Menon, S. N. and Hall, C. L. and McCue, S. W. and McElwain, D. L. S. (2012) A model for one-dimensional morphoelasticity and its application to fibroblast-populated collagen lattices. Journal of Mathematical Biology . (Submitted)
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.
To demonstrate the utility of our approach, we derive and analyse a system of coupled partial differential equations that describes the deformation of fibroblast-populated collagen lattices. These lattices are rearranged by the fibroblast cells that inhabit them, and can subsequently contract to as little as 10% of their initial lateral (or vertical) extent. It has been observed that when this reorganisation is interrupted, the lattices re-expand slightly but do not return to their original size. We find that our morphoelastic model captures several important observed features of this process and that numerical values for the initial stiffness and viscosity of the collagen gel, obtained by fitting our model to previously obtained data, compare well with the results of rheological experiments.
|Subjects:||D - G > General|
|Research Groups:||Oxford Centre for Collaborative Applied Mathematics|
|Deposited By:||Peter Hudston|
|Deposited On:||26 Jul 2012 09:17|
|Last Modified:||26 Jul 2012 09:17|
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