Comprehensive Physiology Wiley Online Library
For Librarians For Authors Editors About

Gas Mixing in the Airways and Airspaces

Sylvia Verbanck, Manuel Paiva

10.1002/cphy.c100018

Source: Volume 1, Issue 2, April 2011

Published online: April 2011

Full Article on Wiley Online Library



Abstract

Basic physical concepts of diffusion, convection, and dispersion pertaining to gas transport in the human airways are reviewed. Their incorporation into quantitative models of gas mixing is presented, also illustrating the crucial interaction of gas transport equations with the model geometry. Model simulations are confronted with the available experimental gas mixing indices such as the phase III slope obtained in normal human lungs, with some pertinent examples in laboratory animals and in human lung disease. The use of inert gases with differing diffusion coefficients and their associated phase III slope provides invaluable experimental information on gas mixing in the lungs, with the concept of the diffusion front playing a central role. Sources of inter‐ and intraregional ventilation heterogeneity can be related to the location of the diffusion front, which offers the possibility to distinguish between ventilation heterogeneity proximal to the diffusion front (driven by convection between lung units larger than acini) and more peripheral ventilation heterogeneity (driven by diffusion‐convection interaction mainly within the acinus). While specific ventilation distribution and flow asynchrony co‐act to generate convection‐dependent ventilation heterogeneity, local structural asymmetry of the acinar air spaces is sufficient to generate diffusion‐convection‐dependent ventilation heterogeneity. The remaining hiatus in our understanding of ventilation heterogeneity in the human lung is described, together with some potential perspectives for its investigation. © 2011 American Physiological Society. Compr Physiol 1:809‐834, 2011.

Comprehensive Physiology offers downloadable PowerPoint presentations of figures for non-profit, educational use, provided the content is not modified and full credit is given to the author and publication.

Download a PowerPoint presentation of all images


Figure 1. Figure 1.

Cross‐sectional area (solid line, no symbol; right axis) obtained by summing the airway cross section of all parallel airways in any given airway generation for a lung model based on Weibel 23 and Haefeli‐Bleuer 29 morphometric data, at mid‐inspiration of a 1 liter inspiration starting from a lung volume of 3.7 liter. For an inspiratory flow of 0.5 liters/s in this model, Peclet numbers are computed for a large (SF6) and a low (He) molecular mass of the inflowing gas.
Figure 2. Figure 2.

Concentration profiles in a tube, in the cases of piston flow (grey area), with superimposed axial diffusion (dotted line), and in cases of fully developed laminar flow with average velocity , without (solid line) or with radial diffusion (dashed line).
Figure 3. Figure 3.

Ratios of bolus half widths of He and particles (HHe/HPa), of SF6 and particles (HSF6/HPa), and of He and SF6 (HHe/HSF6) plotted versus volumetric lung depth in three dogs; individual data points; from 49.
Figure 4. Figure 4.

Panel A: inspiratory O2 concentration profiles simulated in a symmetrical Weibel model plotted at 0.2 s intervals of a 1 s inhalation (at 1 liters/s); replotted from 76. Panel B: end‐inspiratory He and SF6 concentration profiles simulated in a symmetrical Weibel model (at 1 liters/s); replotted from Paiva and Engel 76.
Figure 5. Figure 5.

Panel A: diffusion front simulations in the human acinus. Dashed and dotted lines are averaged SF6 and He concentrations, respectively, at end inspiration of 2 liters starting from 3.7 liters at 1 liters/s; adapted from 30. Panel B: diffusion front simulations in the rat lung. Dashed and dotted lines are averaged SF6 and He concentrations, respectively, at end inspiration of 2 ml starting from 3.7 ml at 1 ml/s; adapted from 30. For clarity, these plots do not include the variability of He or SF6 concentration around the average value in each generation nor that in the terminal units. Therefore these plots could lead to the perception that mass balance for He and SF6 inside the model is in favor of SF6, which is not the case.
Figure 6. Figure 6.

A: Expiratory O2 fractional concentration profiles simulated in a symmetrical Weibel model plotted at 0.2 s intervals of a 1 s exhalation (at 1 liters/s); t = 0 curves corresponds to the end‐inspiratory O2 concentration profile retrieved from Figure 4A; adapted from 76. B: Expiratory N2 concentration profiles recovered at the model exit as a result of simulations of panel A.
Figure 7. Figure 7.

NO concentration profiles in generations 7 to 23 during expiration as a function of axial distance from alveolar end obtained by solving Eq. 13 with source term Eq. 14 with constant expiratory flow (500 ml/s). Curves are shown every 0.2 s (from 0 to 2 s). Curve at t = 0 is the NO profile after 2 s inspiration at 500 ml/s; adapted from 70.
Figure 8. Figure 8.

A two‐trumpet model representing asymmetry of branching where each subunit is represented by a single trumpet. Structural asymmetry can be generated in two ways: by either considering s1 = s2 and S1 = S2 in which case one of the units can be truncated in axial length at the level of a dashed line, or, by considering the same axial length for both trumpets but s1 ≠ s2 and/or S1 ≠ S2; from 92
Figure 9. Figure 9.

Pattern of fractional O2 concentration at 0.2 s intervals during expiration after 1 liters of inspiration of O2 at 1 liters/s into an asymmetrical two‐trumpet model (V2/V1 = 0.12). Dashed lines: shorter unit (volume V2). Solid lines: longer unit (volume V1) and common pathway. Branch point is at x = 2.7 mm. Inset: O2 at model exit plotted against expired volume [from 92].
Figure 10. Figure 10.

Lower left panel: schematic representation of upper (A) and lower (B) lung units at residual volume (continuous circles: VA > VB). For a vital capacity O2 inspiration, the N2 concentration in unit A is higher than in unit B (lower right panel). The pressure volume loop for a vital capacity breath is represented in the upper panel (downward arrows for the expiratory limb). If the vertical gradient of pleural pressure remains constant during expiration, units A and B, respectively, contribute more at end and at the beginning of expiration (respective expired concentrations are represented by a closed circle and a star); from 21.
Figure 11. Figure 11.

Nitrogen concentration and volume tracings as a function of time during a multiple‐breath washout (MBW) test from a normal subject during a bronchoprovocation test with histamine, which greatly increases the slopes, particularly at the end of the MBW. Inset: alveolar slope versus expired volume from breaths 1 and 20 plotted using an equivalent scaling with respect to mean expired nitrogen concentrations of breaths 1 and 20, respectively; from 128.
Figure 12. Figure 12.

Normalized phase III slopes as a function of breath number, in a baseline condition (no symbols), with increasing specific ventilation heterogeneity only (squares), increased flow asynchrony between two compartments (circles), or both (triangles). The dashed line frame indicates normal measurement range (see corresponding y‐axis in Fig. 13).
Figure 13. Figure 13.

A typical experimental normalized phase III slope curve obtained in a normal subject (closed circles), and its decomposition into the corresponding the evolution of normalized slope as a function of breath number (or lung turnover) predicted by inter‐ and intraregional ventilation heterogeneities combined (open circles) and by intra‐acinar ventilation heterogeneities (open triangles).
Figure 14. Figure 14.

Panel A: normalized slope (Sn) curves obtained for N2 and SF6‐He slope difference in normal subjects; adapted from 130. Panel B: normalized slope curves obtained for N2, with and without end‐inspiratory breath hold times of either 1 or 4 s; from 146.
Figure 15. Figure 15.

Normalized slope curves obtained in humans, rats, and steers. In contrast to humans, CDI is almost negligible in the lungs of rats and steers; from 30.
Figure 16. Figure 16.

Schematic representation of predicted changes of normalized phase III slopes versus lung turnover or breath number and corresponding changes in MBW indices Sacin and Scond, following structural alterations in the proximal or the peripheral lung.


Figure 1.

Cross‐sectional area (solid line, no symbol; right axis) obtained by summing the airway cross section of all parallel airways in any given airway generation for a lung model based on Weibel 23 and Haefeli‐Bleuer 29 morphometric data, at mid‐inspiration of a 1 liter inspiration starting from a lung volume of 3.7 liter. For an inspiratory flow of 0.5 liters/s in this model, Peclet numbers are computed for a large (SF6) and a low (He) molecular mass of the inflowing gas.



Figure 2.

Concentration profiles in a tube, in the cases of piston flow (grey area), with superimposed axial diffusion (dotted line), and in cases of fully developed laminar flow with average velocity , without (solid line) or with radial diffusion (dashed line).



Figure 3.

Ratios of bolus half widths of He and particles (HHe/HPa), of SF6 and particles (HSF6/HPa), and of He and SF6 (HHe/HSF6) plotted versus volumetric lung depth in three dogs; individual data points; from 49.



Figure 4.

Panel A: inspiratory O2 concentration profiles simulated in a symmetrical Weibel model plotted at 0.2 s intervals of a 1 s inhalation (at 1 liters/s); replotted from 76. Panel B: end‐inspiratory He and SF6 concentration profiles simulated in a symmetrical Weibel model (at 1 liters/s); replotted from Paiva and Engel 76.



Figure 5.

Panel A: diffusion front simulations in the human acinus. Dashed and dotted lines are averaged SF6 and He concentrations, respectively, at end inspiration of 2 liters starting from 3.7 liters at 1 liters/s; adapted from 30. Panel B: diffusion front simulations in the rat lung. Dashed and dotted lines are averaged SF6 and He concentrations, respectively, at end inspiration of 2 ml starting from 3.7 ml at 1 ml/s; adapted from 30. For clarity, these plots do not include the variability of He or SF6 concentration around the average value in each generation nor that in the terminal units. Therefore these plots could lead to the perception that mass balance for He and SF6 inside the model is in favor of SF6, which is not the case.



Figure 6.

A: Expiratory O2 fractional concentration profiles simulated in a symmetrical Weibel model plotted at 0.2 s intervals of a 1 s exhalation (at 1 liters/s); t = 0 curves corresponds to the end‐inspiratory O2 concentration profile retrieved from Figure 4A; adapted from 76. B: Expiratory N2 concentration profiles recovered at the model exit as a result of simulations of panel A.



Figure 7.

NO concentration profiles in generations 7 to 23 during expiration as a function of axial distance from alveolar end obtained by solving Eq. 13 with source term Eq. 14 with constant expiratory flow (500 ml/s). Curves are shown every 0.2 s (from 0 to 2 s). Curve at t = 0 is the NO profile after 2 s inspiration at 500 ml/s; adapted from 70.



Figure 8.

A two‐trumpet model representing asymmetry of branching where each subunit is represented by a single trumpet. Structural asymmetry can be generated in two ways: by either considering s1 = s2 and S1 = S2 in which case one of the units can be truncated in axial length at the level of a dashed line, or, by considering the same axial length for both trumpets but s1 ≠ s2 and/or S1 ≠ S2; from 92



Figure 9.

Pattern of fractional O2 concentration at 0.2 s intervals during expiration after 1 liters of inspiration of O2 at 1 liters/s into an asymmetrical two‐trumpet model (V2/V1 = 0.12). Dashed lines: shorter unit (volume V2). Solid lines: longer unit (volume V1) and common pathway. Branch point is at x = 2.7 mm. Inset: O2 at model exit plotted against expired volume [from 92].



Figure 10.

Lower left panel: schematic representation of upper (A) and lower (B) lung units at residual volume (continuous circles: VA > VB). For a vital capacity O2 inspiration, the N2 concentration in unit A is higher than in unit B (lower right panel). The pressure volume loop for a vital capacity breath is represented in the upper panel (downward arrows for the expiratory limb). If the vertical gradient of pleural pressure remains constant during expiration, units A and B, respectively, contribute more at end and at the beginning of expiration (respective expired concentrations are represented by a closed circle and a star); from 21.



Figure 11.

Nitrogen concentration and volume tracings as a function of time during a multiple‐breath washout (MBW) test from a normal subject during a bronchoprovocation test with histamine, which greatly increases the slopes, particularly at the end of the MBW. Inset: alveolar slope versus expired volume from breaths 1 and 20 plotted using an equivalent scaling with respect to mean expired nitrogen concentrations of breaths 1 and 20, respectively; from 128.



Figure 12.

