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International audience; Little is known about transport throughout the respiration cycle in the conducting airways. It is challenging to appropriately describe the time-dependent number of particles entering back into the model during exhalation. Modeling the entire lung is not feasible; therefore multi-domain methods must be employed. Here, we present a new framework that is designed to simulate particles throughout the respiration cycle, incorporating realistic airway geometry and respiration. This framework is applied for a healthy rat lung exposed to 1 micron diameter particles to facilitate parameterization and validation. The flow field is calculated in the conducting airways (3D domain) by solving the incompressible Navier-Stokes equations with experimentally derived boundary conditions. Particles are tracked throughout inspiration by solving a modified Maxey-Riley equation. Next, we pass the time-dependent particle concentrations exiting the 3D model to the 1D volume conservation and advection-diffusion models (1D domain). Once the 1D models are solved, we prescribe the time-dependent number of particles entering back into the 3D airways to again solve for 3D transport. The coupled simulations highlight that about twice as many particles deposit during inhalation compared to exhalation for the entire lung. In contrast to inhalation, where the majority of particles deposit at the bifurcation zones, particles deposit relatively uniformly on the gravitationally dependent side of the 3D airways during exhalation. Strong agreement to previously collected regional experimental data is shown, as the 1D models account for lobe-dependent morphology. This framework may be applied to investigate dosimetry in other species and pathological lungs.

International audience; In this paper, we present a framework that couples three-dimensional (3D) to one-dimensional (1D) transport models to predict particle deposition in the respiratory airways throughout respiration. During respiration, the time dependent flow rate and particle concentration can be passed between the domains (inspiration: 3D to 1D, expiration: 1D to 3D). This framework enables us to predict particle transport and deposition in the whole lung and throughout both inspiration and expiration.

We consider models which are classically used in the simulation of airflows and blood flows andinvestigate the numerical stability of some discretization strategies. The geometrical complexity of the networksin which air/blood flows leads to a classical decomposition of two areas: a truncated 3D geometry correspondingto the largest contribution of the domain and a 0D part connected to the 3D part, modelling air/blood flowsin smaller airways/vessels. The resulting Navier-Stokes system in the 3D truncated part may involve non-local boundary conditions, deriving from a mechanical model. For various 3D/0D coupled models, differentdiscretization processes are presented and analyzed in terms of numerical stability, highlighting strong differencesaccording to the regimes that are considered. In particular, two main stability issues are investigated: first thecoupling between the 3D and the 0D part for which implicit or explicit strategies are studied and, second, thequestion of estimating the amount of kinetic energy entering the 3D domain because of the artificial boundaries.In particular, we prove new estimates in appropriate norms for the discretized-in-time Navier-Stokes system.These estimates are derived under conditions on the smallness of the data, enlighting the intrinsic difficultyencountered with such systems to perform realistic simulations. We illustrate some of the theoretical resultswith numerical simulations, firstly in a single tube, then in a bifurcation geometry and finally in real geometries.Finally, we discuss the difference between airflows and blood flows in terms of numerical stability related tothe magnitude of the physiological and physical parameters. Stokes and Navier-Stokes equations; coupling ofmodels; numerical stability; numerical computations; finite element method

International audience; Aerosolized particles are commonly used for therapeutic drug delivery as they can be delivered to the body systemically or be used to treat lung diseases. Recent advances in computational resources have allowed for sophisticated pulmonary simulations, however it is currently impossible to solve for airflow and particle transport for all length and time scales of the lung. Instead, multi-scale methods must be used. In our recent work, where computational methods were employed to solve for airflow and particle transport in the rat airways (Oakes et al. (2014), Annals of Biomedical Engineering, 42: 899-914), the number of particles to exit downstream of the 3D domain was determined. In this current work, the time-dependent Lagrangian description of particles was used to numerically solve a 1D convection-diffusion model (trumpet model, Taulbee and Yu (1975), Journal of Applied Physiology, 38: 77-85) parameterized specifically for the lung. The expansion of the airway dimensions was determined based on data collected from our aerosol exposure experiments (Oakes et al. (2014), Journal of Applied Physiology, 116: 1561-8). This 3D-1D framework enables us to predict the fate of particles in the whole lung.

