Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Fractal and Non-Fractal Growth of Biological Cell Systems

  • Chapter
Fractals in Biology and Medicine

Part of the book series: Mathematics and Biosciences in Interaction ((MBI))

  • 475 Accesses

Abstract

The paper gives a general view of the microscopic growth rules and the behaviour of the spatial cell systems which generate fractal and non-fractal structures, with the purpose of finding their mathematical characteristics.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aharony, A.: Fractal growth. In: Fractals and Disordered Systems (A. Bunde, S. Havlin eds), pp. 151–173. Berlin: Springer (1991).

    Chapter  Google Scholar 

  2. Athreya, K.B.: Stochastic iteration of stable processes. In: Stochastic Processes and Related Topics (M.L. Puri ed.), Vol. 1, pp. 239–247. New York: Academic Press (1975).

    Google Scholar 

  3. Barnsley, M.F., Demko, S.: Iterated function systems and the global construction of fractals, Proc. Roy. Soc. (London) A 399, 243–275 (1985).

    Google Scholar 

  4. Bennett, C.H.: Dissipation, information, computational complexity and the definition of organization. In: Emerging Syntheses in Sciences (D. Pines ed.), pp. 215–233. Redwood: Addison-Wesley (1988).

    Google Scholar 

  5. Beran, J.: Statistical methods for data with long-range dependence, Statist. Sci. 7, 404–427 (1992).

    Article  Google Scholar 

  6. Biggins, J.D.: The asymptotic shape of the branching random walk, Adv. Appl. Probab. 10, 62–84 (1978).

    Article  Google Scholar 

  7. Bookstein, F.L.: The Measurement of Biological Shape and Shape Change, Lecture Notes Biomath. 24. Berlin: Springer (1978).

    Book  Google Scholar 

  8. Bramson, M., Griffeath, D.: On the Williams-Bjerknes tumour growth model, I, II, Ann. Probab. 9, 173–185 (1981); Math. Proc. Camb. Phil. Soc. 88, 339–357 (1980).

    Article  Google Scholar 

  9. Burrough, P.A.: Fractal dimensions of landscapes and other environmental data, Nature 294, 240–242 (1981).

    Article  Google Scholar 

  10. Durrett, R.: Lecture Notes on Particle Systems and Percolation. Pacific Grove (CA): Wadsworth & Brooks/Cole (1988).

    Google Scholar 

  11. Durrett, R.: Stochastic growth models: Recent results and open problems. In: Mathematical Approaches to Problems in Resource Management and Epidemiology (C. Castillo-Chaves et al. eds) [Lecture Notes Biomath. 81], pp. 308–312. Berlin: Springer (1989).

    Chapter  Google Scholar 

  12. Durrett, R.: Stochastic growth models: Bounds on critical values, J. Appl. Probab. 29, 11–20 (1992).

    Article  Google Scholar 

  13. Durrett, R., Griffeath, D.: Contact processes in several dimensions, Z. Wahrscheinlichkeitstheorie verw. Gebiete 59, 535–552 (1982).

    Article  Google Scholar 

  14. Durrett, R, Liggett, T.M.: The shape of the limit set in Richardson s growth model, Ann. Probab. 9, 186–193 (1981).

    Article  Google Scholar 

  15. Edelman, G.M.: Topobiology. An Introduction to Molecular Embryology. New York: Basic Books (1988).

    Google Scholar 

  16. Edgar, G.A.: Measure, Topology, and Fractal Geometry. New York: Springer (1990).

    Google Scholar 

  17. Falconer, K.: Fractal Geometry. Mathematical Foundations and Applications. Chichester: Wiley (1990).

    Google Scholar 

  18. Family, F.: Dynamic scaling and phase transitions in interface growth, Physica A 168, 561–580 (1990).

    Article  Google Scholar 

  19. Family, F., Vicsek, T. (eds): Dynamics of Fractal Surfaces. Singapore: World Scientific (1991).

    Google Scholar 

  20. Feller, W.: The asymptotic distribution of the range of independent random variables, Ann. Math. Statist. 22, 427–432 (1951).

    Article  Google Scholar 

  21. Feller, W.: On regular variation and local limit theorems, Proc. 5th Berkeley Symp. Math. Statist. Probab., Vol. II, Part 1, pp. 373–388. Berkeley: Univ. California Press (1967).

    Google Scholar 

  22. Fujikawa, H., Matsushita, M.: Bacterial fractal growth in the concentration field of nutrient, J. Phys. Soc. Japan 60, 88–94 (1991).

    Article  Google Scholar 

  23. Gouyet, J.-F., Rosso, M., Sapoval, B.: Fractal surfaces and interfaces. In: Fractals and Disordered Systems (A. Bunde, S. Havlin eds), pp. 229–261. Berlin: Springer (1991).

    Chapter  Google Scholar 

  24. Granger, C.W.J., Orr, D.: «Infinite variance» and research strategy in time series analysis, J. Amer. Statist. Assoc. 67, 275–285 (1972).

    Google Scholar 

  25. Griffin, P.S.: An integral test for the rate of escape of d-dimensional random walk, Ann. Probab. 11, 953–961 (1983).

    Article  Google Scholar 

  26. Grigorescu, S., Popescu, G.: Random systems with complete connections as a framework for fractals, Stud. Cerc. Mat. 41, 481–489 (1989).

    Google Scholar 

  27. Hammersley, J.M.: Harnesses, Proc. 5th Berkeley Symp. Math. Statist. Probat)., Vol. III, pp. 89–117. Berkeley: Univ. California Press (1967). (Reprinted in [38])

    Google Scholar 

  28. Hart, T.N., Trainor, L.E.H.: Geometrical aspects of surface morphogenesis, J. Theor. Biol. 138, 271–296 (1989).

    Article  PubMed  CAS  Google Scholar 

  29. Hatlee, M.D., Kozak, J.J.: Stochastic flows in integral and fractal dimensions and morphogenesis, Proc. Natl. Acad. Sci. USA 78, 972–975 (1981).

    Article  PubMed  CAS  Google Scholar 

  30. Hughes, B.D., Shlesinger, M.F., Montroll, E.W.: Random walks with self-similar clusters, Proc. Natl. Acad. Sci. USA 78, 3287–3291 (1981).

    Article  PubMed  CAS  Google Scholar 

  31. Iannacone, P.M.: Fractal geometry in mosaic organs: a new interpretation of mosaic pattern, FASEB J. 4, 1508–1512 (1990).

    Google Scholar 

  32. Kaandorp, J.A.: Modelling growth forms of biological objects using fractals, Ph. D. Thesis, Univ. of Amsterdam (1992).

    Google Scholar 

  33. Kesten, H.: Aspects of first passage percolation. In: École d’Áté de Probabilités de Saint-Flour XIV-1984 [Lecture Notes Math. 1180], pp. 125–264. Berlin: Springer (1986).

    Chapter  Google Scholar 

  34. Kesten H.: Percolation theory and first-passage percolation, Ann. Probab. 15, 1231–1271 (1987).

    Article  Google Scholar 

  35. Lamperti, J.: Limiting distributions for branching processes, Proc. 5th Berkeley Symp. Math. Statist. Probab., Vol. II, Part 2, pp. 225–241, Berkeley: Univ. California Press (1967).

    Google Scholar 

  36. Lamperti, J.: Semi-stable Markov processes, Z. Wahrscheinlichkeitstheorie verw. Gebiete 22, 205–225 (1972).

    Article  Google Scholar 

  37. Lawler, G.F., Bramson, M., Griffeath, D.: Internal diffusion limited aggregation, Ann. Probab. 20, 2117–2140 (1992).

    Article  Google Scholar 

  38. Liggett, T.M.: Long range exclusion processes, Ann. Probab. 8, 861–889 (1980).

    Article  Google Scholar 

  39. Liggett, T.M.: Interacting Particle Systems. New York: Springer (1985).

    Book  Google Scholar 

  40. Mandelbrot, B.B.: Fractal geometry: what is it, and what does it do?, Proc. Roy. Soc. (London) A 423, 3–16 (1989).

    Google Scholar 

  41. Mandelbrot, B.B.: Plane DLA is not self-similar; is it a fractal that becomes increasingly compact as it grows?, Physica A 191, 95–107 (1992).

    Google Scholar 

  42. Matsushita, M., Fujikawa, H.: Diffusion-limited growth in bacterial colony formation, Physica A 168, 489–506 (1990).

    Google Scholar 

  43. Matsuura, S., Miyazima, S.: Self-affine fractal growth of Aspergillus oryzae, Physica A 191, 30–34 (1992).

    Google Scholar 

  44. Meakin, P.: A new model for biological pattern formation, J. Theor. Biol. 118, 101–113 (1986).

    Article  PubMed  CAS  Google Scholar 

  45. Meakin, P.: Models for colloidal aggregation, Ann. Rev. Phys. Chem. 39, 237–267 (1988).

    Article  CAS  Google Scholar 

  46. Mollison, D.: Spatial contact models for ecological and epidemic spread, J. Roy. Statist. Soc. B 39, 283–326 (1977).

    Google Scholar 

  47. Ng, Y.-K., Iannaccone, P.M.: Fractal geometry of mosaic pattern demonstrates liver regeneration is a self-similar process, Develop. Biol. 151, 419–430 (1992).

    Article  PubMed  CAS  Google Scholar 

  48. Ng, Y.-K., Iannaccone, P.M.: Experimental chimeras: Current concepts and controversies in normal development and pathogenesis, Current Topics Develop. Biol. 27, 235–274 (1992)

    Article  CAS  Google Scholar 

  49. Nittmann, J., Daccord, G., Stanley, H.E.: When are viscous fingers fractal? In: Fractals in Physics (L. Pietronero, E. Tosatti eds), pp. 193–202. Amsterdam: North Holland (1986).

    Google Scholar 

  50. Othmer, H.G., Pate, E.: Scale-invariance in reaction-diffusion models of spatial pattern formation, Proc. Natl. Acad. Sci USA 77, 4180–4184 (1980).

    Article  PubMed  CAS  Google Scholar 

  51. Pelce, P., Sun, J.: Geometrical models for the growth of unicellular algae, J. Theor. Biol. 160, 375–386 (1993).

    Article  Google Scholar 

  52. Röthinger, B., Tautu, P.: On the genealogy of large cell populations. In: Stochastic Modelling in Biology (P. Tautu ed.), pp. 166–235. Singapore: World Scientific (1990).

    Google Scholar 

  53. Rudnick, J., Gaspari, G.: The shapes of random walks, Science 237, 384–389 (1987).

    Article  PubMed  CAS  Google Scholar 

  54. Schaefer, D.W., Bunker, B.C., Wilcoxon, J.P.: Fractals and phase separation. Proc. Roy. Soc. (London) A 423, 35–53 (1989).

    Google Scholar 

  55. Schrandt, R.G., Ulam, S.M.: On recursively defined geometrical objects and patterns of growth. In: Essays on Cellular Automata (A.W. Burks ed.), pp. 232–243. Urbana: Univ. Illinois Press (1970).

    Google Scholar 

  56. Schürger, K.: On a class of branching processes on a lattice with interactions, Adv. Appl. Probab. 13, 14–39 (1981).

    Article  Google Scholar 

  57. Schürger, K., Tautu, P.: A Markov configuration model for carcinogenesis. In: Mathematical Models in Medicine (J. Berger et al. eds) [Lecture Notes in Biomath. 11], pp. 92–108. Berlin: Springer (1976).

    Chapter  Google Scholar 

  58. Stanley, H.E.: Fractals and multifractals: The interplay of physics and geometry. In: Fractals and Disordered Systems (A. Bunde, S. Havlin eds), pp. 1–49. Berlin: Springer (1991).

    Chapter  Google Scholar 

  59. Tautu, P.: On the qualitative behaviour of interacting biological cell systems. In: Stochastic Processes in Physics and Engineering (S. Albeverio et al. eds), pp. 381–402. Dordrecht: Reidel (1988).

    Chapter  Google Scholar 

  60. Tautu, P.: Interacting biological cell systems. In: Stochastic Modelling in Biology (P. Tautu ed.), pp. 50–90. Singapore: World Scientific (1990).

    Google Scholar 

  61. Todd, P.H.: Gaussian curvature as a parameter of biological surface growth, J. Theor. Biol. 113, 63–68 (1985).

    Article  PubMed  CAS  Google Scholar 

  62. Todd, P.H.: Intrinsic Geometry of Biological Surface Growth. Lecture Notes Biomath. 67. Berlin: Springer (1986).

    Book  Google Scholar 

  63. Tsonis, A.A.: Chaos. From Theory to Applications. New York: Plenum Press (1992).

    Google Scholar 

  64. Vecchia, A.V.: A general class of models for stationary two-dimensional random processes, Biometrika 72, 281–291 (1985).

    Article  Google Scholar 

  65. Vicsek, T.: Fractal Growth Phenomena. Singapore: World Scientific (1989).

    Google Scholar 

  66. Vicsek, T., Cserzö, M., Horváth, V.K.: Self-affine growth of bacterial colonies, Physica A 167, 315–321 (1990).

    Google Scholar 

  67. Whittle, P.: Topographic correlation, power-law covariance functions, and diffusion, Biometrika 49, 305–314 (1962).

    Google Scholar 

  68. Whittle, P.: Systems in Stochastic Equilibrium. Chichester: Wiley (1986).

    Google Scholar 

  69. Williams, D.: Some basic theorems on harnesses. In: Stochastic Analysis (D.G. Kendall, E.F. Harding eds), pp. 349–363. London: Wiley (1973).

    Google Scholar 

  70. Yates, F.E.: Fractal applications in biology: Scaling time in biochemical networks. In: Numerical Computer Methods (L. Brand and M.L. Johnson eds) [Methods in Enzymology 210], pp. 636–675. San Diego: Academic Press (1992).

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Models Research Program Bioinformatics, German Cancer Research Center, D-69009, Heidelberg, Germany

    Petre Tautu

Authors
  1. Petre Tautu
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Mathematische Physik, Universität Ulm, Albert-Einstein-Allee 11, D-89069, Ulm, Germany

    T. F. Nonnenmacher

  2. Laboratorio di Patologia Cellulare, Istituto Cantonale di Patologia, Via in Selva 24, CH-6604, Locarno, Switzerland

    G. A. Losa

  3. Anatomisches Institut, Universität Bern, Bühlstrasse 26, CH-3012, Bern, Switzerland

    E. R. Weibel

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer Basel AG

About this chapter

Cite this chapter

Tautu, P. (1994). Fractal and Non-Fractal Growth of Biological Cell Systems. In: Nonnenmacher, T.F., Losa, G.A., Weibel, E.R. (eds) Fractals in Biology and Medicine. Mathematics and Biosciences in Interaction. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8501-0_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8501-0_7

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9652-8

  • Online ISBN: 978-3-0348-8501-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation