Skip to main content

Hill-Based Muscle Models: A Systems Engineering Perspective

  • Chapter
Multiple Muscle Systems

Abstract

Chapter 1 (Zahalak) provided a brief historical treatment of the early findings that led to the muscle model structure first proposed by A. V. Hill (1938). From a “systems engineering” perspective, this is a phenomenologically based, lumped-parameter model that is based on interpretations of input-output data obtained from controlled experiments. Simply stated, this model consists of a contractile element (CE) that is surrounded, both in series and in parallel, by “passive” connective tissue (Figure 5.1). CE is furthermore characterized by two fundamental relationships: CE tension-length and CE force-velocity. Each of these is modulated by an activation input that is structurally distinct from the location for mechanical coupling between the muscle and the environment (Figure 5.1).

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Abbott, B.C. and Aubert, X.M. (1951) The force exerted by active striated muscle during and after change in length. J. Physiol Lond. 117: 77–86.

    Google Scholar 

  • Abbott, B.C. and Wilkie, D.R. (1973) The relationship between velocity of shortening and the tension-length curve of skeletal muscle, J. Physiol 120: 214–222.

    Google Scholar 

  • Agarwal, G.C. and Golltieb, G.L. (1977) Oscillation of the human ankle joint in response to applied sinusoidal torque on the foot. J. Physiol 268: 151–176.

    CAS  PubMed  Google Scholar 

  • Aubert, X. (1956) Le Couplage Energetique de la Ccontraction Musculaire. Brussels, Editions Arscia.

    Google Scholar 

  • Alexander, R. McN. (1983) Animal Mechanics ( Second Edition ). Blackwell Sci. Publ., Oxford.

    Google Scholar 

  • Alexander, R. McN. and Bennet-Claik, H.C. (1977) Storage of elastic strain energy in muscle and other tissues. Nature 265: 114–117.

    Article  CAS  PubMed  Google Scholar 

  • Audu, M.L. and Davy, D.T. (1985) The influence of muscle model complexity in musculoskeletal motion modeling. J. Biomech. Engng. 107: 147–157.

    Article  CAS  Google Scholar 

  • Bagley, A.M. (1987) Analysis of human response to slow isokinetic movement. M.S. Thesis, Arizona State University.

    Google Scholar 

  • Bahler, A.S. (1967) The series elastic element of mammalian skeletal muscle. Am. J. Physiol 213: 1560–1564.

    CAS  PubMed  Google Scholar 

  • Bach, T.M., Chapman, A.E. and Calvert, T.W. (1983) Mechanical resonance of the human body during voluntary oscillations about the ankle joint. J. Biomech. 16: 85–90.

    Article  CAS  PubMed  Google Scholar 

  • Bahler, A.S. (1968) Modeling of mammalian skeletal muscle. IEEETrans.Biomed.Engng. BME-13: 248–257.

    Google Scholar 

  • Baildon, R.W.A. and Chapman, A.E. (1983) A new approach to the human muscle model. J. Biomech. 16: 803–809.

    Article  CAS  PubMed  Google Scholar 

  • Bennett, M.B., Ker, R.F., Dimery, N.J. and Alexander, R. McN. (1986) Mechanical properties of various mammalian tendons. J. Zool. 209: 537.

    Article  Google Scholar 

  • Bigland, B. and Lippold, O.C.J. (1954) The relation between force, velcoity and integrated electrical activity in human muscles. J. Physiol 1253: 214–224.

    Google Scholar 

  • Bobet, Stein, R.B. and Oguztoreli, M.N. (1990) Mechanisms relating force and high-frequency stiffness in skeletal muscle. J. Biomech., accepted.

    Google Scholar 

  • Bouchaert, J.P., Capellen, L. and de Blende, J. (1930) J. Physiol 69: 473.

    Google Scholar 

  • Butler, D.L., Grood, E.S., Noyes, F.R. and Zernicke, R.F. (1979) Biomechanics of ligaments and tendon. Exer. Sport Sci. Rev. 6: 125–185.

