A modal definition of ideal alveolar oxygen

Abstract In the three‐compartment model of lung ventilation‐perfusion heterogeneity (VA/Q scatter), both Bohr dead space and shunt equations require values for central “ideal” compartment O2 and CO2 partial pressures. However, the ideal alveolar gas equation most accurately calculates mixed (ideal and alveolar dead space) PAO2. A novel “modal” definition has been validated for ideal alveolar CO2 partial pressure, at the VA/Q ratio in a lung distribution where CO2 elimination is maximal. A multicompartment computer model of physiological, lognormal distributions of VA and Q was used to identify modal “ideal” PAO2, and find a modification of the alveolar gas equation to estimate it across a wide range of severity of VA/Q heterogeneity and FIO2. This was then validated in vivo using data from a study of 36 anesthetized, ventilated patients with FIO2 0.35–80. Substitution in the alveolar gas equation of respiratory exchange ratio R with modalR=R–1–PEtCO2/PaCO2 achieved excellent agreement (r 2 = 0.999) between the calculated ideal PAO2 and the alveolar‐capillary Pc'O2 at the modal VO2 point (“modal” Pc'O2), across a range of log standard deviation of VA 0.25–1.75, true shunt 0%–20%, overall VA/Q 0.4–1.6, and FIO2 0.18–1.0, where the modeled PaO2 was over 50 mm Hg. Modal ideal PAO2 can be reliably estimated using routine blood gas measurements.


| INTRODUCTION
Heterogeneity of alveolar ventilation (VA) and blood flow (Q) ratios across the lung (VA ∕Q scatter) is commonly described using the three-compartment or "Riley" model, described over 75 years ago. This simple model depicts all alveolar-capillary lung gas exchange taking place within a theoretical, central, "ideal" lung compartment which has uniform blood flow and ventilation. The ideal compartment sits between an unperfused alveolar dead space compartment and a third, unventilated, venous admixture, or shunt compartment. (Riley et al., 1946) This framework provides two simple and familiar mixing equations that allow V A ∕Q scatter to be quantified, the Bohr dead space equation, and the shunt equation of Berggren (Berggren, 1942;Bohr, 1891).
Solution of the Bohr equation and the shunt equation requires a value for the alveolar partial pressure of carbon dioxide (CO 2 ) or oxygen (O 2 ), respectively, in the ideal compartment. However, there have always been difficulties in defining ideal alveolar gas partial pressure within this model. The Enghoff modification of the Bohr equation substitutes this unknown quantity with arterial CO 2 partial pressure (PaCO 2 ) as a readily measurable approximation for ideal alveolar CO 2 partial pressure, but PaCO 2 in fact represents the combined CO 2 content of the theoretical ideal and shunt compartments (Enghoff, 1938).
Estimation of ideal alveolar O 2 partial pressure (PAO 2 ideal) and blood O 2 content in the ideal compartment customarily uses the alveolar gas equation, (Nunn, 1993;Riley & Cournand, 1949) where PACO 2 ideal is the alveolar CO 2 partial pressure in the ideal compartment, PB barometric pressure, FIO 2 the fractional inspired O 2 concentration, and R the respiratory exchange ratio (the term on the right in square brackets is frequently omitted for simplicity). This convention has been followed in many previous studies measuring shunt fraction (Nunn, 1993;Peyton et al., 2004Peyton et al., , 2005West et al., 2020). Indeed, this equation is still frequently referred to as the "ideal alveolar gas equation" (West et al., 2020). For practicality of measurement, PaCO 2 is often used instead of PACO 2 Since the development by Riley and colleagues of the three-compartment model, (Riley et al., 1946;Riley & Cournand, 1949) advances in technology and computing have allowed us to model and study physiologically realistic, "lognormal" patterns of distribution of V A and Q, and of gas exchange for O 2 and CO 2 and other gases, across the range of V A ∕Q ratios throughout the lung. These models provide much better and more sophisticated understanding of gas exchange for gases of different solubilities, but challenge some existing assumptions of threecompartment theory. For example, it has been shown in patients under inhalational anesthesia how gases with differing solubilities have widely different alveolar dead space volumes, and therefore different effective or "ideal" alveolar ventilation rates, simultaneously within the same V A ∕Q distribution (Peyton et al., 2020).
Derivation of the alveolar gas equation (see Appendix 1C), based on mass balance principles, assumes that all O 2 uptake (VO 2 ) and CO 2 elimination (VCO 2 ) takes place within the same "ideal" compartment. However, due to their widely differing solubilities in blood, V O 2 and V CO 2 distributions are not colocated across the range of V A ∕Q ratios in the lung, and their divergence increases as V A ∕Q scatter worsens (Farhi, 1967;Farhi & Yokoyama, 1967;Peyton et al., 2020). O 2 uptake takes place predominantly in lower V A ∕Q ratio lung units, whereas elimination of CO 2 , which is more highly soluble, takes place largely in well ventilated lung regions. This undermines the fundamental assumption on which the concept of a common central "ideal" compartment containing all gas exchange is based.
Recently, an alternative definition of ideal alveolar gas has been proposed based upon physiological, lognormal distributions of V A and Q, instead of the threecompartment model (Peyton, 2021). Ideal alveolar gas partial pressure for any gas species is defined as that found at the V A ∕Q ratio where the gas exchange rate for that gas is maximal or "modal" across the lung. This V A ∕Q point is determined by the solubility of the gas in blood, and is widely different for different gases.
For CO 2 , with a relatively linear dissociation curve, the alveolar-capillary partial pressure at the V A ∕Q ratio of modal CO 2 elimination (modal ideal PACO 2 ) has been shown to equal the mean of the measured arterial and end-tidal CO 2 partial pressures (Peyton, 2021). This is also the case for any inert gas. Due to its alinear dissociation curve, however, this relationship does not hold for O 2 .
In the current study, data from a physiological, multicompartment computer model of lung V A ∕Q scatter was used to seek a modification of the alveolar gas equation that would provide a practical estimate of modal ideal alveolar O 2 partial pressure (modal ideal PAO 2 ), at the V A ∕Q ratio of maximal (modal) O 2 uptake rate. The mass balance principle described by Equation 1 applies to any V A ∕Q compartment in the lung (see Appendix 1B) if R within that compartment is known. Therefore, this required characterization, within any given distribution of lung V A and Q , of the particular value of R in Equation 1 at the V A ∕Q ratio where the modal ideal PAO 2 is located (modal R). The resulting equation should be generalizable across a wide range of severity of V A ∕Q heterogeneity and of FIO 2 . Its ability to accurately predict the modal ideal PAO 2 was therefore tested across a diverse range of modeled scenarios. The concept was illustrated and subsequently validated with modeling of lung gas exchange in a series of anesthetized, ventilated surgical patients, using in vivo clinical data collected in these patients as input and target output modeling variables.

| Clinical in vivo data collection
Data were used that was collected following Ethics review and approval and informed patient consent (HREC H99/00798 and HREC/16/Austin/419) at the Austin Hospital, Melbourne, in two cohorts of patients with no history of acute or chronic respiratory disease undergoing cardiac surgery during near steady-state conditions in the precardiopulmonary bypass period. The earlier cohort recruited six patients and the later cohort 30 patients. Of these, a total of 30 patients were male. The mean (SD) patient age was 66 (10) years and body mass index was 30.5 (5.3). Delivered concentrations of (1) oxygen were set to achieve an FIO 2 of 0.5-0.6 in the later cohort, and 0.3-0.5 in the earlier cohort in whom a second set of measurements was performed in the postcardiopulmonary bypass period. These were treated as independent measurements for the purposes of statistical analysis, resulting in a total of 42 measurements. The anesthesia management and data collection protocol is detailed in Appendix 3.

| Lung computer modeling
Data collection from multicompartment lung modeling involved two parallel modelling exercises: A. Systematic modeling was done, using arbitrary input data, of distributions in the lung of V A, Q , V A ∕Q scatter and the resulting distributions of partial pressures and gas exchange in each lung compartment n (VCO 2 n, V O 2 n and respiratory exchange ratio Rn). Values were obtained for variables in the alveolar gas equation within these distributions at the V A ∕Q ratio of modal V O 2 , and their relationship across a range of theoretical scenarios was characterized. B. Clinical simulation used the data measurements from patients as inputs and target outputs for the model, to both illustrate the distributions being studied and validate the resulting equation in vivo.
The model of lung gas exchange used physiological distributions of V A and Q across the lung. These distributions were idealized, unimodal, lognormal distributions across 100 lung compartments n, (VAn and Q n) with the degree of V A ∕Q scatter scalable by the log standard deviation (log SD) of V A to represent a wide range of severity of V A ∕Q heterogeneity. In addition, the model allows a proportion of the total pulmonary blood flow to be allocated to an additional "true shunt" compartment where no alveolar ventilation and gas exchange occurs. The structure of the model is summarized in Appendix 2 and has been previously described (Kelman, 1966(Kelman, , 1967Olszowska & Wagner, 1980;Peyton et al., 2001;Siggaard-Andersen, 1974;West, 1969). Within each lung compartment, alveolar and end-capillary partial pressure were considered fully equilibrated (alveolar-capillary partial pressure).
A. Systematic modeling: Theoretical scenarios were modeled using arbitrary input values, where overall R was 0.8. For each of these theoretical scenarios, the log SD of V A was varied in seven increments of 0.25 up to 1.75. The modeling was repeated for scenarios with an overall V A ∕Q ratio of 0.4, 0.8, and 1.6, and for R = 1.0. Two further scenarios were modeled where 10% and 20% of total blood flow Q t was allocated to true shunt, giving a total of 42 different scenarios. For each of these, six different FIO 2 values were modeled, ranging from 0.18 to 1.0.
The outputs calculated by the model were the combined flow-weighted means of the equilibration partial pressures and end-capillary blood gas contents, within all alveolar-capillary lung compartments n (including true shunt), to obtain mixed alveolar gas partial pressure (PAO 2 and PACO 2 ) and arterial partial pressure (PaO 2 and PaCO 2 ) and blood content (CaO 2 and CaCO 2 ) of O 2 and CO 2 , as well as the distributions across all lung compartments of alveolar-capillary equilibration partial pressures and gas exchange of O 2 (Pc ′ O 2 n and V O 2 n) and CO 2 (Pc ′ CO 2 n and V CO 2 n). From the distributions of V CO 2 n and V O 2 n the respiratory exchange ratio in each lung compartment (Rn) was calculated.
Within each compartment n, the same mass balance principle holds as that expressed by the alveolar gas equation for the lung overall (see Appendix 1B)

