Growth laws in morphoelasticity - Erlich, A (2017)
Many living biological tissues are known to grow in response to their mechanical environment, such as changes in the surrounding pressure. This growth response can be seen, for instance, in the adaptation of heart chamber size and arterial wall thickness to changes in blood pressure. Moreover, many living elastic tissues actively maintain a preferred level of mechanical internal (residual) 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 material. The goal of this thesis is to develop mathematical techniques to analyse growth laws that are biologically plausible, and to explore issues of heterogeneity and growth stability. Firstly, we review attempts based on the Second Law of Thermodynamics (Coleman-Noll procedure) concluding that they cannot universally restrict the mathematical form of growth laws. In light of these results we focus on the phenomenological concept of homeostasis. We hypothesize that growth laws are functions of homeostatic stress (homeostasis-driven growth). Secondly, we demonstrate that for a static residually stressed network made of elastic bars connected in series and parallel, the network response has the same functional form as the response of individual bars (similarly to electrical circuits). For a dynamically evolving network, on the other hand, the macroscopic network response is a nonlinear function of microscopic homeostasis-driven growth laws with no electrical counterpart. We characterise the macroscopic growth dynamics as non-monotonic and non-oscillatory. Thirdly, we discuss the growth dynamics of tubular structures, which are very common in biology (e.g. arteries, plant stems, airways). We model the homeostasis-driven growth dynamics of tubes which produces spatially inhomogeneous residual stress. We show that the stability of the homeostatic state nontrivially depends on the anisotropy of the growth response. The key role of anisotropy may provide a foundation for experimental testing of homeostasis-driven growth laws. Fourthly, we apply our theoretical framework to the growth of Ammonites' seashells. We demonstrate how homeostasis-driven growth produces seashell morphology that is consistent with observation and that cannot readily be captured with previous models.