A viscoelastic model with non-local damping application to the human lungs
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 1, p. 201-224
In this paper we elaborate a model to describe some aspects of the human lung considered as a continuous, deformable, medium. To that purpose, we study the asymptotic behavior of a spring-mass system with dissipation. The key feature of our approach is the nature of this dissipation phenomena, which is related here to the flow of a viscous fluid through a dyadic tree of pipes (the branches), each exit of which being connected to an air pocket (alvelola) delimited by two successive masses. The first part focuses on the relation between fluxes and pressures at the outlets of a dyadic tree, assuming the flow within the tree obeys Poiseuille-like laws. In a second part, which contains the main convergence result, we intertwine the outlets of the tree with a spring-mass array. Letting again the number of generations (and therefore the number of masses) go to infinity, we show that the solutions to the finite dimensional problems converge in a weak sense to the solution of a wave-like partial differential equation with a non-local dissipative term.
DOI : https://doi.org/10.1051/m2an:2006009
Classification:  74D05,  74Q10,  76S05,  92B05
Keywords: Poiseuille flow, dyadic tree, kernel operator, damped wave equation, human lungs
@article{M2AN_2006__40_1_201_0,
     author = {Grandmont, C\'eline and Maury, Bertrand and Meunier, Nicolas},
     title = {A viscoelastic model with non-local damping application to the human lungs},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {1},
     year = {2006},
     pages = {201-224},
     doi = {10.1051/m2an:2006009},
     zbl = {pre05038398},
     mrnumber = {2223510},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2006__40_1_201_0}
}
Grandmont, Céline; Maury, Bertrand; Meunier, Nicolas. A viscoelastic model with non-local damping application to the human lungs. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 1, pp. 201-224. doi : 10.1051/m2an:2006009. http://www.numdam.org/item/M2AN_2006__40_1_201_0/

[1] H. Brezis, Analyse fonctionnelle. Théorie et applications. Masson (1993). | MR 697382 | Zbl 0511.46001

[2] C. Grandmont, Y. Maday and B. Maury, A multiscale/multimodel approach of the respiration tree, in Proc. of the International Conference, “New Trends in Continuum Mechanics” 8-12 September 2003, Constantza, Romania Theta Foundation Publications, Bucharest (2005). | Zbl 1187.92030

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

[4] J. Kaye, F.P. Primiano Jr. and D.N. Metaxas, A 3D virtual environment for modeling mechanical cardiopulmonary interactions. Med. Imag. An. 2 (1998) 169-195.

[5] Y. Lanir, Constitutive equations for the lung tissue. J. Biomech Eng. 105 (1983) 374-380.

[6] A. Lefebvre and B. Maury, Micro-macro modelling of arrays of spheres interacting through lubrication forces, Prépublication du Laboratoire de Mathématiques de l'Université Paris-Sud (2005) 46.

[7] J.L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, I. Springer-Verlag, New York (1972). | Zbl 0223.35039

[8] G.N. Maksym and J.H.T. Bates, A distributed nonlinear model of lung tissue elasticity. J. Appl. Phys. 82 (1997) 32-41.

[9] B. Mauroy, M. Filoche, J.S. Andrade Jr. and B. Sapoval, Interplay between flow distribution and geometry in an airway tree. Phys. Rev. Lett. 14 (2003) 90.

[10] B. Mauroy, M. Filoche, E.R. Weibel and B. Sapoval, The optimal bronchial tree is dangerous. Nature 427 (2004) 633-636.

[11] P. Oswald, Multilevel norms for H -1/2 . Computing 61 (1998) 235-255. | Zbl 0930.65120

[12] S.B. Ricci, P. Cluzel, A. Constantinescu and T. Similowski, Mechanical model of the inspiratory pump. J. Biomechanics 35 (2002) 139-145.

[13] J.R. Rodarte, Stress-strain analysis and the lung. Fed. Proc. 41 (1982) 130-135.