i) Modal Rn
For each theoretical scenario modeled, the lung compartment of maximal or modal V O 2 n within the distributions of V O 2 n was identified using a peak detector subroutine. The Pc ′ O 2 n (modal Pc ′ O 2 n) and Rn (modal Rn) in this compartment were recorded. The relationship of modal Rn to overall R in each scenario was examined, and data from all scenarios were combined and plotted. An empirical equation was sought for this relationship (see Results: Equation 3, below), which uses overall R and clinically available gas partial pressure measurements, to provide an estimate of the modal Rn (predicted modal Rn). The agreement of the predicted modal Rn from this equation with the modal Rn identified within each distribution was assessed across all scenarios modeled.
ii) Modal ideal PAO 2 In each scenario, the modal ideal PAO 2 was defined as the Pc ′ O 2 n at the modal V O 2 n point within the distributions of V O 2 n. This value is therefore predicted by the following modification of Equation 1n, using modal Rn (see Equation 3, below) and using PaCO 2 as a substitute for Pc ′ CO 2 n: The agreement of the modal Pc ′ O 2 n identified by peak detection within the distributions with the predicted modal ideal PAO 2 from Equations 2 and 3 (see Results: Equation 2a, below) was assessed across all scenarios combined. The acceptability of use of PaCO 2 as a substitute for Pc ′ CO 2 n in this equation was also assessed. In addition, the threshold of arterial hypoxemia (lowest calculated PaO 2 ) simulated in these scenarios, beyond which agreement deteriorated, was examined.

B. Clinical in vivo modeling:
i) Mean in vivo simulation: To illustrate the nature of the relationship of gas exchange distributions for O 2 and CO 2 to distributions of V A ∕Q scatter, the mean measured values in the patient sample studied were used as input variables for a multicompartment lung model scenario. The log SD of V A and true shunt fraction that most closely approximated the mean measured output variables (PAO 2 , PACO 2 , PaO 2 , and PaCO 2 ) were determined. The resulting distributions across 100 lung compartments n of V An, Q n, gas exchange (VO 2 n and V CO 2 n), and alveolar-capillary partial pressures (Pc ′ O 2 n and Pc ′ CO 2 n) across the lung were plotted. The modal ideal PAO 2 and modal ideal PACO 2 were identified. ii) Validation with individual patient modeling: Using the measured FIO 2 as an input variable in each patient, the modal Pc ′ O 2 n for each patient was identified by peak detection using the multicompartment lung model. This was then compared with the predicted modal ideal PAO 2 for each patient calculated from combination of Equations 2 and 3 (see Results: Equation 2a below).

| Statistical analysis
For the purposes of in vivo validation, the primary statistical comparison was between the predicted modal ideal PAO 2 and the modal Pc ′ O 2 n identified by the multicompartment lung model for each patient. Secondary comparisons were made between the PAO 2 from Equation 1 and the measured end-tidal O 2 partial pressure (PE ′ O 2 ) in the patients. Measured end-tidal gas partial pressure (PE ′ O 2 and PE ′ CO 2 ) was considered physiologically equivalent to mixed alveolar gas partial pressure in this population with no history of lung pathology and with no longtitudinal gas flow stratification, manifested as a flat end-expiratory Phase 3 gas concentration plateau (see Supplementary material for an illustrative example). It was hypothesized that there would be no clinically significant difference between these two variables, consistent with the derivation of the alveolar gas equation (Appendix 1C). Comparisons were also made with the mean alveolar O 2 partial pressure from Equation 1a, and with the predicted modal ideal PAO 2 from Equation 2. Comparison of mean values was done with the t-test for paired data for normally distributed or log transformed non-normal data confirmed with the Shapiro-Wilk test. The Pearson correlation coefficient r was used to measure agreement across the range of V A ∕Q relationships and FIO 2 modeled, as well as the median (and 95th centile) relative error between modeled modal values and calculated values for modal Rn and modal ideal PAO 2 . Agreement was also calculated using Bland-Altman analysis for clinical data. All statistical tests were two-tailed, with a threshold of significance of p < 0.05, and analysis was done using Stata 12 (StataCorp, College Station, TX, USA).

