## 3 Finite Element Analysis of Knee Integration Model
### Material Properties and Boundary and Load Condition  
The analysis was conducted using ANSYS Workbench 2024. The meshing of each anatomical structure was performed using a 10-node approach to generate shell meshes with tetrahedral grid cells. The bone mesh size was set to 2.0 mm, while the ACL, being the primary focus, had a mesh size of 0.5 mm. The mesh size for the meniscus, ligaments, and other structures was set to 1.0 mm. Various material models were applied to simulate different structures, with all materials considered linear elastic except for the wrapped soft tissues, ACL, and posterior cruciate ligament [11]. The stress-strain relationship (linear elastic material) is given by the following [34; 35; 36]:  
\[\sigma=\mathrm{E}\epsilon, \tag{2}\]  
In the context of finite element analysis, the Newton-Raphson method is commonly used to solve non-linear equations, making it particularly suitable for addressing equilibrium equations. This method approximates the roots of an equation through an iterative process, governed by the iteration formula [36; 37]:  
\[\mathrm{x}_{\mathrm{k}+1}=\mathrm{x}_{\mathrm{k}}-(\mathrm{J}(\mathrm{x}_{ \mathrm{k}}))^{-1}\mathrm{F}(\mathrm{x}_{\mathrm{k}}), \tag{3}\]  
To describe the hyperelastic behavior of the ACL, the Neo-Hooke model is employed, involving two key parameters: C\({}_{1}\) and D. The constant C\({}_{1}\) is related to the shear modulus of the material and indicates the linear elastic response under small deformations, with higher values signifying greater rigidity. Conversely, D is associated with the strain rate, primarily used to characterize material behavior under dynamic loading or rapid deformation, reflecting its stiffness and time-dependent effects. In the Neo-Hooke model, the strain energy equation for a first-order polynomial strain can be expressed as follows [36; 38]:  
\[\mathrm{W}=\mathrm{C}_{1}\big{(}\overline{\mathrm{I}}_{1}-3\big{)}+\mathrm{D} \epsilon, \tag{4}\]  
The material properties adopted in the numerical models are summarized in Table 1. Each component was assigned the appropriate mechanical properties based on the experimental data or the established literature to ensure biological relevance and computational accuracy. Hyperelastic materials, such as the skin, soft tissues, and ligaments, were modeled using first- or second-order strain energy functions, as these adequately capture the non-linear, anisotropic behaviors typical of biological tissues. The femur and tibia were modeled as isotropic linear elastic materials, reflecting their cortical bone structure, while the cartilage and meniscus were assigned near-incompressible properties to simulate their physiological behavior. Boundary conditions were applied to replicate realistic physiological scenarios and experimental constraints. The distal end of the tibia was fixed in all degrees of freedom to simulate ground contact during load-bearing tasks, while theproximal femur was assigned a prescribed displacement or load to mimic joint interactions. Ligament insertions were tied to adjacent bones using rigid constraints, and frictionless contact was applied between cartilage layers to simulate synovial joint articulation. Forces and muscle forces applied were aligned with the experimental data to replicate in vivo loading conditions. To ensure comparability and reproducibility, the modeling assumptions, such as hyperelastic parameters and boundary conditions, were derived from validated sources. References for each parameter are listed in Table 1, and specific equations governing hyperelastic material behavior are provided in the Table 1. To ensure the accuracy of the simulation results, the convergence criteria were defined, with a 5% relative change between two neighboring meshes deemed convergent. In addition to material properties, six contact pairs were established to simulate the interactions within the knee joint.