Control of sliding and rolling at natural joints

The planar motion of the human knee joint is modeled, involving the relative motion of the geometry of the contacting surface between the tibia and the femur. The pure gliding motion and the pure rolling motion are formulated including the holonomic and nonholonomic constraints that must be satisfied. A control strategy with two classes of inputs: muscle forces that stabilize and bring about the motion and the ligament forces that maintain the constraints is presented. Finally, the effectiveness of this control structure is demonstrated via digital computer simulations in the pure gliding motion and the pure rolling motion of the knee.     

Control exerted by ligaments

The function of the ligaments as local controllers, independent of the central nervous system, in maintaining the integrity of the joint is demonstrated by modelling the human knee in the sagittal plane, and studying its anterior-posterior motion. In addition to the ligaments, the model includes the characteristic geometry of the joint surface and some muscle groups. The connecting reaction forces at the point of contact between the tibia and the femur are considered to be constraint forces due to three different surface motions--gliding, rolling and combined gliding and rolling. It is demonstrated that the ligamentous structure maintains these holonomic and nonholonomic constraints that describe the joint motion, and that stability of the knee joint is provided mainly by ligaments. Muscular structures further stabilize and contribute to joint movement. Computer simulation of rolling movement of the knee is presented to illustrate the importance of the ligaments for joint integrity and stability.     

The coupled motion of the femur and patella during in vivo weightbearing knee flexion

The movement of the knee joint consists of a coupled motion between the tibiofemoral and patellofemoral articulations. This study measured the six degrees-of-freedom kinematics of the tibia, femur, and patella using dual-orthogonal fluoroscopy and magnetic resonance imaging. Ten normal knees from ten living subjects were investigated during weightbearing flexion from full extension to maximum flexion. The femoral and the patellar motions were measured relative to the tibia. The femur externally rotated by 12.9 deg and the patella tilted laterally by 16.3 deg during the full range of knee flexion. Knee flexion was strongly correlated with patellar flexion (R(2)=0.91), posterior femoral translation was strongly correlated to the posterior patellar translation (R(2)=0.87), and internal-external rotation of the femur was correlated to patellar tilt (R(2)=0.73) and medial-lateral patellar translation (R(2)=0.63). These data quantitatively indicate a kinematic coupling between the tibia, femur, and patella, and provide base line information on normal knee joint kinematics throughout the full range of weightbearing flexion. The data also suggest that the kinematic coupling of tibia, femur, and patella should be considered when investigating patellar pathologies and when developing surgical techniques to treat knee joint diseases.     

On the kinematic modeling and the parameter estimation of the human knee joint.

This paper presents some results on the modeling and the parameter estimation of the human knee joint. Based on the geometric characteristics of the femur condyle and the tibia plateau, a part of femoro-tibial joint model includes an involute-on-plane submodel. Data recorded by camera type device are used to analyze the kinematic characteristics of the knee joint and to estimate the corresponding submodel parameters. Experimental results are presented and the model is further validated.     

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.     

Relative movements between the tibia and femur induced by external plantar shocks are controlled by muscle forces in vivo

The purpose of this study was to investigate the role of muscle activation on the relative motion between tibia and femur. Impacts were initiated under the heels of four volunteers in three different activation levels of muscles crossing the extended knee joint: 0%, 30% and 60% of previously performed maximal voluntary isometric contractions. Impact forces were measured and tibial and femoral accelerations and displacements were determined by means of accelerometry. The accelerometers were mounted on the protruding ends of intracortical pins, inserted into the distal aspect of the femur and proximal aspect of the tibia. Under the 0%-condition the impact force (475±64N) led to 2.3±1.2mm knee compression and to 2.4±1.9mm medio-lateral and 4.4±1.1mm antero-posterior shear. The impact forces increased significantly with higher activation levels (619±33N (30%), 643±147N (60%)), while the knee compression (1.5±1.2, 1.4±1.3mm) and both medio-lateral shear (1.8±1.4, 1.5±1.1mm) and antero-posterior shear (2.6±1.3, 1.5±1.1mm) were significantly reduced. This study indicated that muscles are effective in controlling the relative motion between tibia and femur when the knee is subjected to external forces.     

Three-dimensional kinematics of the human knee during walking.

