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Description
Epithelial tissues at a pre-tumoral stage exhibit morphological changes: in particular epithelial ducts depart from the cylindrical shape, showing invaginations and evagination in the regions of the surface with malignant cells. Experiments report that at the inner and outer boundary of the epithelial sheets are concentrated molecular motors able to generate a surface active tension, that can vary between healthy and tumor cells. The mechanical origin of such morphology can be mathematically tackled by a continuum mechanical model \cite{ambrosi_1} able to relate, also quantitatively, the role of the impaired surface tensions.
The mathematical model, derived from first principles, accounts for the competition between the bulk elasticity of the epithelium and the surface tension of the apical and basal boundaries. The variation of the energy functional yields the Euler-Lagrange equations to be numerically integrated. The numerical results reproduce a variety of morphological shapes, from invagination to evagination, depending on the ratio between bulk and surface energy at variance of the length of the section. In particular, using parameters independently measured, we are able to reproduce experimental data reported for a ring partially made of transformed cells.
The numerical results obtained with a mathematical model that accounts, in a suitable way, for the thickness of the epithelial wall, prompt us to a deeper mathematical characterization that we address exploiting the Euler Elastica. In this framework we study the variety of possible shapes that a planar inextensible closed rod can take because of a piecewise inhomogeneity in its natural curvature. On the basis of numerical simulations, perturbation analysis and geometrical arguments, we are able to devise three morphological regimes and we provide a first order approximation of the curves that separate the shape regimes in the (k, s0 ) plane, k being the jump in natural curvature in the relative curvilinear coordinate s0. Our perturbation analysis, supported by geometrical arguments, compares well with the numerical results based on the fully nonlinear theory.
Bibliography
@article{ambrosi_1,
title = {Oncogenic transformation of tubular epithelial ducts: How mechanics affects morphology},
journal = {European Journal of Mechanics - A/Solids},
volume = {117},
pages = {105984},
year = {2026},
author = {D. Ambrosi and A. Favata and R. Paroni and G. Tomassetti},
}
