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Biomechanics and Modeling in Mechanobiology

, Volume 15, Issue 1, pp 245–258 | Cite as

Contraction dynamics and function of the muscle-tendon complex depend on the muscle fibre-tendon length ratio: a simulation study

  • Falk MörlEmail author
  • Tobias Siebert
  • Daniel Häufle
Original Paper

Abstract

Experimental studies show different muscle-tendon complex (MTC) functions (e.g. motor or spring) depending on the muscle fibre-tendon length ratio. Comparing different MTC of different animals examined experimentally, the extracted MTC functions are biased by, for example, MTC-specific pennation angle and fibre-type distribution or divergent experimental protocols (e.g. influence of temperature or stimulation on MTC force). Thus, a thorough understanding of variation of these inner muscle fibre-tendon length ratios on MTC function is difficult. In this study, we used a hill-type muscle model to simulate MTC. The model consists of a contractile element (CE) simulating muscle fibres, a serial element (SE) as a model for tendon, and a parallel elastic element (PEE) modelling tissue in parallel to the muscle fibres. The simulation examines the impact of length variations of these components on contraction dynamics and MTC function. Ensuring a constant overall length of the MTC by \(L_\mathrm{MTC} = L_\mathrm{SE} + L_\mathrm{CE}\), the SE rest length was varied over a broad physiological range from 0.1 to 0.9 MTC length. Five different MTC functions were investigated by simulating typical physiological experiments: the stabilising function with isometric contractions, the motor function with contractions against a weight, the capability of acceleration with contractions against a small inertial mass, the braking function by decelerating a mass, and the spring function with stretch-shortening cycles. The ratio of SE and CE mainly determines the MTC function. MTC with comparably short tendon generates high force and maximal shortening velocity and is able to produce maximal work and power. MTC with long tendon is suitable to store and release a maximum amount of energy. Variation of muscle fibre-tendon ratio yielded two peaks for MTC’s force response for short and long SE lengths. Further, maximum work storage capacity of the SE is at long \(\mathrm{rel}L_\mathrm{SE,0}\). Impact of fibre-tendon length ratio on MTC functions will be discussed. Considering a constant set of MTC parameters, quantitative changes in MTC performance (work, stiffness, force, energy storage, dissipation) depending on varying muscle fibre-tendon length ratio were provided, which enables classification and grading of different MTC designs.

Keywords

Tendon length Biomechanics Simulation Direct dynamics Muscle model Energy storage 

Notes

Acknowledgments

The authors thank Michael Günther for the fruitful discussions and his comments on the manuscript. The study was partially supported by the Deutsche Forschungsgemeinschaft (DFG SI841/6,7 to TS and SCH2392/5-1).

Conflict of interest

None.

