Individual modeling of oxygen capture by the human lungs.

It has been proposed that oxygen capture by the human lungs depends on four determinants: ventilation, cardiac output, oxygen partial pressure in the inspired air and the venous blood. Indeed, the theoretical-numerical model proposed recently by Kang et al. was able to interpret the known empirical relation between the average of the determinants and the average oxygen capture called VO2. This method is tested here at the individual level in a group of 31 subjects submitted to standard pulmonary function testing and cardiopulmonary exercise testing. For this, an inverse method is used in which individual cardiac output is predicted from the clinical test data. Comparison to the cardiac output deduced from Fick principle confirms that the dynamic model is a "microscopic" justification of the "macroscopic" Fick principle. It shows that in addition to the four determinants, two secondary determinants, namely hemoglobin concentration and Bohr effect, expressed here through P50, play significant roles.

Diffusion Reaction of Carbon Monoxide in the Human Lung.

The capture of CO, a standard lung function test, results from diffusion-reaction processes of CO with hemoglobin inside red blood cells (RBCs). In its current understanding, suggested by Roughton and Forster in 1957, the capture is represented by two independent resistances in series, one for diffusion from the gas to the RBC periphery, the second for internal diffusion reaction. Numerical studies in 3D model structures described here contradict the independence hypothesis. This results from two different theoretical reasons: (i) The RBC peripheries are not equi-concentrations; (ii) diffusion times in series are not additive.

The Roughton-Forster equation for DLCO and DLNO re-examined.

Roughton and Forster (RF) proposed to split the lung diffusing capacity into two contributions describing first, diffusion to red blood cells (RBC), and second, capture by diffusion from the RBC surface and reaction with haemoglobin. Solving the diffusion-reaction equations for simplified capillary-RBC structures, we investigate the RF interpretation. This reveals first that the conventional extrapolation to zero pressure of 1/DLCO on PO2 is not a correct measure of the diffusive component. Consequently the capillary volumes deduced from this extrapolation are erroneous. Secondly, capture mechanisms are different for CO and NO: while DLCO characterizes "volume absorption" in the RBC and is correlated with hematocrit, DLNO quantifies "surface absorption" and provide information about the morphology of the space between the alveolar surface and the RBC surfaces. In conclusion, the RF approach may lead to erroneous physiological interpretations of DLCO; nevertheless, the measurement of DLCO and DLNO bring different types of information that give the potential for a better understanding of respiratory diseases.

A first principles calculation of the oxygen uptake in the human pulmonary acinus at maximal exercise.

It has recently been shown that the acinus can have a reduced efficiency due to a "screening effect" governed by the ratio of oxygen diffusivity to membrane permeability, the gas flow velocity, as well as the size and configuration of the acinus. We present here a top to bottom calculation of the functioning of a machine acinus at exercise that takes this screening effect into account. It shows that, given the geometry and the breathing dynamics of real acini, respiration can be correlated to a single equivalent parameter that we call the integrative permeability. In particular we find that both V(O(2,max)) and PA(O(2)) depend on this permeability in a non-linear manner. Numerical solutions of dynamic convection-diffusion equations indicate that only a narrow range of permeability values is compatible with the experimental measurements of PA(O(2)) and V(O(2,max)). These permeability values are significantly smaller than those found in the literature. In a second step, we present a new type of evaluation of the diffusive permeability, yielding values compatible with the top to bottom approach, but smaller than the usual morphometric value.

Faster diffusion across an irregular boundary.

We investigate how the shape of a heat source may enhance global heat transfer at short time. An experiment is described that allows us to obtain a direct visualization of heat propagation from a prefractal radiator. We show, both experimentally and numerically, that irregularly shaped passive coolers rapidly dissipate at short times, but their efficiency decreases with time. The de Gennes scaling argument is shown to be only a large scale approximation, which is not sufficient to describe adequately the temperature distribution close to the irregular frontier. This work shows that radiators with irregular surfaces permit increased cooling of pulsed heat sources.

An anatomical and functional model of the human tracheobronchial tree.

The human tracheobronchial tree is a complex branched distribution system in charge of renewing the air inside the acini, which are the gas exchange units. We present here a systematic geometrical model of this system described as a self-similar assembly of rigid pipes. It includes the specific geometry of the upper bronchial tree and a self-similar intermediary tree with a systematic branching asymmetry. It ends by the terminal bronchioles whose generations range from 8 to 22. Unlike classical models, it does not rely on a simple scaling law. With a limited number of parameters, this model reproduces the morphometric data from various sources (Horsfield K, Dart G, Olson DE, Filley GF, Cumming G. J Appl Physiol 31: 207-217, 1971; Weibel ER. Morphometry of the Human Lung. New York: Academic Press, 1963) and the main characteristics of the ventilation. Studying various types of random variations of the airway sizes, we show that strong correlations are needed to reproduce the measured distributions. Moreover, the ventilation performances are observed to be robust against anatomical variability. The same methodology applied to the rat also permits building a geometrical model that reproduces the anatomical and ventilation characteristics of this animal. This simple model can be directly used as a common description of the entire tree in analytical or numerical studies such as the computation of air flow distribution or aerosol transport.

