Free Access
Issue
Math. Model. Nat. Phenom.
Volume 7, Number 1, 2012
Cancer modeling
Page(s) 3 - 28
DOI https://doi.org/10.1051/mmnp/20127102
Published online 25 January 2012
  1. A. Angelle. Pancreatic cancer shown to be surprisingly slow killer. Live Science, October 27, 2010.
  2. N. Armstrong, K. Painter, J. Sherratt. A continuum approach to modeling cell-cell adhesion. J. Theor. Biol., 243 (1), 98–113. [CrossRef] [PubMed]
  3. B.P. Ayati, G.F. Webb, A.R.A. Anderson. Computational methods and results for structured multiscale methods of tumor invasion. Multiscale Model. Simul., 5 (2006), 1–20. [CrossRef] [MathSciNet] [PubMed]
  4. S. Aznavoorian, M. Stracke, H. Krutzsch, E. Schiffmann, L. Liotta. Signal transduction for chemotaxis and haptotaxis by matrix molecules in tumor cells. J. Cell Biol., 110(4), (1990), 1427–1438. [CrossRef] [PubMed]
  5. B. Bazaliy, A. Friedman. Global existence and stability for an elliptic-parabolic free boundary problem : Application to a model with tumor growth. Indiana Univ. Math. J., 52 (2003), 1265–1304. [CrossRef] [MathSciNet]
  6. B.V. Bazaliy, A. Friedman. A free boundary problem for an elliptic-parabolic system : Application to a model of tumor growth. Comm. in PDE, 28 (2003), 627.
  7. S. Bunimovich-Mendrazitsky, E. Shochat, L. Stone. Mathematical Model of BCG immuno- therapy in superficial bladder cancer. Bull. Math. Biol., 69 (2007), 1847–1870. [CrossRef] [MathSciNet] [PubMed]
  8. S. Bunimovich-Mendrazitsky, J.C. Gluckman, J. Chaskalovich. A mathematical model of combined bacillus Calmette-Guerin (BCG) and interleuken (IL)-2 immunotherapy of superficial bladder cancer. J. Theor. Biol, 277 (2011), 27–40. [CrossRef] [PubMed]
  9. H.M. Byrne, M.A.J. Chaplain. Growth of necrotic tumors in the presence and absence of inhibitors. Math. Biosci., 135 (1996), 187–216. [CrossRef] [PubMed]
  10. A. Campbell, T. Sivakumaran, M. Davidson, M. Lock, E. Wong. Mathematical modeling of liver metastases tumour growth and control with radiotherapy. Phys. Med. Biol., 53 (2008), 7225–7239. [CrossRef] [PubMed]
  11. X. Chen, A. Friedman. A free boundary problem for elliptic-hyperbolic system : An application to tumor growth. SIAM J. Math. Anal., 35 (2003), 974–986. [CrossRef] [MathSciNet]
  12. X. Chen, S. Cui, A. Friedman. A hyperbolic free boundary problem modeling tumor growth : Asymptotic behavior. Trans. Amer. Math. Soc., 357 (2005), 4771–4804. [CrossRef] [MathSciNet]
  13. S. Cui, A. Friedman. Analysis of a mathematical model of the growth of necrotic tumors. J. Math. Anal. & Appl., 255 (2001), 636–677. [CrossRef] [MathSciNet]
  14. S. Cui, A. Friedman. A free boundary problem for a singular system of differential equations : An application to a model of tumor growth. Trans. Amer. Math. Soc., 355 (2003), 3537–3590. [CrossRef] [MathSciNet]
  15. S. Cui, A. Friedman. A hyperbolic free boundary problem modeling tumor growth. Interfaces & Free Boundaries, 5 (2003), 159–182. [CrossRef]
  16. S.E. Eikenberry, J.D. Nagy, Y. Kuang. The evolutionary impact of androgen levels on prostate cancer in a multi-scale mathematical model. Biol. Direct, 5 (2010), 24–52. [CrossRef] [PubMed]
  17. S.E. Eikenberry, T. Sankar, M.C. Preul, E.J. Kostelich, C.J. Thalhauser, Y. Kuang. Virtual glioblastoma : growth, migration and treatment in a three-dimensional mathematical model. Cell Prolif., 42 (2009), 511–528. [CrossRef] [PubMed]
  18. S. Eikenberry, C. Thalhauser, Y. Kuang. Mathematical modeling of melanoma. PLoS Comput Biol., 5 :e1000362 (2009).
  19. S. Eikenberry, C. Thalhauser, Y. Kuang. Tumor-immune interaction, surgical treatment, and cancer recurrence in a mathematical model of melanoma. PLoS Comput Biol., 5 :e1000362 (2009), Epub 2009, April 24.
  20. M.A. Fontelos, A. Friedman. Symmetry-breaking bifurcations of free boundary problems in three dimensions. Asymptotic Anal., 35 (2003), 187–206.
  21. S.J.H. Franks, H.M. Byrne, J.P. King, J.C.E. Underwood, C.E. Lewis. Modeling the early growth of ductal carcinoma in situ of the breast. J. Math. Biol., 47 (2003), 424–452. [CrossRef] [MathSciNet] [PubMed]
  22. S.J.H. Franks, H.M. Byrne, J.P. King, J.C.E. Underwood, C.E. Lewis. Modeling the growth of comedo ductal carcinoma in situ. Math. Med. & Biol., 20 (2003), 277–308. [CrossRef]
  23. S.J.H. Franks, H.M. Byrne, J.C.E. Underwood, C.E. Lewis. Biological inferences from a mathematical model of comedo ductal carcinoma in situ of the breast. J. Theor. Biol., 232 (2005), 523–543. [CrossRef] [PubMed]
  24. S.J.H. Franks, J.P. King. Interactions between a uniformly proliferating tumor and its surroundings : Uniform material properties. Math. Med. & Biol., 20 (2003), 47–89. [CrossRef]
  25. A. Friedman. A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth. Interfaces and Free Boundaries, 8 (2006), 247–261. [CrossRef] [MathSciNet]
  26. A. Friedman. A multiscale tumor model. Interfaces & Free Boundaries, 10 (2008), 245–262. [CrossRef] [MathSciNet]
  27. A. Friedman. Free boundary value problems associated with multiscale tumor models. Mathematical Modeling of Natural Phenomena, 4 (2009), 134–155. [CrossRef] [EDP Sciences]
  28. A. Friedman, B. Hu. Bifurcation from stability to instability for a free boundary problem arising in tumor model. Arch. Rat. Mech. Anal., 180 (2006), 293–330. [CrossRef]
  29. A. Friedman, B. Hu. Asymptotic stability for a free boundary problem arising in a tumor model. J. Diff. Eqs., 227 (2006), 598–639. [CrossRef]
  30. A. Friedman, B. Hu. Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation. Math. Anal & Appl., 327 (2007), 643–664. [CrossRef] [MathSciNet]
  31. A. Friedman, B. Hu. Bifurcation for a free boundary problem modeling tumor growth by Stokes equation. SIAM J. Math. Anal., 39 (2007), 174–194. [CrossRef] [MathSciNet]
  32. A. Friedman, B. Hu. Stability and instability of Liapounov-Schmidt and Hopf bifurcations for a free boundary problem arising in a tumor model. Trans. Amer. Math. Soc., 360 (2008), 5291–5342. [CrossRef] [MathSciNet]
  33. A. Friedman, B. Hu. The role of oxygen in tissue maintenance : Mathematical modeling and qualitative analysis. Math. Mod. Meth. Appl. Sci., 18 (2008), 1–33. [CrossRef]
  34. A. Friedman, B. Hu, C-Y. Kao. Cell cycle control at the first restriction point and its effect on tissue growth. J. Math. Biol., 60 (2010), 881–907. [CrossRef] [MathSciNet] [PubMed]
  35. A. Friedman, Y. Kim. Tumor cells-proliferation and migration under the influence of their microenvironment. Math Biosci. & Engin., 8 (2011), 373–385.
