Mechanics of the passive knee joint. Part 2: interaction between the ligaments and the articular surfaces in guiding the joint motion

The aim of this study was to examine how the interaction between ligament tensions and contact forces guides the knee joint through its specific pattern of passive motion. A computer model was built based on cadaver data. The passive motion and the ligament lengthening and force patterns predicted by the model were verified with data from the literature. The contribution of each ligament and contact force was measured in terms of the rotational moment that it produced about the tibial medial plateau and the anterior-posterior (AP) force that it exerted on the tibia. The high tension of the anterior cruciate ligament (ACL) and the geometric constraints of the anterior horns of the menisci were found to be key features that stabilized the knee at full extension. The mutual effect of the cruciates was found as the reason for the screw-home mechanism at early flexion. Past 300, the AP component of contact force on the convex geometry of the lateral tibial plateau and tension of the lateral collateral ligament (LCL) were identified as elements that control the joint motion. From 60 degrees to 90 degrees, reduction in the tension of the ACL was determined as a reason for continuation of the tibial anterior translation. From 90 degrees to 120 degrees, increase in the tension of the posterior cruciate ligament and the AP component of the contact force on the convex geometry of the lateral tibial plateau pushed the tibia more anteriorly. This anterior translation was limited by the constraining effects of the ACL tension and the AP component of the contact force on the medial meniscus. The important guiding role observed for the LCL suggests that it should not be overlooked in knee models.     

A model of human knee ligaments in the sagittal plane. Part 2: Fibre recruitment under load.

A mathematical model of the knee ligaments in the sagittal plane is used to study the forces in the cruciate and collateral ligaments produced by anterior/posterior tibial translation. The model is based on ligament fibre functional architecture. Geometric analysis of the deformed configurations of the model ligaments provides the additional compatibility conditions necessary for calculation of the statically indeterminate distributions of strain and stress within the ligaments and the sharing of load between ligaments. The investigation quantifies the process of ligament fibre recruitment, which occurs when fibres made slack by passive flexion/extension of the knee stretch and change their spatial positions in order to resist applied loads. The calculated ligament forces are in reasonable agreement with experimental results reported in the literature. The model explains some subtleties of ligament function not incorporated in models that represent the ligaments by a small number of lines.     

The geometry of the knee in the sagittal plane

A geometric model of the tibio-femoral joint in the sagittal plane has been developed which demonstrates the relationship between the geometry of the cruciate ligaments and the geometry of the articular surfaces. The cruciate ligaments are represented as two inextensible fibres which, with the femur and the tibia, are analysed as a crossed four-bar linkage. The directions of the ligaments at each position of flexion are calculated. The instant centre, where the flexion axis crosses the parasagittal plane through the joint, lies at the intersection of the cruciates. It moves relative to each of the bones during flexion and extension. The successive positions of the flexion axis relative to a fixed femur and to a fixed tibia are deduced. The shapes of articular surfaces which would allow the bones to flex and extend while maintaining the ligaments each at constant length are calculated and are found to agree closely with the shapes of the natural articular surfaces. The calculated movements of the contact point between the femur and the tibia during flexion also agree well with measurements made on cadaver specimens. The outcome is a geometric simulation of the tibio-femoral joint in the sagittal plane which illustrates the central role played by the cruciate ligaments in the kinematics of the knee and which can be used for the analysis of ligament and contact forces.     

Mechanics of the passive knee joint. Part 1: The role of the tibial articular surfaces in guiding the passive motion

The motion of the unloaded knee is associated with tibial internal rotation and femoral posterior translation. Although it is known that the passive motion is the result of the interaction between the articular surfaces and the ligaments, the mechanism through which the particular pattern of motion is guided is not completely understood. The goal of this study was to focus on the tibial geometry and to identify the roles that its geometric features have in guiding the passive knee motion. The method used in this study simplified the geometry of the tibial plateaux and the menisci into basic features that could be eliminated individually. The generated tibial geometry was implemented in a computer model to simulate the passive motion. Different parts of the geometry were eliminated individually and the comparison between the simulation results was used to identify the role that each part of the geometry had in guiding the passive motion. The medial meniscus was found as the feature that promoted the tibial internal rotation and restrained the femoral posterior translation. The lateral meniscus and the medial aspect of the tibial eminence, on the other hand, were found as the elements that confined the tibial internal rotation.     

Muscle-ligament interactions at the human knee (II)

In part I of this paper, a three-dimensional model of the human knee joint was incorporated into a detailed human musculoskeletal lower extremity model. The model was used to determine the muscle, ligament, and articular contact forces transmitted at the knee as humans extend/flex in an isometric state. Part II investigates the sensitivity of the model calculations to changes in the parameters which describe the mechanical behavior of cartilage and the ligaments of the knee. The ligament function in the real knee was most sensitive to changes in ligament reference lengths or strains, less sensitive to changes in cartilage stiffness, and least sensitive to changes in ligament stiffness.     

