Abstract
Recent computed tomography studies show that inspired gas composition affects the development of anesthesiarelated atelectasis. This suggests that gas absorption plays an important role in the genesis of the atelectasis. A mathematical model was developed that combined models of gas exchange from an ideal lung compartment, peripheral gas exchange, and gas uptake from a closed collapsible cavity. It was assumed that, initially, the lung functioned as an ideal lung compartment but that, with induction of anesthesia, the airways to dependent areas of lung closed and these areas of lung behaved as a closed collapsible cavity. The main parameter of interest was the time the unventilated area of lung took to collapse; the effects of preoxygenation and of different inspired gas mixtures during anesthesia were examined. Preoxygenation increased the rate of gas uptake from the unventilated area of lung and was the most important determinant of the time to collapse. Increasing the inspired O_{2} fraction during anesthesia reduced the time to collapse. Which inert gas (N_{2} or N_{2}O) was breathed during anesthesia had minimal effect on the time to collapse.
 nitrogen splinting
 nitrous oxide
 pulmonary collapse
 perioperative atelectasis
atelectasis develops with the induction of anesthesia. It is visible on computed tomography (CT) scans in 90% of healthy subjects as dependent lung densities (1, 610, 2427). Clinical studies indicate that absorption atelectasis plays a key role in the genesis of anesthesiarelated atelectasis (22).
Theoretical studies examining the effect of inspired gases on absorption atelectasis have suggested atelectasis is promoted by using N_{2}O, instead of N_{2}, in the inspired gas mixture; critical ventilationperfusion (V˙a/Q˙) is higher (3, 31), and, if complete airway occlusion occurs, then the time to collapse is shorter (11, 30). In contrast, the limited data available from clinical studies suggest that whether N_{2} or N_{2}O is breathed during anesthesia has little effect on the amount of atelectasis (5). The theoretical studies assume that mixed venous inert gas concentrations are constant. This is valid at “steady state” but not during the induction of anesthesia, when inert gas concentrations in mixed venous blood change rapidly, particularly if N_{2}O is administered.
This paper presents a theoretical mathematical model of the kinetics of absorption atelectasis during anesthesia that incorporates a model of peripheral inert gas exchange. This enables the kinetics to be studied during the early stages of anesthesia.
METHODS
General description of the model.
The model consists of two major compartments, lung and tissue. Before the induction of anesthesia, the lungs were modeled as an ideal lung, with normal ventilation. At the induction of anesthesia, the lung compartment is divided into two subcompartments, one of which continues to ventilate whereas the other is unventilated (see Fig.1). The ventilated lung compartment is modeled as an ideal lung, whereas the unventilated lung is modeled as a closed collapsible cavity (11). The tissue compartment models peripheral inert gas exchange, tissue O_{2}extraction, and CO_{2} production. It consists of four peripheral tissue subcompartments as described by Wagner (29).
Fig. 1.
Overview of model. Before induction (left), lung is modeled as an ideal lung, perfused by entire cardiac output (Q˙_{tot}). After induction (right), lung is divided into a ventilated lung compartment and an unventilated lung compartment. Four tissue compartments extract O_{2}, produce CO_{2}, and exchange inert gases, to produce mixed venous blood.Q˙l andQ˙p: lung and pocket Q˙, respectively;V˙co
_{2}, CO_{2} production; VRG, vesselrich group; MG, muscle group; FG, fat group; VPG, vesselpoor group.
Initial conditions are set according to the scenario to be modeled. Gas exchange from each compartment is described by a series of differential equations. Integration with respect to time allows the gas contents of the various compartments to be plotted against time; of particular interest are the changes in volume within the unventilated lung compartment.
Preinduction phase.
The preinduction phase allows the effects of preoxygenation to be examined. Anesthesia has not been induced, so airway closure has not occurred. The lungs were modeled as an ideal lung, consisting of two subcompartments, the alveolar compartment and the lung tissue compartment (see Fig. 2). The volume of the alveolar compartment (Va; 3,000 mlbtps) and that of the lung tissue compartment (Vl,ti, 600 ml) were maintained constant. Instantaneous equilibration of the gases between these two compartments is assumed. Gases dissolve in the lung tissue compartment according to their individual Ostwald solubility coefficients (λl,ti). Inspired gas reaches the lung compartment at a rateV˙ai, and alveolar gas leaves at a rateV˙ae, whereV˙ is ventilation. Mixed venous blood perfuses the lung compartment, and perfusion limitation of gas uptake is assumed. Blood flow to the lung compartment and to the peripheral tissues was maintained constant at 6 l/min, giving a blood flowtoalveolar volume ratio of 2:1. The tissue compartment consists of four peripheral tissue subcompartments (see Fig. 1): vesselrich group (VRG), muscle group (MG), fat group (FG), and vesselpoor group (VPG) as described by Wagner (29). Each has its individual blood flow, volume, and solubility for each inert gas. Perfusionlimited gas transfer of inert gases between blood and the tissue compartments is assumed. The partial pressures of the inert gases in mixed venous blood are determined by the mixing of blood leaving the four tissue subcompartments. The tissue compartment extracts 250 ml O_{2}
stpd/min and produces 200 ml CO_{2}
stpd/min.
Fig. 2.
Ventilated lung compartment. This compartment consists of alveolar gas and lung tissue subcompartments. Alveolar gas subcompartment receives inspired gas at a constant inspired alveolar ventilation (V˙ai) and is perfused by mixed venous blood. Gas exchange occurs between alveolar gas subcompartment and lung tissue subcompartment, which has a constant volume and known solubilities for each gas. Instantaneous equilibration of gases between these 2 subcompartments is assumed.V˙ae, expired alveolar ventilation; Pi, Pa, and
: inspired, alveolar, and mixed venous partial pressure, respectively; Va, volume of alveolar compartment; Vl,ti and λl,ti, volume and Ostwald solubility coefficient of lung tissue compartment, respectively.
Initial conditions are set assuming that air has been breathed until equilibration has been reached (see appendix
). The alveolar partial pressure of CO_{2}(
PACO2
) is set at 40 Torr, and other alveolar partial pressures are calculated by the ideal alveolar gas equation. The inspired ventilation to the lung compartment (i.e.,V˙ai) that satisfies these initial conditions is calculated and then maintained constant (4.376 l/min btps in the standard version of the model). Inert gas partial pressures in arterial and venous blood, and in the tissue compartments, are then set equal to alveolar values. At the start of the preinduction phase, the inspired gas composition is set either at air (no preoxygenation scenario) or to an inspired O_{2} fraction (
FIO2
) of 1.0 (preoxygenation scenario). Once initial conditions are set, the model is an “initialvalue problem,” whereby gas exchange at the different compartments is determined by a series of 12 differential equations (see appendix
). The changes in the system with time were solved with Gill’s modification of the RungeKutta method (23). The duration of the preinduction phase is set according to the scenario to be modeled; in the standard version of the model, this period was 3 min.
Postinduction phase.
At the induction of anesthesia, two changes are made to the model. First, the lung compartment is further divided into two subcompartments, one of which continues to ventilate whereas the other is unventilated (see Fig. 1). Second, the inspired gas mixture is changed according to the scenario to be modeled.
The unventilated lung compartment (or pocket) represents dependent areas of lung, the airway to which closes at induction of anesthesia. The unventilated lung compartment is modeled as a closed collapsible cavity, where gas composition and volume vary with gas uptake but total pressure is maintained constant at barometric pressure (11). A lung tissue subcompartment was not included, because of previous work showing that including such a compartment makes minimal difference to the time the unventilated lung takes to collapse (11). Mixed venous blood perfuses the pocket, and perfusion limitation of gas exchange is assumed. The ventilated lung compartment and the peripheral tissue compartments are modeled in a similar manner as before induction of anesthesia.
Initial conditions for the postinduction phase depend on the scenario to be modeled (see appendix
). The initial partial pressures of gases in the pocket and the ventilated lung compartment are set equal to the alveolar partial pressures at the end of the preinduction phase. Alveolar volume in the ventilated lung compartment (i.e., Va) is reduced by the initial volume of the pocket (300 ml, which is 10% of preinduction alveolar volume in the standard version of the model), then maintained constant. Lung tissue volume in the ventilated lung compartment (i.e., Vl,ti) is reduced so that the ratio of lung tissue volume to alveolar volume in this compartment is unchanged. The initial partial pressures of gases in the four tissue compartments are set equal to the pressures in the corresponding compartments at the end of the preinduction phase. Total lung blood flow is maintained constant at 6 l/min, but the blood flow per unit volume to the nonventilated lung is initially set at 1.5 times that of the lung as a whole. This gives a blood flowtoalveolar volume ratio for the nonventilated lung compartment of 3:1 and that for the lung compartment (ventilated + nonventilated) of 2:1. This blood flow is then maintained constant, except where hypoxic pulmonary vasoconstriction (HPV) is incorporated into the model (seeappendix
).V˙ai is maintained constant at the preinduction value. Once initial conditions are set, the model is an initialvalue problem, with 16 differential equations (see appendix
), which is solved with Gill’s modification of the RungeKutta method (23).
Scenarios modeled.
The changes in volume and composition within the pocket were examined for a variety of inspired gas mixtures during anesthesia. Each gas mixture consisted of O_{2} and a single inert gas, either N_{2} or N_{2}O. A range of
FIO2
from 0.21 to 1.0 was modeled, both with and without preoxygenation for 3 min. The effects of including or not including HPV in the model, varying the initial volume of the pocket from 1 to 30% of preinduction alveolar volume, and varying the duration of preoxygenation from 0 to 60 min were also examined. The program was written by using Think Pascal (Symantec) and run on a Macintosh LC with a Motorola 68882 math coprocessor.
RESULTS
Unless stated otherwise, the results presented in Figs.35and Table 1, and described in the following paragraphs are for the standard version of the model, with the initial volume of the pocket set at 10% of the preinduction alveolar volume, HPV incorporated, and with a preinduction time of 3 min. The pattern of results described here was consistent over all versions of the model.
Fig. 3.
Time to collapse of unventilated lung compartment. PreO_{2}, 3 min of preoxygenation. N_{2} or N_{2}O is inert gas breathed after induction. Collapse occurred faster with preoxygenation than without. The higher the inspired O_{2}fraction (
) after induction, the faster the collapse. Time to collapse is largely independent of whether inspired gas mixture after induction contains N_{2} or N_{2}O.
Fig. 4.
Changes in composition and volume (vol; dotdashed lines) of unventilated lung compartment. A: no preoxygenation, with O_{2} and N_{2}O breathed after induction.B: no preoxygenation, with O_{2} and N_{2} breathed after induction.C: preoxygenation for 3 min, with O_{2} and N_{2}O breathed after induction.D: preoxygenation for 3 min, with O_{2} and N_{2} breathed after induction. Results for postinduction
of 0.4 is shown.
,
, and
: pocket partial pressure of N_{2}, N_{2}O, and O_{2}, respectively (solid lines);
,
, and
: mixed venous partial pressure of N_{2}, N_{2}O, and O_{2}, respectively (dotted lines). Pocket and mixed venous CO_{2}partial pressures are not shown. Time to collapse is much faster with preoxygenation than without; which inert gas is breathed after induction has only a small effect.
