Modélisation mathématique du poumon humain

1 Analyse numérique et équation aux dérivées partielles
LM-Orsay - Laboratoire de Mathématiques d'Orsay
Abstract : The present work adresses some theoretical issues raised by the modelling of the bronchial tree and its interactions with the parenchyma in which it is embedded. The bronchial tree is represented by a resistive dyadic tree with 23 generations, through which the air flow is described by the Stokes equations. According to Poiseuille's law, the flow rate through a pipe is proportional to the drop in pressure between its ends. This tree is embedded in a viscoelastic medium which is intending to represent the parenchyma. The ventilation process is caused by negative pressures, due to diaphragm contractions at the alveolar level. In the first part, we treat the bronchial tree as an infinite resistive tree letting the number of generations go to infinity. With the help of trace theorems, the ventilation process can be described by a Dirichlet-Neumann operator, which relates the pressure field at the boundary of the tree to outward flux. The second part explores some models of the parenchyma. The complexity of this viscoelastic medium derives from the fact that the tree connects all the alveolae. This coupled system causes some non local dissipation effects and calls for non-standard damping models. We first study a one-dimensional model of the parenchyma involving a wave-like partial differential equation with non-local dissipative term. We specially focus on the long time behavior of this system. We finally propose a new model for higher dimensions taking into account both the elasticity of the parenchyma and the resistive tree.
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• HAL Id : tel-00739462, version 1

Citation

Vannier Christine. Modélisation mathématique du poumon humain. Analyse numérique [math.NA]. Université Paris Sud - Paris XI, 2009. Français. ⟨tel-00739462⟩

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