## Growth and Homeostasis in Morphoelasticity

Many living biological tissues are known to grow in response to their mechanical environment, such as changes in the surrounding pressure. This 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 internal mechanical stresses, called the homeostasis. My long term goal is to better understand pathological conditions like cardiovascular diseases and tumour growth, which can interpreted as failures of the homeostasis mechanism.

On a theoretical level, I am interested in homeostasis from a mechanical and thermodynamical perspective within the framework of morphoelasticity which is a theory based on continuum mechanics and nonlinear elasticity. In morphoelasticity, the deformation gradient \(\bf{F}\) of a tissue is separated into an active growth contribution \(\bf{G}\) and the elastic accommodation \(\bf{A}\) of the added material. To incorporate the concept of homeostasis, tissue growth is modeled as a function of its local mechanical stresses: $$\dot{\bf{G}}=\bf{K}:\left(\bf{T}-\bf{T}^{*}\right)\bf{G}$$ where \(\bf{T}\) is the Cauchy stress tensor, \(\bf{T}^{*}\) is the homeostatic stress, \(\bf{K}\) is a coefficient matrix and the dot denotes time differentiation. This modeling framework allows us to quantify the exact conditions under which homeostasis produces physiological (healthy) or pathological (uncontrolled) tissue growth.

**Left**: Schematic of deformation gradient decomposition which is central to morphoelasticity. **Right**: Cross section of \(N\) bonded cylinders, a model system for homeostatic growth control.

## Seashell morphogenesis

The growth of Ammonites’ shells is a very nice application for my theoretical framework. It allows us to re-interpret and add insights into the morphology of the beautiful spiral coiling and oscillatory ribbing pattern of the seashells in the light of biomechanics and homeostasis.

A simulation of the morphogenesis of Ammonites’ shells, based on our morphomechanical model of seashell growth.