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Documents published in Scipedia

• Diffusion-reaction model for Drosophila embryo development

  R. Allena, J. Muñoz, D. Aubry

  Comput. Methods Appl. Mech. Engrg., (2011). Vol. 16 (3), pp. 235-248

  
  

  Abstract

  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.

  Abstract
  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. [...]

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• Stress-dependent morphogenesis: continuum mechanics and truss systems

  J. Muñoz, V. Conte, M. Miodownik

  Biomechanics and Modeling in Mechanobiology (2010). Vol. 9 (4), pp. 451-467

  
  

  Abstract

  A set of equilibrium equations is derived for the stress-controlled shape change of cells due to the remodelling and growth of their internal architecture. The approach involves the decomposition of the deformation gradient into an active and a passive component; the former is allowed to include a growth process, while the latter is assumed to be hyperelastic and mass-preserving. The two components are coupled with a control function that provides the required feedback mechanism. The balance equations for general continua are derived and, using a variational approach, we deduce the equilibrium equations and study the effects of the control function on these equations. The results are applied to a truss system whose function is to simulate the cytoskeletal network constituted by myosin microfilaments and microtubules, which are found experimentally to control shape change in cells. Special attention is paid to the conditions that a thermodynamically consistent formulation should satisfy. The model is used to simulate the multicellular shape changes observed during ventral furrow invagination of the Drosophila melanogaster embryo. The results confirm that ventral furrow invagination can be achieved through stress control alone, without the need for other regulatory or signalling mechanisms. The model also reveals that the yolk plays a distinct role in the process, which is different to its role during invagination with externally imposed strains. In stress control, the incompressibility constraint of the yolk leads, via feedback, to the generation of a pressure in the ventral zone of the epithelium that eventually eases its rise and internalisation.

  Abstract
  A set of equilibrium equations is derived for the stress-controlled shape change of cells due to the remodelling and growth of their internal architecture. The approach involves [...]

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• Robust mechanisms of ventral furrow invagination require the combination of cellular shape changes

  V. Conte, J. Muñoz, B. Baum, M. Miodownik

  Physical Biology (2009). Vol. 6 (1), pp. 16010-16021

  
  

  Abstract

  Ventral furrow formation in Drosophila is the first large-scale morphogenetic movement during the life of the embryo, and is driven by co-ordinated changes in the shape of individual epithelial cells within the cellular blastoderm. Although many of the genes involved have been identified, the details of the mechanical processes that convert local changes in gene expression into whole-scale changes in embryonic form remain to be fully understood. Biologists have identified two main cell deformation modes responsible for ventral furrow invagination: constriction of the apical ends of the cells (apical wedging) and deformation along their apical–basal axes (radial lengthening/shortening). In this work, we used a computer 2D finite element model of ventral furrow formation to investigate the ability of different combinations of three plausible elementary active cell shape changes to bring about epithelial invagination: ectodermal apical–basal shortening, mesodermal apical–basal lengthening/shortening and mesodermal apical constriction. We undertook a systems analysis of the biomechanical system, which revealed many different combinations of active forces (invagination mechanisms) were able to generate a ventral furrow. Two important general features were revealed. First that combinations of shape changes are the most robust to environmental and mutational perturbation, in particular those combining ectodermal pushing and mesodermal wedging. Second, that ectodermal pushing plays a big part in all of the robust mechanisms (mesodermal forces alone do not close the furrow), and this provides evidence that it may be an important element in the mechanics of invagination in Drosophila.

  Abstract
  Ventral furrow formation in Drosophila is the first large-scale morphogenetic movement during the life of the embryo, and is driven by co-ordinated changes in the [...]

