Modeling of the oxygen transfer in the respiratory process∗
1 INRIA Paris Rocquencourt, REO project
– BP 105, 78153
Le Chesnay cedex,
2 Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, 91405 Orsay cedex, France.
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.
Mathematics Subject Classification: 35Q92 / 76Z05 / 76R50 / 92C35 / 92C50
Key words: Oxygen transfer / ventilation / lung diffusion capacity / advection-diffusion equation
© EDP Sciences, SMAI, 2013