Physiologically Structured Cell Population Dynamic Models with Applications to Combined Drug Delivery Optimisation in Oncology

J. Clairambault¹ ; O. Fercoq²

¹Sorbonne Universités, Inria, UPMC Univ. Paris 06, Lab. J.-L. Lions UMR CNRS 7598, 4, pl. Jussieu, b. c. 187, F75252 Paris Cedex 05, France
²LTCI, CNRS, Télécom ParisTech, Université Paris-Saclay, F75634 Paris Cedex 13, France

Mathematical modelling of natural phenomena, Tome 11 (2016) no. 6, pp. 45-70

Résumé

In this paper, we introduce a model for drug delivery optimisation in a chronotherapeutics framework. We present a pharmacokinetics and pharmacodynamics model for oxaliplatin and 5-Fluorouracil, a classic therapeutic association in the treatment of colorectal cancer. We derive the pharmacokinetics model using law of mass action and enzyme kinetics. We design an age-structured cell cycle PDE model with drug damage and repair phases to account for the effect of the drugs on proliferating cell populations, with different parameters for healthy and cancer cell populations focused on their different synchronisation responses to circadian clock triggering . Our goal is to minimise the growth rate of cancerous cells while maintaining the growth rate of healthy cells above a given toxicity threshold. We numerically optimise the drug delivery schedules under our model and obtain theoretically efficient infusion schemes in a chronotherapeutics framework, with as well as without circadian clock involvement in the molecular pharmacological model.

Détail

DOI : 10.1051/mmnp/201611604

Bibliographie

[1] A. Altinok, D. Gonze, F. Lévi, A. Goldbeter An automaton model for the cell cycle Interface focus 36 47 2011

[2] A. Altinok, F. Lévi, A. Goldbeter A cell cycle automaton model for probing circadian patterns of anticancer drug delivery Adv. Drug Deliv. Rev. 1036 1010 2007

[3] Altinok, A., Lévi, F., Goldbeter, A.: Optimizing temporal patterns of anticancer drug delivery by simulations of a cell cycle automaton. In: M. Bertau, E. Mosekilde, H. Westerhoff (eds.) Biosimulation in Drug Development, pp. 275–297. Wiley (2008)

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

[5] C. Basdevant, J. Clairambault, F. Lévi Optimisation of time-scheduled regimen for anti-cancer drug infusion Mathematical Modelling and Numerical Analysis 1069 1086 2006

[6] A. Berman, R.J. Plemmons Nonnegative matrices in the mathematical sciences Amer. Math. Soc. 1994

[7]

[8] F. Billy, J. Clairambault Designing proliferating cell population models with functional targets for control by anti-cancer drugs Discrete and Continuous Dynamical Systems - Series B 865 889 2013

[9] F. Billy, J. Clairambault, F. Delaunay, C. Feillet, N. Robert Age-structured cell population model to study the influence of growth factors on cell cycle dynamics Mathematical Biosciences and Engineering 1 17 2013

[10] Billy, F., Clairambault, J., Fercoq, O.: Optimisation of cancer drug treatments using cell population dynamics. In: A. Friedman, E. Kashdan, U. Ledzewicz, H. Schättler (eds.) Mathematical Models and Methods in Biomedicine, Lecture Notes on Mathematical Modelling in the Life Sciences, pp. 265–309. Springer (2013)

[11] F. Billy, J. Clairambaultt, O. Fercoq, S. Gaubert, T. Lepoutre, T. Ouillon, S. Saito Synchronisation and control of proliferation in cycling cell population models with age structure Mathematics and Computers in Simulation 66 94 2014

[12] G. Bocci, R. Danesi, A.D. Paolo, F. Innocenti, G. Allegrini A.F.: Comparative pharmacokinetic analysis of 5-fluorouracil and its major metabolite 5- fluoro-5,6-dihydrouracil after conventional and reduced test dose in cancer patients Clin. Cancer Res. 3032 37 2000

[13] C. Bokemeyer, I. Bondarenko, A. Makhson, J.T. Hartmann, J. Aparicio, F. De Braud, S. Donea, H. Ludwig, G. Schuch, C. Stroh, A.H. Loos, A. Zubel, P. Koralewski Fluorouracil, leucovorin, and oxaliplatin with and without cetuximab in the first-line treatment of metastatic colorectal cancer J Clin Oncol 663 671 2009

[14] R.H. Chisholm, T. Lorenzi, J. Clairambault Cell population heterogeneity and evolution towards drug resistance in cancer: Biological and mathematical assessment, theoretical treatment optimisation Biochimica et Biophysica Acta (BBA) - General Subjects 2627 2645 2016

