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Menon SN, Hall CL, McCue SW, McElwain DLS. A model for one-dimensional morphoelasticity and its application to fibroblast-populated collagen lattices. Biomech Model Mechanobiol 2017; 16:1743-1763. [PMID: 28523375 DOI: 10.1007/s10237-017-0917-3] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/28/2016] [Accepted: 05/03/2017] [Indexed: 11/26/2022]
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.
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Affiliation(s)
- Shakti N Menon
- The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai, 600113, India
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, 4001, Australia
- Institute of Health and Biomedical Innovation, Queensland University of Technology, Brisbane, QLD, 4001, Australia
| | - Cameron L Hall
- Mathematics Applications Consortium with Science and Industry, University of Limerick, Castletroy, Limerick, V94 T9PX, Ireland
- Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford, OX1 3LB, UK
| | - Scott W McCue
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, 4001, Australia.
| | - D L Sean McElwain
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, 4001, Australia
- Institute of Health and Biomedical Innovation, Queensland University of Technology, Brisbane, QLD, 4001, Australia
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