End-tidal to Arterial Gradients and Alveolar Deadspace for Anesthetic Agents

We conducted an observational study combining retrospective data obtained from a previous investigation in patients undergoing inhalational anesthesia for cardiac surgery with prospectively collected data in a similar population, forming two series. In both series, arterial and end-tidal carbon dioxide partial pressures and blood oxygen content were measured simultaneously with the volatile anesthetic partial pressure cascade (the declining gradient in partial pressure between inspired gas and mixed venous blood) for four groups of patients, each administered a different volatile anesthetic agent spanning a range of blood-gas partition coefficients, during near steady-state phase inhalational anesthesia.

In both series, data were obtained from patients undergoing elective cardiac or thoracic aortic surgery, with no history of acute or chronic respiratory disease, who were recruited with written informed patient consent. With institutional review board approval (METC97/114 A/mF, 1997), the first series was conducted at Catharina Hospital, Eindhoven, The Netherlands, and Onze-Lieve-Vrouw (OLV) Hospital, Aalst, Belgium, from 1997 to 2000, and recruited 32 patients who were sequentially allocated to receive maintenance phase inhalational general anesthesia with halothane (n = 11), sevoflurane (n = 11), or isoflurane (n = 10). This study was conducted to measure the time course of alveolar-capillary uptake rate of these volatile agents using the direct Fick method, and results of this study from 28 of these patients were previously published.¹⁶  The second series was conducted at the Austin Hospital, Melbourne, Australia, from 2017 to 2019, after ethics review and approval (HREC/16/Austin/419, October 21, 2016), and recruited 30 patients who received sevoflurane (n = 10), desflurane (n = 10), or isoflurane (n = 10) in similar fashion, with all data collected prospectively for the purposes of the current study.

In both series, anesthesia management consisted of routine patient cardiovascular monitoring with peripheral arterial and pulmonary artery catheters, along with continuous tidal gas concentration monitoring and arterial and mixed venous blood gas analysis on at least one occasion in the precardiopulmonary bypass period. After intravenous induction of anesthesia and tracheal intubation, the patient was connected to a circle breathing system with fresh carbon dioxide absorber to eliminate carbon dioxide rebreathing. Tidal volume was set at 7 to 10 ml/kg with 5 cm H₂O positive end-expiratory pressure, and respiratory rate set so as to achieve an end-tidal carbon dioxide partial pressure (Petco₂) of 25 to 35 mmHg, with an inspired to expired ratio that ensured a smooth and flat phase 3 plateau on the capnogram. Maintenance anesthesia consisted of inhalational agent without nitrous oxide. Delivered concentrations of oxygen and volatile agent were set to achieve a fractional inspired oxygen tension of 50 to 60%, and end-tidal concentrations of 0.5 to 0.75 minimum alveolar concentration (MAC), with or without supplementary intravenous propofol infusion and opioid at the anesthesiologist’s discretion to maintain adequate depth of anesthesia determined by clinical assessment and/or processed electroencephalograph monitoring.

During near steady-state maintenance phase inhalational anesthesia, between 30 and 90 min postinduction of anesthesia and before cardiopulmonary bypass, and during a period of stability (less than 10% change over at least a 5-min period of time) in ventilation settings, heart rate, mean arterial blood pressure, and end-expired concentration of carbon dioxide and volatile agent, cardiac output was measured by right heart thermodilution, and simultaneous paired arterial and mixed venous blood samples were collected, along with recording of inspired and end-tidal concentrations of oxygen, carbon dioxide, and volatile agent G (Pig and Petg) by side-stream gas sampling using a Datex-Ohmeda M-CAiOVX gas analyzer (GE Healthcare, USA; Series 1), or a Datex-Ohmeda Capnomac Ultima gas analyzer (GE Healthcare) connected to a notebook computer (Macbook Air, Apple Corp., USA) running LabVIEW 2011 software (National Instruments, USA), via an analog-digital converter card (USB 6009, National Instruments; Series 2).

