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Mechanobiology and morphogenesis in living matter: a survey
Abstract
Morphogenesis in living tissues is the paramount example of a time- and space-dependent orchestration of living matter where shape and order emerge from undifferentiated initial conditions. The genes encode the protein expression that eventually drives the emergence of the phenotype, while energy supply and cell-to-cell communication mechanisms are necessary to such a process. The overall control of the system likely exploits the laws of chemistry and physics through robust and universal processes. Even if the identification of the communication mechanisms is a question of fundamental nature, a long-standing investigation settled in the realm of chemical factors only (also known as morphogens) faces a number of apparently unsolvable questions. In this paper, we investigate at what extent mechanical forces, alone or through their biological feedbacks, can direct some basic aspects of morphogenesis in development biology. In this branch of mechano-biology, we discuss the typical rheological regimes of soft living matter and the related forces, providing a survey on how local mechanical feedbacks can control global size or even gene expression. We finally highlight the pivotal role of nonlinear mechanics to explain the emergence of complex shapes in living matter.
... The Cauchy stress (9) then takes the form 6 We can see that the dissipation inequality simplifies by inserting (9) into (8). This leaves Θ = B0 |G| ρ r G : ...
... Shi-Lei et al. [109] Ambrosi et al. [15] Walker et al. [107] present residual stress as in experiments × only at periphery necrotic core is present × oxygen diffusion is modeled × derived from thermodynamics × × Table 1: Comparison of tumor models. The category "residual stress as in experiments" refers to tensile hoop stress at the periphery and compressive hoop stress near the core, and compressive radial stress throughout, which would lead to an opening of the spheroid upon incision [94,8,26,63]. In [107], this appears to be true only very closely to the edge of the spheroid, while in the bulk the profile is reversed. ...
... In order to evaluate the dissipation inequality (8), we must first evaluate the expressions ∂f /∂A and ∂f /∂ |G|. To this end, the following identities are very useful, see Matrix Cookbook ( [85]) equations (49), (108) and (52), respectively: ∂I1 ∂A = 2A, ∂I3 ∂A = 2I 3 A −T and ∂|A| ∂A = |A| A −T . ...
... A temporal function is also often used (Ambrosi and Mollica, 2002;Papastavrou et al., 2013;Holland et al., 2013) instead of Eq. (2.8) to control unbounded exponential growth or resorption, ...
... The variable x can be a specific physical variable such as the temperature T. Then, the function k(w, T) represents the amount of energy supplied by the biochemical sources in the tissue that can be transformed into growth (Ambrosi et al., 2017). It can also be a more abstract field such as the "biological availability" q in the tissue, which represents the amount of growth the tissue's metabolism can sustain with the available nutrients (Bellomo et al., 2011). ...
... The additional variable can be easily introduced as a static field with a spatial inhomogeneous distribution or with a given predefined evolution. However, coupling this variable to a diffusion model allows enriching the growth model by incorporating the driving forces for the distribution of biochemical growth drivers in the tissue, e.g., to study the influence of nutrient diffusion and chemical signaling in a growing tumor (Ambrosi and Mollica, 2002;Ciarletta et al., 2012;Ramírez-Torres et al., 2015). In specific applications, researchers have proposed expressions for the evolution of growth that do not exactly fit the representative function described in Eq. (2.6). ...
Growth of the skeletal system (both bone and cartilage) is influenced by mechanical loads. Here we demonstrate how we can explore the mechanobiology of skeletal growth using finite element analysis in four applications: (1) muscle growth, (2) cartilage growth, (3) endochondral growth and ossification (long bone growth), and (4) periosteal growth (cross-sectional adaptation). For each of these growth processes, we use finite element analysis to determine potential mechanical stimuli (stress, strain, or fluid flow), implement mechanobiological rules that govern how the physical stimulus influences the growth, and simulate growth with expansion of the elements in a particular direction. Using finite element analysis allows us to explore different potential stimuli by comparing temporal and spatial growth responses. We can also explore the effects of altering loading conditions to alter growth patterns. Understanding the role of loading in development can help inform why deformities occur and how to mitigate or prevent them..
... Papers [5][6][7] address the topic of muscles and mechanobiology. In particular, Naldi discusses a chemo-mechanical model for the single myofibril in striated muscle contraction exploiting nonlinear nonlocal equations [5]. ...
... In particular, Naldi discusses a chemo-mechanical model for the single myofibril in striated muscle contraction exploiting nonlinear nonlocal equations [5]. Ambrosi, Beloussov, and Ciarletta review the progress and the open challenges of continuum mechanics theories of morphogenesis in living matter [6]. Sabato, Panzetta, and Netti investigate the regulation dynamics of focal adhesion in cells with respect to their capability of sensing the substrate stiffness [7]. ...
... It was shown in [184] that the primary periodicity may be of spatial rather than temporal origin and that the required synchronization may be due to mechanical, rather than biochemical, signalling. Mechanical instabilities can lead to the development of stress inhomogeneities in spatially distant material points thus generating a robust number of segments without dependence on genetic oscillations [9,184]. ...
The phenomenon of oscillatory instability called ‘flutter’ was observed in aeroelasticity and rotor dynamics about a century ago. Driven by a series
of applications involving nonconservative elasticity theory at different physical scales, ranging from nanomechanics to the mechanics of large space
structures and including biomechanical problems of motility and growth, research on flutter is experiencing a new renaissance. A review is presented of the most notable applications and recent advances in fundamentals, both theoretical and experimental aspects, of flutter instability and Hopf bifurcation. Open problems, research gaps, and new perspectives for investigations are indicated.
... Relaxation of the tissue tension in an embryo of the xenopus laevis at the blastula stage resulted in complete disorganization of development. At the gastrula stage, such manipulations caused a disruption in the development of axial structures (Ambrosi et al., 2017). This direction of research L.V. Belousov named morphomechanics. ...
... In recent years, biomechanical engineering has been developed to better understand the biological behavior of living organisms [1,2]. Determination of mechanical behavior and properties of organs such as the brain, bone and veins is one of the most interesting fields for biomechanical researchers. ...
In this paper, a micromechanical approach is employed to propose a cerebral cortex tissue representative volume element (RVE) and simulate the mechanical behavior of this type of tissue in different loadings. In this regard, a MATLAB code is developed to homogenize a random distribution of neurons in the extracellular matrix. To create the RVE, different inputs including the size of RVE, the number of neurons, the radius of the neuron cell body, the coordinates of the axon and dendrites of a neuron, the radius of the axon and dendrites, and the minimum distance between the cellular volumes are considered. Then, a PYTHON code is developed which generates the desired RVE in ABAQUS employing the outputs of the MATLAB code. Also, a viscoelastic material model is considered for material components of the cerebral cortex tissue in this study. To analyze the developed RVE, some relaxation tests are performed on the RVE. Strain rate, neuron volume fraction (NVF), loading time and neuron distribution are investigated in different stress relaxation tests on the developed RVEs. Considering the NVF of 1, 2 and 3%, it is concluded that the maximum tensile and compressive stresses rise by increasing the NVF. Also, the results demonstrate that different irregular distributions of neurons have no effect on the bulk mechanical properties of the tissue for a constant NVF and only affect the distribution of local stresses (and even the maximum stress) in the tissue. Finally, the numerical simulations revealed that the developed RVE is a robust element which can be employed in realistic model of the brain tissue in different loading conditions such as trauma.
... If we could measure the forces generated by the cells and how cells react to those forces, it would allow us to distinguish cause and effect for cell sheet movements and cell differentiation (Beloussov and Harris, 2006;Beloussov, 1994;Beloussov et al., 2015;Gordon, 1999). How cells react to the forces placed on them plays a major part in how an embryo develops (Ambrosi et al., 2017;Fleury and Gordon, 2012 (Brodland et al., 2010b(Brodland et al., , 2014Cranston et al., 2010) based on a method initiated by Stein and Gordon (1982). Such direct observations of forces during the major malformations of spina bifida and anencephaly (Björklund and Gordon, 2006;Brodland et al., 2010a;Gordon, 1985) and heart defects (cf. ...
We have designed and constructed a Flipping Stage for a light microscope that can view the whole exterior surface of a 2 mm diameter developing axolotl salamander embryo. It works by rapidly inverting the bottom-heavy embryo, imaging it as it rights itself. The images are then montaged to reconstruct the whole 3D surface versus time, for a full 4D record of the surface. Imaging early stage axolotl development will help discover how cell differentiation and movement takes place in the early embryo. For example, the switch from ectodermal to neural plate cells takes place on the top, animal surface portion the egg/embryo and can be observed using the flipping stage microscope. Detailed pictures of the whole surface need to be obtained so that cell tracking and event histories, such as cell divisions and participation in differentiation waves, of individual cells can be recorded. Imaging the whole exterior of the eggs/embryos will allow for the analysis of cell behavior and the forces the cells experience in their natural setting in the intact or manipulated embryo. This will give insights into embryogenesis, development, developmental disruptions, birth defects, cell differentiation and tissue engineering.
The phenomenon of oscillatory instability called 'flutter' was observed in aeroelasticity and rotor dynamics about a century ago. Driven by a series of applications involving non-conservative elasticity theory at different physical scales, ranging from nanomechanics to the mechanics of large space structures and including biomechanical problems of motility and growth, research on flutter is experiencing a new renaissance. A review is presented of the most notable applications and recent advances in fundamentals, both theoretical and experimental aspects, of flutter instability and Hopf bifurcation. Open problems, research gaps and new perspectives for investigations are indicated.
The chapter presents an analytic description of evolutionary and developmental morphogenetic events in Metazoa using concepts of self-organization, morphological and molecular–genetic data, and the topological approach to the analysis. Biological objects are complex systems capable of dynamic self-organization at all levels of biological complexity. Some examples of self-organization in cyanobacteria, metazoan cells in vitro (chick embryo myogenic cells, molluscan hemocytes, sea urchin embryo cells), and animal communities of some vertebrates are shown. Following René Thom, a topological interpretation of some evolutionary and developmental transformations is presented using well-known mathematical concepts. Toroidal forms are considered as examples of functionally optimized biological design and attractors in metazoan morphogenesis. Molecular–genetic evidence of genomic–phenomic correlations determining the body plan and evolutionary trajectories in Metazoa is discussed. Gene regulatory networks and whole metazoan genomes are interpreted as self-organizing network systems dynamically transforming in development and evolution. Symmetry breaking, topological discontinuities and catastrophes, and body plan transformations are fundamental phenomena in metazoan development and evolution.
This chapter aims at presenting a concise picture of the research in Biomechanics developed by researchers of various disciplinary areas that refer to AIMETA. The research collectors are the international journal Meccanica, published by Springer, and the AIMETA congresses including publications of studies presented therein. This chapter is devoted to studies related to AIMETA activity and mainly refers to the topics from the above-mentioned sources. Final comments and future developments are outlined.KeywordsBiomechanicsBiological tissuesArticular biomechanicsMedical devicesBiological fluid mechanicsCardiovascular mechanics
We have designed and constructed a Flipping Stage for a light microscope that can view the whole exterior surface of a 2 mm diameter developing axolotl salamander embryo. It works by rapidly inverting the bottom-heavy embryo, imaging it as it rights itself. The images are then montaged to reconstruct the whole 3D surface versus time, for a full 4D record of the surface. Imaging early stage axolotl development will help discover how cell differentiation and movement takes place in the early embryo. For example, the switch from ectodermal to neural plate cells takes place on the top, animal surface portion the egg/embryo and can be observed using the flipping stage microscope. Detailed pictures of the whole surface need to be obtained so that cell tracking and event histories, such as cell divisions and participation in differentiation waves, of individual cells can be recorded. Imaging the whole exterior of the eggs/embryos will allow for the analysis of cell behavior and the forces the cells experience in their natural setting in the intact or manipulated embryo. This will give insights into embryogenesis, development, developmental disruptions, birth defects, cell differentiation and tissue engineering.
