**4. Discussion**

The present study investigated the relationship between the optimum angle of force production and growth age. The main finding of the present study was that the relationship between the optimum angle and growth age was regressed by a cubic curve. To the best of our knowledge, this is the first study investigating the change in the optimum angle of force production during growth.

The PHV age was 13.3 ± 0.9 in this study and this result was the same as that reported by a previous study for Japanese subjects [11]. However, the PHV age was 0.2 to 0.8 years earlier than previous studies for Europeans and North Americans [4,12,13]. It is considered that race might influence the inter-study differences in PHV age.

A significant linear regression relationship was obtained between the peak torque and growth age. The relationship between optimum angle and growth age was significant when analyzed using the cubic regression model. These results support our hypothesis, and sugges<sup>t</sup> that the optimum angle temporarily changes, despite the peak torque of force production developed with growth. The optimum angle of knee extension force production was temporarily smaller at approximately 1 year before PHV because the growth age on the minimal local value was −1.0 years. According to previous studies, it is reported that the increase in femur length of male adolescent peaks at 1.1 years before PHV [14]. The growth age when the optimum angle reaches its minimal value in this study nearly coincided with the increase in femur length in the previous study. In addition, it has been reported in other studies of the Japanese population that the flexibility of the quadriceps femoris muscle decreases in around 12-year-old individuals [5]. The optimum angle of force production in the present study was shown to be the minimum local around 12 years old when converted to age. Considering this, the increase of bone length and decrease of muscle flexibility could be related to temporary changes of the optimum angle of the knee extension muscle force production.

The results of the present study showed that the optimum angle of knee extension muscle force production temporarily decreases with growth. The muscle lengthens by an increase in bone length [7], and muscle length (joint angle) affects the force production [9,15]. Therefore, muscle elongation due to an increase of bone length could affect the force–angle relationship. In particular, because the quadriceps femoris muscle is elongated when the knee is flexed, muscle elongation due to growth is considered to

cause the same change as knee flexion. Based on the above, the optimum angle of knee extension force production was considered to temporarily decrease due to growth.

The growth age of the maximum local value on the regression curve was 2.3 years. The peak of the temporary decrease in BMD was almost the same as that of PHV [3], which was earlier than the timing at which the optimum angle became the original value after the temporary change. In addition, because the average PHV age was 13.3 years, the optimum angle of force production was shown to be a maximum local at 15 to 16 years old when converted to age. In a previous study in Europeans and North Americans, the incidence of sports injuries in adolescents increased until 15 to 16 years of age and decreased after this period [1,2]. Although the PHV age in the present study was 0.2 to 0.8 years earlier than previous studies for Europeans and North Americans [4,12,13], the maximum local of the optimum was just before or almost the same as the timing when the incidence of sports injury begins to decline. Therefore, adolescent sports injuries could be a ffected by changes in force production characteristics due to growth.

There are a few limitations in this study. First, because the knee joint angle during force production was not measured, the joint angle in this study is di fferent from the actual joint angle. Therefore, we do not mention the specific joint angle. However, the knee joint angle was defined using the same method in all subjects. Thus, this limitation cannot a ffect the result that the optimum angle of force production changes due to growth in adolescents. Second, because the range of the subjects' age was limited, it is not clear when the change of the optimal angle begins. A previous study showed that the muscle flexibility of the quadriceps muscle decreased in boys aged 11 years [5]. In addition, there is a sex di fference in the incidence of Osgood–Schlatter disease, which is related to the quadricep muscle flexibility [16,17]. Therefore, further research needs to include at least 11-year-old boys and investigate the sex di fference of the optimum angle change.
