Research

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  Skin tumour modeling

• "The radial growth phase of malignant melanoma: multiphase modeling, numerical simulations and linear stability analysis."

  P. Ciarletta, L. Forêt and M. Ben Amar. to be published. Abstract Cutaneous melanoma is disproportionately lethal despite its relatively low incidence and its po- tential for cure in the early stages. The aim of the this study is to foster understanding of the micro- structural changes occurring in diseased skin during melanoma evolution. The authors propose a biomechanical analysis of its radial growth phase, investigating the role of intercellular/stromal con- nections on the initial stages of epidermis invasion. The radial growth phase of a primary melanoma is modeled within the multiphase theory of mixtures, reproducing the mechanical behavior 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 keratyno- cytes, consequent to a downregulation of E-cadherin, are the fundamental biomechanical features for promoting tumor initiation. Finally, the authors provide the mathematical proof of a short wave- length instability of the tumor front during the early stages of melanoma invasion. This result opens the perspective to correlate the early morphology of a growing melanoma with the biomechanical characteristics of its micro-environment.

  Biomechanics and animal morphogenesis.

• "Continuum model of epithelial morphogenesis during C. elegans embryonic elongation.", Phil. Trans. R. Soc. A, 367 (2009) 3379-3400 .

  P. Ciarletta, M. Ben Amar et M. Labouesse. Abstract The purpose of this work is to provide a biomechanical model to investigate the interplay between cellular structures and the mechanical force distribution during the elongation process of Caenorhabditis elegans embryos. Epithelial morphogenesis drives the elongation process of an ovoid embryo to become a worm-shaped embryo about four times longer and three times thinner. The overall anatomy of the embryo is modelled in the continuum mechanics framework from the structural organization of the subcellular filaments within epithelial cells. The constitutive relationships consider embryonic cells as homogeneous materials with an active behaviour, determined by the non-muscle myosin II molecular motor, and a passive viscoelastic response, related to the directional properties of the filament network inside cells. The axisymmetric elastic solution at equilibrium is derived by means of the incompressibility conditions, the continuity conditions for the overall embryo deformation and the balance principles for the embryonic cells. A particular analytical solution is proposed from a simplified geometry, demonstrating the mechanical role of the microtubule network within epithelial cells in redistributing the stress from a differential contraction of circumferentially oriented actin filaments. The theoretical predictions of the biomechanical model are discussed within the biological scenario proposed through genetic analysis and pharmacological experiments.

• "A finite dissipative theory of temporary interfibrillar bridges in the extra-cellular matrix of ligaments and tendons.", J. R. Soc. Interface, 6 (2009) 909-924

  P. Ciarletta et M. Ben Amar. Abstract 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.

  Vegetal morphogenesis.

• "Morphogenesis of growing soft tissues.", Phys. Rev. Lett., 101 (2008) 068101.

  J. Dervaux and M. Ben Amar. Abstract Recently, much attention has been given to a noteworthy property of some soft tissues: their ability to grow. Many attempts have been made to model this behavior in biology, chemistry, and physics. Using the theory of finite elasticity, Rodriguez has postulated a multiplicative decomposition of the geometric deformation gradient into a growth-induced part and an elastic one needed to ensure compatibility of the body. In order to fully explore the consequences of this hypothesis, the equations describing thin elastic objects under finite growth are derived. Under appropriate scaling assumptions for the growth rates, the proposed model is of the Fo¨ppl–von Ka´rma´n type. As an illustration, the circumferential growth of a free hyperelastic disk is studied.

• "Elastic growth in thin geometries.", Origins of Life: Self-Organization and/or Biological Evolution? ,(2009) 79-94.

  J. Dervaux and M. Ben Amar. Abstract Generation of shapes in biological tissues is a complex multiscale phenomenon. Biochemical details of cell proliferation, death and mobility can be incorporated within a continuum mechanical framework by specifying locally the amplitude and direction of growth. For tissues exhibiting an elastic behavior, equilibrium shapes of growing bodies can be evaluated through the minimization of an appropriate energy. This model is applied to thin shells and plates, a geometry relevant to nuts and pollen grains but also leaves, petals and algae.

• "Morphogenesis of thin hyperelastic plates: A constitutive theory of biological growth in the Föppl-von Karman limit.", J. Mech. Phys. Solids, 57 (2009) 458-471.

  J. Dervaux and M. Ben Amar. 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 Föppl–von Kármán 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.

  Instability in swelling hydrogel.

• "Swelling instability of surface-attached gels as a model of soft tissue growth under geometric constraints.", J. Mech. Phys. Solids, 58 (2010) 935–954.

