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The nonlinear viscoelasticity of hyaluronic acid and its role in joint lubrication Zhenhuan Zhang and Gordon F. Christopher* Hyaluronic acid solutions have been widely studied due to their relevance to the rheological behavior of synovial fluid and joint lubrication. Ambulatory joint motion is typically large oscillatory deflections; therefore, large amplitude oscillatory shear strain experiments are used to examine the relevant nonlinear viscoelastic properties of these solutions. Using the sequence of physical processes method to analyze data provides time dependent viscoelastic moduli, which exhibit a clear physiologically relevant behavior to hyaluronic acids non-linear viscoelasticity. In particular, it is seen that during peak strain/ acceleration, the time dependent elastic modulus peaks and the loss modulus is at a minimum. The
Received 16th January 2015 Accepted 9th February 2015
hyaluronic acid can provide an immediate elastic response to sudden forces, acting like a shock absorber
DOI: 10.1039/c5sm00131e
during sudden changes in direction of motion or maximum deflection. However, during peak rate, the elastic modulus is at a minimum and the loss modulus is at a maximum, which provides greater efficacy
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to hydrodynamic shear lubrication.
Introduction Synovial uid within joints is the lubricant responsible for the skeletal system's mobility. In synovial joints, cartilage capped bones meet and are encapsulated by the synovial membrane to create a cavity lled with a thin lm of synovial uid that is up to 50 mm thick, depending on physiological and mechanical factors.1–6 The majority of joint lubrication research has focused on boundary lubrication which occurs when surfaces are in direct contact aer long periods of motion and is primarily affected by cartilage viscoelasticity and individual proteins/ polyelectrolytes.7–16 However, during early stages of motion when cartilage surfaces are separated by thicker synovial uid lms, lubrication is hydrodynamic and/or elasto-hydrodynamic, relying heavily on synovial uid viscoelasticity. In these regimes, synovial uid's high viscosity and shear thinning effectively lubricate joints during exion, separating cartilage surfaces and reducing friction as speed increases. Therefore, characterizing synovial uid rheology is important to understanding the mechanical function of joints, treating the effects of diseases such as arthritis, and developing effective joint replacements.1–6,17–19 In previous studies of synovial uid, measurements of viscosity,11,12,14,20–25 friction coefficient,6–16 or linear viscoelastic moduli5,25 have been commonplace. These tests apply steady shear or small amplitude oscillatory shear deformations in order to measure rheological parameters relevant to lubrication. Steady shear and friction coefficient tests would be akin
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409, USA. E-mail: [email protected]
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to constant rotation of the knee in a single direction, and depending on their length and shear rate would create nonphysiological strains. Small amplitude oscillatory shear tests are typically done at frequencies within the range of human motion, but represent joint deection of less than a degree. However, joint motion is typically large amplitude oscillatory deection, especially in joints associated with ambulatory movements such as the knee; for example, deformation of synovial uids inside the synovial membrane can be up to 1000% during running with the varying frequency from 1 rad s1 to 9 rad s1.26 Under these types of deformation, viscoelastic materials exhibit non-linear properties that cannot be extrapolated from linear viscoelasticity or viscosity. These non-linear rheological properties are the parameters that dictate the efficacy and functionality of joint lubrication. Therefore, lubricating abilities would be better evaluated based on non-linear viscoelastic properties that are representative of the material response to large amplitude oscillatory shear. In this work, experiments are designed to simulate human knee movements, providing non-linear moduli relevant to physiological motion. Nonlinear rheological properties of a model synovial uid composed of hyaluronic acid are characterized by large amplitude oscillatory shear (LAOS) strain experiments using a rotational rheometer. Using a number of analysis techniques, the non-linear behavior of hyaluronic acid solutions is found to provide both elastic and viscous response that are relevant to different stages during a single cycle of oscillatory motion. These results may provide guidance for arthritis treatments and human prostheses design in the future.
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Background
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Hyaluronic acid solution linear viscoelastic moduli and viscosity Synovial uid is viscoelastic and primarily composed of a polyelectrolyte polysaccharide, hyaluronic acid, which under normal physiological conditions can vary in length from 0.8 mega-Daltons up to 10 mega-Daltons. In addition, there are a number of proteins in synovial uid.5 Due to the difficulty of obtaining large quantities of human synovial uid, the steady shear rheology of hyaluronic acid solutions and model synovial uid composed of hyaluronic acid, and a combination of relevant proteins have been widely studied to understand synovial uid rheology. Within the literature, there has been active debate as to whether hyaluronic acid alone determines the rheology of synovial uid or if interaction between the various proteins and hyaluronic acid create microstructure that provides enhanced viscoelasticity. A number of recent papers have concluded that hyaluronic acid is the only functional component in determining the rheology of synovial uid.20,22 In particular, Bingol and coworkers did an extensive study of the steady shear rheology of hyaluronic acid solutions, hyaluronic acid combined with bovine serum albumin and g-globulin, and human synovial uid. They saw no difference between pure hyaluronic acid solutions and synovial uids when concentration and molecular weight of hyaluronic acid were equivalent; hyaluronic acid alone was contributing to steady shear behavior.20 Recent work by the authors of this paper has shown conclusively that the behavior in steady sheer and in small amplitude oscillatory shear of model synovial uids are identical to pure hyaluronic acid solutions when tests are properly conducted and effects of interfacial rheology are removed from the results.25,27 As shown in Fig. 1a, hyaluronic acid solutions and model synovial uid composed of hyaluronic acid, bovine serum albumin and g-globulin, behave identically under steady shear when interfacial rheology and surface tension effects are removed by applying a surfactant to the interface to displace adsorbed proteins which create a viscoelastic interface. Both solutions are extremely shear thinning and present a zero shear viscosity near 15 mPa s. Similarly in small amplitude oscillatory shear (Fig. 1b), both solutions are primarily viscous for all strain amplitudes, with small storage modulus that is an order of magnitude smaller than loss moduli. At strain amplitudes lower than 0.01, an increase in elasticity of the model synovial uid is observed that does not occur for the hyaluronic acid which could be attributable to a number of factors, including but not limited to the sensitivity of the rheometer at low strains, possible bulk protein aggregation affects, possible residual interfacial rheological response, or increased surface tension effects at low strains.
Large amplitude oscillatory shear Dynamic oscillatory shear tests are common in rheology and have been extensively used to investigate so matter's linear viscoelastic properties. Typically, a small amplitude sinusoidal
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Fig. 1 (a) Steady shear viscosity of model synovial fluid solution composed of hyaluronic acid, bovine serum albumin and g-globulin (,) and pure hyaluronic acid solution (B). Concentration and molecular weight of hyaluronic acid are identical in both solutions. (b) Storage moduli (solid symbols) and loss moduli (hollow symbols) of the same two solutions. As can be seen except at extremely small strains, the behavior of these solutions is identical; therefore, hyaluronic acid is the primary determinant of synovial fluid viscosity.
strain or stress is applied to a sample. As long as the material remains in the linear regime, the sample's response stress or strain will be sinusoidal but out of phase with the input. The inphase portion of the signal can be related to elastic properties through a storage modulus, and the out of phase portion can be related to viscous properties through a loss modulus.28 LAOS tests can be performed on a material, however the analysis methodology from small amplitude oscillatory tests cannot be simply extended into the nonlinear regime. This is due to the materials response incorporating higher harmonics, which results in a multi-modal sinusoidal signal that cannot be easily decomposed into elastic and viscous components. Because of this, analysis of LAOS data is not uniquely dened, and there has been debate about how to appropriately analyze LAOS data. There are several methods available including Bowditch–Lissajous curves,29,30 Fourier transform rheology (FT-rheology),31,32 sequence of physical processes,33,34 and a range of others.29 Bowditch–Lissajous curves Researchers have found the use of Bowditch–Lissajous curves can provide a physical interpretation to LAOS data.29,30 Bowditch–Lissajous curves plot strain, strain rate and stress in deformation space. This 3D curve is projected onto strain–stress plane or strain rate–stress plane. In the strain–stress plane, pure viscous uids become a circle and purely elastic materials become a straight line. Viscoelastic materials appear as a
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rotated ellipse in the linear regime, and take more complicated shapes in the nonlinear regime. These curves can be explored over a wide phase space to create a ngerprint of a material over a range of applied deformations. Although this method enables a physical interpretation of nonlinear viscoelastic response, it does not result in useful parameters that can be extrapolated to a wider range of deformations. FT-rheology During LAOS strain experiments, a sinusoidal strain is applied to tested solutions, but the resulting stress is non-sinusoidal and therefore incorporates higher harmonics. FT-rheology converts LAOS non-sinusoidal stress signal from the time domain into the frequency domain using a Fourier transform. The resultant magnitude of the peaks in the frequency space can be used to characterize a material's non-linearity. Because of symmetry in the stress and strain direction, the higher harmonic terms only contain odd times of base frequency: s ¼ I1 sin(ut) + I3 sin(3ut) + I5 sin(5ut) + .
(1)
A tested material can be characterized by examining the intensity of each higher harmonic. In this study FT-rheology analysis focuses on comparing the ratio of I3 to I1 because intensity of higher harmonic terms is under noise oor. In the linear regime, this ratio should be zero and will increase as response becomes non-linear. The value of I3/I1 can be easily used to distinguish the boundary between the two regimes and a characterization of overall non-linearity.31,32 Methods such as stress decomposition,35 power function expansion,36,37 or Chevbyshev polynomials38 have been used to extend FT rheology analysis by nding non-linear viscous and elastic moduli; the resultant moduli are time independent. Time dependent moduli The sequence of physical process method can be used to nd instantaneous, non-linear viscoelastic moduli.33,34 In deformation space, the strain–strain rate-stress curve, r(t) ¼ [g(t)g0 (t)s(t)] of a viscoelastic material in the linear regime is a 3D planar curve with a unique bi-normal vector dened by the cross product of tangent and normal vectors of the curve (eqn (2) and Fig. 2a). The complex modulus can be described by the angle 4 formed by bi-normal vector and stress axis (eqn (3)). Meanwhile the phase angle d is equal to the angle formed by strain axis and projection of negative bi-normal vector: : :: r r * BðtÞ ¼ Bg ; Bg: ; Bs ¼ : (2) :: r r Bs ¼ G* tanð4Þ ¼ tan cos1 jBj
(3)
G0 ¼ cos d tan 4
(4)
G0 ¼ sin d tan 4
(5)
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Fig. 2 (a) Stress, strain, and strain rate planar curve created by a viscoelastic material plotted in the deformation space when tested in the linear regime. (b) Stress, strain, and strain rate 3D curve created by a viscoelastic material plotted in the deformation space when tested in the non-linear regime.
When material undergoes large amplitude strain, the trajectory of r(t) becomes a 3D twisted curve (Fig. 2b). However, a time dependent bi-normal vector at each point can be dened. This instantaneous bi-normal vector can be decomposed into a storage and loss moduli identically to the method above, resulting in time dependent storage and loss moduli. The resulting moduli have the same physical interpretation as linear viscoelastic moduli; however, they are time dependent due to the material's non-linear response.
Materials and methods Materials A hyaluronic acid (Sigma-Aldrich 53747) solution is used to model synovial uid. It is composed of hyaluronic acid at 3.4 mg ml1, which is based on the level of hyaluronic acid in healthy synovial uid.5 The hyaluronic acid has a molecular weight of 1.6 mega-Daltons and is synthesized from streptococcus equi bacteria with protein content less than 1% protein. A second model synovial uid solution consisting of identical hyaluronic acid with bovine serum albumin (10 mg ml1, Sigma-Aldrich
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A3059) and gamma-globulin (0.5 mg ml1, Sigma-Aldrich G5009) is also made to serve as control. All solutions are prepared in a 10 mM phosphate buffered saline solution with pH 7.4 and stored in refrigerator and used within several days of preparation in order to avoid bacterial contamination or sample degradation.
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Rheometry All LAOS-strain experiments are conducted using an AR-G2 TA Instruments stress controlled rheometer with a double wall cylinder geometry (rotor inner diameter 32 mm, rotor outer diameter 35 mm, and rotor height 53 mm) operating in strain control mode. This instrument and geometry are selected due to their sensitivity to low viscosity materials. From research on the human gait,26 the deection angle of knees ranges from 0 to 50 during walking, which causes a strain amplitude of up to 1000%. Typical human ambulatory frequency varies from 1 rad s1 to 8 rad s1. To simulate these conditions, a group of LAOS strain experiments are applied at strain amplitudes of 1–300% at frequencies ranging from 1 rad s1 to 8 rad s1 at a temperature 25 C. The maximum strain amplitudes are lower than 1000%, because above strain amplitudes of 300% noise and other experimental difficulties do not allow accurate measurement. For each strain input, 10 cycles of data of stress output signals are collected in transient mode with sampling frequency as 1 kHz. In order to remove interfacial rheology effects on bulk rheology,25,27 the surfactant polysorbate 80 (Sigma-Aldrich) is added to tested solutions in the concentration of 0.01 mM l1.
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furthermore, only report results in this study for those measurements in which inertial torque is less than 20% of the measured signal. To nd the stress response to the sinusoidal strain, the raw torque data and displacement data from the experiments are used to calculate viscoelastic moduli; this analysis is done using custom written code in Matlab. Numerical derivatives of the displacement are taken to nd strain rate. The strain rate and system variables are used to calculate inertial torque, which is then subtracted from total system torque to obtain sample torque. This is then converted to stress using the stress factor. Due to low viscosity and the removal of inertial torque, the resultant material stress is noisy; this is also true of the strain rate. Therefore, Fourier transforms of these curves are taken. Only the rst and third harmonic intensities are used to construct clean response curves without the high frequency noise. The strain, reconstructed strain rate and reconstructed stress signals are used for all subsequent analysis of the data.
Results LAOS responses of hyaluronic acid compared to model synovial uid A group of LAOS-strain experiments are applied to both the hyaluronic acid solution and the model synovial uids solution. The stress outputs from both solutions are nominally identical within measurement error (Fig. 3). The hyaluronic solution dominates the nonlinear rheological properties of model synovial uid during LAOS. Only hyaluronic acid solution is used in further experiments because it is less interfacially active and can be stored longer.
Data analysis methodology Typically LAOS-strain experiments are done using a strain controlled rheometer; these experiments provide a non-sinusoidal stress response to a given sinusoidal strain. Replicating these experiments with a stress controlled rheometer has typically been considered impossible; however, due to the typically low viscosities of hyaluronic acid solutions, strain controlled rheometers do not provide the necessary sensitivity to conduct experiments desired in this study. Although LAOS-stress experiments can be performed, the lack of developed analytical methods developed for LAOS-stress restrict its usefulness. Therefore, all LAOS-strain experiments were done on a stress controlled rheometer operating in strain control, but a number of important steps were taken to make these measurements provide accurate results. Recent studies have shown that if conducted and analyzed appropriately, LAOS-strain on a stress controlled rheometer produces identical results to LAOS-strain on a strain controlled rheometer;39 in particular, if inertial torque from the rheometer is less than 20% of the total torque, then results between stress and strain controlled rheometers running LAOS-strain are identical. If inertial torque exceeds this limit, removing the inertial torque from the measurement will correct any errors in stress controlled LAOS-strain experiments. Therefore, we remove inertial torque from all measurements, and
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Bowditch–Lissajous curves and FT-rheology In order to qualitatively analyze non-linear viscoelasticity of hyaluronic acid, the reconstructed data is plotted in the stress– strain plane to create a Bowditch–Lissajous curve; a Pipkin space of these results is shown in Fig. 4. The hyaluronic acid has clear elliptical curve signifying viscoelasticity for the 1 rad s1 10% strain amplitude test. As strain is increased, curves become circular and the material is dominated by viscosity. Both 4 rad
Reconstructed stress of hyaluronic acid and model synovial fluids by first and third harmonic term of their raw stress response under strain amplitude 300% and angular frequency 4.5 rad s1. Red solid line: hyaluronic acid, black dash line: model synovial fluid. Fig. 3
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Fig. 4 Bowditch–Lissajous curves for LAOS-strain data at a range of applied strains and frequencies. The y-axis is stress, non-dimensionalized by max stress, and the x-axis is strain, non-dimensionalized by max strain. Red dashed lines are perfect circles representing an ideal viscous response. Black lines represent hyaluronic acid solution response.
s1 and 8 rad s1 exhibit similar behavior, but are more viscous at 10% strain. However, both maintain more elliptical curve shapes at larger strains, indicating a greater retention of elasticity. Overall, we conclude that the hyaluronic acid solutions are primarily viscous from these results. Nonlinear response in LAOS is also examined by FT-rheology using the ratio of the magnitudes of the third harmonic term to the rst harmonic term (Fig. 5). The transition from linear to nonlinear regions can be clearly distinguished by observing when this ratio becomes larger than 5%. At u ¼ 1 rad s1, the sample stays in the linear regime for strain amplitude smaller than 10%, followed by a transient area up to 30% then steps into nonlinear region nally. Similar behavior is observed for 4 rad s1 and 8 rad s1; however the transition to non-linear response occurs at smaller strain amplitudes, of 5% and 2% respectively and the transition region becomes smaller.
Percentage ratio of third to first harmonic of hyaluronic acid solutions for all strains tested at 1, 4, and 8 rad s1. Based on results we can see the transition to non-linear behavior when this ratio is above 1%. Error bars are found using the sampling frequency of the rheometer (1 kHz), which creates an error of 7% on the intensities which is propagated to their ratio. Fig. 5
storage and loss moduli at all frequencies tested. The degree of variations in the moduli clearly increases with strain amplitudes. Furthermore, the degree of variation increases with the applied frequency.
Time dependent moduli Using the sequence of physical process method, the instantaneous time dependent loss and storage moduli are shown in Fig. 6 for 1 non-dimensional period. Looking at both the loss and storage moduli at all amplitudes, it is observed that moduli magnitude increases with frequency. Overall, the magnitudes of the moduli do not increase with strain amplitude, and we see nominally identical average values of the moduli at all strain amplitudes for a given frequency. Examining the time dependent behavior, we start rst with the 10% strain amplitude data. For 1 rad s1 and 4 rad s1, the time dependent moduli response is essentially at; at these frequencies and strain amplitudes, the system is responding with single moduli during all stages of oscillation. There is some variation of the moduli in the 8 rad s1 data. Examining the 180% and 300% strain amplitude data, we see signicant variation during oscillation. Time dependence is seen for both
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Time dependent loss moduli (left column) and storage moduli (right column) plotted vs. time, non-dimensionalized by period, for a single cycle of LAOS for 1, 4, and 8 rad s1. As strain increases, there is a clear increase in the variation of the moduli during the cycle. Frequency also clearly increases the magnitude of these moduli. Fig. 6
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Examining the behavior in more detail, there is a clear periodic response, most evident in the 300% strain amplitude data. This periodicity is seen in both moduli; however, the two moduli do not behave identically or in sync. The loss moduli shows a simpler behavior with 2 peaks and 2 valleys and a small plateau region. The storage moduli show a more complicated response with 2 maximum peaks and 2 smaller sub-peaks between the true maximums. It is important to note that in essence all of the responses at varying frequencies for the 300% strain amplitude show the same basic behavior, but with increasing magnitude and degree of variation with frequency. Looking at the tan(d) data (Fig. 7), we can further understand how these samples react to LAOS. At small strain amplitude 10%, all tan(d) data are close to constant, however periodic variation of tan(d) starts to appear as frequency is increased. The values of tan(d) also decrease with frequency, signifying that the solutions are becoming more elastic with increasing frequency. However, the magnitude of all tan(d) is well above 1; hence, the solutions are predominately viscous. As strain amplitude increases, we see increasing variation in tan(d) with time, with clear peaks of high viscosity, and valleys of increased elasticity. At no point are the solutions ever predominately elastic. Furthermore, we continue to see that as frequency is increased for all strain amplitudes, the value of tan(d) decreases, signifying increased elasticity. Overall, it is clear that the solution behaves most like a Newtonian viscous uid at low frequencies and small strain amplitudes, and shows increasing elastic behavior with increasing frequency, and increasing non-linear/ time dependent behavior with increasing strain amplitude. In order to further explore the nature of these time dependent moduli and their relation to applied deformation, the time-dependent dimensionless storage modulus normalized by their maximum value is compared to dimensionless strain
Fig. 7 Tan(d) vs. time, non-dimensionalized by period, for a single
cycle of LAOS for a range of applied strains and frequencies. As strain increases we see increased variation in tan(d). Furthermore, as frequency increases, the value of tan(d) decreases, signifying increased overall elasticity of the solutions.
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acceleration, while the loss modulus is compared to strain rate in Fig. 8. Looking at the storage moduli, timing of the peak elasticity occurs a few moments before the maximum imposed strain acceleration during oscillation, which for the sinusoidal signal applied is identical to the maximum imposed strain during oscillation. This behavior is evident for all frequencies tested, and is consistent throughout the non-linear regime. Examining the loss moduli, its peak values during oscillation do not align with those of the storage moduli; instead, they align with peak values of the imposed strain rate during oscillation. We can see that the loss moduli are in fact at a minimum (nearly zero) during the moment of peak imposed strain/strain acceleration when the storage moduli are at their maximum. Correspondingly, the max loss moduli occur with the minimum storage moduli.
Discussion In examining the non-linear behavior of the hyaluronic acid solutions, the FT rheology does not provide signicant physiological insight. As frequency is increased, the boundary of linear and nonlinear response regime appears at smaller strain amplitudes. Fig. 5 clearly shows the strain amplitudes where the transition to non-linearity occurs are far below typical motion of a walking knee, which is approximately 1000%. Therefore we can conclude that for any typical joint motion, synovial uid will be in the non-linear regime, and therefore the non-linear viscoelastic moduli are the values that should be considered to evaluate lubrication. The Bowditch–Lissajous curves provide some insight into the behavior of the tested solutions. For large strain amplitudes, the hyaluronic acid provides primarily a viscous response. As
Fig. 8 Time dependent loss moduli (left column) and storage moduli (right column) for 300% strain non-dimensionalized by their maximum value plotted vs. time non-dimensionalized by period for a single cycle of LAOS for 1, 4, and 8 rad s1. Strain rate and acceleration included on figures as well to show that max loss modulus occurs at max rate and max storage modulus occurs during max acceleration.
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suspected, solution viscosity is primarily responsible for lubrication during hydrodynamic regimes based on the dominance of the viscous response. However, small elasticity in the solution is observed with increasing strain amplitudes at higher frequencies; this indicates that the elasticity of the synovial uid may still play some role in knee mechanics at these increased strain amplitudes and frequencies. Examination of Fig. 6, 7 and 8, which use of the sequence of physical processes methodology, allow us to fully understand the role of the small elasticity observed in the Bowditch–Lissajous curves. Noting peaks in elasticity align with the largest imposed strain/acceleration during oscillation demonstrates an important time dependent role of elasticity when deection reaches its maximum position and/or during sudden acceleration. Indeed, during largest imposed strain/acceleration the elasticity of the synovial uid increases dramatically. This allows the uid to provide an immediate response to a force and act as shock absorber and provide enhanced protection to the knee and the synovial cavity; during this phase the hyaluronic acid shows much less viscous response. Due to the applied sinusoidal signal, it is impossible to deconvolve whether the response comes from the strain amplitude or acceleration. During peak velocity, when lubrication properties are most important, the viscosity of the solution peaks and elasticity goes to zero in order to provide the best shear lubrication mechanism for the knee. We see similar behavior by examining tan(d) of the system; its value shows a more gel like state of hyaluronic acid when strain approaches its peak which can provide more elasticity to synovial joints in order to immediately respond to force during peak strain amplitude/acceleration.
Conclusions Previous analysis of linear and steady shear properties of hyaluronic acid have ignored the typical large oscillation behavior of joints like the knee. Our results demonstrate that the nonlinear rheological properties that can be observed with techniques like LAOS-strain, are in fact the properties most relevant to actual human ambulation, since synovial uid is always in a non-linear viscoelastic regime during typical ambulation. This non-linear viscoelasticity of hyaluronic acid solution provides multiple functions to joints during large oscillatory deformation. Using LAOS-strain, the behavior of hyaluronic acid based model synovial uid is shown to have a physiologically relevant non-linear, time dependent storage and loss modulus that are timed to the moments of peak strain amplitude/acceleration and shear rate of the knee respectively. These moments of peak elasticity provide immediate response to sudden forces that may cause joints damage. During peak shear rates, hydrodynamic lubrication is most important to avoid joints damage; synovial uid displays peak viscosity to separate cartilage surfaces and reduce friction during these times. These behaviors apply only to the synovial uid and impact the synovial cavity. The viscoelasticity of cartilage, surrounding ligaments, and tissues cannot be concluded to behave in a similar way; however, these results indicate that time dependent moduli
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may be an appropriate way to examine the behaviors of these tissues as well. This information is clearly useful to better understanding the role of hyaluronic acid in protecting and lubricating the knee, and provides insight into how to better treat joint diseases that degrade the rheological behavior of synovial uid in future. Furthermore by applying a range of analysis techniques to understand LAOS-strain behavior of hyaluronic acid solution, we have shown that it is important to consider a range of analysis techniques when examining LAOS data. All methods can provide relevant and interesting information about nonlinear material properties; however, it is clear that by pursuing only a single methodology relevant information can be missed. In the case of this work, both Bowditch–Lissajous curves and FT-rheology did not exhibit any particularly interesting behaviors that would help explain the physiological function of synovial uid. However, using the sequence of physical processes method, we found clear increased elasticity and physiological signicance in the non-linear response of the hyaluronic acid solution.
Acknowledgements The authors are indebted to Dr Simon. A. Rogers for discussions on the use of sequence of physical process methods in this research.
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25 Z. Zhang, S. Barman and G. F. Christopher, So Matter, 2014, 10, 5965–5973. 26 T. F. Novacheck, Gait Posture, 1998, 7, 77–95. 27 Z. H. Zhang, S. Barman and G. F. Christopher, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2014, 89, 4. 28 C. W. Macosko and R. G. Larson, Rheology : principles, measurements, and applications, VCH, New York, NY, 1994. 29 K. Hyun, M. Wilhelm, C. O. Klein, K. S. Cho, J. G. Nam, K. H. Ahn, S. J. Lee, R. H. Ewoldt and G. H. McKinley, Prog. Polym. Sci., 2011, 36, 1697–1753. 30 R. H. Ewoldt and G. H. McKinley, Rheol. Acta, 2010, 49, 213– 219. 31 M. Wilhelm, Macromol. Mater. Eng., 2002, 287, 83–105. 32 M. Wilhelm, P. Reinheimer and M. Ortseifer, Rheol. Acta, 1999, 38, 349–356. 33 S. A. Rogers, J. Rheol., 2012, 56, 1129–1151. 34 S. A. Rogers and M. P. Lettinga, J. Rheol., 2012, 56, 1–25. 35 K. S. Cho, K. Hyun, K. H. Ahn and S. J. Lee, J. Rheol., 2005, 49, 747–758. 36 R. H. Ewoldt and N. A. Bharadwaj, Rheol. Acta, 2013, 52, 201– 219. 37 A. K. Gurnon and N. J. Wagner, J. Rheol., 2012, 56, 333–351. 38 R. H. Ewoldt, A. E. Hosoi and G. H. McKinley, J. Rheol., 2008, 52, 1427–1458. 39 D. Merger and M. Wilhelm, Rheol. Acta, 2014, 53, 621–634.
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The nonlinear viscoelasticity of hyaluronic acid and its role in joint lubrication Zhenhuan Zhang and Gordon F. Christopher* Hyaluronic acid solutions have been widely studied due to their relevance to the rheological behavior of synovial fluid and joint lubrication. Ambulatory joint motion is typically large oscillatory deflections; therefore, large amplitude oscillatory shear strain experiments are used to examine the relevant nonlinear viscoelastic properties of these solutions. Using the sequence of physical processes method to analyze data provides time dependent viscoelastic moduli, which exhibit a clear physiologically relevant behavior to hyaluronic acids non-linear viscoelasticity. In particular, it is seen that during peak strain/ acceleration, the time dependent elastic modulus peaks and the loss modulus is at a minimum. The
Received 16th January 2015 Accepted 9th February 2015
hyaluronic acid can provide an immediate elastic response to sudden forces, acting like a shock absorber
DOI: 10.1039/c5sm00131e
during sudden changes in direction of motion or maximum deflection. However, during peak rate, the elastic modulus is at a minimum and the loss modulus is at a maximum, which provides greater efficacy
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to hydrodynamic shear lubrication.
Introduction Synovial uid within joints is the lubricant responsible for the skeletal system's mobility. In synovial joints, cartilage capped bones meet and are encapsulated by the synovial membrane to create a cavity lled with a thin lm of synovial uid that is up to 50 mm thick, depending on physiological and mechanical factors.1–6 The majority of joint lubrication research has focused on boundary lubrication which occurs when surfaces are in direct contact aer long periods of motion and is primarily affected by cartilage viscoelasticity and individual proteins/ polyelectrolytes.7–16 However, during early stages of motion when cartilage surfaces are separated by thicker synovial uid lms, lubrication is hydrodynamic and/or elasto-hydrodynamic, relying heavily on synovial uid viscoelasticity. In these regimes, synovial uid's high viscosity and shear thinning effectively lubricate joints during exion, separating cartilage surfaces and reducing friction as speed increases. Therefore, characterizing synovial uid rheology is important to understanding the mechanical function of joints, treating the effects of diseases such as arthritis, and developing effective joint replacements.1–6,17–19 In previous studies of synovial uid, measurements of viscosity,11,12,14,20–25 friction coefficient,6–16 or linear viscoelastic moduli5,25 have been commonplace. These tests apply steady shear or small amplitude oscillatory shear deformations in order to measure rheological parameters relevant to lubrication. Steady shear and friction coefficient tests would be akin
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409, USA. E-mail: [email protected]
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to constant rotation of the knee in a single direction, and depending on their length and shear rate would create nonphysiological strains. Small amplitude oscillatory shear tests are typically done at frequencies within the range of human motion, but represent joint deection of less than a degree. However, joint motion is typically large amplitude oscillatory deection, especially in joints associated with ambulatory movements such as the knee; for example, deformation of synovial uids inside the synovial membrane can be up to 1000% during running with the varying frequency from 1 rad s1 to 9 rad s1.26 Under these types of deformation, viscoelastic materials exhibit non-linear properties that cannot be extrapolated from linear viscoelasticity or viscosity. These non-linear rheological properties are the parameters that dictate the efficacy and functionality of joint lubrication. Therefore, lubricating abilities would be better evaluated based on non-linear viscoelastic properties that are representative of the material response to large amplitude oscillatory shear. In this work, experiments are designed to simulate human knee movements, providing non-linear moduli relevant to physiological motion. Nonlinear rheological properties of a model synovial uid composed of hyaluronic acid are characterized by large amplitude oscillatory shear (LAOS) strain experiments using a rotational rheometer. Using a number of analysis techniques, the non-linear behavior of hyaluronic acid solutions is found to provide both elastic and viscous response that are relevant to different stages during a single cycle of oscillatory motion. These results may provide guidance for arthritis treatments and human prostheses design in the future.
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Hyaluronic acid solution linear viscoelastic moduli and viscosity Synovial uid is viscoelastic and primarily composed of a polyelectrolyte polysaccharide, hyaluronic acid, which under normal physiological conditions can vary in length from 0.8 mega-Daltons up to 10 mega-Daltons. In addition, there are a number of proteins in synovial uid.5 Due to the difficulty of obtaining large quantities of human synovial uid, the steady shear rheology of hyaluronic acid solutions and model synovial uid composed of hyaluronic acid, and a combination of relevant proteins have been widely studied to understand synovial uid rheology. Within the literature, there has been active debate as to whether hyaluronic acid alone determines the rheology of synovial uid or if interaction between the various proteins and hyaluronic acid create microstructure that provides enhanced viscoelasticity. A number of recent papers have concluded that hyaluronic acid is the only functional component in determining the rheology of synovial uid.20,22 In particular, Bingol and coworkers did an extensive study of the steady shear rheology of hyaluronic acid solutions, hyaluronic acid combined with bovine serum albumin and g-globulin, and human synovial uid. They saw no difference between pure hyaluronic acid solutions and synovial uids when concentration and molecular weight of hyaluronic acid were equivalent; hyaluronic acid alone was contributing to steady shear behavior.20 Recent work by the authors of this paper has shown conclusively that the behavior in steady sheer and in small amplitude oscillatory shear of model synovial uids are identical to pure hyaluronic acid solutions when tests are properly conducted and effects of interfacial rheology are removed from the results.25,27 As shown in Fig. 1a, hyaluronic acid solutions and model synovial uid composed of hyaluronic acid, bovine serum albumin and g-globulin, behave identically under steady shear when interfacial rheology and surface tension effects are removed by applying a surfactant to the interface to displace adsorbed proteins which create a viscoelastic interface. Both solutions are extremely shear thinning and present a zero shear viscosity near 15 mPa s. Similarly in small amplitude oscillatory shear (Fig. 1b), both solutions are primarily viscous for all strain amplitudes, with small storage modulus that is an order of magnitude smaller than loss moduli. At strain amplitudes lower than 0.01, an increase in elasticity of the model synovial uid is observed that does not occur for the hyaluronic acid which could be attributable to a number of factors, including but not limited to the sensitivity of the rheometer at low strains, possible bulk protein aggregation affects, possible residual interfacial rheological response, or increased surface tension effects at low strains.
Large amplitude oscillatory shear Dynamic oscillatory shear tests are common in rheology and have been extensively used to investigate so matter's linear viscoelastic properties. Typically, a small amplitude sinusoidal
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Fig. 1 (a) Steady shear viscosity of model synovial fluid solution composed of hyaluronic acid, bovine serum albumin and g-globulin (,) and pure hyaluronic acid solution (B). Concentration and molecular weight of hyaluronic acid are identical in both solutions. (b) Storage moduli (solid symbols) and loss moduli (hollow symbols) of the same two solutions. As can be seen except at extremely small strains, the behavior of these solutions is identical; therefore, hyaluronic acid is the primary determinant of synovial fluid viscosity.
strain or stress is applied to a sample. As long as the material remains in the linear regime, the sample's response stress or strain will be sinusoidal but out of phase with the input. The inphase portion of the signal can be related to elastic properties through a storage modulus, and the out of phase portion can be related to viscous properties through a loss modulus.28 LAOS tests can be performed on a material, however the analysis methodology from small amplitude oscillatory tests cannot be simply extended into the nonlinear regime. This is due to the materials response incorporating higher harmonics, which results in a multi-modal sinusoidal signal that cannot be easily decomposed into elastic and viscous components. Because of this, analysis of LAOS data is not uniquely dened, and there has been debate about how to appropriately analyze LAOS data. There are several methods available including Bowditch–Lissajous curves,29,30 Fourier transform rheology (FT-rheology),31,32 sequence of physical processes,33,34 and a range of others.29 Bowditch–Lissajous curves Researchers have found the use of Bowditch–Lissajous curves can provide a physical interpretation to LAOS data.29,30 Bowditch–Lissajous curves plot strain, strain rate and stress in deformation space. This 3D curve is projected onto strain–stress plane or strain rate–stress plane. In the strain–stress plane, pure viscous uids become a circle and purely elastic materials become a straight line. Viscoelastic materials appear as a
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rotated ellipse in the linear regime, and take more complicated shapes in the nonlinear regime. These curves can be explored over a wide phase space to create a ngerprint of a material over a range of applied deformations. Although this method enables a physical interpretation of nonlinear viscoelastic response, it does not result in useful parameters that can be extrapolated to a wider range of deformations. FT-rheology During LAOS strain experiments, a sinusoidal strain is applied to tested solutions, but the resulting stress is non-sinusoidal and therefore incorporates higher harmonics. FT-rheology converts LAOS non-sinusoidal stress signal from the time domain into the frequency domain using a Fourier transform. The resultant magnitude of the peaks in the frequency space can be used to characterize a material's non-linearity. Because of symmetry in the stress and strain direction, the higher harmonic terms only contain odd times of base frequency: s ¼ I1 sin(ut) + I3 sin(3ut) + I5 sin(5ut) + .
(1)
A tested material can be characterized by examining the intensity of each higher harmonic. In this study FT-rheology analysis focuses on comparing the ratio of I3 to I1 because intensity of higher harmonic terms is under noise oor. In the linear regime, this ratio should be zero and will increase as response becomes non-linear. The value of I3/I1 can be easily used to distinguish the boundary between the two regimes and a characterization of overall non-linearity.31,32 Methods such as stress decomposition,35 power function expansion,36,37 or Chevbyshev polynomials38 have been used to extend FT rheology analysis by nding non-linear viscous and elastic moduli; the resultant moduli are time independent. Time dependent moduli The sequence of physical process method can be used to nd instantaneous, non-linear viscoelastic moduli.33,34 In deformation space, the strain–strain rate-stress curve, r(t) ¼ [g(t)g0 (t)s(t)] of a viscoelastic material in the linear regime is a 3D planar curve with a unique bi-normal vector dened by the cross product of tangent and normal vectors of the curve (eqn (2) and Fig. 2a). The complex modulus can be described by the angle 4 formed by bi-normal vector and stress axis (eqn (3)). Meanwhile the phase angle d is equal to the angle formed by strain axis and projection of negative bi-normal vector: : :: r r * BðtÞ ¼ Bg ; Bg: ; Bs ¼ : (2) :: r r Bs ¼ G* tanð4Þ ¼ tan cos1 jBj
(3)
G0 ¼ cos d tan 4
(4)
G0 ¼ sin d tan 4
(5)
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Fig. 2 (a) Stress, strain, and strain rate planar curve created by a viscoelastic material plotted in the deformation space when tested in the linear regime. (b) Stress, strain, and strain rate 3D curve created by a viscoelastic material plotted in the deformation space when tested in the non-linear regime.
When material undergoes large amplitude strain, the trajectory of r(t) becomes a 3D twisted curve (Fig. 2b). However, a time dependent bi-normal vector at each point can be dened. This instantaneous bi-normal vector can be decomposed into a storage and loss moduli identically to the method above, resulting in time dependent storage and loss moduli. The resulting moduli have the same physical interpretation as linear viscoelastic moduli; however, they are time dependent due to the material's non-linear response.
Materials and methods Materials A hyaluronic acid (Sigma-Aldrich 53747) solution is used to model synovial uid. It is composed of hyaluronic acid at 3.4 mg ml1, which is based on the level of hyaluronic acid in healthy synovial uid.5 The hyaluronic acid has a molecular weight of 1.6 mega-Daltons and is synthesized from streptococcus equi bacteria with protein content less than 1% protein. A second model synovial uid solution consisting of identical hyaluronic acid with bovine serum albumin (10 mg ml1, Sigma-Aldrich
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A3059) and gamma-globulin (0.5 mg ml1, Sigma-Aldrich G5009) is also made to serve as control. All solutions are prepared in a 10 mM phosphate buffered saline solution with pH 7.4 and stored in refrigerator and used within several days of preparation in order to avoid bacterial contamination or sample degradation.
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Rheometry All LAOS-strain experiments are conducted using an AR-G2 TA Instruments stress controlled rheometer with a double wall cylinder geometry (rotor inner diameter 32 mm, rotor outer diameter 35 mm, and rotor height 53 mm) operating in strain control mode. This instrument and geometry are selected due to their sensitivity to low viscosity materials. From research on the human gait,26 the deection angle of knees ranges from 0 to 50 during walking, which causes a strain amplitude of up to 1000%. Typical human ambulatory frequency varies from 1 rad s1 to 8 rad s1. To simulate these conditions, a group of LAOS strain experiments are applied at strain amplitudes of 1–300% at frequencies ranging from 1 rad s1 to 8 rad s1 at a temperature 25 C. The maximum strain amplitudes are lower than 1000%, because above strain amplitudes of 300% noise and other experimental difficulties do not allow accurate measurement. For each strain input, 10 cycles of data of stress output signals are collected in transient mode with sampling frequency as 1 kHz. In order to remove interfacial rheology effects on bulk rheology,25,27 the surfactant polysorbate 80 (Sigma-Aldrich) is added to tested solutions in the concentration of 0.01 mM l1.
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furthermore, only report results in this study for those measurements in which inertial torque is less than 20% of the measured signal. To nd the stress response to the sinusoidal strain, the raw torque data and displacement data from the experiments are used to calculate viscoelastic moduli; this analysis is done using custom written code in Matlab. Numerical derivatives of the displacement are taken to nd strain rate. The strain rate and system variables are used to calculate inertial torque, which is then subtracted from total system torque to obtain sample torque. This is then converted to stress using the stress factor. Due to low viscosity and the removal of inertial torque, the resultant material stress is noisy; this is also true of the strain rate. Therefore, Fourier transforms of these curves are taken. Only the rst and third harmonic intensities are used to construct clean response curves without the high frequency noise. The strain, reconstructed strain rate and reconstructed stress signals are used for all subsequent analysis of the data.
Results LAOS responses of hyaluronic acid compared to model synovial uid A group of LAOS-strain experiments are applied to both the hyaluronic acid solution and the model synovial uids solution. The stress outputs from both solutions are nominally identical within measurement error (Fig. 3). The hyaluronic solution dominates the nonlinear rheological properties of model synovial uid during LAOS. Only hyaluronic acid solution is used in further experiments because it is less interfacially active and can be stored longer.
Data analysis methodology Typically LAOS-strain experiments are done using a strain controlled rheometer; these experiments provide a non-sinusoidal stress response to a given sinusoidal strain. Replicating these experiments with a stress controlled rheometer has typically been considered impossible; however, due to the typically low viscosities of hyaluronic acid solutions, strain controlled rheometers do not provide the necessary sensitivity to conduct experiments desired in this study. Although LAOS-stress experiments can be performed, the lack of developed analytical methods developed for LAOS-stress restrict its usefulness. Therefore, all LAOS-strain experiments were done on a stress controlled rheometer operating in strain control, but a number of important steps were taken to make these measurements provide accurate results. Recent studies have shown that if conducted and analyzed appropriately, LAOS-strain on a stress controlled rheometer produces identical results to LAOS-strain on a strain controlled rheometer;39 in particular, if inertial torque from the rheometer is less than 20% of the total torque, then results between stress and strain controlled rheometers running LAOS-strain are identical. If inertial torque exceeds this limit, removing the inertial torque from the measurement will correct any errors in stress controlled LAOS-strain experiments. Therefore, we remove inertial torque from all measurements, and
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Bowditch–Lissajous curves and FT-rheology In order to qualitatively analyze non-linear viscoelasticity of hyaluronic acid, the reconstructed data is plotted in the stress– strain plane to create a Bowditch–Lissajous curve; a Pipkin space of these results is shown in Fig. 4. The hyaluronic acid has clear elliptical curve signifying viscoelasticity for the 1 rad s1 10% strain amplitude test. As strain is increased, curves become circular and the material is dominated by viscosity. Both 4 rad
Reconstructed stress of hyaluronic acid and model synovial fluids by first and third harmonic term of their raw stress response under strain amplitude 300% and angular frequency 4.5 rad s1. Red solid line: hyaluronic acid, black dash line: model synovial fluid. Fig. 3
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Fig. 4 Bowditch–Lissajous curves for LAOS-strain data at a range of applied strains and frequencies. The y-axis is stress, non-dimensionalized by max stress, and the x-axis is strain, non-dimensionalized by max strain. Red dashed lines are perfect circles representing an ideal viscous response. Black lines represent hyaluronic acid solution response.
s1 and 8 rad s1 exhibit similar behavior, but are more viscous at 10% strain. However, both maintain more elliptical curve shapes at larger strains, indicating a greater retention of elasticity. Overall, we conclude that the hyaluronic acid solutions are primarily viscous from these results. Nonlinear response in LAOS is also examined by FT-rheology using the ratio of the magnitudes of the third harmonic term to the rst harmonic term (Fig. 5). The transition from linear to nonlinear regions can be clearly distinguished by observing when this ratio becomes larger than 5%. At u ¼ 1 rad s1, the sample stays in the linear regime for strain amplitude smaller than 10%, followed by a transient area up to 30% then steps into nonlinear region nally. Similar behavior is observed for 4 rad s1 and 8 rad s1; however the transition to non-linear response occurs at smaller strain amplitudes, of 5% and 2% respectively and the transition region becomes smaller.
Percentage ratio of third to first harmonic of hyaluronic acid solutions for all strains tested at 1, 4, and 8 rad s1. Based on results we can see the transition to non-linear behavior when this ratio is above 1%. Error bars are found using the sampling frequency of the rheometer (1 kHz), which creates an error of 7% on the intensities which is propagated to their ratio. Fig. 5
storage and loss moduli at all frequencies tested. The degree of variations in the moduli clearly increases with strain amplitudes. Furthermore, the degree of variation increases with the applied frequency.
Time dependent moduli Using the sequence of physical process method, the instantaneous time dependent loss and storage moduli are shown in Fig. 6 for 1 non-dimensional period. Looking at both the loss and storage moduli at all amplitudes, it is observed that moduli magnitude increases with frequency. Overall, the magnitudes of the moduli do not increase with strain amplitude, and we see nominally identical average values of the moduli at all strain amplitudes for a given frequency. Examining the time dependent behavior, we start rst with the 10% strain amplitude data. For 1 rad s1 and 4 rad s1, the time dependent moduli response is essentially at; at these frequencies and strain amplitudes, the system is responding with single moduli during all stages of oscillation. There is some variation of the moduli in the 8 rad s1 data. Examining the 180% and 300% strain amplitude data, we see signicant variation during oscillation. Time dependence is seen for both
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Time dependent loss moduli (left column) and storage moduli (right column) plotted vs. time, non-dimensionalized by period, for a single cycle of LAOS for 1, 4, and 8 rad s1. As strain increases, there is a clear increase in the variation of the moduli during the cycle. Frequency also clearly increases the magnitude of these moduli. Fig. 6
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Examining the behavior in more detail, there is a clear periodic response, most evident in the 300% strain amplitude data. This periodicity is seen in both moduli; however, the two moduli do not behave identically or in sync. The loss moduli shows a simpler behavior with 2 peaks and 2 valleys and a small plateau region. The storage moduli show a more complicated response with 2 maximum peaks and 2 smaller sub-peaks between the true maximums. It is important to note that in essence all of the responses at varying frequencies for the 300% strain amplitude show the same basic behavior, but with increasing magnitude and degree of variation with frequency. Looking at the tan(d) data (Fig. 7), we can further understand how these samples react to LAOS. At small strain amplitude 10%, all tan(d) data are close to constant, however periodic variation of tan(d) starts to appear as frequency is increased. The values of tan(d) also decrease with frequency, signifying that the solutions are becoming more elastic with increasing frequency. However, the magnitude of all tan(d) is well above 1; hence, the solutions are predominately viscous. As strain amplitude increases, we see increasing variation in tan(d) with time, with clear peaks of high viscosity, and valleys of increased elasticity. At no point are the solutions ever predominately elastic. Furthermore, we continue to see that as frequency is increased for all strain amplitudes, the value of tan(d) decreases, signifying increased elasticity. Overall, it is clear that the solution behaves most like a Newtonian viscous uid at low frequencies and small strain amplitudes, and shows increasing elastic behavior with increasing frequency, and increasing non-linear/ time dependent behavior with increasing strain amplitude. In order to further explore the nature of these time dependent moduli and their relation to applied deformation, the time-dependent dimensionless storage modulus normalized by their maximum value is compared to dimensionless strain
Fig. 7 Tan(d) vs. time, non-dimensionalized by period, for a single
cycle of LAOS for a range of applied strains and frequencies. As strain increases we see increased variation in tan(d). Furthermore, as frequency increases, the value of tan(d) decreases, signifying increased overall elasticity of the solutions.
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acceleration, while the loss modulus is compared to strain rate in Fig. 8. Looking at the storage moduli, timing of the peak elasticity occurs a few moments before the maximum imposed strain acceleration during oscillation, which for the sinusoidal signal applied is identical to the maximum imposed strain during oscillation. This behavior is evident for all frequencies tested, and is consistent throughout the non-linear regime. Examining the loss moduli, its peak values during oscillation do not align with those of the storage moduli; instead, they align with peak values of the imposed strain rate during oscillation. We can see that the loss moduli are in fact at a minimum (nearly zero) during the moment of peak imposed strain/strain acceleration when the storage moduli are at their maximum. Correspondingly, the max loss moduli occur with the minimum storage moduli.
Discussion In examining the non-linear behavior of the hyaluronic acid solutions, the FT rheology does not provide signicant physiological insight. As frequency is increased, the boundary of linear and nonlinear response regime appears at smaller strain amplitudes. Fig. 5 clearly shows the strain amplitudes where the transition to non-linearity occurs are far below typical motion of a walking knee, which is approximately 1000%. Therefore we can conclude that for any typical joint motion, synovial uid will be in the non-linear regime, and therefore the non-linear viscoelastic moduli are the values that should be considered to evaluate lubrication. The Bowditch–Lissajous curves provide some insight into the behavior of the tested solutions. For large strain amplitudes, the hyaluronic acid provides primarily a viscous response. As
Fig. 8 Time dependent loss moduli (left column) and storage moduli (right column) for 300% strain non-dimensionalized by their maximum value plotted vs. time non-dimensionalized by period for a single cycle of LAOS for 1, 4, and 8 rad s1. Strain rate and acceleration included on figures as well to show that max loss modulus occurs at max rate and max storage modulus occurs during max acceleration.
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suspected, solution viscosity is primarily responsible for lubrication during hydrodynamic regimes based on the dominance of the viscous response. However, small elasticity in the solution is observed with increasing strain amplitudes at higher frequencies; this indicates that the elasticity of the synovial uid may still play some role in knee mechanics at these increased strain amplitudes and frequencies. Examination of Fig. 6, 7 and 8, which use of the sequence of physical processes methodology, allow us to fully understand the role of the small elasticity observed in the Bowditch–Lissajous curves. Noting peaks in elasticity align with the largest imposed strain/acceleration during oscillation demonstrates an important time dependent role of elasticity when deection reaches its maximum position and/or during sudden acceleration. Indeed, during largest imposed strain/acceleration the elasticity of the synovial uid increases dramatically. This allows the uid to provide an immediate response to a force and act as shock absorber and provide enhanced protection to the knee and the synovial cavity; during this phase the hyaluronic acid shows much less viscous response. Due to the applied sinusoidal signal, it is impossible to deconvolve whether the response comes from the strain amplitude or acceleration. During peak velocity, when lubrication properties are most important, the viscosity of the solution peaks and elasticity goes to zero in order to provide the best shear lubrication mechanism for the knee. We see similar behavior by examining tan(d) of the system; its value shows a more gel like state of hyaluronic acid when strain approaches its peak which can provide more elasticity to synovial joints in order to immediately respond to force during peak strain amplitude/acceleration.
Conclusions Previous analysis of linear and steady shear properties of hyaluronic acid have ignored the typical large oscillation behavior of joints like the knee. Our results demonstrate that the nonlinear rheological properties that can be observed with techniques like LAOS-strain, are in fact the properties most relevant to actual human ambulation, since synovial uid is always in a non-linear viscoelastic regime during typical ambulation. This non-linear viscoelasticity of hyaluronic acid solution provides multiple functions to joints during large oscillatory deformation. Using LAOS-strain, the behavior of hyaluronic acid based model synovial uid is shown to have a physiologically relevant non-linear, time dependent storage and loss modulus that are timed to the moments of peak strain amplitude/acceleration and shear rate of the knee respectively. These moments of peak elasticity provide immediate response to sudden forces that may cause joints damage. During peak shear rates, hydrodynamic lubrication is most important to avoid joints damage; synovial uid displays peak viscosity to separate cartilage surfaces and reduce friction during these times. These behaviors apply only to the synovial uid and impact the synovial cavity. The viscoelasticity of cartilage, surrounding ligaments, and tissues cannot be concluded to behave in a similar way; however, these results indicate that time dependent moduli
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may be an appropriate way to examine the behaviors of these tissues as well. This information is clearly useful to better understanding the role of hyaluronic acid in protecting and lubricating the knee, and provides insight into how to better treat joint diseases that degrade the rheological behavior of synovial uid in future. Furthermore by applying a range of analysis techniques to understand LAOS-strain behavior of hyaluronic acid solution, we have shown that it is important to consider a range of analysis techniques when examining LAOS data. All methods can provide relevant and interesting information about nonlinear material properties; however, it is clear that by pursuing only a single methodology relevant information can be missed. In the case of this work, both Bowditch–Lissajous curves and FT-rheology did not exhibit any particularly interesting behaviors that would help explain the physiological function of synovial uid. However, using the sequence of physical processes method, we found clear increased elasticity and physiological signicance in the non-linear response of the hyaluronic acid solution.
Acknowledgements The authors are indebted to Dr Simon. A. Rogers for discussions on the use of sequence of physical process methods in this research.
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