Are homeostatic states stable? Dynamical stability in morphoelasticity

Biological growth is often driven by mechanical cues, such as changes in external pressure or tensile loading. Moreover, it is well known that many living tissues actively maintain a preferred level of mechanical internal stress, called the mechanical homeostasis. The tissue-level feedback mechanism by which changes of the local mechanical stresses affect growth is called a growth law within the theory of morphoelasticity, a theory for understanding the coupling between mechanics and geometry in growing and evolving biological materials. This coupling between growth and mechanics occurs naturally in macroscopic tubular structures, which are common in biology (e.g. arteries, plant stems, airways). We study a continuous tubular system with spatially heterogeneous residual stress via a novel discretisation approach which allows us to obtain precise results about the stability of equilibrium states of the homeostasis-driven growing dynamical system. This method allows us to show explicitly that the stability of the homeostatic state depends nontrivially on the anisotropy of the growth response. The key role of anisotropy may provide a foundation for experimental testing of homeostasis-driven growth laws.