Optimal Poiseuille flow in a finite elastic dyadic tree
Mauroy, Benjamin ; Meunier, Nicolas
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008), p. 507-533 / Harvested from Numdam
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In this paper we construct a model to describe some aspects of the deformation of the central region of the human lung considered as a continuous elastically deformable medium. To achieve this purpose, we study the 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 use of a fixed point theorem in order to find the optimal flow associated to a deformed tree. We also give some numerical results with interesting consequences on human lung deformation during expiration, particularly concerning the localization of the equal pressure point (EPP).

Publié le : 2008-01-01
DOI : https://doi.org/10.1051/m2an:2008015
Classification:  74D05,  74Q10,  76S05,  92B05 
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     author = {Mauroy, Benjamin and Meunier, Nicolas},
     title = {Optimal Poiseuille flow in a finite elastic dyadic tree},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {42},
     year = {2008},
     pages = {507-533},
     doi = {10.1051/m2an:2008015},
     mrnumber = {2437772},
     zbl = {pre05318499},
     zbl = {1203.74033},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2008__42_4_507_0}
}

Mauroy, Benjamin; Meunier, Nicolas. Optimal Poiseuille flow in a finite elastic dyadic tree. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008) pp. 507-533. doi : 10.1051/m2an:2008015. http://gdmltest.u-ga.fr/item/M2AN_2008__42_4_507_0/

[1] P. Dejours, Principles of Comparative Respiratory Physiology. Elsevier/North-Holland Biomedical Press (1982).

[2] P. Feynman, Electromagnétisme 2. InterEditions (1979).

[3] C. Grandmont, B. Maury and N. Meunier, A viscoelastic model with non-local damping application to the human lungs. ESAIM: M2AN 40 (2006) 201-224. | Numdam | MR 2223510 | Zbl pre05038398

[4] B. Housset, Pneumologie. Masson (1999).

[5] 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

[6] B. Mauroy, M. Filoche, J.S. Andrade, Jr., and B. Sapoval, Interplay between geometry and flow distribution in an airway tree. Phys. Rev. Lett. 90 (2003) 148101.

[7] B. Mauroy, M. Filoche, E.R. Weibel and B. Sapoval, An optimal bronchial tree may be dangerous. Nature 427 (2004) 633-636.

[8] B. Maury and C. Vannier, Une modélisation du poumon humain par un arbre infini. CANUM (2006).

[9] M.L. Oelze, R.J. Miller and J.P. Blue, Jr., Impedance measurements of ex vivo rat lung at different volumes of inflation. J. Acoust. Soc. Am. 114 (2003) 3384-3393.

[10] F. Preteux, C. Fetita, A. Capderou and P. Grenier, Modeling, segmentation, and caliber estimation of bronchi in high-resolution computerized tomography. J. Electron. Imaging 8 (1999) 36-45.

[11] F.G. Salerno and M.S. Ludwig, Elastic moduli of excised constricted rat lungs. J. Appl. Physiol. 86 (1999) 66-70.

[12] E.R. Weibel, Morphometry of the Human Lung. Springer, Verlag (1963).

[13] E.R. Weibel, The Pathway for Oxygen. Harvard University Press (1984).

[14] G.B. West, J.H. Brown and B.J. Enquist, A general model for the origin of allometric scaling laws in biology. Science 276 (1997) 122-126.

[15] M.S. Zach, The physiology of forced expiration. Paed. Resp. Review 1 (2000) 36-39.