Acta Mechanica

, Volume 230, Issue 6, pp 2125–2135 | Cite as

A visco-hyperelastic model of brain tissue incorporating both tension/compression asymmetry and volume compressibility

  • Zhongmeng Zhu
  • Chengkai Jiang
  • Han JiangEmail author
Original Paper


The understanding of the mechanical behavior of the brain tissue is essential to prevent the occurrence of potential brain damage, such as traumatic brain injury. Recent experimental results showed that brain tissue exhibits significant tension/compression asymmetry. Due to the migration and diffusion of interstitial fluid, brain tissue also shows a moderate volume compressibility during loading. These mechanical characteristics have a strong impact on the deformation response of brain tissue. In this paper, a visco-hyperelastic constitutive model incorporating both tension/compression asymmetry and volume compressibility is proposed to describe the mechanical behavior of brain tissue under various loading modes. An Ogden-type model with the addition of a viscoelastic part is used to characterize the tension/compression asymmetry as well as the viscoelastic properties. Poisson’s ratio was introduced as a phenomenological index to represent the total volume change as well as the compressibility/recoverability. The mechanical responses of brain tissue under uniaxial tension, unconfined compression, stress relaxation, and cyclic compression were reproduced with a good capture of the tension/compression asymmetry, volume compressibility, significant viscoelastic properties, and cyclic hysteresis behaviors. The good agreement with the experimental data implies that the proposed model has a strong capability to describe the complex mechanical performance of brain tissue under a variety of loading conditions.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This work was supported by the National Natural Science Foundation of China (Grant No. 11472231, 11872322) and partially supported by Doctoral Innovation Fund Program of Southwest Jiaotong University (Grant No. D-CX201835).


  1. 1.
    Taylor, C.A., Bell, J.M., Breiding, M.J., Xu, L.: Traumatic brain injury–related emergency department visits, hospitalizations, and deaths-united states, 2007 and 2013. MMWR Surveill Summ. 66(SS-9), 1–16 (2017)Google Scholar
  2. 2.
    Miller, K., Chinzei, K.: Mechanical properties of brain tissue in tension. J. Biomech. 35(4), 483–490 (2002)CrossRefGoogle Scholar
  3. 3.
    Zhang, W., Zhang, R.R., Wu, F., Feng, L.I., Yu, S.B., CW, Wu: Differences in the viscoelastic features of white and grey matter in tension. J. Biomech. 49(16), 3990–3995 (2016)CrossRefGoogle Scholar
  4. 4.
    Labus, K.M., Puttlitz, C.M.: Viscoelasticity of brain corpus callosum in biaxial tension. J. Mech. Phys. Solids 96, 591–604 (2016)CrossRefGoogle Scholar
  5. 5.
    Pervin, F., Chen, W.W.: Dynamic mechanical response of bovine gray matter and white matter brain tissues under compression. J. Biomech. 42(6), 731–735 (2009)CrossRefGoogle Scholar
  6. 6.
    Prevost, T.P., Balakrishnan, A., Suresh, S., Socrate, S.: Biomechanics of brain tissue. Acta Biomat. 7(1), 83–95 (2011)CrossRefGoogle Scholar
  7. 7.
    Bilston, L.E., Liu, Z., Phan-Thien, N.: Linear viscoelastic properties of bovine brain tissue in shear. Biorheology 34(6), 377–385 (1997)CrossRefGoogle Scholar
  8. 8.
    Rashid, B., Destrade, M., Gilchrist, M.D.: Mechanical characterization of brain tissue in simple shear at dynamic strain rates. J. Mech. Behav. Biomed. 28, 71–85 (2013)CrossRefGoogle Scholar
  9. 9.
    Moran, R., Smith, J.H., García, J.J.: Fitted hyperelastic parameters for human brain tissue from reported tension, compression, and shear tests. J. Biomech. 47(15), 3762–3766 (2014)CrossRefGoogle Scholar
  10. 10.
    Destrade, M., Gilchrist, M., Murphy, J.G., Rashid, B., Saccomandi, G.: Extreme softness of brain matter in simple shear. Int. J. Nonlin. Mech. 75, 54–58 (2015)CrossRefGoogle Scholar
  11. 11.
    Li, G., Zhang, J., Wang, K., Wang, M., Gao, C., Ma, C.: Experimental research of mechanical behavior of porcine brain tissue under rotational shear stress. J. Mech. Behav. Biomed. 57, 224–234 (2016)CrossRefGoogle Scholar
  12. 12.
    Balakrishnan, A.: Development of novel dynamic indentation techniques for soft tissue applications. Ph.D. thesis. (2007)
  13. 13.
    Van Dommelen, J., Van der Sande, T., Hrapko, M., Peters, G.: Mechanical properties of brain tissue by indentation: interregional variation. J. Mech. Behav. Biomed. Mater. 3(2), 158–166 (2010)CrossRefGoogle Scholar
  14. 14.
    Prevost, T.P., Jin, G., De Moya, M.A., Alam, H.B., Suresh, S., Socrate, S.: Dynamic mechanical response of brain tissue in indentation in vivo, in situ and in vitro. Acta Biomat. 7(12), 4090–4101 (2011)CrossRefGoogle Scholar
  15. 15.
    Budday, S., Nay, R., de Rooij, R., Steinmann, P., Wyrobek, T., Ovaert, T.C., Kuhl, E.: Mechanical properties of gray and white matter brain tissue by indentation. J. Mech. Behav. Biomed. Mater. 46, 318–330 (2015)CrossRefGoogle Scholar
  16. 16.
    MacManus, D.B., Pierrat, B., Murphy, J.G., Gilchrist, M.D.: A viscoelastic analysis of the P56 mouse brain under large-deformation dynamic indentation. Acta Biomat. 48, 309–318 (2017)CrossRefGoogle Scholar
  17. 17.
    Feng, Y., Lee, C.H., Sun, L., Ji, S., Zhao, X.: Characterizing white matter tissue in large strain via asymmetric indentation and inverse finite element modeling. J. Mech. Behav. Biomed. Mater. 65, 490–501 (2017)CrossRefGoogle Scholar
  18. 18.
    Budday, S., Sommer, G., Birkl, C., Langkammer, C., Haybaeck, J., Kohnert, J., Bauer, M., Paulsen, F., Steinmann, P., Kuhl, E.: Mechanical characterization of human brain tissue. Acta Biomat. 48, 319–340 (2017)CrossRefGoogle Scholar
  19. 19.
    Pogoda, K., Chin, L., Georges, P.C., Byfield, F.J., Bucki, R., Kim, R., Weaver, M., Wells, R.G., Marcinkiewicz, C., Janmey, P.A.: Compression stiffening of brain and its effect on mechanosensing by glioma cells. New J. Phys. 16(7), 075002 (2014)CrossRefGoogle Scholar
  20. 20.
    Mihai, L.A., Chin, L., Janmey, P.A., Goriely, A.: A comparison of hyperelastic constitutive models applicable to brain and fat tissues. J. R. Soc. Interface 12(110), 20150486 (2015)CrossRefGoogle Scholar
  21. 21.
    Mihai, L.A., Budday, S., Holzapfel, G.A., Kuhl, E., Goriely, A.: A family of hyperelastic models for human brain tissue. J. Mech. Phys. Solids 106, 60–79 (2017)CrossRefGoogle Scholar
  22. 22.
    Lodish, H.: Molecular Cell Biology, 6th edn. Macmillan, New York (2008)Google Scholar
  23. 23.
    de Rooij, R., Kuhl, E.: Constitutive modeling of brain tissue: current perspectives. Appl. Mech. Rev. 68(1), 010801 (2016)CrossRefGoogle Scholar
  24. 24.
    Shuck, L.Z., Advani, S.H.: Rheological response of human brain tissue in shear. J. Basic Eng. 94, 905–911 (1972)CrossRefGoogle Scholar
  25. 25.
    Jenson, D., Unnikrishnan, V.U.: Energy dissipation of nanocomposite based helmets for blast-induced traumatic brain injury mitigation. Compos. Struct. 121, 211–216 (2015)CrossRefGoogle Scholar
  26. 26.
    Hrapko, M., Van Dommelen, J., Peters, G., Wismans, J.: The mechanical behaviour of brain tissue: large strain response and constitutive modelling. Biorheology 43(5), 623–636 (2006)Google Scholar
  27. 27.
    Bilston, L.E., Liu, Z., Phan-Thien, N.: Large strain behaviour of brain tissue in shear: some experimental data and differential constitutive model. Biorheology 38(4), 335–345 (2001)Google Scholar
  28. 28.
    Cloots, R., Van Dommelen, J., Nyberg, T., Kleiven, S., Geers, M.: Micromechanics of diffuse axonal injury: influence of axonal orientation and anisotropy. Biomech. Model. Mechan. 10(3), 413–422 (2011)CrossRefGoogle Scholar
  29. 29.
    Laksari, K., Shafieian, M., Darvish, K.: Constitutive model for brain tissue under finite compression. J. Biomech. 45(4), 642–646 (2012)CrossRefGoogle Scholar
  30. 30.
    Ogden, R.W.: Large deformation isotropic elasticity-on the correlation of theory and experiment for incompressible rubberlike solids. Proc. R. Soc. Lond. A 326, 565–584 (1972)CrossRefzbMATHGoogle Scholar
  31. 31.
    Mendis, K., Stalnaker, R., Advani, S.: A constitutive relationship for large deformation finite element modeling of brain tissue. J. Biomech. Eng. 117(3), 279–285 (1995)CrossRefGoogle Scholar
  32. 32.
    Miller, K., Chinzei, K.: Constitutive modelling of brain tissue: experiment and theory. J. Biomech. 30(11), 1115–1121 (1997)CrossRefGoogle Scholar
  33. 33.
    Prange, M.T., Margulies, S.S.: Regional, directional, and age-dependent properties of the brain undergoing large deformation. J. Biomech. Eng. 124(2), 244–252 (2002)CrossRefGoogle Scholar
  34. 34.
    Budday, S., Sommer, G., Hayback, J., Steinmann, P., Holzapfel, G.A., Kuhl, E.: Rheological characterization of human brain tissue. Acta Biomat. 60, 315–329 (2017)CrossRefGoogle Scholar
  35. 35.
    Budday, S., Sommer, G., Holzapfel, G.A., Steinmann, P., Kuhl, E.: Viscoelastic parameter identification of human brain tissue. J. Mech. Behav. Biomed. Mater. 74, 463–476 (2017)CrossRefGoogle Scholar
  36. 36.
    Haslach Jr., H.W., Leahy, L.N., Riley, P., Gullapalli, R., Xu, S., Hsieh, A.H.: Solid-extracellular fluid interaction and damage in the mechanical response of rat brain tissue under confined compression. J. Mech. Behav. Biomed. Mater. 29, 138–150 (2014)CrossRefGoogle Scholar
  37. 37.
    Grandjean, A.C., Grandjean, N.R.: Dehydration and cognitive performance. J. Am. Coll. Nutr. 26(5), 549S–554S (2007)CrossRefGoogle Scholar
  38. 38.
    Yoon, J., Cai, S., Suo, Z., Hayward, R.C.: Poroelastic swelling kinetics of thin hydrogel layers: comparison of theory and experiment. Soft Matter 6(23), 6004–6012 (2010)CrossRefGoogle Scholar
  39. 39.
    Wang, Q.-M., Mohan, A.C., Oyen, M.L., Zhao, X.-H.: Separating viscoelasticity and poroelasticity of gels with different length and time scales. Acta Mech. Sinica-PRC 30(1), 20–27 (2014)CrossRefzbMATHGoogle Scholar
  40. 40.
    Hu, Y., Suo, Z.: Viscoelasticity and poroelasticity in elastomeric gels. Acta Mech. Solida Sin. 25(5), 441–458 (2012)CrossRefGoogle Scholar
  41. 41.
    Kaczmarek, M., Subramaniam, R.P., Neff, S.R.: The hydromechanics of hydrocephalus: steady-state solutions for cylindrical geometry. B. Math. Biol. 59, 295–323 (1997)CrossRefzbMATHGoogle Scholar
  42. 42.
    Kyriacou, S.K., Mohamed, A., Miller, K., Neff, S.: Brain mechanics for neurosurgery: modeling issues. Biomech. Model. Mechan. 1, 151–164 (2002)CrossRefGoogle Scholar
  43. 43.
    Ames, N.M.: An internal variable theory for isotropic visco-elastic-plastic solids: application to indentation of amorphous polymeric solids. Master thesis. (2003)
  44. 44.
    Anand, L., Ames, N.: On modeling the micro-indentation response of an amorphous polymer. Int. J. Plasticity 22(6), 1123–1170 (2006)CrossRefzbMATHGoogle Scholar
  45. 45.
    Voyiadjis, G.Z., Samadi-Dooki, A.: Hyperelastic modeling of the human brain tissue: effects of no-slip boundary condition and compressibility on the uniaxial deformation. J. Mech. Behav. Biomed. Mater. 83, 63–78 (2018)CrossRefGoogle Scholar
  46. 46.
    Kleiven, S.: Predictors for traumatic brain injuries evaluated through accident reconstructions (No. 2007-22-0003). SAE Technical Paper. (2007)Google Scholar
  47. 47.
    Wang, R., Sarntinoranont, M.: Biphasic analysis of rat brain slices under creep indentation shows nonlinear tension-compression behavior. J. Mech. Behav. Biomed. Mater. 89, 1–8 (2019)CrossRefGoogle Scholar
  48. 48.
    Samadidooki, A.: Experimental, Analytical, and Numerical Evaluation of the Mechanical Properties of the Brain Tissue. Doctoral thesis. (2018)
  49. 49.
    Abdulhafez, M., Kadry, K., Zaazoue, M., Goumnerova, L.C., Bedewy, M.: Biomechanical root cause analysis of complications in head immobilization devices for pediatric neurosurgery. In: ASME 2018 13th International Manufacturing Science and Engineering Conference (pp. V001T05A007-V001T05A007). American Society of Mechanical Engineers. (2018)Google Scholar
  50. 50.
    Viano, D.C., Casson, I.R., Pellman, E.J., Zhang, L., King, A.I., Yang, K.H.: Concussion in professional football: brain responses by finite element analysis: part 9. Neurosurgery 57(5), 891–916 (2005)CrossRefGoogle Scholar
  51. 51.
    Gu, L., Chafi, M.S., Ganpule, S., Chandra, N.: The influence of heterogeneous meninges on the brain mechanics under primary blast loading. Compos. Part B- Eng. 43(8), 3160–3166 (2012)CrossRefGoogle Scholar
  52. 52.
    Nolan, D., Gower, A., Destrade, M., Ogden, R., McGarry, J.: A robust anisotropic hyperelastic formulation for the modelling of soft tissue. J. Mech. Behav. Biomed. Mater. 39, 48–60 (2014)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and EngineeringSouthwest Jiaotong UniversityChengduPeople’s Republic of China

Personalised recommendations