A theoretical model of the knee and ACL: theory and experimental verification.

A three-dimensional mathematical model of the human knee joint was developed to examine the role of single ligaments, such as an anterior cruciate ligament (ACL) graft in ACL reconstruction, on joint motion and tissue forces. The model is linear and valid for small motions about an equilibrium position. The knee joint is modeled as two rigid bodies (the femur and the tibia) interconnected by deformable structures, including the ACL or ACL graft, the cartilage layer, and the remainder of the knee tissues (modeled as a single element). The model was demonstrated for the equilibrium condition of the knee in extension with an anterior tibial force, causing anterior drawer and hyperextension. The knee stiffness matrix for this condition was measured for a human right knee in vitro. Predicted model response was compared with experimental observations. Qualitative agreement was found between model and experiment, validating the model and its assumptions. The model was then used to predict the change in graft and cartilage forces and joint motion of the knee due to an increment of load in the normal joint both after ACL removal and with various altered states simulating ACL reconstructions. Results illustrate the interdependence between loads in the ACL graft, other knee structures, and contact force. Stiffer grafts and smaller maximum unloaded length of the ligament lead to higher graft and contact forces. Changes in cartilage stiffness alter load sharing between ACL graft and other joint tissues.     

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.     

A new one-DOF fully parallel mechanism for modelling passive motion at the human tibiotalar joint

Knowledge on how ligaments and articular surfaces guide passive motion at the human ankle joint complex is fundamental for the design of relevant surgical treatments. The paper presents a possible improvement of this knowledge by a new kinematic model of the tibiotalar articulation. Passive motion, i.e. in virtually unloaded conditions, was captured in vitro in four lower leg specimens by means of a surgical navigation system with cluster of active markers attached to the tibia and talus. The anatomical geometry of the passive structures, i.e. articular surfaces and attachment areas of the ligaments, were taken by digitisation with a pointer. An equivalent spatial mechanism for the passive motion simulation was defined by three sphere-to-sphere contact points and two rigid links. These contact points were identified at the lateral talo-fibular articulation and at the medial and lateral aspects of the articulation between tibial mortise and trochlea tali. The two rigid links were identified by the isometric fibres at the calcaneofibular and tibiocalcaneal ligaments. An optimisation algorithm was developed for the identification of the final geometrical parameters resulting from an iterative refining process, which targets best matching between model predictions and corresponding experimental measurements of the spatial motion. The specimen-specific equivalent spatial mechanisms replicated the original passive motion very well, with mean discrepancies in position smaller than 2.5 mm and in rotation smaller than 1°. The study demonstrates that the articular surfaces and the ligaments, acting together as a mechanism, control the passive kinematics of the ankle joint.     

A Three-Dimensional Musculoskeletal Model of the Human Knee Joint. Part 1: Theoretical Construction

A three-dimensional model of the knee is developed to study the interactions between the muscles, ligaments, and bones during activity. The geometry of the distal femur, proximal tibia, and patella is based on cadaver data reported for an average-size knee. The shapes of the femoral condyles are represented by high-order polynomials: the tibial plateaux and patellar facets are approximated as flat surfaces. The contacting surfaces of the femur and tibia are modeled as deformable, while those of the femur and patella are assumed to be rigid. Interpenetration of the femur and tibia is taken into account by modeling cartilage as a thin, linear, elastic layer, mounted on rigid bone. Twelve elastic elements 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 modeled as a three-element muscle in series with tendon. The path of each muscle is approximated as a straight line, except where it contacts and wraps around bone and other muscles; changes in muscle paths are taken into account using data obtained from MRI. In the first part of this paper, the model is used to simulate passive knee flexion. Quantitative comparisons of the model results with experimental data reported in the literature indicate that the relative movements of the bones and the geometry of the ligaments and muscles in the model are similar to those evident in the real knee. In Part II, the model is used to describe knee-ligament function during anterior-posterior draw, axial rotation, and isometric knee-extension and knee-flexion exercises.     

The interaction of muscle moment arm,knee laxity,and torque in a multi-scale musculoskeletal model of the lower limb

