# Reducing the Complexity of Musculoskeletal Models Using Gaussian Process Emulators
## 4 Discussion
Conversely, the 50 simulations each for SGM and SSM needed to train the emulator could be run in only 25 min, a speedup of 1200%. Overall, the SA found a correspondence between the muscles deemed as influential for a certain joint and the real anatomical and functional role of those muscles. For what concern SSM, we found a strong dependence of hip JCF on Gluteus Medius F\({}_{\text{max}}\), particularly in the determination of push off peak (also referred to as Peak 2), with a moderate correlation between these parameters. A recently published study showed that the hip loading is significantly affected by modifications in the strength of Gluteus Medius [6,32]. However, in our study Gluteus Medius' F\({}_{\text{max}}\) had an even larger effect on the resulting knee JCF, particularly on load acceptance peak force (Peak 1), where we also observed a strong negative correlation between the parameters. This can be ascribed to the strict relationship between hip and knee movement and hypothetical crosstalk due to the limited knee motion (1 degree of freedom) that is compensated at the hip level through non-sagittal motion [15]. Knee JCF was also dependent on Tensor Fasciae Latae, Rectus Femoris, and Gastrocnemius Medialis F\({}_{\text{max}}\), with the latter having a significant positive correlation with push off peak force. A previous study found that hip-crossing muscles' F\({}_{\text{max}}\) has a significant effect on the knee JCF [43] and [PERSON] et al. [20] particularly highlighted the role of Gluteus Medius and Gastrocnemius Medialis as major players in the determination of the peak knee contact force. Interestingly, the correlation coefficients between pennation angle and peak knee JCF reported by them match the R values obtained here for the correlation between peak knee JCF and F\({}_{\text{max}}\) but have opposite sign. The pennation angle appears, through its cosine, in the theoretical definition of F\({}_{\text{max}}\)[22]. While this parameter is not explicitly present in our model, it is possible that the generation of the virtual population for SA drew samples from the parameter space whose effect was equivalent to the sampling of the pennation angle, thus partially explaining the similar correlations observed. However, further refinement of the input space are needed to clarify this aspect. Lastly, Tibialis Posterior and Anterior and Soleus F\({}_{\text{max}}\) had an impact on the determination of ankle JCF, with a strong negative correlation between the F\({}_{\text{max}}\) of soleus and ankle peak force. Interestingly, some unexpected relationships were found too, such as the effect of Gastrocnemius Medialis on the hip or the Rectus Femoris on the ankle. They are likely due to higher order interactions, which cause the effect of one muscle to depend on the value of other muscles' F\({}_{\text{max}}\). This hypothesis should be further investigated in future studies by reducing the number of variables in the sensitivity analysis. In the case of SGM, hip and knee JCFs were particularly sensitive to the F\({}_{\text{max}}\) of Tensor Fasciae Latae, Sartorius, Rectus Femoris, and Gracilis. These muscles do indeed play a role in the actuation of hip and knee joints, particularly in the sagittal and frontal plane. Interestingly, previous sensitivity studies either ignored the role of Gracilis [12,20] or found its effect negligible towards the estimate of knee JCF [19]. Our results showed a dominant effect of this muscle, particularly on the knee and can be explained by the input F\({}_{\text{max}}\) values associated with Gracilis, which varied by over 35% (second largest variation amongst the analysed muscles) according to literature measurements [23]. This figure was measured over a small cohort of eleven older women and therefore may not be representative of a larger young mixed-gender population. Nonetheless, it offers a meaningful insight on the dynamic behaviour of subject-specific MSKMs.