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Departments of 1 Physiology and Biophysics and 2 Chemical Engineering, University of Washington, Seattle, Washington 98195
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ABSTRACT |
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Top
Abstract
Introduction
References
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We reported changes in alveolar-arterial PO2 gradient, ventilation-perfusion heterogeneity, and arterial-alveolar PCO2 gradient during partial liquid ventilation (PLV) in healthy piglets (E. A. Mates, P. Tarczy-Hornoch, 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 steady-state (SS) gas exchange conditions during PLV and to estimate the contribution of diffusion limitation to SS arterial-alveolar 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. Time-dependent solutions of both models show that SS is established for various inert and respiratory gases within 5-150 s. In fluid-filled unventilated terminal units, all times to SS increased sometimes by hours, e.g., SF6 exceeded 4 h. SS solutions for the ventilated spherical model predicted minor end-capillary disequilibrium of inert gases and significant disequilibrium of respiratory gases, which could explain a large portion of the arterial-alveolar PCO2 gradient measured during PLV (14). We conclude that, during PLV, diffusion gradients for gases are generally small, except for CO2.
liquid breathing; perfluorocarbon liquids; mathematical model; gas exchange
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INTRODUCTION |
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Top
Abstract
Introduction
References
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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 O2-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 arterial-alveolar PO2
and PCO2 gradients
[(A-a)DO2 and
(a-A)DCO2] (12-14). Compared
with conventional gas ventilation with 100% O2, there was
a 50% increase in ventilation-perfusion (
A/
) heterogeneity and
a 50% increase in O2 shunt, both of which can contribute
to the alveolar-arterial 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 alveolar-arterial differences in
healthy animals during PLV. To test the feasibility of this hypothesis,
we developed two mathematical models of gaseous diffusion in partially
PFC-filled lung subunits.
In our experimental studies we used the multiple inert gas elimination
technique (MIGET) to measure
A/
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., Pa/
and
PE/,
where Pa, , and
PE are arterial, mixed venous, and expired
pressures, respectively) does not change with time and there is no
further storage or net loss of mass within the lung over time. Using a
very simple model, we showed previously that the time to steady state
for SF6 (a gas used in MIGET to estimate shunt) was
prohibitively long because of its high solubility in PFC vs. blood
(13). This required us to modify MIGET by eliminating SF6
from the analysis, inasmuch as it did not satisfy the underlying
assumption that steady-state conditions exist (13). With the more
sophisticated models described here, we were able to refine and verify
these original predictions and further explore the effects of PFC on
attainment of steady-state gas exchange for the remaining five inert
gases as well as O2 and CO2. We are also able
to explore the effect of PFC dose on diffusion-limited gas transport in
the alveolus.
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 liquid-filled lungs. Many of the equations in traditional gas exchange theory are based on the assumptions that steady-state 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 fluid-filled lung. Despite mild increases in (A-a)DO2 and (a-A)DCO2 during PLV in healthy animals, oxygenation and ventilation can be achieved surprisingly well through a liquid-filled lung. The success of PLV in a clinical setting may depend on altering our thinking about shunt and dead space when we add a high-solubility fluid with diffusion resistance to the air space of the lung. The models described here have been helpful in exploring these ideas.
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MATHEMATICAL MODELS |
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In prior publications we presented two different models of gas exchange during PLV: 1) a two-compartment well-mixed model used to estimate times to steady state (13) and 2) a spherical gas exchange model used to estimate steady state arterial-alveolar differences across a PFC diffusion barrier (14). Here we expand on both models, adding a gas compartment to the one-dimensional well-mixed model, providing time- and space-dependent 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 O2, CO2, 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 PFC-filled alveolus probably includes some features of both. The well-stirred 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 · Torr1)
C
Concentration of a tracer gas in a solvent (ml gas/ml solvent)
D
Molecular diffusion coefficient (cm2/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
Blood flow (ml/s)
RR
Respiratory rate (min1)
r
Radial distance from center of gas compartment (cm)
rc
Radius of gas exchange unit at the capillary boundary (cm)
rg
Radius of gas compartment (cm)
t
Time (s)
Time constant (s)
T
Temperature (°C or K)
T98
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 PFC-gas interface are
negligible. Because the presence of the PFC in the alveolar space does
not affect gas exchange properties of the alveolar-capillary membrane,
the assumption of complete equilibrium across the membrane is as valid
as in the gas-filled lung. Blood flow and ventilation are assumed
continuous and nonpulsatile (i.e., and
A are constant).
of tracer gases was assumed
to be constant, and variation in body tissue partial pressures was
assumed to be negligible. In the experimental situation,
of the inert gases will vary slightly with time as the body comes to a new steady state after a
perturbation in gas exchange. We believed that this variation was
small, inasmuch as the body tissues were previously equilibrated with
inert gas and the recirculated component is a small fraction of the
total . The error introduced
by this assumption will lead to a slight underestimation of the true
time to reach steady state.
