Instantaneous and Equilibrium Responses of the Brain Tissue by Stress Relaxation and Quasi-Linear Viscoelasticity Theory

Document Type : Article


1 Department of Mechanical Engineering, North Dakota State University, Fargo, ND 58108-6050, USA.

2 Department of Physiology and Biomedical Engineering, Mayo Clinic, 200 First Street, S.W., Rochester, MN 55905, USA.


Human brain and brainstem tissues have viscoelastic characteristics and their behaviours are functions of strains, as well as strain rates. Determination of the equilibrium and instantaneous stresses happening at low and high strain rates provide insights into a better understanding of the behaviour of such tissues. In this manuscript we present the results of a series of stress relaxation tests, at six different values of strains conducted on porcine brainstem tissue samples to indirectly measure the equilibrium and instantaneous stresses. The equilibrium stresses at low strain rates are measured from long-term responses of the stress relaxation test. The instantaneous stresses at high strain rates are determined using Quasi-Linear Viscoelasticity (QLV) theory at six strains. The results show that the instantaneous stresses are much larger (almost 11 times) than the equilibrium stresses and across all the strains. It can be concluded that the instantaneous response can be reasonably estimated from the long-term response which can be easily measured experimentally. The experimental results also show that the reduced relaxation moduli, estimated from the QLV theory, vary for the six strains tested.


