laboratoire de physique statistique
 
 
laboratoire de physique statistique

Publications

Rechercher
 
2012
DOI
21
Pattern formation in fiber-reinforced tubular tissues: Folding and segmentation during epithelial growth - Ciarletta, P. and Ben Amar, M.
JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS 60525-537 (2012) 
LPS


Abstract : Constrained growth processes in living materials result in a complex distribution of residual strains, which in certain geometries may induce a bifurcation in the elastic stability. In this work, we investigate the combined effects of growth and material anisotropy in the epithelial pattern formation of tubular tissues. In order to represent the structural organization of most organs, we adopt a strain energy density which accounts for the presence of a nonlinear reinforcement made of cross-ply fibers distributed inside a ground matrix. Using a canonical transformation in mixed polar coordinates, we transform the nonlinear elastic boundary value problem into a variational formulation, performing a straightforward derivation of the Euler-Lagrange equations for perturbations in circumferential and longitudinal directions. The corresponding curves of marginal stability are obtained numerically: the results demonstrate that both the three-dimensional distribution of residual strains and the mechanical properties of fiber reinforcements within the tissue are fundamental to determine the emergence of a specific instability pattern. In particular, different proportions of axial and circumferential residual strains can model the epithelial formation of mucosal folds in the esophagus and of plicae circulares in the small intestine. The theoretical predictions are compared with morphological data for embryonic intestinal tissues, suggesting that the volumetric growth of the epithelium can also drive the early stages of villi morphogenesis. (C) 2011 Elsevier Ltd. All rights reserved.
DOI
22
Growth instabilities and folding in tubular organs: A variational method in non-linear elasticity - Ciarletta, P. and Ben Amar, M.
INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS 47248-257 (2012) 
LPS


Abstract : Morphoelastic theories have demonstrated that elastic instabilities can occur during the growth of soft materials, initiating the transition toward complex patterns. Within the framework of non-linear elasticity, the theory of incremental elastic deformations is classically employed for solving stability problems with finite strains. In this work, we define a variational method to study the bifurcation of growing cylinders with circular section. Accounting for a constant axial pre-stretch, we define a set of canonical transformations in mixed polar coordinates, providing a locally isochoric mapping. Introducing a generating function to derive an implicit gradient form of the mixed variables, the incompressibility constraint for the elastic deformation is solved exactly. The canonical representation allows to transform a generic boundary value problem, characterized by conservative body forces and surface traction loads, into a completely variational formulation. The proposed variational method gives a straightforward derivation of the linear stability analysis, which would otherwise require lengthy manipulations on the governing incremental equations. The definition of a generating function can also account for the presence of local singularities in the elastic solution. Bifurcation analysis is performed for few constrained growth problems of biomechanical interests, such as the mucosal folding of tubular tissues and surface instabilities in tumor growth. In a concluding section, the theoretical results are discussed for clarifying how anisotropy, residual strains and external constraints can affect the stability properties of soft tissues in growth and remodeling processes. (C) 2011 Elsevier Ltd. All rights reserved.
DOI
23
Papillary networks in the dermal-epidermal junction of skin: A biomechanical model - Ciarletta, Pasquale and Ben Amar, Martine
MECHANICS RESEARCH COMMUNICATIONS 4268-76 (2012) 
LPS


Abstract : Complex networks of finger-like protrusions characterize the dermal-epidermal junction of human skin. Although formed during the foetal development, such dermal papillae evolve in adulthood, often in response to a pathological condition. The aim of this work is to investigate the emergence of biaxial papillary networks in skin from a mechanical perspective. For this purpose, we define a biomechanical model taking into account the volumetric growth and the microstructural properties of the dermis and the epidermis. A scalar stream function is introduced to generate incompressible transformations, and used to define a variational formulation of the boundary value elastic problem. We demonstrate that incompatible growth of the layers can induce a bifurcation of the elastic stability driving the formation of dermal papillae. Such an interfacial instability is found to depend both on the geometrical constraints and on the mechanical properties of the skin components. The results provide a mechanical interpretation of skin morphogenesis, with possible applications for micropattern fabrication in soft layered materials. (C) 2011 Elsevier Ltd. All rights reserved.
DOI
24
Petal shapes of sympetalous flowers: the interplay between growth, geometry and elasticity - Ben Amar, Martine and Mueller, Martin Michael and Trejo, Miguel
NEW JOURNAL OF PHYSICS 14 (2012) 
LPS