Blood samples were processed for measurement of respiratory blood gases on a Radiometer ABL 700 (Series 1) or a Radiometer ABL 800 gas analyzer (Series 2) calibrated according to the manufacturer’s specifications (Radiometer, Denmark). Volatile agent partial pressures for each gas G in arterial (Pag) and mixed venous (Pvg) blood were measured by gas analysis after equilibration with 10 ml of headspace gas in gas-tight glass containers. In Series 1 headspace gas analysis was performed using gas chromatography on a Carlo Erba GC 6300 gas chromatograph (Vega Interscience, The Netherlands) with flame ionization detector. In Series 2, headspace gas analysis was performed by the same infrared gas analyzer used to measure tidal gas concentrations, using a method previously described.¹⁷  The linearity of both devices was confirmed by construction of calibration curves using stock solutions of sevoflurane, isoflurane, and halothane encompassing the clinical range of partial pressures (5 to 80 mcg/l; Series 1), and using eightfold serial 50% dilutions of prepared mixtures of each agent in oxygen/air (Series 2). The infrared rapid gas analyzers used to measure both carbon dioxide and volatile agent had 10 to 90% response times of approximately 360 ms, allowing accurate measurement of end inspired and expired gas partial pressures in tidal gas sampling at the respiratory frequencies used (10 to 14 breaths per min), and of headspace gas sampling for measurement in blood samples.¹⁷  All measured partial pressures were corrected to body temperature, pressure and water vapor saturated. Partial pressures were calculated from those measured after equilibration using the mass balance principle assuming blood/gas partition coefficients for each agent G (λ_b/gg) equal to published values at 37°C, in a population of similar age (halothane = 2.4, isoflurane = 1.45, sevoflurane = 0.74, desflurane = 0.57).¹⁸ 

Calculated variables were obtained using measured input values for Petg, Pag, Pvg, and inspired (Pig) partial pressures for each volatile agent G, as well as measured partial pressures of carbon dioxide and blood content of oxygen. The methods used followed that used by Landon et al.⁹  employing the three-compartment model of scatter, depicted as a continuous flow model in figure 1. The equations used in the calculations are given in the appendix.

A “dilution factor” equivalent to alveolar deadspace fraction for carbon dioxide (Vda/Vaco₂) was calculated from generalization of the Bohr–Enghoff mixing principle, using measured Petco₂ and Paco₂. Alveolar deadspace fraction for each volatile agent G (Vda/Vag) was calculated from the measured Petg, Pig, and Pag using the same mixing principle as for calculation of Vda/Vaco₂.

Alveolar ventilation rate () was calculated from Petco₂ and the carbon dioxide elimination rate (co₂), estimated from the measured oxygen uptake rate (o₂, calculated from the direct Fick principle) assuming a respiratory quotient (RQ) of 0.8 (as co₂ was not directly measured in the study). Effective (i.e., non-deadspace) alveolar ventilation rate (effective ) was then calculated for each agent G and for carbon dioxide from and their calculated alveolar deadspace fraction.

The ideal Pag was calculated from measured Petg corrected for the contribution of alveolar deadspace to end-tidal gas partial pressure, using the “dilution factor” (Vda/Vaco₂). Pc’g was calculated from Pag corrected for the contribution of calculated venous admixture (Qs/Q) and measured Pvg.

Using a previously published multicompartment computer model of scatter,^19,20  predicted values for each volatile agent G of Petg, Pag and gas uptake rate (g ) were calculated from flow weighted means of outputs of 50 parallel lung compartments modeling continuous flow in each lung compartment n of pulmonary blood flow (n) and expired alveolar ventilation (en). Generic log normal distributions of n and en were generated, with the ability to control the degree of scatter across the 50 lung compartments quantified by the log SD of the distributions according to the method described by West.¹³  Steady-state gas exchange in each lung compartment was computed from mass balance according to the Fick principle simultaneously for every alveolar gas using an iterative method, as described by Olszowska and Wagner,¹⁵  as given in the appendix, incorporating the dissociation curve for oxygen and carbon dioxide according to the equations of Kelman.^21,22 

Input values to the model for PIg, Pio₂, PIn₂, Pvg, and for summated o₂, , and e across all compartments matched the mean measured values listed in table 1 for these variables, and matched mean and measured in each volatile agent group. Input co₂ assumed an overall RQ of 0.8. λ_b/gg for each volatile agent G were as listed above and λ_b/gn₂ was 0.014, and total nitrogen flux in the lung was assumed to be zero. Using a further iterative step, the simulated logSD of and was found so as to generate values for both Petco₂ and Paco₂ (and therefore Vda/Vaco₂) that matched the mean of values measured in each volatile agent group. Using this logSD, the final output values for Petg and Pag were obtained, and Vda/Vag for each volatile agent predicted from the multicompartment model was then obtained using equation A6 (see appendix).