[15] R.H. Chisholm, T. Lorenzi, A. Lorz, A.K. Larsen, L.N. De Almeida, A. Escargueil, J. Clairambault Emergence of drug tolerance in cancer cell populations: an evolutionary outcome of selection, nongenetic instability, and stress-induced adaptation Cancer Res 930 939 2015

[16] J. Clairambault Modeling oxaliplatin drug delivery to circadian rhythm in drug metabolism and host tolerance Adv. Drug Deliv. Rev. 1054 1068 2007

[17] J. Clairambault Modelling physiological and pharmacological control on cell proliferation to optimise cancer treatments Mathematical Modelling of Natural Phenomena 12 67 2009

[18] J. Clairambault Optimizing cancer pharmacotherapeutics using mathematical modeling and a systems biology approach Personalized Medicine 271 286 2011

[19] J. Clairambault, S. Gaubert, T. Lepoutre Comparison of Perron and Floquet eigenvalues in age-structured cell division models Mathematical Modelling of Natural Phenomena 183 209 2009

[20] J. Clairambault, S. Gaubert, T. Lepoutre Circadian rhythm and cell population growth Mathematical and Computer Modelling 1558 1567 2011

[21] J. Clairambault, S. Gaubert, 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 549 554 2007

[22] Clairambault, J., Laroche, B., Mischler, S., Perthame, B.: A mathematical model of the cell cycle and its control. Tech. rep., Number 4892, INRIA, Domaine de Voluceau, BP 105, 78153 Rocquencourt, France (2003). URL http://hal.inria.fr/inria-00071690

[23] J. Clairambault, P. Michel, B. Perthame Circadian rhythm and tumour growth C. R. Acad. Sci. (Paris) Ser. I Mathématique (Équations aux dérivées partielles) 17 22 2006

[24] Clairambault, J., Michel, P., Perthame, B.: A model of the cell cycle and its circadian control. In: A. Deutsch, L. Brusch, H. Byrne, G. de Vries, J. Herzel (eds.) Mathematical Modeling of Biological Systems, Volume I: Cellular Biophysics, Regulatory Networks, Development, Biomedicine, and Data Analysis, pp. 239–251. Birkhäuser, Boston (2007)

[25] A. De Gramont, A. Figer, M. Seymour, M. Homerin, A. Hmissi, J. Cassidy, C. Boni, H. Cortes-Funes, A. Cervantes, G. Freyer, D. Papamichael, N. Le Bail, C. Louvet, D. Hendler, F. De Braud, C. Wilson, F. Morvan, A. Bonetti Leucovorin and fluorouracil with or without oxaliplatin as first-line treatment in advanced colorectal cancer J Clin Oncol 2938 2947 2000

[26] R. Diasio, B. Harris Clinical pharmacology of 5-fluorouracil Clin. Pharmacokinet. 215 237 1989

[27] L. Dimitrio, J. Clairambault, R. Natalini A spatial physiological model for p53 intracellular dynamics J Theor Biol 9 24 2013

[28] J. Eliaš, J. Clairambault Reaction-diffusion systems for spatio-temporal intracellular protein networks: A beginner’s guide with two examples Comput Struct Biotechnol J 12 22 2014

[29] J. Eliaš, L. Dimitrio, J. Clairambault, R. Natalini The dynamics of p53 in single cells: physiologically based ode and reaction-diffusion pde models Phys Biol 2014

[30] J. Eliaš, L. Dimitrio, J. Clairambault, R. Natalini The p53 protein and its molecular network: modelling a missing link between dna damage and cell fate Biochim Biophys Acta - Proteins and Proteomics 232 247 2014

[31] S. Faivre, D. Chan, R. Salinas, B. Woynarowska, J.M. Woynarowski DNA strand breaks and apoptosis induced by oxaliplatin in cancer cells Biochem Pharmacol 225 237 2003

[32] O. Fercoq Perron vector optimization applied to search engines Applied Numerical Mathematics 77 99 2014

[33] E. Filipski, P.F. Innominato, M. Wu, X.M. Li, S. Iacobelli, L.J. Xian, F. Lévi Effects of light and food schedules on liver and tumor molecular clocks in mice J Natl Cancer Inst 507 517 2005

[34] E. Filipski, V.M. King, X. Li, T.G. Granda, M.C. Mormont, X. Liu, B. Claustrat, M.H. Hastings, F. Lévi Host circadian clock as a control point in tumor progression J Natl Cancer Inst 690 697 2002

[35] J. Fischel, M. Etienne, P. Formento, G. Milano Search for the optimal schedule for the oxaliplatin/5-fluorouracil association modulated or not by folinic acid: preclinical data Clinical Cancer Research 2529 1998

[36] N.C. Fonville, D. Bates, P.J. Hastings, P.C. Hanawalt, S.M. Rosenberg Role of RecA and the SOS response in thymineless death in Escherichia coli PLoS Genet e1000,865 2010

[37] P. Gabriel, S.P. Garbett, D.R. Tyson, G.F. Webb, V. Quaranta The contribution of age structure to cell population responses to targeted therapeutics Journal of Theoretical Biology 19 27 2012

[38] R. Gatenby A change of strategy in the war on cancer Nature 508 509 2009

[39] R. Gatenby, A. Silva, R. Gillies, B. Friden Adaptive therapy Cancer Research 4894 4903 2009

[40] C. Gérard, A. Goldbeter Temporal self-organization of the cyclin/Cdk network driving the mammalian cell cycle Proc Natl Acad Sci U S A 21,643 21,648 2009

[41] C. Gérard, A. Goldbeter From simple to complex patterns of oscillatory behavior in a model for the mammalian cell cycle containing multiple oscillatory circuits Chaos 045,109 2010

[42] C. Gérard, A. Goldbeter A skeleton model for the network of cyclin-dependent kinases driving the mammalian cell cycle Interface Focus 24 35 2011

[43] A. Gréchez-Cassiau, B. Rayet, F. Guillaumond, M. Teboul, F. Delaunay The circadian clock component BMAL1 is a critical regulator of p21(WAF1/CIP1) expression and hepatocyte proliferation J. Biol. Chem. 4535 42 2008

[44] C. Gérard, A. Goldbeter Entrainment of the mammalian cell cycle by the circadian clock: modeling two coupled cellular rhythms PLoS Comput Biol e1002,516 2012

[45] P. Hinow, S. Wang, C. Arteaga, G. Webb A mathematical model separates quantitatively the cytostatic and cytotoxic effects of a her2 tyrosine kinase inhibitor Theoretical Biology and Medical Modelling 14 2007

[46] Kato, T.: Perturbation Theory for Linear Operators. Springer-Verlag Berlin and Heidelberg GmbH Co. K (1966)

[47] F. Lévi Cancer chronotherapeutics Special issue of Chronobiology International 1 19 2002

[48] F. Lévi Chronotherapeutics: the relevance of timing in cancer therapy Cancer Causes Control 611 621 2006

[49] F. Lévi The circadian timing system: A coordinator of life processes. implications for the rhythmic delivery of cancer therapeutics IEEE-EMB Magazine 17 20 2008

[50] F. Lévi, A. Altinok, J. Clairambault, A. Goldbeter Implications of circadian clocks for the rhythmic delivery of cancer therapeutics Phil. Trans. Roy. Soc. A 3575 3598 2008

[51] F. Lévi, A. Karaboué, L. Gorden, P.F. Innominato, R. Saffroy, S. Giacchetti, D. Hauteville, C. Guettier, R. Adam, M. Bouchahda Cetuximab and circadian chronomodulated chemotherapy as salvage treatment for metastatic colorectal cancer (mCRC): safety, efficacy and improved secondary surgical resectability Cancer Chemother Pharmacol 339 348 2011

[52] F. Lévi, A. Okyar, S. Dulong, P. Innominato, J. Clairambault Circadian timing in cancer treatments Annual Review of Pharmacology and Toxicology 377 421 2010

[53] F. Lévi, U. Schibler Circadian rhythms: Mechanisms and therapeutic implications Ann. Rev. Pharmacol. Toxicol. 493 528 2007

[54] A.S. Lewis, M.L. Overton Eigenvalue optimization Acta Numerica 149 190 1996

[55] X. Li, G. Metzger, E. Filipski, G. Lemaigre, F. Lévi Modulation of nonprotein sulphydryl compounds rhythm with buthionine sulphoximine: relationship with oxaliplatin toxicity in mice Arch. Toxicol. 574 579 1998

[56] D.B. Longley, D.P. Harkin, P.G. Johnston 5-fluorouracil: mechanisms of action and clinical strategies Nat Rev Cancer 330 338 2003

[57] T. Lorenzi, R.H. Chisholm, J. Clairambault Tracking the evolution of cancer cell populations through the mathematical lens of phenotype-structured equations Biol Direct 2016

[58] A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil, B. Perthame Modeling the effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors Bull Math Biol 1 22 2015

[59] A. Lorz, T. Lorenzi, M.E. Hochberg, J. Clairambault, B. Perthame Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies ESAIM: Mathematical Modelling and Numerical Analysis 377 399 2013

