Squeeze-film lubrication of the human ankle joint with synovial fluid filtrated by articular cartilage with the superficial zone worn out
Introduction
Articular cartilage is a layered medium composed of several structural zones (Lane and Weiss, 1975): superficial, middle and deep. The superficial zone of the normal cartilage is the most distinct and reaches up to 200 μm. Its upper most layer, only about 3 μm thick, is called “lamina splendens” and is made of non-banded, randomly arranged fine collagen filaments 4–12 nm in diameter. Below this layer there are sheets of a well-identified period, each sheet with predominantly parallel, banded collagen fibrils, their diameter increasing with depth. They remind of industrially produced angle-ply laminates. The collagen fibres of the middle zone are randomly oriented and homogeneously dispersed. In the deep zone, the fibres form larger, radially oriented fibre bundles that insert into the calcified zone across the tidemark and furcate to anchor the tissue to the subchondral bone. Proteoglycan occupies the interfibrilar space of the tissue and its concentration increases from the surface to a maximum in the middle zone and then diminishes again. Within each zone, regions of specific cellular organization exist. This composite solid matrix is swollen by water in which a variety of mobile electrolytes are distributed.
The collagen fibres limit the swelling of the proteoglycan gel and experience tensile stresses giving the cartilage its tensile stiffness. Kempson (1973) found that the normal articular cartilage shows large differences in tensile stiffness and strength, depending on the place and orientation. The tensile stiffness and strength are low in the middle zone, independent of orientation parallel to the surface. However, they are high in the superficial zone. A decrease in the tensile stiffness, as well as disruption and disorganization of the collagen network of the superficial zone, have been reported in a canine model of experimental osteoarthritis generated by transection of the anterior cruciate ligament of the knee (Stockwell et al., 1983; Guilak et al., 1994).
It is the surface layer that is much important for synovial fluid filtration by cartilage. For the normal cartilage the surface layer is formed by the intact superficial zone that has mechanical properties quite different from the surface layer of a cartilage in an early stage of osteoarthritis. In the latter case, the intact superficial zone is either already disrupted (with a decreased stiffness in tension when compared to its originally higher value) or even worn out so that the surface layer may be considered with the same stiffness in tension as the bulk material. Thus, an arthritic cartilage might be roughly considered homogeneous throughout, as it is the case in the models analysed so far (Hou et al., 1992; Jin et al., 1992; Hlaváček (1993a), Hlavác̀ek (1993b), Hlaváèek (1995); Hlaváček and Novák, 1995; Ateshian et al (1994), Ateshian and Wang (1995)). However, the homogeneous model can hardly describe lubrication for the intact articular cartilage.
The present paper gives an analysis of the squeeze-film lubrication of the human ankle joint under stationary step loading in normal approach motion and plane strain with the articular cartilage biphasic and homogeneous, i.e. with the superficial zone already disrupted or worn out. As distinct from Hlaváček and Novák (1995) where the fluid-film profiles were obtained for axial symmetry and for very low loading only (calculations for physiologic loading broke), in the present paper the inverse method of elastohydrodynamic lubrication theory (Blok, 1964) is used, with the use of the dry contact pressure profile instead of the fluid pressure of the lubricated contact, in order to obtain the film thickness profile even for high, physiologic loading.
Section snippets
Geometric model of the human ankle joint
The human ankle joint can be simply taken as cylindrical (Medley et al., 1983), enabling rotation in the saggital plane only. The joint is represented by two rigid circular cylinders in the inner contact (a cylinder encased in a cylindrical cavity), coated with thin deformable layers — cartilages of constant thickness H under plane strain conditions (Fig. 1). Some joints show wider, some deeper sockets than the corresponding joint heads. For the effective radius R of the cylindrical contact (in
Results
Eq. (14) gives for the parameters (22) with the value of , i.e. we set a′=a. Thus, the contact spreads over the whole tibial arc for all three values of R and p(x1) in Eq. (13a) approximates well the SF pressure in the whole contact (including the contact edges).
Fig. 2 shows the contact pressure distributions for , and ∞ from Eq. (13a). Apparently, the effect of R is low when a′=a. Fig. 3 shows the central film thickness distributions for R→∞ and w=0,2,3, as
Discussion
The present model indicates that for normal approach motion of the articular surfaces in the loaded human ankle joint with the superficial zone of the articular cartilage worn out a small jump in the fluid pressure at the articular surface occurs at the moment of a step-load application. This jump (by two orders lower than the fluid pressure) makes water imbibe intensively into the cartilage and the SF film is quickly depleted in the majority of the contact. Within about half a second, SF turns
Acknowledgments
This study has been partly sponsored by the Grant Agency of the Czech Republic (103/00/0008).
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