Mathematical Models and Numerical Methods for Analysis of Mechanical and Chemical Loading in Articular Cartilage

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Title: Mathematical Models and Numerical Methods for Analysis of Mechanical and Chemical Loading in Articular Cartilage
Author: Schugart, Richard Charles
Advisors: Mansoor Haider, Committee Chair
Farshid Guilak, Committee Member
Ralph Smith, Committee Member
Sharon Lubkin, Committee Member
Abstract: Articular cartilage is the primary load-bearing soft tissue in joints such as the knee, shoulder, and hip. 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 the normal and osteoarthritic mechanical and physico-chemical states in the tissue. In this dissertation, three studies were conducted involving the development of numerical methods and mathematical models pertaining to the cells and extracellular matrix of articular cartilage. In the first investigation, an accelerated numerical method for the continuous spectrum biphasic poroviscoelastic model of articular cartilage deformation was developed. A common constitutive law for modeling the intrinsic dissipation in cartilage extracellular matrix is the theory of quasi-linear viscoelasticity, in which the solid matrix stress depends on the strain rate via a hereditary integral with a continuous relaxation spectrum. The proposed numerical method was based on an alternate formulation of the viscoelastic law that was implemented using Gaussian quadrature time integration in combination with quadratic interpolation of the strain history. The accuracy and cost of the numerical method were evaluated and compared to a theoretical solution using a finite difference implementation of the 1-D confined compression stress-relaxation problem. Comparisons were also made between the accelerated numerical method and a discrete spectrum method, which is commonly used to overcome the cost of evaluating the hereditary stress-strain integral via an exponential series approximation of the continuous spectrum relaxation function. The second study consisted of the formulation and application of a triphasic me-chano-chemical model to analyze osmotic loading experiments for an isolated articular cartilage cell. The cell was modeled as a charged-hydrated mixture of three phases (solid, fluid, ionic). The model was formulated under the hypothesis that the cell membrane was permeable to both water and ions. Under osmotic loading, isolated cartilage cells exhibit a passive volumetric response, which is that of an ideal osmometer. The triphasic mechano-chemical model was analyzed for consistency with the Boyle-van't Hoff law, which represents the ideal response. The resulting triphasic model suggested that Donnan osmotic pressure alone was not sufficient to balance the elastic stress at equilibrium. A non-zero chemical-expansion stress, which is a measure of the charge-to-charge repulsive forces within the cell, was required to balance the elastic stresses. Since the existence of an intracellular chemical-expansion stress is not well established, it was hypothesized that the triphasic 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 used for analysis of 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 and the cell was modeled as an ideal osmometer. The PCM was modeled as a triphasic continuum mixture with a fixed charge density arising from the negatively-charged proteoglycans that are characteristic of cartilage extracellular matrix. Both parametric and asymptotic analyses were conducted to compare cell, PCM, and chondron deformation under osmotic loading.
Date: 2005-06-27
Degree: PhD
Discipline: Mathematics

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