Multiscale modelling of sound propagation through the lung parenchyma^∗

¹ UniversitéPierre et Marie Curie-Paris 6, UMR 7598, Laboratoire J.-L. Lions, 75005 Paris, France
² Inria Projet REO, Rocquencourt, BP 105, 78153 Le Chesnay Cedex, France
cazeaux@ann.jussieu.fr
³ Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA
JanHesthaven@brown.edu

Received: 27 September 2012
Revised: 23 April 2013

Abstract

In this paper we develop and study numerically a model to describe some aspects of sound propagation in the human lung, considered as a deformable and viscoelastic porous medium (the parenchyma) with millions of alveoli filled with air. Transmission of sound through the lung above 1 kHz is known to be highly frequency-dependent. We pursue the key idea that the viscoelastic parenchyma structure is highly heterogeneous on the small scale ε and use two-scale homogenization techniques to derive effective acoustic equations for asymptotically small ε. This process turns out to introduce new memory effects. The effective material parameters are determined from the solution of frequency-dependent micro-structure cell problems. We propose a numerical approach to investigate the sound propagation in the homogenized parenchyma using a Discontinuous Galerkin formulation. Numerical examples are presented.