Quelques modèles mathématiques homogénéisés appliqués à la modélisation du parenchyme pulmonaire

Paul Cazeaux 1, 2
2 REO - Numerical simulation of biological flows
LJLL - Laboratoire Jacques-Louis Lions, Inria Paris-Rocquencourt, UPMC - Université Pierre et Marie Curie - Paris 6
Abstract : We present macroscopic models of the mechanical behavior of the human lung's parenchyma obtained by the two--scale homogenization method. The first part focuses on the coupling between parenchyma and bronchial tree. We begin by describing a model for the parenchyma deformation. We model (i) the parenchyma as a linear elastic material, (ii) the alveoli as periodically distributed cavities of size epsilon in the macroscopic parenchyma domain and (iii) the bronchial tree as a dyadic resistive tree. We obtain by homogenization when epsilon goes to zero a macroscopic description of the parenchyma as a viscoelastic material, under some conditions on the irrigation of the domain by the tree, which we study next. The tree induces a spatially non--local dissipation. We illustrate the previous work by numerical simulations. The second part focuses on the sound wave propagation in the parenchyma. We do not take into account the effect of the bronchial tree in this case. We homogenize in the frequency domain a first model coupling the linearized elasticity equations in the parenchyma and the acoustic equation in the air. We obtain a linearized elastic macroscopic behavior law away from a set of resonant frequencies. Then we homogenize a second model, which takes into account the viscoelastic and heterogeneous nature of the parenchyma at the microscopic level. The macroscopic material obtained exhibits some new memory effects compared to its individual components that we study numerically.
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https://tel.archives-ouvertes.fr/tel-00765331
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Submitted on : Sunday, March 2, 2014 - 12:20:00 PM
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  • HAL Id : tel-00765331, version 2

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Paul Cazeaux. Quelques modèles mathématiques homogénéisés appliqués à la modélisation du parenchyme pulmonaire. Equations aux dérivées partielles [math.AP]. Université Pierre et Marie Curie - Paris VI, 2012. Français. ⟨tel-00765331v2⟩

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