On the existence of spatially patterned dormant malignancies in a model for the growth of non-necrotic vascular tumors

Mathematical Models and Methods in Applied Sciences

Volumn 11, Issue 4, 2001, Pages 601-625

  Friedman, Avner ^a   Reitich, Fernando ^a  

^a UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Free boundary problems; Parabolic equations; Tumors

Indexed keywords

EID: 0035634058     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S021820250100101X     Document Type: Article

References (19)

1. J. A. Adam, General aspects of modeling tumor growth and immune response, in A Survey of Models for Tumor-Immune System Dynamics, eds. J. A. Adam and N. Bellomo (Birkhäuser, 1996), pp. 15-87.
2. J. A. Adam and S. A. Maggelakis, Diffusion regulated growth characteristics of a spherical prevascular carcinoma, Bull. Math. Biol. 52 (1990) 549-582.
3. O. P. Bruno and P. Laurence, Existence of three-dimensional toroidal MHD equilibria with nonconstant pressure, Comm. Pure Appl. Math. 49 (1996) 717-764.
4. O. P. Bruno and F. Reitich, Solution of a boundary value problem for the Helmholtz equation via variation of the boundary into the complex domain, Proc. R. Soc. Edinburgh, Sec. A 122 (1992) 317-340.
5. H. M. Byrne, The importance of intercellular adhesion in the development of carcinomas, IMA J. Math. Appl. Med. Biol. 14 (1997) 305-323.
6. H. M. Byrne, The effect of time delays on the dynamics of avascular tumor growth, Math. Biosci. 144 (1997) 83-117.
7. H. M. Byrne, A weakly nonlinear analysis of a model of avascular solid tumour growth, J. Math. Biol. 39 (1999) 59-89.
8. H. M. Byrne and M. A. J. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci. 130 (1995) 151-181.
9. H. M. Byrne and M. A. J. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci. 135 (1996) 187-216.
10. 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.
11. A. Friedman, On the regularity of solutions of nonlinear elliptic and parabolic systems of partial differential equations, J. Math. Mech. 7 (1958) 43-60.
12. A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol. 38 (1999) 262-284.
13. 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.
14. H. P. Greenspan, Models for the growth of a solid tumor by diffusion, Studies Appl. Math. 52 (1972) 317-340.
15. H. P. Greenspan, On the growth and stability of cell cultures and solid tumors, Theor. Biol. 56 (1976) 229-242.
16. S. A. Maggelakis and J. A. Adam, Mathematical model for prevascular growth of a spherical carcinoma, Math. Comp. Modelling 13 (1990) 23-38.
17. Math. Mod. Methods Appl. Sci., Special Issue, 9 (4) (1999).
18. D. L. S. McElwain and L. E. Morris, Apoptosis as a volume loss mechanism in mathematical models of solid tumor growth, Math. Biosci. 39 (1978) 147-157.
19. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge Univ. Press, 1944), 2nd edition.