Biomechanics and Modeling in Mechanobiology

, Volume 14, Issue 5, pp 1119–1128 | Cite as

Measuring the linear and nonlinear elastic properties of brain tissue with shear waves and inverse analysis

Original Paper

Abstract

We use supersonic shear wave imaging (SSI) technique to measure not only the linear but also the nonlinear elastic properties of brain matter. Here, we tested six porcine brains ex vivo and measured the velocities of the plane shear waves induced by acoustic radiation force at different states of pre-deformation when the ultrasonic probe is pushed into the soft tissue. We relied on an inverse method based on the theory governing the propagation of small-amplitude acoustic waves in deformed solids to interpret the experimental data. We found that, depending on the subjects, the resulting initial shear modulus \(\mu _0 \) varies from 1.8 to 3.2 kPa, the stiffening parameter \(b\) of the hyperelastic Demiray–Fung model from 0.13 to 0.73, and the third- \((A)\) and fourth-order \((D)\) constants of weakly nonlinear elasticity from \(-\)1.3 to \(-\)20.6 kPa and from 3.1 to 8.7 kPa, respectively. Paired \(t\) test performed on the experimental results of the left and right lobes of the brain shows no significant difference. These values are in line with those reported in the literature on brain tissue, indicating that the SSI method, combined to the inverse analysis, is an efficient and powerful tool for the mechanical characterization of brain tissue, which is of great importance for computer simulation of traumatic brain injury and virtual neurosurgery.

Keywords

Supersonic shear wave imaging technique Inverse method Brain tissue Elastic and hyperelastic properties 

Notes

Acknowledgments

Supports from the National Natural Science Foundation of China (Grant No. 11172155), Tsinghua University (2012Z02103) and 973 Program of MOST (2010CB631005) are gratefully acknowledged. We also thank the referees for helping us improve greatly previous versions of the article.

Conflict of interest

The authors have no financial and personal relationships that could inappropriately influence or bias this work.

Supplementary material

10237_2015_658_MOESM1_ESM.gif (3.1 mb)
Supplementary material 1 (gif 3222 KB)
10237_2015_658_MOESM2_ESM.gif (1.2 mb)
Supplementary material 2 (gif 1211 KB)

References

  1. Atay SM, Kroenke CD, Sabet A, Bayly PV (2008) Measurement of the dynamic shear modulus of mouse brain tissue in vivo by magnetic resonance elastography. J Biomech Eng 130:021013CrossRefGoogle Scholar
  2. Bercoff J, Tanter M, Fink M (2004a) Supersonic shear imaging: a new technique for soft tissue elasticity mapping. IEEE Trans Ultrason Ferroelectr Freq Control 51:396–409CrossRefGoogle Scholar
  3. Bercoff J, Tanter M, Muller M, Fink M (2004b) Sonic boom in soft materials: the elastic Cerenkov effect. Appl Phys Lett 84:2202–2204CrossRefGoogle Scholar
  4. Bercoff J, Tanter M, Muller M, Fink M (2004c) The role of viscosity in the impulse diffraction field of elastic waves induced by the acoustic radiation force. IEEE Trans Ultrason Ferroelectr Freq Control 51:1523–1536CrossRefGoogle Scholar
  5. Brillouin L (1946) Les Tenseurs en Mécanique et en Elasticité. Dover Publications, New YorkMATHGoogle Scholar
  6. Chatelin S, Constantinesco A, Willinger R (2010) Fifty years of brain tissue mechanical testing: from in vitro to in vivo investigations. Biorheology 47:255–276Google Scholar
  7. Demiray H (1972) A note on the elasticity of soft biological tissues. J Biomech 5:309–311CrossRefGoogle Scholar
  8. Destrade M, Gilchrist MD, Murphy JG (2010a) Onset of non-linearity in the elastic bending of blocks. ASME J Appl Mech 77:061015CrossRefGoogle Scholar
  9. Destrade M, Gilchrist MD, Saccomandi G (2010b) Third- and fourth-order constants of incompressible soft solids and the acousto-elastic effect. J Acoust Soc Am 127:2759–2763CrossRefGoogle Scholar
  10. Destrade M, Gilchrist MD, Ogden RW (2010c) Third- and fourth-order elasticity of biological soft tissues. J Acoust Soc Am 127:2103–2106CrossRefGoogle Scholar
  11. Destrade M, Ogden RW (2010) On the third- and fourth-order constants of incompressible isotropic elasticity. J Acoust Soc Am 128:3334–3343CrossRefGoogle Scholar
  12. Donnelly DR, Medige J (1997) Shear properties of human brain tissue. ASME J Biomech Eng 119:423–432CrossRefGoogle Scholar
  13. Gefen A, Gefen N, Zhu Q, Raghupathi R, Margulies SS (2003) Age-dependent changes in material properties of the brain and braincase of the rat. J Neurotrauma 20:1163–1177CrossRefGoogle Scholar
  14. Gefen A, Margulies SS (2004) Are in vivo and in situ brain tissues mechanically similar? J Biomech 37:1339–1352CrossRefGoogle Scholar
  15. Gennisson JL, Rénier M, Catheline S, Barrière C, Bercoff J, Tanter M, Fink M (2007) Acoustoelasticity in soft solids: assessment of the nonlinear shear modulus with the acoustic radiation force. J Acoust Soc Am 122:3211–3219CrossRefGoogle Scholar
  16. Green MA, Bilston LE, Sinkus R (2008) In vivo brain viscoelastic properties measured by magnetic resonance elastography. NMR Biomed 21:755–764CrossRefGoogle Scholar
  17. Hamilton MF, Ilinskii YA, Zabolotskaya EA (2004) Separation of compressibility and shear deformation in the elastic energy density. J Acoust Soc Am 116:41–44CrossRefGoogle Scholar
  18. Hrapko M, Van Dommelen JAW, Peters GWM, Wismans JSHM (2006) The mechanical behaviour of brain tissue: large strain response and constitutive modelling. Biorheology 43:623–636Google Scholar
  19. Jiang Y, Li GY, Qian LX, Hu XD, Liu D, Liang S, Cao YP (2015) Characterization of the nonlinear elastic properties of soft tissues using the supersonic shear imaging (SSI) technique: inverse method, ex vivo and in vivo experiments. Med Image Anal 20:97–111CrossRefGoogle Scholar
  20. Karimi A, Navidbakhsh M, Haghi AM, Faghihi S (2013) Measurement of the uniaxial mechanical properties of rat brains infected by Plasmodium berghei ANKA. J Eng Med 227:609–614CrossRefGoogle Scholar
  21. Kaster T, Sack I, Samani A (2011) Measurement of the hyperelastic properties of ex vivo brain tissue slices. J Biomech 44:1158–1163CrossRefGoogle Scholar
  22. Klatt D, Hamhaber U, Asbach P, Braun J, Sack I (2007) Noninvasive assessment of the rheological behavior of human organs using multifrequency MR elastography: a study of brain and liver viscoelasticity. Phys Med Biol 52:7281CrossRefGoogle Scholar
  23. Kleiven S, Hardy WN (2002) Correlation of an FE model of the human head with local brain motion-consequences for injury prediction. Stapp Car Crash J 46:123–144Google Scholar
  24. Kruse SA, Rose GH, Glaser KJ, Manduca A, Felmlee JP, Jack JCR, Ehman RL (2008) Magnetic resonance elastography of the brain. NeuroImage 39:231–237CrossRefGoogle Scholar
  25. Latorre-Ossa H, Gennisson JL, De Brosses E, Tanter M (2012) Quantitative imaging of nonlinear shear modulus by combining static elastography and shear wave elastography. IEEE Trans Ultrason Ferroelectr Freq Control 59:833–839CrossRefGoogle Scholar
