# Patellar Tendon Force Differs Depending on Jump-Landing Tasks and Estimation Methods
## 2 Materials and Methods
### Data Processing  
A 23 degree of freedom (DOF) full-body musculoskeletal model, with 92 muscles actuators (43 per leg and 6 at the torso) was used to compute muscle forces, joint kinematics and moments through OpenSim v4.0 [23]. The trunk had 3-DOF and the lower extremity was modeled as seven body segments: pelvis, femur, patella, tibia/fibula, talus, foot, and toes. The pelvis segment had 6-DOF and was able to rotate and translate in all three dimensions with respect to the ground. The knee was modeled as a simple 1-DOF joint. The hip and the ankle joints had respectively 3 and 2-DOF. The metatarsophalangeal joint had 1-DOF. The generic model was scaled to match the anthropometry of each participant as closely as possible. Three-dimensional marker coordinates were low pass filtered at 10 Hz using a zero time-lag fourth order Butterworth filter and used to calculate the hip, knee and ankle joint angles using OpenSim's inverse kinematics tool. Joint kinematics conventions were defined such as hip flexion, knee flexion, and ankle dorsiflexion and were assigned to be positive. Then, the combination of ground reaction force and joint kinematic data was used to calculate net internal joint moments from OpenSim's inverse dynamics tool. To overcome inaccuracies in assessment of joint moment calculation, force plate data were filtered in the same way as kinematic data using a cutoff frequency of 10 Hz and a zero time-lag fourth order Butterworth filter [24]. Individual muscle forces were estimated from a static optimization procedure using OpenSim's static optimization tool [25]. The muscular load sharing problem was solved for each time step, by minimizing a function composed of the sum squared of muscle activations subject to the constraints of force-length-velocity properties of muscles. The individual muscular moment is calculated from the muscle force (the unknown of the optimization problem) and muscle moment arms, which are based on musculoskeletal anatomy and may or may not depend on joint angles. The maximum possible muscle forces are limited by physiological values, which constitutes an additional constraint. The quadriceps force was calculated by summing the force of four muscles of the musculoskeletal model: rectus femoris, vastus lateralis, vastus intermedius, vastus intermedius, vastus intermediusis. The PTF estimated with the inverse dynamic method was calculated following the equation:  
\[PTF=\frac{Net\;knee\;joint\;moment}{Patellar\; tendon\;moment\;arm} \tag{1}\]  
where the patellar tendon moment arm was calculated as a function of the knee flexion angle [26; 27]. Finally, the PTF estimated with the static optimization method was calculated following the equation:  
\[PTF=Quadriceps\;force\times Ratio_{PTF}/Quadriceps\;Force \tag{2}\]  
where the ratio of PTF and quadriceps force was estimated by a regression equation using the knee flexion angle [28]. The landing phase was defined as the time interval from initial contact to the maximum knee flexion. Initial contact was determined when the vertical ground reaction force (GRF) first exceeded 10 N. Peaks of PTF from both methods were determined as the maximum PTF during the landing phase. Loading rates of PTF from both methods were calculated by dividing the PTF peak minus the PTF at initial contact by the time interval between initial contact to the time of the PTF peak. Peak of quadriceps force was determined as the maximum quadriceps force during the landing phase. PTF peaks, PTF loading rates and peak of quadriceps force were normalized to body weight. Additional kinetic and kinematic dependent variables were also analyzed to understand the effects of jump-landing tasks and PTF estimation methods. Kinetic dependent variables were composed of the magnitude of the peak of vertical GRF and the peak of anteroposterior GRF, the loading rate of peak vertical GRF, and the peak knee moment in the sagittal plane. The loading rate was defined as the average slope of the force-time curve from 20% to 80% before the corresponding peak [29].