Modeling of the oxygen transfer in the respiratory process
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 4, p. 935-960

In this article, we propose an integrated model for oxygen transfer into the blood, coupled with a lumped mechanical model for the ventilation process. Objectives. We aim at investigating oxygen transfer into the blood at rest or exercise. The first task consists in describing nonlinear effects of the oxygen transfer under normal conditions. We also include the possible diffusion limitation in oxygen transfer observed in extreme regimes involving parameters such as alveolar and venous blood oxygen partial pressures, capillary volume, diffusing capacity of the membrane, oxygen binding by hemoglobin and transit time of the red blood cells in the capillaries. The second task consists in discussing the oxygen concentration heterogeneity along the path length in the acinus. Method. A lumped mechanical model is considered: a double-balloon model is built upon physiological properties such as resistance of the branches connecting alveoli to the outside air, and elastic properties of the surrounding medium. Then, we focus on oxygen transfer: while the classical [F.J. Roughton and R.E. Forster, J. Appl. Physiol. 11 (1957) 290-302]. approach accounts for the reaction rate with hemoglobin by means of an extra resistance between alveolar air and blood, we propose an alternate description. Under normal conditions, the Hill's saturation curve simply quantifies the net oxygen transfer during the time that venous blood stays in the close neighborhood of alveoli (transit time). Under degraded and/or exercise conditions (impaired alveolar-capillary membrane, reduced transit time, high altitude) diffusion limitation of oxygen transfer is accounted for by means of the nonlinear equation representing the evolution of oxygen partial pressure in the plasma during the transit time. Finally, a one-dimensional model is proposed to investigate the effects of longitudinal heterogeneity of oxygen concentration in the respiratory tract during the ventilation cycle, including previous considerations on oxygen transfer. Results. This integrated approach allows us to recover the right orders of magnitudes in terms of oxygen transfer, at rest or exercise, by using well-documented data, without any parameter tuning or curve fitting procedure. The diffusing capacity of the alveolar-capillary membrane does not affect the oxygen transfer rate in the normal regime but, as it decreases (e.g. because of emphysema) below a critical value, it becomes a significant parameter. The one-dimensional model allows to investigate the screening phenomenon, i.e. the possibility that oxygen transfer might be significantly affected by the fact that the exchange area in the peripheral acinus poorly participates to oxygen transfer at rest, thereby providing a natural reserve of transfer capacity for exercise condition. We do not recover this effect: in particular we show that, at rest, although the oxygen concentration is slightly smaller in terminal alveoli, transfer mainly occurs in the acinar periphery.

DOI : https://doi.org/10.1051/m2an/2012052
Classification:  35Q92,  76Z05,  76R50,  92C35,  92C50
Keywords: oxygen transfer, ventilation, lung diffusion capacity, advection-diffusion equation
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author = {Martin, S\'ebastien and Maury, Bertrand},
title = {Modeling of the oxygen transfer in the respiratory process},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {47},
number = {4},
year = {2013},
pages = {935-960},
doi = {10.1051/m2an/2012052},
zbl = {06198325},
mrnumber = {3082284},
language = {en},
url = {http://www.numdam.org/item/M2AN_2013__47_4_935_0}
}

Martin, Sébastien; Maury, Bertrand. Modeling of the oxygen transfer in the respiratory process. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 4, pp. 935-960. doi : 10.1051/m2an/2012052. http://www.numdam.org/item/M2AN_2013__47_4_935_0/

[1] E. Agostoni and R.E. Hyatt, Static behavior of the respiratory system, in Handbook of physiology, edited by S.R. Geiger, 2nd edition. American Physiological Society, Bethesda (1986) 113-130.

[2] D.V. Bates, C.J. Varvis, R.E. Donevan and R.V. Christie, Variations in the pulmonary capillary blood volume and membrane diffusion component in health and disease. J. Clin. Invest. 39 (1960) 1401-1412.

[3] R. Begin, A.D. Renzetti Jr., A.H. Bigler and S. Watanabe, Flow and age dependence of airway closure and dynamic compliance. J. Appl. Physiol. 38 (1975) 199-207.

[4] A. Ben-Tal, Simplified models for gas exchange in the human lungs. J. Theor. Biol. 238 (2006) 474-495.

[5] C. Brighenti, G. Gnudi and G. Avanzolini, A simulation model of the oxygen alveolo-capillary exchange in normal and pathological conditions. Physiol. Meas. 24 (2003) 261-275.

