Understanding the growth of branching organs is an important scientific endeavour. It has crucial applications, from saving premature newborns, to repairing or even regenerating organs. Despite differences in timing and shape, branching morphogenesis of all branching organs or glands (lung, kidney, salivary, lachrymal, mammary glands, sebaceous and sweat glands, prostate, guts papillae etc.) is similar: an epithelial sheet of cells, forming a 2D layer, penetrates into a 3D mass of mesenchymal cells. Inside the epithelium, a lumen is filled with fluid. As the epithelium grows, it evolves into a branched structure. The pattern of branches is in some cases stereotypic, deterministic, and it has memory effects. We present a simple line of reasoning that predicts that viscous fingering of biological tissue will exhibit all of these features. The line of reasoning is based on the idea that surface tension selects the shape of a growing branch, as is well known in the context of moving boundary problems, except that in this case, the surface is akin to a liquid-crystal. The anisotropy of the surface tension comes from a symmetry breaking by collagen and fibroblasts. The equilibrium shape of the corresponding boundary is that of an actual fingertip, and the out-of-equilibrium shape is that of branched organs, such as the lung.

La morphogenèse des organes branchés et, plus généralement, des structures tubulaires dans le vivant, y compris des doigts, est mal comprise. On a bien identifié lexpression dun certain nombre de gènes au voisinage de ces tubes, en cours de morphogenèse, mais ces découvertes ne permettent pas de construire la croissance dun tube dune forme donnée. Le fait même que ces structures soient des tubes nest pas expliqué ni pris en compte par ces études biologiques. Nous présentons un modèle qui explique largement les propriétés morphologiques de la formation de structures tubulaires, des organes arborisés, et qui prédit, en particulier, la forme des doigts et le principe itératif de construction des reins, poumons, et autres organes ou glandes arborisés.