# A model for one-dimensional morphoelasticity and its application to fibroblast-populated collagen lattices

## TL;DR

This paper introduces a novel one-dimensional morphoelastic model for biological tissue remodelling, capturing structural changes and dynamics such as contraction and re-expansion in collagen lattices, bridging classical mechanics and plasticity theory.

## Contribution

It presents a new morphoelastic framework based on deformation gradient decomposition, modeling tissue remodelling from discrete cells to continuum, applicable to collagen lattice behaviour.

## Key findings

- Successfully models collagen lattice contraction and re-expansion.
- Bridges classical elasticity with plasticity theory for tissue remodelling.
- Provides qualitative agreement with observed biological tissue dynamics.

## Abstract

The mechanical behaviour of solid biological tissues has long been described using models based on classical continuum mechanics. However, the classical continuum theories of elasticity and viscoelasticity cannot easily capture the continual remodelling and associated structural changes of biological tissues. Furthermore, models drawn from plasticity theory are difficult to apply and interpret in this context, where there is no equivalent of a yield stress or flow rule. In this work, we describe a novel one-dimensional mathematical model of tissue remodelling based on the multiplicative decomposition of the deformation gradient. We express the mechanical effects of remodelling as an evolution equation for the 'effective strain', a measure of the difference between the current state and a hypothetical mechanically-relaxed state of the tissue. This morphoelastic model combines the simplicity and interpretability of classical viscoelastic models with the versatility of plasticity theory. A novel feature of our model is that while most models describe growth as a continuous quantity, here we begin with discrete cells and develop a continuum representation of lattice remodelling based on an appropriate limit of the behaviour of discrete cells. To demonstrate the utility of our approach, we use this framework to capture qualitative aspects of the continual remodelling observed in fibroblast-populated collagen lattices, in particular its contraction and its subsequent sudden re-expansion when remodelling is interrupted.