Background:
Although the role of endogenous mechanical stresses in regulating morphogenetic movements and cell differentiation is now well established, many aspects of mechanical stress generation and transmission in developing embryos remain unclear and require quantitative studies.
Results:
By measuring stress-bearing linear deformations (caused by differences in cell movement rates) in the outer cell layer of blastula - early tail-bud Xenopus embryos, we revealed a set of long-term tension-generating gradients of cell movement rates, modulated by short-term cell-cell displacements much increasing the rates of local deformations. Experimental relaxation of tensions distorted the gradients but preserved and even enhanced local cell-cell displacements. During development, an incoherent mode of cell behavior, characterized by extensive cell-cell displacements and poorly correlated cell trajectories, was exchanged for a more coherent regime with the opposite characteristics. In particular, cell shifts became more synchronous and acquired a periodicity of several dozen minutes.
Conclusions:
Morphogenetic movements in Xenopus embryos are mediated by mechanically stressed dynamic structures of two different levels: extended gradients and short-term cell-cell displacements. As development proceeds, the latter component decreases and cell trajectories become more correlated. In particular, they acquire common periodicities, making morphogenesis more coherent.
Understanding the mechanisms regulating development requires a quantitative characterization of cell divisions, rearrangements, cell size and shape changes, and apoptoses. We developed a multiscale formalism that relates the characterizations of each cell process to tissue growth and morphogenesis. Having validated the formalism on computer simulations, we quantified separately all morphogenetic events in the Drosophila dorsal thorax and wing pupal epithelia to obtain comprehensive statistical maps linking cell and tissue scale dynamics. While globally cell shape changes, rearrangements and divisions all significantly participate in tissue morphogenesis, locally, their relative participations display major variations in space and time. By blocking division we analyzed the impact of division on rearrangements, cell shape changes and tissue morphogenesis. Finally, by combining the formalism with mechanical stress measurement, we evidenced unexpected interplays between patterns of tissue elongation, cell division and stress. Our formalism provides a novel and rigorous approach to uncover mechanisms governing tissue development.
Living matter can functionally adapt to external physical factors by developing internal tensions, easily revealed by cutting experiments. Nonetheless, residual stresses intrinsically have a complex spatial distribution, and destructive techniques cannot be used to identify a natural stress-free configuration. This work proposes a novel elastic theory of pre-stressed materials. Imposing physical compatibility and symmetry arguments, we define a new class of free energies explicitly depending on the internal stresses. This theory is finally applied to the study of arterial remodelling, proving its potential for the non-destructive determination of the residual tensions within biological materials.
This book outlines a unified theory of embryonic development, assuming morphogenesis to be a multi-level process including self-organizing steps while also obeying general laws. It is shown how molecular mechanisms generate mechanical forces, which in the long run lead to morphological changes. Questions such as how stress-mediated feedback acts at the cellular and supra-cellular levels and how executive and regulatory mechanisms are mutually dependent are addressed, while aspects of collective cell behavior and the morphogenesis of plants are also discussed. The morphomechanical approach employed in the book is based on the general principles of self-organization theory.
This study investigates the buckling of a uniaxially compressed neo-Hookean thin film bonded to a neo-Hookean substrate. Previous studies have shown that the elastic bifurcation is supercritical if r ≡ μf/μs > 1.74 and subcritical if r < 1.74, where μf and μs are the shear moduli of the film and substrate, respectively. Moreover, existing numerical simulations of the fully nonlinear post-buckling behaviour have all been focused on the regime r > 1.74. In this paper, we consider instead a subset of the regime r < 1.74, namely when r is close to unity. Four near-critical regimes are considered. In particular, it is shown that, when r > 1 and the stretch is greater than the critical stretch (the subcritical regime), there exists a localized solution that arises as the limit of modulated periodic solutions with increasingly longer and longer decaying tails. The evolution of each modulated periodic solution is followed as r is decreased, and it is found that there exists a critical value of r at which the deformation gradient develops a discontinuity and the solution becomes a static shock. The semi-analytical results presented could help future numerical simulations of the fully nonlinear post-buckling behaviour. © 2015 The Author(s) Published by the Royal Society. All rights reserved.
Significance
Animals have muscles to act on their environment. The molecules endowing them with this faculty are actin and myosin and are also present in nonmuscle cells. Our modelling breaks down the motor-like properties of the actomyosin network in single nonmuscle cells, which show striking similarity to the properties of muscles. In particular, an internal friction sets the maximum speed of contraction of both cells and muscles when myosins do not have time to detach after pulling, just as rowers lifting their oars too slowly after their stroke. The same modelling explains cell-scale mechanosensing: the combination of myosin-driven contraction and actin polymerization-driven protrusivity regulates cell length in a force-dependent manner, making response to rigidity a property of the very material of the cell cortex.
While a general consensus exists that the morphogenesis of living organisms has its roots in genetically encoded information, there is a big debate about the physical mechanisms that actually mediate its control. In embryo development, cells stop proliferating at homeostasis, a target state in terms of physical conditions that can represent, for instance, the shape and size of an organ. However, while control of mitosis is local, the spatial dimension of a tissue is a global information. How do single cells get aware of that at the same time? Which is their communication mechanism? While morphogen factors are demonstrated to play a key role in morphogenesis, and in particular for shape emergence, they seem unable to produce a global control on size by themselves and, conversely, many recent experiments suggest that active mechanics plays a role. Here we focus on a paradigmatic larval structure: the imaginal disk that will become the wing of the fruit fly. By a formalization of theoretical conjectures in terms of simple mathematical models, we show that inhomogeneous stress, likely dictated by morphogenetic patterns, is an admissible mechanism to convey locally the global information of organ size.
