From: Leendert Blankevoort , Passive motion characteristics of the human knee joint -- Experiments and computer simulations (ISBN 90-9004358-6). PhD-thesis, 9 October 1991, Catholic University Nijmegen, The Netherlands CHAPTER X Summary and discussion INTRODUCTION The knee joint is a complex three-dimensional mechanical system. Its motions during activities depend on the passive freedom of motion characteristics on the one hand, and the loading conditions and muscle activity during a particular function on the other. The articular surfaces and the ligamentous structures determine the passive motion characteristics. Understanding the relationships between the geometrical and mechanical properties of the anatomic structures and the passive motion characteristics is important for the evaluation of diagnostic and surgical procedures with respect to injuries of the knee and for design and testing of total knee prostheses. For example, the technical result of an ACL reconstruction procedure can be quantified by evaluating the passive motion characteristics in terms of flexion range and anterior laxity, because the goal of an ACL reconstruction is to restore the normal passive motion feasibilities. As a passive mechanical system, the knee can be described in terms of the relationship between forces and motions. The external forces are the effects of muscles, gravity and accelerations. The relationship between forces and motions is governed by two feed-back loops (Fig. X.1), one through the ligaments and one through the articular surfaces, both generating internal forces in the system, which interfere with dynamic equilibrium. The ligament forces depend on the relative motions of the joint via the ligament insertion and ligament properties. The contact forces also depend on the motions, via the articular geometry and the properties of the articular cartilage. The experiments and mathematical model studies described in this thesis can be put into perspective using the schematic model of Fig. X.1. The experimental evaluations in Chapter II (The envelope of passive knee-joint motion) addressed the relationship between forces and motions. In Chapter III (Recruitment of knee- joint ligaments) the ligament length patterns were analyzed. An alternative way of representing knee motions by helical axes was described in Chapter IV (Helical axes of passive knee-joint motions). The three chapters dealing with the knee model and its validation relative to the experimental data, transform the sche- matic model of Fig. X.1 into a mathematical model, which is used for computer simulations of the experiments. Chapter V (Articular contact in a three-dimensional model of the knee) describes the representation of the articular surfaces, the contact between the surfaces in the knee model, and the effects of the articular geometry and the cartilage properties on the motion characteristics. Although not represented in the sche- matic model of Fig. X.1, a description of a ligament wrapping around a bony surface is incorporated in the model for the medial collateral ligament, interacting with the edge of the tibia. The effect of this interaction on the passive motion characteristics was evaluated in Chapter VI (Ligament-bone interaction in a three-dimensional model of the knee). The confrontation of the knee models with the experimental data (one model for each experimental knee) was performed through a parametric analysis. The capacity of the computer model to simulate, realistically, the passive motion characteristics was assessed in Chapter VII (Parametric validation of a three-dimensional model of the knee). Finally, the computer model was applied to investigate a biomechanical and a surgical problem. The biomechanical problem concerned the contribution of the ligaments and the articular geometry to the restraints to axial rotation of the tibia, relative to the femur. Hence, in fact Chapter VIII (Rotation restraints in the knee joint) explored the relative importance of both feed-back loops in the schematic model of Fig. X.1. The surgical problem addressed with the knee model was the sensitivity of the functional result of an ACL reconstruction procedure to the femoral insertion location of the ACL graft, in Chapter IX (ACL reconstruction: simply a matter of isometry?). This concluded the evaluation of the knee model as a tool in the analysis of knee biomechanics, diagnosis of knee ligament injuries and evaluation of surgical procedures. SUMMARY OF CHAPTERS II-IX II. The envelope of passive knee-joint motion Four knee-joint specimens were moved through flexion in a motion- and loading rig under several combinations of external loading, including internal-external moments, axial compressive forces and anterior-posterior forces. Euler rotation angles and translation vectors, describing the relative spatial motions of the joint, were measured using an accurate Roentgenstereophotogrammetric system. Conceptually the joint was considered as a two degrees-of-freedom of motion mechanism (flexion and tibial rotation), whereby the limits of internal and external rotation were defined at torques of ù3 Nm. The motion pathways along these limits were defined as the envelopes of passive knee-joint motion. It was found that these envelope pathways were consistent and hardly influenced by additional axial forces up to 300 N and AP-forces of 30 N. Within the envelope of motion, however, the motion patterns are highly susceptible to small changes in the external load configuration. It was shown that the external tibial rotation during extension (`screw-home mechanism') was not an obligatory effect of the passive joint characteristics, but a direct result of the external loads. Anatomical differences notwithstanding, the inter-individual discrepancies in the motion patterns of the four specimens tested, showed to be relatively small in a qualitative sense. Quantitative differences were explained by small differences in the alignment of the coordinate systems relative to the joint anatomy and by differences in rotatory laxity. III. Recruitment of knee-joint ligaments On the basis of the kinematics of the passive knee-joint motions of the four knee specimens, the length changes of ligament fiber bundles were determined by using the points of insertion on tibia and femur. Like the kinematic data, the insertions of the ligaments were obtained by using Roentgenstereophotogrammetry. Different fiber bundles of the anterior and posterior cruciate ligaments and the medial and lateral collateral ligaments were identified. On the basis of an assumption for the maximal strain of each ligament fiber bundle during the experiments, the minimal recruitment length and the probability of recruitment were defined and determined for the motions and loading conditions described previously. The ligament length and recruitment patterns were found to be consistent for some ligament bundles and less consistent for other ligament bundles. The most posterior bundle of each ligament was recruited in extension and the lower flexion angles, whereas the anterior bundle was recruited for the higher flexion angles. External rotation generally recruited the collateral ligaments and internal rotation recruited the cruciate ligaments. However, the anterior bundle of the posterior cruciate ligament was recruited with external rotation at the higher flexion angles. At the lower flexion angles, the anterior cruciate and the lateral collateral ligaments were recruited with an anterior force. The recruitment of the posterior cruciate ligament with a posterior force showed that neither its most anterior nor its most posterior bundle was recruited at the lower flexion angles. Hence, the posterior restraint must have been provided by the intermediate fiber bundles, which were not considered in the experiment. At the higher flexion angles, the anterior bundles of the anterior cruciate ligament and the posterior cruciate ligament were found to be recruited with anterior and posterior forces, respectively. The minimal recruitment length and the recruitment probability of ligament fiber bundles can be useful parameters for the evaluation of ligament length changes in those experiments where no other method can be used to determine the zero strain lengths, ligaments strains and tensions. IV. Helical axes of passive knee-joint motions Finite helical axes were determined for step-by-step flexion motions of the knees tested. The descriptive value of the finite helical axes was evaluated with respect to consistency and reproducibility. In addition to the original four knees, a fifth specimen was used to study the effects of some of the experimental conditions on the axes parameters. The axes were determined for flexion motions along the envelope of passive motion, i.e. flexion motion with either an internal or an external torque of 3 Nm on the tibia. The positions and orientations of the axes were described relative to the insertions of the four major ligaments and the geometry of the articular surfaces of the femur, and also as intersections with a medial and a lateral sagittal plane. The three-dimensional patterns of the helical axes of the four knee specimens were found to be highly reproducible and consistent for each of the two motion pathways. The axes patterns were not unique, but reflected the particular combination of flexion and axial rotation for each particular motion pathway. Although small, the helical translations indicated medial motions of the tibia relative to the femur. This medial helical translation was more pronounced for the internal pathway as compared to the external pathway. V. Articular contact in a three-dimensional model of the knee An analysis was made of articular contact in a three- dimensional mathematical model of the human knee-joint. In particular the effect of articular contact on the passive motion characteristics was assessed in relation to experimentally obtained joint kinematics. Two basically different mathematical contact descriptions were compared for this purpose. One description was for rigid contact and one for deformable contact. The description for deformable contact was based on a simplified theory for contact of a thin elastic layer on a rigid foundation. The articular cartilage was either described as a linear elastic material or as a non-linear elastic material. The contact descriptions were introduced in a mathematical model of the knee. The locations of the ligament insertions and the geometry of the articular surfaces were obtained from a joint specimen of which experimentally determined kinematic data was available, and were used as input for the model. The ligaments were described by non-linear elastic line elements. The mechanical properties of the ligaments and the articular cartilage were derived from literature data. Parametric model evaluations showed that relative to rigid articular contact, the incorporation of deformable contact did not alter the motion characteristics in a qualitative sense and that the quantitative changes were small. Variation of the elasticity of the elastic layer revealed that decreasing the surface stiffness caused the ligaments to relax and, as a consequence, increased the joint laxity, particularly for axial rotation. The difference between the linear and the non-linear deformable contact in the knee model was very small for moderate loading conditions. The motion characteristics simulated with the knee model compared very well with the experiments. It is concluded that for simulation of the passive motion characteristics of the knee the simplified description for contact of a thin linear elastic layer on a rigid foundation is a valid approach when aiming at the study of the motion characteristics for moderate loading conditions. With deformable contact in the knee model, geometric conformity between the surfaces can be modelled as opposed to rigid contact which assumed only point contact. VI. Ligament-bone interaction in a three-dimensional model of the knee In mathematical knee-joint models, the ligaments are usually represented by straight line elements, connecting the insertions of the femur and tibia. Such a model may not be valid if a ligament is bent in its course over bony surfaces, particularly not if the resulting redirection of the ligament force has a considerable effect on the laxity or motion characteristics of the knee-joint model. In the present study, a model for wrapping of a ligament around bone was incorporated in a three-dimensional mathematical model of the human knee. The bony edge was described by a curved line on which the contact point of the line element representing a ligament bundle was located. Frictionless contact between the ligament bundle and the bone was assumed. This model was applied to the medial collateral ligament (MCL) interacting with the bony edge of the tibia. It was found that, in comparison with the original model without bony interactions, the bony edge redirected the ligament force of the MCL in such a way that it counterbalanced valgus moments on the tibia more effectively. The effect of the bony interaction with the MCL on the internal-external rotation laxity, however, was negligible. VII. Parametric validation of a three-dimensional model of the knee Validation of three-dimensional mathematical models of the human knee was previously based on comparisons with general motion or laxity characteristics as reported in literature. In this study, the validation of the knee model was performed through a direct specimen-related comparison with the passive motion characteristics of four joints from which the geometry data was used as input of the model. The knee model was quasi- static and was based on equilibrium of internal and external loads. The femur was considered to move relative to the tibia and both bones were connected through non-linear elastic line elements representing the ligament fiber bundles and by linear elastic articular contacts. From each joint the passive freedom- of-motion characteristics, the insertions of ligament fiber bundles and the geometry of the articular surfaces were available. The mechanical properties of the ligaments and articular cartilage were estimated on the basis of literature data. Since virtually no data was available on the so-called reference strain in the ligament bundles, which is the ligament strain for the reference extension position of the joint, an optimization method was used to estimate this parameter on the basis of a comparison between the kinematics of the model and the kinematics of the experiments. A good match between the model and the experimental kinematics could be obtained, for one specimen better than for the other. The validation process provided the means for assessing the quality of each individual knee model relative to the experimental data. The main limitation of the knee model in its present form is the limited representation of the cruciate ligaments by only two line elements. A confrontation of the four joint models with the anterior-posterior laxity values as reported in the literature, revealed a good agreement at both 20 and 90 degrees flexion, although for each individual joint some underestimation or overestimation of the laxity occurred. Despite the gross simplifications relative to the complex anatomy of the knee, the present knee model can realistically simulate the passive motion characteristics of the human knee-joint. With this specimen- based validation, the three-dimensional mathematical model of the human knee has matured to a powerful simulation tool for the evaluation of diagnostic and surgical procedures in knee-joint surgery. VIII. Rotation restraints in the knee joint Ligament function in restraining axial rotation of the tibia relative to the femur can not be revealed by the analysis of ligament recruitment or ligament forces alone. The action of the articular surfaces should be taken into account as well. A three-dimensional mathematical model of the human knee-joint was used to simulate axial rotation, whereby the forces in the ligaments and articular contact were calculated together with their contribution to the restraint moment, which was required to maintain an internally or an externally rotated position. In external rotation, the direct axial restraint was provided for by the collateral ligaments. In internal rotation, where predominantly the cruciate ligaments were loaded, the direct restraint moment resulting from the ligament forces was not sufficient to balance the applied moment. The articular contact forces, as caused by the ligaments compressing the joint surfaces together, contributed considerably to the internal rotation restraint. Depending on the flexion angle, the contact forces provided for approximately 50% to 85% of the internal rotation restraint, whereas the external rotation restraint was by 95% to 100% accounted for by the ligament forces. IX. ACL reconstruction: simply a matter of isometry? The restoration of normal knee laxity after surgical reconstruction of the anterior cruciate ligament (ACL) depends on a number of technical factors, of which the location of the graft insertion is considered of importance. This study addresses the requirement for isometry of the tibial and femoral graft insertions. `Isometry' refers to a constant length during knee flexion, implying that tension variations will be small during flexion. The question is whether an isometrically placed ACL substitute will result in a restoration of normal knee laxity. This question is addressed in this study by using a computer model of the knee in which the ACL reconstruction procedure was simulated. For 11 different femoral insertion locations, graft tension, anterior laxity, internal-external rotation (IE) laxity and the anterior-posterior malpositioning of the tibia relative to the femur (AP error) were evaluated after the ACL reconstruction, whereby the two-bundled ACL was replaced by a singled bundled graft. The model simulations were performed twice for each alternative femoral insertion, one with the graft stiffness equal to the ACL stiffness and one with the graft stiffness equal to half the ACL-stiffness. The result of the ACL reconstruction was far more sensitive to the femoral insertion location than to the graft stiffness. Anterior-posterior variations of the graft insertion were more critical than inferior-superior variations. ACL reconstruction was not simply a matter of isometry, since both isometric and non-isometric placement could lead to normal or nearly normal anterior laxity. DISCUSSION It is evident that the knee is a complex mechanical system. A certain level of abstraction is needed to understand the complex relationships between forces and motions. This abstraction is provided for by the three-dimensional computer simulation model of the knee. In the model, the knee is stripped to its essential structures, i.e. the articular surfaces of femur and tibia with deformable contact, and the four major ligaments represented by nonlinear elastic line elements. This limited number of structural elements make the model comprehensible as a mechanism, without violating the essential characteristics of the system it represents, where it concerns its passive motion feasibilities. The strength of the present knee model as compared to its predecessors, is its firm quantitative basis on experimental data. The anatomic variations between the knees are dealt with by using the geometry of the articular surfaces and the ligaments of each knee as input for the model and subsequently evaluating the validity of the four knee models, by a comparison of the motion characteristics of each individual knee model with the experimental motion data from the same knee. When applying the knee model to biomechanical or clinical problems, there are now models of different knees available, with which at least an impression can be obtained of the effects of anatomic variability on the behavior of the knee. There are limitations of the knee model, in particular with respect to the loading conditions and the role of the menisci. The knee model was designed to simulate the quasi-static passive motion characteristics of the knee for low or moderate loading conditions, much the same way as clinicians evaluate knee laxity in different directions. In some in vitro experiments, where the joint is loaded in combination with many degrees of freedom of motion, the loading conditions can only be moderate because of the difficulty of controlling the experiment for high loads and the weakness of the mechanical connections between the loading frame and the joint. Also here the model can be applied as a simulation tool. The model can not be applied, or only to a limited extent, for high or dynamic loading conditions as for example during gait or running where loads on the knee can rise to several times body weight. For these conditions the menisci play a considerable role in stabilizing the joint and in transferring the load between tibia and femur. Also the mechanics of load transfer through the articular surfaces can not be addressed in detail by the knee model because of the simplified elastic representation of the articular cartilage, neglecting the time-dependent bi-phasic behavior, and the absence of the menisci. The knee model, in fact, represents a meniscectomized knee, in which the role of the menisci in determining the passive motion characteristics is taken over by the ligaments to some extent. The validation method as applied by using the optimization technique, offers the possibility for parametric analyses while applying the knee model to address biomechanical and clinical problems and for developments of future enhancements of the knee model. Two examples of the application of the knee model were already given in chapters VIII and IX. Other possible applications can be considered. For instance, the evaluation of artificial ligaments, the effects of pretension of an artificial ligament on the laxity characteristics, the consequences of the (mal)positioning of knee prostheses in which the goal is to preserve the ligaments of the knee after placement of the prosthesis. In fact, due to the absence of menisci in knee prostheses and the relatively well defined mechanical properties of their bearing surfaces, the knee model would be excellent to study the effects of joint surface geometry on laxity, contact surface areas and stresses, friction and wear. These alternative applications for the knee model refer to direct clinical questions with respect to surgical problems. The knee model can also be used to prepare and define experimental studies. Through parametric analyses, the most important experimental parameters can be determined before the actual experiments take place. The experiment can then be designed optimally, such that the extent of the experiment can be limited to only the most important aspects of the problem. For example, when evaluating the consequences of the positioning of the substitute in an ACL- reconstruction procedure by in vitro experiments, the range of the variations of insertions and pretensions can be limited to those values which can be expected to give satisfactory results, as based on the model analyses. Further development of the knee model may include a representation of the muscles, through active line elements, and an improvement of the ligament representation of the ligaments. By using the same mathematical formulations as for the tibiofemoral joint, the patellofemoral joint can be included in the model, thus giving the possibility to analyze the effects of the quadriceps muscle on joint stability. The enhancement of the ligament representation, for example by increasing the number of line elements per ligament, should be performed through experiments guided by model evaluations, again integrating the knee model with experiments. In conclusion, with its strength in some aspects and limitations in others, the three-dimensional mathematical model of the human knee, as implemented in a computer program, offers great opportunities in advancing the field of knee biomechanics and our knowledge of the basic mechanisms involved in clinical evaluation of knee joint injuries, surgical treatment and knee prosthesis design.