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

Abstract

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.

A formulation of volumetric growth as a mechanical problem subjected to non-holonomic and rheonomic constraint

“…where r n is the rate at which the nutrients are absorbed by the tumor, and D : = D ^ ° ( F , K ) is the diffusivity tensor (see literature [60, 93] for possible constitutive expressions of D ). Moreover, following literature [66, 86, 90, 92], we can equip equation (5) with boundary and initial conditions of the type…”

“…Within a line of thought similar to the one followed by Cermelli et al [52], the idea of the extended kinematics summarized above was adopted by DiCarlo and Quiligotti [54] in the context of biomechanics for addressing growth and remodeling. These processes are both anelastic and consist of the variation of mass and change of material properties of biological tissues [55] or cellular complexes [56][57][58][59][60], respectively. In fact, in several biologically relevant situations, both growth and remodeling are described by having recourse to the BKL decomposition [61], or to similar decompositions [62], and the factor of the decomposition employed that accounts for the anelastic distortions accompanying growth or remodeling is sometimes referred to as growth tensor or remodeling tensor.…”

An efficient algorithm for biomechanical problems based on a fully implicit nested Newton solver

“…For elastic and hyperelastic contact problems, there exist solution techniques for small strain and obstacle problems [44,45], and more recently, also for large deformation contact problems, cf., e.g., [21,41,62,91,92]. However, to the best of our knowledge, the robust solution of problems combining contact scenarios and finite strains [21,41,44,45,57,63,66,67,75,79,88,92] are limited, at most, to isotropic models.…”

“…In the present work, we study remodeling only, thereby neglecting phenomena such as growth or mass transfer, as the latter occur at time-scales sharply separated from those of remodeling. Such a situation is typical, for example of biological systems such as cellular aggregates [21], in which the term remodeling accounts for processes like cell re-orientation and re-organization of the cell adhesion bonds [21,41,42,47,49], and fiber-reinforced soft tissues, like articular cartilage [15,19,46] or arteries [73], in which the structural changes of the extra-cellular matrix (ECM) feature an evolution of the fiber pattern. Other examples of remodeling phenomena are related to cell-matrix interactions mediated by focal adhesions, in terms of creation and rupture of bonds between cells and the adhesion plaque, as well as the adaptation of the protein structure of the latter [18,20], and to bone tissue, explained in terms of irreversible formation of micro-cracks [38,39,74].…”