The a priori primary endpoint was the relationship between measured Vda/Vag for each volatile agent G (equation A6) and its log blood/gas partition coefficient (λ_b/gg), determined by linear regression analysis.

Preplanned secondary endpoints were as follows:

• (a)Difference (dVda/Vag-co₂) between simultaneous measurements in individual patients of Vda/Vaco₂ and Vda/Vag (equation 1 and equation A6) using Bland–Altman analysis and Pitman’s test of difference in variance, and its relationship to λ_b/gg, using linear regression analysis

• (b)Difference between calculated Vda/Vag and that predicted by multicompartment computer modeling

• (c)Difference between calculated ideal Pag (equation A2) and Pc’g (equation A5) to assess the validity of the three-compartment model applied to each volatile agent

Comparison of mean values with the t test was used where distributions of data were consistent with a normal distribution according to the Shapiro–Wilk test, reporting SD or 95% CI. Otherwise, median values and interquartile range and nonparametric statistical tests (Kruskal–Wallis equality-of-populations rank test) were used. All statistical comparisons were two-tailed, with a threshold of significance of P < 0.05, and analysis was done using Stata 12 (StataCorp, USA). No a priori statistical power calculation was made, and the sample size in Series 2 was based on estimation of expected a-a partial pressure differences observed in previous published studies, and predicted values generated by generic multicompartment modeling of typical input data.^9–11,20  All complete sets of data available from patients in Series 1 were included, and clinical data from the two series were collated and analyzed for the purposes of the current study after completion of Series 2.

In Series 1, mixed venous blood gas samples or cardiac output measurements at the required point in time in the precardiopulmonary bypass period were not available for 10 patients, and these were therefore excluded from the preplanned analysis. A complete set of retrospective data was therefore available from 22 patients (sevoflurane [n = 6], isoflurane [n = 7], and halothane [n = 9]) in Series 1, and prospective data from all 30 patients in Series 2 (sevoflurane [n = 10], isoflurane [n = 10], and desflurane [n = 10]), totaling 52 patients.

The mean ± SD age of included patients was 66 ± 11 yr, and body mass index was 29 ± 6 kg/m². Vda/Vaco₂ was similar overall between patients in the two series (mean difference [CI], –0.02 [–0.06 to 0.01], P = 0.212 on the unpaired t test) as was dVda/Vag-co₂ (mean difference [CI], 0.01 [–0.05 to 0.07], P = 0.629 on the unpaired t test). Data from the two series were therefore combined, resulting in four groups of patients each administered one of the four volatile anesthetic agents. Mean MAC achieved at time of sampling was 0.69 for halothane, 0.56 for isoflurane, 0.54 for sevoflurane, and 0.57 for desflurane. Median (interquartile range) time of sample collection was 55 (45, 60) min after induction of anesthesia overall and was similar in the four volatile agent groups (P = 0.561 on linear regression analysis).

Measured mean ± SD values for , hemoglobin concentration, Paco₂, and Petco₂ are shown in table 1. Calculated values for co₂ and , and Qs/Q in each group are also shown, along with the alveolar-capillary uptake rate of the agent in each group (g) calculated from measured rate , Pag, P g, and λ_b/gg for each agent using the direct Fick method. Table 2 shows the measured partial pressure cascade (Pig, Petg, Pag, and Pvg) for all four volatile agents.

Alveolar deadspace fractions for volatile agent used (Vda/Vag) and for carbon dioxide (Vda/Vaco₂) in each of the four groups are given in table 3 and plotted in figure 2 in relation to blood gas partition coefficient for each agent (λ_b/gg). First order best fit curves obtained from least squares analysis for both Vda/Vag and Vda/Vaco₂ are plotted (solid lines).

Alveolar deadspace fraction for volatile agent was strongly inversely related to blood solubility (Vda/Vagvs. λ_b/gg slope [CI], –0.10 [–0.13 to –0.06], P < 0.001 on linear regression analysis; fig. 2). Vda/Vaco₂ was weakly inversely related to λ_b/gg (slope [CI], –0.04 [–0.07 to –0.01], P = 0.014 on linear regression analysis; fig. 2). Mean ± SD of Vda/Vaco₂ overall was 0.23 ± 0.07. Alveolar deadspace was significantly larger for each volatile agent than for carbon dioxide. Mean difference (CI) between Vda/Vaco₂ and Vda/Vag overall was –0.34 (–0.37 to –0.31), Pitman’s test of difference in variance was r = –0.42, and P = 0.009.

