
==== Front
Physiol Rep
Physiol Rep
10.1002/(ISSN)2051-817X
PHY2
physreports
Physiological Reports
2051-817X
John Wiley and Sons Inc. Hoboken

10.14814/phy2.16175
PHY216175
PHYSREP-2024-04-261
*Regular Manuscript
Original Article
Original Article
Multi‐compartment V/Q lung modeling: Log normal distributions of inspired or expired alveolar gas?
Scott and Morgan
Scott Peter H. https://orcid.org/0000-0002-6428-6867
1 2 p.scott1@uq.edu.au

Morgan Thomas J. 3
1 Intensive Care Department Mater Health Services Brisbane Queensland Australia
2 University of Queensland Brisbane Queensland Australia
3 Mater Research and University of Queensland Brisbane Queensland Australia
* Correspondence
Peter H. Scott, Intensive Care Department, Mater Health Services, Stanley Street, South Brisbane, Brisbane, QLD 4101, Australia.
Email: p.scott1@uq.edu.au

01 9 2024
9 2024
12 17 10.14814/phy2.v12.17 e1617523 7 2024
29 4 2024
24 7 2024
© 2024 The Author(s). Physiological Reports published by Wiley Periodicals LLC on behalf of The Physiological Society and the American Physiological Society.
https://creativecommons.org/licenses/by/4.0/ This is an open access article under the terms of the http://creativecommons.org/licenses/by/4.0/ License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Abstract

Using a 50‐compartment Python‐coded mathematical lung model, we compared mixed venous blood flow (Q) distributions and arterial oxygen tension/inspired oxygen fraction (PaO2/FiO2) relationships in lungs modeled with log normal distributions (LND) of inspired (V I) versus expired (V A) alveolar gas volumes. In lungs with normal V/Q heterogeneity, Q versus V A/Q and Q versus V I/Q distributions were similar with either approach, and PaO2/FiO2 sequences remained indistinguishable. In V/Q heterogeneous lungs at high FiO2, V ILND generated low Q versus V A/Q shoulders and some negative V A units, while V ALND preserved Q versus V A/Q log normality by blood flow diversion from low V I/Q units. We managed V ILND‐induced negative V A units either by shunt conversion (V I decreased to 0) or V I redistribution simulating collateral ventilation (V I increased till V A = 0). Comparing oxygen transfer: V ALND > V ILND (redistribution) > V ILND (shunt). In V/Q heterogeneous lungs V ALND and V ILND (redistribution) regained near optimal oxygen transfer on 100% oxygen, while impairment persisted with V ILND (shunt). Unlike V ALND, V ILND (redistribution) produced Q versus V A/Q distributions in V/Q heterogeneity compatible with multiple inert gas (MIGET) reports. V ILND (redistribution) is a physiologically–based MIGET–compatible alternative to West's original V ALND lung modeling approach.

alveolar gas
expired
inspired
log normal
MIGET
V/Q model
Mater HealthN/A source-schema-version-number2.0
cover-dateSeptember 2024
details-of-publishers-convertorConverter:WILEY_ML3GV2_TO_JATSPMC version:6.4.8 mode:remove_FC converted:01.09.2024
Scott, P. H. , & Morgan, T. J. (2024). Multi‐compartment V/Q lung modeling: Log normal distributions of inspired or expired alveolar gas? Physiological Reports, 12 , e16175. 10.14814/phy2.16175
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pmc1 INTRODUCTION

John West's landmark model of pulmonary gas exchange (West, 1969) equilibrates alveolar gas and mixed venous blood in a series of virtual lung units. The model architecture, described by West as “traditional,” is structured around log normal volume distributions of expired alveolar gas (V A) (West & Wagner, 1977).

Assigning log normal volume distributions to expired rather than inspired alveolar gas (V I) is an important detail. West cited theoretical and experimental data to support this approach (Farhi & Rahn, 1955; Lenfant & Okubo, 1968; Rahn, 1949), commenting to the effect that imposing log normal V A distributions up front prevents negative V A values, which he characterized as “physiologically meaningless” (West & Wagner, 1977).