[6] L. Brochard, J. Mancebo, M. Wysocki, F. Lofaso, G. Conti, A. Rauss, G. Simonneau, S. Benito, A. Gasparetto, F. Lemaire, D. Isabey and A. Harf, Noninvasive ventilation for acute exacerbations of chronic obstructive pulmonary disease. N. Engl. J. Med. 333 (1995) 817-822.

[7] J.E. Cotes, D.J. Chinn and M.R. Miller, Lung function: Physiology, Measurement and Application in Medicine, 6th edition. Blackwell Publishing Ltd. (2006).

[8] J.E. Cotes, D.J. Chinn, Ph. Quanjer, J. Roca and J.C. Yernault, Standardization of the measurement of transfer factor (diffusing capacity). Eur. Respir. J. suppl 16 (1993) 41-52.

[9] Crandall, E.D. and R.W. Flumerfelt, Effect of time-varying blood flow on oxygen uptake in the pulmonary capillaries. Appl. Physiol. 23 (1967) 944-953.

[10] The lung: Scientific Foundations, edited by R.G. Crystal, J.B. West, E.R. Weibel and P.J. Barnes, 2nd edition. Lippincott-Raven Press, Philadelphia 2 (1997).

[11] W.A. Eaton, E.R. Henry, J. Hofrichter and A. Mozzarelli, Is cooperative oxygen binding by hemoglobin really understood?. Nat. Struct. Biol. 6 (1999) 351-358.

[12] M. Felici, M. Filoche and B. Sapoval, Diffusional screening in the human pulmonary acinus. J. Appl. Physiol. 94 (2003) 2010-2016.

[13] M. Felici, M. Filoche and B. Sapoval, Renormalized random walk study of oxygen absorption in the human lung. Phys. Rev. Lett. 92 (2004) 068101.

[14] M. Felici, M. Filoche, C. Straus, T. Similowski and B. Sapoval, Diffusional screening in real 3D human acini - a theoretical study. Respir. Physiol. Neurobiol. 145 (2005) 279-293.

[15] M. Filoche and M. Florens, The stationary flow in a heterogeneous compliant vessel network. J. Phys. Conf. Ser. 319 (2011) 012008.

[16] A. Foucquier, Dynamique du transport et du transfert de l'oxygène au sein de l'acinus pulmonaire humain. Ph.D. thesis, École Polytechnique (2010).

[17] P. Gehr, M. Bachofen and E.R. Weibel, The normal human lung: ultrastructure and morphometric estimation of diffusion capacity. Respir. Physiol. 32 (1978) 121-140.

[18] A.C. Guyton and J.E. Hall, Textbook of medical physiology, 9th edition. W.B. Saunders Co, Philadelphia (1996).

[19] M.P. Hlastala and A.J. Berger, Physiology of Respiration, 2nd edition. Oxford University Press, Oxford (2001).

[20] C. Hou, S. Gheorghiu, M.-O. Coppens, V.H. Huxley and P. Pfeifer, Gas diffusion through the fractal landscape of the lung: How deep does oxygen enter the alveolar system? in Fractals in Biology and Medicine, edited by G.A. Losa, D. Merlini, T.F. Nonnenmacher, E.R. Weibel. Basel: Birkhäuser IV (2005) 17-30.

[21] J.M.B. Hughes, Pulmonary gas exchange. in Lung Function Testing, edited by R. Gosselink and H. Stam. European Respiratory Monograph 10 (2005) 106-126.

[22] J. Keener and J. Sneyd, Mathematical Physiology. Interdisciplinary Applied Mathematics. Springer (1998). | MR 1673204 | Zbl 1273.92017

[23] G.R. Kelman, Digital computer subroutine for the conversion of oxygen tension into saturation. J. Appl. Physiol. 21 (1966) 1375-1376.

[24] J.D. Kibble and C. Halsey, Medical Physiology, The Big Picture. McGraw Hill (2009).

[25] C.H. Liu, S.C. Niranjan, J.W. Clark, K.Y. San, J.B. Zwischenberger and A. Bidani, Airway mechanics, gas exchange, and blood flow in a nonlinear model of the normal human lung. J. Appl Physiol. 84 (1998) 1447-1469.

