Normal Contact of Elastic Spheres With Two Elastic Layers as a Model of Joint Articulation A. W. Eberhardt Department of Mechanical Engineering, University of Alabama at Birmingham, Birmingham, AL 32548
J. L. Lewis Department of Orthopaedic Surgery, University of Minnesota, Minneapolis, MN 55455
L. M. Keer Department of Civil Engineering, Northwestern University, Evanston, IL 60208
An analytical model of two elastic spheres with two elastic layers in normal, frictionless contact is developed which simulates contact of articulating joints, and allows for the calculation of stresses and displacements in the layered region of contact. Using various layer/layer/substrate combinations, the effects of variations in layer and substrate properties are determined in relation to the occurrence of tensile and shear stresses as the source of crack initiation in joint cartilage and bone. Vertical cracking at the cartilage surface and horizontal splitting at the tidemark have been observed in joints with primary osteoarthritis. Deep vertical cracks in the calcified cartilage and underlying bone have been observed in blunt trauma experiments. The current model shows that cartilage stresses for a particular system are a function of the ratio of contact radius to total layer thickness (a/h). Surface tension, which is observed for a/h small, is alleviated as a/h is increased due to increased load, softening and/or thinning of the cartilage layer. Decreases in a/h due to cartilage stiffening lead to increased global compressive stresses and increased incidence ofsUljace tension, consistent with impact-induced surface cracks. Cartilage stresses are not significantly affected by variations in stiffness of the underlying material. Tensile radial strains in the cartilage layer approach one-third ofthe normal compressive strains, and increase significantly with cartilage softening. For cases where the middle layer stiffness exceeds that of the underlying substrate, tensile stresses occur at the base of the middle layer, consistent with impact induced cracks in the zone of calcified cartilage and subchondral bone. The presence of the superficial tangential zone appears to have little effect on underlying cartilage stresses.
Introduction Osteoarthritis (OA) is a degenerative disease which involves deterioration of the articular cartilage of synovial joints and alteration of the underlying calcified cartilage and subchondral bone [I]. It may be categorized into three types, depending on the assumed cause, including traumatic OA, induced by mechanical trauma to the joint, instability OA, caused by loss of ligaments and assumed excess forces and motions at the joint surfaces, and primary OA, the most common form of the disease of which the causes are unknown. Although each type has distinctive features, they all exhibit cracking of the uncalcified cartilage in the region of contact. In primary OA, the cracks generally appear to begin at the cartilage surface and propagate vertically down into the zone of calcified cartilage (ZCC) and underlying bone [2], although horizontal cracks have also been observed at the tidemark, the interface between the uncalcified cartilage and the ZCC [3]. The resulting state is shown in Fig. I, taken from [2], which shows a histological section through the cartilage/ZCC/bone of a human joint with primary OA. In more severe forms of the disease, increasing
Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division January 26, 1990; revised manuscript received May 15, 1991.