Normalized phase III slopes as a function of breath number, in a baseline condition (no symbols), with increasing specific ventilation heterogeneity only (squares), increased flow asynchrony between two compartments (circles), or both (triangles). The dashed line frame indicates normal measurement range (see corresponding y‐axis in Fig. 13).



Figure 13.

A typical experimental normalized phase III slope curve obtained in a normal subject (closed circles), and its decomposition into the corresponding the evolution of normalized slope as a function of breath number (or lung turnover) predicted by inter‐ and intraregional ventilation heterogeneities combined (open circles) and by intra‐acinar ventilation heterogeneities (open triangles).



Figure 14.

Panel A: normalized slope (Sn) curves obtained for N2 and SF6‐He slope difference in normal subjects; adapted from 130. Panel B: normalized slope curves obtained for N2, with and without end‐inspiratory breath hold times of either 1 or 4 s; from 146.



Figure 15.

Normalized slope curves obtained in humans, rats, and steers. In contrast to humans, CDI is almost negligible in the lungs of rats and steers; from 30.



Figure 16.

Schematic representation of predicted changes of normalized phase III slopes versus lung turnover or breath number and corresponding changes in MBW indices Sacin and Scond, following structural alterations in the proximal or the peripheral lung.

References
 1. Rauwerda PE. Unequal ventilation of different parts of the lung and the determination of cardiac output.” PhD Thesis. University of Groningen, The Netherlands. 1946.
 2. Georg J, Lassen NA, Mellemgaard K, Vinther A. Diffusion in the gas phase of the lungs in normal and emphysematous subjects. Clin Sci 29(3): 525–532, 1965.
 3. Lacquet LM. Convection and diffusion in the airways of the lung. Bull Physiopathol Respir (Nancy) 8(1): 152–154, 1972.
 4. Kawashiro T, Sikand RS, Adaro F, Takahashi H, Piiper J. Study of intrapulmonary gas mixing in man by simultaneous wash‐out of helium and sulphur hexafluoride. Respir Physiol 28(2): 261–275, 1976.
 5. Okubo T, Piiper J. Intrapulmonary gas mixing in excised dog lung lobes studied by simultaneous washout of two inert gases. Respir Physiol 21(2): 223–239, 1974.
 6. Scheid P, Hlastala MP, Piiper J. Inert gas elimination from lungs with stratified inhomogeneity: Theory. Respir Physiol 44(3): 299–309, 1981.
 7. Dutrieue B, Verbanck S, Darquenne C, Prisk GK. Airway closure in microgravity. Respir Physiol Neurobiol 148(1‐2): 97–111, 2005.
 8. Prisk GK, Elliott AR, Guy HJ, Verbanck S, Paiva M, West JB. Multiple‐breath washin of helium and sulfur hexafluoride in sustained microgravity. J Appl Physiol 84(1): 244–252, 1998.
 9. Lauzon AM, Prisk GK, Elliott AR, Verbanck S, Paiva M, West JB. Paradoxical helium and sulfur hexafluoride single‐breath washouts in short‐term vs. sustained microgravity. J Appl Physiol 82(3): 859–865, 1997.
 10. Prisk GK, Lauzon AM, Verbanck S, Elliot AR, Guy HJ, Paiva M, West JB. Anomalous behavior of helium and sulfur hexafluoride during single‐breath tests in sustained microgravity. J Appl Physiol 80(4): 1126–1132, 1996.
 11. Chang HK, Farhi LE. Ternary diffusion and effective diffusion coefficients in alveolar spaces. Respir Physiol 40(2): 269–279, 1980.
 12. Chang H, Tai RC, Farhi LE. Some implications of ternary diffusion in the lung. Respir Physiol 23(1): 109–120, 1975.
 13. Scheid P, Pipper J. Intrapulmonary gas mixing and stratification. In: West JB, editor. Pulmonary Gas Exchange, Vol 1. New York: Academic Press, 1980, p. 87–130.
 14. Worth H, Piiper J. Diffusion of helium, carbon monoxide and sulphur hexafluoride in gas mixtures similar to alveolar gas. Respir Physiol 32(2): 155–166, 1978.
 15. Worth H, Piiper J. Model experiments on diffusional equilibration of oxygen and carbon dioxide between inspired and alveolar gas. Respir Physiol 35(1): 1–7, 1978.
 16. Piiper J, Worth H. Value and limits of Graham's law for prediction of diffusivities of gases in gas mixtures. Respir Physiol 41(3): 233–240, 1980.
 17. Chang HK. General concepts of molecular diffusion. In: Engel LA, Paiva M, editors. Gas Mixing and Distribution in the Lung. Lung Biology in Health and Disease, vol. 25. New York: Marcel Dekker, 1985, p 1–22.
 18. Reid RC, Sherwood TK. Diffusion coefficients. In: The Properties of Gases and Liquids. New York. McGraw Hill, 1966, p. 520–570.
 19. van Liew HD, Thalmann ED, Sponholtz DK. Diffusion‐dependence of pulmonary gas mixing at 5.5 and 9.5 ATA. Undersea Biomed Res 6(3): 251–258, 1979.
 20. Dutrieue B, Paiva M, Verbanck S, Le Gouic M, Darquenne C, Prisk GK. Tidal volume single‐breath washin of SF6 and CH4 in transient microgravity. J Appl Physiol 94(1): 75–82, 2003. Epub 2002 Sep 6.
 21. Paiva M, Engel LA. Gas mixing in the lung periphery. In: Chang HK, Paiva M, editors. Respiratory Physiology: An Analytical Approach. New York: Dekker, 1989, p. 245–276.
 22. Paiva M, Engel LA. Theoretical studies of gas mixing and ventilation distribution in the lung. Physiol Rev 67: 750–796, 1987.
