Sensitivity of calibrated muscle–tendon parameters to methodological choices in children with cerebral palsy

Abstract

Introduction & Purpose

Musculoskeletal modelling estimates muscle and joint contact forces not directly measurable in-vivo (Kainz & Schwartz, 2021). Calibration of muscle–tendon parameters can yield more physiologically plausible outputs (Davico et al., 2022) but requires methodological choices (Davico et al., 2020). Sensitivity to calibration settings and trial selection remains insufficiently investigated (Bennett et al., 2022). Muscle–tendon morphology and properties may differ from generic models, particularly in athletes and pathological cases such as children with cerebral palsy (CP) (Kubo et al., 2011; Veerkamp et al., 2023). We investigated how calibration settings and trial selection influence calibrated muscle–tendon parameters in children with CP.

Methods

Three gait cycles from each of three children with CP were retrospectively analysed (Kainz et al., 2021). Motion capture and electromyography data from eight lower limb muscles were processed. A generic musculoskeletal model (Lerner et al., 2015; Rajagopal et al., 2016; Uhlrich et al., 2022) was scaled to each participant, muscle–tendon parameters optimized (Modenese et al., 2016), and muscle moment arms checked for discontinuities (Koller et al., 2025). OpenSim-derived joint moments and muscle–tendon lengths (Delp et al., 2007) were used for CEINMS calibration (Hambly et al., 2025; Pizzolato et al., 2015). Optimal fibre length, tendon slack length, and strength coefficient were calibrated. Three loss-function weightings—moment error (ME), excitations squared (ES), and synergy extraction (SE)—were varied between 10, 100, and 1000 to test sensitivity across different relative contributions, yielding 27 combinations. Included gait trials (t) varied between one (t1,t2), two (t1&t2), and three (t1&t2&t3). Weighting-related sensitivity was assessed within trial  configurations and trial-related sensitivity between configurations using matched weighting values. Bounds followed previous CP modelling: 95–105% for optimal fibre and tendon slack lengths to maintain physiological plausibility and 50–150% for strength coefficients due to greater uncertainty in maximum isometric force (Veerkamp et al., 2019). Parameter changes were normalized to their permitted ranges for comparability. Six right lower-limb muscles were analysed.

Results

Weighting-related sensitivity was low: across subjects, 76.9% of parameters fell within an interval spanning 5% of the permitted calibration range. Tendon slack length was the most sensitive parameter, with m. rectus femoris showing highest sensitivity. Trial-related median changes were 3.2%, 0.5%, and 1.1% of the permitted calibration range for selecting another, adding a second, and adding a third trial, respectively. Despite this, selecting another trial shifted median rectus femoris optimal fibre length by 97.5% of the permitted calibration range in one subject, nearly spanning the calibration limits (CP3 in Figure 1). Optimal fibre length showed the highest trial-related sensitivity.

Discussion

Parameters were generally robust to weighting and trial choice. Trial selection affected them more than adding trials. With three participants, findings are exploratory and may not represent broader CP populations. Narrower length bounds may have limited observable variability, whereas wider bounds could allow physiologically implausible values. The selected bounds therefore balanced sensitivity to parameter variation against physiological plausibility. Further studies should include more participants and movements.

Conclusion

Parameters were robust; selecting different trials caused greater variability than adding trials.

References

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