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The function of organs such as lungs, kidneys and mammary glands relies on the three-dimensional geometry of their epithelium. To adopt shapes such as spheres, tubes and ellipsoids, epithelia generate mechanical stresses that are generally unknown. Here we engineered curved epithelial monolayers of controlled size and shape and mapped their state of stress. We designed pressurized epithelia with circular, rectangular and ellipsoidal footprints. We developed a computational method to map the stress tensor in these epithelia. This method establishes a direct correspondence between epithelial shape and mechanical stress without assumptions of material properties. In epithelia with spherical geometry spanning more than one order of magnitude in radius, we show that stress weakly increases with areal strain in a size-independent manner. In epithelia with rectangular and ellipsoidal cross-section we found pronounced stress anisotropies consistent with the asymmetric distribution of tractions measured at the cell-substrate contact line. In these anisotropic profiles, cell shape tended to align with the direction of maximum principal stress but this alignment was non-universal and depended on epithelial geometry. Besides interrogating the fundamental mechanics of epithelia over a broad range of sizes and shapes, our approach will enable a systematic study of how geometry and stress influence epithelial fate and function in three-dimensions.

Living tissues are active, multifunctional materials capable of generating, sensing, withstanding and responding to mechanical stress. These capabilities enable tissues to adopt complex shapes during development, to sustain those shapes during homeostasis and to restore them during healing and regeneration. Abnormal stress is associated with a broad range of pathological conditions, including developmental defects, inflammatory diseases, tumour growth and metastasis. A number of techniques are available to measure mechanical stress in living tissues at cellular and subcellular resolution. 2D techniques that map stress in cultured cell monolayers provide the highest resolution and accessibility, and include 2D traction force microscopy, micropillar arrays, monolayer stress microscopy and monolayer stretching between flexible cantilevers. Mapping stresses in tissues cultured in 3D can be achieved using 3D traction force microscopy and the microbulge test. Techniques for measuring stress in vivo include servo-null methods for measuring luminal pressure, deformable inclusions, Forster resonance energy transfer tension sensors, laser ablation and computational methods for force inference. Although these techniques are far from becoming everyday tools in biomedical laboratories, their rapid development is fostering key advances in our understanding of the role of mechanics in morphogenesis, homeostasis and disease. Methods for measuring stress in living cells, tissues and organs are advancing steadily and are increasingly being used for biomedical applications. In this Review, we discuss the concept of tissue stress and the techniques available to measure it in 2D and 3D cell and tissue cultures and in vivo.