Thoughts on the mechanics of growth: some remarks on how the inhomogeneity and heterogeneity of tissues may impact their evolution

Alfio Grillo (DISMA, Politecnico di Torino, Italy)

Thursday 12th May, 2022 15:30-16:15 Maths 311B AND ZOOM


In this talk, I would like to review the main results of two papers [1, 2] that address the mechanics of growth of biological tissues, and aim at showing how the process of growth may be influenced by the structural properties of the bodies in which growth itself occurs, such as their inhomogeneity [1] and heterogeneity [2].

I will concentrate, in particular, on the growth of tumors in the stages preceding the formation of blood vessels, i.e., on the avascular growth [3], and, as a point of departure for my talk, I will present a model, discussed in detail in [4], in which some of the most relevant aspects of tumor growth mechanics are summarized. Two of these aspects are, in fact, the multiplicative decomposition of the tumor’s deformation gradient tensor into an elastic part and a purely growth part, and an advection-diffusion-reaction model for the chemical substances that are responsible for the activation and the deactivation of growth.

In the first part of my talk, I will report on a study that, on the basis of the model put forward in [4], considers some differential geometry concepts associated with the distortions induced by growth [1]. These are described through a non-integrable tensor field, referred to as growth tensor, that constitutes the growth part of the multiplicative decomposition of the tumor’s deformation gradient, and generates a Riemannian metric tensor by means of which the curvature tensor and the scalar curvature of the theory are computed. Under the simplifying hypothesis that the growth tensor is spherical, the scalar curvature is interpreted as a descriptor of the distribution of the material inhomogeneities inside the tumor. Hence, by adopting standard arguments, and extending an idea taken from [5], the scalar curvature is assumed to contribute to the evolution of the growth-induced distortions, thereby giving rise to a “self-induced growth” [1].

In the second part of the talk, I will focus on another elaboration of the model presented in [4], whose results have been summarized in [2]. On the basis of the consideration according to which the diffusion of chemical agents may be non-local in media with pronouncedly heterogeneous internal structure [6], it is hypothesized that the mass flux vector of such agents is related to their concentration gradient through an integro-differential law of fractional type. In particular, by working out some tools of Fractional Calculus taken from [7], a “non-local diffusivity tensor” is defined [2], and the impact of a fractional law of diffusion on the growth of an idealized tumor is studied through a dedicated benchmark problem. This is done in a fully non-linear regime, in which diffusion is coupled with the growth-induced distortions, with the change of shape of the tumor and with the evolution of the pressure of the tumor’s interstitial fluid, as growth proceeds [2].


[1] Di Stefano, S., Ramírez-Torres, A., Penta, R., Grillo, A. International Journal of Non-Linear Mechanics, 106 (2018) 174-187.
[2] Ramírez-Torres, A., Di Stefano, S., Grillo, A. Mathematics and Mechanics of Solids, 26(9) (2021) 1264-1293.
[3] Chaplain, M.A.J. Mathematical and Computer Modelling, 23(6) (1996) 47-87.
[4] Mascheroni, P., Carfagna, M., Grillo, A., Boso, D.P., Schrefler, B.A. Mathematics and Mechanics of Solids, 23(4) (2018) 686-712.
[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] Tarasov, V.E. Annals of Physics 323 (2008) 2756-2778.
[7] Atanackovic, T.M., Pilipovic, S., Stankovic, B., Zorica, D. Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles. London: Wiley 2014.


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