**Abstract**The focus of my talk is an overview on some recent models of tumour growth in avascular stage.
After mentioning a few biological aspects [1, 2], such as the mass exchange among the constituents
of a tumour and the transport of chemical species, I will concentrate on some purely mechanical and
geometrical features of tumour growth. More specifically, taking inspiration from [3, 4], I will adopt
Differential Geometry to show how the Riemannian curvature arising from the inelastic distortions
accompanying growth can be joined together with the concept of material inhomogeneities [5].
Then, I will describe a way of contextualising this geometric framework within a “minimal”
chemo-mechanical model of tumour growth [6, 7], in which the evolving structure of a tumour
concurs to modulate the diffusion and mass exchange processes necessary for growth to occur [7].
In particular, I will discuss the case in which one or more chemical species undergo anomalous
diffusion, and I will compare this situation with the one of standard Fickian diffusion. This last part
of the talk summarises some preliminary results of a work in progress done in collaboration with
Ariel Ramírez-Torres and Salvatore Di Stefano.

References

[1] Chaplain, M.A.J., Benson, D.L., Maini, P.K., “Nonlinear Diffusion of a Growth Inhibitory
Factor in Multicell Spheroids”, Mathematical Biosciences, 121 (1994) 1–13.

[2] Byrne, H., Preziosi, L: “Modelling solid tumour growth using the theory of mixtures”, Math.
Med. Biol., 20(4) (2003) 341–366 http://dx.doi.org/10.1093/imammb/20.4.341.

[3] Yavari, A., Goriely, A.: “Weyl geometry and the nonlinear mechanics of distributed
point defects”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 468 (2012) 3902–3922
DOI:10.1098/rspa.2012.0342

[4] Goriely, A.: The mathematics and mechanics of biological growth, Springer, New York, 2016.

[5] Epstein, M.: “Self-driven continuous dislocations and growth”, in: M.G. Steinmann P. (Ed.),
Mechanics of Material Forces, in: Advances in Mechanics and Mathematics, vol. 11, Springer,
Boston, MA, 2005, pp. 129–139.

[6] Mascheroni, P., Carfagna, M., Grillo, A., Boso, D.P., Schrefler, B.A.: “An avascular tumor
growth model based on porous media mechanics and evolving natural states”, 23(4) (2018)
686–712 DOI: 10.1177/1081286517711217

[7] Di Stefano, S., Ramírez-Torres, A., Penta, R., Grillo, A.: “Self-influenced growth through
evolving material inhomogeneities”, Int. J. Non-Linear Mechanics, 106 (2018) 174–187
https://doi.org/10.1016/j.ijnonlinmec.2018.08.003.

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