Fig. 5.
Effect of varying duration of preoxygenation. Time to collapse for a range of preoxygenation times (middle; min) when a mixture of N_{2}O and O_{2}(
= 0.21–1.0) was breathed postinduction (left) and when a mixture of N_{2} and O_{2} was breathed postinduction (right) is shown. Longer preoxygenation decreased time to collapse, with most of this effect occurring as preoxygenation time was increased from 0 to 3 min.
Table 1.
Prediction by the model of the time after induction of anesthesia that the unventilated area of lung takes to collapse
The main result centered on the time that the pocket took to collapse. Preoxygenation for 3 min increased the rate of collapse substantially. For any given inspired gas composition after induction, collapse was at least four times faster with preoxygenation for 3 min than without. With preoxygenation for 3 min, collapse was complete in ∼0.5 h even when air was breathed postinduction; when
FIO2
was 1.0 postinduction, collapse was complete in <10 min. In contrast, without preoxygenation, collapse took >4 h when air was breathed; collapse took >0.5 h when
FIO2
was 1.0 postinduction. Both with and without preoxygenation, the higher the
FIO2
after induction, the faster the rate of collapse. Without preoxygenation, the time to collapse was largely independent of whether N_{2} or N_{2}O was breathed after induction. With preoxygenation for 3 min, breathing N_{2}O rather than N_{2} after induction reduced the time to collapse by no more than 31%. Including HPV in the model prolonged the time to collapse, and this effect was greater without preoxygenation. These findings suggest that the presence or absence of preoxygenation is the most important determinant of the kinetics of absorption atelectasis during anesthesia, that the
FIO2
after induction plays an important though lesser role, and that whether N_{2} or N_{2}O is breathed after induction is unimportant.
The second result of interest was the changes in composition and volume of the pocket (see Fig. 4, AD). In all scenarios, the partial pressure of CO_{2} in the pocket (
PpCO2
) equilibrated rapidly with the mixed venous partial pressure of CO_{2}(
Pv¯CO2
). The partial pressure of O_{2} in the pocket (
PpO2
) always equilibrated with the mixed venous partial pressure of O_{2}(
Pv¯O2
) before the pocket had collapsed, but if preoxygenation had been performed and then high
FIO2
breathed, equilibration was not reached until late in the time course of pocket collapse.
With no preoxygenation, equilibration of
PpO2
with
Pv¯O2
took ∼2 min.
If N_{2}O was breathed after induction (see Fig. 4
A) the mixed venous partial pressure of N_{2}(
Pv¯
N2
) fell and the mixed venous partial pressure of N_{2}O (
Pv¯
N2O
) rose, favoring movement of N_{2} out of the pocket and N_{2}O into it. Because of its higher solubility, N_{2}O moved in faster than N_{2} moved out, so the pocket expanded. The partial pressure of N_{2}O in the pocket (
PpN2O
) followed
Pv¯
N2O
closely, except that the
PpN2O

Pv¯
N2O
gradient sometimes widened during rapid fluxes of O_{2} and CO_{2}. The partial pressure of N_{2} in the pocket
(PpN2
) never approached equilibration with
Pv¯
N2
. After the early rapid gas fluxes, gas left the pocket at a relatively rapid rate. Volume fell slightly as O_{2} and CO_{2} equilibrated with mixed venous blood, rose as N_{2}O entered the pocket, then fell as a gradient for both N_{2} and N_{2}O to leave the pocket was established.
If N_{2} was breathed after induction (see Fig. 4
B), volume fell rapidly as O_{2} and CO_{2} equilibrated with mixed venous blood. A state of constant composition was soon reached, where
PpN2
substantially exceeded
Pv¯
N2
, and gas left the pocket at a relatively slow rate, determined mainly by the
PpN2

Pv¯
N2
gradient and the relatively low solubility of N_{2}.
For the same
FIO2
, which inert gas was breathed after induction did not greatly affect the
PpN2

Pv¯
N2
gradient after the rapid early gas fluxes.
With preoxygenation for 3 min, the pocket initially contained mainly O_{2}, so despite rapid initial uptake from the pocket, equilibration of
PpO2
with
Pv¯O2
took 5–7 min.
If N_{2}O was breathed after induction (see Fig. 4
C),
Pv¯N2O
rose, causing influx of N_{2}O into the pocket, and
PpN2O
roughly followed
Pv¯
N2O
.
Pv¯
N2
fell slowly with N_{2}washout from the tissue compartment.
PpN2
rose initally as N_{2} was concentrated by uptake of other gases, then fell slowly. Volume fell rapidly until
PpO2
equilibrated with
Pv¯O2
, then fell more slowly.
If N_{2} was breathed after induction (see Fig. 4
D),
Pv¯
N2
was maintained or rose, depending on the
FIO2
. In both cases,
Pv¯
N2
was at first higher than
PpN2
, so N_{2} moved into the pocket;
PpN2
rose because of flux into the pocket and the concentration of N_{2} in the pocket by O_{2} uptake; when
PpN2
rose above
Pv¯
N2
, N_{2} flux out of the pocket began. Volume fell rapidly until
PpO2
equilibrated with
Pv¯O2
, then fell more slowly at a constant rate determined mainly by the
PpN2

Pv¯
N2
gradient and the relatively low solubility of N_{2}.
For the same
FIO2
, which inert gas was breathed after induction made little difference to the
PpN2

Pv¯
N2
gradient after the rapid early gas fluxes.