Supplementary material

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References

  1. Ahn AN, Full RJ (2002) A motor and a brake: two leg extensor muscles acting at the same joint manage energy differently in a running insect. J Exp Biol 205(Pt 3):379–389Google Scholar
  2. Alexander RM, Bennet-Clark HC (1977) Storage of elastic strain energy in muscle and other tissues. Nature 265(5590):114–117CrossRefGoogle Scholar
  3. Allen DG, Lamb GD, Westerblad H (2008) Skeletal muscle fatigue: cellular mechanisms. Physiol Rev 88(1):287–332. doi: 10.1152/physrev.00015.2007 CrossRefGoogle Scholar
  4. Ariano MA, Armstrong RB, Edgerton VR (1973) Hindlimb muscle fiber populations of five mammals. J Histochem Cytochem 21(1):51–55CrossRefGoogle Scholar
  5. Asmussen G, Maréchal G (1989) Maximal shortening velocities, isomyosins and fibre types in soleus muscle of mice, rats and guinea-pigs. J Physiol 416:245–254CrossRefGoogle Scholar
  6. Bennett AF (1985) Temperature and muscle. J Exp Biol 115:333–344Google Scholar
  7. Biewener AA, Baudinette RV (1995) In vivo muscle force and elastic energy storage during steady-speed hopping of tammar wallabies (macropus eugenii). J Exp Biol 198(Pt 9):1829–1841Google Scholar
  8. Biewener AA (1998) Muscle function in vivo: a comparison of muscles used for elastic energy savings versus muscles used to generate mechanical power. Am Zool 38:703–717CrossRefGoogle Scholar
  9. Biewener AA, Gillis GB (1999) Dynamics of muscle function during locomotion: accommodating variable conditions. J Exp Biol 202(Pt 23):3387–3396Google Scholar
  10. Biewener AA, Konieczynski DD, Baudinette RV (1998) In vivo muscle force-length behavior during steady-speed hopping in tammar wallabies. J Exp Biol 201(Pt 11):1681–1694Google Scholar
  11. Biewener AA, McGowan C, Card GM, Baudinette RV (2004) Dynamics of leg muscle function in tammar wallabies (m. eugenii) during level versus incline hopping. J Exp Biol 207(Pt 2):211–223CrossRefGoogle Scholar
  12. Bobbert MF (2001) Dependence of human squat jump performance on the series elastic compliance of the triceps surae: a simulation study. J Exp Biol 204(Pt 3):533–542Google Scholar
  13. Brown I, Scott S, Loeb G (1995) Preflexes. In: Programmable, high-gain, zero-delay intrinsic responses of perturbed musculoskeletal systems. Society neuroscience abstract, vol 21Google Scholar
  14. Davies AS, Gunn HM (1972) Histochemical fibre types in the mammalian diaphragm. J Anat 112(Pt 1):41–60Google Scholar
  15. Ettema GJ, Huijing PA (1994) Effects of distribution of muscle fiber length on active length-force characteristics of rat gastrocnemius medialis. Anat Rec 239(4):414–420. doi: 10.1002/ar.1092390408 CrossRefGoogle Scholar
  16. Faulkner J, Claflin D, McCully K (1986) Power output of fast and slow fibres from human skeletal muscles. In: Jones NL, McCartney N, McComas AJ (eds) Human muscle power. Human Kinetics, 332 ppGoogle Scholar
  17. Fitts RH (1994) Cellular mechanisms of muscle fatigue. Physiol Rev 74(1):49–94Google Scholar
  18. Gillis GB, Biewener AA (2001) Hindlimb muscle function in relation to speed and gait: in vivo patterns of strain and activation in a hip and knee extensor of the rat (rattus norvegicus). J Exp Biol 204(Pt 15):2717–2731Google Scholar
  19. Günther M, Ruder H (2003) Synthesis of two-dimensional human walking: a test of the lambda-model. Biol Cybern 89(2):89–106. doi: 10.1007/s00422-003-0414-x CrossRefzbMATHGoogle Scholar
  20. Günther M, Schmitt S, Wank V (2007) High-frequency oscillations as a consequence of neglected serial damping in hill-type muscle models. Biol Cybern 97(1):63–79. doi: 10.1007/s00422-007-0160-6 CrossRefzbMATHGoogle Scholar
  21. Günther M, Röhrle O, Haeufle DFB, Schmitt S (2012) Spreading out muscle mass within a hill-type model: a computer simulation study. Comput Math Methods Med 2012:848,630. doi: 10.1155/2012/848630 MathSciNetCrossRefzbMATHGoogle Scholar
  22. Granata KP, Marras WS (2000) Cost-benefit of muscle cocontraction in protecting against spinal instability. Spine 25(11):1398–1404CrossRefGoogle Scholar
