Mathematical models for the mechanics of soft tissues: from linear elasticity to morpho-poroelasticity

Biological issues are often subjected to forces. In many cases, such as tumor growth or skin contraction, it is crucially important to model the state of tissues that are exposed to forces in order to improve or optimize therapies for different pathologies. The simplest models use linear elasticity as a constitutive law. This linearity enables the use of the superposition principle and the use of fundamental solutions to analyze the influence of multiple points of action of forces. A clear illustration of this principle is the immersed interface method. In this presentation, we discuss this principle in terms of convergence properties.

However, in real-life tissues, the use of linear elasticity is too restrictive due to the presence of moisture and the porous structure of biological tissues. Furthermore, in various biomedical cases, the microstructure of the tissue changes due to cellular activity. For this reason, we construct and use a model that consists of elasticity, porosity and microstructural changes. The mathematical framework is referred to as morpho-visco-poroelasticity. This framework is original and for this reason, we analyze this framework in terms of stability around equilibria. Since numerical solutions can be characterized by spurious oscillations, we provide conditions for monotonicity by mathematical analysis. Furthermore, we propose a numerical stabilization method to avoid spurious oscillations on the forehand.