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Bulletin of Mathematical Biology

, Volume 38, Issue 3, pp 219–237 | Cite as

Distribution of end-points of a branching network with decaying branch length

  • William H. Warner
  • Theodore A. Wilson
Article
  • 34 Downloads

Abstract

The geometry of the human bronchial tree has been described as a network formed by successive dichotomous branching with constant branching angles and geometrically decaying branch lengths. Models having these properties and with randomly distributed branching planes are constructed. The distribution of the end points of the model networks are described by computing the variance of the distributions in the direction of the axis of the network and in the transverse directions. It is found that, for a given decay ratio, there is a branching angle for which the volume filled by the end points is a maximum. The advantages of the network with the decay ratio and branching angle of the human bronchial tree are discussed.

Keywords

Dead Space Recursion Relation Bronchial Tree Daughter Branch Dimensionless Volume 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Literature

  1. Feller, W. 1957.An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd Ed., New York: John Wiley and Sons, Inc.Google Scholar
  2. Hess, W. R. 1914. “Das Prinzip des kleinsten Kraftverbrauches im Dienste hämodynamischer Forschung”.Archiv f. physiol., 1–62.Google Scholar
  3. Hoppin, F. G., J. M. B. Hughes and J. Mead. 1974. “What Makes Airways Lengthen?”Proc. Fed. Am. Soc. Exp. Biol.,33, 303.Google Scholar
  4. Horsfield, K. and G. Cumming. 1967. “Angles of Branching and Diameters of Branches in the Human Bronchial Tree.”Bull. Math. Biophys.,29, 245–259.Google Scholar
  5. —, G. Dart, D. Olson, G. Filley, and G. Cumming. 1971. “Models of the Human Bronchial Tree.”J. Appl. Physiol.,31, 207–217.Google Scholar
  6. Kac, M. 1959.Probability and Related Topics in Physical Sciences. New York: Interscience Publishers, Inc.Google Scholar
  7. Roux, W. 1878. “Ueber die Verzweigungen der Blutgefasse.”Jenaische Zeitschs. f. Naturwiss. 12, 206–266.Google Scholar
  8. Schleier, J. 1919. “Der Energieverbrauch in der Blutbahn.”Pflüger's Archiv f. Physiol. 173, 172–204.CrossRefGoogle Scholar
  9. Weibel, E. R. 1963.Morphometry of the Human Lung, Berlin: Springer-Verlag.Google Scholar
  10. Wilson, T. A. 1967. “Design of the Bronchial Tree.”Nature,213, 668–669.CrossRefGoogle Scholar
  11. — and K-H. Lin. 1970. “Convection and diffusion in the Airways and the Design of the Bronchial Tree.” InAirway Dynamics: Physiology andPharmacology. A. Bouhuys, ed. Springfield: Charles C. Thomas.Google Scholar

Copyright information

© Society for Mathematical Biology 1976

Authors and Affiliations

  • William H. Warner
    • 1
  • Theodore A. Wilson
    • 1
  1. 1.Department of Aerospace Engineering and MechanicsUniversity of MinnesotaMinneapolisU.S.A.

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