Multiscale modelling of sound propagation through the lung parenchyma
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 1, p. 27-52
The full text of recent articles is available to journal subscribers only. See the article on the journal's website

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.

DOI : https://doi.org/10.1051/m2an/2013093
Classification:  93A30,  35B27,  35B40,  74D05,  65M60
Keywords: 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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {1},
     year = {2014},
     pages = {27-52},
     doi = {10.1051/m2an/2013093},
     zbl = {1285.93014},
     mrnumber = {3177836},
     language = {en},
     url = {http://www.numdam.org/item/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 - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 1, pp. 27-52. doi : 10.1051/m2an/2013093. http://www.numdam.org/item/M2AN_2014__48_1_27_0/

[1] G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482-1518. | MR 1185639 | Zbl 0770.35005

[2] L. Baffico, C. Grandmont, Y. Maday and A. Osses, Homogenization of elastic media with gaseous inclusions. Multiscale Model. Simul. 7 (2008) 432-465. | MR 2399553 | Zbl 1158.93003

[3] M. Baumgaertel and H.H. Winter, Determination of discrete relaxation and retardation time spectra from dynamic mechanical data. Rheologica Acta 28 (1989) 511-519.

[4] A. Blasselle and G. Griso, Mechanical modeling of the skin. Asymptotic Analysis 74 (2011) 167-198. | MR 2868051 | Zbl 1281.35087

[5] S. Boyaval, Reduced-basis approach for homogenization beyond the periodic setting. Multiscale Model. Simul. 7 (2008) 466-494. | MR 2438574 | Zbl 1156.74358

[6] R. Burridge and J. Keller, Biot's poroelasticity equations by homogenization, in Macroscopic Properties of Disordered Media, vol. 154 of Lecture Notes in Physics. Springer (1982) 51-57. | MR 674958 | Zbl 0487.73001

[7] J.P. Butler, J.L. Lehr and J.M. Drazen, Longitudinal elastic wave propagation in pulmonary parenchyma. J. Appl. Phys. 62 (1987) 1349-1355.

[8] J. Clegg and M.P. Robinson, 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] F. Dunn, Attenuation and speed of ultrasound in lung: Dependence upon frequency and inflation. J. Acoust. Soc. Am. 80 (1986) 1248-1250.

[10] M. Fabrizio and A. Morro, Mathematical problems in linear viscoelasticity, vol. 12 of SIAM Studies in Applied Mathematics. SIAM, Philadelphia, PA (1992). | MR 1153021 | Zbl 0753.73003

[11] M. Fang, R.P. Gilbert and X. Xie, Deriving the effective ultrasound equations for soft tissue interrogation. Comput. Math. Appl. 49 (2005) 1069-1080. | MR 2141251 | Zbl 1236.74239

[12] R.P. Gilbert and A. Mikelić, Homogenizing the acoustic properties of the seabed. I. Nonlinear Anal. 40 (2000) 185-212. | MR 1768409 | Zbl 0958.35108

[13] Q. Grimal, A. Watzky and S. Naili, A one-dimensional model for the propagation of transient pressure waves through the lung. J. Biomech. 35 (2002) 1081-1089.

[14] A. Hanygan, Viscous dissipation and completely monotonic relaxation moduli. Rheologica Acta 44 (2005) 614-621.

[15] F. Hecht, FreeFem++ manual (2012).

[16] J.S. Hesthaven and T. Warburton, Nodal discontinuous Galerkin methods, vol. 54 of Texts in Applied Mathematics. Springer, New York (2008). | MR 2372235 | Zbl 1134.65068

[17] A. Kanevsky, M.H. Carpenter, D. Gottlieb and J.S. Hesthaven, Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes. J. Comput. Phys. 225 (2007) 1753-1781. | MR 2349202 | Zbl 1123.65097

[18] D.F. Kelley, T.J. Destan and R.J. Luebbers, Debye function expansions of complex permittivity using a hybrid particle swarm-least squares optimization approach. Antennas Propagation IEEE Trans. 55 (2007) 1999-2005. | MR 2338772

[19] C.A. Kennedy and M.H. Carpenter, Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Appl. Numer. Math. 44 (2003) 139-181. | MR 1951292 | Zbl 1013.65103

[20] A. Kloeckner, Hedge: Hybrid and Easy Discontinuous Galerkin Environment. http://www.cims.]nyu.edu/˜kloeckner/ (2010).

