Skip to main content
Log in

Nonlinearities make a difference: comparison of two common Hill-type models with real muscle

Biological Cybernetics Aims and scope Submit manuscript

Abstract

Compared to complex structural Huxley-type models, Hill-type models phenomenologically describe muscle contraction using only few state variables. The Hill-type models dominate in the ever expanding field of musculoskeletal simulations for simplicity and low computational cost. Reasonable parameters are required to gain insight into mechanics of movement. The two most common Hill-type muscle models used contain three components. The series elastic component is connected in series to the contractile component. A parallel elastic component is either connected in parallel to both the contractile and the series elastic component (model [CC+SEC]), or is connected in parallel only with the contractile component (model [CC]). As soon as at least one of the components exhibits substantial nonlinearities, as, e.g., the contractile component by the ability to turn on and off, the two models are mechanically different. We tested which model ([CC+SEC] or [CC]) represents the cat soleus better. Ramp experiments consisting of an isometric and an isokinetic part were performed with an in situ cat soleus preparation using supramaximal nerve stimulation. Hill-type models containing force–length and force–velocity relationship, excitation–contraction coupling and series and parallel elastic force–elongation relations were fitted to the data. To test which model might represent the muscle better, the obtained parameters were compared with experimentally determined parameters. Determined in situations with negligible passive force, the force–velocity relation and the series elastic component relation are independent of the chosen model. In contrast to model [CC+SEC], these relations predicted by model [CC] were in accordance with experimental relations. In conclusion model [CC] seemed to better represent the cat soleus contraction dynamics and should be preferred in the nonlinear regression of muscle parameters and in musculoskeletal modeling.

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

References

  • Abbot BC, Aubert XM (1952) The force exerted by active striated muscle during and after change of length. J Physiol 117: 77–86

    Google Scholar 

  • Ahearn TS, Staff RT, Redpath TW, Semple SI (2005) The use of the Levenberg–Marquardt curve-fitting algorithm in pharmacokinetic modelling of DCE-MRI data. Phys Med Biol 50(9): 85–92

    Article  Google Scholar 

  • Blickhan R, Wagner H, Seyfahrt A (2003) Brain or muscles?. Recent Res Dev Biomech 1: 215–245

    Google Scholar 

  • Brown IE, Scott SH, Loeb GE (1996) Mechanics of feline soleus: II. Design and validation of a mathematical model. J Muscle Res Cell Motil 17: 221–233

    Article  PubMed  CAS  Google Scholar 

  • Brown IE, Cheng EJ, Loeb GE (1999) Measured and modeled properties of mammalian skeletal muscle. II. The effects of stimulus frequency on force–length and force–velocity relationships. J Muscle Res Cell Motil 20(7): 627–643

    Article  PubMed  CAS  Google Scholar 

  • Burkholder TJ, Lieber RL (2001) Sarcomere length operating range of vertebrate muscles during movement. J Exp Biol 204(9): 1529–1536

    PubMed  CAS  Google Scholar 

  • Curtin NA, Gardner-Medwin AR, Woledge RC (1998) Predictions of the time course of force and power output by dogfish white muscle fibres during brief tetani. J Exp Biol 201: 103–114

    PubMed  CAS  Google Scholar 

  • Epstein M, Wong M, Herzog W (2006) Should tendon and aponeurosis be considered in series?. J Biomech 39(11): 2020–2025

    Article  PubMed  Google Scholar 

  • Ettema GJ, Meijer K (2000) Muscle contraction history: modified Hill versus an exponential decay model. Biol Cybern 83(6): 491–500

    Article  PubMed  CAS  Google Scholar 

  • Fung YC (1993) Biomechanics: mechanical properties of living tissues. Springer, New York

    Google Scholar 

  • Gareis H, Solomonow M, Baratta R, Best R, D’Ambrosia R (1992) The isometric length-force models of nine different skeletal muscles. J Biomech 25(8): 903–916

    Article  PubMed  CAS  Google Scholar 

  • Geyer H, Seyfarth A, Blickhan R (2003) Positive force feedback in bouncing gaits?. Proc Biol Sci 270: 2173–2183

    Article  PubMed  Google Scholar 

  • Hardin EC, Su A, Bogert AJ (2004) Foot and ankle forces during an automobile collision: the influence of muscles. J Biomech 37(5): 637–644

    Article  PubMed  CAS  Google Scholar 

  • Herzog W, Leonard TR (1997) Depression of cat soleus-forces following isokinetic shortening. J Biomech 30(9): 865–872

    Article  PubMed  CAS  Google Scholar 

  • Herzog W, Leonard TR (2000) The history dependence of force production in mammalian skeletal muscle following stretch-shortening and shortening stretch cycles. J Biomech 33(5): 531–545

    Article  PubMed  CAS  Google Scholar 

  • Herzog W, Kamal S, Clarke HD (1992) Myofilament lengths of cat skeletal muscle: theoretical considerations and functional implications. J Biomech 25(8): 945–948

    Article  PubMed  CAS  Google Scholar 

  • Herzog W, Leonard TR, Stano A (1995) A system for studying the mechanical properties of muscles and the sensorimotor control of muscle forces during unrestrained locomotion in the cat. J Biomech 28(2): 211–218

    Article  PubMed  CAS  Google Scholar 

  • Hill AV (1938) The heat of shortening and the dynamic constants of muscle. Proc R Soc Lond 126: 136–195

    Article  Google Scholar 

  • Klein Breteler MD, Spoor CW, Van der Helm FC (1999) Measuring muscle and joint geometry parameters of a shoulder for modeling purposes. J Biomech 32(11): 1191–1197

    Article  PubMed  CAS  Google Scholar 

  • Marechal G, Plaghki L (1979) The deficit of the isometric tetanic tension redeveloped after a release of frog muscle at a constant velocity. J Gen Physiol 73(4): 453–467

    Article  PubMed  CAS  Google Scholar 

  • Muraoka T, Kawakami Y, Tachi M, Fukunaga T (2001) Muscle fiber and tendon length changes in the human vastus lateralis during slow pedaling. J Appl Physiol 91(5): 2035–2040

    PubMed  CAS  Google Scholar 

  • Otten E (1987) A myocybernetic model of the jaw system of the rat. J Neurosci Methods 21: 287–302

    Article  PubMed  CAS  Google Scholar 

  • Proske U, Morgan DL (1984) Stiffness of cat soleus muscle and tendon during activation of part of muscle. J Neurophysiol 52: 459–468

    PubMed  CAS  Google Scholar 

  • Rack PMH, Westbury DR (1969) The effects of length and stimulus rate on tension in the isometric cat soleus muscle. J Physiol 204(2): 443–460

    PubMed  CAS  Google Scholar 

  • Rode C, Siebert T, Herzog W, Blickhan R (2007) The effects of parallel and series elastic components on estimated active cat soleus muscle force. Acta Physiol (submitted)

  • 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: 207–219

    Article  PubMed  CAS  Google Scholar 

  • Siebert T, Sust M, Thaller S, Tilp M, Wagner H (2007) An improved method to determine neuromuscular properties using force laws – From single muscle to applications in human movements. Hum Mov Sci 26: 320–341

    Article  PubMed  CAS  Google Scholar 

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

    Article  PubMed  Google Scholar 

  • van Leeuwen JJ (1992) Muscle function in locomotion. In: Alexander RMcN (ed) Comparative and environmental Physiology. Mechanics of animal locomotion, vol 11. Springer, Berlin, pp 191–250

    Google Scholar 

  • Soest AJ, Bobbert MF (1993) The contribution of muscle properties in the control of explosive movements. Biol Cybern 69(3): 195–204

    Article  PubMed  Google Scholar 

  • Soest AJ, Haenen WP, Rozendaal LA (2003) Stability of bipedal stance: the contribution of cocontraction and spindle feedback. Biol Cybern 88(4): 293–301

    Article  PubMed  Google Scholar 

  • Wagner H, Blickhan R (2003) Stabilizing function of antagonistic neuromusculoskeletal systems: an analytical investigation. Biol Cybern 199: 163–179

    Google Scholar 

  • Wagner H, Siebert T, Ellerby DJ, Marsh RL, Blickhan R (2005) ISOFIT—A model-based method to measure muscle-tendon properties simultaneously. J Biomech Model Mechanobiol 4: 10–19

    Article  CAS  Google Scholar 

  • Wagner H, Thaller S, Dahse R, Sust M (2006) Biomechanical muscle properties and angiotensin-converting enzyme gene polymorphism: a model-based study. Eur J Appl Physiol 98(5): 507–515

    Article  PubMed  CAS  Google Scholar 

  • Winters JM, Woo SLY (1990) Multiple muscle system. Springer, New York

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tobias Siebert.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Siebert, T., Rode, C., Herzog, W. et al. Nonlinearities make a difference: comparison of two common Hill-type models with real muscle. Biol Cybern 98, 133–143 (2008). https://doi.org/10.1007/s00422-007-0197-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00422-007-0197-6

Keywords

Navigation