  23. Haeufle DFB, Günther M, Bayer A, Schmitt S (2014) Hill-type muscle model with serial damping and eccentric force-velocity relation. J Biomech 47(6):1531–1536. doi: 10.1016/j.jbiomech.2014.02.009 CrossRefGoogle Scholar
  24. Hill AV (1938) The heat of shortening and the dynamic constants of muscle. Proc R Soc Lond Biol Sci 126(843):136–195CrossRefGoogle Scholar
  25. Hill AV (1950) The dimensions of animals and their muscular dynamics. Sci Prog 38(150):209–230Google Scholar
  26. John CT, Anderson FC, Higginson JS, Delp SL (2013) Stabilisation of walking by intrinsic muscle properties revealed in a three-dimensional muscle-driven simulation. Comput Methods Biomech Biomed Eng 16(4):451–462. doi: 10.1080/10255842.2011.627560 CrossRefGoogle Scholar
  27. Jones AM, Campbell IT, Pringle JSM (2004a) Influence of muscle fibre type and pedal rate on the vo2-work rate slope during ramp exercise. Eur J Appl Physiol 91(2–3):238–245. doi: 10.1007/s00421-003-0971-7 CrossRefGoogle Scholar
  28. Jones DA, Round J, de Haan A (2004b) Skeletal muscle: from molecules to movement. Churchill Livingstone, EdinburghGoogle Scholar
  29. Josephson RK (1985) Mechanical power output from striated muscle during cyclic contraction. J Exp Biol 114:493–512Google Scholar
  30. Kaya M, Leonard T, Herzog W (2003) Coordination of medial gastrocnemius and soleus forces during cat locomotion. J Exp Biol 206(Pt 20):3645–3655CrossRefGoogle Scholar
  31. Kistemaker DA, van Soest AJ, Bobbert MF (2006) Is equilibrium point control feasible for fast goal-directed single-joint movements? J Neurophysiol 95(5):2898–2912CrossRefGoogle Scholar
  32. Lloyd DG, Besier TF (2003) An EMG-driven musculoskeletal model to estimate muscle forces and knee joint moments in vivo. J Biomech 36(6):765–776CrossRefGoogle Scholar
  33. Lutz GJ, Rome LC (1996) Muscle function during jumping in frogs. ii. mechanical properties of muscle: implications for system design. Am J Physiol 271(2 Pt 1):C571–C578Google Scholar
  34. Maas H, Huijing PA (2005) Myofascial force transmission in dynamic muscle conditions: effects of dynamic shortening of a single head of multi-tendoned rat extensor digitorum longus muscle. Eur J Appl Physiol 94(5–6):584–592. doi: 10.1007/s00421-005-1367-7 CrossRefGoogle Scholar
  35. McMahon TA (1984) Muscles, reflexes and locomotion. Princeton University Press, PrincetonGoogle Scholar
  36. Millard M, Uchida T, Seth A, Delp SL (2013) Flexing computational muscle: modeling and simulation of musculotendon dynamics. J Biomech Eng 135:021,005. doi: 10.1115/1.4023390 CrossRefGoogle Scholar
  37. Miller RH, Umberger BR, Hamill J, Caldwell GE (2011) Evaluation of the minimum energy hypothesis and other potential optimality criteria for human running. Proc Biol Sci R Soc 1498–1505. doi: 10.1098/rspb.2011.2015
  38. Mörl F, Siebert T, Schmitt S, Blickhan R, Günther M (2012) Electro-mechanical delay in hill-type muscle models. JMMB 12(5). doi: 10.1142/S0219519412500856
  39. Prilutsky BI, Herzog W, Allinger TL (1996) Mechanical power and work of cat soleus, gastrocnemius and plantaris muscles during locomotion: possible functional significance of muscle design and force patterns. J Exp Biol 199(Pt 4):801–814Google Scholar
  40. Rack P, Westbury D (1969) The effects of length and stimulus rate on tension in the isometric cat soleus muscle. J Physiol 204(2):443–460CrossRefGoogle Scholar
  41. Ranatunga KW (1984) The force-velocity relation of rat fast- and slow-twitch muscles examined at different temperatures. J Physiol 351:517–529CrossRefGoogle Scholar
  42. Ranatunga KW, Thomas PE (1990) Correlation between shortening velocity, force-velocity relation and histochemical fibre-type composition in rat muscles. J Muscle Res Cell Motil 11(3):240–250CrossRefGoogle Scholar
  43. Roberts TJ, Azizi E (2010) The series-elastic shock absorber: tendons attenuate muscle power during eccentric actions. J Appl Physiol 109(2):396–404. doi: 10.1152/japplphysiol.01272.2009 (Bethesda, Md: 1985)CrossRefGoogle Scholar
  44. Roberts TJ, Azizi E (2011) Flexible mechanisms: the diverse roles of biological springs in vertebrate movement. J Exp Biol 214(Pt 3):353–361. doi: 10.1242/jeb.038588 CrossRefGoogle Scholar
  45. Roberts TJ, Marsh RL, Weyand PG, Taylor CR (1997) Muscular force in running turkeys: the economy of minimizing work. Science 275(5303):1113–1115CrossRefGoogle Scholar
  46. Rode C, Siebert T, Blickhan R (2009) Titin-induced force enhancement and force depression: A ’sticky-spring’ mechanism in muscle contractions? J Theor Biol 259(2):350–360. doi: 10.1016/j.jtbi.2009.03.015 CrossRefGoogle Scholar
  47. Rode C, Siebert T, Herzog W, Blickhan R (2009b) The effects of parallel and series elastic components on the active cat soleus force-length relationship. JMMB 9(1):105–122. doi: 10.1142/S0219519409002870
  48. Rohen JW, Yokochi C (1993) Anatomie des Menschen. Schattaeur, StuttgartGoogle Scholar
  49. Rome LC, Funke RP, Alexander RM, Lutz G, Aldridge H, Scott F, Freadman M (1988) Why animals have different muscle fibre types. Nature 335(6193):824–827. doi: 10.1038/335824a0 CrossRefGoogle Scholar
  50. Rome LC, Sosnicki AA, Goble DO (1990) Maximum velocity of shortening of three fibre types from horse soleus muscle: implications for scaling with body size. J Physiol 431:173–185CrossRefGoogle Scholar
  51. Rupp TK, Ehlers W, Karajan N, Günther M, Schmitt S (2015) A forward dynamics simulation of human lumbar spine flexion predicting the load sharing of intervertebral discs, ligaments, and muscles. Biomech Model Mechanobiol. doi: 10.1007/s10237-015-0656-2
  52. Sandercock TG, Heckman CJ (1997) Force from cat soleus muscle during imposed locomotor-like movements: experimental data versus Hill-type model predictions. J Neurophysiol 77(3):1538–1552Google Scholar
  53. Scott SH, Winter DA (1991) A comparison of three muscle pennation assumptions and their effect on isometric and isotonic force. J Biomech 24(2):163–167CrossRefGoogle Scholar
  54. Scott SH, Brown IE, Loeb GE (1996) Mechanics of feline soleus: I. effect of fascicle length and velocity on force output. J Muscle Res Cell Motil 17(2):207–219CrossRefGoogle Scholar
  55. Shadwick RE (1990) Elastic energy storage in tendons: mechanical differences related to function and age. J Appl Physiol 68(3):1033–1040CrossRefGoogle Scholar
  56. Siebert T, Rode C (2014) Computational modeling of muscle biomechanics. Woodhead Publishing, Elsevier, AmsterdamCrossRefGoogle Scholar
  57. Siebert T, Rode C, Herzog W, Till O, Blickhan R (2008) Nonlinearities make a difference: comparison of two common hill-type models with real muscle. Biol Cybern 98(2):133–143. doi: 10.1007/s00422-007-0197-6 MathSciNetCrossRefzbMATHGoogle Scholar
  58. Siebert T, Till O, Stutzig N, Günther M, Blickhan R (2014) Muscle force depends on the amount of transversal muscle loading. 47(8):1822–1828. doi: 10.1016/j.jbiomech.2014.03.029
  59. van Soest AJ, Bobbert MF (1993) The contribution of muscle properties in the control of explosive movements. Biol Cybern 69(3):195–204CrossRefGoogle Scholar
  60. Wilson A, Lichtwark G (2011) The anatomical arrangement of muscle and tendon enhances limb versatility and locomotor performance. Philos Trans R Soc Lond B Biol Sci 366(1570):1540–1553. doi: 10.1098/rstb.2010.0361 CrossRefGoogle Scholar
  61. Woittiez RD, Huijing PA, Rozendal RH (1983) Influence of muscle architecture on the length-force diagram. A model and its verification. Pflugers Arch 397(1):73–74CrossRefGoogle Scholar
  62. Zajac FE (1989) Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control. Crit Rev Biomed Eng 17(4):359–411Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Forschungsgesellschaft für Angewandte Systemsicherheit und Arbeitsmedizin mbH, Zentrum für BewegungstherapieErfurtGermany
  2. 2.Institute of Sport and Motion ScienceUniversity of StuttgartStuttgartGermany
  3. 3.Human Movement Simulation Lab, Institute of Sport and Motion ScienceUniversity of StuttgartStuttgartGermany

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