Multiscale Modelling of Sound Propagation through the Lung Parenchyma

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 frequencydependent 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. Mathematics Subject Classification. 93A30, 35B27, 35B40, 74D05, 65M60. Received September 27, 2012. Revised April 23, 2013. Published online November 15, 2013. Understanding the acoustic properties of the lungs is of great interest in a range of medical domains, from the diagnostic of pathologies such as bronchitis and asthma to recently introduced imaging methods of breathing sounds. Medical doctors have been commonly using the analysis of lung sounds for the diagnostic of pathologies, mainly relying on the stethoscope, and have a good empirical understanding of the relationship between the characteristics of lung sounds and underlying pathologies. However, far less is known about the detailed physical mechanisms of sound generation and transmission in the respiratory system. More advanced models could also explain lung contusions due to a blunt, high-velocity impact to the chest as well as exposure to explosions or, for small mammals, to medical ultrasound. There is extensive experimental data, both in vitro and in vivo, and a relatively good theoretical understanding of the propagation through lung tissue of relatively low-frequency sound, in the audible range. In the frequency band between roughly a few Hz to 1 kHz, the two main components of the lung tissue, air and connective tissue, are acoustically strongly coupled and the resulting mixture behaves like a homogeneous material with high