Strain-rate Dependent Stiffness of Articular Cartilage in Unconfined Compression

Transcription

1 L. P. Li* Biosyntech Inc., 475 Arand-Frappier Blvd., Park of Science and High Technology Laval, Quebec, Canada H7V 4B3 M. D. Buschann Departent of Cheical Engineering and Institute of Bioedical Engineering, Ecole Polytechnique of Montreal, Canada H3C 3A7 A. Shirazi-Adl Departent of Mechanical Engineering and Institute of Bioedical Engineering, Ecole Polytechnique of Montreal, Canada H3C 3A7 Strain-rate Dependent Stiffness of Articular Cartilage in Unconfined Copression The stiffness of articular cartilage is a nonlinear function of the strain aplitude and strain rate as well as the loading history, as a consequence of the flow of interstitial water and the stiffening of the collagen fibril network. This paper presents a full investigation of the interplay between the fluid kinetics and fibril stiffening of unconfined cartilage disks by analyzing over 200 cases with diverse aterial properties. The lower and upper elastic liits of the stress (under a given strain) are uniquely established by the instantaneous and equilibriu stiffness (obtained nuerically for finite deforations and analytically for sall deforations). These liits could be used to deterine safe loading protocols in order that the stress in each solid constituent reains within its own elastic liit. For a given copressive strain applied at a low rate, the loading is close to the lower liit and is ostly borne directly by the solid constituents (with little contribution fro the fluid). In contrast, however in case of faster copression, the extra loading is predoinantly transported to the fibrillar atrix via rising fluid pressure with little increase of stress in the nonfibrillar atrix. The fibrillar atrix absorbs the loading increent by self-stiffening: the quicker the loading the faster the fibril stiffening until the upper elastic loading liit is reached. This self-protective echanis prevents cartilage fro daage since the fibrils are strong in tension. The present work deonstrates the ability of the fibril reinforced poroelastic odels to describe the strain rate dependent behavior of articular cartilage in unconfined copression using a echanis of fibril stiffening ainly induced by the fluid flow. DOI: / Keywords: Cartilage Mechanics, Fibril Reinforceent, Finite Eleent Analysis, Nonlinear Bioechanics, Poroelasticity *Corresponding author s current address: LePing Li, Faculty of Kinesiology, University of Calgary, 2500 University Drive, N.W., Calgary, Alberta, Canada T2N 1N4. Tel ; Fax Eail: poroelasticity@yahoo.co Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received March 2001; revised anuscript received Noveber Associate Editor: G. A. Ateshian. 1 Introduction Articular cartilage of the knee joint, for exaple is typically subjected to physiological loading involving large strains at high rates. At sufficiently large copressive strains e.g. 20%, the cartilage stiffness at high-speed loadings can reach ties as high as that for static loadings. It is essential to understand how the cartilage behavior changes with the strain aplitude and strain rate as well as the iplications of such changes in cartilage functioning. Much experiental work has been done in observing the consequences of ipact loadings, including echanical failure, changes in biocheical coposition and cartilage pathology due to the resulting strains 1 8. It has been found that the strain stress rates as well as the strain stress aplitudes are iportant factors influencing the noral function of cartilage However, previous odel studies, which consider the fibrils and proteoglycans as one constituent, have experienced difficulties in predicting the relevant echanical behavior especially that of the nonlinear transient response. For exaple, the widely used isotropic biphasic odel predicts a axiu transient stiffness of unconfined cartilage disk at only 1.5 ties of the equilibriu stiffness 13. Obviously such odels are not adequate to study cartilage response to high rate loadings. This is probably why extensive analyses have not yet been reported in correlating the transient stiffness of cartilage to the strain aplitude and strain rate, and hence in deterining the individual role of each of the three constituents in the transient response. The fibril reinforced poroelastic odels recently developed distinguish the role of the collagen fibrillar network fro that of the proteoglycan atrix and thus have the potential to odel the interactions aong three aterial constituents: collagen fibrils, proteoglycan atrix and water containing diffusible ions. One of the advantages of such odeling is the ability of the odels to predict the nonlinear transient response in unconfined