The initial volume of the pocket varied from 1 to 30% of preinduction alveolar volume (f = 0.01–0.3) and compared with the standard model (f = 0.1). The overall pattern of results was similar: preoxygenation for 3 min substantially increased the rate of collapse; breathing high
FIO2
postinduction increased the rate of collapse, but this effect was smaller than that of preoxygenation; and which inert gas was breathed after induction made little difference. However, the size of the initial volume of the pocket did affect the absolute time to collapse. For most scenarios, as the size of the pocket increased, so did the time to collapse, but there were some exceptions. If the difference between the time to collapse at f = 0.1 and the time to collapse at the f of interest are expressed as a percentage of the time to collapse at f = 0.1, then the maximum difference was −12% at f = 0.01, −7% at f = 0.05, 19% atf = 0.20, and 49% at f = 0.3.
The duration of preoxygenation varied from 0–60 min (see Fig. 5). Longer preoxygenation decreased the time to collapse. This effect was greater at low postinduction
FIO2
. Most of the effect occurred as the preoxygenation time was increased from 0 to 3 min; longer preoxygenation produced relatively small reductions in the time to collapse.
DISCUSSION
With the induction of anesthesia, functional residual capacity (FRC) falls by 20%, and atelectasis is visible on CT scans as dependent lung densities. The amount by which FRC is reduced, and the size of the area of atelectasis, is independent of whether intravenous or inhalational anesthesia or muscle relaxation is used, intermittent positive pressure ventilation is used, or spontaneous ventilation is maintained (7). The average amount of atelectasis seen on CT scans after induction corresponds to 8–10% of the whole lung (21), so not all the decrease in volume is due to atelectasis.
Atelectasis during anesthesia could be caused by three basic mechanisms (20): compression atelectasis, loss of surfactant atelectasis, or absorption atelectasis. Initially, it was thought that compression atelectasis was the major mechanism (1), but more recent work has shown that very little atelectasis develops during anesthesia if preoxygenation is avoided and O_{2}and N_{2} with an
FIO2
of 0.3 is breathed after induction (22). This argues strongly for gas absorption being the main mechanism. Absorption atelectasis can occur by either complete airway occlusion (16, 19) or by reduction of the inspiredV˙a/Q˙to below a critical level (3).
With the induction of anesthesia, diaphragmatic tone is reduced and FRC falls (27). If FRC is reduced below closing capacity, airway closure will occur. Beyond the site of airway closure, gas will be trapped during at least part of the respiratory cycle, with a predisposition to absorption atelectasis. The importance of muscle tone and changes in FRC in the genesis of atelectasis is illustrated by ketamine anesthesia. With ketamine anesthesia, muscle tone is maintained, FRC does not change, and atelectasis does not develop; only if muscle paralysis is added do FRC fall and atelectasis develop (27).
Theoretical and clinical studies.
Gunnarson et al. (5) examined the amount of atelectasis that developed on CT scans after induction of anesthesia and muscle paralysis. Two groups were examined: one breathed a mixture of N_{2}O and O_{2} with an
FIO2
of 0.4 after induction, and the other received a mixture of N_{2} and O_{2} with an
FIO2
of 0.4. Atelectasis was present on scans 10 min after induction and progressively increased on subsequent scans, with no difference between the two groups. Dantzker et al. (3) calculated the effect of inert gas solubility and
FIO2
on criticalV˙a/Q˙with a theoretical model. He found that, when a mixture of O_{2} and an inert gas with an
FIO2
of 0.4 is breathed, criticalV˙a/Q˙was 20 times greater when the inert gas was N_{2}O than when it was N_{2}. This suggests that more extensive atelectasis should develop when N_{2}O instead of N_{2} is breathed. Joyce et al. (11) calculated the time that an area of lung takes to collapse, when the airway to it becomes occluded. If the same gas mixture was breathed before and after the occlusion, the calculated times for collapse were 214 min (11) for 30% O_{2}70% N_{2} and 8 min (11) for 30% O_{2}70% N_{2}O. Webb and Nunn (30) calculated that, if air was breathed before the occlusion and 30% O_{2}70% N_{2}O afterward, complete absorption took just over 100 min. This suggests that atelectasis develops more rapidly when N_{2}O is breathed instead of N_{2} and is consistent with the results from an experimental dog model (2, 12). However, neither the calculations nor the experimental models were designed to mimic the gas fluxes during the early phases of N_{2}O uptake during anesthesia, as they assume that mixed venous gas partial pressures remain constant. This explains the difference between the results of these studies and the results of Gunnarson et al. (5). Our model (HPV not incorporated) predicts that if there is no preoxygenation, and a mixture of O_{2} and an inert gas with an
FIO2
of 0.4 is breathed after induction of anesthesia, complete collapse will take 87.6 min if the inert gas is N_{2} and 84.1 min if it is N_{2}O. The predictions are in agreement with the findings of Gunnarson et al. of progressive development of atelectasis over 90 min, with no difference between the N_{2}O and N_{2} groups.
Rothen et al. (22) examined the amount of atelectasis visible on CT scans 20 min after induction of anesthesia and muscle paralysis. One group (high
FIO2
group) was ventilated with an
FIO2
of 1.0 during induction, then subsequently with N_{2} and O_{2} with an
FIO2
of 0.4. The other group (low
FIO2
group) was ventilated with N_{2} and O_{2} with an
FIO2
of 0.3 during and after induction. In the high
FIO2
group, there was more atelectasis on the 20min scan than on a control scan before induction, whereas atelectasis was minimal on both scans in the low
FIO2
group. Some of the low
FIO2
group were also scanned at 70 min, and there was more atelectasis than earlier. Our model predicts that with preoxygenation and breathing of O_{2} and N_{2} with an
FIO2
of 0.4, complete collapse will take <20 min, whereas without preoxygenation and breathing O_{2} and N_{2} with an
FIO2
of 0.3, complete collapse will take >2 h. If ventilation with an
FIO2
of 1.0 during induction of anesthesia is analogous to preoxygenation in our model, then our predictions agree with the findings of Rothen et al.
