter ∼91% of the diameter of human alveoli (656 × 0.91 = 597 μm). On the basis of these data we chose an endinspiratory r _{c} of 300 μm. Surface area and volume of a single spherical unit are therefore 0.0113 cm^{2}and 0.000113 cm^{3}, respectively. The number (n) of terminal sacs or gas exchange units in a piglet lung was then determined by the ratio of endinspiratory lung volume [(48 ml/kg) × (2.5 kg) = 120 ml at r _{c} = 300 μm] to gas exchange unit volume (1.13 × 10^{−5} ml): 1,062,000 units/lung, which we rounded to 1 × 10^{6}. Endinspiratory lung volume was determined as the sum of functional residual capacity lung volume (30 ml/kg), tidal volume (Vt, 15 ml/kg), and 3 ml/kg associated with positive endexpiratory pressure of 5 cmH_{2}O used in all our experimental work (12). If there are 20 alveoli per terminal gas exchange unit, there would be ∼20 × 10^{6} alveoli/piglet. Lung volume is obviously not constant throughout the respiratory cycle. We evaluated the steadystate model (Eqs. 10 and 11 ) for several lung volumes in the range of tidal breathing, i.e.,r _{c} of 270 and 300 μm, to illustrate the impact of lung volume on (aa)Do _{2} and (aa)Dco _{2}. We did not simulate tidal breathing in the sense of secondtosecond variation inV˙_{g}.
Ventilation per gas exchange unit (V˙a) was determined using our typical experimental Vt of 15 ml/kg (12), estimated dead space (Vd) of 4.5 ml/kg, RR of 20 min^{−1}, M of 2.5 kg, and n as follows:V˙a = (Vt − Vd) ⋅ RR ⋅ M/n= 8.74 × 10^{−6} ml/s. Blood flow per gas exchange unit (Q˙) was derived from average piglet cardiac output of 500 ml/min (12) divided by n: 8.33 × 10^{−6} ml/s. The capillary blood volume was derived on the basis of anatomic data that show pulmonary capillaries to cover 75% of the alveolar surface (i.e., capillary surface area per spherical model unit = 0.75 × 0.0113 cm^{2}) and have a thickness equivalent to the red cell diameter (5 μm), giving a V_{c} per unit of 4.24 × 10^{−6} ml. The volume of PFC per gas exchange unit (V_{pfc}) was determined from the total dose of PFC divided by n. For example, a dose of 30 ml/kg in a 2.5kg piglet results in a total dose of 75 ml, or V_{pfc} of 7.5 × 10^{−5} ml/unit. PFC layer thickness is dependent on r _{c} and the volume of PFC present, with the assumption that PFC is distributed as a spherical shell with a gas hole in the center (Fig. 2). Normal parameter values for the spherical model under matchedV˙a/Q˙ and V_{pfc}of 30 ml/kg are summarized in Table 1.
Values of β_{b} and β_{pfc} for inert gases were obtained from experimental measurements of gas solubility in pig blood and in the PFC perflubron (C_{8}F_{17}Br, LiquiVent, Alliance Pharmaceutical, San Diego, CA) (12). The “solubility” of a tracer gas in the gas phase (β_{g}) is defined in the classic paper by Piiper et al. (16) as 0.00132 Torr^{−1} (=1/760 at sea level).
The solubility of O_{2} and CO_{2} in blood was determined by the slope of the curve of gas content vs. partial pressure. This relationship is nonlinear over the physiological range of partial pressures of these gases because of chemical binding in the blood. O_{2} combines with Hb, resulting in an Sshaped concentration vs. pressure curve in the partial pressure range 0–150 Torr. For Po
_{2} >150 Torr, the concentration vs. partial pressure curve is linear, because Hb is saturated, and for O_{2}, β_{b} is the same as in plasma: 0.003 ml ⋅ 100 ml solvent^{−1} ⋅ Torr^{−1}. For Po
_{2} <150 Torr, β_{b} for O_{2} is much higher; e.g., at Po
_{2} of 40 Torr it is 0.06 ml ⋅ 100 ml solvent^{−1} ⋅ Torr^{−1} as determined by the slope of the O_{2} content (Co
_{2}, ml O_{2}/100 ml blood) vs. Po
_{2} (Torr) curve generated by the subroutines of Olszowka and Farhi (15). For the steadystate partial pressure differences calculated using Eqs. 10
and
11
, we used only β_{b} for O_{2} of 0.003, because for all the experimental data against which we are comparing model results arterial Po
_{2}(
Few molecular diffusion coefficients (D _{pfc}) of dissolved gases in PFC are precisely known. Tham et al. (19) measuredD _{pfc} of O_{2} and CO_{2} in three perfluorochemicals (CaroxinD, CaroxinF, and FC80), finding the average diffusion coefficient for O_{2} in PFC to be 5.61 × 10^{−5} cm^{2}/s at 37°C with a range of 5.57–5.65 × 10^{−5} cm^{2}/s and for CO_{2} in PFC at 37°C to be 4.36 × 10^{−5}cm^{2}/s with a range of 4.21–4.48 × 10^{−5}cm^{2}/s. The diffusion coefficients of O_{2} and CO_{2} in H_{2}O at 37°C are 3.3 × 10^{−5} and 2.6 × 10^{−5} cm^{2}/s, respectively (6).