Although effective, this strategy introduces a cause/effect conflict. The V A output from each lung unit is a dependent variable subject to the input volumes of V I and mixed venous blood (Q) and their subsequent equilibration. Mandating a unimodal log normal V A distribution (V ALND) across lung units is an imposed independent outcome requiring specific adjustments of these proximal “feed in” factors.

The multiple inert gas technique (MIGET) is an investigative tool derived from the V A/Q model (Wagner, 2008; Wagner et al., 1974; 1977; 1978). MIGET readouts often report V A/Q distributions which deviate from unimodal log normality. Examples include Q versus V A/Q “shoulders” in low V A/Q lung regions, and bimodal distributions which typify MIGET reports in conditions such as the acute respiratory distress syndrome and asthma (Dantzker et al., 1979; Wagner et al., 1974, 1978; West, 1977).

When our group adapted West's V/Q lung model for a machine learning project (Morgan, Langley, et al., 2023; Morgan, Scott, et al., 2023), we chose log normal distributions of inspired gas (V ILND) as a more intuitive modeling format, knowing that with this approach V A distributions can deviate from log normal, and that in some instances there will be negative V A outputs from low V I/Q units (Morgan, Scott, et al., 2023). For simulation purposes, we redefined units with negative V A outputs as having undergone atelectasis due to denitrogenation and remodeled them as shunt compartments (V I/Q = 0).

V I redistribution was an alternative approach. Rather than reducing V I/Q ratios to zero to mimic alveolar collapse, V I redistribution can be modeled by diverting small volumes of inspired gas from the total pool to increase the V I of affected units, boosting negative V A values to zero or higher (V A/Q ≥ 0). Redistributions along these lines simulate collateral ventilation, a phenomenon thought to occur via the Pores of Kohn (Desplechain et al., 1983) and facilitated by mechanical support from surrounding ventilated alveoli, a mechanism known as “alveolar interdependence” (Daly et al., 1975).

Accordingly, we tested the hypothesis that a V ILND approach incorporating the V I redistribution option has physiological fidelity when set against the original West V ALND approach. The V ILND “shunt conversion” option was also included in the comparison.

2 METHODS

We devised two scenarios for hypothesis testing. In the first, focusing on pulmonary blood flow distribution (Q vs V/Q; log scale), we ascertained the effects at low and high FiO2 of the unmodified V ILND approach on post‐equilibration Q versus V A/Q relationships, comparing these with the corresponding retroactive influence of the West V ALND approach on pre‐equilibration Q versus V I/Q relationships. In the second scenario, we compared the oxygen transfer outcomes of both policies, dealing with negative V A units induced by V ILND by trialing the two supplementary options: In Option 1, all negative V A units were subjected to shunt conversion by reducing V I to zero (V I/Q = 0) to mimic across the board denitrogenation‐induced collapse.

In Option 2, the V I inputs of affected units were boosted from the inspired gas pool until V A/Q = 0, to model a limited V I redistribution.

A 50‐compartment Python program was developed capable of switching between West's V ALND approach and the V ILND approach without changing inputs (see Appendix S1).

Fixed model inputs were:

Blood hemoglobin concentration (Hb) 15 g/dL.

Arterial PCO2 40 mm Hg.

Base excess 0 mEq/L (Siggaard‐Andersen, 1977).

CO2 production (VCO2) 200 mL/min.

O2 consumption (VO2) 250 mL/min.

Standard P50 26.8 mm Hg (Morgan, 1999).

Cardiac output 5 L/min.

Shunt 0 L/min.

FiO2 0.3 or 0.9 (Scenario 1 only).

2.1 Scenario 1

Graphical representations (log scale) were created of Q versus the relevant V/Q ratios of the two policies at FiO2 = 0.3 and 0.9 for:

Log standard deviation (Log SD) = 0.4.

