A report on some recent studies on growth and remodelling

Alfio Grillo (Politecnico di Torino)

Thursday 21st March, 2019 14:00-15:00 Maths 311B

Abstract

I would like to report on some results, summarised in [1, 2], that address the issues of growth and remodelling from the point of view of theoretical biomechanics. For the sake of clarity, I divide my talk into three parts. The first part is devoted to a general formulation of the mechanics of porous media in the biomechanical context. To this end, I re-obtain the fundamental balance laws on which the subsequent models are constructed. In the second part, I specialise the introduced set-up to tumour growth. In particular, I focus on a recent paper [1], in which the evolution of a tumour is studied by having recourse to some tools of Continuum Mechanics and Differential Geometry. For this purpose, the growth of the tumour is viewed as an inelastic process, and is framed under the light shed by the multiplicative decomposition of the tumour’s deformation gradient tensor.
After reviewing some results on this topic [3, 4], and considering the tumour as a biphasic solid-fluid medium, growth is assumed to occur in the solid phase of the tumour in response to its exchange of mass with the fluid phase. Such transfer of matter, and its redistribution in the tumour, is described by a “growth tensor”, F\gamma = \gamma I. The scalar-valued function is referred to as “growth parameter” and is assumed to be characterised by two main properties: (1) it is such that F cannot be expressed as a deformation gradient; (2) the metric tensor C\gamma induces a nontrivial Ricci curvature tensor, R\gamma, and a nonzero scalar curvature.
The fundamental idea of the presented work is that the scalar curvature contributes to the evolution of \gamma, which is required to solve a generalised diffusion-reaction-diffusion equation. This picture is adapted from a seminal work of Epstein [5] on the evolution of inhomogeneities in solids, and distinguishes itself from the classical theories of growth, which determine the growth parameter by solving an ordinary differential equation. In the third part of the talk, I discuss an ongoing work [2], in which the coupling between growth and remodelling is studied in the case of an avascular tumour. The remodelling of the tumour is identified with the breakage and restoration of the cells’ adhesion bonds, and is described as a plastic-like distortion. Furthermore,
following [6], remodelling is assumed to occur on two different length scales, and a strain-gradient approach is proposed in order to account for the distortions that are produced by remodelling close to interfaces or internal boundaries. Finally, the results of some numerical simulations and the limitations of the considered model are commented.

References

[1] Di Stefano, S., Ramírez-Torres, A., Penta, R., Grillo, A. “Self-influenced growth through evolving material
inhomogeneities”. International Journal of Nonlinear Mechanics, 106 (2018) 174–187.

[2] Grillo, A., Di Stefano, S., Ramírez-Torres, A., Loverre, M. “A study of growth and remodelling in isotropic tissues, based on the Anand-Aslan-Chester theory of strain-gradient plasticity”. Submitted.

[3] Goriely, A. “The Mathematics and Mechanics of Biological Growth”, Springer, New York, 2016.

[4] Sadik, S., Yavari, A. “On the origins of the idea of the multiplicative decomposition of the deformation gradient”. Mathematics and Mechanics of Solids, 22(4) (2017) 771–772.

[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] Anand, L., Aslan, O. Chester, S.A. “A large-deformation gradient theory for elastic-plastic materials: Strain softening and regularization of shear bands”. International Journal of Plasticity, 30-31 (2012) 116–143.

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