Abstract
Skeletal muscles usually wrap over multiple anatomical features, and their mass moves along the curved muscle paths during human locomotion. However, existing musculoskeletal models simply lump the mass of muscles to the nearby body segments without considering the effect of mass flow, which has been shown to induce non-negligible errors. A mass-variable multibody formulation is proposed here to simultaneously characterize muscle wrapping and mass flow effects. To achieve this goal, a novel cable element of the muscle–tendon unit, which integrates the mass flow feature with a typical Hill-type constitutive relationship, was developed based on an arbitrary Lagrangian–Eulerian description. In addition, sliding joints were used to constrain the elements to move over the underlying bone geometries. After validating the proposed modeling method using two benchmark samples, it was applied to build a large-scale lower limb musculoskeletal model, where knee joint moments were calculated and compared with isokinetic dynamometry measurements of 12 healthy males. The results of the comparison confirm that muscular mass distribution play an important role in the force transmission of muscle wrapping, and the proposed mass-variable formulation provides a better way of predicting and understanding the dynamics of musculoskeletal systems.
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Acknowledgements
This work was partially supported by the National Natural Science Foundation of China (Grant No. 11872221 and 11302114), the Beijing Nova Program Interdisciplinary Cooperation Project (xxjc201705), the Fund of Clinical Key Projects of Peking University Third Hospital (BYSY2017012) and the Key Laboratory of Photoelectronic Imaging Technology and System, Beijing Institute of Technology, Ministry of Education of China (2017OEIOF08).
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Jianqiao Guo and Hongshi Huang contributed equally to this work as first authors. Correspondence and requests for materials should be addressed to Zhihua Zhao, Gexue Ren or Yingfang Ao. Jianqiao Guo, Zhihua Zhao, and Gexue Ren designed and developed the software used in analysis; Jianqiao Guo, Hongshi Huang, and Yingfang Ao conceived and designed the experiments; Jianqiao Guo, Hongshi Huang, Yuanyuan Yu, and Zixuan Liang performed the experiments; Jianqiao Guo, Zhihua Zhao, and Hongshi Huang wrote the paper; Yuanyuan Yu, Zixuan Liang, Jorge Ambrósio, Gexue Ren, and Yingfang Ao revised the paper. All authors gave final approval for publication.
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Appendix: Appe Analytical equations of the sliding hyoid model
Appendix: Appe Analytical equations of the sliding hyoid model
Two assumptions should be adopted to simplify the system:
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1.
The inertia force and weight of the digastric muscle are neglected.
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2.
The hyoid bone always contacts the muscle without friction.
According to Assumption 1, the two muscle bellies are both straight-line segments, and the two segments generate the same amount of tensile force \(F^{\mathrm{mus}}\) owing to Assumption 2.
As shown in Fig. 14, the origin point \(O\) is defined at the middle of the two suspension points \(A(-0.25l^{\mathrm{mus}}_{0}, 0)\) and \(B(0.25l^{\mathrm{mus}}_{0}, 0)\), and the location of the sliding mass is given as \(C(x,y)\). Thus, the deformed length of the muscle \(l^{\mathrm{mus}}\) is:
where \(l_{1}\), \(l_{2}\) are the deformed lengths of each muscle segment, and the length change rate \(\dot{l}^{\mathrm{mus}}\) can be obtained by the time derivative of \(l^{\mathrm{mus}}\).
The governing dynamic equations of the whole system are derived based on the Newton–Euler formulation:
with
Here, \(m_{\mathrm{hyo}}\) is the sliding mass, \(g\) is the gravity, and \(\theta _{1}\), \(\theta _{2}\) are the angles between the two muscles and the horizontal line, respectively.
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Guo, J., Huang, H., Yu, Y. et al. Modeling muscle wrapping and mass flow using a mass-variable multibody formulation. Multibody Syst Dyn 49, 315–336 (2020). https://doi.org/10.1007/s11044-020-09733-1
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DOI: https://doi.org/10.1007/s11044-020-09733-1