Mathematical Biosciences and Engineering, 2015, 12(6): 1257-1275. doi: 10.3934/mbe.2015.12.1257.

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Dynamics and control of a mathematical model for metronomic chemotherapy

1. Dept. of Mathematics and Statistics, Southern Illinois University, Edwardsville, Il 62025
2. Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo 63130

A $3$-compartment model for metronomic chemotherapy that takes intoaccount cancerous cells, the tumor vasculature and tumorimmune-system interactions is considered as an optimal controlproblem. Metronomic chemo-therapy is the regular, almost continuousadministration of chemotherapeutic agents at low dose, possibly withsmall interruptions to increase the efficacy of the drugs. Thereexists medical evidence that such administrations of specificcytotoxic agents (e.g., cyclophosphamide) have both antiangiogenicand immune stimulatory effects. A mathematical model for angiogenicsignaling formulated by Hahnfeldt et al. is combined with theclassical equations for tumor immune system interactions byStepanova to form a minimally parameterized model to capture theseeffects of low dose chemotherapy. The model exhibits bistablebehavior with the existence of both benign and malignant locallyasymptotically stable equilibrium points. In this paper, thetransfer of states from the malignant into the benign regions isused as a motivation for the construction of an objective functionalthat induces this process and the analysis of the correspondingoptimal control problem is initiated.
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Keywords cancer treatment; angiogenic signaling; Optimal control; tumor immune-system interactions.; metronomic chemotherapy

Citation: Urszula Ledzewicz, Behrooz Amini, Heinz Schättler. Dynamics and control of a mathematical model for metronomic chemotherapy. Mathematical Biosciences and Engineering, 2015, 12(6): 1257-1275. doi: 10.3934/mbe.2015.12.1257

References

  • 1. Nature Reviews Clinical Oncology, 11 (2014), 413-431.
  • 2. Future Oncology, 7 (2011), 385-394.
  • 3. Mathematical Modeling of Natural Phenomena, 7 (2012), 306-336.
  • 4. J. Theoretical Biology, 335 (2013), 235-244.
  • 5. Cancer Research, 62 (2002), 6938-6943.
  • 6. Springer Verlag, Series: Mathematics and Applications, 2003.
  • 7. American Institute of Mathematical Sciences, Springfield, Mo, 2007.
  • 8. Cancer Research, 60 (2000), 1878-1886.
  • 9. Mathematical Biosciences and Engineering-MBE, 10 (2013), 565-578.
  • 10. Springer, New York, 1983.
  • 11. Mathematical Biosciences and Engineering, 2 (2005), 511-525.
  • 12. J. of Biological Systems, 14 (2006), 13-30.
  • 13. Cancer Research, 69 (2009), 4894-4903.
  • 14. J. of Biological Systems, 22 (2014), 199-217.
  • 15. Springer Verlag, New York, 1983.
  • 16. J. of Theoretical Biology, 220 (2003), 545-554.
  • 17. Cancer Research, 59 (1999), 4770-4775.
  • 18. J. Clinical Investigations, 105 (2000), 1045-1047.
  • 19. Cancer Letters, 354 (2014), 220-226.
  • 20. J. Clinical Oncology, 18 (2000), 2935-2937.
  • 21. Bulletin of Mathematical Biology, 72 (2010), 1029-1068.
  • 22. J. Clinical Investigations, 105 (2000), R15-R24.
  • 23. Bulletin of Mathematical Biology, 56 (1994), 295-321.
  • 24. J. of Mathematical Biology, 64 (2012), 557-577.
  • 25. Mathematical Biosciences and Engineering - MBE, 10 (2013), 787-802.
  • 26. J. of Optimization Theory and Applications - JOTA, 114 (2002), 609-637.
  • 27. J. of Biological Systems, 10 (2002), 183-206.
  • 28. Discrete and Continuous Dynamical Systems, Series B, 6 (2006), 129-150.
  • 29. SIAM J. on Control and Optimization, 46 (2007), 1052-1079.
  • 30. Mathematical Modeling of Natural Phenomena, 9 (2014), 131-152.
  • 31. J. of Biological Systems, 22 (2014), 177-197.
  • 32. in Mathematical Oncology 2013 (eds. A. d'Onofrio and A. Gandolfi), Springer, (2014), 295-334.
  • 33. Nature Reviews|Clinical Oncology, 7 (2010), 455-465.
  • 34. Newsletter of the Society for Mathematical Biology, 26 (2013), 9-10.
  • 35. J. of Clinical Oncology, 23 (2005), 939-952.
  • 36. MacMillan, New York, 1964.
  • 37. Bioequivalence and Bioavailability, 3 (2011), p4.
  • 38. Springer Verlag, 2012.
  • 39. J. of Mathematical Biology, published online June 19, 2015.
  • 40. Bulletin of Mathematical Biology, 48 (1986), 253-278.
  • 41. Biophysics, 24 (1980), 917-923.
  • 42. J. Clinical Investigations, 117 (2007), 1137-1146.
  • 43. Mathematical Biosciences, 101 (1990), 237-284.
  • 44. IMACS Ann. Comput. Appl. Math., 5 (1989), 51-53.
  • 45. J. of Biological Systems, 3 (1995), 41-54.
  • 46. Nonlinear Analysis, 47 (2001), 375-386.
  • 47. Int. J. Applied Mathematics and Computer Science, 13 (2003), 357-368.
  • 48. J. of Theoretical Biology, 227 (2004), 335-348.
  • 49. J. of Clinical Oncology, 11 (1993), 820-821.
  • 50. Boston-Philadelphia: Hilger Publishing, 1988.

 

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