Establishment of the knee joint model
The knee joint is one of the most complex joints in the human body, with a complex anatomic structure and biomechanical properties. The traditional mechanical method to study its biomechanical functions usually involves the application of extra-articular loads and use of mechanical measuring instruments , and it is difficult to investigate the stress distribution within the joints and other issues using this method. Therefore, establishing knee joint models such as the crossed four-link physical model and the two-dimensional mathematical model of the sagittal knee joint as well as the three-dimensional model of dynamic response of the knee joint has become an important measure for further studying the biomechanical characteristics of knee joints . Since its first application in orthopedic biomechanics by Brekelmans et al.  in 1972, the FEM has been widely used in modeling teeth, artificial limbs, spine, etc.  and has been gradually applied to the biomechanics of ankles, knees, wrists, and other joints [13, 14].
Single-mode CT or MR images are typically unable to provide a clear contrast for intact knee joints, leading to difficulty in accurately constructing an FEM of knee joints containing multiple anatomic structures. Studies have found that although CT image data alone can be used to accurately construct bone structure models, it cannot be used to accurately simulate the cartilage, ligament, meniscus, and other soft tissues . In contrast, MR imaging data alone can be used to accurately construct the anatomic structure models of various soft tissues including knee joints, while it cannot be used to accurately simulate bone structures . Thus, using CT or MR imaging alone will significantly decrease the accuracy of these models, leading to inaccurate mechanical analysis of the knee joints.
Yao et al.  accurately constructed FEMs of the femoral cartilage, tibial cartilage, and medial meniscus using MATLAB and Hypermesh software, but not of other structures of knee joints. Therefore, a single software often has some limitations, and the constructed FEM fails to truly represent the anatomic characteristics of knee joints; thus, the FEMs of knee joints can be accurately constructed only by collaborative application of a variety of modeling software.
In this study, a variety of modes of CT and MR imaging as well as MIMICS 14.0, Geomagic Studio 12.0, and ANSYS modeling software were applied to construct a three-dimensional FEM of knee joints using the reverse engineering (RE) principle. In the MIMICS software, the structures in CT and MR images were assembled according to the human anatomy. However, this assembled model was very coarse due to the presence of interference surfaces. This issue could be addressed using Geomagic Studio 12.0, in which the interference surfaces in the model were removed. However, this software was not able to generate ANSYS pre-processing files; therefore, the repaired models were imported again into the 3-matic module of MIMICS software for initial meshing before being imported into ANSYS for finite element analysis (FEA). In addition, since this study focused on the mechanical analyses of the ligaments, their mesh size was refined at 1 mm, and the bone structures were set as a solid body with a mesh size of 4 mm, reflecting the different focuses of subjects in FEA. Because the meshing quality determined the accuracy of the FEA results, an even finer mesh would be required to analyze non-linear contacts. Local meshes with poor quality were optimized, and interactive meshes with satisfactory 6-node pentahedrons and 8-node hexahedrons were obtained based on the high-quality area meshes, in order to achieve more accurate results compared to those obtained using meshes with 10-node tetrahedron elements. Ultimately, a three-dimensional model of the human right knee joint containing a variety of anatomic structures, such as the middle and upper segments of the femur, middle and upper segments of the tibia, fibula, patella, meniscus, ACL, PCL, MCL, lateral collateral ligament, and patellar ligament, was constructed. Meanwhile, high-quality volume meshes were developed, satisfying the requirements for FEA of biomechanics of knee joints. This FEM can be used to analyze the stress distribution of ligaments, contacts of tibiofemoral joints, stress distribution on articular surface, changes of stress distribution under different ligament deficiencies, and other biomechanical studies, as well as to simulate the effects of surgical results on the biomechanics of knee joints under different surgical conditions, and conduct biomechanical analyses of surgical fixations.
Biomechanical analyses of medial collateral ligament of the knee joint
The anatomy of MCL has been extensively studied . In this experiment, based on the human anatomy, an FEM of the knee joint was established to simulate the anterior-posterior translation, valgus-varus rotation, and internal-external rotation of the knee joint, so as to study the biomechanical functions of its superficial and deep MCLs, in which the knee joint varus was excluded because the knee MCL is completely relaxed in this condition. In the experiment, a gradually increasing color grading from blue to red color indicated gradually increasing von Mises stress, which represented a greater load on the ligament and a greater role of the site and likelihood of damage.
