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A 3-Compartment Model for Chemotherapy of Heterogeneous Tumor Populations

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Abstract

We consider a mathematical model for cancer chemotherapy with a single agent that distinguishes three levels of sensitivities calling the subpopulations ‘sensitive’, ‘partially sensitive’ and ‘resistant’. We analyze the dynamic properties of the system under what could be considered metronomic (continuous, low-dose, constant) chemotherapy and, more generally, also consider the optimal control problem of minimizing the tumor burden over a prescribed therapy interval. Interestingly, when several levels of chemotherapeutic sensitivities are taken into account in the model, lower time-varying dose rates as they are given by singular controls become a treatment option. This is only the case once a significant residuum of resistant cells has been created in a simpler 2-compartment model that only considers sensitive and resistant cells. For heterogeneous tumor populations, a more modulated approach that varies the dose rates of the drugs may be more beneficial than the classical maximum tolerated dose approach pursued in medical practice.

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References

  1. André, N., Padovani, L., Pasquier, E.: Metronomic scheduling of anticancer treatment: the next generation of multitarget therapy? Future Oncol. 7(3), 385–394 (2011)

    Article  Google Scholar 

  2. Bocci, G., Nicolaou, K., Kerbel, R.S.: Protracted low-dose effects on human endothelial cell proliferation and survival in vitro reveal a selective antiangiogenic window for various chemotherapeutic drugs. Cancer Res. 62, 6938–6943 (2002)

    Google Scholar 

  3. Bonnard, B., Chyba, M.: Singular Trajectories and Their Role in Control Theory. Mathématiques & Applications, vol. 40. Springer, Paris (2003)

    MATH  Google Scholar 

  4. Bressan, A., Piccoli, B.: Introduction to the Mathematical Theory of Control. American Institute of Mathematical Sciences, Springfield (2007)

    MATH  Google Scholar 

  5. Eisen, M.: Mathematical Models in Cell Biology and Cancer Chemotherapy. Lecture Notes in Biomathematics, vol. 30. Springer, Berlin (1979)

    MATH  Google Scholar 

  6. Friedman, A.: Cancer as multifaceted disease. Math. Model. Nat. Phenom. 7, 1–26 (2012)

    Article  Google Scholar 

  7. Goldie, J.H.: Drug resistance in cancer: a perspective. Cancer Metastasis Rev. 20, 63–68 (2001)

    Article  Google Scholar 

  8. Goldie, J.H., Coldman, A.: Drug Resistance in Cancer. Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  9. Greene, J., Lavi, O., Gottesman, M., Levy, D.: The impact of cell density and mutations in a model of multidrug resistance in solid tumors. Bull. Math. Biol. (2014). doi:10.1007/s11538-014-9936-8

    MathSciNet  Google Scholar 

  10. Hahnfeldt, P., Hlatky, L.: Cell resensitization during protracted dosing of heterogeneous cell populations. Radiat. Res. 150, 681–687 (1998)

    Article  Google Scholar 

  11. Hahnfeldt, P., Folkman, J., Hlatky, L.: Minimizing long-term burden: the logic for metronomic chemotherapy dosing and its angiogenic basis. J. Theor. Biol. 220, 545–554 (2003)

    Article  Google Scholar 

  12. Hanahan, D., Bergers, G., Bergsland, E.: Less is more, regularly: metronomic dosing of cytotoxic drugs can target tumor angiogenesis in mice. J. Clin. Invest. 105(8), 1045–1047 (2000)

    Article  Google Scholar 

  13. Kamen, B., Rubin, E., Aisner, J., Glatstein, E.: High-time chemotherapy or high time for low dose? J. Clin. Oncol. 18(16), 2935–2937 (2000), Editorial

    Google Scholar 

  14. Klement, G., Baruchel, S., Rak, J., Man, S., Clark, K., Hicklin, D.J., Bohlen, P., Kerbel, R.S.: Continuous low-dose therapy with vinblastine and VEGF receptor-2 antibody induces sustained tumor regression without overt toxicity. J. Clin. Invest. 105(8), R15–R24 (2000)

    Article  Google Scholar 

  15. Lavi, O., Greene, J.M., Levy, D., Gottesman, M.M.: The role of cell density and intratumoral heterogeneity in multidrug resistance. Cancer Res. 73(24), 7168–7175 (2013)

    Article  Google Scholar 

  16. Ledzewicz, U., Schättler, H.: Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy. J. Optim. Theory Appl. 114, 609–637 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ledzewicz, U., Schättler, H.: Drug resistance in cancer chemotherapy as an optimal control problem. Discrete Contin. Dyn. Syst., Ser. B 6, 129–150 (2006)

    MATH  MathSciNet  Google Scholar 

  18. Ledzewicz, U., Schättler, H., Reisi Gahrooi, M., Mahmoudian Dehkordi, S.: On the MTD paradigm and optimal control for combination cancer chemotherapy. Math. Biosci. Eng. 10(3), 803–819 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  19. Loeb, L.A.: A mutator phenotype in cancer. Cancer Res. 61, 3230–3239 (2001)

    Google Scholar 

  20. Martin, R., Teo, K.L.: Optimal Control of Drug Administration in Cancer Chemotherapy. World Scientific, Singapore (1994)

    MATH  Google Scholar 

  21. Norton, L., Simon, R.: Tumor size, sensitivity to therapy, and design of treatment schedules. Cancer Treat. Rep. 61, 1307–1317 (1977)

    Google Scholar 

  22. Norton, L., Simon, R.: The Norton–Simon hypothesis revisited. Cancer Treat. Rep. 70, 41–61 (1986)

    Google Scholar 

  23. Pasquier, E., Kavallaris, M., André, N.: Metronomic chemotherapy: new rationale for new directions. Nat. Rev. Clin. Oncol. 7, 455–465 (2010)

    Article  Google Scholar 

  24. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. MacMillan, New York (1964)

    MATH  Google Scholar 

  25. Schättler, H., Ledzewicz, U.: Geometric Optimal Control. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  26. Schimke, R.T.: Gene amplification, drug resistance and cancer. Cancer Res. 44, 1735–1742 (1984)

    Google Scholar 

  27. Swan, G.W.: Role of optimal control in cancer chemotherapy. Math. Biosci. 101, 237–284 (1990)

    Article  MATH  Google Scholar 

  28. Swierniak, A.: Optimal treatment protocols in leukemia—modelling the proliferation cycle. In: Proc. of the 12th IMACS, vol. 4, pp. 170–172. World Congress, Paris (1988)

    Google Scholar 

  29. Swierniak, A.: Cell cycle as an object of control. J. Biol. Syst. 3, 41–54 (1995)

    Article  Google Scholar 

  30. Swierniak, A., Smieja, J.: Cancer chemotherapy optimization under evolving drug resistance. Nonlinear Anal. 47, 375–386 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

This material is based upon work supported by the National Science Foundation under collaborative research Grants Nos. DMS 1311729/1311733. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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Authors and Affiliations

  1. Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL, 62026-1653, USA

    Urszula Ledzewicz & Kenneth Bratton

  2. Dept. of Electrical and Systems Engineering, Washington University, St. Louis, MO, 63130-4899, USA

    Heinz Schättler

Authors
  1. Urszula Ledzewicz
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  2. Kenneth Bratton
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  3. Heinz Schättler
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Corresponding author

Correspondence to Urszula Ledzewicz.

Additional information

This material is based upon work supported by the National Science Foundation under collaborative research Grants Nos. DMS 1311729/1311733. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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Ledzewicz, U., Bratton, K. & Schättler, H. A 3-Compartment Model for Chemotherapy of Heterogeneous Tumor Populations. Acta Appl Math 135, 191–207 (2015). https://doi.org/10.1007/s10440-014-9952-6

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  • DOI: https://doi.org/10.1007/s10440-014-9952-6

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