June  2013, 18(4): 865-889. doi: 10.3934/dcdsb.2013.18.865

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

1. 

INRIA Paris-Rocquencourt, Domaine de Voluceau, Rocquencourt, B.P. 105, F-78153 Le Chesnay Cedex

Received  July 2012 Revised  September 2012 Published  February 2013

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.
Citation: Frédérique Billy, Jean Clairambault. Designing proliferating cell population models with functional targets for control by anti-cancer drugs. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 865-889. doi: 10.3934/dcdsb.2013.18.865
References:
[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.   Google Scholar

[2]

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.  doi: 10.1016/j.nonrwa.2009.11.001.  Google Scholar

[3]

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

[4]

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).   Google Scholar

[5]

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.  doi: 10.1016/S0022-5193(03)00244-3.  Google Scholar

[6]

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.  doi: 10.1016/j.jtbi.2004.04.016.  Google Scholar

[7]

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

[8]

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

[9]

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.   Google Scholar

[10]

A. Altinok, F. Lévi and A. Goldbeter, Optimizing temporal patterns of anticancer drug delivery by simulations of a cell cycle automaton,, In M. Bertau, (2008), 275.   Google Scholar

[11]

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.   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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.  doi: 10.1016/j.mbs.2008.11.008.  Google Scholar

[16]

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.   Google Scholar

[17]

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.  doi: 10.1051/m2an:2005052.  Google Scholar

[18]

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.  doi: 10.1007/s00285-003-0203-0.  Google Scholar

[19]

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.  doi: 10.1007/s00285-003-0254-2.  Google Scholar

[20]

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.   Google Scholar

[21]

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.  doi: 10.1016/j.bulm.2004.10.003.  Google Scholar

[22]

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.  doi: 10.1007/s11538-006-9185-6.  Google Scholar

[23]

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.  doi: 10.1016/j.mcm.2007.06.008.  Google Scholar

[24]

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.  doi: 10.1007/s00285-007-0147-x.  Google Scholar

[25]

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

[26]

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.   Google Scholar

[27]

A. Bellouquid and M. Delitala, "Modelling Complex Multicellular Systems - A Kinetic Theory Approach,", Birkhäuser, (2006).   Google Scholar

[28]

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.  doi: 10.1051/mmnp/20127114.  Google Scholar

[29]

F. Billy, J. Clairambault and O. Fercoq, "Optimisation of Cancer Drug Treatments Using Cell Population Dynamics,", In A. Friedman, (2013), 265.   Google Scholar

[30]

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).   Google Scholar

[31]

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.   Google Scholar

[32]

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.  doi: 10.3934/dcdsb.2009.11.613.  Google Scholar

[33]

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

[34]

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.  doi: 10.1137/070708895.  Google Scholar

[35]

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

[36]

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.  doi: 10.1093/imammb/dqp021.  Google Scholar

[37]

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

[38]

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

[39]

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

[40]

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.  doi: 10.1051/mmnp/20094308.  Google Scholar

[41]

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

[42]

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.  doi: 10.1016/j.crma.2007.10.001.  Google Scholar

[43]

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.  doi: 10.1016/j.crma.2005.10.029.  Google Scholar

[44]

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

[45]

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

[46]

A. d'Onofrio, Rapidly acting antitumoral antiangiogenic therapies,, Phys. Rev. E. Stat. Nonlin. Soft Matter Phys., 76 (2007).   Google Scholar

[47]

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.  doi: 10.1016/j.mbs.2004.06.003.  Google Scholar

[48]

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

[49]

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

[50]

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.   Google Scholar

[51]

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

[52]

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

[53]

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.   Google Scholar

[54]

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.   Google Scholar

[55]

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

[56]

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

[57]

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

[58]

T. Haferlach, Molecular genetic pathways as therapeutic targets in acute myeloid leukemia,, Hematology, (2008), 400.   Google Scholar

[59]

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.   Google Scholar

[60]

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.   Google Scholar

[61]

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).   Google Scholar

[62]

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.   Google Scholar

[63]

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), 1155.  doi: 10.1142/S0218202506001492.  Google Scholar

[64]

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.  doi: 10.1016/S0025-5564(00)00065-1.  Google Scholar

[65]

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.   Google Scholar

[66]

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.  doi: 10.3934/mbe.2011.8.307.  Google Scholar

[67]

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.   Google Scholar

[68]

A. Lorz, T. Lorenzi, J. Clairambault and B. Perthame, Dimorphism in cancer cell populations evolving under drug pressure,, In preparation., ().   Google Scholar

[69]

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).   Google Scholar

[70]

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

[71]

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

[72]

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.   Google Scholar

[73]

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.   Google Scholar

[74]

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

[75]

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

[76]

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

[77]

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

[78]

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.   Google Scholar

[79]

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

[80]

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, (2008), 2050.   Google Scholar

[81]

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

[82]

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

[83]

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.   Google Scholar

[84]

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

[85]

B. Perthame, "Transport Equations in Biology,", Frontiers in Mathematics series. Birkhäuser, (2007).   Google Scholar

[86]

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.  doi: 10.1016/j.jtbi.2012.05.015.  Google Scholar

[87]

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.  doi: 10.1016/j.jtbi.2006.07.013.  Google Scholar

[88]

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.   Google Scholar

[89]

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.   Google Scholar

[90]

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.   Google Scholar

[91]

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.   Google Scholar

[92]

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.  doi: 10.1007/s11538-012-9725-1.  Google Scholar

[93]

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

[94]

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.   Google Scholar

[95]

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

[96]

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.  doi: 10.3934/dcdsb.2004.4.323.  Google Scholar