1. Ratajczak, M., Ptak, M., Chybowski, L., Gawdzi_nska, K., and B_edzi_nski, R. Material and structural modeling aspects of brain tissue deformation under dynamic loads", Materials, 12(2), p. 271 (2019). 2. Eslaminejad, A., Hosseini Farid, M., Ziejewski, M., and Karami, G. Brain tissue constitutive material models and the _nite element analysis of blast-induced traumatic brain injury", Scientia Iranica, 25, pp. 3141- 3150 (2018). 3. Khalid, G.A., Jones, M., Prabhu, R., Mason-Jones, A., Whittington, W., Bakhtiarydavijani, P., and Theobald, P. Development of a paediatric head model for the computational analysis of head impact interactions", Int. J. Math. Comput. Phys., Electr. Comput. Eng., 11, pp. 113-116 (2017). 4. Hosseini-Farid, M., Ramzanpour, M., Eslaminejad, A., Ziejewski, M., and Karami, G. Computational simulation of brain injury by golf ball impacts in adult and children", Biomedical Sciences Instrumentation, 54(1), pp. 369-376 (2018). 5. Ghajari, M., Hellyer, P.J., and Sharp, D.J. Computational modelling of traumatic brain injury predicts the location of chronic traumatic encephalopathy pathology", Brain, 140(2), pp. 333-343 (2017). 6. Ramzanpour, M., Eslaminejad, A., Hosseini-Farid, M., Ziejewski, M., and Karami, G. Comparative study of coup and contrecoup brain injury in impact induced TBI", Biomedical Sciences Instrumentation, 54(1), pp. 76-82 (2018). 7. Galford, J.E. and McElhaney, J.H. A viscoelastic study of scalp, brain, and dura", Journal of Biomechanics, 3(2), pp. 211-221 (1970). 8. Farid, M.H., Eslaminejad, A., Ziejewski, M., and Karami, G. A study on the e_ects of strain rates on characteristics of brain tissue", ASME 2017 International Mechanical Engineering Congress and Exposition, pp. V003T04A003-V003T04A003 (2017). 9. Farahmand, F. and Ahmadian, M. A novel procedure for micromechanical characterization of white matter constituents at various strain rates", Scientia Iranica, (In Press). 10. Saboori, P. and Sadegh, A. Material modeling of the head's subarachnoid space", Scientia Iranica, 18(6), pp. 1492-1499 (2011). 11. Hosseini-Farid, M., Ramzanpour, M., Ziejewski, M., and Karami, G. A compressible hyper-viscoelastic material constitutive model for human brain tissue and the identi_cation of its parameters", International Journal of Non-Linear Mechanics, 116, pp. 147-154 (2019). 12. Javid, S., Rezaei, A., and Karami, G. A micromechanical procedure for viscoelastic characterization of the axons and ECM of the brainstem", Journal of the Mechanical Behavior of Biomedical Materials, 30, pp. 290-299 (2014). 13. Goriely, A., Geers, M.G.D., Holzapfel, G.A.W., Jayamohan, J., J_erusalem, A., Sivaloganathan, S., Squier, W., van Dommelen, J.A.W., Waters, S., and Kuhl, E. Mechanics of the brain: perspectives, challenges, and opportunities", Biomechanics and Modeling in Mechanobiology, 14(5), pp. 931-965 (2015). 14. Tamura, A., Hayashi, S., Watanabe, I., Nagayama, K., and Matsumoto, T. Mechanical characterization of brain tissue in high-rate compression", Journal of Biomechanical Science and Engineering, 2(3), pp. 115- 126 (2007). 15. Miller, K. and Chinzei, K. Constitutive modeling of brain tissue: experiment and theory", Journal of Biomechanics, 30, pp. 1115-1121 (1997). 16. Rashid, B., Destrade, M., and Gilchrist, M.D. Mechanical characterization of brain tissue in compression at dynamic strain rates", Journal of the Mechanical Behavior of Biomedical Materials, 10, pp. 23-38 (2012). 17. Darvish, K. and Crandall, J. Nonlinear viscoelastic e_ects in oscillatory shear deformation of brain tissue", Medical Engineering & Physics, 23(9), pp. 633-645 (2001). 18. Chatelin, S., Constantinesco, A., and Willinger, R. Fifty years of brain tissue mechanical testing: from in vitro to in vivo investigations", Biorheology, 47(5- 6), pp. 255-276 (2010). 19. Zhao, H., Yin, Z., Li, K., Liao, Z., Xiang, H., and Zhu, F. Mechanical characterization of immature porcine brainstem in tension at dynamic strain rates", Medical Science Monitor Basic Research, 22, p. 6 (2016). 20. Moran, R., Smith, J.H., and Garc__a, J.J. Fitted hyperelastic parameters for human brain tissue from M. Hosseini-Farid et al./Scientia Iranica, Transactions A: Civil Engineering 26 (2019) 2047{2056 2055 reported tension, compression, and shear tests", Journal of Biomechanics, 47(15), pp. 3762-3766 (2014). 21. Destrade, M., Gilchrist, M., Murphy, J.G., Rashid, B., and Saccomandi, G. Extreme softness of brain matter in simple shear", International Journal of Non-Linear Mechanics, 75, pp. 54-58 (2015). 22. El Sayed, T., Mota, A., Feraternali, F., and Ortiz, M. A variational constitutive model for soft biological tissues", Journal of Biomechanics, 41, pp. 1458-1466 (2008). 23. Prevost, T.P., Balakrishnan, A., Suresh, S., and Socrate, S. Biomechanics of brain tissue", Acta Biomaterialia, 7(1), pp. 83-95 (2011). 