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Designing proliferating cell population models with functional targets for control by anti-cancer drugs

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  • We review the main types of mathematical models that have been designed to represent and predict the evolution of a cell population under the action of anti-cancer drugs that are in use in the clinic, with effects on healthy and cancer tissue growth, which from a cell functional point of view are classically divided between ``proliferation, death and differentiation''. We focus here on the choices of the drug targets in these models, aiming at showing that they must be linked in each case to a given therapeutic application. We recall some analytical results that have been obtained in using models of proliferation in cell populations with control in recent years. We present some simulations performed when no theoretical result is available and we state some open problems. In view of clinical applications, we propose possible ways to design optimal therapeutic strategies by using combinations of drugs, cytotoxic, cytostatic, or redifferentiating agents, depending on the type of cancer considered, acting on different targets at the level of cell populations.
    Mathematics Subject Classification: Primary: 92-02; Secondary: 92D25, 92C50, 35Q92.


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  • [1]

    M. Adimy, F. Crauste, and A. El Abdllaoui, Discrete maturity-structured model of cell differentiation with applications to acute myelogenous leukemia, Journal of Biological Systems, 16 (2008), 395-424.


    M. Adimy, F. Crauste and C. Marquet, Asymptotic behavior and stability switch for a mature-immature model of cell differentiation, Nonlinear Analysis: Real World Applications, 11 (2010), 2913-2929.doi: 10.1016/j.nonrwa.2009.11.001.


    Z. Agur, R. Hassin and S. Levy, Optimizing chemotherapy scheduling using local search heuristics, Operations Research, 54 (2006), 829-846.


    C. A. Aktipis, V. S. Kwan, K. A. Johnson, S. L. Neuberg and C. C. Maley, Overlooking evolution: A systematic analysis of cancer relapse and therapeutic resistance research, PLoS One, 6 (2011), e26100.


    T. Alarcón, , H. M. Byrne and P. K. Maini, A cellular automaton model for tumour growth in inhomogeneous environment, J. Theor. Biol., 225 (2003), 257-274.doi: 10.1016/S0022-5193(03)00244-3.


    T. Alarcón, H. Byrne and P. Maini, A mathematical model of the effects of hypoxia on the cell-cycle of normal and cancer cells, Journal of Theoretical Biology, 229 (2004), 395-411.doi: 10.1016/j.jtbi.2004.04.016.


    T. Alarcón, H. Byrne and P. Maini, A multiple scale model for tumor growth, Multiscale Model. Simul., 3 (2005), 440-475.doi: 10.1137/040603760.


    A. Altinok, D. Gonze, F. Lévi and A. Goldbeter, An automaton model for the cell cycle, Interface focus, 1 (2011), 36-47.


    A. Altinok, F. Lévi and A. Goldbeter, A cell cycle automaton model for probing circadian patterns of anticancer drug delivery, Adv. Drug Deliv. Rev., 59 (2007), 1036-1053.


    A. Altinok, F. Lévi and A. Goldbeter, Optimizing temporal patterns of anticancer drug delivery by simulations of a cell cycle automaton, In M. Bertau, E. Mosekilde and H. Westerhoff, editors, Biosimulation in Drug Development, 275-297, Wiley, 2008.


    A. Altinok, F. Lévi and A. Goldbeter, Identifying mechanisms of chronotolerance and chronoefficacy for the anticancer drugs 5-fluorouracil and oxaliplatin by computational modeling, Eur. J. Pharm. Sci., 36 (2009), 20-38.


    O. Arino, A survey of structured cell population dynamics, Acta. Biotheor., 43 (1995), 3-25.


    O. Arino and M. Kimmel, Comparison of approaches to modeling of cell population dynamics, SIAM J. Appl. Math., 53 (1993), 1480-1504.doi: 10.1137/0153069.


