A Cellular Potts Model of single cell migration in presence of durotaxis.

Cell migration is a fundamental biological phenomenon during which cells sense their surroundings and respond to different types of signals. In presence of durotaxis, cells preferentially crawl from soft to stiff substrates by reorganizing their cytoskeleton from an isotropic to an anisotropic distribution of actin filaments. In the present paper, we propose a Cellular Potts Model to simulate single cell migration over flat substrates with variable stiffness. We have tested five configurations: (i) a substrate including a soft and a stiff region, (ii) a soft substrate including two parallel stiff stripes, (iii) a substrate made of successive stripes with increasing stiffness to create a gradient and (iv) a stiff substrate with four embedded soft squares. For each simulation, we have evaluated the morphology of the cell, the distance covered, the spreading area and the migration speed. We have then compared the numerical results to specific experimental observations showing a consistent agreement.

Diffusion model to describe osteogenesis within a porous titanium scaffold.

In this study, we develop a two-dimensional finite element model, which is derived from an animal experiment and allows simulating osteogenesis within a porous titanium scaffold implanted in ewe's hemi-mandible during 12 weeks. The cell activity is described through diffusion equations and regulated by the stress state of the structure. We compare our model to (i) histological observations and (ii) experimental data obtained from a mechanical test done on sacrificed animal. We show that our mechano-biological approach provides consistent numerical results and constitutes a useful tool to predict osteogenesis pattern.

Mechanical link between durotaxis, cell polarity and anisotropy during cell migration.

Cell migration, a fundamental mechanobiological process, is highly sensitive to the biochemical and mechanical properties of the environment. Efficient cell migration is ensured by the intrinsic polarity of the cell, which triggers a transition from an isotropic to an anisotropic configuration of the acto-mysion filaments responsible for the protrusion-contraction movement of the cell. Additionally, polarity may be highly influenced by the substrate rigidity, which results in a phenomenon called durotaxis. In the present work, we propose a two-dimensional finite element model able to capture three main features of cell migration: durotaxis, cell polarity and anisotropy. The cell is modelled as a continuum able to develop cyclic active strains regulated by the polymerization and depolymerization of the acto-myosin filaments and synchronized with the adhesion forces between the cell and the substrate underneath. A generalized Maxwell model is used to describe the viscoelastic behaviour of the cell constituted by a solid anisotropic branch with active strains (i.e. the acto-myosin filaments) and a fluid viscoelastic branch (i.e. the cytoplasm). Several types of substrate have been tested which are homogeneously soft or stiff or include both regions. The numerical results have been qualitatively compared with experimental observations showing a good agreement and have allowed us to find the mechanical link between durotaxis, cell polarity and anisotropy.

Healthy vs. osteoarthritic hips: a comparison of hip, pelvis and femoral parameters and relationships using the EOS® system.

BACKGROUND: Osteoarthritis is a debilitating disease, for which the development path is unknown. Hip, pelvis and femoral morphological and positional parameters relate either to individual differences or to changes in the disease state, both of which should be taken into account when diagnosing and treating patients. These have not yet been comprehensively quantified. Previous imaging studies have been limited by a number of factors: supine rather than standing measurements; high radiation dose; a limited field of view; and 2D rather than 3D measurements. EOS®, a new radiographic imaging modality that acquires simultaneous frontal and lateral (sagittal) X-ray images of the full body, allows 3D reconstruction of the hip, pelvis and lower limb. The aim of the study was to explore similarities and differences between healthy and osteoarthritis groups. METHODS: Two groups of subjects, 30 healthy and 30 with hip osteoarthritis, were assessed and compared for pelvic, acetabular and femoral parameters in the standing position. FINDINGS: There were not only significant differences between groups but also considerable overlap amongst the individuals. Sacral slope, acetabular angle of Idelberger and Frank, femoral mechanical angle and femoral head eccentricity as well as right-left asymmetries in centre-edge acetabular angle and femoral head diameter were higher on average in osteoarthritic patients compared to healthy subjects, whereas acetabular abduction was lower in the osteoarthritic group (P<0.05). Correlations were identified between key parameters in both groups. INTERPRETATION: Differences between the groups suggest either degenerative changes over time or inherent differences between individuals that may contribute to the disease progression. These data provide a basis for longitudinal and post-surgery studies. Due to the considerable variability amongst individuals and the considerable overlap between groups, patients should be evaluated individually and at multiple joints when planning hip, knee and spine surgery.

