Tumor Development Under Combination Treatments with Anti-angiogenic Therapies

  • Urszula Ledzewicz
  • Alberto d’Onofrio
  • Heinz Schättler
Part of the Lecture Notes on Mathematical Modelling in the Life Sciences book series (LMML)


Tumors are a family of high-mortality diseases, each differing from the other, but all exhibiting a derangement of cellular proliferation and characterized by a remarkable lack of symptoms [52] and by time courses that, in a broad sense, may be classified as nonlinear. As a consequence, despite the enormous strides in prevention and, to a certain extent, cure, cancer is one of the leading causes of death worldwide, and, unfortunately, is likely to remain so for many years to come [4, 53].


Optimal Control Problem Optimal Protocol Optimal Synthesis Singular Control Angiogenic Inhibitor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We would like to thank an anonymous referee for his careful reading of our chapter and several suggestions that we incorporated into the final version. The research of A.  d’Onofrio has been done in the framework of the Integrated Project “p-medicine—from data sharing and integration via VPH models to personalized medicine,” project identifier: 270089, which is partially funded by the European Commission under the 7th framework program. The research of U. Ledzewicz and H. Schättler has been partially supported by the National Science Foundation under collaborative research grant DMS 1008209/1008221.


  1. 1.
    Agur, Z., Arakelyan, L., Daugulis, P., Ginosar, Y.: Hopf point analysis for angiogenesis models. Discrete Continuous Dyn. Syst. B 4, 29–38 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Anderson, A.R.A., Chaplain, M.A.: Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull. Math. Biol. 60, 857–899 (1998)zbMATHCrossRefGoogle Scholar
  3. 3.
    Bellomo, N., Maini, P.K.: Introduction to “special issue on cancer modelling’. Math. Models Meth. Appl. Sci. 15, iii (2005)Google Scholar
  4. 4.
    Boyle, P., d’Onofrio, A., Maisonneuve, P., Severi, G., Robertson, C., Tubiana, M., Veronesi, U.: Measuring progress against cancer in Europe: has the 15% decline targeted for 2000 come about? Ann. Oncol. 14, 1312–1325 (2003)Google Scholar
  5. 5.
    Brenner, D.J., Hall, E.J., Huang, Y., Sachs, R.K.: Optimizing the time course of brachytherapy and other accelerated radiotherapeutic protocols. Int. J. Radiat. Oncol. Biol. Phys. 29, 893–901 (1994)CrossRefGoogle Scholar
  6. 6.
    Browder, T., Butterfield, C.E., Kraling, B.M., Shi, B., Marshall, B., O’Reilly, M.S., Folkman, J.: Antiangiogenic scheduling of chemotherapy improves efficacy against experimental drug-resistant cancer. Canc. Res. 60, 1878–86 (2000)Google Scholar
  7. 7.
    Colleoni, M., Rocca, A., Sandri, M.T., Zorzino, L., Masci, G., Nole, F., Peruzzotti, G., Robertson, C., Orlando, L., Cinieri, S., de Braud, F., Viale, G., Goldhirsch, A.: Low-dose oral methotrexate and cyclophosphamide in metastatic breast cancer: antitumour activity and correlation with vascular endothelial growth factor levels. Ann. Oncol. 13, 73–80 (2002)CrossRefGoogle Scholar
  8. 8.
    Ergun, A., Camphausen, K., Wein, L.M., Optimal scheduling of radiotherapy and angiogenic inhibitors, Bull. of Math. Biology, 65, 407–424 (2003)CrossRefGoogle Scholar
  9. 9.
    Folkman, J.: Tumor angiogenesis: Therapeutic implications. New Engl. J. Med. 295, 1182–1196 (1971)Google Scholar
  10. 10.
    Folkman, J.: Antiangiogenesis: New concept for therapy of solid tumors. Ann. Surg. 175, 409–416 (1972)CrossRefGoogle Scholar
  11. 11.
    Folkman, J.: Opinion - Angiogenesis: an organizing principle for drug discovery? Nature Rev. Drug. Disc. 6, 273–286 (2007)CrossRefGoogle Scholar
  12. 12.
    Fowler, J.F.: The linear-quadratic formula and progress in fractionated radiotherapy, Br. J. Radiol. 62, 679–694 (1989)CrossRefGoogle Scholar
  13. 13.
    Frame, D.: New strategies in controlling drug resistance. J. Manag. Care Pharm. 13, 13–17 (2007)Google Scholar
  14. 14.
