Free Access
Issue
Math. Model. Nat. Phenom.
Volume 7, Number 1, 2012
Cancer modeling
Page(s) 78 - 104
DOI https://doi.org/10.1051/mmnp/20127105
Published online 25 January 2012
  1. M. Abercrombie. The crawling movement of cells. Proc. R. Soc. London B., 207 (1980), 129–147. [CrossRef]
  2. T. Alarcon, H. Byrne, P. Maini. A cellular automaton model for tumour growth in inhomogeneous environment. J. Theor. Biol., 225 (2003), 257–274. [CrossRef] [PubMed]
  3. B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, P. Walter. Molecular Biology of the Cell, 4th ed. Garland Science, New York, 2002.
  4. A. Anderson, A. Weaver, P. Commmings, V. Quaranta. Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment. Cell, 127 (2006), 905–915. [CrossRef] [PubMed]
  5. A. Anderson. A hybrid mathematical model of solid tumour invasion : the importance of cell adhesion. Math. Med. Biol., 22 (2005), 163–186. [CrossRef] [PubMed]
  6. R. Araujo, D. McElwain. A history of the study of solid tumour growth : the contribution of mathematical modelling. Bull. Math. Biol., 66 (2004), 1039–1091. [CrossRef] [MathSciNet] [PubMed]
  7. A. Balter, R. M. H. Merks, N. J. Poplawski, M. Swat, J. A. Glazier. The Glazier-Graner-Hogeweg model : extensions, future directions, and opportunities for further study. In A. R. A. Anderson, M. A. J. Chaplain, and K. A. Rejniak editors, Single-Cell-Based Models in Biology and Medicine, Mathematics and Biosciences in Interactions, Birkaüser, 151–167, 2007.
  8. N. Bellomo, N. K. Li, P. K. Maini. On the foundations of cancer modelling : selected topics, speculations, and perspectives. Math. Models Methods Appl. Sci., 18 (2008), 593–646. [CrossRef] [MathSciNet] [PubMed]
  9. J. M. Bock, L. L. Sinclair, N. S. Bedford, R. E. Jackson, J. H. Lee, D. K. Trask. Modulation of cellular invasion by VEGF-C expression in squamous cell carcinoma of the head and neck. Arch. Otolaryngol. Head. Neck. Surg., 134 (2008), No. 4, 355–362. [CrossRef] [MathSciNet] [PubMed]
  10. J. M. Brown. Tumor microenvironment and the response to anticancer therapy. Cancer Biol. Ther., 1 (2002), 453–458. [CrossRef] [MathSciNet] [PubMed]
  11. A. Bru, S. Albertos, J. L. Subiza, J. L. García-Asenjo, I. Bru. The universal dynamics of tumor growth. Bioph. J., 85 (2003), No. 5, 2948–2961. [CrossRef] [PubMed]
  12. Cancer modeling and simulation, L. Preziosi editor, Mathematical Biology and Medicine Sciences, Chapman and Hall/CRC, 2003.
  13. H. Byrne, T. Alarcon, M. Owen, S. Webb, P. Maini. Modeling aspects of cancer dynamics : a review. Philos. Trans. R. Soc. A., 364 (2006), 1563–1578. [CrossRef]
  14. M. A. J. Chaplain, A. R. A. Anderson. Mathematical modelling of tissue invasion. In L. Preziosi editor, Cancer Modelling and Simulation, Chapman Hall/CRC, 269–297, 2003.
