Mathematical analysis and challenges arising from models of tumor growth

Mathematical Models and Methods in Applied Sciences

Volumn 17, Issue SUPPL. 1, 2007, Pages 1751-1772

  Friedman, Avner ^a  

^a OHIO STATE UNIVERSITY   (United States)

Author keywords

Asymptotic stability; Bifurcation; Free boundary problems; Multiscale models; Tumor growth

Indexed keywords

EID: 35348819870     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202507002467     Document Type: Article

References (39)

1. R. P. Aranjo and D. L. S. McElwain, A history of the study of solid tumor growth: The contribution of mathematical modelling, Bull Math. Biol. 66 (2003) 1039-1091.
2. B. P. Ayati, G. F. Webb and A. R. A. Anderson, Computational methods and results for structured mutliscale methods of tumor invasion, Multiscale Mod. Simul. 5 (2006) 1-20.
3. B. Bazaliy and A. Friedman, A free boundary problem for an elliptic-parabolic system: Application to a model of tumor growth, Comm. Partial Diff. Eqns. 28 (2003) 517-560.
4. B. Bazaliy and A. Friedman, Global existence and stability for an elliptic-parabolic free boundary problem: An application of a model of tumor growth. Indiana Univ. Math. J. 52 (2003) 1265-1304.
5. B. Bazaliy and A. Friedman, The Hele-Shaw problem with surface tension in a halfplane, J. Diff. Eqns. 216 (2005) 439-469.
6. B. Bazaliy and A. Friedman, The Hele-Shaw problem with surface tension in a halfplane: A model problem, J. Diff. Eqns. 216 (2005) 387-483.
7. A. Borisovich and A. Friedman, Symmetry breaking bifurcations for free boundary problems, Indiana Univ. Math J. 54 (2005) 927-947.
8. H. M. Byrne, The importance of intercellular adhesion in the development of carcinomas, I MA J. Math. Appl. Med. Biol. 14 (1997) 305-323.
9. H. M. Byrne, A weakly nonlinear analysis of a model of vascular solid tumor, J. Math. Biol. 39 (1999) 59-89.
10. H. M. Byrne and M. A. J. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci. 130 (1995) 130-151.
11. H. M. Byrne and M. A. J. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci. 135 (1996) 187-216.
12. H. M. Byrne and M. A. J. Chaplain, Modelling the role of cell-cell adhesion in the growth and development of carcinomas, Math. Comput. Mod. 24 (1996) 1-17.
13. H. M. Byrne and M. A. J. Chaplain, Free boundary value problems associated with growth and development of multicellular spheroids, Euro. J. Appl. Math. 8 (1997) 358-639.
14. M. A. J. Chaplain, The development of a spatial pattern in a model for cancer growth, In Experimental and Theoretical Advances in Biological Pattern Formation, eds. H. G. Othmer, P. K. Maini and J. D. Murray (Plenum Press, 1993), pp. 45-60.
15. X. Chen, S. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior, Trans. Amer. Math. Soc. 357 (2005) 4771-4804.
16. X. Chen and A. Friedman, A free boundary problem for elliptic-hyperbolic system: An application to tumor growth, SIAM J. Math. Anal. 35 (2003) 974-986.
17. M. G. Crandall and R. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971) 321-340.
18. S. Cui, Existence of a stationary solution for the modified Ward-King tumor growth model, Adv. Appl. Math. 36 (2006) 421-445.
19. S. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth, Interfaces Free Bound. 5 (2003) 159-182.
20. 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. Amer. Math. Soc. 355 (2003) 3537-3590.
21. J. Escher, Classical solutions to a moving boundary problem for an elliptic-parabolic system, Interfaces and Free Boundaries 6 (2004) 175-193.
22. M. A. Fontelos and A. Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymp. Anal. 35 (2003) 187-206.
23. M. A. Fontelos and A. Friedman, Symmetry-breaking bifurcations of charged drops, Arch. Rat. Mech. Anal. 172 (2004) 267-294.
24. S. J. H. Franks, H. M. Byrne, J. P. King, J. C. E. Underwood and C. E. Lewis, Modelling the early growth of ductal carcinoma in situ of the breast, J. Math. Biol. 47 (2003) 424-452.
25. S. J. H. Franks, H. M. Byrne, J. P. King, J. C. E. Underwood and C. E. Lewis, Modelling the growth of comedo ductal carcinoma in situ, Math. Med. Biol. 20 (2003) 277-308.
26. S. J. H. Franks, H. M. Byrne, J. C. E. Underwood and 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.
27. S. J. H. Franks and J. P. King, Interactions between a uniformly proliferating tumour and its surroundings: Uniform material properties, Math. Med. Biol. 20 (2003) 47-89.
28. A. Friedman, A free boundary problem for coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth, Interface and Free Boundaries 8 (2006) 247-261.
29. A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem arising in tumor model, Arch. Rat. Mech. Anal. 180 (2006) 293-330.
30. A. Friedman and B. Hu, Asymptotic stability for a free boundary problem arising in a tumor model. J. Diff. Eqns. 227 (2006) 598-639.
31. A. Friedman and B. Hu, Stability and instability of Liapounov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model, Trans. Amer. Math. Soc., in press.
32. A. Friedman and 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.
33. A. Friedman and B. Hu, Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal., in press.
34. A. Friedman and F. Reitich, Analysis of a mathematical model for growth of tumors, J. Math. Biol. 38 (1999) 262-284.
35. 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 (2001) 1587-1634.
36. H. Greenspan, Models for the growth of solid tumor by diffusion, Stud. Appl. Math. 51 (1972) 317-340.
37. H. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theor. Biol. 56 (1976) 229-242.
38. G. J. Pettet, C. P. Please, M. J. Tindall and D. L. S. McElwain, The migration of cells in multicell tumor spheroids, Bull. Math. Biol. 63 (2001) 231-257.
39. B. Ribba, T. Colin and S. Schnell, A multiscale model of cancer, and its use in analyzing irradiation therapies. Theor. Bio. Med. Mod. 3 (2006) 1-19.