Bianchini, Lorenzo; Fasano, Antonio A model combining acid-mediated tumour invasion and nutrient dynamics. (English) Zbl 1163.92319 Nonlinear Anal., Real World Appl. 10, No. 4, 1955-1975 (2009). Summary: We illustrate a mathematical model for the evolution of multicellular tumour spheroids in a host tissue, including the effect of the excess H\(^{+}\) ions and nutrient dynamics. Both the avascular and the vascular case are considered. The model is a nontrivial generalization of the simple scheme proposed by K. Smallbone et al. [The role of acidity in solid tumour growth and invasion. J. Theor. Biol. 235, 476–484 (2005)]. Many different situations may occur, depending on the values of the physical parameters involved. Existence of solutions and qualitative properties are investigated. Cited in 10 Documents MSC: 92C50 Medical applications (general) 35R35 Free boundary problems for PDEs Keywords:mathematical models of cancer PDF BibTeX XML Cite \textit{L. Bianchini} and \textit{A. Fasano}, Nonlinear Anal., Real World Appl. 10, No. 4, 1955--1975 (2009; Zbl 1163.92319) Full Text: DOI References: [1] Smallbone, K.; Gavaghan, D. J.; Gatenby, R. A.; Maini, P. K., The role of acidity in solid tumour growth and invasion, J. Theoret. Biol., 235, 476-484 (2005) · Zbl 1445.92082 [2] Gatenby, R. A.; Gawlinski, E. T., A reaction-diffusion model of cancer invasion, Cancer Res., 56, 5745-5753 (1996) [4] Warburg, O., On the origin of cancer cells, Science, 123, 309-314 (1956) [5] Venkatasubramanian, R.; Henson, M. A.; Forbes, N. S., Incorporating energy metabolism into a growth model of multicellular tumour spheroids, J. Theoret. Biol., 242, 8, 440-453 (2006) · Zbl 1447.92108 [7] Preziosi, L.; Astanin, S., Modelling the formation of capillaries, (Quarteroni, A.; Formaggia, L.; Veneziani, A., Complex Systems in Biomedicine (2006), Springer-Verlag: Springer-Verlag Italia), 109-145 · Zbl 1387.92012 [8] Anderson, A. R.A., Mathematical modelling of flow in 2D and 3D vascular networks: Applications to anti-angiogenic and chemotherapeutic drug strategies, Math. Comput. Modelling, 41, 1137-1156 (2005) · Zbl 1080.92050 [9] Martinez-Zaguilan, R.; Seftor, E. A.; Seftor, R. E.; Chu, Y. W.; Gillies, R. J.; Hendrix, M. J., Acidic pH enhances the invasive behavior of human melanoma cells, Clin. Exp. Metastasis, 14, 176-186 (1996) [10] Freyer, J. P.; Sutherland, R. M., Regulation of growth saturation and development of necrosis in \(E M T 6 / R_0\) multicellular spheroids by the glucose and oxygen supply, Cancer Res., 46, 3504-3512 (1986) [11] Folkman, J.; Hochberg, M., Self-regulation of growth in three dimensions, J. Exp. Med., 138, 745-753 (1973) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.