# Reducing the Complexity of Musculoskeletal Models Using Gaussian Process Emulators
## 4 Discussion  
This study proposes a methodology for comprehensive sensitivity analyses of MSKMs' outputs to changing model parameters such as bone geometry, joint type and degrees of freedom, and muscle parameters and geometry at the same time. This can offer a clearer insight into the model response and which input parameters must be personalised. To test the methodology, the feasibility of using GPs to emulate the JCF prediction of lower-limb MSK models was investigated, under the hypothesis that a trained GP could be used to identify the muscles whose F\({}_{\text{max}}\) is most influential to the estimate of JCFs. Notably, the methodology presented in this study has the potential to be applied to any input parameters given their statistical distributions. In the first part of the study, we obtained JCF curves from 200 sets of OpenSim simulations (both for scaled-generic and subject-specific models) while varying F\({}_{\text{max}}\) of 29 limb muscles by 21 \(\pm\) 6%, based on literature measurements [23]. Resulting simulated curves were overall similar in shape to literature reference curves obtained from MSKMs [7, 17, 20]. When comparing scaled-generic and subject-specific models, some qualitative differences were found between the scaled-generic and the subject-specific models. SGM had higher late stance peaks (for all joints) and higher knee early stance peak, while SSM had higher hip early stance peak. These were likely associated with differences in the definition of the joint axes and muscle paths, leading to different joint kinematics as well as moment arms and associated actuation strategies, as already hypothesised in previous literature [13, 16, 42]. As a consequence of changing F\({}_{\text{max}}\), we found variations up to 104% of baseline values (knee joint of SSM). The peak JCF of hip and knee corresponding to the push off phase of gait was particularly affected by the variations of F\({}_{\text{max}}\), in line with previous findings [7, 20, 43]. The sensitivity analysis was performed on data points that were artificially generated by the emulator, whose validation is of paramount importance to the credibility of the SA and the consequent model reduction strategy. In our case, the emulator proved able to capture the entire variability in JCFs of the validation dataset using only 50 training points. The excellent behaviour on a validation dataset three times larger than the training dataset clearly shows that the emulator training has been effective, and the risk of overfitting has been averted [44]. When applied to SSM and SGM this sensitivity analysis identified two distinct sets of muscles whose F\({}_{\text{max}}\) contribute to the determination of total JCFs. This was likely due to the above-mentioned differences in joints and muscle path definition and in the values assigned to F\({}_{\text{max}}\), leading to different actuation strategies. This also confirmed the existence of a component of model-dependency in SA outcomes, as hypothesised in previous literature [13, 20], and showed that different sets of muscle F\({}_{\text{max}}\) are influential in determining total JCFs when using generic-scaled or subject-specific models as a template for the generation of virtual populations. Therefore, ad hoc sensitivity analyses should be conducted whenever developing a new model. The results from this study support the idea that, because of their computationally expensive nature, such sensitivity studies will benefit  
Figure 5: RMSD (**left**) and \(\Delta\text{max}\) (**right**) between fully personalised model SSM and reduced models SSM\({}_{\text{m}}\), as a function of the number of personalised muscles. from the adoption of statistical emulators to produce Sobol points. Conducting this analysis using deterministic OpenSim would have required 30,000 model evaluations each for SGM and SSM. With the typical processing times of 30 s per simulations this would amount to more than 20 days of continuous computing on single core machines.