Eur. Phys. J. E (2015) 38: 33
https://doi.org/10.1140/epje/i2015-15033-4

Colloquium

Colloquium: Mechanical formalisms for tissue dynamics

¹ Laboratoire Matière et Systèmes Complexes, Université Denis Diderot - Paris 7, CNRS UMR 7057, 10 rue Alice Domon et Léonie Duquet, F-75205, Paris Cedex 13, France
² Laboratoire Physico-Chimie Curie, Institut Curie, Université Marie et Pierre Curie - Paris 6, CNRS UMR 168, 26 rue d’Ulm, F-75248, Paris Cedex 05, France
³ Laboratoire Charles Coulomb, Univ. Montpellier II, CNRS UMR 5221, Place Eugène Bataillon, CC070, F-34095, Montpellier Cedex 5, France
⁴ Institut de Génomique Fonctionnelle, Univ. Montpellier I, Univ. Montpellier II, 141 rue de la Cardonille, CNRS UMR 5203, INSERM UMR S 661, F-34094, Montpellier Cedex 05, France
⁵ Laboratoire Jean Kuntzmann, Université Joseph Fourier - Grenoble I, CNRS UMR 5524, BP 53, F-38041, Grenoble Cedex, France
⁶ Academy of Bradylogists, Paris Cedex 13, France

^* e-mail: cyprien.gay@univ-paris-diderot.fr
^** e-mail: francois.graner@univ-paris-diderot.fr

Received: 27 September 2013
Revised: 22 December 2014
Accepted: 9 March 2015
Published online: 13 May 2015

Abstract

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 developed 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.