numbers of fissures appear and the cartilage begins to fragment and wear away. Hypertrophy of bone also occurs. Instability OA demonstrates surface fissures as well. Traumatic OA is
Fig. 1 Histological section of osteoarthritic cartilage showing vertical crack from cartilage surface, and horizontal splitting at the tidemark (1) (2)
Transactions of the ASM E
410 I Vol. 113, NOVEMBER 1991
Copyright © 1991 by ASME Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 05/10/2015 Terms of Use: http://asme.org/terms
subchondral bone Flg.2 Histological section of impacted patella showing vertical cracks in ZCC and bone
more variable, but animal models have shown horizontal splitting at the tidemark [4], and in blunt impact studies, the authors have observed impact-induced vertical cracks which appear to initiate in the ZCC and underlying bone. Figure 2 shows a histological section of a canine patella immediately after impact, taken from tests done by the authors, where a planeended platen has impacted the anesthetized intact patella-femoral joint of the animal. As the impact force is increased, cracks first develop in the ZCC and underlying bone, perpendicular to the tidemark. As impact force is further increased, these cracks extend into the cartilage, and another system of cracks is formed at the cartilage surface. With instability OA and primary OA, the cause of cracking is not certain, and there is controversy as to whether (i) the normal joint is first overloaded and cartilage fails mechanically, and then further degenerates due to biological processes, or (ii) the tissues are initially degenerated biologically, and then fail under relatively "normal" loads. Poor understanding of joint loads and stress distributions within the joint system, as well as ignorance of the mechanical processes involved in tissue degeneration, contribute to this uncertainty. The present study is an extension of earlier models [5, 6] for which the predicted stress fields did not correlate well with experimental observations of cracking in primary or traumatic OA. The question to be addressed is whether or not the new model predicts stresses necessary to generate the type of tissue failure seen. The failure criteria of these materials is not known in detail, but it seems reasonable that tensile stresses would be an essential element in the formation of the vertical cracks. If this is the case, an accurate contact model should predict tension along planes at which these cracks are observed if it correctly simulates the physics of these conditions. A negative result would indicate either the cracks are not mechanically initiated, they are not a result of tensile stresses, or the model is not an accurate simulation of joint contact. Either conclusion would aid in understanding of articulating joint contact and the etiology of the OA process. Articular cartilage has a long history of study. Early' 'elastic" studies include those of Hirsch [7], Zarek and Edwards [8] and Sokoloff [9]. Hayes et al. [10] developed the mathematical solution for indentation of an elastic layer on a rigid halfspace to determine the shear modulus, p" assuming Poisson's ratio, 11. Hori and Mockros [11], using measurements from independent confined compression and torsion tests taken one second after loading, obtained a range of "short-time" values of 0.46 MPa:::;p,:::;3.47 MPa, and 0.42:::;11:::;0.49 for healthy and degenerated cartilage. Journal of Biomechanical Engineering
The response of cartilage to the sudden application of a static load is characterized by a rapid initial compressive phase, followed by a slow, time-dependent creep process toward equilibrium. Studies which examine time-dependent effects include those of Kempson et al. [12], Coletti et al. [13], and Parsons and Black [14]. The biphasic model for articular cartilage was begun by McCutchen [15] and further developed by Mow and coworkers [16, 17]. In an effort to determine cartilage stresses in-situ, Askew and Mow [18] modeled the cartilage-bone systems as a nonhomogeneous, anisotropic, layered continuum model subject to a prescribed normal surface stress. Their model predicted significant shear stresses and tensile strains, however they concluded "the articular surface is not exposed to continuum tensile stresses." In [5], the authors simulated contact of an articulating joint using an analytical model of normal, frictionless contact between two elastic spheres (bone), each with a single elastic layer (cartilage). The resulting stress fields from this model showed surface tension consistent with vertical surface cracks as observed in primary OA only for cases where a/h is small ("" 1). The inclusion of a frictional surface loading [6] showed significant surface tension only when friction coefficients are exceptionally high, or, as in the frictionless model, for a/h small. Actual a/h values which occur in-vivo are uncertain, but it is believed that a/h values would generally be greater than one due to conformity of contacting surfaces. Using Fuji film, however, Chin-Purcell [19] measured the contact area of impacted canine patellas, showing values