The difference between alveolar deadspace for each volatile agent and carbon dioxide (dVda/Vag-co₂) is shown as a function of λ_b/gg in figure 3 with the best fit curve, and in table 3. There was a significant inverse relationship (slope [CI], –0.06 [–0.10 to –0.02], P = 0.008 on linear regression analysis), which reflects that in figure 2, but controls for differences between individuals in scatter measured using Vda/Vaco₂.

The effective alveolar ventilation rate for each volatile agent increased with blood solubility. Calculated from equation A9, mean effective was 1.0 l/min for desflurane, 1.4 l/min for sevoflurane, 1.4 l/min for isoflurane, and 1.6 l/min for halothane. Effective for carbon dioxide in each group (from equation A8) was considerably larger, ranging between 2.1 and 2.6 l/min (table 3).

The measured partial pressure cascade for each volatile agent is shown in figure 4 and table 2. Ideal Pag and Pc’g for each agent calculated from equations A2 and A5 are included. A significant difference between ideal PaG and Pc’g was found overall and for each of the agents except desflurane (halothane, P < 0.001, isoflurane, P < 0.001, sevoflurane, P = 0.004, desflurane, P = 0.396; table 2; for all four agents combined, median [interquartile range], Pag 5.1 [3.7, 8.9] vs. Pc’g 4.0 [2.5, 6.2], P = 0.011 on the Kruskal–Wallis test). There was no significant difference in calculated venous admixture (Qs/Q) between groups (table 1).

The values for input variables in each group for the multicompartment model ( and measured ,co₂, Hb, Petco₂, Paco₂, Pig, and Pvg) are listed in tables 1 and 2, and those for output variables (predicted Petg, Pag, Vda/Vag, and dVda/Vag-co₂) are shown in table 4.

Theoretical distributions generated by the model of alveolar-capillary uptake rates for the volatile agent in each group are shown in figure 5, with displayed values scaled for convenience. For each agent, the area under the curve across the 50 lung compartments is the predicted mean gas alveolar-capillary uptake rate of the agent in each group (g), which is also listed in table 4. Distributions for carbon dioxide and oxygen exchange and pulmonary blood flow and expired alveolar ventilation are also plotted using as inputs the overall (n = 52) mean values for o₂, co₂, , and measured in the study. The log SD of the distribution of and plotted is 1.5, which was the mean of values across the four groups.

As the blood solubility of the agent declined, the predicted distribution of gas uptake shifted further leftward relative to the distribution of carbon dioxide elimination, toward lower ratio lung compartments, reducing effective alveolar ventilation rate for the agent and therefore increasing alveolar deadspace ventilation. A lower effective for volatile agents is explained by the greater reliance of less soluble gases on perfusion-driven uptake in lower lung compartments, with greater “wasted ventilation” in higher lung compartments than is the case for carbon dioxide, which more closely follows the distribution of alveolar ventilation.

The predicted values from the multicompartment model of alveolar deadpace fraction Vda/Vag and dVda/Vag-co₂ for each agent are shown in figures 2 and 3, respectively, with a best fit curve (broken line). The slope of the best fit line closely paralleled that of Vda/Vag based on equation A6 found in our patients (solid line). However, predicted Vda/Vag underestimated it for all four agents by a similar amount. The predicted (broken) best fit line lay below the 95% CI (dotted line) for the observed (solid) best fit line (generated using the “lfitci” function in Stata 12) at all points. Consistent with this, predicted g (table 4) from the model was greater than mean g measured in each group (table 1). This indicated a contribution to measured alveolar deadspace and inefficiency of gas exchange of anesthetic agents other than that predicted by the effect of solubility alone from equation A10 (appendix).

We found that alveolar deadspace for inhaled anesthetic agents, calculated from the three-compartment model of scatter, is much larger than that calculated simultaneously for carbon dioxide, and its magnitude is linearly related to the blood/gas partition coefficient of the agent, increasing substantially as blood solubility declines, while reducing their effective alveolar ventilation rate.

Our data show that the custom of calculating alveolar deadspace using measured A-a partial pressure gradients for carbon dioxide (equation 1) is misleading in predicting the A-a gradients, alveolar deadspace, and effective ventilation rate of other gases with different blood solubilities, such as inhaled agents used in anesthesia, and unhelpful in understanding anesthetic gas exchange. Indeed, our data show how there is no “definitive” pulmonary alveolar deadspace. In fact, alveolar deadspace fraction for modern anesthetic gases is much greater, and their effective is considerably lower than for carbon dioxide, which is more soluble in blood (table 3). Our data show how we must conceive of a range of different ideal and alveolar deadspace compartments simultaneously for gases of different solubilities in the same patient.

These concepts and their clinical implications are poorly understood in the field of anesthesiology and reflect the limitations of the concept of alveolar deadspace as it is customarily measured in helping us understand the kinetics of inhalational agents in the lung. However, the relationship of a-a gradients and alveolar deadspace to blood solubility is predicted by a multicompartment model of scatter using physiologically realistic log normal distributions of and of the kind shown in figure 5. Despite such physiologic models of gas exchange being first described many years ago by researchers in respiratory physiology,^13–15  commentators in anesthesiology have rarely engaged them in trying to measure or explain observed gas exchange during inhalational anesthesia.

A good example was a long-running misunderstanding of the mechanism and magnitude of the “second gas effect,” where the rapid alveolar-capillary uptake of nitrous oxide powerfully concentrates the alveolar and arterial partial pressures of the accompanying volatile agent.²³  This concentrating effect is greatest where the effective is low relative to perfusion, as it is for volatile agents as we have shown, as well as for nitrous oxide, but failure to understand this led some commentators to dismiss the second gas effect as insignificant or implausible.^12,24–29 

Our findings have implications for the reliability of measurements of anesthetic gas uptake rates made at the level of the mouthpiece in a breathing circuit. Where end-expired gas partial pressures are measured, an estimate of is needed,^30,31  which is traditionally obtained by subtraction from minute ventilation of anatomical deadspace ventilation measured from nitrogen exhalation (Fowler’s method).³²  However, our data show how this measurement of will not account for the differences in effective (non-deadspace) between gases of different solubilities in blood. Attempts to estimate volatile agent lung uptake rates this way can be widely inconsistent with definitive measurements made invasively from the Fick principle (from measured pulmonary blood flow and arterial and mixed venous partial pressures).³³ 

More generally, failure by simple models of lung gas exchange to account for realistic A-a gradients due to scatter makes them potentially misleading.¹⁶  Such models, while useful teaching tools, can be unreliable tools for physiologic modeling and research purposes. This simplistic approach is encouraged by our clinical experience, where reliance on tidal gas concentration monitoring as a routine tool for assessing anesthetic delivery and depth is reinforced by the concept of MAC, which focuses entirely on end-expired gas concentration measurements. Indeed, MAC effectively achieves its clinical purpose while ignoring the existence of an A-a gradient, which is typically 15 to 30% of end-tidal volatile agent partial pressure.^9–12,33  Although clearly vital clinical tools, end-tidal measurements can be insufficient as investigative tools when considering the physiology and pharmacokinetics of inhalational anesthesia. In contrast, the concept of substantial A-a gradients, which are different for different gases, has been familiar to respiratory physiologists for many years, in particular to advocates of the Multiple Inert Gas Elimination Technique, which uses expired to arterial partial pressure gradients measured for several gases across a wide range of blood solubilities to recover, in reverse fashion, distributions in the lung.^34,35 

Our data show a substantial difference between ideal Pag and Pc’g calculated from alveolar deadspace based on carbon dioxide measurements, and the shunt equation based on oxygen content measurements. This was true for the agents we studied overall, and individually except desflurane (where greater variability in inspired partial pressure in this patient subgroup at the time of sampling produced excessive scatter in other measured partial pressures). This difference between ideal Pag and Pc’g was first demonstrated by Landon et al. for isoflurane.⁹  These authors suggested diffusion limitation of these heavy molecules within the terminal airways, or at the alveolar-capillary barrier, not considered by three-compartment theory, to explain this. They did not discuss that published multicompartment models using physiologic distributions of scatter predicted that gas solubility in blood strongly determines A-a gradients, and would explain much of their findings.^13–15  Our data from a multicompartment model confirm this, and furthermore accurately predict the difference between agents (slope) in Vda/Vag based on differences in blood solubility (figs. 2 and 3).

