# American Institute of Mathematical Sciences

March  2019, 14(1): 131-147. doi: 10.3934/nhm.2019007

## On the role of tumor heterogeneity for optimal cancer chemotherapy

 1 Institute of Mathematics, Lodz University of Technology, 90-924 Lodz, Poland 2 Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Il, 62026-1653, USA 3 Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo, 63130, USA 4 Department of Mechanical and Aerospace Engineering, University of Texas at Arlington, Arlington, TX, 76010, USA

* Corresponding author: Urszula Ledzewicz

Received  April 2018 Revised  October 2018 Published  January 2019

We review results about the influence tumor heterogeneity has on optimal chemotherapy protocols (relative to timing, dosing and sequencing of the agents) that can be inferred from mathematical models. If a tumor consists of a homogeneous population of chemotherapeutically sensitive cells, then optimal protocols consist of upfront dosing of cytotoxic agents at maximum tolerated doses (MTD) followed by rest periods. This structure agrees with the MTD paradigm in medical practice where drug holidays limit the overall toxicity. As tumor heterogeneity becomes prevalent and sub-populations with resistant traits emerge, this structure no longer needs to be optimal. Depending on conditions relating to the growth rates of the sub-populations and whether drug resistance is intrinsic or acquired, various mathematical models point to administrations at lower than maximum dose rates as being superior. Such results are mirrored in the medical literature in the emergence of adaptive chemotherapy strategies. If conditions are unfavorable, however, it becomes difficult, if not impossible, to limit a resistant population from eventually becoming dominant. On the other hand, increased heterogeneity of tumor cell populations increases a tumor's immunogenicity and immunotherapies may provide a viable and novel alternative for such cases.

Citation: Urszula Ledzewicz, Heinz Schättler, Shuo Wang. On the role of tumor heterogeneity for optimal cancer chemotherapy. Networks & Heterogeneous Media, 2019, 14 (1) : 131-147. doi: 10.3934/nhm.2019007
##### References:
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Discrete and Continuous Dynamical Systems, Series B, 22 (2017), 763-781. doi: 10.3934/dcdsb.2017037. Google Scholar [6] B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Mathématiques & Applications, vol. 40, Springer Verlag, Paris, 2003. Google Scholar [7] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, 2007. Google Scholar [8] M. Delitalia and T. Lorenzi, Formations of evolutionary patterns in cancer dynamics, in: Pattern Formation in Morphogenesis: Problems and Mathematical Issues, (V. Capasso et al., Eds.), Springer Proceedings in Mathematics, 15 (2013), 179–190. doi: 10.1007/978-3-642-20164-6_15. Google Scholar [9] M. Delitalia and T. Lorenzi, Recognition and learning in a mathematical model for immune response against cancer, Discr. and Cont. Dyn. Syst., Series B, 18 (2013), 891-914. doi: 10.3934/dcdsb.2013.18.891. Google Scholar [10] M. Delitalia and T. Lorenzi, Mathematical modelling of cancer under target therapeutic actions: Selection, mutation and drug resistance, in: Managing Complexity, Reducing Perplexity in Biological Systems, (M. Delitala and G. Ajmone Marsan Eds.), Springer Proceedings in Mathematics & Statistics, 2014, 81–99.Google Scholar [11] G. P. Dunn, A. T. Bruce, H. Ikeda, L. J. Old and R. D. Schreiber, Cancer immunoediting: From immunosurveillance to tumor escape, Nat. Immunol., 3 (2002), 991-998. doi: 10.1038/ni1102-991. Google Scholar [12] H. Easwaran, H. C. Tsai and S. B. Baylin, Cancer epigenetics: Tumor Heterogeneity, Plasticity of Stem-like States, and Drug Resistance, Molecular Cell, 54 (2014), 716-727. doi: 10.1016/j.molcel.2014.05.015. Google Scholar [13] M. Eisen, Mathematical Models in Cell Biology and Cancer Chemotherapy, Lecture Notes in Biomathematics, Vol. 30, Springer Verlag, Berlin, 1979. Google Scholar [14] R. A. Gatenby, A change of strategy in the war on cancer, Nature, 459 (2009), 508-509. doi: 10.1038/459508a. Google Scholar [15] R. A. Gatenby, A. S. Silva, R. J. Gillies and B. R. Frieden, Adaptive therapy, Cancer Research, 69 (2009), 4894-4903. doi: 10.1158/0008-5472.CAN-08-3658. Google Scholar [16] J. H. Goldie, Drug resistance in cancer: A perspective, Cancer and Metastasis Review, 20 (2001), 63-68. Google Scholar [17] J. H. Goldie and A. Coldman, A model for resistance of tumor cells to cancer chemotherapeutic agents, Mathematical Biosciences, 65 (1983), 291-307. Google Scholar [18] R. Grantab, S. Sivananthan and I. F. Tannock, The penetration of anticancer drugs through tumor tissue as a function of cellular adhesion and packing density of tumor cells, Cancer Research, 66 (2006), 1033-1039. doi: 10.1158/0008-5472.CAN-05-3077. Google Scholar [19] J. Greene, O. Lavi, M. M. Gottesman and D. Levy, The impact of cell density and mutations in a model of multidrug resistance in solid tumors, Bull. Math. Biol., 74 (2014), 627-653. doi: 10.1007/s11538-014-9936-8. Google Scholar [20] P. Hahnfeldt and L. Hlatky, Cell resensitization during protracted dosing of heterogeneous cell populations, Radiation Research, 150 (1998), 681-687. doi: 10.2307/3579891. Google Scholar [21] P. Hahnfeldt, J. Folkman and L. Hlatky, Minimizing long-term burden: the logic for metronomic chemotherapeutic dosing and its angiogenic basis, J. of Theoretical Biology, 220 (2003), 545-554. doi: 10.1006/jtbi.2003.3162. Google Scholar [22] D. Hanahan, G. Bergers and E. Bergsland, Less is more, regularly: Metronomic dosing of cytotoxic drugs can target tumor angiogenesis in mice, J. Clinical Investigations, 105 (2000), 1045-1047. doi: 10.1172/JCI9872. Google Scholar [23] M. Kimmel and A. Swierniak, An optimal control problem related to leukemia chemotherapy, Scientific Bulletins of the Silesian Technical University, 65 (1983), 120-130. Google Scholar [24] O. Lavi, J. Greene, D. Levy and M. Gottesman, The role of cell density and intratumoral heterogeneity in multidrug resistance, Cancer Research, 73 (2013), 7168-7175. doi: 10.1158/0008-5472.CAN-13-1768. Google Scholar [25] U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy, J. of Optimization Theory and Applications - JOTA, 114 (2002), 609-637. doi: 10.1023/A:1016027113579. Google Scholar [26] U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy, J. of Biological Systems, 10 (2002), 183-206. doi: 10.1142/S0218339002000597. Google Scholar [27] U. Ledzewicz and H. Schättler, Drug resistance in cancer chemotherapy as an optimal control problem, Discrete and Continuous Dynamical Systems, Series B, 6 (2006), 129-150. doi: 10.3934/dcdsb.2006.6.129. Google Scholar [28] U. Ledzewicz and H. Schättler, Tumor microenvironment and anticancer therapies: An optimal control approach, in: Mathematical Oncolgy 2013, (A. d'Onofrio and A. Gandolfi, Eds.), Birkhäuser, New York, 2013, 295–334. Google Scholar [29] U. Ledzewicz, K. Bratton and H. Schättler, A $3$-compartment model for chemotherapy of heterogeneous tumor populations, Acta Applicandae Matematicae, 135 (2015), 191-207. doi: 10.1007/s10440-014-9952-6. Google Scholar [30] U. Ledzewicz, M. S. Faraji Mosalman and H. Schättler, Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost, Discrete and Continuous Dynamical Systems, Series B, 18 (2013), 1031-1051. doi: 10.3934/dcdsb.2013.18.1031. Google Scholar [31] U. Ledzewicz and H. Schättler, On optimal chemotherapy for heterogeneous tumors, J. of Biological Systems, 22 (2014), 177-197. doi: 10.1142/S0218339014400014. Google Scholar [32] U. Ledzewicz and H. Schättler, A review of optimal chemotherapy protocols: From MTD towards metronomic therapy, Math. Model. Nat. Phenom., 9 (2014), 131-152. doi: 10.1051/mmnp/20149409. Google Scholar [33] U. Ledzewicz and H. Schättler, Optimizing chemotherapeutic anti-cancer treatment and the tumor microenvironment: an analysis of mathematical models, in: Systems Biology of Tumor Microenvironment, Advances in Experimental Medicine and Biology Vol. 936, (K.A. Rejniak (ed.)), 2016, 209–223. doi: 10.1007/978-3-319-42023-3_11. Google Scholar [34] A. Lorz, T. Lorenzi, M. E. Hochberg, J. Clairambault and B. Berthame, Population adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 