copression. Previous work on these odels has been focused on forulation and verification of the ethod. It has been shown analytically that the fibril reinforceent plays a predoinant role in the transient behavior of cartilage in unconfined copression. The objectives of the present work are to explore the echanis of the transient response, especially at high strain rates not previously considered, and to deterine the strain aplitude/rate dependent stiffness of cartilage in unconfined copression by eploying the fibril reinforced odels. The work is likely helpful in describing the experiental findings, such as cartilage daage is loading rate dependent 10, that ay not be explainable if the solid skeleton of cartilage is forulated as one single constituent. It is hypothesized that the collagen fibrils play an iportant role in supporting high rate loadings whereas the proteoglycan atrix is relatively ore crucial in supporting low rate or equilibriu loadings. 2 Methods In an experiental study 17, 3 diaeter bovine articular cartilage thick disks with underlying bone thick were copressed in unconfined geoetry with copression aplitudes ranging fro 12.5 to 300 and speeds fro 0.5 to 50 /s. The copressive stiffness of the disks during the transient was found to vary non-onotonically with the copression aplitude and to increase substantially with the copression speed. In a preliinary odel study, the cartilage and bone were both assued as hoogeneous aterials 18. In another Journal of Bioechanical Engineering Copyright 2003 by ASME APRIL 2003, Vol. 125 Õ 161

2 Fig. 1 Axial copressive stiffness of cartilage disks with bone attached, the nonhoogeneous odel versus experients eanásd, nä4. The aterial paraeters of cartilage used for siulation are as follows: at the articular surface, E Ä0.20 MPa, Ä0.12, the void ratio is 3.6; at the cartilage and bone interface E Ä0.80 MPa, Ä0.42, the void ratio is 2.88; kˆ Ä ÕNs and MÄ13; E f Ä2800 MPa and E 0 f Ä3 MPa at the surface. For the underlying bone, EÄ18 MPa, Ä0.25, k Æ ÕNs, the void ratio is 0.5. attept to siulate the experiental data, a nonhoogeneous odel involving depth-varying aterial properties 15 was eployed in the present work to reduce the extent of aterial discontinuity at the cartilage/bone interface, in an effort to iprove the nuerical results. In the current siulation, the disk radius was taken to be 1.5 and the thickness of cartilage and bone were taken to be 1.1 and 0.4 respectively which were representative diensions for the explants used in the tests 17. Both bone and cartilage were considered as poroelastic. The bone was defined by the Young s odulus E, Poisson s ratio and a constant pereability k. The cartilage skeleton was considered to consist of a fibrillar atrix collagenous and a nonfibrillar atrix ainly proteoglycans. The fibrillar atrix was presented by the Young s odulus E f (E f fibril strain E f 0 ), where was a function of cartilage depth 15, 1 for hoogeneous odel. The nonfibrillar atrix was defined by E, and k kˆ exp(m Dilatation). Linear depth variations were assued for E and, as well as the initial void ratio 15. The paraeters used for siulation given in the caption of Fig. 1 were chosen to be copatible with easureents reported and our previous odeling work. The finite eleent esh axisyetric for continuu eleents was in the axial direction: 5 for the cartilage layer, 2 for the bone. Further analyses focused on exploring the behavior of cartilage without the presence of the bone. Thus only the cartilage layer was considered in finite eleent analysis. A hoogeneous odel 14 was then eployed to reduce coputation tie. This siplification would not produce qualitative changes in the results, since only the overall response of the disks was of interest in this study deterined by the total load and displaceent aplitudes. The depth dependence of cartilage properties was previously found to have little influence on the load pattern when the bone was not attached 15. The radius and thickness of the cartilage disks were taken to be 1.5 and 1.0 respectively. Only half the thickness of the disk was eshed axisyetric 14 4). In each case, five copression rates were investigated: the static response zero copression speed, 0.05%, 0.5% and 5% copressive axial strain per second and the instantaneous response infinite copression speed. The copression was applied continuously at one of the axial strain rates fro the undefored state to 20% copressive noinal strain. This loading protocol was different fro that of the experients 17 where a sequence of copressionrelaxation-release raps of desired strain aplitudes was iposed, accopanied by interspersed witness raps to onitor possible changes in cartilage properties during the loading cycles. Such repeated loadings were not necessary in coputation since the aterial properties were assued to be unaffected by the loading. It is to be noted that the strain aplitudes and strain rates adopted in the purely theoretical analyses were higher than those used in the experiental data fitting. Various cases with diverse aterial properties were investigated Table 1, concerns will be addressed in the discussion. Considering the five copression speeds, 224 cases in total were involved, noting that the static and instantaneous responses are independent of M. The fibril s nonlinear property (E f 0), the pereability nonlinearity (M 0) and the effect of finite deforation were considered except when specifically noted otherwise. The static equilibriu and instantaneous responses are fluidflow independent. Accordingly, the equilibriu stiffness was obtained by eploying elastic continuu eleents, or by assuing the disks were drained. The instantaneous stiffness, on the other hand, was obtained when the fluid was assued to be trapped, i.e. all fluid diffusion boundaries were sealed and geoetrical boundary conditions were applied in no tie. Alternatively, analytical solutions for both static and instantaneous copressions are feasible. The sall deforation solutions are provided in Appendix A. Table 1 The aterial paraeters for various cases coputed. For all cases, the void ratio is 3.5 and the initial pereability is kˆ Ä ÕNs käkˆ exp MÃDilatation. E and are, respectively, the Young s odulus and Poisson s ratio of the nonfibrillar atrix; E f is the Young s odulus of the fibrillar atrix. The diension for E f, E and the aggregate odulus H A is MPa. Each of the seven groups of E and, together with other paraeters, constitutes a case for coputation. E f ( E f f E 0 f ) H A E (1 )/(1 )(1 2 ) E f 0 E f M E 0.26 E E 0.36 E 1.07 E E 0.50 E Õ Vol. 125, APRIL 2003 Transactions of the ASME

3 Fig. 2 Axial copressive stiffness of cartilage when no fluid flow is feasible. For all cases, E 0 f Ä3 MPa. a Elastic static stiffness or equilibriu stiffness: Ä0.38 and E f Ä1600 MPa except when specified in the legend otherwise. b Instantaneous stiffness for E f Ä1600, 1200 and 800 for which the fluid is trapped: Ä0.38 and E is as specified in the legend. 3 Results The axial copressive stiffness of the cartilage disks and the radial strain logarithic at the center are presented versus the copressive noinal axial strain. For convenience, the noinal odulus, defined as the ratio of the total noinal axial stress over noinal axial strain, is generally adopted to represent the stiffness exceptions in Fig. 9. During the transient, the stresses are not unifor in the radial direction and the noinal axial stress is calculated using the total load and the initial cross-sectional area of the disk. The odel captures the ajor trends of the experiental data Fig. 1. The siulation involves odeling the underlying bone, whose aterial properties are assued to be hoogeneous and independent of strains. It is seen that the predicted growth of the transient stiffness slows down at larger strains for the copression rate 50 /s Fig. 1. Should the bone be assued to stiffen with the strain while a saller E f be used, the odel results would atch the experiental data better. This is indicated by a faster growth in Fig. 3a 5%/s where only cartilage is considered. Moreover, the underlying bone is actually nonhoogeneous: between cartilage and subchondral bone lies a layer of calcified cartilage 3 8% cartilage thickness 19 that has a transitional Fig. 3 Transient copressive stiffness of cartilage at various strain rates as specified in the legends noinal copressive axial strain per second. For all cases, E 0 f Ä3 MPa; Ä0.38 and E takes either 0.36 or 2.14 MPa. a Stiffness for E f Ä1600. b Deviations of the stiffness when E f is reduced fro 1600 sae as shown in figure a to 1200 shown with sybols. Larger deviations are observed for the cases of higher strain rates. For clarity of the figure, the curves for the strain rate 5%Õs are oitted. odulus 20. Pure subchondral bone has been reported to have very high transient stiffness at the order of 1 4 GPa e.g. 20. However, static easureents by ultrasound obtained a stiffness of 19.8 MPa for noral subchondral bone fro huan feoral head 21. Obviously a great deal of work is required to deterine the bone and cartilage interactions. For this reason, we only considered cartilage in the reaining part of this paper Figs We first consider the cases involving full nonlinearity Figs The equilibriu stiffness is not strongly dependent on either the axial strain or the fibrillar odulus. In the present unconfined copressions (E and as shown, it is ainly deterined by the aggregate odulus H A i.e. the confined stiffness, with less dependence on E or varying individually Fig. 2a. The instantaneous stiffness is uch higher and grows very quickly with the axial strain Fig. 2b. Obviously the elastic properties of the drained nonfibrillar atrix (E and ) have very little influence on it. The instantaneous stiffness is predoinantly affected by the fibrillar odulus E f, and its growth is deterined by the fibril stiffening represented by E f ) Fig. 2b. The transient stiffness is largely strain-rate and load-history dependent Fig. 3. When the Journal of Bioechanical Engineering APRIL 2003, Vol. 125 Õ 163