[26] S. Martin, T. Similowski, C. Straus and B. Maury, Impact of respiratory mechanics model parameter on gas exchange efficiency. ESAIM Proc. 23 (2008) 30-47. | MR 2509204 | Zbl 1156.92310

[27] B. Mauroy and P. Bokov, Influence of variability on the optimal shape of a dichotomous airway tree branching asymmetrically. Phys. Biol. 7 (2010) 016007.

[28] B. Mauroy, M. Filoche, J.S. Andrade Jr. and B. Sapoval, Interplay between flow distribution and geometry in an airway tree. Phys. Rev. Lett. 90 (2003) 14.

[29] B. Mauroy, M. Filoche, E.R. Weibel, and B. Sapoval, An optimal bronchial tree may be dangerous. Nature 427 (2004) 633-636.

[30] B. Mauroy and N. Meunier, Optimal Poiseuille flow in a finite elastic dyadic tree. ESAIM: M2AN 42 (2008) 507-534. | Numdam | MR 2437772 | Zbl 1203.74033

[31] M. Paiva and L.A. Engel, Model analysis of gas distribution within human lung acinus. J. Appl. Physiol. 56 (1984) 418-425.

[32] J. Piiper and P. Scheid, Respiration: alveolar gas exchange. Annu. Rev. Physiol. 33 (1971) 131-154.

[33] F.J. Roughton and R.E. Forster, Relative importance of diffusion and chemical reaction rates in determining rate of exchange of gases in the human lung, with special reference to true diffusing capacity of pulmonary membrane and volume of blood in the lung capillaries. J. Appl. Physiol. 11 (1957) 290-302.

[34] B. Sapoval and M. Filoche, Role of diffusion screening in pulmonary diseases. Adv. Exp. Med. Biol. 605 (2008) 173-178.

[35] B. Sapoval, M. Filoche and E.R. Weibel, Smaller is better − but not too small: a physical scale for the design of the mammalian pulmonary acinus. Proc. Natl. Acad. Sci. USA 99 (2002) 10411.

[36] T. Similowski and J.H.T. Bates, Two-compartment modelling of respiratory system mechanics at low frequencies: gas redistribution or tissue rheology? Eur. Respir. J. 4 (1991) 353-358.

[37] T.T. Soong, P. Nicolaides, C.P. Yu and S.C. Soong, A statistical description of the human tracheobronchial tree geometry. Respir. Physiol. 37 (1979) 161-72.

[38] A.J. Swan and M.H. Tawhai, Evidence for minimal oxygen heterogeneity in the healthy human pulmonary acinus. J. Appl. Physiol. 110 (2011) 528-537.

[39] J. Sznitman, Convective gas transport in the pulmonary acinus: comparing roles of convective and diffusive lengths. J. Biomech. 42 (2009) 789-792.

[40] C. Tantucci, A. Duguet, P. Giampiccolo, T. Similowski, M. Zelter and J.-P. Derenne, The best peak expiratory flow is flow-limited and effort-independent in normal subjects. Am. J. Respir. Crit. Care Med. 165 (2002) 1304-1308.

[41] M.H. Tawhai and P.J. Hunter, Characterising respiratory airway gas mixing using a lumped parameter model of the pulmonary acinus. Respir. Physiol. 127 (2001) 241-248.

[42] E.R. Weibel, Morphometry of the human lung, Springer Verlag and Academic Press, Berlin, New York (1963).

[43] E.R. Weibel, The pathway for oxygen, Harvard University Press (1984).

[44] E.R. Weibel, Design and morphometry of the pulmonary gas exchanger, in The lung: scientific foundations, 2nd edition, edited by R.G. Crystal, J.B. West, E.R. Weibel, P.J. Barnes. Lippincott-Raven Press, Philadelphia 1 (1997) 1147-1157.

[45] E.R. Weibel, B. Sapoval and M. Filoche, Design of peripheral airways for efficient gas exchange. Resp. Phys. Neur. 148 (2005) 3-21.

[46] E.R. Weibel, How does lung structure affect gas exchange? Chest 83 (1983) 657-665.

[47] J.B. West, Respiratory physiology: the essentials, Baltimore: Williams and Wilkins (1974).

[48] J.P. Whiteley, D.J. Gavaghan and C.E. Hahn, Some factors affecting oxygen uptake by red blood cells in the pulmonary capillaries. Math. Biosci. 169 (2001) 153-172. | MR 1818484 | Zbl 0977.92006