[97]

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.   Google Scholar

[98]

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

[99]

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

[100]

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

[101]

G. Webb, A non linear cell population model of periodic chemotherapy treatment,, Recent Trends Ordinary Differential Equations, (1992), 569.   Google Scholar

[102]

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

show all references

References:
[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.   Google Scholar

[2]

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.  doi: 10.1016/j.nonrwa.2009.11.001.  Google Scholar

[3]

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

[4]

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).   Google Scholar

[5]

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.  doi: 10.1016/S0022-5193(03)00244-3.  Google Scholar

[6]

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.  doi: 10.1016/j.jtbi.2004.04.016.  Google Scholar

[7]

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

[8]

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

[9]

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.   Google Scholar

[10]

A. Altinok, F. Lévi and A. Goldbeter, Optimizing temporal patterns of anticancer drug delivery by simulations of a cell cycle automaton,, In M. Bertau, (2008), 275.   Google Scholar

[11]

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.   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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.  doi: 10.1016/j.mbs.2008.11.008.  Google Scholar

[16]

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.   Google Scholar

[17]

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.  doi: 10.1051/m2an:2005052.  Google Scholar

[18]

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.  doi: 10.1007/s00285-003-0203-0.  Google Scholar

[19]

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.  doi: 10.1007/s00285-003-0254-2.  Google Scholar

[20]

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.   Google Scholar

[21]

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.  doi: 10.1016/j.bulm.2004.10.003.  Google Scholar

[22]

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.  doi: 10.1007/s11538-006-9185-6.  Google Scholar

[23]

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.  doi: 10.1016/j.mcm.2007.06.008.  Google Scholar

[24]

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.  doi: 10.1007/s00285-007-0147-x.  Google Scholar

[25]

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

[26]

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.   Google Scholar

[27]

A. Bellouquid and M. Delitala, "Modelling Complex Multicellular Systems - A Kinetic Theory Approach,", Birkhäuser, (2006).   Google Scholar

[28]

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.  doi: 10.1051/mmnp/20127114.  Google Scholar

[29]

F. Billy, J. Clairambault and O. Fercoq, "Optimisation of Cancer Drug Treatments Using Cell Population Dynamics,", In A. Friedman, (2013), 265.   Google Scholar

[30]

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).   Google Scholar

[31]

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.   Google Scholar

[32]

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.  doi: 10.3934/dcdsb.2009.11.613.  Google Scholar

[33]

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

[34]

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.  doi: 10.1137/070708895.  Google Scholar

[35]

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

[36]

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.  doi: 10.1093/imammb/dqp021.  Google Scholar

[37]

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

[38]

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

[39]

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

[40]

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.  doi: 10.1051/mmnp/20094308.  Google Scholar

[41]

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

[42]

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.  doi: 10.1016/j.crma.2007.10.001.  Google Scholar

[43]

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.  doi: 10.1016/j.crma.2005.10.029.  Google Scholar

[44]

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

[45]

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

[46]

A. d'Onofrio, Rapidly acting antitumoral antiangiogenic therapies,, Phys. Rev. E. Stat. Nonlin. Soft Matter Phys., 76 (2007).   Google Scholar

[47]

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.  doi: 10.1016/j.mbs.2004.06.003.  Google Scholar

[48]

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

[49]

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

[50]

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.   Google Scholar

[51]

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

[52]

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

[53]

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.   Google Scholar

[54]

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.   Google Scholar

[55]

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

[56]

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

[57]

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

[58]

T. Haferlach, Molecular genetic pathways as therapeutic targets in acute myeloid leukemia,, Hematology, (2008), 400.   Google Scholar

[59]

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.   Google Scholar

[60]

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.   Google Scholar

[61]

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).   Google Scholar

[62]

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.   Google Scholar

[63]

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), 1155.  doi: 10.1142/S0218202506001492.  Google Scholar

[64]

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.  doi: 10.1016/S0025-5564(00)00065-1.  Google Scholar

[65]

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.   Google Scholar

[66]

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.  doi: 10.3934/mbe.2011.8.307.  Google Scholar

[67]

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.   Google Scholar

[68]

A. Lorz, T. Lorenzi, J. Clairambault and B. Perthame, Dimorphism in cancer cell populations evolving under drug pressure,, In preparation., ().   Google Scholar

[69]

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).   Google Scholar

[70]

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

[71]

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

[72]

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.   Google Scholar

[73]

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.   Google Scholar

[74]

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

[75]

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

[76]

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

[77]

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

[78]

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.   Google Scholar

[79]

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

[80]

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, (2008), 2050.   Google Scholar

[81]

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

[82]

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

[83]

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.   Google Scholar

[84]

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

[85]

B. Perthame, "Transport Equations in Biology,", Frontiers in Mathematics series. Birkhäuser, (2007).   Google Scholar

[86]

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.  doi: 10.1016/j.jtbi.2012.05.015.  Google Scholar

[87]

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.  doi: 10.1016/j.jtbi.2006.07.013.  Google Scholar

[88]

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.   Google Scholar

[89]

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.   Google Scholar

[90]

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.   Google Scholar

[91]

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.   Google Scholar

[92]

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.  doi: 10.1007/s11538-012-9725-1.  Google Scholar

[93]

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

[94]

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.   Google Scholar

[95]

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

[96]

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.  doi: 10.3934/dcdsb.2004.4.323.  Google Scholar

[97]

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.   Google Scholar

[98]

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

[99]

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

[100]

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

[101]

G. Webb, A non linear cell population model of periodic chemotherapy treatment,, Recent Trends Ordinary Differential Equations, (1992), 569.   Google Scholar

[102]

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

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