 23. Weibel ER. Morphometry of the Human Lung. Berlin, New York: Springer Verlag and Academic Press, 1963.
 24. Altshuler B. Behavior of airborne particles in the respiratory tract. In: Wolstenholme GEW, Knight J, Editors. Circulatory and Respiratory Mass Transport. London, Churchill, 1969, p. 215–231.
 25. Paiva M. Gaseous diffusion in an alveolar duct simulated by a digital computer. Comput Biomed Res. 7(6): 533–543, 1974.
 26. Verbanck S, Paiva M. Effective axial diffusion in an expansile alveolar duct model. Respir Physiol 73(2): 273–278, 1988.
 27. Federspiel WJ, Fredberg JJ. Axial dispersion in respiratory bronchioles and alveolar ducts. J Appl Physiol 64(6): 2614–2621, 1988.
 28. Davidson MR. Further considerations in a theoretical description of gas transport in lung airways. Bull Math Biol. 43(5): 517–548, 1981.
 29. Haefeli‐Bleuer B, Weibel ER. Morphometry of the human pulmonary acinus. Anat Rec 220: 401–414, 1988.
 30. Verbanck S, Weibel ER, Paiva M. Simulations of washout experiments in postmortem rat lungs. J Appl Physiol 75(1): 441–451, 1993.
 31. Rodriguez M, Bur S, Favre A, Weibel ER. The pulmonary acinus: Geometry and morphometry of the peripheral airway system in rat and rabbit. Am J Anat 180: 143–155, 1987.
 32. Taylor GI. Dispersion of soluble matter in solvent flowing slowly through a tube, Proc Roy Soc A 219, 186–203, 1953.
 33. Aris R. On the dispersion of a solute in a fluid flowing through a tube. Proc Roy Soc Lond A Math Phys Sci 235: 67–77, 1956.
 34. Lacquet LM, Van Der Linden LP, Paiva M. Transport of H2 and SF6 in the lung. Respir Physiol 25(2): 157–173, 1975.
 35. Horsfield K, Davies A, Cumming G. Role of conducting airways in partial separation of inhaled gas mixtures. J Appl Physiol 43(3): 391–396, 1977.
 36. Hogg W, Brunton J, Kryger M, Brown R, Macklem P. Gas diffusion across collateral channels. J Appl Physiol 33(5): 568–575, 1972.
 37. van Ertbruggen C, Hirsch C, Paiva M. Anatomically based three‐dimensional model of airways to simulate flow and particle transport using computational fluid dynamics. J Appl Physiol 98(3): 970–980, 2005.
 38. Schroter RC, Sudlow MF. Flow patterns in models of the human bronchial airways. Respir Physiol 7(3): 341–355, 1969.
 39. Gill WN, Ananthakrishnan V, Nunge RJ. Dispersion in developing velocity fields. Am Inst Chem Eng J 14: 939–946, 1968
 40. Gill WN, Sankarasubramanian R. Exact analysis of unsteady convective diffusion. Proc Roy Soc Lond A 316: 341–350, 1970.
 41. Gill WN, Sankarasubramanian R. Dispersion of a non‐uniform slug in time dependent flow. Proc Roy Soc London A 322: 101–117, 1971.
 42. Weaver DW, Ultman JS. Axial dispersion through tube constrictions. Am Inst Chem Eng J 26: 9–15, 1980.
 43. Pack A, Hooper MB, Nixon W, Taylor JC. A computational model of pulmonary gas transport incorporating effective diffusion. Respir Physiol 29(1): 101–123, 1977.
 44. Ultman JS, Thomas MW. Longitudinal mixing in pulmonary airways: Comparison of inspiration and expiration. J Appl Physiol 46(4): 799–805, 1979.
 45. Scherer PW, Shendalman LH, Greene NM, Bouhuys A. Measurement of axial diffusivities in a model of the bronchial airways. J Appl Physiol 38(4): 719–723, 1975.
 46. Ultman JS, Blatman HS. A compartmental dispersion model for the analysis of mixing in tube networks. Am Inst Chem Eng J 23: 169–176, 1977.
 47. Ultman JS, Blatman HS. Longitudinal mixing in pulmonary airways. Analysis of inert gas dispersion in symmetric tube network models. Respir Physiol 30(3): 349–367, 1977.
 48. Ultman JS, Doll BE, Spiegel R, Thomas MW. Longitudinal mixing in pulmonary airways—normal subjects respiring at a constant flow. J Appl Physiol 44(2): 297–303, 1978.
 49. Schulz H, Heilmann P, Hillebrecht A, Gebhart J, Meyer M, Piiper J, Heyder J. Convective and diffusive gas transport in canine intrapulmonary airways. J Appl Physiol 72(4): 1557–1562, 1992.
 50. Brand P, Rieger C, Schulz H, Beinert T, Heyder J. Aerosol bolus dispersion in healthy subjects. Eur Respir J 10: 460–467, 1997.
 51. Darquenne C, West JB, Prisk GK. Dispersion of 0.5‐ to 2‐m aerosol in μG and hypergravity as a probe of convective inhomogeneity in the lung. J Appl Physiol 86: 1402–1409, 1999.
 52. Fukuchi Y, Cosio M, Kelly S, Engel LA. Influence of pericardial fluid on cardiogenic gas mixing in the lung. J Appl Physiol 42(1): 5–12, 1977.
 53. Engel LA. Dynamic distribution of gas flow. In: Macklem RT, Mead J, editors. Handbook of Physiology. Section 3: The Respiratory System, Vol. 3: Mechanics of Breathing. Bethesda, MD: American Physiological Society, 1986, p. 575–593.
 54. McGrath MW, Hugh‐Jones P. Some observations on the distribution of gas flow in the human bronchial tree. Clin Sci 24: 209–222, 1963.
 55. West JB, Hugh‐Jones P. Pulsatile gas flow in bronchi caused by the heart beat. J Appl Physiol 16: 697–702, 1961.
 56. Colebatch HJ, Ng CK, Maccioni FJ. Inspiratory gas flow induced by cardiac systole. Respir Physiol 105(1‐2): 103–108, 1996.
 57. Engel LA, Menkes H, Wood LD, Utz G, Joubert J, Macklem PT. Gas mixing during breath holding studied by intrapulmonary gas sampling. J Appl Physiol 35(1): 9–17, 1973.
 58. Engel LA, Wood LD, Utz G, Macklem PT. Gas mixing during inspiration. J Appl Physiol 35(1): 18–24, 1973.
 59. Fukuchi Y, Roussos CS, Macklem PT, Engel LA. Convection, diffusion and cardiogenic mixing of inspired gas in the lung; an experimental approach. Respir Physiol 26(1): 77–90, 1976.
 60. Zhang S, Saltzman AR, Klocke RA. Influence of cardiac action on gas mixing in closed‐chest dogs. J Appl Physiol 79(1): 113–120, 1995.
 61. Drechsler DM, Ultman JS. Cardiogenic mixing in the pulmonary conducting airways of man? Respir Physiol 56(1): 37–44, 1984.
 62. Paiva M, Engel LA. Influence of bronchial asymmetry on cardiogenic gas mixing in the lung. Respir Physiol 49(3): 325–338, 1982.
 63. Paiva M. Computation of the boundary conditions for diffusion in the human lung. Comput Biomed Res 5(6): 585–595, 1972.
 64. Scherer PW, Shendalman LH, Greene NM. Simultaneous diffusion and convection in single breath lung washout. Bull Math Biophys 34(3): 393–412, 1972.
 65. Baker LG, Ultman JS, Rhoades RA. Simultaneous gas flow and diffusion in a symmetric airway system: A mathematical model. Respir Physiol 21(1): 119–138, 1974.
 66. Davidson MR, Fitz‐Gerald JM. Transport of O2 along a model pathway through the respiratory region of the lung. Bull Math Biol 36(3): 275–303, 1974.
 67. Scherer PW, Neff JD, Baumgardner JE, Neufeld GR. The importance of a source term in modeling multibreath inert gas washout. Respir Physiol 103(1): 99–103, 1996.
 68. Tawhai MH, Hunter PJ. Multibreath washout analysis: Modelling the influence of conducting airway asymmetry. Respir Physiol 127(2‐3): 249–258, 2001.
 69. Paiva M, Engel LA. Model analysis of intra‐acinar gas exchange. Respir Physiol 62(2): 257–272, 1985.
 70. Van Muylem A, Noël C, Paiva M. Modeling of impact of gas molecular diffusion on nitric oxide expired profile. J Appl Physiol 94(1): 119–127, 2003.
 71. Shin HW, George SC. Impact of axial diffusion on nitric oxide exchange in the lungs. J Appl Physiol 93: 2070–2080, 2002.
 72. Taulbee DB, Yu CP. A theory of aerosol deposition in the human respiratory tract. J Appl Physiol 38(1): 77–85, 1975.
 73. Taulbee DB, Yu CP, Heyder J. Aerosol transport in the human lung from analysis of single breaths. J Appl Physiol. 44(5): 803–812, 1978.
 74. Darquenne C, Paiva M. One‐dimensional simulation of aerosol transport and deposition in the human lung. J Appl Physiol 77(6): 2889–2898, 1994.
 75. Darquenne C, Paiva M. Two‐ and three‐dimensional simulations of aerosol transport and deposition in alveolar zone of human lung. J Appl Physiol 80: 1401–1414, 1996.
 76. Paiva M, Engel LA. Pulmonary interdependence of gas transport. J Appl Physiol 47(2): 296–305, 1979.
 77. Wilson A, Lin K.‐H. Convection and diffusion in the airways and the design of the bronchial tree. In: Bouhuys A, Editor. Airway Dynamics. Springfield, IL: Thomas, 1970, p. 5–19.
 78. Paiva M, Lacquet LM, Van Der Linden LP. Gas transport in a model derived from Hansen‐Ampaya anatomical data of the human lung. J Appl Physiol 41(1): 115–119, 1976.
 79. Engel LA, Paiva M, Siegler DI, Fukuchi Y. Dual tracer single breath studies of gas transport in the lung. Respir Physiol 36(2): 103–119, 1979.
 80. Paiva M, Engel LA. Model analysis of gas distribution within human lung acinus. J Appl Physiol 56(2): 418–425, 1984.
 81. Engel LA, Paiva M. Analyses of sequential filling and emptying of the lung. Respir Physiol 45(3): 309–321, 1981.
 82. Dutrieue B, Vanholsbeeck F, Verbanck S, Paiva M. A human acinar structure for simulation of realistic alveolar plateau slopes. J Appl Physiol 89: 1859–1867, 2000.
 83. Horsfield K, Davies A, Mills C, Cumming G. Effect of flow oscillations on the stationary concentration front in a hollow cast of the airways. Lung 157(2): 103–111, 1980.
 84. Hobbs SH, Lightfoot EN. A Monte Carlo simulation of convective dispersion in the large airways. Respir Physiol 37(3): 273–292, 1979.
 85. Fowler WS. Lung function studies. II. The respiratory dead space. Am J Physiol Lond 106: 405–416, 1948.
 86. Bohr C. Über die Lungenatmung. Scand Arch Physiol 2: 236–268, 1891.
 87. Siebeck F. Ueber den Gasaustausch zwischen AuBenluft und den Alvoolen. II Mitteilung. Ueber die Bedeutung und Bestimmung des “schadlichen Raumes” bei der Atmung. Skand Arch Physiol 25: 81–95, 1911.
 88. Krogh A, Lindhard J. The volume of the dead space in breathing, and the mixing of gases in the lungs of man. J Physiol Lond 51: 59–90, 1917.
 89. Paiva M. Gas transport in the human lung. J Appl Physiol 35(3): 401–410, 1973.
 90. Mon E, Ultman JS. Monte Carlo simulation of simultaneous gas flow and diffusion in an asymmetric distal pulmonary airway model. Bull Math Biol 38(2): 161–192, 1976.
 91. Parker H, Horsfield K, Cumming G. Morphology of distal airways in the human lung. J Appl Physiol 31(3): 386–391, 1971.
 92. Paiva M, Engel LA. The anatomical basis for the sloping N2 plateau. Respir Physiol 44(3): 325–337, 1981.
 93. Luijendijk SC, Zwart A, de Vries WR, Salet WM. The sloping alveolar plateau at synchronous ventilation. Pflugers Arch 384(3): 267–277, 1980.
 94. Paiva M. Diffusion and convection in a branching tube. Comput Biomed Res 16(2): 190–198, 1983.
 95. González Mangado N, Peces‐Barba G, Verbanck S, Paiva M. Single‐breath washout experiments in rat lungs. J Appl Physiol 71(3): 855–862, 1991.
 96. de Vries WR, Luijendijk SC, Zwart A. Helium and sulfur hexafluoride washout in asymmetric lung models. J Appl Physiol 51(5): 1122–1130, 1981.
 97. Bowes C, Cumming G, Horsfield K, Loughhead J, Preston S. Gas mixing in a model of the pulmonary acinus with asymmetrical alveolar ducts. J Appl Physiol 52(3): 624–633, 1982.
 98. Bowes CL, Richardson JD, Cumming G, Horsfield K. Effect of breathing pattern on gas mixing in a model with asymmetrical alveolar ducts. J Appl Physiol 58(1): 18–26, 1985.
 99. Davidson MR, Engel LA. Gas transport in an asymmetrical acinus. Bull Eur Physiopathol Respir 18(2): 203–214, 1982.
 100. Paiva M. Model analysis of inert gas mixing in the lung. Bull Eur Physiopathol Respir 18(2): 189–201, 1982.
 101. Kelly S, Cohen C, Powell E, Paiva M, Engel LA. Gas mixing in the lungs of dogs and pigs. Respir Physiol 47(3): 341–349, 1982.
 102. Kelly S, Paiva M, Engel LA. Bronchoconstriction and gas mixing in canine and pig lungs. Bull Eur Physiopathol Respir 18(2): 229–237, 1982.
 103. Van Muylem A, Antoine M, Yernault JC, Paiva M, Estenne M. Inert gas single‐breath washout after heart‐lung transplantation. Am J Respir Crit Care Med 152(3): 947–952, 1995.
 104. Sapoval B, Filoche M, Weibel ER. Smaller is better—but not too small: A physical scale for the design of the mammalian pulmonary acinus. Proc Natl Acad Sci U S A 99(16): 10411–10416, 2002.
 105. Weibel ER, Sapoval B, Filoche M. Design of peripheral airways for efficient gas exchange. Respir Physiol Neurobiol 148(1‐2): 3–21, 2005.
 106. Weibel ER. The Pathway for Oxygen: Structure and Function in the Mammalian Respiratory System. Cambridge, MA: Harvard University Press, 1984.
 107. Sapoval B. Transfer to and across irregular membranes modeled by fractal geometry. In: Nonnenmacher TF, Losa GA, Weibel ER, editors. Fractals in Biology and Medicine. Basel: Birkhauser, 1994, p. 241–249.
 108. Swan AJ, Clark AR, Tawhai MH. Spatial and temporal variation in Pao2 in the pulmonary acinus. Am J Respir Crit Care Med 181: A3645, 2010.
 109. Woods JC, Yablonskiy DA, Choong CK, Chino K, Pierce JA, Hogg JC, Bentley J, Cooper JD, Conradi MS, Macklem PT. Long‐range diffusion of hyperpolarized 3He in explanted normal and emphysematous human lungs via magnetization tagging. J Appl Physiol 99: 1992–1997, 2005.
 110. Verbanck S, Paiva M. Simulation of the apparent diffusion of helium‐3 in the human acinus. J Appl Physiol 103(1): 249–54, 2007. Epub 2007 Mar 22.
 111. Verbanck SA, Paiva M. Acinar determinants of the apparent diffusion coefficient for Helium‐3. J Appl Physiol 108: 793–799, 2010. [Epub ahead of print]
 112. Bartel SE, Haywood SE, Woods JC, Chang YV, Menard C, Yablonskiy DA, Gierada DS, Conradi MS. Role of collateral paths in long‐range diffusion in lungs. J Appl Physiol 104: 1495–1503, 2008.
 113. Pérez‐Sánchez JM, Rodríguez I, Ruiz‐Cabello J. Random walk simulation of the MRI apparent diffusion coefficient in a geometrical model of the acinar tree. Biophys J 97(2): 656–664, 2009.
 114. Ball WC, Stewart PB, Newsham LGS, Bates DV. Regional pulmonary function study with Xenon133. J Clin Invest 41: 519–531, 1962.
 115. Dollfuss RE, Milic‐Emili J, Bates DV. Regional ventilation of lung, studied with boluses of 133Xenon. Respir Physiol 2: 234–246, 1967.
 116. Robertson PC, Anthonisen NR, Ross D. Effect of inspiratory flow rate on regional distribution of inspired gas. J Appl Physiol 26(4): 438–443, 1969.
 117. Milic‐Emili J, Henderson JA, Dolovich MB, Trop D, Kaneko K. Regional distribution of inspired gas in the lung. J Appl Physiol 21(3): 749–759, 1966.
 118. Anthonisen NR, Robertson PC, Ross WR. Gravity‐depende sequential emptying of lung regions. J Appl Physiol 28(5): 589–595, 1970.
 119. Verbanck S, Linnarsson D, Prisk GK, Paiva M. Specific ventilation distribution in microgravity. J Appl Physiol 80(5): 1458–1465, 1996.
 120. Young AC, Martin CJ. The sequence of lobar emptying in man. Respir Physiol 1(4): 372–381, 1966.
 121. Engel LA, Utz G, Wood LD, Macklem PT. Ventilation distribution in anatomical lung units. J Appl Physiol 37(2): 194–200, 1974.
 122. Olson LE, Rodarte JR. Regional differences in expansion in excised dog lung lobes. J Appl Physiol 57(6): 1710–1714, 1984.
 123. Robertson HT, Hlastala MP. Microsphere maps of regional blood flow and regional ventilation. J Appl Physiol 102(3): 1265–1272, 2007.
 124. Robertson HT, Kreck TC, Krueger MA. The spatial and temporal heterogeneity of regional ventilation: Comparison of measurements by two high‐resolution methods. Respir Physiol Neurobiol 148: 85–95, 2005.
 125. Porra L, Monfraix S, Berruyer G, Le Duc G, Nemoz C, Thomlinson W, Suortti P, Sovijärvi AR, Bayat S. Effect of tidal volume on distribution of ventilation assessed by synchrotron radiation CT in rabbit. J Appl Physiol 96(5): 1899–1908, 2004.