B.(i) Mean in vivo simulation:
This modeling scenario is presented first to illustrate the relationships being described in the study. The required modeling input variable are listed in Table 1. The measured mean patient data for these input variables are shown. FIO 2 was 0.53 with balance gas nitrogen, hemoglobin concentration was 12.6 g/L, body temperature 35.4°C, with V O 2 of 190 mL/min with overall R of 0.94. Total Q (Qt) was set to 4.6 L/min and V A 4.2 L/min to match the measured values in the patient sample. Input values for the model were nominated which were equal to the mean values in Table 1. A simulation with a log SD of V A of 1.17 and true shunt fraction of 8.5% was found by an iterative trial and error process to produce outputs from the model (PaO 2 , PAO 2 , PaCO 2 , and PACO 2 ) that most closely approximated the measured mean values in the patients.
The resulting lung distributions of V An, Q n, V CO 2 n, and V O 2 n across the range of V A ∕Q ratios for this scenario are shown in Figure 1. Also plotted are the corresponding alveolar-capillary partial pressures (Pc ′ O 2 n and Pc ′ CO 2 n) and the compartmental respiratory exchange ratio (Rn). The modal V O 2 n point and modal V CO 2 n point in this simulation are indicated.
The position of the Pc ′ O 2 n at the modal V O 2 n point (modal ideal PAO 2 ) was 321.4 mm Hg in this simulation, and is indicated in Figure 1, along with the position of the modal ideal PACO 2 . The Pc ′ CO 2 n at the V A ∕Q ratio of the modal Pc ′ O 2 n point (43.5 mm Hg) more closely approximated the PaCO 2 (40.3 mm Hg) than did the PE ′ CO 2 (30.5 mm Hg), and the PaCO 2 was subsequently used in seeking a modification of the alveolar gas equation to calculate the modal ideal PAO 2 .

A.(i) Characterization of modal Rn:
Across the range of scenarios modeled, the Rn value at the modal V O 2 n point (modal Rn) was empirically found to approximate the following relationship to the overall R, which includes a term which reflects the effect of increasing V A ∕Q scatter, using the calculated alveolar-arterial CO 2 partial pressure gradient for each scenario, The accuracy of Equation 3 to predict the modal Rn deteriorated progressively where the PaO 2 calculated in the scenario was below 50 mm Hg. Rn becomes relatively fixed in the low V An ∕Qn range, as is evident in Figure 1, which is due to the alinearity of the O 2 dissociation curve relative to that for CO 2 . This was seen in several scenarios where an overall V A ∕Q ratio of 0.4 was modeled, and other scenarios at lower FIO 2 where a severe degree of V A ∕Q scatter (log SD of V A of 1.5 or more) or large true shunt fraction (20%) were modeled. Scenarios where the PaO 2 calculated was below 50 mm Hg was therefore subsequently excluded from the analysis.
The accuracy of Equation 3 is illustrated in Figure 2 across the range of theoretical scenarios modeled. The predicted modal Rn from Equation 3 is plotted against the modal Rn identified from the distributions in each scenario. The error is summarized in Table 2 at each FIO 2 modeled.
A.(ii) Modal ideal PAO 2 : Substitution of R in Equation 2 with modal Rn from Equation 3 gives a modified form of the alveolar gas equation that predicts the modal ideal PAO 2 .
The agreement between the modal Pc ′ O 2 n identified from the distributions from the model and the modal ideal PAO 2 predicted from Equation 2a was examined across the range of theoretical scenarios modeled and is plotted in Figure 2b. The agreement (relative error and standard deviation) is summarized in Table 3. Overall relative median error was less than 1%, with 95% of measurements lying within 17.5% of the target value.

B.(ii) Individual patient modeling:
Forty-one complete sets of measurements were obtained. Data from one patient in the second cohort was excluded from the analysis as there was evidence of inadvertent dilution of the mixed venous blood sample, making calculation of V O 2 unreliable in that patient. FIO 2 ranged from 0.35 to 0.80, with a mean (SD) of 0.53 (0.11). V O 2 and V CO 2 were 0.190 (0.061) l/min and 0.178 (0.048) L/ min, respectively. Alveolar dead space fraction V DA ∕VA using the Bohr-Enghoff equation was 24.4 (7.6)% and venous admixture Q s ∕Qt was 12.4 (6.8)%. Table 4 shows the calculated alveolar O 2 partial pressures in the patients.
Primary endpoint: The predicted modal ideal PAO 2 calculated from Equation 2a was similar in magnitude to the modal Pc ′ O 2 n identified from the model (mean (SD) 321.3 (82.7) versus 323.5 (84.6), p = 0.4887). The comparison for all patients is shown in Figure 3a,b. r 2 was 0.973, and the median (95th centile) relative error was 1.5 (9.0)%. Mean bias was 1.4 mm Hg with upper and lower 95% limits of agreement of 6.7 and − 3.7 mm Hg, respectively. This confirmed in vivo the agreement demonstrated by the theoretical modeling data shown in Figure 2b and Table 3.
Secondary Endpoints: Mean PAO 2 from the alveolar gas equation (Equation 1) using measured PE ′ CO 2 as PACO 2 was not different to the measured PE ′ O 2 (mean (SD) 341.1 (79.6) versus 343.3 (82.5), p = 0.6566), which confirmed the basis for the derivation of alveolar gas equation given in Appendix 1. Substitution of PACO 2 with measured PaCO 2 (Equation 1a) resulted in a calculated mean PAO 2 which was 10.4 mm Hg lower. Both these variables were significantly higher than the mean modal Pc ′ O 2 n identified from the multicompartment model (323.5 mm Hg) for each patient (p < 0.0001).

