Parametric and Sensitivity Analysis of Hill’s Three-Element Muscle Model Using the Finite Element Method: Influence of Material Parameters on Mechanical Response

Accurately capturing muscle behavior remains a challenging task in computational biomechanics, primarily due to the nonlinear response, anisotropy, and time-dependent characteristics of muscle tissue. In this context, finite element methods have proven to be a suitable framework for representing such complex mechanical behavior. Among the available constitutive approaches, Hill’s three-element model continues to be widely adopted, largely because it offers a reasonable balance between physiological interpretability and computational efficiency. In this work, a parametric and sensitivity-oriented analysis of the Hill three-element muscle model is performed within a finite element formulation originally proposed by Kojić, Mijailović, and Zdravković (1998) and implemented in the PAK software environment. The analysis considers five key parameters, which are varied independently: the stiffness parameter of the series elastic element (α), the corresponding stress scaling parameter (β), the modulus of the parallel elastic element (E), the activation level (a), and the length ratio constant (k). To enable comparison between parameters of different physical nature, normalized sensitivity indices are used. The results show that the activation parameter a has the strongest influence on active force generation, with an increase of 36.4% at the highest considered activation level. In contrast, parameters α and β primarily affect the behavior of the series elastic component, with variations on the order of ±15–18%. It can also be observed that the influence of individual parameters depends on the deformation regime. At lower deformation levels, the response is mainly governed by the parameter E, while α and β become more relevant in the intermediate nonlinear range. At higher deformation levels, the activation parameter a becomes dominant. From a modeling perspective, these findings suggest a structured approach to parameter calibration in Hill-type finite element models. In addition, they provide further insight into the sensitivity characteristics of such formulations within computational biomechanics.