We used the average value of the CO_{2} diffusion coefficient as measured by Tham et al. (19) to estimate D _{pfc} of each respiratory gas in perflubron, the PFC used in our experiments. There are no experimental data available measuring diffusivity in PFC of the six inert gases used in MIGET (9, 20). Their diffusivities in H_{2}O at 37°C are 1.63 × 10^{−5}cm^{2}/s for SF_{6}, 1.96 × 10^{−5}cm^{2}/s for ethane, 1.84 × 10^{−5}cm^{2}/s for cyclopropane, 1.28 × 10^{−5}cm^{2}/s for halothane, 0.85 × 10^{−5}cm^{2}/s for ether, and 1.62 × 10^{−5}cm^{2}/s for acetone (17, 21). Because their diffusivities in H_{2}O are only slightly less than those of CO_{2} in H_{2}O, we chose the value of D _{pfc} for CO_{2} in PFC to represent the diffusivity of the six inert gases in the absence of experimental data.
RESULTS
Solutions for both of the models were well behaved with no instances of negative results or mass imbalance. Partial pressures at the boundaries between compartments were continuous. The numerically integrated time and spacedependent solutions for the spherical model converged on the analytic steadystate solutions. For each of the eight gases simulated, the time to steadystate equilibrium was estimated by two independent models, and the times generated by both models were within 30% of each other and usually within 10%.
Time to reach steadystate equilibrium.
Figure 3 illustrates the time rate of change of partial pressure of the eight gases in the simpler wellmixed threecompartment model with V_{pfc} of 30 ml/kg after a step change in the input partial pressure of each gas. For O_{2}this involved setting P_{gi} at 650 Torr and
Figure 4 demonstrates the time and space rate of change in the spherical gas exchange unit with 30 ml/kg PFC and matched V˙a and Q˙ (as described in parameter estimates). Figure 4
A shows successive time traces of Pco
_{2} vs. radial distance from the capillary through PFC to the central gas region. After a step change in
Table 3 reports the T _{98}for eight gases in each of the two models with V_{pfc} of 30 ml/kg. T _{98} values were defined slightly differently for the two models. In the wellmixed model T _{98} was defined as 4τ in Eq. 3; for the spherical model it was the time at which the timedependent solutions (Eqs. 46 ) converged to 98% of the analytic steadystate solutions (Eqs.10 and 11 ). We evaluated the model for three conditions to illustrate the range of T _{98} likely to be encountered in the lung partially filled with PFC: matchedV˙a and Q˙,V˙a approximately zero withQ˙ normal (shunt), and Q˙ near zero with V˙a normal (dead space).
For V˙a and Q˙ wellmatched (V˙a/Q˙ = 1), all times to steady state were <3 min. The gas with the longest time to steady state was cyclopropane followed by ether, SF_{6}, halothane, O_{2}, CO_{2}, and acetone. Under shunt conditions all times to steady state were prolonged (except for acetone, which is insensitive to shunt), with SF_{6} having the longest times at ∼5 h. The time to steady state for O_{2} was also markedly prolonged at ∼27 min, whereas that for CO_{2} remained short at 15–20 s. Under dead space conditions the times were intermediate, with the longest being for acetone at ∼26 min. CO_{2} equilibration times were mildly prolonged under these conditions, ∼95 s.
Steadystate gas exchange in a PFCfilled spherical shell.