Three-dimensional kinematics of the tibiofemoral joint were studied during normal walking. Target markers were fixed to tibia and femur by means of intra-cortical traction pins. Radiographs of the lower limb were obtained to compute the position of the target markers relative to internal anatomical structures. High-speed cine cameras were used to measure three-dimensional coordinates of the target markers in five subjects walking at a speed of 1.2 m s-1. Relative motion between tibia and femur was resolved according to a joint coordinate system (JCS). The measurements have identified that substantial angular and linear motions occur about and along each of the JCS axes during walking. The results do not, however, support the traditional view that the so-called 'screw home' mechanism of the knee joint operates during gait.     

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 automatic 2D-3D image matching method for reproducing spatial knee joint positions using single or dual fluoroscopic images

Fluoroscopic image technique, using either a single image or dual images, has been widely applied to measure in vivo human knee joint kinematics. However, few studies have compared the advantages of using single and dual fluoroscopic images. Furthermore, due to the size limitation of the image intensifiers, it is possible that only a portion of the knee joint could be captured by the fluoroscopy during dynamic knee joint motion. In this paper, we presented a systematic evaluation of an automatic 2D-3D image matching method in reproducing spatial knee joint positions using either single or dual fluoroscopic image techniques. The data indicated that for the femur and tibia, their spatial positions could be determined with an accuracy and precision less than 0.2?mm in translation and less than 0.4° in orientation when dual fluoroscopic images were used. Using single fluoroscopic images, the method could produce satisfactory accuracy in joint positions in the imaging plane (in average up to 0.5?mm in translation and 1.3° in rotation), but large variations along the out-plane direction (in average up to 4.0?mm in translation and 2.2° in rotation). The precision of using single fluoroscopic images to determine the actual knee positions was worse than its accuracy obtained. The data also indicated that when using dual fluoroscopic image technique, if the knee joint outlines in one image were incomplete by 80%, the algorithm could still reproduce the joint positions with high precisions.     

Automatic determination of anatomical coordinate systems for three-dimensional bone models of the isolated human knee

The combination of three-dimensional (3-D) models with dual fluoroscopy is increasingly popular for evaluating joint function in vivo. Applying these modalities to study knee motion with high accuracy requires reliable anatomical coordinate systems (ACSs) for the femur and tibia. Therefore, a robust method for creating ACSs from 3-D models of the femur and tibia is required. We present and evaluate an automated method for constructing ACSs for the distal femur and proximal tibia based solely on 3-D bone models. The algorithm requires no observer interactions and uses model cross-sectional area, center of mass, principal axes of inertia, and cylindrical surface fitting to construct the ACSs. The algorithm was applied to the femur and tibia of 10 (unpaired) human cadaveric knees. Due to the automated nature of the algorithm, the within specimen variability is zero for a given bone model. The algorithm’s repeatability was evaluated by calculating variability in ACS location and orientation across specimens. Differences in ACS location and orientation between specimens were low (<1.5 mm and <2.5°). Variability arose primarily from natural anatomical and morphological differences between specimens. The presented algorithm provides an alternative method for automatically determining subject-specific ACSs from the distal femur and proximal tibia.     

Tibiofemoral kinematics and condylar motion during the stance phase of gait

Accurate knowledge of the dynamic knee motion in-vivo is instrumental for understanding normal and pathological function of the knee joint. However, interpreting motion of the knee joint during gait in other than the sagittal plane remains controversial. In this study, we utilized the dual fluoroscopic imaging technique to investigate the six-degree-of-freedom kinematics and condylar motion of the knee during the stance phase of treadmill gait in eight healthy volunteers at a speed of 0.67 m/s. We hypothesized that the 6DOF knee kinematics measured during gait will be different from those reported for non-weightbearing activities, especially with regards to the phenomenon of femoral rollback. In addition, we hypothesized that motion of the medial femoral condyle in the transverse plane is greater than that of the lateral femoral condyle during the stance phase of treadmill gait. The rotational motion and the anterior–posterior translation of the femur with respect to the tibia showed a clear relationship with the flexion–extension path of the knee during the stance phase. Additionally, we observed that the phenomenon of femoral rollback was reversed, with the femur noted to move posteriorly with extension and anteriorly with flexion. Furthermore, we noted that motion of the medial femoral condyle in the transverse plane was greater than that of the lateral femoral condyle during the stance phase of gait (17.4±2.0 mm vs. 7.4±6.1 mm, respectively; p<0.01). The trend was opposite to what has been observed during non-weightbearing flexion or single-leg lunge in previous studies. These data provide baseline knowledge for the understanding of normal physiology and for the analysis of pathological function of the knee joint during walking. These findings further demonstrate that knee kinematics is activity-dependent and motion patterns of one activity (non-weightbearing flexion or lunge) cannot be generalized to interpret a different one (gait).     

A procedure for estimating the relevant forces in the human knee using a four-bar mechanism

Knee injuries, especially those that affect the cruciate and lateral ligaments, are one of the most serious and frequent pathologies that affect the lower human extremity. Hence, the aim of this study is to develop a dynamic model for the lower extremity capable of estimating forces, forces in the cruciate and collateral ligaments and those normal to the articular cartilage, generated in the knee. The proposed model considers a four-bar mechanism in the knee, a spherical joint in the pelvis and a revolute one in the ankle. The four-bar mechanism is obtained by a synthesis process. The dynamic model includes the inertial properties of the femur, tibia, patella and the foot, the ground reaction force and the most important muscles in the knee. Muscle forces are estimated using an optimisation technique. Results from the application of the model on a real human task are presented.     

Anterior cruciate ligament injury induced by internal tibial torsion or tibiofemoral compression

The knee is one of the most frequently injured joints in the human body. Approximately 91% of ACL injuries occur during sporting activities, usually from a non-contact event. The most common kinetic scenarios related with ACL injuries are internal twisting of the tibia relative to the femur or combined torque and compression during a hard landing. The hypothesis of this study was that the magnitudes and types of motion observed after ACL rupture would significantly change from the relative joint displacements present just before ACL injury. Compression or torsion experiments were conducted on 7 pairs of knee joints with repetitive tests at increasing intensity until catastrophic failure. ACL injury was documented in all cases at 5.4±2 kN of TF compression or 33±13 Nm of internal tibial torque. The femur displaced posteriorly relative to the tibia in pre-failure and with a higher magnitude in failure tests under both loading conditions. In compression experiments there was internal rotation of the tibia in pre-failure tests, but external rotation of the tibia after the ACL failed. In torsion experiments, failure occurred at 58±19° of internal tibial rotation, and valgus rotation of the femur increased significantly after ACL injury. These new data show that the joint motions can vary in magnitude and direction before and after failure of the ACL. Video-based studies consistently document external rotation of the tibia combined with valgus knee bending as the mechanism of ACL injury although these motions could be occurring after ACL rupture.     

An automatic 2D–3D image matching method for reproducing spatial knee joint positions using single or dual fluoroscopic images

Fluoroscopic image technique, using either a single image or dual images, has been widely applied to measure in vivo human knee joint kinematics. However, few studies have compared the advantages of using single and dual fluoroscopic images. Furthermore, due to the size limitation of the image intensifiers, it is possible that only a portion of the knee joint could be captured by the fluoroscopy during dynamic knee joint motion. In this paper, we presented a systematic evaluation of an automatic 2D–3D image matching method in reproducing spatial knee joint positions using either single or dual fluoroscopic image techniques. The data indicated that for the femur and tibia, their spatial positions could be determined with an accuracy and precision less than 0.2 mm in translation and less than 0.4° in orientation when dual fluoroscopic images were used. Using single fluoroscopic images, the method could produce satisfactory accuracy in joint positions in the imaging plane (in average up to 0.5 mm in translation and 1.3° in rotation), but large variations along the out-plane direction (in average up to 4.0 mm in translation and 2.2° in rotation). The precision of using single fluoroscopic images to determine the actual knee positions was worse than its accuracy obtained. The data also indicated that when using dual fluoroscopic image technique, if the knee joint outlines in one image were incomplete by 80%, the algorithm could still reproduce the joint positions with high precisions.     

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.     

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.     

Automatic registration of MRI-based joint models to high-speed biplanar radiographs for precise quantification of in vivo anterior cruciate ligament deformation during gait

Understanding in vivo joint mechanics during dynamic activity is crucial for revealing mechanisms of injury and disease development. To this end, laboratories have utilized computed tomography (CT) to create 3-dimensional (3D) models of bone, which are then registered to high-speed biplanar radiographic data captured during movement in order to measure in vivo joint kinematics. In the present study, we describe a system for measuring dynamic joint mechanics using 3D surface models of the joint created from magnetic resonance imaging (MRI) registered to high-speed biplanar radiographs using a novel automatic registration algorithm. The use of MRI allows for modeling of both bony and soft tissue structures. Specifically, the attachment site footprints of the anterior cruciate ligament (ACL) on the femur and tibia can be modeled, allowing for measurement of dynamic ACL deformation. In the present study, we demonstrate the precision of this system by tracking the motion of a cadaveric porcine knee joint. We then utilize this system to quantify in vivo ACL deformation during gait in four healthy volunteers.     

Using relative velocity vectors to reveal axial rotation about the medial and lateral compartment of the knee

A new technique is presented that utilizes relative velocity vectors between articulating surfaces to characterize internal/external rotation of the tibio-femoral joint during dynamic loading. Precise tibio-femoral motion was determined by tracking the movement of implanted tantalum beads in high-speed biplane X-rays. Three-dimensional, subject-specific CT reconstructions of the femur and tibia, consisting of triangular mesh elements, were positioned in each analyzed frame. The minimum distance between subchondral bone surfaces was recorded for each mesh element comprising each bone surface, and the relative velocity between these opposing closest surface elements was determined in each frame. Internal/external rotation was visualized by superimposing tangential relative velocity vectors onto bone surfaces at each instant. Rotation about medial and lateral compartments was quantified by calculating the angle between these tangential relative vectors within each compartment. Results acquired from 68 test sessions involving 23 dogs indicated a consistent pattern of sequential rotation about the lateral condyle (approximately 60 ms after paw strike) followed by rotation about the medial condyle (approximately 100 ms after paw strike). These results imply that axial knee rotation follows a repeatable pattern within and among subjects. This pattern involves rotation about both the lateral and medial compartments. The technique described can be easily applied to study human knee internal/external rotation during a variety of activities. This information may be useful to define normal and pathologic conditions, to confirm post-surgical restoration of knee mechanics, and to design more realistic prosthetic devices. Furthermore, analysis of joint arthrokinematics, such as those described, may identify changes in joint mechanics associated with joint degeneration.     

A planar model of the knee joint to characterize the knee extensor mechanism

A simple planar static model of the knee joint was developed to calculate effective moment arms for the quadriceps muscle. A pathway for the instantaneous center of rotation was chosen that gives realistic orientations of the femur relative to the tibia. Using the model, nonlinear force and moment equilibrium equations were solved at one degree increments for knee flexion angles from 0 (full extension) to 90 degrees, yielding patellar orientation, patellofemoral contact force and patellar ligament force and direction with respect to both the tibial insertion point and the tibiofemoral contact point. The computer-derived results from this two-dimensional model agree with results from more complex models developed previously from experimentally obtained data. Due to our model's simplicity, however, the operation of the patellar mechanism as a lever as well as a spacer is clearly illustrated. Specifically, the thickness of the patella was found to increase the effective moment arm significantly only at flexions below 35 degrees even though the actual moment arm exhibited an increase throughout the flexion range. Lengthening either the patella or the patellar ligament altered the force transmitted from the quadriceps to the patellar ligament, significantly increasing the effective moment arm at flexions greater than 25 degrees. We conclude that the levering action of the patella is an essential mechanism of knee joint operation at moderate to high flexion angles.     

Measurement of in vivo anterior cruciate ligament strain during dynamic jump landing

Despite recent attention in the literature, anterior cruciate ligament (ACL) injury mechanisms are controversial and incidence rates remain high. One explanation is limited data on in vivo ACL strain during high-risk, dynamic movements. The objective of this study was to quantify ACL strain during jump landing. Marker-based motion analysis techniques were integrated with fluoroscopic and magnetic resonance (MR) imaging techniques to measure dynamic ACL strain non-invasively. First, eight subjects' knees were imaged using MR. From these images, the cortical bone and ACL attachment sites of the tibia and femur were outlined to create 3D models. Subjects underwent motion analysis while jump landing using reflective markers placed directly on the skin around the knee. Next, biplanar fluoroscopic images were taken with the markers in place so that the relative positions of each marker to the underlying bone could be quantified. Numerical optimization allowed jumping kinematics to be superimposed on the knee model, thus reproducing the dynamic in vivo joint motion. ACL length, knee flexion, and ground reaction force were measured. During jump landing, average ACL strain peaked 55±14 ms (mean and 95% confidence interval) prior to ground impact, when knee flexion angles were lowest. The peak ACL strain, measured relative to its length during MR imaging, was 12±7%. The observed trends were consistent with previously described neuromuscular patterns. Unrestricted by field of view or low sampling rate, this novel approach provides a means to measure kinematic patterns that elevate ACL strains and that provide new insights into ACL injury mechanisms.