Marcel Filoche

Smaller is better—but not too small: A physical scale for the design of the mammalian pulmonary acinus

The transfer of oxygen from air to blood in the lung involves three processes: ventilation through the airways, diffusion of oxygen in the air phase to the alveolar surface, and finally diffusion through tissue into the capillary blood. The latter two steps occur in the acinus, where the alveolar gas-exchange surface is arranged along the last few generations of airway branching. For the acinus to work efficiently, oxygen must reach the last branches of acinar airways, even though some of it is absorbed along the way. This “screening effect” is governed by the relative values of physical factors like diffusivity and permeability as well as size and design of the acinus. Physics predicts that efficient acini should be space-filling surfaces and should not be too large. It is shown that the mammalian acini fulfill these requirements, small mammals being more efficient than large ones both at rest and in exercise.

Design of peripheral airways for efficient gas exchange

Peripheral airways combine branched tubes for ventilation with the gas exchanging alveoli in the pulmonary acini, defined as the complex of airways supplied by one first order respiratory or transitional bronchiole. In this part, the replenishment of oxygen at the alveolar surface occurs by a combination of convective air flow with diffusion of oxygen in the air. The transition between convection and diffusion depends on the morphometric properties of the airways. The design of the peripheral airways in the acinus of the human lung is described quantitatively on the basis of measurements obtained on casts of the acinar airways. Comparable data for rat and rabbit are also discussed. On the basis of this morphometric information, a typical path model for human acinar airways is derived. These studies also form the basis for advanced modeling studies of gas exchange and ventilation. In particular the problems occurring because of diffusional screening and the design conditions for minimizing this effect are discussed.

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.

Universal mechanism for Anderson and weak localization

Localization of stationary waves occurs in a large variety of vibrating systems, whether mechanical, acoustical, optical, or quantum. It is induced by the presence of an inhomogeneous medium, a complex geometry, or a quenched disorder. One of its most striking and famous manifestations is Anderson localization, responsible for instance for the metal-insulator transition in disordered alloys. Yet, despite an enormous body of related literature, a clear and unified picture of localization is still to be found, as well as the exact relationship between its many manifestations. In this paper, we demonstrate that both Anderson and weak localizations originate from the same universal mechanism, acting on any type of vibration, in any dimension, and for any domain shape. This mechanism partitions the system into weakly coupled subregions. The boundaries of these subregions correspond to the valleys of a hidden landscape that emerges from the interplay between the wave operator and the system geometry. The height of the landscape along its valleys determines the strength of the coupling between the subregions. The landscape and its impact on localization can be determined rigorously by solving one special boundary problem. This theory allows one to predict the localization properties, the confining regions, and to estimate the energy of the vibrational eigenmodes through the properties of one geometrical object. In particular, Anderson localization can be understood as a special case of weak localization in a very rough landscape.

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.

Optimal villi density for maximal oxygen uptake in the human placenta.

We present a stream-tube model of oxygen exchange inside a human placenta functional unit (a placentone). The effect of villi density on oxygen transfer efficiency is assessed by numerically solving the diffusion-convection equation in a 2D+1D geometry for a wide range of villi densities. For each set of physiological parameters, we observe the existence of an optimal villi density providing a maximal oxygen uptake as a trade-off between the incoming oxygen flow and the absorbing villus surface. The predicted optimal villi density 0.47±0.06 is compatible to previous experimental measurements. Several other ways to experimentally validate the model are also proposed. The proposed stream-tube model can serve as a basis for analyzing the efficiency of human placentas, detecting possible pathologies and diagnosing placental health risks for newborns by using routine histology sections collected after birth.

The role of morphology in mathematical models of placental gas exchange

The performance of the placenta as a gas exchanger has a direct impact on the future health of the newborn. To provide accurate estimates of respiratory gas exchange rates, placenta models need to account for both the physiology of exchange and the organ morphology. While the former has been extensively studied, accounting for the latter is still a challenge. The geometrical complexity of placental structure requires use of carefully crafted approximations. We present here the state of the art of respiratory gas exchange placenta modeling and demonstrate the influence of the morphology description on model predictions. Advantages and shortcomings of various classes of models are discussed, and experimental techniques that may be used for model validation are summarized. Several directions for future development are suggested.

Particle capture into the lung made simple

Understanding the impact distribution of particles entering the human respiratory system is of primary importance as it concerns not only atmospheric pollutants or dusts of various kinds but also the efficiency of aerosol therapy and drug delivery. To model this process, current approaches consist of increasingly complex computations of the aerodynamics and particle capture phenomena, performed in geometries trying to mimic lungs in a more and more realistic manner for as many airway generations as possible. Their capture results from the complex interplay between the details of the aerodynamic streamlines and the particle drag mechanics in the resulting flow. In contrast, the present work proposes a major simplification valid for most airway generations at quiet breathing. Within this context, focusing on particle escape rather than capture reveals a simpler structure in the entire process. When gravity can be neglected, we show by computing the escape rates in various model geometries that, although still complicated, the escape process can be depicted as a multiplicative escape cascade in which each elementary step is associated with a single bifurcation. As a net result, understanding of the particle capture may not require computing particle deposition in the entire lung structure but can be abbreviated in some regions using our simpler approach of successive computations in single realistic bifurcations. Introducing gravity back into our model, we show that this multiplicative model can still be successfully applied on up to nine generations, depending on particle type and breathing conditions.

Catalytic effectiveness of irregular interfaces and rough pores: the land surveyor approximation

Abstract We apply the concept of active zone in Laplacian transport to investigate the steady state mass transfer of a diffusive species towards an arbitrarily irregular catalytic interface. By means of a recently proposed coarse-graining technique, it is possible to compute the reactive flux on the catalytic interface from its geometry alone, without solving the general Laplace problem. As a result, we demonstrate by direct numerical simulation of molecular diffusion and first-order reaction that this method allows one to predict the catalytic effectiveness of a slit-shaped pore with an arbitrary rough geometry and over a wide range of diffusion–reaction conditions. It is found that, contrary to the traditional pseudo-homogeneous approach, the effect of the irregular morphology at the mesoscopic pore level is to modify the reaction rate and not the effective diffusion coefficient. We show that, for all practical situations where the reactant penetration in the pore is significant, a simplified picture of a smooth pore with an effective reactivity K eff can be used to describe the efficiency of a rough pore. Remarkably, K eff is the product of the intrinsic reactivity by a screening factor S , which has an elementary geometrical meaning, namely, the ratio between the real and the apparent surface area.

Three-dimensional model of surfactant replacement therapy

Significance To our knowledge, this paper presents the first structural model of surfactant replacement therapy (SRT), the delivery of liquid bolus surfactant mixtures into the lung airways. SRT has succeeded in premature neonates with surfactant deficiency but mostly failed in adults with acute respiratory distress syndrome, which claims ∼74,500 lives annually in the United States at a 39% mortality rate. Our model incorporates the fluid mechanics of the liquid bolus propagating through a branching airway tree with gravity effects. It reveals well-mixed distributions in the neonatal lung, but poorly mixed, highly nonlinear distribution features in adult lungs. The model suggests adult SRT failures are likely due to inadequate delivery, while providing a rational basis for successful protocols. Surfactant replacement therapy (SRT) involves instillation of a liquid-surfactant mixture directly into the lung airway tree. It is widely successful for treating surfactant deficiency in premature neonates who develop neonatal respiratory distress syndrome (NRDS). However, when applied to adults with acute respiratory distress syndrome (ARDS), early successes were followed by failures. This unexpected and puzzling situation is a vexing issue in the pulmonary community. A pressing question is whether the instilled surfactant mixture actually reaches the adult alveoli/acinus in therapeutic amounts. In this study, to our knowledge, we present the first mathematical model of SRT in a 3D lung structure to provide insight into answering this and other questions. The delivery is computed from fluid mechanical principals for 3D models of the lung airway tree for neonates and adults. A liquid plug propagates through the tree from forced inspiration. In two separate modeling steps, the plug deposits a coating film on the airway wall and then splits unevenly at the bifurcation due to gravity. The model generates 3D images of the resulting acinar distribution and calculates two global indexes, efficiency and homogeneity. Simulating published procedural methods, we show the neonatal lung is a well-mixed compartment, whereas the adult lung is not. The earlier, successful adult SRT studies show comparatively good index values implying adequate delivery. The later, failed studies used different protocols resulting in very low values of both indexes, consistent with inadequate acinar delivery. Reasons for these differences and the evolution of failure from success are outlined and potential remedies discussed.