Skip to main content
Log in

Modeling muscle wrapping and mass flow using a mass-variable multibody formulation

  • Published:
Multibody System Dynamics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Al Nazer, R., Klodowski, A., Rantalainen, T., Heinonen, A., Sievänen, H., Mikkola, A.: Analysis of dynamic strains in tibia during human locomotion based on flexible multibody approach integrated with magnetic resonance imaging technique. Multibody Syst. Dyn. 20(4), 287–306 (2008)

    Article  MATH  Google Scholar 

  2. Baldwin, M.A., Clary, C., Maletsky, L.P., Rullkoetter, P.J.: Verification of predicted specimen-specific natural and implanted patellofemoral kinematics during simulated deep knee bend. J. Biomech. 42(14), 2341–2348 (2009)

    Article  Google Scholar 

  3. Benjamin, M.: The fascia of the limbs and back – a review. J. Anat. 214(1), 1–18 (2009)

    Article  Google Scholar 

  4. Blankevoort, L., Huiskes, R.: Ligament-bone interaction in a three-dimensional model of the knee. J. Biomech. Eng. 113(3), 263–269 (1991)

    Article  Google Scholar 

  5. Czaplicki, A., Silva, M., Ambrósio, J., Jesus, O., Abrantes, J.: Estimation of the muscle force distribution in ballistic motion based on a multibody methodology. Comput. Methods Biomech. Biomed. Eng. 9(1), 45–54 (2006)

    Article  Google Scholar 

  6. Delp, S.L., Loan, J.P., Hoy, M.G., Zajac, F., Topp, E., Rosen, J.: An interactive graphics-based model of the lower extremity to study orthopedic surgical procedures. IEEE Trans. Biomed. Eng. 37(8), 757–767 (1990)

    Article  Google Scholar 

  7. Earp, J.E., Newton, R.U., Cormie, P., Blazevich, A.J.: Knee angle-specific EMG normalization: the use of polynomial based EMG-angle relationships. J. Electromyogr. Kinesiol. 23(1), 238–244 (2013)

    Article  Google Scholar 

  8. El-Rich, M., Shirazi-Adl, A.: Effect of load position on muscle forces, internal loads and stability of the human spine in upright postures. Comput. Methods Biomech. Biomed. Eng. 8(6), 359–368 (2005)

    Article  Google Scholar 

  9. Ernst, H., Gerhard, W.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 14th edn. Springer Berlin Heidelberg, Berlin, Heidelberg (1991)

    MATH  Google Scholar 

  10. Fu, K., Zhao, Z., Ren, G., Xiao, Y., Feng, T., Yang, J., Gasbarri, P.: From multiscale modeling to design of synchronization mechanisms in mesh antennas. Acta Astronaut. 159, 156–165 (2019)

    Article  Google Scholar 

  11. Gantoi, F.M., Brown, M.A., Shabana, A.A.: ANCF finite element/multibody system formulation of the ligament/bone insertion site constraints. J. Comput. Nonlinear Dyn. 5(3), 31,006–31,009 (2010)

    Article  Google Scholar 

  12. Garner, B.A., Pandy, M.G.: The obstacle-set method for representing muscle paths in musculoskeletal models. Comput. Methods Biomech. Biomed. Eng. 3(1), 1–30 (2000)

    Article  Google Scholar 

  13. Gérard, J.M., Ohayon, J., Luboz, V., Perrier, P., Payan, Y.: Indentation for estimating the human tongue soft tissues constitutive law: application to a 3D biomechanical model. Lect. Notes Comput. Sci. 3078, 77–83 (2004). https://hal.archives-ouvertes.fr/hal-00081929

    Article  Google Scholar 

  14. Gollapudi, S.K., Lin, D.C.: Experimental determination of sarcomere force–length relationship in type-I human skeletal muscle fibers. J. Biomech. 42(13), 2011–2016 (2009)

    Article  Google Scholar 

  15. Han, M., Hong, J., Park, F.C.: Musculoskeletal dynamics simulation using shape-varying muscle mass models. Multibody Syst. Dyn. 33(4), 367–388 (2015)

    Article  MathSciNet  Google Scholar 

  16. Hannam, A.G., Stavness, I., Lloyd, J.E., Fels, S.: A dynamic model of jaw and hyoid biomechanics during chewing. J. Biomech. 41(5), 1069–1076 (2008)

    Article  Google Scholar 

  17. Heers, G., O’Driscoll, S.W., Halder, A.M., Zhao, C., Mura, N., Berglund, L.J., Zobitz, M.E., An, K.N.: Gliding properties of the long head of the biceps brachii. J. Orthop. Res. 21(1), 162–166 (2003)

    Article  Google Scholar 

  18. Hong, D., Ren, G.: A modeling of sliding joint on one-dimensional flexible medium. Multibody Syst. Dyn. 26(1), 91–106 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hong, D., Tang, J., Ren, G.: Dynamic modeling of mass-flowing linear medium with large amplitude displacement and rotation. J. Fluids Struct. 27(8), 1137–1148 (2011)

    Article  Google Scholar 

  20. Horsman, M.K., Koopman, H., van der Helm, F., Prosé, L.P., Veeger, H.: Morphological muscle and joint parameters for musculoskeletal modelling of the lower extremity. Clin. Biomech. 22(2), 239–247 (2007)

    Article  Google Scholar 

  21. Huang, H., Guo, J., Yang, J., Jiang, Y., Yu, Y., Müller, S., Ren, G., Ao, Y.: Isokinetic angle-specific moments and ratios characterizing hamstring and quadriceps strength in anterior cruciate ligament deficient knees. Sci. Rep. 7(1), 7269 (2017)

    Article  Google Scholar 

  22. Huberti, H.H., Hayes, W.C.: Patellofemoral contact pressures. The influence of q-angle and tendofemoral contact. J. Bone Jt. Surg., Am. 66(5), 715–724 (1984)

    Article  Google Scholar 

  23. Joyce, G.C., Rack, P.M.H., Westbury, D.R.: The mechanical properties of cat soleus muscle during controlled lengthening and shortening movements. J. Physiol. 204(2), 461–474 (1969)

    Article  Google Scholar 

  24. Kłodowski, A., Rantalainen, T., Mikkola, A., Heinonen, A., Sievänen, H.: Flexible multibody approach in forward dynamic simulation of locomotive strains in human skeleton with flexible lower body bones. Multibody Syst. Dyn. 25(4), 395–409 (2011)

    Article  MATH  Google Scholar 

  25. Kłodowski, A., Valkeapää, A., Mikkola, A.: Pilot study on proximal femur strains during locomotion and fall-down scenario. Multibody Syst. Dyn. 28(3), 239–256 (2012)

    Article  MathSciNet  Google Scholar 

  26. Koolstra, J.H., van Eijden, T.: Combined finite-element and rigid-body analysis of human jaw joint dynamics. J. Biomech. 38(12), 2431–2439 (2005)

    Article  Google Scholar 

  27. Krylow, A.M., Sandercock, T.G.: Dynamic force responses of muscle involving eccentric contraction. J. Biomech. 30(1), 27–33 (1997)

    Article  Google Scholar 

  28. Lloyd, D.G., Besier, T.F.: An EMG-driven musculoskeletal model to estimate muscle forces and knee joint moments in vivo. J. Biomech. 36(6), 765–776 (2003)

    Article  Google Scholar 

  29. Marieb, E.N.: Human Anatomy & Physiology: Pearson New International Edition. Pearson Education, UK (2013)

    Google Scholar 

  30. Mashima, H.: Force-velocity relation and contractility in striated muscles. Jpn. J. Phys. 34(1), 1–17 (1984)

    Article  Google Scholar 

  31. Mason, J.J., Leszko, F., Johnson, T., Komistek, R.D.: Patellofemoral joint forces. J. Biomech. 41(11), 2337–2348 (2008)

    Article  Google Scholar 

  32. Michael, G., Röhrle, O., Daniel, F.B. Haeufle, Schmitt, S.: Spreading out muscle mass within a Hill-type model: a computer simulation study. Comput. Math. Methods Med. 2012, 848,630 (2012)

    MathSciNet  MATH  Google Scholar 

  33. Millard, M., Uchida, T., Seth, A., Delp, S.L.: Flexing computational muscle: modeling and simulation of musculotendon dynamics. J. Biomech. Eng. 135(2), 21,005 (2013)

    Article  Google Scholar 

  34. Miller, R.K., Murray, D.W., Gill, H.S., O’Connor, J.J., Goodfellow, J.W.: In vitro patellofemoral joint force determined by a non-invasive technique. Clin. Biomech. 12(1), 1–7 (1997)

    Article  Google Scholar 

  35. Mohamed, A.N.A., Brown, M.A., Shabana, A.A.: Study of the ligament tension and cross-section deformation using nonlinear finite element/multibody system algorithms. Multibody Syst. Dyn. 23(3), 227–248 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Nazer, R.A., Rantalainen, T., Heinonen, A., Sievänen, H., Mikkola, A.: Flexible multibody simulation approach in the analysis of tibial strain during walking. J. Biomech. 41(5), 1036–1043 (2008)

    Article  MATH  Google Scholar 

  37. Nedel, L.P., Thalmann, D.: Real time muscle deformations using mass–spring systems. In: Proceedings of the Computer Graphics International 1998. IEEE Computer Society, Washington, DC, USA (1998)

    Google Scholar 

  38. Pai, D.K.: Muscle mass in musculoskeletal models. J. Biomech. 43(11), 2093–2098 (2010)

    Article  MathSciNet  Google Scholar 

  39. Peng, Y., Wei, Y., Zhou, M.: Efficient modeling of cable-pulley system with friction based on arbitrary-Lagrangian–Eulerian approach. Appl. Math. Mech. 38(12), 1785–1802 (2017)

    Article  MathSciNet  Google Scholar 

  40. Peng, Y., Zhao, Z., Zhou, M., He, J., Yang, J., Xiao, Y.: Flexible multibody model and the dynamics of the deployment of mesh antennas. J. Guid. Control Dyn. 40(6), 1499–1510 (2017)

    Article  Google Scholar 

  41. Piazza, S.J., Delp, S.L.: Three-dimensional dynamic simulation of total knee replacement motion during a step-up task. J. Biomech. Eng. 123(6), 599–606 (2001)

    Article  Google Scholar 

  42. Pizzolato, C., Lloyd, D.G., Sartori, M., Ceseracciu, E., Besier, T.F., Fregly, B.J., Reggiani, M.: CEINMS: a toolbox to investigate the influence of different neural control solutions on the prediction of muscle excitation and joint moments during dynamic motor tasks. J. Biomech. 48(14), 3929–3936 (2015)

    Article  Google Scholar 

  43. Quental, C., Folgado, J., Ambrósio, J., Monteiro, J.: A multibody biomechanical model of the upper limb including the shoulder girdle. Multibody Syst. Dyn. 28(1–2), 83–108 (2012)

    Article  MathSciNet  Google Scholar 

  44. Sartori, M., Farina, D., Lloyd, D.G.: Hybrid neuromusculoskeletal modeling to best track joint moments using a balance between muscle excitations derived from electromyograms and optimization. J. Biomech. 47(15), 3613–3621 (2014)

    Article  Google Scholar 

  45. Scholz, A., Sherman, M., Stavness, I., Delp, S., Kecskeméthy, A.: A fast multi-obstacle muscle wrapping method using natural geodesic variations. Multibody Syst. Dyn. 36(2), 195–219 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  46. Serpas, F., Yanagawa, T., Pandy, M.: Forward-dynamics simulation of anterior cruciate ligament forces developed during isokinetic dynamometry. Comput. Methods Biomech. Biomed. Eng. 5(1), 33–43 (2002)

    Article  Google Scholar 

  47. Siebert, T., Till, O., Blickhan, R.: Work partitioning of transversally loaded muscle: experimentation and simulation. Comput. Methods Biomech. Biomed. Eng. 17(3), 217–229 (2014)

    Article  Google Scholar 

  48. Silva, M.P.T., Ambrósio, J.A.C.: Solution of redundant muscle forces in human locomotion with multibody dynamics and optimization tools. Mech. Based Des. Struct. Mach. 31(3), 381–411 (2003)

