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Optimal Poiseuille flow in a finite elastic dyadic tree

Published online by Cambridge University Press:  27 May 2008

Benjamin Mauroy  and
Nicolas Meunier
Benjamin Mauroy
Affiliation:
Laboratoire MSC, Université Paris 7 (Denis Diderot), 2 place Jussieu, building 33/34, 75251 Paris Cedex 05, France. [email protected]
Nicolas Meunier
Affiliation:
Laboratoire de Mathématiques MAP5, Université Paris 5 (R. Descartes), 45 rue des Saints Pères, 75006 Paris, France. [email protected]
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Abstract

In this paper we construct a model to describe someaspects of the deformation of the central region of the human lung considered as acontinuous elastically deformable medium. To achieve this purpose, we studythe interaction between the pipes composing the tree and the fluid that goes through it. We use a stationary model to determine the deformed radius of each branch. Then, we solve a constrained minimization problem, so as to minimize the viscous (dissipated) energy in the tree. The key feature of our approach is the useof a fixed point theorem in order to find the optimal flow associatedto a deformed tree. We also give some numerical results withinteresting consequences on human lung deformation during expiration, particularlyconcerning the localization of the equal pressure point (EPP).

Keywords

Fixed pointPoiseuille flowfinite treeelastic walllungsequal pressure point.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 4 , July 2008 , pp. 507 - 533
Copyright
© EDP Sciences, SMAI, 2008

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References

P. Dejours, Principles of Comparative Respiratory Physiology. Elsevier/North-Holland Biomedical Press (1982).
P. Feynman, Electromagnétisme 2. InterEditions (1979).
Grandmont, C., Maury, B. and Meunier, N., A viscoelastic model with non-local damping application to the human lungs. ESAIM: M2AN 40 (2006) 201–224. CrossRef
B. Housset, Pneumologie. Masson (1999).
B. Mauroy, Hydrodynamique dans le poumon, relations entre flux et géométries. Ph.D. thesis, ENS de Cachan (2004), http://www.cmla.ens-cachan.fr/~mauroy/mauroy_these.pdf
Mauroy, B., Filoche, M., Andrade, J.S., Jr., and B. Sapoval, Interplay between geometry and flow distribution in an airway tree. Phys. Rev. Lett. 90 (2003) 148101. CrossRef
Mauroy, B., Filoche, M., Weibel, E.R. and Sapoval, B., An optimal bronchial tree may be dangerous. Nature 427 (2004) 633–636. CrossRef
B. Maury and C. Vannier, Une modélisation du poumon humain par un arbre infini. CANUM (2006).
Oelze, M.L., Miller, R.J. and Blue, J.P., Impedance, Jr. measurements of ex vivo rat lung at different volumes of inflation. J. Acoust. Soc. Am. 114 (2003) 3384–3393. CrossRef
Preteux, F., Fetita, C., Capderou, A. and Grenier, P., Modeling, segmentation, and caliber estimation of bronchi in high-resolution computerized tomography. J. Electron. Imaging 8 (1999) 36–45. CrossRef
Salerno, F.G. and Ludwig, M.S., Elastic moduli of excised constricted rat lungs. J. Appl. Physiol. 86 (1999) 66–70.
E.R. Weibel, Morphometry of the Human Lung. Springer, Verlag (1963).
E.R. Weibel, The Pathway for Oxygen. Harvard University Press (1984).
West, G.B., Brown, J.H. and Enquist, B.J., A general model for the origin of allometric scaling laws in biology. Science 276 (1997) 122–126. CrossRef
Zach, M.S., The physiology of forced expiration. Paed. Resp. Review 1 (2000) 36–39.