M. Ben Amar

Fission of a multiphase membrane tube

A common mechanism for intracellular transport is the use of controlled deformations of the membrane to create spherical or tubular buds. While the basic physical properties of homogeneous membranes are relatively well known, the effects of inhomogeneities within membranes are very much an active field of study. Membrane domains enriched in certain lipids, in particular, are attracting much attention, and in this Letter we investigate the effect of such domains on the shape and fate of membrane tubes. Recent experiments have demonstrated that forced lipid phase separation can trigger tube fission, and we demonstrate how this can be understood purely from the difference in elastic constants between the domains. Moreover, the proposed model predicts time scales for fission that agree well with experimental findings.

Theory of dendritic growth in a weakly undercooled melt

We analyse the integro-differential equation giving the shape of steady dendrites growing from a weakly undercooled melt in two spatial dimensions. A generalized WKB method shows that solutions at small velocities can exist with anisotropic surface tension only, a result completely out of reach of regular perturbation theory.

INSTABILITIES IN ELASTOMERS AND IN SOFT TISSUES

Summary Biological soft tissues exhibit elastic properties that can be dramatically different from rubber-type materials (elastomers). To gain a better understanding of the role of constitutive relationships in determining material responses under loads we compare three different types of instabilities (two in compression, one in extension) in hyperelasticity for various forms of strain-energy functions typically used for elastomers and for soft tissues. Surprisingly, we find that the strain-hardening property of soft tissues does not always stabilize the material. In particular we show that the stability analyses for a compressed half-space and for a compressed spherical thick shell can lead to opposite conclusions: a soft tissue material is more stable than an elastomer in the former case and less stable in the latter case.

The radial growth phase of malignant melanoma: multi-phase modelling, numerical simulations and linear stability analysis.

Cutaneous melanoma is disproportionately lethal despite its relatively low incidence and its potential for cure in the early stages. The aim of this study is to foster understanding of the role of microstructure on the occurrence of morphological changes in diseased skin during melanoma evolution. The authors propose a biomechanical analysis of its radial growth phase, investigating the role of intercellular/stromal connections on the initial stages of epidermis invasion. The radial growth phase of a primary melanoma is modelled within the multi-phase theory of mixtures, reproducing the mechanical behaviour of the skin layers and of the epidermal–dermal junction. The theoretical analysis takes into account those cellular processes that have been experimentally observed to disrupt homeostasis in normal epidermis. Numerical simulations demonstrate that the loss of adhesiveness of the melanoma cells both to the basal laminae, caused by deregulation mechanisms of adherent junctions, and to adjacent keratynocytes, consequent to a downregulation of E-cadherin, are the fundamental biomechanical features for promoting tumour initiation. Finally, the authors provide the mathematical proof of a long wavelength instability of the tumour front during the early stages of melanoma invasion. These results open the perspective to correlate the early morphology of a growing melanoma with the biomechanical characteristics of its micro-environment.

A finite dissipative theory of temporary interfibrillar bridges in the extracellular matrix of ligaments and tendons

The structural integrity and the biomechanical characteristics of ligaments and tendons result from the interactions between collagenous and non-collagenous proteins (e.g. proteoglycans, PGs) in the extracellular matrix. In this paper, a dissipative theory of temporary interfibrillar bridges in the anisotropic network of collagen type I, embedded in a ground substance, is derived. The glycosaminoglycan chains of decorin are assumed to mediate interactions between fibrils, behaving as viscous structures that transmit deformations outside the collagen molecules. This approach takes into account the dissipative effects of the unfolding preceding fibrillar elongation, together with the slippage of entire fibrils and the strain-rate-dependent damage evolution of the interfibrillar bridges. Thermodynamic consistency is used to derive the constitutive equations, and the transition state theory is applied to model the rearranging properties of the interfibrillar bridges. The constitutive theory is applied to reproduce the hysteretic spectrum of the tissues, demonstrating how PGs determine damage evolution, softening and non-recoverable strains in their cyclic mechanical response. The theoretical predictions are compared with the experimental response of ligaments and tendons from referenced studies. The relevance of the proposed model in mechanobiology research is discussed, together with several applications from medical practice to bioengineering science.

An elastica problem: instabilities of an elastic arch

Abstract We analyze the modes of instability of an elastic homogeneous arch that is loaded at its center. We study first the dynamical linear stability of the symmetric elastic arch with respect to small perturbations. We also perform a constrained minimization of the static energy of the system. The numerical resolution of both the dynamical and the static problems allows the determination of the phase diagram corresponding to the behavior of the arch. For this simple elastica problem, the phase diagram shows a very rich structure that is in agreement with the experimental results.

Emergence of microstructural patterns in skin cancer: a phase separation analysis in a binary mixture

Clinical diagnosis of skin cancers is based on several morphological criteria, among which is the presence of microstructures (e.g. dots and nests) sparsely distributed within the tumour lesion. In this study, we demonstrate that these patterns might originate from a phase separation process. In the absence of cellular proliferation, in fact, a binary mixture model, which is used to represent the mechanical behaviour of skin cancers, contains a cell?cell adhesion parameter that leads to a governing equation of the Cahn?Hilliard type. Taking into account a reaction?diffusion coupling between nutrient consumption and cellular proliferation, we show, with both analytical and numerical investigations, that two-phase models may undergo a spinodal decomposition even when considering mass exchanges between the phases. The cell?nutrient interaction defines a typical diffusive length in the problem, which is found to control the saturation of a growing separated domain, thus stabilizing the microstructural pattern. The distribution and evolution of such emerging cluster morphologies, as predicted by our model, are successfully compared to the clinical observation of microstructural patterns in tumour lesions.

Budding and fission of a multiphase vesicle

Abstract.We present a model of bi-phasic vesicles in the limit of large surface tension. In this regime, the vesicle is completely stretched and well described by two spherical caps with a fold, which concentrates the membrane stress. The conservation laws and geometric constraints restrict the space of possible shapes to a pair of solutions labeled by a parameter

FINGER BEHAVIOR OF A SHEAR THINNING FLUID IN A HELE-SHAW CELL

\tilde{{\sigma}}

Viscous fingering in complex fluids

given by line tension/pressure. For a given value of