Abstract : The growth of a thin elastic sheet imposes constraints on its geometry such as its Gaussian curvature KG. In this paper, we construct the shapes of sympetalous bell-shaped flowers with a constant Gaussian curvature. Minimizing the bending energies of both the petal and the veins, we are able to predict quantitatively the global shape of these flowers. We discuss two toy problems where the Gaussian curvature is either negative or positive. In the former case, the axisymmetric pseudosphere turns out to mimic the correct shape before edge curling; in the latter case, singularities of the mathematical surface coincide with strong veins. Using a variational minimization of the elastic energy, we find that the optimal number for the veins is either four, five or six, a number that is deceptively close to the statistics on real flowers in nature.
 
2011
DOI
25
Effective line tension and contact angles between membrane domains in biphasic vesicles - Trejo, M. and Ben Amar, M.
EUROPEAN PHYSICAL JOURNAL E 34 (2011) 
LPS


Abstract : Inhomogeneities in membranes give rise to localized interactions at the interface between domains in two-component vesicles. The corresponding energy is expressed as a line tension between the two phases. In this paper we study the implications of the thickness mismatch between domains which has been experimentally reported to be of order 20-30\% and the conditions under which the induced line tension can destabilize the domains in inhomogeneous vesicles. For asymmetric lipidic membranes we prove an increase of the line tension and the existence of a contact angle. Adsorption of impurities is also examined, our scope being the extension of the Canham-Helfrich model to describe elastic deformations and chemical interactions arising at microscopic scales. This mismatch effect may have important consequences for the stability of very small domains.
DOI
26
Mutual Adaptation of a Faraday Instability Pattern with its Flexible Boundaries in Floating Fluid Drops - Pucci, G. and Fort, E. and Ben Amar, M. and Couder, Y.
PHYSICAL REVIEW LETTERS 106 (2011) 
LPS


Abstract : Hydrodynamic instabilities are usually investigated in confined geometries where the resulting spatiotemporal pattern is constrained by the boundary conditions. Here we study the Faraday instability in domains with flexible boundaries. This is implemented by triggering this instability in floating fluid drops. An interaction of Faraday waves with the shape of the drop is observed, the radiation pressure of the waves exerting a force on the surface tension held boundaries. Two regimes are observed. In the first one there is a coadaptation of the wave pattern with the shape of the domain so that a steady configuration is reached. In the second one the radiation pressure dominates and no steady regime is reached. The drop stretches and ultimately breaks into smaller domains that have a complex dynamics including spontaneous propagation.
DOI
27
Cell motility: A viscous fingering analysis of active gels - Ben Amar, M. and Manyuhina, O. V. and Napoli, G.
EUROPEAN PHYSICAL JOURNAL PLUS 126 (2011) 
LPS


Abstract : The symmetry breaking of the actin network from radial to longitudinal symmetry has been identified as the major mechanism for keratocytes (fish cells) motility on solid substrate. For strong friction coefficient, the two-dimensional actin flow which includes the polymerisation at the edge and depolymerisation in the bulk can be modelled as a Darcy flow, the cell shape and dynamics being then modelled by standard complex analysis methods. We use the theory of active gels to describe the orientational order of the filaments which varies from the border to the bulk. We show analytically that the reorganisation of the cortex is enough to explain the motility of the cell and find the velocity as a function of the orientation order parameter in the bulk.
DOI
28
Buckling condensation in constrained growth - Dervaux, Julien and Ben Amar, Martine
JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS 59538-560 (2011) 
LPS


