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We present a mechanical model of tissue homeostasis that is specialised tothe intestinal crypt. Growth and deformation of the crypt, idealised as a lineof cells on a substrate, are modelled using morphoelastic rod theory.Alternating between Lagrangian and Eulerian mechanical descriptions enables usprecisely to characterise the dynamic nature of tissue homeostasis, whereby theproliferative structure and morphology are static in the Eulerian frame, butthere is active migration of Lagrangian material points out of the crypt.Assuming mechanochemical growth, we identify the necessary conditions forhomeostasis, reducing the full, time-dependent system to a static boundaryvalue problem characterising a spatially heterogeneous "treadmilling" state. Weextract essential features of crypt homeostasis, such as the morphology, theproliferative structure, the migration velocity, and the sloughing rate. Wealso derive closed-form solutions for growth and sloughing dynamics inhomeostasis, and show that mechanochemical growth is sufficient to generate theobserved proliferative structure of the crypt. Key to this is the concept ofthreshold-dependent mechanical feedback, that regulates an established Wntsignal for biochemical growth. Numerical solutions demonstrate the importanceof crypt morphology on homeostatic growth, migration, and sloughing, andhighlight the value of this framework as a foundation for studying the role ofmechanics in homeostasis.

The role of mechanics in the growth and homeostasis of the intestinal crypt