Fibre and extracellular matrix contributions to passive forces in human skeletal muscles: An experimental based constitutive law for numerical modelling of the passive element in the classical Hill-type three element model

Autoři: Lorenzo Marcucci aff001;  Michela Bondì aff001;  Giulia Randazzo aff001;  Carlo Reggiani aff001;  Arturo N. Natali aff002;  Piero G. Pavan aff002
Působiště autorů: Department of Biomedical Sciences, University of Padova, Padova, Italy aff001;  Centre for Mechanics of Biological Materials, University of Padova, Padova, Italy aff002;  Kinesiology Research Center, Garibaldijeva, Koper, Slovenia aff003;  Department of Industrial Engineering, University of Padova, Padova, Italy aff004;  Fondazione Istituto di Ricerca Pediatrica Città della Speranza, Corso Stati Uniti 4, Padova, Italy aff005
Vyšlo v časopise: PLoS ONE 14(11)
Kategorie: Research Article
doi: 10.1371/journal.pone.0224232


The forces that allow body movement can be divided into active (generated by sarcomeric contractile proteins) and passive (sustained by intra-sarcomeric proteins, fibre cytoskeleton and extracellular matrix (ECM)). These are needed to transmit the active forces to the tendon and the skeleton. However, the relative contribution of the intra- and extra- sarcomeric components in transmitting the passive forces is still under debate. There is limited data in the literature about human muscle and so it is difficult to make predictions using multiscale models, imposing a purely phenomenological description for passive forces. In this paper, we apply a method for the experimental characterization of the passive properties of fibres and ECM to human biopsy and propose their clear separation in a Finite Element Model. Experimental data were collected on human single muscle fibres and bundles, taken from vastus lateralis muscle of elderly subjects. Both were progressively elongated to obtain two stress-strain curves which were fitted to exponential equations. The mechanical properties of the extracellular passive components in a bundle of fibres were deduced by the subtraction of the passive tension observed in single fibres from the passive tension observed in the bundle itself. Our results showed that modulus and tensile load bearing capability of ECM are higher than those of fibres and defined their quantitative characterization that can be used in macroscopic models to study their role in the transmission of forces in physiological and pathophysiological conditions.

Klíčová slova:

Behavior – Extracellular matrix – Muscle components – Muscle fibers – Muscle proteins – Skeletal muscles – Tangents


1. Huxley AF. Muscle Structure and Theories of Contraction. Prog Biophys Biophys Chem. 1957;7: 255–318. 13485191

2. Street SF. Lateral transmission of tension in frog myofibers: a myofibrillar network and transverse cytoskeletal connections are possible transmitters. J Cell Physiol. 1983;114: 346–364. doi: 10.1002/jcp.1041140314 6601109

3. Huijing PA, Baan GC, Rebel GT. Non-myotendinous force transmission in rat extensor digitorum longus muscle. J Exp Biol. 1998;201: 683–691.

4. Ramaswamy KS, Palmer ML, van der Meulen JH, Renoux A, Kostrominova TY, Michele DE, et al. Lateral transmission of force is impaired in skeletal muscles of dystrophic mice and very old rats. J Physiol. 2011;589: 1195–1208. doi: 10.1113/jphysiol.2010.201921 21224224

5. Moore AZ, Caturegli G, Metter EJ, Makrogiannis S, Resnick SM, Harris TB, et al. Difference in muscle quality over the adult life span and biological correlates in the Baltimore Longitudinal Study of Aging. J Am Geriatr Soc. 2014;62: 230–236. doi: 10.1111/jgs.12653 24438020

6. Virgilio KM, Martin KS, Peirce SM, Blemker SS. Multiscale models of skeletal muscle reveal the complex effects of muscular dystrophy on tissue mechanics and damage susceptibility. Interface Focus. 2015;5: 20140080. doi: 10.1098/rsfs.2014.0080 25844152

7. Magid A, Law DJ. Myofibrils bear most of the resting tension in frog skeletal muscle. Science. 1985;230: 1280–1282. doi: 10.1126/science.4071053 4071053

8. Nocella M, Cecchi G, Bagni MA, Colombini B. Force enhancement after stretch in mammalian muscle fiber: no evidence of cross-bridge involvement. Am J Physiol-Cell Physiol. 2014;307: C1123–C1129. doi: 10.1152/ajpcell.00290.2014 25298425

9. Granzier HL, Irving TC. Passive tension in cardiac muscle: contribution of collagen, titin, microtubules, and intermediate filaments. Biophys J. 1995;68: 1027–1044. doi: 10.1016/S0006-3495(95)80278-X 7756523

10. Brown SHM, Carr JA, Ward SR, Lieber RL. Passive mechanical properties of rat abdominal wall muscles suggest an important role of the extracellular connective tissue matrix. J Orthop Res Off Publ Orthop Res Soc. 2012;30: 1321–1326. doi: 10.1002/jor.22068 22267257

11. Gillies AR, Smith LR, Lieber RL, Varghese S. Method for decellularizing skeletal muscle without detergents or proteolytic enzymes. Tissue Eng Part C Methods. 2011;17: 383–389. doi: 10.1089/ten.TEC.2010.0438 20973753

12. Prado LG, Makarenko I, Andresen C, Krüger M, Opitz CA, Linke WA. Isoform diversity of giant proteins in relation to passive and active contractile properties of rabbit skeletal muscles. J Gen Physiol. 2005;126: 461–480. doi: 10.1085/jgp.200509364 16230467

13. Smith LR, Barton ER. Collagen content does not alter the passive mechanical properties of fibrotic skeletal muscle in mdx mice. Am J Physiol Cell Physiol. 2014;306: C889–898. doi: 10.1152/ajpcell.00383.2013 24598364

14. Lieber RL, Runesson E, Einarsson F, Fridén J. Inferior mechanical properties of spastic muscle bundles due to hypertrophic but compromised extracellular matrix material. Muscle Nerve. 2003;28: 464–471. doi: 10.1002/mus.10446 14506719

15. Ward SR, Tomiya A, Regev GJ, Thacker BE, Benzl RC, Kim CW, et al. Passive mechanical properties of the lumbar multifidus muscle support its role as a stabilizer. J Biomech. 2009;42: 1384–1389. doi: 10.1016/j.jbiomech.2008.09.042 19457491

16. Meyer GA, Lieber RL. Elucidation of extracellular matrix mechanics from muscle fibers and fiber bundles. J Biomech. 2011;44: 771–773. doi: 10.1016/j.jbiomech.2010.10.044 21092966

17. Smith LR, Lee KS, Ward SR, Chambers HG, Lieber RL. Hamstring contractures in children with spastic cerebral palsy result from a stiffer extracellular matrix and increased in vivo sarcomere length. J Physiol. 2011;589: 2625–2639. doi: 10.1113/jphysiol.2010.203364 21486759

18. Brynnel A, Hernandez Y, Kiss B, Lindqvist J, Adler M, Kolb J, et al. Downsizing the molecular spring of the giant protein titin reveals that skeletal muscle titin determines passive stiffness and drives longitudinal hypertrophy. Ha T, Chakraborty AK, editors. eLife. 2018;7: e40532. doi: 10.7554/eLife.40532 30565562

19. Meyer G, Lieber RL. Muscle fibers bear a larger fraction of passive muscle tension in frogs compared with mice. J Exp Biol. 2018;221: jeb182089. doi: 10.1242/jeb.182089 30237238

20. Wood LK, Kayupov E, Gumucio JP, Mendias CL, Claflin DR, Brooks SV. Intrinsic stiffness of extracellular matrix increases with age in skeletal muscles of mice. J Appl Physiol. 2014;117: 363–369. doi: 10.1152/japplphysiol.00256.2014 24994884

21. Silldorff MD, Choo AD, Choi AJ, Lin E, Carr JA, Lieber RL, et al. Effect of Supraspinatus Tendon Injury on Supraspinatus and Infraspinatus Muscle Passive Tension and Associated Biochemistry. J Bone Joint Surg Am. 2014;96. doi: 10.2106/JBJS.M.01315 25320205

22. Gillies AR, Lieber RL. Structure and function of the skeletal muscle extracellular matrix. Muscle Nerve. 2011;44: 318–331. doi: 10.1002/mus.22094 21949456

23. Mijailovich SM, Stojanovic B, Kojic M, Liang A, Wedeen VJ, Gilbert RJ. Derivation of a finite-element model of lingual deformation during swallowing from the mechanics of mesoscale myofiber tracts obtained by MRI. J Appl Physiol Bethesda Md 1985. 2010;109: 1500–1514. doi: 10.1152/japplphysiol.00493.2010 20689096

24. Pato MPM, Santos NJG, Areias P, Pires EB, Carvalho M de, Pinto S, et al. Finite element studies of the mechanical behaviour of the diaphragm in normal and pathological cases. Comput Methods Biomech Biomed Engin. 2011;14: 505–513. doi: 10.1080/10255842.2010.483683 21082461

25. Zhang G, Chen X, Ohgi J, Miura T, Nakamoto A, Matsumura C, et al. Biomechanical simulation of thorax deformation using finite element approach. Biomed Eng OnLine. 2016;15: 1–18. doi: 10.1186/s12938-016-0132-y 26852020

26. Marcucci L, Reggiani C, Natali AN, Pavan PG. From single muscle fiber to whole muscle mechanics: a finite element model of a muscle bundle with fast and slow fibers. Biomech Model Mechanobiol. 2017;16: 1833–1843. doi: 10.1007/s10237-017-0922-6 28584973

27. Tang CY, Zhang G, Tsui CP. A 3D skeletal muscle model coupled with active contraction of muscle fibres and hyperelastic behaviour. J Biomech. 2009;42: 865–872. doi: 10.1016/j.jbiomech.2009.01.021 19264310

28. Lu YT, Zhu HX, Richmond S, Middleton J. A visco-hyperelastic model for skeletal muscle tissue under high strain rates. J Biomech. 2010;43: 2629–2632. doi: 10.1016/j.jbiomech.2010.05.030 20566197

29. Gindre J, Takaza M, Moerman KM, Simms CK. A structural model of passive skeletal muscle shows two reinforcement processes in resisting deformation. J Mech Behav Biomed Mater. 2013;22: 84–94. doi: 10.1016/j.jmbbm.2013.02.007 23587721

30. Bleiler C, Ponte Castañeda P, Röhrle O. A microstructurally-based, multi-scale, continuum-mechanical model for the passive behaviour of skeletal muscle tissue. J Mech Behav Biomed Mater. 2019;97: 171–186. doi: 10.1016/j.jmbbm.2019.05.012 31125890

31. Spyrou LA, Agoras M, Danas K. A homogenization model of the Voigt type for skeletal muscle. J Theor Biol. 2017;414: 50–61. doi: 10.1016/j.jtbi.2016.11.018 27884495

32. Blemker SS, Pinsky PM, Delp SL. A 3D model of muscle reveals the causes of nonuniform strains in the biceps brachii. J Biomech. 2005;38: 657–665. doi: 10.1016/j.jbiomech.2004.04.009 15713285

33. Sharafi B, Blemker SS. A mathematical model of force transmission from intrafascicularly terminating muscle fibers. J Biomech. 2011;44: 2031–2039. doi: 10.1016/j.jbiomech.2011.04.038 21676398

34. Lieber RL, Ward SR. Skeletal muscle design to meet functional demands. Philos Trans R Soc B Biol Sci. 2011;366: 1466. doi: 10.1098/rstb.2010.0316 21502118

35. Pišot R, Marusic U, Biolo G, Mazzucco S, Lazzer S, Grassi B, et al. Greater loss in muscle mass and function but smaller metabolic alterations in older compared with younger men following 2 wk of bed rest and recovery. J Appl Physiol Bethesda Md 1985. 2016;120: 922–929. doi: 10.1152/japplphysiol.00858.2015 26823343

36. Rejc E, Floreani M, Taboga P, Botter A, Toniolo L, Cancellara L, et al. Loss of maximal explosive power of lower limbs after 2 weeks of disuse and incomplete recovery after retraining in older adults. J Physiol. 2018;596: 647–665. doi: 10.1113/JP274772 29266264

37. NIH Image to ImageJ: 25 years of image analysis | Nature Methods [Internet]. [cited 19 Feb 2019]. Available:

38. Metzger JM, Moss RL. Calcium-sensitive cross-bridge transitions in mammalian fast and slow skeletal muscle fibers. Science. 1990;247: 1088–1090. doi: 10.1126/science.2309121 2309121

39. Hill AV. The Heat of Shortening and the Dynamic Constants of Muscle. Proc R Soc Lond B Biol Sci. 1938;126: 136–195. doi: 10.1098/rspb.1938.0050

40. Rehorn MR, Schroer AK, Blemker SS. The passive properties of muscle fibers are velocity dependent. J Biomech. 2014;47: 687–693. doi: 10.1016/j.jbiomech.2013.11.044 24360198

41. Gao Y, Wineman AS, Waas AM. Mechanics of muscle injury induced by lengthening contraction. Ann Biomed Eng. 2008;36: 1615–1623. doi: 10.1007/s10439-008-9547-3 18686034

42. Motulsky HJ, Ransnas LA. Fitting curves to data using nonlinear regression: a practical and nonmathematical review. FASEB J Off Publ Fed Am Soc Exp Biol. 1987;1: 365–374.

43. He ZH, Bottinelli R, Pellegrino MA, Ferenczi MA, Reggiani C. ATP consumption and efficiency of human single muscle fibers with different myosin isoform composition. Biophys J. 2000;79: 945. doi: 10.1016/S0006-3495(00)76349-1 10920025

44. Larsson L, Moss RL. Maximum velocity of shortening in relation to myosin isoform composition in single fibres from human skeletal muscles. J Physiol. 1993;472: 595–614. doi: 10.1113/jphysiol.1993.sp019964 8145163

45. Wang K, McCarter R, Wright J, Beverly J, Ramirez-Mitchell R. Regulation of skeletal muscle stiffness and elasticity by titin isoforms: a test of the segmental extension model of resting tension. Proc Natl Acad Sci U S A. 1991;88: 7101–7105. doi: 10.1073/pnas.88.16.7101 1714586

46. Chen X, Sanchez GN, Schnitzer MJ, Delp SL. Changes in sarcomere lengths of the human vastus lateralis muscle with knee flexion measured using in vivo microendoscopy. J Biomech. 2016;49: 2989–2994. doi: 10.1016/j.jbiomech.2016.07.013 27481293

47. Lieber RL, Loren GJ, Fridén J. In vivo measurement of human wrist extensor muscle sarcomere length changes. J Neurophysiol. 1994;71: 874–881. doi: 10.1152/jn.1994.71.3.874 8201427

48. Horowits R. Passive force generation and titin isoforms in mammalian skeletal muscle. Biophys J. 1992;61: 392–398. doi: 10.1016/S0006-3495(92)81845-3 1547327

49. Tirrell TF, Cook MS, Carr JA, Lin E, Ward SR, Lieber RL. Human skeletal muscle biochemical diversity. J Exp Biol. 2012;215: 2551–2559. doi: 10.1242/jeb.069385 22786631

50. Hwang W, Kelly NG, Boriek AM. Passive mechanics of muscle tendinous junction of canine diaphragm. J Appl Physiol Bethesda Md 1985. 2005;98: 1328–1333. doi: 10.1152/japplphysiol.00816.2004 15772060

51. Sharafi B, Blemker SS. A micromechanical model of skeletal muscle to explore the effects of fiber and fascicle geometry. J Biomech. 2010;43: 3207–3213. doi: 10.1016/j.jbiomech.2010.07.020 20846654

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