## Section: New Results

### Respiration tree modeling

#### Airway flow and related environmental issues

Participants : Laurent Boudin, Julien Castelneau, Céline Grandmont, Michaël Grasseau, Driss Yakoubi.

L. Boudin, D. Götz (Berlin & Darmstadt) and B. Grec (Paris 5) have studied a model of air (and aerosol) diffusion in the lower part of the lung, the so-called Maxwell-Stefan law for multicomponent mixture, which is to be compared with the standard Fick model classically used for this study. They point out that in some situations, Fick's model does not hold anymore. This work is about to be submitted, and can be seen as a preliminary step on the mathematical and numerical study of the Maxwell-Stefan law. In 2010, it should also involve Prof. Francesco Salvarani (Univ. Pavia, Italy), who is an expert on diffusive models.

What happens outside the human lung also really matters for the respiratory system. In the framework of the ANR “PiTAC”, A. Blouza (Rouen), L. Boudin and S.M. Kaber (UPMC) [35] designed suitable parallel in time algorithms coupled with reduction methods for the stiff differential systems integration arising in chemical kinetics. They consider linear as well as nonlinear systems. Numerical efficiency of our approach is illustrated by a realistic ozone production model, which is a key problem in the atmosphere.

Different kind of fluid numerical schemes have been implemented by D. Yakoubi to improve the numerical method introduced in [28] to discretize the multiscale system proposed in [42] and describing the air flow in the proximal part of the bronchial tree.

S. Martin and D. Yakoubi in collaboration with A. Devys and B. Maury studied the diffusion of oxygen in the bronchial tree and the gas exchange capacity of the human lungs [46] .

#### Double-layered airway surface fluid

Participant : Marc Thiriet.

Human conducting airways are mostly lined with a pseudostratified, secretory, and ciliated epithelium that comprises 3 major cell types – ciliated, secretory, and basal cells – with submucosal glands and eventual cartilaginous elements. Mucus is a viscoelastic fluid (thickness 2–5 m ) secreted by the respiratory epithelium that protects tracheobronchial tree mucosa from dehydration and traps inhaled particles (allergens, carcinogens, dust, micro-organisms, and inflammatory debris) that come into contact with it to clear them from airways. Mucus cleans airways, as it flows from either the tracheobronchial tree or upper airways toward the pharynx, where it is swallowed (or expectorated). Ciliary motions propulse mucus. Mucus is more or less continuously secreted, shed, and recycled, discarded, or degraded.

A preliminary study has been carried out this year to investigate the flow of air and mucus in a small 2D exploration domain that incorporates the periciliary fluid and mucus layers as well as the boundary layer of flowing air using a bi-fluid Navier-Stokes equation and the vorticity-velocity formulation. The cilium was assumed to have during the propulsion phase a sinusoidal movement only (at rest during rest and recovery phases) with a frequency that ranges between 5 and 20 Hz. After splitting, a semi-implicite Runge-Kutta 2 scheme and an explicite and implicit Euler scheme were used for the advection step, gravitational component, and diffusion stage, respectively.

The simulations show that when cilia are at rest, the mucus velocity is null even when the air velocity is maximal. But when the mucus velocity is finite, momentum is slightly transfered from mucus to air close to the mucus layer. Consequently, the bidirectional 3D air flow do not strongly interfere with the slow unidirectional mucus motion that cleans airways.

#### Modeling of aerosol and spray

Participants : Laurent Boudin, Céline Grandmont, Michaël Grasseau, Ayman Moussa, Marc Thiriet, Driss Yakoubi.

The model mathematically studied in [15] fits the behavior of an aerosol in the airflow inside the lung. This Vlasov-Navier-Stokes system belongs to the class of the so-called fluid-kinetic models, which are currently the object of numerous studies. Along with B. Boutin (CEA Saclay & UPMC), B. Fornet (ONERA Toulouse), T. Goudon (INRIA SIMPAF), P. Lafitte (Lille 1 & INRIA SIMPAF), F. Lagoutière (Paris 7 & Paris 11) and B. Merlet (Paris 13), during Cemracs '08, L. Boudin studied an asymptotic analysis of a fluid-particles coupled model, in the bubbling regime. From a theoretical point of view, they extended the analysis done in [59] for the case of an isentropic gas to the case of an ideal gas, thus adding the internal energy, or temperature, which is unknown. they formally derived the bubbling limit system in the same way as in [59] and propose a numerical scheme to solve this limit system. The numerical resolution of the non-limit system, and the numerical analysis of the asymptotic properties of the scheme (e.g. the asymptotic preserving property), as performed in [59] , is under study.

