Skip to main content
SpringerLink
Log in

A nonlinear structured population model of tumor growth with quiescence

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

A nonlinear structured cell population model of tumor growth is considered. The model distinguishes between two types of cells within the tumor: proliferating and quiescent. Within each class the behavior of individual cells depends on cell size, whereas the probabilities of becoming quiescent and returning to the proliferative cycle are in addition controlled by total tumor size. The asymptotic behavior of solutions of the full nonlinear model, as well as some linear special cases, is investigated using spectral theory of positive simigroup of operators.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arendt, W., Grabosch, A., Greiner, G., Groh, U., Lotz, H.P., Moustakis, U., Nagel, R., Neubrander, F., Schlotterbeck, U.: One-parameter semigroups of positive operators. In: Nagel, R. (ed.) (Lect. Notes Math., vol. 1184) Berlin Heidelberg New York Tokyo: Springer 1986

    Google Scholar 

  2. Arino, O., Kimmel, M.: Asymptotic analysis of a cell-cycle model based on unequal division. SIAM J. Appl. Math. 47, 128–145 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  3. Arino, O., Kimmel, M.: Asymptotic behavior of a nonlinear functional-integral equation of cell kinetics with unequal division. J. Math. Biol. 27, 341–354 (1989)

    MATH  MathSciNet  Google Scholar 

  4. Bell, G. I., Anderson, E. C.: Cell growth and division. I. A mathematical model with applications to cell volume distributions in mammalian suspension cultures. Biophys. J. 7, 329–351 (1967)

    Article  Google Scholar 

  5. Burton, A. C.: Rate of growth of solid tumours as a problem of diffusion. Growth 30, 157–176 (1966)

    Google Scholar 

  6. Clément, Ph., Heijmans, H. J. A. M., Angenent, S., van Duijn, C. J., de Pagter, B.: One-parameter semigroups. Amsterdam New York: North-Holland 1987

    MATH  Google Scholar 

  7. Diekmann, O., Heijmans, H. J. A. M., Thieme, H. R.: On the stability of the cell size distribution. J. Math. Biol. 19, 227–248 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  8. Diekmann, O., Heijmans, H. J. A. M., Thieme, H. R.: On the stability of the cell size distribution II. In: Witten, M. (ed.) Hyperbolic partial differential equations III. Inter. Series in Modem Appl. Math. Computer Science, vol. 12, pp. 491–512. Oxford New York: Pergamon Press 1986

    Google Scholar 

  9. Eisen, M.: Mathematical models in cell biology and cancer chemotherapy. (Lect. Notes Biomath., vol. 30) Berlin Heidelberg New York: Springer 1979

    Google Scholar 

  10. Frenzen, C., Murray, J.: A cell kinetics justification for Gompertz' equation. SIAM J. Appl. Math. 46, 614–629 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  11. Greiner, G., Nagel, R.: Growth of cell populations via one-parameter semigroups of positive operators, pp. 79–105. In: Mathematics Applied to Science. New York: Academic Press 1985

    Google Scholar 

  12. Gyllenberg, M.: The size and scar distributions of the yeast saccharomyces cerevisiae. J. Math. Biol. 24, 81–101 (1986)

    MATH  MathSciNet  Google Scholar 

  13. Gyllenberg, M., Webb, G. F.: Age-size structure in populations with quiescence. Math. Biosci. 86, 67–95 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gyllenberg, M., Webb, G. F.: Quiescence as an explanation of Gompertzian tumor growth. Growth, Development, and Aging 53, 25–33 (1989)

    Google Scholar 

  15. Kimmel, M., Darzynkiewicz, Z., Arino, O., Traganos, F.: Analysis of a model of cell cycle based on unequal division of mitotic constituents to daughter cells during cytokinesis. J. Theor. Biol. 101, 637–664 (1984)

    Google Scholar 

  16. Lasota, A., Mackey, M. C.: Globally asymptotic properties of proliferating cell populations. J. Math. Biol. 19, 43–62 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  17. Martin, R. H.: Nonlinear operators and differential equations in Banach spaces. New York: Wiley 1976