  15. V. Cristini, H. Frieboes, R. Gatenby, S. Caserta, M. Ferrari, J. Sinek. Morphologic instability and cancer invasion. Clin. Cancer Res., 11 (2005), 6772–6779. [CrossRef] [PubMed]
  16. V. Cristini, J. Lowengrub, Q. Nie. Nonlinear simulation of tumor growth. J. Math. Biol., 46 (2003), 191–224. [CrossRef] [MathSciNet] [PubMed]
  17. S. S. Cross. Fractals in pathology. J. Pathol., 182 (1997), 1–8. [PubMed]
  18. A. De Luca, N. Arena, L. M. Sena, E. Medico. Met overexpression confers HGF-dependent invasive phenotype to human thyroid carcinoma cells in vitro. J. Cell Physiol., 180 (1999), 365 –371. [CrossRef] [PubMed]
  19. M. F. Di Renzo, M. Oliviero, R. P. Narsimhan, S. Bretti, S. Giordano, E. Medico, P. Gaglia, P. Zara, P. M. Comoglio. Expression of the Met/HGF receptor in normal and neoplastic human tissues. Oncogene, 6 (1991), 1997–2003. [PubMed]
  20. A. Engler, L. Bacakova, C. Newman, A. Hategan, M. Griffin, D. Discher. Substrate compliance versus ligand density in cell on gel responses. Biophys. J., 86 (2004), 617–628. [CrossRef] [PubMed]
  21. R. Gatenby, K. Smallbone, P. Maini, F. Rose, J. Averill, R. Nagle, L. Worrall, R. Gillies. Cellular adaptations to hypoxia and acidosis during somatic evolution of breast cancer. Br. J. Cancer, 97 (2007), 646–653. [CrossRef] [PubMed]
  22. C. Gaudet, W. Marganski, S. Kim, C. T. Brown, V. Gunderia, M. Dembo, J. Wong. Influence of type I collagen surface density on fibroblast spreading, motility, and contractility. Biophys. J., 85 (2003), 3329–3335. [CrossRef] [PubMed]
  23. P. Gerlee, A. Anderson. An evolutionary hybrid cellular automaton model of solid tumor growth. J. Theor. Biol., 246 (2007), 583–603. [CrossRef] [PubMed]
  24. P. Gerlee, A. Anderson. Stability analysis of a hybrid cellular automaton model of cell colony growth. Phys. Rev. E, 75 (2007), 051911. [CrossRef]
  25. C. Giverso, M. Scianna, L. Preziosi, N. Lo Buono, A. Funaro. Individual cell-based model for in-vitro mesothelial invasion of ovarian cancer. Math. Model. Nat. Phenom., 5 (2010), No. 1, 203–223. [CrossRef] [EDP Sciences]
  26. J. A. Glazier, A. Balter, N. J. Poplawski. Magnetization to morphogenesis : A brief history of the Glazier-Graner-Hogeweg model. In A. R. A. Anderson, M. A. J. Chaplain, and K. A. Rejniak editors, Single-Cell-Based Models in Biology and Medicine, Mathematics and Biosciences in Interactions, Birkaüser, 79–106, 2007.
  27. J. A. Glazier, F. Graner. Simulation of the differential adhesion driven rearrangement of biological cells. Physical. Rev. E, 47 (1993), 2128–2154. [CrossRef] [PubMed]
  28. F. Graner, J. A. Glazier. Simulation of biological cell sorting using a two dimensional extended Potts model. Phys. Rev. Lett., 69 (1992), 2013–2017. [CrossRef] [PubMed]
  29. H. Hatzikirou, A. Deutsch, C. Schaller, M. Simon, K. Swanson. Mathematical modeling of glioblastoma tumour development : a review. Math. Models Methods Appl. Sci., 15 (2005), 1779–1794. [CrossRef] [MathSciNet] [PubMed]
  30. B. Hegedus, F. Marga, K. Jakab, K. L. Sharpe-Timms, G. Forgacs. The interplay of cell-cell and cell-matrix interactions in the invasive properties of brain tumors. Biophys. J., 91 (2006), 2708–2716. [CrossRef] [PubMed]
  31. C. Hogea, B. Murray, J. Sethian. Simulating complex tumor dynamics from avascular to vascular growth using a general level-set method. J. Math. Biol., 53 (2006), 86–134. [CrossRef] [MathSciNet] [PubMed]
  32. S. Huang, D. E. Ingber. The structural and mechanical complexity of cell-growth control. Nat. Cell Biol., 1 (1999), 131–138. [CrossRef]
  33. E. Ising. Beitrag zur theorie des ferromagnetismus. Z. Physik., 31 (1925), 253. [CrossRef]
  34. Y. Jiang, J. Pjesivac-Grbovic, C. Cantrell, J. Freyer. A multiscale model for avascular tumor growth. Biophys. J., 89 (2005), 3884–3894. [CrossRef] [PubMed]
  35. H. A. Kenny, S. Kaur, L. M. Coussens, E. Lengyel. The initial steps of ovarian cancer cell metastasis are mediated by MMP-2 cleavage of vitronectin and fibronectin. J. Clin. Invest., 118 (2008), 1367–1379. [CrossRef] [PubMed]
  36. G. Landini, Y. Hirayama, T. J. Li, M. Kitano. Increased fractal complexity of the epithelialŰ connective tissue interface in the tongue of 4NQ0-treated rats. Pathol. Res. Pract., 196 (2000), 251–258. [CrossRef] [PubMed]
  37. X. Li, V. Cristini, Q. Nie, J. Lowengrub. Nonlinear three-dimensional simulation of solid tumor growth. Discrete Dyn. Continuous Dyn. Syst. B, 7 (2007), 581–604. [CrossRef]
  38. J. Lowengrub, V. Cristini, H. B. Frieboes, X. Li, P. Macklin, S.Sanga, S. M. Wise, X. Zheng. Nonlinear modeling and simulation of tumor growth, In N. Bellomo, M. Chaplain and E. DeAngelis Modeling and Simulation in Science, Birkaüser, in press, 2011.
