Abstract
The process of biological growth and the associated generation of residual stress has previously been considered as a driving mechanism for tissue buckling and pattern selection in numerous areas of biology. Here, we develop a two-dimensional thin plate theory to simulate the growth of cultured intestinal epithelial cells on a deformable substrate, with the goal of elucidating how a tissue engineer might best recreate the regular array of invaginations (crypts of Lieberkühn) found in the wall of the mammalian intestine. We extend the standard von Kármán equations to incorporate inhomogeneity in the plate’s mechanical properties and surface stresses applied to the substrate by cell proliferation. We determine numerically the configurations of a homogeneous plate under uniform cell growth, and show how tethering to an underlying elastic foundation can be used to promote higher-order buckled configurations. We then examine the independent effects of localised softening of the substrate and spatial patterning of cellular growth, demonstrating that (within a two-dimensional framework, and contrary to the predictions of one-dimensional models) growth patterning constitutes a more viable mechanism for control of crypt distribution than does material inhomogeneity.
Highlights
This article addresses the mechanism by which biological growth may generate a build-up of residual stresses within a tissue, the relief of which drives deformations
We present the configurations attained by a homogeneous plate buckling under the influence of a uniformly growing cell layer, and show how these configurations are affected by attachment to a supporting elastic foundation, localised softening of the substrate, and localised cellular growth
We have developed a model to examine the mechanisms of growth-induced buckling in a two-dimensional tissue
Summary
This article addresses the mechanism by which biological growth may generate a build-up of residual stresses within a tissue, the relief of which drives deformations. Considering a typical cell culture substrate as a thin plate, we present an extension to the standard von Kármán equations to incorporate (i) surface stresses induced by proliferating cells upon the substrate’s upper surface, and a supporting foundation below, and (ii) spatial variations in the plate’s mechanical properties. We present the configurations attained by a homogeneous plate buckling under the influence of a uniformly growing cell layer, and show how these configurations are affected by attachment to a supporting elastic foundation, localised softening of the substrate, and localised cellular growth. In (6a), gðX; YÞ describes the distribution of stresses associated with the flat configuration; as the cells grow, g increases and the layer must deform in order to relieve the induced in-plane compression.
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