Diffusion models of multicomponent mixtures in the lung

In this work, we are interested in two different diffusion models for multicomponent mixtures. We numerically recover experimental results underlining the inadequacy of the usual Fick diffusion model, and the importance of using the Maxwell-Stefan model in various situations. This model nonlinearly couples the mole fractions and the fluxes of each component of the mixture. We then consider a subregion of the lower part of the lung, in which we compare the two different models. We first recover the fact that the Fick model is enough to model usual air breathing. In the case of chronic obstructive bronchopneumopathies, a mixture of helium and oxygen is often used to improve a patient’s situation. The Maxwell-Stefan model is then necessary to recover the experimental behaviour, and to observe the benefit for the patient, namely an oxygen peak. Introduction The bronchial tree, which has a dyadic structure, can be schematically divided into two parts [15]. In the upper part of the airways (generations 0 to 14), the main phenomenon putting the air in motion is the convection, and the air can be described by the Navier-Stokes or the Stokes equations. In the lower part of the airways (generations 15 to 23), the gas behaviour is mainly diffusive, since the global velocity of the air is almost zero. We here consider this second distal part of the respiratory system, where the gaseous exchanges between the air and the blood take place. For the sake of simplicity, we neglect the convection in the distal part of the bronchial tree, and only focus on the diffusive effects. Diffusion can be defined in a general way as the physical process that brings matter from one region of a system to another using random motions at a molecular level, see [4], for instance. The macroscopic diffusion model which is most commonly used is the Fick model (see, among many references, [6] in the framework of the lung), where the flux of a given species is directly proportional to the concentration gradient of this species. At this level, the relevant unknowns appear to be the concentrations of each species (or any other proportional quantities). There are several modelling issues in the diffusion phenomenon in the lung, which can be investigated: the diffusion model itself, which we tackle here, but also the geometry of interest, which cannot be determined by ∗ This work was partially funded by the ANR-08-JCJC-013-01 project headed by C. Grandmont. 1 UPMC Paris 06, UMR 7598 LJLL, Paris, F-75005, France; e-mail: [email protected] 2 INRIA Paris-Rocquencourt, REO Project team, BP 105, F-78153 Le Chesnay Cedex, France; 3 TU Darmstadt, Fachbereich Mathematik, IRTG 1529, Schloßgartenstr. 7, D-64289 Darmstadt, Germany; e-mail: [email protected] 4 MAP5, CNRS UMR 8145, Université Paris Descartes, F-75006 Paris, France; e-mail: [email protected] 5 Institut Camille Jordan, CNRS UMR 5208, Université Claude Bernard Lyon 1, F-69622 Villeurbanne, France c © EDP Sciences, SMAI 2010 2 ESAIM: PROCEEDINGS biological imaging, and has a great importance for the space and time scales, or the mechanism of gas exchange between the alveoli and the blood vessels through the alveolar membrane. As far as the geometry is concerned, Kitaoka et al. [9] proposed a three-dimensional model of the human pulmonary acinus as a gas exchange unit built with a labyrinthine algorithm generating branching ducts, that completely fill a given space. Alveoli are attached to the inner walls of the space on study. They obtain physically relevant values for the total alveolar surface and the numbers of alveolar ducts, alveolar sacs, and alveoli. Mauroy also used the Kitaoka model of acinus in his PhD thesis [11]. Sapoval et al. [12] found out that efficient acini should be space-filling surfaces and should remain quite small. The efficiency depends on both the acini design and the values of physical factors like diffusivities of the gaseous components of the air in the airways and permeability of the membrane between the blood vessels and the alveoli. In this work, we aim to check the relevance of two different diffusion models. The first one is the Fick model, which we already briefly presented. The main ideas of the second one were developed following independent pioneering works of Maxwell and Stefan, and we shall then refer to the Maxwell-Stefan model, as in [10]. This model takes into account the friction effects between different species. We here show the shortcomings of the Fick model in some specific, but still realistic, situations in the lung. We provide numerical comparisons between Fick and Maxwell-Stefan’s laws for the diffusion processes of various mixtures in the lower airways. In the case when the respiratory system is healthy, we recover the facts that both Fick and Maxwell-Stefan models are relevant, and that they provide results which are very close to each other. On the other hand, when one deals with patients who suffer from a severe airway obstruction, in the cases of chronic obstructive bronchopneumopathies (COBP), the ventilation has to happen faster. Thiriet et al. [14] lead in vivo experiments and used a binary mixture of helium (79 %) and oxygen (21 %), which is usually called heliox, to speed up the oxygen transport towards the lower airways, so that the diffusion and the gaseous exchanges can more rapidly happen. We here point out, as suggested in [14], that when heliox is involved, the Fick law does not apply anymore. In [1], Chang obtains the same kind of result using some considerations on the model. In our first section, we present the two diffusion processes of a multicomponent mixture, using the definition of the fluxes by Fick’s and Maxwell-Stefan’s laws. Then, in Section 2, we recall and numerically repeat the Duncan and Toor experiment that brought to light in the first place the interest of this more intricate model than Fick’s. Eventually, we go back to the framework of the lung in Section 3 to compare the Fick and MaxwellStefan laws for the air in a two-dimensional branch, and point out some cases when the Fick law does not hold anymore. 1. Maxwell-Stefan vs. Fick’s laws Consider a gaseous mixture with M components, and assume they constitute an ideal gas mixture. The concentration ci of species i, 1 ≤ i ≤ M , depends on time t ≥ 0 and space location X ∈ R N , 1 ≤ N ≤ 3. If we denote ctot = ∑ cj the total concentration, the mole fraction xi of species i is defined by xi = ci/ctot. It satisfies the following mass conservation equation: ∂txi +∇X · Ni = 0, (1) where Ni ∈ R N is the molar flux of species i. The expression of Ni with respect to the mole fractions (xj)1≤j≤M depends on the diffusion model one chooses. Both cases are detailed in Subsection 1.2. We first present a brief physical derivation of Maxwell-Stefan’s law in a one-dimensional setting. 1.1. Derivation of the Maxwell-Stefan law in one dimension For the sake of simplicity, we momentarily assume N = 1 and consider a one-dimensional diffusion following the axis X := X1. The force acting on species i in a control volume is given by − dpi/ dX , where pi is the partial pressure of species i in the mixture. ESAIM: PROCEEDINGS 3 Taking into account the ideal gas law pi = RTci, where R is the ideal gas constant and T the absolute temperature, yields (RT/ci)(− dci/ dX) for the force per mole of species i. At the equilibrium, this force is balanced by the drag (or friction) forces exerted by the other species in the mixture. It is standard to assume that the drag force is proportional to the relative velocity as well as to the mole fraction of the other components. The friction force between species i and j acting on species i then writes (RT/Dij)xj(ui − uj). Here, ui, respectively uj, denote the molar velocity of species i, respectively j, and Dij is the Maxwell-Stefan diffusivity (or binary diffusion coefficient) between the two components. The constant RT/Dij may then be seen as a drag coefficient. Altogether, this gives the force balance for species i