Skip to main content
SpringerLink
Log in

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

  • Original Paper
  • Published:
Biomechanics and Modeling in Mechanobiology Aims and scope Submit manuscript

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.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • 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 130:021013

    Article  Google Scholar 

  • Bercoff J, Tanter M, Fink M (2004a) Supersonic shear imaging: a new technique for soft tissue elasticity mapping. IEEE Trans Ultrason Ferroelectr Freq Control 51:396–409

    Article  Google Scholar 

  • Bercoff J, Tanter M, Muller M, Fink M (2004b) Sonic boom in soft materials: the elastic Cerenkov effect. Appl Phys Lett 84:2202–2204

    Article  Google Scholar 

  • 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 51:1523–1536

    Article  Google Scholar 

  • Brillouin L (1946) Les Tenseurs en Mécanique et en Elasticité. Dover Publications, New York

    MATH  Google Scholar 

  • Chatelin S, Constantinesco A, Willinger R (2010) Fifty years of brain tissue mechanical testing: from in vitro to in vivo investigations. Biorheology 47:255–276

    Google Scholar 

  • Demiray H (1972) A note on the elasticity of soft biological tissues. J Biomech 5:309–311

    Article  Google Scholar 

  • Destrade M, Gilchrist MD, Murphy JG (2010a) Onset of non-linearity in the elastic bending of blocks. ASME J Appl Mech 77:061015

    Article  Google Scholar 

  • 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 127:2759–2763

    Article  Google Scholar 

  • Destrade M, Gilchrist MD, Ogden RW (2010c) Third- and fourth-order elasticity of biological soft tissues. J Acoust Soc Am 127:2103–2106

    Article  Google Scholar 

  • Destrade M, Ogden RW (2010) On the third- and fourth-order constants of incompressible isotropic elasticity. J Acoust Soc Am 128:3334–3343

    Article  Google Scholar 

  • Donnelly DR, Medige J (1997) Shear properties of human brain tissue. ASME J Biomech Eng 119:423–432

    Article  Google Scholar 

  • 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 20:1163–1177

    Article  Google Scholar 

  • Gefen A, Margulies SS (2004) Are in vivo and in situ brain tissues mechanically similar? J Biomech 37:1339–1352

    Article  Google Scholar 

  • 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 122:3211–3219

    Article  Google Scholar 

  • Green MA, Bilston LE, Sinkus R (2008) In vivo brain viscoelastic properties measured by magnetic resonance elastography. NMR Biomed 21:755–764

    Article  Google Scholar 

  • Hamilton MF, Ilinskii YA, Zabolotskaya EA (2004) Separation of compressibility and shear deformation in the elastic energy density. J Acoust Soc Am 116:41–44

    Article  Google Scholar 

  • Hrapko M, Van Dommelen JAW, Peters GWM, Wismans JSHM (2006) The mechanical behaviour of brain tissue: large strain response and constitutive modelling. Biorheology 43:623–636

    Google Scholar 

  • 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 20:97–111

    Article  Google Scholar 

  • 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 227:609–614

    Article  Google Scholar 

  • Kaster T, Sack I, Samani A (2011) Measurement of the hyperelastic properties of ex vivo brain tissue slices. J Biomech 44:1158–1163

    Article  Google Scholar 

  • 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 52:7281

    Article  Google Scholar 

  • 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 46:123–144

    Google Scholar 

  • Kruse SA, Rose GH, Glaser KJ, Manduca A, Felmlee JP, Jack JCR, Ehman RL (2008) Magnetic resonance elastography of the brain. NeuroImage 39:231–237

    Article  Google Scholar 

  • 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 59:833–839

    Article  Google Scholar 

  • 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 30:550–558

    Article  Google Scholar 

  • Miga MI, Paulsen KD (2000) In vivo quantification of a homogeneous brain deformation model for updating preoperative images during surgery. IEEE Trans Biomed Eng 47:266–273

    Article  Google Scholar 

  • Miller K, Chinzei K (1997) Constitutive modelling of brain tissue: experiment and theory. J Biomech 30:1115–1121

    Article  Google Scholar 

  • Miller K (1999) Constitutive model of brain tissue suitable for finite analysis of surgical procedures. J Biomech 32:531–537

    Article  Google Scholar 

  • Miller K, Chinzei K, Orssengo G, Bednarz P (2000) Mechanical properties of brain tissue in-vivo: experiment and computer simulation. J Biomech 33:1369–1376

    Article  Google Scholar 

  • Miller K, Chinzei K (2002) Mechanical properties of brain tissue in tension. J Biomech 35:483–490

    Article  Google Scholar 

  • Nicolle S, Lounis M, Willinger R, Palierne JF (2005) Shear linear behavior of brain tissue over a large frequency range. Biorheology 42:209–223

    Google Scholar 

  • 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 41:314–325

    Article  Google Scholar 

  • 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 1–26

  • Pervin F, Chen WW (2009) Dynamic mechanical response of bovine gray matter and white matter brain tissues under compression. J Biomech 42:731–735

    Article  Google Scholar 

  • Prange MT, Margulies SS (2002) Regional, directional, and age-dependent properties of the brain undergoing large deformation. J Biomech Eng 124:244–252

    Article  Google Scholar 

  • 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 7:4090–4101

    Article  Google Scholar 

  • Rashid B, Destrade M, Gilchrist MD (2013a) Influence of preservation temperature on the measured mechanical properties of brain tissue. J Biomech 46:1276–1281

    Article  Google Scholar 

  • Rashid B, Destrade M, Gilchrist MD (2013b) Mechanical characterization of brain tissue in simple shear at dynamic strain rates. J Mech Behav Biomed Mater 28:71–85

    Article  Google Scholar 

  • 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 93:101912

    Article  Google Scholar 

  • 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 45:1199–1206

    Article  Google Scholar 

  • 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 46:652–657

    Article  Google Scholar 

  • Saraf H, Ramesh KT, Lennon AM, Merkle AC, Roberts JC (2007) Mechanical properties of soft human tissues under dynamic loading. J Biomech 40:1960–1967

    Article  Google Scholar 

  • 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 24:385–392

    Google Scholar 

  • 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 45:369–394

    Google Scholar 

  • Zhang MG, Cao YP, Li GY, Feng XQ (2014a) Spherical indentation method for determining the constitutive parameters of hyperelastic soft materials. Biomech Model Mechanobiol 13:1–11

    Article  Google Scholar 

  • Zhang MG, Cao YP, Li GY, Feng XQ (2014b) Pipette aspiration of hyperelastic compliant materials: theoretical analysis, simulations and experiments. J Mech Phys Solids 68:179–196

    Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Institute of Biomechanics and Medical Engineering, AML, Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, People’s Republic of China

    Yi Jiang, Guoyang Li & Yanping Cao

  2. Department of Ultrasound, Beijing Friendship Hospital, Capital Medical University, Beijing, 100050, People’s Republic of China

    Lin-Xue Qian & Si Liang

  3. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, University Road, Galway, Ireland

    Michel Destrade

Authors
  1. Yi Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guoyang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lin-Xue Qian
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Si Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Michel Destrade
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Yanping Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Michel Destrade or Yanping Cao.

Additional information

Yi Jiang and Guoyang Li have contributed equally to this study.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (gif 3222 KB)

Supplementary material 2 (gif 1211 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiang, Y., Li, G., Qian, LX. et al. Measuring the linear and nonlinear elastic properties of brain tissue with shear waves and inverse analysis. Biomech Model Mechanobiol 14, 1119–1128 (2015). https://doi.org/10.1007/s10237-015-0658-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10237-015-0658-0

Keywords

Search

Navigation