Time-dependent gas exchange in a well-stirred three-compartment
model.
Figure 1 schematically describes this
model, in which blood is delivered to the capillary compartment at a
flow rate (ml/s) and ventilation through the gas
compartment occurs at a rate A (ml/s). A
tracer gas may enter the gas exchange unit dissolved in blood at
partial pressure or via
ventilation at partial pressure Pgi. Mass balance for the
tracer in three compartments is given by Eq. 1 with the
assumption that the PFC layer is well mixed. Thus the gas partial
pressures in all compartments are equal (i.e.,
Pc = Ppfc = Pg) at time
t
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(1) |
· P) to partial pressures (P),
applying the assumption Pc = Ppfc = Pg (i.e., well-mixed with no diffusion gradients), and
rearranging into the standard form for a first-order differential
equation · 
+ P = K (where
is the time constant and K is the steady-state asymptotic
value of P)
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(2) |
can be expressed as follows
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(3) |
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(3b) |
A · Pgi · (g/b) + · ]/[A
· (g/b) + ].
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, the time for the exponential term to decrease by
63%. At 4, steady-state equilibrium is >98% complete. The standard MIGET theory assumes that Pc = Pg = constant; i.e., after a change in the infusate, the time at which gas
exchange measurements are taken is much longer than , so the
exponential term in Eq. 3 becomes negligible.
Equation 3 shows that when PFC is present in the alveolus and
pfc > b, is prolonged, especially
if pfc is greater than both b and
g. For gases in which this holds true, larger volumes of
PFC result in longer times to equilibrium. For O2, is
actually prolonged in the absence of PFC, because g > pfc. It is also prolonged as
A approaches zero (i.e., shunt), because
PFC must equilibrate to a higher final value, i.e., input partial
pressures or
Pgi. When A is nonzero, the
steady-state partial pressure (K) is less than input partial
pressure and is accordingly shorter. Increasing
A or
shortens for all gases.
Time-dependent 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 gas-filled center. The branching, space-filling nature of lung architecture is too complex for small-scale 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 surface-to-volume 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.
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r2c. The area of the inner gas
space (4r2g) depends on the
volume of PFC administered and on total lung volume. PFC is assumed to
distribute uniformly as a spherical shell with the ventilated gas
"hole" in the center. As the hole radius approaches zero, the
unit becomes "flooded" with PFC. As rg
approaches rc the model represents a gas-filled
lung with no diffusion gradient (see PARAMETER ESTIMATES
for description of actual dimensions used).
We use three coupled differential equations to describe mass flux
between blood, PFC, and gas. Equation 4 represents the rate of
change of mass ( · 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. Equation 5 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 air-liquid interface from the PFC layer adjacent
to the
compartment
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(4) |
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(5) |
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(6) |
Steady-state gas exchange in a spherical shell with radial diffusion. Under steady-state 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) steady-state 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 blood-gas partial pressure differences
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(7) |
pfc · dPpfc/dr at
r = rc and also at r = rg, and then substituting into Eqs. 4 and 6 gives
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(8) |
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(9) |
· (rc · rg)/(rc 
rg) · pfc/g
and solving for Pc and Pg
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(10) |
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(11) |
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(12) |
b/(b + A/),
where b = b/g.
Notice that the capillary-to-gas partial pressure
(Pc Pg) difference is dependent on the
absolute values of A and
.
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PARAMETER ESTIMATES |
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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. Haefeli-Bleuer
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
end-inspiratory rc of 300 µm. Surface area and
volume of a single spherical unit are therefore 0.0113 cm2
and 0.000113 cm3, respectively. The number (n) of
terminal sacs or gas exchange units in a piglet lung was then
determined by the ratio of end-inspiratory lung volume [(48
ml/kg) × (2.5 kg) = 120 ml at rc = 300
µm] to gas exchange unit volume (1.13 × 105 ml):
1,062,000 units/lung, which we rounded to 1 × 106.
End-inspiratory 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
end-expiratory pressure of 5 cmH2O used in all our
experimental work (12). If there are 20 alveoli per terminal gas
exchange unit, there would be ~20 × 106 alveoli/piglet.
