Modélisation mathématique et numérique du poumon humain

Abstract : The aim of this thesis is to give a multi-compartment model of the human lung. We propose a decomposition of the respiratory tree into three stages: a proximal part, the trachea and the first generations of bronchial tubes (around the fifth or the sixth one). In this part, we make direct simulations of the Navier-Stokes equations. A medium part, corresponding to the complex geometrical part of the bronchial tree. This part is a network of tubes of low diameters and the flow is linear, viscous and incompressible, described by the Stokes equations and regulated by the pressure gradient between the inlets and the outlets. This makes it possible to condense this part and to replace it by an equivalent single tube. The last compartment corresponds to the alveolar part and we model the diaphragm action by a spring displacement. We make direct simulations of Navier-Stokes equations in the upper part and condense the last two parts in a new boundary condition, so this multi-compartment approach avoids to mesh the complex geometrical part of the tree. First of all, we study the coupling of the two first compartments in the particular case of Stokes equations, we explain how to give a condensed equivalent to the medium part and make numerical simulations to validate this coupling. Then, we generalize the study to the Navier-Stokes equations. The main difficulty is to control the kinetic energy flux; we introduce particular boundary conditions, dissipative essential conditions, and prove the existence of weak solutions, locally in time for large data, and globally in time for small enough data. In the context of natural boundary conditions, the existence of locally in time solutions for small data, and globally in time solutions for smaller data, are proved, but in the two-dimensional case only. However, if we handle with a more regular class of solutions, we prove the existence and uniqueness of a weak solution locally in time for large data, and the existence globally in time for small data. For the global model incluging the spring we prove the existence of weak solutions, locally in time for large data in the case of essential boundary conditions, while we prove the existence of weak solutions, locally in time for small data in the two-dimensial case for the natural boundary conditions. Finally, we give a time discretization of the globally coupled model and prove that the discrete energy balance is of order one. We present some two-dimensional simulations for both a healthy and an ill lung (asthma).
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https://tel.archives-ouvertes.fr/tel-00207495
Contributor : Assia Soualah Alila <>
Submitted on : Thursday, January 17, 2008 - 3:09:07 PM
Last modification on : Thursday, January 11, 2018 - 6:12:18 AM
Long-term archiving on : Tuesday, April 13, 2010 - 4:51:30 PM

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Assia Soualah Alila. Modélisation mathématique et numérique du poumon humain. Mathématiques [math]. Université Paris Sud - Paris XI, 2007. Français. ⟨tel-00207495⟩

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