International audience

International audience; In this chapter we deal with some specific existence and numerical results applied to a 2D/1D fluid-structure coupled model, for an incompressible fluid and a thin elastic structure. We will try to underline some of the mathematical and numeri- cal difficulties that one may face when studying this kind of problems such as the geometrical nonlinearities or the added mass effect. In particular we will try to underline the link between the strategies of proof of weak or strong solutions and the possible algorithms to discretize these type of coupled problems

WOS:000317040000096; International audience

International audience; In the first part of this paper, we prove hölderian and logarithmic stability estimates associated to the unique continuation property for the Stokes system. The proof of these results is based on local Carleman inequalities. In the second part, these estimates on the fluid velocity and on the fluid pressure are applied to solve an inverse problem: we consider the Stokes system completed with mixed Neumann and Robin boundary conditions and we want to recover the Robin coefficient (and obtain stability estimate for it) from measurements available on a part of the boundary where Neumann conditions are prescribed. For this identification parameter problem, we obtain a logarithmic stability estimate under the assumption that the velocity of a given reference solution stays far from 0 on a part of the boundary where Robin conditions are prescribed.

International audience; The respiratory system realizes the transfer of oxygen from the outside air to the alveolar membrane, through which it diffuses into the blood. As pure diffusion is far from being sufficient to realize that transfer, most of it is of advective type, and this advection is triggered by inflation-deflation cycles of the parenchyma. The mechanical part of the lungs can then be seen as a tree-like domain (conducting airways) embedded in an elastic medium. The flow in the upper part is inertial (incompressible Navier-Stokes equations), whereas inertia can be neglected for deeper branches (Stokes equations), which allows to use Poiseuille's law for each branch, and consequently Darcy like equations on the corresponding subtrees. We address here the delicate issues in terms of theory, numerics, and modeling, raised by the coupling of those models (Navier-Stokes, Darcy equations on a network, elasticity equations).

Dans le cadre de l'ANR M3RS (ANR-08-JCJC-0013-01); Les travaux présentés dans ce mémoire s'articulent essentiellement autour de deux thèmes : l'étude mathématique et numérique des phénomènes d'interaction fluide-structure et la modélisation de l'appareil respiratoire humain. C'est au cours de ma thèse que j'ai commencé à travailler sur les phénomènes d'interaction fluide-structure. Ces travaux se sont poursuivis par la suite au CEREMADE (Université Paris Dauphine) puis au sein du projet REO à l'INRIA. Je me suis essentiellement intéressée au cas des fluides newtoniens, visqueux, incompressibles satisfaisant aux équations de Navier-Stokes. La structure est, quant à elle, mobile : rigide ou déformable. On se place dans le cas ou les déplacements de la structure ont une amplitude telle que l'on ne peut pas les supposer infinitésimaux, ce qui va faire apparaitre des non linéarités géométriques. J'ai étudie ces problèmes aussi bien d'un point de vue théorique (existence de solutions faibles ou fortes pour des systèmes stationnaires ou instationnaires) que d'un point de vue numérique (étude de la stabilité et de la convergence des schémas, conception d'algorithmes performants). Les applications ou de tels phénomènes apparaissent sont nombreuses : tant en aérodynamique qu'en biomécanique. Nous pourrions citer beaucoup d'exemples : écoulement autour d'un bateau, écoulement sanguin dans les artères, pour l'hydrodynamique (fluide en phase liquide), écoulement autour d'ailes d'avion, étude de l'influence des vents sur le tablier d'un pont, pour l'aéroélasticité (fluide en phase gazeuse). En particulier, ils apparaissent dans l'étude des écoulements biologiques : écoulement sanguin dans les artères, écoulement de l'air dans l'appareil respiratoire. Ces dernières années je me suis particulièrement intéressée à ce type d'applications et plus spécifiquement à la modélisation mathématique et numérique de l'appareil respiratoire, tout d'abord dans le cadre d'une ACI nouvelle interface des mathématiques le poumon vous dis-je, puis au sein du projet REO. Je m'intéresse à la modélisation de l'écoulement de l'air dans les voies aériennes, la modélisation du transport et dépôt de particules ainsi qu'à la modélisation des tissus élastiques pulmonaire. Ces applications nécessitent le développement de modèles mathématiques nouveaux et de méthodes numériques spécifiques et adaptées.