of a/h of approximately 2. This uncertainty merits further study. In addition to the lack of agreement with cracking in primary OA, the one layer results are not consistent with impact-induced cracks originating in the ZCC or underlying bone. In the present study, therefore, a second elastic layer is added to each contacting sphere, and the stress analysis is performed to determine for which two layer systems tensile stresses are induced by normal, frictionless loading. The model is used to analyze three distinct layer/layer/substrate combinations: (i) cartilage/ZCC/bone, (ii) cartilage/subchondral plate/cancellous bone, and (iii) STZ/cartilage/bone. In (i) and (iii) a shear modulus of p,= 1.5 MPa and a Poisson's ratio of 11=0.475, within a range determined in I9], are used in this study to represent "normal" cartilage, and bone is assigned properties p, = 580 MPa and 11= 0.3, as obtained in [20]. These values were used in the one layer studies [5, 6]. The actual stiffness of the ZCC is not known, therefore in (i) the ZCC is assigned stiffness values of either one-half or four times that of the underlying bone. Since the stiffness properties of the ZCC have never been measured, it is not known whether the stiffer ZCC is realistic, however this possibility is testable. Properties for cartilage (p, = 2.41 MPa, 11 = 0.475), the subchondral plate (SP), (p,=769 MPa, 11=0.3) and cancellous bone (p,=26.9 MPa, 11 = 0.3), used in (ii) are taken from I21]. In (iii) the STZ is modeled as having a stiffness four times that of the underlying cartilage. Important features of the layered contacting system, as well as effects of alterations in system parameters, such as stiffness and layer thickness, are analyzed using variations of these layer and substrate combinations. In [5], using recently published results of creep and stressrelaxation tests, the authors show that for consideration of cartilage deformation under functional loads such as walking and running, and for impact loads, it is reasonable to ignore time-dependency of the cartilage deformational response. Cartilage response was assumed elastic, and in the present study this assumption is maintained, therefore stresses and displacements predicted by the model may be considered representative of peak values obtained during short-time loadings such as impact. The new feature of this solution relative to previous work is that it is a multi-layered contact problem where the contact area and resulting stress distributions are integral parts of the solution. NOVEMBER 1991, Vol. 113/411
Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 05/10/2015 Terms of Use: http://asme.org/terms
r2 = n2¥i
r i = m'R
Cartilage/ZCC/boJ3£ (v 0 =.3 VI =.3 V2 = 475) T2 ri Y 0.0052 0.5 Case I: 0.9 0.0O065 4.0 Case 2: 0.9 0.0052 0.5 Case 3: 0.5 0.00065 4.0 Case 4: 0.5 20.0 0.00065 Case 5: 0.9
Canilape/SP/cancellous bone (vo= .3. vi = .3, V2= .475) Y H Ti Case 6: 0.8 28.6 0.0031 STZ/Cartilaee/bone (V()= .3, vi = .475, vj= .475) y ri T2 Case 7: 0.1 0.0052 4.0
Fig. 3 Two layer analytical model of joint contact
4>o(s)-
J
O n z = 0; ^
= 7*2 = 0
0
J. L. Lewis Department of Orthopaedic Surgery, University of Minnesota, Minneapolis, MN 55455
L. M. Keer Department of Civil Engineering, Northwestern University, Evanston, IL 60208
An analytical model of two elastic spheres with two elastic layers in normal, frictionless contact is developed which simulates contact of articulating joints, and allows for the calculation of stresses and displacements in the layered region of contact. Using various layer/layer/substrate combinations, the effects of variations in layer and substrate properties are determined in relation to the occurrence of tensile and shear stresses as the source of crack initiation in joint cartilage and bone. Vertical cracking at the cartilage surface and horizontal splitting at the tidemark have been observed in joints with primary osteoarthritis. Deep vertical cracks in the calcified cartilage and underlying bone have been observed in blunt trauma experiments. The current model shows that cartilage stresses for a particular system are a function of the ratio of contact radius to total layer thickness (a/h). Surface tension, which is observed for a/h small, is alleviated as a/h is increased due to increased load, softening and/or thinning of the cartilage layer. Decreases in a/h due to cartilage stiffening lead to increased global compressive stresses and increased incidence ofsUljace tension, consistent with impact-induced surface cracks. Cartilage stresses are not significantly affected by variations in stiffness of the underlying material. Tensile radial strains in the cartilage layer approach one-third ofthe normal compressive strains, and increase significantly with cartilage softening. For cases where the middle layer stiffness exceeds that of the underlying substrate, tensile stresses occur at the base of the middle layer, consistent with impact induced cracks in the zone of calcified cartilage and subchondral bone. The presence of the superficial tangential zone appears to have little effect on underlying cartilage stresses.
