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A viscoelastic model with non-local damping application to the human lungs

Published online by Cambridge University Press:  23 February 2006

Céline Grandmont
Affiliation:
Université Paris Dauphine, 75775 Paris Cedex 16 & INRIA, France.
Bertrand Maury
Affiliation:
Laboratoire de Mathématiques, Université Paris-Sud, 91405 Orsay Cedex, France. bertrand.maury@math.u-psud.fr
Nicolas Meunier
Affiliation:
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 75252 Paris Cedex 05, France.
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Abstract

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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2006

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References

H. Brezis, Analyse fonctionnelle. Théorie et applications. Masson (1993).
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).
Grimal, Q., Watzky, A. and Naili, S., A one-dimensional model for the propagation of pressure waves through the lung. J. Biomechanics 35 (2002) 10811089. CrossRef
Kaye, J., Primiano Jr, F.P.. and D.N. Metaxas, A 3D virtual environment for modeling mechanical cardiopulmonary interactions. Med. Imag. An. 2 (1998) 169195. CrossRef
Y. Lanir, Constitutive equations for the lung tissue. J. Biomech Eng. 105 (1983) 374–380.
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.
J.L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, I. Springer-Verlag, New York (1972).
Maksym, G.N. and Bates, J.H.T., A distributed nonlinear model of lung tissue elasticity. J. Appl. Phys. 82 (1997) 3241.
Mauroy, B., Filoche, M., Andrade Jr, J.S.. and B. Sapoval, Interplay between flow distribution and geometry in an airway tree. Phys. Rev. Lett. 14 (2003) 90.
Mauroy, B., Filoche, M., Weibel, E.R. and Sapoval, B., The optimal bronchial tree is dangerous. Nature 427 (2004) 633636. CrossRef
Oswald, P., Multilevel norms for H-1/2 . Computing 61 (1998) 235255. CrossRef
Ricci, S.B., Cluzel, P., Constantinescu, A. and Similowski, T., Mechanical model of the inspiratory pump. J. Biomechanics 35 (2002) 139145. CrossRef
Rodarte, J.R., Stress-strain analysis and the lung. Fed. Proc. 41 (1982) 130135.