Skip to main content
Log in

A mathematical model of periodically pulsed chemotherapy: Tumor recurrence and metastasis in a competitive environment

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

Abstract

A competition model describing tumor-normal cell interaction with the added effects of periodically pulsed chemotherapy is discussed. The model describes parameter conditions needed to prevent relapse following attempts to remove the tumor or tumor metastasis. The effects of resistant tumor subpopulations are also investigated and recurrence prevention strategies are explored.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Agur, Z., R. Arnon and B. Schechter. 1988. Reduction of cytotoxicity to normal tissues by new regimens of cell-cycle phase-specific drugs.Math. Biosci. 92, 1–15.

    Article  MATH  Google Scholar 

  • Aroesty, J., T. Lincoln, N. Shapiro and G. Boccia. 1973. Tumor growth and chemotherapy: Mathematical methods, computer simulations, and experimental foundations.Math. Biosci. 17, 243–300.

    Article  MATH  Google Scholar 

  • Bellomo, N. and G. Forni. 1994. Dynamics of tumor interaction with the host immune system.Math. Comput. Modelling 20, 107–122.

    Article  MATH  Google Scholar 

  • Berenbaum, M. C. 1969. Dose-response curves for agents that impair cell reproductive integrity.Br. J. Cancer 23, 434–445.

    Google Scholar 

  • Birkhead, B. G. and W. M. Gregory, 1984. A mathematical model of the effects of drug resistance in cancer chemotherapy.Math. Biosci. 72, 59–69.

    Article  MATH  Google Scholar 

  • Burger, R. A., E. A. Grosen, G. R. Ioli, M. E. Van Eden, H. D. Brightbill, M. Gatanaga, P. J. DiSaia, G. A. Granger and T. Gatanaga. 1994. Host-tumor interaction in ovarian cancer spontaneous release of tumor necrosis factor and interleukin-1 inhibitors by purified cell populations from human ovarian carcinoma in vitro.Gynecologic Oncology 55, 294–303.

    Article  Google Scholar 

  • Cojocaru, L. and Z. Agur. 1992. A theoretical analysis of interval drug dosing for cell-cycle-phase-specific drugs.Math. Biosci. 109, 85–97.

    Article  MATH  Google Scholar 

  • Cornil, I., D. Theodorescu, S. Man, M. Herlyn, J. Jambrosic and R. S. Kerbel. 1991. Fibroblast cell interactions with human melanoma cells affect tumor cell growth as a function of tumor progression.Proc. Natl. Acad. Sci. USA 88, 6028–6032.

    Article  Google Scholar 

  • Dotto, G. P., A. Weinberg and A. Ariza. 1988. Malignant transformation of mouse primary keratinocytes by Harvey sarcoma virus and its modulation by surrounding normal cells.Proc. Natl. Acad. Sci. USA 85, 6389–6393.

    Article  Google Scholar 

  • Eisen, M. 1979.Mathematical Models in Cell Biology and Cancer Chemotherapy. Lecture Notes in Biomathematics, Vol. 30. New York: Springer-Verlag.

    Google Scholar 

  • Fisher, B. and E. R. Fisher. 1959. Experimental studies of factors influencing hepatic matastases.Cancer 12, 929–932.

    Article  Google Scholar 

  • Gatenby, R. A. 1991. Population ecology issues in tumor growth.Cancer Res. 51, 2542–2547.

    Google Scholar 

  • Gatenby, R. A. 1994. Population ecology models of neoplastic growth: Implications for tumor biology and treatment. Private communication with Dr. John Adam.

  • Goldie, J. H. and A. J. Coldman. 1979. A mathematical model for relating the drug sensitivity of tumors to their spontaneous mutation rate.Cancer Treat. Rep. 63, 1727–1733.

    Google Scholar 

  • Gyori, I., S. Michelson and J. Leith. 1988. Time-dependent subpopulation induction in heterogeneous tumors.Bull. Math. Biol. 50, 681–696.

    Article  MATH  MathSciNet  Google Scholar 

  • Jansson, B. and L. Révész. 1977. Cell ecology: Deductive and dynamic models for proliferation, differentiation and competition of tumor cell populations.J. Theoret. Biol. 68, 43–51.

    Article  Google Scholar 

  • Knolle, H. 1988.Cell Kinetic Modelling and the Chemotherapy of Cancer. Lecture Notes in Biomathematics, Vol. 75. New York: Springer-Verlag.

    Google Scholar 

  • Kot, M. and E. Funasaki. 1993. Invasion and chaos in a periodically pulsed mass-action chemostat.Theoret. Population Biol. 44, 203–224, 1992.

    Article  MATH  Google Scholar 

  • La Rocca, S. A., M. Grossi, G. Falcone, S. Alemà and F. Tatò. 1989. Interaction with normal cells suppresses the transformed phenotype of v-myc-transformed quail muscle cells.Cell 58, 123–131.

    Article  Google Scholar 

  • Leith, J. T., G. Padfield and S. Michelson. 1992. Effects of partial hepatectomy on the growth characteristics and hypoxic fractions of xenografted DLD-2 human colon cancers.Rad. Res. 132, 263–268.

    Google Scholar 

  • Liotta, A. L. 1992. Cancer cell invasion and metastasis.Scientific American February, 54–63.

    Article  Google Scholar 

  • Martin, R. B., M. E. Fisher, R. F. Michin and K. L. Teo. 1992a. Low-intensity combination chemotherapy maximizes host survival time for tumors containing drug-resistant cells.Math. Biosci. 110, 221–252.

    Article  MATH  Google Scholar 

  • Martin, R. B., M. E. Fisher, R. F. Michin and K. L. Teo. 1992b. Optimal control of tumor size used to maximize survival time when cells are resistant to chemotherapy.Math. Biosci. 110, 201–219.

    Article  MATH  Google Scholar 

  • Michelson, S. and J. T. Leith. 1988. Unexpected equilibria resulting from differing growth rates of subpopulations within heterogeneous tumors.Math. Biosci. 91, 119–129.

    Article  MATH  MathSciNet  Google Scholar 

  • Michelson, S. and J. T. Leith. 1991. Autocrine and paracrine growth factors in tumor growth: A mathematical model.Bull. Math. Biol. 53, 639–656.

    Google Scholar 

  • Michelson, S. and J. T. Leith. 1993a. Growth factors and growth control of heterogeneous cell populations.Bull. Math. Biol. 55, 993–1011.

    Article  MATH  Google Scholar 

  • Michelson, S. and J. T. Leith. 1993b. Tumor heterogeneity: A review of the theory.Drug News & Perspectives 6, 655–661.

    Google Scholar 

  • Michelson, S. and J. T. Leith. 1995. Interlocking triads of growth control in tumors.Bull. Math. Biol. 57, 345–366.

    Article  MATH  Google Scholar 

  • Michelson, S., B. E. Miller, A. S. Glicksman and J. T. Leith. 1987. Tumor micro-ecology and competitive interactions.J. Theoret. Biol. 128, 233–246.

    Article  MathSciNet  Google Scholar 

  • Miller, F. R., D. Medina and G. H. Heppner. 1981. Preferential growth of mammary tumors in intact mammary fatpads.Cancer Res. 41, 3863–3867.

    Google Scholar 

  • Murray, J. M. 1990. Some optimal control problems in cancer chemotherapy with a toxicity limit.Math. Biosci. 100, 49–67.

    Article  MATH  MathSciNet  Google Scholar 

  • Panetta, J. C. and J. A. Adam. 1995. A mathematical model of chemotherapy: Cycle-specific therapy.Math. Comput. Modelling.22 (2), 67–82.

    Article  MATH  MathSciNet  Google Scholar 

  • Paschkis, K. E., A. Cantarow, J. Stasney and J. H. Hobbs. 1955. Tumor growth in partially hepatectomized rats.Cancer Res. 15, 579–582.

    Google Scholar 

  • Swan, G. W. 1981.Optimization of Human Cancer Radiotherapy. Lecture Notes in Biomathematics, Vol. 42. New York: Springer-Verlag.

    Google Scholar 

  • Waltman, P. 1983.Competition Models in Population Biology, Vol. 45. Society for Industrial and Applied Mathematics, Philadelphia, PA.

    Google Scholar 

  • Webb, F. G. 1992a. A cell population model of periodic chemotherapy treatment.Biomedical Modeling and Simulation, pp. 83–92. New York: Elsevier Science Publishers.

    Google Scholar 

  • Webb, G. F. 1992b. A nonlinear cell population model of periodic chemotherapy treatment.Recent Trends in Ordinary Differential Equations. Series in Applicable analysis, Vol. 1, pp. 569–583. Singapore: World Scientific Publishing Company.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Panetta, J.C. A mathematical model of periodically pulsed chemotherapy: Tumor recurrence and metastasis in a competitive environment. Bltn Mathcal Biology 58, 425–447 (1996). https://doi.org/10.1007/BF02460591

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02460591

Keywords

Navigation