Abstract
In the mammalian lung acini, O_{2} diffuses into quasistatic air toward the alveolar membrane, where the gas exchange with blood takes place. The O_{2} flux is then influenced by the O_{2}diffusivity, the membrane permeability, and the acinus geometric complexity. This phenomenon has been recently studied in an abstract geometric model of the acinus, the Hilbert acinus (Sapoval B, Filoche M, and Weibel ER, Proc Natl Acad Sci USA 99: 10411, 2002). This is extended here to a more realistic geometry originated from the morphological model of Kitaoka et al. (Kitaoka K, Tamura S, and Takaki R, J Appl Physiol 88: 2260–2268, 2000). Twodimensional numerical simulations of the steadystate diffusion equation with mixed boundary conditions are used to quantify the process. The alveolar O_{2} concentration, or partial pressure, and the O_{2} flux are computed and show that diffusional screening exists at rest. These results confirm that smaller acini are more efficient, as suggested for the Hilbert acini.
 oxygen diffusion
 acinar gas mixing
 mathematical modeling
the human pulmonary acinus, which is the functional unit of gas exchange, has a complicated morphological structure. It comprises the last five to six generations of the bronchial tree, namely the alveolated ducts, originating from one respiratory bronchiole (2, 12). The arrangement in the threedimensional space of the alveoli is such that the alveolar membrane, where gas exchange takes place, forms a unique, highly irregular surface. O_{2} can reach this surface through the alveolated ducts, which form a branched system. It is known that, at this stage, O_{2} transport is achieved mainly through diffusion.
In the human lung, at rest conditions, the transition between convective and diffusive transport extends over a zone situated between the first transitional bronchiole and the first generations of the acinus (10). In the diffusiondominated region, which roughly corresponds to what has been called the ⅛subacinus by HaefeliBleuer and Weibel (2), O_{2} diffuses, driven by a partial pressure gradient. This gradient is created by the absorption of O_{2} molecules on the alveolar membrane containing the arterious capillaries. The law governing this phenomenon is the wellknown Fick's law. The global process includes four main processes (4): 1) air convective flow;2) O_{2} diffusion in air; 3) diffusion through membrane and plasma; and 4) final binding of O_{2} with erythrocytes hemoglobin.
At each of these steps, there is associated a resistance to O_{2} transport, depending on the morphology and on the physicochemical transport coefficients. In this work, we study in some detail step 2, O_{2} diffusion in air, andstep 3, diffusion through membrane and plasma, and we use an approximate way to take care of convective flow and O_{2}binding.
Convection is supposed to become negligible here at the entry of ⅛subacinus. O_{2} transport is then described by the stationary state diffusion into quasistatic air. The rate of binding of O_{2} to hemoglobin, which is actually related to the O_{2} concentration (or partial pressure) in alveolar air in a nonlinear way through the hemoglobin saturation curve, is here considered to be very large, so that no additional resistance is considered on the blood side. This assumption can be considered as a working hypothesis to be discussed later.
This approach is justified by the recent results that have permitted establishment of a direct link between diffusivity and permeability and the size of the acinus of several mammals from the mouse to the human (10). These last results were obtained through a numerical study of a simplified geometric model of the acinus, called the Hilbert acinus. Our goal here is to study the same type of phenomenon in the more realistic acinus geometry proposed by Kitaoka et al. (3).
The arguments that permit discussion of this question are based on extensive studies carried out on the problem of diffusional transport toward irregular reacting surfaces. This problem is common to a wide class of physical systems and has been addressed in recent years for several cases by means of simple physical models (6, 8). The main concept one has to introduce for a better understanding of such systems is the concept of diffusional screening. Screening means that the particles diffusing toward an irregular absorbing surface are very unlikely to ever reach some deeper regions of the surface because they collide with the nearest and protruding regions of the absorbing surface first. These nearest and protruding regions then act like a “screen” behind which are “hidden” the less accessible regions. The consequence is that only a fraction of the surface is really active, with the relative efficiency (η) of the system being, therefore, diminished. Of course, if the molecules are not absorbed at the first hits on the membrane, they gain a fair chance to explore the deeper regions, which then work and contribute to the overall system efficiency. Because of the serial arrangement of alveoli along the last generations of alveolated ducts, diffusional screening is likely to occur in the pulmonary acinus, which contains 1,000 alveoli. In the following paragraphs, we show that this is the case, at least for rest conditions.
As shown in Refs. 1 and 7, the essential quantity that determines how the system works is a physical parameter, called the “unscreened perimeter length” (Λ). This length, given by the ratio of the O_{2} diffusivity to the membrane permeability, is shown to be the unique parameter governing the efficiency of the system for a given geometry.