Log SD = 1.1.

Log SD = 1.8.

2.2 Scenario 2

Arterial PO2 (PaO2) values and venous admixture calculations were computed from FiO2 = 0.2 to FiO2 = 1.0 at FiO2 intervals of 0.05 for the above three log SD values via: The West V ALND approach

The V ILND approach modified so that:

Units with negative V A values were converted to shunt (V I/Q = 0) (Option 1).

V I/Q ratios of negative V A units were increased by redistributing inspired gas so that V A/Q = 0 (Option 2).

3 RESULTS

3.1 Scenario 1

Figure 1 demonstrates that at FiO2 = 0.3 the West V ALND approach produces almost identical Q versus V A/Q and Q versus V I/Q relationships at log SD = 0.4, with slight reduction in concordance as log SD increases.

FIGURE 1 Effects of West's V ALND approach on pre‐equilibration Q versus V I/Q relationships at a low inspired oxygen concentration (FiO2 = 0.3). Closed circles: Q versus V A/Q maintained as a log normal distribution. Open circles: Q versus V I/Q: Remains close to log normal as log SD increases, but with some unilateral rightward shift on the left side of the log normal mean.

Similarly, as shown in Figure 2, the V ILND approach at FiO2 = 0.3 produces almost identical Q versus V A/Q and Q versus V I/Q distributions at log SD = 0.4 and 1.1. At log SD = 1.8 blood flow to low V A/Q units has more clearly increased, and now includes flow to 7 negative V A units of the 50‐compartment model.

FIGURE 2 Effects of the V ILND approach on post‐equilibration Q versus V A/Q relationships at FiO2 = 0.3. Open and closed circles as in Figure 1, but with Q versus V I/Q (open circles) maintained as log normal. There is an elevated Q versus V A/Q ‘tail’ on the left of the log normal mean as log SD increases, indicating increased blood flow to low V A/Q units. At log SD = 1.8 there is also blood flow to 7 negative V A units.

Figure 3 shows that the West V ALND approach operating at FiO2 = 0.9 causes a unilateral rightward shift in pre‐equilibration blood flow away from low V I/Q values as log SD increases.

FIGURE 3 Effects of West's V ALND approach on Q versus V I/Q pre‐equilibration relationships at a high inspired oxygen concentration (FiO2 = 0.9). Open and closed circles as in Figure 1 with Q versus V A/Q (closed circles) maintained as log normal. As log SD increases there is now significant and progressive unilateral deviation from a log normal Q versus V I/Q relationship by rightward shifts on the left of the log normal mean.

Figure 4 shows that with the V ILND approach operating at FiO2 = 0.9, there is unilateral upward displacement of Q versus V A/Q relationships (on the low V/Q side) as log SD increases, indicating a diversion of blood flow to low V A/Q units. At log SD = 1.1, blood is also flowing to 7 negative V A units, amounting to 1.3% of cardiac output. At log SD = 1.8, 18 of the 50 compartments have become negative V A units, receiving nearly 16% of cardiac output in total.

FIGURE 4 Effects of the V ILND approach on Q versus V A/Q post‐equilibration relationships at high FiO2 (0.9). Open and closed circles as in Figure 1, with Q versus V I/Q (open circles) maintained as log normal. There is a progressive Q versus V A/Q “shoulder” on the left of the log normal mean as log SD increases, indicating escalating blood flow to low V A/Q units. There are now major flows to negative V A units.

3.2 Scenario 2

Figure 5 shows that at log SD = 0.4 the three strategies: V ALND, V ILND with Option 1 (shunt conversion of negative V A units), and V ILND with Option 2 (limited V I redistribution to negative V A units) display superimposed PaO2 versus FiO2 relationships throughout the FiO2 range. At log SD = 1.1, concordance is reduced, and is lost at log SD =1.8, with West's V ALND approach producing consistently higher PaO2 values across the FiO2 range. At log SD = 1.8 and higher FiO2 settings, PaO2 values achieved by the V ILND with the Option 2 redistribution strategy recover toward the West PaO2/FiO2 curve, reaching near parity at FiO2 = 1.0. As a result, both achieve venous admixture values close to zero at this FiO2 (Figure 6). By contrast shunt conversion (Option 1) causes persistently low PaO2 values which diverge progressively from those of the other two methods as FiO2 increases, while corresponding venous admixture calculations remain significant at around 20%.

FIGURE 5 Relationships between FiO2 and PaO2 for log SD = 0.4, 1.1 and 1.8. Closed circles: West's V ALND approach. Open triangles: V ILND approach with conversion of negative V A units to shunt (V I/Q = 0) (Option 1). Open circles: V ILND approach with redistribution of inspired gas to negative V A units so that V A/Q = 0 (Option 2). PaO2 values vs FiO2 are close to identical at log SD = 0.4 but diverge for all policies at higher log SD values, with West's V ALND approach achieving the highest oxygen transfer at each FiO2.

FIGURE 6 Relationships between FiO2 and venous admixture for log SD = 0.4, 1.1 and 1.8. Closed circles: West's V ALND policy; Open triangles: V ILND policy with conversion of negative V A units to shunt (V I/Q = 0) (Option 1); Open circles: V ILND policy with redistribution of inspired gas to negative V A units so that V A/Q = 0 (Option 2). Venous admixture values are close to identical for all policies at log SD = 0.4 and log SD = 1.1, and diverging at log SD = 1.8. For log SD = 1.8 and FiO2 = 1.0, venous admixture persists at approximately 20% for Option 1, but is verging on zero for V ILND plus Option 2 and for West's V ALND policy.

4 DISCUSSION

We explored two scenarios in which the V ILND lung modeling approach, versions of which were used previously to simulate bedside diagnostic data for machine learning (Morgan, Langley, et al., 2023; Morgan, Scott, et al., 2023), was evaluated in parallel with West's original V ALND approach (West & Wagner, 1977). For PaO2 calculations, negative V A units encountered with V ILND underwent V I redistribution (V I increased till V A = 0) or shunt conversion (V I decreased to 0).

In contrast to the West approach which fixes expired alveolar gas distributions “post‐equilibration” as log normal, the V ILND adaptation is applied “pre‐equilibration” to fix inspired alveolar gas distributions as log normal. The Python program developed for the comparison can alternate between the two policies, so that a log normal distribution of alveolar gas can be switched to either pole of the V I/V A axis. The distribution at the non‐dominant pole becomes the dependent variable.

In this evaluation, several aspects impacting the physiological fidelity of both approaches are evident. With either approach operative at log SD = 0.4 (representing the low V/Q heterogeneity of healthy lungs), Q versus V I/Q and Q versus V A/Q distributions remain undistorted close copies at both low and high FiO2 (Figures 1, 2, 3, 4). No negative V A units are produced with V ILND. Consequently, as shown in Figure 5, V ALND/V ILND switches at this healthy V/Q distribution make little difference to gas exchange across the FiO2 range. It is thus reasonable to assume that inspired and expired alveolar gas display similar distributions when modeling healthy lungs whether using V ALND or V ILND. Given previous findings (Farhi & Rahn, 1955; Lenfant & Okubo, 1968), these are likely to be log normal or close approximations.

However, Figures 1, 2, 3, 4, 5 show that V ALND/V ILND interchangeability is lost at higher log SD values characteristic of conditions with increased V/Q heterogeneity such as chronic obstructive pulmonary disease. With either approach operative, loss of concordance between Q versus V I/Q and Q versus V A/Q distributions is most evident in the current data at FiO2 = 0.9 (Figures 3 and 4), along with a separation of V ALND and V ILND–generated PaO2/FiO2 sequences (Figure 5).

As outlined in the Introduction, West's V ALND approach eliminates negative V A units by the direct imposition of unimodal log normal expired gas distributions across lung units. How this modifies the “feed in” pre‐equilibration factors can be seen in Figures 1 and 3, where Q distributions are selectively shifted from lower V I/Q units, most prominently at high log SD scenarios at the higher FiO2 setting of 0.9. Put simply, West's V ALND approach has the effect of directing mixed venous blood away from poorly ventilated alveoli, whereas under similar conditions the V ILND approach does the opposite by directing blood flow towards low (and negative) V A/Q units.

In some respects, West's V ALND approach resembles the feedback action of hypoxic pulmonary vasoconstriction (HPV), since HPV operates on the same broad principle of restricting mixed venous blood flow to poorly ventilated units (Dantzker et al., 1974; Grant et al., 1976). One difference is that HPV alters pulmonary blood flow distribution via a vasoconstrictive loop sensitive to a composite alveolar/mixed venous PO2 value (Marshall & Marshall, 1983). Another is that whereas higher FiO2 settings reduce rather than magnify HPV‐induced pulmonary blood flow redistribution (Marshall & Marshall, 1983), the opposite occurs with the V ALND approach (Figure 3). In short, West's V ALND approach is not a model of HPV.

As mentioned, the effect of V ILND on blood flow distribution (Figures 2 and 4) is such that at higher log SD values flow increases through low V A/Q units in configurations similar to MIGET “shoulders” reported in the lungs of older subjects (Wagner et al., 1974). At the same time some mixed venous blood perfuses negative V A units, more so at high FiO2. Although V ALND abolishes “physiologically meaningless” negative V A values (West & Wagner, 1977), it does so by forcing idiosyncratic reconfigurations of incoming V I distributions (Figure 3), in effect transferring biological implausibility “upstream.” An advantage of the V ILND approach is that negative V A values when encountered can be reconfigured directly by responses more consistent with known physiology.

Two examples were evaluated in the present study. Option 1 (converting negative V A units into shunt) causes a severe gas exchange deficit (Figure 5) and persistently high venous admixture (Figure 6) all the way to 100% oxygen. In contrast, with Option 2 (increasing V A/Q to zero by V I redistribution) an initial deficit in oxygen transfer relative to the West approach is followed by a steady recovery with increasing FiO2 so that both achieve near zero venous admixture at FiO2 = 1.0 (Figure 6).

Of note, MIGET analysis using the standard inert gas mix, which includes sulfur hexafluoride as the least soluble gas, will report blood flows to units with V A/Q < 0.005 as shunt (V I/Q = 0) (Wagner et al., 1974). Hence negative V A units redistributed to V A/Q = 0 as in Option 2 fall into the MIGET “shunt” category, whereas more extensive V I redistribution generating V A/Q values ≥0.005 (Figure 7) and subjected to MIGET's smoothing algorithm (Wagner, 2008) would likely be reported as bimodal or even trimodal Q versus V A/Q distribution patterns. This raises the possibility that multi‐modal MIGET reports are flagging extensive alveolar gas redistribution.

FIGURE 7 Illustration of extensive V I redistribution. Log SD = 1.8. FiO2 = 0.9. Prior to redistribution 20% of mixed venous flow was to negative V A units. V I redistribution to these units was randomized to achieve V A/Q values between zero and 0.01. The result is 8% mixed venous blood flow to units with V A/Q < 0.005 (all data points to the left of the Y axis), interpretable by MIGET as shunt flow. The rest (all data points to the right of the Y axis) would potentially be processed by the MIGET smoothing algorithm as a bimodal distribution.

Our group originally adopted Option 1 for Machine Learning data simulations (Morgan, Langley, et al., 2023; Morgan, Scott, et al., 2023). However converting negative V A units into shunt across the board fails to replicate reported improvements in oxygen transfer as FiO2 increases in V/Q heterogeneous lungs (Wagner et al., 1977). These are more closely reproduced with Option 2 (V I redistribution).