    O. Arino and E. Sanchez, A survey of cell population dynamics, J. Theor. Med., 1 (1997), 35-51.


    D. Barbolosi, A. Benabdallah, F. Hubert and F. Verga, Mathematical and numerical analysis for a model of growing metastatic tumors, Math. Biosci., 218 (2009), 1-14.doi: 10.1016/j.mbs.2008.11.008.


    D. Barbolosi and A. Iliadis, Optimizing drug regimens in cancer chemotherapy: A simulation study using a PK-PD model, Comput. Biol. Med., 31 (2001), 157-172.


    C. Basdevant, J. Clairambault and F. Lévi, Optimisation of time-scheduled regimen for anti-cancer drug infusion, Mathematical Modelling and Numerical Analysis, 39 (2006), 1069-1086.doi: 10.1051/m2an:2005052.


    B. Basse, B. C. Baguley, E. S. Marshall, W. R. Joseph, B. van Brunt, G. Wake and D. J. N. Wall, A mathematical model for analysis of the cell cycle in cell lines derived from human tumors, J. Math. Biol., 47 (2003), 295-312.doi: 10.1007/s00285-003-0203-0.


    B. Basse, B. C. Baguley, E. S. Marshall, W. R. Joseph, B. van Brunt, G. Wake and D. J. N. Wall, Modelling cell death in human tumour cell lines exposed to the anticancer drug paclitaxel, J. Math. Biol., 49 (2004), 329-357.doi: 10.1007/s00285-003-0254-2.


    B. Basse, B. C. Baguley, E. S. Marshall, G. C. Wake and D. J. N. Wall, Modelling cell population growth with applications to cancer therapy in human tumour cell lines, Prog. Biophys Mol. Biol., 85 (2004), 353-368.


    B. Basse, B. C. Baguley, E. S. Marshall, G. C. Wake and D. J. N. Wall, Modelling the flow cytometric data obtained from unperturbed human tumour cell lines: Parameter fitting and comparison, Bull. Math. Biol., 67 (2005), 815-830.doi: 10.1016/j.bulm.2004.10.003.


    B. Basse and P. Ubezio, A generalised age- and phase-structured model of human tumour cell populations both unperturbed and exposed to a range of cancer therapies, Bull. Math. Biol., 69 (2007), 1673-1690.doi: 10.1007/s11538-006-9185-6.


    F. Bekkal Brikci, J. Clairambault and B. Perthame, Analysis of a molecular structured population model with possible polynomial growth for the cell division cycle, Mathematical and Computer Modelling, 47 (2008), 699-713.doi: 10.1016/j.mcm.2007.06.008.


    F. Bekkal Brikci, J. Clairambault, B. Ribba and B. Perthame, An age-and-cyclin-structured cell population model for healthy and tumoral tissues, Journal of Mathematical Biology, 57 (2008), 91-110.doi: 10.1007/s00285-007-0147-x.


    N. Bellomo, "Modelling Complex Living Systems - A Kinetic Theory and Stochastic Game Approach," Birkhäuser, 2008.


    N. Bellomo and M. Delitala, From the mathematical kinetic, and stochastic game theory to modeling mutations, onset, progression and immune competition of cancer cells, Phys. Life Rev., 5 (2008), 183-206.


    A. Bellouquid and M. Delitala, "Modelling Complex Multicellular Systems - A Kinetic Theory Approach," Birkhäuser, Boston, 2006.


    S. Benzekry, N. André, B. Assia, C. Joseph, C. Faivre, H. Florence and D. Barbolosi, Modeling the impact of anticancer agents on metastatic spreading, Mathematical Modelling of Natural Phenomena, 7 (2012), 306-336.doi: 10.1051/mmnp/20127114.


    F. Billy, J. Clairambault and O. Fercoq, "Optimisation of Cancer Drug Treatments Using Cell Population Dynamics," In A. Friedman, E. Kashdan, U. Ledzewicz and H. Schättler, editors, Mathematical Models and Methods in Biomedicine, Part 4, 265-309, Springer, New-York, 2013.


