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.     

Ligament-bone interaction in a three-dimensional model of the knee

In mathematical knee-joint models, the ligaments are usually represented by straight-line elements, connecting the insertions of the femur and tibia. Such a model may not be valid if a ligament is bent in its course over bony-surfaces, particularly not if the resulting redirection of the ligament force has a considerable effect on the laxity or motion characteristics of the knee-joint model. In the present study, a model for wrapping of a ligament around bone was incorporated in a three-dimensional mathematical model of the human knee. The bony edge was described by a curved line on which the contact point of the line element representing a ligament bundle was located. Frictionless contact between the ligament bundle and the bone was assumed. This model was applied to the medial collateral ligament (MCL) interacting with the bony edge of the tibia. It was found that, in comparison with the original model without bony interactions, the bony edge redirected the ligament force of the MCL in such a way that it counterbalanced valgus moments on the tibia more effectively. The effect of the bony interaction with the MCL on the internal-external rotation laxity, however, was negligible.     

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.     

Effect of changes in cruciate ligaments pretensions on knee joint laxity and ligament forces

The knee joint cruciate ligaments are reconstructed with the rationale to avoid joint instability, recurrent injury, damage to soft tissues and osteoarthritis. Wide range of procedures with different stiffness, pretension, orientation and insertion locations have been proposed with the primary goal to restore the joint laxity. Apart from the general lack of success in preservation of force in the reconstructed ligament itself, the concern, not yet addressed, arises as to the effect of such perturbation on the other intact cruciate ligament. The interaction between cruciate ligament forces is hypothesized in this work. Using a 3-D nonlinear finite element model of the tibiofemoral joint, we examined this hypothesis by quantifying the extent of coupling between cruciate ligaments while varying the prestrain in each ligament under flexion with and without anterior-posterior (A-P) loads. A remarkable coupling was predicted between cruciate ligament forces in flexion thus confirming the hypothesis; forces in both cruciate ligaments increased as initial strain or pretension in one of them increased whereas they both diminished as one of them became slack. Moreover, changes in laxity and in ligament forces as a cruciate ligament prestrained or pretensioned varied with flexion angle and external loads. These findings have important consequences in joint functional biomechanics following a ligament injury or replacement surgery and in selection of laxity matched or ligament force matched pretensioning protocols.     

A three-dimensional study of the kinematics of the human knee

This paper represents a three-dimensional study of the human knee-joint and studies kinematic effects of the cruciate ligaments. Two methods were used for our studies, one method was preferred. This method used a time lapse photograph and strobe light to give us a plot of reference points to carry out our analysis using the method of Rouleaux applied to three dimensions. Five cadaver joints were used, each of which was used for three series of experiments, including the joint with capsule intact, with one of the ligaments cut and with the remaining ligament cut. Both lateral and medial studies were conducted to provide data for a three-dimensional study.

It was found that the cruciate ligaments had little effect on the kinematics of the knee, and that the knee motion remained unchanged after cutting one or both of the cruciate ligaments. It was concluded that the motion of the knee was due to the geometry of the bones and perhaps the collateral ligaments, and that the joint could be replaced with a prosthesis having a three dimensional axis of rotation with a fixed center.     

Ligaments and articular contact guide passive knee flexion

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.     

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.     

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.     

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.     

The contribution of the cruciate ligaments to the load-displacement characteristics of the human knee joint

Human knee specimens were subjected to anterior-posterior, medial-lateral, varus-valgus, and torsional displacement tests. Loads were recorded for the intact joint and for the joint with all soft tissues cut except for the cruciate ligaments. The effect of condylar interference was determined for anterior-posterior, medial-lateral, and torsional displacements. The variation in load with flexion angle was considerable for medial-lateral (0-90-deg flexion) displacements, and less for varus-valgus (0-45-deg flexion) displacements. The cruciates were found to carry almost the entire anterior-posterior load; they carried a significant percentage of the medial-lateral load which varied considerably with flexion angle. A small, but not insignificant percentage of the varus-valgus load was carried by the cruciates and the variations with flexion angle were small. In torsion, the cruciates resisted only internal rotation. In the tested displacement ranges, condylar interference had a small effect on the medial-lateral load but did not affect anterior-posterior or torsional loads.     

