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Cancer as Multifaceted Disease

Published online by Cambridge University Press:  25 January 2012

A. Friedman
A. Friedman*
Affiliation:
Department of Mathematics, The Ohio State University, 43221 Columbus, OH USA
*
⋆ E-mail: afriedman@math.ohio-state.edu
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Abstract

Cancer has recently overtaken heart disease as the world’s biggest killer. Cancer is initiated by gene mutations that result in local proliferation of abnormal cells and their migration to other parts of the human body, a process called metastasis. The metastasized cancer cells then interfere with the normal functions of the body, eventually leading to death. There are two hundred types of cancer, classified by their point of origin. Most of them share some common features, but they also have their specific character. In this article we review mathematical models of such common features and then proceed to describe models of specific cancer diseases.

Keywords

tumor modelsfree boundary problemshyperbolic equationsparabolic equationsStokes equation
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 1: Cancer modeling , 2012 , pp. 3 - 28
Copyright
© EDP Sciences, 2012

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References

A. Angelle. Pancreatic cancer shown to be surprisingly slow killer. Live Science, October 27, 2010.
Armstrong, N., Painter, K., Sherratt, J.. A continuum approach to modeling cell-cell adhesion. J. Theor. Biol., 243 (1), 98–113. CrossRef
Ayati, B.P., Webb, G.F., Anderson, A.R.A.. Computational methods and results for structured multiscale methods of tumor invasion. Multiscale Model. Simul., 5 (2006), 1–20. CrossRefGoogle Scholar
Aznavoorian, S., Stracke, M., Krutzsch, H., Schiffmann, E., Liotta, L.. Signal transduction for chemotaxis and haptotaxis by matrix molecules in tumor cells. J. Cell Biol., 110(4), (1990), 1427–1438. CrossRefGoogle Scholar
Bazaliy, B., Friedman, A.. 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. Google Scholar
Bazaliy, B.V., Friedman, A.. A free boundary problem for an elliptic-parabolic system : Application to a model of tumor growth. Comm. in PDE, 28 (2003), 627. CrossRefGoogle Scholar
Bunimovich-Mendrazitsky, S., Shochat, E., Stone, L.. Mathematical Model of BCG immuno- therapy in superficial bladder cancer. Bull. Math. Biol., 69 (2007), 1847–1870. CrossRefGoogle Scholar
Bunimovich-Mendrazitsky, S., Gluckman, J.C., Chaskalovich, J.. 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. CrossRefGoogle Scholar
Byrne, H.M., Chaplain, M.A.J.. Growth of necrotic tumors in the presence and absence of inhibitors. Math. Biosci., 135 (1996), 187–216. CrossRefGoogle ScholarPubMed
Campbell, A., Sivakumaran, T., Davidson, M., Lock, M., Wong, E.. Mathematical modeling of liver metastases tumour growth and control with radiotherapy. Phys. Med. Biol., 53 (2008), 7225–7239. CrossRefGoogle ScholarPubMed
Chen, X., Friedman, A.. A free boundary problem for elliptic-hyperbolic system : An application to tumor growth. SIAM J. Math. Anal., 35 (2003), 974–986. CrossRefGoogle Scholar
Chen, X., Cui, S., Friedman, A.. A hyperbolic free boundary problem modeling tumor growth : Asymptotic behavior. Trans. Amer. Math. Soc., 357 (2005), 4771–4804. CrossRefGoogle Scholar
Cui, S., Friedman, A.. Analysis of a mathematical model of the growth of necrotic tumors. J. Math. Anal. & Appl., 255 (2001), 636–677. CrossRefGoogle Scholar
Cui, S., Friedman, A.. 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. CrossRefGoogle Scholar
Cui, S., Friedman, A.. A hyperbolic free boundary problem modeling tumor growth. Interfaces & Free Boundaries, 5 (2003), 159–182. CrossRefGoogle Scholar
Eikenberry, S.E., Nagy, J.D., Kuang, Y.. The evolutionary impact of androgen levels on prostate cancer in a multi-scale mathematical model. Biol. Direct, 5 (2010), 24–52. CrossRefGoogle Scholar
Eikenberry, S.E., Sankar, T., Preul, M.C., Kostelich, E.J., Thalhauser, C.J., Kuang, Y.. Virtual glioblastoma : growth, migration and treatment in a three-dimensional mathematical model. Cell Prolif., 42 (2009), 511–528. CrossRefGoogle Scholar
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.
Fontelos, M.A., Friedman, A.. Symmetry-breaking bifurcations of free boundary problems in three dimensions. Asymptotic Anal., 35 (2003), 187–206. Google Scholar
Franks, S.J.H., Byrne, H.M., King, J.P., Underwood, J.C.E., Lewis, C.E.. Modeling the early growth of ductal carcinoma in situ of the breast. J. Math. Biol., 47 (2003), 424–452. CrossRefGoogle Scholar
Franks, S.J.H., Byrne, H.M., King, J.P., Underwood, J.C.E., Lewis, C.E.. Modeling the growth of comedo ductal carcinoma in situ. Math. Med. & Biol., 20 (2003), 277–308. CrossRefGoogle Scholar
Franks, S.J.H., Byrne, H.M., Underwood, J.C.E., Lewis, C.E.. Biological inferences from a mathematical model of comedo ductal carcinoma in situ of the breast. J. Theor. Biol., 232 (2005), 523–543. CrossRefGoogle ScholarPubMed
Franks, S.J.H., King, J.P.. Interactions between a uniformly proliferating tumor and its surroundings : Uniform material properties. Math. Med. & Biol., 20 (2003), 47–89. CrossRefGoogle ScholarPubMed
Friedman, A.. 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. CrossRefGoogle Scholar
Friedman, A.. A multiscale tumor model. Interfaces & Free Boundaries, 10 (2008), 245–262. CrossRefGoogle Scholar
Friedman, A.. Free boundary value problems associated with multiscale tumor models. Mathematical Modeling of Natural Phenomena, 4 (2009), 134–155. CrossRefGoogle Scholar
Friedman, A., Hu, B.. Bifurcation from stability to instability for a free boundary problem arising in tumor model. Arch. Rat. Mech. Anal., 180 (2006), 293–330. CrossRefGoogle Scholar
Friedman, A., Hu, B.. Asymptotic stability for a free boundary problem arising in a tumor model. J. Diff. Eqs., 227 (2006), 598–639. CrossRefGoogle Scholar
Friedman, A., Hu, B.. Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation. Math. Anal & Appl., 327 (2007), 643–664. CrossRefGoogle Scholar
Friedman, A., Hu, B.. Bifurcation for a free boundary problem modeling tumor growth by Stokes equation. SIAM J. Math. Anal., 39 (2007), 174–194. CrossRefGoogle Scholar
Friedman, A., Hu, B.. 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. CrossRefGoogle Scholar
Friedman, A., Hu, B.. The role of oxygen in tissue maintenance : Mathematical modeling and qualitative analysis. Math. Mod. Meth. Appl. Sci., 18 (2008), 1–33. CrossRefGoogle Scholar
Friedman, A., Hu, B., Kao, C-Y.. Cell cycle control at the first restriction point and its effect on tissue growth. J. Math. Biol., 60 (2010), 881–907. CrossRefGoogle ScholarPubMed
Friedman, A., Kim, Y.. Tumor cells-proliferation and migration under the influence of their microenvironment. Math Biosci. & Engin., 8 (2011), 373–385. CrossRefGoogle Scholar
Friedman, A., Reitich, F.. Analysis of a mathematical model for growth of tumors. J. Math. Biol., 38 (1999), 262–284. CrossRefGoogle ScholarPubMed
Friedman, A., Reitich, F.. 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. CrossRefGoogle Scholar
Friedman, A., Tao, Y.. Analysis of a model of virus that replicates selectively in tumor cells. J. Math. Biol., 47 (2003), 391–423. CrossRefGoogle ScholarPubMed
Friedman, A., Tian, J.J., Fulci, G., Chiocca, E.A., Wang, J.. Glioma virotherapy : The effects of innate immune suppression and increased viral replication capacity. Cancer Research, 66 (2006), 2314–2319. CrossRefGoogle Scholar
Fulci, G., Breymann, L., Gianni, D., Kurozomi, K., Rhee, S., Yu, J., Kaur, B., Louis, D., Weissleder, R., Caligiuri, M., Chiocca, E.A.. Cyclophosphamide enhances glioma virotherapy by inhibiting innate immune responses. PNAS, 103 (2006), 12873–12878. CrossRefGoogle ScholarPubMed
DeGiorgi, V., Massai, D., Gerlini, G., Mannone, F., Quercioli, E., 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. Google Scholar
Green, J.E.F., Waters, S.L., Shakesheff, K.M., Byrne, H.M.. A Mathematical Model of Liver Cell Aggregation In Vitro. Bull. Math. Biol., 71 (2009), 906–930. CrossRefGoogle ScholarPubMed
Green, J.E.F., Waters, S.L., Whiteley, J.P., Edelstein-Keshet, L., Shakesheff, K.M., Byrne, H.M.. Nonlocal models for the formation of hepatocyte-stellate cell aggregates. J. Theor. Biol., 267 (2010), 106–120. CrossRefGoogle Scholar
Harper, P.R., Jones, S.K.. Mathematical models for the early detection and treatment of colo-rectal cancer. Health Care Management Science, 8 (2005), 101–109. CrossRefGoogle Scholar
Harpold, H., Ec, J., Swanson, K.. The evolution of mathematical modeling of glioma proliferation and invasion. J. Neuropathol. Exp. Neurol., 66 (1) (2007), 1–9. CrossRefGoogle ScholarPubMed
Ideta, A., Tanaka, G., Takeuchi, T., Aihara, K.. A Mathematical model of intermittent androgen suppression for prostate cancer. J. Nonlinear Sci., 18 (2008), 593–614. CrossRefGoogle Scholar
Jackson, T.L.. A mathematical model of prostate tumor growth and androgen-independent relapse. Discrete Cont. Dyn-B, 4 (2004), 187–201. CrossRefGoogle Scholar
Jackson, T.L.. A mathematical investigation of the multiple pathways to recurrent prostate cancer : comparison with experimental data. Neoplasia, 6 (2004), 697–704. CrossRefGoogle ScholarPubMed
H.V. Jain, S. Clinton, A. Bhinder, A. Friedman. Mathematical model of hormone treatment for prostate cancer, to appear.
Jiang, Y., Pjesivac-Grbovic, J., Cantrell, C., Freyer, J.P.. A multiscale model for avascular tumor growth. Biophy. J., 89 (2005), 3884–3894. CrossRefGoogle Scholar
Jones, J.B., Song, J.J., Hempen, P.M., Parmigiani, G., Hruban, R.H., Kern, S.E.. 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. Google ScholarPubMed
Kim, Y., Friedman, A.. Interaction of tumor with its microenvironment : a mathematical model. Bull. Math. Biol., 72 (2010), 1029–1068. CrossRefGoogle Scholar
Kim, Y., Lawler, S., Nowicki, M.O., Chiocca, E.A., Friedman, A.. A mathematical model of brain tumor : pattern formation of glioma cells outside the tumor spheroid core. J. Theor. Biol., 260 (2009), 359–371. CrossRefGoogle ScholarPubMed
