Cell differentiation, proliferation and migration are essential processes in tissue regeneration. Experimental evidence confirms that cell differentiation or proliferation can be regulated according to the extracellular matrix stiffness. For instance, mesenchymal stem cells (MSCs) can differentiate to neuroblast, chondrocyte or osteoblast within matrices mimicking the stiffness of their native substrate. However, the precise mechanisms by which the substrate stiffness governs cell differentiation or proliferation are not well known. Therefore, a mechano-sensing computational model is here developed to elucidate how substrate stiffness regulates cell differentiation and/or proliferation during cell migration. In agreement with experimental observations, it is assumed that internal deformation of the cell (a mechanical signal) together with the cell maturation state directly coordinates cell differentiation and/or proliferation. Our findings indicate that MSC differentiation to neurogenic, chondrogenic or osteogenic lineage specifications occurs within soft (0.1-1 kPa), intermediate (20-25 kPa) or hard (30-45 kPa) substrates, respectively. These results are consistent with well-known experimental observations. Remarkably, when a MSC differentiate to a compatible phenotype, the average net traction force depends on the substrate stiffness in such a way that it might increase in intermediate and hard substrates but it would reduce in a soft matrix. However, in all cases the average net traction force considerably increases at the instant of cell proliferation because of cell-cell interaction. Moreover cell differentiation and proliferation accelerate with increasing substrate stiffness due to the decrease in the cell maturation time. Thus, the model provides insights to explain the hypothesis that substrate stiffness plays a key role in regulating cell fate during mechanotaxis.

Between other parameters, cell migration is partially guided by the mechanical properties of its substrate. Although many experimental works have been developed to understand the effect of substrate mechanical properties on cell migration, accurate 3D cell locomotion models have not been presented yet. In this paper, we present a novel 3D model for cells migration. In the presented model, we assume that a cell follows two main processes: in the first process, it senses its interface with the substrate to determine the migration direction and in the second process, it exerts subsequent forces to move. In the presented model, cell traction forces are considered to depend on cell internal deformation during the sensing step. A random protrusion force is also considered which may change cell migration direction and/or speed. The presented model was applied for many cases of migration of the cells. The obtained results show high agreement with the available experimental and numerical data.

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Data Science has burst into simulation-based engineering sciences with an impressive impulse. However, data are never uncertainty-free and a suitable approach is needed to face data measurement errors and their intrinsic randomness in problems with well-established physical constraints. As in previous works, this problem is here faced by hybridizing a standard mathematical modeling approach with a new data-driven solver accounting for the phenomenological part of the problem, with the aim of finding a solution point, satisfying some constraints, that minimizes a distance to a given data-set. However, unlike such works that are established in a deterministic framework, we use the Mahalanobis distance in order to incorporate statistical second order uncertainty of data in computations, i.e. spread and correlations. We develop the underlying stochastic theoretical framework and establish the fundamental mathematical and statistical properties. The performance of the resulting reliability-based data-driven procedure performance is evaluated in a simple but illustrative unidimensional problem as well as in a more realistic solution of a 3D structural problem with a material with intrinsically random constitutive behavior as concrete. The results show, in comparison with other data-driven solvers, better * Corresponding author Email address: mdoblare@unizar.es (Manuel Doblaré) convergence, higher accuracy, clearer interpretation, and major flexibility besides the relevance of allowing uncertainty management, with low computational demand.

The processes in which cardiac cells are reorganized for tissue regeneration is still unclear. It is a complicated process that is orchestrated by many factors such as mechanical, chemical, thermal, and/or electrical cues. Studying and optimizing these conditions in-vitro is complicated and time costly. In such cases, in-silico numerical simulations can offer a reliable solution to predict and optimize the considered conditions for the cell culture process. For this aim, a 3D novel and enhanced numerical model has been developed to study the effect of the mechanical properties of the extracellular matrix (ECM) as well as the applied external forces in the process of the cell differentiation and proliferation for cardiac muscle tissue regeneration. The model has into account the essential cellular processes such as migration, cell-cell interaction, cell-ECM interaction, differentiation, proliferation and/or apoptosis. It has employed to study the initial stages of cardiac muscle tissue formation within a wide range of ECM stiffness (8-50 kPa). The results show that, after cell culture within a free surface ECM, cells tend to form elongated aggregations in the ECM center. The formation rate, as well as the aggregation morphology, have been found to be a function of the ECM stiffness and the applied external force. Besides, it has been found that the optimum ECM stiffness for cardiovascular tissue regeneration is in the range of 29-39 kPa, combined with the application of a mechanical stimulus equivalent to deformations of 20-25%.

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Computational methods for the advection-diffusion-reaction transport equation are still a challenge. Although there exist globally stable methods, oscillations around sharp layers such as boundary, inner and outflow layers, are typical in multi-dimensional flows. In this paper a variational formulation that combines two types of stabilization integrals is proposed, namely an adjoint stabilization and a gradient adjoint stabilization. Two free parameters are chosen by imposing one-dimensional superconvergence. Then, when applied to multi-dimensional flows, the method presents better local stability than the present stabilized methods. Furthermore, in the advective-diffusive limit and for piecewise linear functional spaces, the method recovers the classical SUPG method.

Cell migration is a vital process in many biological phenomena ranging from wound healing to tissue regeneration. Over the past few years, it has been proven that in addition to cell-cell and cell-substrate mechanical interactions (mechanotaxis), cells can be driven by thermal, chemical and/or electrical stimuli. A numerical model was recently presented by the authors to analyse single cell migration in a multi-signalling substrate. That work is here extended to include multi-cell migration due to cell-cell interaction in a multi-signalling substrate under different conditions. This model is based on balancing the forces that act on the cell population in the presence of different guiding cues. Several numerical experiments are presented to illustrate the effect of different stimuli on the trajectory and final location of the cell population within a 3D heterogeneous multi-signalling substrate. Our findings indicate that although multi-cell migration is relatively similar to single cell migration in some aspects, the associated behaviour is very different. For instance, cell-cell interaction may delay single cell migration towards effective cues while increasing the magnitude of the average net cell traction force as well as the local velocity. Besides, the random movement of a cell within a cell population is slightly greater than that of single cell migration. Moreover, higher electrical field strength causes the cell slug to flatten near the cathode. On the other hand, as with single cell migration, the existence of electrotaxis dominates mechanotaxis, moving the cells to the cathode or anode pole located at the free surface. The numerical results here obtained are qualitatively consistent with related experimental works.