Nonlinear simulation of tumor necrosis, neo-vascularization and tissue invasion via an adaptive finite-element/level-set method

Bulletin of Mathematical Biology

Volumn 67, Issue 2, 2005, Pages 211-259

  Zheng, X ^a   Wise, S M ^a,b   Cristini, V ^a,b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b University of California at Irvine   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 13744255578     PISSN: 00928240     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.bulm.2004.08.001     Document Type: Article

References (52)

1. Spheroids in Cancer Research H. Acker J. Carlsson R. Durand R. Sutherland 1984 Springer Berlin, Heidelberg
2. B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter Molecular Biology of the Cell 4th edition 2002 Garland Science New York
3. D. Ambrosi, and F. Mollica On the mechanics of a growing tumor Int. J. Eng. Sci. 40 2002 1297
4. A. Anderson, and M. Chaplain Continuous and discrete mathematical models of tumor-induced angiogenesis Bull. Math. Biol. 60 1998 857 900
5. R. Araujo, and D. McElwain A history of the study of solid tumour growth: the contribution of mathematical modelling Bull. Math. Biol. 2003 (in press)
6. D. Arnold, F. Brezzi, and M. Fortin A stable finite-element method for the stokes equations Calcolo 21 1984 337
7. D. Ausprunk, and J. Folkman Migration and proliferation of endothelial cells in preformed and newly formed blood vessels during tumour angiogenesis Microvasc. Res. 14 1977 53 65
8. T. Barth, and J. Sethian Numerical schemes for the Hamilton-Jacobi level set equations on triangulated domains J. Comput. Phys. 145 1998 1
9. G. Batchelor An Introduction to Fluid Dynamics 1967 Cambridge University Press Cambridge
10. N. Bellomo, and L. Preziosi Modelling and mathematical problems related to tumor evolution and its interaction with the immune system Math. Comput. Modelling 32 2000 413 542
11. D. Braess Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics 1997 Cambridge University Press Cambridge
12. C. Breward, H. Byrne, and C. Lewis A multiphase model describing vascular tumour growth Bull. Math. Biol. 65 2003 609 640
13. F. Bussolino, M. Arese, E. Audero, E. Giraudo, S. Marchiò, S. Mitola, L. Primo, and G. Serini Biological aspects of tumour angiogenesis Cancer Modelling and Simulation 2003 Chapman and Hall/CRC London 1 22 (Chapter 1)
14. H. Byrne, and M. Chaplain Growth of nonnecrotic tumors in the presence and absence of inhibitors Math. Biosci. 130 1995 151 181
15. H. Byrne, and M. Chaplain Growth of necrotic tumors in the presence and absence of inhibitors Math. Biosci. 135 1996 187 216
16. H. Byrne, and M. Chaplain Free boundary value problems associated with the growth and development of multicellular spheroids Eur. J. Appl. Math. 8 1997 639
17. M. Chaplain Avascular growth, angiogenesis and vascular growth in solid tumours: the mathematical modelling of the stages of tumour development Math. Comput. Modelling 23 1996 47 87
18. M. Chaplain, and A. Anderson Mathematical modelling of tissue invasion Cancer Modelling and Simulation 2003 Chapman and Hall/CRC London 269 298 (Chapter 10)
19. M. Chaplain, and A. Stuart A model mechanism for the chemotactic response of endothelial cells to tumor angiogenesis factor IMA J. Math. Appl. Med. Biol. 10 1993 149 168
20. B. Cockburn Discontinuous galerkin methods for convection-dominated problems High-Order Methods for Computational Physics Lecture Notes in Computational Science and Engineering vol. 9 1999 Springer 69 224
21. B. Cockburn, and C. Dawson Some extensions of the local discontinuous Galerkin method for convection-diffusion equations in multidimensions J. Whiteman The Proceedings of the Conference on the Mathematics of Finite Elements and Applications: MAFELAP X 2000 Elsevier 225 238