Kim, Y., Stolarska, M., Othmer, H.. A hybrid model for tumor spheroid growth in vitro I : theoretical development and early results. Math. Mod. Meth. Appl. Sci., 17 (2007), 1773–1798. CrossRefGoogle Scholar
Kim, Y., Wallace, J., Li, F., Ostrowski, M., Friedman, A.. Transformed epithelial cells and fibroblasts/myofibroblasts interaction in breast tumor : a mathematical model and experiments. J. Math. Biol., 61 (2010), 401–421. CrossRefGoogle Scholar
Komarova, N.L., Lengauer, C., Vogelstein, B., Nowak, M.. Dynamics of genetic instability in sporadic and familial colorectal cancer. Cancer Biology & Therapy, 1 (2002), 685–692. CrossRefGoogle ScholarPubMed
Levine, H.A., Nilsen-Hamilton, M.. Angiogenesis-A biochemical/mathematical perspective. Lecture Notes Math., 1872 (2006), 23–76, Springer-Verlag, Berlin-Heidelberg. CrossRefGoogle Scholar
Levine, H.A., Pamuk, S.L., Sleeman, B.D., Nilsen-Hamilton, M.. Mathematical modeling of capillary formation and development in tumor angiogenesis : penetration into the stroma. Bull. Math. Biol., 63 (2001), 801–863. CrossRefGoogle ScholarPubMed
Mandonnet, E., Delattre, J., Tanguy, M., Swanson, K., Carpentier, A., Duffau, H., Cornu, P., Effenterre, R., Ec, J., Capelle, L.J.. Continuous growth of mean tumor diameter in a subset of grade ii gliomas. Ann. Neurol., 53 (4) (2003), 524–528. CrossRefGoogle Scholar
Mantzaris, N., Webb, S., Othmer, H.G.. Mathematical modeling of tumor angiogenesis. J. Math. Biol., 49 (2004), 111–187. CrossRefGoogle Scholar
Perumpanani, A., Byrne, H.. Extracellular matrix concentration exerts selection pressure on invasive cells. Eur. J. Cancer, 35(8) (1999), 1274–1280. CrossRefGoogle ScholarPubMed
Pettet, G.J., Please, C.P., Tindall, M.J., McElwain, D.L.S.. The migration of cells in multicell tumor spheroids. Bull. Math. Biol., 63 (2001), 231–257. CrossRefGoogle ScholarPubMed
Potter, L.K., Zagar, M.G., Barton, H.A.. 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. CrossRefGoogle ScholarPubMed
Ribba, R., Colin, T., Schnell, S.. A multiscale model of cancer, and its use in analyzing irradiation therapies. Theor. Biol. & Med. Mod., 3 (2006), 7, 1–19. Google ScholarPubMed
Ribba, B., Sant, O., Colin, T., Bresch, D., Grenien, E., Boissel, J.P.. A multiscale model of avascular tumor growth to investigate agents. J. Theor. Biol., 243 (2006), 532–541. CrossRefGoogle Scholar
Sherratt, J., Gourley, S., Armstrong, N., Painter, K.. 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. CrossRefGoogle Scholar
Swanson, K., Ec, J., Murray, J.. A quantitative model for differential motility of gliomas in grey and white matter. Cell Prolif., 33 (5) (2000), 317–329. CrossRefGoogle ScholarPubMed
van Leeuwen, I.M.M., Byrne, H.M., Jensen, O.E., King, J.R.. Crypt dynamics and colorectal cancer : advances in mathematical modeling. Cell Prolif., 39 (2006), 157–181. CrossRefGoogle Scholar
Wu, J.T., Byrne, H.M., Kirn, D.H., Wein, L.M.. Modeling and analysis of a virus that replicates selectively in tumor cells. Bull. Math. Biol., 63 (2001), 731–768. CrossRefGoogle ScholarPubMed
Wu, J., Cui, S.. 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. CrossRefGoogle Scholar
Wu, J.T., Kirn, D.H., Wein, L.M.. Analysis of a three-way race between tumor growth, a replication-competent virus and an immune response. Bull. Math. Biol., 66 (2004), 605–625. CrossRefGoogle ScholarPubMed