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February  2017, 14(1): 217-235. doi: 10.3934/mbe.2017014

On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach

1. 

Institute of Mathematics, Lodz University of Technology, 90-924 Lodz, Poland

2. 

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

3. 

Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo, 63130, USA

4. 

Service d'hématologie et Oncologie Pédiatrique, Centre Hospitalo-Universitaire Timone Enfants, AP-HM, Marseille, UMR S_911 CRO2 Aix Marseille Université, Marseille, France

5. 

Metronomics Global Health Initiative, Marseille, France

6. 

Childrens Cancer Institute Australia, Lowy Cancer Research Centre, University of New South Wales, Randwick, NSW, Australia

* Corresponding author

Received  December 2015 Accepted  June 30, 2016 Published  October 2016

Effects that tumor heterogeneity and drug resistance have on the structure of chemotherapy protocols are discussed from a mathematical modeling and optimal control point of view. In the case when two compartments consisting of sensitive and resistant cells are considered, optimal protocols consist of full dose chemotherapy as long as the relative proportion of sensitive cells is high. When resistant cells become more dominant, optimal controls switch to lower dose regimens defined by so-called singular controls. The role that singular controls play in the structure of optimal therapy protocols for cell populations with a large number of traits is explored in mathematical models.

Citation: Urszula Ledzewicz, Shuo Wang, Heinz Schättler, Nicolas André, Marie Amélie Heng, Eddy Pasquier. On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach. Mathematical Biosciences & Engineering, 2017, 14 (1) : 217-235. doi: 10.3934/mbe.2017014
References:
[1]

E. Afenya, Using mathematical modeling as a resource in clinical trials, Math. Biosci. and Engr., (MBE), 2 (2005), 421-436.  doi: 10.3934/mbe.2005.2.421.  Google Scholar

[2]

N. AndréS. AbedD. OrbachC. Armari AllaL. PadovaniE. PasquierJ. C. Gentet and A. Verschuur, Pilot study of a pediatric metronomic 4-drug regimen, Oncotarget, 2 (2011), 960-965.   Google Scholar

[3]

N. AndréL. Padovani and E. Pasquier, Metronomic scheduling of anticancer treatment: The next generation of multitarget therapy?, Future Oncology, 7 (2011), 385-394.   Google Scholar

[4]

D. BarbolosiJ. CiccoliniB. LacarelleF. Barlési and N. André, Computational oncology-mathematical modelling of drug regimens for precision medicine, Nat. Rev. Clin. Oncol., 13 (2016), 242-254.  doi: 10.1038/nrclinonc.2015.204.  Google Scholar

[5]

J. Bellmunt, J. M. Trigo, E. Calvo, J. Carles, J. L. Pérez-Garci, J. Rubió, J. A. Virizuela, R. López, M. L´azaro and J. Albanell, Activity of a multitargeted chemo-switch regimen (sorafenib, gemcitabine, and metronomic capecitabine) in metastatic renal-cell carcinoma: a phase 2 study (SOGUG-02-06), Lancet Oncol., 11 (2010), 350-357, http://www.ncbi.nlm.nih.gov/pubmed/20163987. doi: 10.1016/S1470-2045(09)70383-3.  Google Scholar

[6]

G. BocciK. Nicolaou and R. S. Kerbel, Protracted low-dose effects on human endothelial cell proliferation and survival in vitro reveal a selective antiangiogenic window for various chemotherapeutic drugs, Cancer Research, 62 (2002), 6938-6943.   Google Scholar

[7]

B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Series: Mathematics and Applications, Springer-Verlag, Berlin, 2003.  Google Scholar

[8]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, 2007.  Google Scholar

[9]

T. BrowderC. E. ButterfieldB. M. KrälingB. ShiB. MarshallM. S. O'Reilly and J. Folkman, Antiangiogenic scheduling of chemotherapy improves efficacy against experimental drug-resistant cancer, Cancer Research, 60 (2000), 1878-1886.   Google Scholar

[10]