 126. Fukuchi Y, Cosio M, Murphy B, Engel LA. Intraregional basis for sequential filling and emptying of the lung. Respir Physiol 41(3): 253–266, 1980.
 127. Verbanck S, Paiva M. Model simulations of gas mixing and ventilation distribution in the human lung. J Appl Physiol 69(6): 2269–2279, 1990.
 128. Verbanck S, Schuermans D, Van Muylem A, Paiva M, Noppen M, Vincken W. Ventilation distribution during histamine provocation. J Appl Physiol 83(6): 1907–1916, 1997.
 129. Paiva M. Two new pulmonary functional indexes suggested by a simple mathematical model. Respiration 32(5): 389–403, 1975.
 130. Crawford AB, Makowska M, Paiva M, Engel LA. Convection‐ and diffusion‐dependent ventilation maldistribution in normal subjects. J Appl Physiol 59(3): 838–846, 1985.
 131. Becklake MR. A new index of the intrapulmonary mixture of inspired air. Thorax 7(1): 111–116, 1952.
 132. Edelman NH, Mittman C, Norris AH, Shock NW. Effects of respiratory pattern on age differences in ventilation uniformity. J Appl Physiol 24(1): 49–53, 1968.
 133. Saniie J, Saidel GM, Chester EH. Real‐time moment analysis of pulmonary nitrogen washout. J Appl Physiol 46(6): 1184–1190, 1979.
 134. Fowler WS, Cornish ER, Ketty, SS. Lung function studies. VIII. Analysis of alveolar ventilation by pulmonary N2 clearance curves. J Clin Invest 31: 40–50, 1952.
 135. Cumming G, Guyatt AR. Alveolar gas mixing efficiency in the human lung. Clin Sci (Lond) 62(5): 541–547, 1982.
 136. Engel LA. Intraregional gas mixing and distribution. In: Paiva M, editor. Gas Mixing and Distribution in the Lung. New York: Marcel Dekker, 1985, p. 287–358.
 137. Fuchs SI, Eder J, Ellemunter H, Gappa M. Lung clearance index: Normal values, repeatability, and reproducibility in healthy children and adolescents. Pediatr Pulmonol 44(12): 1180–1185, 2009.
 138. Robinson PD, Goldman MD, Gustafsson PM. Inert gas washout: Theoretical background and clinical utility in respiratory disease. Respiration 78(3): 339–355, 2009. Epub 2009 Jun 12. Review.
 139. Aurora P, Gustafsson P, Bush A, Lindblad A, Oliver C, Wallis CE, Stocks J. Multiple breath inert gas washout as a measure of ventilation distribution in children with cystic fibrosis. Thorax 59(12): 1068–1073, 2004.
 140. Macleod KA, Horsley AR, Bell NJ, Greening AP, Innes JA, Cunningham S. Ventilation heterogeneity in children with well controlled asthma with normal spirometry indicates residual airways disease. Thorax 64(1): 33–37, 2009. Epub 2008 Aug 4.
 141. Van Muylem A, Scillia P, Knoop C, Paiva M, Estenne M. Single‐breath test in lateral decubitus reflects function of single lungs grafted for emphysema. J Appl Physiol 100(3): 834–838, 2006. Epub 2005 Nov 23.
 142. Gillis HL, Lutchen KR. How heterogeneous bronchoconstriction affects ventilation distribution in human lungs: A morphometric model. Ann Biomed Eng 27(1): 14–22, 1999. Erratum in: Ann Biomed Eng 1999 May‐Jun;27(3): 411.
 143. Nucci G, Suki B, Lutchen K. Modeling airflow‐related shear stress during heterogeneous constriction and mechanical ventilation. J Appl Physiol 95(1): 348–356, 2003. Epub 2003 Mar 21.
 144. Verbanck S, Schuermans D, Van Malderen S, Vincken W, Thompson B. The effect of conductive ventilation heterogeneity on diffusing capacity measurement. J Appl Physiol 104(4): 1094–1100, 2008. Epub 2008 Feb 14.
 145. Suresh V, Shelley DA, Shin HW, George SC. Effect of heterogeneous ventilation and nitric oxide production on exhaled nitric oxide profiles. J Appl Physiol 104(6): 1743–1752, 2008. Epub 2008 Mar 20.
 146. Crawford AB, Makowska M, Kelly S, Engel LA. Effect of breath holding on ventilation maldistribution during tidal breathing in normal subjects. J Appl Physiol 61(6): 2108–2115, 1986.
 147. Tsang JY, Emery MJ, Hlastala MP. Ventilation inhomogeneity in oleic acid‐induced pulmonary edema. J Appl Physiol 82(4): 1040–1045, 1997.
 148. Verbanck S, González Mangado N, Peces‐Barba G, Paiva M. Multiple‐breath washout experiments in rat lungs. J Appl Physiol 71(3): 847–854, 1991.
 149. Rollin F, Desmecht D, Verbanck S, Van Muylem A, Lekeux P, Paiva M. Multiple‐breath washout and washin experiments in steers. J Appl Physiol 81(2): 957–963, 1996.
 150. De Matteo R, Snibson K, Thompson B, Koumoundouros E, Harding R. Lung function in developing lambs: Is it affected by preterm birth? J Appl Physiol 107(4): 1083–1088, 2009. Epub 2009 Aug 13.
 151. Cosio M, Ghezzo H, Hogg JC, Corbin R, Loveland M, Dosman J, Macklem PT. The relations between structural changes in small airways and pulmonary‐function tests. N Engl J Med 298(23): 1277–1281, 1978.
 152. Dosman JA, Cotton DJ, Graham BL, Hall DL, Li R, Froh F, Barnett GD. Sensitivity and specificity of early diagnostic tests of lung function in smokers. Chest 79(1): 6–11, 1981.
 153. Olofsson J, Bake B, Svardsudd K, Skoogh BE. The single breath N2‐test predicts the rate of decline in FEV1. The study of men born in 1913 and 1923. Eur J Respir Dis 69(1): 46–56, 1986.
 154. Stanescu DC, Rodenstein DO, Hoeven C, Robert A. “ Sensitive tests” are poor predictors of the decline in forced expiratory volume in one second in middle‐aged smokers. Am Rev Respir Dis 135(3): 585–590, 1987.
 155. Buist AS, Vollmer WM, Johnson LR, McCamant LE. Does the single‐breath N2 test identify the smoker who will develop chronic airflow limitation? Am Rev Respir Dis 137(2): 293–301, 1988.
 156. Stanescu D, Sanna A, Veriter C, Robert A. Identification of smokers susceptible to development of chronic airflow limitation: A 13‐year follow‐up. Chest 114(2): 416–425, 1998.
 157. Bourdin A, Paginan F, Préfaut C, Kieseler D, Godard P, Chanez P. Nitrogen washout slope in poorly controlled asthma. Allergy 61: 85–89, 2006.
 158. Battaglia S, den Hertog H, Timmers MC, Lazeroms SP, Vignola AM, Rabe KF, Bellia V, Hiemstra PS, Sterk PJ. Small airways function and molecular markers in exhaled air in mild asthma. Thorax 60(8): 639–644, 2005.
 159. Van Muylem A, De Vuyst P, Yernault JC, Paiva M. Inert gas single‐breath washout and structural alteration of respiratory bronchioles. Am Rev Respir Dis. 146(5 Pt 1): 1167–1172, 1992.
 160. Estenne M, Van Muylem A, Knoop C, Antoine M. Detection of obliterative bronchiolitis after lung transplantation by indexes of ventilation distribution. Am J Respir Crit Care Med 162(3 Pt 1): 1047–1051, 2000.
 161. Gustafsson PM, Ljungberg HK, Kjellman B. Peripheral airway involvement in asthma assessed by single‐breath SF6 and He washout. Eur Respir J 21(6): 1033–1039, 2003.
 162. Ljungberg HK, Gustafsson PM. Peripheral airway function in childhood asthma, assessed by single‐breath He and SF6 washout. Pediatr Pulmonol 36(4): 339–347, 2003.
 163. Verbanck S, Schuermans D, Noppen M, Vincken W, Paiva M. Methacholine versus histamine: Paradoxical response of spirometry and ventilation distribution. J Appl Physiol 91(6): 2587–2594, 2001.
 164. Verbanck S, Schuermans D, Noppen M, Van Muylem A, Paiva M, Vincken W. Evidence of acinar airway involvement in asthma. Am J Respir Crit Care Med 159(5 Pt 1): 1545–1550, 1999.
 165. Downie SR, Salome CM, Verbanck S, Thompson B, Berend N, King GG. Ventilation heterogeneity is a major determinant of airway hyperresponsiveness in asthma, independent of airway inflammation. Thorax 62(8): 684–689, 2007. Epub 2007 Feb 20.
 166. Verbanck S, Schuermans D, Vincken W. Inflammation and airway function in the lung periphery of patients with stable asthma. J Allergy Clin Immunol 125(3): 611–616, 2010. Epub 2010 Feb 4.
 167. Verbanck S, Schuermans D, Paiva M, Vincken W. Nonreversible conductive airway ventilation heterogeneity in mild asthma. J Appl Physiol 94(4): 1380–1386, 2003. Epub 2002 Dec 6.
 168. Verbanck S, Schuermans D, Meysman M, Paiva M, Vincken W. Noninvasive assessment of airway alterations in smokers: The small airways revisited. Am J Respir Crit Care Med 170(4): 414–419, 2004. Epub 2004 May 6.
 169. Venegas JG, Winkler T, Musch G, Vidal Melo MF, Layfield D, Tgavalekos N, Fischman AJ, Callahan RJ, Bellani G, Harris RS. Self‐organized patchiness in asthma as a prelude to catastrophic shifts. Nature 434(7034): 777–782, 2005. Epub 2005 Mar 16.
 170. Nielsen J, Dahlqvist M, Welinder H, Thomassen Y, Alexandersson R, Skerfving S. Small airways function in aluminium and stainless steel welders. Int Arch Occup Environ Health 65: 101–105, 1993.
 171. Scano G, Stendardi L, Bracamonte M, de Coster A, Sergysels R. Site of action of inhaled histamine in asymptomatic asthmatic patients. Clin Allergy 12: 281–288, 1982.
 172. Harris EA, Buchanan PR, Whitlock RML. Human alveolar gas mixing efficiency for gases of differing diffusivity in health and airflow limitation. Clin Sci 73: 351–359, 1987.
 173. Langley F, Horsfield K, Burton G, Seed WA, Parker S, Cumming G. Effect of inhaled methacholine on gas mixing efficiency. Clin Sci 74: 187–192, 1988.
 174. Tawhai MH, Burrowes KS. Developing integrative computational models of pulmonary structure. Anat Rec B New Anat 275(1): 207–218, 2003. Review.
 175. Kitaoka H, Takaki R, Suki B. A three‐dimensional model of the human airway tree. J Appl Physiol 87(6): 2207–2217, 1999.
 176. Mercer RR, Laco JM, Crapo JD. Three‐dimensional reconstruction of alveoli in the rat lung for pressure‐volume relationships. J Appl Physiol 62(4): 1480–1487, 1987.
 177. Thiberville L, Salaün M, Lachkar S, Dominique S, Moreno‐Swirc S, Vever‐Bizet C, Bourg‐Heckly G. Human in vivo fluorescence microimaging of the alveolar ducts and sacs during bronchoscopy. Eur Respir J 33(5): 974–985, 2009. Epub 2009 Feb 12.
 178. Behzad AR, McDonough JE, Walker DC, Seyednejad N, Sanchez PG, Cooper JD, Elliott WM, Horng D, Gefter WB, Wright AC, Hogg JC. Early destruction of alveolar walls in human COPD. Am J Respir Crit Care Med 179: A2080, 2009.

Related Articles:

Microgravity
Spatial Distribution of Ventilation and Perfusion: Mechanisms and Regulation
Airway Gas Flow
Transport of Gases between the Environment and Alveoli—Theoretical Foundations

Contact Editor

Submit a note to the editor about this article by filling in the form below.

* Required Field

Article Title: Gas Mixing in the Airways and Airspaces
 

How to Cite

Sylvia Verbanck, Manuel Paiva. Gas Mixing in the Airways and Airspaces. Compr Physiol 2011, 1: 809-834. doi: 10.1002/cphy.c100018