Introduction Osteoarthritis (OA) is a degenerative disease which involves deterioration of the articular cartilage of synovial joints and alteration of the underlying calcified cartilage and subchondral bone [I]. It may be categorized into three types, depending on the assumed cause, including traumatic OA, induced by mechanical trauma to the joint, instability OA, caused by loss of ligaments and assumed excess forces and motions at the joint surfaces, and primary OA, the most common form of the disease of which the causes are unknown. Although each type has distinctive features, they all exhibit cracking of the uncalcified cartilage in the region of contact. In primary OA, the cracks generally appear to begin at the cartilage surface and propagate vertically down into the zone of calcified cartilage (ZCC) and underlying bone [2], although horizontal cracks have also been observed at the tidemark, the interface between the uncalcified cartilage and the ZCC [3]. The resulting state is shown in Fig. I, taken from [2], which shows a histological section through the cartilage/ZCC/bone of a human joint with primary OA. In more severe forms of the disease, increasing
Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division January 26, 1990; revised manuscript received May 15, 1991.
numbers of fissures appear and the cartilage begins to fragment and wear away. Hypertrophy of bone also occurs. Instability OA demonstrates surface fissures as well. Traumatic OA is
Fig. 1 Histological section of osteoarthritic cartilage showing vertical crack from cartilage surface, and horizontal splitting at the tidemark (1) (2)
Transactions of the ASM E
410 I Vol. 113, NOVEMBER 1991
Copyright © 1991 by ASME Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 05/10/2015 Terms of Use: http://asme.org/terms
subchondral bone Flg.2 Histological section of impacted patella showing vertical cracks in ZCC and bone
more variable, but animal models have shown horizontal splitting at the tidemark [4], and in blunt impact studies, the authors have observed impact-induced vertical cracks which appear to initiate in the ZCC and underlying bone. Figure 2 shows a histological section of a canine patella immediately after impact, taken from tests done by the authors, where a planeended platen has impacted the anesthetized intact patella-femoral joint of the animal. As the impact force is increased, cracks first develop in the ZCC and underlying bone, perpendicular to the tidemark. As impact force is further increased, these cracks extend into the cartilage, and another system of cracks is formed at the cartilage surface. With instability OA and primary OA, the cause of cracking is not certain, and there is controversy as to whether (i) the normal joint is first overloaded and cartilage fails mechanically, and then further degenerates due to biological processes, or (ii) the tissues are initially degenerated biologically, and then fail under relatively "normal" loads. Poor understanding of joint loads and stress distributions within the joint system, as well as ignorance of the mechanical processes involved in tissue degeneration, contribute to this uncertainty. The present study is an extension of earlier models [5, 6] for which the predicted stress fields did not correlate well with experimental observations of cracking in primary or traumatic OA. The question to be addressed is whether or not the new model predicts stresses necessary to generate the type of tissue failure seen. The failure criteria of these materials is not known in detail, but it seems reasonable that tensile stresses would be an essential element in the formation of the vertical cracks. If this is the case, an accurate contact model should predict tension along planes at which these cracks are observed if it correctly simulates the physics of these conditions. A negative result would indicate either the cracks are not mechanically initiated, they are not a result of tensile stresses, or the model is not an accurate simulation of joint contact. Either conclusion would aid in understanding of articulating joint contact and the etiology of the OA process. Articular cartilage has a long history of study. Early' 'elastic" studies include those of Hirsch [7], Zarek and Edwards [8] and Sokoloff [9]. Hayes et al. [10] developed the mathematical solution for indentation of an elastic layer on a rigid halfspace to determine the shear modulus, p" assuming Poisson's ratio, 11. Hori and Mockros [11], using measurements from independent confined compression and torsion tests taken one second after loading, obtained a range of "short-time" values of 0.46 MPa:::;p,:::;3.47 MPa, and 0.42:::;11:::;0.49 for healthy and degenerated cartilage. Journal of Biomechanical Engineering
The response of cartilage to the sudden application of a static load is characterized by a rapid initial compressive phase, followed by a slow, time-dependent creep process toward equilibrium. Studies which examine time-dependent effects include those of Kempson et al. [12], Coletti et al. [13], and Parsons and Black [14]. The biphasic model for articular cartilage was begun by McCutchen [15] and further developed by Mow and coworkers [16, 17]. In an effort to determine cartilage stresses in-situ, Askew and Mow [18] modeled the cartilage-bone systems as a nonhomogeneous, anisotropic, layered continuum model subject to a prescribed normal surface stress. Their model predicted significant shear stresses and tensile strains, however they concluded "the articular surface is not exposed to continuum tensile stresses." In [5], the authors simulated contact of an articulating joint using an analytical model of normal, frictionless contact between two elastic spheres (bone), each with a single elastic layer (cartilage). The resulting stress fields from this model showed surface tension consistent with vertical surface cracks as observed in primary OA only for cases where a/h is small ("" 1). The inclusion of a frictional surface loading [6] showed significant surface tension only when friction coefficients are exceptionally high, or, as in the frictionless model, for a/h small. Actual a/h values which occur in-vivo are uncertain, but it is believed that a/h values would generally be greater than one due to conformity of contacting surfaces. Using Fuji film, however, Chin-Purcell [19] measured the contact area of impacted canine patellas, showing values of a/h of approximately 2. This uncertainty merits further study. In addition to the lack of agreement with cracking in primary OA, the one layer results are not consistent with impact-induced cracks originating in the ZCC or underlying bone. In the present study, therefore, a second elastic layer is added to each contacting sphere, and the stress analysis is performed to determine for which two layer systems tensile stresses are induced by normal, frictionless loading. The model is used to analyze three distinct layer/layer/substrate combinations: (i) cartilage/ZCC/bone, (ii) cartilage/subchondral plate/cancellous bone, and (iii) STZ/cartilage/bone. In (i) and (iii) a shear modulus of p,= 1.5 MPa and a Poisson's ratio of 11=0.475, within a range determined in I9], are used in this study to represent "normal" cartilage, and bone is assigned properties p, = 580 MPa and 11= 0.3, as obtained in [20]. These values were used in the one layer studies [5, 6]. The actual stiffness of the ZCC is not known, therefore in (i) the ZCC is assigned stiffness values of either one-half or four times that of the underlying bone. Since the stiffness properties of the ZCC have never been measured, it is not known whether the stiffer ZCC is realistic, however this possibility is testable. Properties for cartilage (p, = 2.41 MPa, 11 = 0.475), the subchondral plate (SP), (p,=769 MPa, 11=0.3) and cancellous bone (p,=26.9 MPa, 11 = 0.3), used in (ii) are taken from I21]. In (iii) the STZ is modeled as having a stiffness four times that of the underlying cartilage. Important features of the layered contacting system, as well as effects of alterations in system parameters, such as stiffness and layer thickness, are analyzed using variations of these layer and substrate combinations. In [5], using recently published results of creep and stressrelaxation tests, the authors show that for consideration of cartilage deformation under functional loads such as walking and running, and for impact loads, it is reasonable to ignore time-dependency of the cartilage deformational response. Cartilage response was assumed elastic, and in the present study this assumption is maintained, therefore stresses and displacements predicted by the model may be considered representative of peak values obtained during short-time loadings such as impact. The new feature of this solution relative to previous work is that it is a multi-layered contact problem where the contact area and resulting stress distributions are integral parts of the solution. NOVEMBER 1991, Vol. 113/411
Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 05/10/2015 Terms of Use: http://asme.org/terms
r2 = n2¥i
r i = m'R
Cartilage/ZCC/boJ3£ (v 0 =.3 VI =.3 V2 = 475) T2 ri Y 0.0052 0.5 Case I: 0.9 0.0O065 4.0 Case 2: 0.9 0.0052 0.5 Case 3: 0.5 0.00065 4.0 Case 4: 0.5 20.0 0.00065 Case 5: 0.9
Canilape/SP/cancellous bone (vo= .3. vi = .3, V2= .475) Y H Ti Case 6: 0.8 28.6 0.0031 STZ/Cartilaee/bone (V()= .3, vi = .475, vj= .475) y ri T2 Case 7: 0.1 0.0052 4.0
Fig. 3 Two layer analytical model of joint contact
4>o(s)-
J
O n z = 0; ^
= 7*2 = 0
0