| DISCUSSION
This study describes and validates a mathematical definition of ideal alveolar oxygen partial pressure based on consideration of realistic, physiological distributions of F I G U R E 1 Clinical in vivo mean modeling scenario: Distributions of alveolar ventilation V An, pulmonary blood flow Q n and oxygen uptake V O 2 n and carbon dioxide elimination V CO 2 n across the range of lung V A ∕Q ratios generated by the model, to simulate the mean clinical scenario. Hundred lung compartments n were modeled. Input data were the means measured from the patients of variables listed in Table 1. Total Q (Qt) was 4.6 L/min and V A 4.2 L/min. The log SD of V A was 1.17 with true shunt fraction of 8.5%, which produced outputs from the model (PaO 2 , PAO 2 , PaCO 2 , and PACO 2 ) that most closely approximated the measured mean values in Table 1. Also plotted are the corresponding distributions of alveolar-capillary partial pressures (Pc ′ O 2 n and Pc ′ CO 2 n) and the respiratory exchange ratio (Rn) in each lung compartment. The position of the Pc ′ O 2 n (321.4 mm Hg) at the modal V O 2 n in this scenario is indicated (modal ideal PAO 2 ), as well as the modal ideal PACO 2 . The position of the arterial, mixed venous and end-tidal PCO2 values are shown, as is the V A ∕Q ratio where Rn = overall lung respiratory exchange ratio R of 0.94.
ventilation, blood flow, and respiratory gas exchange in the lung. As shown by modeling of distributions using both theoretical and in vivo data, it can be calculated with adequate precision from a relatively simple modification of the alveolar gas equation which accounts for wide variation in V A ∕Q throughout the lung, and in FIO 2 , using readily measured clinical variables (FIO 2 , and end-tidal and arterial CO 2 partial pressures). This adds to recent work providing a modal definition of ideal alveolar gas for CO 2 and inert gases, and further reconciles the concept of an ideal alveolar partial pressure with modern understanding of physiological distributions of ventilation, blood flow, and gas exchange in the lung.
The O 2 partial pressure at the V A ∕Q ratio where the distribution of V O 2 in a lung is maximal (modal) is a rational definition of "ideal" alveolar O 2 within any lung with a given degree of V A ∕Q scatter. On either side of this point, increasing or decreasing V A ∕Q ratios result in falling effectiveness of O 2 uptake. In this sense, the concept shares the same principle as the central compartment of the three-compartment model of V A ∕Q heterogeneity (Peyton, 2021). However, by contrast, it does not propose a single, uniform compartment where all lung gas exchange is assumed to take place. Instead, it identifies an "ideal" point on a more physiologically realistic continuum of V A ∕Q ratios and gas exchange in the lung.
The traditional theoretical "ideal" compartment of the three-compartment model is defined as containing all V O 2 and V CO 2 . However, Figure 1, which plots realistic, if idealized, distributions of V A, Q and resulting gas exchange, throws this concept into doubt. Due to the very different solubilities of O 2 and CO 2 in blood, V O 2 and V CO 2 distributions are not colocated in the lung. Figure 1 shows that when V A ∕Q scatter becomes substantial, distributions of V O 2 and V CO 2 diverge across a wider range of real V A ∕Q ratios. The assumption that O 2 and CO 2 share a common "ideal" compartment, and the same alveolar dead space fraction, is thus not supported by modeling of physiological distributions. This has been shown to be the case in anesthetized patients, where measured alveolar dead space, and conversely "effective" or ideal alveolar volume, for CO 2 and a range of inert anesthetic gases varied widely, in inverse relationship to their solubility in blood (Peyton et al., 2020). Similar contradictions are encountered in consideration of ideal alveolar gas for shunt fraction calculation with increasing V A ∕Q scatter using the threecompartment model.
By contrast, the mass balance calculation employed in the Riley model to derive Equation 1 is given in Appendix F I G U R E 2 (a) Systematic modeling scenarios: The relationship of the respiratory exchange ratio (Rn) identified by modeling in the lung compartment at the modal V O 2 n (modal Rn) to Rn predicted by Equation 3 across the range of FIO 2 modeled from 0.18 to 1.0. At each FIO 2 , the log standard deviation (log SD) of V A was varied in seven increments of 0.25 up to 1.75, repeated for scenarios with an overall V A ∕Q ratio of 0.4, 0.8 and 1.6, R = 0.8 and 1.0, and true shunt fraction of 0%, 10% and 20%. Scenarios resulting in PaO 2 of less than 50 mmHg were excluded. (b) Systematic modeling scenarios: The relationship of the Pc ′ O 2 n identified by modeling in the lung compartment at the modal V O 2 n (modal Pc ′ O 2 ) to modal ideal PAO 2 predicted by Equation a across the range of FIO 2 modeled from 0.18 to 1.0. At each FIO 2 , the log standard deviation (log SD) of V A was varied in seven increments of 0.25 up to 1.75, repeated for scenarios with an overall V A ∕Q ratio of 0.4, 0.8 and 1.6, R = 0.8 and 1.0, and true shunt fraction of 0%, 10% and 20%. Scenarios resulting in PaO 2 of less than 50 mm Hg were excluded.
