Loading [MathJax]/jax/output/SVG/jax.js

Trace theorems for trees and application to the human lungs

  • Received: 01 February 2008 Revised: 01 February 2009
  • 05C05, 46E35, 46E39.

  • The aim of this paper is to develop a model of the respiratory system. The real bronchial tree is embedded within the parenchyma, and ventilation is caused by negative pressures at the alveolar level. We aim to describe the series of pressures at alveolae in the form of a function, and to establish a sound mathematical framework for the instantaneous ventilation process. To that end, we treat the bronchial tree as an infinite resistive tree, we endow the space of pressures at bifurcating nodes with the natural energy norm (rate of dissipated energy), and we characterise the pressure field at its boundary (i.e. set of simple paths to infinity). In a second step, we embed the infinite collection of leafs in a bounded domain Ω\RRd, and we establish some regularity properties for the corresponding pressure field. In particular, for the infinite counterpart of a regular, healthy lung, we show that the pressure field lies in a Sobolev space Hs(Ω), with s0.45. This allows us to propose a model for the ventilation process that takes the form of a boundary problem, where the role of the boundary is played by a full domain in the physical space, and the elliptic operator is defined over an infinite dyadic tree.

    Citation: Bertrand Maury, Delphine Salort, Christine Vannier. Trace theorems for trees and application to the human lungs[J]. Networks and Heterogeneous Media, 2009, 4(3): 469-500. doi: 10.3934/nhm.2009.4.469

    Related Papers:

    [1] Frédéric Bernicot, Bertrand Maury, Delphine Salort . A 2-adic approach of the human respiratory tree. Networks and Heterogeneous Media, 2010, 5(3): 405-422. doi: 10.3934/nhm.2010.5.405
    [2] Bertrand Maury, Delphine Salort, Christine Vannier . Trace theorems for trees and application to the human lungs. Networks and Heterogeneous Media, 2009, 4(3): 469-500. doi: 10.3934/nhm.2009.4.469
    [3] Boris Muha . A note on the Trace Theorem for domains which are locally subgraph of a Hölder continuous function. Networks and Heterogeneous Media, 2014, 9(1): 191-196. doi: 10.3934/nhm.2014.9.191
    [4] Caihong Gu, Yanbin Tang . Global solution to the Cauchy problem of fractional drift diffusion system with power-law nonlinearity. Networks and Heterogeneous Media, 2023, 18(1): 109-139. doi: 10.3934/nhm.2023005
    [5] Andrea Braides, Valeria Chiadò Piat . Non convex homogenization problems for singular structures. Networks and Heterogeneous Media, 2008, 3(3): 489-508. doi: 10.3934/nhm.2008.3.489
    [6] Boris Andreianov, Mohamed Karimou Gazibo . Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws. Networks and Heterogeneous Media, 2016, 11(2): 203-222. doi: 10.3934/nhm.2016.11.203
    [7] Patrick Joly, Maryna Kachanovska, Adrien Semin . Wave propagation in fractal trees. Mathematical and numerical issues. Networks and Heterogeneous Media, 2019, 14(2): 205-264. doi: 10.3934/nhm.2019010
    [8] Joachim von Below, José A. Lubary . Isospectral infinite graphs and networks and infinite eigenvalue multiplicities. Networks and Heterogeneous Media, 2009, 4(3): 453-468. doi: 10.3934/nhm.2009.4.453
    [9] Ciro D’Apice, Umberto De Maio, T. A. Mel'nyk . Asymptotic analysis of a perturbed parabolic problem in a thick junction of type 3:2:2. Networks and Heterogeneous Media, 2007, 2(2): 255-277. doi: 10.3934/nhm.2007.2.255
    [10] Hirotada Honda . On a model of target detection in molecular communication networks. Networks and Heterogeneous Media, 2019, 14(4): 633-657. doi: 10.3934/nhm.2019025
  • The aim of this paper is to develop a model of the respiratory system. The real bronchial tree is embedded within the parenchyma, and ventilation is caused by negative pressures at the alveolar level. We aim to describe the series of pressures at alveolae in the form of a function, and to establish a sound mathematical framework for the instantaneous ventilation process. To that end, we treat the bronchial tree as an infinite resistive tree, we endow the space of pressures at bifurcating nodes with the natural energy norm (rate of dissipated energy), and we characterise the pressure field at its boundary (i.e. set of simple paths to infinity). In a second step, we embed the infinite collection of leafs in a bounded domain Ω\RRd, and we establish some regularity properties for the corresponding pressure field. In particular, for the infinite counterpart of a regular, healthy lung, we show that the pressure field lies in a Sobolev space Hs(Ω), with s0.45. This allows us to propose a model for the ventilation process that takes the form of a boundary problem, where the role of the boundary is played by a full domain in the physical space, and the elliptic operator is defined over an infinite dyadic tree.