[21] S.S. Kraman, Speed of low-frequency sound through lungs of normal men. J. Appl. Phys. (1983) 1862-1867.

[22] R.J. Leveque, Numerical methods for conservation laws. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (1990). | MR 1077828 | Zbl 0847.65053

[23] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, vol. 1 of Travaux et Recherches Mathématiques. Dunod, Paris (1968). | Zbl 0165.10801

[24] M. Lourakis, levmar: Levenberg-Marquardt nonlinear least squares algorithms in C/C++. http://www.ics.forth.gr/˜lourakis/levmar/ (2004).

[25] Y. Maday, N. Morcos and T. Sayah, Reduced basis numerical homogenization for scalar elliptic equations with random coefficients: application to blood micro-circulation. Submitted to SIAM J. Appl Math. (2012).

[26] N. Morcos, Modélisation mathématique et simulation de systèmes microvasculaires. Ph.D. thesis, Université Pierre et Marie Curie (2011).

[27] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608-623. | MR 990867 | Zbl 0688.35007

[28] M.R. Owen and M.A. Lewis, The mechanics of lung tissue under high-frequency ventilation. SIAM J. Appl. Math. 61 (2001) 1731-1761. | MR 1824522 | Zbl 0994.76105

[29] H. Pasterkamp, S.S. Kraman and G.R. Wodicka, Respiratory sounds. advances beyond the stethoscope. Am. J. Respir. Crit. Care Med. 156 (1997) 974.

[30] D.A. Rice, Sound speed in pulmonary parenchyma. J. Appl. Physiol. 54 (1983) 304-308.

[31] E. Roan and M.W. Waters, What do we know about mechanical strain in lung alveoli? Am. J. Physiol. Lung Cell Mol. Physiol. 301 (2011) 625-635.

[32] D. Rueter, H.P. Hauber, D. Droeman, P. Zabel and S. Uhlig, Low-frequency ultrasound permeates the human thorax and lung: a novel approach to non-invasive monitoring. Ultraschall Med. 31 (2010) 53-62.

[33] E. Sanchez-Palencia, 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] R.A. Schapery, A simple collocation method for fitting viscoelastic models to experimental data. GALCIT SM 63 (1961) 23.

[35] M. Siklosi, O.E. Jensen, R.H. Tew and A. Logg. Multiscale modeling of the acoustic properties of lung parenchyma. ESAIM: Proc. 23 (2008) 78-97. | MR 2509207 | Zbl 1156.92307

[36] J. Sorvari and J. Hämäläinen, Time integration in linear viscoelasticity - a comparative study. Mech. Time-Dependent Mater. 14 (2010) 307-328

[37] B. Suki, S. Ito, D. Stamenović, K.R. Lutchen and E.P. Ingenito, Biomechanics of the lung parenchyma: critical roles of collagen and mechanical forces. J. Appl. Physiol. 98 (2005) 1892-1899.

[38] P. Suquet, 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 902007 | Zbl 0619.00027

[39] L. Tartar, The general theory of homogenization. A personalized introduction, vol. 7 of Lecture Notes of the Unione Matematica Italiana. Springer (2009). | MR 2582099 | Zbl 1188.35004

[40] Y.-M. Yi, S.-H. Park and S.-K. Youn, Asymptotic homogenization of viscoelastic composites with periodic microstructures. Int. J. Solids Struct. 35 (1998) 2039-2055. | MR 1616148 | Zbl 0933.74054