A one-degree-of-freedom spherical mechanism for human knee joint modelling

In-depth comprehension of human knee kinematics is necessary in prosthesis and orthosis design and in surgical planning but requires complex mathematical models. Models based on one-degree-of-freedom equivalent mechanisms have replicated well the passive relative motion between the femur and tibia, i.e. the knee joint motion in virtually unloaded conditions. In these mechanisms, fibres within the anterior and posterior cruciate and medial collateral ligaments were taken as isometric and anatomical articulating surfaces as rigid. A new one-degree-of-freedom mechanism is analysed in the present study, which includes isometric fibres within the two cruciates and a spherical pair at the pivot point of the nearly spherical motion as measured for this joint. Bounded optimization was applied to the mechanism to refine parameter first estimates from experimental measurements on four lower-limb specimens and to best-fit the experimental motion of these knees. Relevant results from computer simulations were compared with those from one previous equivalent mechanism, which proved to be very accurate in a former investigation. The spherical mechanism represented knee motion with good accuracy, despite its simple structure. With respect to the previous more complex mechanism, the less satisfactory results in terms of replication of natural motion were counterbalanced by a reduction of computational costs, by an improvement in numerical stability of the mathematical model, and by a reduction of the overall mechanical complexity of the mechanism. These advantages can make the new mechanism preferable to the previous ones in certain applications, such as the design of prostheses, orthoses, and exoskeletons, and musculoskeletal modelling of the lower limb.     

An optimal control model for determining articular contact forces at the human knee during rising from a static squat position

A two-dimensional dynamic model of the knee joint was incorporated into a four-segment, eight-muscle model of the human body to determine the muscle, ligament, and articular contact forces transmitted at the knee as humans stand up from a static squatting position. Our optimal control model predicted peak tibiofemoral contact forces 8 times as high as body weight. Furthermore, ligament forces, especially those in the anterior-cruciate, were nearly body weight as knee flexion approached 90 degrees. Ligament and tibiofemoral contact loads were dominated by the forces exerted by muscles during the movement.     

Muscle-ligament interactions at the human knee (I)

A three-dimensional dynamic model of the knee was developed to study the interactions between the articulating surfaces of the bones and the geometrical and mechanical properties of the ligaments. The contact-surface geometry of the distal femur, proximal tibia, and patella was modeled by fitting polynomials to each of the eight articular surfaces. Twelve elastic elements were used to describe the function of the ligamentous and capsular structures of the knee. The origin and insertion sites of each model ligament were obtained from cadaveric data reported for an average-size knee. The response of the model to both anterior-posterior drawer and axial rotation suggests that the geometrical and mechanical properties of the model ligaments approximate the behavior of real ligaments in the intact knee. Comparison of the model’s response with experimental data obtained from cadaveric knee extension indicate further that the three-dimensional model reproduces the response of the real knee during movement.     

A sequentially-defined stiffness model of the knee

A new procedure is proposed to obtain a stiffness model of the knee, i.e. a model of the joint when static external loads are applied to the tibia and femur. A sequential approach is used to generalise a kinematic model of the passive motion of the articulation previously presented by the authors. The procedure is devised in such a way that the restraining function of the articular components which guide the passive motion of the joint is preserved after generalisation. As a result, the new stiffness model can replicate the passive motion with the same accuracy as the previous kinematic model, and it can also replicate the relative motion of the tibia and femur when the knee is loaded by static external loads. The proposed procedure is applied to a specimen and the relevant stiffness model is devised. The motion of the model under several loading conditions is then compared with the original motion of the specimen and with data obtained from the literature, in order to show the accuracy of the model and the potential of the proposed procedure.     

Design forms of total knee replacement

The starting point of this article is a general design criterion applicable to all types of total knee replacement. This criterion is then expanded upon to provide more specifics of the required kinematics, and the forces which the total knee must sustain. A characteristic which differentiates total knees is the amount of constraint which is required, and whether the constraint is translational or rotational. The different forms of total knee replacement are described in terms of these constraints, starting with the least constrained unicompartments to the almost fully constrained fixed and rotating hinges. Much attention is given to the range of designs in between these two extreme types, because they constitute by far the largest in usage. This category includes condylar replacements where the cruciate ligaments are preserved or resected, posterior cruciate substituting designs and mobile bearing knees. A new term, 'guided motion knees', is applied to the growing number of designs which control the kinematics by the use of intercondylar cams or specially shaped and even additional bearing surfaces. The final section deals with the selection of an appropriate design of total knee for specific indications based on the design characteristics.     

Muscle-ligament interactions at the knee during walking.

A two-dimensional mathematical model of the knee is used with gait analysis to calculate muscle, cruciate ligament and tibio-femoral contact forces developed at the knee during normal level walking. Ten normal adult subjects--four females and six males--participated. The knee model is based upon a four-bar linkage comprising the femur, tibia and two cruciate ligaments. It takes account of the rolling and sliding of the femur on the tibia during flexion/extension and the changes in direction of the ligaments and muscle tendons. We considered forces transmitted by six elements: quadriceps, hamstrings, gastrocnemius, anterior and posterior cruciate ligaments, and tibio-femoral contact. The equations of mechanics can be used to determine the absolute values of only three of the knee forces simultaneously, so that twenty limiting solutions of three of the six forces were considered. A limiting solution was rejected if any of the three forces were negative, corresponding to compressive muscle or ligament forces, or tensile contact forces. These constraints always reduced and at times removed the redundancy of the knee structures. The high incidence of predicted single muscle activity, supported by electromyography, suggested that the ligaments play a significant role in load transmission during gait. The temporal patterns of muscle and ligament activity and ligament force magnitudes were sensitive to the choice of model parameters. The analysis showed that each of four possible minimum principles of muscle selection--minimal muscle force, muscle stress, ligament force and contact force--was unlikely to be valid throughout the walking cycle.     

Validation of the soft tissue restraints in a force-controlled knee simulator

In vitro testing of total knee replacements (TKRs) is important both at the design stage and after the production of the final components. It can predict long-term in vivo wear of TKRs. The two philosophies for knee testing are to drive the motion by displacement or to drive the motion by force. Both methods have advantages and disadvantages. For force control an accurate simulation of soft tissue restraints is required. This study was devised to assess the accuracy of the soft tissue restraints of the force-controlled Stanmore knee simulator in simulating the restraining forces of the anterior cruciate ligament (ACL) and posterior cruciate ligament (PCL). In order to do this, human cadaver knee joints were subjected to the ISO Standard Walking Cycle. The resulting kinematics were monitored when the soft tissue structures were intact, when the ACL and PCL were resected, and when they were simulated by springs positioned anteriorly and posteriorly. The stiffness of the springs was determined from the literature. Two different stiffnesses of springs were used which were 7.24 N/mm (designated as soft springs) and 33.8 N/mm (designated as hard springs). All the intact knees showed displacements that were within the range of the machine. Cutting the ACL and PCL resulted in anterior and posterior motion and internal external rotation that were significantly greater than the intact knee. Results showed that when the ACL and PCL were cut hard springs positioned anterior and posterior to the knee returned the knee to near normal anterior-posterior (AP) motion. Using hard springs in the posterior position in any condition reduced rotational displacements. Therefore using springs in a force-controlled simulator is a compromise. More accuracy may be obtained using springs that are of intermediate stiffness.     

Three-dimensional contact dynamic model of the human knee joint during walking

It is well known that the geometry of the articular surface has a major role in determining the position of articular contact and the lines of action for the contact forces. The contact force calculation of the knee joint under the effect of sliding and rolling is one of the most challenging issues in this field. We present a 3-D human knee joint model including sliding and rolling motions and major ligaments to calculate the lateral and medial condyle contact forces from the recovered total internal reaction force using inverse dynamic contact modeling and the Least-Square method. As results, it is believed that the patella, muscles and tendon affect a lot for the internal reaction forces at the initial heel contact stage. With increasing flexion angles during gait, the decreasing contact area is progressively shifted to the posterior direction on the tibia plateau. In addition, the medial side contact force is larger than the lateral side contact force in the knee joint during normal human walking. The total internal forces of the knee joint are reasonabe compared to previous studies.     

Three-dimensional dynamic model of the knee

A three-dimensional dynamic model of the knee was developed to study the interactions between the articulating surfaces of the bones and the geometrical and mechanical properties of the ligaments. The contact-surface geometry of the distal femur, proximal tibia, and patella was modeled by fitting polynomials to each of the eight articular surfaces. Twelve elastic elements were used to describe the function of the ligamentous and capsular structures of the knee. The origin and insertion sites of each model ligament were obtained from cadaveric data reported for an average-size knee. The response of the model to both anterior-posterior drawer and axial rotation suggests that the geometrical and mechanical properties of the model ligaments approximate the behavior of real ligaments in the intact knee. Comparison of the model’s response with experimental data obtained from cadaveric knee extension indicate further that the three-dimensional model reproduces the response of the real knee during movement.     

往复运动速度规划算法对曲线磨削的影响研究

往复加工常见于磨削和刨削等金属切削中,加工过程的平稳性直接影响到产品的加工质量、能源效率乃至机床的寿命。首先探究往复(冲程)运动速度规划对运动过程平稳性的影响,在分析了常见速度规划运动学性能的基础上,针对磨削冲程运动的特点,提出通过改变急动度空间分布来降低柔性冲击对加工表面质量的影响,并设计了两种基于急动度连续的速度规划算法。在此基础上,研究了速度规划算法和磨削力平稳性、加工表面粗糙度和加工能耗间的关系,提出了通过改变加速度空间分布来降低加工能耗的方法。试验结果表明,往复运动速度规划和磨削力平稳性、加工表面粗糙度以及加工能耗均相关,通过改变急动度和加速度空间分布提高了磨削力平稳性和加工表面质量,降低了加工能耗。所提出的Ⅱ型速度规划综合表现优于其他规划,与梯形速度规划相比,切削力波动、加工表面粗糙度和电机驱动能耗均有较为显著的下降。     

A model of human knee ligaments in the sagittal plane. Part 1: Response to passive flexion.

The development of a mathematical model of the knee ligaments in the sagittal plane is presented. Essential features of the model are (a) the representation of selected cruciate ligament fibres as isometric links in a kinematic mechanism that controls passive knee flexion and (b) the mapping of all other ligament fibres between attachments on the tibia and femur. Fibres slacken and tighten as the ligament attachment areas on the bones move relative to each other. The model is used to study the shape and fibre length changes of the cruciate and collateral ligaments in response to passive flexion/extension of the knee. The model ligament shape and fibre length changes compare well qualitatively with experimental results reported in the literature. The results suggest that when designing and implanting a ligament replacement with the aim of reproducing the natural fibre strain patterns, the surgeon must not only implant through the natural attachment areas but must also maintain the natural fibre mapping and render all fibres just tight at the appropriate flexion angle.     

Knee and hip joint forces--sensitivity to the degrees of freedom classification at the knee

Previous research has demonstrated that the number of degrees of freedom (DOF) modelled at a given joint affects the antagonistic muscle activity predicted by inverse dynamics optimization techniques. This higher level of muscle activity in turn results in greater joint contact forces. For instance, modelling the knee as a 3 DOF joint has been shown to result in higher hip and knee joint forces commensurate with a higher level of muscular activity than when the knee is modelled with 1 DOF. In this study, a previously described musculoskeletal model of the lower limb was used to evaluate the sensitivity of the knee and hip joint contact forces to the DOF at the knee during vertical jumping in both a 1 and a 3 DOF knee model. The 3 DOF knee was found to predict higher tibiofemoral and hip joint contact forces and lower patellofemoral joint contact forces. The magnitude of the difference in hip contact force was at least as significant as that found in previous research exploring the effect of subject-specific hip geometry on hip contact force. This study therefore demonstrates a key sensitivity of knee and hip joint contact force calculations to the DOF at the knee. Finally, it is argued that the results of this study highlight an important physiological question with practical implications for the loading of the structures of the knee; that is, the relative interaction of muscular, ligamentous, and articular structures in creating moment equilibrium at the knee.     

A purpose-built dynamometer to objectively measure static and dynamic knee torque

This paper reports the development of a purpose-built knee dynamometer (PBKD) to evaluate passive range of motion (ROM) and isometric muscle strength measurements of the knee. The PBKD uses a TorqSense rotary torque transducer and objectively measures isometric knee muscle strength in a valid and reliable manner and passive resistance to motion through range. The device and all associated instrumentation underwent dynamic and static calibration to ensure consistent and accurate measurements were obtained in terms of knee joint angular position, passive torque measures, and isometric torque measures. Eleven healthy male participants performed a knee flexion and extension task designed to evaluate knee function. The validation of the PBKD entailed measuring the consistency of measurement and accuracy of measurement. Accuracy of the PBKD was determined by comparing peak isometric muscle strength measurements against a KIN-COM machine. No significant differences were observed both passively and isometrically between cycles and between trials. This device can have widespread applications within the rehabilitation and clinical environment and could be used as a functional outcome measuring tool to distinguish pathological from non-pathological knees. The presented preliminary results indicate that reliable and accurate measurements of knee ROM and muscle strength can be obtained.     

An objective tool for assessing the outcome of total knee replacement surgery

The need for an objective tool to assess the outcome of total knee replacement (TKR) surgery is widely recognized. This study investigates the potential of an objective diagnostic tool for assessing the outcome of TKR surgery based on motion analysis techniques. The diagnostic tool has two main elements: collection of data using motion analysis, and the assessment of knee function using a classifier that is based around the Dempster-Shafer theory of evidence. The tool was used to analyse the knee function of nine TKR subjects preoperatively and at three stages post-operatively. Using important measurable characteristics of the knee, the tool was able to establish the level of benefit achieved by surgery and to enable a comparison of subjects. No subject recovered normal knee function following TKR surgery. This has important implications for knee implant designs.