Self-influenced growth through evolving material inhomogeneities

Abstract

We reformulate a model of avascular tumour growth in which the tumour tissue is studied as a biphasic medium featuring an interstitial fluid and a solid phase. The description of growth relies on two fundamental features: One of those is given by the mass transfer among the constituents of the phases, which is taken into account through source and sink terms; the other one is the multiplicative decomposition of the deformation gradient tensor of the solid phase, with the introduction of a growth tensor, which represents the growth-induced structural changes of the tumour. In general, such tensor is non-integrable, and it may allow to define a Levi-Civita connection with non-trivial curvature. Moreover, its evolution is related to the source and sink of mass of the solid phase through an evolution equation. Our goal is to study how growth can be influenced by the inhomogeneity of the growth tensor. To this end, we study the evolution of the latter, as predicted by two different models. In the first one, the dependence of the growth tensor on the tumour's material points is not explicitly considered in the evolution equation. In the second model, instead, the inhomogeneity of the growth tensor is resolved explicitly by introducing the curvature associated with it into the evolution equation. Through numerical simulations, we compare the results produced by these two models, and we evaluate a possible role of the material inhomogeneities on growth.

A study of growth and remodeling in isotropic tissues, based on the Anand‐Aslan‐Chester theory of strain‐gradient plasticity

“…The problem under investigation involves the motion of the solid phase, the motion of the fluid phase, the distortions related to growth, and plastic‐like distortions, which are associated with the reorganization of the tissue's internal structure. The definitions supplied in this section can be encountered in many works addressing mixture theory, and have been recently used for establishing the theoretical framework of previous works of some of us . Such framework, in turn, has been adapted from the kinematic description of biphasic mixtures as developed by Quiligotti and Quiligotti et al…”

“…Following Mascheroni et al and Di Stefano et al, the solid phase of the tissue is assumed to comprise only two types of cells, that is, the proliferating cells and the necrotic ones. Their presence in the tissue is measured by the mass densities ϕ s ρ s c p and ϕ s ρ s c n , respectively, where ϕ s is the volumetric fraction of the solid phase, ρ s is its true mass density, while c p and c n are the cells' mass fractions, compelled to satisfy the constraint c p + c n = 1, everywhere in and .…”

“…Note that the indices “p” and “n” stand for “proliferating” and “necrotic,” respectively. Once the composition of the solid phase is specified, it is possible to characterize the mass balance of the solid phase by writing one balance law for each cell population, that is, As reported by Mascheroni et al and Di Stefano et al, r fp describes the transfer of mass from the fluid phase to the solid phase, r nf measures the dissolution per unit time of the necrotic cells in the fluid, r pn is the rate at which the proliferating cells become necrotic, and the last two terms r p γ and r n γ have been introduced by Di Stefano et al to evaluate how the growth‐induced structural transformations of the tissue influence the local density changes of the solid constituents. Both terms, however, are assumed to be identically zero in the present work.…”

“…To characterize the evolution of the nutrients, we introduce the nutrients' mass fraction , c N , and the mass density ϕ f ρ f c N , where ϕ f and ρ f indicate the volumetric fraction and the mass density of the fluid phase, respectively. In addition, we require that the tissue obeys the saturation condition, that is, ϕ f = 1 − ϕ s , and we consider the mass balance laws of the nutrients and of the fluid phase as a whole, In and , v f is the velocity of the fluid phase, y N is the mass flux vector associated with the motion of the nutrients relative to v f , r Np is the rate at which the nutrients are consumed by the proliferating cells, and the right‐hand‐side of is taken equal to the negative of r s in order to ensure the local conservation of mass for the biphasic mixture under study.…”

“…More in detail, the novelties of the present study with respect to previous publications of some of us are the following: (a) we analyze the coupling between growth and remodeling both theoretically and computationally, and we resolve the remodeling at two different length scales; (b) with the aid of the theory developed by Anand et al, we formulate remodeling within a strain‐gradient framework, thereby generalizing our past approaches, which were of “grade zero” in the remodeling variables.…”