INDENTATION TESTS OF HUMAN ARTICULAR CARTILAGE R. Y. HORI and L. F. MOCKROS Department of Civil Engineering. Northwestern
Umversity. Evanston. IL 60201. U.S.A.
Abstract-The short-time shear and bulk moduli of articular cartilage were determined experimentall) using torsional shear and uniaxial confined compression tests. These results are compared to the values of shear moduli predicted by using data from indenter tests and the theoretical solution for the identation of a rigid punch pressed into an elastic layer bonded to a rigid halfspace. To verify the effectiveness of predicting moduli from indenter tests. control experiments were performed on polyurethane rubber. Although they accurately predict the elastic properties for rubber. the nonlinearity. anisotropy ;~nd non-homogeneity of articular cartilage make indenter tests less appropriate for use with cartilage. The prcd~cted and measured values correlate but with considerable dispersion. The short-time shear modulus of articular cartilage. including both healthy and diseased samples. were found to vary over the range 435 x lo5 N/m* and the short-time bulk modulus over the range 9-170 x lo6 N/m’. The means of tissue storage was found to have a major effect on the measured mechanical properties.
Osteoarthritis or degenera.tive joint disease. a common disorder affecting diarthrodial joints. is characterized by the deterioration. degradation and abrasion of articular cartilage and also by the formation of new bone at the articular surface. Although the aetiology of osteoarthritis is not yet fully understood, mechanical factors are often cited as initiating or aggravating agents (Sokoloff, 1969). One of the functions of articular cartilage is the transmission of loads through joints. Such loads may, at times. peak at 4-7 times body weight (Paul. 1966; Morrison. 1970). Accordingly. the mechanical properties of articular cartilage have been of great interest to researchers for some time. Although simple compression (Camosso and Marotti. 1962) and tension (Kempson et al.. 1968) tests have been attempted. the most common experiment used to study the mechanical properties of articular cartilage has been the indenter test in which a punch under constant load is pressed normally into the artitular surface. Using such a test in a qualitative manner. some of the viscoelactic characteristics of articular cartilage (Hirsch. 1944) have been examined. The influence of immersion (Elmore et al.. 1963). age (Sokoloff. 1966) and degenerative state of the tissue (Hirsch. 1944). the effect of various ionic solutions (Sokoloff. 1963). the relationship between mechanical stiffness and the muco-polysaccharide content of the tissue (Kempson it al.. 1970) and the variation of the deformability of cartilage over the articular surface (Kempson rr al.. 1971b) all have been investigated using comparative data from indenter tests. In order to obtain quantitative data. however. a mathematical model of the test is required. Solutions to this problem have been sought using the theories of classical elasticity. Although articular cartilage is
a viscoelastic material, its short-time response to load can be approximated by an elastic model. Hirsch (1944) was one of the first to appeal to the Hertz solution for contact between two elastic spheres to model the contact between a rigid spherical indenter and the curved articular surface. Zarek and Edwards (1963) use the Hertz solution generalized for the case of contact between a rigid sphere and the flat surface of an elastic half-space as their model. The solution for the indentation of an elastic halfspace by a plane cylindrical indenter. as given by Timoshenko and Goodier ( 1951). was used by Sokoloff (1966). In each of these cases, the layered geometry of the articular joint. an important consideration, was ignored. Furthermore, all the formulae used by these researchers relate punch indentation to two constitutive properties of the elastic test material. namely the elastic modulus and Poisson’s ratio. In every case. knowingly or not, these workers have taken Poisson’s ratio to be equal to 0.3. that for steel. and proceeded to calculate the elastic modulus. In order to account for the layered geometry of the articular joint, Kempson et al. (1971) used as a model the results of indentation tests on thin rubber sheets lying on a rigid surface as reported by Waters (1965). Unfortunately, the Waters solution says little about the boundary conditions at the elastic-rigid interface: the naturally high coefficient of friction of rubber was assumed to carry any shear forces that might occur at the interface but by no means was the elastic rubber layer bonded to the rigid support as is the case in the articular joint. On the basis of some tensile tests, the Poisson’s ratio of articular cartilage (average v = 0.48) was demonstrated to be close to that of rubber (v = 05). Simon (1971) reported he was unable to obtain consistant results with the theoretical models of either Kempson et a/. (1971) or Sokoloff (1966).
R. Y. HORIand L. F.
260
Recently. the contact problem of the indentation by either a spherical or a cylindrical punch pressed normally into an elastic layer bonded to a rigid halfspace was solved by Hayes er al. (1972). Their solution. based on classical theory of elasticity. assumed material isotropy and homogeneity. Using the values for the elastic materiai properties of mildly osteoarthritic cartilage previously obtained (Hayes and Mockros. 1971). these authors were able to get a correspondance between indentations as predicted by their model and the results of Hirsch (1944). The present study consists of a number of indenter tests on samples of human articular cartilage. the independent determination of the material properties of the same cartilage samples. and an attempt at establishing a correlation between the material properties of the cartilage as determined by the independent experiments and those same properties as predicted by the Hayes et al. (1972) solution and the indenter tests. The load was applied suddenly in all tests and the response measured one second after load application. As previously reported (see, e.g. Hayes and Mockros, 1971). the typical response of articular cartilage to suddenly applied loads is an “instantaneous” elastic response, followed by a long-term (l&30 min) viscoelastic response. The present study is only concerned with the “instantaneous” or “shorttime” elastic response.
METHODS
AND MATERIALS
Two series of experiments were carried out, the major difference between them being the method of storing the test samples of human articular cartilage. In series A. twelve necropsy specimens of cartilage, harvested from the proximal head of the tibia. were obtained 624 hr post mortem and stored at -20°C. The specimens were tested within six weeks of collection. Several investigators (Maroudas. 1968; Kempson et 01.. 1971a) have indicated that such storage does not materially affect the mechanical properties of artrcular cartilage. Other investigators (Viidik and Lewin. 1966). however. feel that freezing biological materials can substantially alter the properties of tissue. Accordingly. a more gentle method of tissue preservation was tried to determine the effect, if any. of storage procedures. Thus, in series B. an additional eight necropsy specimens were taken and stored immersed in Ringer’s solutions (Krebs, 1950) at 4°C. These specimens were tested within 72 hr after excision. Viidik and Lewin (1966), however. are also critical of this method of tissue storage and have shown some alterations in the mechanical properties of rabbit cruciate ligaments can occur as a result. Unfortunately. harvesting and testing the cartilage specimens immediately after death, as recommended by Viidik and Lewin. was not technically feasible. On the other hand. since the main purpose of the present
MOCKROS
study is to compare mechanical properties inferred from indenter tests with those determined from independent tests. whether or not the values measured are truly representative of irr rice mechanical properties is not a crucial question. The test specimens were obtained by cutting 19.05 mm dia. cylindrical samples of cartilage and underlying-bone from tibia) condyles using a coring tool at lOCKIrpm in a vertical milling machine. Although care was taken to ensure that the axis of the coring tool was perpendicular to the cartilage surface, this was not always possible as the condyles of the tibia1 plateau are not perfectly flat but are slightly concave. Therefore. the perpendicularity criterion was strictly applied only to the central portion of the core. During coring the cartilage was bathed by a continuous stream of Ringer’s solution to minimize heating of the articular tissue. The cancellous bone was trimmed from the sample using a double-fluted milling tool operating at 3250 rpm until a sample length of approx. 5 mm was obtained. The resulting test specimen consisted of a right circular cylinder approx. 5 mm in length composed of the following layers: the articular cartilage, the underlying dense subchondral plate and some cancellous bone. After the indenter tests using spherical punches, a 6.35 mm dia. cylindrical core was cut, using techniques described in Hayes and Mockros (1971). from the central portion of the test specimen. This smaller specimen was used for material property determinations. The degenerative state of the cartilage was graded visually under a disecting microscope at 30x power using the macroscopic classification system of Ballet. et al. (1963). The degenerative state of the cartilage samples is given in Table 3. Apparatus and procedures Materials properties determirmtion. AS the test apparatus. tissue preparation and experimental procedure for the determination of the constitutive properties of the articular cartilage specimens were virtually identical to those reported by Hayes and Mockros (1971) they will not be repeated in detail here; a complete description of the test procedure is given elsewhere (Hori, 1973). In brief, the elastic shear modulus of the tissue sample was determined using at torsional shear test. A confined, uniaxial compression test was used to obtain the elastic bulk modulus. Zndenter tests. Prior to performing the material properties tests. the indentation tests were performed using spherical punches with radii of curvature 30.4 and IO.0 mm (series A) or 30-4, 20.0 and 10.0 mm (series B). The theoretical analysis used to predict the experimental result concerns an infinite elastic layer bonded to a rigid half-space. Thus, a large area of unloaded cartilage surrounding the loaded contact
indentation
tests of human
Displacement /transducer
Balanced load arm
Loading pad Rigid cavity
ILoad
Loading wd with SpherIcal punch
Arbcular cartllaqe
articular
‘-61
cartilage
In each test,care was taken to ensure that the indenter contacted the center of the 19.05 mm dia. test specimen. In some cases of badly diseased cartilage. however. the surface was so irregular that the indenter naturally seated at some other point slightly offcenter. In these cases. the 6.35 mm diameter core was cut to fully include this area. After all the load experiments. depth measurements were taken using a needle gauge. The depth gauge was manufactured by machining a size B sewing needle to 0.25 mm dia., grinding its end Bat and attaching it to a standard depth micrometer. A large number of measurements were taken. The mean. standard diviation and coefficient of variation of these measurements were computed and those specimens with statistically significant coefficients of thickness variation were rejected for further use in the analysis. Control experiments
Fig. 1. Experimental apparatus for indenter tests with load frame and experimental geometry.
patch was desirable. The 19.05 mm diameter test specimen represented a practical limit to the size of flat cartilage that could be removed from the tibia1 plateau. The apparatus used to perform the confined. uniaxial compression test was modified to perform the indenter tests. The test specimens were held in a rigid die and the assembly was immersed in a bath of Ringer’s solution held at a constant temperature of 37” + 1°C. The indenter was brought lightly to bear normal to the cartilage surface by lowering the balance arm, whose cross-hinge was designed for a minimum of friction, and applying a tare load of 0.18 N. The loads were applied by placing weights on the balance arm tray which had a 6: 1 mechanical advantage over the indenter (see Figure 1). In each test, care was taken to ensure that the inapplied in the following progression: 1.8. 3.0. 4.1. 5.9, 8.8, 11.8 N by placing weights suddenly, but gently. on the balance arm tray. Care was taken to avoid any dynamic loading of the cartilage. A step load is very difficult to apply manually. Although the load was applied quickly, it was not applied instantaneously. Consequently, the “instantaneous” response to load was taken as that occuring one second after the initial load application. The full sequence of loads was run for one indenter at a time. As only the instantaneous indentation was of interest, the loads were applied for only a few seconds and then removed. At each load level, the test was repeated four to seven times. After each toad application. the cartilage was allowed to rest at least twice the loading time to ensure full recovery. Further, length measurements were taken from time to time to check that full recovery had occurred, but this check was not made after every test.
In order to verify the effectiveness of the technique. control experiments were performed on polyurethane rubber (CP-4485, Hysol Corp., Olean, NY). a linear isotropic, homogeneous material. Examination of tests for detenniniug the material properties. Using simple tension and uniaxial, unconfined compression tests. the Young’s modulus. E. for the rubber was determined to be 3.76 x lo6 N/m’. A right cylinder, 6.35 mm dia. x 2.5 mm long, was machined from the same stock of material and bonded to an aluminum base 6.28 mm dia. x 2 mm long. This unit was tested in uniaxial confined compression and in torsional shear as described above. Stress-strain curves of the tests are plotted in Figs. 2 and 3. Polyurethane rubber is known to be a linear material, at least for small strains, as evidenced by the linear torsional shear stress-strain curve. Thus the non-linearity of the stress-strain curve for the confined compression test clearly shows that at low loads,
I 0
I
I
1234567 Twist,
rod/m xIO-~
Fig. 2. Control experiments. Torsional shear test on polyurethane rubber sample 2.50 mm thick x 6.35 mm dia.
262
R. Y. HORI and L. F. M~CKROS IO
Table 2. Shear moduh calculated using the Hayes et al. (1972) theory and measurements from indenter tests on polyurethane rubber sample 2.66 mm thick, bonded to a rigid base Data Uncorrectedfor Tare Indentation Shear mdulur (Experimental G = 1.31 x lo6 N/m2) 60.6 mn 0. Indenter 20.0 fmn0. Indenter (N/m2) (N/m2)
Applied Load (Newtons)
Y)
1.79 2.95 4.14 5.69 6.65 11.77
6-
1.56 x 106 1.46 x 10; :::: : ;:t ::::
;%
of both
oatacorrected
1.62 x 1.67 x 1.55 x 1.46 x 1.43 x 1.45 x
106 106 106 105 106 106
1.51 x 106 N/m2
for Tare Indentation Shear bdulus (Experimental G . 1.31 x 106 ~/m2) 60.6 "," (N/ 0.f ndentcr 20.0 nr;)(~0.2;ndcnter
1
Specimen aluminum
bonded bose
to
-
1.79 2.95 4.14 5.69 6.85 11.77
kg:;4 1.32 x 1.32 x 1.37 x 1.32 x
6 106 106 106 106
1.32 x 1.43 x 1.37 x 1.34 x 1.32 x 1.36 x
106 106 106 106 106 106
Mean for both Indentertests 1.34 x 106 N/m2
Strain,
m/mx
IO2
Fig. 3. Control experiments. Confined compression test on polyurethane rubber sample 250 mm thick x 6.35 mm dia; specimen bonded to aluminum base.
compression was not obtained. Even though the cup, punch and sample were carefully machined to fit tightly, sufficient free space apparently was left to allow the lightly loaded rubber to deform and fill these voids. After a stress level of 2 x 10’ N/m2 is reached and with these voids completely filled, a true state of confined compression is attained, the stress-strain curve becomes linear and the slope of the curve can be taken as a measure of the bulk modulus of polyurethane rubber. In designing the confined compression test, the assumption was made that by carefully cutting out the cartilage, the die would be completely filled. Accordingly the bond at the cartilage-bone interface would have little effect on the stress distributions in the cartilage. In light of the non-linear character of the confined compression stress-strain curve above, a second confined compression test was performed true confined
1. The values of Young’s modulus for polyurethane rubber as determined by direct measurement using simple tension and compression tests and by calculation using torsional and uniaxial confined compression tests
Table
ExpM&tal Experiment
Young'* B!odulu*
Tenston
E - 3.81 x 106 N/o?
E . 3.131x 106 n/m2
Compression
E . 3.52 )Ilo6 N/d E - 3.96 x 106 N/m2
E = 3.52 x 106 N/m2 E = 3.96 x 106 N/m2
Toque
G = 1.13 x lo6 N/m2 G - 1.49 x 106 n/m2
mean E = 3.76 x 106 N/d Test
mean G - 1.31 x 106 N/m2 Confined Ccnnpression \1: Ui6x bonded to base confined Compression not bonded to base
lo' N/f
K . 4.41 x 10' N/m2 Y - 0.486
E f 3.90 x lo6 N/m2 E = 3.89 x 106 N/m2
of rubber 635 mm dia. x 24 mm long resting unbonded to an aluminum base 621 mm dia. x 2 mm long. Both the rubber-base and punch-rubber interfaces were lubricated with a thin film of white silicone grease to eliminate shear forces. The stress-strain curve obtained was virtually identical to Fig. 3. Apparently, the effect of the bond at the rubber-base or the cartilage bone interface is minimal on this test. The rubber cylinder was later bonded to its aluminum base and tested in torsional shear. The Young’s modulus of polyurethane rubber as determined by simple uniaxial tension and unconfined compression tests are compared with the results bf the confined compression and torsional shear tests in Table 1. Verification of the Theoretical Solution: A 200 mm dia. x 2.66 mm thick sample of polyurethane rubber, drawn from the same stock of material discussed previously, was bonded to an aluminum base and indented with the a75 mm diameter and the 2OM) mm dia. spherical indenters in the manner discussed above. In performing the indenter tests, a small tare load (0.12 N) is used to ensure light contact between the punch and the surface of the elastic layer. The amount of indentation caused by this tare load cannot be considered inconsequential as some previous researchers have assumed. The tare indentation was estimated by plotting the load vs indentation curve for the indenter tests and extrapolating the displacement at the tare load level. This result was added to the displacements measured for the applied indenter loads and with these corrected results, a set of predicted shear moduli were calculated using the indenter theory of Hayes et al., 1972. The results are given in Table 2. All the predicted values for the shear modulus, G, of polyurethane rubber using the uncorrected data on a right cylinder
263
Indentation tests of human articular cartilage
The tests on rubber serve (il to experimentally verify the theoretical solution given by Hayes rf al. (1972) and (ii) to examine the experimental procedure used for cartilage tests. These tests also indicate the importance of correcting indenter test-data for displacements due to tare loads. RESULTS Material properties detrrmimtions
/ 0
Specimen
I IO
/
I
20
30
Twist,
No.El21-I_2
I
40
50
60
rod/m
Fig. 4. Typical torque-twist curve of torsional shear test on human articular cartilage.
are high: the mean predicted value of G is 1.51 x IO6 N/m2 vs an experimental G of 1.31 x lo6 N/m’. This represents an error of some 15”,$ A notable trend in the predicted shear moduli is also evident. the highest values being associated with the smaller applied loads. The predicted values for the shear modulus using the corrected data are very close to the experimentally measured G; the mean predicted value of G is 1.34 x lo6 N/m2. The difference is less than 3:
Umversity. Evanston. IL 60201. U.S.A.
Abstract-The short-time shear and bulk moduli of articular cartilage were determined experimentall) using torsional shear and uniaxial confined compression tests. These results are compared to the values of shear moduli predicted by using data from indenter tests and the theoretical solution for the identation of a rigid punch pressed into an elastic layer bonded to a rigid halfspace. To verify the effectiveness of predicting moduli from indenter tests. control experiments were performed on polyurethane rubber. Although they accurately predict the elastic properties for rubber. the nonlinearity. anisotropy ;~nd non-homogeneity of articular cartilage make indenter tests less appropriate for use with cartilage. The prcd~cted and measured values correlate but with considerable dispersion. The short-time shear modulus of articular cartilage. including both healthy and diseased samples. were found to vary over the range 435 x lo5 N/m* and the short-time bulk modulus over the range 9-170 x lo6 N/m’. The means of tissue storage was found to have a major effect on the measured mechanical properties.
Osteoarthritis or degenera.tive joint disease. a common disorder affecting diarthrodial joints. is characterized by the deterioration. degradation and abrasion of articular cartilage and also by the formation of new bone at the articular surface. Although the aetiology of osteoarthritis is not yet fully understood, mechanical factors are often cited as initiating or aggravating agents (Sokoloff, 1969). One of the functions of articular cartilage is the transmission of loads through joints. Such loads may, at times. peak at 4-7 times body weight (Paul. 1966; Morrison. 1970). Accordingly. the mechanical properties of articular cartilage have been of great interest to researchers for some time. Although simple compression (Camosso and Marotti. 1962) and tension (Kempson et al.. 1968) tests have been attempted. the most common experiment used to study the mechanical properties of articular cartilage has been the indenter test in which a punch under constant load is pressed normally into the artitular surface. Using such a test in a qualitative manner. some of the viscoelactic characteristics of articular cartilage (Hirsch. 1944) have been examined. The influence of immersion (Elmore et al.. 1963). age (Sokoloff. 1966) and degenerative state of the tissue (Hirsch. 1944). the effect of various ionic solutions (Sokoloff. 1963). the relationship between mechanical stiffness and the muco-polysaccharide content of the tissue (Kempson it al.. 1970) and the variation of the deformability of cartilage over the articular surface (Kempson rr al.. 1971b) all have been investigated using comparative data from indenter tests. In order to obtain quantitative data. however. a mathematical model of the test is required. Solutions to this problem have been sought using the theories of classical elasticity. Although articular cartilage is
a viscoelastic material, its short-time response to load can be approximated by an elastic model. Hirsch (1944) was one of the first to appeal to the Hertz solution for contact between two elastic spheres to model the contact between a rigid spherical indenter and the curved articular surface. Zarek and Edwards (1963) use the Hertz solution generalized for the case of contact between a rigid sphere and the flat surface of an elastic half-space as their model. The solution for the indentation of an elastic halfspace by a plane cylindrical indenter. as given by Timoshenko and Goodier ( 1951). was used by Sokoloff (1966). In each of these cases, the layered geometry of the articular joint. an important consideration, was ignored. Furthermore, all the formulae used by these researchers relate punch indentation to two constitutive properties of the elastic test material. namely the elastic modulus and Poisson’s ratio. In every case. knowingly or not, these workers have taken Poisson’s ratio to be equal to 0.3. that for steel. and proceeded to calculate the elastic modulus. In order to account for the layered geometry of the articular joint, Kempson et al. (1971) used as a model the results of indentation tests on thin rubber sheets lying on a rigid surface as reported by Waters (1965). Unfortunately, the Waters solution says little about the boundary conditions at the elastic-rigid interface: the naturally high coefficient of friction of rubber was assumed to carry any shear forces that might occur at the interface but by no means was the elastic rubber layer bonded to the rigid support as is the case in the articular joint. On the basis of some tensile tests, the Poisson’s ratio of articular cartilage (average v = 0.48) was demonstrated to be close to that of rubber (v = 05). Simon (1971) reported he was unable to obtain consistant results with the theoretical models of either Kempson et a/. (1971) or Sokoloff (1966).
R. Y. HORIand L. F.
260
Recently. the contact problem of the indentation by either a spherical or a cylindrical punch pressed normally into an elastic layer bonded to a rigid halfspace was solved by Hayes er al. (1972). Their solution. based on classical theory of elasticity. assumed material isotropy and homogeneity. Using the values for the elastic materiai properties of mildly osteoarthritic cartilage previously obtained (Hayes and Mockros. 1971). these authors were able to get a correspondance between indentations as predicted by their model and the results of Hirsch (1944). The present study consists of a number of indenter tests on samples of human articular cartilage. the independent determination of the material properties of the same cartilage samples. and an attempt at establishing a correlation between the material properties of the cartilage as determined by the independent experiments and those same properties as predicted by the Hayes et al. (1972) solution and the indenter tests. The load was applied suddenly in all tests and the response measured one second after load application. As previously reported (see, e.g. Hayes and Mockros, 1971). the typical response of articular cartilage to suddenly applied loads is an “instantaneous” elastic response, followed by a long-term (l&30 min) viscoelastic response. The present study is only concerned with the “instantaneous” or “shorttime” elastic response.
METHODS
AND MATERIALS
Two series of experiments were carried out, the major difference between them being the method of storing the test samples of human articular cartilage. In series A. twelve necropsy specimens of cartilage, harvested from the proximal head of the tibia. were obtained 624 hr post mortem and stored at -20°C. The specimens were tested within six weeks of collection. Several investigators (Maroudas. 1968; Kempson et 01.. 1971a) have indicated that such storage does not materially affect the mechanical properties of artrcular cartilage. Other investigators (Viidik and Lewin. 1966). however. feel that freezing biological materials can substantially alter the properties of tissue. Accordingly. a more gentle method of tissue preservation was tried to determine the effect, if any. of storage procedures. Thus, in series B. an additional eight necropsy specimens were taken and stored immersed in Ringer’s solutions (Krebs, 1950) at 4°C. These specimens were tested within 72 hr after excision. Viidik and Lewin (1966), however. are also critical of this method of tissue storage and have shown some alterations in the mechanical properties of rabbit cruciate ligaments can occur as a result. Unfortunately. harvesting and testing the cartilage specimens immediately after death, as recommended by Viidik and Lewin. was not technically feasible. On the other hand. since the main purpose of the present
MOCKROS
study is to compare mechanical properties inferred from indenter tests with those determined from independent tests. whether or not the values measured are truly representative of irr rice mechanical properties is not a crucial question. The test specimens were obtained by cutting 19.05 mm dia. cylindrical samples of cartilage and underlying-bone from tibia) condyles using a coring tool at lOCKIrpm in a vertical milling machine. Although care was taken to ensure that the axis of the coring tool was perpendicular to the cartilage surface, this was not always possible as the condyles of the tibia1 plateau are not perfectly flat but are slightly concave. Therefore. the perpendicularity criterion was strictly applied only to the central portion of the core. During coring the cartilage was bathed by a continuous stream of Ringer’s solution to minimize heating of the articular tissue. The cancellous bone was trimmed from the sample using a double-fluted milling tool operating at 3250 rpm until a sample length of approx. 5 mm was obtained. The resulting test specimen consisted of a right circular cylinder approx. 5 mm in length composed of the following layers: the articular cartilage, the underlying dense subchondral plate and some cancellous bone. After the indenter tests using spherical punches, a 6.35 mm dia. cylindrical core was cut, using techniques described in Hayes and Mockros (1971). from the central portion of the test specimen. This smaller specimen was used for material property determinations. The degenerative state of the cartilage was graded visually under a disecting microscope at 30x power using the macroscopic classification system of Ballet. et al. (1963). The degenerative state of the cartilage samples is given in Table 3. Apparatus and procedures Materials properties determirmtion. AS the test apparatus. tissue preparation and experimental procedure for the determination of the constitutive properties of the articular cartilage specimens were virtually identical to those reported by Hayes and Mockros (1971) they will not be repeated in detail here; a complete description of the test procedure is given elsewhere (Hori, 1973). In brief, the elastic shear modulus of the tissue sample was determined using at torsional shear test. A confined, uniaxial compression test was used to obtain the elastic bulk modulus. Zndenter tests. Prior to performing the material properties tests. the indentation tests were performed using spherical punches with radii of curvature 30.4 and IO.0 mm (series A) or 30-4, 20.0 and 10.0 mm (series B). The theoretical analysis used to predict the experimental result concerns an infinite elastic layer bonded to a rigid half-space. Thus, a large area of unloaded cartilage surrounding the loaded contact
indentation
tests of human
Displacement /transducer
Balanced load arm
Loading pad Rigid cavity
ILoad
Loading wd with SpherIcal punch
Arbcular cartllaqe
articular
‘-61
cartilage
In each test,care was taken to ensure that the indenter contacted the center of the 19.05 mm dia. test specimen. In some cases of badly diseased cartilage. however. the surface was so irregular that the indenter naturally seated at some other point slightly offcenter. In these cases. the 6.35 mm diameter core was cut to fully include this area. After all the load experiments. depth measurements were taken using a needle gauge. The depth gauge was manufactured by machining a size B sewing needle to 0.25 mm dia., grinding its end Bat and attaching it to a standard depth micrometer. A large number of measurements were taken. The mean. standard diviation and coefficient of variation of these measurements were computed and those specimens with statistically significant coefficients of thickness variation were rejected for further use in the analysis. Control experiments
Fig. 1. Experimental apparatus for indenter tests with load frame and experimental geometry.
patch was desirable. The 19.05 mm diameter test specimen represented a practical limit to the size of flat cartilage that could be removed from the tibia1 plateau. The apparatus used to perform the confined. uniaxial compression test was modified to perform the indenter tests. The test specimens were held in a rigid die and the assembly was immersed in a bath of Ringer’s solution held at a constant temperature of 37” + 1°C. The indenter was brought lightly to bear normal to the cartilage surface by lowering the balance arm, whose cross-hinge was designed for a minimum of friction, and applying a tare load of 0.18 N. The loads were applied by placing weights on the balance arm tray which had a 6: 1 mechanical advantage over the indenter (see Figure 1). In each test, care was taken to ensure that the inapplied in the following progression: 1.8. 3.0. 4.1. 5.9, 8.8, 11.8 N by placing weights suddenly, but gently. on the balance arm tray. Care was taken to avoid any dynamic loading of the cartilage. A step load is very difficult to apply manually. Although the load was applied quickly, it was not applied instantaneously. Consequently, the “instantaneous” response to load was taken as that occuring one second after the initial load application. The full sequence of loads was run for one indenter at a time. As only the instantaneous indentation was of interest, the loads were applied for only a few seconds and then removed. At each load level, the test was repeated four to seven times. After each toad application. the cartilage was allowed to rest at least twice the loading time to ensure full recovery. Further, length measurements were taken from time to time to check that full recovery had occurred, but this check was not made after every test.
In order to verify the effectiveness of the technique. control experiments were performed on polyurethane rubber (CP-4485, Hysol Corp., Olean, NY). a linear isotropic, homogeneous material. Examination of tests for detenniniug the material properties. Using simple tension and uniaxial, unconfined compression tests. the Young’s modulus. E. for the rubber was determined to be 3.76 x lo6 N/m’. A right cylinder, 6.35 mm dia. x 2.5 mm long, was machined from the same stock of material and bonded to an aluminum base 6.28 mm dia. x 2 mm long. This unit was tested in uniaxial confined compression and in torsional shear as described above. Stress-strain curves of the tests are plotted in Figs. 2 and 3. Polyurethane rubber is known to be a linear material, at least for small strains, as evidenced by the linear torsional shear stress-strain curve. Thus the non-linearity of the stress-strain curve for the confined compression test clearly shows that at low loads,
I 0
I
I
1234567 Twist,
rod/m xIO-~
Fig. 2. Control experiments. Torsional shear test on polyurethane rubber sample 2.50 mm thick x 6.35 mm dia.
262
R. Y. HORI and L. F. M~CKROS IO
Table 2. Shear moduh calculated using the Hayes et al. (1972) theory and measurements from indenter tests on polyurethane rubber sample 2.66 mm thick, bonded to a rigid base Data Uncorrectedfor Tare Indentation Shear mdulur (Experimental G = 1.31 x lo6 N/m2) 60.6 mn 0. Indenter 20.0 fmn0. Indenter (N/m2) (N/m2)
Applied Load (Newtons)
Y)
1.79 2.95 4.14 5.69 6.65 11.77
6-
1.56 x 106 1.46 x 10; :::: : ;:t ::::
;%
of both
oatacorrected
1.62 x 1.67 x 1.55 x 1.46 x 1.43 x 1.45 x
106 106 106 105 106 106
1.51 x 106 N/m2
for Tare Indentation Shear bdulus (Experimental G . 1.31 x 106 ~/m2) 60.6 "," (N/ 0.f ndentcr 20.0 nr;)(~0.2;ndcnter
1
Specimen aluminum
bonded bose
to
-
1.79 2.95 4.14 5.69 6.85 11.77
kg:;4 1.32 x 1.32 x 1.37 x 1.32 x
6 106 106 106 106
1.32 x 1.43 x 1.37 x 1.34 x 1.32 x 1.36 x
106 106 106 106 106 106
Mean for both Indentertests 1.34 x 106 N/m2
Strain,
m/mx
IO2
Fig. 3. Control experiments. Confined compression test on polyurethane rubber sample 250 mm thick x 6.35 mm dia; specimen bonded to aluminum base.