Under the load of the 134-N forward force, the tibial displacement changed from 4.89 mm at intact MCL to 5.17, 5.04, and 5.17 mm at SMCL deficiency, DMCL deficiency, and overall MCL deficiency, respectively. A greater variation of tibial displacement at overall MCL deficiency indicated that MCL plays a role in limiting the forward translation of the tibia. Meanwhile, the tibial displacement showed a greater variation at SMCL deficiency compared with that at DMCL deficiency, suggesting that SMCL has a greater effect than the DMCL. During this process, the stress at ACL maintained a maximum value, suggesting that ACL plays the most important role in limiting the tibial anterior translation. Moreover, a greater stress at SMCL than that at DMCL indicated that the SMCL has a greater effect. The stress nephogram showed that the peak stresses at ACL and SMCL were mainly located at the femoral end point, indicating that during tibial anterior translation, injury to the femoral end point is most likely to occur at ACL and SMCL, and less likely to occur at DMCL.
Under the load of the 134-N backward force, the tibial translation showed a very small variation with MCL deficiency, during which the stress at PCL maintained a maximum value, while the stresses at SMCL and DMCL were relatively small, suggesting that PCL plays the most important role in constraining the tibial posterior translation, while the effects of SMCL and DMCL are very small. Meanwhile, the peak stress at PCL occurred at the tibial start and end points, suggesting that in tibial posterior translation, injury is most likely to occur at the femoral start and end points at PCL, while the risk of injury at SMCL and DMCL is small.
Under the load of the 10-N m valgus torque, the tibial valgus angle showed a large variation with MCL deficiency, which changed from 4.06° at intact MCL to 6.08°, 4.86°, and 6.22° at SMCL deficiency, MCL deficiency, and overall MCL deficiency, respectively, suggesting that MCL tends to resist the valgus motion of knee joints. Meanwhile, the stress was the largest at SMCL followed by that at DMCL, indicating that SMCL plays the most important role in limiting the valgus motion and the effect of DMCL is relatively smaller. As evident in the stress nephogram, the peak stress at SMCL occurred at the end point and anterior part of the femur, indicating that injury is most likely to occur at the end point and anterior part of the femur in valgus motion of knee joints at SMCL. In contrast, the peak stress at DMCL occurred at the femoral start and end points, suggesting that they are prone to injury at DMCL.
Under the load of the 10-N m external rotation torque, the tibial external rotation angle changed from 5.92° at intact MCL to 5.95°, 5.94°, and 6.10° at SMCL deficiency, DMCL deficiency, and overall MCL deficiency, respectively. The tibial external rotation angle showed a large variation at overall MCL deficiency, suggesting that MCL tends to resist the external rotation of the knee joints. Although the tibial external rotation angle did not show significant difference between SMCL and DMCL deficiencies, the stress at SMCL was larger than that at DMCL, indicating that SMCL plays a more significant role in limiting the external rotation of the knee joint than the DMCL. As observed in the stress nephogram, the peak stress at SMCL was mainly located at the femoral end point and posterior part, indicating that they are prone to injury at SMCL during external rotation of knee joints, while the injury at DMCL was smaller.
Under the load of the 10-N m internal rotation torque, the tibial internal rotation angle changed from 6.64° at intact MCL to 7.48°, 6.72°, and 7.57° at SMCL deficiency, DMCL deficiency, and overall MCL deficiency, respectively. The tibial internal rotation angle showed a larger variation than the tibial external rotation angle with MCL deficiency, suggesting that knee joint MCL has a greater effect on limiting the internal rotation than the external rotation. Similarly, greater stress at SMCL than that at DMCL indicated that the SMCL has a greater effect on limiting the internal rotation of knee joints than the DMCL. As observed in the stress nephogram, the peak stress at SMCL occurred at the femoral end point, indicating that the femoral end point was prone to injury at SMCL during internal rotation of knee joints, while the injury at DMCL was smaller.
The above analyses show that in the extended position of knee joints, the main effect of MCL is to resist the valgus motion of knee joints, along with limiting the tibial forward displacement as well as the internal and external rotations of knee joints. The SMCL plays the most important role in the structure of the MCL of knee joints, while the effects of DMCL are relatively lesser. In various motions of knee joints, the femoral end point at SMCL is the most prone to injury. The anterior part of the femur is more prone to injury in resisting valgus motion, and the posterior part in resisting external rotation at SMCL. However, injury is less likely to occur at DMCL, and when it does occur, it occurs at the femoral start and end points.