Passivation of irregular surfaces accessed by diffusion.

We investigate the process of progressive passivation of irregular surfaces accessed by diffusion. More precisely, we quantify through numerical simulations how the activity of the von Koch surface is gradually transferred from its initially active (or absorbing) regions to its less accessible regions. We show that in three dimensions, in sharp contrast with the two-dimensional case, the size of the successive active zones steadily decreases during the passivation process, even though a large quantity of alive surface remains available. As a consequence, in three dimensions, the evolution of the efficiency of a surface accessed by diffusion (i.e., by a Laplacian field) can exhibit long-tail behaviors that, unlike in two dimensions, strongly depend on its specific geometry. This fact has important implications for the design of heterogeneous catalysts under deactivation conditions, for the performance of heat exchangers subjected to passivation by "fouling," and for changes in the behavior of the digestive system, where the activity of the absorbing intestinal membrane can be substantially affected by inflammatory disorders.

Screening effects in flow through rough channels.

A surprising similarity is found between the distribution of hydrodynamic stress on the wall of an irregular channel and the distribution of flux from a purely Laplacian field on the same geometry. This finding is a direct outcome of numerical simulations of the Navier-Stokes equations for flow at low Reynolds numbers in two-dimensional channels with rough walls presenting either deterministic or random self-similar geometries. For high Reynolds numbers, the distribution of wall stresses on deterministic and random fractal rough channels becomes substantially dependent on the microscopic details of the walls geometry. Finally, the effects on the flow behavior of the channel symmetry and aspect ratio are also investigated.

Restricted diffusion in a model acinar labyrinth by NMR: theoretical and numerical results.

A branched geometrical structure of the mammal lungs is known to be crucial for rapid access of oxygen to blood. But an important pulmonary disease like emphysema results in partial destruction of the alveolar tissue and enlargement of the distal airspaces, which may reduce the total oxygen transfer. This effect has been intensively studied during the last decade by MRI of hyperpolarized gases like helium-3. The relation between geometry and signal attenuation remained obscure due to a lack of realistic geometrical model of the acinar morphology. In this paper, we use Monte Carlo simulations of restricted diffusion in a realistic model acinus to compute the signal attenuation in a diffusion-weighted NMR experiment. We demonstrate that this technique should be sensitive to destruction of the branched structure: partial removal of the interalveolar tissue creates loops in the tree-like acinar architecture that enhance diffusive motion and the consequent signal attenuation. The role of the local geometry and related practical applications are discussed.

Brownian flights over a fractal nest and first-passage statistics on irregular surfaces.

The diffusive motion of Brownian particles near irregular interfaces plays a crucial role in various transport phenomena in nature and industry. Most diffusion-reaction processes in confining interfacial systems involve a sequence of Brownian flights in the bulk, connecting successive hits with the interface (Brownian bridges). The statistics of times and displacements separating two interface encounters are then determinant in the overall transport. We present a theoretical and numerical analysis of this complex first-passage problem. We show that the bridge statistics is directly related to the Minkowski content of the surface within the usual diffusion length. In the case of self-similar or self-affine interfaces, we show and check numerically that the bridge statistics follows power laws with exponents depending directly on the surface fractal dimension.

Mathematical basis for a general theory of Laplacian transport towards irregular interfaces.

The theory of Laplacian transport towards and across irregular surfaces is reformulated in terms of the Dirichlet-to-Neumann operator and its spectral characteristics. This permits us to obtain an exact equivalent circuit for the impedance of a working interface of arbitrary shape. The important result is that only very few eigenmodes of this operator do govern the entire response of a macroscopic system. This property drastically simplifies the understanding of irregular or prefractal interfaces. The results can be applied in electrochemistry, physiology and chemical engineering, fields where exchange processes across surfaces with complex geometry are ubiquitous.

Multifractal properties of the harmonic measure on Koch boundaries in two and three dimensions.

The multifractal properties of the harmonic measure on quadratic and cubic Koch boundaries are studied with the help of a new fast random walk algorithm adapted to these fractal geometries. The conjectural logarithmic development of local multifractal exponents is guessed for regular fractals and checked by extensive numerical simulations. This development allows one to compute the multifractal exponents of the harmonic measure with high accuracy, even with the first generations of the fractal. In particular, the information dimension in the case of the concave cubic Koch surface embedded in three dimensions is found to be slightly higher than its value D1 =2 for a smooth boundary.