In this paper, we provide a convergence analysis of a projection semi-implicit scheme for the simulation of fluid-structure systems involving an incompressible viscous fluid. The error analysis is performed on a fully discretized linear coupled problem: a finite element approximation and a semi-implicit time-stepping strategy are respectively used for space and time discretization. The fluid is described by the Stokes equations, the structure by the classical linear elastodynamics equations and all changes of geometry are neglected. We derive an error estimate in finite time and we prove that the time discretization error for the coupling scheme is at least $\sqrt{\delta t}$. Finally, some numerical experiments that confirm the theoretical analysis are presented.

International audience; We study the asymptotic behavior of a system modeling a composite material made of an elastic periodically perforated support, with period ε > 0, and a perfect gas placed in each of these perforations, as ε goes to zero. The model we use is linear corresponding to deformations around a reference configuration. We apply both two-scale asymptotic expansion and two-scale convergence methods in order to identify the limit behaviors as ε goes to 0. We state that in the limit, we get a two-scale linear elasticity–like boundary value problem. From this problem, we identify the corresponding homogenized and periodic cell equations which allows us to find the first corrector term. The analysis is performed both in the case of an incompressible and compressible material. We derive some mechanical properties of the limit materials by studying the homogenized coefficients. Finally, we numerically calculate the homogenized coefficients in the incompressible case, for different types of elastic materials.

We address the numerical simulation of fluid-structure systems involving an incompressible viscous fluid. This issue is particularly difficult to face when the fluid added-mass acting on the structure is strong, as it happens in hemodynamics for example. Indeed, several works have shown that, in such situations, implicit coupling seems to be necessary in order to avoid numerical instabilities. Although significant improvements have been achieved during the last years, solving implicit coupling often exhibits a prohibitive computational cost. In this work, we introduce a semi-implicit coupling scheme which remains stable for a reasonable range of the discretization parameters. The first idea consists in treating implicitly the added-mass effect, whereas the other contributions (geometrical non-linearities, viscous and convective effects) are treated explicitly. The second idea, relies on the fact that this kind of explicit-implicit splitting can be naturally performed using a Chorin-Temam projection scheme in the fluid. We prove (conditional) stability of the scheme for a fully discrete formulation. Several numerical experiments point out the efficiency of the present scheme compared to several implicit approaches.

This PhD thesis is devoted to the mathematical modeling and simulation of two existing physical methods to overcome the biological barriers to drug delivery. In the first part, several ways to model electroporation are considered, from the cell scale to the tissue scale. Existing phenomenological models of tissue electroporation are presented and numerically compared. Then a macroscopic model of electroporation is derived from a well-established model of cell elecroporation using homogenization techniques. In the second part, a new poroelastic model for the flows in biological tissues is presented to account for tissue degradation after an enzymatic treatment. To finish, an optimization algorithm is suggested in attempt to determine an optimal protocol when considering enzyme based therapies., Cette thèse présente des travaux concernant la modélisation mathématique de deux méthodes physiques existantes pour surmonter les barrières biologiques s’opposant à l’administration efficace de médicaments. Dans la première partie, plusieurs manières de modéliser l’électroporation sont exposées, aux échelles tissulaire et cellulaire. Des modèles phénoménologiques existants d’électroporation tissulaire sont présentés et comparés numériquement. Puis un modèle macroscopique d’électroporation est déduit d’un modèle d’électroporation cellulaire bien établi en utilisant des techniques d’homogénéisation. Dans la seconde partie, un nouveau modèle poroélastique est introduit pour décrire les écoulements dans un tissu biologique. Celui-ci prend en compte la dégradation tissulaire consécutive à un traitement par enzyme. Pour finir, un algorithme d’optimisation est proposé dans le but de déterminer un protocole optimal pour effectuer un traitement enzymatique.