Cancer modelling
Free Access
Issue
Math. Model. Nat. Phenom.
Volume 15, 2020
Cancer modelling
Article Number 14
Number of page(s) 22
DOI https://doi.org/10.1051/mmnp/2019027
Published online 12 March 2020
  1. N. Ahuja, A.R. Sharma and S.B. Baylin, Epigenetic therapeutics: a new weapon in the war against cancer. Annu. Rev. Med. 67 (2016) 73–89. [CrossRef] [PubMed] [Google Scholar]
  2. L. Almeida, P. Bagnerini, G. Fabrini, B.D. Hughes and T. Lorenzi, Evolution of cancer cell populations under cytotoxic therapy and treatment optimisation: insight from a phenotype-structured model. ESAIM: M2AN 53 (2019) 1157–1190. [CrossRef] [EDP Sciences] [Google Scholar]
  3. P.M. Altrock, L.L. Liu and F. Michor, The mathematics of cancer: integrating quantitative models. Nat. Rev. Cancer 15 (2015) 730. [Google Scholar]
  4. A.R. Anderson and M. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull. Math. Biol. 60 (1998) 857–899. [Google Scholar]
  5. A.R. Anderson and V. Quaranta, Integrative mathematical oncology. Nat. Rev. Cancer 8 (2008) 227. [Google Scholar]
  6. A.R.A. Anderson and P.K. Maini, Mathematical oncology. Bull. Math. Biol. 80 (2018) 945–953. [Google Scholar]
  7. N. Beerenwinkel, C.D. Greenman and J. Lagergren, Computational cancer biology: an evolutionary perspective. PLOS Comput. Biol. 12 (2016) e1004717. [Google Scholar]
  8. M.V. Blagosklonny, Target for cancer therapy: proliferating cells or stem cells. Leukemia 20 (2006) 385–391. [CrossRef] [PubMed] [Google Scholar]
  9. A. Bouchnita, F.-E. Belmaati, R. Aboulaich, M. Koury and V. Volpert, A hybrid computation model to describe the progression of multiple myeloma and its intra-clonal heterogeneity. Computation 5 (2017) 16. [CrossRef] [Google Scholar]
  10. A. Bouchnita, N. Eymard, T.K. Moyo, M.J. Koury and V. Volpert, Bone marrow infiltration by multiple myeloma causes anemia by reversible disruption of erythropoiesis. Am. J. Hematol. 91 (2016) 371–378. [CrossRef] [PubMed] [Google Scholar]
  11. I. Bozic, B. Allen and M. A. Nowak, Dynamics of targeted cancer therapy. Trends Mol. Med. 18 (2012) 311–316. [CrossRef] [PubMed] [Google Scholar]
  12. I. Bozic, J.G. Reiter, B. Allen, T. Antal, K. Chatterjee, P. Shah, Y.S. Moon, A. Yaqubie, N. Kelly, D.T. Le, et al., Evolutionary dynamicsof cancer in response to targeted combination therapy. Elife 2 (2013) e00747. [CrossRef] [PubMed] [Google Scholar]
  13. A. Brock, H. Chang and S. Huang, Non-genetic heterogeneity—a mutation-independent driving force for the somatic evolution of tumours, Nat. Rev. Genet. 10 (2009) 336–342. [CrossRef] [PubMed] [Google Scholar]
  14. R. Brown, E. Curry, L. Magnani, C. S. Wilhelm-Benartzi and J. Borley, Poised epigenetic states and acquired drug resistance in cancer. Nat. Rev. Cancer 14 (2014) 747. [Google Scholar]
  15. A.E. Burgess, T. Lorenzi, P.G. Schofield, S.F. Hubbard and M.A. Chaplain, Examining the role of individual movement in promoting coexistence in a spatially explicit prisoner’s dilemma. J. Theor. Biol. 419 (2017) 323–332. [CrossRef] [PubMed] [Google Scholar]
  16. A.E. Burgess, P.G. Schofield, S.F. Hubbard, M.A. Chaplain, and T. Lorenzi, Dynamical patterns of coexisting strategies in a hybrid discrete-continuum spatial evolutionary game model. MMNP 11 (2016) 49–64. [Google Scholar]
  17. H.M. Byrne, Dissecting cancer through mathematics: from the cell to the animal model. Nat. Rev. Cancer 10 (2010) 221–230. [Google Scholar]
  18. K. Camphausen and P.J. Tofilon, Inhibition of histone deacetylation: a strategy for tumor radiosensitization. J. Clin. Oncol. 25 (2007) 4051–4056. [CrossRef] [PubMed] [Google Scholar]