It is important to notice that screening has the same effect of what has been called in lung physiology “stratified inhomogeneity,” or “stratification” (5), but it is due to a quite different phenomenon. Stratification is related to the idea that the inspired gas has no time to diffuse completely during a normal respiratory cycle so that concentration gradients persist in the lung. On the contrary, diffusional screening characterizes stationary diffusion, that is, the infinite time limit of the diffusion equation. This means that, even after infinite time, the O_{2}concentration or partial pressure cannot be completely uniform in the acinus.
THE UNSCREENED PERIMETER LENGTH AND DIFFUSIONAL SCREENING
Let us define the conductance (Y) of the acinus as the flux of O_{2} into blood per unit concentration at the entry. For systems in which the final transfer flux is the result of transport in the volume and reaction (here permeation) on the walls, the global Y depends on two contributions: the Y to reach the surface (Y_{reach}) (transport) and the Y to cross the surface (Y_{cross}) (reaction). For the acinus, these two terms depend, respectively, on the diffusion coefficient of O_{2} in air (Do
_{2,air}) and on the permeability of the alveolar membrane with respect to O_{2}(W
_{O2}). TheW
_{O2} is defined as the flux across a membrane of unit area per unit concentration difference between the two sides. The relative importance of the two contributions depends also on the shape of the surface, that is, the geometric characteristics of the system. Let us define the diameter of a surface of total areaA, length L, as the diameter of the smallest sphere embedding the surface. The diameter represents then the overall size of the system. One can write
Therefore, the condition Y_{reach} = Y_{cross}amounts to A/L = Do _{2,air}/W _{O2}= Λ , where Λ is the ratio of the diffusion coefficient by the permeability. The last condition then reads L_{p} = Λ . Although Λ is a length given by the physical parameters of the system, L_{p} depends uniquely on the morphology, shape, and size of the surface.
It is now possible to better understand the concept of screening. Screening occurs when a portion of the surface is not reached. This happens if Y_{reach} < Y_{cross} (Λ <L_{p} ). In this case, the Y of the system is limited by the Y to reach the surface. On the contrary, if Y_{reach} > Y_{cross} (Λ >L_{p} ), the surface works uniformly, and the total Y is limited by the Y to cross the surface.
If one considers a region of the surface with a perimeter smaller than Λ , for that region Y_{reach} ≥ Y_{cross}. This region then works uniformly, and screening is not effective. This is the reason why Λ is called unscreened perimeter length. On the contrary, a region with a perimeter greater than Λ does not work in a uniform way. In a different language, the length Λ measures, through its perimeter, the size of the region of the exchange surface that a molecule of O_{2} explores, on average, before being absorbed.
This applies to irregular surfaces that present no deep pores (9). Consider, for example, a pore of length Land diameter d, for which L_{p} = 2L. It has an access Y of orderDd ^{2}/L and a surface Y of WπdL. They are equal if Λ = πL ^{2}/d =L_{p} · (πL/2d). If Λ is smaller than this last length, there is screening, and the pore walls do not work uniformly. Then screening depends both on the “perimeter” L_{p} of the pore and the quantity (πL/2d), which is the pore aspect ratio (always >1). With the acinus being a deeply porous structure, one expects that screening will depend both on the acinus perimeter and on a factor (always >1) related to the porous nature of the geometry.
Screening phenomenon has so far been neglected in the study of gas mixing in the pulmonary acinus, although the idea was qualitatively described in Weibel's book (12). This problem has been precisely formulated from a physical point of view in Refs.710.
THE MATHEMATICAL MODEL
The acinus is a compact bounded structure. There is one inlet, from which fresh air enters the structure. The empty space in the ducts is bounded by a connected gas exchange surface with an average total area of 8.63 cm^{2} (2). These data refer to the ⅛subacinus, which is usually considered as the actual gasexchange unit for the human lung.
In the steadystate condition, the spacedependent concentration C(x, y, z) of O_{2} in air obeys the Laplace equation
The boundary conditions at the surface can be obtained by imposing the conservation of current across the membrane −Do
_{2,air} ∂C/∂n =W
_{O2}C, where n refers to the direction normal to the interface. As a boundary condition for the entry of the acinus, we consider a fixed concentration C_{0}. This means that the source of diffusion, i.e., the diffusion front, is supposed to sit at the entrance of the subacinus. We then neglect convection from there on. The concentration of O_{2} in the acinus is described by the following set of equations
The value of Do _{2,air} = 0.178 cm^{2}/s (12). The value of the permeability depends on the solubility of O_{2} in physiological tissue, considered to be water, and the diffusion barrier between air and the erythrocytes (including the thickness of the membrane and the thickness of the plasma layer). Taking the values from literature (13), it is found that Λ ≈ 30 cm.
The Geometric Model
This mathematical model, illustrated in Eq. 5 , is applied to the geometric model of the pulmonary acinus recently proposed by Kitaoka et al. (3). The acinar structure is simulated by using a labyrinthine algorithm. The ⅛subacinus is approximated to be a set of cubic cells with a side dimension of 0.5 mm; alveoli are generated by attaching alveolar septa 0.25 mm long and 0.1 mm wide to the inner walls of each such cell. The algorithm is designed to construct a threedimensional branching tree without loops that starts from one entrance and passes through all cells, filling completely the given space. Random variables enter in the construction of the intraacinar pathways so that it is possible to obtain different structures within the same global size, having the same mean path length and the same total area. The values of mean path length, surface area, and numbers of ducts and sacs are chosen to fit the morphological data reported in the literature (2). A twodimensional (2D) realization of such a geometry is shown in Fig.1.
NUMERICAL SIMULATIONS
Equation 5 has been solved in 2D in the Kitaoka 2D geometry by means of finiteelement computations by using MATLAB PDE Toolbox. Different values of Λ have been considered, as well as acini with several sizes.
The computation has been carried on for the two types of geometry shown in Fig. 1: first, with type A geometry in which alveolar septa are present on the ducts walls (Fig. 1 A), and second, on a simplified type B geometry in which the septa are missing (Fig. 1 B). As shown below, the presence of the septa does not affect the overall behavior of the system, if one properly rescales the unit length. Therefore, most of the computations have then been performed on the simplified type B geometry.
The result for one acinus of size 20 × 20 (400 cells) is shown in Fig. 2, in which the O_{2}concentration distribution is imaged for four values of Λ . It can be seen qualitatively that, for values of Λ smaller than the perimeter, only a fraction of the surface actually works (red regions in Fig. 2).
To characterize in a quantitative manner the effect of screening, we introduce the acinus efficiency (η), which measures the fraction of the surface that is really active. The η is defined as the ratio of the total flux Φ across the absorbing membrane to the flux in ideal conditions (infinite diffusivity or infinite Λ). The inlet flux is given by
The ideal flux Φ_{id} is the total flux that one would have if the O_{2} had perfectly spread in the volume. In that case, the concentration is equal to C_{0} everywhere in the system and
It should be emphasized that larger efficiency does not imply larger flux. An “efficient” acinus is “better used” in the sense that the whole acinar surface is working for transfer. The introduction of the efficiency concept is necessitated by the fact that the overall functioning of the acinus depends on three independent parameters: the diffusivity, the permeability, and the geometry. As shown below, the problem can be simply understood in terms of only two parameters, the length Λ and the geometry. The flux itself increases with Do
_{2,air}, with everything else constant, and with W
_{O2}, with everything else constant. The dependence of the flux on the system size for constant Do
_{2,air} andW
_{O2} is related to the efficiency. The flux can be written as
In Fig. 4, the efficiencies resulting from the computation in the two types of geometries, with and without septa, are compared: the behavior of η with Λ varying is exactly the same if one rescales Λ with the perimeter, which has different values in the two cases. This means that not only the behavior of the physical quantities in type A geometry can be inferred from the computations on geometry of type B but also the quantitative results for type A can be obtained from those of type B through a simple rescaling. This insensitivity to geometric details is a general property of diffusional processes, due to the smoothing character of the Laplace operator (1). Moreover, this indicates that the proper parameter governing the system is indeed the ratio of the Λ to the total perimeter.
The total number of alveoli in one ⅛subacinus is ∼1,000. In the 2D Kitaoka model, each square cell comprises four alveoli (two on each side), and the alveolus size is 0.25 mm, corresponding to (1/2)ℓ, where ℓ is the square cell size (we put ℓ = 1). In our units, the size of a square containing 1,000 alveoli is thenL ≈ 20 while Λ ≈ 600. As the perimeter of the 20 × 20 acinus is L_{p} = 800, we are in the regime Λ ≤ L_{p} , where screening is effective. For these values, the efficiency is found to be on the order of 25%. As these are the realistic conditions for a human subacinus at rest, this result indicates that screening is effective and reduces significantly the lung efficiency in rest conditions.
If one considers the efficiency as a function of the size, smaller acini are found to be more efficient. This is shown in Fig.5, which gives the dependence of η on size. In fact, for a given Λ , the absolute size of the active region is approximately constant, and an increase of the total perimeter length only has the effect of diminishing the fraction of effective surface and thus the efficiency. Therefore, optimization of diffusional transport requires small acini. This result is general and would be obtained whatever the details of the acinus morphology were. On the other hand, as the effectiveness of convective flow is limited by the diameter of the ducts (hydrodynamic resistance growing as the fourth power of the inverse diameter), the conducting ducts should not be too narrow. These considerations suggest that the actual size of the acini might be the result of a compromise between these two requirements.
DISCUSSION
In this frame, the acinus is only partly efficient at rest. However, it is known that, at exercise, the total uptake of O_{2} can be increased by a factor of 10. It is generally admitted that the pulmonary system is working with maximal efficiency at exercise. Along the above lines, the reason for it is that the transition between convective flow and diffusional transport occurs deeper in the acinus at exercise, due to the increase of the inlet velocity of air (10). This means that, in our model, at exercise, the source of diffusion, where the concentration is fixed at C_{0}, should be put inside the acinus. The regions that are not active at rest then become accessible to diffusing O_{2}, and the global efficiency may rise up to 100%. This “uncovering” or “unscreening” of portions of alveolar surface nonactive at rest might then be one of the reasons explaining the rise in the metabolic rate from rest to exercise.
It may be useful to discuss briefly the concept of stratification, which is a subject of much controversy. We define stratification here as the presence of concentration gradients during the inspiration transitory regime. As a matter of fact, with diffusion speed being finite, for small periods of time, O_{2} cannot reach the regions that are “too far.” In this sense, stratification can be seen as being really the dynamic screening. Inspiration covers a timet _{I}, and O_{2} needs a timet_{t} < t _{I}, wheret_{t} is the time to reach the stationary state from the beginning of inspiration. Before timet_{t} , O_{2} profile evolves in time, penetrating more and more into the acinus; therefore, O_{2}concentration is not homogeneous in the structure because of stratification. After reaching the stationary state, O_{2}profile does not change any more, but concentration gradients are present because of diffusional screening, as it has been shown above. Given the diffusion coefficient of O_{2} and the linear dimensions of the acinus, t_{t} can be evaluated to be on the order of 1 s. This means that 1) the transitory regime is not totally negligible in the respiratory cycle, and 2) concentration gradients persist even in the stationary state. O_{2} concentration is then never homogeneous within the acinus.
Note that the fact that the computation was done in a 2D structure should not be considered as a serious defect of the results, because the real acinus is a tree structure. The diffusion paths must then follow the tree branches, whatever the dimensionality. In that case, the behavior of the system depends directly on its topology. This is why it is important to consider a geometric model that reproduces the topological features of the real system.
The model presented here addresses the problem of gas mixing for the whole acinus to investigate the effect of the morphological characteristics on its functioning. This model focuses on the diffusional transport mechanism and deals with the other phenomena, as convective flow and binding dynamics, in an approximated way. In this approach, the binding rate of O_{2} by hemoglobin has been supposed to be large enough so that it does not limit the net O_{2} transfer.
In exercise conditions, when the increase in airflow velocity allows convection to push the diffusion source inside the acinus, as discussed above, screening should be negligible. The O_{2} partial pressure should then be uniform, and an average value for O_{2} binding could be used. Conversely, the average O_{2} binding could be deduced from the comparison between physiology and geometry.
The results show evidence that the acinus is not at maximal efficiency at rest because of screening. The effect is far from being negligible, with the computed efficiency being on the order of 25%. The size of the acinus is shown on general grounds to be limited by the existence of a finite Λ . Our results generalize those recently obtained in the artificial picture of the socalled Hilbert acinus (10) to the more realistic Kitaoka models of the human pulmonary acinus (3).
In particular, it is interesting to notice that, as a consequence of screening, small animals should be more efficient than big ones, which have larger acini. It is indeed a wellknown fact that the O_{2} uptake per unit body mass decreases with animal mass, as known from the allometric law of the metabolic rates.
Further studies of the O_{2} flux at rest should include a nonlinear binding function of O_{2} concentration, related to the hemoglobin saturation curve. The main limitations of this model appear in that convection is neglected, with the transition from convection to diffusion being modeled here as being sharp. Because of the stationarity assumption, the changes in the acinar volume occurring during the respiratory cycle are not accounted for as well. Further studies on more detailed models, including nonlinear O_{2}binding, should investigate the interplay between the different phenomena.
Acknowledgments
The Centre de Mathématiques et de leurs Applications and the Laboratoire de Physique de la Matière Condensée are Unité Mixte de Recherches du Centre National de la Recherche Scientifique nos. 8536 and 7643, respectively.
Footnotes

Address for reprint requests and other correspondence: M. Felici, Laboratoire de Physique de la Matière Condensée, CNRS, Ecole Polytechnique, 91128 Palaiseau, France (Email:maddalena.felici{at}polytechnique.fr).

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10.1152/japplphysiol.00913.2002
 Copyright © 2003 the American Physiological Society