An optimal bronchial tree may be dangerous


The geometry and dimensions of branched structures such as blood vessels or airways are important factors in determining the efficiency of physiological processes. It has been shown that fractal trees can be space filling1 and can ensure minimal dissipation2,3,4. The bronchial tree of most mammalian lungs is a good example of an efficient distribution system with an approximate fractal structure5,6. Here we present a study of the compatibility between physical optimization and physiological robustness in the design of the human bronchial tree. We show that this physical optimization is critical in the sense that small variations in the geometry can induce very large variations in the net air flux. Maximum physical efficiency therefore cannot be a sufficient criterion for the physiological design of bronchial trees. Rather, the design of bronchial trees must be provided with a safety factor and the capacity for regulating airway calibre. Paradoxically, our results suggest that bronchial malfunction related to asthma is a necessary consequence of the optimized efficiency of the tree structure.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Geometry of a dichotomic and homothetic tree structure.
Figure 2: Homothety ratios for length and diameters of deep bronchi in the human lung (6 ≤ Z ≤ 16).
Figure 3: Dependence on the homothety ratio (h) of the resistance (R) and volume (V) of an 11-generation homothetic tree.
Figure 4: Dependence of tree resistance on bronchial constriction expressed as a percentage of reduction in diameter.


  1. 1

    Mandelbrot, B. The Fractal Geometry of Nature (W. H. Freeman, San Francisco, CA, 1982)

    Google Scholar 

  2. 2

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

    CAS  Article  Google Scholar 

  3. 3

    Brown, J. H., West, G. B. & Enquist, B. J. Scaling in Biology (Oxford Univ. Press, Oxford, UK, 2000)

    Google Scholar 

  4. 4

    Bejan, A. Shape and Structure, From Engineering to Nature (Cambridge Univ. Press, Cambridge, UK, 2000)

    Google Scholar 

  5. 5

    Nelson, T. R. & Manchester, D. K. Modeling of lung morphogenesis using fractal geometries. IEEE Trans. Med. Imaging 7, 321–327 (1988)

    CAS  Article  Google Scholar 

  6. 6

    West, B. J., Barghava, V. & Goldberger, A. L. Beyond the principle of similitude: renormalization in the bronchial tree. J. Appl. Physiol. 60, 1089–1097 (1986)

    CAS  Article  Google Scholar 

  7. 7

    Weibel, E. R. The Pathway for Oxygen (Harvard Univ. Press, Cambridge, MA, 1984)

    Google Scholar 

  8. 8

    Mauroy, B., Filoche, M., Andrade, J. S. & Sapoval, B. Interplay between geometry and flow distribution in an airway tree. Phys. Rev. Lett. 90, 1–4 (2003)

    Article  Google Scholar 

  9. 9

    Hess, W. R. Das Prinzip des kleinsten Kraftverbrauchs im Dienste hämodynamischer Forschung. Archiv Anat. Physiol. 1914, 1–62 (1914)

    Google Scholar 

  10. 10

    Murray, C. D. The physiological principle of minimum work. I. The vascular system and the cost of blood. Proc. Natl Acad. Sci. USA 12, 207–214 (1926)

    ADS  CAS  Article  Google Scholar 

  11. 11

    Weibel, E. R. in The Lung: Scientific Foundations 2nd edn Vol. 1 (eds Crystal, R. G., West, J. B., Weibel, E. R. & Barnes, P. J.) 1061–1071 (Lippincott-Raven, Philadelphia, PA, 1997)

    Google Scholar 

  12. 12

    Que, C. L., Kenyon, C. M., Olivenstein, R., Macklem, P. T. & Maksym, G. N. Homeokinesis and short-term variability of human airway caliber. J. Appl. Physiol. 91, 1131–1141 (2001)

    CAS  Article  Google Scholar 

  13. 13

    Sapoval, B. Universalités et Fractales (Flammarion, Paris, 1997)

    Google Scholar 

  14. 14

    Kitaoka, H., Ryuji, T. & Suki, B. A three-dimensional model of the human airway tree. J. Appl. Physiol. 87, 2207–2217 (1999)

    CAS  Article  Google Scholar 

  15. 15

    Sapoval, B., Filoche, M. & Weibel, E. R. Smaller is better—but not too small: a physical scale for the design of the mammalian pulmonary acinus. Proc. Natl Acad. Sci. USA 99, 10411–10416 (2002)

    ADS  CAS  Article  Google Scholar 

  16. 16

    Wilber, R. L. et al. Incidence of exercise-induced bronchospasm in Olympic winter sport athletes. Med. Sci. Sports Exerc. 32, 732–737 (2000)

    CAS  Article  Google Scholar 

  17. 17

    Weibel, E. R. Symmorphosis (Harvard Univ. Press, Cambridge, MA, 2000)

    Google Scholar 

  18. 18

    Hoppeler, H. & Fluck, M. Plasticity of skeletal muscle mitochondria: structure and function. Med. Sci. Sports Exerc. 35, 95–104 (2003)

    CAS  Article  Google Scholar 

  19. 19

    Weibel, E. R. Morphometry of the Human Lung (Springer, Berlin, 1963)

    Google Scholar 

Download references


The authors wish to thank J. M. Morel and M. Bernot for useful discussions.

Author information



Corresponding author

Correspondence to B. Sapoval.

Ethics declarations

Competing interests

The authors declare that they have no competing financial interests.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mauroy, B., Filoche, M., Weibel, E. et al. An optimal bronchial tree may be dangerous. Nature 427, 633–636 (2004).

Download citation

Further reading


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.