Injuries to the knee ligaments are common, accounting for a large proportion of knee injuries.1 The biomechanical properties of knee ligaments are extremely important because this information can provide orthopedic surgeons with a theoretical basis for the prevention, diagnosis, and treatment of ligament deficiencies.
Simulation of knee flexion is important for the biomechanics of ligaments and can be a challenge with computational methods. Many finite element analyses have used models of the knee joint under single flexion angles and have shown limitations that cannot simulate knee movement.2–4 Some studies achieved different knee flexion angles by rotating the femur or tibia along 1 axis on the sagittal plane but were not precise enough to replicate the complex movement of the knee joint.5,6
To simulate this complex motion, some models have used force or torque applied in the mechanical testing of cadaver specimens.7 The motion of the knee joint was driven by building complete lower limb muscles and bones and using the muscle strength of the lower limbs iteratively calculated from gait data.8–10 An alternate method used a motion capture system, a force plate, and an inverse dynamics method to obtain muscle force and joint contact force, with input of these into finite element models.11,12
Some researchers developed displacement-driven models, and some of them used computed tomography or magnetic resonance imaging scans of the knee joint at several positions. Tibial models of position were registered and fixed, and movements of femoral models were recorded.13,14 Other researchers used flexion-extension, internal-external, and varus-valgus movements of the femur and did not consider translational movements.15,16 These displacement-driven models depended on traditional imaging and did not reflect physiologic weight-bearing positions.
Fluoroscopic imaging has been used widely to acquire knee kinematic data.17 Because the flexion axis is important in knee kinematic measurement, researchers used this technique to study the knee flexion axis.18–21 An in vivo study showed that the geometric center axis (GCA) was closer to the functional flexion axis than to the other flexion axis and could offer an alternative way to determine femoral rotation during lunge.22 Other investigators used this technique to observe morphologic changes in ligaments, such as elongation,23,24 or to measure ligament displacement patterns and calculate ligament strain.25 To date, no study has combined fluoroscopic kinematic data and finite element analysis to research the stress state of knee ligaments.
To investigate the biomechanical behavior of ligaments based on in vivo fluoroscopic kinematic data, the current authors used GCA to represent and quantify knee kinematics, and they conducted a displacement controlled finite element analysis of the knee joint during lunge.
Materials and Methods
The GCA was seen as the functional flexion axis. It has almost no translational degrees of freedom in the vertical direction and almost no rotational degrees of freedom in the forward-backward direction. Therefore, the GCA can be considered to have only 4 degrees of freedom during knee flexion. This study used lunge simulation because it offers a larger range of flexion angles and fluoroscopic kinematic data are easier to obtain during lunge motion.
Setting up the Geometric Center Axis
The geometric model of the knee joint was imported into modeling software (Materialise Mimics; Materialise NV). Two virtual balls were inserted and adjusted until they were fully fitted to the femoral condyles (Figure 1A). The coordinates of the ball centers were recorded.
The geometric center axis was set up by inserting 2 balls fitted to the femoral condyles (a). Equations for translation of the medial condyle in the anterior-posterior (A-P) and proximal-distal (PD) directions (b, left). Equation for the tibia internal rotation (b, right).
Equation for Degrees of Freedom
The current study used fluoroscopic data for the GCA, based on the research of Feng et al.20,21 In their studies, the average flexion angle during lunge was nearly 120°. Relative to the starting position, the medial condyle first moved 9.4 mm forward from full extension to 60° flexion and then moved 3.1 mm backward to 120° flexion. Simultaneously, the medial condyle moved 2.6 mm proximally from full extension to 30° flexion and then moved 1.9 mm distally to 120° flexion. The tibia was continuously rotating internally to an average angle of 26.8°. The current authors used these data to establish equations for the flexion angle and the other 3 degrees of freedom. Positive rotation was defined by the right-hand rule. The forward and distal directions were considered positive directions, and piecewise linear equations were used (Figure 1B).
Finite Element Model
The bone and ligament models used in this analysis were obtained from the Open Knee project ( https://simtk.org/projects/openknee) and were verified and used widely.3,26 Different bundles or portions of ligaments were divided and considered individually.
Bones are considered as shell and defined as rigid body. Ligaments are considered as solid and defined as nearly incompressible, transversely isotropic hyperelastic neo-Hookean materials with the strain-energy function (Figure 2).27
The strain-energy function, where C1 is the neo-Hookean constant and D is the inverse of the bulk modulus k=1/D.
The material properties of ligaments are shown in Table 1. Bonded contact between the ligaments and bones and frictionless contact between the anterior cruciate ligament (ACL) and posterior cruciate ligament (PCL) were defined.
Material Properties of Ligaments
The tibial coordinate system was defined as the global coordinate system. The original point of the femoral coordinate system and the femur reference point was the medial endpoint of the GCA. A coordinate axis of the femoral coordinate system was overlapped with the GCA, and the other 2 coordinate axes were free because they were not used in this study. The tibia reference point was the projection point of the medial endpoint of the GCA on the transverse plane of the global coordinate system. In finite element analysis software (Abaqus FEA; Dassault Systèmes SE), flexion rotation was applied to the femur; anterior-posterior translation, distal-proximal translation, and internal rotation were applied to the tibia (Figure 3). These movement data were obtained from the previously discussed equations for degrees of freedom. Changes in the length and stress state of the ACL, PCL, lateral collateral ligament (LCL), and medial collateral ligament (MCL) at every 5° of knee flexion were computed.
Displacement and rotation are applied to the finite element model: A, medial endpoint of the geometric center axis (femur reference point); B, lateral endpoint of the geometric center axis; C, projection point of A (tibia reference point); A-T, anterior translation; P-T, posterior translation; p-t, proximal translation; d-t, distal translation; F-R, flexion rotation; I-R, internal rotation (a). X-Y-Z coordinate system is the global coordinate system; y axis of the x-y-z coordinate system is overlapped with the geometric center axis (b).
The models at different flexion angles (5°, 30°, 60°, 90°, and 120°) are shown in Figure 4. The length of the double bundles of the ACL decreased gradually during knee flexion. The length of the anteromedial portion of the ACL decreased gradually to 120° flexion by approximately 16.2%, whereas the length of the posterolateral portion of the ACL decreased from full extension to 120° flexion by approximately 23.7%. The length of the double bundles of the PCL first increased and then decreased when the maximum length appeared at approximately 60° flexion. At 60° flexion, the length of the anterolateral and posteromedial portions of the PCL increased 12.2% and 15.4%, respectively. The length of the anterior-posterior portion of the MCL increased from full extension to 40° flexion by approximately 7.7%. Beyond 80°, the length of the anterior-posterior portion of the MCL decreased consistently and reached 16.7% at 120° flexion. The length of the medial-lateral portion of the MCL increased from full extension to 25° flexion by approximately 2.6%. Beyond 50°, the length of the medial-lateral portion of the MCL decreased and reached 25.3% at 120° flexion. The length of the proximal-distal portion of the MCL was reduced consistently along the flexion path and reached 31.0% at 120° flexion. The length of the anterior-posterior portion of the LCL was almost constant from full extension to 25° flexion. Beyond 25°, the length of the anterior-posterior portion of the LCL increased consistently and reached 11.0%. The length of both the medial-lateral and proximal-distal portions of the LCL decreased consistently, with maximum rates of change of 7.5% and 17.7%, respectively, at approximately 90° flexion. Length change rates of 30°, 60°, 90°, and 120° flexion were compared with similar studies.23,24,28–30 Predictions were in general agreement with the results of earlier studies (Figures 5–6).
Anterior (top) and lateral (bottom) views of models at several specific flexion angles.
Length of different bundles or portions of the anterior cruciate ligament (a), posterior cruciate ligament (b), medial collateral ligament (c), and lateral collateral ligament (d) at different flexion angles. Abbreviations: AL, anterolateral; AM, anteromedial; AP, anterior-posterior; MP, medial-lateral; PL, postero-lateral; PM, posteromedial; PP, proximal-distal.
Comparison of the length change rate of different bundles or portions of the anterior cruciate ligament (ACL) (a), posterior cruciate ligament (PCL) (b), medial collateral ligament (MCL) (c), and lateral collateral ligament (LCL) (d) at different flexion angles. Abbreviations: AL, anterolateral; AM, anteromedial; AP, anterior-posterior; MP, medial-lateral; PL, posterolateral; PM, posteromedial; PP, proximal-distal.
Figure 7 shows the stress distribution of each ligament at different flexion angles (5°, 30°, 60°, 90°, and 120°). The stress concentration point of the ACL was transferred from the body of the posterolateral bundle to the body of the anteromedial bundle and finally appeared on the femoral root of the anteromedial bundle. The posteromedial bundle of the PCL was under consistently high stress. The maximum stress on the ACL, MCL, and LCL increased gradually during knee flexion and was approximately 6.2 MPa, 3.3 MPa, and 5.6 MPa, respectively, at 120° flexion. The maximum stress on the PCL was similar to that for the other ligaments from full extension to 50° flexion, and then it increased dramatically from 50° to 60° flexion to the maximum stress of approximately 8.6 MPa at 75° flexion. After that, it decreased slightly and reached 5.8 MPa at the end. Similarly, the increasing rate of stress on the ACL, MCL, and LCL increased gradually from the beginning and reached approximately 28.6%, 11.0%, and 14.2%, respectively, at 120° flexion. The maximum stress on the PCL increased from full extension to 50° flexion by approximately 8.5%. Then it increased sharply from 50° to 60° flexion and had the maximum rate of change of approximately 18.3% at 75° flexion (Figure 8).
Stress distribution and stress concentration point of each ligament at 5°, 30°, 60°, 90°, and 120° of flexion. Abbreviations: ACL, anterior cruciate ligament; LCL, lateral collateral ligament; MCL, medial collateral ligament; PCL, posterior cruciate ligament.
Maximum von Mises stress (a) and stress change rate (b) for each ligament during knee flexion. Abbreviations: ACL, anterior cruciate ligament; LCL, lateral collateral ligament; MCL, medial collateral ligament; PCL, posterior cruciate ligament.
The authors performed displacement controlled finite element analysis of the knee joint to simulate physiologic lunge activity. Because it is difficult to record the precise stress state of knee ligaments during lunge with in vivo or cadaveric studies, the current results were validated by the data on the change in ligament length in previous in vivo and cadaveric studies.23,24,28–30 The change in length of the knee ligaments was similar to that reported in previous studies, which verified the kinematic accuracy of the model. In this study, the length of the double bundles of the ACL decreased gradually, which was consistent with the literature.31 The length of the double bundles of the PCL increased at first and then decreased when the maximum length appeared at 60° flexion. The literature reported that the length of the double bundles of the PCL increased gradually from 0° to 90° flexion.23 For the MCL and LCL, the length of the anterior portion increased, the length of the middle portion did not change significantly, and the length of the posterior portion decreased gradually. These findings were consistent with previous results.24
The stress concentration point on the ACL was transferred from the body of the posterolateral bundle to the body of the anteromedial bundle and finally appeared on the femoral root of the anteromedial bundle. With the increase in flexion angle, the posterolateral bundle gradually relaxed and the anteromedial bundle gradually tensed.32 The results of this study were consistent with this conclusion. In addition, stress concentration on the anteromedial bundle increased more sharply with the increase in flexion angle, which means that the anteromedial bundle of the ACL was more easily damaged at high flexion. Partial injuries of the ACL account for nearly half of all knee ligament injuries.33 Clinicians may pay greater attention and treatment may have the best outcomes for these patients, who have a greater demand for forward lunge activity. The maximum stress of the ACL, MCL, and LCL increased gradually and showed similar trends for range of motion. The maximum stress of the PCL was similar to that of the other ligaments from full extension to 50° flexion, and then it increased significantly from 50° to 60° flexion, when maximum stress appeared at 60° flexion. A previous study reported that PCL loads were higher than ACL loads during isokinetic or isometric flexion and were greater after 50° flexion during squats.34 Papannagari et al35 noted the importance of the posteromedial bundle of the PCL in resisting posterior tibial translation during lunge. In this study, the high stress on the posteromedial bundle of the PCL can be explained by the kinematics of the knee joint. During lunge, the tibia rotates internally and has a backward trend relative to the femur. The PCL underwent a greater change in length. All of these factors increase the stress level on the PCL. Soft tissue balance is crucial in total knee arthroplasty and has not been indicated beyond 90° flexion in previous total knee arthroplasty procedures.36 The current finding shows that the stress state of the MCL and LCL did not completely coincide, showing a greater difference with the increase in flexion angle. This finding provides valuable information for soft tissue balancing at a higher flexion angle during total knee arthroplasty.
Several methods are used to study kinematics through skin and soft tissue, and some errors occur.37,38 Radiostereometric analysis is highly invasive, involving dual radiographs and the insertion of tantalum beads into bone.39 Computed tomography and magnetic resonance imaging do not consider the effect of gravity because of the supine position. Fluoroscopic imaging solves these problems and eliminates the soft tissue error, and this technique was widely used to study the spatial position of the GCA during lunge.
Earlier studies used displacement-driven finite element models to analyze soft tissue stress in the knee joint.40 The boundary condition was extracted from cadaver studies, which were obviously different from in vivo studies. Fluoroscopic imaging can determine the in vivo movement of the knee joint under physiologic loading quickly and accurately. The spatial position of the flexion axis accurately describes the motion track of the femoral condyles. A combination of these approaches can be used to develop a displacement-driven finite element model to analyze the biomechanics of the knee joint under physiologic loading.
Finite element models of the knee joint involving different flexion angles are mostly driven by forces or torques. Researchers use these complex finite element models because they are predictive of knee biomechanics and can be used to study the effect of changing loading conditions on knee kinematics. The process of calculating and simulating lower limb muscle strength is particularly complex. Displacement controlled finite element analysis provides a way to avoid these limitations. The method used in the current study is useful because displacement data are obtained directly from the image registration models. If the motion of the knee joint can be obtained accurately with imaging, the steps of muscle force simulation do not need to be considered and the modeling can be simplified.
This study had some limitations. Only 4 degrees of freedom of the GCA were considered, and 2 degrees of freedom with small amplitude were ignored. The equations for degrees of freedom were linear, but the variation for each degree of freedom for the knee joint was not uniform. Initial prestressing of ligaments was not considered. The individual model used the average kinematic data. In the future, 6 degrees of freedom of the knee joint could be considered and the biomechanical behavior of the knee joint could be evaluated individually.
A displacement controlled finite element analysis of the knee joint based on fluoroscopic kinematic data for the GCA was conducted to investigate the biomechanics of knee ligaments. Significant differences were found in the elongation and stress state of knee ligaments during lunge. This study may help surgeons to understand the functionalities of knee ligaments and improve the methods for finite element analysis simulation in orthopedic research.
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Material Properties of Ligamentsa
|Part||C1||D||Type of element|
|Anterior cruciate ligament||1.95||0.00687||Hexahedral element, C3D8RH|
|Posterior cruciate ligament||3.25||0.00410||Hexahedral element, C3D8RH|
|Medial collateral ligament||1.44||0.00126||Hexahedral element, C3D8RH|
|Lateral collateral ligament||1.44||0.00126||Hexahedral element, C3D8RH|