22. B. Cockburn, and C.-W. Shu The local discontinuous Galerkin method for time-dependent convection-diffusion systems SIAM J. Numer. Anal. 35 1998 2440 2463
23. B. Cockburn, and C.-W. Shu Runge-Kutta discontinuous Galerkin methods for convection-dominated problems J. Sci. Comput. 16 2001 173 261
24. V. Cristini, J. Blawzdziewicz, and M. Loewenberg An adaptive mesh algorithm for evolving surfaces: simulations of drop breakup and coalescence J. Comput. Phys. 168 2001 445 463
25. V. Cristini, J. Lowengrub, and Q. Nie Nonlinear simulation of tumor growth J. Math. Biol. 46 2003 191 224
26. H. Elman, and G. Golub Inexact and preconditioned Uzawa algorithms for saddle point problems SIAM J. Numer. Anal. 31 1994 1645 1661
27. M. Fortin, and F. Brezzi Mixed and Hybrid Finite Elements 1991 Springer New York
28. Frieboes, H., Cristini, V., 2004. Diffusional instability as a mechanism for glioblastoma growth and invasion (in preparation)
29. P. Friedl, E. Brocker, and K. Zanker Integrins, cell matrix interactions and cell migration strategies: fundamental differences in leukocytes and tumor cells Cell Adhe. Commun. 6 1998 225
30. A. Friedman, and F. Reitich Analysis of a mathematical model for the growth of tumors J. Math. Biol. 38 1999 262
31. R. Gatenby, and E. Gawlinski A reaction-diffusion model of cancer invasion Cancer Res. 56 1996 5745
32. M. Gimbrone, R. Cotran, S. Leapman, and J. Folkman Tumor growth and neovascularization: an experimental model using the rabbit cornea J. Natl. Cancer Inst. 52 1974 413 427
33. H. Greenspan On the growth and stability of cell cultures and solid tumors J. Theor. Biol. 56 1976 229 242
34. D. Hanahan, and R. Weinberg The hallmarks of cancer Cell 100 2000 57 70
35. E. Isaacson, and H. Keller Analysis of Numerical Methods 1966 Wiley New York
36. R. LeVeque Numerical Methods for Conservation Laws 1992 Birkhäuser Berlin
37. V. Lubarda, and A. Hoger On the mechanics of solids with a growing mass Int. J. Solids Structures 39 2002 4627
38. E. Maher, F. Furnari, R. Bachoo, D. Rowitch, D. Louis, W. Cavenee, and R. De-Pinho Malignant glioma: genetics and biology of a grave matter Genes Dev. 15 2001 1311
39. Mantzaris, N., Webb, S., Othmer, H., 2003. Mathematical modelling of tumor-induced angiogenesis. Technical Report: School of Mathematics, University of Minnesota, Minneapolis, MN, USA
40. S. McDougall, A. Anderson, M. Chaplain, and J. Sherratt Mathematical modelling of flow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy strategies Bull. Math. Biol. 64 2002 673 702
41. M. Natarajan, T. Hacker, and C. Gladson Fak signaling in anaplastic astrocytoma and glioblastoma tumors Cancer J. 9 2003 127
42. J. Ortega, and W. Reinbolt Iterative Solution of Nonlinear Equations in Several Variable 1970 Academic Press San Diego
43. S. Osher Riemann solvers, the entropy condition, and difference approximations SIAM J. Numer. Anal. 21 1984 217 235
44. S. Osher, and R. Fedkiw Level Set Methods and Dynamic Implicit Surfaces 2003 Springer New York
45. S. Osher, and J. Sethian Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulation J. Comput. Phys. 79 1988 12
46. S. Paku, and N. Paweletz First step of tumor-related angiogenesis Lab. Invest. 65 1991 334 346
47. N. Paweletz, and M. Knierim Tumor-related angiogenesis Crit. Rev. Oncol. Hematol. 9 1989 197 242
48. G. Strang On the construction and comparison of difference schemes SIAM J. Numer. Anal. 5 1968 506 517
49. M. Sussman, P. Smereka, and S. Osher A level set approach for computing solutions to incompressible two-phase flow J. Comput. Phys. 114 1994 146
50. W. Tiller Migration of a liquid zone through a solid. 3 J. Appl. Phys. 36 1965 261
51. G. Truskey, F. Yuan, and D. Katz Transport Phenomena in Biological Systems 2004 Pearson Prentice Hall Upper Saddle River, NJ
52. R. Tyson, L. Stern, and R. LeVeque Fractional step methods applied to a chemotaxis model J. Math. Biol. 41 2000 455 475