Rheumatology 2001; 40: 274-284
© 2001 British Society for Rheumatology
The relationship of the compressive modulus of articular cartilage with its deformation response to cyclic loading: does cartilage optimize its modulus so as to minimize the strains arising in it due to the prevalent loading regime?
Rheumatology and Rehabilitation Research Unit, University of Leeds, 36 Clarendon Road, Leeds LS2 9NZ, UK
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Abstract |
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Aim. To investigate the relationship of the instantaneous compressive modulus with its deformation response to cyclic loading typical of that encountered at the knee joint during level walking.
Method. The study was performed on 24 osteochondral plugs taken from three unembalmed cadaveric knees. As the compressive modulus of cartilage has been shown to vary topographically across the knee in an established manner, the specimens were taken from specific sites on the femur and tibia of each knee. All the cartilage specimens were immersed in Hanks’ salt solution at 37°C and were subjected to the same cyclic loading regimen that was representative of a typical walking cycle in a specialized indentation apparatus, for over 1 h.
Results and conclusion. The viscous and elastic components of matrix strain, the creep rate and the cartilage compressive modulus were measured. The latter was found to be significantly related to the strain response of cartilage to cyclic loading. Elastic strain varied exponentially with the compressive modulus; specimens with a modulus less than 4 MPa experienced elastic strains in the range 0.18–0.36, whereas stiffer specimens experienced strains between 0.05 and 0.13. Viscous strain varied linearly with cartilage stiffness and was as low as 0.02 at the lower values of the compressive modulus but increased to 0.22 for a compressive modulus of 18 MN/m2. The rate of creep under cyclic load was inversely linearly related to cartilage stiffness. The strain response of soft specimens approached steady state by 200 cycles but that of stiff specimens did not approach it until 1300 cycles. It was hypothesized that the viscous strain response of cartilage can be explained in terms of differences in permeability between specimens of different compressive modulus, stiffer cartilage having a lower permeability than soft cartilage.
KEY WORDS: Articular cartilage, Compressive modulus, Cyclic loading, Strain, Fluid flow.
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Introduction |
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The deformation of cartilage has been studied for the most part under static loading, and the various studies have shown that cartilage exhibits viscoelastic behaviour [1–5]. The viscoelasticity of cartilage tissue has been explained in terms of interstitial fluid flow within the cartilage matrix and of the intrinsic viscoelasticity of the matrix itself [6–9].
During locomotion and exercise, cartilage is subjected to cyclic loading and the loads are applied at fast rates, the rise times being 10–150 ms [10], and act for short durations (10–400 ms) [11, 12, 34]. The deformation response of cartilage under cyclic loading conditions has received limited attention. Basic experimental measurements have been attempted [14–19], while various models have been used to predict cyclic strain behaviour [20–22]. Maroudas [23] made predictions as to the likely deformation response of cartilage under cyclic loading conditions. She postulated that recovery between load cycles is likely to be incomplete because the recovery rate would be governed by the swelling pressure, which would be much lower than the physiological stress caused by the cyclic load that is acting.
The deformation of cartilage and rate of flow of interstitial fluid through its matrix must each be a function of the parameters of the duty cycle as well as of cartilage properties. Parameters of the duty cycle include: (i) the loading rate; (ii) the frequency of load application, (iii) the amplitude of the applied stress; (iv) the ratio of the loading period to the recovery period within a loading cycle; and (iv) the number of loading cycles. The property of cartilage most relevant to its deformation would be its compressive modulus.
This study focuses on the effect of the compressive modulus on cartilage deformation under cyclic loading conditions—particularly those arising during level walking, which is the predominant human activity. One important reason for focusing this investigation on the effect of the compressive modulus on cartilage deformation rather than on the other parameters mentioned above is the greater variation in the magnitude of the compressive modulus of cartilage compared with the other parameters. The modulus ranges from 1 to 20 MN/m2, whereas the frequency of loading falls within a narrow band, between 1 Hz (during slow walking) and 2.25 Hz (during sprinting). Furthermore, the average stress in the knees during walking is 1–1.5 MPa.