Tubular organs display a wide variety of surface morphologies including circumferential and longitudinal folds, square and hexagonal undulations, and finger-type protrusions. Surface morphology is closely correlated to tissue function and serves as a clinical indicator for physiological and pathological conditions, but the regulators of surface morphology remain poorly understood. Here, we explore the role of geometry and elasticity on the formation of surface patterns. We establish morphological phase diagrams for patterns selection and show that increasing the thickness or stiffness ratio between the outer and inner tubular layers induces a gradual transition from circumferential to longitudinal folding. Our results suggest that physical forces act as regulators during organogenesis and give rise to the characteristic circular folds in the esophagus, the longitudinal folds in the valves of Kerckring, the surface networks in villi, and the crypts in the large intestine.
The changing mass of biomaterials can either be modelled at the constitutive level or at the kinematic level. This contribution attends on the description of growth at the kinematic level. The deformation gradient will be multiplicatively split into a growth part and an elastic part. Hence, in addition to the material and the spatial configuration, we consider an intermediate configuration or grown configuration without any elastic deformations. With such an ansatz at hand, contrary to the modelling of mass changes at the constitutive level, both a change in density and a change in volume can be modelled. The algorithmic realisation of this framework within a finite element setting constitutes the main contribution of this paper. To this end the key kinematic variable, i.e., the isotropic stretch ratio, is introduced as internal variable at the integration point level. The consistent linearisation of the stress update based on an implicit time integration scheme is developed. Basic features of the model are illustrated by means of representative numerical examples.
The present contribution is dedicated to the computational modeling of growth phenomena typically encountered in modern biomechanical applications. We set the basis by critically reviewing the relevant literature and classifying the existing models. Next, we introduce a geometrically exact continuum model of growth which is not a priori restricted to applications in hard tissue biomechanics. The initial boundary value problem of biomechanics is primarily governed by the density and the deformation problem which render a nonlinear coupled system of equations in terms of the balance of mass and momentum. To ensure unconditional stability of the required time integration procedure, we apply the classical implicit Euler backward method. For the spatial discretization, we suggest two alternative strategies, a node-based and an integration point-based approach. While for the former, the discrete balance of mass and momentum are solved simultaneously on the global level, the latter is typically related to a staggered solution with the density treated as internal variable. The resulting algorithms of the alternative solution techniques are compared in terms of stability, uniqueness, efficiency and robustness. To illustrate their basic features, we elaborate two academic model problems and a typical benchmark example from the field of biomechanics.
The model of volumetric material growth is introduced in the framework of finite elasticity. The state variables include the deformations, temperature and the transplant matrix function. The wellposedness of the proposed model is shown. The existence of local in time classical solutions for the quasistatic deformations boundary value problem coupled with the energy balance and the growth evolution of the transplant is obtained. The new mathematical results for a broad class of growth models in mechanics and biology are presented with complete proofs.
Not long ago it was thought that the most important characteristics of the mechanics of soft tissues were their complex mechanical properties: they often exhibit nonlinear, anisotropic, nearly incompressible, viscoelastic behavior over finite strains. Indeed, these properties endow soft tissues with unique structural capabilities that continue to be extremely challenging to quantify via constitutive relations. More recently, however, we have come to appreciate an even more important characteristic of soft tissues, their homeostatic tendency to adapt in response to changes in their mechanical environment. Thus, to understand well the biomechanical properties of a soft tissue, we must not only quantify their structure and function at a given time, we must also quantify how their structure and function change in response to altered stimuli. In this paper, we introduce a new constrained mixture theory model for studying growth and remodeling of soft tissues. The model melds ideas from classical mixture and homogenization theories so as to exploit advantages of each while avoiding particular difficulties. Salient features include the kinetics of the production and removal of individual constituents and recognition that the neighborhood of a material point of each constituent can have a different, evolving natural (i.e. stress-free) configuration.
Organ size is controlled by the concerted action of biochemical and physical processes. Although mechanical forces are known to regulate cell and tissue behavior, as well as organogenesis, the precise molecular events that integrate mechanical and biochemical signals to control these processes are not fully known. The recently delineated Hippo-tumor suppressor network and its two nuclear effectors, YAP and TAZ, shed light on these mechanisms. YAP and TAZ are proto-oncogene proteins that respond to complex physical milieu represented by the rigidity of the extracellular matrix, cell geometry, cell density, cell polarity and the status of the actin cytoskeleton. Here, we review the current knowledge of how YAP and TAZ function as mechanosensors and mechanotransducers. We also suggest that by deciphering the mechanical and biochemical signals controlling YAP/ TAZ function, we will gain insights into new strategies for cancer treatment and organ regeneration.
Progressive airflow obstruction is a classical hallmark of chronic lung disease, affecting more than one fourth of the adult population. As the disease progresses, the inner layer of the airway wall grows, folds inwards, and narrows the lumen. The critical failure conditions for airway folding have been studied intensely for idealized circular cross-sections. However, the role of airway branching during this process is unknown. Here, we show that the geometry of the bronchial tree plays a crucial role in chronic airway obstruction and that critical failure conditions vary significantly along a branching airway segment. We perform systematic parametric studies for varying airway cross-sections using a computational model for mucosal thickening based on the theory of finite growth. Our simulations indicate that smaller airways are at a higher risk of narrowing than larger airways and that regions away from a branch narrow more drastically than regions close to a branch. These results agree with clinical observations and could help explain the underlying mechanisms of progressive airway obstruction. Understanding growth-induced instabilities in constrained geometries has immediate biomedical applications beyond asthma and chronic bronchitis in the diagnostics and treatment of chronic gastritis, obstructive sleep apnea and breast cancer.
