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.     

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.     

Kinematic and dynamic analysis of an anatomically based knee joint

This paper presents a knee-joint model to provide a better understanding on the interaction between natural joints and artificial mechanisms for design and control of rehabilitation exoskeletons. The anatomically based knee model relaxes several commonly made assumptions that approximate a human knee as engineering pin-joint in exoskeleton design. Based on published MRI data, we formulate the kinematics of a knee-joint and compare three mathematical approximations; one model bases on two sequential circles rolling a flat plane; and the other two are mathematically differentiable ellipses-based models with and without sliding at the contact. The ellipses-based model taking sliding contact into accounts shows that the rolling-sliding ratio of a knee-joint is not a constant but has an average value consistent with published measurements. This knee-joint kinematics leads to a physically more accurate contact-point trajectory than methods based on multiple circles or lines, and provides a basis to derive a knee-joint kinetic model upon which the effects of a planar exoskeleton mechanism on the internal joint forces and torque during flexion can be numerically investigated. Two different knee-joint kinetic models (pin-joint approximation and anatomically based model) are compared against a condition with no exoskeleton. The leg and exoskeleton form a closed kinematic chain that has a significant effect on the joint forces in the knee. Human knee is more tolerant than pin-joint in negotiating around a singularity but its internal forces increase with the exoskeleton mass-to-length ratio. An oversimplifying pin-joint approximation cannot capture the finite change in the knee forces due to the singularity effect.     

The use of basis functions in modelling joint articular surfaces: application to the knee joint

This article introduces a new method to represent bone surface geometry for simulations of joint contact. The method uses the inner product of two basis functions to provide a mathematical representation of the joint surfaces. This method guarantees a continuous transition in the direction of the surface normals, an important property for computation of joint contact. Our formulation handles experimental data that are not evenly distributed, a common characteristic of digitized data of musculoskeletal morphologies. The method makes it possible to represent highly curved surfaces, which are encountered in many anatomical structures. The accuracy of this method is demonstrated by modeling the human knee joint. The mean relative percentage error in the representation of the patellar track surface was 0.25% (range 0-1.56%) which corresponded to an absolute error of 0.17mm (range 0-0.16mm).     

The influence of ligament modelling strategies on the predictive capability of finite element models of the human knee joint

In finite element (FE) models knee ligaments can represented either by a group of one-dimensional springs, or by three-dimensional continuum elements based on segmentations. Continuum models closer approximate the anatomy, and facilitate ligament wrapping, while spring models are computationally less expensive. The mechanical properties of ligaments can be based on literature, or adjusted specifically for the subject. In the current study we investigated the effect of ligament modelling strategy on the predictive capability of FE models of the human knee joint. The effect of literature-based versus specimen-specific optimized material parameters was evaluated. Experiments were performed on three human cadaver knees, which were modelled in FE models with ligaments represented either using springs, or using continuum representations. In spring representation collateral ligaments were each modelled with three and cruciate ligaments with two single-element bundles. Stiffness parameters and pre-strains were optimized based on laxity tests for both approaches. Validation experiments were conducted to evaluate the outcomes of the FE models.Models (both spring and continuum) with subject-specific properties improved the predicted kinematics and contact outcome parameters. Models incorporating literature-based parameters, and particularly the spring models (with the representations implemented in this study), led to relatively high errors in kinematics and contact pressures. Using a continuum modelling approach resulted in more accurate contact outcome variables than the spring representation with two (cruciate ligaments) and three (collateral ligaments) single-element-bundle representations. However, when the prediction of joint kinematics is of main interest, spring ligament models provide a faster option with acceptable outcome.     

