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Mater Sci Eng C Mater Biol Appl. Author manuscript; available in PMC 2012 May 10.

Published in final edited form as:

PMCID: PMC3087607

NIHMSID: NIHMS260009

The publisher's final edited version of this article is available at Mater Sci Eng C Mater Biol Appl

Articular cartilage is the load bearing soft tissue that covers the contacting surfaces of long bones in articulating joints. Healthy cartilage allows for smooth joint motion, while damaged cartilage prohibits normal function in debilitating joint diseases such as osteoarthritis. Knowledge of cartilage mechanical function through the progression of osteoarthritis, and in response to innovative regeneration treatments, requires a comprehensive understanding of the molecular nature of interacting extracellular matrix constituents and interstitial fluid. The objectives of this study were therefore to (1) examine the timescale of cartilage stress-relaxation using different mechanistic models and (2) develop and apply a novel (termed “sticky”) polymer mechanics model to cartilage stress-relaxation based on temporary binding of constituent macromolecules. Using data from calf cartilage samples, we found that different models captured distinct timescales of cartilage stress-relaxation: monodisperse polymer reptation best described the first second of relaxation, sticky polymer mechanics best described data from ~1-100 seconds of relaxation, and a model of inviscid fluid flow through a porous elastic matrix best described data from 100 seconds to equilibrium. Further support for the sticky polymer model was observed using experimental data where cartilage stress-relaxation was measured in either low or high salt concentration. These data suggest that a complete understanding of cartilage mechanics, especially in the short time scales immediately following loading, requires appreciation of both fluid flow and the polymeric behavior of the extracellular matrix.

Articular cartilage is a complex macromolecular biopolymer [1-4]. The tissue constituents include type II collagen, the aggregating proteoglycan (aggrecan), and numerous other molecules such as decorin, superficial zone protein, and numerous collagens. Further, the hydrated tissue has a distinct zonal structure with defined orientations of collagen and precise localization of proteoglycans. Macroscopically, the cartilage ultrastructure gives rise to a unique deformable and load-bearing solid with low friction and wear characteristics at the articular surface. Unfortunately, disruption of the tissue structure during the course of degenerative diseases, such as osteoarthritis, results in altered mechanics and severe tissue wear.

Historically, cartilage viscoelasticity has been separated into flow-dependent and flow-independent contributions [5, 6]. The flow-dependent contribution results from fluid flowing through a permeable solid matrix and occurs on a long timescale as a result of the low permeability of cartilage [7]. The flow-independent portion of cartilage relaxation includes viscoelasticity not resulting from fluid flow and is often modeled phenomenologically using Fung's quasi-linear viscoelastic model [8, 9]. Previously, we demonstrated that the relaxation function derived for simple (monodisperse) polymeric solutions (i.e. reptation), where all polymers are assumed to have the same molecular weight, was able to model the early stress relaxation of articular cartilage in unconfined compression [10]. Although a generally-accepted mechanistic explanation of flow-independent viscoelasticity remains to be determined, this work remains [10] as one candidate mechanism.

While our previous work modeled cartilage stress relaxation by monodisperse reptation, it is well-known that cartilage macromolecules are polydisperse (e.g. [11]) and likely exhibit complex molecular interactions. Considering the complex nature of cartilage structure, it is possible that a polydisperse polymeric solution, where the polymers are assumed to have a distribution of molecular weights, and thus a more complex set of interactions, provides a better description of relaxation data. Additionally, the incorporation of fluid flow in the numerical modeling is essential considering the flow-dependent contributions to viscoelasticity for the hydrated tissue. However the extent to which any of these mechanistic models describes the time course of relaxation is unknown. Thus, our knowledge of cartilage viscoelasticity, especially for timescales immediately following loading, is largely incomplete.

The purpose of this paper is to investigate the relative contributions of monodisperse and polydisperse polymeric models, as well as fluid flow, to cartilage viscoelasticity. Herein, we first compare monodisperse and polydisperse models for short-term stress relaxation, then combine the polymer model with the well known KLM fluid flow model [5, 12] to separate relaxation caused by flow-dependent and flow-independent mechanisms. The objectives of this study were to (1) examine the timescale of cartilage stress-relaxation using different mechanistic models and (2) develop and apply a novel (termed “sticky”) polymer mechanics model, based on temporary binding of constituent macromolecules, to cartilage stress-relaxation. We further tested the sticky polymer model using experimental measurements of cartilage stress relaxation at both low and high salt concentrations, under the hypothesis that high salt concentration would screen temporary electrostatic bonds.

Many biological molecules are long relative to their diameters. A large length-to-diameter ratio is characteristic of structural molecules in connective tissues [13] and many engineering polymers [14-17]. Mechanical properties in a monodisperse system, or one where all polymers are assumed to have the same molecular weight, are governed by molecules that are long, flexible, and continuously changing conformation [18]. Reptation describes this constant motion as “every [polymer] chain, at a given instant, is confined within a ‘tube’ as it cannot intersect the neighboring chains. The chain thus moves inside the tube like a snake.” [18] (Figure 1). Stress relaxation based on reptation theory assumes that the stress relaxes with the movement of the polymer chains in their tubes [14]. After creation of the tube, the chain moves from the original tube by diffusion. For a stress relaxation experiment it is assumed that the stress is proportional to the *fraction of the chain remaining in the original tube*. For a monodisperse, reptating system, the fraction of the chain remaining in the initial tube (equivalent to the relaxation function) is

A system of polydisperse polymers permits constrained motion of constituent molecules. Lateral entanglement constraints (due to, for example, molecules shown as dots and oriented along *z*) allow polymers to move most easily along their own length, a motion **...**

$$\psi (t;{\tau}_{d})=\frac{8}{{\pi}^{2}}\sum _{p=1,3,5,\dots}\frac{1}{{p}^{2}}\text{exp}(-{p}^{2}t/{\tau}_{d})$$

Equation 1

where *τ _{d}* is the characteristic disengagement time for the chain to escape from its tube,

Cartilage is a heterogeneous material made of multiple polydisperse polymers with various binding interactions [19-21]. Subsequently, we demonstrate how three additional complexities in a polymer model separately result in a stretched exponential stress relaxation function:
$\psi (t;{\tau}_{\mathit{\text{K}}WW},\beta )=\text{exp}(-{(t/{\tau}_{\mathit{\text{K}}WW})}^{\beta})$. The stretching parameter (*β*) depends on the polymer mechanism being modeled and may provide insight into active polymer mechanisms in articular cartilage.

The polymer relaxation function (Equation 1) was derived assuming that all of the molecules are linear and of the same length (i.e. a monodisperse system). In a natural system such as cartilage there are different-sized biopolymers interacting to result in macroscale mechanical properties [22]. The issue of a *polydisperse* reptating polymer system was examined by de Gennes who determined the relaxation function for a system with an exponential molecular weight distribution [23]

$$M(t)\approx B(t)\text{exp}(-{(ft)}^{\beta})$$

Equation 2

where, *f* is an average relaxation rate, *t* is time, and *β*=*x/(x*+*1)* where *x*>*0* is the weight distribution exponent. De Gennes' derivation was for broad classes of reptating systems and demonstrated that for exponential weight distributions that a stretched exponential (or Kolrausch-Williams-Watt, abbreviated KWW) relaxation function is expected for temperatures above the glass transition temperature. In the original derivation, the exponent *β* was predicted to be temperature-independent with a range between 0.25 and 0.33 for the specific weight distributions considered.

Stretched exponential behavior also can arise from mechanisms other than reptation of a polydisperse system. Another source of stretched exponential behavior occurs when there are barriers to molecular motion (e.g. steric interference or temporary bonding). For example, Edwards and Vilgis [24] examined the case of long rod molecules that interfere with each others' motions and demonstrated that stress relaxation behaved as
$\sigma ={\sigma}_{0}\text{exp}\left[-{(t/\tau )}^{1/2}\right]$, where *σ _{0}* is the initial stress and

Motivated by the work of Edwards and Vilgis, we examined the case of a linear polymer having *N _{T}* temporary binding sites with probability

$${N}_{m}=\left(\frac{{(1-p)}^{2}{p}^{m}}{1-{p}^{{N}_{T}}}\right){N}_{T}\phantom{\rule{1em}{0ex}}m=1,2,\dots ,{N}_{T}$$

Equation 3

For *N _{T}* of a reasonable size, there will always be at least

$$\varphi (t)=\sum _{m}\frac{{N}_{m}}{\text{\u2211}_{m}{N}_{m}}\Psi (t,{L}_{m})=\frac{1-p}{\left(1-{p}^{{N}_{T}}\right)p}\sum _{m}{p}^{m}\Psi (t,{L}_{m})$$

Equation 4

In Equation 3, the number of sites of length *L _{m}* is seen to be proportional to the probability of the sequence “0111…1110” with “

$$\varphi (t)=\frac{(1-p)}{(1-{p}^{{N}_{T}})p}\sum _{m=1}^{{N}_{T}}{p}^{m}\text{exp}(-at/{L}_{m})$$

Equation 5

This can be fit well by a stretched exponential function with 0.58<*β*<0.69 for many combinations of parameters (Electronic Supplement, Appendix 2).

In summary, the different complex polymer models result in distinct predictions for the stretched exponential stretching parameter *β*. The polydispersity model of de Gennes resulted in a prediction for the stretched exponential exponent of 0.25<*β*<0.33, the Edwards and Vilgis “steric interference” model makes the prediction *β* = 0.5, and our model for temporary bonding results in the range 0.58<*β*<0.69. With these disparate results it is appropriate to test whether experimental cartilage stress relaxation data is consistent with any of the models.

The flow of interstitial fluid in cartilage caused by pressure gradients describes flow-dependent stress relaxation. For unconfined compression of a linear poroelastic cylinder of radius *r* filled with an inviscid fluid, the relaxation kernel is

$${\psi}_{\mathit{\text{K}}LM}(t;\upsilon ,\frac{{H}_{a}k}{{r}^{2}})=\sum _{n=1}^{\infty}{A}_{n}(\upsilon )\text{exp}\left(-\frac{{\alpha}_{n}\cdot {H}_{a}\cdot k\cdot t}{{r}^{2}}\right)$$

Equation 6

where the coefficients of the series are *A _{n}(ν)* = ((1-

Benchtop and *in silico* experiments were performed to investigate the relative contributions of monodisperse and polydisperse polymeric models, as well as fluid flow, to cartilage viscoelasticity. Monodisperse and polydisperse (i.e. stretched exponential) models were first evaluated using short-term stress relaxation data on a timescale (less that 60 seconds) that is considerably shorter than expected for fluid flow [7, 27]. Second, an elastic-matrix—inviscid-fluid-flow—stretched-exponential model evaluated full-term stress relaxation data (to 1800 seconds) to determine (1) the time scales where fluid flow and polymer dynamics were dominant and (2) whether the stretching parameter (*β*) of the model was consistent with any of the specific theoretical polymer mechanisms. Finally, we evaluated the ability of the sticky polymer model to predict results under either low or high ionic concentration [28].

Five mm diameter osteochondral plugs were aseptically harvested within 6 hours of slaughter from load bearing regions of the lateral femoral condyles of bovine calf stifle joints using a cork borer. The subchondral bone was removed, and the cartilage plugs for the short-time tests were equilibrated at 37° C for 5 days in chemically-defined media composed of DMEM/F-12 supplemented with 0.1% (v/v) BSA, 100 units/ml penicillin, 100 μg/ml streptomycin, and 50 μg/ml ascorbate-2-phosphate [29]. On the day of short-time testing, tissue thickness (4.39±0.0254 mm) was measured with a dial indicator (Harbor Freight, Camarillo, CA Model 623-0VGA). All samples were the same thickness within the precision (25.4 microns) of our measuring equipment, which resulted in less than 3% error in the level of applied strain. Mechanical testing was performed in unconfined compression (Enduratec ELF 3200, Bose Electroforce, Eden Prarie, MN) using polished stainless steel platens. For the long-time experiments, samples were equilibrated under a 5N preload before application of a 5% nominal compression for 1800 s. For the short-time experiments, a 20% compression was applied for 1 minute.

Stress-relaxation tests were performed in unconfined compression. A 20% nominal compressive strain was applied with polished stainless steel platens for 1 minute in a media bath using an Instron 8511 Materials Testing System. Short-time (60 s) stress-relaxation data were obtained from twelve independent samples taken from twelve separate joints. Strain was applied in approximately 0.1 seconds, and force was recorded at 200 Hz for 60 seconds following the application of strain. The relaxation portion of the data (stress data following the peak stress) was used to determine the applicability of the relaxation models, described below. Next, data were fit to two different models, a monodisperse relaxation function

$$\sigma (t)={S}_{0}{\sum _{p,\mathit{\text{odd}}}\frac{8}{{p}^{2}{\pi}^{2}}e}^{-{p}^{2}t/{\tau}_{d}}+{S}_{1}$$

Equation 7

where *p*=1,3… is an odd index counting the terms used to approximate the infinite sum of the reptation relaxation function, *τ _{d}* is the characteristic disengagement time, and the term

$$\sigma (t)={S}_{0}{e}^{-{(t/{\tau}_{\mathit{\text{K}}WW})}^{\beta}}+{S}_{1}$$

Equation 8

where *S _{0}* and

The 5 mm diameter samples were collected in the same manner as described above and equilibrated in the same chemically-defined medium for 30 minutes after harvest and prior to testing. A total of 10 samples were tested in this experiment, coming from the medial and lateral patellofemoral grooves midway between the proximal and distal boundaries from 5 independent joints. Testing was performed using polished stainless steel platens in unconfined compression with an Enduratec ELF 3200 electromechanical testing system. The specimens were submerged in culture medium during the compression. A 5N compressive preload was applied for 10 minutes after which the initial height of the sample was measured. Subsequently, a 5% compression was applied at a nominal rate of 5 mm/s (load was completed in 0.2-0.3 s). Force data were sampled at 200 Hz for the first minute of relaxation, and 60 Hz for the following 29 minutes. From comparison of the monodisperse and stretched exponential models it was determined that the latter was better for modeling the full 60 seconds of relaxation. We therefore decided to determine the relative contributions of the stretched exponential, elastic matrix and inviscid fluid flow models to 1800 seconds of relaxation data.

The inviscid fluid-elastic model was composed as the KLM relaxation kernel plus a constant term:

$$\sigma (t){S}_{\infty}\left(1+\sum _{n=1}^{\infty}{A}_{n}(\upsilon ){e}^{-{\alpha}_{n}Bt}\right)$$

Equation 9

where, *S _{∞}, B*, and

$$\sigma (t)-{\sigma}_{\mathit{\text{K}}LM}\phantom{\rule{0.1em}{0ex}}(t)={S}_{\mathit{\text{K}}WW}\phantom{\rule{0.1em}{0ex}}{e}^{{-(t/{\tau}_{\mathit{\text{K}}WW})}^{\beta}}$$

Equation 10

where, *S _{KWW}* >

The sticky polymer model predicts that stress-relaxation will proceed faster when the probability of temporary binding is decreased. We sought to validate this model using cartilage stress-relaxation data obtained in either low or high ionic strength solutions, assuming that the high ion concentration would minimize temporary bonds associated with the anionic glycosaminoglycans chains of aggrecan. Cartilage explants were subjected to repeated stress-relaxation tests [28]. The first test occurred in low-ionic strength solution (either 0.15M NaCl or 0.075M CaCl_{2}, *n*=8 samples). The second test occurred after equilibration in high concentration solution (either 1M NaCl or 0.5M CaCl_{2}).