A. Friedman and Y. Kim, Tumor cell proliferation and migration under the influence of their microenvironment, Mathematical Biosciences and Engineering -MBE, 8 (2011), 371-383.  doi: 10.3934/mbe.2011.8.371.  Google Scholar

[11]

R. A. GatenbyA. S. SilvaR. J. Gillies and B. R. Frieden, Adaptive therapy, Cancer Research, 69 (2009), 4894-4903.  doi: 10.1158/0008-5472.CAN-08-3658.  Google Scholar

[12]

R. A. Gatenby, A change of strategy in the war on cancer, Nature, 459 (2009), 508-509.  doi: 10.1038/459508a.  Google Scholar

[13]

J. H. Goldie, Drug resistance in cancer: A perspective, Cancer and Metastasis Review, 20 (2001), 63-68.   Google Scholar

[14] J. H. Goldie and A. Coldman, Drug Resistance in Cancer, Cambridge University Press, 1998.  doi: 10.1017/CBO9780511666544.  Google Scholar
[15]

R. GrantabS. 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

[16]

J. GreeneO. LaviM. 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

[17]

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

[18]

P. HahnfeldtD. PanigrahyJ. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Research, 59 (1999), 4770-4775.   Google Scholar

[19]

P. HahnfeldtJ. 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

[20]

D. HanahanG. 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

[21]

Y. B. HaoS. Y. YiJ. RuanL. Zhao and K. J. Nan, New insights into metronomic chemotherapy-induced immunoregulation, Cancer Letters, 354 (2014), 220-226.  doi: 10.1016/j.canlet.2014.08.028.  Google Scholar

[22]

L.E. Harnevo and Z. Agur, Drug resistance as a dynamic process in a model for multistep gene amplification under various levels of selection stringency, Cancer Chemotherapy and Pharmacology, 30 (1992), 469-476.  doi: 10.1007/BF00685599.  Google Scholar

[23]

B. KamenE. RubinJ. Aisner and E. Glatstein, High-time chemotherapy or high time for low dose?, J. Clinical Oncology, editorial, 18 (2000), 2935-2937.   Google Scholar

[24]

G. KlementS. BaruchelJ. RakS. ManK. ClarkD.J. HicklinP. Bohlen and R.S. Kerbel, Continuous low-dose therapy with vinblastine and VEGF receptor-2 antibody induces sustained tumor regression without overt toxicity, J. Clinical Investigations, 105 (2000), R15-R24.   Google Scholar

[25]

O. LaviJ. GreeneD. 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

[26]

U. LedzewiczB. Amini and H. Schättler, Dynamics and control of a mathematical model for metronomic chemotherapy, Math. Biosci. and Engr., (MBE), 12 (2015), 1257-1275.  doi: 10.3934/mbe.2015.12.1257.  Google Scholar

[27]

U. LedzewiczK. Bratton and H. Schättler, A 3-compartment model for chemotherapy of heterogeneous tumor populations, Acta Applicanda Mathematicae, 135 (2015), 191-207.  doi: 10.1007/s10440-014-9952-6.  Google Scholar

[28]

U. LedzewiczH. Maurer and H. Schättler, Minimizing tumor volume for a mathematical model of anti-angiogenesis with linear pharmacokinetics, in: Recent Advances in Optimization and its Applications in Engineering, M. Diehl, F. Glineur, E. Jarlebring and W. Michiels, Eds., (2010), 267-276.  doi: 10.1007/978-3-642-12598-0_23.  Google Scholar

[29]

U. Ledzewicz and H. Schättler, Drug resistance in cancer chemotherapy as an optimal control problem, Discr. Cont. Dyn. Syst., Ser. B, 6 (2006), 129-150.   Google Scholar

[30]

U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. Contr. Optim., 46 (2007), 1052-1079.  doi: 10.1137/060665294.  Google Scholar

[31]

U. Ledzewicz and H. Schättler, Singular controls and chattering arcs in optimal control problems arising in biomedicine, Control and Cybernetics, 38 (2009), 1501-1523.   Google Scholar

[32]

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

[33]