Effect of knee joint flexion and femur rotation on retropatellar contact of the human knee joint]

The aim of the present study was to evaluate retropatellar contact characteristics at different angles of flexion of the knee joint. To this end, 6 cadaveric legs were examined using pressure sensitive film (Fuji Prescale type "super low") at angles of flexion of 45 degrees, 60 degrees, 90 degrees and 120 degrees both in neutral rotation and 10 degrees internal and external rotation of the femur in the same knee joints. A force of 140 N was applied to both the vastus medialis and lateralis, and a comparison made with a medially and a laterally dominating muscle force. The contact areas decreased with increasing angles of flexion. The medially dominating muscle traction increased the contact area. Comparison between internal and external rotation revealed a decrease in contact area on internal rotation. The pressure measurements were comparable in all loading situations. Comparison between neutral and medial traction revealed significant differences in contact area, pressure and force. The influence of femoral rotation showed no significant difference. A comparison of the different angles of flexion revealed only few significant differences. To prevent the development of retropatellar arthrosis, maximum contact areas are necessary. The study has shown an advantage for medially dominating muscle traction, and external rotation of the femur.     

Interlimb symmetry of dynamic knee joint stiffness and co-contraction is maintained in early stage knee osteoarthritis

Individuals with knee OA often exhibit greater co-contraction of antagonistic muscle groups surrounding the affected joint which may lead to increases in dynamic joint stiffness. These detrimental changes in the symptomatic limb may also exist in the contralateral limb, thus contributing to its risk of developing knee osteoarthritis. The purpose of this study is to investigate the interlimb symmetry of dynamic knee joint stiffness and muscular co-contraction in knee osteoarthritis.Muscular co-contraction and dynamic knee joint stiffness were assessed in 17 subjects with mild to moderate unilateral medial compartment knee osteoarthritis and 17 healthy control subjects while walking at a controlled speed (1.0 m/s). Paired and independent t-tests determined whether significant differences exist between groups (p < 0.05).There were no significant differences in dynamic joint stiffness or co-contraction between the OA symptomatic and OA contralateral group (p = 0.247, p = 0.874, respectively) or between the OA contralateral and healthy group (p = 0.635, p = 0.078, respectively). There was no significant difference in stiffness between the OA symptomatic and healthy group (p = 0.600); however, there was a slight trend toward enhanced co-contraction in the symptomatic knees compared to the healthy group (p = 0.051).Subjects with mild to moderate knee osteoarthritis maintain symmetric control strategies during gait.     

A validated three-dimensional computational model of a human knee joint

This paper presents a three-dimensional finite element tibio-femoral joint model of a human knee that was validated using experimental data. The geometry of the joint model was obtained from magnetic resonance (MR) images of a cadaveric knee specimen. The same specimen was biomechanically tested using a robotic/universal force-moment sensor (UFS) system and knee kinematic data under anterior-posterior tibial loads (up to 100 N) were obtained. In the finite element model (FEM), cartilage was modeled as an elastic material, ligaments were represented as nonlinear elastic springs, and menisci were simulated by equivalent-resistance springs. Reference lengths (zero-load lengths) of the ligaments and stiffness of the meniscus springs were estimated using an optimization procedure that involved the minimization of the differences between the kinematics predicted by the model and those obtained experimentally. The joint kinematics and in-situ forces in the ligaments in response to axial tibial moments of up to 10 Nm were calculated using the model and were compared with published experimental data on knee specimens. It was also demonstrated that the equivalent-resistance springs representing the menisci are important for accurate calculation of knee kinematics. Thus, the methodology developed in this study can be a valuable tool for further analysis of knee joint function and could serve as a step toward the development of more advanced computational knee models.     

Dynamic and static control of the human knee joint in abduction-adduction

It is unclear whether humans can voluntarily control dynamic and static properties in knee abduction-adduction, which may be important in performing functional tasks and preventing injuries, whether the main load is about the abduction axis or not. A joint-driving device was used to perturb the knee in abduction-adduction at full knee extension under both passive (muscle relaxed) and active (muscle contracted in abduction or adduction) conditions. Dynamic control properties in knee abduction-adduction were characterized by joint stiffness, viscosity, and limb inertia, and quasi-static knee torque-angle relationship was characterized by knee abduction-adduction laxity and quasi-static stiffness (at a 20Nm moment). It was found that the subjects were capable of generating net abduction and adduction moment through differential co-contraction of muscles crossing the medial and lateral sides of the knee, which helped to reduce the abduction-adduction joint laxity (p< or =0.01) and increase stiffness (p<0.027) and viscous damping. Knee abduction laxity was significantly lower than adduction laxity (p=0.043) and the quasi-static abduction stiffness was significantly higher than adduction stiffness (p<0.001). The knee joint showed significantly higher stiffness and viscosity in abduction-adduction than their counterparts in knee flexion-extension at comparable levels of joint torque (p<0.05). Similar to dynamic flexion-extension properties, the system damping ratio remained constant over different levels of contraction, indicating simplified control tasks for the central nervous system; while the natural undamped frequency increased considerably with abduction-adduction muscle contraction, presumably making the knee a quicker system during strenuous tasks involving strong muscle contraction.     

Validation of a non-invasive fluoroscopic imaging technique for the measurement of dynamic knee joint motion

The accurate measurement of the in vivo knee joint kinematics in six degrees-of-freedom (6DOF) remains a challenge in biomedical engineering. We have adapted a dual fluoroscopic imaging system (DFIS) to investigate the various in vivo dynamic knee joint motions. This paper presents a thorough validation of the accuracy and repeatability of the DFIS system when used to measure 6DOF dynamic knee kinematics. First, the validation utilized standard geometric spheres made from different materials to demonstrate the capability of the DFIS technique to determine the object positions under changing speeds. The translational pose of the spheres could be recreated to less than 0.15±0.09 mm for velocities below 300 mm/s. Next, tantalum beads were inserted into the femur and tibia of two fresh frozen cadaver knees to compare the dynamic kinematics measured by matching knee models to the kinematics from the tantalum bead matching—a technique similar to Roentgen stereophotogrammetric analysis (RSA). Each cadaveric knee was attached to the crosshead of a tensile testing machine and vertically translated at a rate of 16.66 mm/s while images were captured with the DFIS. Subsequently, the tibia was held fixed and the femur manually flexed from full extension to 90° of flexion, as the DFIS acquired images. In vitro translation of the cadaver knee using the tensile testing machine deviated from predicted values by 0.08±0.14 mm for the matched knee models. The difference between matching the knee and tantalum bead models during the dynamic flexion–extension motion of the knee was 0.1±0.65°/s in flexion speed; 0.24±0.16 mm in posterior femoral translation; and 0.16±0.61° in internal–external tibial rotation. Finally, we applied the method to investigate the knee kinematics of a living subject during a step ascent and treadmill gait. High repeatability was demonstrated for the in vivo application. Thus, the DFIS provides an easy and powerful tool for accurately determining 6DOF positions of the knee when performing daily functional activities.     

The tidemark of the chondro-osseous junction of the normal human knee joint

Summary The chondro-osseous junction includes the junction between calcified and non-calcified cartilage matrices often referred to as the tidemark. A detailed knowledge of the structure, function and pathophysiology of the chondro-osseous junction is essential for an understanding both of the normal elongation of bones and of the pathogenesis of osteoarthrosis. In this study the molecular anatomy of the tidemark was studied using histochemical techniques, including lectin histochemistry, on blocks of normal cartilage from human knee joints. The tidemark stained with H&E, picro-sirius red, toluidine blue, safranin O and methyl green, but not with alcian blue in the presence of magnesium chloride at 0.05 M or above. It stained with only four lectins, those from Datura stramonium, Maclura pomifera, Erythrina crystagalli and Helix pomatia, out of the 19 used. Therefore, it is rich in collagen and contains hyaluronan, but appears to lack the glycosaminoglycans of conventional proteoglycans and it expresses a very limited and distinctive lectin staining glycoprofile, which is probably attributable to specific glycoproteins. In addition, the tidemark had a distinct microanatomical trilaminate appearance. From all of these results it is clear that this part of the chondro-osseous junctional region is chemically more complex and distinctive than has previously been described.     

Recruitment of knee joint ligaments

On the basis of earlier reported data on the in vitro kinematics of passive knee-joint motions of four knee specimens, the length changes of ligament fiber bundles were determined by using the points of insertion on the tibia and femur. The kinematic data and the insertions of the ligaments were obtained by using Roentgenstereophotogrammetry. Different fiber bundles of the anterior and posterior cruciate ligaments and the medial and lateral collateral ligaments were identified. On the basis of an assumption for the maximal strain of each ligament fiber bundle during the experiments, the minimal recruitment length and the probability of recruitment were defined and determined. The motions covered the range from extension to 95 degrees flexion and the loading conditions included internal or external moments of 3 Nm and anterior or posterior forces of 30 N. The ligament length and recruitment patterns were found to be consistent for some ligament bundles and less consistent for other ligament bundles. The most posterior bundle of each ligament was recruited in extension and the lower flexion angles, whereas the anterior bundle was recruited for the higher flexion angles. External rotation generally recruited the collateral ligaments, while internal rotation recruited the cruciate ligaments. However, the anterior bundle of the posterior cruciate ligament was recruited with external rotation at the higher flexion angles. At the lower flexion angles, the anterior cruciate and the lateral collateral ligaments were recruited with an anterior force. The recruitment of the posterior cruciate ligament with a posterior force showed that neither its most anterior nor its most posterior bundle was recruited at the lower flexion angles. Hence, the posterior restraint must have been provided by the intermediate fiber bundles, which were not considered in the experiment. At the higher flexion angles, the anterior bundles of the anterior cruciate ligament and the posterior cruciate ligament were found to be recruited with anterior and posterior forces, respectively. The minimal recruitment length and the recruitment probability of ligament fiber bundles are useful parameters for the evaluation of ligament length changes in those experiments where no other method can be used to determine the zero strain lengths, ligament strains and tensions.     

Reduced knee joint moment in ACL deficient patients at a cost of dynamic stability during landing

The current study aimed to examine the effect of anterior cruciate ligament deficiency (ACLd) on joint kinetics and dynamic stability control after a single leg hop test (SLHT). Twelve unilateral ACLd patients and a control subject group (n=13) performed a SLHT over a given distance with both legs. The calculation of joint kinetics was done by means of a soft-tissue artifact optimized rigid full-body model. Margin of stability (MoS) was quantified by the difference between the base of support and the extrapolated center of mass. During landing, the ACLd leg showed lower external knee flexion moments but demonstrated higher moments at the ankle and hip compared to controls (p<0.05). The main reason for the joint moment redistribution in the ACLd leg was a more anterior position of the ground reaction force (GRF) vector, which affected the moment arms of the GRF acting about the joints (p<0.05). For the ACLd leg, trunk angle was more flexed over the entire landing phase compared to controls (p<0.05) and we found a significant correlation between moment arms at the knee joint and trunk angle (r2 = 0.48;p<0.01). The consequence of this altered landing strategy in ACLd legs was a more anterior position of the center of mass reducing the MoS (p<0.05). The results illustrate the interaction between trunk angle, joint kinetics and dynamic stability during landing maneuvers and provide evidence of a feedforward adaptive adjustment in ACLd patients (i.e. more flexed trunk angle) aimed at reducing knee joint moments at the cost of dynamic stability control.     

Development and validation of a 3-D model to predict knee joint loading during dynamic movement

The purpose of this study was to develop a subject-specific 3-D model of the lower extremity to predict neuromuscular control effects on 3-D knee joint loading during movements that can potentially cause injury to the anterior cruciate ligament (ACL) in the knee. The simulation consisted of a forward dynamic 3-D musculoskeletal model of the lower extremity, scaled to represent a specific subject. Inputs of the model were the initial position and velocity of the skeletal elements, and the muscle stimulation patterns. Outputs of the model were movement and ground reaction forces, as well as resultant 3-D forces and moments acting across the knee joint. An optimization method was established to find muscle stimulation patterns that best reproduced the subject's movement and ground reaction forces during a sidestepping task. The optimized model produced movements and forces that were generally within one standard deviation of the measured subject data. Resultant knee joint loading variables extracted from the optimized model were comparable to those reported in the literature. The ability of the model to successfully predict the subject's response to altered initial conditions was quantified and found acceptable for use of the model to investigate the effect of altered neuromuscular control on knee joint loading during sidestepping. Monte Carlo simulations (N = 100,000) using randomly perturbed initial kinematic conditions, based on the subject's variability, resulted in peak anterior force, valgus torque and internal torque values of 378 N, 94 Nm and 71 Nm, respectively, large enough to cause ACL rupture. We conclude that the procedures described in this paper were successful in creating valid simulations of normal movement, and in simulating injuries that are caused by perturbed neuromuscular control.