Skip to main content
SpringerLink
Log in

Analysis for a free boundary problem modeling tumor therapy

  • Published:
Applied Mathematics-A Journal of Chinese Universities Aims and scope Submit manuscript

Abstract

This paper is devoted to studying a free boundary problem modeling the effects of drug resistance and vasculature on the response of solid tumors to therapy. The model consists of a system of partial differential equations governing intra-tumoral drug concentration and cancer cell density. By applying the L p theory of parabolic equations and the Banach fixed point theorem, it is proved that this problem has a unique global classical solution.

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. Baxter L, Yuan F, Jain R. Pharmacokinetic analysis of the perivascular distribution of bifunctional antiboies and haptens: comparison with experimental data, Cancer Res, 1992, 52:5838–5864.

    Google Scholar 

  2. Byrne H, Chaplain M. Growth of necrotic tumors in the presence and absence of inhibitors, Math Biosci, 1996, 135: 187–216.

    Article  MATH  Google Scholar 

  3. Byrne H, Chaplain M. Free boundary value problems associated with the growth and development of multicellular spheroids. Euro J Appl Math, 1997, 8: 639–658.

    Article  MATH  MathSciNet  Google Scholar 

  4. Cui S. Analysis of a mathematical model for the growth of tumors under the action of external inhibitors, J Math Biol, 2002, 44: 395–426.

    Article  MATH  MathSciNet  Google Scholar 

  5. Cui S. Global existence of solutions for a free boundary problem modelling the growth of necrotic tumors, Interfaces Free Bound, 2005, 7(2): 147–159.

    MATH  MathSciNet  Google Scholar 

  6. Cui S. Analysis of a free boundary problem modelling tumor growth, Acta Math Sinica (English Series), 2005, 21 (5): 1071–1082.

    Article  MATH  Google Scholar 

  7. Cui S. Existence of a stationary solution for the modified Wark-King tumor growth model, Adv Appl Math, to appear.

  8. Cui S, Friedman A. Analysis of a mathematical model of the effect of inhibitors on the growth of tumors, Math. Biosci, 2000, 164: 103–137.

    Article  MATH  MathSciNet  Google Scholar 

  9. Cui S, Friedman A. Analysis of a mathematical model of the growth of necrotic tumors, J Math Anal Appl, 2001, 255: 636–677.

    Article  MATH  MathSciNet  Google Scholar 

  10. Cui S, Friedman A. A free boundary problem for a singular system of differential equations: an application to a model of tumor growth, Trans Amer Math Soc, 2002, 355: 3537–3590.

    Article  MathSciNet  Google Scholar 

  11. Cui S, Friedman A. A hyperbolic free boundary problem modelling tumor growth, Interfaces Free Bound, 2003, 5: 159–181

    Article  MATH  MathSciNet  Google Scholar 

  12. Cui S, Wei X. Global existence for parabolic-hyperbolic free boundary problem modelling tumor growth, Acta Math Appl Sinica (English Serious), 2005, 21 (4): 597–614.

    Article  MATH  MathSciNet  Google Scholar 

  13. Friedman A, Reitich F. Analysis of a mathematical model for the growth of tumors, J Math Biol, 1999, 38: 262–284.

    Article  MATH  MathSciNet  Google Scholar 

  14. Friedman A, Reitich F. Symmetry-breaking bifurcation of analytic solutions to free boundary problems, Trans Amer Math Soc, 2000, 353: 1587–1634.

    Article  MathSciNet  Google Scholar 

  15. Friedman A, Reitech F. On the existence of spatially patterned dormant malignancies in the model for the growth of non-necrotic vascular tumor, Math Models Appl Sci, 2000, 11: 601–625.

    Article  Google Scholar 

  16. Greenspan H. Models for the growth of solid tumors by diffusion, Stud Appl Math, 1972, 51: 317–340.

    MATH  Google Scholar 

  17. Greenspan H. On the growth and stability of cell cultures and solid tumors, J Theor Biol, 1976, 56: 229–242.

    Article  MathSciNet  Google Scholar 

  18. Jackson T, Byrne H. A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy, Math Biosci, 2000, 164: 17–38.

    Article  MATH  MathSciNet  Google Scholar 

  19. Jain R. Transport of macromolecules across tumor vasculature. Cancer Metastasis Rev, 1987, 6: 559–561.

    Article  Google Scholar 

  20. Kerr D, Schreiber G, Vrudhula V, et al. Regressions and cures of melanoma xenografts following treatment with monoclonal antibody β-lactamase conjugates in combination with anticancer prodrugs, Cancer Res, 1995, 55: 3558–3563.

    Google Scholar 

  21. McElwain D, Pettet G. Cell migration in multicell spheroids: swimming against the tide, Bull Math Biol, 1993, 55: 655–674.

    MATH  Google Scholar 

  22. Pettet G, Please C, McElwaim M. The migration of cells in multicell tumor spheroids, Bull Math Biol, 2001, 63: 231–257.

    Article  Google Scholar 

  23. Sherrat J, Chaplain M. A new mathematical model for vascular tumor growth, J Math Biol, 2001, 43: 291–312.

    Article  MathSciNet  Google Scholar 

  24. Thompson K, Byrne H. Modelling the internalization of labeled cells in tumor spheroids. Bull Math Biol, 1999, 61: 601–623.

    Article  Google Scholar 

  25. Ward J, King J. Mathematical modelling of vascular tumor growth II, IMA J Math Appl Biol Med, 1997, 14: 53–75.

    Google Scholar 

  26. Ward J, King, J. Mathematical modelling of vascular tumor growth, IMA J Math Appl Biol Med, 1998, 15: 1–42

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Sun Yat-Sen University, 510275, Guangzhou, China

    Zhou Fujun

Authors
  1. Zhou Fujun
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported by the National Natural Science Foundation of China (10471157).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fujun, Z. Analysis for a free boundary problem modeling tumor therapy. Appl. Math.- J. Chin. Univ. 21, 143–151 (2006). https://doi.org/10.1007/BF02791351

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02791351

MR Subject Classification

Keywords

Search

Navigation