The modulation of developmental biochemical pathways by mechanical cues is an emerging feature of animal development, but its evolutionary origins have not been explored. Here we show that a common mechanosensitive pathway involving β-catenin specifies early mesodermal identity at gastrulation in zebrafish and Drosophila. Mechanical strains developed by zebrafish epiboly and Drosophila mesoderm invagination trigger the phosphorylation of β-catenin-tyrosine-667. This leads to the release of β-catenin into the cytoplasm and nucleus, where it triggers and maintains, respectively, the expression of zebrafish brachyury orthologue notail and of Drosophila Twist, both crucial transcription factors for early mesoderm identity. The role of the β-catenin mechanosensitive pathway in mesoderm identity has been conserved over the large evolutionary distance separating zebrafish and Drosophila. This suggests mesoderm mechanical induction dating back to at least the last bilaterian common ancestor more than 570 million years ago, the period during which mesoderm is thought to have emerged.
The stability of the wrinkling mode experienced by a compressed half-space of neo-Hookean material is investigated using analytica and numerical methods to study the post-bifurcation behaviour of periodic solutions. It is shown that wrinkling is highl unstable owing to the nonlinear interaction among the multiple modes associated with the critical compressive state. Concomitantly wrinkling is sensitive to exceedingly small initial imperfections that significantly reduce the compressive strain at whic the instability occurs. The study provides insight into the connection between wrinkling and an alternative surface mode the finite amplitude crease or sulcus. The shape of the critical combination of wrinkling modes has the form of an incipien crease, and a tiny initial imperfection can trigger a wrinkling instability that collapses into a crease.
Several finite element methods for large deformation elastic problems in the nearly incompressible and purely incompressible regimes are considered. In particular, the method ability to accurately capture critical loads for the possible occurrence of bifurcation and limit points, is investigated. By means of a couple of 2D model problems involving a very simple neo-Hookean constitutive law, it is shown that within the framework of displacement/pressure mixed elements, even schemes that are inf-sup stable for linear elasticity may exhibit problems when used in the finite deformation regime. The roots of such troubles are identified, but a general strategy to cure them is still missing. Furthermore, a comparison with displacement-based elements, especially of high order, is presented.
Villi are ubiquitous structures in the intestine of all vertebrates, originating from the embryonic development of the epithelial mucosa. Their morphogenesis has similar stages in living organisms but different forming mechanisms. In this work, we model the emergence of the bi-dimensional undulated patterns in the intestinal mucosa from which villi start to elongate. The embryonic mucosa is modelled as a growing thick-walled cylinder, and its mechanical behaviour is described using an hyperelastic constitutive model, which also accounts for the anisotropic characteristics of the reinforcing fibres at the microstructural level. The occurrence of surface undulations is investigated using a linear stability analysis based on the theory of incremental deformations superimposed on a finite deformation. The Stroh formulation of the incremental boundary value problem is derived, and a numerical solution procedure is implemented for calculating the growth thresholds of instability. The numerical results are finally discussed with respect to different growth and materials properties. In conclusion, we demonstrate that the emergence of intestinal villi in embryos is triggered by a differential growth between the mucosa and the mesenchymal tissues. The proposed model quantifies how both the geometrical and the mechanical properties of the mucosa drive the formation of previllous structures in embryos.
On the basis of the nonlinear theory of elasticity, the general constitutive
equation for an isotropic hyperelastic solid in the presence of initial stress
is derived. This derivation involves invariants that couple the deformation
with the initial stress and in general, for a compressible material, it
requires 10 invariants, reducing to 9 for an incompressible material.
Expressions for the Cauchy and nominal stress tensors in a finitely deformed
configuration are given along with the elasticity tensor and its specialization
to the initially stressed undeformed configuration. The equations governing
infinitesimal motions superimposed on a finite deformation are then used to
study the combined effects of initial stress and finite deformation on the
propagation of homogeneous plane waves in a homogeneously deformed and
initially stressed solid of infinite extent. This general framework allows for
various different specializations, which make contact with earlier works. In
particular, connections with results derived within Biot's classical theory are
highlighted. The general results are also specialized to the case of a small
initial stress and a small pre-deformation, i.e. to the evaluation of the
acoustoelastic effect. Here the formulas derived for the wave speeds cover the
case of a second-order elastic solid without initial stress and subject to a
uniaxial tension [Hughes and Kelly, Phys. Rev. 92 (1953) 1145] and are
consistent with results for an undeformed solid subject to a residual stress
[Man and Lu, J. Elasticity 17 (1987) 159]. These formulas provide a basis for
acoustic evaluation of the second- and third-order elasticity constants and of
the residual stresses. The results are further illustrated in respect of a
prototype model of nonlinear elasticity with initial stress, allowing for both
finite deformation and nonlinear dependence on the initial stress.
Active reactions of embryonic tissues to mechanical forces play an important role in morphogenesis. To study these reactions, experimental models that enable to evaluate the applied forces and the deformations of the tissues are required. A model based upon the active intrusion of a living early gastrula Xenopus embryo into a tube half the embryo in diameter is described. The intrusion is initially triggered by a suction force of several dozen Pa but then continues in the absence of external driving force, stopping immediately after the entire embryo has penetrated into the tube. The process can be stopped by cytoskeletal drugs or by the damage of the part of the embryo still non-aspirated and is associated with the transversal contraction and meridional elongation of the non-aspirated part of the embryo surface and quasi-periodic longitudinal contractions/extensions of the cells within the part already aspirated. We suggest that this reaction is an active response to the embryo deformation and discuss its morphogenetic role. The problem of estimating the elastic modules of embryonic tissues is also discussed.
Within the context of finite deformation elasticity theory the problem of deforming an open sector of a thick-walled circular cylindrical tube into a complete circular cylindrical tube is analyzed. The analysis provides a means of estimating the radial and circumferential residual stress present in an intact tube, which is a problem of particular concern in dealing with the mechanical response of arteries. The initial sector is assumed to be unstressed and the stress distribution resulting from the closure of the sector is then calculated in the absence of loads on the cylindrical surfaces. Conditions on the form of the elastic strain-energy function required for existence and uniqueness of the deformed configuration are then examined. Finally, stability of the resulting finite deformation is analyzed using the theory of incremental deformations superimposed on the finite deformation, implemented in terms of the Stroh formulation. The main results are that convexity of the strain energy as a function of a certain deformation variable ensures existence and uniqueness of the residually-stressed intact tube, and that bifurcation can occur in the closing of thick, widely opened sectors, depending on the values of geometrical and physical parameters. The results are illustrated for particular choices of these parameters, based on data available in the biomechanics literature.
