A comparison of hyperelastic constitutive models applicable to brain and fat tissues

In some soft biological structures such as brain and fat tissues, strong experimental evidence suggests that the shear modulus increases significantly under increasing compressive strain, but not under tensile strain, while the apparent Young’s elastic modulus increases or remains almost constant when compressive strain increases. These tissues also exhibit a predominantly isotropic, incompressible behaviour. Our aim is to capture these seemingly contradictory mechanical behaviours, both qualitatively and quantitatively, within the framework of finite elasticity, by modelling a soft tissue as a homogeneous, isotropic, incompressible, hyperelastic material and comparing our results with available experimental data. Our analysis reveals that Fung and Gent models, which are typically used to model soft tissues, are inadequate for the modelling of brain or fat under combined stretch and shear, and so are the classical neo-Hookean and Mooney-Rivlin models used for elastomers. However, a sub-class of Ogden hyperelastic models are found to be in excellent agreement with the experiments. Our findings provide explicit models suitable for integration in large-scale finite element computations

Emergence of tissue-like mechanics from fibrous networks confined by close-packed cells

The viscoelasticity of the crosslinked semiflexible polymer networks—such as the internal cytoskeleton and the extracellular matrix—that provide shape and mechanical resistance against deformation is assumed to dominate tissue mechanics. However, the mechanical responses of soft tissues and semiflexible polymer gels differ in many respects. Tissues stiffen in compression but not in extension1,2,3,4,5, whereas semiflexible polymer networks soften in compression and stiffen in extension6,7. In shear deformation, semiflexible polymer gels stiffen with increasing strain, but tissues do not1,2,3,4,5,6,7,8. Here we use multiple experimental systems and a theoretical model to show that a combination of nonlinear polymer network elasticity and particle (cell) inclusions is essential to mimic tissue mechanics that cannot be reproduced by either biopolymer networks or colloidal particle systems alone. Tissue rheology emerges from an interplay between strain-stiffening polymer networks and volume-conserving cells within them. Polymer networks that soften in compression but stiffen in extension can be converted to materials that stiffen in compression but not in extension by including within the network either cells or inert particles to restrict the relaxation modes of the fibrous networks that surround them. Particle inclusions also suppress stiffening in shear deformation; when the particle volume fraction is low, they have little effect on the elasticity of the polymer networks. However, as the particles become more closely packed, the material switches from compression softening to compression stiffening. The emergence of an elastic response in these composite materials has implications for how tissue stiffness is altered in disease and can lead to cellular dysfunction9,10,11. Additionally, the findings could be used in the design of biomaterials with physiologically relevant mechanical properties

Normal and Fibrotic Rat Livers Demonstrate Shear Strain Softening and Compression Stiffening: A Model for Soft Tissue Mechanics.

Tissues including liver stiffen and acquire more extracellular matrix with fibrosis. The relationship between matrix content and stiffness, however, is non-linear, and stiffness is only one component of tissue mechanics. The mechanical response of tissues such as liver to physiological stresses is not well described, and models of tissue mechanics are limited. To better understand the mechanics of the normal and fibrotic rat liver, we carried out a series of studies using parallel plate rheometry, measuring the response to compressive, extensional, and shear strains. We found that the shear storage and loss moduli G' and G" and the apparent Young's moduli measured by uniaxial strain orthogonal to the shear direction increased markedly with both progressive fibrosis and increasing compression, that livers shear strain softened, and that significant increases in shear modulus with compressional stress occurred within a range consistent with increased sinusoidal pressures in liver disease. Proteoglycan content and integrin-matrix interactions were significant determinants of liver mechanics, particularly in compression. We propose a new non-linear constitutive model of the liver. A key feature of this model is that, while it assumes overall liver incompressibility, it takes into account water flow and solid phase compressibility. In sum, we report a detailed study of non-linear liver mechanics under physiological strains in the normal state, early fibrosis, and late fibrosis. We propose a constitutive model that captures compression stiffening, tension softening, and shear softening, and can be understood in terms of the cellular and matrix components of the liver

Normal and fibrotic livers demonstrate strain softening and compression stiffening.

