MATHEMATICAL MODEL OF CANCER GROWTH CONTROLED BY METRONOMIC CHEMOTHERAPIES

We propose in this article to compare the efficiency of two chemotherapeutic schedules: the traditional and the metronomic. For this, we develop a new mathematical model describing the growth dynamics of tumor and endothelial cells as well as the impact of molecules as oxygen or vascular endothelial growth factor on this dynamics. The model construction: biological assumptions, description of the equations and their discretization, constitutes the core of the article. Numerical experiments illustrate the expected behavior of the disease under the two chemotherapeutic schedules.

[1]  R. Kerbel,et al.  Inhibition of tumor angiogenesis as a strategy to circumvent acquired resistance to anti‐cancer therapeutic agents , 1991, BioEssays : news and reviews in molecular, cellular and developmental biology.

[2]  M. Plank,et al.  A mathematical model of tumour angiogenesis, regulated by vascular endothelial growth factor and the angiopoietins. , 2004, Journal of theoretical biology.

[3]  A Swierniak,et al.  Optimal control problems arising in cell‐cycle‐specific cancer chemotherapy , 1996, Cell proliferation.

[4]  M Tubiana,et al.  Klaas Breur Medal lecture 1985. The growth and progression of human tumors: implications for management strategy. , 1986, Radiotherapy and oncology : journal of the European Society for Therapeutic Radiology and Oncology.

[5]  Helen M Byrne,et al.  A multiphase model describing vascular tumour growth , 2003, Bulletin of mathematical biology.

[6]  H M Byrne,et al.  Modelling the Role of Angiogenesis and Vasculogenesis in Solid Tumour Growth , 2007, Bulletin of mathematical biology.

[7]  Durjoy Majumder,et al.  PATHO-PHYSIOLOGICALLY BASED LOGISTICS FOR TREATMENT OF CANCER , 2006 .

[8]  J. Folkman.  What is the evidence that tumors are angiogenesis dependent? , 1990, Journal of the National Cancer Institute.

[9]  J. Folkman,et al.  Anti‐Angiogenesis: New Concept for Therapy of Solid Tumors , 1972, Annals of surgery.

[10]  M. Chaplain,et al.  A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor. , 1993, IMA journal of mathematics applied in medicine and biology.

[11]  R J Jarvis,et al.  A mathematical analysis of a model for tumour angiogenesis , 1995, Journal of mathematical biology.

[12]  H. Othmer,et al.  Mathematical modeling of tumor-induced angiogenesis , 2004, Journal of mathematical biology.

[13]  Dominique Barbolosi,et al.  Optimizing drug regimens in cancer chemotherapy: a simulation study using a PK-PD model , 2001, Comput. Biol. Medicine.

[14]  D. Hanahan,et al.  Patterns and Emerging Mechanisms of the Angiogenic Switch during Tumorigenesis , 1996, Cell.

[15]  Stéphane Gaubert,et al.  Synchronisation and control of proliferation in cycling cell population models with age structure , 2014, Math. Comput. Simul..

[16]  Alberto Gandolfi,et al.  A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy. , 2008, Mathematical medicine and biology : a journal of the IMA.

[17]  Florence Hubert,et al.  A new mathematical model for optimizing the combination between antiangiogenic and cytotoxic drugs in oncology , 2012 .

[18]  L. Preziosi,et al.  ADVECTION-DIFFUSION MODELS FOR SOLID TUMOUR EVOLUTION IN VIVO AND RELATED FREE BOUNDARY PROBLEM , 2000 .

[19]  M. A. J. Chaplain,et al.  The mathematical modelling of tumour angiogenesis and invasion , 1995, Acta biotheoretica.

[20]  Mark A. J. Chaplain,et al.  A mathematical model of vascular tumour growth and invasion , 1996 .

[21]  J. Folkman.  Tumor angiogenesis: therapeutic implications. , 1971, The New England journal of medicine.

[22]  P. Kevrekidis,et al.  Minimal model for tumor angiogenesis. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  A Iliadis,et al.  New Adaptive Method for Phase I Trials in Oncology , 2008, Clinical pharmacology and therapeutics.

[24]  Dominique Barbolosi,et al.  Optimizing Drug Regimens in Cancer Chemotherapy by an Efficacy-Toxicity Mathematical Model , 2000, Comput. Biomed. Res..

[25]  Athanassios Iliadis,et al.  A mechanistic model predicting hematopoiesis and tumor growth to optimize docetaxel + epirubicin (ET) administration in metastatic breast cancer (MBC): Phase I trial , 2007 .

[26]  P. Hahnfeldt,et al.  Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. , 1999, Cancer research.

[27]  Long Chen.  FINITE VOLUME METHODS , 2011 .

[28]  Nicolas André,et al.  Metronomic scheduling of anticancer treatment: the next generation of multitarget therapy? , 2011, Future oncology.

[29]  Christian Faivre,et al.  Modeling the Impact of Anticancer Agents on Metastatic Spreading , 2012 .

[30]  S. Baruchel,et al.  Safety and pharmacokinetics of temozolomide using a dose-escalation, metronomic schedule in recurrent paediatric brain tumours. , 2006, European journal of cancer.

[31]  H M Byrne,et al.  A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy. , 2000, Mathematical biosciences.

[32]  Juxiang Chen,et al.  The efficacy of temozolomide for recurrent glioblastoma multiforme , 2013, European journal of neurology.

[33]  H M Byrne,et al.  Growth of nonnecrotic tumors in the presence and absence of inhibitors. , 1995, Mathematical biosciences.

[34]  J. Sethian,et al.  Simulating complex tumor dynamics from avascular to vascular growth using a general level-set method , 2006, Journal of mathematical biology.

[35]  Gabriele Bergers,et al.  Less is more, regularly: metronomic dosing of cytotoxic drugs can target tumor angiogenesis in mice. , 2000, The Journal of clinical investigation.

[36]  Durjoy Majumder,et al.  TUMOR ANGIOGENESIS BASED ANALYTICAL MODEL FOR THE ASSESSMENT OF MCT AND MTD CHEMOTHERAPEUTIC STRATEGIES IN CANCER , 2010 .

[37]  Christian Ingvar,et al.  Ki67 proliferation in core biopsies versus surgical samples - a model for neo-adjuvant breast cancer studies , 2011, BMC Cancer.

[38]  H. Friedman,et al.  Phase 2 study of temozolomide in children and adolescents with recurrent central nervous system tumors , 2007, Cancer.

[39]  D. Bresch,et al.  Computational Modeling of Solid Tumor Growth: The Avascular Stage , 2010, SIAM J. Sci. Comput..

[40]  Y. Tao,et al.  Nonlinear analysis of a model of vascular tumour growth and treatment , 2004 .

[41]  Nicolas André,et al.  Metronomic chemotherapy: new rationale for new directions , 2010, Nature Reviews Clinical Oncology.

[42]  B. Zetter,et al.  Angiogenesis and tumor metastasis. , 1998, Annual review of medicine.

[43]  Alberto Gandolfi,et al.  Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al. (1999). , 2004, Mathematical biosciences.

[44]  A Iliadis,et al.  A new model for determining the MTD during phase-I trials in pediatric oncology. , 2012, Current topics in medicinal chemistry.

[45]  Abhik Mukherjee,et al.  Mathematical modelling for the assessment of the effect of drug application delays in metronomic chemotherapy of cancer due to physiological constraints , 2008, Biosyst..

[46]  L. Wein,et al.  Optimal scheduling of radiotherapy and angiogenic inhibitors , 2003, Bulletin of mathematical biology.

[47]  Trachette L. Jackson,et al.  Vascular tumor growth and treatment: Consequences of polyclonality, competition and dynamic vascular support , 2002, Journal of mathematical biology.

[48]  S. Baruchel,et al.  Continuous low-dose therapy with vinblastine and VEGF receptor-2 antibody induces sustained tumor regression without overt toxicity. , 2000, The Journal of clinical investigation.

[49]  Alberto d'Onofrio,et al.  Rapidly acting antitumoral antiangiogenic therapies. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[50]  D. Balding,et al.  A mathematical model of tumour-induced capillary growth. , 1985, Journal of theoretical biology.

[51]  Walter J Curran,et al.  Dose-dense temozolomide for newly diagnosed glioblastoma: a randomized phase III clinical trial. , 2013, Journal of clinical oncology : official journal of the American Society of Clinical Oncology.

[52]  P. Hahnfeldt,et al.  Minimizing long-term tumor burden: the logic for metronomic chemotherapeutic dosing and its antiangiogenic basis. , 2003, Journal of theoretical biology.

[53]  J. Folkman,et al.  Antiangiogenic scheduling of chemotherapy improves efficacy against experimental drug-resistant cancer. , 2000, Cancer research.

[54]  Mauro Ferrari,et al.  Prediction of drug response in breast cancer using integrative experimental/computational modeling. , 2009, Cancer research.

[55]  Didier Bresch,et al.  A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy. , 2009, Journal of theoretical biology.

[56]  G. W. Swan.  Role of optimal control theory in cancer chemotherapy. , 1990, Mathematical biosciences.

[57]  H M Byrne,et al.  Mathematical models for tumour angiogenesis: numerical simulations and nonlinear wave solutions. , 1995, Bulletin of mathematical biology.

[58]  V. Cristini,et al.  Nonlinear simulation of tumor necrosis, neo-vascularization and tissue invasion via an adaptive finite-element/level-set method , 2005, Bulletin of mathematical biology.

[59]  Mitsuhiro Hayashi,et al.  Ki-67 as a prognostic marker according to breast cancer subtype and a predictor of recurrence time in primary breast cancer. , 2010, Experimental and therapeutic medicine.