3
On the friction coefficient of synovial fluid in knee joint
ON THE FRICTION COEFFICIENT OF SYNOVIAL FLUID IN KNEE JOINT
Valeria MOSNEGUTU1, Veturia CHIROIU1, Rodica IOAN1
1 Institute of Solid Mechanics of the Romanian Academy
C-tin Mille Street, No.15, 010141 Bucharest 1, ROMANIA
E-mail: , http://www.imsar.ro
In this paper the friction coefficient for synovial fluid is estimated. The synovial fluid of normal joints is a biological lubricant, which provides low-friction and low-wear to articular cartilage surfaces. We show that the frictional coefficient is lower than the frictional coefficient of ice to ice.
Keywords: synovial fluid, friction coefficient.
1. INTRODUCTION
From medical dictionary, the synovial fluid is a clear thixotropic fluid, the main function of which being to serve as a lubricant in a joint. The fluid forms a thin microscopic layer of fluid at subatmospheric pressure of about 50at the surface of cartilage, filling all empty spaces of joint (Fig. 1). So, the synovial fluid of joints is a natural lubricant, providing low-friction and low-wear in cartilage surfaces. This is achieved through contributions of proteoglycan 4 (PRG4) at a concentration 0.05-0.35 [mg/ml], hyaluronic acid (HA) at a concentration 1-4 [mg/ml], and surface active phospholipids (SAPL) at a concentration 0.1 [mg/ml].
These lubricants are secreted by chondrocytes in articular cartilage and synoviocytes in synovium, and concentrated in the synovial space by the semi-permeable synovial membrane. A deficiency in this lubricating system may contribute to the erosion of articulating cartilage surfaces in conditions of disease [1].
It is known that the articular cartilage is its primary bearing material and it approves important frictional properties which cannot be explained by classical lubrication theories, despite the presence of synovial fluid in the joint which had led to early assumptions of a fluid film lubrication mechanism [3]-[6].
In biotribology researches of diarthrodial joints [7]-[10] it is admitted that pressurization of the cartilage interstitial fluid may contribute predominantly to reducing the friction coefficient at the contact interface of its articular layers. It is also known that there exists a Donnan osmotic pressure difference between cartilage and its external environment, due to the negatively charged proteoglycans enmeshed within the collagen matrix [11]-[12]. Therefore, the role of Donnan osmotic pressure in the frictional response of cartilage can be significant.
Another phenomenon is the disparity between the tensile and compressive moduli of articular cartilage which is acting to enhance the magnitude of interstitial fluid pressurization in compression [13]. This tension-compression nonlinearity stems from the fibrillar collagenous nature of the solid matrix of articular cartilage, with tensile equilibrium moduli specifically ranging from 5 to 40MPa and compressive equilibrium moduli ranging from 0.1 to 1MPa. The Ateshian’s paper has incorporated both osmotic effects and tension-compression nonlinearity of the solid matrix into the formulation of a model to predict the frictional response of articular cartilage.
Running up from the Ateshian’s model [2], we estimate, in this paper, the friction coefficient in human synovial fluid. The model is based on the theory of mixtures for soft hydrated charged tissues and the mixture consists from matrix solid, water, cations and anions, saturated and each phase of the mixture. We develop a human knee joint model with 12 freedom degree for later determination of the friction coefficient. Considering the tibia-femoral configuration with 3 degree of freedom in translation, and the patella-femoral configuration with three 3 degree of freedom in translation, we calculate the shear deformation. Finally, we estimate the static friction coefficient for synovial fluid.
Fig. 1. Cut section view of knee joint.
2. THE MODEL
We developed a mathematical model of the knee joint that describes motion in 12 generalized coordinates as a function of the externally motion [14], [15]. The model is based on the patellar track geometry experimental data. The surface of the patellar track is modeled by using the n-ellipsoid model.
The dynamical equations of the knee motion are:
, / (1)where is the mass matrix, is a vector containing the Coriolis and centrifugal forces and torques arising from the motion of the thigh, is a vector of forces applied by two muscles (the quadriceps tendon and muscle), is a matrix describing the moment arms of applied muscle forces, is a vector containing the forces applied by four ligaments and two tendons, is a matrix describing the moment arms of the knee ligament and tendon forces, and is an vector of external torques applied at the joints. It is supposed that the patellar tendon is inextensible and the interpenetration between the boundaries of the patellar femur can be neglected. These two assumptions define three holonomic constraints for movement of patella on the femur. These three constraints can be combined with the six forces and moment equilibrium equations for the patella, to yield a set of six non-linear algebraic equations for patella-femoral mechanics. The set of six non-linear algebric equations are:
, , / (2)where it is an vector of the force applied by the quadriceps tendon to the patella, is Kroneker delta , and the coefficients and are experimentally found. From (1) and (2) we calculate the unknowns, namely the vector of generalized coordinates, .
The expression of these coordinates is obtained by using the linear equivalent method.
/ (3)The constants are conforming to the associate inverse problem.
Ateshian’s paper [2] gives the constitutive law for synovial fluid
, / (4)where the constants and are experimentally determined, and is the shear strain. This law is represented in Fig. 2, for different bathing solution salt concentrations. Because the law (4) is an elastic one, we intend to incorporate (4) into generalized Kelvin-Voight model given by
, / (5)where is relaxation time of stress for constant strain, is relaxation time of strain for constant stress, and is relaxation elastic modulus. Therefore, the new law for synovial fluid is
, / (6)or
. / (7)If it denotes with the pressure (inclusive of osmotic effects) and with the ambient pressure, integrating this fluid pressure, minus the ambient pressure, over a contact interface produces a fluid load of
. / (8)where the total normal load at the interface is
, / (9)and represents the unit normal at . The static friction coefficient is , for , where is friction force. The friction coefficient is given by
. / (10)Having the coordinates , , for and flexion, and , , for , and flexion, the difference will define the infinitesimal displacement , and . The variation with respect to time for the strain is
, , / (11)where the comma represents the partial differentials with respect to coordinates, and .
To determinate the friction static coefficient of synovial fluid we utilize the experimental results reported by Ateshian [2] (figs. 2 and 3).
Fig. 2. The variation relative to the strain for different values of concentration [2].
Fig. 3. The variation relative to the normal force at surface [2].
3. Conclusions
For a concentration at 0.6M it is possible to excerpt, from Ateshian’s results, using an identification process and (5), the values , , and . The static frictional coefficient for synovial fluid/cartilage is estimated to be
. / (10)As a conclusion, this coefficient is lower than the friction coefficient for ice to ice (0.003), and it is maximum for a concentration of 0.0015M, the value being of 0.0009344755.
Acknowledgement. The authors acknowledge the financial support of the PN2 project nr.106/2007, CNCSIS code 247/2007.
REFERENCES
1. Blewis, ME., Nugent-Derfus, GE., Schmidt, TA., Schumacher BL. and Sah RL., A model of synovial fluid lubricant composition in normal and injured joints, European Cells and Materials 13, 26-39, 2007.
2. Ateshian, G. A., SOLTZ, M.A., MAUCK, R.L., BASALO, I.M., HUNG, C.T., LAI, W.M., The role of osmotic pressure and tension-compression nonlinearity in the frictional response of articular cartilage, Transport in Porous Media 50: 5-33, 2003.
3. MacCONAILL, M. A, The function of intra-articular fibrocartilages, with special references to the knee and inferior radio-ulnar joints, J. Anat. 66, 210–227, 1932.
4. DINTENFASS, L., Lubrication in synovial joints, Nature (London) 197, 496–497, 1963.
5. TANNER, R. I., An alternative mechanism for the lubrication of synovial joints, Phys. Med. Biol. 11, 119–127, 1966.
6. DOWSON, D., Modes of lubrication in human joints, Proc. Inst. Mech. Eng. [H] 181, 45–54, 1967.
7. McCUTCHEN, C. W., The frictional properties of animal joints, Wear 5, 1–17, 1962.
8. MALCOM, L. L., An experimental investigation of the frictional and deformational responses of articular cartilage interfaces to static and dynamic loading, PhD Thesis, University of California, San Diego, 1976.
9. FORSTER, H. AND FISHER, J., The influence of loading time and lubricant on the friction of articular cartilage, Proc. Inst. Mech. Eng. [H] 210, 109–119, 1996.
10. ATESHIAN, G. A., A theoretical formulation for boundary friction in articular cartilage, J. Biomech. Eng. 119, 81–86, 1997.
11. MAROUDAS, A., Physicochemical properties of cartilage in the light of ion exchange theory, Biophys. J. 8, 575–595. 1968.
12. MAROUDAS, A., Physicochemical properties of articular cartilage, In: M. A.R.Freeman (ed.), Adult Articular Cartilage, 2nd edn., Tunbridge Wells, England, Pitman Medical, 215–290, 1979.
13. SOLTZ, M. A. AND ATESHIAN, G. A., A conewise linear elasticity mixture model for the analysis of tension-compression nonlinearity in articular cartilage, J. Biomech. Eng. 122, 576–586, 2000.
14. MOŞNEGUTU, V., CHIROIU, V., CĂPITANU, L., POPESCU M., On the dynamic model of the human knee, Recent Advances in Biology and Biomedicine, A Series of Reference Books and Textbooks, Mathematics and Computers in Biology and Chemistry, Proceedings of the 9th WSEAS Int. Conf. on Mathematics & Computers in Biology & Chemistry (MCBC '08), Bucharest, June 24-26, 76–81, 2008.
15. MOŞNEGUTU, V., CHIROIU, V., CĂPITANU, L., POPESCU M., The influence of the geometry and the material properties on the behavior of the human knee, Wseas Transactions of Mathematics, 7, 6, 417–429, 2008.