The velocities and directions of movements of individual outer ectodermal cells of Xenopus embryos in the course of normal development from the blastula to the early tail-bud stage, as well as after mechanical relaxation in the early gastrula, were measured. An alternation of the periods of directed movements of large cell masses and local cell wanderings was detected. In both cases, the trajectories of individual cells consisted primarily of orthogonal segments. Cell movements were measured on two scales. At a smallscale consideration (time intervals of the order of several hours and distances of the order of tens of microns), fairly slight linear stretching and compressive deformations were detected, which looked like gentle smooth gradients along which the upward morphogenetic movements of cells were directed. At a large-scale consideration (time intervals of the order of tens of minutes and distances of the order of microns), quasi-periodic fluctuations of velocities of individual cells partly correlated in time were found. The differences between these velocities generated microdeformations, which reached several tens of percent and developed within time intervals not more than 10 min. Measurements of relative magnitudes of mechanical forces influencing the cell walls suggests that microdeformations generate local stretching and compressive deformations modulating smoother tension gradients.
Stress distribution through the wall thickness of the canine carotid artery was analyzed on the basis of the uniform strain hypothesis in which the wall circumferential strain was assumed to be constant over the wall cross-section under physiological loading condition. A newly proposed logarithmic type of strain energy density function was used to describe the wall properties. In contrast with other studies, this hypothesis gave almost uniform distribution of wall stresses under the physiological condition and non-zero residual stresses when all external forces were removed.
It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.
A clear demarcation between various processes of material evolution is established and the implications of the symmetry type on our ability to distinguish between them are investigated. The general features of the various types of material evolution are emphasized by establishing a spatio-temporal analogy between material uniformity and processes of material evolution.
The ability of living tissues to grow and remodel in response to altered loads has long been considered an important characteristic in biomechanics. However, in the last few decades most studies have focussed on hard tissues such as bones. It is important to emphasize that a major difference between hard and soft tissues lies in the magnitudes of the deformations that they can sustain. While infinitesimal deformation theories can reasonably be applied to hard tissues, the deformation of soft tissues requires a nonlinear theory that allows for large deformations. One of the first attempts to formulate a mathematical description of growth in soft tissues was that of Skalak et al. (1982). The first formal framework for soft tissue that accounts for volumetric growth only appeared with the work of Rodriguez et al. (1994). Since then, amongst others, Taber (1995), Rachev et al. (1998) and Rachev (2003) have contributed to our understanding of this growth behaviour. See also the paper by Lubarda and Hoger (2002) and references contained therein. In the present paper we adopt the definition used by many researchers that considers growth as a change in mass and geometry. This distinguishes growth from remodelling, which is often regarded as a rearrangement of the microstructure in the tissue and is not considered here. This paper is divided into two main sections. In Section 2, we focus on a general formulation of the mechanics of growth. Balance equations of mass, linear and angular momentum and energy incorporating volumetric and surface sources are reviewed. General constitutive equations that include the effect of residual stresses are also described. The theory is illustrated with a simple example in Section 3, in which the problem of extension and inflation of an artery is considered. The artery is modelled as a residually stressed mechanically incompressible circular cylindrical tube. Residual stresses prior to growth are calculated on the assumption that the circumferential stress is uniform at typical physiological pressure. Growth in the thickness of the artery wall due to increased blood pressure is then discussed briefly along with the evolution of residual stress. Because of limitations on space detailed derivations of the equations and the results for the specific problem are not given here but will be the subject of a separate paper.
Knowledge of stress distributions in the arterial wall is important for many reasons. In the study of the propagation of pulse waves, one must know the incremental modulus of the elasticity that changes with the stress level. In the study of circulation control, the action of the vascular smooth muscle, which depends on its local stress level (see review in Fung, 1984), must be evaluated. In the study of atherogenesis, one must know the stress distribution in the vessel wall because the tensile and shear stress can alter the local wall permea-bility and pressure gradient which is the force that drives the fluid in or out of the vessel wall (Chuong and Fung, 1983). Accurate evaluation of stress distributions in the arterial wall is therefore an important step toward a better understanding of various physiological functions and pathological mechanisms associated with the circulatory system.
Significance
The convolutions of the human brain are a symbol of its functional complexity and correlated with its information-processing capacity. Conversely, loss of folds is correlated with loss of function. But how did the outer surface of the brain, the layered cortex of neuronal gray matter, get its folds? Guided by prior experimental observations of the growth of the cortex relative to the underlying white matter, we argue that these folds arise due to a mechanical instability of a soft tissue that grows nonuniformly. Numerical simulations and physical mimics of the constrained growth of the cortex show how compressive mechanical forces sculpt it to form characteristic sulci and gyri, consistent with observations across species in both normal and pathological situations.
Segmentation is a characteristic feature of the vertebrate body plan. The prevailing paradigm explaining its origin is the ‘clock and wave-front’ model, which assumes that the interaction of a molecular oscillator (clock) with a traveling gradient of morphogens (wave) pre-defines spatial periodicity. While many genes potentially responsible for these processes have been identified, the precise role of molecular oscillations and the mechanism leading to physical separation of the somites remain elusive. In this paper we argue that the periodicity along the embryonic body axis anticipating somitogenesis is controlled by mechanical rather than bio-chemical signaling. Using a prototypical model we show that regular patterning can result from a mechanical instability induced by differential strains developing between the segmenting mesoderm and the surrounding tissues. The main ingredients of the model are the assumptions that cell–cell adhesions soften when overstretched, and that there is an internal length scale defining the cohesive properties of the mesoderm. The proposed mechanism generates a robust number of segments without dependence on genetic oscillations.