U. LedzewiczH. SchättlerM. Reisi Gahrooi and S. Mahmoudian Dehkordi, On the MTD paradigm and optimal control for multi-drug cancer chemotherapy, Math. Biosci. and Engr. (MBE), 10 (2013), 803-819.  doi: 10.3934/mbe.2013.10.803.  Google Scholar

[34] D. Liberzon, Calculus of Variations and Optimal Control Theory, Princeton University Press, Princeton, NJ, 2012.   Google Scholar
[35]

A. LorzT. LorenziM. E. HochbergJ. 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

[36]

A. LorzT. LorenziJ. ClairambaultA. 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

[37]

P. S. Malik, V. Raina and N. André, Metronomics as maintenance treatment in oncology: Time for chemo-switch, Front. Oncol., 10 (2014), 1-7, http://www.ncbi.nlm.nih.gov/pubmed/24782987. doi: 10.3389/fonc.2014.0007.  Google Scholar

[38]

N. McGranahan and C. Swanton, Biological and therapeutic impact of intratumor heterogeneity in cancer evolution, Cancer Cell, 27 (2015), 15{26, http://www.ncbi.nlm.nih.gov/pubmed/25584892 Google Scholar

[39]

L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treatment Reports, 61 (1977), 1307-1317.   Google Scholar

[40]

L. Norton and R. Simon, The Norton-Simon hypothesis revisited, Cancer Treatment Reports, 70 (1986), 41-61.   Google Scholar

[41]

E. PasquierM. Kavallaris and N. André, Metronomic chemotherapy: New rationale for new directions, Nature Reviews|Clinical Oncology, 7 (2010), 455-465.  doi: 10.1038/nrclinonc.2010.82.  Google Scholar

[42]

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

[43]

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

[44]

H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer Verlag, 2012. doi: 10.1007/978-1-4614-3834-2.  Google Scholar

[45] Schättler and Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies, Springer Publishing Co., New York, USA, 2015.  doi: 10.1007/978-1-4939-2972-6.  Google Scholar
[46]

H. SchättlerU. Ledzewicz and B. Amini, Dynamical properties of a minimally parametrized mathematical model for metronomic chemotherapy, J. of Math. Biol., 72 (2016), 1255-1280.  doi: 10.1007/s00285-015-0907-y.  Google Scholar

[47]

C. Swanton, Cancer evolution: The final frontier of precision medicine? Ann. Oncol., 25 2014), 549-551, http://www.ncbi.nlm.nih.gov/pubmed/24567514. doi: 10.1093/annonc/mdu005.  Google Scholar

[48]

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

[49]

S. Wang and H. Schättler, Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity, Math. Biosci. and Engr. -MBE, 13 (2016), 1223-1240.  doi: 10.3934/mbe.2016040.  Google Scholar

[50]

J. WaresJ. CrivelliC. YunI. ChoiJ. Gevertz and P. Kim, Treatment strategies for combining immunostimulatory oncolytic virus therapeutics with dendritic cell injections, Math. Biosci. and Engr. -MBE, 12 (2015), 1237-1256.  doi: 10.3934/mbe.2015.12.1237.  Google Scholar

[51]

S. D. WeitmanE. Glatstein and B. A. Kamen, Back to the basics: the importance of concentration × time in oncology, J. of Clinical Oncology, 11 (1993), 820-821.   Google Scholar

show all references

References:
[1]

E. Afenya, Using mathematical modeling as a resource in clinical trials, Math. Biosci. and Engr., (MBE), 2 (2005), 421-436.  doi: 10.3934/mbe.2005.2.421.  Google Scholar

[2]

N. AndréS. AbedD. OrbachC. Armari AllaL. PadovaniE. PasquierJ. C. Gentet and A. Verschuur, Pilot study of a pediatric metronomic 4-drug regimen, Oncotarget, 2 (2011), 960-965.   Google Scholar

[3]

N. AndréL. Padovani and E. Pasquier, Metronomic scheduling of anticancer treatment: The next generation of multitarget therapy?, Future Oncology, 7 (2011), 385-394.   Google Scholar

