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

Céline Grandmont; Bertrand Maury; Nicolas Meunier

ESAIM: Mathematical Modelling and Numerical Analysis (2006)

  • Volume: 40, Issue: 1, page 201-224
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Grandmont, Céline, Maury, Bertrand, and Meunier, Nicolas. "A viscoelastic model with non-local damping application to the human lungs." ESAIM: Mathematical Modelling and Numerical Analysis 40.1 (2006): 201-224. <http://eudml.org/doc/249617>.

@article{Grandmont2006,
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. },
author = {Grandmont, Céline, Maury, Bertrand, Meunier, Nicolas},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Poiseuille flow; dyadic tree; kernel operator; damped wave equation; human lungs.; human lungs},
language = {eng},
month = {2},
number = {1},
pages = {201-224},
publisher = {EDP Sciences},
title = {A viscoelastic model with non-local damping application to the human lungs},
url = {http://eudml.org/doc/249617},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Grandmont, Céline
AU - Maury, Bertrand
AU - Meunier, Nicolas
TI - A viscoelastic model with non-local damping application to the human lungs
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2006/2//
PB - EDP Sciences
VL - 40
IS - 1
SP - 201
EP - 224
AB - 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.
LA - eng
KW - Poiseuille flow; dyadic tree; kernel operator; damped wave equation; human lungs.; human lungs
UR - http://eudml.org/doc/249617
ER -

References

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  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. Nature427 (2004) 633–636.  
  11. P. Oswald, Multilevel norms for H-1/2. Computing61 (1998) 235–255.  
  12. S.B. Ricci, P. Cluzel, A. Constantinescu and T. Similowski, Mechanical model of the inspiratory pump. J. Biomechanics35 (2002) 139–145.  
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