Limitations of the model.
The model assumes that atelectasis develops because of complete airway closure at induction of anesthesia, with subsequent absorption of trapped gas. With complete airway closure, atelectasis is inevitable; altering inspired gas composition can only affect time to collapse. With ventilation at a lowV˙a/Q˙, altering inspired gas composition affects not only time to collapse but also criticalV˙a/Q˙, which may determine whether atelectasis develops.
The timing of airway closure during anesthesia has not been well defined by experimental studies. If airway closure does not occur until several minutes after induction, then gas in the alveoli at induction will be largely replaced by gas breathed subsequently. The gas breathed before induction would have little effect on gas uptake from unventilated lung. There is evidence that airway closure occurs very early during anesthesia. First, the gas breathed during induction is critical in determining the amount of atelectasis that develops (22). This argues that airway closure occurs before washout of alveoli with the postinduction inspired gas. Second, atelectasis can be demonstrated on CT scans at 10 min after induction. Even with complete denitrogenation by prolonged breathing of an
FIO2
of 1.0 before airway closure, a lung unit will take ∼8 min to collapse when an
FIO2
of 1.0 is breathed after airway closure (11). This suggests that airway closure must have occurred very early, to allow sufficient time for collapse to occur.
Perfusion limitation of inert gas uptake from the lung is assumed, but this is widely accepted in the physiological literature (28). Although equilibration of O_{2} and CO_{2} between gas in the unventilated area of lung and the blood perfusing it may not be complete under some circumstances, this should not introduce significant error into calculations of gas uptake (11). Before induction, the lung has been modeled as an ideal lung, and after induction the ventilated lung compartment has been modeled as an ideal compartment. This is unlikely to result in significant error if the lungs are normal.
The model assumes that cardiac output, O_{2} consumption, CO_{2} production, and inspired alveolar ventilation remain constant. Many anesthetic agents reduce cardiac contractility, and cardiac output often falls with the induction of anesthesia. O_{2}consumption and CO_{2} production usually fall by ∼10%. Most anesthetic agents depress respiratory drive, and minute ventilation falls in the spontaneously breathing subject. In the mechanically ventilated subject, minute ventilation will be maintained. The model does not incorporate the tidal nature of ventilation or the pulsatile nature of cardiac output.
The distribution of blood flow between the ventilated and unventilated lung compartments will vary with the vascular resistances of the two compartments. Because of HPV, these resistances will vary with the alveolar partial pressure of O_{2}(
PAO2
),
PpO2
, and
Pv¯O2
. The adjustment for HPV is given in appendix
and is based on data from Marshall et al. (18). N_{2}O obtunds the HPV response (4). There are insufficient data to quantify the effect of HPV in the presence of N_{2}O, but the effect should lie between the extremes of normal HPV and no HPV. The results for these extremes are presented in this study.
If a circulation time delay between peripheral tissues and the lung is not included in models of inert gas uptake, significant errors are only present in the first 2 min of uptake or elimination (17). The lack of such a time delay in our model is unlikely to introduce significant error, because the shortest time to collapse found by the model was over 7 min. Varying the volume of any of the four peripheral tissue subcompartments by ±20% changed the time to collapse by <4%.
In normal lung, diffusion equilibrium exists between the lung tissue and gas in the alveoli, but this is not necessarily the case as an area of lung collapses. An analysis of gas uptake from unventilated lung has shown that this does not result in >10% error in predicted times to collapse of an area of unventilated lung (11).
Explanation of the findings of the model.
First, we consider the standard version of the model when air is breathed before induction. Because of the low solubility of N_{2}, the limiting factor determining how long the pocket takes to collapse is N_{2} uptake. When the airway closes, the amount of trapped N_{2} is the same regardless of what is breathed afterward. Uptake of N_{2} is determined by N_{2} solubility and the
PpN2

Pv¯
N2
gradient. For the same postinduction
FIO2
, which inert gas was breathed postinduction made little difference in the time to collapse or in the
PpN2

Pv¯
N2
gradient.
This may be explained by the following argument. After the early rapid gas fluxes,
PpO2
and
PpCO2
have equilibrated with
Pv¯O2
and
Pv¯CO2
, whereas barometric pressure (Pb) and the saturated vapor pressure of water (Ph
_{2}
o) are constant; the sum of the partial pressures of inert gases in the pocket equals Pb − Ph
_{2}
o−
Pv¯O2
−
Pv¯CO2
, which is constant for a given
FIO2
regardless of which inspired gas mixture is breathed. When no N_{2}O is breathed,
PpN2
= Pb − Ph
_{2}
o−
Pv¯O2
−
Pv¯CO2
. When N_{2}O is breathed,
PpN2O
closely approximates
Pv¯
N2O
, so
PpN2
= Pb − Ph
_{2}
o−
Pv¯O2
−
Pv¯CO2
−
Pv¯
N2O
. Thus when N_{2}O instead of N_{2} is breathed,
PpN2
is reduced by an amount approximately equal to
Pv¯
N2O
. Therefore, the
PpN2

Pv¯
N2
gradient is the same whether N_{2}O or N_{2} is breathed postinduction, provided that any change in
Pv¯
N2
is matched by an opposite change in
Pv¯
N2O
. Consider the two extremes, “induction” and “equilibration of tissue gas exchange.” At induction, only air has been breathed, so
Pv¯
N2
is independent of whether N_{2}O or N_{2} will be breathed later. At equilibration of tissue gas exchange, the inert gas partial pressures in the tissues, mixed venous blood, and the ventilated lung compartment are equal (ignoring the small effect of blood perfusing the pocket); Pb = Ph
_{2}
o+
PACO2
+
PAO2
+
Pv¯
N2
+
Pv¯
N2O
; at a given
FIO2
, Ph
_{2}
o,
PACO2
, and
PAO2
are not affected by which inert gas is breathed postinduction; thus any difference in
Pv¯
N2
between breathing N_{2} or N_{2}O postinduction must be matched by an opposite change in
Pv¯
N2O
. Therefore, at these two extremes, the
PpN2

Pv¯
N2
gradient will be the same regardless of which inert gas is breathed, providing the
FIO2
is the same. Between these points, the
PpN2

Pv¯
N2
gradient will be the same only if the time course of changes in the partial pressures of N_{2} and N_{2}O are similar in the various compartments. Washin of gas into the ventilated lung compartment, and equilibration of the vesselrich compartment with the ventilated lung compartment, is rapid for both N_{2}and N_{2}O, largely complete within 5 min. The halflife (min) for changes in N_{2} in the tissue groups are VRG (1.02), MG (22.7), FG (169), and VPG (113) and for N_{2}O are VRG (0.74), MG (26.1), FG (75), and VPG (113) (29). The only major difference between the two gases is in the FG, which receives only 5% of cardiac output. Thus for a given
FIO2
, once the initial rapid equilibration of
PpO2
and
PpCO2
with mixed venous blood has occurred, the
PpN2

Pv¯
N2
gradient, and therefore the time to collapse, will be similar whether N_{2}O or N_{2} is breathed postinduction.
Second, consider the standard version of the model when preoxygenation is performed. Most of the gas in the pocket is O_{2}, which is rapidly taken up until
PpO2
equilibrates with
Pv¯O2
. The effect on time to collapse of whatever inert gas is breathed postinduction can only be relatively small. During the slow uptake phase, N_{2} uptake limits the total uptake from the pocket; for a given
FIO2
, the N_{2} uptake is the same regardless of whether N_{2}O or N_{2} is breathed postinduction (see above); thus the main determinant of the duration of this phase is the amount of N_{2} in the pocket at the start of this phase. When N_{2} is breathed postinduction,
Pv¯
N2
is maintained or rises, depending on the
FIO2
; there is a greater flux of N_{2} into the pocket during the rapid phase of gas uptake than if N_{2}O was breathed, so the amount of N_{2} in the pocket at the start of the slow uptake phase is greater and time to collapse is slightly longer.
Finally, consider the effect of variations from the standard model. The main effect of preoxygenation is to wash N_{2} out of the lung compartment, reducing the initial amount of N_{2}in the pocket. This washout is virtually complete within 3 min. Further preoxygenation will wash N_{2} out of other compartments, reducing
Pv¯
N2
and hence time to collapse, but this effect is much smaller.
Once
PpO2
equilibrated with
Pv¯O2
, it was lower than
PO2
in the ventilated lung compartment. If HPV was included in the model, blood flow was then diverted away from the pocket, so collapse took longer than without HPV. With preoxygenation, the equilibration of
PpO2
with
Pv¯O2
did not occur until most of the gas had left the pocket, so the effect of HPV was relatively small.
Most of the N_{2} uptake from the pocket occurred after equilibration of
PpO2
with
Pv¯O2
. After this equilibration,
PpN2
always exceeded alveolar partial pressure of N_{2}. Blood from the pocket and the ventilated lung compartment combines to form arterial blood. As the pocket size increases, the fraction of cardiac output passing to it increases; alveolar partial pressure of N_{2} and, hence,
Pv¯
N2
rise, reducing the
PpN2

Pv¯
N2
gradient, so collapse takes longer. This general pattern does not apply to all scenarios, because gas fluxes before equilibration of
PpO2
with
Pv¯O2
may override this effect.
Conclusion.
Preoxygenation before the induction of anesthesia will promote atelectasis. After induction, addition of N_{2} or N_{2}O to the inspired gas mixture will retard atelectasis. Our model predicts that whether the inert gas is N_{2}O or N_{2} will have little effect on the development of atelectasis. This prediction is quite the opposite of what has been predicted previously on theoretical grounds, but it is entirely in keeping with the limited experimental evidence to date. Further experimental studies addressing this question are awaited with interest.
Footnotes

Address for reprint requests and other correspondence: C. J. Joyce, Intensive Care Unit, Princess Alexandra Hospital, Ipswich Rd., Woolloongabba, Queensland 4102, Australia (Email address:c.joyce{at}mailbox.uq.edu.au).
 Copyright © 1999 the American Physiological Society
Appendix
Calculation of Initial Conditions for the Preinduction Phase
The
FIO2
is set to 0.21, inspired N_{2} fraction (
FIN2
) to 0.79, and the inspired N_{2}O fraction (
FIN2O
) to 0. The inspired partial pressures (
PIO2
,
PIN2
, and
PIN2O
) are calculated by PIO2= FIO2⋅(PB−PH2O)
PIN2= FIN2⋅(PB−PH2O)
PIN2O= FIN2O⋅(PB−PH2O)
PACO2
is set at 40 Torr. Equilibration with air is assumed, so the ideal alveolar gas equation applies.
The respiratory quotient (RQ) is 0.8, soPAO2=PIO2−(PACO2/RQ)⋅[1−FIO2(1−RQ)]
Because the CO_{2} efflux in the expired alveolar ventilation must equal CO_{2} production (V˙co
_{2}) atbtps
V˙AE=V˙CO2⋅CF⋅(PB−PH2O)/PACO2
where CF is thestpdtobtpscorrection factor.
