A theoretical solution for the frictionless rolling contact of cylindrical biphasic articular cartilage layers

Abstract

Previous studies have shown that interstitial fluid pressurization plays an important role in the load support mechanism of articular cartilage under normal step loading. These studies have demonstrated that interstitial fluid pressurization decreases with time if the applied load is maintained constant. In the present study, a theoretical solution is obtained for another common loading of articular cartilage, namely the contact of surfaces in rolling motion. Pure rolling of symmetrical frictionless cylindrical cartilage layers is analyzed under steady state. The linear biphasic model of Mow et al. [J. Biomech. Engng 102, 73-84 (1980)] is used to describe the mechanical response of articular cartilage. The solution of this contact problem reduces to simultaneously solving a set of four integral equations, akin to the dual integral problem of elastic contact. It is found that the solution is dependent on four non-dimensional parameters: Rh = Vb/HAk, W/2 mu b, R/b, and v, where V is the surface velocity, b the cartilage thickness, HA the aggregate modulus, mu the shear modulus, v Poisson's ratio, k the permeability, R the radius of cylindrical surfaces, and W the applied load per unit cylinder length. For Rh << 1, interstitial fluid pressurization is found to be negligible, and all the applied load is supported by the solid collagen-proteoglycan phase of the tissue, thus causing significant cartilage deformation. As Rh increases to a physiological level (Rh approximately 10(4)), interstitial fluid pressurization may support more than 90% of the total applied load, shielding the solid matrix from high effective stresses and reducing matrix strains and deformation. The protective effect of interstitial fluid pressurization is observed to increase with increasing joint congruence (R/b); similarly, as the applied load (W/2 mu b) is increased, a greater proportion of it is supported by the fluid. In degenerative cartilage, Rh may drop by one or more orders of magnitude, primarily as a result of increased permeability. Thus, the protective stress-shielding effect of interstitial fluid pressurization may become less effective in diseased tissue, possibly setting a pathway for further tissue degeneration.