Fibril Stiffening Accounts for Strain-dependent Stiffness of Articular Cartilage in Unconfined Compression

Introduction: Articular cartilage is typically subjected to physiological loading at high strain rates. The mechanical response to this type of load is mainly governed by interactions among three material phases: collagen fibrils, proteoglycan matrix and water containing diffusible ions. Previous model studies, which have adopted the same constitutive laws for the fibrils and the proteoglycans, have experienced difficulties in describing the transient or dynamic response of cartilage. On the other hand, fibril-reinforced poroelastic models that recognize the distinct roles of the fibrillar and proteoglycan matrices have been able to explain several experimentally observed phenomena not previously understood [1,2]. The objective of the present work is to explore the mechanism of the transient response of cartilage in unconfined compression using a fibril reinforced model. The results are supported by experimental data. Methods: In a previous experimental study [3], 3 mm diameter bovine articular cartilage (0.9-1.2mm thick) disks with a layer of subchondral bone (0.3-0.6mm thick) were compressed in unconfined geometry using amplitudes ranging from 12.5-300μm and speeds from 0.5 to 50 μm/s. The compressive modulus of the disks at equilibrium and at the transient peak (the mean nominal axial stress over the mean nominal axial strain) were investigated. The test data used for comparison here were averaged over four samples. In computation, we used a disk radius of 1.5mm and that the thickness of cartilage and bone were taken as 1.1 and 0.4 mm respectively. Since only the overall stiffness of the disks was desired, the homogeneous fibril reinforced model [1] rather than the heterogeneous model [2] was employed in order to reduce computation time. For the cartilage layer, the nonfibrillar and fibrillar matrices were modeled as poroelastic and elastic, respectively. The bone was taken to be elastic. The finite element mesh (axisymmetric) for continuum elements was 14×6 (6 in the axial direction: 4 for the cartilage, 2 for the bone). The material properties used for the cartilage are as follows: Em = 0.26 MPa and νm = 0.42 for the nonfibrillar matrix; the fibrillar modulus Ef = (1600 ×fibrilstrain+3) MPa except when specifically noted otherwise; the hydraulic permeability k = 0.003 exp(M×dilatation) mm/Ns. Two groups of Young’s moduli and Poisson’s ratios are assumed for the bone: 8MPa and 0.35 for the elements adjacent to the cartilage, and 40MPa and 0.25 for the remaining bone elements. These parameters are compatible with those of other studies. The instantaneous modulus was obtained when the fluid was assumed to be trapped; difficulties in convergence were then encountered when a large deformation was applied in one step due to discontinuous elastic properties. Thus the instantaneous modulus for strains greater than 5% was obtained by assuming the same material properties for the bone as for the cartilage. Results and Discussion: The compressive modulus at equilibrium (Fig.1) varies much less with the strain compared to the transient modulus (Fig.2). The predicted transient modulus approximately matches the test data, i.e. stiffening with high strains and high compression speeds (50μm/s). Kinetics of fluid flow is an important factor. For example, at a low compression rate, the disk can appear softer when the strain increases since the fluid dissipates relatively quickly with respect to the rate of matrix deformation. However, as compression further increases, the permeability is reduced, flow becomes more difficult and the disk stiffens, explaining the decrease in transient modulus followed by an increase with the strain (Fig.2, 0.5μm/s). On the other hand, the collagen fibrils stiffen quickly with the flow-dependent lateral expansion, and therefore the transient modulus grows quickly with the strain if the compression speed is high when the fluid has little time to dissipate (Fig.2, 50μm/s), possibly approaching the instantaneous modulus. Thus the transient modulus can first increase, then decrease and again increase with the axial strain for an intermediate compression speed (Fig.2, 5μm/s), depending on the combination of the dissipation speed and fibril stiffening rate. The interplay between fluid kinetics and fibril stiffening is further demonstrated in Figures 3 and 4. If the permeability were constant (M=0) with other conditions remaining unchanged, the transient modulus for 5μm/s would decrease after reaching its peak at about 4% strain (Fig.3). (The initial rise is due to difficulty in initiating the fluid flow in a short time.) However, even if the permeability varies with the matrix dilatation (M=10), but the fibrils do not stiffen with its strain (Ef=constant), the transient modulus will not increase much with the strain, and a higher constant Ef will not result in a stronger increase (Fig.4). In conclusion, fibril stiffening is a main factor that can produce a stiffening transient compressive modulus at higher axial strains. The fibril stiffening, on the other hand, is caused by lateral expansion, which is influenced by fluid dissipation that in turn is affected by compression speeds and amplitudes (in addition to the matrix elasticity and permeability). Thus the transient compressive stiffness does not always increase monotonically with the axial strain, unless compression speeds are high. The ability of cartilage to stiffen under high loading amplitudes and speeds is of physiological importance. Models accounting for these phenomena are helpful to reveal load-bearing mechanisms as well as thresholds for degradation and physical factors involved in guiding chondrocytes to remodel their matrix.