    F. Billy, J. Clairambault, O. Fercoq, S. Gaubert, T. Lepoutre, T. Ouillon and S. Saito, "Synchronisation and Control of Proliferation in Cycling Cell Population Models with Age Structure," Math. Comp. Simul., 2012, in press, available on line Apr. 2012.


    F. Billy, B. Ribba, O. Saut, H. Morre-Trouilhet, T. Colin, D. Bresch, J.-P. Boissel, E. Grenier and J.-P. Flandrois, A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy, Journal of Theoretical Biology, 260 (2009), 545-562.


    R. Borges, A. Calsina and S. Cuadrado, Equilibria of a cyclin structured cell population model, Discrete and Continuous Dynamical Systems Series B, 11 (2009), 613-627.doi: 10.3934/dcdsb.2009.11.613.


    R. Borges, A. Calsina and S. Cuadrado, Oscillations in a molecular structured cell population model, Nonlinear Analysis: Real World Applications, 12 (2011), 1911-1922.doi: 10.1016/j.nonrwa.2010.12.007.


    D. Bresch, T. Colin, E. Grenier, B. Ribba and O. Saut, Computational modeling of solid tumor growth: The avascular stage, SIAM J. Sci. Comput., 32 (2010), 2321-2344.doi: 10.1137/070708895.


    H. Byrne and D. Drasdo, Individual-based and continuum models of growing cell populations: A comparison, Journal of Mathematical Biology, 58 (2009), 657-687.doi: 10.1007/s00285-008-0212-0.


    A. Chauvière, L. Preziosi and H. Byrne, A model of cell migration within the extracellular matrix based on a phenotypic switching mechanism, Mathematical Medicine and Biology, 27 (2010), 255-281.doi: 10.1093/imammb/dqp021.


    J. Clairambault, Modelling oxaliplatin drug delivery to circadian rhythm in drug metabolism and host tolerance, Adv. Drug Deliv. Rev., 59 (2007), 1054-1068.


    J. Clairambault, Modelling physiological and pharmacological control on cell proliferation to optimise cancer treatments, Mathematical Modelling of Natural Phenomena, 4 (2009), 12-67.doi: 10.1051/mmnp/20094302.


    J. Clairambault, Optimising cancer pharmacotherapeutics using mathematical modelling and a systems biology approach, Personalized Medicine, 8 (2011), 271-286.


    J. Clairambault, S. Gaubert and T. Lepoutre, Comparison of Perron and Floquet eigenvalues in age structured cell division models, Mathematical Modelling of Natural Phenomena, 4 (2009), 183-209.doi: 10.1051/mmnp/20094308.


    J. Clairambault, S. Gaubert and T. Lepoutre, Circadian rhythm and cell population growth, Mathematical and Computer Modelling, 53 (2011), 1558-1567.doi: 10.1016/j.mcm.2010.05.034.


    J. Clairambault, S. Gaubert and B. Perthame, An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age-structured equations, C. R. Acad. Sci. (Paris) Ser. I Mathématique, 345 (2007), 549-554.doi: 10.1016/j.crma.2007.10.001.


    J. Clairambault, P. Michel and B. Perthame, Circadian rhythm and tumour growth, C. R. Acad. Sci. (Paris) Ser. I Mathématique (Équations aux dérivées partielles), 342 (2006), 17-22.doi: 10.1016/j.crma.2005.10.029.


    M. Delitala and T. Lorenzi, A mathematical model for the dynamics of cancer hepatocytes under therapeutic actions, Journal of Theoretical Biology, 297 (2012), 88-102.doi: 10.1016/j.jtbi.2011.11.022.


    L. Dimitrio, J. Clairambault and R. Natalini, A spatial physiological model for p53 intracellular dynamics, J. Theor. Biol., 316 (2013), 9-24.


    A. d'Onofrio, Rapidly acting antitumoral antiangiogenic therapies, Phys. Rev. E. Stat. Nonlin. Soft Matter Phys., 76 (2007), (3 Pt 1), 031920.


