Buckling instability in growing tumor spheroids

International audienceA growing tumor is subjected to intrinsic physical forces, arising from the cellular turnover in a spatially constrained environment. This work demonstrates that such residual solid stresses can provoke a buckling instability in heterogeneous tumor spheroids. The growth rate ratio between the outer shell of proliferative cells and the inner necrotic core is the control parameter of this instability. The buckled morphology is found to depend both on the elastic and the geometric properties of the tumor components, suggesting a key role of residual stresses for promoting tumor invasiveness. DOI: 10.1103/PhysRevLett.110.15810

Shear instability in skin tissue

We propose two toy-models to describe, predict, and interpret the wrinkles appearing on the surface of skin when it is sheared. With the first model, we account for the lines of greatest tension present in human skin by subjecting a layer of soft tissue to a pre-stretch, and for the epidermis by endowing one of the layer's faces with a surface tension. For the second model, we consider an anisotropic model for the skin, to reflect the presence of stiff collagen fibres in a softer elastic matrix. In both cases, we find an explicit bifurcation criterion, linking geometrical and material parameters to a critical shear deformation accompanied by small static wrinkles, with decaying amplitudes normal to the free surface of skin

Shape transitions in a soft incompressible sphere with residual stresses

Residual stresses may appear in elastic bodies, owing to the formation of misfits in the microstructure, driven by plastic deformations or thermal or growth processes. They are especially widespread in living matter, resulting from dynamic remodelling processes aimed at optimizing the overall structural response to environmental physical forces. From a mechanical viewpoint, residual stresses are classically modelled through the introduction of a virtual incompatible state that collects the local relaxed states around each material point. In this work, we employ an alternative approach based on a strain energy function that constitutively depends only on the deformation gradient and the residual stress tensor. In particular, our objective is to study the morphological stability of an incompressible sphere, made of a neo-Hookean material, and subjected to given distributions of residual stresses. The boundary value elastic problem is studied with analytic and numerical tools. Firstly, we perform a linear stability analysis on the prestressed solid sphere using the method of incremental deformations. The marginal stability conditions are given as a function of a control parameter, which is the dimensionless variable that represents the characteristic intensity of the residual stresses. Secondly, we perform finite-element simulations using a mixed formulation in order to investigate the postbuckling morphology in the fully nonlinear regime. Considering different initial distributions of the residual stresses, we find that different morphological transitions occur around the material domain, where the hoop residual stress reaches its maximum compressive value. The loss of spherical symmetry is found to be controlled by the mechanical and geometrical properties of the sphere, as well as the spatial distribution of the residual stress. The results provide useful guidelines for designing morphable soft spheres, for example by controlling residual stresses through active deformations. They finally suggest a viable solution for the nondestructive characterization of residual stresses in soft tissues, such as solid tumours

Morphoelasticity and mechano-transduction in living matter

The emergence of shapes in living matter is the final result of a series of complex interactions relating the biochemical processes driving changes of microstructure to the macroscopic reorganization of the material architecture. Understanding the basic principles and the key mechanisms coordinating such a crosstalk at different lengthscales is one of the main challenges in developmental biology. Moving from a biomechanical perspective, in this talk I will show how the combination of physical and mechanical theories/methods can foster understanding on the role played by the nonlinear elasticity in the formation of a specific pattern, with the aim to identify the key mechanical feedbacks regulating growth and remodeling in biological materials. In continuum mechanics, a considerable amount of modeling research has been performed in the last decade to accommodate the evolution laws of mass variation inside a material, written as a volumetric change of bulk material or an accretion/resorption at a surface. A seminal idea has been borrowed from dislocation theory, assuming a separability principle between volumetric growth (i.e. smooth change in a single-phase material) and elastic deformation. Such a multiplicative decomposition hypothesis is the fundamental assumption of many morphoelastic theories, although encountering several limitations in biomechanical applications, often requiring a number of simplifications for providing analytical results of complex boundary value problems for biological materials (i.e. generally viscoelastic, anisotropic, incompressible bodies). Moreover, even if stressdriven growth processes can be taken into account, continuum theories of growth and remodeling often exclude ab-initio the possibility to include a diffusive mass flow inside the body, being not suitable to describe morphogenetic events where mass transport and chemo-mechanical coupling are of paramount importance. In the first part, I will show how complex biological patterns can arise from the loss of elastic stability due to geometrical incompatibilities in the growth processes. In particular, I we will focus on the bifurcation analysis of soft growing materials, with the aim to investigate the influence both of growth rate and of external constraints (intended in terms of applied traction loads and/or spatial confinement). The cylindrical geometry is chosen to describe the grown pattern of multilayered tubular organs (e.g. airways, esophagus, intestine and blood vessels), where incompatible growth between the layers results in a complex transmural distribution of residual strains. Introducing a generating function to derive the implicit gradient form of displacement fields, the incompressibility constraint on the elastic deformation tensor is solved exactly using a canonical transformation. Therefore, a generic boundary value problem, with conservative applied volume forces and traction, can be completely transformed into a variational problem: the elastic solution is given by the value of the scalar generating function minimizing the total potential energy of the growing continuum, respecting the given boundary constraints. The variational formulation is able to provide a straightforward derivation of the linear stability analysis, which would otherwise require lengthy manipulations on the governing incremental equations. In addition, the use of a generating function allows accounting for the presence of local singularities in the elastic solution. The proposed variational method is applied in the bifurcation analysis of few growth problems of biomechanical interest. The theoretical results show that folding of the inner mucosal layer can appear in physiological growth conditions, even in absence of external loading. Furthermore, surface instability can occur at the outer boundary of a thin proliferating ring in contact with a soft inner core, as observed in the early development of skin cancers. The effects of external constraints, material properties and growth characteristics on the creation of a non-circular shapes are discussed in comparison with a wide variety of experimental studies. The theoretical predictions indicate that the transition towards a more complex pattern for soft materials is often initiated by a mechanical instability. Other than the mentioned clinical interest, a quantification of the effects of growth on the stability of soft materials is important for applications in bioengineering and in biomaterials. In the second part, I will focus on the kinematic description and the main balance equations of a novel thermo-mechanical growth theory for a second-gradient continuum. The aim of such a modeling is to include the role of diffusing morphogens in the determination of spatial patterns of cell differentiation, providing a thermodynamicallybased coupling for modeling the stress-driven feedback mechanisms that regulate growth and orchestrate shape during morphogenetic events. The viewpoint assumes that genes carry specific biochemical instructions for the creation of biological matter, while the biomechanical and biochemical interactions with the environment generate the pattern emergence. In the framework of a second gradient hyperelastic theory, the first singlephase continuum theory accounting both for volumetric growth and for mass transport phenomena is formulated. Mass changes are defined by a material isomorphism where growth processes act as local rearrangements of the material inhomogeneities: a firstorder uniformity transplant determines the extent of volumetric growth, while a secondorder transplant takes into account the curvature effects induced by a local differential deformation. The diffusion of biochemical species (e.g. morphogens, nutrients, migration signals) inside the biological matter is considered using the theory of configurations forces with internal variables. Mass transport phenomena are found to depend both on the first- and on the second-order material connections, and their driving forces can be written in terms of covariant material derivatives based on the two connections, reflecting in a purely geometrical manner the presence of a (first-order) torsion and a (second-order) curvature. The expression of remodeling evolution laws for first- and second-order material inhomogeneities is given, having a great importance for a consistent thermodynamical description of morphogenetic events. The proposed constitutive theory is applied for modeling the effects of an Eshelbian coupling on volumetric growth and mass transport in two biomechanical examples. First, the avascular development of a ductal carcinoma is considered, illustrating how both mechano-transduction and spatial limitation of nutrient diffusion can inhibit growth. Secondly, the proposed evolution equations of material inhomogeneities are used to analyze the remodeling laws driving homeostasis in blood vessels, both in healthy and in pathological conditions. The results indicate that diffusive mass fluxes play a fundamental role in the active regulations of homeostatic conditions, possibly being involved in the integral feedback mechanisms driving local growth rates. Finally, the biomechanical quantification of curvature-dependent effects on growth can help not only in understanding the growth patterns of cellular aggregates, but also has important applications for optimizing scaffolds in regenerative medicine and tissue engineering