    MATH  Google Scholar 

  18. Martinez, A. O., Griego, R. J.: Growth dynamics of multicell spheroids from three murine tumors. Growth 44, 112–122 (1980)

    Google Scholar 

  19. Metz, J. A. J., Diekmann, O.: The dynamics of physiologically structured populations. (Lect. Notes Biomath, vol. 68) Berlin Heidelberg New York Tokyo: Springer 1986

    Google Scholar 

  20. Nagel, R.: Well-posedness and positivity for systems of linear evolution equations. Conferenze del Siminario di Matematica Bara 203, 1–29 (1985)

    Google Scholar 

  21. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Berlin Heidelberg New York: Springer 1983

    MATH  Google Scholar 

  22. Rotenberg, M.: Correlations, asymptotic stability and the G 0 theory of the cell cycle. In: Valleron, A.-J., Macdonald, P. D. M. (eds.) Biomathematics and cell kinetics, development in cell Biology, vol. 2, pp. 59–69. Amsterdam New York: Elsevier/North-Holland Biomedical Press 1978

    Google Scholar 

  23. Rotenberg, M.: Theory of distributed quiescent state in the cell cycle. J. Theor. Biol. 96, 495–509 (1982)

    Article  MathSciNet  Google Scholar 

  24. Rubinow, S.: Age-structured equations in the theory of cell populations. In: Levin, S. A. (ed.) Studies in mathematical biology, vol. 16, part II. Populations and Communities, pp. 389–410. The Mathematical Association of America 1978

  25. Sinko, J. W., Streifer, W.: A model for population reproducing by fission. Ecology 52, 330–335 (1971)

    Article  Google Scholar 

  26. Tannock, I. F.: The relation between cell proliferation and the vascular system in a transplanted mouse mammary tumour. Br. J. Cancer 22, 258–273 (1968)

    Google Scholar 

  27. Tsuchiya, H. M., Fredrickson, A. G., Aris, R.: Dynamics of microbial cell populations, Advan. Chem. Engineering 6, 125–198 (1966)

    Article  Google Scholar 

  28. Tubiana, M.: The kinetics of tumour cell proliferation and radiotherapy. Br. J. Radiology 44, 325–347 (1971)

    Article  Google Scholar 

  29. Tucker, S. L., Zimmerman, S. O. A nonlinear model of population dynamics containing an arbitrary number of continuous variables. SIAM J. Appl. Math. 48, 549–591 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  30. Tyson, J., Hannsgen, K.: Cell growth and division: global asymptotic stability of the size distribution in probabilistic models of the cell cycle. J. Math. Biol. 23, 231–246 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  31. van der Mee, C., Zweifel, P.: A Fokker-Planck equation for growing cell populations. J. Math. Biol. 25, 61–72 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  32. Webb, G. F.: Random transition, size control, and inheritance in cell population dynamics. Math. Biosci. 85, 71–91 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  33. Webb, G., Grabosch, A.: Asynchronous exponential growth in transition probability models of the cell cycle. SIAM J. Math. Anal. 18, 897–907 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  34. White, R.: A review of some mathematical models in cell kinetics. In: Rotenbewrg, M. (ed.) Biomathematics of cell kinetics, developments in cell biology, vol. 8, pp. 243–261. Amsterdam New York: Elsevier/North-Holland Biomedical Press 1981

    Google Scholar 

  35. Witten, M.: Modeling cellular systems and aging processes: II. Some thoughts on describing an asynchronously dividing cellular system, pp. 1023–1035. Nonlinear Phenomena in Mathematical Sciences. New York: Academic Press 1982

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Luleå University of Technology, S-95187, Luleå, Sweden

    M. Gyllenberg

  2. Department of Mathematics, Vanderbilt University, 37235, Nashville, TN, USA

    G. F. Webb

Authors
  1. M. Gyllenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. F. Webb
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported in part by the National Science Foundation under Grant No. DMS-8722947

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gyllenberg, M., Webb, G.F. A nonlinear structured population model of tumor growth with quiescence. J. Math. Biol. 28, 671–694 (1990). https://doi.org/10.1007/BF00160231

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00160231

Key words

Search

Navigation