    A. d'Onofrio and A. Gandolfi, Tumour eradication by antiangiogenic therapy: Analysis and extensions of the model by Hahnfeldt et al. (1999), Math. Biosci., 191 (2004), 159-184.doi: 10.1016/j.mbs.2004.06.003.


    A. d'Onofrio and A. Gandolfi, A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy, Math. Med. Biol., 26 (2009), 63-95.


    M. Doumic, Analysis of a population model structured by the cells molecular content, Mathematical Modelling of Natural Phenomena, 3 (2007), 121-152.doi: 10.1051/mmnp:2007006.


    B. Druker, M. Talpaz, D. Resta, B. Peng, E. Buchdunger, J. Ford, N. Lydon, H. Kantarjian, R. Capdeville, S. Ohno-Jones and C. Sawyers, Efficacy and safety of a specific inhibitor of the BCR-ABL tyrosine kinase in chronic myeloid leukemia, N. Engl. J. Med., 344 (2001), 1031-1037.


    A. Ergun, K. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors, Bull. Math. Biol., 65 (2003), 407-424.


    C. Foley and M. C. Mackey, Dynamic hematological disease: A review, Journal of Mathematical Biology, 58 (2009), 285-322.doi: 10.1007/s00285-008-0165-3.


    H. Frieboes, M. Edgerton, J. Fruehauf, F. Rose, L. Worrall, R. Gatenby, M. Ferrari and V. Cristini, Prediction of drug response in breast cancer using integrative experimental/computational modeling, Cancer Research, 69 (2009), 4484-4492.


    P. Gabriel, S. P. Garbett, D. R. Tyson, G. F. Webb and V. Quaranta, The contribution of age structure to cell population responses to targeted therapeutics, Journal of Theoretical Biology, 311 (2012), 19-27.


    R. Gatenby, A change of strategy in the war on cancer, Nature, 459 (2009), 508-509.


    R. Gatenby, A. Silva, R. Gillies and B. Friden, Adaptive therapy, Cancer Research, 69 (2009), 4894-4903.


    M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence, J Math Biol, 28 (1990), 671-694.doi: 10.1007/BF00160231.


    T. Haferlach, Molecular genetic pathways as therapeutic targets in acute myeloid leukemia, Hematology, (2008), 400-411. Am. Soc. Hematol. Educ. Program.


    P. Hahnfeldt, J. Folkman and L. Hlatky, Minimizing long-term tumor burden: the logic for metronomic chemotherapeutic dosing and its antiangiogenic basis, J. Theor. Biol., 220 (2003), 545-554.


    P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Res., 59 (1999), 4770-4775.


    P. Hinow, S. E. Wang, C. L. Arteaga and G. F. Webb, A mathematical model separates quantitatively the cytostatic and cytotoxic effects of a HER2 tyrosine kinase inhibitor, Theoretical Biology and Medical Modelling, 4 (2007), 14.


    K. Iwata, K. Kawasaki and N. Shigesada, A dynamical model for the growth and size distribution of multiple metastatic tumors, J. Theor. Biol., 203 (2000), 177-186.


    Y. Kheifetz, Y. Kogan and Z. Agur, Long-range predictability in models of cell populations subjected to phase-specific drugs: growth-rate approximation using properties of positive compact operators, Math. Models Methods Appl. Sci., 16 (2006) (7, suppl.), 1155-1172.doi: 10.1142/S0218202506001492.


    F. Kozusko, P. Chen, S. G. Grant, B. W. Day and J. C. Panetta, A mathematical model of in vitro cancer cell growth and treatment with the antimitotic agent curacin A, Math Biosci, 170 (2001), 1-16.doi: 10.1016/S0025-5564(00)00065-1.


    G. Lahav, N. Rosenfeld, A. Sigal, N. Geva-Zatorsky, A. J. Levine, M. B. Elowitz and U. Alon, Dynamics of the p53-mdm2 feedback loop in individual cells, Nature Genetics, 36 (2004), 147-150.


