Determining the contribution of glycosaminoglycans to tendon mechanical properties with a modified shear-lag model

Abstract

Tendon has a complex hierarchical structure composed of both a collagenous and a non-collagenous matrix. Despite several studies that have aimed to elucidate the mechanism of load transfer between matrix components, the roles of glycosaminoglycans (GAGs) remain controversial. Thus, this study investigated the elastic properties of tendon using a modified shear-lag model that accounts for the structure and non-linear mechanical response of the GAGs. Unlike prior shear-lag models that are solved either in two dimensions or in axially symmetric geometries, we present a closed-form analytical model for three-dimensional periodic lattices of fibrils linked by GAGs. Using this approach, we show that the non-linear mechanical response of the GAGs leads to a distinct toe region in the stress-strain response of the tendon. The critical strain of the toe region is shown to decrease inversely with fibril length. Furthermore, we identify a characteristic length scale, related to microstructural parameters (e.g. GAG spacing, stiffness, and geometry) over which load is transferred from the GAGs to the fibrils. We show that when the fibril lengths are significantly larger than this length scale, the mechanical properties of the tendon are relatively insensitive to deletion of GAGs. Our results provide a physical explanation for the insensitivity for the mechanical response of tendon to the deletion of GAGs in mature tendons, underscore the importance of fibril length in determining the elastic properties of the tendon, and are in excellent agreement with computationally intensive simulations.

Keywords: Decorin; Extracellular matrix; Fibril length; Modeling; Proteoglycan.

Fig. 1

The shear-lag model consists of fibrils (cyan) interconnected by GAGs (springs). A unit cell (red) with two neighboring fibrils is also shown. The force (F), elongation (Δ) behavior of the GAGs is also shown. Here, for Δ ≤ Δ_c, the stiffness is negligible, while K_G denotes the stiffness when Δ > Δ_c.

Fig. 2

Distribution of the (a) normal and (b) shear strain along the fibril obtained from our modified shear-lag model and from the numerical work of Redaelli et al., 2003. x = 0 and x = L denote the center and the end of the fibril, respectively. The applied strain is ε = 0.05. The maximum normal strains at the point x = 0 from the analytical and numerical methods are 5.61% and 4.93%, respectively while minimum shear strains at point x = L / 2 are 8.05 and 8.30, respectively. (c) Stress-strain curves obtained from our modified shear-lag model and Redaelli et al., 2003. The parameters used in the simulations are l₀ = d_f = 100 nm, R_f = 90 nm, 2L = 100 μm, α = 4, β = 60°, E_f = 2 GPa.

Fig. 3

Comparison between the current model, experiments and FEM analysis (Fessel and Snedeker, 2011) on both intact and 80% GAG depleted rat tail tendon samples. For the intact model, we use the parameters R_f = 90 nm, 2L = 400 μm, α = 6, E_f = 1.0 GPa, β = 60° and ϕ = 0.6 For the GAG depleted model, the spacing between GAGs has increased to d_G = 340 nm.

Fig. 4

Stress-strain curve for tendon with fibril lengths (a) 100 μm and (b) 10000 μm as a function of L_c (we setl₀ = d_f = 100nm).

Fig. 5

Variation of E / E_fφ of the tendon with different fibril lengths at fixed applied strain, ε = 0.1 (we setl₀ = d_f = 100nm).

Fig. 6

The distribution of normal stress along the short and long fibrils for intact and GAG depleted models. The normal stress generated in the short fibrils is smaller compared to the in long fibrils. In addition, most of the GAGs in the long fibrils are unstrained, while for short fibrils all of the GAGs carry significant load and are elongated. As a result, deletion of GAGs in short fibrils results in removal of load-bearing elements and a reduction in the modulus, E. Alternatively, GAG deletion in long fibrils does not result in removal of significant load bearing elements over the lengths of the fibrils.

Fig. 7

Stress-strain curve for tendon with (a) long 2L = 10000 μm and (b) short 2L = 100 μm fibrils. The parameters L_c = 3.1 μm and l₀ = d_f = 100nm are held fixed for all the curves.

Fig. 8

Variation of the tendon Young's modulus for different fibril length as a function of force-free strain, γ_G. The parameters L_c = 3.1 μm and l₀ = d_f = 100nm are held fixed and the applied strain is ε = 0.1.