  19. N. Champagnat, R. Ferrière and G. Ben Arous The canonical equation of adaptive dynamics: a mathematical view. Selection 2 (2002) 73–83. [CrossRef] [Google Scholar]
  20. N. Champagnat, R. Ferrière and S. Méléard, Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models. Theor. Populat. Biol. 69 (2006) 297–321. [CrossRef] [PubMed] [Google Scholar]
  21. W. Chen, T.K. Cooper, C.A. Zahnow, M. Overholtzer, Z. Zhao, M. Ladanyi, J.E. Karp, N. Gokgoz, J.S. Wunder, I.L. Andrulis, A.J. Levine, J.L. Mankowski and S.B. Baylin, Epigenetic and genetic loss of hic1 function accentuates the role of p53 in tumorigenesis. Cancer Cell 6 (2004) 387–398. [CrossRef] [PubMed] [Google Scholar]
  22. R.H. Chisholm, T. Lorenzi and J. Clairambault, Cell population heterogeneity and evolution towards drug resistance in cancer: biological and mathematical assessment, theoretical treatment optimisation. Biochim. Biophys. Acta 1860 (2016) 2627–2645. [CrossRef] [PubMed] [Google Scholar]
  23. R.H. Chisholm, T. Lorenzi, L. Desvillettes and B.D. Hughes, Evolutionary dynamics of phenotype-structured populations: from individual-level mechanisms to population-level consequences. Z. Angew. Math. Phys. 67 (2016) 1–34. [Google Scholar]
  24. R.H. Chisholm, T. Lorenzi and A. Lorz, Effects of an advection term in nonlocal lotka–volterra equations. Commun. Math. Sci. 14 (2016) 1181–1188. [Google Scholar]
  25. R.H. Chisholm, T. Lorenzi, A. Lorz, A.K. Larsen, L.N. De Almeida, A. Escargueil and J. Clairambault, Emergence of drug tolerance in cancer cell populations: an evolutionary outcome of selection, nongenetic instability and stress-induced adaptation. Cancer Res. 75 (2015) 930–939. [Google Scholar]
  26. H. Cho and D. Levy, Modeling the dynamics of heterogeneity of solid tumors in response to chemotherapy. Bull. Math. Biol. 79 (2017) 2986–3012. [Google Scholar]
  27. H. Cho and D. Levy, Modeling the chemotherapy-induced selection of drug-resistant traits during tumor growth. J. Theor. Biol. 436 (2018) 120–134. [CrossRef] [PubMed] [Google Scholar]
  28. D.D. De Carvalho, S. Sharma, J.S. You, S.-F. Su, P.C. Taberlay, T.K. Kelly, X. Yang, G. Liang and P.A. Jones, DNA methylation screening identifies driver epigenetic events of cancer cell survival. Cancer Cell 21 (2012) 655–667. [CrossRef] [PubMed] [Google Scholar]
  29. M. Delitala and T. Lorenzi, A mathematical model for the dynamics of cancer hepatocytes under therapeutic actions. J. Theor. Biol. 297 (2012) 88–102. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  30. M. Esteller, Epigenetics in cancer. N. Engl. J. Med. 358 (2008) 1148–1159. [Google Scholar]
  31. N. Eymard, V. Volpert, P. Kurbatova, V. Volpert, N. Bessonov, K. Ogungbenro, L. Aarons, P. Janiaud, P. Nony, A. Bajard, et al., Mathematical model of t-cell lymphoblastic lymphoma: disease, treatment, cure or relapse of a virtual cohort of patients. Math. Med. Biol. 35 (2016) 25–47. [Google Scholar]
  32. A.P. Feinberg, M.A. Koldobskiy and A. Göndör, Epigenetic modulators, modifiers and mediators in cancer aetiology and progression. Nat. Rev. Genet. 17 (2016) 284. [CrossRef] [PubMed] [Google Scholar]
  33. A.P. Feinberg and B. Tycko, The history of cancer epigenetics. Nat. Rev. Cancer 4 (2004) 143. [CrossRef] [PubMed] [Google Scholar]
  34. L.C. Franssen, T. Lorenzi, A.E. Burgess and M.A. Chaplain, A mathematical framework for modelling the metastatic spread of cancer. Bull. Math. Biol. (2018) 1–46. [Google Scholar]
  35. A. Ganesan, Epigenetic drug discovery: a success story for cofactor interference. Philos. Trans. R. Soc. B: Biol. Sci. 373 (2018) 20170069. [Google Scholar]
  36. R.A. Gatenby and P.K. Maini, Mathematical oncology: cancer summed up. Nature 421 (2003) 321. [Google Scholar]
  37. R.A. Gatenby, A.S. Silva, R.J. Gillies and B.R. Frieden, Adaptive therapy. Cancer Res. 69 (2009) 4894–4903. [Google Scholar]
  38. R. Glasspool, J.M. Teodoridis and R. Brown, Epigenetics as a mechanism driving polygenic clinical drug resistance. Br. J. Cancer 94 (2006) 1087–1092. [CrossRef] [PubMed] [Google Scholar]
  39. M. Greaves and C.C. Maley, Clonal evolution in cancer. Nature 481 (2012) 306–313. [Google Scholar]
  40. S. Hamis, P. Nithiarasu and G.G. Powathil, What does not kill a tumour may make it stronger: in silico insights into chemotherapeutic drug resistance. J. Theor. Biol. 454 (2018) 253–267. [CrossRef] [PubMed] [Google Scholar]
  41. S. Heerboth, K. Lapinska, N. Snyder, M. Leary, S. Rollinson and S. Sarkar, Use of epigenetic drugs in disease: an overview. Genet. Epigenet. 6 (2014) 9. [CrossRef] [PubMed] [Google Scholar]
  42. G. Housman, S. Byler, S. Heerboth, K. Lapinska, M. Longacre, N. Snyder and S. Sarkar, Drug resistance in cancer: an overview. Cancers 6 (2014) 1769–1792. [CrossRef] [PubMed] [Google Scholar]
  43. S. Huang, Genetic and non-genetic instability in tumor progression: link between the fitness landscape and the epigenetic landscape of cancer cells. Cancer Metas. Rev. 32 (2013) 423–448. [CrossRef] [PubMed] [Google Scholar]
  44. P.A. Jones and P.W. Laird, Cancer-epigenetics comes of age. Nat. Genet. 21 (1999) 163. [Google Scholar]
  45. M.R. Junttila and F.J. de Sauvage, Influence of tumour micro-environment heterogeneity on therapeutic response. Nature 501 (2013) 346–354. [Google Scholar]
  46. K.S. Korolev, J.B. Xavier and J. Gore, Turning ecology and evolution against cancer. Nat. Rev. Cancer 14 (2014) 371. [Google Scholar]
  47. S. Kumar, R.K. Srivastav, D.W. Wilkes, T. Ross, S. Kim, J. Kowalski, S. Chatla, Q. Zhang, A. Nayak, M. Guha, et al., Estrogen-dependent dll1-mediated notch signaling promotes luminal breast cancer. Oncogene (2018) 1. [Google Scholar]
  48. P. Kurbatova, S. Bernard, N. Bessonov, F. Crauste, I. Demin, C. Dumontet, S. Fischer and V. Volpert, Hybrid model of erythropoiesis and leukemia treatment with cytosine arabinoside. SIAM J. Appl. Math. 71 (2011) 2246–2268. [Google Scholar]
  49. A.A. Lane and B.A. Chabner, Histone deacetylase inhibitors in cancer therapy. J. Clin. Oncol. 27 (2009) 5459–5468. [CrossRef] [PubMed] [Google Scholar]
  50. O. Lavi, J.M. Greene, D. Levy and M.M. Gottesman, The role of cell density and intratumoral heterogeneity in multidrug resistance. Cancer Res. 73 (2013) 7168–7175. [Google Scholar]
  51. O. Lavi, J.M. Greene, D. Levy and M.M. Gottesman, Simplifying the complexity of resistance heterogeneity in metastasis. Trends Molec. Med. 20 (2014) 129–136. [CrossRef] [PubMed] [Google Scholar]
  52. T. Lorenzi, R.H. Chisholm and J. Clairambault, Tracking the evolution of cancer cell populations through the mathematical lens of phenotype-structured equations. Biol. Direct 11 (2016) 43. [PubMed] [Google Scholar]
  53. T. Lorenzi, R.H. Chisholm, L. Desvillettes and B.D. Hughes, Dissecting the dynamics of epigenetic changes in phenotype-structured populations exposed to fluctuating environments. J. Theor. Biol. 386 (2015) 166–176. [CrossRef] [PubMed] [Google Scholar]
  54. T. Lorenzi, C. Venkataraman, A. Lorz and M.A. Chaplain, The role of spatial variations of abiotic factors in mediating intratumour phenotypic heterogeneity. J. Theor. Biol. 451 (2018) 101–110. [CrossRef] [PubMed] [Google Scholar]
  55. A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil and B. Perthame, Modeling the effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors. Bull. Math. Biol. 77 (2015) 1–22. [Google Scholar]
  56. A. Lorz, T. Lorenzi, M.E. Hochberg, J. Clairambault and B. Perthame, Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies. ESAIM: M2AN 47 (2013) 377–399. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  57. D.E. Matei and K.P. Nephew, Epigenetic therapies for chemoresensitization of epithelial ovarian cancer. Gynecolog. Oncol. 116 (2010) 195–201. [CrossRef] [Google Scholar]
  58. L.M. Merlo, J.W. Pepper, B.J. Reid and C.C. Maley, Cancer as an evolutionary and ecological process. Nat. Rev. Cancer 6 (2006) 924–935. [Google Scholar]
  59. J. Miller, Parabolic cylinder functions, in Handbook of Mathematical Functions, U.S. Government Printing Office, Washington, DC (1964) 686–720. [Google Scholar]
  60. R.L. Momparler, Cancer epigenetics. Oncogene 22 (2003) 6479. [Google Scholar]
  61. P.C. Nowell, The clonal evolution of tumor cell populations. Science 194 (1976) 23–28. [Google Scholar]
  62. A. Olivier and C. Pouchol, Combination of direct methods and homotopy in numerical optimal control: application to the optimization of chemotherapy in cancer. J. Optim. Theory Appl. (2018). [Google Scholar]
  63. J. Otwinowski and J.B. Plotkin, Inferring fitness landscapes by regression produces biased estimates of epistasis. Proc. Natl. Acad. Sci. 111 (2014) E2301–E2309. [CrossRef] [Google Scholar]
  64. P. Peltomäki, Mutations and epimutations in the origin of cancer. Exp. Cell Res. 318 (2012) 299–310. [PubMed] [Google Scholar]
  65. B. Perthame, Transport equations in biology, Birkhäuser, Basel, 2006. [Google Scholar]
  66. S.X. Pfister and A. Ashworth, Marked for death: targeting epigenetic changes in cancer. Nat. Rev. Drug Disc. 16 (2017) 241. [CrossRef] [Google Scholar]
  67. G. Piazzi, L. Fini, M. Selgrad, M. Garcia, Y. Daoud, T. Wex, P. Malfertheiner, A. Gasbarrini, M. Romano, R.L. Meyer, et al., Epigenetic regulation of delta-like controls notch activation in gastric cancer. Oncotarget 2 (2011) 1291. [CrossRef] [PubMed] [Google Scholar]
  68. A. Pisco and S. Huang, Non-genetic cancer cell plasticity and therapy-induced stemness in tumour relapse:‘what does not kill me strengthens me’. Br. J. Cancer 112 (2015) 1725–1732. [CrossRef] [PubMed] [Google Scholar]
  69. A.O. Pisco, A. Brock, J. Zhou, A. Moor, M. Mojtahedi, D. Jackson and S. Huang, Non-darwinian dynamics in therapy-induced cancer drug resistance. Nat. Commun. 4 (2013) 2467. [CrossRef] [PubMed] [Google Scholar]
  70. F.J. Poelwijk, D.J. Kiviet, D.M. Weinreich and S.J. Tans, Empirical fitness landscapes reveal accessible evolutionary paths. Nature 445 (2007) 383. [Google Scholar]
  71. C. Pouchol, J. Clairambault, A. Lorz and E. Trélat, Asymptotic analysis and optimal control of an integro-differential system modelling healthy and cancer cells exposed to chemotherapy. J. Math. Pures Appl. 116 (2018) 268–308. [Google Scholar]
  72. Y. Pu, F. Zhao, H. Wang and S. Cai, Mir-34a-5p promotes multi-chemoresistance of osteosarcoma through down-regulation of the dll gene. Sci. Reports 7 (2017) 44218. [CrossRef] [Google Scholar]
  73. D.F. Quail and J.A. Joyce, Microenvironmental regulation of tumor progression and metastasis. Nat. Med. 19 (2013) 1423–1437. [CrossRef] [PubMed] [Google Scholar]