It is interesting that the relationship between cartilage deformation and its modulus has not been subject to any detailed experimental study. This might well be due to an expectation that such a relationship would obviously be of an inverse, perhaps monotonic nature and therefore in itself would not be of interest. However, since cartilage viscoelastic behaviour, as mentioned earlier, is contributed to by the intrinsic viscoelastic nature of the cartilage matrix and by interstitial fluid flow in undefined proportions, cartilage behaviour under cyclic loading is complex and may not be as predictable as was at first thought.
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Materials and methods |
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Materials
We used eight cartilage specimens from each of three unembalmed human cadaveric knee joints. When these had been procured they were immediately stored at -20°C, and before use they were thawed overnight at room temperature. Each knee was dissected to remove all soft tissue and to expose the cartilage surfaces of the femoral condyles and tibial plateaux. Meachim's Indian ink test [24] was performed to identify any areas of fibrillated cartilage, and only healthy areas of cartilage were selected for testing. Osteochondral plugs 12 mm in diameter and 12 mm long were removed for testing by the use of a specially designed reamer. Eight such specimens were harvested from each knee joint from specific sites on the femur and tibia, as illustrated in Fig. 1
. Cartilage specimens from these sites have been shown to have very different stiffnesses [25, 26], which allowed the effect of cartilage stiffness on its deformation response to be studied.
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Methods
Specimen alignment.
The osteochondral specimen, which was 12 mm in height, was secured in a specimen- holder with dental cement (Fig. 2a), leaving the cartilage layer exposed and protruding above the upper surface of the specimen-holder (the cartilage was thus unconstrained at its edges). The cartilage specimen was then positioned beneath the tip of the indenter in the test apparatus. Using both the naked eye and an arthroscopic camera (and monitor) to view the indenter tip and cartilage surface, fine adjustment of the alignment was made using three screws on the base of the specimen-holder (Fig. 2b). Throughout the alignment procedure, the cartilage specimen was kept moist by irrigating it with Hanks’ balanced salt solution (HBSS). The accuracy of the alignment technique was better than 1°. Once perpendicular alignment of the cartilage surface to the indenter had been achieved, a heating stage was fixed to the specimen-holder. This heating stage also formed a chamber around the cartilage specimen, so that it could be immersed in HBSS maintained at 37°C by circulating heated water in the heating element.
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Measurement of cartilage modulus.
The stiffness of each specimen was measured in terms of its 50 ms compressive modulus, which is representative of the physiological loading rate encountered during walking. It is based on the deformation of cartilage 50 ms after initial contact of the indenter tip with the cartilage surface. The calculation of the modulus required three measurements to be obtained: the cartilage deformation after 50 ms of loading, the applied load after 50 ms of loading and the cartilage thickness. The indentation and load measurements were performed twice on each specimen, once before and once after performing the cyclic loading experiment. This was done in order to check if the repetitive loading had damaged the specimens. As the rate of loading in these modulus measurements was high (rise time 50 ms or less) and fluid flow (if any) would not be significant within such intervals (the measurement was undertaken in a single indentation), it was justified to treat cartilage as an elastic material and to use an impervious indenter. This latter was hemispherical and had a radius of 1.5 mm. It is appropriate to comment on the effect of the curvature of the cartilage of the various osteochondral plugs tested on the calculated values of the cartilage modulus. The equation from which such values are calculated assumes contact between a hemispherical rigid indenter and a flat cartilage surface. The curvature of the specimens will vary from 20 mm on the femoral condyles to 70 mm on the tibial plateaux (the medial being concave and the lateral convex). At the most, the equivalent radius based on the measurements would be lower by 7.5% for a surface of 20 mm radius, such as the femoral condyle, and by 2% for a surface of a 70 mm radius, such as that of a tibial plateau. Such variations in contact geometry will result in a small error in the calculated value of the compressive modulus. The test parameters were identical to those used in two previous studies from our laboratory [26, 27], which induced physiological stresses.
The cartilage thickness was measured using a technique that employed a needle, which caused disruption of the surface of the cartilage specimen. For this reason, the thickness measurement was left until all other testing had been completed so as not to violate the cartilage surface. This technique is described by Swann and Seedhom [25].
Deformation response under cyclic loading.
This was investigated immediately after we had measured the indentation for the determination of the modulus. Therefore, before the cyclic load was applied, the specimen was left for 30 min to ensure full recovery from the indentation caused during the modulus measurement. The hemispherical indenter used for this measurement was then replaced with a plane-ended impervious indenter of the same diameter (3 mm), in order to ensure that all cartilage specimens were subjected to the same contact stress. The duty cycle parameters were as follows: amplitude of applied stress 1.4 MPa (typical of the stress occurring during walking); frequency of load application 1 Hz; load rise time 20 ms; and ratio of loading duration to recovery duration 1:2 (loading period 330 ms, recovery period 670 ms) (Fig. 3). Load and displacement traces were recorded simultaneously during each of the first 64 consecutive cycles and thereafter during one complete cycle every 64 cycles, for a little over 1 h (these numbers were dictated by the electronics of the data acquisition system). In total, data were collected from 4160 loading cycles.
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Data analysis
Strain components.
The various load and displacement traces were then analysed using a Fortran programme producing two sets of data; one set consisted of the position of the cartilage surface at the instant of contact between indenter and cartilage at the beginning of each loading cycle. The other data set consisted of the position of the cartilage surface prior to the instant of cessation of contact between cartilage and the indenter, for each cycle, as described by Barker and Seedhom [13]. A typical set of these data points is shown in Fig. 4a and is described here in order to define the variables analysed. The curves exhibited two characteristic phases of the cartilage deformation under cyclic loading. The first phase occurs within the first 1000 cycles and the data show that the cartilage surface does not fully recover its deformation between individual loading cycles. This implies that more water is being exuded during the loading phase than is being imbibed during the load-free recovery phase. The curve is steepest during the first 200 loading cycles but the gradient reduces with the number of cycles, indicating a reduction in the amount of fluid being lost. The second phase occurs from 1000 cycles onwards. The curve heads asymptotically towards a steady state, during which cartilage is being depressed and then allowed to recover fully during each cycle. Analysis of the load and displacement during this steady state shows that there are two processes occurring. The first process, which accounts for most of the deformation, is that of instantaneous deformation of the solid matrix during the loading phase and an equal, instantaneous recovery during the recovery phase. These deformations are equal and opposite after the specimen has reached a steady state. The second process, which accounts for a small fraction of the total deformation, is that of fluid exudation during loading and fluid uptake during recovery. These small volumes of fluid flow are also equal. Hence, the overall deformation response at this steady state is termed ‘elastic’, because the cartilage surface position at the beginning of each cycle (i.e. when the indenter comes into contact with the cartilage surface) remains unchanged.
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The cyclic strain thus comprised two components: a viscous component and an elastic one (Fig. 4a
). The higher curve represents the strain that is purely due to the cumulative loss of fluid from the loaded region, which has been termed ‘viscous strain’ (
viscous). The difference between the upper and lower curves is the elastic strain of the solid cartilage matrix (elastic). The total matrix strain (total) experienced during a particular loading cycle, which is the sum of the viscous and elastic strain components, is represented by the lower of the two curves.
Cartilage creep.
The creep rate of each specimen throughout the test was also investigated. A two-term exponential curve (equation 1) was fitted to the total strain curve (total) of each specimen (Fig. 4b). The two data curves were very similar in shape. However, the lower one was used for the curve fit because the points generally displayed a smoother curve. The slight variation in the points of the upper curve was due to the inaccuracies that arise in measuring the surface position of the cartilage at such high approach velocities. These inaccuracies were not present on the lower curve as the indenter is virtually static at this point. The form of the curve fit was:
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total is total tissue strain, x is the number of cycles, and C0, C1, C2, t1 and t2 are constants. For each specimen, equation 1 was differentiated to investigate the gradient of the strain curve and hence the rate of cumulative strain:
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To obtain a measure of the number of load cycles applied to the specimen before reaching steady state, equation 1 was differentiated to yield equation 2. Solving equation 2 for C3=0, the value of x, provides a measure of the number of cycles applied until the gradient of the fitted creep curve (1) became zero (or steady state). Some samples were still undergoing creep after the 1 h of loading applied during this test series, so C3=0 could not be used. Instead, a value of C3=-0.003 was found to provide a solution across all curve fits and represented a gradient of creep curve very nearly approaching the steady state. The solution x was in fact the cycle number (Nss) at which this gradient occurred and was therefore calculated for each specimen. Figure 4b shows the exponential curve, which was fitted to the total data in Fig. 4a. The differential of this curve is also displayed. In this case it took 600 cycles before the steady state was reached.