Quantifying in vivo laxity in the anterior cruciate ligament and individual knee joint structures

A custom knee loading apparatus (KLA), when used in conjunction with magnetic resonance imaging, enables in vivo measurement of the gross anterior laxity of the knee joint. A numerical model was applied to the KLA to understand the contribution of the individual joint structures and to estimate the stiffness of the anterior-cruciate ligament (ACL). The model was evaluated with a cadaveric study using an in situ knee loading apparatus and an ElectroForce test system. A constrained optimization solution technique was able to predict the restraining forces within the soft-tissue structures and joint contact. The numerical model presented here allowed in vivo prediction of the material stiffness parameters of the ACL in response to applied anterior loading. Promising results were obtained for in vivo load sharing within the structures. The numerical model overestimated the ACL forces by 27.61–92.71%. This study presents a novel approach to estimate ligament stiffness and provides the basis to develop a robust and accurate measure of in vivo knee joint laxity.     

A three-dimensional finite element analysis of the combined behavior of ligaments and menisci in the healthy human knee joint

We present here a three-dimensional FE model of the healthy human knee that included the main structures of the joint: bones, all the relevant ligaments and patellar tendon, menisci and articular cartilages. Bones were considered to be rigid, articular cartilage and menisci linearly elastic, isotropic and homogeneous and ligaments hyperelastic and transversely isotropic. Initial strains on the ligaments and patellar tendon were also considered. This model was validated using experimental and numerical results obtained by other authors. Our main goal was to analyze the combined role of menisci and ligaments in load transmission and stability of the human knee. The results obtained reproduce the complex, nonuniform stress and strain fields that occur in the biological soft tissues involved and the kinematics of the human knee joint under a physiological external load.     

关节镜下前交叉韧带重建术后早期康复训练研究

目的:探讨关节镜下前交叉韧带重建术后早期进行康复训练的康复疗效。方法:回顾性分析单侧膝关节前交叉韧带断裂患者60例治疗康复情况,通过对比手术前后膝关节稳定性、活动范围、Lysholm评分及IKDC评分来评估疗效。结果:术后60例患者均获随访,早期康复训练组患者术后6月膝关节稳定性与活动度均良好,本体感觉得到恢复,可进行正常生活、工作。结论:关节镜下前交叉韧带重建术后早期进行康复训练可以恢复膝关节满意稳定性及活动度,疗效可靠。     

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.     

The Effect of Graft Strength on Knee Laxity and Graft In-Situ Forces after Posterior Cruciate Ligament Reconstruction

Surgical reconstruction is generally recommended for posterior cruciate ligament (PCL) injuries; however, the use of grafts is still a controversial problem. In this study, a three-dimensional finite element model of the human tibiofemoral joint with articular cartilage layers, menisci, and four main ligaments was constructed to investigate the effects of graft strengths on knee kinematics and in-situ forces of PCL grafts. Nine different graft strengths with stiffness ranging from 0% (PCL rupture) to 200%, in increments of 25%, of an intact PCL’s strength were used to simulate the PCL reconstruction. A 100 N posterior tibial drawer load was applied to the knee joint at full extension. Results revealed that the maximum posterior translation of the PCL rupture model (0% stiffness) was 6.77 mm in the medial compartment, which resulted in tibial internal rotation of about 3.01°. After PCL reconstruction with any graft strength, the laxity of the medial tibial compartment was noticeably improved. Tibial translation and rotation were similar to the intact knee after PCL reconstruction with graft strengths ranging from 75% to 125% of an intact PCL. When the graft’s strength surpassed 150%, the medial tibia moved forward and external tibial rotation greatly increased. The in-situ forces generated in the PCL grafts ranged from 13.15 N to 75.82 N, depending on the stiffness. In conclusion, the strength of PCL grafts have has a noticeable effect on anterior-posterior translation of the medial tibial compartment and its in-situ force. Similar kinematic response may happen in the models when the PCL graft’s strength lies between 75% and 125% of an intact PCL.     