Lung volume is obviously not constant throughout the respiratory cycle.
We evaluated the steady-state model (Eqs. 10 and 11)
for several lung volumes in the range of tidal breathing, i.e.,
rc of 270 and 300 µm, to illustrate the impact
of lung volume on (A-a)DO2 and
(a-A)DCO2. We did not simulate
tidal breathing in the sense of second-to-second variation in
g.
Ventilation per gas exchange unit (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 min1, M of 2.5 kg, and n as follows:
A = (VT VD) · RR · M/n = 8.74 × 106 ml/s. Blood flow per gas exchange unit
() was derived from average piglet cardiac output of
500 ml/min (12) divided by n: 8.33 × 106 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 cm2) and have a thickness
equivalent to the red cell diameter (5 µm), giving a
Vc per unit of 4.24 × 106 ml. The volume
of PFC per gas exchange unit (Vpfc) was determined from the
total dose of PFC divided by n. For example, a dose of 30 ml/kg
in a 2.5-kg piglet results in a total dose of 75 ml, or
Vpfc of 7.5 × 105 ml/unit. PFC layer
thickness is dependent on rc 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 matched
A/ and Vpfc
of 30 ml/kg are summarized in Table 1.
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Values of b and pfc for inert gases were
obtained from experimental measurements of gas solubility in pig blood
and in the PFC perflubron (C8F17Br,
LiquiVent, Alliance Pharmaceutical, San Diego, CA) (12). The
"solubility" of a tracer gas in the gas phase (g)
is defined in the classic paper by Piiper et al. (16) as 0.00132 Torr1 (=1/760 at sea level).
The solubility of O2 and CO2 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. O2 combines with Hb, resulting in an S-shaped
concentration vs. pressure curve in the partial pressure range
0-150 Torr. For PO2 >150 Torr, the concentration vs. partial pressure curve is linear, because Hb is
saturated, and for O2, b is the same as in
plasma: 0.003 ml · 100 ml
solvent1 · Torr1. For
PO2 <150 Torr, b for
O2 is much higher; e.g., at PO2 of 40 Torr it is 0.06 ml · 100 ml
solvent1 · Torr1 as
determined by the slope of the O2 content
(CO2, ml O2/100 ml blood) vs.
PO2 (Torr) curve generated by the subroutines
of Olszowka and Farhi (15). For the steady-state partial pressure differences calculated using Eqs. 10 and 11, we used
only b for O2 of 0.003, because for all the
experimental data against which we are comparing model results arterial
PO2
(PaO2) was >150 Torr (12).
The solubility of CO2 in blood is a function of dissolved CO2 as well as CO2 converted to
HCO3. The content (CCO2) vs. PCO2 curve
is approximately linear within 40-80 Torr PCO2. With use of the blood-gas routines of
Olszowka and Farhi, b for CO2 was determined
from the slope of CCO2 vs.
PCO2 over this range and was found to be 0.779 ml · 100 ml
blood1 · Torr1.
O2 and CO2 solubilities in PFC were provided by
Alliance Pharmaceutical (Table 2).
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Few molecular diffusion coefficients (Dpfc) of
dissolved gases in PFC are precisely known. Tham et al. (19) measured
Dpfc of O2 and CO2 in three
perfluorochemicals (Caroxin-D, Caroxin-F, and FC-80), finding the
average diffusion coefficient for O2 in PFC to be 5.61 × 105 cm2/s at 37°C with a range of
5.57-5.65 × 105 cm2/s and for
CO2 in PFC at 37°C to be 4.36 × 105
cm2/s with a range of 4.21-4.48 × 105
cm2/s. The diffusion coefficients of O2 and
CO2 in H2O at 37°C are 3.3 × 105 and 2.6 × 105 cm2/s,
respectively (6).
We used the average value of the CO2 diffusion coefficient
as measured by Tham et al. (19) to estimate Dpfc 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
H2O at 37°C are 1.63 × 105
cm2/s for SF6, 1.96 × 105
cm2/s for ethane, 1.84 × 105
cm2/s for cyclopropane, 1.28 × 105
cm2/s for halothane, 0.85 × 105
cm2/s for ether, and 1.62 × 105
cm2/s for acetone (17, 21). Because their diffusivities in
H2O are only slightly less than those of CO2 in
H2O, we chose the value of Dpfc for
CO2 in PFC to represent the diffusivity of the six inert
gases in the absence of experimental data.