[60] L. Ma, J. Wagner, J.J. Rice, W. Hu, A.J. Levine, G.A. Stolovitzky A plausible model for the digital response of p53 to DNA damage Proc. Natl. Acad. Sci 266 71 2005

[61] T. Matsuo, S. Yamaguchi, S. Mitsuia, A. Emi, F. Shimoda, H. Okamura Control mechanism of the circadian clock for timing of cell division in vivo Science 255 259 2003

[62] A. Mckendrick Applications of mathematics to medical problems Proc. Edinburgh Math. Soc. 98 130 1926

[63] P. Montagnier, R.J. Spiteri, J. Angeles The control of linear time-periodic systems using Floquet-Lyapunov theory International Journal of Control 472 490 2004

[64] M.C. Mormont, F. Levi Cancer chronotherapy: principles, applications, and perspectives Cancer 155 169 2003

[65] T. Oguri, Y. Bessho, H. Achiwa, H. Ozasa, K. Maeno, H. Maeda, S. Sato, R. Ueda MRP8/ABCC11 directly confers resistance to 5-fluorouracil Mol Cancer Ther 122 127 2007

[66] M. Overton, R. Womersley On minimizing the spectral radius of a nonsymmetric matrix function: optimality conditions and duality theory SIAM Journal on Matrix Analysis and Applications 473 498 1988

[67] Perthame, B.: Transport Equations in Biology. Frontiers in Mathematics series. Birkhäuser, Boston (2007)

[68] G. Peters, J. Lankelma, R. Kok, P. Noordhuis, C. Van Groeningen, C. Van Der Wilt Prolonged retention of high concentrations of 5-fluorouracil in human and murine tumors as compared with plasma Cancer Chemother. Pharmacol. 269 276 1993

[69] Polak, E.: Optimization: algorithms and consistent approximations, vol. 124. Springer Science Business Media (2012)

[70] Pontryagin, L.S., Boltyanski, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The mathematical theory of optimal processes. Interscience Publishers (1962). Translated from the Russian by K.N. Trirogoff

[71] B.P. Porsin, J.L. Formento, E. Filipski, M.C. Etienne, M. Francoual, N. Renée, N. Magné, F. Lévi, G. Milano Dihydropyrimidine dehydrogenase circadian rhythm in mouse liver: comparison between enzyme activity and gene expression Eur. J. Cancer 822 828 2003

[72] Pouchol, C., Clairambault, J., Lorz, A., Trélat, E.: Asymptotic study and optimal control of integrodifferential systems modelling healthy and cancer cells exposed to chemotherapy (2016). In review

[73] D.C. Rees, E. Johnson, O. Lewinson ABC transporters: the power to change Nat Rev Mol Cell Biol 218 227 2009

[74] Schättler, H., Ledzewicz, U.: Geometric Optimal Control, Interdisciplinary Applied Mathematics, vol. 38. Springer (2012)

[75] Schättler, H., Ledzewicz, U.: Optimal Control for Mathematical Models of Cancer Therapies, An Application of Geometric Methods, Interdisciplinary Applied Mathematics, vol. 42. Springer (2015)

[76] Scilab: http://www.scilab.org/en. Free open source software for numerical computation

[77] T. Soussi The p53 tumor suppressor gene: from molecular biology to clinical investigation Ann N Y Acad Sci 121 37 2000

[78] Y. Touitou, C. Touitou, A. Bogdan, A. Reinberg, A. Auzéby, H. Becck, P. Guillet Differences between young and elderly subjects in seasonal and circadian variations of total plasmaproteins and blood volume as reflected by hemoglobin, hematocrit, and erythrocyte counts Clinical Che 801 804 1986

[79] J.E. Trosko Mechanisms of tumor promotion: possible role of inhibited intercellular communication Eur J Cancer Clin Oncol 599 601 1987

[80] J.E. Trosko A conceptual integration of extra-, intra- and gap junctional- intercellular communication in the evolution of multi-cellularity and stem cells: How disrupted cell-cell communication during development can affect diseases later in life International Journal of Stem Cell Research & Therapy 021 2016

[81] S. William-Faltaos, D. Rouillard, P. Lechat, G. Bastian Cell cycle arrest and apoptosis induced by oxaliplatin (L-OHP) on four human cancer cell lines Anticancer Res 2093 2099 2006

[82] S. William-Faltaos, D. Rouillard, P. Lechat, G. Bastian Cell cycle arrest by oxaliplatin on cancer cells Fundam Clin Pharmacol 165 172 2007

[83] P. Wood, J. Du-Quiton, S. You, W. Hrushesky Circadian clock coordinates cancer cell cycle progression, thymidylate synthase, and 5-fluorouracil therapeutic index Mol Cancer Ther 2023 2033 2006