This study aimed to investigate the influence of sex and knee joint rotation on PFJ stress. We found that these two factors affect PFJ stress through a mathematical modelling study. To elucidate the reason for this sex difference in the incidence of PFPS, previous studies [7, 8] have investigated sex differences in PFJ stress during running. Additionally, patients with PFPS exhibit excessive knee rotation during running . However, they [7, 8] did not consider sex-specific and knee rotation-specific PFJ contact area regardless of PFJ contact area differs between males and females, among knee joint rotation conditions . To the best of our knowledge, the present study is the first to simultaneously investigate the influence of sex and knee joint rotation alignment on PFJ stress.
In this study, PFJ stress was maximum when the knee joint extension moment and flexion angle were at the maximum, regardless of sex and knee rotation conditions (Figure 3). These findings are consistent with those of previous studies [27, 28]. An increase in knee joint extension moment is associated with an increase in the quadriceps force and PFJ reaction force, which ultimately leads to an increase in PFJ stress. Moreover, an increase in knee joint flexion angle leads to an increase in the demand for the quadriceps muscles. Lenhart et al.  showed that the PFJ reaction force is low when the knee joint angle is low during running; thus, the peak knee flexion angle is a good predictor of the PFJ reaction force (R2 = 0.68). In contrast, the knee joint angle was not the minimum value (i.e. 10°) when PFJ stress was the minimum value. A previous study  discussed that PFJ stress is not necessarily low when the knee joint flexion angle is at its maximum value, which supports the results of the present study.
In all knee joint rotation conditions, females showed increased maximum PFJ stress compared to males (Figure 4), and these results were consistent with our hypothesis. The observed differences between males and females in the present study are likely due to the PFJ contact area. Based on a previous study data set , the maximum simulation range in the knee joint extension moment was set at 240 Nm and 220 Nm in males and females, respectively. The PFJ reaction is calculated based on the knee joint extension moment, and it is associated with the joint moment magnitude. Thus, the PFJ reaction force in females was lower than that in males at all knee flexion angles (Figure 5). However, PFJ stress is influenced not only by the PFJ reaction force but also by the PFJ contact area, and the PFJ contact area in females was lower than that in males  (Figure 5). Therefore, even if the PFJ reaction force was small in females, the PFJ contact area was also small in females, and ultimately, PFJ stress was considered to be high in females. Female recreational runners had a higher incidence of PFPS than males [2, 6], but no agreement of PFJ stress during running for males and females has been reported in previous studies [7, 8]. Since these previous studies did not consider the sex-specific PFJ contact area, the discrepancy in the results between previous studies may be explained by the results of the present study. Additionally, the results of the present study may be useful in understanding the reason behind the sex difference in the incidence of PFPS.
Additionally, PFJ stress maximum values differed among knee joint rotation conditions for males and females, which is also considered to be due to the PFJ contact area as well as the reason for sex differences in PFJ stress. Figure 5 indicates that the PFJ contact area is different among knee joint rotation conditions, and previous studies [10, 12, 13] have also reported that the PFJ contact area is influenced by knee joint rotation. PFJ stress is the highest in the NT condition for both males and females, which is thought to be due to the small PFJ contact area at the knee joint maximum angle (45°). Previous studies have shown that knee rotation during running differs between patients with and without PFPS  and between sexes [19, 29]. The present results of PFJ stress considering knee rotation will be useful for understanding the treatment of PFPS and the mechanism of PFPS development.
Our findings may be beneficial for reducing symptoms in patients with PFPS and/or preventing the occurrence of this condition in runners. Elevated PFJ stress is considered the cause of PFP in runners. Power et al.  reported that wearing a knee brace when fast walking reduces pain (56% reduction by a 10-point visual analogue scale) in patients with PFP because of the decreased PFJ stress. Furthermore, a previous study  showed that pain decreased when wearing a knee brace due to a 1 MPa change in PFJ stress. In the present study, for example, sex difference of PFJ stress was at least 2.3 MPa (IR condition), and difference of PFJ stress between NT and ER conditions were 1.6 MPa and 1.0 MPa in males and females, respectively (Figure 4). Hence, the difference in PFJ stress between sexes and among knee joint rotation in this study may be meaningful for reducing pain.
The present study has some limitations. First, the model utilised to estimate PFJ stress was a simplified planar model; thus, this model does not take into consideration the individual three-dimensional patella. A previous study reported that patellar kinematics during a loading task is different between patients with and without PFPS . It is possible that individual alterations in the patellar kinematics may also affect the contact area, thereby affecting PFJ stress. Second, this study calculated the specific (sex and knee rotation) PFJ contact area, but the contact area was based on cadaver data from a previous study . Therefore, it is different from the contact area in vivo; consequently, PFJ stress may also be different from the present results. However, since PFJ stress cannot be directly measured in vivo, most previous studies are using contact area of cadaver data as in the present study. These limitations need to be addressed in future studies.