However, while the slope of the relationship of Vda/Vag to blood-gas partition coefficient was predicted, the magnitude of the difference in Vda/Va between each agent and carbon dioxide was underestimated by the multicompartment model (figs. 2 and 3). As equation A10 (appendix) indicates, blood solubility was the only characteristic of the inert gas computed by multicompartment models such as ours. The residual difference between calculated dVda/Vag-co₂ and that predicted by the model implies a contribution to the Vda/Vag observed in our patients other than solubility alone. A diffusion-limited basis for this residual difference in alveolar deadspace fraction, involving limitation to longitudinal gas mixing in terminal airways, and possibly to alveolar-capillary gas transfer, remains highly plausible given the high molecular weight of these agents compared to carbon dioxide.^36,37  The fact that this residual difference we have demonstrated (approximately 10% of Vda/Vag) was similar in magnitude for all four agents supports diffusion-limitation as a mechanism, given that they have comparable molecular weights (between 168 and 200) and would be expected to exhibit roughly similar diffusion characteristics in the terminal airways and alveolus. Further research to distinguish the contributions of blood solubility and diffusion-limitation to A-a partial pressure gradients and inefficiency of gas exchange of inhalational anesthetic agents is indicated.

We found a weak relationship between Vda/Vaco₂ and volatile agent group in our study, which was an unexpected but interesting finding, given that the agents were administered at similar proportions of MAC but different inspired concentrations. The suggestion that the agents might exert a differential effect on overall matching in the lung related to either blood solubility or alveolar concentration, but not to anesthetic depth, is interesting but is unsupported by the similar mean venous admixture calculated across all four groups, suggesting a similar degree of impairment of hypoxic pulmonary vasoconstriction by the agents.¹ 

A limitation of our study is that we used generic log normal distributions for the purposes of our multicompartment modeling, and did not collect data using, for example, the Multiple Inert Gas Elimination Technique, to measure individualized patient distributions, which may, for instance, be multimodal in shape. However, this modeling simulated log SD values that predicted identical A-a partial pressure gradients for carbon dioxide to the mean of those measured in each patient group we studied (table 1) and therefore is likely to represent the mean predicted behavior of each agent relative to carbon dioxide with acceptable precision for each patient group.

The combining of retrospective and prospective data collected from the two series conducted at different times can be seen as both a limitation and a strength of this study. Importantly, it allowed combination of data on halothane and desflurane, with widely different blood solubilities. Although the clinical protocols of the two studies were similar, the equipment and methodologies for measurement of anesthetic agent in blood in the two series were different. However, the similar findings in these two series of Vda/Vag for both isoflurane and sevoflurane relative to Vda/Vaco₂ reinforce the reproducibility and validity of our overall findings.

In conclusion, measurement of the partial pressure cascade for inhaled anesthetic agents shows that pulmonary alveolar deadspace fraction for these agents is much larger than that conventionally calculated from arterial to end-tidal carbon dioxide partial pressure gradients, and that this is a function of the different solubility of the agents in blood. This makes estimates of the effect of scatter on inhalational anesthetic gas kinetics in the lungs, using measurements of scatter made this way according to the traditional three-compartment model, misleading. Physiologically realistic multicompartment modeling of scatter describes the relative differences in A-a partial pressure gradients and alveolar deadspace of inhalational agents accurately, but also suggests a residual component of the A-a gradient that could potentially be explained by longitudinal diffusion limitation for these heavy molecules.

This work was supported by a project grant from the Australian and New Zealand College of Anaesthetists Research Foundation, Melbourne, Australia, and by an internal research grant of the Catharina Hospital, Eindhoven, The Netherlands, and an internal research grant of the Onze-Lieve-Vrouw (OLV) Hospital, Aalst, Belgium.

Dr. Peyton has received research consultancy payments from Maquet Critical Care/Getinge (Solna, Sweden). Dr. Hendrickx has received lecture support, travel reimbursements, equipment loans, consulting fees, and meeting organizational support from AbbVie (Chicago, Illinois), Acertys (Aartselaar, Belgium), Air Liquide (Paris, France), Allied Healthcare (St. Louis, Missouri), Armstrong Medical (United Kingdom), Baxter (Lessines, Belgium), Dräger (Lübeck, Germany), GE (Madison, Wisconsin), Getinge (Solna, Sweden), Hospithera (Brussels, Belgium), Lowenstein (Karlsruhe, Germany), Intersurgical (United Kingdom), Maquet (Solna, Sweden), MDMS (Loos, France), MEDEC (Aalst, Belgium), Micropore (Elkton, Maryland), Molecular Products (United Kingdom), Philips (Amsterdam, The Netherlands), Piramal (Mumbai, India), and Quantium Medical (Barcelona, Spain). The other authors declare no competing interests.

