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

Abstract

Motivated by the increasing interest of the biomechanical community towards the employment of strain‐gradient theories for solving biological problems, we study the growth and remodeling of a biological tissue on the basis of a strain‐gradient formulation of remodeling. Our scope is to evaluate the impact of such an approach on the principal physical quantities that determine the growth of the tissue. For our purposes, we assume that remodeling is characterized by a coarse and a fine length scale and, taking inspiration from a work by Anand, Aslan, and Chester, we introduce a kinematic variable that resolves the fine scale inhomogeneities induced by remodeling. With respect to this variable, a strain‐gradient framework of remodeling is developed. We adopt this formulation in order to investigate how a tumor tissue grows and how it remodels in response to growth. In particular, we focus on a type of remodeling that manifests itself in two different, but complementary, ways: on the one hand, it finds its expression in a stress‐induced reorganization of the adhesion bonds among the tumor cells, and, on the other hand, it leads to a change of shape of the cells and of the tissue, which is generally not recovered when external loads are removed. To address this situation, we resort to a generalized Bilby‐Kröner‐Lee decomposition of the deformation gradient tensor. We test our model on a benchmark problem taken from the literature, which we rephrase in two ways: microscale remodeling is disregarded in the first case, and accounted for in the second one. Finally, we compare and discuss the obtained numerical results.

Effective balance equations for electrostrictive composites

“…Such a situation would imply, on the one hand, the revision of the constitutive framework adopted so far and, on the other hand, the study of new cell problems to investigate the role played by each field in the determination of the effective quantities of the composite. Indeed, by activating the magnetic field H, these quantities would require to resolve the interactions among the three fields u, E and H, thereby giving rise to a "game among three players", in the jargon of [19] (see also [33]).…”

“…where we used the hypothesis of macroscopic uniformity reported in Eqs. (33) and (34), the expression of the symmetrised y-gradient of u (1) α in Eq. 53, and the fact that u 0 depends on x only.…”

Influence of non-local diffusion in avascular tumour growth

“…One is related to the changes of the tissue’s internal structure due to the gain or loss of mass, and the other one to distortions of purely elastic nature (note that, here and in the sequel, we shall use the terms “tumour” and “tissue” interchangeably). We refer to [17, 19, 22–25], and to the references therein, for a more complete discussion of the BKL multiplicative decomposition.…”

“…For our purposes, we adopt part of the modelling assumptions outlined in [19, 25, 30, 64]. Specifically, we study the tumour as a mixture comprising a fluid phase and a solid phase, and we identify its growth with the gain or loss of mass of the solid phase at the expense or advantage of the fluid one.…”

“…In particular, the model we employ predicts a gain of mass for a sufficiently high concentration of chemical agents (in fact, nutrients) and a loss of mass when the concentration of these falls below a certain threshold [30, 65]. Moreover, in the case of mass uptake of the solid phase, the model accounts for mechanotransduction [25, 30, 65, 66], thereby allowing a modulation of growth by means of stress [30, 65], whereas, for both positive and negative growth, the onset of structural transformations and their related inelastic distortions are considered. In the remainder of this work, we address only the most pertinent considerations and equations, while we refer the reader to [19, 30, 64] for further details.…”