Mechanical formalism for tissue dynamics
Résumé
The understanding of morphogenesis in living organisms has been renewed by tremendous progress
in experimental techniques that provide access to cell-scale, quantitative information both on the
shapes of cells within tissues and on the genes being expressed. This information suggests that our
understanding of the respective contributions of gene expression and mechanics, and of their crucial
entanglement, will soon leap forward. Biomechanics increasingly benefits from models, which assist
the design and interpretation of experiments, point out the main ingredients and assumptions, and
ultimately lead to predictions. The newly accessible local information thus calls for a reflection
on how to select suitable classes of mechanical models. We review both mechanical ingredients
suggested by the current knowledge of tissue behaviour, and modelling methods that can help
generate a rheological diagram or a constitutive equation. We distinguish cell scale (“intra-cell”)
and tissue scale (“inter-cell”) contributions. We recall the mathematical framework developped
for continuum materials and explain how to transform a constitutive equation into a set of partial
differential equations amenable to numerical resolution. We show that when plastic behaviour is
relevant, the dissipation function formalism appears appropriate to generate constitutive equations;
its variational nature facilitates numerical implementation, and we discuss adaptations needed in the
case of large deformations. The present article gathers theoretical methods that can readily enhance
the significance of the data to be extracted from recent or future high throughput biomechanical
experiments.
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