    Google Scholar 

  • Cavagna, G.A., Komarek, L., Citterio, G. and Margaria, R. (1971) Power output of the previously stretch muscle. Med. Sport (Biomech. II) 6: 159–167.

    Google Scholar 

  • Cecchi, G., Griffiths, P.J. and Taylor, S.R. (1984a) The kinetics of crossbridge attachment studies by high frequency stiffness measurements. In Contractile Mechanisms in Muscle (Pollack, G.H. and Sugi, H., eds.), Cecchi, G., Griffiths, P.J. and Taylor, S.R, pp. 641–655, New York.

    Google Scholar 

  • Cecchi, G., Lombardi, V. and Menchetti, G. (1984b) Development of force-velcoity relation and rise of isometric tetanic tension measure the time course of different processes. Pflugers Arch. 401: 396–405.

    Article  CAS  PubMed  Google Scholar 

  • Chapman, A.E. (1985) The mechancial properties of human muscle. Exer. Sci. Sport Rev. 13: 443–501.

    CAS  Google Scholar 

  • Chapman, A.E. and Calvert, T.W. (1979) Estimations of active-state from EMG recordings of human muscular contraction. Electromyogr. Clin. Neurophys. 19: 199–222.

    CAS  Google Scholar 

  • Close, R.I. (1972) Dynamic properties of mammalian skeletal muscles. Physiol. Rev. 52: 129–197.

    CAS  PubMed  Google Scholar 

  • Cook, G. and Stark, L. (1968) The human eye movement mechanism: experiments, modelling and model testing. Arch. Ophthal. 79: 428–436.

    CAS  PubMed  Google Scholar 

  • Cordo, P.J. and Rymer, W.Zev (1982) Contributions of motor-unit recruitment and rate modulation to compensation for muscle yielding, J. Neurophys. 47: 797–809.

    CAS  Google Scholar 

  • Crowe, A., Van Atteveldt, H. an Groothedde, H. (1980) Simulation studies of contracting skeletal muscles during mechanical strech. J. Biomech. 13: 333–340.

    CAS  Google Scholar 

  • Dern, R.J., Levine, J.M. and Blair, H.A. (1947) Forces exerted at different velocities in human arm movement Am. J. Physiol 151: 415–437.

    CAS  PubMed  Google Scholar 

  • Ebashi, S.M. and Endo, M. (1968) Calcium ion and muscle contraction. Prog. Biophys. Mol. Biol. 18: 123.

    Article  CAS  PubMed  Google Scholar 

  • Edman, K.A.P., Mulieri, L.A. and Scubon-Mulieri, B. (1976) Non-hyperbolic force-velocity relationship in single muscle fibres, Acta Physiol. Scand. 98: 143–156.

    Article  CAS  PubMed  Google Scholar 

  • Edman, K.A.P., Elzinga, G. and Noble, M.I.M. (1978) Enhancement of mechanical performance by stretch during tetanic contractions of vertebrate skeletal muscle fibres. J. Physiol. 280: 139–155.

    Google Scholar 

  • Elliott, D.H. (1965) Structure and function of mammalian tendon. Biol. Rev. 40: 392–425.

    Article  CAS  PubMed  Google Scholar 

  • Fenn, W.O. and Marsh, B.S. (1935) Muscular force at different wpeeds of shortening. J. Physiol. (Lond.) 85: 277–297.

    CAS  Google Scholar 

  • Hitney, F.W. and Hirst, D.G. (1978) Crossbridge detachment and sarcomere “give” during stretch in active frog’s muscle. J. Physiol. 276: 449–465.

    Google Scholar 

  • Ford, L.E., Huxley, A.F., and Simmons, R.M. (1977) Tension responses to sudden length change in stimulated frog muscle fibers near slack length. J. Physiol. 269: 441–515.

    CAS  PubMed  Google Scholar 

  • Fuchs, F. (1977) Cooperative interactions between calcium-binding sites on glycerinated muscle fibers-the influence of corss-bridge attachment Biochim. Biophys. Acta 462: 314–322.

