Free boundary problems associated with multiscale tumor models

Mathematical Modelling of Natural Phenomena

Volumn 4, Issue 3, 2009, Pages 134-155

  Friedman, A ^a  

^a OHIO STATE UNIVERSITY   (United States)

Author keywords

cell cycle; free boundary problems; system of PDEs; tumor

Indexed keywords

CELL CYCLE; COUPLED SYSTEMS; CYCLE TIME; EXISTENCE AND UNIQUENESS THEOREM; FREE BOUNDARY; FREE-BOUNDARY PROBLEMS; GENE MUTATIONS; HYPERBOLIC EQUATIONS; INDIVIDUAL CELLS; MULTISCALES; NONLOCAL TERM; POPULATION DENSITIES; SYSTEM OF PDES; TUMOR CELLS; TUMOR MODELS; TWO TIME SCALE;

BOUNDARY VALUE PROBLEMS; EULER EQUATIONS; GENES; NONLINEAR EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; POPULATION STATISTICS;

TUMORS;

EID: 70350567515     PISSN: 09735348     EISSN: 17606101     Source Type: Journal    
DOI: 10.1051/mmnp/20094306     Document Type: Article

References (29)

1. B.P. Ayati, G.F. Webb, A.R.A Anderson. Computational methods and results for structured mutliscale methods of tumor invasion. Multiscale Model. Simul., 5 (2006), 1-20. (Pubitemid 46388873)
2. H.M. Byrne. The importance of intercellular adhesion in the development of carcinomas. I. MA J. Math. Appl. Med. Biol., 14 (1997), 305-323. (Pubitemid 127463786)
3. H.M. Byrne, M.A.J. Chaplain. Growth of nonnecrotic tumors in the presence and absence of inhibitors. Math. Biosci., 130 (1995), 130-151.
4. H.M. Byrne, M.A.J. Chaplain. Modelling the role of cell-cell adhesion in the growth and development of carcinomas. Math. Comput. Modeling, 24 (1996), 1-17. (Pubitemid 27118822)
5. X. Chen, S. Cui, A. Friedman. A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior. Trans. Amer. Math. Soc., 357 (2005), 4771-4804. (Pubitemid 41787744)
6. 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.
7. 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. (Pubitemid 37052684)
8. S. Cui, A. Friedman. A hyperbolic free boundary problem modeling tumor growth. Interfaces Free Bound., 5 (2003) , 159-182. (Pubitemid 36669109)
9. S.J.H. Franks, H.M. Byrne, J.P. King, J.C.E. Underwood, C.E. Lewis. Modelling the early growth of ductal carcinoma in situ of the breast. J. Math. Biology, 47 (2003), 424-452. (Pubitemid 38261106)
10. S.J.H. Franks, H.M. Byrne, J.P. King, J.C.E. Underwood, C.E. Lewis. Modelling the growth of comedo ductal carcinoma in situ. Mathematical Medicine & Biology, 20 (2003), 277-308. (Pubitemid 37526838)
11. 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. Theoretical Biology, 232 (2005), 523-543. (Pubitemid 39575095)
12. S.J.H. Franks, J.P. King. Interactions between a uniformly proliferating tumour and its surroundings: Uniform material properties. Mathematical Medicine & Biology, 20 (2003), 47-89.
13. A. Friedman. Cancer models and their mathematical analysis. In: Tutorials in Mathematical Biosciences III. Lecture Notes in Mathematics, 1872, 223-246. Springer, Berlin, 2006.
14. 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. (Pubitemid 44231232)
15. A. Friedman. Mathematical analysis and challenges arising from models of tumor growth. Math. Models & Methods in Applied Sciences, 17 (2007), 1751-1772.
16. A. Friedman. A multiscale tumor model. Interfaces and Free Boundaries, 10 (2008), 245-262.
17. 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.
18. A. Friedman, B. Hu, C-Y. Kao. Cell cycle control at the first restriction point and its effect on tissue growth. Submitted for publication.
19. A. Friedman, F. Reitich. Quasi-static motion of a capillary drop, II: The three-dimensional case. J. Diff. Eqs., 186 (2002), 509-557. (Pubitemid 36006081)
20. Y. Jiang, J. Pjesivac-Grbovic, C. Cantrell, J.P. Freyer. A multiscale model for avascular tumor growth. Biophysical Journal, 89 (2005), 3884-3894. (Pubitemid 41725610)
21. N. Komaraova. Stochastic modeling of loss- and gain-of-function mutation in cancer. Bull. Math. Biology, 17 (2007), 1647-1673.
22. 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. Biology, 63 (2001), 801-863. (Pubitemid 32834844)
23. G. Lolas. Mathematical modelling of proteolysis and cancer cell invasion of tissue. In: Tutorials in Mathematical Biosciences III, Lecture Notes in Mathematics, 1872, 77-130. Springer, Berlin, 2006.
24. N. Mantzaris, S. Webb, H.G. Othmer. Mathematical modeling of tumor-induced angiogenesis. J. Math. Biol., 49 (2004), 87-111.
25. M.A. Nowak, K. Sigmund. Evolutionary dynamics of biological games. Science, 303 (2004), 793-799. (Pubitemid 38174649)
26. 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. (Pubitemid 32577707)
27. R. Ribba, T. Colin, S. Schnell. A multiscale model of cancer, and its use in analyzing irradiation therapies. Theoretical Biology and Medical Modeling, 3 (2006), No. 7, 1-19.
28. V.A. Solonnikov. On quasistationary approximation in the problem of a capillary drop. In: J. Escher & G. Simonett (Eds.), Progress in Nonlinear Differential Equations and Their Applications, 35, 643-671. BirkhÄauser Verlag, Basel, 1999.
29. M.M. Vainberg, V.A. Trenogin. Theory of branching solutions of non-linear equations. Nordhoff International Publishing, Leyden, 1974.