A Three-Dimensional Musculoskeletal Model of the Human Knee Joint. Part 2: Analysis of Ligament Function

A three-dimensional model of the knee is used to study ligament function during anterior-posterior (a-p) draw, axial rotation, and isometric contractions of the extensor and flexor muscles. The geometry of the model bones is based on cadaver data. The contacting surfaces of the femur and tibia are modeled as deformable; those of the femur and patella are assumed to be rigid. Twelve elastic elements are used to describe the geometry and mechanical properties of the cruciate ligaments, the collateral ligaments, and the posterior capsule. The model is actuated by thirteen musculotendinous units, each unit represented as a three-element muscle in series with tendon. The calculations show that the forces applied during a-p draw are substantially different from those applied by the muscles during activity. Principles of knee-ligament function based on the results of in vitro experiments may therefore be overstated. Knee-ligament forces during straight a-p draw are determined solely by the changing geometry of the ligaments relative to the bones: ACL force decreases with increasing flexion during anterior draw because the angle between the ACL and the tibial plateau decreases as knee flexion increases; PCL force increases with increasing flexion during posterior draw because the angle between the PCL and the tibial plateau increases. The pattern of ligament loading during activity is governed by the geometry of the muscles spanning the knee: the resultant force in the ACL during isometric knee extension is determined mainly by the changing orientation of the patellar tendon relative to the tibia in the sagittal plane; the resultant force in the PCL during isometric knee flexion is dominated by the angle at which the hamstrings meet the tibia in the sagittal plane.     

A Three-Dimensional Musculoskeletal Model of the Human Knee Joint. Part 2: Analysis of Ligament Function

A three-dimensional model of the knee is used to study ligament function during anterior-posterior (a-p) draw, axial rotation, and isometric contractions of the extensor and flexor muscles. The geometry of the model bones is based on cadaver data. The contacting surfaces of the femur and tibia are modeled as deformable; those of the femur and patella are assumed to be rigid. Twelve elastic elements are used to describe the geometry and mechanical properties of the cruciate ligaments, the collateral ligaments, and the posterior capsule. The model is actuated by thirteen musculotendinous units, each unit represented as a three-element muscle in series with tendon. The calculations show that the forces applied during a-p draw are substantially different from those applied by the muscles during activity. Principles of knee-ligament function based on the results of in vitro experiments may therefore be overstated. Knee-ligament forces during straight a-p draw are determined solely by the changing geometry of the ligaments relative to the bones: ACL force decreases with increasing flexion during anterior draw because the angle between the ACL and the tibial plateau decreases as knee flexion increases; PCL force increases with increasing flexion during posterior draw because the angle between the PCL and the tibial plateau increases. The pattern of ligament loading during activity is governed by the geometry of the muscles spanning the knee: the resultant force in the ACL during isometric knee extension is determined mainly by the changing orientation of the patellar tendon relative to the tibia in the sagittal plane; the resultant force in the PCL during isometric knee flexion is dominated by the angle at which the hamstrings meet the tibia in the sagittal plane.     

A spatial mechanism with higher pairs for modelling the human knee joint

By generalizing a previous model proposed in the literature, a new spatial kinematic model of the knee joint passive motion is presented. The model is based on an equivalent spatial parallel mechanism which relies upon the assumption that fibers within the anterior cruciate ligament (ACL), the medial collateral ligament (MCL) and the posterior cruciate ligament (PCL) can be considered as isometric during the knee flexion in passive motion (virtually unloaded motion). The articular surfaces of femoral and tibial condyles are modelled as 3-D surfaces of general shapes. In particular, the paper presents the closure equations of the new mechanism both for surfaces represented by means of scalar equations that have the Cartesian coordinates of the points of the surface as variables and for surfaces represented in parametric form. An example of simulation is presented in the case both femoral condyles are modelled as ellipsoidal surfaces and tibial condyles as spherical surfaces. The results of the simulation are compared to those of the previous models and to measurements. The comparison confirms the expectation that a better approximation of the tibiofemoral condyle surfaces leads to a more accurate model of the knee passive motion.     

Three-dimensional knee model: constrained by isometric ligament bundles and experimentally obtained tibio-femoral contacts

The purpose of this study is to investigate the effect of anterior portion of anterior cruciate ligament, posterior cruciate ligament, anterior and deep portions of medial collateral ligament and the tibio-femoral articular contacts on passive knee motion. A well-accepted reference model for a normal tibio-femoral joint is reconstructed from the literature. The proposed three-dimensional dynamic tibio-femoral model includes the isometric fascicles, ligament bundles and irregularly shaped medial-lateral contact surfaces. With the approach we aim to analyze bone shape and ligament related abnormalities of knee kinematics. The rotations, translations and the contact forces during passive knee flexion were compared against a reference model and the results were found in close accordance. This study demonstrated that isometric ligament bundles play an important role in understanding the femur shape from contact points on tibia. Femoral condyles are not necessarily spherical. The surgical treatments should consider both ligament bundle lengths and contact surface geometries to achieve a problem free knee kinematics after a knee surgery.     

A constraint-based approach to modelling the mobility of the human knee joint

A model of knee mobility able to predict the range and pattern of movement in the unloaded joint was proposed by Wilson et al. (J. Biomech. 31 (1998) 1127-1136). The articular surfaces in the lateral and medial compartments and isometric fascicles in three of the knee ligaments were represented as five constraints on motion between the femur and tibia in a single degree-of-freedom parallel spatial mechanism. The path of movement of the bones during passive flexion was found by solving the forward kinematics of the mechanism using an iterative method. The present paper shows that such a mechanism-based solution approach can lead to an underestimation of the flexion range. This is due to the mechanism reaching a 'stationary configuration' and 'locking'. A new, constraint-based approach to the solution of the model joint displacement is proposed. It avoids the representation of ligaments and articular surfaces by kinematically equivalent chains of one degree-of-freedom pairs which are prone to singularities. It relies instead on a numerical solution of five non-linear constraint equations to find the relative positions of the bones at a series of flexion angles. The method is successful both in its ability to predict motion through a physiological range and in its efficiency with a solution rate forty times faster than the original algorithm. The new approach may be extended to include more complex joint surface geometry, allowing a study of the effects of articular surface shape and ligament arrangement on joint kinematics.     

Cruciate coupling and screw-home mechanism in passive knee joint during extension--flexion

The screw-home mechanism and coupling between forces in cruciate ligaments during passive knee joint flexion were investigated for various boundary conditions, flexion axis alignments and posterior cruciate ligaments (PCL)/anterior cruciate ligament (ACL) conditions. A developed non-linear 3D finite element model was used to perform detailed elasto-static response analyses of the human tibiofemoral joint as a function of flexion angle varying from 10 degrees hyper-extension to 90 degrees flexion. The tibia rotated internally as the femur flexed and externally as the femur extended. The re-alignment of the flexion axis by +/-5 degrees rotation about the axial (distal-proximal) axis, transection of the ACL and changes in cruciate ligament initial strains substantially influenced the 'screw-home' motion. On the other hand, restraint on this coupled rotation diminished ACL forces in flexion. A remarkable coupling was predicted between ACL and PCL forces in flexion; forces in both cruciate ligaments increased as the initial strain or pretension in one of them increased whereas they both diminished as one of them was cut or became slack. This has important consequences in joint functional biomechanics following a ligament injury or replacement surgery and, hence, in the proper management of joint disorders.     

3-D anatomically based dynamic modeling of the human knee to include tibio-femoral and patello-femoral joints

An anatomical dynamic model consisting of three body segments, femur, tibia and patella, has been developed in order to determine the three-dimensional dynamic response of the human knee. Deformable contact was allowed at all articular surfaces, which were mathematically represented using Coons' bicubic surface patches. Nonlinear elastic springs were used to model all ligamentous structures. Two joint coordinate systems were employed to describe the six-degrees-of-freedom tibio-femoral (TF) and patello-femoral (PF) joint motions using twelve kinematic parameters. Two versions of the model were developed to account for wrapping and nonwrapping of the quadriceps tendon around the femur. Model equations consist of twelve nonlinear second-order ordinary differential equations coupled with nonlinear algebraic constraint equations resulting in a Differential-Algebraic Equations (DAE) system that was solved using the Differential/Algebraic System Solver (DASSL) developed at Lawrence Livermore National Laboratory. Model calculations were performed to simulate the knee extension exercise by applying non-linear forcing functions to the quadriceps tendon. Under the conditions tested, both "screw home mechanism" and patellar flexion lagging were predicted. Throughout the entire range of motion, the medial component of the TF contact force was found to be larger than the lateral one while the lateral component of the PF contact force was found to be larger than the medial one. The anterior and posterior fibers of both anterior and posterior cruciate ligaments, ACL and PCL, respectively, had opposite force patterns: the posterior fibers were most taut at full extension while the anterior fibers were most taut near 90 degrees of flexion. The ACL was found to carry a larger total force than the PCL at full extension, while the PCL carried a larger total force than the ACL in the range of 75 degrees to 90 degrees of flexion.     

