Chapter 13 - A phenomenological dashpot model for morphoelasticity for the contraction of scars

Abstract

Mechanical forces exerted by cells and external bodies in combination with growth or shrinkage lead to permanent deformations of tissues. In order to mathematically model this phenomenon, one can use a morphoelastic framework, which entails a system of two differential equations. The equations involve a momentum balance combined with an evolution equation for the strain. This equation takes into account movement of the coordinates due to deformation and growth or shrinkage, as well as the possibility of rotation, by, respectively, the material and Jaumann derivative. The current manuscript analyzes most of the crucial features of morphoelasticity by considering an ordinary differential equation that is motivated by a momentum balance in combination with a permanent deformation. The model is based on a spring–dashpot system and serves to be illustrative, and to be used as a phenomenological model. The current manuscript aims at an accessible presentation of principles behind morphoelasticity using simple physical and simple mathematical principles. Besides being illustrative, the current model also provides a computationally cheap alternative to the expensive numerical solution of systems of partial differential equations.