The main aim of the paper is to embed the experimental results recently obtained studying the detachment force of single adhesion bonds in a multiphase model developed in the framework of mixture theory. In order to do that the microscopic information is upscaled to the macroscopic level to describe the dependence of some crucial terms appearing in the PDE model on the sub-cellular dynamics involving, for instance, the density of bonds on the membrane, the probability of bond rupture and the rate of bond formation. In fact, adhesion phenomena influence both the interaction forces among the constituents of the mixtures and the constitutive equation for the stress of the cellular components. Studying the former terms a relationship between interaction forces and relative velocity is found. The dynamics presents a behavior resembling the transition from epithelial to mesenchymal cells or from mesenchymal to ameboid motion, though the chemical cues triggering such transitions are not considered here. The latter terms are dealt with using the concept of evolving natural configurations consisting in decomposing in a multiplicative way the deformation gradient of the cellular constituent distinguishing the contributions due to growth, to cell rearrangement and to elastic deformation. This allows the description of situations in which if in some points the ensemble of cells is subject to a stress above a threshold, then locally some bonds may break and some others may form, giving rise to an internal reorganization of the tissue that allows to relax exceedingly high stresses.

A number of biological phenomena are interlaced with classical mechanics. In this review we discuss the role of mechanics in tumor growth, namely the avascular phase of solid tumors. While a growing mass produces a traction of the surrounding tissues, a feedback mechanism controls the proliferation of the malignant cells depending on the tensional state. The formalism of continuum mechanics, possibly accompanied by numerical simulations, is able to shed light on biological controversial subjects. The converse is also true: non-standard mechanical problems suggest new challenging theoretical questions.

The emergence of residual stress as due to growth and remodeling of soft biological tissues is considered in the framework of the mixture theory. The focus is on mixtures composed by one elastic solid component and several fluid ones. It is shown that the standard theory is unable to predict residual stresses unless enriched by a suitable descriptor of growth. Both the introduction of a dependence of the free energy on the density of the solid component and the Kroner-Lee multiplicative decomposition of the gradient of deformation are effective in this respect, with different levels of generality. When adopting a multiplicative decomposition of the tensor gradient of deformation, thermodynamical arguments suggest constitutive laws for the evolution of the growth tensor that point out the role of the concentration of fluid species in driving the emergence of residual stress thanks to inhomogeneous growth.

a b s t r a c tForce traction microscopy is an inversion method that allows one to obtain the stress field applied by a living cell on the environment on the basis of a pointwise knowledge of the displacement produced by the cell itself. This classical biophysical problem, usually addressed in terms of Green functions, can be alternatively tackled using a variational framework and then a finite elements discretization. In such a case, a variation of the error functional under suitable regularization is operated in view of its minimization. This setting naturally suggests the introduction of a new equation, based on the adjoint operator of the elasticity problem. In this paper we illustrate the rigorous theory of the two-dimensional and three-dimensional problems, involving in the former case a distributed control and in the latter case a surface control. The pointwise observations require one to exploit the theory of elasticity extended to forcing terms that are Borel measures.

Force Traction Microscopy is an inversion method that allows to obtain the stress field applied by a living cell on the environment on the basis of a pointwise knowledge of the displacement produced by the cell itself. This classical biophysical problem, usually addressed in terms of Green's functions, can be alternatively tackled in a variational framework. In such a case, a variation of the error functional under suitable regularization is operated in view of its minimization. This setting naturally suggests the introduction of a new equation, based on the adjoint operator of the elasticity problem. In this paper, we illustrate a numerical strategy of the inversion method that discretizes the partial differential equations associated to the optimal control problem by finite elements. A detailed discussion of the numerical approximation of a test problem (with known solution) that contains most of the mathematical difficulties of the real one, allows a precise evaluation of the degree of confidence that one can achieve by in the numerical results.