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Multibody System Dynamics

, Volume 36, Issue 2, pp 195–219 | Cite as

A fast multi-obstacle muscle wrapping method using natural geodesic variations

  • Andreas ScholzEmail author
  • Michael Sherman
  • Ian Stavness
  • Scott Delp
  • Andrés Kecskeméthy
Article
First Online:

Abstract

Musculoskeletal simulation has become an essential tool for understanding human locomotion and movement disorders. Muscle-actuated simulations require methods that continuously compute musculotendon paths, their lengths, and their rates of length change to determine muscle forces, moment arms, and the resulting body and joint loads. Musculotendon paths are often modeled as locally length minimizing curves that wrap frictionlessly over moving obstacle surfaces representing bone and tissue. Biologically accurate wrapping surfaces are complex, and a single muscle path may wrap around many obstacles. However, state-of-the-art muscle wrapping methods are either limited to analytical results for a pair of simple surfaces, or they are computationally expensive. In this paper, we introduce the Natural Geodesic Variation (NGV) method for the fast and accurate computation of a musculotendon’s shortest path across an arbitrary number of general smooth wrapping surfaces, and an explicit formula for the path’s exact rate of length change. The total path is regarded as a concatenation of straight-line segments between local surface geodesics, where each geodesic is naturally parameterized by its starting point, direction, and length. The shortest path is computed by finding the root of a global path-error constraint equation that enforces that the geodesics connect collinearly with adjacent straight-line segments. High computational speed is achieved using Newton’s method to zero the path error, with an explicit, banded Jacobian that maps natural variations of the geodesic parameters to path-error variations. Three simulation benchmarks demonstrate that the NGV method computes high-precision solutions for path length and rate of length change, allows for wrapping over biologically accurate surfaces, and is capable of simulating muscle paths over hundreds of surfaces in real time. We thus believe the NGV method will facilitate the development of more accurate yet very efficient musculoskeletal models.

Keywords

Muscle wrapping Musculotendon path Shortest path Geodesic Geodesic variation Jacobi field 
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Notes

Acknowledgements

The authors gratefully acknowledge the support of Leonidas Guibas, Adrian Butscher, and Justin Solomon when discussing Jacobi fields; Matthew Millard for his valuable feedback on the method, the manuscript and the figures; and Francisco Geu Flores for his support in benchmarking.