  26. Macé E, Cohen I, Montaldo G, Miles R, Fink M, Tanter M (2011) In vivo mapping of brain elasticity in small animals using shear wave imaging. IEEE Trans Med Imaging 30:550–558CrossRefGoogle Scholar
  27. Miga MI, Paulsen KD (2000) In vivo quantification of a homogeneous brain deformation model for updating preoperative images during surgery. IEEE Trans Biomed Eng 47:266–273CrossRefGoogle Scholar
  28. Miller K, Chinzei K (1997) Constitutive modelling of brain tissue: experiment and theory. J Biomech 30:1115–1121CrossRefGoogle Scholar
  29. Miller K (1999) Constitutive model of brain tissue suitable for finite analysis of surgical procedures. J Biomech 32:531–537CrossRefGoogle Scholar
  30. Miller K, Chinzei K, Orssengo G, Bednarz P (2000) Mechanical properties of brain tissue in-vivo: experiment and computer simulation. J Biomech 33:1369–1376CrossRefGoogle Scholar
  31. Miller K, Chinzei K (2002) Mechanical properties of brain tissue in tension. J Biomech 35:483–490CrossRefGoogle Scholar
  32. Nicolle S, Lounis M, Willinger R, Palierne JF (2005) Shear linear behavior of brain tissue over a large frequency range. Biorheology 42:209–223Google Scholar
  33. O’Donnell M, Skovoroda AR, Shapo BM, Emelianov SY (1994) Internal displacement and strain imaging using ultrasonic speckle tracking. IEEE Trans Ultrason Ferroelectr Freq Control 41:314–325CrossRefGoogle Scholar
  34. Ogden RW (2007) Incremental statics and dynamics of pre-stressed elastic materials. In: Destrade M, Saccomandi G (eds) Waves in nonlinear pre-stressed materials. Springer, Vienna, pp 1–26Google Scholar
  35. Pervin F, Chen WW (2009) Dynamic mechanical response of bovine gray matter and white matter brain tissues under compression. J Biomech 42:731–735CrossRefGoogle Scholar
  36. Prange MT, Margulies SS (2002) Regional, directional, and age-dependent properties of the brain undergoing large deformation. J Biomech Eng 124:244–252CrossRefGoogle Scholar
  37. Prevost TP, Jin G, De Moya MA, Alam HB, Suresh S, Socrate S (2011) Dynamic mechanical response of brain tissue in indentation in vivo, in situ and in vitro. Acta Biomater 7:4090–4101CrossRefGoogle Scholar
  38. Rashid B, Destrade M, Gilchrist MD (2013a) Influence of preservation temperature on the measured mechanical properties of brain tissue. J Biomech 46:1276–1281CrossRefGoogle Scholar
  39. Rashid B, Destrade M, Gilchrist MD (2013b) Mechanical characterization of brain tissue in simple shear at dynamic strain rates. J Mech Behav Biomed Mater 28:71–85CrossRefGoogle Scholar
  40. Rénier M, Gennisson JL, Barrière C, Royer D, Fink M (2008) Fourth-order shear elastic constant assessment in quasi-incompressible soft solids. Appl Phys Lett 93:101912CrossRefGoogle Scholar
  41. Roberts DW, Miga MI, Hartov A, Eisner S, Lemery JM, Kennedy FE, Paulsen KD (1999) Intraoperatively updated neuroimaging using brain modeling and sparse data. Neurosurgery 45:1199–1206CrossRefGoogle Scholar
  42. Sack I, Beierbach B, Wuerfel J, Klatt D, Hamhaber U, Papazoglou S, Braun J (2009) The impact of aging and gender on brain viscoelasticity. Neuroimage 46:652–657CrossRefGoogle Scholar
  43. Saraf H, Ramesh KT, Lennon AM, Merkle AC, Roberts JC (2007) Mechanical properties of soft human tissues under dynamic loading. J Biomech 40:1960–1967CrossRefGoogle Scholar
  44. Streitberger KJ, Wiener E, Hoffmann J, Freimann FB, Klatt D, Braun J, Sack I (2011) In vivo viscoelastic properties of the brain in normal pressure hydrocephalus. NMR Biomed 24:385–392Google Scholar
  45. Zhang L, Yang KH, Dwarampudi R, Omori K, Li T, Chang K, Hardy WN, Khalil TB, King AI (2001) Recent advances in brain injury research: a new human head model development and validation. Stapp Car Crash J 45:369–394Google Scholar
  46. Zhang MG, Cao YP, Li GY, Feng XQ (2014a) Spherical indentation method for determining the constitutive parameters of hyperelastic soft materials. Biomech Model Mechanobiol 13:1–11CrossRefGoogle Scholar
  47. Zhang MG, Cao YP, Li GY, Feng XQ (2014b) Pipette aspiration of hyperelastic compliant materials: theoretical analysis, simulations and experiments. J Mech Phys Solids 68:179–196CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institute of Biomechanics and Medical Engineering, AML, Department of Engineering MechanicsTsinghua UniversityBeijingPeople’s Republic of China
  2. 2.Department of Ultrasound, Beijing Friendship HospitalCapital Medical UniversityBeijingPeople’s Republic of China
  3. 3.School of Mathematics, Statistics and Applied MathematicsNational University of Ireland GalwayGalwayIreland

Personalised recommendations