[4]

D. BarbolosiJ. CiccoliniB. LacarelleF. Barlési and N. André, Computational oncology-mathematical modelling of drug regimens for precision medicine, Nat. Rev. Clin. Oncol., 13 (2016), 242-254.  doi: 10.1038/nrclinonc.2015.204.  Google Scholar

[5]

J. Bellmunt, J. M. Trigo, E. Calvo, J. Carles, J. L. Pérez-Garci, J. Rubió, J. A. Virizuela, R. López, M. L´azaro and J. Albanell, Activity of a multitargeted chemo-switch regimen (sorafenib, gemcitabine, and metronomic capecitabine) in metastatic renal-cell carcinoma: a phase 2 study (SOGUG-02-06), Lancet Oncol., 11 (2010), 350-357, http://www.ncbi.nlm.nih.gov/pubmed/20163987. doi: 10.1016/S1470-2045(09)70383-3.  Google Scholar

[6]

G. BocciK. Nicolaou and R. S. Kerbel, Protracted low-dose effects on human endothelial cell proliferation and survival in vitro reveal a selective antiangiogenic window for various chemotherapeutic drugs, Cancer Research, 62 (2002), 6938-6943.   Google Scholar

[7]

B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Series: Mathematics and Applications, Springer-Verlag, Berlin, 2003.  Google Scholar

[8]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, 2007.  Google Scholar

[9]

T. BrowderC. E. ButterfieldB. M. KrälingB. ShiB. MarshallM. S. O'Reilly and J. Folkman, Antiangiogenic scheduling of chemotherapy improves efficacy against experimental drug-resistant cancer, Cancer Research, 60 (2000), 1878-1886.   Google Scholar

[10]

A. Friedman and Y. Kim, Tumor cell proliferation and migration under the influence of their microenvironment, Mathematical Biosciences and Engineering -MBE, 8 (2011), 371-383.  doi: 10.3934/mbe.2011.8.371.  Google Scholar

[11]

R. A. GatenbyA. S. SilvaR. J. Gillies and B. R. Frieden, Adaptive therapy, Cancer Research, 69 (2009), 4894-4903.  doi: 10.1158/0008-5472.CAN-08-3658.  Google Scholar

[12]

R. A. Gatenby, A change of strategy in the war on cancer, Nature, 459 (2009), 508-509.  doi: 10.1038/459508a.  Google Scholar

[13]

J. H. Goldie, Drug resistance in cancer: A perspective, Cancer and Metastasis Review, 20 (2001), 63-68.   Google Scholar

[14] J. H. Goldie and A. Coldman, Drug Resistance in Cancer, Cambridge University Press, 1998.  doi: 10.1017/CBO9780511666544.  Google Scholar
[15]

R. GrantabS. 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

[16]

J. GreeneO. LaviM. 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

[17]

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

[18]

P. HahnfeldtD. PanigrahyJ. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Research, 59 (1999), 4770-4775.   Google Scholar

[19]

P. HahnfeldtJ. 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

[20]

D. HanahanG. 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

[21]

Y. B. HaoS. Y. YiJ. RuanL. Zhao and K. J. Nan, New insights into metronomic chemotherapy-induced immunoregulation, Cancer Letters, 354 (2014), 220-226.  doi: 10.1016/j.canlet.2014.08.028.  Google Scholar

[22]

L.E. Harnevo and Z. Agur, Drug resistance as a dynamic process in a model for multistep gene amplification under various levels of selection stringency, Cancer Chemotherapy and Pharmacology, 30 (1992), 469-476.  doi: 10.1007/BF00685599.  Google Scholar

[23]

B. KamenE. RubinJ. Aisner and E. Glatstein, High-time chemotherapy or high time for low dose?, J. Clinical Oncology, editorial, 18 (2000), 2935-2937.   Google Scholar

[24]

G. KlementS. BaruchelJ. RakS. ManK. ClarkD.J. HicklinP. Bohlen and R.S. Kerbel, Continuous low-dose therapy with vinblastine and VEGF receptor-2 antibody induces sustained tumor regression without overt toxicity, J. Clinical Investigations, 105 (2000), R15-R24.   Google Scholar

