Shape and Size in Biology and Medicine

  • Vincenzo Capasso


As D’Arcy Thompson pointed out in his pioneering book on Growth and Form, “THERE IS AN IMPORTANT RELATIONSHIP BETWEEN THE FORM OR SHAPE OF A BIOLOGICAL STRUCTURE AND HIS FUNCTION”. Shape analysis deals with the statistical analysis of a family of “objects” in presence of stochastic fluctuations; stochastic geometry deals with the analysis of geometric aspects of “objects” subject to stochastic fluctuations. The scope of this presentation is to introduce relevant mathematical concepts and methods of shape analysis and of stochastic geometry, thus providing a guided tour in a selected bibliography. A relevant aspect of stochastic geometry is the analysis of the spatial structure of objects which are random in location and shape. In this case we may simply say that the mathematical interest is in spatial occupation. In various cases, described by specific examples (Birth-and-growth model; Boolean Model; a tumor growth model based on an inhomogeneous Boolean model), spatial occupation occurs via a random tessellation of the available space region. Hence a quantitative description of a random closed set can be obtained in terms of mean densities of volumes, surfaces, edges, vertices, etc., at the various Hausdorff dimensions.


Hausdorff Dimension Boolean Model Stochastic Fluctuation Stochastic Geometry Mathematical Interest 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Barndorff-Nielsen O.E., Kendall W.S., Van Lieshout M.N.M. Eds., Stochastic Geometry. Likelihood and Computation, Chapman and Hall-CRC, Boca Raton 1999.Google Scholar
  2. [2]
    Bookstein F.L., The Measurement of Biological Shape and Shape Change, Lecture Notes Biomath 24, Springer-Verlag, Heidelberg 1978.Google Scholar
  3. [3]
    Braghieri E., Teoria Matematica della Shape Analysis (in Italian), Thesis, Mathematics, University of Milano 1999–2000.Google Scholar
  4. [4]
    Capasso V. ED., Mathematical Modelling for Polymer Processing. Polymerization, Crystallization, Manufacturing,Springer-Verlag, Heidelberg, in press.Google Scholar
  5. [5]
    Capasso V., Micheletti A., Local spherical contact distribution function and local mean densities for inhomogeneous random sets, Stochastics and Stoch. Rep. 71 (2000), 51–67.Google Scholar
  6. [6]
    Capasso V., Micheletti A., Stochastic geometry of spatially structured birth-and-growth processes. Application to crystallization processes,in:“Spatial Stochastic Processes”, E. Merzbach ed., Lecture Notes in Mathematics - CIME Subseries, Springer-Verlag, Heidelberg, in press.Google Scholar
  7. [7]
    Capasso V., Micheletti A., BURGER M., Densities of n-facets of incomplete Johnson-Mehl tessellations generated by inhomogeneous birth-and-growth processes, preprint 2001.Google Scholar
  8. [8]
    Cressie N., Statistics for Spatial Data, Wiley, New York 1993.Google Scholar
  9. [9]
    Dryden I.L., Mardia K.V., Statistical Shape Analysis, Wiley, New York 1998.Google Scholar
  10. [10]
    Johnson W.A., Mehl R.F., Reaction Kinetics in processes of nucleation and growth, Trans. A.I.M.M.E. 135 (1939), 416–458.Google Scholar
  11. [ll]
    Kolmogorov A.N., Foundations of the Theory of Probability, Chelsea Pub. Co., New York 1956.Google Scholar
  12. [12]
    Matheron G., Random Sets and Integral Geometry, Wiley, New York 1975.Google Scholar
  13. [13]
    Moller J., Random Johnson-Mehl tessellations,Adv. Appl. Prob. 24 (1992) 814–844.Google Scholar
  14. [14]
    Murray J.D., Mathematical Biology, Springer-Verlag, Heidelberg 1989.CrossRefGoogle Scholar
  15. [15]
    Serra J., Image Analysis and Mathematical Morphology, Academic Press, London 1982.Google Scholar
  16. [16]
    Shilov G.E., Gurevich B.L., Integral, Measure and Derivative: a Unified Approach, Dover Pub., New York 1977.Google Scholar
  17. [17]
    Small C.G., The Statistical Theory of Shapes, Springer, New York 1996.CrossRefGoogle Scholar
  18. [18]
    Stoyan D., Kendall W.S., Mecke J., Stochastic Geometry and its Application, John Wiley and Sons, New York 1995.Google Scholar
  19. [19]
    Thompson D.W., On Growth and Form (1917), Cambridge University Press, Cambridge 1970.Google Scholar
  20. [20]
    Turing A.M., The mechanical basis of morphogenesis,Phil. Trans. Roy. Soc. Lond. 237 (1952), 37–72.Google Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Vincenzo Capasso
    • 1
  1. 1.MIRIAM and Dipartimento di MatematicaUniversità di MilanoMilanoItaly

Personalised recommendations