Morpho-elasticity of solid tumours
Davide Riccobelli (SISSA, Italy)
Thursday 14th May, 2020 14:00-15:00 ZOOM (https://uofglasgow.zoom.us/j/93092976377)
Please Note: The ZOOM meeting will be password protected, as per the e-mail you received with subject: "Applied Mathematics Seminar ZOOM password".
During the last years, many applied mathematician and engineers have developed novel mathematical models of solid tumours to provide insight on its growing process and possible therapies for cancer treatment. The experimental evidence that a feedback exists between growth and stress in tumours poses challenging questions. First, the rheological properties (the “constitutive equations”) of aggregates of malignant cells are still a matter of debate. Secondly, the feedback law (the “growth law”) that relates stress and mitotic-apoptotic rate is far to be identified. Furthermore, solid tumours have the ability to assemble their own vascular network for optimizing their access to the vital nutrients. These new capillaries are morphologically different from normal physiological vessels. In particular,they have a much higher spatial tortuosity forcing an impaired flow within the peritumoral area. This is a major obstacle for the efficient delivery of antitumoral drugs.
In this talk, we address these questions on the basis of a theoretical analysis of in vitro and in vivo experiments. We show that solid tumours exhibit several mechanical features of a poroelastic material, where the cellular component behaves like an elastic solid. When the solid component of the spheroid is loaded at the boundary, the cellular aggregate grows up to an asymptotic volume that depends on the exerted compression. Residual stress shows up when solid tumours are radially cut, highlighting a peculiar tensional pattern.
In a second part of the talk, we propose a morpho–elastic model of the tumour vessels. A tumour capillary is considered as a growing hyperelastic tube that is spatially constrained by a linear elastic environment, representing the interstitial matter. We study the morphological stability of the capillary by means of the method of incremental deformations superposed on finite strains. The incompatible axial growth of the straight capillary is found to control the onset of a bifurcation towards a tortuous shape. The post-buckling morphology is studied using a mixed finite element formulation in the fully nonlinear regime.