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

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 in 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.

Uncertainty quantification on a spatial Markov-chain model for the progression of skin cancer

“…Their approach is based on a system of ordinary differential equation to describe the displacement of the spatial coordinate in the tissue in a one-dimensional configuration. Their model contains some similarities with the morpho-elastic models, which were applied in Koppenol and Vermolen (2017) and Menon et al (2017). The approach is entirely deterministic and one-dimensional, contrary to our current model which is entirely stochastic and three-dimensional.…”

Stability of a two-dimensional biomorphoelastic model for post-burn contraction

“… 2006 ; Koppenol 2017 ; Menon et al. 2017 ). We can divide most of these models into continuum hypothesis-based, discrete cell-based, and hybrid models (Koppenol 2017 ).…”

Stability of a one-dimensional morphoelastic model for post-burn contraction

“…Various studies report on mathematical models to predict the behavior of experimental and clinical wounds and to gain insight into which elements of the wound healing response might have a substantial influence on the contraction (Tranquillo and Murray 1992;Olsen et al 1995;Barocas and Tranquillo 1997;Dallon et al 1999;McDougall et al 2006;Koppenol 2017;Menon et al 2017) to name a few. This study uses the morphoelastic model for burn wound contraction that has been developed by Koppenol and Vermolen (2017).…”