Stress dependent morphogenesis: continuum mechanics and system of trusses

The mechanical analysis of soft tissues in biomechanics has experienced increasing progress during the last decade. Part of this success is due to the development and application of some techniques of continuum mechanics, in particular, the decomposition of the deformation gradient, and the introduction of mass, density or volume changes in the reference configuration. Resorting to the common terminology employed in the literature, the changes in biomechanical processes may be classified as growth (change of mass), remodelling (change of density or other material properties such as fibre orientation) or morphogenesis (change of shape). Although the use of those concepts in bone and cardiovascular analysis is well extended, their use in morphogenesis during embryo development has been far less studied. The reasons for this fact may be found in the large shape changes encountered during this process, or the complexity of the material changes involved. In this chapter we develop a general framework for the modelling of morphogenesis by introducing a growth process in the structural elements of the cell, which in turn depends on the stress state of the tissue. Some experimental observations suggest this feedback mechanism during embryo development, and only very recently this behaviour has started to be simulated. We here derive the necessary equilibrium equations of a stress controlled growth mechanism in the context of continuum mechanics. In these derivations we assume a free energy source which is responsible for the active forces during the elongation process, and a passive hyperelastic response of the material. In addition, we write the necessary conditions that the active elongation law must satisfy in order to be thermodynamically consistent. We particularise these equations and conditions for the relevant elements of the cytoskeleton, namely, microfilaments and microtubules. We apply the model to simulate the shape changes observed during embryo morphogenesis in truss element. As a salient result, the model reveals that by imposing boundary stress conditions, unbounded elongation would be obtained. Therefore, either prescribed displacements or cross-links between fibres are necessary to reach a homeostatic state.