Finite Element Analysis of Elbow Joint Stability by Different Flexion Angles of the Annular Ligament

Objective The injury of the annular ligament can change the stress distribution and affect the stability of the elbow joint, but its biomechanical mechanism is unclear. The present study investigated the biomechanical effects of different flexion angles of the annular ligament on elbow joint stability. Methods A cartilage and ligament model was constructed using SolidWorks software according to the magnetic resonance imaging results to simulate the annular ligament during normal, loosened, and ruptured conditions at different buckling angles (0°, 30°, 60°, 90°, and 120°). The fixed muscle strengths were 40 N (F1), 20 N (F2), 20 N (F3), 20 N (F4), and 20 N (F5) for the triceps, biceps, and brachial tendons and the base of the medial collateral ligament and lateral collateral ligament. The different elbow three‐dimensional (3D) finite element models were imported into ABAQUS software to calculate and analyze the load, contact area, contact stress, and stress of the medial collateral ligament of the olecranon cartilage. Results The results showed that the stress value of olecranon cartilage increased under different conditions (normal, loosened, and ruptured annular ligament) with elbow extension, and the maximum stress value of olecranon cartilage was 2.91 ± 0.24 MPa when the annular ligament was ruptured. The maximum contact area of olecranon cartilage was 254  mm2 with normal annular ligament when the elbow joint was flexed to 30°, while the maximum contact area of loosened and ruptured annular ligament was 283 and 312 mm2 at 60° of elbow flexion, and then decreased gradually. The maximum stress of the medial collateral ligament was 6.52 ± 0.23, 11.51 ± 0.78, and 18.74 ± 0.94 MPa under the different conditions, respectively. Conclusion When the annular ligament ruptures, it should be reconstructed as much as possible to avoid the elevation of stress on the surface of the medial collateral ligament of the elbow and the annular cartilage, which may cause clinical symptoms.


Introduction
T he elbow is a complex joint, and its stability plays an important role in maintaining daily activities. Elbow instability is caused by damage to the bone joint surface and the ligament structure of the elbow joint. It is a common disease secondary to acute fracture dislocation and chronic exercise strain 1,2 . Elbow dysfunction is often accompanied by ligament damage, which results in elbow instability. The medial collateral, lateral collateral, and annular ligaments maintain the stability of the elbow 3-5 . Frangiamore et al. 3 found that the ulnar collateral ligament was the primary stabilizer of valgus stress in the elbow and provided a clear understanding of the anatomical relationships of the static and dynamic stabilizers of the elbow. Wang et al. 4 demonstrated that surgical reattachment of the lateral collateral ligament (LCL) complex to the lateral epicondyle was helpful for elbow stability using relevant biomechanical studies. Anderson et al. 5 demonstrated the role of the annular ligament in providing proximal radial head stability based on knowledge of forearm biomechanics. The annular ligament is vulnerable to injury in cases of elbow dislocation or fracture, such as Montage's fracture, radial head fracture, and radial head dislocation, especially in severe cases. There is great controversy surrounding the treatment of the annular ligament. Some scholars believe that the elbow joint functions well without annular ligament repair [6][7][8] . Chen et al. 6 concluded that patients maintained elbow stability using a surgical strategy to reduce the radial head without annual ligament reconstruction for neglected radial head dislocation following Monteggia fracture dislocation. Han et al. 8 found that annular ligament repair was not essential in the operative treatment of isolated radial head fractures when the lateral collateral ligament was intact. Other scholars believe that annular ligament repair makes the elbow joint more stable and reduces the pain and injury caused by elbow joint instability in later periods 9-13 . Bhaskar 10 found that the need for annular ligament reconstruction was based on intraoperative findings of radial head instability for missed Monteggia fractures. Nwoko et al. 11 showed that incompetence of the annular ligament caused persistent instability of the proximal radius requiring ligament reconstruction. Therefore, further understanding of the role of the annular ligament in elbow joint stability is particularly important. Most biomechanical studies focused on the influence of elbow joint dislocation, deformity, and fracture on elbow joint stability, but the role of the annular ligament in maintaining elbow joint stability has been ignored, and there are few studies on the biomechanics of the annular ligament 14 . Tan et al. 14 constructed a finite element analysis (FEA) model of the ulna and radius with the annular ligament to simulate Monteggia fracture and explain the annular ligament pathology. Therefore, a considerable number of patients do not receive effective and timely treatment, and the late elbow joint instability that occurs due to the annular ligament injury is not given sufficient attention.
The present study intends to establish finite element models of elbow bone, medial and lateral collateral ligaments, and annular ligaments through CT and magnetic resonance imaging (MRI) images of elbow joint to simulate the mechanical effects of different states of annular ligaments (normal, loosened, and ruptured) on elbow joint tissues in the process of flexion, and to analyze the stress distribution among different tissues.

3D Elbow Joint Model Establishment
The ethics committee of the local institution approved this study, and all protocols were performed in accordance with relevant guidelines and regulations. Written informed consent was obtained from the participants. This model was validated in our previous research 15 . The surface grid editing tool in Geomagic 2013 software was used to analyze the 3D models of each part of the elbow. Necessary editing and modifications were made to the 3D model reconstructed from MRI scan data to make the model smoother and more compliant. A high-quality surface model was achieved. Closed-space non-uniform rational B-spline (NURBS) was simulated and exported in step format. 3D soft-tissue surface models based on two different modal data sets were registered for alignment in SolidWorks 2012 software. The positions of the medial collateral ligament (MCL), LCL, annular ligaments, and other 3D models reconstructed from the MRI scan data were converted to the CT scan data space, and the MCL and LCL, annular ligaments, articular surface cartilage, and other tissues were constructed using the direct modeling method. The model was imported into ABAQUS, and the ranges of bones, ligaments, and cartilage were defined differently based on the difference between the surface definition and the internal definition. The material parameters and attributes were assigned appropriately. The finite element mesh was divided into the model, and the finite element model was established, as shown in Figure 1. All of the structures were simulated using tetrahedral elements. The modeling period was greatly shortened by simplifying the bone tissue and articular surface cartilage in the model to homogeneous and isotropic elastomer material. To better reflect the biomechanical response of ligaments, ligaments were defined as linear elastic materials and used 3D solid elements according to relevant literature 16 . The material parameters are shown in Table 1.

Boundary and Loading Conditions
According to the previous literature 16,17 , the interaction between the cartilage surfaces was simulated using the penalty function method with implementation of "surface-tosurface contact." The cartilage thickness was assumed to be a constant 1 mm on all surfaces. The contact between cartilage and subchondral bone was also modeled. The coefficient of friction between the contact pairs was set to μ = 0.1. The contact area between the cartilage surface of the distal humerus and the olecranon was calculated in this paper. The MCL, LCL, and annular ligaments of the elbow joint played an important role in maintaining stability during joint movement by connecting bones and restricting joint movement, which made the model closer to SolidWorks. The two ends of each major ligament and its anatomical attachment point were set as the common node contact connection. The internal surface of the articular cartilage was set to be fixed with the surface of bone tissue. Mechanical analysis was performed according to previous literature 16

Constructed Different Condition Models
To meet the needs of the stress-strain state study, normal, loosened, and ruptured models of the annular ligament were established. According to the anatomical structure and previous literature 17 , the normal MCL, LCL, and annular ligament were established using computer-aided design (CAD) modeling. The contact area between the annular ligament and the proximal radius was moved 2 mm outward in the loosened annular ligament model; otherwise, it was unchanged. The ruptured annular ligament model was constructed by removing the annular ligament. The geometric models were simulated from 0 to 120 of flexion, with a 30 at intervals, and the position of the humerus was assumed to remain unchanged during flexion and extension of the elbow joint 16 .

Grid Convergence
To verify the sufficiency of the mesh, we tested the convergence of the mesh in the elbow joint model. Mesh sizes with different buckling angles were set as 0.5, 1.0, 1.5, and 2.0 mm, as shown in Figure 3. The boundary and loading conditions are shown in Figure 2. To evaluate the predictive capacity of the finite element (FE) models, a few criteria were used, including the multiple correlation coefficients (R2), the root mean squared error (RMSE), and the mean absolute error (MAE) 18 . R Ç is a measure of the variation around the mean that the regression model produced, and RMSE and MAE are methods that attempt to determine the relationships between input variables and one or more response variables. The criteria were defined as follows.
where y i are the predicted values obtained from the finite element model, x i are actual values, and n is the number of data points that were analyzed, as shown in Equations (1) and (2). e i are the residual sum, and SS res and SS tot are called the residual sum of squares and total sum of squares, respectively, as shown in Equations (3)(4)(5). To compare the various errors in predicting outputs in this case, MAE, RMSE, and R 2 were calculated from the normalized data (0-1). A mesh quality assessment in ABAQUS showed that 0.94% of the elements had an aspect ratio of less than 5.0 (maximum 10.0), and 96% and 97% of all shell and solid elements had Jacobian values larger than 0.6 (minimum 0.24).

Observational Factors
The loading conditions and boundary conditions were the same in the 15 elbow joint models. The mean and standard deviation of the contact stress in the olecranon cartilage, the contact area of olecranon cartilage, and the medial collateral ligament were calculated.

Grid Convergence
The maximum contact stress value of the olecranon cartilage surface was calculated and compared, as shown in Table 2.

Validation of the Finite Element Model
The contact area of the olecranon cartilage was simulated and extracted, as shown in Table 3.
The resulting values of R Ç (0.973) were close to 1, which indicated that these models had good predictive capacity. Relative to the experimental and FE values, the MAE (4.800) and RMSE (4.939) values were minor, which indicated that the present FE model possessed good predictive capacity. The elbow joint was flexed at 0,30,60,90 and 120 degrees and subjected to the corresponding load. The stress values of the olecranon cartilage surface are shown in Fig. 4 and 5. The values of olceranon cartilage surface contact stress when the annular ligament was under different conditions (normal, loosened, and ruptured) with elbow extension were 2.13 AE 0.18, 2.41 AE 0.35, 2.91 AE 0.24 MPa, respectively. The stress value decreased as the angle increased. With 120 degrees of elbow flexion, the minimum stress values were 0.92 AE 0.12, 1.15 AE 0.38 and 1.23 AE 0.29 MPa under normal, loosened and ruptured, respectively. This result suggests that the annular ligament plays a role in maintaining elbow stability. The radial head is prone to instability in cases of annular ligament rupture, which led to increased ulnar pressure and overload.
The contact area of the olecranon cartilage significantly increased and reached a maximum value of 254 mm 2 in Figure 6 when the contact surface of the olecranon was flexed from 0 to 30 degrees. The results when the annular ligament was loosened and ruptured were different from the normal annular ligament. The maximum values for loosened and ruptured were reached at 60 of elbow flexion and were 283 and 312 mm 2 , respectively. The contact area gradually decreased as the angle increased. These results showed that the contact area of the olecranon reached a maximum value when the annular ligament was completely ruptured. This result occurred because movement of the radial head led to stress migration after annular ligament fracture, and the load was transferred through the olecranon, which led to    overload. This result also suggests that the radial head plays an important role in maintaining elbow stability.
The changes in the stress value of the medial collateral ligament with the changes in the angle of the elbow joint are shown in Figure 7. Because the medial collateral ligament is under a tensed condition in elbow extension and the annular ligament is under a different condition, the maximum stress values were 6.52 AE 0.23, 11.51 AE 0.78 and 18.74 AE 0.94 MPa under normal, loosened, and ruptured conditions, respectively. The ligament state changed from tension to relaxation with the increase in the angle, and there was a decline in the different degrees of stress. The different states of the annular ligament corresponded to stress values of 2.81 AE 0.18, 4.83 AE 0.56 and 6.22 AE 0.72 MPa, respectively, at 120 of elbow flexion. As shown in Figure 7, stress significantly increased with annular ligament relaxation or fracture, and the medial ligament and extended annular ligament rupture. The maximum value was 26.42 MPa. Ligament stress and ultimate tensile strength were relatively unclear, but a long period of stress may occur in clinical pain and other clinical symptoms.

Discussion
I n this study, the bone, articular cartilage, and lateral collateral ligament were constructed by the elbow CT and MRI images, and the finite element model was established to simulate the biomechanics of annular ligament injury and analyze the stability of the elbow joint after annular ligament injury. Based on the above results, we found that the stress of cartilage and collateral ligament increased significantly after annular ligament injury, which had a significant impact on the stability of elbow joint.

Advantages of the FE Method at the Annular Ligament
The densely packed parallel fiber arrangement and few elastic fibers of the AUCL, RCL, and AL indicate a strong biomechanically stabilizing function 19 . Hayami et al. 20 performed biomechanical studies on the rupture and reconstruction of the annular ligament in five cadavers and found that anatomical reconstruction of the annular ligament provided  multidirectional stability to the radial head. However, due to the individual differences of cadavers, it is difficult to obtain the uniformity and diversity of samples through cadaver biomechanical experiments with the loosened and ruptured annular ligaments. The repeatability of finite element analysis is one of the common mechanical analysis methods.
Previous studies primarily focused on the ulna, radial head, cartilage, and collateral ligament [21][22][23] , and few biomechanical studies were performed on elbow instability caused by annular ligament injury. There are no reports on the effects of annular ligament fracture on the annular ligament and cartilage using FEA. The FE method effectively simulates and analyzes models under different conditions for the mechanical analysis of problems that cannot be solved using traditional biomechanics or in cases where the structure is complex or a biological experiment cannot be performed 24,25 .

Role of the Annular Ligament in Maintaining Elbow Stability
The present study established a 3D elbow finite element model that included the humerus, ulna, radius, articular cartilage, MCL, and other relevant structures and a buckling process simulation model to study the effects of different states of the annular ligament and other anatomical parameters. The variation trends of stress and strain on the cartilage surface of the MCL and LCL with different flexion angles of the olecranon were analyzed. The results showed that the stability of the radial head was damaged, with annular ligament rupture, and dislocation was more likely to occur, which resulted in poor contact of the ulnar and radial joints and stability damage and led to an increase in the maximum contact stress of the articular cartilage of the olecranon. When the annular ligament ruptured in an extended position, the maximum stress of the articular surface of the olecranon was up to 2.91 AE 0.24 MPa. According to previous literature results 26 , the cartilage matrix may be damaged when stress reaches 3-5 MPa. Compared to the results of this study, the olecranon cartilage surface was more prone to cartilage damage. Sandman et al. 27 used biomechanical studies and showed that reconstruction of the anatomical structure between bones alone was not sufficient to maintain the corresponding relationship of the ulnar and radial joints, and reconstruction of the annular ligament was of great significance for the long-term stability of the ulnar and radial joints, which effectively reduced the instability of the elbow joint and the excessive stress on the ulna. Hayami et al. 20 found that anatomical reconstruction of the annular ligament provided multidirectional stability of the radial head. Radial head instability likely resulted when the annular ligament was fractured, which led to a significant increase in the probability of radial head dislocation.

Role of the MCL to the Elbow
The annular ligament is a strong fibrous band around the radial head that contacts the radial notch of the ulna. Lapner et al. 28 found that the annular ligament was an important component of the proximal radial joint, the radial humeral joint, and adjacent muscles and ligaments. When the annular ligament was loosened, the stress of the medial collateral ligament increased, and the maximum stress value was 18.74 AE 0.94 MPa when the annular ligament was ruptured. The table shows that elbow joint instability significantly increases at this time, which indicates that the MCL plays an important role in maintaining elbow joint stability, similar to a previous study.
Morrey et al. demonstrated that the MCL played a crucial role in elbow joint stability. Rahman et al. 29 studied the effect of different degrees of MCL deficiency on elbow joint stability using biomechanics, which led to elbow joint instability when the medial collateral ligament was completely removed. Seiber et al. 30 showed that the anterior fasciculus varus of the elbow joint had more than twice the effect on stability compared to the LCL in cadaver studies. The medial muscle tissue of the elbow joint primarily affected elbow joint stability, which emphasizes its role as a secondary stabilizer and is consistent with our results.
Simple annular ligament rupture is relatively rare clinically. It is generally caused by trauma and often accompanied by fracture and dislocation. It is more common in Montsillar fractures and radial head dislocation. Whether the annular ligament should be repaired remains controversial. Chen et al. and Kawoosa et al. 6,7 concluded that reducing the radial height effectively achieved reduction without open reduction, and annular ligament reconstruction restored elbow function and improved elbow pain and stability. Canton et al. 31 indicated that annular ligament rupture affected elbow joint biomechanics and resulted in radial head dislocation. Previous studies indicated the importance of anatomical reconstruction of the annular ligament, which indicates that the annular ligament plays a crucial role in radial head stability. Previous research results showed that annular ligament loosening or rupture had a great impact on elbow joint stability, which significantly increased the stress of the medial collateral ligament and led to pain and other symptoms in later stages. Due to the increased stress on the annular cartilage surface, osteoarthritis may occur in severe cases. Therefore, the integrity of the annular ligament plays an important role in elbow joint stability.

Limitations
There are other limitations in this study. First, it is necessary to further simulate the joint capsule, muscle, skin, and other tissues, and the model is limited to flexion. Second, when the finite element model of the elbow joint was established, the bone, soft tissues, and ligaments were assumed to be isotropic linear elastic materials, which has certain limitations in terms of physiological conditions. Third, the model simulates only the static mechanics of the elbow joint at different flexion angles, and the dynamic flexion of the elbow joint is not reflected.

Conclusion
The present study established a successful 3D FE model of the normal structures of the elbow joint and included the MCL, LCL, annular ligament, and cartilage surface. Data analysis revealed that annular ligament loosening or rupture led to an increase in lateral collateral ligament and ulna olecranon articular cartilage surface stress, which demonstrates that the annular ligament plays an important role in maintaining elbow joint stability. The annular ligament should be reconstructed upon rupture as much as possible to avoid the elevation of surface stress on the MCL of the elbow and the annular surface cartilage, which may cause clinical symptoms.