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

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.