Abstract : The multiple complexities inherent to living objects have motivated the search for abiotic substitutes, able to mimic some of their relevant physical properties. Hydrogels provide a highly monitorable counterpart and have thus found many applications in medicine and bioengineering. Recently, it has been recognized that their ability to swell could be used to unravel some of the universal physical processes at work during biological growth. However, it is yet unknown how the microscopic distinctions between swelling and biological growth affect macroscopic changes (shape, stresses) induced by volume variations. To answer this question, we focus on a clinically motivated example of growth. Some solid tumors such as melanoma or glioblastoma undergo a shape transition during their evolution. This bifurcation appears when growth is confined at the periphery of the tumor and is concomitant with the transition from the avascular to the vascular stage of the tumor evolution. To model this phenomenon, we consider in this paper the deformation of an elastic ring enclosing a core of different stiffness. When the volume of the outer ring increases, the system develops a periodic instability. We consider two possible descriptions of the volume variation process: either by imposing a homogeneous volumetric strain (biological growth) or through migration of solvent molecules inside a solid network (swelling). For thin rings, both theories are in qualitative agreement. When the interior is soft, we predict the emergence of a large wavelength buckling. Upon increasing the stiffness of the inner disc, the wavelength of the instability decreases until a condensation of the buckles occurs at the free boundary. This short wavelength pattern is independent of the stiffness of the disc and is only limited by the presence of surface tension. For thicker rings, two scenarios emerge. When a volumetric strain is prescribed, compressive stresses accumulate in the vicinity of the core and the deformation localizes itself at the boundary between the disc and the ring. On the other hand, swelling being an instance of stress-modulated growth, elastic stretches near the core saturate and the instability occurs primarily at the free boundary. Besides its implications for the mechanical stability of avascular tumors, this work provides important results concerning layered tissues growth and the role of hydrogels as biological tissues substitutes. (C) 2010 Elsevier Ltd. All rights reserved.
DOI
29
The radial growth phase of malignant melanoma: multi-phase modelling, numerical simulations and linear stability analysis - Ciarletta, P. and Foret, L. and Ben Amar, M.
JOURNAL OF THE ROYAL SOCIETY INTERFACE 8345-368 (2011) 
LPS


Abstract : 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.
DOI
30
Contour Instabilities in Early Tumor Growth Models - Ben Amar, M. and Chatelain, C. and Ciarletta, P.
PHYSICAL REVIEW LETTERS 106 (2011) 
LPS


Abstract : Recent tumor growth models are often based on the multiphase mixture framework. Using bifurcation theory techniques, we show that such models can give contour instabilities. Restricting to a simplified but realistic version of such models, with an elastic cell-to-cell interaction and a growth rate dependent on diffusing nutrients, we prove that the tumor cell concentration at the border acts as a control parameter inducing a bifurcation with loss of the circular symmetry. We show that the finite wavelength at threshold has the size of the proliferating peritumoral zone. We apply our predictions to melanoma growth since contour instabilities are crucial for early diagnosis. Given the generality of the equations, other relevant applications can be envisaged for solving problems of tissue growth and remodeling.
DOI
31
Shape Transition in Artificial Tumors: From Smooth Buckles to Singular Creases - Dervaux, Julien and Couder, Yves and Guedeau-Boudeville, Marie-Alice and Ben Amar, Martine
PHYSICAL REVIEW LETTERS 107 (2011) 
LPS


Abstract : Using swelling hydrogels, we study the evolution of a thin circular artificial tumor whose growth is confined at the periphery. When the volume of the outer proliferative ring increases, the tumor loses its initial symmetry and bifurcates towards an oscillatory shape. Depending on the geometrical and elastic parameters, we observe either a smooth large-wavelength undulation of the swelling layer or the formation of sharp creases at the free boundary. Our experimental results as well as previous observations from other studies are in very good agreement with a nonlinear poroelastic model.
DOI
32
Emergence of microstructural patterns in skin cancer: a phase separation analysis in a binary mixture - Chatelain, C. and Balois, T. and Ciarletta, P. and Ben Amar, M.
NEW JOURNAL OF PHYSICS 13 (2011) 
LPS


Abstract : 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.
DOI
33
Morphological changes in early melanoma development: Influence of nutrients, growth inhibitors and cell-adhesion mechanisms - Chatelain, Clement and Ciarletta, Pasquale and Ben Amar, Martine
JOURNAL OF THEORETICAL BIOLOGY 29046-59 (2011) 
LPS