The ideal alveolar partial pressure for volatile agent G (ideal Pag) was calculated from measured Petg corrected for the contribution of alveolar deadspace to end-tidal gas partial pressure, using a “dilution factor” equivalent to alveolar deadspace fraction for carbon dioxide (Vda/Vaco₂) from generalization of the mixing principle of the Bohr–Enghoff equation to the alveolar gas compartment (fig. 1). This uses end-tidal (alveolar) carbon dioxide partial pressure (Petco₂), and also uses arterial blood (Paco₂) carbon dioxide partial pressure as an approximation of ideal Paco₂, which ignores the contribution of shunt to Paco₂, because the mixed venous-arterial Pco₂ difference is small. Accordingly,⁹ 

Assuming that no rebreathing of carbon dioxide occurs from the breathing system and inspired carbon dioxide partial pressure (PI_CO2) is zero, this becomes

Ideal P_AG was then calculated from⁹ 

End-capillary partial pressure (Pc’g) was calculated from Pa_G corrected for the contribution of venous admixture (Qs/Q) and Pvg.

Qs/Q was calculated using measurements of oxygen content in blood according to the shunt equation of Berggren

where Cao₂ and Cvo₂ are arterial and mixed venous blood oxygen content measurements calculated from the measured oxygen hemoglobin saturations and oxygen partial pressures, and end-capillary oxygen content (Cc’o₂) was calculated from ideal Pao₂ obtained from the ideal alveolar gas equation and an assumed respiratory quotient (RQ) of 0.8,

and where it is assumed that ideal Pao₂ = Pc’o₂.

Cc’o₂ was calculated from Pc’o₂ and the oxyhemoglobin dissociation curve using the equations of Kelman.²¹ 

Pc’g was then calculated from⁹ 

The contribution of shunt in measured Paco₂ used in equations 1 and A4 to error in the calculation of ideal P_AGand Pc’g using equations A2 and A5 has been shown to have a negligible influence on these calculations for isoflurane, with an estimated relative adjustment to the alveolar “dilution factor” of approximately 2%, and to calculated ideal PAG and PcG for isoflurane of only 0.4% and 0.05%, respectively, where mean shunt fraction is similar in magnitude to our study.⁹ 

Alveolar deadspace fraction for each volatile agent G (Vda/Vag) was calculated from the measured Pa_g using the same mixing principle as equation 1:

Alveolar ventilation rate () was calculated from Petco₂ and the carbon dioxide elimination rate (co₂), estimated from the measured O₂ uptake rate (o₂, calculated from the direct Fick principle) assuming a respiratory quotient RQ of 0.8) according to

where Pb was barometric pressure at body temperature and pressure saturated.

Effective (i.e., non-deadspace) alveolar ventilation rate (effective ) was then calculated for each agent G and for carbon dioxide from and the calculated alveolar deadspace fraction from equations 1 and A6:

and

Using generic log normal distributions of pulmonary blood flow (n) and expired alveolar ventilation (en) (where = Σen and = Σn), with a given log SD (logSD) generated as described by West,¹³  for each gas g (oxygen, carbon dioxide, volatile agent G, and balance gas nitrogen N₂) in each lung compartment n,

where

assuming in each compartment that Pc’gn = Pagn for all gases, where inert gas (volatile agent and N₂) content of end-capillary, arterial, and mixed venous blood (Cc’g, Cag, and Cvg) is a product of its partial pressure (Pc’g, Pag, and Pvg)/Pb and λ_b/g for g, according to Henry’s law. The relationships in each lung compartment between partial pressures and blood content of oxygen and carbon dioxide in end-capillary, arterial, and mixed venous blood were calculated from the dissociation curve for oxygen and carbon dioxide according to the equations of Kelman,^21,22  incorporating corrections for measured hemoglobin concentration in each group, and correcting for the wide variation in acid-base balance with Paco₂n and Va/Q ratio across the 50 compartments modeled.²⁰ 

Flow-weighted mean values for mixed alveolar (end-tidal) partial pressure (Petg) and mixed end-capillary (arterial) gas content (Cag) were calculated from outputs from all lung compartments:

and

and gas uptake rate for each gas g was