# Team : reo

## Section: New Results

### Respiration tree modelling

Participants : Laurent Boudin, Céline Grandmont, Yvon Maday, Bertrand Maury, Marc Thiriet.

#### Modelling

In [24], our interest is, starting from a simple, rather naive, model of the acini, to show how an upstream model of the respiration tree can be hooked up and result in a well posed coupled system that will allow for simulations.

##### Work in progress

In colaboration with N. Meunier, we are looking at a simplified model of the respiration tree. This work has been initiated at CEMRACS 2004. In a first part we assume that we have a viscous fluid which flows through a tree connected pipes, each of which being characterized by its resistance. We establish the relation between pressures and fluxes at the outlets and investigate the convergence of this operator as the height of the tree goes to infinity. In a second part we couple the tree model with a mass–spring chain and study the convergence of it as the number of outlets (i. e. the height of the tree) tends to infinity. We obtain a wave equation with an additional non local dissipative term This term represents the influence of the tree. In collaboration with Yves Capdeboscq we are studying, in order to obtain a simplified model of the acini, the homogenization of an incompressible elastic structure with holes. These holes are filled with an incompressible fluid that can be evacuated through possibly connected pipes. In the same spirit, we are looking at the homogenization of an incompressible elastic structure containing small compressible gaz inclusions.

#### Modeling of biosprays for upper airways

A work initiated during the CEMRACS 2004 [30] is the first step of the study of sprays in the upper airways. In the following kinetic equation, which is similar to the one in [35]

_{t}fv

_{x}ffuv

_{r}f

_{v}f

the probability density function of dropletsf depends on time t, position x, velocity v, radius r. The quantity describes the evolution of the radius of the droplets, and u denotes the velocity field of the ambient air. This equation is coupled with standard Navier-Stokes equations.

Our work consists in investigating the behaviour of the 2D numerical solution of this fluid-kinetic problem, which is solved with a finite volume method for the fluid part and a particle method for the spray. For the moment, the throat is roughly modelled as a set of two rectangles (horizontal and vertical), the injected droplets have high velocities, the inhalated sprays are almost motionless. The first numerical tests will be presented in [30]. In the future, the geometrical description of the throat will be improved.

#### Airway flow

The flow in the proximal airways has been studied numerically on image-derived computational domains. Ventilation distribution has been studied at different phases of the respiratory cycle (mid decelerating phase of inspiration, mid accelerating phase of expiration, peak expiration, mid decelerating phase of expiration, mid accelerating phase of inspiration, and peak inspiration). But the results of ventilation distribution are questionable because the boundary conditions remains nowadays non-suitable and modelling is in progress to take into account adjoining parts of the pipe network. An example of fluid particle trajectories are plotted at selected phases of the inspiration in Fig. 1.

Surprisingly, between mesh comparisons, using meshes of 173788, 521468 and 1340723 elements, did not exhibit large variations in the flow fields. On an image-based model of upper airways, mass conservation and cycle reproducibility were checked. Our initialisation procedure by solving a Stokes problem did not disturb too much the flow results in the tracheobronchial tree. However, less good reproducibility is observed at flow inversion phases.

Results have been compared using either different meshes and fluid solvers (GAMBIT and FIDAP or INRIA softwares YAMS, GHS3D and NSI3IFS). We did not observed great discrepencies between the tests.