IntroductionMusculoskeletal modeling allows insight into the interaction of muscle force and knee joint kinematics that cannot be measured in the laboratory. However, musculoskeletal models of the lower extremity commonly use simplified representations of the knee that may limit analyses of the interaction between muscle forces and joint kinematics. The goal of this research was to demonstrate how muscle forces alter knee kinematics and consequently muscle moment arms and joint torque in a musculoskeletal model of the lower limb that includes a deformable representation of the knee.MethodsTwo musculoskeletal models of the lower limb including specimen-specific articular geometries and ligament deformability at the knee were built in a finite element framework and calibrated to match mean isometric torque data collected from 12 healthy subjects. Muscle moment arms were compared between simulations of passive knee flexion and maximum isometric knee extension and flexion. In addition, isometric torque results were compared with predictions using simplified knee models in which the deformability of the knee was removed and the kinematics at the joint were prescribed for all degrees of freedom.ResultsPeak isometric torque estimated with a deformable knee representation occurred between 45° and 60° in extension, and 45° in flexion. The maximum isometric flexion torques generated by the models with deformable ligaments were 14.6% and 17.9% larger than those generated by the models with prescribed kinematics; by contrast, the maximum isometric extension torques generated by the models were similar. The change in hamstrings moment arms during isometric flexion was greater than that of the quadriceps during isometric extension (a mean RMS difference of 9.8 mm compared to 2.9 mm, respectively).DiscussionThe large changes in the moment arms of the hamstrings, when activated in a model with deformable ligaments, resulted in changes to flexion torque. When simulating human motion, the inclusion of a deformable joint in a multi-scale musculoskeletal finite element model of the lower limb may preserve the realistic interaction of muscle force with knee kinematics and torque.     

A Three-Dimensional Musculoskeletal Model of the Human Knee Joint. Part 1: Theoretical Construct

A three-dimensional model of the knee is developed to study the interactions between the muscles, ligaments, and bones during activity. The geometry of the distal femur, proximal tibia, and patella is based on cadaver data reported for an average-size knee. The shapes of the femoral condyles are represented by high-order polynomials; the tibial plateaux and patellar facets are approximated as flat surfaces. The contacting surfaces of the femur and tibia are modeled as deformable, while those of the femur and patella are assumed to be rigid. Interpenetration of the femur and tibia is taken into account by modeling cartilage as a thin, linear, elastic layer, mounted on rigid bone. Twelve elastic elements 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 modeled as a three-element muscle in series with tendon. The path of each muscle is approximated as a straight line, except where it contacts and wraps around bone and other muscles; changes in muscle paths are taken into account using data obtained from MRI. In the first part of this paper, the model is used to simulate passive knee flexion. Quantitative comparisons of the model results with experimental data reported in the literature indicate that the relative movements of the bones and the geometry of the ligaments and muscles in the model are similar to those evident in the real knee. In Part II, the model is used to describe knee-ligament function during anterior-posterior draw, axial rotation, and isometric knee-extension and knee-flexion exercises.     

Two-dimensional dynamic modelling of human knee joint

A mathematical dynamic model of the two-dimensional representation of the knee joint is presented. The profiles of the joint surfaces are determined from X-ray films and they are represented by polynomials. The joint ligaments are modelled as nonlinear elastic springs of realistic stiffness properties. Nonlinear equations of motion coupled with nonlinear constraint conditions are solved numerically. Time derivatives are approximated by Newmark difference formulae and the resulting nonlinear algebraic equations are solved employing the Newton-Raphson iteration scheme. Several dynamic loads are applied to the center of mass of the tibia and the ensuing motion is investigated. Numerical results on ligament forces, contact point locations between femur and tibia, and the orientation of tibia relative to femur are presented. The results are shown to be consistent with the anatomy of the knee joint.     

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.     

Comparison of deformable and elastic foundation finite element simulations for predicting knee replacement mechanics

Rigid body total knee replacement (TKR) models with tibiofemoral contact based on elastic foundation (EF) theory utilize simple contact pressure-surface overclosure relationships to estimate joint mechanics, and require significantly less computational time than corresponding deformable finite element (FE) methods. However, potential differences in predicted kinematics between these representations are currently not well understood, and it is unclear if the estimates of contact area and pressure are acceptable. Therefore, the objectives of the current study were to develop rigid EF and deformable FE models of tibiofemoral contact, and to compare predicted kinematics and contact mechanics from both representations during gait loading conditions with three different implant designs. Linear and nonlinear contact pressure-surface overclosure relationships based on polyethylene material properties were developed using EF theory. All other variables being equal, rigid body FE models accurately estimated kinematics predicted by fully deformable FE models and required only 2% of the analysis time. As expected, the linear EF contact model sufficiently approximated trends for peak contact pressures, but overestimated the deformable results by up to 30%. The nonlinear EF contact model more accurately reproduced trends and magnitudes of the deformable analysis, with maximum differences of approximately 15% at the peak pressures during the gait cycle. All contact area predictions agreed in trend and magnitude. Using rigid models, edge-loading conditions resulted in substantial overestimation of peak pressure. Optimal nonlinear EF contact relationships were developed for specific TKR designs for use in parametric or repetitive analyses where computational time is paramount. The explicit FE analysis method utilized here provides a unique approach in that both rigid and deformable analyses can be run from the same input file, thus enabling simple selection of the most appropriate representation for the analysis of interest.     

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.     

A constraint-based approach to modelling the mobility of the human knee joint

A model of knee mobility able to predict the range and pattern of movement in the unloaded joint was proposed by Wilson et al. (J. Biomech. 31 (1998) 1127-1136). The articular surfaces in the lateral and medial compartments and isometric fascicles in three of the knee ligaments were represented as five constraints on motion between the femur and tibia in a single degree-of-freedom parallel spatial mechanism. The path of movement of the bones during passive flexion was found by solving the forward kinematics of the mechanism using an iterative method. The present paper shows that such a mechanism-based solution approach can lead to an underestimation of the flexion range. This is due to the mechanism reaching a 'stationary configuration' and 'locking'. A new, constraint-based approach to the solution of the model joint displacement is proposed. It avoids the representation of ligaments and articular surfaces by kinematically equivalent chains of one degree-of-freedom pairs which are prone to singularities. It relies instead on a numerical solution of five non-linear constraint equations to find the relative positions of the bones at a series of flexion angles. The method is successful both in its ability to predict motion through a physiological range and in its efficiency with a solution rate forty times faster than the original algorithm. The new approach may be extended to include more complex joint surface geometry, allowing a study of the effects of articular surface shape and ligament arrangement on joint kinematics.     

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.     

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.     

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.     

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.     

Experimental evaluation of an elastic foundation model to predict contact pressures in knee replacements

Computational wear prediction is an attractive concept for evaluating new total knee replacement designs prior to physical testing and implementation. An important hurdle to such technology is the lack of in vivo contact pressure predictions. To address this issue, this study evaluates a computationally efficient simulation approach that combines the advantages of rigid and deformable body modeling. The hybrid method uses rigid body dynamics to predict body positions and orientations and elastic foundation theory to predict contact pressures between general three-dimensional surfaces. To evaluate the method, we performed static pressure experiments with a commercial knee implant in neutral alignment using flexion angles of 0, 30, 60, and 90 degrees and loads of 750, 1500, 2250, and 3000N. Using manufacturer CAD geometry for the same implant, an elastic foundation model with linear or nonlinear polyethylene material properties was implemented within a commercial multibody dynamics software program. The model's ability to predict experimental peak and average contact pressures simultaneously was evaluated by performing dynamic simulations to find the static configuration. Both the linear and nonlinear material models predicted the average contact pressure data well, while only the linear material model could simultaneously predict the trends in the peak contact pressure data. This novel modeling approach is sufficiently fast and accurate to be used in design sensitivity and optimization studies of knee implant mechanics and ultimately wear.     

Computationally efficient finite element evaluation of natural patellofemoral mechanics

Finite element methods have been applied to evaluate in vivo joint behavior, new devices, and surgical techniques but have typically been applied to a small or single subject cohort. Anatomic variability necessitates the use of many subject-specific models or probabilistic methods in order to adequately evaluate a device or procedure for a population. However, a fully deformable finite element model can be computationally expensive, prohibiting large multisubject or probabilistic analyses. The aim of this study was to develop a group of subject-specific models of the patellofemoral joint and evaluate trade-offs in analysis time and accuracy with fully deformable and rigid body articular cartilage representations. Finite element models of eight subjects were used to tune a pressure-overclosure relationship during a simulated deep flexion cycle. Patellofemoral kinematics and contact mechanics were evaluated and compared between a fully deformable and a rigid body analysis. Additional eight subjects were used to determine the validity of the rigid body pressure-overclosure relationship as a subject-independent parameter. There was good agreement in predicted kinematics and contact mechanics between deformable and rigid analyses for both the tuned and test groups. Root mean square differences in kinematics were less than 0.5 deg and 0.2 mm for both groups throughout flexion. Differences in contact area and peak and average contact pressures averaged 5.4%, 9.6%, and 3.8%, respectively, for the tuned group and 6.9%, 13.1%, and 6.4%, respectively, for the test group, with no significant differences between the two groups. There was a 95% reduction in computational time with the rigid body analysis as compared with the deformable analysis. The tuned pressure-overclosure relationship derived from the patellofemoral analysis was also applied to tibiofemoral (TF) articular cartilage in a group of eight subjects. Differences in contact area and peak and average contact pressures averaged 8.3%, 11.2%, and 5.7% between rigid and deformable analyses in the tibiofemoral joint. As statistical, probabilistic, and optimization techniques can require hundreds to thousands of analyses, a viable platform is crucial to component evaluation or clinical applications. The computationally efficient rigid body platform described in this study may be integrated with statistical and probabilistic methods and has potential clinical application in understanding in vivo joint mechanics on a subject-specific or population basis.     

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.     

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.