4 Fig. 5 Transient copressive stiffness of cartilage when the pereability is taken to be constant kæ ÕNs,MÄ0, Ä0.38, E f Ä1600 MPa and E 0 f Ä3 MPa. If E f is reduced to 1200 MPa, the deviations will be invisible with sae 2 digits for the stiffness at the copression rate 0.05%Õs and alost invisible after 10% axial strain at the rate 0.5%Õs. The deviations are relatively significant at the copression rate 5%Õs. Fig. 4 Radial strain at the center of cartilage disk for E f Ä1600 vs E f Ä1200 while E 0 f Ä3 MPa. For the case of E f Ä1200, the strain is shown with unconnected sybols. a The radial strain at equilibriu, for which E and are as specified in the legend. b The radial strain during the transient for different strain rates, for which E Ä1.07 MPa and Ä0.38. The curves for E Ä0.78 and Ä0.42 and for E Ä1.48 and Ä0.30 would be hard to distinguish fro the corresponding curves while other paraeters are the sae, were they drawn very little difference at the low speed, alost no difference at the high speeds. when the two curves for different E f are nearly superiposed Fig. 4b, since the radial strain is independent of aterial properties when the volue is conserved. On the other hand, when E f is reduced fro 1600 to 1200, the transient radial strain Fig. 4b increases less significantly than does the equilibriu strain Fig. 4a, but it produces larger differences in the stiffness at high strain rates Fig. 3b vs Fig. 2a. Now consider the cases for which the nonlinearity in aterial properties partially disappears, possibly due to cartilage pathology for exaple Figs If the pereability is constant while the fibril nonlinearity reains unchanged Fig. 5, the transient stiffness increases with the axial strain, onotonically for the case with copression rate of 5%/s. In all these cases, the stiffness at high strains is substantially reduced due to loss of nonlinear per- copression rate is low 0.05%/s, the transient stiffness initially increases for a short tie, then decreases and again increases with the axial strain. Altering the elastic properties of the nonfibrillar atrix has relatively larger ipact on the transient stiffness at the lower rates 0.05% and 0.5%/s. In contrast, the fibril stiffening with its tensile strain has little ipact on the disk stiffness when copression rate is as low as 0.05%/s Fig. 3b. The radial strain logarithic is shown against the axial noinal strain Fig. 4. The logarithic axial strain is slightly larger than the noinal one, e.g. when the noinal is 10% and 20%, the logarithic strain is, respectively, 10.5% and 22.3%. The radial strain at equilibriu is very sall copared with the axial strain. At the beginning of copression, the radial strain during transient is close to but less than half the axial strain, representing a very sall rate of volue change. The radial strain deviates fro half the axial strain when the copression continues. The higher the copression strain rate, the closer the radial strain to half the axial strain and the larger the axial strain till which this trend lasts. For the high copression rate 5%/s, the bulk volue changes little at the center of disk until around 8% axial strain. This is observed Fig. 6 Transient radial strain at the center of disk for E Ä1.07 MPa, Ä0.38 and E 0 f Ä3 MPa. The strain for MÄ10 nonlinear pereability is copared with that for MÄ0 constant pereability while E f Ä1600. The strain is shown for M Ä0 and E f Ä1200 with the fine dots. 164 Õ Vol. 125, APRIL 2003 Transactions of the ASME

5 Fig. 7 Transient stiffness of cartilage with constant fibrillar odulus E f Æ20 MPa for MÄ10 vs MÄ0 while E Ä0.36 and Ä0.38. No significant qualitative changes in the stiffness are observed for a larger E and thus the results for the coparison group are not included. Fig. 9 Significance of different easures in quantifying cartilage stiffness, for E Ä0.36 MPa, Ä0.38, E 0 f Ä1600 and E f Ä3 MPa. The tangent odulus is defined as d z Õd z and the secant odulus is defined as z Õ z, where z is the Cauchy stress in the axial direction and z is the logarithic strain. The noinal odulus, generally adopted elsewhere in this paper, is the ratio of noinal axial stress over noinal axial strain. For the convenience of coparison, the oduli are shown as functions of the noinal axial strain. a Instantaneous oduli as obtained by eploying the finite deforation FD and sall deforation SD theories. b Transient oduli for the copression rate 0.5%Õs MÄ10, FD theory only. Fig. 8 Axial copressive stiffness of cartilage for constant E f Æ20 MPa and Ä0.38 when no fluid flow is feasible. The finite deforation and sall deforation SD theories are eployed respectively. In all cases, the stiffness extracted fro the SD theory is not axial strain dependent horizontal lines. a Elastic static stiffness for E Ä0.36, 1.07 and 2.14 MPa respectively. b Instantaneous stiffness for which the fluid is trapped. eability Fig. 5 vs Fig. 3. When E f is reduced by 25% not shown, there is no change negligible in the stiffness for the strain rate 0.05%/s and alost no change for the strain rate 0.5%/s after 10% axial strain. Again E or ) has a ore significant role in the stiffness when the strain rate is low Fig. 5. The ipact of nonlinear pereability is deonstrated by the radial strain. The pereability nonlinearity akes no differences in the radial strain at low axial strains, but large differences at high axial strains Fig. 6. Furtherore, the higher the copression rate, the larger the axial strain at which the pereability nonlinearity starts to ake a difference Fig. 6. On the other hand, aong the three copression rates, the largest differences in the radial strain produced by the pereability nonlinearity occur at 0.5%/s Fig. 6. This is so because when the strain rate is too high or too low, the cartilage behavior is ainly elastic; the pereability plays the ost significant role in the stiffness at a oderate strain rate. These trends due Journal of Bioechanical Engineering APRIL 2003, Vol. 125 Õ 165

6 to alteration in the pereability patterns reain when E f is reduced the fine dots in Fig. 6 vs the data sybols in Fig. 4b. When E f is constant, a weak disk stiffening is possible only if the copression rate is high 5%/s, Fig. 7. Otherwise the transient stiffness decreases significantly with the axial strain at low and oderate strains, and increases slightly at high strains for M 10 nonlinear pereability ; the stiffness decreases onotonically if M 0 Fig. 7 except for the case with E 2.14 and copression rate 0.05%/s for which a very slight increase at high strains is found not shown. The significance of eploying the finite deforation theory is shown for the cases with linear aterial properties Fig. 8. The equilibriu and instantaneous stiffness extracted by the sall deforation theory are given by equations A-11 and A-4 respectively. Eploying the sall deforation theory results in saller errors in the equilibriu stiffness. The results would be soewhat different if the easure for the stiffness defined in the first paragraph of this section were altered. For the exaple deonstrated Fig. 9, nonlinear aterial properties, the stiffness represented by the solid curves has been shown earlier Figs. 2b and 3. The oduli are defined as in the caption of the figure, where the Cauchy stress for the transient is obtained by the total copression load and the real cross-sectional area. The instantaneous oduli deterined by the sall deforation theory Fig. 9a are calculated by equations A-4 and A-5 respectively; they are linear functions of the axial strain. When the axial strain is not so large e.g. 8%, the finite deforation theory yields nearly the sae results Fig. 9a. The secant odulus can be replaced by the noinal odulus when the axial strain is sall for both transient and instantaneous responses. The tangent odulus for the transient case is found quite different, which was obtained by differentiating the nuerical results of the stress and strain with 800 strain intervals 8000 tie steps were eployed in the finite eleent coputation. However, the results for 9 out of 10 steps were not saved due to large output by ABAQUS. 4 Discussion It has been shown that the fluid flow doinates the transient response of cartilage. In general, for the fully nonlinear case, there is an initial rise in the stiffness due to tissue resistance to iediate fluid exodus, witnessed by little volue change, i.e. the radial strain is nearly half the axial strain at the beginning of copression Fig. 4b. This trend of volue conservation will be lost sooner or later depending on the strain rate. Then the disk stiffening diinishes significantly Figs. 3b and 4b, or even the disk appears softer if the copression rate is sufficiently low since the fluid dissipates relatively quickly with respect to the rate of atrix deforation. However, as the copression strain further increases at the low copression rate, the pereability is substantially reduced and the flow becoes ore difficult, expanding the disk