  36. A. Friedman, F. Reitich. Analysis of a mathematical model for growth of tumors. J. Math. Biol., 38 (1999), 262–284. [CrossRef] [MathSciNet] [PubMed]
  37. A. Friedman, F. Reitich. Symmetry-breaking bifurcation of analytic solutions to free boundary problems : An application to a model of tumor growth. Trans. Amer. Math. Soc., 353 (2001), 1587–1634. [CrossRef] [MathSciNet]
  38. A. Friedman, Y. Tao. Analysis of a model of virus that replicates selectively in tumor cells. J. Math. Biol., 47 (2003), 391–423. [CrossRef] [MathSciNet] [PubMed]
  39. A. Friedman, J.J. Tian, G. Fulci, E.A. Chiocca, J. Wang. Glioma virotherapy : The effects of innate immune suppression and increased viral replication capacity. Cancer Research, 66 (2006), 2314–2319. [CrossRef] [PubMed]
  40. G. Fulci, L. Breymann, D. Gianni, K. Kurozomi, S. Rhee, J. Yu, B. Kaur, D. Louis, R. Weissleder, M. Caligiuri, E.A. Chiocca. Cyclophosphamide enhances glioma virotherapy by inhibiting innate immune responses. PNAS, 103 (2006), 12873–12878. [CrossRef]
  41. V. DeGiorgi, D. Massai, G. Gerlini, F. Mannone, E. Quercioli, et al. Immediate local and regional recurrence after the excision of a polypoid melanoma : Tumor dormancy or tumor activation. Dermatol. Surg., 29 (2003), 664–667. [CrossRef] [PubMed]
  42. J.E.F. Green, S.L. Waters, K.M. Shakesheff, H.M. Byrne. A Mathematical Model of Liver Cell Aggregation In Vitro. Bull. Math. Biol., 71 (2009), 906–930. [CrossRef] [MathSciNet] [PubMed]
  43. J.E.F. Green, S.L. Waters, J.P. Whiteley, L. Edelstein-Keshet, K.M. Shakesheff, H.M. Byrne. Nonlocal models for the formation of hepatocyte-stellate cell aggregates. J. Theor. Biol., 267 (2010), 106–120. [CrossRef] [PubMed]
  44. P.R. Harper, S.K. Jones. Mathematical models for the early detection and treatment of colo-rectal cancer. Health Care Management Science, 8 (2005), 101–109. [CrossRef] [PubMed]
  45. H. Harpold, J. Ec, K. Swanson. The evolution of mathematical modeling of glioma proliferation and invasion. J. Neuropathol. Exp. Neurol., 66 (1) (2007), 1–9. [CrossRef] [PubMed]
  46. A. Ideta, G. Tanaka, T. Takeuchi, K. Aihara. A Mathematical model of intermittent androgen suppression for prostate cancer. J. Nonlinear Sci., 18 (2008), 593–614. [CrossRef]
  47. T.L. Jackson. A mathematical model of prostate tumor growth and androgen-independent relapse. Discrete Cont. Dyn-B, 4 (2004), 187–201. [CrossRef]
  48. T.L. Jackson. A mathematical investigation of the multiple pathways to recurrent prostate cancer : comparison with experimental data. Neoplasia, 6 (2004), 697–704. [CrossRef] [PubMed]
  49. H.V. Jain, S. Clinton, A. Bhinder, A. Friedman. Mathematical model of hormone treatment for prostate cancer, to appear.
  50. Y. Jiang, J. Pjesivac-Grbovic, C. Cantrell, J.P. Freyer. A multiscale model for avascular tumor growth. Biophy. J., 89 (2005), 3884–3894. [CrossRef] [PubMed]
  51. J.B. Jones, J.J. Song, P.M. Hempen, G. Parmigiani, R.H. Hruban, S.E. Kern. Detection of mitochondrial DNA mutations in pancreatic cancer offers a “Mass"-ive advantage over detection of nuclear DNA mutations. Cancer Research, 61 (2001), 1299–1304. [PubMed]
  52. Y. Kim, A. Friedman. Interaction of tumor with its microenvironment : a mathematical model. Bull. Math. Biol., 72 (2010), 1029–1068. [CrossRef] [MathSciNet] [PubMed]
  53. Y. Kim, S. Lawler, M.O. Nowicki, E.A. Chiocca, A. Friedman. A mathematical model of brain tumor : pattern formation of glioma cells outside the tumor spheroid core. J. Theor. Biol., 260 (2009), 359–371. [CrossRef] [PubMed]