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RESULTS |
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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 space-dependent solutions for the spherical model converged on the analytic steady-state solutions. For each of the eight gases simulated, the time to steady-state 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 steady-state equilibrium.
Figure 3 illustrates the time rate of
change of partial pressure of the eight gases in the simpler well-mixed
three-compartment model with Vpfc of 30 ml/kg after a step
change in the input partial pressure of each gas. For O2
this involved setting Pgi at 650 Torr and
at 40 Torr and for the
remaining 7 gases Pgi at 0 Torr and
at 1 Torr at time 0. Figure 3A illustrates the application of Eq. 3 for
normal conditions of matched A and
(A/ = 1).
Figure 3B illustrates the same for near-zero ventilation (shunt conditions). Because O2 is delivered by ventilation,
PO2 was not simulated for shunt conditions.
Whenever A is negligible, Eq. 3
shows that the final value is always
, and the time constants are
lengthened. Both features are apparent in Fig. 3B. Gases with the lowest b/pfc ratio (i.e.,
SF6) take the longest to equilibrate, because PFC acts as a
large capacitor that fills slowly when there is great disparity in
solubilities.
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A and (as
described in PARAMETER ESTIMATES). Figure 4A shows successive time traces of PCO2 vs. radial
distance from the capillary through PFC to the central gas region.
After a step change in from
0 to 40 Torr, PCO2 increases in the gas
exchange unit until it converges on the steady-state value. Figure
4B shows similar successive time traces of
PO2 vs. radial distance through the PFC after a
step change in Pgi from 0 to 650 Torr.
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in Eq. 3; for the spherical model it was the
time at which the time-dependent solutions (Eqs. 4-6)
converged to 98% of the analytic steady-state solutions (Eqs.
10 and 11). We evaluated the model for three conditions
to illustrate the range of T98 likely to be
encountered in the lung partially filled with PFC: matched
A and ,
A approximately zero with
normal (shunt), and near zero
with A normal (dead space).
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A and well-matched
(A/ = 1), all times
to steady state were <3 min. The gas with the longest time to steady
state was cyclopropane followed by ether, SF6, halothane,
O2, CO2, and acetone. Under shunt conditions
all times to steady state were prolonged (except for acetone, which is
insensitive to shunt), with SF6 having the longest times at
~5 h. The time to steady state for O2 was also markedly
prolonged at ~27 min, whereas that for CO2 remained short
at 15-20 s. Under dead space conditions the times were
intermediate, with the longest being for acetone at ~26 min.
CO2 equilibration times were mildly prolonged under these
conditions, ~95 s.
Steady-state gas exchange in a PFC-filled spherical shell.
Steady-state partial pressure differences of inert and respiratory
gases were calculated from Eqs. 10 and 11.
SF6 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
Pc-Pg differences of seven gases
normalized by input partial pressure
( for the 5 inert gases and
CO2 and Pgi for O2) vs. rg for A of 8.74 × 106 ml/s and of 8.33 × 106 ml/s. An rg of 0 corresponds to
a flooded terminal sac with no gas compartment, and
rg of 300 µm corresponds to a gas exchange unit
with no PFC. Values of rg equal to 210, 250, and
280 µm correspond to the three doses of PFC used in our experimental
work: 30, 20, and 10 ml/kg, respectively (12). CO2 shows
the largest difference at all values of rg, with
the Pc-Pg difference nearly 10% of the input
pressure at rg of 210 µm. The partial pressure
gradient of O2 is very low at the same dose (<1% of
Pgi), rising only when rg becomes very
small as the gas exchange unit becomes flooded with PFC. The inert
gases also show a negligible Pc-Pg difference for rg of 210 µm, with halothane having the
largest Pc-Pg difference at 3% of
followed by cyclopropane,
ethane, acetone, and ether. As the PFC layer increases in thickness,
the Pc-Pg difference rises exponentially,
approaching values for
CO2 and the inert gases and Pgi for
O2. For these simulations,
for O2 was set to zero and Pgi to 1 for the sake of comparison.
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of 40 Torr and 30 ml/kg PFC in the lung varied from 3.7 Torr at the
large lung volume to 9.6 Torr at the lower lung volume. At the small
PFC dose of 10 ml/kg, the Pc-Pg difference for
CO2 varied from 0.8 to 1.2 with the change in lung volume.
We previously showed (a-A)DCO2 in
healthy animals with 30 ml/kg PFC in the lungs to be 12 Torr (12). The
difference for Pc-Pg difference for O2 varied in a similar manner with an increase from 6 to 18 Torr as the gas exchange unit volume decreased with 30 ml/kg PFC in the
lung and Pgi of 650 Torr.
