Continuum mechanical modeling of developing epithelial tissues with anisotropic surface growth

In this study, we address the computational modeling of biological soft tissue growth, focusing on the development of epithelial tissues. The formulation of the corresponding constitutive growth law using non-linear continuum mechanics and its implementation within a total-Lagrangian-type finite element method are described. In describing the growth law, we use multiplicative decomposition of the deformation gradient into a growth part and an elastic part. We propose two surface growth deformation gradients; isotropic surface growth and anisotropic surface growth with relative shrinkage in the principal direction of maximum stress at the initial state. We first apply our laws to a hollow thick-walled hemiellipsoid that idealizes a structure generally observed in the early development of epithelial tissues. Our simulation shows the following: (i) under isotropic surface growth, the hemiellipsoid becomes sphere-like, i.e., the ratio between the longest and shortest axial length tends to 1; (ii) in contrast, under anisotropic surface growth, the tissue elongates and flattens, i.e., the ratio between the longest and shortest axial length is enhanced. These results motivate us to apply the latter growth law to the realistic example of a vertebrate limb bud, which shows similar elongation and flattening during development. As expected, our numerical simulation succeeded in reproducing the essential aspects of morphological change in the limb bud, providing a new hypothesis for the vertebrate limb development.