Branching instability in expanding bacterial colonies

International audienceSelf-organization in developing living organisms relies on the capability of cells to duplicate and perform a collective motion inside the surrounding environment. Chemical and mechanical interactions coordinate such a cooperative behaviour, driving the dynamical evolution of the macroscopic system. In this work, we perform an analytical and computational analysis to study pattern formation during the spreading of an initially circular bacterial colony on a Petri dish. The continuous mathematical model addresses the growth and the chemotactic migration of the living monolayer, together with the diffusion and consumption of nutrients in the agar. The governing equations contain four dimensionless parameters, accounting for the interplay among the chemotactic response, the bacteria-substrate interaction and the experimental geometry. The spreading colony is found to be always linearly unstable to perturbations of the interface, whereas branching instability arises in finite-element numerical simulations. The typical length scales of such fingers, which align in the radial direction and later undergo further branching, are controlled by the size parameters of the problem, whereas the emergence of branching is favoured if the diffusion is dominant on the chemotaxis. The model is able to predict the experimental morphologies, confirming that compact (resp. branched) patterns arise for fast (resp. slow) expanding colonies. Such results, while providing new insights into pattern selection in bacterial colonies, may finally have important applications for designing controlled patterns

Coupling solid and fluid stresses with brain tumour growth and white matter tract deformations in a neuroimaging-informed model

Brain tumours are among the deadliest types of cancer, since they display a strong ability to invade the surrounding tissues and an extensive resistance to common therapeutic treatments. It is therefore important to reproduce the heterogeneity of brain microstructure through mathematical and computational models, that can provide powerful instruments to investigate cancer progression. However, only a few models include a proper mechanical and constitutive description of brain tissue, which instead may be relevant to predict the progression of the pathology and to analyse the reorganization of healthy tissues occurring during tumour growth and, possibly, after surgical resection. Motivated by the need to enrich the description of brain cancer growth through mechanics, in this paper we present a mathematical multiphase model that explicitly includes brain hyperelasticity. We find that our mechanical description allows to evaluate the impact of the growing tumour mass on the surrounding healthy tissue, quantifying the displacements, deformations, and stresses induced by its proliferation. At the same time, the knowledge of the mechanical variables may be used to model the stress-induced inhibition of growth, as well as to properly modify the preferential directions of white matter tracts as a consequence of deformations caused by the tumour. Finally, the simulations of our model are implemented in a personalized framework, which allows to incorporate the realistic brain geometry, the patient-specific diffusion and permeability tensors reconstructed from imaging data and to modify them as a consequence of the mechanical deformation due to cancer growth

Morpho-elasticity of intestinal villi

In the context of morphogenetic processes in animal biology, in particular for those related to soft tissues, a morpho-elastic model for the embryonal development of intestinal villi, is presented. Villi originate from the embryonic development of the epithelial layer in the intestinal mucosa. Since the first stages of development, a bi-dimensional pattern starts to affect the intestinal epithelium, as a consequence of growth and residual stresses inside the tissue. It's from this undulated pattern, that villi will start to elongate. The embryonic mucosa is modeled as a growing thick-walled cylinder, and its mechanical behavior is described using an hyperelastic constitutive model, which also accounts for the anisotropic characteristics of the reinforcing fibers at the micro-structural 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, obtained from different growth scenarios are finally compared with the existing experimental results, showing that the geometrical and mechanical properties of the tissue and the differential growth between epithelium and mesenchyma, can drive the formation of intestinal villi in embryos

Morphogenesis in space offers challenges and opportunities for soft matter and biophysics

Abstract The effects of microgravity on soft matter morphogenesis have been documented in countless experiments, but physical understanding is still lacking in many cases. Here we review how gravity affects shape emergence and pattern formation for both inert matter and living systems of different biological complexities. We highlight the importance of building physical models for understanding the experimental results available. Answering these fundamental questions will not only solve basic scientific problems, but will also enable several industrial applications relevant to space exploration