Because at equilibration the only gas exchange is O_{2} and CO_{2}
V˙AI=V˙AE+CF⋅(V˙O2−V˙CO2)
whereV˙o
_{2} is O_{2}uptake, this value of V˙ai is maintained constant throughout the program.
Because there is no inert gas exchangePAN2=PIN2⋅V˙AI/V˙AE
andPAN2O=PIN2O⋅V˙AI/V˙AE
where Pa
_{N2} and Pa
_{N2}
_{O} are the alveolar partial pressures of N_{2}and N_{2}O, respectively.
Because equilibration is present and an ideal lung model is usedPaO2=PAO2
PaCO2=PACO2
where Pa_{O2} and Pa_{CO2} are the arterial partial pressures of O_{2} and CO_{2}, respectively.
Arterial O_{2} and CO_{2} content (
CaO2
and
CaCO2
, respectively) are calculated by using West’s modifications (31) of Kelman’s subroutines (1315)Cv¯O2=CaO2−V˙O2/Q˙
Cv¯CO2=CaCO2+V˙CO2/Q˙
PAN2=PaN2=PVRGN2=PMGN2=PFGN2=PVPGN2=Pv¯N2
PAN2O=PaN2O=PVRGN2O=PMGN2O=PFGN2O
=PVPGN2O=Pv¯N2O
where
Cv¯O2 and Cv¯CO2
are the mixed venous O_{2} and CO_{2} content, respectively.
Appendix
Procedure Followed by the Program To Solve the Differential Equations Used in the RungeKutta Method During the Preinduction Phase
Calculation of arterial and venous partial pressures.
PAO2
,
PACO2
,
PAN2
, and
PAN2O
are known. Arterial partial pressures
(PaO2
and
PaCO2
) are set equal to alveolar partial pressures.
CaO2
and
CaCO2
are calculated from
PaO2
and
PaCO2
.
Cv¯
O2
and
Cv¯
CO2
are calculated byCv¯O2=CaO2−V˙O2/Q˙
andCv¯CO2=CaCO2+V˙CO2/Q˙
Venous partial pressures of inert gas are calculated byPv¯N2=(Q˙VRG⋅PVRGN2+Q˙MG⋅PMGN2+Q˙FG⋅PFGN2
+Q˙VPG⋅PVPGN2)/Q˙tot
and Pv¯N2O=(Q˙VRG⋅PVRGN2O+Q˙MG⋅PMGN2O+Q˙FG⋅PFGN2O
+Q˙VPG⋅PVPGN2O)/Q˙tot
where Q˙_{tot} is totalQ˙.
Differential equations governing gas uptake from the lung compartment.
where β is the solubility of inert gas in blood expressed as milliliters btps of gas per deciliter of blood per millimeters Hg at 37°C, and CF is defined as in appendix
dPAO2/dt+dPACO2/dt+dPAN2/dt+dPAN2O/dt=0
so there are five equations with five unknowns, which are initially solved for V˙ae by using the falseposition method, then Eqs.EB1EB4
are solved by substitution.
Differential equations governing gas exchange in the peripheral tissues.
dPVRGN2/dt=(PaN2−PVRGN2)⋅Q˙VRG/(λVRGN2⋅VVRG)
Equation B5
dPMGN2/dt=(PaN2−PMGN2)⋅Q˙MG/(λMGN2⋅VMG)
Equation B6
dPFGN2/dt=(PaN2−PFGN2)⋅Q˙FG/(λFGN2⋅VFG)
Equation B7
dPVPGN2/dt=(PaN2−PVPGN2)⋅Q˙VPG/(λVPGN2⋅VVPG)
Equation B8
dPVRGN2O/dt
=(PaN2O−PVRGN2O⋅Q˙VRG/(λVRGN2O⋅VVRG)
Equation B9
dPMGN2O/dt
=(PaN2O−PMGN2O)⋅Q˙MG/(λVMGN2O⋅VMG)
Equation B10
dPFGN2O/dt=(PaN2O−PFGN2O)⋅Q˙FG/(λFGN2O⋅VFG)
Equation B11
dPVPGN2O/dt
=(PaN2O−PVPGN2O)⋅Q˙VPG/(λVPGN2O⋅VVPG)
Equation B12
Appendix
Initial Conditions in the Postinduction Phase
At induction, the lung compartment is divided into an unventilated lung compartment (pocket volume Vp) and a ventilated lung compartment (alveolar volume Va and lung tissue volume Vl,ti). Partial pressures in the ventilated and unventilated lung compartments are set equal to alveolar partial pressures at the end of the preinduction phase. Partial pressures in the four tissue compartments are set equal to the pressures in the corresponding compartments at the end of the preinduction phase. Call the alveolar volume of the lung compartment before induction Va
_{Pre}, and the lung tissue volume of the lung compartment prior to induction Vl,ti_{Pre}.
The operator sets the ratio (f ), wheref = Vp/Va
_{Pre}
Vp=f⋅VAPre
The volumes in the pocket corresponding to each gas (btps) are calculatedVpO2= Vp⋅PAO2/(PB−PH2O)
VpCO2= Vp⋅PACO2/(PB−PH2O)
VpN2= Vp⋅PAN2/(PB−PH2O)
VpN2O=Vp⋅PAN2O/(PB−PH2O)
V˙aiis maintained constant at the preinduction valueVA= (1−f )⋅VAPre
VL,ti= (1−f )⋅VL,tiPre
Appendix
Procedure Followed by the Program to Solve the Differential Equations Used in the RungeKutta Method During the Postinduction Phase
Calculation of arterial and venous partial pressures.
