A mathematical model of the patellofemoral joint

A mathematical model of the patellofemoral joint taking into account movements and forces in the sagittal plane is described. The system parameters of the model are the locations of the attachments of the quadriceps muscle and the patellar ligament, the length of the patellar ligament, the dimensions of the patella and the geometry of the articulating surfaces. They were obtained from ten autopsy knees. The model enables calculation of the relative position of the patella, patellar ligament and quadriceps tendon, the location of the patellofemoral contact point and the magnitude of the patellofemoral compression force and the force in the patellar ligament as a function of the location of the tibial tuberosity at different flexion-extension angles of the knee. The model is validated by comparing model data with experimentally determined data.     

The orientation of the distal part of the quadriceps femoris muscle as a function of the knee flexion-extension angle

Lateral view radiographs of ten autopsy knees were used to determine the orientation of the patellar ligament, patella and quadriceps tendon relative to tibia and femur at different flexion-extension angles (0-120 degrees) of the knee. The results show a linear relationship between the angle of flexion and the movement of the patellar ligament relative to the tibia and of the movement of the patella relative to tibia and femur. There is a non-linear relationship between angle of flexion and the movement of the quadriceps tendon relative to the patellar ligament, patella and femur. The angular changes between patella and patellar ligament are negligible. The complicated movements of the distal part of the quadriceps femoris muscle may significantly influence biomechanical parameters such as the forces acting at the patella and tibial tuberosity.     

Geometrical changes of knee ligaments and patellar tendon during passive flexion

Patterns of fibre elongation and orientation for the cruciate and collateral ligaments of the human knee joint and for the patellar tendon have not yet been established in three-dimensions. These patterns are essential for understanding thoroughly the contribution of these soft tissues to joint function and of value in surgical treatments for a more conscious assessment of the knee status. Measurements from 10 normal cadaver knees are here reported using an accurate surgical navigation system and consistent anatomical references, over a large flexion arc, and according to current recommended conventions. The contours of relevant sub-bundles were digitised over the corresponding origins and insertions on the bones. Representative fibres were calculated as the straight line segments joining the centroids of these attachment areas. The most isometric fibre was also taken as that whose attachment points were at the minimum change in length over the flexion arc. Changes in length and orientation of these fibres were reported versus the flexion angle. A good general repeatability of intra- and inter-specimens was found. Isometric fibres were found in the locations reported in the literature. During knee flexion, ligament sub-bundles slacken in the anterior cruciate ligament, and in the medial and lateral collateral ligaments, whereas they tighten in the posterior cruciate ligament. In each cruciate ligament the two compounding sub-bundles have different extents for the change in fibre length, and also bend differently from each other on both tibial planes. In the collateral ligaments and patellar tendon all fibres bend posteriorly. Patellar tendon underwent complex changes in length and orientation, on both the tibial sagittal and frontal planes. For the first time thorough and consistent patterns of geometrical changes are provided for the main knee ligaments and tendons after careful fibre mapping.     

Comparison of material properties in fascicle-bone units from human patellar tendon and knee ligaments

The fascicle material properties in bone-fascicle-bone units were determined for the anterior and posterior cruciate ligaments (ACL, PCL), the lateral collateral ligament (LCL) and the patellar tendon (PT) from three young human donor knees. Groups of fascicles from each tissue were isolated with intact bone ends and failed at a high strain rate in a saline bath at 37 degrees C. In each knee tested the load related material properties (linear modulus, maximum stress and energy density to maximum stress) for the patellar tendon were significantly larger than corresponding values for the cruciate and collateral ligaments. Bundles from different ligaments in the same knee were similar to each other in their mechanical behavior. In addition, no significant differences were present in the maximum strains recorded for any of the four tissue types examined. The results presented have implications in studies of ligament injury. They are also important in the design and use of synthetic and biological ligament replacements and in tissue and whole knee modeling.     

两种石磺齿舌形态差异分析

显微观察了瘤背石磺(Onchidiumstruma)和石磺(O. verruculatum)齿舌的形态结构。运用差异系数法对两种石磺齿舌参数进行比较分析。利用SPSS10.0对瘤背石磺、石磺齿舌参数(齿舌长、齿舌头宽、齿舌中宽、齿舌尾宽、横列数、每排最少齿片数和每排最多齿片数)与个体参数(体长、体宽、体高、足长、足宽和体重)作回归分析。结果表明,两种石磺齿舌都很发达,外观呈长统靴状;齿片排成许多横列,每一横列均有中央齿一枚,侧齿若干无缘齿;两种石磺的齿舌头宽、齿舌中宽和齿舌尾宽差异极显著,但差异系数小于1.28,认为两种石磺的齿片形态存在明显的种间差异,但齿舌参数不适合作为石磺属贝类的分类依据;瘤背石磺的体宽和石磺的体重在评估各自齿舌生物学性状方面起到比较重要的作用。     

A comparison of patellofemoral cartilage morphology and deformation in anterior cruciate ligament deficient versus uninjured knees

Anterior cruciate ligament (ACL) deficient patients have an increased rate of patellofemoral joint (PFJ) osteoarthritis (OA) as compared to the general population. Although the cause of post-injury OA is multi-factorial, alterations in joint biomechanics may predispose patients to cartilage degeneration. This study aimed to compare in vivo PFJ morphology and mechanics between ACL deficient and intact knees in subjects with unilateral ACL ruptures. Eight male subjects underwent baseline MRI scans of both knees. They then performed a series of 60 single-legged hops, followed by a post-exercise MRI scan. This process was repeated for the contralateral knee. The MR images were converted into three-dimensional surface models of cartilage and bone in order to assess cartilage thickness distributions and strain following exercise. Prior to exercise, patellar cartilage was significantly thicker in intact knees as compared to ACL deficient knees by 1.8%. In response to exercise, we observed average patellar cartilage strains of 5.4 ± 1.1% and 2.5 ± 1.4% in the ACL deficient and intact knees, respectively. Importantly, the magnitude of patellar cartilage strain in the ACL deficient knees was significantly higher than in the intact knees. However, while trochlear cartilage experienced a mean strain of 2.4 ± 1.6%, there was no difference in trochlear cartilage strain between the ACL deficient and uninjured knees. In summary, we found that ACL deficiency was associated with decreased patellar cartilage thickness and increased exercise-induced patellar cartilage strain when compared to the uninjured contralateral knees.     

A mathematical model for the evaluation of the behaviour during flexion of condylar-type knee prostheses

A 3D knee model was developed in order to evaluate the mechanical behaviour during flexion of condylar-type knee prosthesis. Based on the total energy minimization principle, it takes into account the articular surfaces (the tibial surface being deformable), the body weight, and the patello femoral joint. It generates the kinematics of the joint, the motion of the centre of contact, the quadriceps forces, the pressure distribution on the tibial plateau, and ligament lengths and forces between 0 and 120 degrees of flexion. The results for ten digitized knees and the commercially available prostheses are presented. They are in general agreement with experimental results published in the literature. It is concluded that this computer program may be, within its limitations, a useful tool in the preliminary evaluation of new condylar-type knee prosthesis designs.     

Kinematic motion of the anterior cruciate ligament deficient knee during functionally high and low demanding tasks

The purpose of this study was to determine whether mechanical adaptations were present in patients with anterior cruciate ligament (ACL)-deficient knees during high-demand activities. Twenty-two subjects with unilateral ACL deficiency (11 males and 11 females, 19.6 months after injury) performed five different activities at a comfortable speed (level walking, ascending and descending steps, jogging, jogging to a 90-degree side cutting toward the opposite direction of the tested side). Three-dimensional knee kinematics for the ACL-deficient knees and uninjured contralateral knees were evaluated using the Point Cluster Technique. There was no significant difference in knee flexion angle, but an offset toward the knee in less valgus and more external tibial rotation was observed in the ACL-deficient knee. The tendency was more obvious in high demand motions, and a significant difference was clearly observed in the side cutting motions. These motion patterns, with the knee in less valgus and more external tibial rotation, are proposed to be an adaptive movement to avoid pivot shift dynamically, and reveal evidence in support of a dynamic adaptive motion occurring in ACL-deficient knees.     

A three-dimensional anatomical model of the human patello-femoral joint,for the determination of patello-femoral motions and contact characteristics

The object of this study is to develop a three-dimensional mathematical model of the patello-femoral joint, which is modelled as two rigid bodies representing a moving patella and a fixed femur. Two-point contact was assumed between the femur and patella at the medial and lateral sides and in the analysis, the femoral and patellar articular surfaces were mathematically represented using Coons' bicubic surface patches. Model equations include six equilibrium equations and eleven constraints: six contact conditions, four geometric compatibility conditions, and the condition of a rigid patellar ligament; the model required the solution of a system of 17 nonlinear equations in 17 unknowns, its response describing the six-degress-of-freedom patellar motions and the forces acting on the patella. Patellar motions are described by six motion parameters representing the translations and rotations of the patella with respect to the femur. The forces acting on the patella include the medial and lateral component of patello-femoral contact and the patellar ligament force, all of which were represented as ratios to the quadriceps tendon force. The model response also includes the locations of the medial and lateral contact points on the femur and the patella. A graphical display of its response was produced in order to visualize better the motion of the components of the extensor mechanism.Model calculations show good agreement with experimental results available from the literature. The patella was found to move distally and posteriorly on the femoral condyles as the knee was flexed from full extension. Results indicate that the relative orientation of the patellar ligament with respect to the patella remains unchanged during this motion. The model also predicts a patellar flexion which always lagged knee flexion.Our calculations show that as the angle of knee flexion increased, the lateral contact point moved distally on the femur without moving significantly either medially or laterally. The medial contact point also moved distally on the femur but moved medially from full extension to about 40° of knee flexion, then laterally as the knee flexion angle increased. The lateral contact point on the patella did not change significantly in the medial and lateral direction as the knee was flexed; however, this point moved proximally toward the basis of the patella with knee flexion. The medial contact point also moved proximally on the patella with knee flexion, and in a similar manner the medial contact point on the patella moved distally with flexion from full extension to about 40° of flexion. However, as the angle of flexion increased, the medial contact point did not move significantly in the medial-lateral direction.Model calculations also show that during the simulated knee extension exercise, the ratio of the force in the patellar ligament to the force in the quadriceps tendon remains almost unchanged for the first 30° of knee flexion, then decreases as the angle of knee flexion increases. Furthermore, model results show that the lateral component of the patello-femoral contact force is always greater than the medial component, both components increasing with knee flexion.     

The effects of knee brace hinge design and placement on joint mechanics

A computer model of 23 knees was obtained by embedding, slicing and digitizing the bone outlines and ligament co-ordinates. Using co-ordinate transformations, various three-dimensional motions were imposed on the knees, and calculations made of femoral-tibial contact error, contact point locations and ligament lengths. Significant deviations in these parameters were noted for abnormal motions including the elimination of internal-external rotation and a-p displacement and the misplacement of a hinge producing correct motion. The resulting mismatch could result in shear in soft tissues, cuff-to-skin slippage and inaccurate ligament length patterns.     

In vivo patellofemoral forces in high flexion total knee arthroplasty

This study compares the in vivo patellofemoral contact forces generated in high flexion fixed bearing posterior cruciate retaining Nexgen CR-Flex (PCR) and high flexion posterior stabilized Nexgen LPS-Flex (LPS) TKAs with that of normal knees from full knee extension to maximum weight bearing flexion. Ten patients with the PCR total knee arthroplasty (TKA), ten with the LPS TKA and seven patients having normal knees were fluoroscoped while performing a deep knee bend activity. In vivo femorotibial kinematics, obtained from 3D-to-2D registration technique, and patellar kinematics obtained by direct measurements from the fluoroscopic images were entered into a 3D inverse dynamics mathematical model to determine the in vivo contact forces at the knee. The variation in the patellofemoral and quadriceps forces with flexion were found to be similar across the three groups-increasing from full extension to 90 degrees of flexion, reaching a maximum between 90 degrees and 120 degrees of flexion and then decreasing until maximum flexion. At maximum knee flexion, these forces were found to be significantly lower in the normal knees than in the TKAs. The patellar ligament to quadriceps force ratio decreased with the increase in knee flexion while the patellofemoral to quadriceps force ratio increased. A strong correlation was found to exist between the patellofemoral forces, the femorotibial contact forces and the forces in the extensor mechanism. The PCR TKA in this study exhibited greater resemblance to the normal patients with respect to the patellofemoral forces than the LPS TKA though significant differences in the two implant types were not observed.     

