# Reducing the Complexity of Musculoskeletal Models Using Gaussian Process Emulators
## 1 Introduction  
Musculoskeletal models (MSKMs) are a commonly adopted solution to estimate biomechanical parameters otherwise not directly measurable, to predict the outcome of interventions, to inform rehabilitation planning, or to test complex scientific hypotheses within clinical context [1; 2; 3]. MSKMs can be divided into two main categories: generic, and subject-specific. Generic models are constructed by scaling a reference model using anthropometric measures [1; 4], while subject-specific models rely on medical images and their segmentation for the personalisation of anatomical features and input parameters [5; 6; 7; 8]. They are known to provide accurate estimates of specific biomechanical quantities for an individual [8; 9]. Amongst these quantities, individual muscle forces and joint contact forces (JCFs) are of particular interest in the assessment of joint loading and have been linked to the onset and progression of degenerative diseases of the musculoskeletal system (i.e., juvenile idiopathic arthritis [3] and osteoarthritis [10; 11]). The creation of a subject-specific model is a time-consuming and operator-dependent process, which limits the applicability of imaging-based MSK modelling protocols. Overcoming these limitations would widen their usability across the biomechanical community, and ultimately in the clinical practice. Sensitivity studies can identify the parameters that influence models' output, and inform strategies that guide model personalisation and limit pre-processing time to the minimum while still guaranteeing the accuracy of the predictions [12]. Several studies reported differences in model outputs when using MRI-based or generic anatomy [10, 13, 14], different joint types and degrees of freedom [15, 16, 17], or different muscle geometry and musculotendon parameters [6, 7, 12]. Since musculotendon parameters are used to describe the Hill-type behaviour of each muscle when generating force [18], they play a crucial role in the solution of the optimisation problem (which depends on the chosen minimisation criterion) and the determination of individual muscle forces and associated JCFs. With regard to muscle parameters in particular, existing sensitivity studies conducted on MSKMs reported that tendon slack length and optimal fibre length are amongst the most important parameters for characterising muscle behaviour and influencing JCFs, followed by maximal isometric force (F\({}_{\text{max}}\)) and others [12, 19, 20]. While tendon slack length and optimal fibre length can only be characterised through cadaveric measurements and ad hoc imaging protocols [8], respectively, F\({}_{\text{max}}\) of a muscle can be easily estimated from MRI [21]. In fact, the maximal force that a muscle can express is known to be proportional to its cross-sectional area and specific tension [22]. High resolution MRI can be segmented to extract a measure of individual muscle cross-sectional area and data from young adults [21] have been previously used to inform MSKMs' sensitivity analyses [4, 7]. A further dataset of measurements was made available by our group collecting data from older women [23] representing a novel resource compared to existing data. In addition, sensitivity analyses on models with different characteristics may lead to contrasting findings or results that are not directly comparable and hence raise the question whether a more comprehensive study should include different types of models and participants to test model- and subject-dependency of model sensitivity. [PERSON] and co-workers [20] conducted a sensitivity study combining several layers of input uncertainties and compared results across three participants to test subject-dependency. However, they based the analysis on a generic model and used a Monte Carlo approach relying on a limited number of simulations. In fact, to test the sensitivity of a model with \(N\) input parameters O(1000 \(\times\)\(N\)) model evaluations are typically required [24], a task that can prove prohibitive even for relatively small models.