Cancer as Multifaceted Disease
Mathematical Modelling of Natural Phenomena (2012)
- Volume: 7, Issue: 1, page 3-28
- ISSN: 0973-5348
Access Full Article
topAbstract
topHow to cite
topReferences
top- A. Angelle. Pancreatic cancer shown to be surprisingly slow killer. Live Science, October 27, 2010.
- N. Armstrong, K. Painter, J. Sherratt. A continuum approach to modeling cell-cell adhesion. J. Theor. Biol., 243 (1), 98–113.
- 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.
- 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.
- 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.
- 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.
- S. Bunimovich-Mendrazitsky, E. Shochat, L. Stone. Mathematical Model of BCG immuno- therapy in superficial bladder cancer. Bull. Math. Biol., 69 (2007), 1847–1870.
- 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.
- H.M. Byrne, M.A.J. Chaplain. Growth of necrotic tumors in the presence and absence of inhibitors. Math. Biosci., 135 (1996), 187–216.
- 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.
- 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.
- X. Chen, S. Cui, A. Friedman. A hyperbolic free boundary problem modeling tumor growth : Asymptotic behavior. Trans. Amer. Math. Soc., 357 (2005), 4771–4804.
- S. Cui, A. Friedman. Analysis of a mathematical model of the growth of necrotic tumors. J. Math. Anal. & Appl., 255 (2001), 636–677.
- 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.
- S. Cui, A. Friedman. A hyperbolic free boundary problem modeling tumor growth. Interfaces & Free Boundaries, 5 (2003), 159–182.
- 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.
- 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.
- S. Eikenberry, C. Thalhauser, Y. Kuang. Mathematical modeling of melanoma. PLoS Comput Biol., 5 :e1000362 (2009).
- 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.
- M.A. Fontelos, A. Friedman. Symmetry-breaking bifurcations of free boundary problems in three dimensions. Asymptotic Anal., 35 (2003), 187–206.
- 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.
- 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.
- 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.
- 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.
- 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.
- A. Friedman. A multiscale tumor model. Interfaces & Free Boundaries, 10 (2008), 245–262.
- A. Friedman. Free boundary value problems associated with multiscale tumor models. Mathematical Modeling of Natural Phenomena, 4 (2009), 134–155.
- 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.
- A. Friedman, B. Hu. Asymptotic stability for a free boundary problem arising in a tumor model. J. Diff. Eqs., 227 (2006), 598–639.
- 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.
- A. Friedman, B. Hu. Bifurcation for a free boundary problem modeling tumor growth by Stokes equation. SIAM J. Math. Anal., 39 (2007), 174–194.
- 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.
- 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.
- 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.
- A. Friedman, Y. Kim. Tumor cells-proliferation and migration under the influence of their microenvironment. Math Biosci. & Engin., 8 (2011), 373–385.
- A. Friedman, F. Reitich. Analysis of a mathematical model for growth of tumors. J. Math. Biol., 38 (1999), 262–284.
- 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.
- A. Friedman, Y. Tao. Analysis of a model of virus that replicates selectively in tumor cells. J. Math. Biol., 47 (2003), 391–423.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- T.L. Jackson. A mathematical model of prostate tumor growth and androgen-independent relapse. Discrete Cont. Dyn-B, 4 (2004), 187–201.
- T.L. Jackson. A mathematical investigation of the multiple pathways to recurrent prostate cancer : comparison with experimental data. Neoplasia, 6 (2004), 697–704.
- H.V. Jain, S. Clinton, A. Bhinder, A. Friedman. Mathematical model of hormone treatment for prostate cancer, to appear.
- Y. Jiang, J. Pjesivac-Grbovic, C. Cantrell, J.P. Freyer. A multiscale model for avascular tumor growth. Biophy. J., 89 (2005), 3884–3894.
- 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.
- Y. Kim, A. Friedman. Interaction of tumor with its microenvironment : a mathematical model. Bull. Math. Biol., 72 (2010), 1029–1068.
- 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.
- 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.
- 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.
- 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.
- H.A. Levine, M. Nilsen-Hamilton. Angiogenesis-A biochemical/mathematical perspective. Lecture Notes Math., 1872 (2006), 23–76, Springer-Verlag, Berlin-Heidelberg.
- 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.
- 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.
- N. Mantzaris, S. Webb, H.G. Othmer. Mathematical modeling of tumor angiogenesis. J. Math. Biol., 49 (2004), 111–187.
- A. Perumpanani, H. Byrne. Extracellular matrix concentration exerts selection pressure on invasive cells. Eur. J. Cancer, 35(8) (1999), 1274–1280.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.