*2.1. Subjects*

Eighty-eight healthy junior high school boys (age: 13.6 ± 1.0 years, height: 157.9 ± 9.2 cm, body mass: 47.1 ± 8.8 kg; means ± SDs) were recruited in this study. The inclusion criteria were male junior high school soccer players who participated in a medical examination in their team. Exclusion criteria included history of lower extremity orthopedic surgeries or orthopedic disorders in the lower extremities on measurement day. Written informed consent was obtained from parents of the subjects prior to participation. This study was approved by an institutional human research ethics committee (2013-167) and was carried out in accordance with the declaration of Helsinki.

### *2.2. Measurement of the Optimum Angle*

Isokinetic knee extension muscle strength of the dominant leg was recorded using an electric dynamometer (BIODEX System 3, BIODEX, Shirley, NY, USA). The subjects were seated on the dynamometer with their trunk, pelvic, and dominant thigh secured to a dynamometer by using non-elastic straps. The subjects' dominant leg was secured to an attachment and rotation axis of the knee joint was matched with the rotation axis of the dynamometer in a drooped position of the leg. The angle at which the thigh and leg of the subject were perpendicular was set as 90◦ of the dynamometer. At the measurement, the maximal isokinetic contraction of the knee extension was conducted at 60◦/s in the range of maximal knee flex position to maximal knee extension position. The measurement was taken 3 times. We calculated the average value of the maximal knee extension torque (peak torque). The optimal angle of the force production was calculated by imposing the obtained scatter diagram of the joint angle and the exerted torque with a cubic curve and obtaining the coordinates of the local maximum point.

### *2.3. Estimation of the Growth Age*

To calculate the subjects PHV age, height history in primary school was determined by listing annual physical measurement data on a questionnaire. In addition, the subjects' height in junior high school was recorded using a height meter. PHV age was estimated from the acquired height history using analysis software (AUXAL 3.1, Scientific Software International, Skokie, IL, USA). Finally, we calculated the di fference between age at measurement and PHV, as a growth age.