Forces acting on the patella during maximal voluntary contraction of the quadriceps femoris muscle at different knee flexion/extension angles

From knee extension moments measured with a dynamometer, the quadriceps muscle force, the patellar ligament force and the reaction force in the patellofemoral joint at various knee angles (0-90 degrees) were estimated. The information needed to calculate the combined effect of both patellofemoral and tibiofemoral joint on the mechanical advantage of the muscle was obtained from lateral-view radiographs of autopsy knees. The results show that the smallest quadriceps force (2,000 N) is exerted at maximal extension, and the largest force (8,000 N) at about 75 degrees of flexion. The patellar ligament force reaches a maximum (5,000 N) at 60 degrees. The reaction force in the patellofemoral joint is the smallest (1,000 N) at extension and is of the same values as the muscle force in a range from 75 to 90 degrees. Especially at large flexion angles, the value of the estimated forces is considerably larger (by 100%) than reported in the literature. This difference is attributed to the influence of the patellofemoral joint on the mechanical advantage of the muscle, which has not been taken into account in other studies.     

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.     

Prediction of total knee motion using a three-dimensional computer-graphics model

Twenty-three knees were sectioned, digitized, and standardized to determine the 'average' three-dimensional bony geometry and ligamentous attachments. Data on normal knee motion were obtained from a cadaveric study. An algorithm was written to simulate three-dimensional patella motion. Verification of the knee model was achieved by determining femoro-tibial and patello-femoral contact locations, as well as ligament length patterns, and comparing the results with published data. The criterion for maximum predicted knee motion with a prosthesis in place was the length of the posterior cruciate ligament. Three total knee replacement surfaces were mathematically generated: flat, laxity and conforming. A greater flexion angle was obtained with a flat tibial surface than for the laxity or conforming. Posterior tibial component displacement increased the range of motion, but only slightly. For all tibial surfaces, increased range of motion was achieved with a 10 degrees posterior tilt of the tibial tray. Anterior femoral component displacement increased motion due to reduction in posterior cruciate tension during flexion. The results are applicable to the design and surgical technique of total knee replacement.     

Increasing quadriceps loads affect the lengths of the ligaments and the kinematics of the knee.

The relationships between the lengths of the ligaments and kinematics of the knee and quadriceps load, for low to physiologic levels of quadriceps loads, have not previously been studied. We investigated the effects of increasing levels of quadriceps force, necessary to balance increasing levels of externally applied flexion moments, on the kinematics of the tibiofemoral joint and on the separation distances between insertions of selected fibers of the major ligaments of the knee in twelve cadavera. Static measurements were made using a six-degree-of-freedom digitizer for flexion angles ranging from 0 to 120 deg in 15 deg increments. Quadriceps generated extension of the knee was performed by applying loads to the quadriceps tendon to equilibrate each of four magnitudes of external flexion moments equivalent to 8.33, 16.67, 25.00, and 33.33 percent of values previously reported for maximum isometric extension moments. The magnitude of quadriceps force increased linearly (p < 0.0001) as external flexion moment increased throughout the entire range of flexion. Anterior translation, internal rotation, and abduction of the tibia increased linearly (p < 0.0001, p < 0.001, p < 0.001) as external flexion moment and, hence, quadriceps load increased. For the fibers studied, the anterior cruciate ligament (p < 0.0076), posterior cruciate ligament (p < 0.0001), and medial collateral ligament (p < 0.0383) lengthened linearly while the lateral collateral ligament (p < 0.0124) shortened linearly as quadriceps load increased. Based on these results for low to physiologic levels of quadriceps loads, it is reasonable to assume that the ligament lengths or knee kinematics expected with higher quadriceps loads can be extrapolated.     

In vivo tracking of the human patella using cine phase contrast magnetic resonance imaging

Improper patellar tracking is often considered to be the cause of patellar-femoral pain. Unfortunately, our knowledge of patellar-femoral-tibial (knee) joint kinematics is severely limited due to a lack of three-dimensional, noninvasive, in vivo measurement techniques. This study presents the first large-scale, dynamic, three-dimensional, noninvasive, in vivo study of nonimpaired knee joint kinematics during volitional leg extensions. Cine-phase contrast magnetic resonance imaging was used to measure the velocity profiles of the patella, femur, and tibia in 18 unimpaired knees during leg extensions, resisted by a 34 N weight. Bone displacements were calculated through integration and then converted into three-dimensional orientation angles. We found that the patella displaced laterally, superiorly, and anteriorly as the knee extended. Further, patellar flexion lagged knee flexion, patellar tilt was variable, and patellar rotation was fairly constant throughout extension.     

Results of the surgical reconstruction of the anterior cruciate ligament

Results of the surgical reconstruction of the anterior cruciate ligament (ACL), using as a graft fourfold hamstring tendons (gracilis and semitendinosus) and middle third of the patellar ligament, were compared. In all patients that were participating in this study clinical examination and magnetic resonance showed ACL rupture, and apart from the choice of the graft, surgical technique was identical. We evaluated 112 patients with implemented patellar ligament graft and fourfold hamstring tendons graft six months after the procedure. Both groups were similar according to age, sex, activity level, knee instability level and rehabilitation program. The results showed that there was no significant difference between groups regarding Lysholm Knee score, IKDC 2000 score, activity level, musculature hypotrophy, and knee joint stability 6 months after the surgery. Anterior knee pain incidence is significantly higher in the group with patellar ligament graft (44% vs. 21%). Both groups had a significant musculature hypotrophy of the upper leg of the knee joint that was surgically treated, six months after the procedure. Both grafts showed good subjective and objective results.     

Three-dimensional mathematical model analysis of the patellofemoral joint

This paper is concerned with a mathematical model analysis of the patellofemoral joint in the human knee, taking into account the articular surface geometry and mechanical properties of the ligament. It was made by the application of a computer-aided design theory (previously studied) and it was possible to express the articular surface geometries in a mathematical formulation and hence elucidate the joint movement mechanics. This method was then applied to a three-dimensional geometrical model of the patellofemoral joint. For the modelling of tendofemoral contact at large angles of knee flexion, the geodestic line theory was adopted. Applying the Newton-Raphson method and the Runge-Kutta Gil method to the model, variables such as patellar attitudes, patellofemoral contact force and tensile force of the patellar ligament for various knee flexion angles were computed. Applying the Hertzian elastic theory, contact stress was also computed. These results showed good agreement with the previously reported experimental results. As an application for the model, some parameter analyses were performed in terms of the contact stress variations and compared with those of the normal knee. The simulation results indicated that both the Q-angle increase and decrease increased contact stress, the patella alta showed undulating variations of stress while the patella infera showed little change of stress, and the tibial tuberositas elevation showed 20-30% reduction of stress.     

Moment arm of the patellar tendon in the human knee

The moment arm of the knee-extensor mechanism is described by the moment arm of the patellar tendon calculated with respect to the screw axis of the tibia relative to the femur. The moment arm may be found once the line of action of the patellar tendon and the position and orientation of the screw axis are known. In this study, the orientation of the patellar tendon and the position and orientation of the finite screw axis of the tibia relative to the femur were calculated from measurements of the three-dimensional positions of the bones obtained from fresh cadaver specimens. Peak values of the patellar tendon moment arm ranged from 4-6 cm for the six knees tested; the moment arm was maximum near 45 degrees of knee flexion. The moment arm of the patellar tendon was nearly equal to the shortest (perpendicular) distance between the line of action of the patellar tendon and the axis of rotation of the knee at all flexion angles, except near full extension. Near full extension, the angle between the patellar tendon and the screw axis was significantly less than 90 degrees, and the magnitude of the moment arm was then less than the perpendicular distance between these two lines. The patellar tendon moment arm remained roughly constant across individuals when normalized by femoral condyle width, suggesting that anatomical differences play a large role in determining the moment arm of the extensor mechanism.     

Differences in tibial rotation during walking in ACL reconstructed and healthy contralateral knees

This study tested the hypotheses that in patients with a successful anterior cruciate ligament (ACL) reconstruction, the internal–external rotation, varus–valgus, and knee flexion position of reconstructed knees would be different from uninjured contralateral knees during walking. Twenty-six subjects with unilateral ACL reconstructions (avg 31 years, 1.7 m, 68 kg, 15 female, 24 months past reconstruction) and no other history of serious lower limb injury walked at a self-selected speed in the gait laboratory, with the uninjured contralateral knee as a matched control. Kinematic measurements of tibiofemoral motion were made using a previously-described point-cluster technique. Repeated-measures ANOVA (α=0.017) was used to compare ACL-reconstructed knees to their contralateral knees at four distinct points during the stance phase of walking. An offset towards external tibial rotation in ACL-reconstructed knees was maintained over all time points (95%CI 2.3±1.3°). Twenty-two out of twenty-six individuals experienced an average external tibial rotation offset throughout stance phase. Varus–valgus rotation and knee flexion were not significantly different between reconstructed and contralateral knees. These findings show that differences in tibial rotation during walking exist in ACL reconstructed knees compared to healthy contralateral knees, providing a potential explanation why these patients are at higher risk of knee osteoarthritis in the long-term.