L. Boudin, C. Grandmont and A. Moussa are tackling a problem close to the one of [15] , but in a moving domain (ALE), i.e. a situation more realistic for the airways. They also try to enhance the numerical scheme in the ALE case, because some difficult points still remains to be solved, like the deposition criterion itself.

L. Boudin, C. Grandmont, B. Grec (Paris 5) and D. Yakoubi extended the FreeFEM++ code developed in [57] to study the influence of the aerosol on the airflow (the retroaction term). In addition to the fact that big particles (radius>30 m) have a significant effect on the fluid, they find out that they also have an effect on the deposition phenomenon. This work is about to be submitted.

L. Boudin, C. Grandmont, M. Grasseau, A. Moussa, M. Thiriet and D. Yakoubi are currently studying in collaboration with P. Diot and L. Vecellio (Inserm Tours U618) the behavior of an aerosol inside an experimental device U618 developed. They investigate the aerosol deposition inside the device with respect to the size distribution and average velocities of the aerosol. Preliminary results were presented during ISAM '09 (International Conference on Aerosol in Medicine) by A. Moussa.

L. Boudin, S.M. Kaber (UPMC), C. Majoral (Air Liquide) and L. Vecellio (Inserm Tours) are also working on the way to accurately and rapidly determine the radius distribution of a given aerosol nebulizer using some specific properties of pseudo-invertible matrices.

#### Lung tissue modeling

Participants : Paul Cazeaux, Céline Grandmont.

Lung parenchyma is a foam-like material consisting of millions of alveoli. Sound transmission through the lung plays an important role in the non-invasive diagnosis of many lung diseases. The firts step is to derive new 3D homogenized viscoelastic models taking into account the damping effect of the bronchial tree. It should be noted that, to our knowledge, in most of the studies, the mechanical behaviour of the parenchyma is described by an elastic law or as a porous media. Our choice is here to derive, through homogenization techniques, macroscopic models for a composite material made of an elastic body filled with gaseous bubbles connected through a diadic tree. Once this 3D homogenized viscoelastic model is obtained the questions are: how the obtained PDE can be discretized? How can we couple it with the rest of the respiratory tract? How sound is transmitted is such a media? This is the topic of the just starting PhD thesis of P. Cazeaux, supervised by C. Grandmont and Y. Maday. The thesis and in particular the study of sound propagation will be done in collaboration with Brown university.

**Work in progress**

P. Cazeaux and C. Grandmont have derived new 3D homogenized viscoelastic models taking into account the non local damping effect of the bronchial tree. They are currently studying the behavior of the solution (for instance its long time behavior) though a theoretical study as well as a numerical study based on Freefem++.

#### Inverse problem for air flow modeling

Participants : Muriel Boulakia, Anne-Claire Egloffe, Céline Grandmont.

One interest of the multiscale model proposed in [42] (based on the Navier-Stokes equations coupled with an ODE representing the motion of the diaphragm muscle) is that there are only a few parameters to fit. Our goal is to parametrize the ODE model. In particular one question of interest is: are we able to recover the stiffness of the lung and the resistance of the small airways via partial easily accessible measurements of volume and flux at the mouth? From a theoretical point of view we would like to investigate the stability of parameters with respect to measurements and from a numerical point of view we would like to try to recover these coefficients by optimization procedures.

**Work in progress**

A.-C. Egloffe started to work on the topic during her master internship and is now doing her PhD thesis supervised by M. Boulakia and C. Grandmont. She has obtained theoretical stability results for the heat equation with non-standard boundary conditions and encouraging 2D numerical results to recover parameters of the multiscale model.