A computational mechanics approach to assess the link between cell morphology and forces during confined migration.

Confined migration plays a fundamental role during several biological phenomena such as embryogenesis, immunity and tumorogenesis. Here, we propose a two-dimensional mechanical model to simulate the migration of a HeLa cell through a micro-channel. As in our previous works, the cell is modelled as a continuum and a standard Maxwell model is used to describe the mechanical behaviour of both the cytoplasm (including active strains) and the nucleus. The cell cyclically protrudes and contracts and develops viscous forces to adhere to the substrate. The micro-channel is represented by two rigid walls, and it exerts an additional viscous force on the cell boundaries. We test four channels whose dimensions in terms of width are i) larger than the cell diameter, ii) sub-cellular, ii) sub-nuclear and iv) much smaller than the nucleus diameter. The main objective of the work is to assess the necessary conditions for the cell to enter into the channel and migrate through it. Therefore, we evaluate both the evolution of the cell morphology and the cell-channel and cell-substrate surface forces, and we show that there exists a link between the two, which is the essential parameter determining whether the cell is permeative, invasive or penetrating.

Reaction-diffusion finite element model of lateral line primordium migration to explore cell leadership.

Collective cell migration plays a fundamental role in many biological phenomena such as immune response, embryogenesis and tumorigenesis. In the present work, we propose a reaction-diffusion finite element model of the lateral line primordium migration in zebrafish. The population is modelled as a continuum with embedded discrete motile cells, which are assumed to be viscoelastic and able to undergo large deformations. The Wnt/ß-catenin-FGF and cxcr4b-cxcr7b signalling pathways inside the cohort regulating the migration are described through coupled reaction-diffusion equations. The coupling between mechanics and the molecular scenario occurs in two ways. Firstly, the intensity of the protrusion-contraction movement of the cells depends on the cxcr4b concentration. Secondly, the intra-synchronization between the active deformations and the adhesion forces inside each cell is triggered by the cxcr4b-cxcr7b polarity. This influences the inter-synchronization between the cells and results in two main modes of migration: uncoordinated and coordinated. The main objectives of the work were (i) to validate our assumptions with respect to the experimental observations and (ii) to decipher the mechanical conditions leading to efficient migration of the primordium. To achieve the second goal, we will specifically focus on the role of the leader cells and their position inside the population.

Mechanical modelling of confined cell migration across constricted-curved micro-channels.

Confined migration is a crucial phenomenon during embryogenesis, immune response and cancer. Here, a two-dimensional finite element model of a HeLa cell migrating across constricted-curved micro-channels is proposed. The cell is modelled as a continuum with embedded cytoplasm and nucleus, which are described by standard Maxwell viscoelastic models. The decomposition of the deformation gradient is employed to define the cyclic active strains of protrusion and contraction, which are synchronized with the adhesion forces between the cell and the substrate. The micro-channels are represented by two rigid walls and exert an additional viscous force on the cell boundaries. Five configurations have been tested: 1) top constriction, 2) top-bottom constriction, 3) shifted top-bottom constriction, 4) embedded obstacle and 5) bending micro-channel. Additionally, for the first four micro-channels both sub-cellular and sub-nuclear constrictions have been obtained, while for the fifth micro-channel three types of bending have been investigated ('curved', 'sharp' and 'sharper'). For each configuration, several parameters such as the cell behaviour, the covered distance, the migration velocity, the ratio between the cell and the nucleus area as well as the cell-substrate and cell-channel surfaces forces have been evaluated. The results show once more the fundamental role played by mechanics of both the cell and the environment.