  M. Ben Amar and P. Ciarletta 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.

  Elastic singularities

• "Conical Defects in Growing Sheets.", Phys. Rev. Lett., 101 (2008) 156104.

  M. M. Müller, M. Ben Amar et J. Guven Abstract A growing or shrinking disc will adopt a conical shape, its intrinsic geometry characterized by a surplus angle fe at the apex. If growth is slow, the cone will find its equilibrium. Whereas this is trivial if fe=0, the disc can fold into one of a discrete infinite number of states if fe>0. We construct these states in the regime where bending dominates and determine their energies and how stress is distributed in them. For each state a critical value of fe is identified beyond which the cone touches itself. Before this occurs, all states are stable; the ground state has twofold symmetry.

• "Self-contact and instabilities in the anisotropic growth of elastic membranes."

  N. Stoop, F. K. Wittel, M. Ben Amar, M. M. Müller et H. J. Herrmann. submitted to Phys.Rev.Lett. Abstract We investigate the morphology of thin discs and rings growing in 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 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.

• "Localized growth of layered tissues.", IMA Journal of Applied Mathematics, (2010) sous presse.

  J. Dervaux and M. Ben Amar. 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.

• "Hamiltonian formulation of surfaces with constant Gaussian curvature.", J. Phys. A: Math. Theor., 42 (2009) 425204.

  M. Trejo, M. Ben Amar and Martin Michael Müller. Abstract Dirac’s method for constrained Hamiltonian systems is used to describe surfaces of constant Gaussian curvature. A geometrical free energy, for which these surfaces are equilibrium states, is introduced and interpreted as an action. An equilibrium surface can then be generated by the evolution of a closed space curve. Since the underlying action depends on second derivatives, the velocity of the curve and its conjugate momentum must be included in the set of phasespace variables. Furthermore, the action is linear in the acceleration of the curve and possesses a local symmetry—reparametrization invariance—which implies primary constraints in the canonical formalism. These constraints are incorporated into the Hamiltonian through Lagrange multiplier functions that are identified as the components of the acceleration of the curve. The formulation leads to four first-order partial differential equations, one for each canonical variable. With the appropriate choice of parametrization, only one of these equations has to be solved to obtain the surface which is swept out by the evolving space curve. To illustrate the formalism, several evolutions of pseudospherical surfaces are discussed.

  Biological membranes.

• "Local Membrane Mechanics of Pore-Spanning Bilayers.", J. Am. Chem. Soc., 131 (2009) 7031.

  I. Mey, M. Stephan, E. K. Schmitt, M. M. Müller, M. Ben Amar, C. Steinem et A. Janshoff 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.

• "Periodic lipid membrane tubes.", Europhys. Lett., 77 (2007) 38006.

  F. Campelo, J.-M. Allain et M. Ben Amar. Abstract We investigate the formation of two-phase lipidic tubes of membrane in the framework of the Canham-Helfrich model. The two phases have different elastic moduli (bending and Gaussian rigidity), different tensions and a line tension prevents the mixing. For a set of parameters close to experimental values, periodic patterns with arbitrary wavelength can be found numerically. A wavelength selection is detected via the existence of an energy minimum. When the chemical composition induces an important enough size disequilibrium between both phases, a segregation into two half infinite tubes is preferred to a periodic structure.

• "Stokes instability in inhomogeneous membranes: Application to lipoprotein suction of cholesterol-enriched domains.", Phys. Rev. Lett., 99 (2007) 044503.

  M. Ben Amar, J.-M. Allain, N. Puff et M. Angelova. Abstract We examine the time-dependent distortion of a nearly circular viscous domain in an infinite viscous sheet when suction occurs. Suction, the driving force of the instability, can occur everywhere in the two phases separated by an interface. The model assumes a two-dimensional Stokes flow; the selection of the wavelength at short times is determined by a variational procedure. Contrary to the viscous fingering instability, undulations of the boundary may be observed for enough pumping, whatever the sign of the viscosity contrast between the two fluids involved. We apply our model to the suction by lipoproteins of cholesterol-enriched domains in giant unilamellar vesicles. Comparison of the number of undulations given by the model and by the experiments gives reasonable values of physical quantities such as the viscosities of the domains.

• "Suction in Darcy and Stokes interfacial flows: Maximum growth rate versus minimum dissipation.", Eur. Phys. J. Special Topics, 166 (2009) 83.

  M. Ben Amar et A. Boudaoud. 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 retracts 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.

  Instability in this liquid crystal films.

  Faraday instability.

• "The interplay of an instability pattern with its flexible boundaries.", (2010) submitted to PRL.

  G. Pucci, E. Fort, M. Ben Amar and Y. Couder. Abstract