1C. The "ideal" compartment envisaged in the Riley model, with ventilation V Aideal, is considered to contain all V O 2 and V CO 2 . However, Figure 1 shows that V O 2 occurs predominantly across a lower range of values of V A ∕Q ratios than V CO 2 . Thus, any "ideal" compartment which captures all V CO 2 must contain higher V A ∕Q ratios that would form part of alveolar dead space for O 2 . This means that PAO 2 ideal calculated in Equation 1 must be contaminated with alveolar dead space gas for O 2 (with partial pressure PIO 2 ).
In fact Figure 1 shows how, in a lung with significant V A ∕Q scatter such as studied here, the volume of the alveolar compartment that captures all V O 2 and V CO 2 most closely approximates that of total alveolar ventilation V A. The data in Table 4 from the patient sample studied here, showing the equivalence of PE ′ O 2 with PAO 2 ideal calculated from Equation 1, is consistent with this. The derivation in Appendix 1A appropriately reflects this and calculates mixed alveolar gas partial pressure, which includes the entire content of the alveolar dead space compartment as well as the ideal compartment. This manifests as end-tidal partial pressure in healthy lungs with no longitudinal gas flow stratification and a flat end-expiratory Phase 3 gas concentration plateau. Note that in the absence of V A ∕Q scatter, where there are no significant alveolar-arterial partial pressure gradients for CO 2 , Equation 2a will indeed simply approximate Equation 1a.
Thus, the Riley model does not provide a satisfactory basis for identifying or measuring the content of a central "ideal" compartment, even though this is fundamental to calculation of venous admixture and dead space using the shunt and Bohr equations, (Berggren, 1942;Bohr, 1891) and the alveolar gas equation is still commonly referred to as the "ideal alveolar gas" equation (West et al., 2020). In their seminal manuscript of 1949, Riley and Cournand presented a "Concept of 'ideal' alveolar air" using a representation of the three-compartment lung model (Riley & Cournand, 1949). This incorporated a pulmonary shunt and an "ideal" uniform gas exchanging compartment, but considered only a single dead space compartment that included serial (anatomic) dead space. Indeed, the authors pointed out that their derivation of the mass balance alveolar gas equation arising from this model was predicated upon homogenous lung gas exchange. Their model effectively ignored the possibility of a separate alveolar dead space compartment arising from V A ∕Q heterogeneity. This can be identified by distinguishing the content of end-tidal gas from mixed expired gas, something that was technically difficult to do in that day, but is now readily done during routine clinical monitoring of tidal gas concentrations using rapid response gas analyzers.
The common substitution of PE ′ CO 2 with PaCO 2 in the alveolar gas equation (Equation 1a) only increased this confusion by further distancing the use of the alveolar gas equation from its original derivation. Interestingly, however, the use of PaCO 2 in the numerator of Equation 2a makes sense when employing the modal definition, as it more closely approximates the Pc ′ CO 2 n at the modal V O 2 n point as can be seen in Figure 1. T A B L E 2 Systematic modeling: Agreement (variance r 2 , relative median error (%) and 95th centile) between the respiratory exchange ratio Rn at the V A ∕Q ratio of modal V O 2 n identified from distributions of V O 2 calculated by the multicompartment lung model (modal Rn), and the predicted modal Rn from Equation 3, across theoretical scenarios modeled where the resulting PaO 2 was 50 mm Hg or greater.  Riley's approach was further constrained by the implicit assumption made by all authors in the field that alveolar O 2 and CO 2 partial pressures shared not just a common central "ideal" compartment, but a common ideal point on the "O 2 -CO 2 " or Fenn diagram, which describes the range of all possible alveolar partial pressures for O 2 and CO 2 that can be present for any given combination of inspired and mixed venous partial pressures, at all possible V A ∕Q ratios between zero and infinity. In contrast to the three-compartment model, the Fenn diagram models a continuum of possible gas exchange states across the lung (Fenn et al., 1946). The assumption of a common "ideal point" for the two respiratory gases has heavily influenced thinking on gas exchange theory. Riley's theory proposed that this ideal V A ∕Q ratio was located where Rn is equal to overall lung R. This point is indicated in Figure 1 and lies between the modal V O 2 n and V CO 2 n points. There is no feature to suggest a common "ideal" characteristic of the partial pressures for both gases at this V A ∕Q ratio, as has been recently demonstrated by other authors using a simpler 2-compartment model of V A ∕Q mismatch (Wagner et al., 2022).

