Mathematical Modeling of Synovial Fluids Flow

The aim of this article is to model the synovial fluid flow under specific conditions where it can be described as an incompressible viscous non-Newtonian fluid. We propose two different models for viscosity to capture the shear-thinning property and the viscosity dependence on the concentration of the hyaluronan present in the synovial fluid. First we determine the unknown material parameters of the models by fit to the experimental data. Then we numerically solve the two-dimensional prototypical example and compare the solutions of both models. The model with shear-thinning index directly dependent on the concentration of hyaluronan seems to be more appropriate. Introduction Human moveable joints have coefficient of friction much smaller than any man-made machine, they even can be exposed to extreme shock loads of many tons as for example in gymnastic performance. This efficiency is due to perfect combination of amazing materials elastic cartilage and viscoelastic synovial fluid. Not always this combination remains perfect, more than 20% of population older than 45 years is affected by some kind of arthritis, even worse sometimes patient’s joint has to be replaced by artificial one. It is obvious that for a successful treatment of these world wide illnesses we need sufficient information about biochemical, physical and mechanical properties and all the processes involved. In this paper we study most important biochemical and rheological properties of synovial fluid and we simulate the synovial fluid flow under some special physical conditions. Composition and rheological properties of synovial fluid Synovial fluid is a biological fluid filling the synovial joint cavity several micrometers thick layer between the interstitial cartilages. It supports the joint by high effective cartilage lubrication and it acts as a transport medium of nutrients/metabolic waste to/from avascular cartilage [Fung, 1993]. Its composition and visual characteristic are strongly dependent on the state of health of the patient, without any disorders the synovial fluid has egg white like consistency and colour. Synovial fluid is secreted to the cavity by its inner membrane called synovium [Coleman et al., 1997]. The main component of synovial fluid is an ultrafiltrate of the blood plasma devoid of high-molecular proteins, blood cells and aggressors. While the most important component is actively added lubricin called hyaluronan/hyaluronic acid [Voet and Voet, 2004]. Several studies showed that the viscoelastic properties of synovial fluid are due to presence of hyaluronan [Laurent et al., 1995; Ogston and Stanier, 1953]. Hyaluronan is natively present in synovial fluid in relatively high concentrations 2.5 − 4.7 mg/ml [Decker et al., 1957], chemically it belongs to the class of non-sulphated glycosaminoglycans. The molecule comprises a long (up to one half of the size of a blood platelet) unbranched polymer of repeating dissacharide units (namely N-acetylglucosamine linked to D-glucuronic acid via repeating β-bond). In a solution the hyaluronate natively adopts α-helix conformation and forms an expanded coiled configuration with radius of ≈ 100 nm [Voet and Voet, 2004; Laurent et al., 1996]. The size of polymeration is very important because the volume of the hyaluronan random coils plays a significant role in the viscoelastic properties of the synovial fluid [Coleman et al., 2000]. Mechanically speaking if the solution is exposed to the fast oscillatory deformations, the 32 WDS'08 Proceedings of Contributed Papers, Part III, 32–37, 2008. ISBN 978-80-7378-067-8 © MATFYZPRESS