Steadystate partial pressure differences of inert and respiratory gases were calculated from Eqs. 10
and
11
. SF_{6} was left out of the following discussion, since it was not included in our experimental MIGET analysis (12, 13) because of its prohibitively long time to reach steady state under shunt conditions. Figure 5 shows P_{c}P_{g} differences of seven gases normalized by input partial pressure (
Figure 6 illustrates the effect of gas exchange unit volume (“lung volume”) on partial pressure difference of O_{2} and CO_{2}. Although our model does not incorporate features of tidal breathing, we explored the effect of varying the gas exchange unit volume between the extremes of end inspiration (r
_{c} = 300 μm) and end expiration (r
_{c} = 270 μm). This might be equivalent to breathholding maneuvers at the extremes of cyclic breathing. For both gases, the P_{c}P_{g} difference increased at the lower lung volume for all PFC doses. The percent increase in the P_{c}P_{g} difference was greater with larger doses of PFC. The P_{c}P_{g} difference for CO_{2} with a
We examined the impact of varying V˙a andQ˙ independently on the P_{c}P_{g}difference for inert gases, O_{2}, and CO_{2}. At PFC thicknesses up to 100 μm (PFC dose ∼30 ml/kg), varyingV˙a and Q˙ had a small impact on MIGET gas P_{c}P_{g} differences. At PFC thicknesses >100 μm, the gradients increased exponentially, as in the case of matched V˙a andQ˙ (Fig. 5). The P_{c}P_{g}difference for the inert gases never exceeded 10% of
DISCUSSION
Evaluation of model assumptions.
We had two specific questions in mind when developing these models of gas exchange in a terminal sac filled with PFC: 1) Do gases that are exchanged in a PFCcontaining alveolus reach steady state at usual respiratory rates? 2) How large are the alveolararterial differences as a result of diffusion across PFC barriers? Two different models were developed in an attempt to answer these questions. The wellmixed threecompartment model provided a simple approach to estimating time to steady state. Its major assumptions are that neither diffusion times in the PFC nor the geometry of a gasexchanging subunit significantly affect the solutions. By contrast, our spherical model explicitly incorporated the diffusion gradients and more realistic geometry but, despite major mathematical differences, the results showed very close agreement with the wellmixed compartment model predictions of time to steady state (Table 3).
Both models depict gas exchange in a single terminal alveolar sac. Parameters such as Q˙, V˙a, and V_{pfc} were arrived at by partitioning an equal amount ofQ˙, V˙t, and V_{pfc} to all terminal sacs in the lung. The lung is not homogeneous in its distribution of any of these parameters, and application of model results to interpretation of experimental data must be done with this in mind. In reality, there will be a heterogeneous distribution of gas exchange units ranging from completely PFC filled to partially PFC filled to completely gas filled that are ventilated and perfused in some heterogeneous distribution. Measured arterial and expired gas partial pressures are weighted averages of gas exchange subunits. Model predictions of gas exchange in a single terminal sac help us explore the range of possible alveolar P_{c}P_{g} differences due to diffusion limitation and provide a gross approximation to overall lung arterialalveolar differences.
Additionally, the choice of a spherical shape of our gas exchange unit to approximate the terminal alveolar sac likely overestimates the diffusion barrier somewhat. A terminal sac is not a smooth sphere but, rather, a cluster of cupshaped alveoli opening up to a common duct. There are sheets of perfused alveolarcapillary membrane extending inward toward the duct that increase the surface area for exchange and bring those parts of the membrane close to the PFCgas interface. This would be equivalent to “thinning” the PFC spherical shell in our model and decreasing the P_{c}P_{g} gradient for ventilated units. On the other hand, regions of shunt whereV˙a is zero probably behave similarly to the model as the PFC pool equilibrates with mixed venous blood and geometry becomes irrelevant.
We feel justified in our choice of inert gas diffusion coefficients on the basis of the fact that the inert gases and CO_{2} had similar diffusion coefficients in H_{2}O and that all should have increased diffusivity in perflubron because it is a nonpolar solvent. The rate of diffusion of a molecule through a fluid medium depends on the “effective radius” of the molecule, a function of molecular size and van der Waals interactions with neighboring molecules. Increased D _{pfc} for O_{2} and CO_{2} in PFC compared with H_{2}O suggests that the molecules have smaller effective volumes in PFC because of reduced van der Waals interactions. Although there are certain to be discrepancies between the true diffusion coefficients of these gases in PFC and our approximated D _{pfc}, model results show little dependence of our time or spacedependent solutions on diffusive resistance. As we demonstrate, the disparity in P_{c}P_{g} gradients for different gases with the same D _{pfc} (i.e., CO_{2} vs. ether) supports the conclusion that minor variations inD _{pfc} will not significantly affect our model results.
Time to reach steady state.
