Mechanical forces and deformations represent fast and long-range information carriers for living cells. By propagating through elastic substrates, they drive indirect cell-cell communication and affect the self-assembly of functional multicellular structures during tissue morphogenesis, growth and healing. Here, we investigate how substrate-mediated cell-cell mechanical interactions regulate the reorientation dynamics of non-confluent cell populations, by interpreting this process –identified as mechanotropism– as result of a cooperative/non-cooperative game in which cells act as players aiming to attain optimal arrangements guarantying them minimal elastic energy expense while preserving homeostatic levels of contractile forces at focal adhesions. By translating the problem into a biophysically coherent mathematical formulation, equilibria of pairs and clusters of adherent cells are derived as the free response of a non-linear dynamical system with multiple degrees of freedom, the resulting collective rotational motion posing as a mechanobiological analogous of the n-body problem studied in classical physics, albeit with qualitative differences in the outcomes. Formation of strings, rings, polygons and branched networks is theoretically predicted for cells on incompressible substrates, thus retracing experimental observations of literature, which show dance-like motives and nearly regular topologies in different types of cell cultures in vitro. Unprecedented outcomes are finally obtained as a function of the Poisson’s ratio of the substrate, symmetry breaking and chiral patterns emerging for auxetic materials. These results might contribute to improve the current knowledge on mechanobiology of tissue formation and repair, and help to conceive novel engineered scaffolds for biomedical applications, as well as bioinspired self-organizing materials and active systems.
Durotactic cell migration plays a fundamental role in many processes, like tumor invasion, wound healing, and tissue engineering applications. This work presents a novel computational model for durotaxis-mediated cell migration within enzyme-sensitive scaffolds. Cell motion is described as an advective-diffusive process through a multi-phase-field formulation and is driven by the mechanical interplay between cells and substrate. The model couples cell migration with enzyme production and diffusion, which cause hydrogel degradation and the associated spatio-temporal evolution of viscosity and stiffness. Mechanical sensing is triggered by cell contraction, which generates a stress field in the surrounding scaffold that is then evaluated to identify the preferred migration direction. The resulting durotactic cue regulates the transition from random motion to a mechanically biased migration regime, while cell velocity is derived from a force balance involving stochastic protrusion, stress-driven polarization, and traction forces generated at the cell-matrix interface. The model is implemented in a finite-element framework and applied to several representative scenarios, in which cell durotaxis is induced by spatial variations in substrate stiffness or/and by externally applied mechanical loads, remarkably reconciled within a single modeling framework. The numerical results show that the model captures the occurrence of directional migration toward mechanically favorable regions, predicts cell speed and traction force values consistent with experimental evidence, and provides quantitative insights into the coupled evolution of cell motion, substrate degradation, and stress redistribution.
Homogenized continuum models are widely used to describe wave propagation and band-gap behavior in mechanical metamaterials without explicitly resolving their microstructure. Their validity, however, typically relies on the classical separation of scales assumption, according to which the wavelength of the propagating disturbance is much larger than the characteristic size of the unit cell. In finite-sized metamaterial samples and at higher frequencies, this assumption progressively breaks down, and the dynamic response becomes strongly influenced by the way the microstructure is truncated at the external boundaries.In this work we introduce a concept in the homogenized description of mechanical metamaterials: the inertial contribution of macroscopic interfaces. We show that different truncations of the same lattice generate boundaries with distinct mass distributions, which lead to measurable differences in the dynamic response of finite-sized specimens. To capture this complex mechanism in a homogenized framework, we extend the relaxed micromorphic model by introducing a kinetic surface energy defined on the boundary of the considered body. This generates an additional inertial term in the boundary conditions that can be seen as the homogenized counterpart of the interface inertia produced by the truncation of the microstructure. As a result, the homogenized model can now distinguish between finite-sized specimens that share identical bulk properties but differ only in the configuration of their interfaces.The proposed formulation preserves the variational structure of the relaxed micromorphic model while enabling the continuum to reproduce boundary-dependent responses observed in fully resolved simulations, particularly in frequency regimes where scale separation no longer holds.This result opens the door to predictive homogenized models, enabling the upscaling towards larger-scale structures assembled from finite metamaterial building blocks, a step that would be computationally infeasible using fully resolved microstructured models.
Typically, there is a challenge for developing explicit, interpretable and physically admissible anisotropic hyperelastic constitutive models, especially if only a limited number of multi-mode mechanical data are available. This work suggests a novel physics‑constrained symbolic regression model empowered with data augmentation techniques using neural networks to uncover the strain‑energy functions involving invariants. The material is represented as a transversely isotropic hyperelastic solid with dependence on the first isotropic invariant and the anisotropic invariant that takes into consideration fibre dependency. All neural networks are utilized only as mode-specific surrogate interpolators to create dense data sets for optimization and independent validation from sparse data sets of reference responses and the product, the numerical outputs, are finally formulated as closed-form symbolic relations. One of the ingredients of the proposed approach is analytical and semi analytical derivation of stress–deformation relationships for four test modes: uniaxial tension along the fiber direction, transverse uniaxial tension, simple shear and biaxial tension. In both, incorporating incompressibility constraints, the Lagrange multiplier of incompressibility is always removed with chosen traction conditions, giving us direct mappings of candidate strain energy functions into nominal stress responses. This stress mapping approach eliminates the use of numerical differentiation, the need for multiple numerical solution procedures, and facilitates effective symbolic evaluation of symbolic candidates in the multi mode regression process. Using these as the basis, a library of 30 candidate strain energy functions, which have been calibrated and ranked in order of accuracy, model complexity, performance in the validation phase and physical admissibility after fitting, has been developed. The analysis shows the best compromise between accuracy and complexity is reached by using compact polynomial formulations with fibre contributions represented by a quadratic term and the use of shifted invariant variables. The PLQI QF model yielded better fitting error, while the simpler PLI QF model had similar validation accuracy but used fewer parameters and was less complex. Independent validation provided a confirmation of uniaxial, shear and biaxial responses were well reproduced; however, the biaxial case was the most restrictive one. We believe that the proposed framework provides a data efficient and mechanically consistent way to extract and identify interpretable constitutive models of an repeatable and predictable form for an anisotropic hyperelastic material.