  • This article has been cited by:

    1. Yang Xu, Huachao Mao, Cenyi Liu, Zhengyu Du, Weijia Yan, Zhuoyuan Yang, Jouni Partanen, Yong Chen, Hopping Light Vat Photopolymerization for Multiscale Fabrication, 2023, 19, 1613-6810, 10.1002/smll.202205784
    2. Patrick Joly, Maryna Kachanovska, Transparent boundary conditions for wave propagation in fractal trees: convolution quadrature approach, 2020, 146, 0029-599X, 281, 10.1007/s00211-020-01145-9
    3. Robert Carlson, Dirichlet to Neumann maps for infinite quantum graphs, 2012, 7, 1556-1801, 483, 10.3934/nhm.2012.7.483
    4. Jonathan Sarhad, Robert Carlson, Kurt E. Anderson, Population persistence in river networks, 2014, 69, 0303-6812, 401, 10.1007/s00285-013-0710-6
    5. Paul Cazeaux, Céline Grandmont, Homogenization of a multiscale viscoelastic model with nonlocal damping, application to the human lungs, 2015, 25, 0218-2025, 1125, 10.1142/S0218202515500293
    6. Nicolas Pozin, Spyridon Montesantos, Ira Katz, Marine Pichelin, Irene Vignon-Clementel, Céline Grandmont, A tree-parenchyma coupled model for lung ventilation simulation, 2017, 33, 20407939, e2873, 10.1002/cnm.2873
    7. Patrick Joly, Adrien Semin, Mathematical and numerical modeling of wave propagation in fractal trees, 2011, 349, 1631073X, 1047, 10.1016/j.crma.2011.09.008
    8. Fabio Punzo, Alberto Tesei, Blow-up on metric graphs and Riemannian manifolds, 2023, 0, 1531-3492, 0, 10.3934/dcdsb.2023016
    9. Delio Mugnolo, 2014, Chapter 2, 978-3-319-04620-4, 11, 10.1007/978-3-319-04621-1_2
    10. Patrick Joly, Maryna Kachanovska, Local Transparent Boundary Conditions for Wave Propagation in Fractal Trees (I). Method and Numerical Implementation, 2021, 43, 1064-8275, A3760, 10.1137/20M1362334
    11. Patrick Joly, Maryna Kachanovska, Local Transparent Boundary Conditions for Wave Propagation in Fractal Trees (II). Error and Complexity Analysis, 2022, 60, 0036-1429, 529, 10.1137/20M1357524
    12. Serge Nicaise, Adrien Semin, Density and trace results in generalized fractal networks, 2018, 52, 0764-583X, 1023, 10.1051/m2an/2018021
    13. Alberto Tesei, 2024, Chapter 12, 978-3-031-60772-1, 241, 10.1007/978-3-031-60773-8_12
    14. Valentina Franceschi, Kiyan Naderi, Konstantin Pankrashkin, Embedded trace operator for infinite metric trees, 2024, 0025-584X, 10.1002/mana.202300574
  • Reader Comments
  • © 2009 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3794) PDF downloads(104) Cited by(12)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog