Section: New Results

Lung and respiration modeling

Participants : Laurent Boudin, Paul Cazeaux, Bérénice Grec, Muriel Boulakia, Anne-Claire Egloffe, Benoit Fabreges, Miguel Ángel Fernández Varela, Jean-Frédéric Gerbeau, Céline Grandmont, Stéphane Liwarek, Sébastien Martin, Ayman Moussa.

• [59] , [60] :We are concerned here with identifiability, stability properties and estimates for the inverse problem of identifying a Robin coefficient on some non accessible part of the boundary from available data on the other part of boundary corresponding to solutions of the Stokes equations. In [59] , we first consider a steady state two-dimensional Stokes problem and study the identifiability of Robin coefficient and then we establish a stability estimate of logarithm type using a global Carleman inequality. We then consider the unsteady problem. In [60] :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 the inverse problem of identifying a Robin coefficient. 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.

• In [61] we are interested in the mathematical modeling of the propagation of sound waves in the lung parenchyma, which is a foam–like elastic material containing millions of air– filled alveoli. In this study, the parenchyma is governed by the linearized elasticity equations and the air by the acoustic wave equations. The geometric arrangement of the alveoli is assumed to be periodic with a small period ε > 0. We consider the time–harmonic regime forced by vibrations induced by volumic forces. We use the two–scale convergence theory to study the asymptotic behavior as ε goes to zero and prove the convergence of the solutions of the coupled fluid–structure problem to the solution of a linear–elasticity boundary value problem.

• In [53] we develop and study numerically a model to describe some aspects of sound propagation in the human lung, considered as a deformable and viscoelastic porous medium (the parenchyma) with millions of alveoli filled with air. Transmission of sound through the lung above 1 kHz is known to be highly frequency–dependent. We pursue the key idea that the viscoelastic parenchyma structure is highly heterogeneous on the small scale ε and use two–scale homogenization techniques to derive effective acoustic equations for asymptotically small ε. This process turns out to introduce new memory effects. The effective material parameters are determined from the solution of frequency–dependent micro–structure cell problems. We propose a numerical approach to investigate the sound propagation in the homogenized parenchyma using a Discontinuous Galerkin formulation. Numerical examples are presented.

• In [22] , we consider the Maxwell-Stefan model of diffusion previously introduced. We provide a qualitative and quantitative mathematical and basic numerical analysis of the model.

• In [65] we propose an integrated model for oxygen transfer into the blood, coupled with a lumped mechanical model for the ventilation process. We aim at investigating oxygen transfer into the blood at rest or exercise. The first task consists in describing nonlinear effects of the oxygen transfer under normal conditions. We also include the possible diffusion limitation in oxygen transfer observed in extreme regimes involving parameters such as alveolar and venous blood oxygen partial pressures, capillary volume, diffusing capacity of the membrane, oxygen binding by hemoglobin and transit time of the red blood cells in the capillaries. The second task consists in discussing the oxygen concentration heterogeneity along the path length in the acinus.

• In [43] we presented preliminary work on a multiscale 3D-0D airflow model to study differences between healthy and emphysema rats. The 0D model parameters were estimated from experimental data. 3D Navier-Stokes simulations were performed in healthy lungs, and in homogenous and heterogeneous emphysema lungs.