Mathematical Modelling of the Biomechanical Properties of Articular Cartilage

Articular cartilage is the translucent, heterogeneous three-component biological load processing gel that overlays the end of the articulating bones of mammalian joints. Normally, healthy intact articular cartilage performs two biomechanical functions very effectively. These are (i) redistribution of stresses due to loads acting on the joint; (ii) act as a near-frictionless interface between contacting bone ends. These principal functions are enabled by its highly elastic properties. Under normal physiological conditions, these essential biomechanical functions are provided over the lifetime of a mammalian joint with little or no degenerative changes. However, certain levels of physiological and traumatic loads and degenerative processes induced by activities such as running, walking, extreme sport, and aging can alter the composition and structure of the tissue, leading to changes in its biomechanical properties. This, inturn, influences its functional characteristics. The most common degenerative change in articular cartilage is osteoarthritis and the management and treatment of this disease is pivotal to all research targeted toward articular cartilage. Several scientific groups around the world have developed models of articular cartilage to predict its fundamental and functional responses to load and altered biochemical conditions through both in vivo and in vitro studies. The most predominant of these models are the biphasic and triphasic models, which are based on the conceptualisation of articular cartilage as a dispersed mixture of its three main components namely collagen fibrils proteoglycan aggregates and water. The triphasic model is an extension of the biphasic model and incorporates swelling as a separate identifiable component of the tissue's biomechanical response. While these models are capable of predicting the elastic and viscoelastic behaviour and certain aspects of the swelling characteristics of articular cartilage, they are incapable of accounting for its short-term responses where the fluid component is the main carrier of the applied pressure. The hydrostatic and swelling components of the fluid content determine the manner of stress-sharing and hence transient load processing within the matrix as stress is transmitted to the underlying structure. Furthermore, the understanding of the nature of this stress-sharing between fluid and solid components of the tissue is fundamental to the comprehension of the nature of degeneration and its biomechanical consequence in the function of the articulating joint. The inability of the biphasic and triphasic theories to predict, in accordance with experimental results, the transient behaviour of the loaded matrix fluid requires a more representative model. This imperative therefore forms the basis for the research work presented in this thesis. In this thesis, a new mathematical model of articular cartilage load carriage is presented which can predict the transient load-induced responses. The model is based on a continuum framework invoking the principle of mechanical consolidation of fluid-saturated, swollen porous elastic materials. The cartilage matrix is conceptualised as a heterogeneous anisotropic fluid-saturated porous material in which its solid component responds to load as a hyperelastic material and whose interaction with the swelling component produces a partially distributed time-varying permeability. In accordance with the principle of consolidation, a phenomenological approach is adopted for developing both analogue/engineering models and mathematical models for the tissue. The models are then used to predict both bulk matrix responses and the properties of the hypothetical layers of the tissue when subjected to physiological loading conditions. Ultimately, the generalized mathematical model is used to analyse the effect of superficial layer laceration on the stress-processing or stress-sharing characteristic of normal healthy articular cartilage. Finally, predicted results are shown to compare with experimental data demonstrating that the new models for swelling deformation, the hyperelastic law for solid skeletal structure and the distributed, time-dependent permeability are representative of the articular cartilage.