Abstract : Current diagnostic methods for skin cancers are based on some morphological characteristics of the pigmented skin lesions, including the geometry of their contour. The aim of this article is to model the early growth of melanoma accounting for the biomechanical characteristics of the tumor microenvironment, and evaluating their influence on the tumor morphology and its evolution. The spatial distribution of tumor cells and diffusing molecules are explicitly described in a three-dimensional multiphase model, which incorporates general cell-to-cell mechanical interactions, a dependence of cell proliferation on contact inhibition, as well as a local diffusion of nutrients and inhibiting molecules. A two-dimensional model is derived in a lubrication limit accounting for the thin geometry of the epidermis. First, the dynamical and spatial properties of planar and circular tumor fronts are studied, with both numerical and analytical techniques. A WIG method is then developed in order to analyze the solution of the governing partial differential equations and to derive the threshold conditions for a contour instability of the growing tumor. A control parameter and a critical wavelength are identified, showing that high cell proliferation, high cell adhesion, large tumor radius and slow tumor growth correlate with the occurrence of a contour instability. Finally, comparing the theoretical results with a large amount of clinical data we show that our predictions describe accurately both the morphology of melanoma observed in vivo and its variations with the tumor growth rate. This study represents a fundamental step to understand more complex microstructural patterns observed during skin tumor growth. Its results have important implications for the improvement of the diagnostic methods for melanoma, possibly driving progress towards a personalized screening. (C) 2011 Elsevier Ltd. All rights reserved.
 
2010
DOI
34
Swelling instability of surface-attached gels as a model of soft tissue growth under geometric constraints - Ben Amar, Martine and Ciarletta, Pasquale
JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS 58935-954 (2010) 
LPS


Abstract : The purpose of this work is to provide a theoretical analysis of the mechanical behavior of the growth of soft materials under geometrical constraints. In particular, we focus on the swelling of a gel layer clamped to a substrate, which is still the subject of many experimental tests. Because the constrained swelling process induces compressive stresses, all these experiments exhibit surface instabilities, which ultimately lead to cusp formation. Our model is based on fixing a neo-Hookean constitutive energy together with the incompressibility requirement for a volumetric, homogeneous mass addition. Our approach is developed mostly, but not uniquely, in the plane strain configuration. We show how the standard equilibrium equations from continuum mechanics have a similarity with the two-dimensional Stokes flows, and we use a nonlinear stream function for the exact treatment of the incompressibility constraint. A free energy approach allows the extension both to arbitrary hyperelastic strain energies and to additional interactions, such as surface energies. We find that, at constant volumetric growth, the threshold for a wavy instability is completely governed by the amount of growth. Nevertheless, the determination of the wavelength at threshold, which scales with the initial thickness of the gel layer, requires the coupling with a surface effect. Our findings, which are valid in proximity of the threshold, are compared to experimental results. The proposed treatment can be extended to weakly nonlinearities within the aim of the theory of bifurcations. (C) 2010 Elsevier Ltd. All rights reserved.
DOI
35
Localized growth of layered tissues - Dervaux, Julien and Ben Amar, Martine
IMA JOURNAL OF APPLIED MATHEMATICS 75571-580 (2010) 
LPS


Abstract : Due to their structural heterogeneity, stratified media usually loose their initial symmetry when they grow. Mismatching expansions between distinct layers create elastic stresses that lead to the emergence of wavy patterns. In the context of morphogenesis, this process of constrained growth has been used to explain patterns in biological soft-layered tissues such as the fingerprints of the skin. Discovery of morphogens has however revealed that growth is not always a homogeneous process and can be highly localized. Spatially concentrated growth processes are thought to be responsible for the formation of various structures, from placodes in the chicken embryo to nevus and skin tumours. On a different ground, the processing of complex textured surfaces requires to understand the connection between the local variation of mass and the resulting surface profile. In this paper, we investigate a model of a locally growing thin sheet bound to a soft infinite substrate. We show that, in two dimensions, bending effects select the shape of the growing sheet, therefore preventing the fabrication of an arbitrary surface. In three dimensions, the high energetic cost of stretching sets a stronger constraint on the attainable patterns and provides a simple relation between the profile of the surface and the local properties of the growth process.
DOI
36
Self-Contact and Instabilities in the Anisotropic Growth of Elastic Membranes - Stoop, Norbert and Wittel, Falk K. and Ben Amar, Martine and Mueller, Martin Michael and Herrmann, Hans J.
PHYSICAL REVIEW LETTERS 105 (2010) 
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Abstract : We investigate the morphology of thin discs and rings growing in the circumferential direction. Recent analytical results suggest that this growth produces symmetric excess cones ( e cones). We study the stability of such solutions considering self-contact and bending stress. We show that, contrary to what was assumed in previous analytical solutions, beyond a critical growth factor, no symmetric e cone solution is energetically minimal any more. Instead, we obtain skewed e cone solutions having lower energy, characterized by a skewness angle and repetitive spiral winding with increasing growth. These results are generalized to discs with varying thickness and rings with holes of different radii.
DOI
37
Instability patterns in ultrathin nematic films: Comparison between theory and experiment - Manyuhina, O. V. and Cazabat, A. -M. and Ben Amar, M.
EPL 92 (2010) 
LPS