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• A 3D finite element model of ventral furrow invagination in the Drosophila melanogaster embryo

  V. Conte, J. Muñoz, M. Miodownik

  Journal of the Mechanical Behavior of Biomedical Materials (2008). Vol. 1 (2), pp. 188-198

  
  

  Abstract

  The paper describes a mechanical model of epithelial tissue development in Drosophila embryos to investigate a buckling phenomenon called invagination. The finite element method is used to model this ventral furrow formation in 3D by decomposing the total deformation into two parts: an imposed active deformation, and an elastic passive deformation superimposed onto the latter. The model imposes as boundary conditions (i) a constant yolk volume and (ii) a sliding contact condition of the cells against the vitelline membrane, which is interpolated as a B-Spline surface. The active deformation simulates the effects of apical constriction and apico-basal elongation of cells. This set of local cellular mechanisms leads to global shape changes of the embryo which are associated with known gene expressions. Using the model we have tested different plausible hypotheses postulated to account for the mechanical behaviour of epithelial tissues. In particular, we conclude that only certain combinations of local cell shape change can successfully reproduce the invagination process. We have quantitatively compared the model with a 2D model and shown that it exhibits a more robust invagination phenomenon. The 3D model has also revealed that invagination causes a yolk flow from the central region to the anterior and posterior ends of the embryo, causing an accordion-like global compression and expansion wave to move through the embryo. Such a phenomenon cannot be described by 2D models.

  Abstract
  The paper describes a mechanical model of epithelial tissue development in Drosophila embryos to investigate a buckling phenomenon called invagination. The finite [...]

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• On the modelling of incompressibility in linear and non-linear elasticity with the master-slave approach

  J. Muñoz

  International Journal for Numerical Methods in Engineering (2008). Vol. 74 (2), pp. 269-293

  
  

  Abstract

  The master–slave approach is adapted to model the kinematic constraints encountered in incompressibility. The method presented here allows us to obtain discrete displacement and pressure fields for arbitrary finite element formulations that have discontinuous pressure interpolations. The resulting displacements satisfy exactly the incompressibility constraints in a weak sense, and are obtained by solving a system of equations with the minimum (independent) degrees of freedom. In linear analysis, the method reproduces the well‐known stability results for inf–sup compliant elements, and permits to compute the pressure modes (physical or spurious) when they exist. By rewriting the equilibrium equations of a hyperelastic material, the method is extended to non‐linear elasticity, while retaining the exact fulfilment of the incompressibility constraints in a weak sense. Problems with analytical solution in two and three dimensions are tested and compared with other solution methods.

  Abstract
  The master–slave approach is adapted to model the kinematic constraints encountered in incompressibility. The method presented here allows us to obtain discrete displacement [...]

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• Modelling unilateral frictionless contact using the null-space method and cubic B-Spline interpolation

  J. Muñoz

  Comput. Methods Appl. Mech. Engrg., (2008). Vol. 197 (9-12), pp. 979-993

  
  

  Abstract

  The analysis of unilateral sliding contact in elasticity is equivalent to a minimisation problem subjected to a set of inequality constraints. However, the presence of boundary discontinuities, such as those stemming from the spatial discretisation, appears as a major problem to determine the set of active constraints. This work introduces a smoothing technique of the master surface resorting to cubic B-Spline interpolation, which is C1 continuous in contact situations between elastic and rigid bodies, and G1 continuous in elastic–elastic contact problems. The technique is applied in conjunction with the null-space method, where the solution is searched in an unconstrained manifold. The resulting formulation eases the contact transition along the master surface, and recovers the quadratic convergence of the iterative Newton–Raphson process. The robustness of the method is demonstrated using 2D and 3D examples.

  Abstract
  The analysis of unilateral sliding contact in elasticity is equivalent to a minimisation problem subjected to a set of inequality constraints. However, the presence of boundary [...]

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• A deformation gradient decomposition method for the analysis of the mechanics of morphogenesis

  J. Muñoz, K. Barrett, M. Miodownik

  Journal of Biomechanics (2007). Vol. 40 (6), pp. 1372-1380

  
  

