Validity of the Cauchy-Born rule applied to discrete cellular-scale models of biological tissues

Y. Davit, J. M. Osborne, H. M. Byrne, D. Gavaghan, and J. Pitt-Francis
Phys. Rev. E 87, 042724 – Published 30 April 2013

Abstract

The development of new models of biological tissues that consider cells in a discrete manner is becoming increasingly popular as an alternative to continuum methods based on partial differential equations, although formal relationships between the discrete and continuum frameworks remain to be established. For crystal mechanics, the discrete-to-continuum bridge is often made by assuming that local atom displacements can be mapped homogeneously from the mesoscale deformation gradient, an assumption known as the Cauchy-Born rule (CBR). Although the CBR does not hold exactly for noncrystalline materials, it may still be used as a first-order approximation for analytic calculations of effective stresses or strain energies. In this work, our goal is to investigate numerically the applicability of the CBR to two-dimensional cellular-scale models by assessing the mechanical behavior of model biological tissues, including crystalline (honeycomb) and noncrystalline reference states. The numerical procedure involves applying an affine deformation to the boundary cells and computing the quasistatic position of internal cells. The position of internal cells is then compared with the prediction of the CBR and an average deviation is calculated in the strain domain. For center-based cell models, we show that the CBR holds exactly when the deformation gradient is relatively small and the reference stress-free configuration is defined by a honeycomb lattice. We show further that the CBR may be used approximately when the reference state is perturbed from the honeycomb configuration. By contrast, for vertex-based cell models, a similar analysis reveals that the CBR does not provide a good representation of the tissue mechanics, even when the reference configuration is defined by a honeycomb lattice. The paper concludes with a discussion of the implications of these results for concurrent discrete and continuous modeling, adaptation of atom-to-continuum techniques to biological tissues, and model classification.

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  • Received 11 June 2012

DOI:https://doi.org/10.1103/PhysRevE.87.042724

©2013 American Physical Society

Authors & Affiliations

Y. Davit1,2,3, J. M. Osborne4, H. M. Byrne1,4, D. Gavaghan4, and J. Pitt-Francis4

  • 1Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, United Kingdom
  • 2Université de Toulouse, INPT, UPS, Institut de Mécanique des Fluides de Toulouse, Allée Camille Soula, F-31400 Toulouse, France
  • 3CNRS, Institut de Mécanique des Fluides de Toulouse, F-31400 Toulouse, France
  • 4Department of Computer Science, University of Oxford, Parks Road, Oxford OX1 3QD, United Kingdom

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Issue

Vol. 87, Iss. 4 — April 2013

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