[25]

O. LaviJ. GreeneD. 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

[26]

U. LedzewiczB. Amini and H. Schättler, Dynamics and control of a mathematical model for metronomic chemotherapy, Math. Biosci. and Engr., (MBE), 12 (2015), 1257-1275.  doi: 10.3934/mbe.2015.12.1257.  Google Scholar

[27]

U. LedzewiczK. Bratton and H. Schättler, A 3-compartment model for chemotherapy of heterogeneous tumor populations, Acta Applicanda Mathematicae, 135 (2015), 191-207.  doi: 10.1007/s10440-014-9952-6.  Google Scholar

[28]

U. LedzewiczH. Maurer and H. Schättler, Minimizing tumor volume for a mathematical model of anti-angiogenesis with linear pharmacokinetics, in: Recent Advances in Optimization and its Applications in Engineering, M. Diehl, F. Glineur, E. Jarlebring and W. Michiels, Eds., (2010), 267-276.  doi: 10.1007/978-3-642-12598-0_23.  Google Scholar

[29]

U. Ledzewicz and H. Schättler, Drug resistance in cancer chemotherapy as an optimal control problem, Discr. Cont. Dyn. Syst., Ser. B, 6 (2006), 129-150.   Google Scholar

[30]

U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. Contr. Optim., 46 (2007), 1052-1079.  doi: 10.1137/060665294.  Google Scholar

[31]

U. Ledzewicz and H. Schättler, Singular controls and chattering arcs in optimal control problems arising in biomedicine, Control and Cybernetics, 38 (2009), 1501-1523.   Google Scholar

[32]

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

[33]

U. LedzewiczH. SchättlerM. Reisi Gahrooi and S. Mahmoudian Dehkordi, On the MTD paradigm and optimal control for multi-drug cancer chemotherapy, Math. Biosci. and Engr. (MBE), 10 (2013), 803-819.  doi: 10.3934/mbe.2013.10.803.  Google Scholar

[34] D. Liberzon, Calculus of Variations and Optimal Control Theory, Princeton University Press, Princeton, NJ, 2012.   Google Scholar
[35]

A. LorzT. LorenziM. E. HochbergJ. 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

[36]

A. LorzT. LorenziJ. ClairambaultA. 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

[37]

P. S. Malik, V. Raina and N. André, Metronomics as maintenance treatment in oncology: Time for chemo-switch, Front. Oncol., 10 (2014), 1-7, http://www.ncbi.nlm.nih.gov/pubmed/24782987. doi: 10.3389/fonc.2014.0007.  Google Scholar

[38]

N. McGranahan and C. Swanton, Biological and therapeutic impact of intratumor heterogeneity in cancer evolution, Cancer Cell, 27 (2015), 15{26, http://www.ncbi.nlm.nih.gov/pubmed/25584892 Google Scholar

[39]

L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treatment Reports, 61 (1977), 1307-1317.   Google Scholar

[40]

L. Norton and R. Simon, The Norton-Simon hypothesis revisited, Cancer Treatment Reports, 70 (1986), 41-61.   Google Scholar

[41]

E. PasquierM. Kavallaris and N. André, Metronomic chemotherapy: New rationale for new directions, Nature Reviews|Clinical Oncology, 7 (2010), 455-465.  doi: 10.1038/nrclinonc.2010.82.  Google Scholar

[42]

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

[43]

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

[44]

H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer Verlag, 2012. doi: 10.1007/978-1-4614-3834-2.  Google Scholar

[45] Schättler and Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies, Springer Publishing Co., New York, USA, 2015.  doi: 10.1007/978-1-4939-2972-6.  Google Scholar
[46]

H. SchättlerU. Ledzewicz and B. Amini, Dynamical properties of a minimally parametrized mathematical model for metronomic chemotherapy, J. of Math. Biol., 72 (2016), 1255-1280.  doi: 10.1007/s00285-015-0907-y.  Google Scholar

