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

  • Received: 01 December 2015 Accepted: 30 June 2016 Published: 01 February 2017
  • MSC : Primary: 49K15, 92B05; Secondary: 93C95

  • 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[J]. Mathematical Biosciences and Engineering, 2017, 14(1): 217-235. doi: 10.3934/mbe.2017014

    Related Papers:

  • 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.


    加载中
    [1] [ E. Afenya, Using mathematical modeling as a resource in clinical trials, Math. Biosci. and Engr., (MBE), 2 (2005): 421-436.
    [2] [ N. André,S. Abed,D. Orbach,C. Armari Alla,L. Padovani,E. Pasquier,J. C. Gentet,A. Verschuur, Pilot study of a pediatric metronomic 4-drug regimen, Oncotarget, 2 (2011): 960-965.
    [3] [ N. André,L. Padovani,E. Pasquier, Metronomic scheduling of anticancer treatment: The next generation of multitarget therapy?, Future Oncology, 7 (2011): 385-394.
    [4] [ D. Barbolosi,J. Ciccolini,B. Lacarelle,F. Barlési,N. André, Computational oncology-mathematical modelling of drug regimens for precision medicine, Nat. Rev. Clin. Oncol., 13 (2016): 242-254.
    [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.
    [6] [ G. Bocci,K. Nicolaou,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.
    [7] [ B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Series: Mathematics and Applications, Springer-Verlag, Berlin, 2003.
    [8] [ A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, 2007.
    [9] [ T. Browder,C. E. Butterfield,B. M. Kräling,B. Shi,B. Marshall,M. S. O'Reilly,J. Folkman, Antiangiogenic scheduling of chemotherapy improves efficacy against experimental drug-resistant cancer, Cancer Research, 60 (2000): 1878-1886.
    [10] [ A. Friedman,Y. Kim, Tumor cell proliferation and migration under the influence of their microenvironment, Mathematical Biosciences and Engineering -MBE, 8 (2011): 371-383.
    [11] [ R. A. Gatenby,A. S. Silva,R. J. Gillies,B. R. Frieden, Adaptive therapy, Cancer Research, 69 (2009): 4894-4903.
    [12] [ R. A. Gatenby, A change of strategy in the war on cancer, Nature, 459 (2009): 508-509.
    [13] [ J. H. Goldie, Drug resistance in cancer: A perspective, Cancer and Metastasis Review, 20 (2001): 63-68.
    [14] [ J. H. Goldie,A. Coldman, null, Drug Resistance in Cancer, Cambridge University Press, 1998.
    [15] [ R. Grantab,S. Sivananthan,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.
    [16] [ J. Greene,O. Lavi,M. M. Gottesman,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.
    [17] [ P. Hahnfeldt,L. Hlatky, Cell resensitization during protracted dosing of heterogeneous cell populations, Radiation Research, 150 (1998): 681-687.
    [18] [ P. Hahnfeldt,D. Panigrahy,J. Folkman,L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Research, 59 (1999): 4770-4775.
    [19] [ P. Hahnfeldt,J. Folkman,L. Hlatky, Minimizing long-term burden: The logic for metronomic chemotherapeutic dosing and its angiogenic basis, J. of Theoretical Biology, 220 (2003): 545-554.
    [20] [ D. Hanahan,G. Bergers,E. Bergsland, Less is more, regularly: Metronomic dosing of cytotoxic drugs can target tumor angiogenesis in mice, J. Clinical Investigations, 105 (2000): 1045-1047.
    [21] [ Y. B. Hao,S. Y. Yi,J. Ruan,L. Zhao,K. J. Nan, New insights into metronomic chemotherapy-induced immunoregulation, Cancer Letters, 354 (2014): 220-226.
    [22] [ L.E. Harnevo,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.
    [23] [ B. Kamen,E. Rubin,J. Aisner,E. Glatstein, High-time chemotherapy or high time for low dose?, J. Clinical Oncology, editorial, 18 (2000): 2935-2937.
    [24] [ G. Klement,S. Baruchel,J. Rak,S. Man,K. Clark,D.J. Hicklin,P. Bohlen,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.