FIO
By contrast, the modal ideal definition accepts different distributions of gas exchange in the lung for gases of differing solubility, and different ideal points. The positions of these gas exchange distributions, skewed toward lower T A B L E 4 In vivo patient data: Measured end-tidal O 2 partial pressure (PE ′ O 2 ) and calculated and modeled variables in patients in the study (n = 41 measurements).

F I G U R E 3 Clinical in vivo individual patient modeling:
The relationship of the Pc ′ O 2 n identified by modeling in the lung compartment at the modal V O 2 n (modal Pc ′ O 2 ) to modal ideal PAO 2 predicted by Equation 2a in the patients studied (n = 41 measurements) as a correlation plot (Figure 3a). FIO 2 ranged from 0.35 to 0.80. The corresponding Bland-Altman plot is shown in Figure 3b.

(a) (b)
VA ∕Q ratios for less soluble gases such as O 2 , represent different degrees of "wasted ventilation" and therefore of alveolar dead space. This results in vastly different alveolar dead space fractions for gases of differing solubility, which has been shown when comparing CO 2 and soluble anesthetic gases in anesthetized patients (Peyton et al., 2020). This implies very different sized ideal compartments and effective alveolar ventilation rates for different gases simultaneously within any given patient, which is not consistent with the concept of a common ideal point or lung compartment for different gases along the V A ∕Q axis. The modal definition of the ideal partial pressure was recently described for CO 2 , and a simple modification of the Bohr equation for calculation of dead space fraction was validated in the same population as the current study (Peyton, 2021). That investigation showed that the modal ideal PACO 2 was equal to the mean of the arterial and mixed alveolar or end-tidal CO 2 partial pressures, across a wide range of V A ∕Q scatter and overall V A ∕Q ratios. This was also the case for all inert gases, which have linear blood gas dissociation curves. Thus, for CO 2 The accuracy of Equation 2a in predicting modal ideal PAO 2 is relatively robust in presence of varying degrees of true shunt. Redistribution of a proportion of total pulmonary blood flow to a true shunt compartment had little a priori effect on the position of the modal Pc ′ O 2 n point in the modeling. However, Equations 3 and 2a use arterial-alveolar CO 2 partial pressure gradients, which are relatively unaffected by true shunt compared to O 2 partial pressures, to estimate Rn from R. However, a limitation of the study is that the accuracy of Equation 3 in predicting modal Rn at the modal V O 2 n point only remained acceptable in the theoretical lung modeling where the resulting PaO 2 of the scenario was 50 mm Hg or more. In scenarios simulating more profound degrees of hypoxemia, this relationship ceased to provide a satisfactory prediction of Rn, due to the flattening of the slope of the O 2 dissociation curve relative to that for CO 2 in an increasing proportion of lower V A ∕Q lung units with very low Pc ′ O 2 n values. This may limit its application in altitude physiology, for instance. It should be noted that the linearity of estimation of alveolar PO 2 by the traditional alveolar gas equation in the presence of severe hypoxemia has not been systematically assessed by modeling in the past, although the variability in calculation of shunt fraction with variation in physiological factors such as PIO 2 or cardiac output has been the subject of a number of studies (Lynch et al., 1979;Quan et al., 1980;Whiteley et al., 2002).
A further limitation of this study is that the clinical validation of the theory presented is conducted in an anesthetized population, where FIO 2 was maintained above 0.35 in accordance with standard safe intraoperative management. Thus, the results presented of modeling at lower FIO 2 (0.18 and 0.21) are not directly confirmed. Nevertheless, when Equation 2a is used for estimation of blood O 2 content in the ideal compartment, it is expected to lead to a smaller calculated shunt fraction (for example, approximately 15% smaller in relative terms for the scenario depicted in Figure 1) than a result based on the customary alveolar gas equation, which will overestimate ideal PAO 2 in comparison due to its inability to distinguish the contributions of "ideal" and alveolar dead space lung compartments. This difference in shunt calculation will be relatively larger again at lower PIO 2 , such as room air breathing or altitude, than in the population studied here.
Despite this limitation, the data in Table 4 illustrate the value of studying an anesthetized, ventilated population with normal underlying lung function, when exploring the effects of V A ∕Q mismatch on alveolar-capillary gas exchange. This population has substantial degrees of V A ∕Q scatter, but with relatively flat end-expired gas concentration curves (see Figure S1). Using modern rapid gas analyzers, mixed alveolar gas partial pressures can therefore be estimated with reasonable precision from end-tidal gas sampling. This technology and study population were not available when many of the seminal works of respiratory physiology of the postwar era were being published, nor was the computing technology that allows distributions such those in Figure 1 to be generated and studied. Riley and Cournand lamented that it was unfortunate at the time that "mixed alveolar air and mixed capillary blood cannot be determined with accuracy" (Riley & Cournand, 1949). Subsequently, reliance on study of healthy awake subjects with little V A ∕Q heterogeneity on the one hand, and of patients with lung disease with disturbed expiratory gas flow on the other, has obscured the anomalies underlying the use of the alveolar gas equation to estimate "ideal" alveolar gas.
With this in mind, it should be noted that the modal definition for ideal alveolar gas is based upon consideration of alveolar-capillary gas exchange distribution in the presence of V A ∕Q heterogeneity. It should not be confused with other factors, such as longtitudinal or "stratified" ventilatory heterogeneity commonly seen in lung disease or airflow obstruction, which distort the expirogram for CO 2 and other gases. These factors, largely concerned with gas flow limitation and heterogeneity in the conducting airways, may prompt controversies about the most appropriate point on the expiratory gas concentration curve to define alveolar partial pressure, for example (Tusman et al., 2012). However, such discussions should be seen as separate to the theory of (4) modal ideal PACO 2 = PaCO 2 + PE � CO 2 ∕2 alveolar-capillary gas exchange on which the modal ideal point is based.
In conclusion, an alternative "modal" definition of ideal alveolar oxygen partial pressure is described, based on realistic physiological distributions of alveolar ventilation and lung blood flow, at the V A ∕Q ratio where the distribution of oxygen uptake rate is maximal or modal. The modal ideal alveolar oxygen partial pressure is able to be determined with accuracy across a wide range of V A ∕Q heterogeneity and FIO 2 , using a simple modification of the alveolar gas equation (Equation 2a) and should be considered as a more coherent solution to the problem of calculation of ideal alveolar oxygen.