24. Kohandel, M., Sivaloganathan, S., Tenti, G., and Drake, J.M. The constitutive properties of the brain parenchyma Part 1. Strain energy approach", Medical Engineering & Physics, 28, pp. 449-454 (2006). 25. Voyiadjis, G.Z. and Samadi-Dooki, A. Hyperelastic modeling of the human brain tissue: E_ects of no-slip boundary condition and compressibility on the uniaxial deformation", Journal of the Mechanical Behavior of Biomedical Materials, 83, pp. 63-78 (2018). 26. Murphy, M., Mun, S., Horstemeyer, M., Baskes, M., Bakhtiary, A., LaPlaca, M.C., Gwaltney, S.R., Williams, L.N., and Prabhu, R. Molecular dynamics simulations showing 1-palmitoyl-2-oleoylphosphatidylcholine (POPC) membrane mechanoporation damage under di_erent strain paths", Journal of Biomolecular Structure and Dynamics, 37(5), pp. 1-14 (2018). 27. Hosseini-Farid, M., Ramzanpour, M., Ziejewski, M., and Karami, G. Estimating the brain strain rates during traumatic brain injury", Biomedical Sciences Instrumentation, 54(1), pp. 361-368 (2018). 28. Farid, M.H., Eslaminejad, A., Ramzanpour, M., Ziejewski, M., and Karami, G. The strain rates of the brain and skull under dynamic loading", ASME 2018 International Mechanical Engineering Congress and Exposition, pp. V003T04A067-V003T04A067 (2018). 29. Cheng, S., Clarke, E.C., and Bilston, L.E. Rheological properties of the tissues of the central nervous system: a review", Medical Engineering & Physics, 30(10), pp. 1318-1337 (2008). 30. Amin, A., Alam, M., and Okui, Y. An improved hyperelasticity relation in modeling viscoelasticity response of natural and high damping rubbers in compression: experiments, parameter identi_cation and numerical veri_cation", Mechanics of Materials, 34(2), pp. 75-95 (2002). 31. Huber, N. and Tsakmakis, C. Finite deformation viscoelasticity laws", Mechanics of Materials, 32(1), pp. 1-18 (2000). 32. Sadeghnejad, S., Elyasi, N., Farahmand, F., Vossoughi, G., and Hosseini, S.M.S. Hyperelastic modeling of sino-nasal tissue for haptic neurosurgery simulation", Scientia Iranica, http://scientiairanica. 21263.html (2019). 33. Babaei, B., Abramowitch, S.D., Elson, E.L., Thomopoulos, S., and Genin, G.M. A discrete spectral analysis for determining quasi-linear viscoelastic properties of biological materials", Journal of The Royal Society Interface, 12(113), p. 20150707 (2015). 34. Garo, A., Hrapko, M., Van Dommelen, J.A.W., and Peters, G.W. Towards a reliable characterisation of the mechanical behaviour of brain tissue: the e_ects of post-mortem time and sample preparation", Biorheology, 44(1), pp. 51-58 (2007). 35. Abbasi, A.A., Ahmadian, M.T., Alizadeh, A., and Tarighi, S. Application of hyperelastic models in mechanical properties prediction of mouse oocyte and embryo cells at large deformations", Scientia Iranica, 25(2), pp. 700-710 (2018). 36. Budday, S., Sommer, G., Holzapfel, G., Steinmann, P., and Kuhl, E. Viscoelastic parameter identi_cation of human brain tissue", Journal of the Mechanical Behavior of Biomedical Materials, 74, pp. 463-476 (2017). 37. Toms, K., Dakin, G.J., Lemons, J.E., and Eberhardt, A.W. Quasi-linear viscoelastic behavior of the human periodontal ligament", Journal of Biomechanics, 35(10), pp. 1411-1415 (2002). 38. Abramowitch, S.D. and Woo, S.L. An improved method to analyze the stress relaxation of ligaments following a _nite ramp time based on the quasilinear viscoelastic theory", Journal of Biomechanical Engineering, 126(1), pp. 92-97 (2004). 39. Laksari, K., Sha_eian, M., and Darvish, K. Constitutive model for brain tissue under _nite compression", Journal of Biomechanics, 45, pp. 642-646 (2012). 40. De Rooij, R. and Kuhl, E. Constitutive modeling of brain tissue: current perspectives", Applied Mechanics Reviews, 68(1), p. 010801 (2016). 41. Nigul, I. and Nigul, U. On algorithms of evaluation of Fung's relaxation function parameters", Journal of Biomechanics, 20(4), pp. 343-352 (1987). 42. Rousseau, E., Sauren, A., Van Hout, M., and Van Steenhoven, A. Elastic and viscoelastic material behaviour of fresh and glutaraldehyde-treated porcine aortic valve tissue", Journal of Biomechanics, 16(5), pp. 339-348 (1983). 43. Sauren, A. and Rousseau, E. A concise sensitivity analysis of the quasi-linear viscoelastic model proposed by Fung", J. Biomech. Eng., 105(1), pp. 92-95 (1983). 44. Hrapko, M., van Dommelen, J.A.W., Peters, G.W.M., and Wismans, J.S.H.M. The mechanical behaviour of brain tissue: large strain response and constitutive modeling", Biorheology, 43, pp. 623-636 (2006). 2056 M. Hosseini-Farid et al./Scientia Iranica, Transactions A: Civil Engineering 26 (2019) 2047{2056 45. Mendis, K.K., Stalnaker, R.K., and Advani, S.H. A constitutive relationship for large deformation _nite element modeling of brain tissue", Journal of Biomechanical Engineering, 117, pp. 279-285 (1995). 46. Pervin, F. and Chen, W.W. Dynamic mechanical response of bovine gray matter and white matter brain tissues under compression",Journal of Biomechanics, 42(6), pp. 731-735 (2009).