A molecular theory of cartilage viscoelasticity

Recent work on the subject of cartilage mechanics has begun to focus on the relationship between the microscopic structure of cartilage and its macroscopic mechanical properties (Bader et al., Biochem. Biophys. Acta, 1116 (1992) 147-154; Buschmann, PhD Thesis, Massachusetts Institute of Technology, 1992; Kovach, Biophys. Chem., 53 (1995) 181-187; Lai et al., J. Biochem. Eng., 113 (1991) 245-248; Armstrong and Mow, J. Bone Jt. Surg., 64A (1982) 88; Jackson and James, Biorheology, 19 (1982) 317-330). This paper reviews recent theoretical developments and presents a comprehensive explanation of the viscoelastic properties of cartilage in terms of molecular structure. In doing this, a closed form hybrid solution to the non-linear, cylindrical Poisson-Boltzmann equation is developed to describe the charge-dependent component of the equilibrium elasticity arising from polysaccharide charge (Benham, J. Chem. Phys., 79 (4) (1983) 1969-1973; Einevoll and Hemmer, J. Phys. Chem., 89 (1) (1988) 474-484; Fixman, J. Chem. Phys., 70 (11) (1979) 4995-5001; Ramanathan and Woodburg, J. Chem. Phys., 82 (3) (1985) 1482-1491; Wennerstrom et al., J. Chem. Phys., 76 (9) (1982) 4665-4670). This solution agrees with numerical solutions found in the literature (Buschmann, PhD Thesis, Massachusetts Institute of Technology, 1992). The charge-independent, entropic contribution to the equilibrium elasticity is explained in a manner similar to that recently presented for concentrated proteoglycan solution (Kovach, Biophys. Chem., 53 (1995) 181-187). This approach exploits a lattice model of the solution, subject to a Bragg-Williams type approximation to derive the volume dependence of polysaccharide configuration entropy (Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY, 1953; Huggins, Some properties of Solutions of Long-chain Compounds, 1941, pp. 151-157; Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, Oxford, 1971). Together, these two contributions accurately reproduce the experimentally determined osmotic pressure of cartilage as previously determined by Maroudas (Maroudas and Bannon, Biorheology, 18 (1981) 619-632). The time-dependent, or creep, phenomena which cartilage exhibits when subject to mechanical load is explained in terms of frictional drag on the polysaccharide chain monomers in terms of a Kirkwood-Riseman type model (Kirkwood and Riseman, J. Chem. Phys., 16 (6) (1948) 573-579). This approach is shown to accurately predict the hydraulic permeability of cartilage as previously determined by Maroudas (Madouras, Ann. Rheum. Dis., 34 (suppl. 3) (1975) 77). By use of a quasi-static approximation (neglecting inertial effects) the time-dependent response to a uniform compressive force is determined and also found to be in good agreement with experimental values from the literature.