  We present a mechanical model of tissue homeostasis that is specialised to
the intestinal crypt. Growth and deformation of the crypt, idealised as a line
of cells on a substrate, are modelled using morphoelastic rod theory.
Alternating between Lagrangian and Eulerian mechanical descriptions enables us
precisely to characterise the dynamic nature of tissue homeostasis, whereby the
proliferative structure and morphology are static in the Eulerian frame, but
there is active migration of Lagrangian material points out of the crypt.
Assuming mechanochemical growth, we identify the necessary conditions for
homeostasis, reducing the full, time-dependent system to a static boundary
value problem characterising a spatially heterogeneous "treadmilling" state. We
extract essential features of crypt homeostasis, such as the morphology, the
proliferative structure, the migration velocity, and the sloughing rate. We
also derive closed-form solutions for growth and sloughing dynamics in
homeostasis, and show that mechanochemical growth is sufficient to generate the
observed proliferative structure of the crypt. Key to this is the concept of
threshold-dependent mechanical feedback, that regulates an established Wnt
signal for biochemical growth. Numerical solutions demonstrate the importance
of crypt morphology on homeostatic growth, migration, and sloughing, and
highlight the value of this framework as a foundation for studying the role of
mechanics in homeostasis.