compression was not obtained. Even though the cup, punch and sample were carefully machined to fit tightly, sufficient free space apparently was left to allow the lightly loaded rubber to deform and fill these voids. After a stress level of 2 x 10’ N/m2 is reached and with these voids completely filled, a true state of confined compression is attained, the stress-strain curve becomes linear and the slope of the curve can be taken as a measure of the bulk modulus of polyurethane rubber. In designing the confined compression test, the assumption was made that by carefully cutting out the cartilage, the die would be completely filled. Accordingly the bond at the cartilage-bone interface would have little effect on the stress distributions in the cartilage. In light of the non-linear character of the confined compression stress-strain curve above, a second confined compression test was performed true confined
1. The values of Young’s modulus for polyurethane rubber as determined by direct measurement using simple tension and compression tests and by calculation using torsional and uniaxial confined compression tests
Table
ExpM&tal Experiment
Young'* B!odulu*
Tenston
E - 3.81 x 106 N/o?
E . 3.131x 106 n/m2
Compression
E . 3.52 )Ilo6 N/d E - 3.96 x 106 N/m2
E = 3.52 x 106 N/m2 E = 3.96 x 106 N/m2
Toque
G = 1.13 x lo6 N/m2 G - 1.49 x 106 n/m2
mean E = 3.76 x 106 N/d Test
mean G - 1.31 x 106 N/m2 Confined Ccnnpression \1: Ui6x bonded to base confined Compression not bonded to base
lo' N/f
K . 4.41 x 10' N/m2 Y - 0.486
E f 3.90 x lo6 N/m2 E = 3.89 x 106 N/m2
of rubber 635 mm dia. x 24 mm long resting unbonded to an aluminum base 621 mm dia. x 2 mm long. Both the rubber-base and punch-rubber interfaces were lubricated with a thin film of white silicone grease to eliminate shear forces. The stress-strain curve obtained was virtually identical to Fig. 3. Apparently, the effect of the bond at the rubber-base or the cartilage bone interface is minimal on this test. The rubber cylinder was later bonded to its aluminum base and tested in torsional shear. The Young’s modulus of polyurethane rubber as determined by simple uniaxial tension and unconfined compression tests are compared with the results bf the confined compression and torsional shear tests in Table 1. Verification of the Theoretical Solution: A 200 mm dia. x 2.66 mm thick sample of polyurethane rubber, drawn from the same stock of material discussed previously, was bonded to an aluminum base and indented with the a75 mm diameter and the 2OM) mm dia. spherical indenters in the manner discussed above. In performing the indenter tests, a small tare load (0.12 N) is used to ensure light contact between the punch and the surface of the elastic layer. The amount of indentation caused by this tare load cannot be considered inconsequential as some previous researchers have assumed. The tare indentation was estimated by plotting the load vs indentation curve for the indenter tests and extrapolating the displacement at the tare load level. This result was added to the displacements measured for the applied indenter loads and with these corrected results, a set of predicted shear moduli were calculated using the indenter theory of Hayes et al., 1972. The results are given in Table 2. All the predicted values for the shear modulus, G, of polyurethane rubber using the uncorrected data on a right cylinder
263
Indentation tests of human articular cartilage
The tests on rubber serve (il to experimentally verify the theoretical solution given by Hayes rf al. (1972) and (ii) to examine the experimental procedure used for cartilage tests. These tests also indicate the importance of correcting indenter test-data for displacements due to tare loads. RESULTS Material properties detrrmimtions
/ 0
Specimen
I IO
/
I
20
30
Twist,
No.El21-I_2
I
40
50
60
rod/m
Fig. 4. Typical torque-twist curve of torsional shear test on human articular cartilage.
are high: the mean predicted value of G is 1.51 x IO6 N/m2 vs an experimental G of 1.31 x lo6 N/m’. This represents an error of some 15”,$ A notable trend in the predicted shear moduli is also evident. the highest values being associated with the smaller applied loads. The predicted values for the shear modulus using the corrected data are very close to the experimentally measured G; the mean predicted value of G is 1.34 x lo6 N/m2. The difference is less than 3:
