Ligaments and articular contact guide passive knee flexion

Journal of Biomechanics 31 (1998) 1127—1136

Ligaments and articular contact guide passive knee flexion D.R. Wilson*, J.D. Feikes, J.J. O’Connor Oxford Orthopaedic Engineering Centre and Department of Engineering Science, University of Oxford, Oxford, U.K. Received 24 February 1997; accepted 23 July 1998

Abstract The aim of this study was to test the hypothesis that the coupled features of passive knee flexion are guided by articular contact and by the isometric fascicles of the ACL, PCL and MCL. A three-dimensional mathematical model of the knee was developed, in which the articular surfaces in the lateral and medial compartments and the isometric fascicles in the ACL, PCL and MCL were represented as five constraints in a one degree-of-freedom parallel spatial mechanism. Mechanism analysis techniques were used to predict the path of motion of the tibia relative to the femur. Using a set of anatomical parameters obtained from a cadaver specimen, the model predicts coupled internal rotation and ab/adduction with flexion. These predictions correspond well to measurements of the cadaver specimen’s motion. The model also predicts posterior translation of contact on the tibia with flexion. Although this is a well-known feature of passive knee flexion, the model predicts more translation than has been reported from experiments in the literature. Modelling of uncertainty in the anatomical parameters demonstrated that the discrepancy between theoretical predictions and experimental measurement can be attributed to parameter sensitivity of the model. This study shows that the ligaments and articular surfaces work together to guide passive knee motion. A principal implication of the work is that both articular surface geometry and ligament geometry must be preserved or replicated by surgical reconstruction and replacement procedures to ensure normal knee kinematics and by extension, mechanics.  1998 Elsevier Science Ltd. All rights reserved. Keywords: Knee; Kinematics; Model; Mechanism; Geometry

1. Introduction The functional anatomy of the knee must be understood when designing or assessing surgical procedures. The mechanical functions of the structures at the tibiofemoral joint include guiding the relative motion of the tibia and femur and transmitting load between these two bones. Surgeons seek to preserve or restore both of these mechanical functions during joint reconstruction or replacement. During surgery, both the geometry of joint structures and their mechanical properties are often changed. In anterior cruciate ligament (ACL) reconstruction, geometric parameters include the sites of origin and insertion of the graft, while in total knee arthroplasty, geometric parameters include the shapes and sizes of the

* Corresponding author. Present address. Orthopedic Biomechanics Laboratory, Beth Israel Deaconess Medical Center and Harvard Medical School, 330 Brookline Ave. RN 115, Boston, MA 02215, USA. Tel.: 1-617-667-5185; fax: 1-617-667-4561; e-mail: [email protected]. harvard.edu

articular surfaces and their locations relative to the ligament and muscle tendon insertions. Intra-operatively, the surgeon flexes the knee passively and observes knee motion to assess component or graft placement. If flexion is resisted or if the knee appears unstable, component alignment or graft length may be adjusted. How geometric changes to the joint structures affect the three-dimensional movement of the joint has not been explained completely. It is not clear from the literature which anatomical structures guide the knee in passive flexion and how their geometric arrangement produces the unique path of passive knee motion. The knee can be moved passively through a full range of flexion/extension (0—130°) with minimal resistance. Both internal rotation (Meyer, 1853) and posterior translation of the femur on the tibia (Weber and Weber, 1836) are coupled to flexion. This movement path must be guided by the anatomical structures of the tibiofemoral joint. Several mathematical models have been developed that predict three-dimensional motion of the knee by solving the equations of mechanical equilibrium (and, sometimes, simultaneously

0021-9290/98/$ — see front matter  1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 9 8 ) 0 0 1 1 9 - 5

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Fig. 1. An isometric ligament fascicle (a) and its equivalent connections using (b) higher pairs and (c) one degree-of-freedom pairs (after (Hunt, 1978)).

Fig. 2. Articular contact (a) and its equivalent connections using (b) higher pairs and (c) one degree-of-freedom pairs (after (Hunt, 1978)).

solving equations of geometric constraint) (Andriacchi et al., 1983; Blankevoort et al., 1991; Essinger et al., 1989; Wismans et al., 1980). However, these models have been developed to study the response of the joint to moderate loads and they have not been used to predict the unique three-dimensional path of motion of the unloaded joint. In a contrasting approach, the knee has been studied as

a mechanical linkage or mechanism (Goodfellow and O’Connor, 1978; Huson 1974; Menschik, 1974; O’Connor et al., 1989; Strasser, 1917). These models predict knee motion from the geometry of the anatomical structures. By comparing model predictions to experimental measurements of knee motion, hypotheses concerning which structures guide knee motion and the influence of changes to the geometry of the structures on knee motion can be tested. Although two-dimensional linkage models have successfully explained posterior translation of the femur on the tibia, no linkage model has yet predicted the simultaneous coupled internal tibial rotation and posterior femoral translation that characterizes passive knee flexion. To explain the relationship between knee anatomy and knee flexion completely, a linkage model must be developed that predicts these simultaneous coupled features of passive knee flexion from the geometry of the anatomical structures. The aim of this study is to test the hypothesis that passive knee motion is guided by the anterior cruciate ligament (ACL), posterior cruciate ligament (PCL) and medial collateral ligament (MCL) and articular contact in the medial and lateral compartments. This article describes how this hypothesis was tested by formulating and analyzing a three-dimensional mechanism model of the knee and by comparing its predictions to measurements of knee motion.

2. Method A three-dimensional mechanism model of the knee was formulated by making assumptions about joint anatomy based on evidence in the literature. Experimental studies suggest that within the ACL and PCL (Fuss, 1989; Rovick et al., 1991; Sidles et al., 1988) and the MCL (Rovick et al., 1991) there is a fascicle that remains nearly isometric through the range of flexion of the knee. The posterior femoral condyles are nearly spherical (Kurosawa and Walker, 1985) and the tibial condyles are

Fig. 3. Parallel spatial mechanism model of the knee: (a) frontal (b) sagittal and (c) transverse views. ACL" Anterior Cruciate Ligament, PCL"Posterior Cruciate Ligament, MCL"Medial Collateral Ligament, MCON"Medial Contact, LCON"Lateral Contact.

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curved slightly in both the sagittal and coronal planes (Kapandji, 1987). As a first approximation, it was assumed that the tibial condyles are planar, the femoral condyles are spherical, the tibia and femur are rigid and contact is maintained continuously in both compartments. These conditions specify a single point of tibiofemoral contact in each compartment. Experimental studies suggest that the other major structures of the knee do not constrain motion in passive flexion. The menisci slide freely in the anteroposterior direction (Kapandji, 1987; Thompson et al., 1990) and the posterior capsule of the joint is taut only near full extension (Bishop, 1977; O’Connor et al., 1990). Although the LCL is taut in full extension, evidence suggests that all fascicles of the LCL go slack in flexion, rendering the ligament unable to constrain knee flexion (Kapandji, 1987; Rovick et al., 1991; Wang et al., 1973). These structures were therefore excluded from this analysis. The isometric fascicles in the ACL, PCL, MCL and contact in the medial and lateral compartments were represented by connections made up of standard kinematic pairs which link one rigid bone to the other. Each fascicle was represented by a rod connecting a spherical pair (ball joint) on the tibia to a spherical pair on the femur. This constraint corresponds to isometricity of the fascicle (Fig. 1a and b). Medial and lateral articular constraints were each represented by a spherical pair at the centre of the femoral condyle connected by a rod to a planar pair at the point of contact on the tibial condyle (Fig. 2a and b). This connection allows rolling and sliding at the contact point. The resulting configuration (Fig. 3) is a parallel spatial mechanism (PSM) similar to a Stewart Platform robotic manipulator (Stewart, 1966), so named because the output link (the femur) is connected to the input (the tibia) by a parallel arrangement of constraints in three dimensions. In the first step of the kinematic analysis, the mobility of the mechanism was determined. The mobility M of a spatial mechanism with u linearly independent constraints is given by (Hunt, 1978) [M"6!u].

1. Reference coordinate systems were defined. One system was fixed to the tibia and had its origin at the most posterior point of the tibial ACL attachment (Fig. 4). In the right knee, the X-axis was oriented posteriorly, the ½-axis was oriented laterally, and the Z-axis was oriented superiorly. The other system was fixed to the femur and, at the starting position, the origins of both systems were coincident and their respective X-, ½- and Z-axes were parallel. 2. The positions of the spherical pairs, the positions and orientations of the planar pairs and the dimensions of the links at the starting position (full extension) were determined from a set of anatomical parameters (appendix). 3. The spherical and planar pairs making up the ligament and articular connections were replaced by kinematically equivalent chains of one DOF pairs (hinges and sliders: Figs. 1b and c and 2b and c). 4. Each one DOF pair was assigned a coordinate system using an established convention (Denavit and Hartenberg, 1955). Four pair quantities, a , s , h and a define G G G G the transformation from the coordinate system in one pair to the coordinate system in an adjacent pair. These four quantities were calculated for each of the 25 one DOF pairs using simple geometry, yielding 100 pair quantities describing the starting position of the

(1)

Because the three ligament connections and the two articular connections each represent one independent constraint, the PSM has one degree of freedom (DOF). One input parameter is therefore sufficient to specify the position of the output link (the position of the femur relative to the tibia is specified fully by the flexion angle). The path of the femur as a function of the flexion angle was found by solving the forward kinematics of the mechanism. The mathematical problem is to determine a motion path which will satisfy the five constraints of this model. Because no closed-form solution to the forward kinematics of this PSM has been found (Lee et al., 1993), an iterative method was used (Uicker et al., 1964) :

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Fig. 4. Reference coordinate system fixed in the tibia.

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mechanism. In the revolute pair, only h changes as the G pair rotates, and in the prismatic pair, only s changes G as the pair slides. Thus, as the mechanism moves, 75 pair parameters remain constant (3 at every pair) and 25 pair variables change. 5. The transformation across any pair can be written as a 4;4 transformation matrix, A , whose elements are GH functions of a , s , h and a (Uicker et al., 1964): G G G G A " GH 1 a cos h G G a sin h G G s G

0 0 0 cos h !sin h cos a sin h sin a G G G G G sin h cos h cos a ! cos h sin a G G G G G 0 sin a cos a G G

.

6. Four independent closed loops were defined, each starting and ending at the tibial insertion of the ACL: ACL-PCL, ACL-MCL, ACL-Medial Contact Normal, ACL-Lateral Contact Normal. 7. The revolute pair fixed to the tibia in the ACL connection was chosen as the input and named pair 1. A small change in the input angle h was applied to  move the mechanism from the starting position. Values of the 24 other pair variables were determined using iterative computations (Uicker et al., 1964). 8. At every increment of the input angle, the transformation matrix between the coordinate system in the femur and the coordinate system in the tibia was calculated. A sequence-independent joint coordinate system (Cole et al., 1993) was used to present model kinematics in a way that is both mathematically correct and anatomically relevant. Flexion was calculated about the ½-axis of the femur-fixed coordinate system, internal/external rotation was calculated about the Z-axis of the tibiafixed coordinate system, and varus/valgus was calculated about the floating (‘third’) axis defined by the cross product of the other two axes. To obtain parameters for the model, measurements were made on a single fresh post-mortem human knee specimen. The specimen was flexed once from 0 to 100° of flexion and back in a test rig designed to apply no constraint (i.e. no force or moment) to the remaining five components of motion (Feikes et al., 1998). Kinematic data were measured and represented in the joint coordinate system described above. After the experiment, the specimen was disarticulated, the menisci were removed and an electromagnetic tracking system (Isotrak II, Polhemus, Colchester VT, USA) was used to digitize the articular surfaces of the two bones and the attachments of the ACL, PCL and MCL, together with the locations of markers defining reference systems in each bone. ACL, PCL and MCL isometric fascicles were chosen based on a search through all possible combinations of measured tibial and femoral at-

tachment points. Those fascicles which experienced the least overall length change (based on distance from tibial to femoral attachment points) throughout the range of passive motion were chosen. Planes were fitted to the medial and lateral tibial condyles and spheres were fitted to each of the posterior femoral condyles with a leastsquares minimization routine. The centre and radius of the sphere representing each condyle was determined. The line joining the two sphere centres defined the flexion axis of the JCS. The starting configuration of the model corresponded to the knee in full extension. In each compartment, the initial contact points on the tibia were defined as the intersection with the tibial plane of the normal to the plane passing through the centre of the corresponding femoral sphere. Parameters are listed in the appendix.

3. Results The model predicts coupled internal rotation of the knee in passive flexion. Motion is predicted from 6° of hyperextension to 106° of flexion. As the knee is flexed, the tibia rotates internally and ab/adducts slightly (Fig. 5). Experimental measurements of internal/external rotation and ab/adduction on the knee from which the model parameters were taken match the model predictions closely (Fig. 5). The model also predicts posterior translation of the femur on the tibia. In both compartments, the contact point on the tibial condyle translates posteriorly with flexion (Fig. 6). Translation on the lateral condyle is nearly twice that on the medial condyle, reflecting coupled internal rotation. Relative motion at the point of contact is characterized by the slip ratio. The slip ratio is defined as the ratio of the distance travelled by the contact point on the tibia to the distance travelled by the contact point on the femur over a specified increment of movement. A slip ratio of zero indicates pure slipping, while a slip ratio of one describes pure rolling. A slip ratio between zero and one indicates partial slipping, while a slip ratio greater than one describes skidding (Fig. 7). For the mechanism model, the slip ratio has been determined for every small increment of the input angle of the mechanism. In both compartments, the slip ratio decreases with flexion (Fig. 7). More slip is observed in the medial compartment than in the lateral compartment. Alternative configurations of the mechanism model show that it is unlikely that other structures guide passive flexion. There are four possible configurations that include the two articular surfaces and three of the four principal ligaments (ACL,PCL,MCL and LCL). Of these, the ACL/PCL/LCL configuration predicts external rotation with flexion and the PCL/MCL/LCL combination has virtually no range of motion (Fig. 8). The ACL/MCL/LCL configuration predicts motion close to

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Fig. 5. Model predictions and experimental measurements of tibial rotation relative to the femur as a function of flexion. Internal rotation and ab/adduction coupled to flexion are observed in experiment and predicted by the model.

features of passive knee flexion. The anatomical parameters define the starting position of the mechanism. The effect of uncertainty was studied by predicting motion for models in which a parameter had been changed by $5 mm (or $5° of orientation of the tibial plateaux). For the tibial attachment of the ACL, errors in the Z-direction have the most influence, while errors in the ½-direction have the least influence (Fig. 9). Although uncertainty of $5 mm or $5° in some parameters has a substantial effect on the maximum internal rotation predicted by the model, coupled internal rotation is always predicted (Fig. 10).

4. Discussion

Fig. 6. Contact points on the tibial condyles at 10° increments of flexion. The tibial attachment of the isometric ACL fascicle is labelled ‘iso—acl’. Tibiofemoral contact in both compartments moves posteriorly with flexion.

that of the original configuration, although the range of motion is reduced. Substantial uncertainty in the anatomical parameters does not affect the model’s prediction of the principal

To test the hypothesis that passive knee motion is guided by the ACL, PCL and MCL and articular contact in the medial and lateral compartments, the knee was modelled as a three-dimensional parallel spatial mechanism. In a novel approach, a systematic method was used to model joint structures as links and pairs in a spatial mechanism. This approach can easily be applied to any joint in which the appropriate assumptions of rigidity and isometricity can be made. The development of the three-dimensional model was motivated by the success of the four-bar linkage model. This two-dimensional model is widely used to explain the functional anatomy of the knee (Kapandji, 1987), and has been applied to guide the design of prostheses

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Fig. 7. Slip ratios in the medial and lateral compartments. The legend to the left shows the velocity profile (normal to the axis of rotation) of femoral condyle points in the plane normal to the axis of rotation and passing through the contact point. The medial compartment slips more than the lateral compartment, and slip increases in both compartments with flexion.

(Goodfellow and O’Connor, 1978), used as a basis for mechanical models for studying the loads in the structures of the knee joint (Collins and O’Connor, 1991; Zavatsky and O’Connor, 1992a,b, 1993), and used to rationalize physical therapy techniques (Zavatsky et al., 1994). The four-bar linkage is limited because it is confined to a plane and therefore cannot predict coupled internal rotation of the knee. This study has demonstrated that the parallel spatial mechanism model is an appropriate tool to prove the initial hypothesis. For the chosen configuration, a path of motion was found which satisfies the ligamentous and articular contact constraints and corresponds to experimental measurements of knee motion. Of all possible configurations tested, the ACL/PCL/MCL combination that was originally identified by making the anatomical assumptions produced motion most resembling that of the natural knee. Model sensitivity to uncertainty in the anatomical parameters shows that variations in knee anatomy on the order of those one would expect within the population produce reasonable variations in knee motion. These results emphasize the validity of the assumptions made during the formulation of the model. A principal limitation of the model is that the geometry of the anatomical structures was simplified. The femoral condyles of the natural knee are aspherical and

the tibial plateaux are curved in the sagittal and coronal planes. Differences between the model and natural knee ligament constraints may be of particular importance. The MCL was modelled as a straight-line segment connecting the tibial and femoral attachments, while the natural MCL wraps around the tibial plateau. Kinematically, the constraint provided by an isometric fascicle ‘bent’ around the tibial plateau is identical to that provided by a straight-line segment connecting the two attachments. The difficulty lies in finding the isometric fascicle. If tibial and femoral attachments are found such that the straight-line segment between them is isometric, the corresponding ligament fascicle may well be anisometric because the point of wrapping on the tibial condyle changes with flexion. It is also possible that the exact locations of the ACL and PCL isometric fascicles were not found. In the identification procedure, it was not stipulated that the most isometric line segment must correspond physically to a real fascicle. The procedure identified a posterior PCL isometric segment, which does correspond physically to a ligament fascicle and to the accepted location for the isometric fascicle (Fuss, 1989; Rovick et al., 1991; Sidles et al., 1988). The posterior MCL isometric segment also corresponds physically to a ligament fascicle but not to an accepted isometric fascicle (Rovick et al., 1991). The femoral anterior/tibial

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Fig. 8. Predictions of internal/external rotation for four configurations of the mechanism model. Coupled internal rotation through the range of flexion of the knee is predicted for only two of the four configurations. No motion is predicted for the PCL/MCL/LCL configuration.

Fig. 9. Model predictions of internal/external rotation when errors of $5 mm are introduced into the tibial attachment parameters of the ACL. Small parameter changes affect the range of motion and the magnitude of automatic rotation, but they do not change the qualitative prediction of coupled internal rotation.

posterior isometric segment in the ACL does not correspond physically to a real fascicle and, obviously, not to an accepted isometric fascicle either (Fuss, 1989; Rovick et al., 1991; Sidles et al., 1988). As an alternative, it is

possible to situate the ligament constraints using published locations of the isometric fascicles, and model predictions using these parameters are similar to those found in the current study (Wilson, 1996). However, the

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Fig. 10. Effects of errors in the model parameters of $50 mm and $5° on the maximum internal rotation of the model. The parameters varied are the X, Y and Z coordinates of the tibial attachment (x y z ) and the femoral attachment (x y z ) of each structure and the rotations of the tibial planes       about the tibial Y-axis (sa) and tibial X-axis (ca) passing through the contact point at the starting position of the mechanism. Small uncertainties in the parameters do not affect the qualitative prediction of automatic rotation.

experimental determination of all parameters is an important part of the technique that will be improved along with the representations of knee geometry in the model. It should be emphasized that the parameter sensitivity study demonstrated that the model is sufficiently robust to make accurate predictions of natural knee motion even when the simplified geometry and potential flaws in the ligament parameters are considered. The uncertainties introduced in this study could correspond to a deviation from the assumed articular geometry, an incorrectly situated isometric fascicle, or a change in length of an isometric fascicle. When uncertainties of a reasonable size were introduced, coupled internal rotation and posterior translation of the contact points were still predicted. Model predictions are generally consistent with studies of knee kinematics in the literature. Coupled internal rotation was first documented by Meyer (Meyer, 1853) and has been measured in many studies of the unloaded human knee (Biden et al., 1984; Markolff et al., 1976; Shoemaker et al., 1993; Trent et al., 1976). Posterior translation of contact areas on the tibia has also been observed in many experiments. Model predictions of posterior translation of the contact points (32.4 mm in the lateral compartment and 16.3 mm in the medial compartment) are higher than reported experimental measurements of 12 and 6 mm (Kapandji, 1987) and 11.2 and 5.1 mm (Thompson et al., 1990) in the lateral and medial compartments, respectively. This inconsistency may be explained by the simplified shapes of the model

articular surfaces. The four-bar linkage model of the knee showed that the slip ratio (and, by extension, posterior translation of the contact point) is sensitive to the assumed shape of the tibial plateaux (O’Connor et al., 1989). Inclusion of more anatomical shapes for the plateaux may yield contact point excursions closer to physiological values. Model predictions are also consistent with results from a three-dimensional physical model (Huson et al., 1989), which predicts posterior translation of the tibiofemoral contact points. However, this model does not predict coupled internal rotation, likely because it does not include the medial collateral ligament constraint. This study has several clinical implications. The model predicts that rupture of a ligament or erosion of an articular surface would reduce the number of constraints to motion and increase the number of degrees of freedom of the joint, which would appear as excessive laxity or instability. This has been extensively reported clinically: joints stricken by advanced osteoarthritis or ligament damage are frequently unstable. If a ligament is reconstructed or replaced with either insufficient or excessive tension, the normal guiding function of the isometric fascicle will be lost and the normal path of motion will be distorted. A total knee prosthesis with conforming articular surfaces will change knee kinematics: the surfaces will provide more constraint than the natural, nonconforming surfaces, which will distort or eliminate the guiding role of the ligaments. The applicability of the spatial

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mechanism model to motion analysis was demonstrated in an earlier article (Wilson and O’Connor, 1997). This study demonstrates that passive knee flexion is guided by the ACL, PCL, MCL and by articular contact in the medial and lateral compartments. These structures guide the knee along its unique path of motion and produce the two well-known coupled features of passive flexion: internal rotation and posterior translation of contact on the tibia. Because of this vital role, the constraining structures must be preserved or replicated after injury or affliction by disease to maintain normal joint function.

Acknowledgements The authors thank Prof. K. H. Hunt, Department of Mechanical and Manufacturing Engineering, University of Melbourne and Dr Ross McAree, Department of Engineering Science, University of Oxford, for their advice on theoretical kinematics, and CAMARC-II, the Arthritis and Rheumatism Council (UK), the Fonds FCAR (Que´bec, Canada), and the Overseas Research Studentship Scheme (UK) for support. Appendix. List of model parameters These parameters define the positions of the spherical pairs and the positions and orientations of the planar pairs making up the ligament and articular contact connections in the mechanism at the starting configuration (full extension) (Table 1). Table 1 Parameter

x (mm)

ACL tibial attachment PCL tibial attachment MCL tibial attachment Medial tibial contact point Lateral tibial contact point ACL femoral attachment PCL femoral attachment MCL femoral attachment Medial femoral condyle centre Lateral femoral condyle centre Medial tibial plane (normal) Lateral tibial plane (normal)

0.0 20.2 !11.4 2.8 !3.3 19.2 16.8 11.2 5.1 3.0 0.1 0.21

y (mm) z (mm) 0.0 12.2 !2.4 !9.5 30.6 16.9 !6.5 !36.1 !15.6 35.5 !0.25 0.16

0.0 !18.3 !53.6 !4.6 !2.1 26.8 10.8 14.8 19.1 27.1 0.96 0.97

References Andriacchi, T.P., Mikosz, R.P., Hampton, S.J., Galante, J.O., 1983. Model studies of the stiffness characteristics of the human knee joint. Journal of Biomechanics 16(1), 23—29. Biden, E., O’Connor, J., Goodfellow, J., 1984. Tibial rotation in the cadaver knee. Transaction of Orthopaedic Research Society Atlanta, 30.

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Bishop, R.E.D., 1977. On the mechanics of the human knee. Engineering in Medicine 6, 46—52. Blankevoort, L., Kuiper, J.H., Huiskes, R., Grootenboer, H.J., 1991. Articular contact in a three-dimensional model of the knee. Journal of Biomechanics 24(11), 1019—1031. Cole, G.K., Nigg, B.M., Ronsky, J.L., Yeadon, M.R., 1993. Application of the three-dimensional joint attitude and movement representation: a standardization proposal. Transactions of ASME — Journal of Biomechanical Engineering 115, 344—349. Collins, J.J., O’Connor, J.J., 1991. Muscle—ligament interactions at the knee during walking. Procceedings, Institute of Mechanical Engineering Part H, Journal of Engineering in Medicine 205, 11—18. Denavit, J., Hartenberg, R.S., 1955. A kinematic notation for lower pair mechanisms based on matrices. Journal of Applied Mechanics, Transactions of ASME 22, 215—221. Essinger, J.R., Leyvraz, P.F., Heegard, J.H., Robertson, D.D., 1989. A mathematical model for the evaluation of the behaviour during flexion of condylar-type knee prostheses. Journal of Biomechanics 22, 1229—1241. Feikes, J., Wilson, D., O’Connor, J., 1998. The unique track of intact passive knee motion as a kinematic baseline. Transactions of Orthopaedic Research Society, New Orleans, 170. Fuss, F.K., 1989. Anatomy of the cruciate ligaments and their function in extension and flexion of the human knee joint. American Journal of Anatomy 184, 165—176. Goodfellow, J.W., O’Connor, J.J., 1978. The mechanics of the knee and prosthesis design. Journal of Bone Joint Surgery [Br] 60-B, 358—69. Hunt, K.H., 1978. Kinematic Geometry of Mechanisms. Oxford University Press, Oxford. Huson, A., 1974. Biomechanische probleme des kniegelenks. Orthopade 3, 119—126. Huson, A., Spoor, C.W., Verbout, A.J., 1989. A model of the human knee, derived from kinematic principles and its relevance for endoprosthesis design. Acta Morphologica Neerlando Scandinavica 27, 45—62. Kapandji, I.A., 1987. The Physiology of the Joints, vol. 2 The Lower Limb, 5th ed. Churchill Livingstone, London (Translated by L.H. Honore). Kurosawa, H., Walker, P.S., Abe, S., Garg, A., Hunter, T., 1985. Geometry and motion of the knee for implant and orthotic design. Journal of Biomechanics 18(7), 487—499. Lee, H.Y., Lin, W., Duffy, J., 1993. A method for forward displacement analysis of in-parallel platform mechanisms. Mechatronics 3(5), 659—669. Markolf, K.L., Mensch, J.S., Amstutz, H.C., 1976. Stiffness and laxity of the knee — the contributions of the supporting structures. a quantitative in-vitro study. Journal of Bone and Joint Surgery [Am], 58-A, 583—593. Menschik, A., 1974. Mechanik des kniegelenks — teil 1. Zeitschrift fuer Orthopaedie 112, 481—495. Meyer, H., 1853. Die mechanik des kniegelinks. Archiv fur Anatomie Physiologie 497—547. O’Connor, J.J., Shercliff, T., FitzPatrick, D., Biden, E., Goodfellow, J., 1990. Mechanics of the knee. In: Daniel, D.M., Akeson, W.H., O’Connor, J.J., (Ed.), Knee Ligaments: Structure, Function, Injury, and Repair, Chap 11, pp 201—38. Raven Press, New York. O’Connor, J.J., Shercliff, T.L., Biden, E., Goodfellow, J.W., 1989. The geometry of the knee in the sagittal plane. Procceedings, Institute of Mechanical Engineering Part H, Journal of Engineering in Medicine 203, 223—233. Rovick, J.S., Reuben, J.D., Schrager, R.J., Walker, P.S., 1991. Relation between knee motion and ligament length patterns. Clinical Biomechanics 6(4), 213—220. Shoemaker, S.C., Adams, D., Daniel, D.M., Woo, S.L., 1993. Quadriceps anterior cruciate graft interaction: An in vitro study of joint kinematics and ACL graft tension. Clinical Orthopaedics 294, 379—390.

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D.R. Wilson et al. / Journal of Biomechanics 31 (1998) 1127—1136

Sidles, J.A., Larson, R.V., Garbini, J.L., Downey, D.J., Matsen III, F.A., 1988. Ligament length relationships in the moving knee. Journal of Orthopaedic Research 6(4), 593—610. Stewart, D., 1966. A platform with six degrees of freedom. Proceedings Institute of Mechanical Engineers, Part 1, 180, 371—386. Strasser, H., 1917. Lehrbuch der Muskel- und Gelenkmechanik. Springer, Berlin. Thompson, W.O., Fu, F.H., Thaete, F.L., Dye, S., 1990. Assessment of tibial meniscal kinematics by 3-d magnetic resonance imaging. Transactions of Orthopaedic Research Society 245. Trent, P.S., Walker, P.S., Wolf, B., 1976. Ligament length patterns, strength and rotational axes of the knee joint. Clinical Orthopaedics 117, 263—270. Uicker, J.J., Denavit, J., Hartenberg, R.S., 1964. An iterative method for the displacement analysis of spatial mechanisms Journal of Applied Mechanics, Transactions of ASME, 309—314. Wang, C.-J., Walker, P.S., Wolf, B., 1973. The effects of flexion and rotation on the length patterns of the ligaments of the knee. Journal of Biomechanics 6(6), 587—596. Weber, W.E., Weber, E.F.W., 1836. Mechanik der menschlichen Gehwerkzeuge. in der Dietrichschen Buchhandlung, Gottingen. Wilson, D.R., O’Connor, J.J., 1996. A three-dimensional geometric model of the knee for the study of joint forces in gait. Gait and Posture 5, 108—115.

Wilson, D.R., 1996. Three-dimensional kinematics of the knee. DPhil Thesis, University of Oxford, England. Wismans, J., Veldpaus, F., Janssen, J., 1980. A three-dimensional mathematical model of the knee-joint. Journal of Biomechanics 13, 677—685. Zavatsky, A.B., 1996. Passive flexion of the knee in a six degree-offreedom testing rig. In European Society of Biomechanics, Leuven, Belgium. Zavatsky, A.B., Beard, D.J., O’Connor, J.J., 1994. Cruciate ligament loading during isometric muscle contractions. A theoretical basis for rehabilitation. American Journal of Sports Medicine 22(3), 418—423. Zavatsky, A.B., O’Connor, J.J., 1992a. A model of human knee ligaments in the sagittal plane: Part I. Response to passive flexion. Procceedings, Institute of Mechanical Engineering Part H, Journal of Engineering in Medicine 206, 125—134. Zavatsky, A.B., O’Connor, J.J., 1992b. A model of human knee ligaments in the sagittal plane: Part II. Fibre recruitment under load. Procceedings, Institute of Mechanical Engineering Part H, Journal of Engineering in Medicine 206, 135—145. Zavatsky, A.B., O’Connor, J.J., 1993. Muscle—ligament interaction at the knee during isometric quadriceps contractions. Procceedings, Institute of Mechanical Engineering Part H, Journal of Engineering in Medicine 207, 7—18.