<p>(A-C) G' was measured for normal and fibrotic (after 2 and 6 weeks of CCl₄) livers by shear rheometry under increasing strain and under variable degrees of compression ranging from 0–25%. Note that the y-axes scales are the same and that G' values increase significantly as fibrosis progresses. Curves are mean +/- SD for 3–5 livers tested for each condition. (D) Normal liver subjected to three rounds of shear rheometry, each round with increasing, then decreasing strain. Livers demonstrated no evidence of significant tissue damage due to measurements. Representative liver of 3 tested is shown. (See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0146588#pone.0146588.s003" target="_blank">S3 Fig</a> for mean curves +/- SD for all three livers tested.)</p

The impact of tissue manipulations on liver mechanics.

<p>Normal livers were subject to the manipulations indicated, and then G' was measured under various strains. (A) G' as a function of compression for treated livers. (B) Strain sweeps of normal liver, disintegrin-treated, amylase-treated, and permeabilized livers at 0% compression. Key as in (A). (C) Strain sweeps of decellularized normal and fibrotic (2 weeks CCl₄) livers at 0% compression, compared to a curve for normal liver (the same as shown in (B)). Note the different shape of the curves for decellularized compared to normal liver. (D) G' measured under 2% oscillatory shear for livers at either 0% or 25% compression. G’ values are after 120 s of relaxation. By two-way ANOVA for the data plotted in (A), G’ for amylase treated livers was significantly reduced from normal and permeabilized livers (brown *, p≤0.02). Disintegrin treatment significantly reduced G’ of normal liver (green *, p = 0) and G’ of decellularized livers was significantly reduced from normal and all other manipulated livers (green *, p = 0). Two-way ANOVA of the data in (D) showed that G’ compared between 0% and 25% compression was significantly different only for normal livers (blue ***, p≤0.001) and permeabilized livers (green ***, p = 0). G’ values at 25% compression were significantly different between normal livers and α-Amylase (*, p≤0.05), VLO4-treated (brown **, p = 0.059), and decellularized livers (brown ***, p = 0). For all graphs, data represent the mean of 3 independent livers +/- SD per condition.</p

The relationship between G' and E in compression.

<p><b>(</b>A) Normal and fibrotic (2 and 6 weeks of CCl₄) livers underwent shear rheometry with measurement of G’ under varying degrees of compression. G’ shown here was taken at 120 s from the start of compression, approximating equilibrium values after viscous effects have largely dissipated. (B) E was determined by calculating the slope from the stress-strain curve, where stress (calculated from the normal force) was taken at 120 s from the start of compression. The slope was determined in between two neighboring points – 0 and 10, 10 and 15, 15 and 20, 20 and 25 for compression. (C) G' and E for normal livers were calculated as described above in both tension and compression and normalized with respect to initial G’ and E values at no compression. For all cases, 3–5 livers were analyzed for each condition and curves represent the mean +/- SD. By two-way ANOVA, G’ for both 2 and 6 week CCl₄ livers are significantly higher than for normal livers (p≤ 0.002) pink *; E for 6 week CCl₄ livers is significantly higher than for normal and 2 weeks CCl₄ livers (p≤ 0.02) blue *.</p

G’ increases linearly with compressional stress, while stress increases nonlinearly with compressional strain for normal and fibrotic liver.

<p>Normal and fibrotic (2 and 6 weeks of CCl₄) livers were subjected to shear rheometry under various degrees of compression. Compressional stress was calculated in mm Hg and plotted against G' or compressional strain (% compression). (A) G' vs. compressional stress, showing a nearly linear relationship between the two conditions in both normal and fibrotic livers. Lines were fit to each curve (in red) and are shown in the graph. (B) Compressional stress vs. compressional strain, shown for normal and fibrotic livers. G’ and compressional stress values are after 120 s of relaxation. Curves reflect mean +/- SD for 3–5 livers per condition.</p

Schematic of the model.

<p>A) Schematic for liver tissue, consisting of cells and ECM. B) When under compression, cells maintain the same volume but begin to contact each other and generate strong resistance, resulting in compression stiffening. C) We propose that there are two kinds of cell-ECM connections: strong connections that sustain large loads and re-formable connections that break under large loads and re-form when there is no external load. When under shear, the re-formable connections break, which results in shear softening.</p

Rheometry methods.

<p>Liver samples were cut to a diameter of 20 mm and placed on a parallel plate rheometer with an upper platen of 25 mm. Tension and compression were generated by applying force in a direction perpendicular to the sample. Shear forces were applied by rotating the bottom plate in a direction parallel to the sample. Values derived from compression and tension studies were corrected to account for the difference in size between the sample and the top platen, and for narrowing at the waist of the sample in tension (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0146588#sec002" target="_blank">methods</a>).</p