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

Abstract

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.

Characterization of the mechanical properties of resected porcine organ tissue using optical fiber photoelastic polarimetry

“…The experimentally determined range of elastic modulus values using the optical fiber PEP method and QLV fit is 0.1-0.5 KPa for liver, 0.15-0.9 KPa for kidney, and 0.2-8.0 KPa for pancreas. Our results are within the range of previously published elastic modulus values [1,2,[20][21][22][23][24]. Previously reported elastic modulus values vary due to several factors, including the animal model, sample preparation method, and mathematical fit used.…”

“…The general trend in the measurements is as the strain increases the elastic modulus increases. This behavior is commonly observed in viscoelastic materials [2,20]. The experimentally determined range of elastic modulus values using the optical fiber PEP method and QLV fit is 0.1-0.5 KPa for liver, 0.15-0.9 KPa for kidney, and 0.2-8.0 KPa for pancreas.…”

Biochemical and Cellular Determinants of Renal Glomerular Elasticity

“…Decellularized glomeruli and liver slices have greatly reduced Young’s moduli (compression, Fig 4) and shear moduli (liver) [44]. Chelation of extracellular Ca reduces the affinity of integrins for their ligands as well as the affinity of cell-cell adhesions molecules such as cadherins.…”

Stochastic isotropic hyperelastic materials: constitutive calibration and model selection

“…Critically, equations (3.11) and (3.12) imply that µ(a 0 ) follows a Gamma distribution (the maximum entropy distribution) [72,73], 13) where the mean value µ(a 0 ) is obtained at step 1. For the random vector (R 1 (a 0 ), · · · , Rn(a 0 )), applying the constraints [23,25] 15) this vector follows a Dirichlet distribution [41,74], D (ξ 1 (a 0 ), · · · , ξn(a 0 )). Then, every random variable Rp(a 0 ), p = 1, · · · , n, follows a standard Beta distribution [40,41], B(ξp(a 0 ), ψp(a 0 )), with ξp(a 0 ) > 0 and ψp(a 0 ) = n q=1,q =p ξq(a 0 ) > 0 satisfying 17) and is calculated from the mean values obtained at step 1.…”

“…For natural and engineered materials, uncertainties in the experimental observations typically arise from the inherent micro-structural inhomogeneity [13,14], sample-to-sample intrinsic variability, or when elastic data are extracted from viscoelastic mechanical tests [15][16][17][18][19]. For these materials, hyperelastic models based on mean data values constitute a starting point for the development of more complex models.…”