  54. Y. Kim, M. Stolarska, H. Othmer. A hybrid model for tumor spheroid growth in vitro I : theoretical development and early results. Math. Mod. Meth. Appl. Sci., 17 (2007), 1773–1798. [CrossRef]
  55. Y. Kim, J. Wallace, F. Li, M. Ostrowski, A. Friedman. Transformed epithelial cells and fibroblasts/myofibroblasts interaction in breast tumor : a mathematical model and experiments. J. Math. Biol., 61 (2010), 401–421. [CrossRef] [MathSciNet] [PubMed]
  56. N.L. Komarova, C. Lengauer, B. Vogelstein, M. Nowak. Dynamics of genetic instability in sporadic and familial colorectal cancer. Cancer Biology & Therapy, 1 (2002), 685–692. [CrossRef] [MathSciNet] [PubMed]
  57. H.A. Levine, M. Nilsen-Hamilton. Angiogenesis-A biochemical/mathematical perspective. Lecture Notes Math., 1872 (2006), 23–76, Springer-Verlag, Berlin-Heidelberg. [CrossRef]
  58. H.A. Levine, S.L. Pamuk, B.D. Sleeman, M. Nilsen-Hamilton. Mathematical modeling of capillary formation and development in tumor angiogenesis : penetration into the stroma. Bull. Math. Biol., 63 (2001), 801–863. [CrossRef] [PubMed]
  59. E. Mandonnet, J. Delattre, M. Tanguy, K. Swanson, A. Carpentier, H. Duffau, P. Cornu, R. Effenterre, J. Ec, L.J. Capelle. Continuous growth of mean tumor diameter in a subset of grade ii gliomas. Ann. Neurol., 53 (4) (2003), 524–528. [CrossRef] [PubMed]
  60. N. Mantzaris, S. Webb, H.G. Othmer. Mathematical modeling of tumor angiogenesis. J. Math. Biol., 49 (2004), 111–187. [MathSciNet] [PubMed]
  61. A. Perumpanani, H. Byrne. Extracellular matrix concentration exerts selection pressure on invasive cells. Eur. J. Cancer, 35(8) (1999), 1274–1280. [CrossRef] [PubMed]
  62. G.J. Pettet, C.P. Please, M.J. Tindall, D.L.S. McElwain. The migration of cells in multicell tumor spheroids. Bull. Math. Biol., 63 (2001), 231–257. [CrossRef] [PubMed]
  63. L.K. Potter, M.G. Zagar, H.A. Barton. Mathematical model for the androgenic regulation of the prostate in intact and castrated adult male rats. Am. J. Physiol. Endocrinol. Metab., 291 (2006), E952–E964. [CrossRef] [PubMed]
  64. R. Ribba, T. Colin, S. Schnell. A multiscale model of cancer, and its use in analyzing irradiation therapies. Theor. Biol. & Med. Mod., 3 (2006), 7, 1–19. [CrossRef]
  65. B. Ribba, O. Sant, T. Colin, D. Bresch, E. Grenien, J.P. Boissel. A multiscale model of avascular tumor growth to investigate agents. J. Theor. Biol., 243 (2006), 532–541. [CrossRef] [PubMed]
  66. J. Sherratt, S. Gourley, N. Armstrong, K. Painter. Boundedness of solutions of a non-local reaction diffusion model for adhesion in cell aggregation and cancer invasion. Eur. J.Appl. Math., 20 (2009), 123–144. [CrossRef]
  67. K. Swanson, J. Ec, J. Murray. A quantitative model for differential motility of gliomas in grey and white matter. Cell Prolif., 33 (5) (2000), 317–329. [CrossRef] [PubMed]
  68. I.M.M. van Leeuwen, H.M. Byrne, O.E. Jensen, J.R. King. Crypt dynamics and colorectal cancer : advances in mathematical modeling. Cell Prolif., 39 (2006), 157–181. [CrossRef] [PubMed]
  69. J.T. Wu, H.M. Byrne, D.H. Kirn, L.M. Wein. Modeling and analysis of a virus that replicates selectively in tumor cells. Bull. Math. Biol., 63 (2001), 731–768. [CrossRef] [PubMed]
  70. J. Wu, S. Cui. Asymptotic stability of stationary solutions of a free boundary problem modeling the growth of tumors with fluid tissues. SIAM J. Math. Anal., 41 (2010), 391–414. [CrossRef]
  71. J.T. Wu, D.H. Kirn, L.M. Wein. Analysis of a three-way race between tumor growth, a replication-competent virus and an immune response. Bull. Math. Biol., 66 (2004), 605–625. [CrossRef] [MathSciNet] [PubMed]

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