A hierarchy of cancer models and their mathematical challenges

Discrete and Continuous Dynamical Systems - Series B

Volumn 4, Issue 1, 2004, Pages 147-159

  Friedman, Avner ^a  

^a OHIO STATE UNIVERSITY   (United States)

Author keywords

Bifurcation; Cancer; Free boundary problems; Proliferating cells

Indexed keywords

EID: 1042279440     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2004.4.147     Document Type: Conference Paper

References (30)

1. J. A. Adam and S. A. Maggelakis, Diffusion regulated growth characteristics of spherical prevascular carcinoma, Bull. Math. Biology, 52 (1990), 549-582.
2. B. V. Bazally and A. Friedman, A free boundary problem for an elliptic-parabolic system: Application to a model of tumor growth, Comm. PDE, 28 (2003), 627-675.
3. B. V. Bazaliy and A. Friedman, Global existence and stability for an elliptic-parabolic free boundary problem; An application to a model of tumor growth, Indiana University Math. J., to appear.
4. N. Bellomo and L. Preziosi, Modelling and mathematical problems related to tumor evolution and its interaction with the immune system, Mathematical and Computer Modelling, 32 (2000), 413-452.
5. N. Britton and M. Chaplain, A qualitative analysis of some models of tissue growth, Math. Biosci. 113(1993), 77-89.
6. H. M. Byrne, A weakly nonlinear analysis of a model of vascular solid tumour growth, J. Math. Biol. 39 (1999), 59-89.
7. H. M. Byrne and M. A. J. Chaplain, Growth of nonnecrotic tumours in the presence and absence of inhibitors, Mathematical Biosciences, 181 (1995), 130-151.
8. H. Byrne and M. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci. 135 (1996), 187-216.
9. H. M. Byrne and M. A. J. Chaplain, Free boundary value problems associated with growth and development of multicellular spheroids, European J. Appl. Math., 8 (1997), 639-358.
10. X. Chen, S. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth: A symptotic behavior, to appear.
11. X. Chen and A. Friedman, A free boundary problem for elliptic-hyperbolic system: An application to tumor growth, SIAM J. Math. Anal., to appear.
12. S. Cui and A. Friedman, Analysis of a mathematical model of the effect of inhibitors on the growth of tumors, Math. Biosci. 164 (2000), 103-137.
13. S. Cui and A. Friedman, Analysis of a mathematical model of the growth of necrotic tumors, J. Math. Anal. Appl. 255 (2001), 636-677.
14. S. Cui and A. Friedman, A free boundary problem for a singular system of differential equations: An application to a model of tumor growth, Trans. AMS, 355 (2003), 3537-3590.
15. S. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth, Interfaces and Free Boundaries, 5 (2003), 159-182.
16. M. Fontelos and A. Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptotic Analysis, to appear.
17. M. Dorie, R. Kallman and M. Coyne, Effect of cytochalasin b, nocodazole and irradiation on migration and internalization of cells and microspheres in tumor cell spheroids, Exp. Cell Res. 166 (1986), 370-378.
18. M. Dorie, R. Kallman, D. Rapacchietta and et al, Migration and internalization of cells and polystrene microspheres in tumor cell spheroids, Exp. Cell Res. 141 (1982), 201-209.
19. A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol. 38 (1999), 262-284.
20. A. Friedman and 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 (2000), 1587-1634.
21. A. Friedman and F. Reitich, On the existence of spatially patterned dormant malignancies in a model for the growth of non-necrotic vascular tumor, Math. Models and Methods in Appl. Sciences, 77 (2001), 1-25.
22. H. Greenspan, Models for the growth of solid tumor by diffusion, Stud. Appl. Math 51 (1972), 317-340.
23. H. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theor. Biol. 56 (1976), 229-242.
24. F. Hughes and C. McCulloch, Quantification of chemotactic response of quiescent and proliferating fibroblasts in boyden chambers by computer-assisted image analysis, J. Histochem. Cytochem., 39 (1991), 243-246.
25. D. McElwain and G. Pettet, Cell migration in multicell spheroids: swimming against the tides, Bull. Math. Biol., 55 (1993), 655-674.
26. J. Palka, B. Adelman-Griff, P. Franz and K. Bayreuter, Differentiation stage and cell cycle position determine the chemotactic response of fibroblasts, Folia Histochem. Cytobiol. 34 (1996), 121-127.
27. G. Pettet, C. P. Please, M. J. Tindall and D. McElwain, The migration of cells in multicell tumor spheroids, Bull. Math. Biol. 63 (2001), 231-257.
28. J. Sherratt and M. Chaplain, A new mathematical model for avascular tumor growth, J. Math. Biol. 43 (2001), 291-312.
29. K. Thompson and H. Byrne, Modelling the internalisation of labelled cells in tumor spheroids, Bull. Math. Biol. 61 (1999), 601-623.
30. J. P. Ward and J. R. King, Mathematical modelling of avascular-tumor growth II: Modelling growth saturation, IMA J. Math. Appl. Med. Biol. 15 (1998), 1-42.