quicker and resulting in fibril stiffening and a rise in pore pressure. This explains the decrease in transient stiffness followed by an increase with respect to the axial strain Fig. 3, 0.05%/s. On the other hand, the collagen fibrils stiffen quickly with the flowdependent lateral expansion. The transient stiffness hence grows quickly with the strain if the copression rate is high so that the fluid has little tie to dissipate, approaching the instantaneous odulus if within the elastic liit. Thus the actual pattern of the transient stiffness versus the axial strain depends on the cobination of the fluid dissipation speed and the rate of fibril stiffening. The stiffness increases onotonically with the axial strain only if the copression rate is sufficiently high Fig. 3a, 5%/s. For the sae reason, the transient response is load-history dependent and therefore, the transient stiffness and strain would be altered should the copression protocol be changed. The interplay between fluid kinetics and fibril stiffening can be further illustrated by the cases of reduced aterial nonlinearity. When the pereability is constant (M 0) with other conditions reaining unchanged, the transient stiffness still increases quickly and onotonically at high copression rates Fig. 5, 5%/s. However, even if the pereability varies with the atrix dilatation (M 10), but the fibrils do not stiffen with its strain (E f 20 MPa, a typical value for ediu strains, the transient odulus does not increase uch with the strain even for the relatively high copression rate Fig. 7, 5%/s. Moreover it has been found that a higher constant E f 50 MPa does not result in a greater increase not shown. This indicates the overwheling ability of fibrillar atrix stiffening to produce an increasing transient stiffness at higher axial strains. The actual role of the fibrillar atrix is however deterined by fluid dissipation that in turn is affected by copression rates and aplitudes in addition to the atrix elasticity and pereability, as well as loading history. The current study does not consider the fibril viscoelasticity that ight play a role in the strain-rate dependence of cartilage unconfined stiffness. The instantaneous stiffness which is the axiu elastic stiffness is certainly of interest in clinical studies. It is unique and deterines the upper elastic liit of loadings at a given strain The lower liit corresponds to the equilibriu stiffness. The instantaneous stress of the nonfibrillar atrix is dependent only on the shear odulus of the atrix when the deforation is sall A-1. This could indicate the failure type of cartilage should the nonfibrillar atrix be overloaded. Incidentally, under an ipact loading cartilage disks were found to crack at a plane approxiately 45 to the articular surface 3. The flow independent responses derived in the appendix for sall deforations can be obtained analytically for large deforations too. Then finite eleent coputations are not required if only the upper and lower liits are desired. Siilar studies have been done previously for poroelastic beas, coluns and plates with linear aterial properties. In those cases, if the deflection is sall, the solutions for the instantaneous responses can be obtained by the corresponding equilibriu solutions after the Young s odulus of the drained aterial is ultiplied by a diensionless constant 22. The aterial paraeters used as reference in the present work were originally based on fitting the data fro ulti-step rap copression tests, for which the noinal axial strain was not greater than 10% and the strain rates were around 0.1%/s 14,15. We had to adopt saller M and E f in order to fit the transient responses for different copression rates when the axial strain was as large as 20% 18. This is because they are not so sensitive in the case of sall deforations and low strain rates. The fibrillar odulus E f, currently taken as a linear function of the strain f, ight not be accurate enough for rather large deforations. Should E f ( f ) be a convex curve while f ( f ) is concave, a saller E f for the case of large deforation and a larger E f for the case of sall deforation ust be taken in order to iprove fittings when E f is approxiately replaced by a secant. Further work is required to test this hypothesis. Probably a hyperelastic constitutive law should be eployed when the axial strain is close to 20%. However, we do not expect uch change in the general patterns of the transient stiffness when the representation of fibrillar stiffness is revised. Alternative aterial paraeters were also considered to exaine their influence on cartilage stiffness. In soe cases, E 2.14 MPa ight be slightly too large to represent articular cartilage. However, the purpose to use large E was to deonstrate