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A and
independently on the Pc-Pg
difference for inert gases, O2, and CO2. At PFC
thicknesses up to 100 µm (PFC dose ~30 ml/kg), varying
A and had a small impact on MIGET gas Pc-Pg differences. At PFC
thicknesses >100 µm, the gradients increased exponentially, as in
the case of matched A and
(Fig. 5). The Pc-Pg
difference for the inert gases never exceeded 10% of
over this range of
A and . Each
gas was affected to a different degree depending on their relative solubilities. Figure 7 illustrates the
effect of varying A and on O2 and CO2. Figure
7A shows minimal effect on the Pg-Pc difference for O2 with varying
A over a range from 0.1 to 10 times the
average ventilation of a terminal alveolar sac
(8.74 × 106 ml/s) with fixed
(8.33 × 106 ml/s). Figure 7B shows an 8-fold
increase in the Pg-Pc
difference for O2 with a 10-fold increase in
. The gradient drops to near zero as
decreases to 0.1 its average value. Changes in
partial pressure differences of CO2 with varying
A and are shown in
Fig. 7, C and D. There is a 3-fold increase in the
Pc-Pg difference with
A 10 times its average value, and the
gradient drops to near zero with A at 0.1 its average value. The Pc-Pg difference for
CO2 drops in half with a decrease in but
is essentially unchanged with a 10-fold increase in .
Comparison of the solutions 10 × A and
0.1 × in Fig. 7, A and B, as well
as 7, C and D, illustrates that the
Pc-Pg gradient is different for each condition, despite equivalent A/
ratios.
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DISCUSSION |
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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 PFC-containing alveolus reach steady state at usual respiratory rates? 2) How large are the alveolar-arterial differences as a result of diffusion across PFC barriers? Two different models were developed in an attempt to answer these questions. The well-mixed three-compartment 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 gas-exchanging 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 well-mixed compartment model predictions of time to steady state (Table 3).
Both models depict gas exchange in a single terminal alveolar sac. Parameters such as, A,
and Vpfc were arrived at by partitioning an equal amount of
, T, and
Vpfc 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
Pc-Pg differences due to diffusion limitation
and provide a gross approximation to overall lung arterial-alveolar 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 cup-shaped alveoli opening up to a common duct.
There are sheets of perfused alveolar-capillary membrane extending
inward toward the duct that increase the surface area for exchange and
bring those parts of the membrane close to the PFC-gas interface. This
would be equivalent to "thinning" the PFC spherical shell in our
model and decreasing the Pc-Pg gradient for
ventilated units. On the other hand, regions of shunt where
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 CO2 had
similar diffusion coefficients in H2O 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 Dpfc for O2 and
CO2 in PFC compared with H2O 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 Dpfc, model results show little
dependence of our time- or space-dependent solutions on diffusive
resistance. As we demonstrate, the disparity in
Pc-Pg gradients for different gases with the
same Dpfc (i.e., CO2 vs. ether)
supports the conclusion that minor variations in Dpfc 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 assess
A/ heterogeneity in
healthy piglets during PLV, we modified the standard protocol (9, 20)
to incorporate a 60-min 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 SF6 under shunt conditions (>4 h). We
showed this previously and eliminated SF6 from MIGET
analyses during PLV (12, 13). The remaining five inert gases reach
steady state within the 1-h time frame. The next longest equilibration time was for acetone, which took ~26 min to come to steady state under "dead space" conditions ( = 0). Except
for one case, O2 and CO2 reached steady state
in <2 min for the range of possible A
and that might exist during PLV. O2 took
26.6 min to reach steady state under shunt conditions because of the
slow delivery rate.
A and in our
model, but this would be a useful extension. It would be interesting to
see if O2 and CO2 reach steady state, despite
breath-to-breath variations in Pg and pulsatile changes in
Pc that occur over a 2- to 4-s time period. Intuition leads
us to think that a gas exchange unit would reach steady state about an
average value of Pc and Pg, filtering out
second-to-second fluctuations.
Partial pressure differences at steady state. PFC acts as a mild diffusion barrier for all gases in the steady state, creating a Pc-Pg 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 steady-state gas exchange has several important consequences. Gas exchange efficiency is reduced with overall arterial-alveolar partial pressure gradients increased compared
or Pgi) with
Vpfc = 30 ml/kg and rc = 300 µm
)
and a low gas-exchange unit volume (
) for