Abstract : Motivated by recent experimental observations (Delabre U. et al., Langmuir, 24 (2008) 3998), we reconsider an instability of ultrathin nematic films, spread on liquid substrates. Within a continuum elastic theory of liquid crystals, in the harmonic approximation, we find an analytical expressions for the critical thickness as well as for the critical wave number, characterizing the onset of instability towards the stripe phase. Comparing theoretical predictions with experimental observations, we establish the utility of surface-like term such as an azimuthal anchoring. Copyright (C) EPLA, 2010
 
2009
DOI
38
Suction in Darcy and Stokes interfacial flows: Maximum growth rate versus minimum dissipation - Ben Amar, M. and Boudaoud, A.
EUROPEAN PHYSICAL JOURNAL-SPECIAL TOPICS 16683-88 (2009) 
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Abstract : Two-dimensional flows with suction or mass loss are investigated within Darcy's or Stokes' framework. Examples include a Hele-Shaw cell with a lifted plate or extraction of lipids from a lipid bilayer. An initially circular patch ret-racts due to the suction and might undergo an instability whereby it becomes undulating. The selection of the wavelength of undulations is investigated with the help of an extremum principle, the minimization of the generalized dissipation, from which derive the flow equations.
DOI
39
Morphogenesis of thin hyperelastic plates: A constitutive theory of biological growth in the Foppl-von Karman limit - Dervaux, Julien and Ciarletta, Pasquale and Ben Amar, Martine
JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS 57458-471 (2009) 
LPS


Abstract : The shape of plants and other living organisms is a crucial element of their biological functioning. Morphogenesis is the result of complex growth processes involving biological, chemical and physical factors at different temporal and spatial scales. This study aims at describing stresses and strains induced by the production and reorganization of the material. The mechanical properties of soft tissues are modeled within the framework of continuum mechanics in finite elasticity. The kinematical description is based on the multiplicative decomposition of the deformation gradient tensor into an elastic and a growth term. Using this formalism, the authors have studied the growth of thin hyperelastic samples. Under appropriate assumptions, the dimensionality of the problem can be reduced, and the behavior of the plate is described by a two-dimensional surface. The results of this theory demonstrate that the corresponding equilibrium equations are of the Foppl-von Karman type where growth acts as a source of mean and Gaussian curvatures. Finally, the cockling of paper and the rippling of a grass blade are considered as two examples of growth-induced pattern formation. (C) 2008 Published by Elsevier Ltd.
DOI
40
Local Membrane Mechanics of Pore-Spanning Bilayers - Mey, Ingo and Stephan, Milena and Schmitt, Eva K. and Mueller, Martin Michael and Ben Amar, Martine and Steinem, Claudia and Janshoff, Andreas
JOURNAL OF THE AMERICAN CHEMICAL SOCIETY 1317031-7039 (2009) 
LPS


Abstract : The mechanical behavior of lipid bilayers; spanning the pores of highly ordered porous silicon substrates was scrutinized by local indentation experiments as a function of surface functionalization, lipid composition, solvent content, indentation velocity, and pore radius. Solvent-containing nano black lipid membranes (nano-BLMs) as well as solvent-free pore-spanning bilayers were imaged by fluorescence and atomic force microscopy prior to force curve acquisition, which allows distinguishing between membrane-covered and uncovered pores. Force indentation curves on pore-spanning bilayers attached to functionalized hydrophobic porous silicon substrates reveal a predominately linear response that is mainly attributed to prestress in the membranes. This is in agreement with the observation that indentation leads to membrane lysis well below 5\% area dilatation. However, membrane bending and lateral tension dominate over prestress and stretching if solvent-free supported membranes obtained from spreading giant liposomes on hydrophilic porous silicon are indented. An elastic regime diagram is presented that readily allows determining the dominant contribution to the mechanical response upon indentation as a function of load and pore radius.