Dynamic simulation of knee joint mechanics: individualized multi-moment finite element modelling of patellar tendon stress during landing

Abstract

Patellar tendinopathy is prevalent in sports requiring high jumping demands, and understanding the in vivo biomechanical behavior of the patellar tendon (PT) during landing is crucial for developing effective injury prevention and rehabilitation strategies. This study investigates the in vivo biomechanical behavior of the PT during the landing phase of a stop-jump task, integrating musculoskeletal modelling, finite element analysis (FEA), and a high-speed dual fluoroscopic imaging system (DFIS). A subject-specific knee joint model was constructed from CT, MRI, and dynamic X-ray data for a 27-year-old male (178 cm, 68 kg) at six time points during landing. Musculoskeletal simulations were used to estimated knee joint moments and quadriceps muscle forces, which were then applied to the finite element models. DFIS ensured accurate 3D spatial alignment of the models. Ridge regression analysis explored the relationship between applied biomechanical loads and the maximum equivalent (von Mises) stress in the PT. Maximum PT stress was observed at the bone attachment sites, with the highest stress (94.44 MPa) at initial ground contact, decreasing to a minimum of 16.37 MPa during landing. Regression analysis demonstrated a significant correlation (R 2 = 0.859, P < 0.001) between knee flexion moments, quadriceps muscle forces, and maximum PT stress, identifying these factors as key determinants of PT loading. This study underscores the importance of knee flexion moments and quadriceps muscle forces in influencing PT stress during landing. Future studies should include larger cohort to validate these results and explore the potential of machine learning for real-time injury risk prediction.

1 Introduction

The knee joint, one of the most complex and heavily loaded joints in the human body, plays a crucial role in daily activities and sports, especially in high-impact exercises like jumping. Among the structures of the knee, the patellar tendon (PT) is particularly important as it transmits forces generated by the quadriceps muscle to the tibia, facilitating knee extension and absorbing impact during dynamic movements ( ^{Magnusson et al., 2008}). Despite its critical role, the PT is frequently injured due to overuse, particularly in athletes who engage in repetitive jumping and landing activities. The prevalence of PT injuries is notably high, reaching up to 45 % in volleyball players and 32 % in basketball players ( ^{Albers et al., 2016}). This condition, known as “jumper’s knee,” is closely linked to the magnitude of the load on the PT, which is a key factor in the development of patellar tendinopathy ( ^{Richards et al., 1996; Janssen et al., 2013; Lin et al., 2022}).

Understanding the biomechanical properties of the PT during landing, including the magnitude and distribution of the loads it bears, is therefore essential for developing effective prevention strategies and therapeutic interventions. Various biomechanical modelling approaches, including musculoskeletal modelling and finite element (FE) analysis have been used in previous studies to investigate knee joint mechanics and PT stress. The FE Method is a critical tool in biomechanical research, enabling precise and cost-effective simulations of complex anatomical structures. It accelerates the research process, delivers quantitative insights, and supports the design and optimization of medical treatments and interventions ( ^{Ammarullah et al., 2022b; Jamari et al., 2022a; Ammarullah et al., 2023b}).

For instance, ^{Lu et al. (2023)} integrated musculoskeletal models with FE analysis to explore the impact of Achilles tendon rupture on knee stress during a counter-movement jump. ^{Yan et al. (2024)} further advanced this approach by using musculoskeletal simulations to drive FE models. However, these methods were limited to simple structures and could not simulate more complex joint mechanics. As a result, these earlier studies often relied on simplified models and did not account for changes in the model throughout the movement cycle, making it difficult to capture the complex interactions between structures or the dynamic nature of high-impact activities like landing. Furthermore, many previous studies have used static knee joint models at fixed angles, neglecting the changes in knee joint angles during the landing movement. However, some studies have employed advanced imaging techniques to provide more accurate in vivo assessments of knee biomechanics during dynamic activities. For example, ^{Peng et al. (2023)} used the dual fluoroscopic imaging system (DFIS) to visualize. In vivo, knee biomechanics during badminton lunges at different distances and different foot positions. ^{Li et al. (2009)} employed DFIS to investigate six degrees of freedom (6DOF) knee kinematics during treadmill gait. By using DFIS to adjust the 3D spatial position of the FE models, it is possible to analyze the load-bearing state of in vivo tissues at multiple time points throughout a movement cycle.

Given these limitations in previous research, our study aims to achieve a more comprehensive and realistic analysis of PT stress during landing. As for the type of stress, this study emphasizes von Mises stress due to its reliability in predicting yielding under multiaxial loading conditions in ductile materials like biological tissues. Unlike Tresca stress, which focuses on shear stress, von Mises stress accounts for distortional energy, making it more suitable for modeling complex stress states in heterogeneous tissues, such as those encountered during landing motions. Additionally, contact stress is more localized, offering limited insight into the overall stress distribution in dynamic motion ( ^{Ammarullah et al., 2023a; Hidayat et al., 2024a; Muchammad et al. 2024}).

This study integrates musculoskeletal modelling, FE analysis, and DFIS. The musculoskeletal model is used to obtain the loads applied to the FE model, which is then aligned in 3D using dynamic X-ray images to construct individualized knee joint models at different angles during the landing phase. Given the high computational cost of FE modelling, we also aim to identify simple and accessible indicators that can predict the peak equivalent stress in the PT. To achieve this, we employ ridge regression to investigate the relationship between applied loads and the peak equivalent stress in the PT. Through this individualized, continuous analysis of in vivo PT biomechanics across the entire landing phase, we aim to uncover the underlying mechanisms of PT injury, establish a foundation for prevention and treatment strategies, and explore the potential for real-time monitoring of PT health.

2 Methods
2.1 Participant information

The study involved a young adult male (age: 27 years, height: 178 cm, weight: 68 kg) with no history of neuromuscular diseases or biomechanical abnormalities. The dominant leg, defined as the leg opposite the preferred shooting arm, was selected for analysis, consistent with the concept of crossed symmetry ( ^{Warden et al., 2021}). Informed consent was obtained, and the study protocol was approved by the University Ethics Committee (RAGH20240105).

2.2 Imaging data acquisition

CT and MRI scans were performed to obtain detailed imaging data of the participant's left knee joint. A Somatom Sensation 16-slice spiral CT scanner (scan reconstruction layer thickness: 0.5 mm; 120 kV, 250–300 mA) and a 3.0 T MRI scanner (GE Discovery-sim 750, with the built-in body coil) were used to capture the knee structures. The MRI scans were conducted in the sagittal plane with a slice thickness of 1 mm and no inter-slice gap.

Dynamic X-ray images were captured during the stop-jump landing phase using a high-speed DFIS at 30 Hz. The DFIS consists of two fluoroscopic imaging systems, each comprising an X-ray generator and an image intensifier. Before testing, the system was calibrated to determine the imaging range, perform spatial calibration, and correct image distortion based on established protocols. The analysis system automatically extracted the images for landing phases (from initial contact to maximum knee flexion).

2.3 Motion capture

The participant began with a 5-minute standard dynamic warm-up, which included both full-body dynamic and static stretching ( ^{Scanlan et al., 2012}) before performing ten stop-jump trials, touching a height marker. The target height for the trials was set at 85 % of the average height achieved during two initial maximal-effort warm-up jumps. For each trial, the participant was to touch the marker with the right hand and land with both feet on the force platform to ensure consistency. Thirty-eight retroreflective markers were affixed according to a previously established protocol ( ^{Song et al., 2023}). Motion capture was conducted using a 10-camera Vicon system at 200 Hz (Oxford Metrics Ltd., Oxford, UK) and ground reaction forces (GRF) were simultaneously measured at 1000 Hz using two floor-embedded force plates (Kisler Instruments AG, Winterthur, Switzerland). Electromyography (EMG) signals from lower limb muscles including Rectus Femoris (RF), Vastus Lateralis (VL), and Vastus Medialis (VM), were recorded at 1000 Hz using the wireless Delsys EMG system (Delsys, Boston, MA, USA). For each trial, data from the moment of initial ground contact to the moment of maximum knee flexion were extracted for further analysis. The motion capture data were uniformly segmented into six corresponding time frames because the DFIS captured six frames during the landing phase. Since a total of ten trials were conducted, each time frame yielded ten data points, resulting in a total of 60 data points. The average of the ten data points at each frame was used as input for the FE analysis, and FE analysis was independently performed for each data point for Ridge Regression modeling. The motion capture process and experimental setup are shown in ^{Fig. 1} A.

2.4 Finite element model Construction

In this study, we reconstructed the knee joint model based on CT and MRI imaging data to capture the geometry of bones, cartilage, and ligaments. While the bone and cartilage geometries were directly extracted from imaging data with high fidelity. The ligament geometries (anterior cruciate ligament (ACL); posterior cruciate ligament,(PCL); medial collateral ligament (MCL); lateral collateral ligament (LCL) were represented as simple bands connecting their anatomical insertion points, which were precisely identified from MRI data ( ^{Firminger et al., 2022; Park et al., 2023}). The cartilage layers were reconstructed from high-resolution MRI slices, with defined thickness and contact surfaces. This ensured computational feasibility and anatomical accuracy after re-aligning the reconstructed bones, a step that required redrawing the ligament geometries due to the positional adjustments ( ^{Prakoso et al., 2023; Putra et al., 2023; Abd Aziz et al., 2024}).

To address challenges in accurately aligning static imaging data with dynamic knee motion, dynamic X-ray images were captured during the stop-jump landing phase using a high-speed DFIS (Taoimage, China) at 30 Hz. These images were calibrated to correct for spatial distortion and provided precise alignment data. By using SolidWorks 2021(Dassault Systèmes, France), the reconstructed knee joint models were manually manipulated in six degrees of freedom to align with the contours of bones in the fluoroscopic images for six distinct landing phases. This alignment ensured that the reconstructed geometries accurately represented the joint angles and positions during the dynamic motion.

The FE model includes the distal femur, proximal tibia, proximal fibula, medial and lateral menisci, femoral and tibial cartilages, and four primary knee ligaments. Mesh generation was performed in Ansys Workbench 2021 R1(ANSYS, Inc., United States), where all bones and soft tissues were meshed using 4-node linear tetrahedral elements (C3D4). The ligaments were also meshed with C3D4 to effectively capture the internal stress and strain states. Element sizes were set to 3 mm for bones, 1 mm for the menisci and cartilage, and 1.5 mm for the ligaments ( ^{Edwards et al., 2013; Song et al., 2024, 2025}). In regions of high stress gradients, mesh refinement was applied to improve local accuracy, ensuring convergence with acceptable differences compared to finer meshes ( ^{Jamari et al., 2021; Tauviqirrahman et al., 2022; Hidayat et al., 2024c}).

Bone is often modeled as a rigid or linear elastic material, which is sufficient for most cases and reduces computational cost while ensuring reliable stress calculations. Similarly, cartilage and ligaments are frequently approximated as isotropic linear elastic materials, as their viscoelastic properties can be neglected under high strain rates or short-duration loading scenarios, making simulations more efficient ( ^{Yan et al., 2024}). Based on this established approach, we assigned homogeneous, isotropic, and linear elastic properties to the tissues in our model to balance computational feasibility and accuracy. This allowed efficient calculation of stresses and strains while preserving sufficient accuracy for biomechanical assessments. Material properties were assigned based on reported values in the literature, as summarized in ^{Table 1} ( ^{Bae et al., 2012}; Zhongxin et al., 2020; ^{Jamari et al., 2022b; Tauviqirrahman et al., 2023; Zhang et al., 2023; Yang et al., 2024}).

2.5 Dynamics and EMG data processing

The joint moments and muscle forces were calculated using the general musculoskeletal multibody Gait2392 model in OpenSim (National Center for Simulation in Rehabilitation Research, Stanford, USA). The model was scaled to the subject's anthropometric measurements to ensure anatomical alignment. Marker data were processed using the OpenSim inverse kinematics tool, where marker trajectories were filtered with a zero time-lag, fourth-order Butterworth filter at a cutoff frequency of 10 Hz, to calculate hip, knee, and ankle joint angles. Subsequently, the inverse dynamics tool in OpenSim combined the ground reaction forces (GRF) and joint kinematic data to compute net internal joint moments. The static optimization tool in OpenSim estimated individual muscle forces for each time step, which were then compared to experimental EMG data to assess consistency.

EMG data were processed using baseline correction and normalization. During a static resting phase, baseline EMG signals were recorded to quantify the mean baseline noise level, which was subtracted from the raw EMG signals to remove non-muscle-related noise. Normalization was performed using the peak amplitude of periodic motion cycles as the normalization factor. Signal filtering included a fourth-order Butterworth band-pass filter with cutoff frequencies of 10 Hz and 450 Hz to eliminate motion artifacts and electrical noise. The filtered signal was then full-wave rectified and smoothed using a low-pass filter with a cutoff frequency of 5 Hz ( ^{Boyer et al., 2022}).

2.6 Validation of the musculoskeletal and FE model

Geers metric and the coefficient of multiple correlation (CMC) were employed to validate the simulated muscle force from OpenSim. Geers metric provides a comprehensive evaluation by decomposing errors into amplitude and phase components, making it suitable for assessing both magnitude and timing discrepancies between measured and simulated values (0 < Geers Metric < 0.15: excellent similarity; 0.15 < Geers Metric < 0.30: very good similarity; 0.30 < Geers Metric < 0.45: good similarity; 0.45 < Geers Metric < 0.60: moderate similarity; Geers Metric > 0.60: no similarity). CMC, on the other hand, quantifies overall waveform similarity and evaluates the global alignment of experimental and simulated muscle activation patterns (CMC < 0.65: no similarity; 0.65 < CMC < 0.75: moderate similarity; 0.75 < CMC < 0.85: good similarity; 0.85 < CMC < 0.95: very good similarity; CMC > 0.95: excellent similarity)( ^{Gaspar et al., 2017; Klemt et al., 2019}).

Validation of the computational simulation results is critical to ensure the accuracy and reliability of the model. In this study, values obtained from simulations were compared against previously published data to confirm model correctness ( ^{Salaha et al., 2023; Hidayat et al., 2024b; Tauviqirrahman et al., 2024}). The FE model in this study was validated under anterior-posterior loading conditions, a widely used approach in biomechanical studies to evaluate ligament behavior and joint mechanics. To validate our model, we applied a 134 N anterior force to the tibia while keeping the femur fixed. The resulting tibial translation, along with the maximum stress in the MCL, LCL, and ACL, was compared with experimental data from similar studies ( ^{Benos et al., 2020; Shao et al., 2022}). Although this validation confirms the reliability of the model under basic loading conditions, it does not extend to knee flexion and compression scenarios. Validation under these conditions presents significant challenges due to the lack of directly comparable experimental data, particularly at the specific flexion angles used in this study. Despite these limitations, the model's predictions for stress distribution under flexion remain valuable for exploring dynamic knee mechanics. Future work will include flexion-specific experimental validations to enhance the comprehensiveness of the model.

2.7 Boundary condition and loading

An isothermal condition was assumed for this study, given the primary focus on the knee joint’s biomechanical response under dynamic loading ( ^{Ammarullah et al., 2022a; Jamari et al., 2022c; Hidayat et al., 2023}). Six transient analyses were conducted to evaluate the in vivo biomechanical state of the PT during the landing phase. A frictionless contact assumption was employed in this study to simplify the modeling of contact interfaces, allowing for a more focused evaluation of stress distribution and load transfer within the knee joint. While knee joints experience friction in vivo, incorporating friction parameters can introduce variability and replicability challenges, as these parameters are challenging to measure precisely and may vary slightly across simulations ( ^{Lamura et al., 2023a; Lamura et al., 2023b; Lamura et al., 2024}). Frictionless contact was defined between the menisci and the tibial and femoral cartilages, while bonded contact was assigned between each articular cartilage and the corresponding bone. The six degrees of freedom of the distal tibia and fibula were constrained. These forces and moments were averaged from ten stop-jump trials, performed by a single participant. To address the challenge of dynamic joint geometry changes during landing, DFIS was utilized to capture accurate knee joint geometries. The quadriceps force was applied to the superior surface of the PT, directed along the anatomical axis of the quadriceps tendon. This force was transmitted to the tibial tuberosity, reflecting realistic loading conditions during landing. The specific implementation flowchart is illustrated in ^{Fig. 1} C.

2.8 Ridge regression and data Fitting

In this study, ridge regression was employed to analyze the relationship between various biomechanical factors and the maximum equivalent (von Mises) stress experienced by the PT during landing.

The regression model is formulated as follows: (1) $y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_4+∊$where y represents the maximum equivalent stress as the dependent variable, $x_1,x_2,\cdots ,x_n$ with four independent variables ( $x_1,x_2,x_3,x_4$): knee flexion moment ( $x_1$), internal-external rotation moment ( $x_2$), varus-valgus moment ( $x_3$), and quadriceps muscle forces ( $x_4$). ${\beta }_0$ is the intercept, ${\beta }_1,{\beta }_2,\cdots ,{\beta }_n$ are the regression coefficients, and $∊$ is the error term.

The regression coefficients were determined by minimizing the following objective function: (2) $\text{Minimize}\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2$where $y_i$ represents the observed values, and ${\widehat{y}}_i$ denotes the predicted values. The significance of the model was evaluated using the F-test (with p-values < 0.001), and the goodness-of-fit was assessed through the adjusted R ² value.

3 Results
3.1 Model validation

^{Fig. 2} A compares muscle activation levels from EMG sensors and predicted values of OpenSim for the VL, RF, and VM muscles. The CMC values were 0.859, 0.658, and 0.849, respectively. Geers Metric values were 0.439, 0.268, and 0.496, respectively, reflecting good agreement in amplitude and phase components across all three muscles.

The FE model was validated by applying a 134 N anterior force to the tibia, which produced an anterior tibial translation of 4.81 mm. This translation is in close agreement with data reported in previous studies under similar loading conditions. Additionally, the maximum stress values observed in the ACL, PCL, MCL, and LCL were 17.45 MPa, 5.51 MPa, 1.94 MPa, and 1.66 MPa, respectively, consistent with the previous studies ( ^{Benos et al., 2020; Shao et al., 2022}). The agreement between our model's predictions and the results of previous studies as shown in ^{Fig. 2} B, confirms the validity of the model in this study.

3.2 Maximum stress and stress distribution in the patellar tendon

During the landing phase, as knee flexion increases, both the maximum stress and its stress distribution within the PT exhibit a distinct trend. ^{Fig. 3} presents the FE analysis results across six time points. Initially, at ground contact (Moment 1), the maximum equivalent stress peaked at 94.44 MPa, primarily concentrated at the tendon’s attachment sites to the bone. As landing progressed, stress decreased to 88.89 MPa (Moment 2), 27.41 MPa (Moment 3), 26.68 MPa (Moment 4), 16.37 MPa (Moment 5), and 21.62 MPa (Moment 6). Notably, there was a marked shift in the location of the concentration of stress. Initially, the stress was predominantly concentrated at the tibial attachment; however, with increasing knee flexion, the concentration of stress gradually shifted toward the patellar attachment, particularly involving the posterior fibers of the tendon. Furthermore, the area over which high stress was concentrated in the posterior region of the patellar tendon was consistently larger than that in the anterior region throughout the landing phase.

3.3 Results of PT maximum equivalent (von Mises) stress prediction

^{Fig. 4} shows the results of our ridge regression analysis. The central scatterplot compares simulated versus predicted stresses for the training and test sets, the vast majority of points lie close to the line without systematic bias. The upper‐left histogram shows that residuals are symmetrically distributed around zero and confined within ±5 MPa. The upper‐center and right‐side histograms display the distributions of simulated and predicted stresses, respectively, for both sets; these largely overlap, confirming consistent model performance across datasets with no indication of overfitting or distributional shift. Statistically, the model demonstrated strong explanatory power (R ² = 0.859, adjusted R ² = 0.851) and is highly statistically significant (F = 160.758, p < 0.001). Among the predictors, knee flexion moment has the strongest effect (β = 0.747, t = 17.461, p < 0.01), followed by quadriceps muscle force (β = 0.181, t = 6.248, p < 0.01), whereas internal-external rotation (β = 0.014, t = 0.417, p = 0.679) and varus-valgus moment (β = 0.084, t = 0.249, p = 0.327) contribute negligibly.

4 Discussion

While previous studies have investigated knee biomechanics during landing, to our knowledge, this is the first study to integrate DFIS with a multi-moment FE modeling framework to quantify in vivo PT stress distribution throughout the entire landing phase.

Our individualized approach, which captures subject-specific anatomical and dynamic joint kinematics, revealed that peak equivalent stress is consistently localized at the PT’s tibial and patellar attachment sites during landing, with posterior stress concentrations exceeding anterior regions. These findings align with clinical imaging studies demonstrating that patellar tendinopathy predominantly affects the proximal and posterior PT, particularly near the inferior pole of the patella ( ^{Johnson et al., 1996; Shalaby and Almekinders, 1999}). Notably, 65 % of jumper’s knee cases involve distal patellar pain, while 10 % affect the tibial tubercle ( ^{Blazina et al., 1973}), a distribution mirrored by our FE-derived stress patterns.

Our FE analysis corroborates this clinical observation, demonstrating elevated stress magnitudes and pronounced stress concentrations in the posterior PT fibers during dynamic loading. These computational findings are consistent with ultrasonographic evidence from ^{Figueroa et al. (2016)}, who reported the presence of hypoechoic lesions, a characteristic feature of tendinopathic degeneration, predominantly found in the posterior PT, near the inferior patellar pole. Furthermore, biomechanical investigations by ^{Basso et al. (2002)} provide mechanistic validation. Cadaveric experiments under quadriceps loading demonstrated significantly greater strain in the posterior PT compared to the anterior region during knee flexion. This differential strain pattern reflects the posterior fibers' critical role in force transmission, as their anatomical orientation and insertion geometry predispose them to higher mechanical demands during dynamic tasks.

The regression analysis identified knee flexion moments and quadriceps forces as the primary drivers of equivalent stress within the patellar PT, while rotational moments exhibited negligible influence. This underscores the dominance of sagittal-plane mechanics in governing PT loading during dynamic tasks. Mechanistically, the PT functions as a force-transmitting chain between the quadriceps and tibial tubercle, with stress concentration patterns dictated by the angle between the PT and the patellar axis, and the femoral contact point position during knee flexion, both of which modulate tendon strain distribution ( ^{Dan et al., 2018}). The limited contribution of rotational moments aligns with their primary role in stabilizing the cruciate and collateral ligaments rather than directly influencing PT mechanics ( ^{Wijdicks et al., 2013; Beel et al., 2024}). For instance, varus-valgus moments induce stress redistribution across the medial and LCL, while internal-external rotations engage the cruciate ligament complex to maintain joint integrity. The limited influence of rotational kinematics on PT loading underscores its anatomical specialization as a unidirectional force transmitter, primarily adapted to sagittal-plane loading.

Unlike traditional biomechanical models, which often rely on generic geometries or static assumptions, this study captures subject-specific anatomical and dynamic characteristics, enabling a more accurate and detailed analysis of tendon loading during complex movements. By identifying key determinants of PT stress, this approach enables real-time predictive modeling based on readily accessible biomechanical parameters.

There are limitations in the study. First, the single-subject design limits generalizability, as gender- and age-related differences in anatomical parameters and kinetics may significantly alter stress distributions ( ^{Onambélé et al., 2007; Peek et al., 2022}). While the current knee joint FE model incorporated necessary simplifications to balance computational efficiency and biomechanical fidelity, a methodological approach widely adopted in complex musculoskeletal modeling. For instance, some necessary simplifications were made in the structural and material representation of the foot and shoe to constrain the model complexity ( ^{Cen et al., 2024; Wang et al., 2015}). Such simplifications, while introducing potential errors in localized strain estimation, enable feasible simulation of whole-joint mechanics during dynamic loading. Additionally, the validation does not directly validate the model under flexion conditions. These factors need further exploration to improve the model's robustness and applicability. Finally, this study focused exclusively on acute biomechanical responses, neglecting the cumulative effects of repetitive loading, which are critical in the pathogenesis of tendinopathy ( ^{Malliaras et al., 2015}). Future research should investigate long-term stress accumulation effects through fatigue failure simulations, allowing for a more comprehensive understanding of overuse injury mechanisms.

5 Conclusion

The findings highlight that maximum equivalent stress in the PT occurs at the tendon’s attachment sites during the initial ground contact, decreasing as knee flexion increases throughout the landing phase. Knee flexion moments and quadriceps muscle forces may potentially play a critical role in PT loading, whereas internal-external rotation and varus-valgus moments showed limited influence. However, it is important to note the study's limitation of using data from a single participant, which may restrict the generalizability of the findings. Future research should involve a larger and more diverse cohort to validate and extend these results. Furthermore, integrating machine learning techniques with FE models could enhance the ability to predict injury risk and develop real-time monitoring systems for athletes, ultimately contributing to more personalized injury prevention and rehabilitation strategies.

CRediT authorship contribution statement

Fengping Li: Writing – original draft, Methodology, Investigation, Conceptualization. Dong Sun: Writing – review & editing, Validation, Formal analysis. Yang Song: Writing – review & editing, Supervision, Methodology. Zhanyi Zhou: Writing – review & editing, Resources, Investigation. Dongxu Wang: Writing – review & editing, Resources, Investigation. Xuanzhen Cen: Writing – review & editing, Investigation. Qiaolin Zhang: Writing – review & editing, Validation, Methodology. Zixiang Gao: Writing – review & editing. Yaodong Gu: Writing – review & editing, Supervision, Funding acquisition, Formal analysis, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This study was sponsored by Zhejiang Province Key R&D program of “Pioneer” and “Leader” ( 2023C03197), Zhejiang Province Science Fund for Distinguished Young Scholars (Grant number: LR22A020002), Ningbo Key Research and Development Program (Grant number: 2022Z196), Zhejiang Rehabilitation Medical Association Scientific Research Special Fund ( ZKKY2023001), Research Academy of Medicine Combining Sports, Ningbo (No. 2023001), the Project of Ningbo Leading Medical &Health Discipline (No. 2022-F15, No. 2022-F22), Ningbo Natural Science Foundation, China (Grant number: 2022J065) and K. C. Wong Magna Fund in Ningbo University, China.

Table 1

Material parameters of the model components.

	Young’s Modulus E (MPa)	Poisson’s ratio (ν)	Mesh Size (mm)
Bone	12 000	0.3	3
Cartilage	5	0.46	1
Meniscus	59	0.45	1
ACL	116	0.3	1.5
PCL	87	0.3	1.5
MCL	48	0.3	1.5
LCL	48	0.3	1.5
PT	87	0.3	1.5