In the early embryo, the primitive heart tube (HT) undergoes the morphogenetic process of c-looping as it bends and twists into a c-shaped tube. Despite intensive study for nearly a century, the physical forces that drive looping remain poorly understood. This is especially true for the bending component, which is the focus of this paper. For decades, experimental measurements of mitotic rates had seemingly eliminated differential growth as the cause of HT bending, as it has commonly been thought that the heart grows almost exclusively via hyperplasia before birth and hypertrophy after birth. Recently published data, however, suggests that hypertrophic growth may play a role in looping. To test this idea, we developed finite-element models that include regionally measured changes in myocardial volume over the HT. First, models based on idealized cylindrical geometry were used to simulate the bending process. With the number of free parameters in the model reduced to the extent possible, stress and strain distributions were compared to those measured in embryonic chick hearts that were isolated and cultured for 24 hr. The results show that differential growth alone yields results that agree reasonably well with the trends in our data, but adding active changes in myocardial cell shape provides closer quantitative agreement with stress measurements. Next, the estimated parameters were extrapolated to a model based on realistic 3-D geometry reconstructed from images of an actual chick heart. This model yields similar results and captures quite well the basic morphology of the looped heart. Overall, our study suggests that differential hypertrophic growth in the myocardium is the primary cause of the bending component of c-looping, with other mechanisms possibly playing lesser roles.
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.
A multiscale analysis integrating biomechanics and mechanobiology is today required for deciphering the crosstalk between biochemistry, geometry and elasticity in living materials. In this paper we derive a unified thermomechanical theory coupling growth processes with mass transport phenomena across boundaries and/or material interfaces. Inside a living system made by two contiguous bodies with varying volumes, an interfacial growth mechanism is considered to force fast but continuous variations of the physical fields inside a narrow volume across the material interface. Such a phenomenon is modelled deriving homogenized surface fields on a growing non-material discontinuity, possibly including a singular edge line. A number of balance laws is derived for imposing the conservation of the thermomechanical properties of the biological system. From thermodynamical arguments we find that the normal displacement of the non-material interface is governed by the jump of a new form of material mechanical-energy flux, also involving the kinetic energies and the mass fluxes. Furthermore, the configurational balance indicates that the surface Eshelby tensor is the tangential stress measure driving the material inhomogeneities on the non-material interface. Accordingly, stress-dependent evolution laws for bulk and interfacial growth processes are derived for both volume and surface fields.
Key cellular decisions, such as proliferation or growth arrest, typically occur at spatially defined locations within tissues. Loss of this spatial control is a hallmark of many diseases, including cancer. Yet, how these patterns are established is incompletely understood. Here, we report that physical and architectural features of a multicellular sheet inform cells about their proliferative capacity through mechanical regulation of YAP and TAZ, known mediators of Hippo signaling and organ growth. YAP/TAZ activity is confined to cells exposed to mechanical stresses, such as stretching, location at edges/curvatures contouring an epithelial sheet, or stiffness of the surrounding extracellular matrix. We identify the F-actin-capping/severing proteins Cofilin, CapZ, and Gelsolin as essential gatekeepers that limit YAP/TAZ activity in cells experiencing low mechanical stresses, including contact inhibition of proliferation. We propose that mechanical forces are overarching regulators of YAP/TAZ in multicellular contexts, setting responsiveness to Hippo, WNT, and GPCR signaling.
Eukaryotic cells and biological materials are described from a rheological point of view. Single cells possess typical microrheological properties which can affect cell behaviour, in close connection with their adhesion properties. Single cell properties are also important in the context of multicellular systems, i.e. in biological tissues. Results from experiments are analyzed and models proposed both at the cellular scale and the macroscopic scale. To cite this article: C. Verdier et al., C. R. Physique 10 (2009).
The growth and remodeling of soft tissues depend on a number of biological, chemical and mechanical factors, including the state of tension. In many cases the stress field plays such a relevant role that “stress-modulated growth” has become a very topical subject. Recent theoretical achievements suggest that, irrespective of the specific biological material at hand, a component of the stress-growth coupling is tissue-independent and reads as an Eshelby-like tensor. In this paper we investigate the mathematical properties and qualitative behavior predicted by equations that specialize that model under few simple assumptions. Constitutive equations that satisfy a suitable dissipation principle are compared with heuristic ones that fit well experimental data. Numerical simulations of the growth of a symmetric annulus are discussed and compared with the predicted qualitative behavior.
The growth of biological tissues is here described at the continuum scale of tissue elements. Relying on a previous work in Ganghoffer and Haussy (2005), the rephrasing of the balance laws for tissue elements under growth in terms of suitable Eshelby tensors is done in the present contribution, considering successively volumetric and surface growth. Balance laws for volumetric growth are written in both compatible and incompatible configurations, highlighting the material forces for growth associated to Eshelby tensors. Evolution laws for growth are written from the expression of the local dissipation in terms of a relation linking the growth velocity gradient to a growth-like Eshelby stress, in the spirit of configurational mechanics. Surface growth is next envisaged in terms of phenomena taking place in a varying reference configuration, relying on the setting up of a surface potential depending upon the surface transformation gradient and to the normal to the growing surface. The balance laws resulting from the stationnarity of the potential energy are expressed, involving surface Eshelby tensors associated to growth. Simulations of surface growth in both cases of fixed and moving generating surfaces evidence the interplay between diffusion of nutrients and the mechanical driving forces for growth.
Expressions for the components of strain and the incompressibility condition, for large deformations, are obtained in a cylindrical polar co-ordinate system. The stress-strain relations, equations of motion and boundary conditions for an incompressible, neo-Hookean material, in such a co-ordinate system, are also obtained and specialized to the case of cylindrical symmetry. These results are applied to the special cases of the simple torsion of a solid cylinder and of a hollow, cylindrical tube and to their combined simple extension and simple torsion. In the case of a solid cylinder, it is found that a state of simple torsion can be maintained by surface tractions applied to the ends of the cylinder only, and these consist of a torsional couple together with a compressive force. The necessary torsional couple is proportional to the amount of torsion and the compressive force to the square of the torsion. In the case of a hollow, cylindrical tube, it is again necessary to exert a torsional couple, proportional to the torsion, and a compressive force, proportional to the square of the torsion, on the plane ends, but it is also necessary to exert a normal surface traction, acting in a positive radial direction, on one or other of the curved surfaces of the tube and proportional to the square of the torsion.