    [25] [ O. Lavi,J. Greene,D. Levy,M. Gottesman, The role of cell density and intratumoral heterogeneity in multidrug resistance, Cancer Research, 73 (2013): 7168-7175.
    [26] [ U. Ledzewicz,B. Amini,H. Schättler, Dynamics and control of a mathematical model for metronomic chemotherapy, Math. Biosci. and Engr., (MBE), 12 (2015): 1257-1275.
    [27] [ U. Ledzewicz,K. Bratton,H. Schättler, A 3-compartment model for chemotherapy of heterogeneous tumor populations, Acta Applicanda Mathematicae, 135 (2015): 191-207.
    [28] [ U. Ledzewicz,H. Maurer,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., null (2010): 267-276.
    [29] [ U. Ledzewicz,H. Schättler, Drug resistance in cancer chemotherapy as an optimal control problem, Discr. Cont. Dyn. Syst., Ser. B, 6 (2006): 129-150.
    [30] [ U. Ledzewicz,H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. Contr. Optim., 46 (2007): 1052-1079.
    [31] [ U. Ledzewicz,H. Schättler, Singular controls and chattering arcs in optimal control problems arising in biomedicine, Control and Cybernetics, 38 (2009): 1501-1523.
    [32] [ U. Ledzewicz,H. Schättler, On optimal chemotherapy for heterogeneous tumors, J. of Biological Systems, 22 (2014): 177-197.
    [33] [ U. Ledzewicz,H. Schättler,M. Reisi Gahrooi,S. Mahmoudian Dehkordi, On the MTD paradigm and optimal control for multi-drug cancer chemotherapy, Math. Biosci. and Engr. (MBE), 10 (2013): 803-819.
    [34] [ D. Liberzon, null, Calculus of Variations and Optimal Control Theory, Princeton University Press, Princeton, NJ, 2012.
    [35] [ A. Lorz,T. Lorenzi,M. E. Hochberg,J. Clairambault,B. Berthame, Population adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013): 377-399.
    [36] [ A. Lorz,T. Lorenzi,J. Clairambault,A. Escargueil,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.
    [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.
    [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
    [39] [ L. Norton,R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treatment Reports, 61 (1977): 1307-1317.
    [40] [ L. Norton,R. Simon, The Norton-Simon hypothesis revisited, Cancer Treatment Reports, 70 (1986): 41-61.
    [41] [ E. Pasquier,M. Kavallaris,N. André, Metronomic chemotherapy: New rationale for new directions, Nature Reviews|Clinical Oncology, 7 (2010): 455-465.
    [42] [ K. Pietras,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.
    [43] [ L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Macmillan, New York, 1964.
    [44] [ H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer Verlag, 2012.
    [45] [ Schättler,Ledzewicz, null, Optimal Control for Mathematical Models of Cancer Therapies, Springer Publishing Co., New York, USA, 2015.
    [46] [ H. Schättler,U. Ledzewicz,B. Amini, Dynamical properties of a minimally parametrized mathematical model for metronomic chemotherapy, J. of Math. Biol., 72 (2016): 1255-1280.
    [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.
    [48] [ A. Swierniak,J. Smieja, Cancer chemotherapy optimization under evolving drug resistance, Nonlinear Analysis, 47 (2000): 375-386.
    [49] [ S. Wang,H. Schättler, Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity, Math. Biosci. and Engr. -MBE, 13 (2016): 1223-1240.
    [50] [ J. Wares,J. Crivelli,C. Yun,I. Choi,J. Gevertz,P. Kim, Treatment strategies for combining immunostimulatory oncolytic virus therapeutics with dendritic cell injections, Math. Biosci. and Engr. -MBE, 12 (2015): 1237-1256.
    [51] [ S. D. Weitman,E. Glatstein,B. A. Kamen, Back to the basics: the importance of concentration × time in oncology, J. of Clinical Oncology, 11 (1993): 820-821.
  • Reader Comments
  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3840) PDF downloads(768) Cited by(16)

Article outline

Figures and Tables

Figures(5)  /  Tables(1)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog