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.
Mots clés : mathematical modeling, periodic homogenization, viscoelastic media, fluid-structure interaction, discontinuous Galerkin methods
@article{M2AN_2014__48_1_27_0,
author = {Cazeaux, Paul and Hesthaven, Jan S.},
title = {Multiscale modelling of sound propagation through the lung parenchyma},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {27--52},
publisher = {EDP-Sciences},
volume = {48},
number = {1},
year = {2014},
doi = {10.1051/m2an/2013093},
mrnumber = {3177836},
zbl = {1285.93014},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2013093/}
}
TY - JOUR AU - Cazeaux, Paul AU - Hesthaven, Jan S. TI - Multiscale modelling of sound propagation through the lung parenchyma JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 27 EP - 52 VL - 48 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2013093/ DO - 10.1051/m2an/2013093 LA - en ID - M2AN_2014__48_1_27_0 ER -
%0 Journal Article %A Cazeaux, Paul %A Hesthaven, Jan S. %T Multiscale modelling of sound propagation through the lung parenchyma %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 27-52 %V 48 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2013093/ %R 10.1051/m2an/2013093 %G en %F M2AN_2014__48_1_27_0
Cazeaux, Paul; Hesthaven, Jan S. Multiscale modelling of sound propagation through the lung parenchyma. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 1, pp. 27-52. doi : 10.1051/m2an/2013093. http://www.numdam.org/articles/10.1051/m2an/2013093/
[1] , Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482-1518. | MR | Zbl
[2] , , and , Homogenization of elastic media with gaseous inclusions. Multiscale Model. Simul. 7 (2008) 432-465. | MR | Zbl
[3] and , Determination of discrete relaxation and retardation time spectra from dynamic mechanical data. Rheologica Acta 28 (1989) 511-519.
[4] and , Mechanical modeling of the skin. Asymptotic Analysis 74 (2011) 167-198. | MR | Zbl
[5] , Reduced-basis approach for homogenization beyond the periodic setting. Multiscale Model. Simul. 7 (2008) 466-494. | MR | Zbl
[6] and , Biot's poroelasticity equations by homogenization, in Macroscopic Properties of Disordered Media, vol. 154 of Lecture Notes in Physics. Springer (1982) 51-57. | MR | Zbl
[7] , and , Longitudinal elastic wave propagation in pulmonary parenchyma. J. Appl. Phys. 62 (1987) 1349-1355.
[8] and , A genetic algorithm used to fit Debye functions to the dielectric properties of tissues. 2010 IEEE Congress on Evolutionary Computation (CEC) (2010) 1-8.
[9] , Attenuation and speed of ultrasound in lung: Dependence upon frequency and inflation. J. Acoust. Soc. Am. 80 (1986) 1248-1250.
[10] and , Mathematical problems in linear viscoelasticity, vol. 12 of SIAM Studies in Applied Mathematics. SIAM, Philadelphia, PA (1992). | MR | Zbl
[11] , and , Deriving the effective ultrasound equations for soft tissue interrogation. Comput. Math. Appl. 49 (2005) 1069-1080. | MR | Zbl
[12] and , Homogenizing the acoustic properties of the seabed. I. Nonlinear Anal. 40 (2000) 185-212. | MR | Zbl
[13] , and , A one-dimensional model for the propagation of transient pressure waves through the lung. J. Biomech. 35 (2002) 1081-1089.
[14] , Viscous dissipation and completely monotonic relaxation moduli. Rheologica Acta 44 (2005) 614-621.
[15] , FreeFem++ manual (2012).
[16] and , Nodal discontinuous Galerkin methods, vol. 54 of Texts in Applied Mathematics. Springer, New York (2008). | MR | Zbl
[17] , , and , Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes. J. Comput. Phys. 225 (2007) 1753-1781. | MR | Zbl
[18] , and , Debye function expansions of complex permittivity using a hybrid particle swarm-least squares optimization approach. Antennas Propagation IEEE Trans. 55 (2007) 1999-2005. | MR
[19] and , Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Appl. Numer. Math. 44 (2003) 139-181. | MR | Zbl
[20] , Hedge: Hybrid and Easy Discontinuous Galerkin Environment. http://www.cims.]nyu.edu/˜kloeckner/ (2010).
[21] , Speed of low-frequency sound through lungs of normal men. J. Appl. Phys. (1983) 1862-1867.
[22] , Numerical methods for conservation laws. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (1990). | MR | Zbl
[23] and , Problèmes aux limites non homogènes et applications, vol. 1 of Travaux et Recherches Mathématiques. Dunod, Paris (1968). | Zbl
[24] M. Lourakis, levmar: Levenberg-Marquardt nonlinear least squares algorithms in C/C++. http://www.ics.forth.gr/˜lourakis/levmar/ (2004).
[25] , and , Reduced basis numerical homogenization for scalar elliptic equations with random coefficients: application to blood micro-circulation. Submitted to SIAM J. Appl Math. (2012).
[26] , Modélisation mathématique et simulation de systèmes microvasculaires. Ph.D. thesis, Université Pierre et Marie Curie (2011).
[27] , A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608-623. | MR | Zbl
[28] and , The mechanics of lung tissue under high-frequency ventilation. SIAM J. Appl. Math. 61 (2001) 1731-1761. | MR | Zbl
[29] , and , Respiratory sounds. advances beyond the stethoscope. Am. J. Respir. Crit. Care Med. 156 (1997) 974.
[30] , Sound speed in pulmonary parenchyma. J. Appl. Physiol. 54 (1983) 304-308.
[31] and , What do we know about mechanical strain in lung alveoli? Am. J. Physiol. Lung Cell Mol. Physiol. 301 (2011) 625-635.
[32] , , , and , Low-frequency ultrasound permeates the human thorax and lung: a novel approach to non-invasive monitoring. Ultraschall Med. 31 (2010) 53-62.
[33] , Vibration of mixtures of solids and fluids, in Non-Homogeneous Media and Vibration Theory, vol. 127 of Lecture Notes in Physics. Springer (1980) 158-190.
[34] , A simple collocation method for fitting viscoelastic models to experimental data. GALCIT SM 63 (1961) 23.
[35] , , and . Multiscale modeling of the acoustic properties of lung parenchyma. ESAIM: Proc. 23 (2008) 78-97. | MR | Zbl
[36] and , Time integration in linear viscoelasticity - a comparative study. Mech. Time-Dependent Mater. 14 (2010) 307-328
[37] , , , and , Biomechanics of the lung parenchyma: critical roles of collagen and mechanical forces. J. Appl. Physiol. 98 (2005) 1892-1899.
[38] , Linear problems. In Homogenization Techniques for Composite Media, vol. 272 of Lecture Notes in Physics. Edited by Enrique Sanchez-Palencia and André Zaoui. Springer (1987) 209-230. | MR | Zbl
[39] , The general theory of homogenization. A personalized introduction, vol. 7 of Lecture Notes of the Unione Matematica Italiana. Springer (2009). | MR | Zbl
[40] , and , Asymptotic homogenization of viscoelastic composites with periodic microstructures. Int. J. Solids Struct. 35 (1998) 2039-2055. | MR | Zbl
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