Of particular interest to us was whether the inert gases used in MIGET would reach steady state during PLV within the time period of our experimental measurements (12, 13). In using MIGET to assessV˙a/Q˙ heterogeneity in healthy piglets during PLV, we modified the standard protocol (9, 20) to incorporate a 60min equilibration period between experimental conditions (15 min is more common). Results from our two models suggest that all gases come to equilibrium well within this time period with the exception of SF_{6} under shunt conditions (>4 h). We showed this previously and eliminated SF_{6} from MIGET analyses during PLV (12, 13). The remaining five inert gases reach steady state within the 1h time frame. The next longest equilibration time was for acetone, which took ∼26 min to come to steady state under “dead space” conditions (Q˙ = 0). Except for one case, O_{2} and CO_{2} reached steady state in <2 min for the range of possible V˙aand Q˙ that might exist during PLV. O_{2} took 26.6 min to reach steady state under shunt conditions because of the slow delivery rate.
We did not incorporate the periodic nature ofV˙a and Q˙ in our model, but this would be a useful extension. It would be interesting to see if O_{2} and CO_{2} reach steady state, despite breathtobreath variations in P_{g} and pulsatile changes in P_{c} that occur over a 2 to 4s time period. Intuition leads us to think that a gas exchange unit would reach steady state about an average value of P_{c} and P_{g}, filtering out secondtosecond fluctuations.
Partial pressure differences at steady state.
PFC acts as a mild diffusion barrier for all gases in the steady state, creating a P_{c}P_{g} difference that increases with volume of liquid in the alveolar space (Fig. 5). Less intuitive is the fact that the partial pressure gradient for each gas is different on the basis of the relative solubility of the gas in blood, PFC, and the gas phase. The presence of a partial pressure gradient in alveoli during steadystate gas exchange has several important consequences. Gas exchange efficiency is reduced with overall arterialalveolar partial pressure gradients increased compared with healthy gasfilled lung. This impacts gas exchange calculations using formulas derived for the gasfilled lung such as the Berggren shunt, Bohr dead space, and the model underlying MIGET. Each will be in error by an amount proportional to the partial pressure gradient in the alveolus.
The alveolar gas exchange model underlying MIGET (9, 20) assumes no diffusion gradient in the alveolus
Further inspection of Eqs. 10 and 11 shows that when PFC is present, differences between P_{c} and P_{g}are a function of solubility in each of the three media: D,V˙a/Q˙, andV˙a independent of Q˙. This last point is a significant departure from the theoretical framework of MIGET as well as our understanding ofV˙a/Q˙ heterogeneity as it affects gas exchange physiology. Our model shows that P_{c}and P_{g} are dependent on the absolute values ofV˙a and Q˙ during PLV. Figure 7 B shows that the P_{c}P_{g}difference for O_{2} is very sensitive to Q˙(at fixed V˙a), with the gradient widening when Q˙ is high and becoming negligible whenQ˙ is very small. Variation ofV˙a produces little change in the O_{2} gradient for given Q˙ (Fig.7 B). Partial pressure differences for CO_{2} show the opposite: increasing with high V˙a, becoming negligible with low V˙a, and changing little with Q˙ (Fig. 7, C andD). This is similar to gas exchange in gasfilled lungs where Pa_{O2} is sensitive to shunt and arterial Pco _{2} (Pa_{CO2}) is sensitive to dead space. The difference is that during PLV the absolute value of V˙a and Q˙independent of V˙a/Q˙ will affect overall gas exchange. This may be the most significant pitfall in the use of MIGET during PLV, inasmuch as it does not incorporate this feature in the basic model. The implications are that regions of higherthanaverage blood flow will have greater (aa)Do _{2} and those with higher than average ventilation result in larger (aa)Dco _{2}. Pooling of PFC in dependent regions of lung that receive a greater proportion of blood flow may exacerbate this effect.