  74. S. Sarkar, G. Horn, K. Moulton, A. Oza, S. Byler, S. Kokolus and M. Longacre, Cancer development, progression and therapy: an epigenetic overview. Int. J. Mol. Sci. 14 (2013) 21087–21113. [Google Scholar]
  75. P. Schofield, M. Chaplain and S. Hubbard, Mathematical modelling of host–parasitoid systems: effects of chemically mediated parasitoid foraging strategies on within-and between-generation spatio-temporal dynamics. J. Theor. Biol. 214 (2002) 31–47. [CrossRef] [PubMed] [Google Scholar]
  76. P.G. Schofield, M.A. Chaplain and S.F. Hubbard, Dynamic heterogeneous spatio-temporal pattern formation in host-parasitoid systems with synchronised generations. J. Math. Biol. 50 (2005) 559–583. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  77. A. Sharma, E.Y. Cao, V. Kumar, X. Zhang, H.S. Leong, A.M.L. Wong, N. Ramakrishnan, M. Hakimullah, H.M.V. Teo, F.T. Chong, et al., Longitudinal single-cell RNA sequencing of patient-derived primary cells reveals drug-induced infidelity in stem cell hierarchy. Nat. Commun. 9 (2018) 4931. [PubMed] [Google Scholar]
  78. S. Sharma, T.K. Kelly and P.A. Jones, Epigenetics in cancer. Carcinogenesis 31 (2010) 27–36. [CrossRef] [PubMed] [Google Scholar]
  79. S.V. Sharma, D.Y. Lee, B. Li, M.P. Quinlan, F. Takahashi, S. Maheswaran, U. McDermott, N. Azizian, L. Zou, M.A. Fischbach, et al., A chromatin-mediated reversible drug-tolerant state in cancer cell subpopulations. Cell 141 (2010) 69–80. [CrossRef] [PubMed] [Google Scholar]
  80. A.S. Silva, Y. Kam, Z.P. Khin, S.E. Minton, R.J. Gillies and R.A. Gatenby, Evolutionary approaches to prolong progression-free survival in breast cancer. Cancer Res. 72 (2012) 6362–6370. [Google Scholar]
  81. V. Singh, P. Sharma and N. Capalash, DNA methyltransferase-1 inhibitors as epigenetic therapy for cancer. Curr. Cancer Drug Targets 13 (2013) 379–399. [PubMed] [Google Scholar]
  82. G. Steel and L. Lamerton, The growth rate of human tumours. Br. J. Cancer 20 (1966) 74. [CrossRef] [PubMed] [Google Scholar]
  83. Y. Tamori and W.-M. Deng, Cell competition and its implications for development and cancer. J. Genetics Genom. 38 (2011) 483–495. [CrossRef] [PubMed] [Google Scholar]
  84. N. Temme, Parabolic cylinder functions, NIST Handbook of Mathematical Functions (2010) 303–319. [Google Scholar]
  85. F. Thomas, D. Fisher, P. Fort, J.-P. Marie, S. Daoust, B. Roche, C. Grunau, C. Cosseau, G. Mitta, S. Baghdiguian, et al., Applyingecological and evolutionary theory to cancer: a long and winding road. Evol. Appl. 6 (2013) 1–10. [CrossRef] [PubMed] [Google Scholar]
  86. O. Trédan, C.M. Galmarini, K. Patel and I.F. Tannock, Drug resistance and the solid tumor microenvironment. J. Natl. Cancer Inst. 99 (2007) 1441–1454. [CrossRef] [PubMed] [Google Scholar]
  87. H.-C. Tsai and S.B. Baylin, Cancer epigenetics: linking basic biology to clinical medicine. Cell Res. 21 (2011) 502. [CrossRef] [PubMed] [Google Scholar]
  88. L. Wagstaff, G. Kolahgar and E. Piddini, Competitive cell interactions in cancer: a cellular tug of war. Trends Cell Biol. 23 (2013) 160–167. [Google Scholar]
  89. C.B. Yoo and P.A. Jones, Epigenetic therapy of cancer: past, present and future. Nat. Rev. Drug Discov. 5 (2006) 37–50. [PubMed] [Google Scholar]
  90. H. Zhang, S. Pandey, M. Travers, H. Sun, G. Morton, J. Madzo, W. Chung, J. Khowsathit, O. Perez-Leal, C.A. Barrero, et al., Targeting cdk9 reactivates epigenetically silenced genes in cancer. Cell 175 (2018) 1244–1258. [CrossRef] [PubMed] [Google Scholar]

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