377-399. doi: 10.1051/m2an/2012031. Google Scholar [35] A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil and B. Perthame, Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors, Bull. Math. Biol., 77 (2015), 1-22. doi: 10.1007/s11538-014-0046-4. Google Scholar [36] L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treatment Reports, 61 (1977), 1307-1317. Google Scholar [37] L. Norton and R. Simon, The Norton-Simon hypothesis revisited, Cancer Treatment Reports, 70 (1986), 41-61. Google Scholar [38] B. Perthame, Transport Equations in Biology, Birkhäuser, Basel, Switzerland, 2007. Google Scholar [39] K. Pietras and D. Hanahan, A multi-targeted, metronomic and maximum tolerated dose "chemo-switch" regimen is antiangiogenic, producing objective responses and survival benefit in a mouse model of cancer, J. of Clinical Oncology, 23 (2005), 939-952. Google Scholar [40] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Macmillan, New York, 1964. Google Scholar [41] C. Pouchol, J. Clairambault, A. Lorz and E. Trélat, Asymptotic analysis and optimal control of integro-differential system modelling healtyh and ccells exposed to chemotherapy, J. de Mathématiques Pures et Appliquées, 2017; arXiv: 1612.04698 [math.OC]Google Scholar [42] E. Ramos, C. Nespoli and P. Ramos, Feedback optimal control for mathematical models for cancer treatment, Preprint, 2018.Google Scholar [43] H. Schättler and U. Ledzewicz, Geometric Optimal Control, Interdisciplinary Applied Mathematics, vol. 38, Springer, 2012. doi: 10.1007/978-1-4614-3834-2. Google Scholar [44] H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies, Interdisciplinary Applied Mathematics, vol. 42, Springer, 2015. doi: 10.1007/978-1-4939-2972-6. Google Scholar [45] R. D. Schreiber, L. J. Old and M. J. Smyth, Cancer immunoediting: Integrating immunity's roles in cancer suppression and promotion, Science, 331 (2011), 1565-1570. doi: 10.1126/science.1203486. Google Scholar [46] S. V. Sharma et al., A chromatin-mediated reversible drug-tolerant state in cancer cell subpopulations, Cell, 141 (2010), 69–80. doi: 10.1016/j.cell.2010.02.027. Google Scholar [47] H. E. Skipper, On mathematical modeling of critical variables in cancer treatment (goals: better understanding of the past and better planning in the future), Bulletin of Mathematical Biology, 48 (1986), 253-278. doi: 10.1007/BF02459681. Google Scholar [48] N. V. Stepanova, Course of the immune reaction during the development of a malignant tumour, Biophysics, 24 (1980), 917-923. Google Scholar [49] G. W. Swan, Applications of Optimal Control Theory in Medicine, Marcel Dekker, New York, 1984. Google Scholar [50] G. W. Swan, General applications of optimal control theory in cancer chemotherapy, IMA J. of Mathematical Applications in Medicine and Biology, 5 (1988), 303-316. doi: 10.1093/imammb/5.4.303. Google Scholar [51] A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle, Biomedical Systems Modelling and Simulation (Paris, 1988), 51–53, IMACS Ann. Comput. Appl. Math., 5, IMACS Trans. Sci. Comput. '88, Baltzer, Basel, 1989. Google Scholar [52] A. Swierniak, Cell cycle as an object of control, Journal of Biological Systems, 3 (1995), 41-54. Google Scholar [53] A. Swierniak, U. Ledzewicz and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy, Int. J. Applied Mathematics and Computer Science, 13 (2003), 357-368. Google Scholar [54] A. Swierniak, A. Polanski and M. Kimmel, Optimal control problems arising in cell-cycle-specific cancer chemotherapy, Cell proliferation, 29 (1996), 117-139. Google Scholar [55] A. Swierniak, A. Polanski, M. Kimmel, A. Bobrowski and J. Smieja, Qualitative analysis of controlled drug resistance model - inverse Laplace and semigroup approach, Control and Cybernetics, 28 (1999), 61-75. Google Scholar [56] A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance, Nonlinear Analysis, 47 (2000), 375-386. doi: 10.1016/S0362-546X(01)00184-5. Google Scholar [57] G. Tonini, M. Imperatori, B. Vincenci, A. M. Frezza and D. Santini, Rechallenge therapy and treatment holiday: Different strategies in management of metastatic colorectal cancer, J. Experimental Clinical Cancer Research, 32 (2013), p92. doi: 10.1186/1756-9966-32-92. Google Scholar [58] S. Wang and H. Schättler, Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity, Mathematical Biosciences and Engineering - MBE, 13 (2016), 1223-1240. doi: 10.3934/mbe.2016040. Google Scholar [59] S. Wang and H. Schättler, Optimal control for cancer chemotherapy under tumor heterogeneity, Discrete and Continuous Dynamical Systems, Series B, 48 (2018).Google Scholar [60] S. D. Weitman, E. Glatstein and B. A. Kamen, Back to the basics: the importance of concentration $\times$ time in oncology, J. of Clinical Oncology, 11 (1993), 820-821. doi: 10.1200/JCO.1993.11.5.820. Google Scholar

show all references

##### References:
 [1] MTD, NCI Dictionary of Cancer Terms, https://www.cancer.gov/publications/dictionaries/cancer-terms/def/mtd, accessed 08/18/18.Google Scholar [2] N. André, L. Padovani and E. Pasquier, Metronomic scheduling of anticancer treatment: The next generation of multitarget therapy?, Future Oncology, 7 (2011), 385-394. Google Scholar [3] F. Billy and J. Clairambault, Designing proliferating cell population models with functional targets for control by anti-cancer drugs, Discr. and Cont. Dyn. Syst., Series B, 18 (2013), 865-889. doi: 10.3934/dcdsb.2013.18.865. Google Scholar [4] F. Billy, J. Clairambault and O. Fercoq, Optimisation of Cancer Drug Treatments Using Cell Population Dynamics, in: Mathematical Methods and Models in Biomedicine, (U. Ledzewicz, H. Schättler, A. Friedman and E. Kashdan, Eds.), Springer, New York, 2013, 265–309. doi: 10.1007/978-1-4614-4178-6_10. Google Scholar [5] O. Bonefon, J. Covile and G. Legendre, Concentration phenomena in some non-local equation, J. Discrete and Continuous Dynamical Systems, Series B, 22 (2017), 763-781. doi: 10.3934/dcdsb.2017037. Google Scholar [6] B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Mathématiques & Applications, vol. 40, Springer Verlag, Paris, 2003. Google Scholar [7] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, 2007. Google Scholar [8] M. Delitalia and T. Lorenzi, Formations of evolutionary patterns in cancer dynamics, in: Pattern Formation in Morphogenesis: Problems and Mathematical Issues, (V. Capasso et al., Eds.), Springer Proceedings in Mathematics, 15 (2013), 179–190. doi: 10.1007/978-3-642-20164-6_15. Google Scholar [9] M. Delitalia and T. Lorenzi, Recognition and learning in a mathematical model for immune response against cancer, Discr. and Cont. Dyn. Syst., Series B, 18 (2013), 891-914. doi: 10.3934/dcdsb.2013.18.891. Google Scholar [10] M. Delitalia and T. Lorenzi, Mathematical modelling of cancer under target therapeutic actions: Selection, mutation and drug resistance, in: Managing Complexity, Reducing Perplexity in Biological Systems, (M. Delitala and G. Ajmone Marsan Eds.), Springer Proceedings in Mathematics & Statistics, 2014, 81–99.Google Scholar [11] G. P. Dunn, A. T. Bruce, H. Ikeda, L. J. Old and R. D. Schreiber, Cancer immunoediting: From immunosurveillance to tumor escape, Nat. Immunol., 3 (2002), 991-998. doi: 10.1038/ni1102-991. Google Scholar [12] H. Easwaran, H. C. Tsai and S. B. Baylin, Cancer epigenetics: Tumor Heterogeneity, Plasticity of Stem-like States, and Drug Resistance, Molecular Cell, 54 (2014), 716-727. doi: 10.1016/j.molcel.2014.05.015. Google Scholar [13] M. Eisen, Mathematical Models in Cell Biology and Cancer Chemotherapy, Lecture Notes in Biomathematics, Vol. 30, Springer Verlag, Berlin, 1979. Google Scholar [14] R. A. Gatenby, A change of strategy in the war on cancer, Nature, 459 (2009), 508-509. doi: 10.1038/459508a. Google Scholar [15] R. A. Gatenby, A. S. Silva, R. J. Gillies and B. R. Frieden, Adaptive therapy, Cancer Research, 69 (2009), 4894-4903. doi: 10.1158/0008-5472.CAN-08-3658. Google Scholar [16] J. H. Goldie, Drug resistance in cancer: A perspective, Cancer and Metastasis Review, 20 (2001), 63-68. Google Scholar [17] J. H. Goldie and A. Coldman, A model for resistance of tumor cells to cancer chemotherapeutic agents, Mathematical Biosciences, 65 (1983), 291-307. Google Scholar [18] R. Grantab, S. Sivananthan and I. F. Tannock, The penetration of anticancer drugs through tumor tissue as a function of cellular adhesion and packing density of tumor cells, Cancer Research, 66 (2006), 1033-1039. doi: 10.1158/0008-5472.CAN-05-3077. Google Scholar [19] J. Greene, O. Lavi, M. M. Gottesman and D. Levy, The impact of cell density and mutations in a model of multidrug resistance in solid tumors, Bull. Math. Biol., 74 (2014), 627-653. doi: 10.1007/s11538-014-9936-8. Google Scholar [20] P. Hahnfeldt and L. Hlatky, Cell resensitization during protracted dosing of heterogeneous cell populations, Radiation Research, 150 (1998), 681-687. doi: 10.2307/3579891. Google Scholar [21] P. Hahnfeldt, J. Folkman and L. Hlatky, Minimizing long-term burden: the logic for metronomic chemotherapeutic dosing and its angiogenic basis, J. of Theoretical Biology, 220 (2003), 545-554. doi: 10.1006/jtbi.2003.3162. Google Scholar [22] D. Hanahan, G. Bergers and E. Bergsland, Less is more, regularly: Metronomic dosing of cytotoxic drugs can target tumor angiogenesis in mice, J. Clinical Investigations, 105 (2000), 1045-1047. doi: 10.1172/JCI9872. Google Scholar [23] M. Kimmel and A. Swierniak, An optimal control problem related to leukemia chemotherapy, Scientific Bulletins of the Silesian Technical University, 65 (1983), 120-130. Google Scholar [24] O. Lavi, J. Greene, D. Levy and M. Gottesman, The role of cell density and intratumoral heterogeneity in multidrug resistance, Cancer Research, 73 (2013), 7168-7175. doi: 10.1158/0008-5472.CAN-13-1768. Google Scholar [25] U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy, J. of Optimization Theory and Applications - JOTA, 114 (2002), 609-637. doi: 10.1023/A:1016027113579. Google Scholar [26] U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy, J. of Biological Systems, 10 (2002), 183-206. doi: 10.1142/S0218339002000597. Google Scholar [27] U. Ledzewicz and H. Schättler, Drug resistance in cancer chemotherapy as an optimal control problem, Discrete and Continuous Dynamical Systems, Series B, 6 (2006), 129-150. doi: 10.3934/dcdsb.2006.6.129. Google Scholar [28] U. Ledzewicz and H. Schättler, Tumor microenvironment and anticancer therapies: An optimal control approach, in: Mathematical Oncolgy 2013, (A. d'Onofrio and A. Gandolfi, Eds.), Birkhäuser, New York, 2013, 295–334. Google Scholar [29] U. Ledzewicz, K. Bratton and H. Schättler, A $3$-compartment model for chemotherapy of heterogeneous tumor populations, Acta Applicandae Matematicae, 135 (2015), 191-207. doi: 10.1007/s10440-014-9952-6. Google Scholar [30] U. Ledzewicz, M. S. Faraji Mosalman and H. Schättler, Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost, Discrete and Continuous Dynamical Systems, Series B, 18 (2013), 1031-1051. doi: 10.3934/dcdsb.2013.18.1031. Google Scholar [31] U. Ledzewicz and H. Schättler, On optimal chemotherapy for heterogeneous tumors, J. of Biological Systems, 22 (2014), 177-197. doi: 10.1142/S0218339014400014. Google Scholar [32] U. Ledzewicz and H. Schättler, A review of optimal chemotherapy protocols: From MTD towards metronomic therapy, Math. Model. Nat. Phenom., 9 (2014), 131-152. doi: 10.1051/mmnp/20149409. Google Scholar [33] U. Ledzewicz and H. Schättler, Optimizing chemotherapeutic anti-cancer treatment and the tumor microenvironment: an analysis of mathematical models, in: Systems Biology of Tumor Microenvironment, Advances in Experimental Medicine and Biology Vol. 936, (K.A. Rejniak (ed.)), 2016, 209–223. doi: 10.1007/978-3-319-42023-3_11. Google Scholar [34] A. Lorz, T. Lorenzi, M. E. Hochberg, J. Clairambault and B. Berthame, Population adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 377-399. doi: 10.1051/m2an/2012031. Google Scholar [35] A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil and B. Perthame, Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors, Bull. Math. Biol., 77 (2015), 1-22. doi: 10.1007/s11538-014-0046-4. Google Scholar [36] L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treatment Reports, 61 (1977), 1307-1317. Google Scholar [37] L. Norton and R. Simon, The Norton-Simon hypothesis revisited, Cancer Treatment Reports, 70 (1986), 41-61. Google Scholar [38] B. Perthame, Transport Equations in Biology, Birkhäuser, Basel, Switzerland, 2007. Google Scholar [39] K. Pietras and D. Hanahan, A multi-targeted, metronomic and maximum tolerated dose "chemo-switch" regimen is antiangiogenic, producing objective responses and survival benefit in a mouse model of cancer, J. of Clinical Oncology, 23 (2005), 939-952. Google Scholar [40] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Macmillan, New York, 1964. Google Scholar [41] C. Pouchol, J. Clairambault, A. Lorz and E. Trélat, Asymptotic analysis and optimal control of integro-differential system modelling healtyh and ccells exposed to chemotherapy, J. de Mathématiques Pures et Appliquées, 2017; arXiv: 1612.04698 [math.OC]Google Scholar [42] E. Ramos, C. Nespoli and P. Ramos, Feedback optimal control for mathematical models for cancer treatment, Preprint, 2018.Google Scholar [43] H. Schättler and U. Ledzewicz, Geometric Optimal Control, Interdisciplinary Applied Mathematics, vol. 38, Springer, 2012. doi: 10.1007/978-1-4614-3834-2. Google Scholar [44] H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies, Interdisciplinary Applied Mathematics, vol. 42, Springer, 2015. doi: 10.1007/978-1-4939-2972-6. Google Scholar [45] R. D. Schreiber, L. J. Old and M. J. Smyth, Cancer immunoediting: Integrating immunity's roles in cancer suppression and promotion, Science, 331 (2011), 1565-1570. doi: 10.1126/science.1203486. Google Scholar [46] S. V. Sharma et al., A chromatin-mediated reversible drug-tolerant state in cancer cell subpopulations, Cell, 141 (2010), 69–80. doi: 10.1016/j.cell.2010.02.027. Google Scholar [47] H. E. Skipper, On mathematical modeling of critical variables in cancer treatment (goals: better understanding of the past and better planning in the future), Bulletin of Mathematical Biology, 48 (1986), 253-278. doi: 10.1007/BF02459681. Google Scholar [48] N. V. Stepanova, Course of the immune reaction during the development of a malignant tumour, Biophysics, 24 (1980), 917-923. Google Scholar [49] G. W. Swan, Applications of Optimal Control Theory in Medicine, Marcel Dekker, New York, 1984. Google Scholar [50] G. W. Swan, General applications of optimal control theory in cancer chemotherapy, IMA J. of Mathematical Applications in Medicine and Biology, 5 (1988), 303-316. doi: 10.1093/imammb/5.4.303. Google Scholar [51] A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle, Biomedical Systems Modelling and Simulation (Paris, 1988), 51–53, IMACS Ann. Comput. Appl. Math., 5, IMACS Trans. Sci. Comput. '88, Baltzer, Basel, 1989. Google Scholar [52] A. Swierniak, Cell cycle as an object of control, Journal of Biological Systems, 3 (1995), 41-54. Google Scholar [53] A. Swierniak, U. Ledzewicz and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy, Int. J. Applied Mathematics and Computer Science, 13 (2003), 357-368. Google Scholar [54] A. Swierniak, A. Polanski and M. Kimmel, Optimal control problems arising in cell-cycle-specific cancer chemotherapy, Cell proliferation, 29 (1996), 117-139. Google Scholar [55] A. Swierniak, A. 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Example of locally optimal controls for a $3$-compartment model with cytotoxic ($u$) and cytostatic ($v$) agents. The initial condition is the normalized (in terms of percentages [44]) steady-state solution of the uncontrolled system and an objective of the type (2) has been minimized
Example of an extremal control and associated states for a bang-singular controlled trajectory
Extremal controls (top), evolution of the total tumor $\bar{N}$ (middle) and profiles $n(20,x)$ at the terminal time $T = 20$ for different mutation rates $\theta$
Example of the phase portraits for the system (19)-(20) with a Gompertzian growth function $F(x) = - \xi \ln \left( \frac{x}{K} \right)$ with tumor growth rate $\xi$ and carrying capacity $K$. The benign equilibrium point is marked with a green star and the malignant one with a red star
Example of numerically computed optimal controls for the system (19)-(20) with a Gompertzian growth function
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