The shear mechanics of enzymatically degraded articular cartilage are predicted by a rigidity percolation model

Abstract

Purpose: An accurate model for the mechanics of cartilage would allow for the prediction of future damage to diseased tissue and would direct the necessary advancements for artificial construct development. Recently it has been shown that orders of magnitude of variation of the shear modulus of articular cartilage are captured by describing the disordered collagen network with a rigidity percolation model. This model suggests that low concentrations of collagen, found near the surface of cartilage, are not sufficiently connected to support a shear load, leading to the mechanical properties in this low concentration regime to be dictated by the background aggrecan gel. This result contrasts with the deeper regions of the tissue where there is a higher concentration of collagen. In this region, the collagen rigidity percolates and the shear modulus of the tissue rapidly increases by several orders of magnitude. This model provides a framework for understanding the effect compositional changes cause to the mechanics of the tissue. The model predicts that the removal of the reinforcing background modulus shifts the rigidity percolation threshold of the collagen concentration, and results in a large decrease in shear modulus for the same collagen concentration. In the surface region of the tissue, as the mechanics are dominated by the aggrecan gel, then the shear modulus of the tissue will greatly decrease when the gel is removed - unlike deep into the tissue where the removal of the aggrecan gel will result in only a small decrease of shear modulus as the collagen network is still sufficiently connected. To test these predictions, we degrade the proteoglycan network of cartilage using the enzyme trypsin. Methods: Cartilage explants (N=24) were harvested from femoral condyles of neonatal calves (Fig 1.A.). The plugs (N=12) were coated by an epoxy surface which protected everywhere except the surface. The samples were then submerged in trypsin-EDTA (0.25%) at 37°C for 300 minutes, which removed proteoglycans as a front from the surface towards the deep zone. The samples were then rinsed with protease inhibitors. These plugs were bisected, and one half was used for measurement of the depth dependent shear modulus and biochemical analysis. The other half was fixed and used for histology and FTIR measurement, then biochemical essays were used to measure proteoglycan and collagen content. We were able to stain the tissue to validate proteoglycan loss, and the FTIR measurement provided a relative concentration of the aggrecan and collagen concentration throughout the tissue. We performed our simulations in Fortran, treating collagen as a random Kagome network with a reinforcing uniform background. We were able to calculate a shear modulus of this network, and then add in the reinforcing background afterwards. Results: The histology staining shows that we have a consistent front progressing with depth (Fig 1.B.). This verifies our FTIR measurements which shows a region of significantly reduced aggrecan concentration, with an unchanged collagen content (Fig 1.D.1.). Our shear measurements showed that the loss of proteoglycans caused an uneven decrease of shear modulus by up to a factor of 5 (Fig 1.D.2.). We then fitted the measurements of both the healthy articular cartilage and the degraded tissue (Fig 1.D.3.), and by scaling the reinforcing background modulus in the mechanical simulations, we can make predictions about the mechanical properties of other tissues for a range of proteoglycan concentrations. Conclusions: The trypsin degradation was able to mostly remove the aggrecan gel but deeper in the tissue there was still a non-zero concentration of aggrecan. Despite the removal of most of the proteoglycan background, the shear modulus of the deep zone was still an order of magnitude larger than that of the healthy surface tissue. This backs the prediction of the model which suggests that at high concentrations of collagen, it is the collagen network which provides most of the tissue’s structural support. This rigidity percolation model captures the mechanics of both healthy and deteriorated articular cartilage. Using this model, full mechanical descriptions of articular cartilage can be described using simple constituent scans of the collagen and proteoglycan content of the tissue. The model can also be used for the diagnosis of early proteoglycan loss through mechanical measurement. Finally, this model provides the understanding to direct the necessary advancements in artificial construct development.