Abstract
We are interested in the mathematical modeling of the deformation of the human lung tissue, called the lung parenchyma, during the respiration process. The parenchyma is a foam-like elastic material containing millions of air-filled alveoli connected by a tree-shaped network of airways. In this study, the parenchyma is governed by the linearized elasticity equations and the air movement in the tree by the Poiseuille law in each airway. The geometric arrangement of the alveoli is assumed to be periodic with a small period ε > 0. We use the two-scale convergence theory to study the asymptotic behavior as ε goes to zero. The effect of the network of airways is described by a nonlocal operator and we propose a simple geometrical setting for which we show that this operator converges as ε goes to zero. We identify in the limit the equations modeling the homogenized behavior under an abstract convergence condition on this nonlocal operator. We derive some mechanical properties of the limit material by studying the homogenized equations: the limit model is nonlocal both in space and time if the parenchyma material is considered compressible, but only in space if it is incompressible. Finally, we propose a numerical method to solve the homogenized equations and we study numerically a few properties of the homogenized parenchyma model.
Highlights
Introduction and MotivationBreathing involves the transport of air through the respiratory tract from its external entries, the nose and the mouth
Our motivation concerns the mathematical modeling of the human respiratory system and our interest here is to provide a simple model for the behavior of the alveolar region coupled with the bronchial tree during the respiration
For the sake of simplicity, we present here a particular construction which is based on the symmetric model of the bronchial tree developed by Weibel [40] and allows us to extend to our multi–dimensional setting the results obtained in [18]
Summary
Breathing involves the transport of air through the respiratory tract from its external entries, the nose and the mouth. The purpose of the present work is to obtain rigorously such a model of reduced complexity for the alveolar region by using the tools of two–scale periodic homogenization, involving fluid–structure interaction in the porous domain and flow of air through the bronchial tree. We propose to use the strong convergence of the sequence of these resistance operators in the space L(L2(Ω)), as ε goes to zero, as an abstract condition to model the convergent behavior of the sequence of trees ventilating our parenchyma domain. This allows us to divide the theoretical analysis in two parts. Given any vector field v, we denote e(v) its symmetrized gradient
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