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From:
http://royalsocietypublishing.org/content/early/2012/10/09/rspa.2012.0418
James MacLaurin (1), Jon Chapman (1), Gareth Wyn Jones (2) and Tiina Roose (3)
(1) Mathematical Institute, University of Oxford, 24–29 St Giles, Oxford OX1 3LB, UK
(2) School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
(3) Bioengineering Sciences Research Group, Faculty of Engineering and Environment, University of Southampton, Southampton SO17 1BJ, UK
“The buckling of capillaries in solid tumours”, Proceedings of the Royal Society A, published online October 10, 2012, DOI: 10.1098/rspa.2012.0418
ABSTRACT: We develop a model of the buckling (both planar and axial) of capillaries in cancer tumours, using nonlinear solid mechanics. The compressive stress in the tumour interstitium is modelled as a consequence of the rapid proliferation of the tumour cells, using a multiplicative decomposition of the deformation gradient. In turn, the tumour cell proliferation is determined by the oxygen concentration (which is governed by the diffusion equation) and the solid stress. We apply a linear stability analysis to determine the onset of mechanical instability, and the Liapunov–Schmidt reduction to determine the postbuckling behaviour. We find that planar modes usually go unstable before axial modes, so that our model can explain the buckling of capillaries, but not as easily their tortuosity. We also find that the inclusion of anisotropic growth in our model can substantially affect the onset of buckling. Anisotropic growth also results in a feedback effect that substantially affects the magnitude of the buckle.
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