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Modeling of Gas Exchange in the Lungs

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Abstract

This overview presents the recent progress in our understanding of gas transfer by the lungs during the respiratory cycle and during breath holding. Different phenomena intervene in gas transfer, convection and diffusion in the gas, dissolution, diffusion across the alveolar‐capillary membrane, diffusion across blood plasma, and finally diffusion and reaction with hemoglobin inside blood cells. The different gases, O2, CO, and NO, have very different reaction times with hemoglobin ranging from a few microseconds to tens of milliseconds. This is leading to different outcomes.

For O2, the solutions to the coupled nonlinear gas and blood equations are obtained at the acinus level. They include the fact that the acinar internal ventilation is strongly heterogeneous due to the arborescent structure. Also, in the dynamic calculation, one takes care of the delay between the start of inhalation and arrival of fresh air in the acinus. This “dead” time is the dynamic equivalent of the dead space ventilation.

The question of the dependence of Vo2 on ventilation and perfusion takes a different form. The results show that Vo2 is not only a function of the ventilation/perfusion ratio but also depends on the variables: acinar ventilation VEac and perfusion Qac. The ratio VEac/Qac roughly determines arterial O2 saturation and arterial and alveolar O2 partial pressure.

The classic Roughton‐Forster interpretation of DLCO (separation between independent membrane and blood resistance) was a mathematical conjecture. It was shown recently that this conjecture was violated. This article presents an alternative interpretation that uses time concepts instead of resistance. © 2021 American Physiological Society. Compr Physiol 11:1289‐1314, 2021.

Figure 1. Figure 1. Penetrations of CO and NO obtained by 2D numerical solutions of Eq.  7 in the RBC morphologies obtained by Weibel 84 (A, alveolar air; EC, RBC; P, plasma; T, tissue barrier). The colors correspond to the concentrations of CO or NO. The dark red regions illustrate the active volumes where the majority of the transfer occurs. The penetration from the surface is approximately represented by the thickness of the dark red region. In the cluster shown here, the region active for CO transfer is the majority of the cluster volume, whereas the region active for NO transfer is the thin dark red layer.
Figure 2. Figure 2. Qualitative schematic of the problem that is solved numerically here: profile of O 2 partial pressures along the airway tract during inspiration and along the perfusion pathway from the pulmonary artery to the capillary units and the pulmonary vein. The gas‐exchange units corresponding to the capillary units are perfused in parallel but ventilated in series. Courtesy of E. R. Weibel.
Figure 3. Figure 3. Due to the short transverse diffusion time, the numerical problem can be mapped on tree made of one‐dimensional branches. The x coordinate follows the tree branches.
Figure 4. Figure 4. Time evolution of blood saturation for a diffusion time across membrane and plasma of 1 ms and for two values of the oxygen partial pressure in the gas: P g  = 70 mmHg and 100 mmHg. Also shown in blue is the saturation evolution predicted by Whiteley 90 from a finite element calculation. The final saturation value is close to 0.96 for P g ( x , t ) = 100 mmHg. The arrows are used to define the approximate saturation time t s . See Kang et al. (2015) for details.
Figure 5. Figure 5. Results of the dynamic oxygen uptake model for rest and heavy exercise after Kang et al. 42 . (A) Spatiotemporal distribution of the local alveolar O 2 partial pressure in the acinus. (B) Spatiotemporal distribution of the local transfer. For visibility, the spatial axis is inversed as indicated by the arrow. Data are computed for a situation where the time of arrival (dead time) is 0.7 s at rest and 0.15 s at exercise.
Figure 6. Figure 6. Schematic representation of the DOT model used by Kang et al. Given the average morphology of an acinus, the dynamic model relates the four principal respiratory determinants (VE, PIO 2 , Q, and PvO 2 ) to the four outcome values of O 2 transfer, namely, Vo 2 , SaO 2 , Pa o 2 , and PAO 2 . The subscript ac indicates that the values are for an acinus.
Figure 7. Figure 7. Functioning of single standard acinus at rest for a range of ventilation and perfusion computed by the DOT model. Left and bottom axes of each map represent acinar ventilation and perfusion values, respectively. Right and top axes indicate global values of ventilation and perfusion, respectively, calculated by multiplying the acinar ventilation and perfusion by the number of acini corresponding to a “standard” subject with TLC = 6.2 liter and dead space volume VD = 0.17 liter. Some labels are omitted due to limited space. The blue arrows in the top‐left map illustrate the inverse approach for retrieving Q ac from VE ac and Vo 2ac (see text). The red lines in these figures represent (isoventilation/perfusion) ratio lines, with VE ac / Q ac  = 1 (solid); VE ac / Q ac  = 1.5 (dotted); and VE ac / Q ac  = 0.5 (broken).
Figure 8. Figure 8. Functioning of single standard acinus at peak exercise for a range of ventilation and perfusion computed by the DOT model. Left and bottom axes of each map represent acinar ventilation and perfusion values, respectively. Right and top axes indicate global values of ventilation and perfusion, respectively, calculated by multiplying the acinar ventilation and perfusion by the number of acini corresponding to a standard subject with TLC = 6.2 liter and dead space volume VD = 0.17 liter. Some labels are omitted due to limited space. The red lines represent isoventilation over perfusion ratio lines, with VE ac / Q ac  = 1 (solid), VE ac / Q ac  = 1.5 (dotted), and VE ac / Q ac  = 2 (broken).
Figure 9. Figure 9. Comparison between individual cardiac output values deduced from the inverse approach and their corresponding values deduced from Fick principle. Each point corresponds to a given individual. Red circles are for rest, and green triangles for exercise. Here, the predicted values have been calculated using, in addition to the four primary determinants, two secondary individual determinants, namely, Hb concentration and individual P 50 .
Figure 10. Figure 10. DOT model predictions of high‐altitude maximal oxygen uptake (Vo 2 max), red curve, and its comparison to experimental data.
Figure 11. Figure 11. Gas exchange under significant diffusion disturbance due to increase in membrane thickness as predicted by the model. Top: for rest. Bottom: for peak exercise.
Figure 12. Figure 12. Dependence of oxygen transfer Vo 2 , saturation SaO 2 , mean alveolar pressure PAO 2 , and arterial pressure Pa o 2 on the arrival (dead) time at which the oxygenated fresh air reaches an acinus at constant VE = 7.5 liter/min (computed for PvO 2  = 40 mmHg and Q  = 5 liter/min). Each value was normalized to its value at zero arrival time t arr  = 0 s. This is why the PA and Pa curves coincide. The vertical dotted line represents a “normal” reference condition of t arr  = 0.7 s. Note that the PAO 2 and Pa o 2 curves are practically the same because the ordinates are in relative units, and PAO 2 ‐Pa o 2 keeps small.
Figure 13. Figure 13. Two‐dimensional (2D) illustration of the distribution of volumes of fresh air crossing generation 12 in the Florens et al. model 17 . There are 212 squares, each of which represents one airway at generation 12. The colors correspond to the logarithm of the distributed volume of fresh air expressed in milliliters. The black squares correspond to airways that are too small (diameter smaller than 0.5 mm) to belong to the tracheobronchial tree. Note the logarithmic color scale.
Figure 14. Figure 14. Time evolution of the spatial distribution of CO concentration. t represents the time after the RBC enters in diffusive contact with the gas. The membrane thickness is chosen to be 1 μm. Concentrations are normalized to the concentration at the gas‐membrane interface C g . The horizontal line represents the membrane‐plasma interface. The bottom figure gives the rate of transfer as a function of time. It reaches a steady state after 2 to 3 ms.
Figure 15. Figure 15. Three simplified geometries of the gas‐membrane‐plasma‐RBC structure. Top A: Horizontal cut of a planar parallel structure of a symmetrical gas‐membrane‐plasma‐RBC geometry. This is an artificial case where the alveolar‐capillary membrane, the plasma layer, and the blood cells are supposed to be planar with thicknesses of L M , L P , and L RBC , respectively. The dotted line at the bottom indicates the plane of symmetry. Top B: Quarter of an axial planar cross section of spherical RBC inside a cylindrical capillary shown as the bottom revolution axis. Top C: Quarter of an axial planar cross section of biconcave RBC in a cylindrical capillary. Bottom: 3D illustrations of each situation.
Figure 16. Figure 16. Spatial distribution of CO and NO concentrations in the planar system. Concentrations at the gas membrane interface have been normalized to 1. For simplicity, the same diffusivity is applied to the entire system. (Computed for D CO  = 3.26 × 10 −5  cm 2 /s; D NO  = 3.18 × 10 −5  cm 2 /s; τ CO  = 0.5 ms; τ NO  = 0.005 ms; L M  = 1 μm; L P  = 2.31 μm; L RBC  = 3.08 μm. See below for these values.)
Figure 17. Figure 17. Results of the exact calculation of 1/DLCO as a function of the reaction time τ ( τ is proportional to PO 2 for PO 2  > 80 mmHg) for the planar system of Figure  16 .
Figure 18. Figure 18. Comparison of the planar model prediction using Eq.  34 and experimental data from Forster et al. 21 . Values of 1/DLCO are rescaled using the values 1/DLCO (Ext.Pa o 2  → 0 mmHg) (Eq.  41 ).


Figure 1. Penetrations of CO and NO obtained by 2D numerical solutions of Eq.  7 in the RBC morphologies obtained by Weibel 84 (A, alveolar air; EC, RBC; P, plasma; T, tissue barrier). The colors correspond to the concentrations of CO or NO. The dark red regions illustrate the active volumes where the majority of the transfer occurs. The penetration from the surface is approximately represented by the thickness of the dark red region. In the cluster shown here, the region active for CO transfer is the majority of the cluster volume, whereas the region active for NO transfer is the thin dark red layer.


Figure 2. Qualitative schematic of the problem that is solved numerically here: profile of O 2 partial pressures along the airway tract during inspiration and along the perfusion pathway from the pulmonary artery to the capillary units and the pulmonary vein. The gas‐exchange units corresponding to the capillary units are perfused in parallel but ventilated in series. Courtesy of E. R. Weibel.


Figure 3. Due to the short transverse diffusion time, the numerical problem can be mapped on tree made of one‐dimensional branches. The x coordinate follows the tree branches.


Figure 4. Time evolution of blood saturation for a diffusion time across membrane and plasma of 1 ms and for two values of the oxygen partial pressure in the gas: P g  = 70 mmHg and 100 mmHg. Also shown in blue is the saturation evolution predicted by Whiteley 90 from a finite element calculation. The final saturation value is close to 0.96 for P g ( x , t ) = 100 mmHg. The arrows are used to define the approximate saturation time t s . See Kang et al. (2015) for details.


Figure 5. Results of the dynamic oxygen uptake model for rest and heavy exercise after Kang et al. 42 . (A) Spatiotemporal distribution of the local alveolar O 2 partial pressure in the acinus. (B) Spatiotemporal distribution of the local transfer. For visibility, the spatial axis is inversed as indicated by the arrow. Data are computed for a situation where the time of arrival (dead time) is 0.7 s at rest and 0.15 s at exercise.


Figure 6. Schematic representation of the DOT model used by Kang et al. Given the average morphology of an acinus, the dynamic model relates the four principal respiratory determinants (VE, PIO 2 , Q, and PvO 2 ) to the four outcome values of O 2 transfer, namely, Vo 2 , SaO 2 , Pa o 2 , and PAO 2 . The subscript ac indicates that the values are for an acinus.


Figure 7. Functioning of single standard acinus at rest for a range of ventilation and perfusion computed by the DOT model. Left and bottom axes of each map represent acinar ventilation and perfusion values, respectively. Right and top axes indicate global values of ventilation and perfusion, respectively, calculated by multiplying the acinar ventilation and perfusion by the number of acini corresponding to a “standard” subject with TLC = 6.2 liter and dead space volume VD = 0.17 liter. Some labels are omitted due to limited space. The blue arrows in the top‐left map illustrate the inverse approach for retrieving Q ac from VE ac and Vo 2ac (see text). The red lines in these figures represent (isoventilation/perfusion) ratio lines, with VE ac / Q ac  = 1 (solid); VE ac / Q ac  = 1.5 (dotted); and VE ac / Q ac  = 0.5 (broken).


Figure 8. Functioning of single standard acinus at peak exercise for a range of ventilation and perfusion computed by the DOT model. Left and bottom axes of each map represent acinar ventilation and perfusion values, respectively. Right and top axes indicate global values of ventilation and perfusion, respectively, calculated by multiplying the acinar ventilation and perfusion by the number of acini corresponding to a standard subject with TLC = 6.2 liter and dead space volume VD = 0.17 liter. Some labels are omitted due to limited space. The red lines represent isoventilation over perfusion ratio lines, with VE ac / Q ac  = 1 (solid), VE ac / Q ac  = 1.5 (dotted), and VE ac / Q ac  = 2 (broken).


Figure 9. Comparison between individual cardiac output values deduced from the inverse approach and their corresponding values deduced from Fick principle. Each point corresponds to a given individual. Red circles are for rest, and green triangles for exercise. Here, the predicted values have been calculated using, in addition to the four primary determinants, two secondary individual determinants, namely, Hb concentration and individual P 50 .


Figure 10. DOT model predictions of high‐altitude maximal oxygen uptake (Vo 2 max), red curve, and its comparison to experimental data.


Figure 11. Gas exchange under significant diffusion disturbance due to increase in membrane thickness as predicted by the model. Top: for rest. Bottom: for peak exercise.


Figure 12. Dependence of oxygen transfer Vo 2 , saturation SaO 2 , mean alveolar pressure PAO 2 , and arterial pressure Pa o 2 on the arrival (dead) time at which the oxygenated fresh air reaches an acinus at constant VE = 7.5 liter/min (computed for PvO 2  = 40 mmHg and Q  = 5 liter/min). Each value was normalized to its value at zero arrival time t arr  = 0 s. This is why the PA and Pa curves coincide. The vertical dotted line represents a “normal” reference condition of t arr  = 0.7 s. Note that the PAO 2 and Pa o 2 curves are practically the same because the ordinates are in relative units, and PAO 2 ‐Pa o 2 keeps small.


Figure 13. Two‐dimensional (2D) illustration of the distribution of volumes of fresh air crossing generation 12 in the Florens et al. model 17 . There are 212 squares, each of which represents one airway at generation 12. The colors correspond to the logarithm of the distributed volume of fresh air expressed in milliliters. The black squares correspond to airways that are too small (diameter smaller than 0.5 mm) to belong to the tracheobronchial tree. Note the logarithmic color scale.


Figure 14. Time evolution of the spatial distribution of CO concentration. t represents the time after the RBC enters in diffusive contact with the gas. The membrane thickness is chosen to be 1 μm. Concentrations are normalized to the concentration at the gas‐membrane interface C g . The horizontal line represents the membrane‐plasma interface. The bottom figure gives the rate of transfer as a function of time. It reaches a steady state after 2 to 3 ms.


Figure 15. Three simplified geometries of the gas‐membrane‐plasma‐RBC structure. Top A: Horizontal cut of a planar parallel structure of a symmetrical gas‐membrane‐plasma‐RBC geometry. This is an artificial case where the alveolar‐capillary membrane, the plasma layer, and the blood cells are supposed to be planar with thicknesses of L M , L P , and L RBC , respectively. The dotted line at the bottom indicates the plane of symmetry. Top B: Quarter of an axial planar cross section of spherical RBC inside a cylindrical capillary shown as the bottom revolution axis. Top C: Quarter of an axial planar cross section of biconcave RBC in a cylindrical capillary. Bottom: 3D illustrations of each situation.


Figure 16. Spatial distribution of CO and NO concentrations in the planar system. Concentrations at the gas membrane interface have been normalized to 1. For simplicity, the same diffusivity is applied to the entire system. (Computed for D CO  = 3.26 × 10 −5  cm 2 /s; D NO  = 3.18 × 10 −5  cm 2 /s; τ CO  = 0.5 ms; τ NO  = 0.005 ms; L M  = 1 μm; L P  = 2.31 μm; L RBC  = 3.08 μm. See below for these values.)


Figure 17. Results of the exact calculation of 1/DLCO as a function of the reaction time τ ( τ is proportional to PO 2 for PO 2  > 80 mmHg) for the planar system of Figure  16 .


Figure 18. Comparison of the planar model prediction using Eq.  34 and experimental data from Forster et al. 21 . Values of 1/DLCO are rescaled using the values 1/DLCO (Ext.Pa o 2  → 0 mmHg) (Eq.  41 ).
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Bernard Sapoval, Min‐Yeong Kang, Anh Tuan Dinh‐Xuan. Modeling of Gas Exchange in the Lungs. Compr Physiol 2020, 11: 1289-1314. doi: 10.1002/cphy.c190019