WDS'08 Proceedings of Contributed Papers, Part III, 32–37, 2008.

ISBN 978-80-7378-067-8 © MATFYZPRESS

Mathematical Modeling of Synovial Fluids Flow P. Pustˇejovsk´a Charles University in Prague, Mathematical Institute, Czech Republic.

Abstract. 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

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hyaluronan chains create crosslinked network which can resist the loads and thus the solution displays substantial elastic solid-like character. On the other hand if we apply slow and prolonged deformation conditions, the chains align and they start to slip by each other which results in viscous flow. With increasing shear the measure of the rearrangement is higher and thus the solution resistance against the flow decreases - this can be interpreted as decrease in viscosity of the solution. Such molecular theory was experimentally confirmed by for example [Ogston and Stanier, 1953; Gibbs et al., 1968; Morris et al., 1980], see Fig.1. From above it is clear that viscoelastic properties of the synovial fluid are dependent on concentration of hyaluronan in the solution and on the physical conditions we apply to the fluid [Thurston and Greiling, 1978; Ogston and Stanier, 1953].

Figure 1. From experiments of Gibbs et al. [1968] using oscillating Couette rheometer. Dynamic storage moduli G’ (elastic responses) and dynamic loss moduli G” (viscous responses) of a synovial fluid from knee joint plotted against frequency. At low frequencies synovial fluid is predominantly viscous-like, and predominantly elastic-like at high frequencies.

Figure 2. From experiments of Ogston and Stanier [1953] using Couette viscosimeter. Relative viscosity against velocity gradient for synovial fluid at different concentrations(mg/ml). It shows that shear-thinning properties of synovial fluid are highly dependent on concentration.

Establishing simplified model Let us focus on such physical conditions under which the synovial fluid behaves dominantly as viscous-like. In such case the synovial fluid exhibits incompressible viscous properties with viscosity depending on shear rate and concentration of hyaluronan. This shear-thinning behaviour was measured by Ogston and Stanier [Ogston and Stanier, 1953], see Fig.2.

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First we create model for viscosity µ which is combination of existing models. We use observation from [Laurent et al., 1995] and [Vocel et al., 1998] that viscosity is exponentially (with power of 3.3) dependent on concentration and the generalized power-law model: h in µ(c, D) = K eαc ǫ + γ |D|2 Model 1. For second model we use information that index of shear-thinning is concentration dependent and that for zero concentration we obtain Newtonian plasma: h ie−αc −1 µ(c, D) = K ǫ + γ |D|2

Model 2,

where c is the concentration of hyaluronan in synovial fluid, D is the symmetric part of velocity gradient, n is the index of shear-thinning attaining values between −0.5 and 0, where model with n = 0 represents Newtonian fluid, and K, α, ǫ, γ, n are unknown material parameters. To obtain values for our unknown parameters K, α, ǫ, γ, n for both models we use least squares method to fit the experimental data taken from Ogston and Stanier [1953]. The resulting fits are shown in Fig.3. For Model 1 we obtain following values: K = 1, α = 0.4 ml/mg, ǫ = 10−7 , γ = 4.5 10−7 s2 , n = −0.28. For second model we get: K = 3, ǫ = 10−7 , γ = 4.5 10−7 s2 , α = 0.08 ml/mg. For comparison the shear-thinning index of the Model 2 for c = 3 mg/ml is −0.21. The exponential dependence of the viscosity on concentration with power of 3.3 is not valid in Model 1 but for Model 2 it is satisfied for a typical shear rate.

(a) Model 1

(b) Model 2

Figure 3. Relative viscosity against shear rate for different concentrations and for both models. Experimental data (squares) from Ogston and Stanier [1953]

Governing equations We describe the synovial fluid as a single continuum because we do not assume any presence of high-molecular proteins, aggressors or inflammation, on the other hand we assume that the mechanical influence of the hyaluronan presence on the fluid is just through the viscosity dependence. Motion of synovial fluid is described by generalized incompressible Navier-Stokes equations with viscosity depending on the shear rate and the concentration. Moreover, we must couple this system with one extra convection-diffusion equation for the concentration of hyaluronan.

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The system can be written in the following form:

div v = 0, ̺

(1)

∂v + ̺ [grad v]v = −grad p + 2 div(µ(c, D)D), ∂t ∂c + (grad c) · v = div F + m, ∂t

(2) (3) (4)

where v is the velocity field, ̺ constant density, p pressure, µ viscosity, c concentration of hyaluronan, D symmetric part of the velocity gradient, F concentration flux and m is the volumetric hyaluronan production. We specify the concentration flux by Fourier law F = Dc (|D|2 ) grad c

(5)

with diffusivity Dc depending on shear rate [Rudraiah et al., 1991; Pitsillides et al., 1999] of the form D |2 Dc = D0 + κ |D

(6)

m = −α1 (c − copt )

(7)

and the concentration production by

reflecting local balance between production and destruction of hyaluronan chains (due to for example enzymatic reactions), copt stands for the natural value of concentration in synovial fluid, κ, α1 , D0 are material constants.

Geometry, initial and boundary conditions For simulations we consider simple two-dimensional rectangular domain as shown in Fig.4. On the walls we prescribe no-slip condition for velocity, the top and side walls are fixed while the bottom one is moving with time dependent velocity v0 = const. sin(πt). The concentration flux is non-zero (constant) only between points A and B. The initial condition is the rest state with c(0) = copt . The simulations are computed for several oscillation periods.

v0 A

B

F0 Figure 4. Geometry of the tested problem. v0 is the velocity of the moving bottom plate, F0 is the influx.

Computational method The discretization of the problem is done by the finite element method. We use quadrilateral mesh and piecewise biquadratic approximations for the velocity and concentration and piecewise linear approximations for the pressure. The resulting nonlinear algebraic system is solved by the damped Newton method and linear subproblems by direct linear solver. All the computations were preformed by using the Comsol Multiphysics software.

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Computed results For each model we plot concentration distribution and the velocity vectors in time t = 1.1, see Fig.5. It means just after the bottom plate reaches zero velocity. One can notice small differences in shapes of the distributions. Fig.6 shows logarithms of viscosity distribution for time t = 0.5. It means the time when the bottom plate has highest velocity. In this picture one can notice big changes in the viscosity magnitude - it changes by more than two orders. To compare the differences between simulations results of these two models we include Fig.7. It shows the viscosity and velocity profiles in the cross-section of the domain in the vertical direction for x = 0. As one can see the prediction of the viscosity magnitude is much higher for the Model 1 than for the Model 2. This results in more significant non-Newtonian behaviour in the first model.

(a) Model 1

(b) Model 2

Figure 5. Concentration distribution over the domain with the velocity vector arrows. Both graphs are plotted for time t = 1.1.

(a) Model 1

(b) Model 2

Figure 6. Logarithms of viscosity distribution, both graphs plotted for time t = 0.5.

Conclusion Even though the synovial fluid is viscoelastic material, it can be described under specific physical conditions as a viscous non-Newtonian fluid. For simulations in such conditions we have established two models for the viscosity. Both models were solved by finite element method. Model 1 predicted higher viscosity values and more significant non-Newtonian behaviour than Model 2. Moreover, Model 1 was not able to capture the correct exponential dependence of viscosity on concentration for the whole range of shear rates. On the other hand the second model shows dependence with the correct power. It even predicted viscosity values with reasonable agreement to the experimental results during the whole simulation. The viscosity in our simulation settings occurs in interval of 150–7900 Pa·s. From numerical point of view using the current method the second model was solvable for wider range of parameter settings and boundary conditions than the Model 1. Acknowledgments.

This research was supported by GAUK - 306-10/252509

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(a) Viscosity profile

(b) Magnitude of velocity profile

Figure 7. Figure a) Viscosity profile in the vertical cross-section in x = 0, time t = 0.5. The lower line belongs to Model 2, the upper one belongs to Model 1. Figure b) Velocity profile in the vertical cross-section in x = 0, time t = 0.5. The flatter non-Newtonian profile belongs to Model 1.

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