Call the volume of the unventilated lung compartment “pocket volume”Pocket volume=VpO2+VpCO2+VpN2+VpN2O
The partial pressures of each gas in the pocket are calculatedPpO2= (PB−PH2O)⋅VpO2/pocket volume
PpCO2= (PB−PH2O)⋅VpCO2/pocket volume
PpN2= (PB−PH2O)⋅VpN2/pocket volume
PpN2O= (PB−PH2O)⋅VpN2O/pocket volume
If HPV is not incorporated, Q˙p = 1.5Q˙_{tot} ⋅ f, and Q˙l =Q˙_{tot} −Q˙p, where Q˙p andQ˙l areQ˙ of the pocket and lung, respectively.
If HPV is incorporated, Q˙p andQ˙l are set as described in appendix
.
PAO2
,
PACO2
,
PpO2
, and
PpCO2
are known. The O_{2} and CO_{2} contents in blood leaving the ventilated lung compartment
(CAO2
and
CACO2
) are calculated from
PAO2
and
PACO2
. The O_{2} and CO_{2} contents in blood leaving the unventilated lung compartment (
CpO2
and
CpCO2
) are calculated from
PpO2
and
PpCO2
CaO2=(Q˙L⋅CAO2+Q˙p⋅CpO2)/Q˙tot
CaCO2=(Q˙L⋅CACO2+Q˙p⋅CpCO2)/Q˙tot
Cv¯O2=CaO2−(V˙O2/Q˙tot)
Cv¯CO2=CaCO2+(V˙CO2/Q˙tot)
PaN2=(Q˙L⋅PAN2+Q˙p⋅PpN2)/Q˙tot
PaN2O=(Q˙L⋅PAN2O+Q˙p⋅PpN2O)/Q˙tot
Pv¯N2=(Q˙VRG⋅PVRGN2+Q˙MG⋅PMGN2
+Q˙FG⋅PFGN2+Q˙VPG⋅PVPGN2)/Q˙tot
Pv¯N2O=(Q˙VRG⋅PVRGN2O+Q˙MG⋅PMGN2O
+Q˙FG⋅PFGN2O+Q˙VPG⋅PVPGN2O)/Q˙tot
Differential equations governing gas uptake from the ventilated lung compartment.
Gas uptake from the ventilated lung compartment is calculated by using the same method as during the preinduction phase (by usingEqs. EB1EB4
).
Differential equations governing gas exchange in the peripheral tissues.
Gas exchange in the peripheral tissues is calculated by using the same method as during the preinduction phase (by using Eqs.EB5EB12
).
Differential equations governing gas exchange from the unventilated lung compartment.
dVpO2/dt=CF⋅Q˙⋅(Cv¯O2−CpO2)
Equation D1
dVpCO2/dt=CF⋅Q˙⋅(Cv¯CO2−CpCO2)
Equation D2
dVpN2/dt= Q˙⋅βN2⋅(Pv¯N2−PpN2)
Equation D3
dVpN2O/dt= Q˙⋅βN2O⋅(Pv¯N2O−PpN2O)
Equation D4
Appendix
Calculation of Blood Flow Incorporating HPV
The ability to incorporate HPV was built into the postinduction phase of the model. Because there was only one lung compartment in the preinduction phase, incorporation of HPV into that phase was unnecessary.
The ratio of actual pulmonary vascular resistance to pulmonary vascular resistance under hyperoxic conditions (rPVR) may be predicted by the following calculations [by using a value of maximal rPVR (rPVR_{max}) = 3.15] (18)PsO2= Pv¯O20.41⋅PAO20.59
r%PVRmax= 100⋅PsO2−2.616/(6.683×10−5+PsO2−2.616)
rPVR= 1+r%PVRmax⋅(rPVRmax−1)/100
Using the
PAO2
in the ventilated lung compartment in these calculations will give rPVRa, the rPVR pertaining to the ventilated lung compartment. If
PpO2
is used instead of
PAO2
, these calculations will give rPVR_{p}, the rPVR pertaining to the unventilated lung compartment.
To allow calculation of the blood flow to the ventilated and unventilated lung compartments, it was considered that under hyperoxic conditions blood flow was distributed according to Q˙p=1.5 Q˙tot f
and Q˙L=Q˙tot−Q˙p
wheref is the ratio derived inappendix
.
Given the condition that both vascular beds must have the same perfusion pressure across them, it can be calculated that at the
Pv¯
O2
,
PAO2
, and
PpO2
of interestQ˙p= Q˙tot/{[(1−1.5 f )/1.5 f ]⋅(rPVRp/rPVRA)+1}
Q˙L= Q˙tot−Q˙p
At the time these calculations are made,
PpO2
and
PAO2
are known, but not
Pv¯
O2
.
Pv¯
O2
is dependent on Q˙p andQ˙l, and vice versa, so an iterative method of solution is required. The solution for
Pv¯
O2
,Q˙l, andQ˙p could be found by using the binary search method, but, in practice, convergence with this method was prohibitively slow. The following method was attempted, with test data over the range of
PpO2
and
PAO2
used in the calculations of the program, and it was found to produce values of Q˙land Q˙p within 0.2% of the values produced by the binary search method: 1)
Pv¯
O2
was calculated as if arterial blood were composed entirely of blood draining from the ventilated lung compartment;2) this value of
Pv¯
O2
used with
PpO2
and
PAO2
was used to to calculateQ˙l and Q˙p; 3) these estimates ofQ˙l andQ˙p were used to calculate a new estimate of
Pv¯
O2
; and 4) this new estimate of
Pv¯
O2
used with
PpO2
and
PAO2
was used to calculate final values ofQ˙l andQ˙p.