An elasto-plastic biphasic model of the compression of multicellular aggregates: the influence of fluid on stress and deformation

We present a mathematical model of the compression of multicellular aggregates, and we specialise it to a compression-release test that is well known in the biological literature. Within the adopted mechanical setting, a multicellular aggregate is studied as a biphasic system consisting of a soft solid porous medium saturated with an interstitial fluid. In particular, together with the deformation of the considered aggregate, the characterisation of the model outlined in this work relies on four fundamental features. First, by assuming the interstitial fluid to be macroscopically inviscid and to evolve according to the Darcian regime, we resolve its flow and determine the associated time dependent pressure distribution. Second, we focus our attention on the remodelling of the compressed aggregate, that is, on the rearrangement of its internal structure in response to the external loads applied to it. Specifically, we look at the way in which such a rearrangement is induced by the considered experiment and at how it affects the mechanical behaviour of the aggregate. Moreover, we introduce a remodelling-dependent permeability tensor with the purpose of visualising a more direct influence of remodelling on the dynamics of the aggregate's interstitial fluid. Finally, we resolve the interactions exchanged between the aggregate and the compressive apparatus. This task necessitates the formulation of an appropriate contact problem, thereby calling for the description of the evolution of the area through which the aggregate and the apparatus exchange mechanical interactions. In particular, the continuity conditions to be applied on such a contact area are discussed. Our numerical simulations show the role played by the different phenomena accounted for in the model and the overall dynamics of the aggregate within the considered experiment.