  Abstract

  A new finite element model is proposed for the analysis of the mechanical aspects of morphogenesis and tested on the biologically well studied gastrulation phenomenon, in particular ventral furrow invagination of the Drosophila melanogaster embryo. A set of mechanisms are introduced in the numerical model, which lead to the observed deformed shapes. We split the total deformation into two parts: an imposed active deformation, and an elastic deformation superimposed onto the latter. The active deformation simulates the effects of apical constriction and apico-basal elongation. These mechanisms are associated with known gene expressions and so in this way we attempt to bridge the well explored signalling pathways, and their associated phenotypes in a mechanical model. While the former have been studied in depth, much less can be said about the forces they produce and the mechanisms involved. From the numerical results, we are able to test different plausible mechanical hypotheses that generate the necessary folding observed in the invagination process. In particular, we conclude that only certain ratios between both modes (apical constriction and apico-basal elongation) can successfully reproduce the invagination process. The model also supports the idea that this invagination requires the contribution of several mechanisms, and that their redundancy provides the necessary robustness.

  Abstract
  A new finite element model is proposed for the analysis of the mechanical aspects of morphogenesis and tested on the biologically well studied gastrulation phenomenon, in [...]

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• On the numerical simulation of fatigue driven delamination with interface elements

  J. Muñoz, U. Galvanetto, P. Robinson

  International Journal of Fatigue (2006). Vol. 28 (10), pp. 1136-1146

  
  

  Abstract

  This paper investigates robustness and efficiency of a method for the numerical simulation of fatigue driven delamination growth in laminated composites. The method is based on the fatigue degradation of a bilinear interface element and the paper studies the dependence of the predicted behaviour on the interface element size, Δl, and the number of cycles per increment, ΔN. It is concluded that the value of ΔN must be small enough to ensure that at least two increments are required to advance the crack by Δl. It is also shown that Δl must be sufficiently small so that at least two interface elements lie in the cohesive zone ahead of the crack tip. The paper also proposes extensions of the model to include a non-zero minimum cyclic load value and an improved algorithm for mixed-mode fatigue driven delamination growth.

  Abstract
  This paper investigates robustness and efficiency of a method for the numerical simulation of fatigue driven delamination growth in laminated composites. The method [...]

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• Sliding joints in 3D beams: conserving algorithms using the master-slave approach

  J. Muñoz, G. Jelenic

  Moltibody System Dynamics (2006). Vol. 16 (3), pp. 137-261

  
  

  Abstract

  This paper proposes two time-integration algorithms for motion of geometrically exact 3D beams under sliding contact conditions. The algorithms are derived using the so-called master–slave approach, in which constraint equations and the related time-integration of a system of differential and algebraic equations are eliminated by design. Specifically, we study conservation of energy and momenta when the sliding conditions on beams are imposed and discuss their algorithmic viability. Situations where the contact jumps to adjacent finite elements are analysed in detail and the results are tested on two representative numerical examples. It is concluded that an algorithmic preservation of kinematic constraint conditions is of utmost importance.

  Abstract
  This paper proposes two time-integration algorithms for motion of geometrically exact 3D beams under sliding contact conditions. The algorithms are derived using the so-called [...]

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• Sliding contact conditions using the master–slave approach with application on geometrically non-linear beams

  J. Muñoz, G. Jelenic

  International Journal of Solid and Structures (2004). Vol. 41 (24-25), pp. 6963-6992

  
  

  Abstract

  Frictionless sliding conditions between two bodies are usually defined using either the method of Lagrangian multipliers or by prescribing an artificial (penalty) stiffness which resists the penetration at the contact point. Both of these methods impose the condition that the contact force should be normal to the contact surface, with the Lagrangian multiplier or the penalty parameter serving as a measure of this force. In this work, an alternative approach is undertaken: the frictionless sliding condition is defined through a relationship between nodal parameters of the virtual displacements of a discretised principle of virtual work. This method, which does not involve additional force parameters or degrees of freedom, is known as the master–slave or the minimum-set method and is particularly convenient for displacement-based finite-element implementation. The method is analysed in detail in context of bilateral sliding constraints characteristic of prismatic and cylindrical joints in flexible beam assemblies undergoing large overall motion. Two numerical examples are presented and assessed against the results in the literature.

  Abstract
  Frictionless sliding conditions between two bodies are usually defined using either the method of Lagrangian multipliers or by prescribing an artificial (penalty) stiffness [...]

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