    U. Ledzewicz, H. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Mathematical Biosciences and Engineering, 8 (2011), 307-323.doi: 10.3934/mbe.2011.8.307.


    R. Lev Bar-Or, R. Maya, L. A. Segel, U. Alon, A. J. Levine and M. Oren, Generation of oscillations by the p53-mdm2 feedback loop: A theoretical and experimental study, Proceedings of the National Academy of Sciences of the United States of America (PNAS), 97 (2000), 11250-11255.


    A. Lorz, T. Lorenzi, J. Clairambault and B. PerthameDimorphism in cancer cell populations evolving under drug pressure, In preparation.


    A. Lorz, T. Lorenzi, M. Hochberg, J. Clairambault and B. Perthame, Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies, Mathematical Modelling and Numerical Analysis, 2012. Accepted, http://hal.archives-ouvertes.fr/hal-00714274.


    M. Mackey, Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis, Blood, 51 (1978), 941-956.


    R. Martin, Optimal control drug scheduling of cancer chemotherapy, Automatica, 28 (1992), 1113-1123.doi: 10.1016/0005-1098(92)90054-J.


    R. B. Martin, M. E. Fisher, R. F. Minchin and K. L. Teo, Low-intensity combination chemotherapy maximizes host survival time for tumors containing drug-resistant cells, Math. Biosci., 110 (1992), 221-252.


    R. B. Martin, M. E. Fisher, R. F. Minchin and K. L. Teo, Optimal control of tumor size used to maximize survival time when cells are resistant to chemotherapy, Math. Biosci., 110 (1992), 201-219.


    J. Metz and O. Diekmann, "The Dynamics of Physiologically Structured Populations," volume 68 of { Lecture notes in biomathematics}, Springer, New York, 1986.


    D. Morgan, "The Cell Cycle: Principles of Control," Primers in Biology series. Oxford University Press, 2006.


    J. Murray, Optimal control for a cancer chemotherapy problem with general growth and loss functions, Math Biosci, 98 (1990), 273-287.doi: 10.1016/0025-5564(90)90129-M.


    J. Murray, Some optimal control problems in cancer chemotherapy with a toxicity limit, Math. Biosci., 100 (1990), 49-67.doi: 10.1016/0025-5564(90)90047-3.


    J. Murray, The optimal scheduling of two drugs with simple resistance for a problem in cancer chemotherapy, IMA J. Math. Appl. Med. Biol., 14 (1997), 283-303.


    H. Ozbay, C. Bonnet, H. Benjelloun and J. Clairambault, Stability analysis of cell dynamics in leukemia, Mathematical Modelling of Natural Phenomena, 7 (2012), 203-234.doi: 10.1051/mmnp/20127109.


    H. Ozbay, C. Bonnet and J. Clairambault, Stability analysis of systems with distributed delays and application to hematopoietic cell maturation dynamics, Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, (2008), 2050-2055.


    J. Panetta and J. Adam, A mathematical model of cell-specific chemotherapy, Math Comput Modelling, 22 (1995), 67.doi: 10.1016/0895-7177(95)00154-T.


    J. C. Panetta, A mathematical model of breast and ovarian cancer treated with paclitaxel, Math. Biosci., 146 (1997), 89-113.doi: 10.1016/S0025-5564(97)00077-1.


    J. C. Panetta, W. E. Evans and M. H. Cheok, Mechanistic mathematical modelling of mercaptopurine effects on cell cycle of human acute lymphoblastic leukaemia cells, Br. J. Cancer, 94 (2006), 93-100.


    E. Pasquier, M. Kavallaris and N. André, N, Metronomic chemotherapy: new rationale for new directions, Nat. Rev. Clin. Oncol., 7 (2010), 455-465.