A: Derivation of the alveolar gas equation for the whole lung
The derivation of the alveolar gas equation is based upon simple mass balance in the lung, and here uses the expanded form of the alveolar gas equation which incorporates adjustment for the differences between inspired and expired alveolar tidal ventilation volumes. Barometric pressure PB is considered to be corrected for saturated water vapor pressure. In a lung ventilated with a given FIO 2 , oxygen uptake rate V O 2 is given by and CO 2 elimination rate V CO 2 is given by where PAO 2 and PACO 2 are mixed alveolar partial pressures of O 2 and CO 2 , V A is expired alveolar ventilation rate and inspired alveolar ventilation rate V AI is Given that the respiratory exchange ratio R is then The term in square brackets is often left out to provide the simplified form of the alveolar gas equation (where the difference between V AI and V A in the lung overall is ignored).
Note that substitution of PACO 2 with PaCO 2 is commonly done in practice:

B: Calculation in any lung compartment
Within any lung compartment n, the same mass balance derivation applies. and CO 2 elimination V CO 2 n is given by where Pc ′ O 2 n and Pc ′ CO 2 n are equilibration alveolarcapillary partial pressures of O 2 and CO 2 , V An is expired alveolar ventilation rate in lung compartment n, and inspired alveolar ventilation rate V AIn is Given that the respiratory exchange ratio Rn for lung compartment n is then In similar fashion to Equation A1, approximation of Pc ′ CO 2 n with Pc ′ CO 2 gives

C: Three-compartment (Riley) model calculation
Within an "ideal" lung compartment, the same mass balance derivation applies where all respiratory gas exchange (VO 2 and V CO 2 ) is considered to take place. and CO 2 elimination V CO 2 is given by Given thaṫ In similar fashion to above, approximation of PACO 2 ideal with PaCO 2 gives APPENDIX 2

Multicompartment model structure
Model of theoretical distributions of V A and Q : The multicompartment computer model of V A ∕Q scatter used in the study generated unimodal idealized lognormal distributions across N lung compartments n of blood flow (Qn) and expired alveolar ventilation V An, with the degree of V A ∕Q scatter quantified by the log standard deviation (log SD) of the distribution of either V A and Q according to the method described by West (Kelman, 1966(Kelman, , 1967Olszowska & Wagner, 1980;Peyton et al., 2020). Hundred lung compartments were used for the current study to minimize quantization error in output variables from the model. A proportion of blood flow could be diverted to an additional true shunt compartment. The model makes no attempt to incorporate longtitudinal inhomogeneity or stratification of ventilation distribution, and focuses solely on alveolar-capillary gas exchange calculations.
For any given set of distributions of V A and Q , steadystate gas exchange assuming a continuous flow principle with full equilibration between blood and alveolar gas 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 (Siggaard-Andersen, 1974), incorporating the dissociation curve for O 2 and CO 2 according to the equations of Kelman (Fenn et al., 1946;Wagner et al., 2022) using a previously published algorithm. This determines interdependent oxygen and carbon dioxide partial pressures across a wide range of V A ∕Q ratios and compartmental acid-base status using arterial base excess (BE) incorporating the formula described by Siggaard-Andersen (Kelman, 1966;Quan et al., 1980).
Inputs to the calculation were inspired and mixed venous partial pressures for each gas G (PIG and PvG). For each gas species, O 2 , CO 2 , and nitrogen, gas exchange in each lung compartment n (VGn) was calculated according to the mass balance principle Where V AIn was the inspired alveolar ventilation rate for compartment n, PcʹGn, and CcʹGn were equilibrated partial pressures of alveolar gas and end-capillary blood and corresponding end-capillary blood gas content, respectively, CvG was mixed venous blood content of G, and where for each inert gas, CcʹGn = λb/gG PcʹGn/PB and CvG = b ∕gG • PvG ∕ PB.
Outputs from the model were the distributions of PcʹGn, CcʹGn, and V Gn of each gas. Mixed alveolar gas partial pressure (PAG) and arterial partial pressure (PaG) and content (CaG) were calculated from the flow weighted means of these outputs from all lung compartments, including mixed venous blood from true shunt, according to and total gas exchange for each gas (VG) waṡ VG = ΣVGn.
Because measurements for total oxygen uptake rate (VO 2 ) and CO 2 elimination rate (VCO 2 ) were made in all patients, a further step was added where total inspired V AI, mixed venous partial pressures PvO 2 , and PvCO 2 were varied iteratively using modified continuous bisection so as to achieve the target values. The model was constructed on a graphical user interface using LabVIEW 2011 (National Instruments, TX) (Kelman, 1966;Olszowska & Wagner, 1980). Shunt fraction (Qs∕Qt) was calculated from O 2 content in blood using the shunt equation of Berggren.
End-capillary oxygen content Cc ′ O 2 was obtained from the equations of Kelman which estimate O 2 hemoglobin saturation ScʹO 2 from oxygen partial pressure, using ideal PAO 2 from Equation A1.