A continuum approach to blood vessel growth: Axisymmetric elastic structures

  • Waxman A
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Abstract

In this paper we discuss the morphogenesis of small blood vessels (venules) as the nautral consequence of physical forces prevailing during endothelial cell division. A physical model is developed in which the blood vessel is treated as a growing, thin elastic shell embedded in a viscous fluid (i.e. the surrounding tissue). It is explained how a pre-existing cylindrical vessel, induced to grow by some promoter, can buckle and thereby develop a spatially periodic structure displaying varicosity, sinuosity, and/or helicity. Growth manifests itself dynamically in terms of a "growth pressure" which disturbs any pre-existing force balance. The governing set of non-linear partial differential equations are derived, and solutions corresponding to uniform dilation are obtained. The buckled structure emerges as an instability of this time dependent basic state of uniform dilation. A linear stability analysis yields the dominant wavelength of the varicose mode; these results compare favorably with crude measurements made from the experimental literature. In the hope of uncovering the mechanism which underlies the selection of sprouting sites along a parent vessel, it is suggested that reaction and diffusion processes (between growth promoting and inhibiting substances) on buckled surfaces be coupled to the dynamical force balances discussed here. © 1981.

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Authors

  • Allen M. Waxman

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