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Register a new userThird- and fourth-order elasticity of biological soft tissues
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In the theory of weakly non-linear elasticity, Hamilton et al. [J. Acoust. Soc. Am. textbf{116} (2004) 41] identified $W = mu I_2 + (A/3)I_3 + D I_2^2$ as the fourth-order expansion of the strain-energy density for incompressible isotropic solids. Subsequently, much effort focused on theoretical and experimental developments linked to this expression in order to inform the modeling of gels and soft biological tissues. However, while many soft tissues can be treated as incompressible, they are not in general isotropic, and their anisotropy is associated with the presence of oriented collagen fiber bundles. Here the expansion of $W$ is carried up to fourth-order in the case where there exists one family of parallel fibers in the tissue. The results are then applied to acoustoelasticity, with a view to determining the second- and third-order nonlinear constants by employing small-amplitude transverse waves propagating in a deformed soft tissue.
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Surface tension governed by differential adhesion can drive fluid particle mixtures to sort into separate regions, i.e., demix. Does the same phenomenon occur in confluent biological tissues? We begin to answer this question for epithelial monolayers with a combination of theory via a vertex model and experiments on keratinocyte monolayers. Vertex models are distinct from particle models in that the interactions between the cells are shape-based, as opposed to distance-dependent. We investigate whether a disparity in cell shape or size alone is sufficient to drive demixing in bidisperse vertex model fluid mixtures. Surprisingly, we observe that both types of bidisperse systems robustly mix on large lengthscales. On the other hand, shape disparity generates slight demixing over a few cell diameters, a phenomenon we term micro-demixing. This result can be understood by examining the differential energy barriers for neighbor exchanges (T1 transitions). Experiments with mixtures of wild-type and E-cadherin-deficient keratinocytes on a substrate are consistent with the predicted phenomenon of micro-demixing, which biology may exploit to create subtle patterning. The robustness of mixing at large scales, however, suggests that despite some differences in cell shape and size, progenitor cells can readily mix throughout a developing tissue until acquiring means of recognizing cells of different types.
The present work presents a density-functional microscopic model of soft biological tissue. The model was based on a prototype molecular structure from experimentally resolved collagen peptide residues and water clusters and has the objective to capture some well-known experimental features of soft tissues. It was obtained the optimized geometry, binding and coupling energies and dipole moments. The results concerning the stability of the confined water clusters, the water-water and water-collagen interactions within the CLBM framework were successfully correlated to some important trends observed experimentally in inflammatory tissues.
When a block made of an elastomer is subjected to large shear, its surface remains flat. When a block of biological soft tissue is subjected to large shear, it is likely that its surface in the plane of shear will buckle (apparition of wrinkles). One factor that distinguishes soft tissues from rubber-like solids is the presence -- sometimes visible to the naked eye -- of oriented collagen fibre bundles, which are stiffer than the elastin matrix into which they are embedded but are nonetheless flexible and extensible. Here we show that the simplest model of isotropic nonlinear elasticity, namely the incompressible neo-Hookean model, suffers surface instability in shear only at tremendous amounts of shear, i.e., above 3.09, which corresponds to a 72 degrees angle of shear. Next we incorporate a family of parallel fibres in the model and show that the resulting solid can be either reinforced or strongly weakened with respect to surface instability, depending on the angle between the fibres and the direction of shear, and depending on the ratio E/mu between the stiffness of the fibres and that of the matrix. For this ratio we use values compatible with experimental data on soft tissues. Broadly speaking, we find that the surface becomes rapidly unstable when the shear takes place against the fibres, and that as E/mu increases, so does the sector of angles where early instability is expected to occur.
Acousto-elasticity is concerned with the propagation of small-amplitude waves in deformed solids. Results previously established for the incremental elastodynamics of exact non-linear elasticity are useful for the determination of third- and fourth-order elastic constants, especially in the case of incompressible isotropic soft solids, where the expressions are particularly simple. Specifically, it is simply a matter of expanding the expression for $rho v^2$, where $rho$ is the mass density and v the wave speed, in terms of the elongation $e$ of a block subject to a uniaxial tension. The analysis shows that in the resulting expression: $rho v^2 = a + be + ce^2$, say, $a$ depends linearly on $mu$; $b$ on $mu$ and $A$; and $c$ on $mu$, $A$, and $D$, the respective second-, third, and fourth-order constants of incompressible elasticity, for bulk shear waves and for surface waves.
The rheology of biological tissues is important for their function, and we would like to better understand how single cells control global tissue properties such as tissue fluidity. A confluent tissue can fluidize when cells diffuse by executing a series of cell rearrangements, or T1 transitions. In a disordered 2D vertex model, the tissue fluidizes when the T1 energy barriers disappear as the target shape index approaches a critical value ($s^*_{0} sim 3.81$), and the shear modulus describing the linear response also vanishes at this same critical point. However, the ordered ground states of 2D vertex models become linearly unstable at a lower value of the target shape index (3.72) [1,2]. We investigate whether the ground states of the 2D vertex model are fluid-like or solid-like between 3.72 and 3.81 $-$ does the equation of state for these systems have two branches, like glassy particulate matter, or only one? Using four-cell and many-cell numerical simulations, we demonstrate that for a hexagonal ground state, T1 energy barriers only vanish at $sim 3.81$, indicating that ordered systems have the same critical point as disordered systems. We also develop a simple geometric argument that correctly predicts how non-linear stabilization disappears at $s^*_{0}$ in ordered systems.
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