[47]

C. Swanton, Cancer evolution: The final frontier of precision medicine? Ann. Oncol., 25 2014), 549-551, http://www.ncbi.nlm.nih.gov/pubmed/24567514. doi: 10.1093/annonc/mdu005.  Google Scholar

[48]

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

[49]

S. Wang and H. Schättler, Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity, Math. Biosci. and Engr. -MBE, 13 (2016), 1223-1240.  doi: 10.3934/mbe.2016040.  Google Scholar

[50]

J. WaresJ. CrivelliC. YunI. ChoiJ. Gevertz and P. Kim, Treatment strategies for combining immunostimulatory oncolytic virus therapeutics with dendritic cell injections, Math. Biosci. and Engr. -MBE, 12 (2015), 1237-1256.  doi: 10.3934/mbe.2015.12.1237.  Google Scholar

[51]

S. D. WeitmanE. Glatstein and B. A. Kamen, Back to the basics: the importance of concentration × time in oncology, J. of Clinical Oncology, 11 (1993), 820-821.   Google Scholar

Figure 1.  Schematic illustration of the distribution of traits around dominant steady states
Figure 2.  Dose rates (left) for the control $u_{\tau}$ defined by equation (26) for $\tau=1$ (top row), $\tau=1.75$ (middle row) and $\tau =2$ (bottom row), and corresponding time evolutions of the concentrations $c$ (middle) and the states $S$ and $R$ (right)
Figure 3.  Evolution of the states S and R if the therapy horizon is extended to 42 days for the control strategies uτ for τ = 1:75 (left) and τ = 2 (right)
Figure 4.  An administration protocol which gives full dose until time $\tau$ and then switches to the dose given by the singular control over the interval $[\tau, 10]$ optimized over $\tau$. The weights in the objective are chosen equal using $\bar{\alpha} = \bar{\beta} = \frac{1}{21}$ and $\gamma = 100$. Shown are the control (top, left), corresponding evolution of the total population $T=\sum_{i=1}^{21} N_i(t)$ (top, right), evolution of the traits $N_i(t)$ for $i=1, \ldots, 21$ (middle) and a comparison of the initial density $n(0, x)\equiv \frac{200}{21}$ and the terminal density $n(10, x)$ shown as red curve (bottom)
Figure 5.  An administration protocol which gives full dose until time $\tau$ and then switches to the dose given by the singular control over the interval $[\tau, 10]$ optimized over $\tau$. The weights in the objective are chosen equal using $\bar{\alpha} = \frac{1}{21}$, $\bar{\beta} = \frac{8}{21}$ and $\gamma = 100$. Shown are the control (top, left), corresponding evolution of the total population $T=\sum_{i=1}^{21} N_i(t)$ (top, right), evolution of the traits $N_i(t)$ for $i=1, \ldots, 21$ (middle) and a comparison of the initial density $n(0, x)\equiv \frac{200}{21}$ and the terminal density $n(10, x)$ shown as red curve (bottom)
Table 1.  Values for the initial data and parameters used in numerical computations
parameters interpretation value
S0 initial condition of sensitive cells 9:4051 × 109
R0 initial condition of resistant cells 0:5949 × 109
r1 growth rate of sensitive population 3:5
r2 growth rate of resistant population 1
θ1 rate at which sensitive cells become resistant 0:15
θ2 rate at which resistant cells become resensitized 0:02
ψ1 log-kill coefficient for sensitive population 5
ψ2 log-kill coefficient for resistant population 1
k clearance rate of drug [Taxol] 2:9706
parameters interpretation value
S0 initial condition of sensitive cells 9:4051 × 109
R0 initial condition of resistant cells 0:5949 × 109
r1 growth rate of sensitive population 3:5
r2 growth rate of resistant population 1
θ1 rate at which sensitive cells become resistant 0:15
θ2 rate at which resistant cells become resensitized 0:02
ψ1 log-kill coefficient for sensitive population 5
ψ2 log-kill coefficient for resistant population 1
k clearance rate of drug [Taxol] 2:9706
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