## 2 Materials and Methods
### FE Model
Finally, Equation (2) gives the residual forces in the anterior direction. The described precalculation in the sagittal plane only considered forces acting parallel to this plane. The forces of the collateral ligaments are initialized using their positions at full extension. Its \(z\)-components are calculated under the assumption that the patellar tendon force is zero. Therefore, an initial force distribution of 55% medial and 45% lateral is used, based on their ratio of ultimate tensile strength [39]. As shown in Figure 3, the same equation could be used to calculate the \(z\)-component of the patella tendon force for subsequent time steps by using the collateral ligament forces \(F^{C,m}\) and \(F^{C,l}\) calculated from kinematic results of previous simulation runs. By using the angle of the patellar tendon given by previous simulation runs, the total patellar-tendon force could be obtained, as well as its \(y\)-component, which is needed to calculate \(F_{y}^{r}\) using Equation (2). The calculation of \(F_{x}^{r}\) was analogous to the described algorithm for \(F_{y}^{r}\). As per [PERSON] et al. [39], who reported no significant difference between medial and lateral collateral ligament, the stiffness of the collateral ligaments was set to 60 N/mm. The four studies in this review, which used moderate elongation rates (\(\leq\)200 mm/min), reported an averaged stiffness between 58 and 65 N/mm [40; 41; 42; 43]. To be able to consider the forces of the collateral ligaments when calculating the input data for the FE model, it was necessary to calculate their length using kinematic data from previous results. Therefore, the simulation was divided into 600 steps, one for every  
Figure 3: Iterative solving process using kinematic data from previous time step. period between two data points in the given dataset from [PERSON] et al. (2018). To calculate the input data (\(F_{x}^{r}\), \(F_{y}^{r}\), and \(F_{z}^{r}\)) for time step \(t\), the kinematic results from step \(t-1\) were used. The deviation in tibial forces, which was caused by the delay of one time step of the kinematic data for calculating the input data, was negligible. To implement the described algorithm, Ansys Workbench (Release 19.2) and Ansys Mechanical APDL were used. Inertial forces of the simulated components have only minor effects on this analysis because the simulated load cycles do not contain high accelerations and can be assumed as quasi-static. The small inertial forces of the body are implicitly included by using input data from instrumented implants. Nevertheless, to achieve a high numerical stability, a full transient analysis type using a sparse matrix direct solver was chosen in this study. The tibial moments are given within the reference frame shown in Figure 1, and the tibiofemoral kinematics given in this study are defined as follows: the flexion angle is given as relative rotation of the femoral to the tibial component around the femoral medio-lateral axis of the implant. A positive femoral anterior-posterior translation (AP translation) is given as femoral movement to posterior parallel to the tibial \(y\)-axis, which is shown in Figure 1. The femoral rotation is the rotation of the femoral component around the tibial proximo-distal axis. Medial rotation is defined to be positive. The patellar kinematics are given as relative movements of the patella with respect to the femoral coordinate system, as shown in Figure 4. The rotations are calculated as cardan sequence using the rotation order flexion-spin-tilt. Therefore, the patellar flexion is given as the relative rotation of the patella with respect to the femur around the femoral flexion axis. The values are negative if the patella flexes less than the femur, which is usually the case. The patellar tilt is described as the rotation around the patellar proximo-distal axis, with a positive sign if the patella rotates towards the medial side. For patellar spin, the axis is given as floating axis orthogonal to the other two.