Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Viscoelasticity of the axon limits stretch-mediated growth

  • 188 Accesses


Understanding how axons fail is critical to preventing brain injury. From stretch experiments, we know how axons respond to forces on the time scales of milliseconds and days. Yet, there is no mechanical model that explains the behavior of the axon at both short and long time scales. Here we propose a constitutive model to study the limits of stretch-mediated axonal disconnection at different time scales. Our model combines viscoelasticity using a neo-Hookean standard linear solid and growth using stress-mediated accelerated elongation. By limiting peak and average membrane tensions, our model predicts critical elongations and elongation rates. Interestingly, the critical elongation rate is not constant, but increases after an acclimation period. Combining viscoelasticity and growth is essential to simulate axonal disconnection in stretch-mediated growth at both short and long time scales. Our model can help optimize axonal stretch experiments and provides insight into the interacting time scales within the axon.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10


  1. 1.

    Abe I, Ochiai N, Ichimura H, Tsujino A, Sun J, Hara Y (2004) Internodes can nearly double in length with gradual elongation of the adult rat sciatic nerve. J Orthop Res 22:571–577

  2. 2.

    Arnaoutoglou CM, Sakellariou A, Vekris M, Mitsionis GI, Korompilias A, Ioakim E, Harhantis A, Beris A (2006) Maximum intraoperative elongation of the rat sciatic nerve with tissue expander: functional neurophysiological, and histological assessment. Microsurgery 26:253–261

  3. 3.

    Bray D (1984) Axonal growth in response to experimentally applied mechanical tension. Dev Biol 102:379–389

  4. 4.

    Budday S, Sommer G, Holzapfel GA, Steinmann P, Kuhl E (2017) Viscoelastic parameter identification of human brain tissue. J Mech Beh Biomed Mat 74:463–476

  5. 5.

    Chada S, Lamoureux P, Buxbaum RE, Heidemann SR (1997) Cytomechanics of neurite outgrowth from chick brain neurons. J Cell Sci 110:1179–1186

  6. 6.

    Dagg AI, Foster JB (1982) The giraffe, its biology, behavior and ecology. Van Nostrand Reinhold, New York

  7. 7.

    Dai J, Sheetz MP (1995) Mechanical properties of neuronal growth cone membranes studied by tether formation with laser optical tweezers. Biophys J 68:988–996

  8. 8.

    Dai J, Sheetz MP, Wan X, Morris CE (1998) Membrane tension in swelling and shrinking molluscan neurons. J Neurosci 18:6681–6692

  9. 9.

    Dennerll TJ, Lamoureux P, Buxbaum RE, Heidemann SR (1989) The cytomechanics of axonal elongation and retraction. J Cell Biol 109:3073–3083

  10. 10.

    de Rooij R, Miller KE, Kuhl E (2017) Modeling molecular mechanisms in the axon. Comput Mech 59:523–537

  11. 11.

    de Rooij R, Kuhl E (2018) Microtubule polymerization and cross-link dynamics explain axonal stiffness and damage. Biophys J 114:201–212

  12. 12.

    de Rooij R, Kuhl E, Miller KE (2018) Modeling the axon as an active partner with the growth cone in axonal elongation. Biophys J 115:1783–1795

  13. 13.

    de Rooij R, Kuhl E (2018) Physical biology of axonal damage. Front Cell Neurosci 12:144

  14. 14.

    Evans E, Heinrich V, Ludwig F, Rawicz W (2003) Dynamic tension spectroscopy and strength of biomembranes. Biophys J 85:2342–2350

  15. 15.

    García JA, Pena JM, McHugh S, Jérusalem A (2012) A model of the spatially dependent mechanical properties of the axon during its growth. CMES Comp Mod Eng Sci 87:411–432

  16. 16.

    García-Grajales JA, Jérusalem A, Goriely A (2017) Continuum mechanical modeling of axonal growth. Comput Methods Appl Mech Eng 314:147–163

  17. 17.

    Goriely A, Budday S, Kuhl E (2015) Neuromechanics: from neurons to brain. Adv Appl Mech 48:79–139

  18. 18.

    Harrison RG (1935) On the origin and development of the nervous system studied by the methods of experimental embryology. Proc R Soc Lond B 118:155–196

  19. 19.

    Hategan A, Law R, Kahn S, Discher DE (2003) Adhesively-tensed cell membranes: lysis kinetics and atomic force microscopy probing. Biophys J 85:2746–2759

  20. 20.

    Hentz VR, Rosen JM, Xiao SJ, McGill KC, Abraham G (1993) The nerve gap dilemma: a comparison of nerves repaired end to end under tension with nerve grafts in a primate model. J Hand Surg Am 18:417–425

  21. 21.

    Hochmuth RM, Buxbaum KL, Evans EA (1980) Temperature dependence of the viscoelastic recovery of red cell membrane. Biophys J 29:177–182

  22. 22.

    Holland MA, Miller KE, Kuhl E (2015) Emerging brain morphologies from axonal elongation. Ann Biomed Eng 43:1640–1653

  23. 23.

    Howarth HM, Alaziz T, Nicolds B, O’Connor S, Shah SB (2019) Redistribution of nerve strain enables end-to-end repair under tension without inhibiting nerve regeneration. Neural Regen Res 14:1280–1288

  24. 24.

    Kolodkin AL, Tessier-Lavigne M (2011) Mechanisms and molecules of neuronal wiring: a primer. Cold Spring Harb Perspect Biol 3:a001727

  25. 25.

    Lowery LA, Van Vactor D (2009) The trip of the tip: understanding the growth cone machinery. Nat Rev Mol Cell Biol 10:332–343

  26. 26.

    Menzel A (2005) Modelling of anisotropic growth in biological tissues. Biomech Mod Mechanobio 3:147–171

  27. 27.

    Menzel A, Kuhl E (2012) Frontiers in growth and remodeling. Mech Res Commun 42:1–14

  28. 28.

    McDonald DS, Bell MS (2010) Peripheral nerve gap repair facilitated by a dynamic tension device. Can J Plast Surg 18:e17–e19

  29. 29.

    O’Toole M, Lamoureux P, Miller KE (2008) A physical model of axonal elongation: force, viscosity, and adhesions govern the mode of outgrowth. Biophys J 94:2610–2620

  30. 30.

    Pannese E (1994) Neurocytology: fine structure of neurons, nerve processes, and neuroglial cells. Thieme, Stuttgart

  31. 31.

    Pfister BJ, Iwata A, Meaney DF (2004) Extreme stretch growth of integrated axons. J Neurosci 24:7978–7983

  32. 32.

    Purohit PK, Smith DH (2016) A model for stretch growth of neurons. J Biomech 49:3934–3942

  33. 33.

    Simpson AHRW, Halliday J, Hamilton DF, Smith M, Mills K (2013) Limb lengthening and peripheral nerve function—factors associated with deterioration of conduction. Acta Orthop 84:579–584

  34. 34.

    Smith DH, Wolf JA, Meaney DF (2001) A new strategy to produce sustained growth of central nervous system axons: continuous mechanical tension. Tissue Eng 7:131–139

  35. 35.

    Sunderland IR, Brenner MJ, Singham J, Rickman SR, Hunter DA, Mackinnon SE (2004) Effect of tension on nerve regeneration in rat sciatic nerve transection model. Ann Plast Surg 53:382–387

  36. 36.

    Terzis J, Faibisoff B, Williams B (1975) The nerve gap: suture under tension vs. graft. Plast Reconstr Surg 56:166–170

  37. 37.

    Van Veen MP, Van Pelt J (1994) Neuritic growth rate described by modeling microtubule dynamics. Bull Math Biol 56:249–273

  38. 38.

    Vaz KM, Brown JM, Shah SB (2014) Peripheral nerve lengthening as a regenerative strategy. Neural Regen Res 9:1498–1501

  39. 39.

    Yousef MAA, Dionigi P, Marconi S, Calligaro A, Cornaglia AI, Alfonsi E, Auricchio F (2015) Successful reconstruction of nerve defects using distraction neurogenesis with a new experimental device. Basic Clin Neurosci 6:253–264

  40. 40.

    Zheng J, Lamoureux P, Santiago V, Dennerll T, Buxbaum RE, Heidemann SR (1991) Tensile regulation of axonal elongation and initiation. J Neurosci 11:1117–1125

Download references


This work was supported by the National Science Foundation Graduate Research Fellowship DGE 1656518 and the Stanford School of Engineering Fellowship to Lucy M. Wang and by the National Science Foundation Grant CMMI 1727268 and the Stanford Bio-X IIP seed Grant to Ellen Kuhl.

Author information

Correspondence to Ellen Kuhl.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, L.M., Kuhl, E. Viscoelasticity of the axon limits stretch-mediated growth. Comput Mech 65, 587–595 (2020).

Download citation


  • Tension
  • Viscosity
  • Growth
  • Axon