Constitutive Models for Active Skeletal Muscle: Review, Comparison, and Application in a Novel Continuum Shoulder Model

TL;DR

This contribution proposes a constitutive description to model active skeletal muscle within complex musculoskeletal systems, focusing on a novel continuum shoulder model and demonstrates the applicability of the proposed material model in a novel continuum mechanical model of the human shoulder.

Abstract

The shoulder joint is one of the functionally and anatomically most sophisticated articular systems in the human body. Both complex movement patterns and the stabilization of the highly mobile joint rely on intricate three-dimensional interactions among various components. Continuum-based finite element models can capture such complexity, and are thus particularly relevant in shoulder biomechanics. Considering their role as active joint stabilizers and force generators, skeletal muscles require special attention regarding their constitutive description. In this contribution, we propose a constitutive description to model active skeletal muscle within complex musculoskeletal systems, focusing on a novel continuum shoulder model. We thoroughly review existing material models before analyzing three selected ones in detail: an active-stress, an active-strain, and a generalized active-strain approach. To establish a basis for comparison, we identify the material parameters based on experimental stress-strain data obtained under various active and passive loading conditions. We discuss the concepts to incorporate active contractile behavior from a mathematical and physiological perspective, address analytical and numerical challenges arising from the mathematical formulations, and analyze the included biophysical principles of force generation in terms of physiological correctness and relevance for human shoulder modeling. Based on these insights, we present an improved constitutive model combining the studied models' most promising and relevant features. Using the example of a fusiform muscle, we investigate force generation, deformation, and kinematics during active isometric and free contractions. Eventually, we demonstrate the applicability of the suggested material model in a novel continuum mechanical model of the human shoulder.

Figures (34)

• Figure 1: Rheological models representing different concepts of muscular activation as in klotz_physiology-guided_2021.
• Figure 2: Schematic overview of the investigated constitutive laws for active skeletal muscle tissue.
• Figure 3: Active force-stretch-dependencies $f_\mathrm{\xi}$ in Eq. \ref{['eq:fxi']} and $f_\mathrm{\xi}^{\mathrm{a}}$ in Eq. \ref{['eq:fxia_blemker']}.
• Figure 4: Adapted passive force-stretch-dependency $f_\mathrm{\xi}^{\mathrm{p}}$ in Eq. \ref{['eq:fxip_blemker']}. The original $f_\mathrm{\xi}^{\mathrm{p}}$ in blemker_three-dimensional_2005 is zero for $\lambda \leq {\lambda_{\mathrm{opt}}^\mathrm{a}}$.
• Figure 5: Time-dependent activation functions ${f_\mathrm{t}^\mathrm{twitch}}$ in Eq. \ref{['eq:Poptft']} and ${f_\mathrm{t}^\mathrm{tanh}}$ in Eq. \ref{['eq:ft_blemker']}.