To fit the temporary binding model to the stress-relaxation data, we used nonlinear optimization in MATLAB. A cost function was constructed using the squared residuals between the data and the model of Eq. 14:

$$\sigma ={S}_{o}(\frac{1-p}{p(1-{p}^{{N}_{T}})}\sum _{i=1}^{100}{p}^{i}{e}^{-ti{l}_{o}/a})+{S}_{\infty}$$

Equation 11

In Eq. 14, *p* represents the probability that each of the *N _{T}* binding sites is unbound,

Both monodisperse and polydisperse models provided good visual fits to the 60 second data (Figure 2). However, the polydisperse model had a smaller mean SSE than the monodisperse model for the entire data range (3.5±1.6 versus 465.5±60.9 MPa^{2} p<0.05, t-test), indicating that the stretched exponential is a better model for 60 seconds of relaxation. Sequential fitting of the sixty seconds of data from the twelve specimens found different results between the two relaxation functions. Consistent with previous results [10], the reptation model showed a minimum error when a short period of data was fitted (0.176±0.061 s; Figure 2B). Additionally, the reptation model had a smaller normalized error than the stretched exponential for times less than approximately one second. The stretched exponential did not show a minimum within the sixty seconds of relaxation (Figure 2B). Therefore, we chose to use the stretched exponential function to test the relative contribution of polymer mechanics and inviscid fluid flow over long term (1800 s) relaxation experiment.

Rapid loading was sufficiently fast to avoid relaxation during compression, as indicated by the linear portion of the relaxation test (r^{2}>0.99) (Figure 3). Further, for all start times, the model (Equation 10) provided good fits to the data (r^{2}>0.975 (Figure 5). However, the KLM portion of the fit showed a rapidly increasing error for start times shorter than ~100 seconds (Figure 4A). Permeability estimated from the KLM portion of the fit (Figure 4D) was dependent upon the fit start time. For start times ~100 seconds and longer, the permeability was similar to literature values [30]. The rapid increase in average error, the presence of an “optimum” start time of approximately 100 seconds, and the convergence of the fitted permeability values for start times greater than 100 seconds, indicated that the KLM model (inviscid fluid—elastic matrix) was able to model the *temporal* majority (from ~100-1800 s) of the relaxation (Figure 4). However, stress relaxation prior to 100 seconds was best fit by the stretched exponential portion of the fit (Table 1, Figure 5). The average quality of the fit (Table 1) was excellent for the total model (coefficient of determination 0.9998±0.0002) with average error for the KLM portion of the same magnitude as the underlying noise in the load cell (~30 mN). The average time constant for the KLM portion of the curve was significantly longer (p>0.05) than that of the polymer component. The average stretching parameter, *β*, of the fits (Table 1) was 0.65±0.04, within the range of the sticky model.

Rapid loading allows observation of fast stress-relaxation dynamics. Example of load-displacement data for 1800 seconds of relaxation (all numbers negative by sign convention). The load application region of the curve is nearly linear because loading **...**

Fluid flow (KLM) model fits long-term cartilage stress-relaxation data. The KLM model can represent the majority of relaxation data (certainly all relaxation after about 100 seconds) but cannot accurately fit early relaxation. This observation is consistent **...**

Polymer models fit short-term and fluid-flow models fit long-term cartilage stress-relaxation. Example of total curve fit to stress relaxation data. Circles represent data downsampled by 200, and line represents the total model including both polymer **...**

Curvefitting the temporary binding model found larger fitted values of unbound probability (*p*) at higher salt concentrations, providing experimental support for the temporary binding mechanism represented by the model. For samples tested in 0.15 M NaCl *p* was smaller than samples tested in 1M NaCl (for both biased and unbiased methods p<0.02, Figure 6A-C). For samples tested in 0.075 M CaCl2 *p* was smaller than samples tested in 0.5M CaCl2 (both p<0.01, Figure 6D-F).

Cartilage is composed of heterogeneous polydisperse biopolymers with multiple complex interactions. To determine the molecular mechanisms of cartilage viscoelasticity, models must be based on specific molecular interactions. In addition, rapid initial loading is necessary to capture the fast relaxation modes that occur on timescales shorter than 100 seconds. In this study, we investigated the relative contributions of monodisperse and polydisperse polymeric models, as well as fluid flow, to cartilage viscoelasticity. Accordingly, the objectives of this study were to (1) examine the timescale of cartilage stress-relaxation using different mechanistic models, and (2) develop and apply a novel (termed “sticky”) polymer mechanics model, based on temporary binding of constituent macromolecules, to cartilage stress-relaxation. These different models necessarily have different numbers of free parameters which reflect the differences in the underlying theories and may result in empirical fitting capabilities that differ between models. More precise separation of the model timescales may be possible using additional methods such as Akaike's Information Criterion [31].

Stress relaxation in articular cartilage is determined by distinct mechanisms, each associated with different time scales and molecular dynamics. At the earliest times (up to ~0.2 s, Figure 2), the monodisperse reptation polymer model provided the best fit to the data, consistent with our previous research [10]. Interestingly, it is possible that inclusion of constraint-release [32] or contour length fluctuation [33] concepts may improve these good fits. This observation supports the hypothesis that for short times after loading molecular motion of cartilage constituents is similar to reptation.

The subsequent timescale (~0.2-100 s) was best described by the stretched exponential model. Most of the stress dissipates during this regime, and one possible mechanism underlying the stretched exponential relaxation is the temporary binding of macromolecules (Equation 11). The finding that curvefits of this model result in larger unbound probabilities for high-salt stress-relaxation provides experimental support for temporary binding mechanisms in cartilage viscoelasticity. The mechanism of temporary binding may be ionic crosslinks between anionic glycosaminoglycans [34] although other salt-induced phenomenon including osmotic pressure may be relevant. Importantly, this model is consistent with other polymer mechanics treatments of temporary binding [35, 36], although further experimental verification, including the possible use of spectroscopic analysis to quantify temporary binding, would further support the relevance of this mechanism to cartilage mechanics.

Finally, the slowest timescale of cartilage stress-relaxation accounted for the temporal majority (~100-1800 s) and was best described by the fluid-flow model. The results of the curvefits using a combined polymer and KLM approach (1) fit the data well (average coefficient of determination 0.9998), (2) provided estimates of the permeability consistent with literature values, and (3) provided insight about which polymer mechanisms might be active in the medium term.

The observation that stress relaxation data separate into a “fluid” and “polymer” regime at approximately 100 seconds is consistent with the experimental results of Wong *et al* [37] who found that for stress relaxation in unconfined compression after rapid loading that apparent stress was proportional to specimen volume only *after* ~60 seconds of relaxation. Theory for a linear elastic porous material interacting with an inviscid fluid (KLM) finds that stress is linearly related to the specimen volume for unconfined compression of a cylinder. Our analysis (Figure 4) demonstrates that KLM becomes less precise when fitting data prior to 100 seconds of relaxation. In fact, we recapitulate the results of Wong *et al* (one minute cutoff point) if we use the optimum total coefficient of determination to define the beginning of the fluid-flow timescales (Figure 4B). The consistent findings of this study and Wong *et al* strongly support the idea that fluid flow occurs on a slow timescale.

The polymer relaxation approach we present here complements other research using finite element models. For example, the paper of Li et al [38] finds that mesoscopic molecular structures (e.g., Benninghoff arcade-like structures) in cartilage result in a tension-compression anisotropy of total relaxation. However, that study uses phenomenological viscoelasticity theory to describe the behavior of the collagen matrix and was not intended to address the underlying molecular nature of viscoelasticity. In contrast, our current study presents candidate mechanisms that, when further studied, could provide additional insight into the molecular nature of flow-independent viscoelasticity in cartilage.

We found that the time course of stress relaxation after rapid loading in articular cartilage is dominated by distinct mechanisms. The first regime (~0-0.2 s) was best fit by a monodisperse polymer model (reptation). The second regime (~0.2-100 s) was best described by a polydisperse polymer model (stretched exponential). One candidate mechanism underlying this model is temporary binding of macromolecules, and was presented herein as a “sticky” polymer model. The third regime (~100-1800 s) accounted for the temporal majority of the stress-relaxation and was consistent with fluid flow through a porous elastic matrix. Neither polymer dynamics alone nor fluid flow alone were sufficient; both models were necessary to describe the experimental data, indicating that both mechanisms are likely critical in cartilage viscoelasticity.

The authors thank Dr. A. Hari Reddi for kind support in the development of cartilage culture methods and for access to his laboratory facility, and Mr. Roger Zauel for implementation of the C++ curvefitting routines. Code is available from the senior author upon request.

** Conflict of Interest:** The authors have no conflict of interest.

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