Conversely, while marked facilitation of oxygen transfer is known to occur when V/Q heterogeneous lungs receive 100% oxygen (Wagner et al., 1977), improvements fall short of the virtual elimination of impairment shown in Figures 5 and 6, where both V ILND plus V I redistribution and West's V ALND approach predict PaO2 >670 mm Hg at FiO2 = 1.0 and close to zero venous admixtures. To our knowledge, even in individuals with healthy lungs 600 mm Hg is the highest documented PaO2 recorded at FiO2 = 1.0, achieved after 35 min equilibration with 100% oxygen and negligible end tidal N2 measurements (Wagner et al., 1974).

Suggested explanations for this apparent underperformance have included incomplete denitrogenation (Wagner et al., 1974), leaks in the delivery system, and negative measurement bias (Wagner et al., 1977). However, the shortfalls could also reflect the contribution of actual physiologic phenomena, for example, new onset atelectasis or “shunt—like” effects from high capillary blood velocity in some units (Domino et al., 1993; Miserocchi et al., 2022). Resultant oxygen transfer shortfalls can be accommodated in West's V ALND approach and the V ILND plus V I redistribution approach by factoring in an additional shunt (V I/Q = 0) sufficient to match the venous admixture.

5 CONCLUSIONS

Based on data generated by a purpose‐designed 50 compartment V/Q lung modeling “switch” program we conclude the following: In lungs with a normal degree of V/Q heterogeneity, the V ILND approach and West's original V ALND model are essentially interchangeable, generating close to identical oxygen transfer outcomes and similar Q versus V A/Q and Q versus V I/Q relationships displaying approximately log normal configurations.

In lungs with increased V/Q heterogeneity, there are important differences.

The V ALND approach avoids negative V A values by direct imposition of unimodal log normal V A distributions, forcing idiosyncratic reconfigurations of incoming Q versus V I/Q distributions. By mandating post‐equilibration V A distribution V ALND cannot accommodate MIGET deviations from Q versus V A/Q log normality or multi‐modal patterns.

By contrast, the V ILND approach under V/Q heterogeneous conditions shifts blood flow towards lower V A/Q units and can cause “non‐physiological” blood flow through negative V A units. These can be resolved by increasing inspired gas distribution to negative V A units until V A ≥0 simulating V I redistribution by collateral ventilation. Such adjustments are consistent with known pathophysiological mechanisms, MIGET reports, and oxygen transfer profiles. Notably: V I redistribution to V A/Q < 0.005 produces Q versus V A/Q distributions which would be reported by MIGET as shunt.

V I redistribution to V A/Q ≥ 0.005 produces Q versus V A/Q distributions potentially reported by MIGET in bimodal or even trimodal formats.

Multimodal Q versus V A/Q MIGET reports may thus in some cases signify considerable redistribution of inspired alveolar gas rather than “true” multi‐modality. In such cases, redistribution at lower levels could have further increased MIGET shunt estimates above actual values.

MIGET interpretations along these lines have as yet unclear implications for the deployment of management strategies such as high flow intranasal therapy, prone positioning, and lung recruitment maneuvers.

A clinical study of bedside MIGET evaluations versus concurrent V ILND (redistribution) lung modeling conducted using Machine Learning methodology (Morgan, Scott, et al., 2023) would be instructive as a first step.

AUTHOR CONTRIBUTIONS

Thomas J. Morgan devised the project and wrote the first draft of the main manuscript text. Peter H. Scott devised the Python V/Q “switch” program which generated the data and wrote the Supplementary Material text. Both revised and approved final versions.

FUNDING INFORMATION

This project was supported by Departmental Funds.

CONFLICT OF INTEREST STATEMENT

The authors declare that they have no competing interests.

Supporting information

Data S1.

ACKNOWLEDGMENTS

The authors wish to thank Mr Robin Barrett for helpful discussions and advice.

DATA AVAILABILITY STATEMENT

Expressions of interest for data availability can be directed to the corresponding author. An online version of the lung model is under development.
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