Mysteries of DevelopmentDevelopmental biologists have found dozens of proteins and genes that play a role in the growth of plants and animals. But how growing organs and organisms can sense their size and know when to stop is still a mystery. Developmental biologists continue to explore that mystery, and the current objects of their attention are imaginal discs, flattened sacs of cells that grow during fruit flies' larval stages. Scientists can also change the rate at which imaginal disc cells divide, prompting either too many or not enough cells to form, but the cell size adjusts so that organ size remains the same. How does a developing organ somehow senses the mechanical forces on its growing and dividing cells?
The interplay between epigenetic modification and chromatin compaction is implicated in the regulation of gene expression, and it comprises one of the most fascinating frontiers in cell biology. Although a complete picture is still lacking, it is generally accepted that the differentiation of embryonic stem (ES) cells is accompanied by a selective condensation into heterochromatin with concomitant gene silencing, leaving access only to lineage-specific genes in the euchromatin. ES cells have been reported to have less condensed chromatin, as they are capable of differentiating into any cell type. However, pluripotency itself-even prior to differentiation-is a split state comprising a naïve state and a state in which ES cells prime for differentiation. Here, we show that naïve ES cells decondense their chromatin in the course of downregulating the pluripotency marker Nanog before they initiate lineage commitment. We used fluorescence recovery after photobleaching, and histone modification analysis paired with a novel, to our knowledge, optical stretching method, to show that ES cells in the naïve state have a significantly stiffer nucleus that is coupled to a globally more condensed chromatin state. We link this biophysical phenotype to coinciding epigenetic differences, including histone methylation, and show a strong correlation of chromatin condensation and nuclear stiffness with the expression of Nanog. Besides having implications for transcriptional regulation and embryonic cell sorting and suggesting a putative mechanosensing mechanism, the physical differences point to a system-level regulatory role of chromatin in maintaining pluripotency in embryonic development.
Growth and remodelling are strictly interlaced in the biological tissues and their unified treatment is a challenge in continuum mechanics, where the evolution of macroscopic quantities, like shape and stresses, are dictated by the non-equilibrium transport and reaction equations of biochemicals at the microscale. Here we derive the kinematic description and the main balance equations of a second gradient theory aimed at modelling both volumetric growth and mass transport inside the living matter. Thermodynamical arguments suggest defining few admissible classes of constitutive theories, which model the diffusive growth processes inside a continuum body. In addition, we propose evolution equations for first and second gradient inhomogeneities, considering different symmetry groups and initial second-order symmetries. Few examples of morphogenetic models will be presented, in order to show the biomechanical importance of coupling volumetric growth, mass transport and internal stress state in physiological and pathological conditions for biological matter.
This work examines critically the various formulations of the balance of linear momentum innonlinear inhomogeneous elasticity. The corresponding variational formulations are presented. From the point of view of the theory of elastic inhomogeneities, the most interesting formulations are those which, being either completely material or mixed-Eulerian, exhibit explicitly the inhomogeneities in the form ofmaterial forces. They correspond to the balance ofpseudomomentum, a material covector which is seldom used but which we show to play a fundamental role in the Hamiltonian canonical formulation of nonlinear elasticity. The flux associated with pseudomomentum is none other than theEshelby material tensor. Applying this formulation to the case of an elastic body containing a crack of finite extent, the notion of suction force acting at the tip of the crack follows while afracture criterion la Griffith can be deduced from a variational inequality. Possible extensions to higher-grade elastic materials and inelastic materials are indicated as well as the role played by pseudomomentum in the quantization of elastic vibrations.
The purpose of this paper is to examine the explicit dependence of the elasticity tensor on residual stress for the case in which the residual stress is produced by an elastic deformation. Constitutive equations appropriate for the description of materials that behave elastically in deformations from a residually stressed configuration have been derived for small displacement gradients [-1, 2] and for small strains with arbitrary rotations [3]. In these derivations no assumption is made concerning the origin of the residual stress. In particular, it is not required that it be the result of prior elastic behavior. The derivation of all of these constitutive equations is carried out by linearizing the finite elastic constitutive equation. In each case, the linearized equation includes a fourth order elasticity tensor similar to that found in the classical linear theory of elasticity. But these elasticity tensors are fundamentally different from their counterpart in the classical theory because they can depend on the residual stress. Some consequences of this dependence were explored in [4]. There it was shown that the elasticity tensor associated with the Cauchy stress is different than that associated with the Piola-Kirchhoff stress; the elasticity tensors are not positive definite in the usual sense; and the material symmetry of the elasticity tensors is dependent on the form of the residual stress. Throughout the work presented in [-4] the dependence of the elasticity tensor on residual stress remained implicit; the functional form of the dependence was unknown.
The basic physical concepts of classical continuum mechanics are body, configuration of a body, and force system acting on a body. In a formal rational development of the subject, one first tries to state precisely what mathematical entities represent these physical concepts: a body is regarded to be a smooth manifold whose elements are the material points; a configuration is defined as a mapping of the body into a three-dimensional Euclidean space, and a force system is defined to be a vector-valued function defined for pairs of bodies1. Once these concepts are made precise one can proceed to the statement of general principles, such as the principle of objectivity or the law of balance of linear momentum, and to the statement of specific constitutive assumptions, such as the assertion that a force system can be resolved into body forces with a mass density and contact forces with a surface density, or the assertion that the contact forces at a material point depend on certain local properties of the configuration at the point. While the general principles are the same for all work in classical continuum mechanics, the constitutive assumptions vary with the application in mind and serve to define the material under consideration.