its insignificance in the transient stiffness when the copression rate is high. For the sae reason, a large range of was considered. Another concern is whether the relevant stress is within the strength of the bulk aterial water saturated. The answer would be positive at least for all the cases with liited copression rates, for which the axiu stress is less than 8 MPa Fig. 3a. The axiu instantaneous stress is 14.5 MPa derived fro Fig. 2b. More evidence is required to know if cartilage fails before reaching this stress, but the instantaneous stiffness presented should be justified for a slightly saller strain e.g. 18% axial. Kerin et al. 4 found that the disks failed at a noinal stress 166 Õ Vol. 125, APRIL 2003 Transactions of the ASME

7 between 14 and 59 MPa, while Repo and Finlay 6 reported the failure at 25 MPa. On the other hand, the current work has not considered intrinsic property changes due to repeated loadings. While such changes do occur, they are not so significant within a load or stress threshold 8,17. The present study has explored the echanis of the strain rate/aplitude dependent cartilage response to external loadings and deonstrated the distinct echanical functions of the three ajor constituents of cartilage. An ipact load is priarily balanced by the fluid pressure restrained by the fibrillar atrix; it is essential that the fibrils be strong in tension and provide extra stiffness whenever a fibril strain increent is produced as a result of rising fluid pressure. The fibrillar stress is then greatly relaxed by water flow, facilitating the physiological activities within cartilage. The stress of the nonfibrillar atrix is relatively invariable under a given strain, a echanis protecting the atrix fro daage, as long as the collagen fibrils have not yet reached their own liit. On the other hand, if the fibrils fail to function properly due to collagen degradation, the cartilage will be easily destroyed by an even sall ipact load. In fact, the iportance of the tensile strength of the fibrils in keeping cartilage integrity has been observed experientally 23. For noral cartilage, the current odel suggests that echanical daage would not occur as long as the loading path strain-varying falls within the upper liit, regardless of the loading rate and history. Acknowledgents The first author received the industrial research fellowships fro the NSERC-Canada. Appendix A: The Instantaneous and Static Responses of Cartilage With Sall Deforation First, consider the instantaneous response of the disk, when the fluid is trapped. Since all constituents of the aterial are incopressible, no volue change is possible. Thus the stress increents are z 2 z p f r 2 E f r p f (A-1) where is a Laé constant, i.e. the shear odulus of the nonfibrillar atrix. The other Laé constant is absent due to zero volue increent. The stress increent for the fibrillar atrix is represented by E f r for which E f E 0 f E f r. It has been assued that the axial fibrils are not stressed in copression. The radial stress increent is required to be zero. Thus the pore pressure increent is deterined as p f 2 E f r (A-2) Noting that the zero volue increent condition requires r z /2, substituting A-2 into the first equation of A-1 yields z f E z (A-3) The radial strain cannot generally be expressed as a siple function of the axial strain. However, if there was no previous loading before the current instantaneous copression, we have r z /2. Therefore the effective odulus ( z / z ) for instantaneous response is presented as axial strain dependent E p f E f E f z z (A-6) It is observed that in the radial direction the pore pressure described by the first ter of A-6 is balanced by the nonfibrillar atrix stress and the difference is balanced by the fibrillar atrix stress. Now consider the static response when the pore pressure is nil. For the nonfibrillar atrix, z 1 E z 2 r r 1 1 E r z (A-7) Here the atrix stress increents r and are identical due to unifor stresses at equilibriu. The first equation of A-7 gives r 1 2 z E z (A-8) The total stress increent in the radial direction ust be zero 1 2 z E z E f r 0 (A-9) Introducing the effective odulus E and Poisson s ratio,as defined by z E z and r z respectively, A-9 and the second equation of A-7 with r replaced per A-8 yield E 2E f E E 2E E 1 (A-10) where (1 )(1 2 ). Hence E and are derived as functions of r E E E E f 1, E E f E (A-11) E E f The initial strains are not considered in the present study, i.e. r 0 when z 0. Then by using d z d r / the axial strain is expressed as a second order polynoial of the radial strain z 1 E E E 0 f 1 2 E f r r (A-12) The positive root of the above algebraic equation gives the radial strain if E f 0 r E E 0 f 2 2E f E z E E 0 f E (A-13) f and