Mathematical models and numerical methods for analysis of mechanical and chemical loading in articular cartilage

Articular cartilage is the primary load-bearing soft tissue in diarthroidal joints. Multiphasic continuum mixture models have been used to describe the relative contribution of effects due to solid, fluid, and ionic phases in cartilage. This research is motivated by the need to quantify differences between normal and osteoarthritic mechanical and physico-chemical states in cartilage. In this dissertation, three studies were conducted involving the development of numerical methods and mathematical models pertaining to the cell and extracellular matrix (ECM) of articular cartilage.; In the first investigation, an accelerated numerical method for the continuous spectrum biphasic poroviscoelastic model of cartilage was developed. The method was based on an alternate formulation of the quasi-linear viscoelastic law, a model for intrinsic dissipation in cartilage ECM in which the solid matrix stress depends on the strain rate via a hereditary integral with a continuous relaxation spectrum. It was implemented using quadrature integration in combination with interpolation of the strain history. The accuracy and cost of the method were compared to a theoretical solution of the 1-D confined compression stress-relaxation problem.; The second study consisted of the formulation and application of a triphasic mechano-chemical model to analyze osmotic loading experiments for an isolated cartilage cell. The model was formulated under the hypothesis that the cell membrane was permeable to both water and ions. A non-zero chemical-expansion stress was required to balance the elastic stress. Since the existence of an intracellular chemical-expansion stress is not well established, it was hypothesized that the cell model should be modified to include a selectively permeable membrane.; The third investigation consisted of the formulation and application of a mechano-chemical model to analyze osmotic loading experiments for an isolated chondron. The chondron is comprised of a cartilage cell and its encapsulating pericellular matrix (PCM). In the chondron, the cell membrane was assumed to be permeable to water, but impermeable to ions. Parametric and asymptotic analyses were conducted to compare cell, PCM, and chondron deformation under osmotic loading.