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Compressive modulus values before and after cyclic loading tests
The compressive modulus for each of these specimens had a value that was consistent with the general pattern of topographical variation in knee cartilage modulus reported previously [25, 26, 28]. Thus, specimens from the femoral condyles were stiffer than specimens from the patellar surface of the femur. Likewise, specimens from the areas of the tibial plateau covered by the meniscus were stiffer than those from the areas that come into direct contact with the femur. In all cases the specimens from the area of the tibial plateau, which come into direct contact with the femur, were the softest. The modulus had a range of values between 1 and 19.5 MN/m2.
Figure 5 and Table 1 show the 50 ms compressive modulus values for each specimen before and after completion of the cyclic loading regimen. In one case the specimen had a significantly higher compressive modulus after testing, indicating that some tissue damage may have occurred to it during the cyclic loading, and this specimen was therefore excluded from further analysis. The sample was probably damaged as a result of the abnormal stress gradients imposed at the edge of the flat-ended indenter. The ‘before’ and ‘after’ 50 ms compressive modulus values determined for all the other specimens fell within the bounds of accuracy of the apparatus, and these specimens were therefore assumed to have been undamaged by the cyclic loading. Therefore 23 specimens were included in this study.
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Cartilage response to cyclic loading
Stiff and soft cartilage.
A comparison between typical deformation responses of soft and stiff cartilage to cyclic loading is shown in Fig. 6. A specimen with a high compressive modulus of 16.8 MN/m2 (Fig. 6a) is compared with a specimen of low compressive modulus of 3.1 MN/m2 (Fig. 6b). The viscous and total strain curves are plotted for each of these two specimens, together with their respective curve fits. The elastic strain is the difference between these two curves. The diagram highlights the significant differences in the cyclic deformation responses of the two specimens. The viscous strain of the softer cartilage is less than that of the stiffer cartilage, whereas its elastic strain during each cycle is much greater. Furthermore, the softer cartilage reaches its steady-state deformation sooner than the stiff cartilage. The results comparing the strain responses of all specimens demonstrate this behaviour consistently.
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Strain components and creep.
Best-fit curves were applied to the elastic strain, viscous strain and creep rate data. The elastic strain data were best fitted by an exponential curve, while the viscous strain and creep rate data were best fitted by linear regression. The equations of all the curve fits, together with their associated Pearson coefficients and significance values, are shown in Table 2. The Pearson coefficient and significance value associated with the exponential curve were determined by plotting the natural logarithm of the strain values against the compressive modulus to give a linear relationship. As shown in Table 2, the P values were all less than 0.001, indicating strong, significant correlations. The stiffness of the cartilage, expressed in terms of its 50 ms (instantaneous) compressive modulus, was thus significantly correlated to both the elastic and viscous strains, and also to the creep rate. As the compressive modulus increased, the elastic strain decreased exponentially, heading towards an asymptotic value of elastic strain between 0.05 and 0.1. Softer specimens with a compressive modulus less than 4 MN/m2 experienced elastic strains in the range 0.18–0.36, while stiffer specimens experienced strains between 0.05 and 0.13. The viscous strains increased linearly with the 50 ms compressive modulus. Soft specimens exhibited low viscous strains (0.02), while those of stiff specimens were up to 0.22. The creep rate was inversely and linearly related to the 50 ms compressive modulus. Soft specimens approached steady state in as few as 200 cycles, while stiff specimens took up to 1300 cycles.
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Data on the compressive modulus, elastic strain, viscous strain and creep rate are summarized in Table 1
and plotted in Fig. 7 for each of the 23 specimens.
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Total strain.
The total strain (sum of the elastic and viscous contributions) incurred by each specimen at the final load cycle is plotted in Fig. 8. The best-fit curve to the data was a second-order polynomial with a minimum at a compressive modulus value of approximately 10 MN/m2.
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Discussion |
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Few experimental studies have measured the deformation and fluid flow of articular cartilage during prolonged cyclic loading [14–19]. Hence, the response of cartilage to cyclic loading has been subject to conjecture and prediction based on experimental data from the application of a single load in uniaxial compression/ recovery experiments, and also on various mathematical models.
In his experiments on intact canine joints, Linn [14] showed that static loading caused an initial deformation followed by a creep response that took over 24 h to reach equilibrium. In contrast, he showed that when the load oscillated the deformation became constant, arriving at a load-specific value in 5–6 min. Simon [15] quoted few results but concluded that for a loading interval of 1 s the cumulative deformation was less than that resulting after continuous loading over the same period. Johnson et al. [16] and Higginson and Snaith [17] performed very similar experiments, and both concluded that the tissue response was elastic, fluid flow playing no part in determining the material properties or the behaviour of the tissue under short-term loading. Lee et al. [18] performed sinusoidal loading from 0.001 to 20 Hz and concluded that fluid flow was significant across the whole range of test frequencies, and showed that, at 1 Hz in confined compression, cartilage did not behave as a linear elastic solid. Torzilli [19] was in agreement with Lee et al. [18], and proposed that the deformation behaviour of articular cartilage due to an oscillatory load would be governed by the oscillatory movement of the interstitial fluid during each load cycle. Few results were quoted, but the observations made were similar to those of Linn [14] and Simon [29], namely that the deformation of the tissue reached a cyclic steady state faster than under a static load. In all cases, the accuracy of the measurement techniques was limited and few results were quoted. Varied loading cycles were employed, none of which closely resembled those experienced physiologically.
Maroudas [23] performed no experimental work in this area, but she postulated that recovery between load cycles was likely to be incomplete because the recovery rate would be governed by the swelling pressure. This pressure is always much lower than the physiological stress arising due to the loads acting.
This study measured for the first time the strain response, to prolonged cyclic loading, of knee cartilage specimens possessing a wide range of compressive modulus values. The data obtained show that the technique used is sensitive enough to measure the small residual deformations occurring over individual cycles. Two aspects of the results call for comment and explanation. The first of these is the marked difference in behaviour of stiff and soft cartilage. The data characterize the history of strain components (viscous and elastic) obtained through the entire test period and which revealed a steady state achieved after periods of loading that varied with the stiffness of the cartilage specimen. The second is the variation in the total strain of cartilage over the range of its compressive modulus. The total strain has a minimum value that corresponds to the mid-range value of the compressive modulus. This is different from what would be predicted intuitively; as the stiffness of a material increases its deformation under load is expected to decrease monotonically.
Strain response of soft and stiff cartilage
Considering the rates of fluid movement during the loading and recovery phases of the loading cycle can lead to an explanation of the observations made in the present experiments. During the loading phase, interstitial fluid moves away from the loaded region towards the unloaded region under the action of the pressure gradient caused by the difference between the applied stress and the cartilage swelling pressure. During the recovery phase, the direction of interstitial fluid flow is reversed, but the flow is driven by a much lower pressure gradient as the swelling pressure during recovery is much lower than the applied stress during the loading phase. Consequently, the recovery of cartilage between successive cycles is not complete, and a residual strain is observed. In the present experiment it was observed that, for a stiffer cartilage specimen, larger residual strains occurred (between individual cycles) than those observed for a softer one. Maroudas found that the stiffness and swelling pressure of cartilage are both directly related to its proteoglycan content [30]. Hence, after cartilage is subjected to compression, the swelling pressure of stiff cartilage during the recovery phase of the loading cycle must be higher than that of softer cartilage. Were both tissues (i.e. the stiff and soft cartilage) to have similar permeability, it would be expected that stiff cartilage should recover faster between successive load cycles than does soft cartilage. However, it was shown in the present study that the converse is true. Furthermore, permeability is directly related to cartilage stiffness, stiff cartilage having lower permeability [31–33]. Therefore, it is more likely that matrix permeability, which also controls interstitial fluid flow, will be dominant in controlling tissue recovery, and may thereby account for the much slower recovery of the stiffer cartilage.
The above explanation is based on the assumption that the solid matrix of cartilage is elastic. However, in reality it may be viscoelastic in part, and the effect of this on the total strain of cartilage is difficult to measure [6]. If this contribution were large it might instead be the dominant factor in the observed behaviour of cartilage. To clarify this ambiguity, an experiment was undertaken in which cartilage was subjected to the same cyclic loading regime, but a small tare load that was 1% of the maximum load was maintained throughout the test. Thus, the surface of the cartilage was not exposed to the fluid during the period of recovery but was in contact throughout the test with the flat-ended indenter. This indenter was of a weight that supplied the tare load, and transmitted the cyclic load to the cartilage surface via its free end. This continuous contact between the indenter and the cartilage surface drastically slowed the recovery of the deformation between cycles and so increased the time to attain the steady state by a factor of at least three [34]. We postulate that the contact between the indenter and cartilage, even under such a small tare load, has almost blocked the path of the fluid being imbibed by the cartilage matrix during the recovery phase of the loading cycle. Furthermore, the fluid path is further restricted by the decreased permeability of the surface layer caused by its significant compression, even under the action of such a small tare load [19]. Were the inherent viscoelasticity of the matrix a major contributor to the observed viscous behaviour, the recovery time would not have been greatly influenced by the presence of the tare load. It appears, therefore, that the interstitial fluid flow, which is governed by the permeability of cartilage, is the dominant factor in its viscous response to load.
The steady-state results similarly reflect the permeability effect. Steady state is attained when sufficient interstitial fluid has been lost from the loaded regions of cartilage, such that the stiffness of the resulting compacted cartilage is sufficient to carry the applied dynamic stresses. Tests showed that all cartilage specimens lost interstitial fluid to attain this steady state, but the softer cartilage reached it sooner than the stiffer. As would be expected of a more permeable tissue, the softer cartilage lost its excess interstitial fluid more readily than did the stiffer, less permeable cartilage.
Optimum stiffness: does cartilage adapt to the loading regime?
The total strain data gave rise to the observation of an optimum range of cartilage stiffness (8–12 MN/m2) within which cartilage incurred minimum strain—about 23%. Specimens with stiffness outside this range experienced higher total strains. Stiffer cartilage outside the range underwent higher total strains that were contributed to by large viscous strains, whereas softer cartilage outside the range underwent higher total strains due to the contribution of the large elastic strains. On the basis of this observation, it may be hypothesized that cartilage adapts its matrix constituents to be least susceptible to damage, as it minimizes the total matrix strain by optimally balancing the viscous and elastic strain contributions. Intuitively, softer cartilage would be prone to early failure because of large elastic strains, while, perhaps more surprisingly, stiffer cartilage may also be prone to failure after many load cycles due to excessive strains caused by viscous losses.
The results may thus have implications for chondrocyte biosynthesis. In softer cartilage the chondrocyte is subject to large elastic matrix strains coupled with little fluid flow, whereas chondrocytes in stiffer cartilage are subject to smaller elastic strains of the surrounding matrix coupled with greater local fluid flow. These differing combinations of matrix strain and fluid flow could be important mechano-transduction factors affecting the the local structure of tissue synthesized by the chondrocyte.
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Conclusions |
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The deformation of cartilage under compressive cyclic loading conditions has a complex relationship with its compressive modulus. The relationship is not an inverse but a bimodal one, and the total strain of cartilage (sum of the elastic and viscous components) had a minimum at the mid-range of the modulus determined at eight predetermined sites on the surfaces of the knee joint. The elastic strain component had an inverse exponential relationship with the modulus, whereas the viscous component increased linearly with increase in the modulus.
The viscoelastic behaviour of cartilage observed under the cyclic loading regime in this study was attributed to the interstitial fluid flow within the cartilage matrix and was explained primarily in terms of the permeability of cartilage rather than the intrinsic viscoelasticity of the matrix itself. This explanation is supported by data from a further experiment in which the total strain of cartilage was measured under the same cyclic loading conditions but in the presence of a small tare load on the cartilage. This delayed the recovery of cartilage between cycles by a factor of at least three, which was attributed to the presence of the tare load, which was transmitted via the flat-ended indenter and which presented the increased resistance in the path of the fluid being imbibed by the cartilage matrix during the recovery period between consecutive load cycles.
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Acknowledgments |
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The authors would like to thank to their technicians Brian Whitham and Michael Pullan for their assistance during the course of this work. This work was supported by a scholarship awarded by the University of Leeds.
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Notes |
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Correspondence to: B. B. Seedhom.
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References |
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