    Article  CAS  PubMed  Google Scholar 

  • Fung, Y.C. (1967) Elasticity of soft tissues in simple elongation. Am. J. Physiol 213: 1532–1544.

    CAS  PubMed  Google Scholar 

  • Fung, Y.C. (1970) Mathematical representation of the mechanical properties of the heart muscle. J. Biomech. 3: 381–404.

    Article  CAS  PubMed  Google Scholar 

  • Fung, Y.C. (1981) Biomechanics. Springer-Verlag, New York.

    Google Scholar 

  • Gasser, H.S. and Hill, A.V. (1924) The dynamics of muscle contraction. Proc. Roy Soc. 96: 398–437.

    Article  Google Scholar 

  • Gielen, C.C.A.M. and Houk, J.C. (1987) A model of the motor servo: incorporating nonlinear spindle receptor and muscle mechanical properties. Biol Cybern. 57: 217–235.

    Article  CAS  PubMed  Google Scholar 

  • Gordon, A.M., Huxley, A.F. and Julian, F.J. (1966) The variation in isometric tension with sarcomere length in vertebrate muscles. J. Physiol 184: 170–192.

    CAS  PubMed  Google Scholar 

  • Goubel, F., Bouisset, S. and Lestinne, F. (1971) Determination of muscular compliance in the course of movement. Med. Sport (Biomech. II) 6: 154–158.

    Google Scholar 

  • Greene, P.R. and McMahon, T.A. (1979) Reflex stiffness of man’s antigravity muscles during kneebends while carrying extra weights. J. Biomech. 12: 881–895.

    Article  CAS  PubMed  Google Scholar 

  • Hanson, J. and Huxley, H.E. (1955) The structural basis of contraction in stiated muscle. Symp. Soc. Exp. Biol 9: 228–264.

    Google Scholar 

  • Hatze, H. (1974) A model of skeletal muscle suitable for optimal motion problems. In: Biomech. IV, pp. 417–422, S. Karger, Basel.

    Google Scholar 

  • Hatze, H. (1977) A myocybernetic control model of skeletal muscle. Biol Cybern. 25: 103–119.

    Article  CAS  PubMed  Google Scholar 

  • Hatze, H. (1981) Myocybernetic Control Models of Skeletal Muscles. Univ. of South Africa.

    Google Scholar 

  • Henneman, E., Somjen, G. and Carpenter, D. (1965) Excitability and inhibitability of motoneurons of different sizes. J. Neurobiol 28: 599–620.

    CAS  Google Scholar 

  • Hill, A.V. (1922) The maximum work and mechanical efficiency of human muscles, and their most economical speed. J. Physiol 56: 19–45.

    CAS  PubMed  Google Scholar 

  • Hill, A.V. (1938) The heat of shortening and the dynamic constants of muscle. Proc. Roy. Soc. B126: 136–195.

    Article  Google Scholar 

  • Hill, A.V. (1949) The abrupt transition from rest to activity in muscle. Proc. Roy. Soc. B126: 399–420.

    Google Scholar 

  • Hill, A.V. (1950) The series elastic component of muscle. Proc. Roy. Soc. 8141: 104–117.

    Google Scholar 

  • Hill, A.V. (1970) First and last experiments in muscle mechanics., Cambridge Univ. Press, Cambridge.

    Google Scholar 

  • Hof, A.L. and Van den Berg, J. (1981) EMG to force processing. I. An electrical analogue of the Hill muscle model. J. Biomech. 14: 747–758.

    Article  CAS  PubMed  Google Scholar 

  • Huxley, A.F. and Simmons, R.M. (1971) Proposed mechanism of force generation in striated muscle. Nature 233: 533–538.

    Article  CAS  PubMed  Google Scholar 

  • Jewell, B.R. and Wilkie, D.R. (1958) An analysis of the mechanical components in frog striated muscle. J. Physiol. 143: 515–540.

    CAS  PubMed  Google Scholar 

  • Joyce, G.C and Rack, P.M.H. (1969) Isotonic lengthening and shortening movements of cat soleus muscle. J. Physiol 204: 475–495.