if E f 0, r is siply proportional to z r E E E 0 f z (A-14) Then the axial stress at equilibriu can be obtained by integrating A-9 z E z 2E f 0 E f r r (A-15) E inst 3E E f E f z In this case, the axial stress is extracted by E inst d z z 3E E f E f z z Siilarly integrating A-2 results in pore pressure (A-4) (A-5) References 1 Borrelli, Jr., J., Torzilli, P. A., Grigiene, R., and Helfet, D. L., 1997, Effect of Ipact Load on Articular Cartilage: Developent of an Intra-articular Fracture Model, J. Orthop. Traua, 11, pp Broo, N. D., 1986, Structural Consequences of Trauatizing Articular Cartilage, Ann. Rheu. Dis., 45, pp Jeffrey, J. E., Gregory, D. W., and Aspden, R. M., 1995, Matrix Daage and Chondrocyte Viability Following a Single Ipact Load on Articular Cartilage, Arch. Bioche. Biophys., 322, pp Journal of Bioechanical Engineering APRIL 2003, Vol. 125 Õ 167

8 4 Kerin, A. J., Wisno, M. R., and Adas, M. A., 1998, The Copressive Strength of Articular Cartilage, Proceedings of the Institution of Mechanical Engineers, Part H-Journal of Engineering in Medicine, 212, pp Quinn, T. M., Grodzinsky, A. J., Hunziker, E. B., and Sandy, J. D., 1998, Effects of Injurious Copression on Matrix Turnover Around Individual Cells in Calf Articular Cartilage Explants, J. Orthop. Res., 16, pp Repo, R. U., and Finlay, J. B., 1977, Survival of Articular Cartilage After Controlled Ipact, J. Bone Jt. Surg., A. Vol., 59, pp Sion, S. R., Radin, E. L., Paul, I. L., and Rose, R. M., 1972, The Response of Joints to Ipact Loading-II In Vivo Behavior of Subchondral Bone, J. Bioech., 5, pp Torzilli, P. A., Grigiene, R., Borrelli, Jr., J., and Helfet, D. L., 1999, Effect of Ipact Load on Articular Cartilage: Cell Metabolis and Viability, and Matrix Water Content, J. Bioech. Eng., 121, pp Chen, C.-T., Burton-Wurster, N., Lust, G., Bank, R. A., and Tekoppele, J. M., 1999, Copositional and Metabolic Changes in Daaged Cartilage are Peakstress, Stress-rate, and Loading-duration Dependent, J. Orthop. Res., 17, pp Ewers, B. J., Dvoracek-Driksna, D., Orth, M. W., and Haut, R. C., 2001, The Extent of Matrix Daage and Chondrocyte Death in Mechanically Trauatized Articular Cartilage Explants Depends on Rate of Loading, J. Orthop. Res., 19, pp Kurz, B., Jin, M., Patwari, P., Lark, M. W., and Grodzinsky, A. J., 2000, Biosynthetic Response and Mechanical Properties of Articular Cartilage After Injurious Copression: The Iportance of Strain Rate, ORS Transactions, 25, p Quinn, T., Allen, R., Schalet, B., Perubuli, P., and Hunziker, E., 2000, Matrix Daage and Cell Injury Caused by Rap Copression of Adult Bovine Articular Cartilage Explants: Effects of Strain Rate and Peak Stress, ORS Transactions, 25, p Arstrong, C. G., Lai, W. M., and Mow, V. C., 1984, An Analysis of the Unconfined Copression of Articular Cartilage, J. Bioech. Eng., 106, pp Li, L. P., Soulhat, J., Buschann, M. D., and Shirazi-Adl, A., 1999, Nonlinear Analysis of Cartilage in Unconfined Rap Copression Using a Fibril Reinforced Poroelastic Model, Clinical Bioechanics, 14, pp Li, L. P., Buschann, M. D., and Shirazi-Adl, A., 2000, A Fibril Reinforced Nonhoogeneous Poroelastic Model for Articular Cartilage: Inhoogeneous Response in Unconfined Copression, J. Bioech., 33, pp Li, L. P., Shirazi-Adl, A., and Buschann, M. D., 2002, Alterations in Mechanical Behavior Of Articular Cartilage Due to Changes in Depth Varying Material Properties A Nonhoogeneous Poroelastic Model Study, Coputer Methods in Bioechanics and Bioedical Engineering, 5, pp Langelier, E., and Buschann, M. D., 1999, Aplitude Dependent Mechanical Alteration and Nonlinearity of Articular Cartilage Material Behavior in Unconfined Copression, ORS Transactions, 24, p Li, L. P., Buschann, M. D., and Shirazi-Adl, A., 2001, Fibril Stiffening Accounts for Strain-dependent Stiffness of Articular Cartilage in Unconfined Copression, ORS Transactions, 26, p Oegea, Jr., T. R., Carpenter, R. J., Hofeister, F., and Thopson, Jr., R. C., 1997, The Interaction of the Zone of Calcified Cartilage and Subchondral Bone in Osteoarthritic, Microsc. Res. Tech., 37, pp Mente, P. L., and Lewis, J. L., 1994, Elastic Modulus of Calcified Cartilage is an Order of Magnitude Less than that of Subchondral Bone, J. Orthop. Res., 12, pp Li, B., and Aspden, R. M., 1997, Mechanical and Material Properties of the Subchondral Bone Plate fro the Feoral Head of Patients with Osteoarthritis or Osteoporosis, Ann. Rheu. Dis., 56, pp Cederbau, G., Li, L. P., and Schulgasser, K., 2000, Poroelastic Structures, Elsevier Science Ltd., New York. 23 McCorack, T., and Mansour, J. M., 1998, Reduction in Tensile Strength of Cartilage Precedes Surface Daage Under Repeated Copression Loading In Vitro, J. Bioech., 31, pp Õ Vol. 125, APRIL 2003 Transactions of the ASME