    B. Perthame, "Transport Equations in Biology," Frontiers in Mathematics series. Birkhäuser, Boston, 2007.


    G. G. Powathil, K. E. Gordon, L. A. Hill and M. A. J. Chaplain, Modelling the effects of cell-cycle heterogeneity on the response of a solid tumour to chemotherapy: Biological insights from a hybrid multiscale cellular automaton model, J. Theor. Biol., 308 (2012), 1-19.doi: 10.1016/j.jtbi.2012.05.015.


    B. Ribba, O. Saut, T. Colin, D. Bresch, E. Grenier and J. P. Boissel, A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents, J. Theor. Biol., 243 (2006), 532-541.doi: 10.1016/j.jtbi.2006.07.013.


    B. Ribba, B. You, M. Tod, P. Girard, B. Tranchand, V. Trillet-Lenoir and G. Freyer, Chemotherapy may be delivered based on an integrated view of tumour dynamics, IET Syst. Biol., 3 (2009), 180-190.


    A. Sakaue-Sawano, H. Kurokawa, T. Morimura, A. Hanyu, H. Hama, H. Osawa, S. Kashiwagi, K. Fukami, T. Miyata, H. Miyoshi, T. Imamura, M. Ogawa, H. Masai and A. Miyawaki, Visualizing spatiotemporal dynamics of multicellular cell-cycle progression, Cell, 132 (2008), 487-498.


    A. Sakaue-Sawano, K. Ohtawa, H. Hama, M. Kawano, M. Ogawa and A. Miyawaki, Tracing the silhouette of individual cells in S/G2/M phases with fluorescence, Chemistry & Biology, 15 (2008), 1243-1248.


    M. Sturrock, A. J. Terry, D. P. Xirodimas, A. M. Thompson and M. A. J. Chaplain, Spatio-temporal modelling of the Hes1 and p53-Mdm2 intracellular signalling pathways, J. Theor. Biol., 273 (2011), 15-31.


    M. Sturrock, A. J. Terry, D. P. Xirodimas, A. M. Thompson and M. A. J. Chaplain, Influence of the nuclear membrane, active transport, and cell shape on the hes1 and p53-mdm2 pathways: Insights from spatio-temporal modelling, Bull. Math. Biol., 74 (2012), 1531-1579.doi: 10.1007/s11538-012-9725-1.


    A. Swierniak, M. Kimmel and J. Smieja, Mathematical modeling as a tool for planning anticancer therapy, European journal of pharmacology, 625 (2009), 108-121.


    A. Swierniak, U. Ledzewicz and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy, Int. J. Appl. Math. Comput. Sci., 13 (2003), 357-368.


    A. Swierniak, A. Polanski and M. Kimmel, Optimal control problems arising in cell-cycle-specific cancer chemotherapy, Cell Prolif., 29 (1996), 117-139.


    P. Ubezio, Unraveling the complexity of cell cycle effects of anticancer drugs in cell populations, Discrete and Continuous Dynamical Systems series B, 4 (2004), 323-335.doi: 10.3934/dcdsb.2004.4.323.


    P. Ubezio, M. Lupi, D. Branduardi, P. Cappella, E. Cavallini, V. Colombo, G. Matera, C. Natoli, D. Tomasoni and M. D'Incalci, Quantitative assessment of the complex dynamics of G1, S, and G2-M, checkpoint activities, Cancer Research, 69 (2009), 5234-5240.


    B. Vogelstein, D. Lane and A. Levine, Surfing the p53 network, Nature, 408 (2000), 307-310.


    G. Webb, Resonance phenomena in cell population chemotherapy models, Rocky Mountain J. Math, 20 (1990), 1195-1216.doi: 10.1216/rmjm/1181073070.


    G. Webb, A cell population model of periodic chemotherapy treatment, Biomedical Modeling and Simulation, (1992), 83-92.


    G. Webb, A non linear cell population model of periodic chemotherapy treatment, Recent Trends Ordinary Differential Equations, Series in Applicable Analysis 1, (1992), 569-583.


    O. Witt, H. Deubzer, T. Milde and I. Oehme, HDAC family: What are the cancer relevant targets? Cancer Letters, 277 (2009), 8-21.

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