    Article  Google Scholar 

  49. Suderman, B.L., Krishnamoorthy, B., Vasavada, A.N.: Neck muscle paths and moment arms are significantly affected by wrapping surface parameters. Comput. Methods Biomech. Biomed. Eng. 15(7), 735–744 (2012)

    Article  Google Scholar 

  50. Sueda, S., Kaufman, A., Pai, D.K.: Musculotendon simulation for hand animation. In: ACM Transactions on Graphics, vol. 27, p. 1 (2008)

    Google Scholar 

  51. Thelen, D.G.: Adjustment of muscle mechanics model parameters to simulate dynamic contractions in older adults. J. Biomech. Eng. 125(1), 70–77 (2003)

    Article  Google Scholar 

  52. Toumi, H., Larguech, G., Filaire, E., Pinti, A., Lespessailles, E.: Regional variations in human patellar trabecular architecture and the structure of the quadriceps enthesis: a cadaveric study. J. Anat. 220(6), 632–637 (2012)

    Article  Google Scholar 

  53. Van Eijden, T.M., Korfage, J.A., Brugman, P.: Architecture of the human jaw-closing and jaw-opening muscles. Anat. Rec. 248(3), 464–474 (1997)

    Article  Google Scholar 

  54. Winters, J.M.: An improved muscle-reflex actuator for use in large-scale neuromusculoskeletal models. Ann. Biomed. Eng. 23(4), 359–374 (1995)

    Article  Google Scholar 

  55. Winters, T.M., Takahashi, M., Lieber, R.L., Ward, S.R.: Whole muscle length–tension relationships are accurately modeled as scaled sarcomeres in rabbit hindlimb muscles. J. Biomech. 44(1), 109–115 (2011)

    Article  Google Scholar 

  56. Yang, C., Cao, D., Zhao, Z., Zhang, Z., Ren, G.: A direct eigenanalysis of multibody system in equilibrium. J. Appl. Math. 2012, 638,546 (2012)

    MathSciNet  MATH  Google Scholar 

  57. Zajac, F.E.: Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control. Crit. Rev. Biomed. Eng. 17(4), 359–411 (1989)

    Google Scholar 

  58. Zatsiorsky, V.M., Prilutsky, B.I.: Biomechanics of Skeletal Muscles. Human Kinetics, Champaign, IL (2012)

    Book  Google Scholar 

Download references

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Zhihua Zhao.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Author contributions

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.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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:

  1. 1.

    The inertia force and weight of the digastric muscle are neglected.

  2. 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:

$$\begin{aligned} \begin{aligned} l^{\mathrm{mus}} &= l_{1} + l_{2}, \\ l_{1} &= \sqrt{(0.25 l^{\mathrm{mus}}_{0}+x)^{2} + y^{2}}, \\ l_{2} &= \sqrt{(0.25 l^{\mathrm{mus}}_{0}-x)^{2} + y^{2}} \end{aligned} \end{aligned}$$
(15)

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 \(l^{\mathrm{mus}}\).

Fig. 14
figure 14

Illustration of the analytical model in Benchmark 2

The governing dynamic equations of the whole system are derived based on the Newton–Euler formulation:

$$\begin{aligned} \begin{aligned} m_{\mathrm{hyo}} \ddot{x} &= F^{\mathrm{mus}} \cos \theta _{2} - F^{\mathrm{mus}} \cos \theta _{1}, \\ m_{\mathrm{hyo}} \ddot{y} &= F^{\mathrm{mus}} \sin \theta _{2} + F^{\mathrm{mus}} \sin \theta _{1} - m_{\mathrm{hyo}}g, \end{aligned} \end{aligned}$$
(16)

with

$$\begin{aligned} \begin{aligned} \sin \theta _{1} &= -\frac{y}{l_{1}}, \quad \cos \theta _{1} = \frac{0.25 l^{\mathrm{mus}}_{0} + x}{l_{1}}, \\ \sin \theta _{2} &= -\frac{y}{l_{2}}, \quad \cos \theta _{2} = \frac{0.25 l^{\mathrm{mus}}_{0} - x}{l_{2}}. \end{aligned} \end{aligned}$$

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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11044-020-09733-1

Keywords

Navigation