    CAS  PubMed  Google Scholar 

  • Joyce, G.C., Rack, P.M.H. and Ross, H.F. (1974) The forces generated at the human elbow joint in response to imposed sinusoidal movements of the forearm. J. Physiol 240: 375–396.

    CAS  PubMed  Google Scholar 

  • Joyce, G.C., Rack, P.M.H. and Westbury, D.R. (1969) The mechanical properties of cat soleus muscle during controlled lengthening and shortening movements, J. Physiol 214: 461–474.

    Google Scholar 

  • Julian, F.J. and Sollins, M.R. (1973) Regulation of force and speed of shortening in muscle contraction. Cold Spring Harbor Symp. Quanrt. Biol 37: 635–646.

    CAS  Google Scholar 

  • Katz, B. (1939) The relation between force and speed in muscular contraction. J. Physiol 96: 45–64.

    CAS  PubMed  Google Scholar 

  • Kleweno, D.K. (1987) Physiological and theoretical analysis of isometric strength curves of the upper limb. M.S. Thesis, Arizona State University.

    Google Scholar 

  • Kleweno, D.K. and Winters, J.M. (1988) Sensitivity of upper-extremity strength curves to 3-D geometry: model results, Adv. in Bioengng., ASME-WAM, BED-8: 53–56.

    Google Scholar 

  • Komi, P.V. (1973) Relationship between muscle tension, EMG and velocity of contraction under concentric and eccentric work. In New Developments in Electromyography and Clinical Neurophysiology, ( Desmedt, J.E., ed.), S. Karger, Basel, pp. 596.

    Google Scholar 

  • Kulig, K., Andrews, J.G. and Hay, J.G. (1984) Human strength curves. Exerc. Sport Sci. Rev. 12: 81–121.

    Article  Google Scholar 

  • Levin, A. and Wyman, J. (1927) The viscous elastic porperties of muscle. Proc. Roy. Soc. B101: 218–243.

    Article  Google Scholar 

  • Lehman, S.L. and Stark., L. (1979) Simulation of linear and nonlinear eye movement models: sensitivity analysis and enumeration studies of time optimal control. J. Cybern. Inform. Sci. 2: 21–43.

    Google Scholar 

  • Lehman, S.L. and Stark, L. (1982) Three algorithms for interpreting models consisting of ordinary differential equations: sensitivity coefficients, sensitivity functions, global optimization. Math. Biosci. 62: 107–122.

    Article  Google Scholar 

  • Ma, S. and Zahalak, G.I. (1987) Activation dynamics for a distribution-momnet model of skeletal muscle. Proc. 6th Int. Conf. Math. Model 11:778–782., St. Louis.

    Google Scholar 

  • McCrorey, H.L., Gale,.H. and Alpert, N.R. (1966) Mechanical properties of the cat tenuissimus muscle. Am. J. Physiol 210: 114–120.

    Google Scholar 

  • McMahon, T.A. (1984) Muscles, Reflexes and Locomotion. Princeton Univ. Press, Princeton.

    Google Scholar 

  • Morgan, D.L. (1977) Separation of active and passive components of short-range stiffness of muscle. Amer. J. Physiol 232: C45–C49.

    CAS  PubMed  Google Scholar 

  • Otten, E. (1988) Concepts and models of functional architecture in skeletal muscle. Exer. Sport Sci. Rev. 89–137.

    Google Scholar 

  • Parmley, W.W., Yeatman, L.A. and Sonnenblick, E.H. (1970) Differences between isotonic and isometric force-velocity relations in cardiac and skeletal muscle. Am. J. Physiol 219: 546–550.

    CAS  PubMed  Google Scholar 

  • Partridge, L.D. (1979) Muscle properties: a problem for the motor controller physiologist. In Posture and Movement ( Talbott, R.E. and Humphery, D.R., eds.), pp. 189–229, Raven Press, New York.

    Google Scholar 

  • Perrine, J.J. and Edgerton, V.R. (1978) Muscle force-velocity and power-velocity relationships under isokinetic loading. Med. Sci. Sports 10: 159–166.

    CAS  PubMed  Google Scholar 

  • Petrofsky, J.S. and Phillips, C.A. (1981) The influence of temperature, initital length and electrical ativity on the force-velocity relationship of the medidal gastrocnemius muscle of the cat. J. Biomech. 14: 297–306.

    Article  CAS  PubMed  Google Scholar 

  • Pertuzon, E. and Bouisset, S. Instantaneous force-velocity relationship in human muscle. Med. Sport, Biomech. III, 8: 230–234.

    Google Scholar 

  • Proske, U. and Morgan, D.L. (1987) Tendon stiffness: methods of measurement and significance for the control of movement. A review. J. Biomech. 20: 75–82.

    Article  CAS  PubMed  Google Scholar 

  • Rack, P.M.H. and Ross, H.F. (1984) The tendon of flexor pollicis longus: its effects on the muscular control of force and position at the thumb. J. Physiol 351: 99–110.

    CAS  PubMed  Google Scholar 

  • Rack, P.M.H. and Westbury, D.R. (1969) The effects of length and stimulus rate on tension in isometric cat soleus muscle. J. Physiol 204: 443–460.

    CAS  PubMed  Google Scholar 

  • Rack, P.M.H. and Westbury, D.R. (1974) The short range stiffness of active mammalian muscle and its effect on mechanical properties. J. Physiol 240: 331–350.

    CAS  PubMed  Google Scholar 

  • Rack, P.M. and Westbury, D.R. (1984) Elastic properties of the cat soleus tendon and their functional importance. J. Physiol 347: 479.

    CAS  PubMed  Google Scholar 

  • Ralston, H.J., Polissart, M.J., Inman, V.T., Close, J.R. and Feinstein, B. (1949) Dynamic features of human isolated voluntary muscle in isometric and free contractions. J. Appl. Physiol 1: 526–533.

    CAS  PubMed  Google Scholar 

  • Ramsey, R.W. and Street, S.F. (1940) The isometric length tension diagram of isolated skeletal muscle fibers of the frog. J. Cell Comp. Physiol 15: 11–34.

    Article  CAS  Google Scholar 

  • Seif-Naraghi, A.H. and Winters, J.M. (1989) Effect of task-specific linearization on musculoskeletal system control strategies. ASME Biomech. Symp., AMD-98: 347–350.

    Google Scholar 

  • Silver-Thorn, M.B. (1987) Muscle imbalance in osteoarthritis of the human knee. M.S. thesis, Arizona State University.

    Google Scholar 

  • Silver-Thorn, M.B. and Winters, J.M. (1988) Muscle imbalance and osteoarthritis of the knee. Adv. in Bioengng. ASME Wint. Ann. Mtng., BED-8: 95–98.

    Google Scholar 

  • Sprigings, E.J. (1986) Simulation of the force enhancement phenomenon in muscle. Comput. Biol. Med. 16: 423–430.

    Article  CAS  PubMed  Google Scholar 

  • Stein, R.B. and Gordon, T. (1986) Nonlinear stiffness- force relationships in whole mammalian skeletal muscles. Can. J. Physiol Pharmacol. 64: 1236–1244.

    Article  CAS  PubMed  Google Scholar 

  • Sugi, H. (1979) The origin of the series elasticity in striated muscle fibers. In Cross-Bridge Mechanism in Muscle Contraction ( Sugi, H. and Pollack, G.H., eds.), pp. 85–102, Univ. of Tokyo Press, Tokyo.

    Google Scholar 

  • Van Atteveldt, H. and Crowe, A. (1980) Active tension changes in frog skeletal muscle during and after mechancial extension. J. Biomech. 13: 323–335.

    Article  PubMed  Google Scholar 

  • Van Dijk, J.H.M. (1978) Simulation of human arm movements controlled by peripheral feedback. Biol. Cybern. 29: 175–186.

    Article  PubMed  Google Scholar 

  • Wells, J.B. (1964) Comparison of mechanical properties between slow and fast mammalian muscles. J. Physiol. 178: 252–269.

    Google Scholar 

  • Wilkie, D.R. (1950) The relation between force and velocity in human muscle. J. Physiol. K110: 248–280.

    Google Scholar 

  • Wilkie, D.R. (1956) The mechanical properties of muscle. Br. Med. Bull. 12: 177–182.

    CAS  PubMed  Google Scholar 

  • Winters, J.M. (1985) Generalized analysis and design of antagonistic muscle models: effect of nonlinear muscle-joint properties on the control of fundamental movements. Ph.D. Dissertation, Univ. of Calif., Berkeley.

    Google Scholar 

  • Winters, J.M. (1988) Improvements within the A.V. Hill model structure: strengths and limitations. Proc. IEEE Engng. Med. Biol., pp. 559–560, New Orleans.

    Google Scholar 

  • Winters, J.M. (1989) A novel approach for modeling transient lengthening with a Hill-based muscle model. XII Int. Congr. Biomech., Abstract 128., Los Angeles.

    Google Scholar 

  • Winters, J.M. and Bagley, A.M. (1987) Biomechanical modelling of muscle-joint systems: why it is useful. IEEE Engng. Med. Biol. 6: 17–21.

    Article  CAS  Google Scholar 

  • Winters, J.M. and Stark, L. (1985) nalysis of fundamental movement patterns through the use of in- depth antagonistic muscle models. IEEE Trans. Biomed. Engng. BME-32: 826–839.

    Google Scholar 

  • Winters, J.M. and Staik, L. (1987) Muscle models: what is gained and what is lost by varying model complexity. Biol. Cybern. 55: 403–420.

    Article  CAS  PubMed  Google Scholar 

  • Winters, J.M. and Stark, L. (1988) Estimated mechanical properties of synergistic muscles involved in movements of a variety of human joints. J. Biomech. 21: 1027–1042.

    Article  CAS  PubMed  Google Scholar 

  • Winters, J.M., Staik, L. and Seif-Naraghi, A.H. (1988) An analysis of the sources of muscle-joint system impedance. J. Biomech. 21: 1011–1026.

    Article  CAS  PubMed  Google Scholar 

  • Woittiez, R.D., Huijing, P.A., Boom, H.B.K. and Rozendal, R.H. (1984) A three-dimensional muscle model: a quantified relation between form and function of skeletal muscles, J. Morphol. 182: 95–113.

    Article  CAS  PubMed  Google Scholar 

  • Yates, J.W. and Kamon, E. (1983) A comparison of peak and constant angle torque-velcoity curves in fast and slow-twitch populations. Eur. J. Appl. Physiol. 51: 67–74.

    Article  CAS  Google Scholar 

  • Zahalak, G.I. (1981) A distribution-moment approximation for kinetic theories of muscular contraction. Math. Biosci. 55: 89–114.

    Article  Google Scholar 

  • Zahalak, G.I., Duffy, J., Stewart, P.A., Litchman, H.M., Hawley, R.H. and Pasley, P.R. (1976) Partially activated human skeletal muscle: an experimental investigation of force, velocity and EMG. J. Appl. Mech. 98: 81–86.

    Article  Google Scholar 

  • Zahalak, G.I. and Heyman, S.J. (1979) A quantitative evaluation of the frequency-response characteristics of active human skeletal muscle in vivo. J. Biomech. Engng. 28: 28–37.

    Article  Google Scholar 

  • Zajac, F. (1989) Muscle and tendon: properties, models, scaling and application to biomechanics and motor control. CRC Crit. Rev. Biomed. Engng. 17: 359–415.

    CAS  Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1990 Springer-Verlag

About this chapter

Cite this chapter

Winters, J.M. (1990). Hill-Based Muscle Models: A Systems Engineering Perspective. In: Winters, J.M., Woo, S.LY. (eds) Multiple Muscle Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9030-5_5

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-9030-5_5

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-9032-9

  • Online ISBN: 978-1-4613-9030-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics