Biomechanics and Modeling in Mechanobiology

, Volume 18, Issue 4, pp 969–981 | Cite as

Homogenization of heterogeneous brain tissue under quasi-static loading: a visco-hyperelastic model of a 3D RVE

  • Morteza Kazempour
  • Majid Baniassadi
  • Hamid Shahsavari
  • Yves Remond
  • Mostafa BaghaniEmail author
Original Paper


Researches, in the recent years, reveal the utmost importance of brain tissue assessment regarding its mechanical properties, especially for automatic robotic tools, surgical robots and helmet producing. For this reason, experimental and computational investigation of the brain behavior under different conditions seems crucial. However, experiments do not normally show the distribution of stress and injury in microscopic scale, and due to various factors are costly. Development of micromechanical methods, which could predict the brain behavior more appropriately, could highly be helpful in reducing these costs. This study presents computational analysis of heterogeneous part of the brain tissue under quasi-static loading. Heterogeneity is created by irregular distribution of neurons in a representative volume element (RVE). Considering time-dependent behavior of the tissue, a visco-hyperelastic constitutive model is developed to predict the RVE behavior more realistically. The RVE is studied in different loads and load rates; 1, 2, 3, 10 and 15% strain load are applied at 0.03 and 0.2 s on the RVE as tensile and shear loads. Due to complexity in geometry, self-consistent approximation method is employed to increase the volume fraction of neurons and analyze RVE behavior in various NVFs. The results show increasing the load rate leads to a raise in the maximum stress that indicates the tissue is more vulnerable at higher rates. Moreover, stiffness of the tissue is enhanced in higher NVFs. Additionally, it is found that axons undergo higher stresses; hence, they are more sensitive in accidents which lead to axonal death and would cause TBI and DAI.


Visco-hyperelastic Traumatic brain injury (TBI) Diffuse axonal injury (DAI) FEM 



  1. Abdel Rahman R et al (2012) An asymptotic method for the prediction of the anisotropic effective elastic properties of the cortical vein: superior sagittal sinus junction embedded within a homogenized cell element. Journal of Mechanics of Materials and Structures 7(6):593–611CrossRefGoogle Scholar
  2. Abolfathi N et al (2009) A micromechanical procedure for modelling the anisotropic mechanical properties of brain white matter. Computer Methods in Biomechanics and Biomedical Engineering 12(3):249–262CrossRefGoogle Scholar
  3. Bergström J, Boyce M (1998) Constitutive modeling of the large strain time-dependent behavior of elastomers. J Mech Phys Solids 46(5):931–954CrossRefzbMATHGoogle Scholar
  4. Bernick KB et al (2011) Biomechanics of single cortical neurons. Acta Biomater 7(3):1210–1219CrossRefGoogle Scholar
  5. Budday S et al (2015) Mechanical properties of gray and white matter brain tissue by indentation. J Mech Behav Biomed Mater 46:318–330CrossRefGoogle Scholar
  6. Christ AF et al (2010) Mechanical difference between white and gray matter in the rat cerebellum measured by scanning force microscopy. J Biomech 43(15):2986–2992CrossRefGoogle Scholar
  7. Cloots RJ et al (2013) Multi-scale mechanics of traumatic brain injury: predicting axonal strains from head loads. Biomech Model Mechanobiol 12(1):137–150CrossRefGoogle Scholar
  8. Coudrillier B et al (2013) Scleral anisotropy and its effects on the mechanical response of the optic nerve head. Biomech Model Mechanobiol 12(5):941–963CrossRefGoogle Scholar
  9. Couper Z, Albermani F (2008) Infant brain subjected to oscillatory loading: material differentiation, properties, and interface conditions. Biomech Model Mechanobiol 7(2):105CrossRefGoogle Scholar
  10. Dréo J et al (2006) Metaheuristics for hard optimization: methods and case studies. Springer, BerlinzbMATHGoogle Scholar
  11. Faul M et al (2010) Traumatic brain injury in the United States: national estimates of prevalence and incidence, 2002–2006. Injury Prevention 16(Suppl 1):A268CrossRefGoogle Scholar
  12. Feng Y et al (2013) Measurements of mechanical anisotropy in brain tissue and implications for transversely isotropic material models of white matter. J Mech Behav Biomed Mater 23:117–132CrossRefGoogle Scholar
  13. Forte AE, Galvan S, Dini D (2018) Models and tissue mimics for brain shift simulations. Biomech Model Mechanobiol 17(1):249–261CrossRefGoogle Scholar
  14. Ganpule S et al (2013) Mechanics of blast loading on the head models in the study of traumatic brain injury using experimental and computational approaches. Biomech Model Mechanobiol 12(3):511–531CrossRefGoogle Scholar
  15. Hashin Z (1983) Analysis of composite materials—a survey. J Appl Mech 50(3):481–505CrossRefzbMATHGoogle Scholar
  16. Hiscox LV et al (2016) Magnetic resonance elastography (MRE) of the human brain: technique, findings and clinical applications. Phys Med Biol 61(24):R401CrossRefGoogle Scholar
  17. Holzapfel GA (2000) Nonlinear solid mechanics: a continuum approach for engineering, 1st edn. Wiley, New YorkzbMATHGoogle Scholar
  18. Horgan CO, Murphy JG (2009) On the volumetric part of strain-energy functions used in the constitutive modeling of slightly compressible solid rubbers. Int J Solids Struct 46(16):3078–3085CrossRefzbMATHGoogle Scholar
  19. Javid S, Rezaei A, Karami G (2014) A micromechanical procedure for viscoelastic characterization of the axons and ECM of the brainstem. J Mech Behav Biomed Mater 30:290–299CrossRefGoogle Scholar
  20. Ji S et al (2014) Head impact accelerations for brain strain-related responses in contact sports: a model-based investigation. Biomech Model Mechanobiol 13(5):1121–1136CrossRefGoogle Scholar
  21. Kanit T et al (2003) Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int J Solids Struct 40(13):3647–3679CrossRefzbMATHGoogle Scholar
  22. Karami G et al (2009) A micromechanical hyperelastic modeling of brain white matter under large deformation. J Mech Behav Biomed Mater 2(3):243–254CrossRefGoogle Scholar
  23. Karami G, Shankar S, Ziejewski M, Azarmi F (2010) Micromechanical hyperelastic modeling of bi-directional oriented axons in brain white matter. In: Proc. ASME. biomedical and biotechnology engineering, pp 619–625Google Scholar
  24. Kleiven S, von Holst H (2002) Consequences of head size following trauma to the human head. J Biomech 35(2):153–160CrossRefGoogle Scholar
  25. Koser DE et al (2015) CNS cell distribution and axon orientation determine local spinal cord mechanical properties. Biophys J 108(9):2137–2147CrossRefGoogle Scholar
  26. Kyriacou SK et al (2002) Brain mechanics for neurosurgery: modeling issues. Biomech Model Mechanobiol 1(2):151–164CrossRefGoogle Scholar
  27. Labus KM, Puttlitz CM (2016) An anisotropic hyperelastic constitutive model of brain white matter in biaxial tension and structural–mechanical relationships. J Mech Behav Biomed Mater 62:195–208CrossRefGoogle Scholar
  28. Laksari K, Shafieian M, Darvish K (2012) Constitutive model for brain tissue under finite compression. J Biomech 45(4):642–646CrossRefGoogle Scholar
  29. Laksari K et al (2015) Computational simulation of the mechanical response of brain tissue under blast loading. Biomech Model Mechanobiol 14(3):459–472CrossRefGoogle Scholar
  30. Lee S et al (2014) Measurement of viscoelastic properties in multiple anatomical regions of acute rat brain tissue slices. J Mech Behav Biomed Mater 29:213–224CrossRefGoogle Scholar
  31. Libertiaux V, Pascon F, Cescotto S (2011) Experimental verification of brain tissue incompressibility using digital image correlation. J Mech Behav Biomed Mater 4(7):1177–1185CrossRefGoogle Scholar
  32. Maltese MR, Margulies SS (2016) Biofidelic white matter heterogeneity decreases computational model predictions of white matter strains during rapid head rotations. Computer methods in biomechanics and biomedical engineering 19:1–12CrossRefGoogle Scholar
  33. Menon DK et al (2010) Position statement: definition of traumatic brain injury. Arch Phys Med Rehabil 91(11):1637–1640CrossRefGoogle Scholar
  34. Miller K (1999) Constitutive model of brain tissue suitable for finite element analysis of surgical procedures. J Biomech 32(5):531–537CrossRefGoogle Scholar
  35. Mura T (2013) Micromechanics of defects in solids. Springer, BerlinGoogle Scholar
  36. Pan Y et al (2013) Finite element modeling of CNS white matter kinematics: use of a 3D RVE to determine material properties. Frontiers in bioengineering and biotechnology 1:19CrossRefGoogle Scholar
  37. Pervin F, Chen WW (2009) Dynamic mechanical response of bovine gray matter and white matter brain tissues under compression. J Biomech 42(6):731–735CrossRefGoogle Scholar
  38. Peter SJ, Mofrad MR (2012) Computational modeling of axonal microtubule bundles under tension. Biophys J 102(4):749–757CrossRefGoogle Scholar
  39. Poli R (2007) An analysis of publications on particle swarm optimization applications. Department of Computer Science, University of Essex, EssexGoogle Scholar
  40. Rashid B, Destrade M, Gilchrist MD (2014) Mechanical characterization of brain tissue in tension at dynamic strain rates. J Mech Behav Biomed Mater 33:43–54CrossRefGoogle Scholar
  41. Rémond Y et al (2016) Homogenization of reconstructed RVE. In: Rémond Y, Ahzi S, Baniassadi M, Garmestani H (eds) Applied RVE reconstruction and homogenization of heterogeneous materials. Wiley, HobokenCrossRefGoogle Scholar
  42. Schmid-Schönbein G et al (1981) Passive mechanical properties of human leukocytes. Biophys J 36(1):243–256CrossRefGoogle Scholar
  43. Shaoning S (2014) Mechanical characterization and modeling of polymer/clay nanocompositesGoogle Scholar
  44. Sheidaei A et al (2013) 3-D microstructure reconstruction of polymer nano-composite using FIB–SEM and statistical correlation function. Composites Science and Technology 80:47–54CrossRefGoogle Scholar
  45. Shulyakov AV et al (2009) Simultaneous determination of mechanical properties and physiologic parameters in living rat brain. Biomech Model Mechanobiol 8(5):415–425CrossRefGoogle Scholar
  46. Sotudeh-Chafi M et al (2008) A multi-scale finite element model for shock wave-induced axonal brain injury. In: ASME 2008 summer bioengineering conference. American Society of Mechanical EngineersGoogle Scholar
  47. Tanielian T et al (2008) Invisible wounds of war. Summary and recommendations for addressing psychological and cognitive injuries. RAND Corp, Santa MonicaCrossRefGoogle Scholar
  48. Torquato S (2013) Random heterogeneous materials: microstructure and macroscopic properties, vol 16. Springer, BerlinzbMATHGoogle Scholar
  49. Vappou J et al (2007) Magnetic resonance elastography compared with rotational rheometry for in vitro brain tissue viscoelasticity measurement. Magn Reson Mater Phys, Biol Med 20(5–6):273CrossRefGoogle Scholar
  50. Wang HC, Wineman AS (1972) A mathematical model for the determination of viscoelastic behavior of brain in vivo—I Oscillatory response. J Biomech 5(5):431–446CrossRefGoogle Scholar
  51. Whitford C et al (2018) A viscoelastic anisotropic hyperelastic constitutive model of the human cornea. Biomech Model Mechanobiol 17(1):19–29CrossRefGoogle Scholar
  52. Yousefi E et al (2017) Effect of nanofiller geometry on the energy absorption capability of coiled carbon nanotube composite material. Composites Science and Technology 153:222–231CrossRefGoogle Scholar
  53. Zhang M-G et al (2014) Spherical indentation method for determining the constitutive parameters of hyperelastic soft materials. Biomech Model Mechanobiol 13(1):1–11MathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.School of Mechanical Engineering, College of EngineeringUniversity of TehranTehranIran
  2. 2.IMFS/CNRSUniversity of StrasbourgStrasbourgFrance

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