On the mechanical interplay between intra- and inter-synchronization during collective cell migration: a numerical investigation.

Collective cell migration is a fundamental process that takes place during several biological phenomena such as embryogenesis, immunity response, and tumorogenesis, but the mechanisms that regulate it are still unclear. Similarly to collective animal behavior, cells receive feedbacks in space and time, which control the direction of the migration and the synergy between the cells of the population, respectively. While in single cell migration intra-synchronization (i.e. the synchronization between the protrusion-contraction movement of the cell and the adhesion forces exerted by the cell to move forward) is a sufficient condition for an efficient migration, in collective cell migration the cells must communicate and coordinate their movement between each other in order to be as efficient as possible (i.e. inter-synchronization). Here, we propose a 2D mechanical model of a cell population, which is described as a continuum with embedded discrete cells with or without motility phenotype. The decomposition of the deformation gradient is employed to reproduce the cyclic active strains of each single cell (i.e. protrusion and contraction). We explore different modes of collective migration to investigate the mechanical interplay between intra- and inter-synchronization. The main objective of the paper is to evaluate the efficiency of the cell population in terms of covered distance and how the stress distribution inside the cohort and the single cells may in turn provide insights regarding such efficiency.

Diffusion-reaction model for Drosophila embryo development.

During the early stages of gastrulation in Drosophila embryo, the epithelial cells composing the single tissue layer of the egg undergo large strains and displacements. These movements have been usually modelled by decomposing the total deformation gradient in an (imposed or strain/stress dependent) active part and a passive response. Although the influence of the chemical and genetic activity in the mechanical response of the cell has been experimentally observed, the effects of the mechanical deformation on the latter have been far less studied, and much less modelled. Here, we propose a model that couples morphogen transport and the cell mechanics during embryogenesis. A diffusion-reaction equation is introduced as an additional mechanical regulator of morphogenesis. Consequently, the active deformations are not directly imposed in the analytical formulation, but they rather depend on the morphogen concentration, which is introduced as a new variable. In this study, we show that strain patterns similar to those observed during biological experiments can be reproduced by properly combining the two phenomena. In addition, we use a novel technique to parameterise the embryo geometry by solving two Laplace problems with specific boundary conditions. We apply the method to two morphogenetic movements: ventral furrow invagination and germ band extension. The matching between our results and the observed experimental deformations confirms that diffusion-reaction of morphogens can actually be controlling large morphogenetic movements.

An extensive numerical simulation of the cephalic furrow formation in Drosophila embryo.

Here, we propose a new finite element model of the Drosophila embryo to simulate the cephalic furrow (CF) formation. Two different 3D geometries are introduced to reproduce the embryo. The two of them are parametrised through a novel system of curvilinear coordinates, well adapted for biological structures, which is obtained by three fictive electrostatic potentials. A gradient decomposition method is used to take into account both the active and the passive deformations occurring to the cells. Four simulations of the CF are proposed, which in turn considers singular and coupled aspects of the problem.

Simulation of multiple morphogenetic movements in the Drosophila embryo by a single 3D finite element model.

The present work describes a 3D finite element model of the Drosophila embryo designed to simulate three morphogenetic movements during early gastrulation: ventral furrow invagination, cephalic furrow formation and germ band extension. The embryo is represented by a regular ellipsoid and only the mesoderm is modeled. Additionally, the parametric description of the biological structure in a special curvilinear system provides mesh-independent endogenous strains. A deformation gradient decomposition is used to couple an active deformation, specific for each morphogenetic movement, together with a passive deformation, which is due to the response of the continuous mesoderm. Boundary conditions such as the rigid contact with the external vitelline membrane and the yolk pressure are also taken into account. The results suggest that the number of active strains responsible for the morphogenetic events can be less than that deduced from direct experimental observations. Finally, the estimation of the non-local pressures induced during morphogenetic movements is in good agreement with the experimental data.