    Goldie, J.H., Coldman, A.J.: A mathematic model for relating the drug sensitivity of tumors to their spontaneous mutation rate. Canc. Treat. Rep. 63, 1727–1733 (1979)Google Scholar
  15. 15.
    Guiot, C., Degiorgis, P.G., Delsanto, P.P., Gabriele, P., Deisboecke, T.S.: Does tumor growth follow a ’universal law’? J. Theor. Biol. 225, 147–151 (2003)CrossRefGoogle Scholar
  16. 16.
    Hahnfeldt, P., Panigrahy, D., Folkman, J., Hlatky, L.: Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy. Canc. Res. 59, 4770–4775 (1999)Google Scholar
  17. 17.
    Hart, D., Shochat, E., Agur, Z.: The growth law of primary breast cancer as inferred from mammography screening trials data. Br. J. Canc. 78, 382–387 (1999)CrossRefGoogle Scholar
  18. 18.
    Jain, R.K., Normalizing tumor vasculature with anti-angiogenic therapy: a new paradigm for combination therapy. Nat. Med. 7, 987–989 (2001)CrossRefGoogle Scholar
  19. 19.
    Jain, R.K., Munn, L.L.: Vascular normalization as a rationale for combining chemotherapy with antiangiogenic agents. Princ. Practical Oncol. 21, 1–7 (2007)Google Scholar
  20. 20.
    Kerbel, R.S.: A cancer therapy resistant to resistance. Nature 390, 335–336 (1997)CrossRefGoogle Scholar
  21. 21.
    Kerbel, R.S.: Tumor angiogenesis: Past, present and near future. Carcinogensis 21, 505–515 (2000)CrossRefGoogle Scholar
  22. 22.
    Kerbel, R., Folkman, J.: Clinical translation of angiogenesis inhibitors. Nat. Rev. Canc. 2, 727–739 (2002)CrossRefGoogle Scholar
  23. 23.
    Kimmel, M., Swierniak, A.: Control theory approach to cancer chemotherapy: benefiting from phase dependence and overcoming drug resistance. In: Tutorials in Mathematical Bioscences III: Cell Cycle, Proliferation, and Cancer, Lecture Notes in Mathematics, vol. 1872, pp. 185–221. Springer, Berlin (2006)Google Scholar
  24. 24.
    Ledzewicz, U., Schättler, H.: Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy. J. Optim. Theor. Appl. - JOTA 114, 609–637 (2002)zbMATHCrossRefGoogle Scholar
  25. 25.
    Ledzewicz, U., Schättler, H.: Analysis of a cell-cycle specific model for cancer chemotherapy, J. Biol. Syst. 10, 183–206 (2002)zbMATHCrossRefGoogle Scholar
  26. 26.
    Ledzewicz, U., Schättler, H.: Anti-angiogenic therapy in cancer treatment as an optimal control problem, SIAM J. Contr. Optim. 46(3), 1052–1079 (2007)zbMATHCrossRefGoogle Scholar
  27. 27.
    Ledzewicz, U., Schättler, H.: Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis. J. Theor. Biol. 252, 295–312 (2008)CrossRefGoogle Scholar
  28. 28.
    Ledzewicz, U., Schättler, H.: Singular controls and chattering arcs in optimal control problems arising in biomedicine. Contr. Cybern. 38, 1501–1523 (2009)zbMATHGoogle Scholar
  29. 29.
    Ledzewicz, U., Schättler, H.: Multi-input optimal control problems for combined tumor anti-angiogenic and radiotherapy treatments. J. Optim. Theor. Appl. - JOTA 153, 195–224 (2012), doi:10.1007/s10957-011-9954-8, published online: 16 November 2011zbMATHCrossRefGoogle Scholar
  30. 30.
    Ledzewicz, U., Munden, J., Schättler, H.: Scheduling of anti-angiogenic inhibitors for Gompertzian and logistic tumor growth models. Discrete Continuous Dyn. Syst. Ser. B 12, 415–439 (2009)zbMATHCrossRefGoogle Scholar
  31. 31.
    Ledzewicz, U., Marriott, J., Maurer, H., Schättler, H.: Realizable protocols for optimal administration of drugs in mathematical models for novel cancer treatments. Math. Med. Biol. 27, 157–179 (2010), doi:10.1093/imammb/dqp012MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Ledzewicz, U., Maurer, H., Schättler, H.: Minimizing tumor volume for a mathematical model of anti-angiogenesis with linear pharmacokinetics. In: Diehl, M., Glineur, F., Jarlebring, E., Michiels, W. (eds.) Recent Advances in Optimization and its Applications in Engineering, pp. 267–276. Springer, Berlin (2010)CrossRefGoogle Scholar
  33. 33.
    Marusic, M., Bajzer, A., Freyer, J.P., Vuk-Povlovic, S.: Analysis of growth of multicellular tumor spheroids by mathematical models. Cell Prolif. 27, 73ff (1994)Google Scholar
  34. 34.
    McDougall, S.R., Anderson, A.R.A., Chaplain, M.A.: Mathematical modelling of dynamic adaptive tumour-induced angiogenesis: Clinical implications and therapeutic targeting strategies. J. Theor. Biol. 241, 564–589 (2006)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Mombach, J.C.M., Lemke, N., Bodmann, B.E.J., Idiart, M.A.P.: A mean-field theory of cellular growth. Europhys. Lett. 59, 923–928 (2002)CrossRefGoogle Scholar
  36. 36.
    Norton, L.: A Gompertzian model of human breast cancer growth. Canc. Res. 48, 7067–7071 (1988)Google Scholar
  37. 37.
    Norton, L., Simon, R.: Growth curve of an experimental solid tumor following radiotherapy. J.  Nat. Canc. Inst. 58, 1735–1741 (1977)Google Scholar
  38. 38.
    Norton, L., Simon R.: The Norton-Simon hypothesis revisited. Canc. Treat. Rep. 70, 163–169 (1986)Google Scholar
  39. 39.
    d’Onofrio, A.: A general framework for modelling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedial inferences. Physica D 208, 202–235 (2005)Google Scholar
  40. 40.
    d’Onofrio, A.: Rapidly acting antitumoral antiangiogenic therapies. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76, 031920 (2007)Google Scholar
  41. 41.
    d’Onofrio, A.: Fractal growth of tumors and other cellular populations: Linking the mechanistic to the phenomenological modeling and vice versa. Chaos, Solitons and Fractals 41, 875–880 (2009)Google Scholar
  42. 42.
    d’Onofrio, A., Gandolfi, A.: Tumour eradication by antiangiogenic therapy: Analysis and extensions of the model by Hahnfeldt et al. Math. Biosci. 191, 159–184 (2004)Google Scholar
  43. 43.
    d’Onofrio, A., Gandolfi, A.: The response to antiangiogenic anticancer drugs that inhibit endothelial cell proliferation, Appl. Math. and Comp. 181, 1155–1162 (2006)Google Scholar
  44. 44.
    d’Onofrio, A., Gandolfi, A.: A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy. Math. Med. Biol. 26, 63–95 (2009)Google Scholar
  45. 45.
    d’Onofrio, A., Gandolfi, A.: Chemotherapy of vascularised tumours: role of vessel density and the effect of vascular “pruning”. J. Theor. Biol. 264, 253–265 (2010)Google Scholar
  46. 46.
    d’Onofrio, A., Gandolfi, A.: Resistance to anti-tumor chemotherapy due to bounded-noise transitions. Phys. Rev. E. 82, (2010) Art.n. 061901, doi:10.1103/PhysRevE.82.061901Google Scholar
  47. 47.
    d’Onofrio, A., Gandolfi, A., Rocca, A.: The dynamics of tumour-vasculature interaction suggests low-dose, time-dense antiangiogenic schedulings. Cell Prolif. 42, 317–329 (2009)Google Scholar
  48. 48.
    d’Onofrio, A., Ledzewicz, U., Maurer, H., Schättler, H.: On optimal delivery of combination therapy for tumors. Math. Biosci. 222, 13–26 (2009), doi:10.1016/j.mbs.2009.08.004Google Scholar
  49. 49.
    d’Onofrio, A., Fasano, A., Monechi, B.: A generalization of Gompertz law compatible with the Gyllenberg-Webb theory for tumour growth Math. Biosci. 230(1), 45–54 (2011)Google Scholar
  50. 50.
    O’Reilly, M.S., Boehm, T., Shing, Y., Fukai, N., Vasios, G., Lane, W.S., Flynn, E., Birkhead, J.R., Olsen, B.R., Folkman, J.: Endostatin: An endogenous inhibitor of angiogenesis and tumour growth. Cell 88, 277–285 (1997)CrossRefGoogle Scholar
  51. 51.
    Panetta, J.C.: A mathematical model of breast and ovarian cancer treated with paclitaxel. Math. Biosci. 146, 89–113 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Peckham, M., Pinedo, H.M., Veronesi, U.: Oxford Textbook of Oncology. Oxford Medical Publications, New York (1995)Google Scholar
  53. 53.
    Quinn, M.J., d’Onofrio, A., Moeller, B., Black, R., Martinez-Garcia, C., Moeller, H., Rahu, M., Robertson, C., Schouten, L.J., La Vecchia, C., Boyle, P.: Cancer mortality trends in the EU and acceding countries up to 2001. Ann. Oncol. 14, 1148–1152 (2003)Google Scholar
  54. 54.
    Ribba, B., Marron, K., Agur, Z., Alarcïn, T., Maini, P.K.: A mathematical model of Doxorubicin treatment efficacy for non-Hodgkin’s lymphoma: Investigation of the current protocol through theoretical modelling results. Bull. Math. Biol. 67, 79–99 (2005)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Schättler, H., Jankovic, M.: A synthesis of time-optimal controls in the presence of saturated singular arcs. Forum Math. 5, 203–241 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Schättler, H., Ledzewicz, U., Cardwell, B.: Robustness of optimal controls for a class of mathematical models for tumor anti-angiogenesis. Math. Biosci. Eng. 8, 355–369 (2011), doi:10.3934/mbe.2011.8.355MathSciNetCrossRefGoogle Scholar
  57. 57.
    Sole, R.V.: Phase transitions in unstable cancer cell populations. Eur. J. Phys. B 35, 117–124 (2003)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Skipper, H.E.: On mathematical modeling of critical variables in cancer treatment (goals: Better understanding of the past and better planning in the future). Bull. Math. Biol. 48, 253–278 (1986)MathSciNetGoogle Scholar
  59. 59.
    Swan, G.W.: Role of optimal control in cancer chemotherapy. Math. Biosci. 101, 237–284 (1990)zbMATHCrossRefGoogle Scholar
  60. 60.
    Swierniak, A.: Optimal treatment protocols in leukemia - modelling the proliferation cycle. In: Proceedings of 12th IMACS World Congress, vol. 4, pp. 170–172, Paris (1988)Google Scholar
  61. 61.
    Swierniak, A.: Cell cycle as an object of control. J. Biol. Syst. 3, 41–54 (1995)CrossRefGoogle Scholar
  62. 62.
    Swierniak, A.: Direct and indirect control of cancer populations. Bull. Pol. Acad. Sci. 56, 367–378 (2008)Google Scholar
  63. 63.
    Swierniak, A.: Comparison of six models of antiangiogenic therapy. Appl. Math. 36, 333–348 (2009)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Swierniak, A., Kimmel, M., Smieja, J.: Mathematical modeling as a tool for planning anti-cancer therapy, Eur. J. Pharmacol. 625, 108–121 (2009)CrossRefGoogle Scholar
  65. 65.
    Swierniak, A., Ledzewicz, U., Schättler, H.: Optimal control for a class of compartmental models in cancer chemotherapy. Int. J. Appl. Math. Comput. Sci. 13, 357–368 (2003)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Thames, H.D., Hendry, J.H.: Fractionation in Radiotherapy. Taylor and Francis, London (1987)Google Scholar
  67. 67.
    Ubezio, P., Cameron, D.: Cell killing and resistance in pre-operative breast cancer chemotherapy. BMC Canc. 8, 201ff (2008)Google Scholar
  68. 68.
    Wheldon, T.E.: Mathematical Models in Cancer Research. Hilger Publishing, Boston-Philadelphia (1988)zbMATHGoogle Scholar
  69. 69.
    Wijeratne, N.S., Hoo, K.A.: Understanding the role of the tumour vasculature in the transport of drugs to solid cancer tumors. Cell Prolif. 40, 283–301 (2007)CrossRefGoogle Scholar
  70. 70.
    Yu, J.L., Rak, J.W.: Host microenvironment in breast cancer development: Inflammatory and immune cells in tumour angiogenesis and arteriogenesis. Breast Canc. Res. 5, 83–88 (2003)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Urszula Ledzewicz
    • 1
  • Alberto d’Onofrio
    • 2
  • Heinz Schättler
    • 3
  1. 1.Department of Mathematics and StatisticsSouthern Illinois University EdwardsvilleEdwardsvilleUSA
  2. 2.Department of Experimental OncologyEuropean Institute of OncologyMilanItaly
  3. 3.Department of Electrical and Systems EngineeringWashington UniversitySt. LouisUSA

Personalised recommendations