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A secondary function of the mammals' respiratory system is during inspiration to heat the air to body temperature and to saturate it with water before it reaches the alveoli. Relying on a mathematical model, we comprehensively analyze this function, considering all the terrestrial mammals (spanning six orders of magnitude of the body mass, $M$) and focusing on the sole contribution of the lungs to this air conditioning. The results highlight significant differences between the small and the large mammals, as well as between rest and effort, regarding the spatial distribution of heat and water exchanges in the lungs, and in terms of regime of mass transfer taking place in the lumen of the airways. Interestingly, the results show that the mammalian lungs appear to be designed just right to fully condition the air at maximal effort: all generations of the bronchial region of the lungs are mobilized for this purpose. For mammals with a mass above a certain threshold ($\simeq 5$ kg at rest and $\simeq 50$ g at maximal effort), it appears that the maximal value of this evaporation rate scales as $M^{-1/8}$ at rest and $M^{-1/16}$ at maximal effort and that around 40\% (at rest) or 50\% (at maximal effort) of the water/heat extracted from the lungs during inspiration is returned to the bronchial mucosa during expiration, independently of the mass, due to a subtle coupling between different phenomena. This last result implies that, above these thresholds, the amounts of water and heat extracted from the lungs by the ventilation scale with the mass such as the ventilation rate (i.e. as $M^{3/4}$ at rest and $M^{7/8}$ at maximal effort). Finally, it is worth mentioning that these amounts appear to remain limited, but not negligible when compared to relevant global quantities, even at maximal effort (4-6\%)., Comment: 21 pages, 7 figures, a supplementary material

We develop a mathematical model of the lung that can estimate independently the air flows and pressures in the upper bronchi. It accounts for the lung multi-scale properties and for the air-tissue interactions. The model equations are solved using the Discrete Fourier Transform, which allows quasi instantaneous solving, in the limit of the model hypotheses. With this model, we explore how the air flow--volume curves are affected by airways obstruction or by change in lung compliance. Our work suggests that a fine analysis of the flow-volume curves might bring information about the inner phenomena occurring in the lung.

The mucus on the bronchial wall forms a thin layer of non-Newtonian fluid. One of the roles of mucus is to protect the lungs by capturing inhaled pollutants. It is transported by mucocilliary clearance toward the tracheo-pharyngeal bifurcation, where it is eliminated. Due to the corrugation of its interface with air, the mucus layer is subject to surface tension forces that interact with its rheology. It is still not clear whether these forces can affect mucus displacement and, if they can, under what conditions and how this displacement can occur. In this work, we model the mucus as a thin Bingham fluid layer located on the wall of idealized, multi-scaled airway bifurcations. We analyze the resulting physical system using lubrication theory and 3D simulations. The theoretical analysis allows us to characterize the nonlinear behavior of the system and determine the geometric conditions under which the Bingham fluid can be moved by surface tension. 3D simulations are then used to quantify the effects in idealized airway bifurcations on a range of scales corresponding to those of bronchial bifurcations. Our results suggest that surface tension effects can displace overly thick mucus layers in airway bifurcations, a typical situation in obstructive lung pathologies (asthma, BPCO, cystic fobrosis, etc.). Moreover, our results indicate that this movement can disrupt mucociliary clearance and the homogeneity of the layer thickness, thus increasing the risk of lung infection.

The purpose of this study is to provide means to physicians for automated and fast recognition of airways diseases. In this work, we mainly focus on measures that can be easily recorded using a spirometer. The signals used in this framework are simulated using the linear bi-compartment model of the lungs. This allows us to simulate ventilation under the hypothesis of ventilation at rest (tidal breathing). By changing the resistive and elastic parameters, data samples are realized simulating healthy, fibrosis and asthma breathing. On this synthetic data, different machine learning models are tested and their performance is assessed. All but the Naive bias classifier show accuracy of at least 99%. This represents a proof of concept that Machine Learning can accurately differentiate diseases based on manufactured spirometry data. This paves the way for further developments on the topic, notably testing the model on real data.

Chest physiotherapy is a set of techniques used to help the draining of the mucus from the lung in pathological situations. The choice of the techniques, and their adjustment to the patients or to the pathologies, remains as of today largely empirical. High Frequency Chest Wall Oscillation (HFCWO) is one of these techniques, performed with a device that applies oscillating pressures on the chest. However, there is no clear understanding of how HFCWO devices interact with the lung biomechanics. Hence, we study idealised HFCWO manipulations applied to a mathematical and numerical model of the biomechanics of the lung. The lung is represented by a fluid--structure interaction model based on an airway tree that is coupled to an homogeneous elastic medium. We show that our model is driven by two dimensionless numbers that drive the effect of the idealised HFCWO manipulation on the model of the lung. Our model allow to analyze the stress applied to an idealised mucus by the air--mucus interaction and by the airway walls deformation. This stress behaves as a buffer and has the effect of reducing the stress needed to overcome the idealised mucus yield stress. Moreover, our model predicts the existence of an optimal range of the working frequencies of HFCWO. This range is in agreement with the frequencies actually used by practitioners during HFCWO maneuvers. Finally, our model suggests that analyzing the mouth airflow during HFCWO maneuvers could allow to estimate the compliance and the hydrodynamic resistance of the lung of a patient.

The airflow in the bronchi applies a shear stress on the bronchial mucus, which can move the mucus. The air--mucus interaction plays an important role in cough and in chest physiotherapy (CP). The conditions under which it induces a displacement of the mucus are still unclear. Yet, the air--mucus interaction justifies common technics of CP used to help the draining of the mucus in prevalent diseases. Hence, the determination of the distribution of the shear stress in the lung is crucial for understanding the effects of these therapies and, potentially, improve their efficiency. We develop a mathematical model to study the distribution of the wall shear stress (WSS) induced by an air flow exiting an airway tree. This model accounts for the main physical processes that determines the WSS, more particularly the compliance of the airways, the air inertia and the tree structure. We show that the WSS distribution in the tree depends on the dynamics of the airways deformation and on the air inertia. The WSS distribution in the tree exhibits a maximum whose amplitude and location depend on the amount of air flow and on the "tissue" pressure surrounding the airways. To characterize the behavior of the WSS at the tree bifurcations, we derive new analytical criteria related to the airway size reduction in the bifurcations. Our results suggest that a tuning of the airflow and of the tissue pressure during a CP maneuver might allow to control, at least partially, the air--mucus interaction in the lung.

A model of optimal control of ventilation has recently been developed for humans. This model highlights the importance of the localization of the transition between a convective and a diffusive transport of respiratory gas. This localization determines how ventilation should be controlled in order to minimize its energetic cost at any metabolic regime. We generalized this model to any mammal, based on the core morphometric characteristics shared by all mammalian lungs and on their allometric scaling from the literature. Since the main energetic costs of ventilation are related to convective transport, we prove that, for all mammals, the localization of the shift from a convective transport to a diffusive transport plays a critical role on keeping this cost low while fulfilling the lung function. Our model predicts for the first time the localization of this transition in order to minimize the energetic cost of ventilation, depending on mammal mass and metabolic regime. From this optimal localization, we are able to predict allometric scaling laws for both tidal volumes and breathing rates, at any metabolic rate. We ran our model for the three common metabolic rates -- basal, field and maximal -- and showed that our predictions reproduce accurately experimental data available in the literature. Our analysis supports the hypothesis that mammals allometric scaling laws of tidal volumes and breathing rates at a given metabolic rate are driven by a few core geometrical characteristics shared by mammalian lungs and by the physical processes of respiratory gas transport., Comment: Version 6 of this preprint has been peer-reviewed and recommended by Peer Community In Mathematical and Computational Biology (https://doi.org/10.24072/pci.mcb.100005)

Mammals have a high metabolism that produces heat proportionally to the power 3/4 of their mass at rest. Any excess of heat has to be dissipated in the surrounding environment to prevent overheating. Most of that dissipation occurs through the skin, but the efficiency of that mechanism decreases with the animal's mass. The role of the other mechanisms for dissipating heat is then raised, more particularly the one linked to the lung that forms a much larger surface area than the skin. The dissipation occurring in the lung is however often neglected, even though there exists no real knowledge of its dynamics, hidden by the complexity of the organ's geometry and of the physics of the exchanges. Here we show, based on an original and analytical model of the exchanges in the lung, that all mammals, independently of their mass, dissipate through their lung the same proportion of the heat they produced, about 6-7 %. We found that the heat dissipation in mammals' lung is driven by a number, universal among mammals, that arises from the dynamics of the temperature of the bronchial mucosa. We propose a scenario to explain how evolution might have tuned the lung for heat exchanges. Furthermore, our analysis allows to define the pulmonary heat and water diffusive capacities. We show in the human case that these capacities follow closely the oxygen consumption. Our work lays the foundations for more detailed analysis of the heat exchanges occurring in the lung. Future studies should focus on refining our understanding of the universal number identified. In an ecological framework, our analysis paves the way to a better understanding of the mammals' strategies for thermoregulation and of the effect of warming environments on mammals' metabolism.

Chest physiotherapy is an empirical technique used to help secretions to get out of the lung whenever stagnation occurs. Although commonly used, little is known about the inner mechanisms of chest physiotherapy and controversies about its use are coming out regularly. Thus, a scientific validation of chest physiotherapy is needed to evaluate its effects on secretions. We setup a quasi-static numerical model of chest physiotherapy based on thorax and lung physiology and on their respective biophysics. We modeled the lung with an idealized deformable symmetric bifurcating tree. Bronchi and their inner fluids mechanics are assumed axisymmetric. Static data from the literature is used to build a model for the lung's mechanics. Secretions motion is the consequence of the shear constraints apply by the air flow. The input of the model is the pressure on the chest wall at each time, and the output is the bronchi geometry and air and secretions properties. In the limit of our model, we mimicked manual and mechanical chest physiotherapy techniques. We show that for secretions to move, air flow has to be high enough to overcome secretion resistance to motion. Moreover, the higher the pressure or the quicker it is applied, the higher is the air flow and thus the mobilization of secretions. However, pressures too high are efficient up to a point where airways compressions prevents air flow to increases any further. Generally, the first effects of manipulations is a decrease of the airway tree hydrodynamic resistance, thus improving ventilation even if secretions do not get out of the lungs. Also, some secretions might be pushed deeper into the lungs; this effect is stronger for high pressures and for mechanical chest physiotherapy. Finally, we propose and tested two adimensional numbers that depend on lung properties and that allow to measure the efficiency and comfort of a manipulation.

In 1926, Murray proposed the first law for the optimal design of blood vessels. He minimized the power dissipation arising from the trade-off between fluid circulation and blood maintenance. The law, based on a constant fluid viscosity, states that in the optimal configuration the fluid flow rate inside the vessel is proportional to the cube of the vessel radius, implying that wall shear stress is not dependent on the vessel radius. Murray's law has been found to be true in blood macrocirculation, but not in microcirculation. In 2005, Alarc\'on et al took into account the non monotonous dependence of viscosity on vessel radius - F{\aa}hr{\ae}us-Lindqvist effect - due to phase separation effect of blood. They were able to predict correctly the behavior of wall shear stresses in microcirculation. One last crucial step remains however: to account for the dependence of blood viscosity on shear rates. In this work, we investigate how viscosity dependence on shear rate affects Murray's law. We extended Murray's optimal design to the whole range of Qu\'emada's fluids, that models pseudo-plastic fluids such as blood. Our study shows that Murray's original law is not restricted to Newtonian fluids, it is actually universal for all Qu\'emada's fluid as long as there is no phase separation effect. When phase separation effect occurs, then we derive an extended version of Murray's law. Our analyses are very general and apply to most of fluids with shear dependent rheology. Finally, we study how these extended laws affect the optimal geometries of fractal trees to mimic an idealized arterial network.

We build an evolutionary scenario that explains how some crucial physiological constraints in the arterial network of mammals - i.e. hematocrit, vessels diameters and arterial pressure drops - could have been selected by evolution. We propose that the arterial network evolved while being constrained by its function as an organ. To support this hypothesis, we focus our study on one of the main function of blood network: oxygen supply to the organs. We consider an idealized organ with a given oxygen need and we optimize blood network geometry and hematocrit with the constraint that it must fulfill the organ oxygen need. Our model accounts for the non-Newtonian behavior of blood, its maintenance cost and F\aa hr\ae us effects (decrease in average concentration of red blood cells as the vessel diameters decrease). We show that the mean shear rates (relative velocities of fluid layers) in the tree vessels follow a scaling law related to the multi-scale property of the tree network, and we show that this scaling law drives the behavior of the optimal hematocrit in the tree. We apply our scenario to physiological data and reach results fully compatible with the physiology: we found an optimal hematocrit of 0.43 and an optimal ratio for diameter decrease of about 0.79. Moreover our results show that pressure drops in the arterial network should be regulated in order for oxygen supply to remain optimal, suggesting that the amplitude of the arterial pressure drop may have co-evolved with oxygen needs., Comment: Shorter version, misspelling correction

The goal of this work is to follow the displacement and possible deformation of a free particle in a fluid flow in 2D axi-symmetry, 2D or 3D using the classical finite elements method without the usual drawbacks finite elements bring for fluid-structure interaction, i.e. huge numerical problems and strong mesh distortions. Working with finite elements is a choice motivated by the fact that finite elements are well known by a large majority of researchers and are easy to manipulate. The method we describe in this paper, called the camera method, is well adapted to the study of a single particle in a network and most particularly when the study focuses on the particle behaviour. The camera method is based on two principles: 1/ the fluid structure interaction problem is restricted to a neighbourhood of the particle, thus reducing drastically the number of degrees of freedom of the problem; 2/ the neighbourhood mesh moves and rotates with the particle, thus avoiding most of the mesh distortions that occur in a standard ALE method. In this article, we present the camera method and the conditions under which it can be used. Then we apply it to several examples from the literature in 2D axi-symmetry, 2D and 3D., Comment: 24 pages, 19 figures

In this paper, we study the role of boundary conditions on the optimal shape of a dyadic tree in which flows a Newtonian fluid. Our optimization problem consists in finding the shape of the tree that minimizes the viscous energy dissipated by the fluid with a constrained volume, under the assumption that the total flow of the fluid is conserved throughout the structure. These hypotheses model situations where a fluid is transported from a source towards a 3D domain into which the transport network also spans. Such situations could be encountered in organs like for instance the lungs and the vascular networks. Two fluid regimes are studied: (i) low flow regime (Poiseuille) in trees with an arbitrary number of generations using a matricial approach and (ii) non linear flow regime (Navier-Stokes, moderate regime with a Reynolds number 100) in trees of two generations using shape derivatives in an augmented Lagrangian algorithm coupled with a 2D/3D finite elements code to solve Navier-Stokes equations. It relies on the study of a finite dimensional optimization problem in the case (i) and on a standard shape optimization problem in the case (ii). We show that the behaviours of both regimes are very similar and that the optimal shape is highly dependent on the boundary conditions of the fluid applied at the leaves of the tree., Comment: \`a para\^itre dans Discrete Contin. Dyn. Syst. (B)

The red blood cells or erythrocytes are biconcave shaped cells and consist mostly in a membrane delimiting a cytosol with a high concentration in hemoglobin. This membrane is highly deformable and allows the cells to go through narrow passages like the capillaries which diameters can be much smaller than red blood cells one. They carry oxygen thanks to hemoglobin, a complex molecule that have very high affinity for oxygen. The capacity of erythrocytes to load and unload oxygen is thus a determinant factor in their efficacy. In this paper, we will focus on the pulmonary capillary where red blood cells capture oxygen. We propose a camera method in order to numerically study the behavior of the red blood cell along a whole capillary. Our goal is to understand how erythrocytes geometrical changes along the capillary can affect its capacity to capture oxygen. The first part of this document presents the model chosen for the red blood cells along with the numerical method used to determine and follow their shapes along the capillary. The membrane of the red blood cell is complex and has been modelled by an hyper-elastic approach coming from Mills et al (2004). This camera method is then validated and confronted with a standard ALE method. Some geometrical properties of the red blood cells observed in our simulations are then studied and discussed. The second part of this paper deals with the modeling of oxygen and hemoglobin chemistry in the geometries obtained in the first part. We have implemented a full complex hemoglobin behavior with allosteric states inspired from Czerlinski et al (1999)., Comment: 17 pages