Partial pressure gradients of O_{2} in the partially PFCfilled gas exchange unit with an inspiratory O_{2}fraction of 650 Torr ranges from 1.2 Torr for a 10 ml/kg dose at high lung volumes to 18 Torr for a 30 ml/kg dose at low lung volumes (Fig.6
A). The P_{g}P_{c} gradient for O_{2} is maximal for an unventilated pool of PFC approaching a P_{gi}
CO_{2} shows the greatest degree of disequilibrium at the level of the terminal sac due, in part, to its relative insolubility in PFC compared with blood (Table 2). CO_{2} retention has not been a problem during PLV, primarily because of ease of adjustment of ventilation to optimize CO_{2} elimination. In our experimental studies we found an increase in Pa_{CO2} during PLV when holding minute ventilation constant (1214). Figures 5, 6 B, and 7,C and D, illustrate the degree of Pco _{2} disequilibrium in the terminal alveolar sac over a range of PFC volumes, lung volumes, andV˙a and Q˙. In an “average” gas exchange unit during PLV, the P_{c}P_{g} gradient for CO_{2} was as much as 10 Torr for a 30 ml/kg dose at low lung volumes and <1 Torr for a 10 ml/kg dose at high lung volumes (Fig. 6 B). Introduction of ventilation heterogeneity broadens the range of partial pressure differences even further with a P_{c}P_{g} gradient for CO_{2} of 13.2 Torr for a 30 ml/kg dose andV˙a 10 times larger than average (Fig.7 C). Large airway mixing and heterogeneity of PFC and ventilation distribution will likely produce global (aa)Dco _{2} somewhere between these extremes. We measured an (aa)Dco _{2}of 12 Torr during PLV with 30 ml/kg PFC in healthy piglets (14). This suggests that diffusion limitation could be responsible for a significant portion of (aa)Dco _{2} in this animal model during PLV.
Dead space ventilation (V˙d, ml/min) is classically determined using mass balance and substitution to arrive at the following equation: Vd/Vt = (Pa_{CO2} − Pe _{CO2})/Pa_{CO2}, where Pe _{CO2} is expired Pco _{2}. The final form of this equation is arrived at by making the assumption that alveolar Pco _{2}(Pa _{CO2}) equals Pa_{CO2}, so that each of the terms on the righthand side of the equation are measurable. This assumption leads to significant overestimation of Vd when a diffusion gradient exists in the alveolus. For example, in a lung with true V˙d of 10% and (aa)Dco _{2} of 10 Torr (e.g., Pa_{CO2} = 40 Torr and Pa _{CO2} = 30 Torr), substitution of Pa_{CO2} for Pa _{CO2} results in an estimated Vd/Vt of 32%! One could argue that the effect of diffusion limitation on (aa)Dco _{2} is equivalent to that of Vd, and we should call this “effective dead space” just as we could call flooded alveoli “shunt” in place of diffusionlimited O_{2} exchange. The advantage of thinking about PFC as a diffusion barrier is that maneuvers can be performed to alter its effects, such as decreasing the total volume of PFC given or rotating the subject to redistribute pooled fluid. This may be preferable to increasing Vt or RR to decrease “dead space” that predisposes to barotrauma.
Summary.
Increased shunt during PLV in healthy animals (12) is due to flooded gas exchange units in which the PFCgas interface is located in small airways throughout the respiratory cycle. Alveoli that are partly filled with PFC (the airliquid interface resides inside the terminal sac) do not contribute significantly to measured (aa)Do _{2}. In contrast, any amount of PFC in alveoli causes a significant increase in (aa)Dco _{2} by virtue of the gas’s low solubility in PFC relative to blood. Thus diffusionlimited gas exchange during PLV is an important mechanism of impaired CO_{2} elimination and less important for oxygenation.
A very interesting result of our modeling effort was the realization that gas exchange during PLV is dependent on the absolute value ofV˙a and Q˙, and not simply their ratio V˙a/Q˙. This increases the complexity of gas exchange analysis and may be the most important reason why MIGET is not applicable in the analysis of gas exchange during PLV. Further work needs to be done to fully investigate this novel situation.
The results of this modeling effort reflect gas exchange in healthy, uncompromised lungs. In diffuse lung injury, gas exchange is improved during PLV (2, 4, 8, 10, 11) because of the combined effects of reduced surface tension and improved delivery of O_{2} to edematous areas of lung. We hope that this study may be used to help optimize the treatment of acute respiratory distress syndrome with PLV by illustrating some of the basic principles and limitations of gas exchange through a fluorocarbon medium. PLV is an exciting new methodology in the treatment of diffuse lung injury, and we hope this modeling effort stimulates further refinement of the technique.
Acknowledgments
This work was supported in part by National Heart, Lung, and Blood Institute Grant HL12174. We are grateful to Alliance Pharmaceutical for also supporting research efforts in this field.
Footnotes

Address for reprint requests: M. P. Hlastala, Div. of Pulmonary and Critical Care Medicine, Box 356522, University of Washington, Seattle, WA 981956522.

Present address of E. M. vanLöbenSels: Dept. of Medicine, Virginia Mason Medical Center, C8IMA, 1100 Ninth Ave., PO Box 900, Seattle, WA 98111.

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 Copyright © 1999 the American Physiological Society