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Multiscale modeling of skeletal muscle properties and experimental validations in isometric conditions


In this article, we describe an approach to model the electromechanical behavior of the skeletal muscle based on the Huxley formulation. We propose a model that complies with a well established macroscopic behavior of striated muscles where force-length, force–velocity, and Mirsky–Parmley properties are taken into account. These properties are introduced at the microscopic scale and related to a tentative explanation of the phenomena. The method used integrates behavior ranging from the microscopic to the macroscopic scale, and allows the computation of the dynamics of the output force and stiffness controlled by EMG or stimulation parameters. The model can thus be used to simulate and carry out research to develop control strategies using electrical stimulation in the context of rehabilitation. Finally, through animal experiments, we estimated model parameters using a Sigma Point Kalman Filtering technique and dedicated experimental protocols in isometric conditions and demonstrated that the model can accurately simulate individual variations and thus take into account subject dependent behavior.

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  1. Benoussaad M, Hayashibe M, Fattal C, Poignet P, Guiraud D (2009) Identification and validation of FES physiological musculoskeletal model in paraplegic subjects. EMBC: Engineering in Medicine and Biology Society, Minneapolis, MN, USA

  2. Benoussaad M, Guiraud D, Poignet P (2009) Physiological musculoskeletal model identification for the lower limbs control of paraplegic under implanted FES. IROS: The IEEE/RSJ international conference on intelligent robots and systems. St. Louis, MO, USA

  3. Bestel J (2000) Modèle différentiel de la contraction musculaire controlée. Application au système cardio-vasculaire. PhD Thesis, University Paris IX Dauphine

  4. Bestel J, Sorine M (2000) A differential model of muscle contraction and applications. In: schloessmann Seminar on mathematical models in biology, chemistry and physics. Max Plank Society, Bad Lausick, Germany, pp 19–23

  5. Crago PE, Peckham PH, Thrope GB (1980) Modulation of muscle force by recruitment during intramuscular stimulation. IEEE Trans Biomed Eng 27(12): 679–684

  6. Davis R, Houdayer T, Andrews B, Emmons S, Patrick J (1997) Paraplegia: prolonged closed-loop standing with implanted nucleus FES-22 stimulator and Andrews’s foot-ankle orthosis”. Proc XIIth World Soc Stereotact Funct Neurosurg 69: 281–287

  7. Ding J, Chou LW, Kesar TM, Lee SCK, Johnston TE, Wexler AS, Macleod SA (2007) Mathematical model that predicts the force-intensity and force-frequency relationships after spinal cord injury. Muscle Nerve 36: 214–222

  8. Donaldson NN, Perkins TA, Worley ACM (1997) Lumbar root stimulation for restoring leg function: stimulator and measurement of muscle actions. Artif Organs 21: 247–249

  9. Durfee KW (1993) Control of standing and gait using electrical stimulation: influence of muscle model complexity on control strategy. Prog Brain Res 97: 369–381

  10. Durfee KW, MacLean KE (1989) Methods for estimating isometric recruitment curves of electrically stimulated muscle. IEEE Trans Biomed Eng 36: 654–667

  11. El Makssoud H (2005) Modélisation et Identification des Muscles Squelettiques sous Stimulation Electrique Fonctionnelle. PhD Thesis, University of Montpellier II

  12. El Makssoud H, Poignet P, Guiraud D (2003) Modeling of the skeletal muscle under functional electrical stimulations. International Functional Electrical Stimulation Society, Australia, pp 206–209

  13. El Makssoud H, Guiraud D, Poignet P (2004) Mathematical muscle model for functional electrical stimulation control strategies, IEEE ICRA. New Orleans, LA, USA, pp 1282–1287

  14. Furman S, Schwedel JB (1959) An intracardiac pacemaker for Stokes-Adams seizures. N Engl J Med 261: 943–948

  15. Guiraud D, Poignet P, Wieber PB, El Makssoud H, Pierrot F, Brogliato B, Fraisse P, Dombre E, Divoux JL, Rabischong P (2003) Modeling of the human paralysed lower limb under FES. IEEE international conference on robotics and automation Taipei, Taiwan, pp 2218–2223

  16. Guiraud D, Stieglitz T, Koch KP, Divoux JL, Rabischong P (2006) An implantable neuroprosthesis for standing and walking in paraplegia: 5-year patient follow-up. J Neural Eng 3: 268–275

  17. Guiraud D, Stieglitz T, Taroni G, Divoux JL (2006) Original electronic design to perform epimysial and neural stimulation in paraplegia. J Neural Eng 3: 276–286

  18. Hatze H (1978) A general myocybernetic control model of skeletal muscle. Biol Cybern 28: 143–157

  19. Hayashibe M, Guiraud D, Poignet P (2009) EMG-to-force estimation with full-scale physiology based muscle model. IROS 2009: The 2009 IEEE/RSJ international conference on intelligent robots and systems, Saint Louis, USA, pp 1621–1626

  20. Hayashibe M, Benoussaad M, Guiraud D, Poignet P, Fattal C (2010) Nonlinear identification method corresponding to muscle property variation in FES—experiments in Paraplegic Patients. IEEE RAS/EMBS international conference on biomedical robotics and biomechatronics, pp 401–406

  21. Hill AV (1938) The heat of shortening and the dynamic constants in muscle. Proc R Soc, London, Sre. B 126: 136–195

  22. Huxley AF (1957) Muscle structure and theories of contraction. Prog Biophys Biophys Chem 7: 255–318

  23. Julier SJ, Uhlmann JK (2004) Unscented filtering and nonlinear estimation. Proc IEEE 92(3): 401–422

  24. Keener J, Sneyd J (1998) Systems physiology. Part II. In: Marsden JE, Sirovich L, Wiggns S (eds) Mathematical physiology. Springer Verlag, NYC, pp 542–578

  25. Kobetic R, Triolo RJ, Marsolais EB (1997) Muscle selection and walking performance of multichannel fes systems for ambulation in Paraplegia. IEEE Trans Rehabil Eng 5: 23–29

  26. Kobetic R, Triolo RJ, Uhlir JP, Bieri C, Wibowo M, Polando G, Marsolais EB, Davis JA (1999) Implanted functional electrical stimulation system for mobility in Paraplegia: a follow-up case report. IEEE Trans Rehabil Eng 7(4): 390–398

  27. Kralj A, Bajd T (1989) Functional electrical stimulation: standing and walking after spinal cord injury. CRC Press Inc, Boca Raton

  28. Leeuwen JL (1991) Optimum power output and structural design of sarcomeres. J Theor Biol 19: 229–256

  29. Liberson WT, Holmquest ME, Scot D, Dow M (1961) Functional electrotherapy: stimulation of the peroneal nerve synchronized with the swing phase of gait of hemiplegic patients. Arch Phys Med Rehabil 42: 101–105

  30. Loeb GE, Peck RA, Moore WH, Hood K (2001) BIONtm system for distributed neural prosthetic interfaces. Med Eng Phys 23: 9–18

  31. Merwe R, Wan E (2003) Sigma-Point Kalman Filters for probabilistic inference in dynamic state-space models. Workshop on Advances in Machine Learning, Montreal

  32. Mirsky I, Parmley WW (1973) Assessment of passive elactic stiffness for isolated heart muscle and intact heart. Circ Res 33: 233–243

  33. Mohammed S, El Makssoud H, Fraisse P, Guiraud D, Poignet P (2005) Robust control law strategy based on high order sliding mode: towards a muscle control. International conference on intelligent robots and systems, Alberta

  34. Mohammed S, Fraisse P, Guiraud D, Poignet P, El Makssoud H (2005) Towards a co-contraction muscle control strategy, Conference on Decision and Control, CDC, pp 7428–7483

  35. Papaiordanidou M, Guiraud D, Varray A (2010a) Does central fatigue exist under low-frequency stimulation of a low fatigue-resistant muscle?. Eur J Appl Physiol 110(4): 815–823

  36. Papaiordanidou M, Guiraud D, Varray A (2010b) Kinetics of neuromuscular changes during low-frequency electrical stimulation. Muscle Nerve 41: 54–62

  37. Perreault EJ, Heckman CJ, Sandercock TG (2003) Hill muscle model errors during movement are greatest within the physiologically relevant range of motor unit firing rates. J Biomech 36: 211–218

  38. Popovic D, Sinkjaer T (2000) Control of movement for the physically disabled: control for rehabilitation technology. Spinger-Verlag, London

  39. Rabischong E, Guiraud D (1993) Determination of fatigue in the electrically stimulated quadriceps muscle and relative effect of ischaemia. J Biomed Eng 15: 443–450

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

  41. Riener R, Quintern J, Psaier E, Schmidt G (1996) Physiological based multi-input model of muscle activation. In: Pedotti A, Ferrarin F, Quintern J, Riener R (eds) Neuroprosthetics from basic research to clinical applications, Chap. 2. Springer-verlag, New York, pp 95–114

  42. Riener R, Fuhr T (1998) Patient-driven control of FES supported standing up: a simulation study. IEEE Trans Rehabil Eng 6: 113–1240

  43. Riener R (1999) Model-based development of neuroprostheses for paraplegic patients. R Soc 354: 877–894

  44. Sainte-Marie J, Chapelle D, Cimrman R, Sorine M (2006) Modeling and estimation of the cardiac electromechanical activity. Comput Struct 84(28): 1743–1759

  45. Shi-Ping M, Zahalak GI (1987) Simple self-consistent distribution-moment model for muscle: chemical energy and heat rates. Math Biosci 84: 211–230

  46. Sommacal L, Dossat A, Melchior P, Petit J, Cabelguen JM, Oustaloup A, Ljspeert AJ (2006) A comparison between two fractional multimodels structures for rat muscle modelling. 6th IFAC symposium on modelling and control in biomedical systems, Reims, France, pp 20–22

  47. Wexler AS, Ding J, Binder Macleod SA (1997) A mathematical model that predicts skeletal muscle force. IEEE Trans Biomed Eng 4(5): 337–348

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

  49. Zahalak GI (1981) A distribution-moment approximation for kinetic theories of muscular contraction. Math Biosci 55: 89–114

  50. Zahalak GI (1992) An overview of muscle modelling. Neural prostheses. In: Stein R, Peckham H, Popovic D (eds) Replacing motor function after disease or disability. Oxford University Press, New York

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Correspondence to David Guiraud.

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An erratum to this article can be found at

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El Makssoud, H., Guiraud, D., Poignet, P. et al. Multiscale modeling of skeletal muscle properties and experimental validations in isometric conditions. Biol Cybern 105, 121 (2011).

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  • Muscle model
  • Parameters identification
  • Kalman filtering
  • Electrical stimulation