Does medio-lateral motion occur in the normal knee? An in-vitro study in passive motion

Medio-lateral translation during knee flexion continues to raise controversy. Small population sizes, small joint flexion ranges, less-reliable measurement techniques and disparate experimental conditions led to inconsistent reports in the past. To study this subject with more accurate and reliable measurements, we carried out femur and tibia tracking in 22 intact cadaver knees during passive joint motion using a state-of-the-art surgical navigation system. Trackers with active light-emitting diodes were fixed onto the femur and tibia, and an instrumented pointer was used to digitize a number of anatomical landmarks. International recommendations were adopted for anatomical-based reference frame definitions and joint kinematic analysis. For the first time, knee joint translations were reported in both the femoral and tibial reference frames, and over a flexion/extension arc as large as 140°. During flexion, in the femoral reference frame, the center of the tibial plateau moved 4.8 ± 2.8mm medially when averaged over the specimens. In the tibial frame, the knee center moved 13.3 ± 5.7 mm laterally. The relative femoral-to-tibial medio-lateral translation was, on average over the specimens, nearly 20% of the width of the tibial plateau, and can be as large as 35%. Medio-lateral translation occurs in the natural normal knee joint.     

The importance of quadriceps and hamstring muscle loading on knee kinematics and in-situ forces in the ACL

This study investigated the effect of hamstring co-contraction with quadriceps on the kinematics of the human knee joint and the in-situ forces in the anterior cruciate ligament (ACL) during a simulated isometric extension motion of the knee. Cadaveric human knee specimens (n = 10) were tested using the robotic universal force moment sensor (UFS) system and measurements of knee kinematics and in-situ forces in the ACL were based on reference positions on the path of passive flexion/extension motion of the knee. With an isolated 200 N quadriceps load, the knee underwent anterior and lateral tibial translation as well as internal tibial rotation with respect to the femur. Both translation and rotation increased when the knee was flexed from full extension to 30 of flexion; with further flexion, these motion decreased. The addition of 80 N antagonistic hamstrings load significantly reduced both anterior and lateral tibial translation as well as internal tibial rotation at knee flexion angles tested except at full extension. At 30 of flexion, the anterior tibial translation, lateral tibial translation, and internal tibial rotation were significantly reduced by 18, 46, and 30%, respectively (p<0.05). The in-situ forces in the ACL under the quadriceps load were found to increase from 27.8+/-9.3 N at full extension to a maximum of 44.9+/-13.8 N at 15 of flexion and then decrease to 10 N beyond 60 of flexion. The in-situ force at 15 was significantly higher than that at other flexion angles (p<0.05). The addition of the hamstring load of 80 N significantly reduced the in-situ forces in the ACL at 15, 30 and 60 of flexion by 30, 43, and 44%, respectively (p<0.05). These data demonstrate that maximum knee motion may not necessarily correspond to the highest in-situ forces in the ACL. The data also suggest that hamstring co-contraction with quadriceps is effective in reducing excessive forces in the ACL particularly between 15 and 60 of knee flexion.     

An improved OpenSim gait model with multiple degrees of freedom knee joint and knee ligaments

Musculoskeletal models are widely used to investigate joint kinematics and predict muscle force during gait. However, the knee is usually simplified as a one degree of freedom joint and knee ligaments are neglected. The aim of this study was to develop an OpenSim gait model with enhanced knee structures. The knee joint in this study included three rotations and three translations. The three knee rotations and mediolateral translation were independent, with proximodistal and anteroposterior translations occurring as a function of knee flexion/extension. Ten elastic elements described the geometrical and mechanical properties of the anterior and posterior cruciate ligaments (ACL and PCL), and the medial and lateral collateral ligaments (MCL and LCL). The three independent knee rotations were evaluated using OpenSim to observe ligament function. The results showed that the anterior and posterior bundles of ACL and PCL (aACL, pACL and aPCL, pPCL) intersected during knee flexion. The aACL and pACL mainly provided force during knee flexion and adduction, respectively. The aPCL was slack throughout the range of three knee rotations; however, the pPCL was utilised for knee abduction and internal rotation. The LCL was employed for knee adduction and rotation, but was slack beyond 20° of knee flexion. The MCL bundles were mainly used during knee adduction and external rotation. All these results suggest that the functions of knee ligaments in this model approximated the behaviour of the physical knee and the enhanced knee structures can improve the ability to investigate knee joint biomechanics during various gait activities.     

Gastrocnemius fascicle length changes with two-joint passive movements

Predicting muscle fascicle length changes during passive movements may lead to a better understanding of muscle function. The purpose of this study was to experimentally compare fascicle length changes in the gastrocnemius during two-joint passive movements with a previously derived kinematic model based on anatomical measures from a cadaver. The ratio of passive ankle to knee motion was manipulated to generate medial gastrocnemius fascicle elongation and lateral gastrocnemius fascicle shortening. Ultrasound images from both heads of the gastrocnemius fascicles were acquired at 10 degrees knee flexion increments and compared with this kinematic model. Our results suggest that the two-joint kinematic model from which we originally based our knee and ankle movements did not adequately reflect fascicle length changes during any of the movement conditions in this study. From our data, we propose that for every degree of ankle motion the medial and lateral gastrocnemius changes 0.42 mm and 0.96 mm, respectively, whereas changes of 0.14 mm and 0.22 mm are observed for the medial and lateral gastrocnemius, respectively, during knee movements.     

Effect of ACL deficiency on MCL strains and joint kinematics

The knee joint is partially stabilized by the interaction of multiple ligament structures. This study tested the interdependent functions of the anterior cruciate ligament (ACL) and the medial collateral ligament (MCL) by evaluating the effects of ACL deficiency on local MCL strain while simultaneously measuring joint kinematics under specific loading scenarios. A structural testing machine applied anterior translation and valgus rotation (limits 100 N and 10 N m, respectively) to the tibia of ten human cadaveric knees with the ACL intact or severed. A three-dimensional motion analysis system measured joint kinematics and MCL tissue strain in 18 regions of the superficial MCL. ACL deficiency significantly increased MCL strains by 1.8% (p<0.05) during anterior translation, bringing ligament fibers to strain levels characteristic of microtrauma. In contrast, ACL transection had no effect on MCL strains during valgus rotation (increase of only 0.1%). Therefore, isolated valgus rotation in the ACL-deficient knee was nondetrimental to the MCL. The ACL was also found to promote internal tibial rotation during anterior translation, which in turn decreased strains near the femoral insertion of the MCL. These data advance the basic structure-function understanding of the MCL, and may benefit the treatment of ACL injuries by improving the knowledge of ACL function and clarifying motions that are potentially harmful to secondary stabilizers.     

Interactions between kinematics and loading during walking for the normal and ACL deficient knee

The relationships between extrinsic forces acting at the knee and knee kinematics were examined with the purpose of identifying specific phases of the walking cycle that could cause abnormal kinematics in the anterior cruciate ligament (ACL) deficient knee. Intersegmental forces and moments in directions that would produce anterior-posterior (AP) translation, internal-external (IE) rotation and flexion-extension (FE) at the knee were compared with the respective translation and rotations of the tibia relative to the femur during four selected phases (heel strike, weight acceptance, terminal extension and swing) of the walking cycle. The kinematic changes associated with loss of the ACL occurred primarily during the terminal portion of swing phase of the walking cycle where, for the ACL deficient knee, the tibia had reduced external rotation and anterior translation as the knee extended prior to heel strike. The kinematic changes during swing phase were associated with a rotational offset relative to the contralateral knee in the average position of the tibia towards internal rotation. The offset was maintained through the entire gait cycle. The abnormal offsets in the rotational position were correlated with the magnitude of the flexion moment (balanced by a net quadriceps moment) during weight acceptance. These results suggest that adaptations to the patterns of muscle firing during walking can compensate for kinematic changes associated with the loss of the ACL. The altered rotational position would cause changes in tibiofemoral contact during walking that could cause the type of degenerative changes reported in the meniscus and the articular cartilage following ACL injury.     

New parameters describing how knee ligaments carry force in situ predict interspecimen variations in laxity during simulated clinical exams

Knee laxity, defined as the net translation or rotation of the tibia relative to the femur in a given direction in response to an applied load, is highly variable from person to person. High levels of knee laxity as assessed during routine clinical exams are associated with first-time ligament injury and graft reinjury following reconstruction. During laxity exams, ligaments carry force to resist the applied load; however, relationships between intersubject variations in knee laxity and variations in how ligaments carry force as the knee moves through its passive envelope of motion, which we refer to as ligament engagement, are not well established. Thus, the objectives of this study were, first, to define parameters describing ligament engagement and, then, to link variations in ligament engagement and variations in laxity across a group of knees. We used a robotic manipulator in a cadaveric knee model (n = 20) to quantify how important knee stabilizers, namely the anterior and posterior cruciate ligaments (ACL and PCL, respectively), as well as the medial collateral ligament (MCL) engage during respective tests of anterior, posterior, and valgus laxity. Ligament engagement was quantified using three parameters: (1) in situ slack, defined as the relative tibiofemoral motion from the neutral position of the joint to the position where the ligament began to carry force; (2) in situ stiffness, defined as the slope of the linear portion of the ligament force–tibial motion response; and (3) ligament force at the peak applied load. Knee laxity was related to parameters of ligament engagement using univariate and multivariate regression models. Variations in the in situ slack of the ACL and PCL predicted anterior and posterior laxity, while variations in both in situ slack and in situ stiffness of the MCL predicted valgus laxity. Parameters of ligament engagement may be useful to further characterize the in situ biomechanical function of ligaments and ligament grafts.     

Coupled motions under compressive load in intact and ACL-deficient knees: a cadaveric study

Knowledge of the coupled motions, which develop under compressive loading of the knee, is useful to determine which degrees of freedom should be included in the study of tibiofemoral contact and also to understand the role of the anterior cruciate ligament (ACL) in coupled motions. The objectives of this study were to measure the coupled motions of the intact knee and ACL-deficient knee under compression and to compare the coupled motions of the ACL-deficient knee with those of the intact knee. Ten intact cadaveric knees were tested by applying a 1600 N compressive load and measuring coupled internal-external and varus-valgus rotations and anterior-posterior and medial-lateral translations at 0 deg, 15 deg, and 30 deg of flexion. Compressive loads were applied along the functional axis of axial rotation, which coincides approximately with the mechanical axis of the tibia. The ACL was excised and the knees were tested again. In the intact knee, the peak coupled motions were 3.8 deg internal rotation at 0 deg flexion changing to -4.9 deg external rotation at 30 deg of flexion, 1.4 deg of varus rotation at 0 deg flexion changing to -1.9 deg valgus rotation at 30 deg of flexion, 1.4 mm of medial translation at 0 deg flexion increasing to 2.3 mm at 30 deg of flexion, and 5.3 mm of anterior translation at 0 deg flexion increasing to 10.2 mm at 30 deg of flexion. All changes in the peak coupled motions from 0 deg to 30 deg flexion were statistically significant (p<0.05). In ACL-deficient knees, there was a strong trend (marginally not significant, p=0.07) toward greater anterior translation (12.7 mm) than that in intact knees (8.0 mm), whereas coupled motions in the other degrees of freedom were comparable. Because the coupled motions in all four degrees of freedom in the intact knee and ACL-deficient knee are sufficiently large to substantially affect the tibiofemoral contact area, all degrees of freedom should be included when either developing mathematical models or designing mechanical testing equipment for study of tibiofemoral contact. The increase in coupled anterior translation in ACL-deficient knees indicates the important role played by the ACL in constraining anterior translation during compressive loading.     

Subject-specific knee joint geometry improves predictions of medial tibiofemoral contact forces

Estimating tibiofemoral joint contact forces is important for understanding the initiation and progression of knee osteoarthritis. However, tibiofemoral contact force predictions are influenced by many factors including muscle forces and anatomical representations of the knee joint. This study aimed to investigate the influence of subject-specific geometry and knee joint kinematics on the prediction of tibiofemoral contact forces using a calibrated EMG-driven neuromusculoskeletal model of the knee. One participant fitted with an instrumented total knee replacement walked at a self-selected speed while medial and lateral tibiofemoral contact forces, ground reaction forces, whole-body kinematics, and lower-limb muscle activity were simultaneously measured. The combination of generic and subject-specific knee joint geometry and kinematics resulted in four different OpenSim models used to estimate muscle–tendon lengths and moment arms. The subject-specific geometric model was created from CT scans and the subject-specific knee joint kinematics representing the translation of the tibia relative to the femur was obtained from fluoroscopy. The EMG-driven model was calibrated using one walking trial, but with three different cost functions that tracked the knee flexion/extension moments with and without constraint over the estimated joint contact forces. The calibrated models then predicted the medial and lateral tibiofemoral contact forces for five other different walking trials. The use of subject-specific models with minimization of the peak tibiofemoral contact forces improved the accuracy of medial contact forces by 47% and lateral contact forces by 7%, respectively compared with the use of generic musculoskeletal model.     

A Three-Dimensional Musculoskeletal Model of the Human Knee Joint. Part 1: Theoretical Construction

A three-dimensional model of the knee is developed to study the interactions between the muscles, ligaments, and bones during activity. The geometry of the distal femur, proximal tibia, and patella is based on cadaver data reported for an average-size knee. The shapes of the femoral condyles are represented by high-order polynomials: the tibial plateaux and patellar facets are approximated as flat surfaces. The contacting surfaces of the femur and tibia are modeled as deformable, while those of the femur and patella are assumed to be rigid. Interpenetration of the femur and tibia is taken into account by modeling cartilage as a thin, linear, elastic layer, mounted on rigid bone. Twelve elastic elements describe the geometry and mechanical properties of the cruciate ligaments, the collateral ligaments, and the posterior capsule. The model is actuated by thirteen musculotendinous units, each unit modeled as a three-element muscle in series with tendon. The path of each muscle is approximated as a straight line, except where it contacts and wraps around bone and other muscles; changes in muscle paths are taken into account using data obtained from MRI. In the first part of this paper, the model is used to simulate passive knee flexion. Quantitative comparisons of the model results with experimental data reported in the literature indicate that the relative movements of the bones and the geometry of the ligaments and muscles in the model are similar to those evident in the real knee. In Part II, the model is used to describe knee-ligament function during anterior-posterior draw, axial rotation, and isometric knee-extension and knee-flexion exercises.     

The interaction of muscle moment arm,knee laxity,and torque in a multi-scale musculoskeletal model of the lower limb

IntroductionMusculoskeletal modeling allows insight into the interaction of muscle force and knee joint kinematics that cannot be measured in the laboratory. However, musculoskeletal models of the lower extremity commonly use simplified representations of the knee that may limit analyses of the interaction between muscle forces and joint kinematics. The goal of this research was to demonstrate how muscle forces alter knee kinematics and consequently muscle moment arms and joint torque in a musculoskeletal model of the lower limb that includes a deformable representation of the knee.MethodsTwo musculoskeletal models of the lower limb including specimen-specific articular geometries and ligament deformability at the knee were built in a finite element framework and calibrated to match mean isometric torque data collected from 12 healthy subjects. Muscle moment arms were compared between simulations of passive knee flexion and maximum isometric knee extension and flexion. In addition, isometric torque results were compared with predictions using simplified knee models in which the deformability of the knee was removed and the kinematics at the joint were prescribed for all degrees of freedom.ResultsPeak isometric torque estimated with a deformable knee representation occurred between 45° and 60° in extension, and 45° in flexion. The maximum isometric flexion torques generated by the models with deformable ligaments were 14.6% and 17.9% larger than those generated by the models with prescribed kinematics; by contrast, the maximum isometric extension torques generated by the models were similar. The change in hamstrings moment arms during isometric flexion was greater than that of the quadriceps during isometric extension (a mean RMS difference of 9.8 mm compared to 2.9 mm, respectively).DiscussionThe large changes in the moment arms of the hamstrings, when activated in a model with deformable ligaments, resulted in changes to flexion torque. When simulating human motion, the inclusion of a deformable joint in a multi-scale musculoskeletal finite element model of the lower limb may preserve the realistic interaction of muscle force with knee kinematics and torque.