  39. J. Lowengrub, V. Cristini. Multiscale modeling of cancer : an integrated experimental and mathematical modeling approach. Cambridge University Press, 2010.
  40. J. Lowengrub, H. B. Frieboes, F. Jin, Y.-L. Chuang, X. Li, P. Macklin, S. M. Wise, V. Cristini. Nonlinear modeling of cancer : bridging the gap between cells and tumors. Nonlinearity, 23 (2010), 1, R1–R91. [CrossRef] [PubMed]
  41. P. Macklin, J. Lowengrub. An improved geometry-aware curvature discretization for level set methods : application to tumor growth. J. Comput. Phys., 215 (2006), 392–401. [CrossRef]
  42. P. Macklin, J. Lowengrub. Nonlinear simulation of the effect of microenvironment on tumor growth. J. Theor. Biol., 245 (2007), No. 4, 677–704. [CrossRef] [PubMed]
  43. A. F. M. Marée, V. A. Grieneisen, P. Hogeweg, P. The Cellular Potts Model and biophysical properties of cells, tissues and morphogenesis. In A. R. A. Anderson, M. A. J. Chaplain, and K. A. Rejniak editors, Single-Cell-Based Models in Biology and Medicine, Mathematics and Biosciences in Interactions, Birkaüser, 107–136, 2007.
  44. R. M. H. Merks, P. Koolwijk. Modeling morphogenesis in silico and in vitro : towards quantitative, predictive, cell-based modeling. Math. Model. Nat. Phenom., 4 (2009), No. 4, 149–171. [CrossRef] [EDP Sciences]
  45. N. Metropolis, A. E. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller. Equation of state calculations by fast computing machines. J. Chem. Phys., 21 (1953), 1087–1092. [NASA ADS] [CrossRef]
  46. E. Montero, C. Abreu, P. Tonino. Relationship between VEGF and p53 expression and tumor cell proliferation in human gastrointestinal carcinomas. Journal of Cancer Research and Clinical Oncology, 134 (2007), No. 2, 193–201. [CrossRef] [PubMed]
  47. W. Mueller-Klieser. Tumor biology and experimental therapeutics. Crit. Rev. Oncol. Hematol., 36 (2002), 123–139. [CrossRef]
  48. G. Murphy, J. Gavrilovic. Proteolysis and cell migration : creating a path ? Curr. Opin. Cell Biol., 11 (1999), 614–621. [CrossRef] [PubMed]
  49. H. Osada, T. Takahashi. Genetic alterations of multiple tumor suppressors and oncogenes in the carcinogenesis and progression of lung cancer. Oncogene, 21 (2002), 7421–7434. [CrossRef] [PubMed]
  50. N. B. Ouchi, J. A. Glazier, J. P. Rieu, A. Upadhyaya, J. Sawada. Improving the realism of the cellular Potts model in simulations of biological cells. Physica A, 329 (2003), 451–458. [CrossRef]
  51. R. B. Potts. Some generalized order-disorder transformations. Proc. Camb. Phil. Soc., 48 (1952), 106–109. [CrossRef] [MathSciNet]
  52. L. Preziosi, A. Tosin. Multiphase modeling of tumor growth and extracellular matrix interaction : mathematical tools and applications, J. Math. Biol., 58 (2007), No. 4-5, 625–656. [CrossRef] [MathSciNet] [PubMed]
  53. L. Preziosi, A. Tosin. Multiphase and multiscale trends in cancer modelling. Math. Model. Nat. Phenom., 4 (2009), No. 3, 1–11. [CrossRef] [EDP Sciences]
  54. V. Quaranta, A. Weaver, P. Cummings, A. Anderson. Mathematical modeling of cancer : The future of prognosis and treatment. Clin. Chim. Acta, 357 (2005), 173–179. [CrossRef] [PubMed]
  55. I. Ramis-Conde, D. Drasdo, A. R. A. Anderson, M. A. J. Chaplain. Modeling the influence of E-cadherin-beta-catenin pathway in cancer cell invasion : a multiscale approach. Biophys. J., 95 (2008), 155–165. [CrossRef] [PubMed]
  56. K. A. Rejniak, R. H. Dillon. A single cell-based model of the ductal tumor microarchitecture. Comp. Math. Meth. Med., 8 (2007), No. 1, 51–69. [CrossRef]
  57. B. Ribba, O. Sautb, T. Colinb, D. Breschc, E. Grenierd, 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. [CrossRef] [PubMed]
  58. C. G. Rolli, T. Seufferlein, R. Kemkemer, J. P. Spatz. Impact of Tumor Cell Cytoskeleton Organization on Invasiveness and Migration : A Microchannel-Based Approach. PLoS ONE, 5 (2010), e8726. doi :10.1371/journal.pone.0008726. [CrossRef] [PubMed]
  59. S. Sanga, J. Sinek, H. Frieboes, M. Ferrari, J. Fruehauf, V. Cristini. Mathematical modeling of cancer progression and response to chemotherapy. Expert. Rev. Anticancer Ther., 6 (2006), 1361–1376. [CrossRef] [PubMed]
  60. N. J. Savill, P. Hogeweg. Modelling morphogenesis : from single cells to crawling slugs. J. Theor. Biol., 184 (1997), 118–124. [CrossRef]
  61. M. Scianna. A multiscale hybrid model for pro-angiogenic calcium signals in a vascular endothelial cell. Bull. Math. Biol., doi : 10.1007/s11538-011-9695-8 (2011), in press.
  62. M. Scianna, L. Preziosi. Multiscale Developments of the Cellular Potts Model. (2010). In revision.
  63. J. A. Smith, L. Martin. Do cells cycle ? Proc. Natl. Acad. Sci. U.S.A., 70 (1973), 1263–1267. [CrossRef] [PubMed]
  64. J. Smolle. Fractal tumor stromal border in a nonequilibrium growth model. Anal. Quant. Cytol. Histol., 20 (1998), 7–13. [PubMed]
  65. I. A. Steele, R. J. Edmondson, H. Y. Leung, B. R. Davies. Ligands to FGF receptor 2-IIIb induce proliferation, motility, protection from cell death and cytoskeletal rearrangements in epithelial ovarian cancer cell lines. Growth Factors, 24 (2006), No. 1, 45–53. [CrossRef] [PubMed]
  66. M. S. Steinberg. Reconstruction of tissues by dissociated cells. Some morphogenetic tissue movements and the sorting out of embryonic cells may have a common explanation. Science, 141 (1963), 401–408. [CrossRef] [PubMed]
  67. M. S. Steinberg. Does differential adhesion govern self-assembly processes in histogenesis ? Equilibrium configurations and the emergence of a hierarchy among populations of embryonic cells. J. Exp. Zool., 173 (1970), No. 4, 395–433. [CrossRef] [PubMed]
  68. W. G. Stetler-Stevenson, S. Aznavoorian, L. A. Liotta. Tumor cell interactions with the extracellular matrix during invasion and metastasis. Ann. Rev. Cell Biol., 9 (1993), 541–573. [CrossRef]
  69. J. L. Su, P. C. Yang, J. Y. Shih, C. Y. Yang, L. H. Wei, M. L. Kuo, et al.. The VEGF-C/Flt-4 axis promotes invasion and metastasis of cancer cells. Cancer Cell, 9 (2006), 209–223. [CrossRef] [PubMed]
  70. P. Tracqui. Biophysical models of tumour growth. Rep. Prog. Phys., 72 (2009), 5, 056701. [CrossRef]
  71. S. Turner, J. A. Sherratt. Intercellular adhesion and cancer invasion : A discrete simulation using the extended Potts model. J. Theor. Biol., 216 (2002), 85–100. [CrossRef] [PubMed]
  72. P. Vaupel, M. Hockel. Blood supply, oxygenation status and metabolic micromilieu of breast cancers : characterization and therapeutic relevance (Review). Int. J. Oncol., 17 (2000), 869–879. [PubMed]
  73. Y. W. Zhang, G. F. Vande Woude. HGF/SF-Met signaling in the control of branching morphogenesis and invasion. J. Cell Biochem., 88 (2003), 408–417. [CrossRef] [PubMed]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.