References

  1. 1.
    Arnold, A.S., Blemker, S.S., Delp, S.L.: Evaluation of a deformable musculoskeletal model for estimating muscle-tendon lengths during crouch gait. Ann. Biomed. Eng. 29, 263–274 (2001) CrossRefGoogle Scholar
  2. 2.
    Kerr, G.H., Selber, P.: Musculoskeletal aspects of cerebral palsy. J. Bone Jt. Surg.—Br. Vol. 85(2), 157–166 (2003) CrossRefGoogle Scholar
  3. 3.
    Arnold, A.S., Anderson, F.C., Pandy, M.G., Delp, S.L.: Muscular contributions to hip and knee extension during the single limb stance phase of normal gait: a framework for investigating the causes of crouch gait. J. Biomech. 38, 2181–2189 (2005) CrossRefGoogle Scholar
  4. 4.
    Hicks, J.L., Schwartz, M.H., Arnold, A.S., Delp, S.L.: Crouched postures reduce the capacity of muscles to extend the hip and knee during the single-limb stance phase of gait. J. Biomech. 41, 960–967 (2008) CrossRefGoogle Scholar
  5. 5.
    Steele, K.M., Seth, A., Hicks, J.L., Schwartz, M.S., Delp, S.L.: Muscle contributions to support and progression during single-limb stance in crouch gait. J. Biomech. 43, 2099–2105 (2010) CrossRefGoogle Scholar
  6. 6.
    Neptune, R.R., Kautz, S.A., Zajac, F.E.: Contributions of the individual ankle plantar flexors to support, forward progression and swing initiation during walking. J. Biomech. 34, 1387–1398 (2001) CrossRefGoogle Scholar
  7. 7.
    Zajac, F.E., Neptune, R.R., Kautz, S.A.: Biomechanics and muscle coordination of human walking. Part 1: introduction to concepts, power transfer, dynamics and simulations. Gait Posture 16, 215–232 (2002) CrossRefGoogle Scholar
  8. 8.
    Zajac, F.E., Neptune, R.R., Kautz, S.A.: Biomechanics and muscle coordination of human walking. Part 2: lessons from dynamical simulations and clinical implications. Gait Posture 17, 1–17 (2003) CrossRefGoogle Scholar
  9. 9.
    Liu, M.Q., Anderson, F.C., Pandy, M.G., Delp, S.L.: Muscles that support the body also modulate forward progression during walking. J. Biomech. 39, 2623–2630 (2006) CrossRefGoogle Scholar
  10. 10.
    van der Krogt, M.M., Delp, S.L., Schwartz, M.H.: How robust is human gait to muscle weakness? Gait Posture 36, 113–119 (2012) CrossRefGoogle Scholar
  11. 11.
    Neptune, R.R., Sasaki, K.: Ankle plantar flexor force production is an important determinant of the preferred walk-to-run transition speed. J. Exp. Biol. 208, 799–808 (2005) CrossRefGoogle Scholar
  12. 12.
    Hamner, S.R., Seth, A., Delp, S.L.: Muscle contributions to propulsion and support during running. J. Biomech. 43, 2709–2716 (2010) CrossRefGoogle Scholar
  13. 13.
    van der Helm, F.C.T.: The shoulder mechanism: a dynamic approach. Ph.D. Thesis. Delft University of Technology (1991) Google Scholar
  14. 14.
    Yu, J., Ackland, D.C., Pandy, M.G.: Shoulder muscle function depends on elbow joint position: an illustration of dynamic coupling in the upper limb. J. Biomech. 44, 1859–1868 (2011) CrossRefGoogle Scholar
  15. 15.
    Sasaki, K., Neptune, R.R.: Individual muscle contributions to the axial knee joint contact force during normal walking. J. Biomech. 43, 2780–2784 (2010) CrossRefGoogle Scholar
  16. 16.
    Lin, Y.-C., Walter, J.P., Banks, S.A., Pandy, M.G., Fregly, B.J.: Simultaneous prediction of muscle and contact forces in the knee during gait. J. Biomech. 43, 945–952 (2010) CrossRefGoogle Scholar
  17. 17.
    Winby, C.R., Lloyd, D.G., Besier, T.F., Kirk, T.B.: Muscle and external load contribution to knee joint contact loads during normal gait. J. Biomech. 42, 2294–2300 (2009) CrossRefGoogle Scholar
  18. 18.
    Moissenet, F., Chèze, L., Dumas, R.: A 3D lower limb musculoskeletal model for simultaneous estimation of musculo-tendon, joint contact, ligament and bone forces during gait. J. Biomech. (2013) Google Scholar
  19. 19.
    Giat, Y., Mizrahl, J., Levine, W.S., Chen, J.: Simulation of distal tendon transfer of the biceps brachii and the brachialis muscles. J. Biomech. 27(8), 1005–1014 (1994) CrossRefGoogle Scholar
  20. 20.
    Hill, A.V.: The mechanics of active muscle. Proc. - Royal Soc., Biol. Sci. 141, 104–117 (1953) CrossRefGoogle Scholar
  21. 21.
    Gordon, A.M., Huxley, A.F., Julian, F.J.: The variation in isometric tension with sarcomere length in vertebrae muscle fibers. J. Physiol. 184, 170–192 (1966) CrossRefGoogle Scholar
  22. 22.
    Bahler, A.S., Fales, J.T., Zierler, K.L.: The dynamic properties of mammalian skeletal muscle. J. Gen. Physiol. 51, 369–384 (1968) CrossRefGoogle Scholar
  23. 23.
    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
  24. 24.
    Millard, M., Uchida, T., Seth, A., Delp, S.L.: Flexing computational muscle: modeling and simulation of musculotendon dynamics. J. Biomech. Eng. 135, 021004-1–021004-11 (2013) Google Scholar
  25. 25.
    Brand, R.A., Crowninshield, R.D., Wittstock, C.E., Pedersen, D.R., Clark, C.R., van Krieken, F.M.: A model of lower extremity muscular anatomy. J. Biomech. Eng. 104(4), 304–310 (1982) CrossRefGoogle Scholar
  26. 26.
    Ackland, D.C., Pandy, M.G.: Lines of action and stabilizing potential of the shoulder musculature. J. Anat. 215, 184–197 (2009) CrossRefGoogle Scholar
  27. 27.
    Blemker, S.S., Asakawa, D.S., Gold, G.E., Delp, S.L.: Image-based musculoskeletal modeling: applications, advances, and future opportunities. J. Magn. Reson. Imaging 25, 441–451 (2007) CrossRefGoogle Scholar
  28. 28.
    Webb, J.D., Blemker, S.S., Delp, S.L.: 3D finite element models of shoulder muscles for computing lines of actions and moment arms. Comput. Methods Biomech. Biomed. Engin., 1–9 (2012) Google Scholar
  29. 29.
    An, K.-A., Berglund, L., Cooney, W.P., Chao, E.Y.S., Kovacevic, N.: Direct in vivo tendon force measurement system. J. Biomech. 23(12), 1269–1271 (1990) CrossRefGoogle Scholar
  30. 30.
    Schuind, F., Garcia-Elias, M., Cooney, W.P. III, An, K.-N.: Flexor tendon forces: in vivo measurements. J. Hand Surg. 17(2), 291–298 (1992) CrossRefGoogle Scholar
  31. 31.
    Garner, B.A., Pandy, M.G.: The obstacle-set method for representing muscle paths in musculoskeletal simulations. Comput. Methods Biomech. Biomed. Eng. 3, 1–30 (1999) CrossRefGoogle Scholar
  32. 32.
    Charlton, I.W., Johnson, G.R.: Application of spherical and cylindrical wrapping algorithms in a musculoskeletal model of the upper limb. J. Biomech. 34, 1209–1216 (2001) CrossRefGoogle Scholar
  33. 33.
    Gao, F., Damsgaard, M., Rasmussen, J., Christensen, S.T.: Computational method for muscle-path representation in musculoskeletal models. Biol. Cybern. 87, 199–210 (2002) CrossRefzbMATHGoogle Scholar
  34. 34.
    Marai, G.E., Laidlaw, D.H., Demiralp, C., Andrews, S., Grimm, C.M., Crisco, J.J.: Estimating joint contact areas and ligaments lengths from bone kinematics and surfaces. IEEE Trans. Biomed. Eng. 51(5), 790–799 (2004) CrossRefGoogle Scholar
  35. 35.
    Carman, A.B., Milburn, P.D.: Dynamic coordinate data for describing muscle-tendon paths: a mathematical approach. J. Biomech. 38, 943–951 (2005) CrossRefGoogle Scholar
  36. 36.
    Blemker, S.S., Delp, S.L.: Rectus femoris and vastus intermedius fiber excursions predicted by three-dimensional muscle models. J. Biomech. 39, 1383–1391 (2006) CrossRefGoogle Scholar
  37. 37.
    Marsden, S.P., Swailes, D.C., Johnson, G.R.: Algorithms for exact multi-object muscle wrapping and application to the deltoid muscle wrapping around the humerus. Proc. Inst. Mech. Eng., H J. Eng. Med. 222(7), 1081–1095 (2008) CrossRefGoogle Scholar
  38. 38.
    Röhrle, O., Davidson, J.B., Pullan, A.J.: Bridging scales: a three-dimensional electromechanical finite element model of skeletal muscle. SIAM J. Sci. Comput. 30(6), 2882–2904 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Audenaert, A., Audenaert, E.: Global optimization method for combined spherical-cylindrical wrapping in musculoskeletal upper limb modeling. Comput. Methods Programs Biomed. 92, 8–19 (2008) CrossRefGoogle Scholar
  40. 40.
    Gatti, C.J., Hughes, R.E.: Optimization of muscle wrapping objects using simulated annealing. Ann. Biomed. Eng. 37(7), 1342–1347 (2009) CrossRefGoogle Scholar
  41. 41.
    Vasavada, A.N., Lasher, R.A., Meyer, T.E., Lin, D.C.: Defining and evaluating wrapping surfaces for MRI-derived spinal muscle paths. J. Biomech. 41, 1450–1457 (2008) CrossRefGoogle Scholar
  42. 42.
    Arnold, E.M., Ward, S.R., Lieber, R.L., Delp, S.L.: A model of the lower limb for analysis of human movement. Ann. Biomed. Eng. 38(2), 269–279 (2010) CrossRefGoogle Scholar
  43. 43.
    Esat, I.I., Ozada, N.: Articular human joint modeling. Robotica 28, 321–339 (2010) CrossRefGoogle Scholar
  44. 44.
    Favre, P., Gerber, C., Snedeker, J.G.: Automated muscle wrapping using finite element detection. J. Biomech. 43, 1931–1940 (2010) CrossRefGoogle Scholar
  45. 45.
    Spyrou, L.A., Aravas, N.: Muscle-driven finite element simulation of human foot movements. Comput. Methods Biomech. Biomed. Eng. 5(9), 925–934 (2012) CrossRefGoogle Scholar
  46. 46.
    Stavness, I., Sherman, M., Delp, S.L.: A general approach to muscle wrapping over multiple surfaces. Florida, USA, 2012. Proc. Amer. Soc. Biomech. (2012) Google Scholar
  47. 47.
    Scholz, A., Stavness, I., Sherman, M., Delp, S.L., Kecskeméthy, A.: Improved muscle wrapping algorithms using explicit path-error Jacobians. Barcelona, Spain, 2012. Comput. Kinematics. (2012) Google Scholar
  48. 48.
    Desailly, E., Sardain, P., Khouri, N., Yepremian, D., Lacouture, P.: The convex wrapping algorithm: a method for identifying muscle paths using the underlying bone mesh. J. Biomech. 43, 2601–2607 (2010) CrossRefGoogle Scholar
  49. 49.
    Struik, D.J.: Lectures on Classical Differential Geometry. Dover, New York (1988) zbMATHGoogle Scholar
  50. 50.
    Strubecker, K.: Differentialgeometrie Band 3: Theorie der Flächenkrümmung. de Gruyter, Berlin (1969) Google Scholar
  51. 51.
    do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice Hall, New York (1976) zbMATHGoogle Scholar
  52. 52.
    Pressley, A.: Elementary Differential Geometry. Springer, Berlin (2010) CrossRefzbMATHGoogle Scholar
  53. 53.
    do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Basel (1993) Google Scholar
  54. 54.
    Strubecker, K.: Differentialgeometrie Band 2: Theorie der Flächenmetrik. de Gruyter, Berlin (1969) Google Scholar
  55. 55.
    Kecskeméthy, A., Hiller, M.: An object-oriented approach for an effective formulation of multibody dynamics. Comput. Methods Appl. Math. 115(3–4), 287–314 (1994) Google Scholar
  56. 56.
    Pai, D.K.: Muscle mass in musculoskeletal models. J. Biomech. 43(11), 2093–2098 (2010) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Andreas Scholz
    • 1
    Email author
  • Michael Sherman
    • 2
  • Ian Stavness
    • 3
  • Scott Delp
    • 4
  • Andrés Kecskeméthy
    • 5
  1. 1.Department of Mechanical EngineeringUniversity of Duisburg-EssenDuisburgGermany
  2. 2.Department of BioengineeringStanford UniversityStanfordUSA
  3. 3.Department of Computer ScienceUniversity of SaskatchewanSaskatoonCanada
  4. 4.Departments of Bioengineering and Mechanical EngineeringStanford UniversityStanfordUSA
  5. 5.Chair of Mechanics and RoboticsUniversity of Duisburg-EssenDuisburgGermany

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