Diffusion-reaction in branched structures: theory and application to the lung acinus.

An exact "branch by branch" calculation of the diffusional flux is proposed for partially absorbed random walks on arbitrary tree structures. In the particular case of symmetric trees, an explicit analytical expression is found which is valid whatever the size of the tree. Its application to the respiratory phenomena in pulmonary acini gives an analytical description of the crossover regime governing the human lung efficiency.

Diffusional screening in real 3D human acini--a theoretical study.

Gas exchange at the acinar level involves several physico-chemical phenomena within a complex geometry. A gas transport model, which takes into account both the diffusion into the acinus and the diffusion across the alveolar membrane, is used to understand gas mixing in realistic systems. It is first shown that the behaviour of the system, computed on model geometries in 3D, only depends on the topological structure of the acinus. Taking advantage of this property, a new efficient method based on random walks on a lattice is used to compute gas diffusion in structures taken from real morphological data. This approach shows that, at rest, the human acinus efficiency is only 30-40%. These results provide a new evidence of the existence of diffusional screening at the acinar level. This implies permanent spatial inhomogeneity of oxygen and carbon dioxide partial pressure. The notion of an "alveolar gas" is reinterpreted as a spatial average of the gas distribution. This model casts new light on the respiratory properties of other gas mixtures, such as helium-oxygen.

Self-stabilized fractality of seacoasts through damped erosion.

Erosion of rocky coasts spontaneously creates irregular seashores. But the geometrical irregularity, in turn, damps the sea waves, decreasing the average wave amplitude. There may then exist a mutual self-stabilization of the wave amplitude together with the irregular morphology of the coast. A simple model of such stabilization is studied. It leads, through a complex dynamics of the earth-sea interface, to the appearance of a stationary fractal seacoast with a dimension close to 4/3. Fractal geometry here plays the role of a morphological attractor directly related to percolation geometry.

Renormalized random walk study of oxygen absorption in the human lung.

The possibility to renormalize random walks is used to study numerically the oxygen diffusion and permeation in the acinus, the diffusion cell terminating the mammalian airway tree. This is done in a 3D tree structure which can be studied from its topology only. The method is applied to the human acinus real morphology as studied by Haefeli-Bleuer and Weibel in order to compute the respiratory efficiency of the human lung. It provides the first quantitative evidence of the role of diffusion screening in real 3D mammalian respiration. The net result of this study is that, at rest, the efficiency of the human acinus is only of order 33%. Application of these results to CO2 clearance provides for the first time a theoretical support to the empirical relation between the O2 and CO2 partial pressures in blood.

An optimal bronchial tree may be dangerous.

The geometry and dimensions of branched structures such as blood vessels or airways are important factors in determining the efficiency of physiological processes. It has been shown that fractal trees can be space filling and can ensure minimal dissipation. The bronchial tree of most mammalian lungs is a good example of an efficient distribution system with an approximate fractal structure. Here we present a study of the compatibility between physical optimization and physiological robustness in the design of the human bronchial tree. We show that this physical optimization is critical in the sense that small variations in the geometry can induce very large variations in the net air flux. Maximum physical efficiency therefore cannot be a sufficient criterion for the physiological design of bronchial trees. Rather, the design of bronchial trees must be provided with a safety factor and the capacity for regulating airway calibre. Paradoxically, our results suggest that bronchial malfunction related to asthma is a necessary consequence of the optimized efficiency of the tree structure.

Transition from Knudsen to molecular diffusion in activity of absorbing irregular interfaces.

We investigate through molecular dynamics the transition from Knudsen to molecular diffusion transport towards two-dimensional absorbing interfaces with irregular geometry. Our results indicate that the length of the active zone decreases continuously with density from the Knudsen to the molecular diffusion regime. In the limit where molecular diffusion dominates, we find that this length approaches a constant value of the order of the system size, in agreement with theoretical predictions for Laplacian transport in irregular geometries. Finally, we show that all these features can be qualitatively described in terms of a simple random-walk model of the diffusion process.

Interplay between geometry and flow distribution in an airway tree.

Uniform flow distribution in a symmetric volume can be realized through a symmetric branched tree. It is shown here, however, by 3D numerical simulation of the Navier-Stokes equations, that the flow partitioning can be highly sensitive to deviations from exact symmetry if inertial effects are present. The flow asymmetry is quantified and found to depend on the Reynolds number. Moreover, for a given Reynolds number, we show that the flow distribution depends on the aspect ratio of the branching elements as well as their angular arrangement. Our results indicate that physiological variability should be severely restricted in order to ensure adequate fluid distribution through a tree.

Diffusional screening in the human pulmonary acinus.

In the mammalian lung acini, O(2) diffuses into quasi-static 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). Two-dimensional numerical simulations of the steady-state 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.