Verification of predicted specimen-specific natural and implanted patellofemoral kinematics during simulated deep knee bend

Verified computational models represent an efficient method for studying the relationship between articular geometry, soft-tissue constraint, and patellofemoral (PF) mechanics. The current study was performed to evaluate an explicit finite element (FE) modeling approach for predicting PF kinematics in the natural and implanted knee. Experimental three-dimensional kinematic data were collected on four healthy cadaver specimens in their natural state and after total knee replacement in the Kansas knee simulator during a simulated deep knee bend activity. Specimen-specific FE models were created from medical images and CAD implant geometry, and included soft-tissue structures representing medial–lateral PF ligaments and the quadriceps tendon. Measured quadriceps loads and prescribed tibiofemoral kinematics were used to predict dynamic kinematics of an isolated PF joint between 10° and 110° femoral flexion. Model sensitivity analyses were performed to determine the effect of rigid or deformable patellar representations and perturbed PF ligament mechanical properties (pre-tension and stiffness) on model predictions and computational efficiency.Predicted PF kinematics from the deformable analyses showed average root mean square (RMS) differences for the natural and implanted states of less than 3.1° and 1.7 mm for all rotations and translations. Kinematic predictions with rigid bodies increased average RMS values slightly to 3.7° and 1.9 mm with a five-fold decrease in computational time. Two-fold increases and decreases in PF ligament initial strain and linear stiffness were found to most adversely affect kinematic predictions for flexion, internal–external tilt and inferior–superior translation in both natural and implanted states. The verified models could be used to further investigate the effects of component alignment or soft-tissue variability on natural and implant PF mechanics.     

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.     

The effect of weightbearing and external loading on anterior cruciate ligament strain

A force balance between the ligaments, articular contact, muscles and body weight maintains knee joint stability. Thus, it is important to study anterior cruciate ligament (ACL) biomechanics, in vivo, under weightbearing conditions. Our objective was to compare the ACL strain response under weightbearing and non-weightbearing conditions and in combination with three externally applied loadings: (1) anterior-posterior shear forces, (2) internal-external torques, and (3) varus-valgus moments. A strain transducer was implanted on the ACL of 11 subjects. All joint loadings were performed with the knee at 20 degrees of flexion. A significant increase in ACL strain was observed as the knee made the transition from non-weightbearing to weightbearing. During anterior shear loading, the strain values produced during weightbearing were greater than those of the non-weightbearing knee (shear loads <40N). At higher shear loads, the strain values became equal. During axial torsion, an internal torque of 10Nm strained the ACL when the knee was non-weightbearing while an equivalent external torque did not. Weightbearing significantly increased ACL strain values in comparison to non-weightbearing with the application of external torques and low internal torques (<3Nm). The strains became equal for higher internal torques. For V-V loading, the ACL was not strained in the non-weightbearing knee. However, weightbearing increased the ACL strain values over the range of moments tested. These data have important clinical ramifications in the development of rehabilitation protocols following ACL reconstruction since weightbearing has been previously thought to provide a protective mechanism to the healing graft.     

A reciprocal connection factor for assessing knee-joint function

In the knee joint, interactions between instantaneous kinetics and kinematics associated with ligamentous and articular tissues are not fully understood. These structures may be represented by the instantaneous screw axis ($) (ISA) and static force vectors ($′). Geometric changes to the joint structure affecting motion have not been fully explained, especially after surgical reconstruction and replacement procedures. The ISA offers a joint-characterisation approach, which is dependent on the combined forces of ligaments, articular contacts and muscles. The standard four-bar linkage model in the sagittal plane demonstrates that the normal contact force and the lines of action of the cruciate ligaments always intersect at the centre of rotation of the joint. A kinematic knee model in which the articular surfaces in the lateral and medial compartments as well as the isometric fascicles in the engaged ligaments may be represented as five constraints in a one-degree-of-freedom parallel spatial mechanism. This study provides a theoretical foundation to elucidate the role of each of these elements in the control of the ISA. A recourse to the principle of virtual work explained through d'Alembert's principle for reducing a dynamics problem to an instantaneous static scenario allows screws to be applied to the biomechanics of human motion. The principle of reciprocity links these approaches together to explain the transmitting load between the tibia and the femur as well as the relative motion within the knee joint. A principal clinical implication of this study is the introduction of the reciprocal connection factor to evaluate knee kinematics and kinetics in one simple term, allowing the quantitative assessment of the outcome of knee-joint treatment and rehabilitation methods.