0
Research Papers

Numerical Simulation of Focused Shock Shear Waves in Soft Solids and a Two-Dimensional Nonlinear Homogeneous Model of the Brain

[+] Author and Article Information
B. Giammarinaro

UMR 7190,
Institut Jean Le Rond d'Alembert,
Sorbonne Universites,
UPMC Univ Paris 06,
Paris F-75005, France
e-mail: giam@dalembert.upmc.fr

F. Coulouvrat

CNRS, UMR 7190,
Institut Jean Le Rond d'Alembert,
Paris F-75005, France
e-mail: francois.coulouvrat@upmc.fr

G. Pinton

Joint Department of Biomedical Engineering,
University of North Carolina at Chapel Hill
and North Carolina State University,
Chapel Hill, NC 27599
e-mail: gia@email.unc.edu

Manuscript received July 31, 2015; final manuscript received January 18, 2016; published online February 19, 2016. Assoc. Editor: Barclay Morrison.

J Biomech Eng 138(4), 041003 (Feb 19, 2016) (12 pages) Paper No: BIO-15-1380; doi: 10.1115/1.4032643 History: Received July 31, 2015; Revised January 18, 2016

Shear waves that propagate in soft solids, such as the brain, are strongly nonlinear and can develop into shock waves in less than one wavelength. We hypothesize that these shear shock waves could be responsible for certain types of traumatic brain injuries (TBI) and that the spherical geometry of the skull bone could focus shear waves deep in the brain, generating diffuse axonal injuries. Theoretical models and numerical methods that describe nonlinear polarized shear waves in soft solids such as the brain are presented. They include the cubic nonlinearities that are characteristic of soft solids and the specific types of nonclassical attenuation and dispersion observed in soft tissues and the brain. The numerical methods are validated with analytical solutions, where possible, and with self-similar scaling laws where no known solutions exist. Initial conditions based on a human head X-ray microtomography (CT) were used to simulate focused shear shock waves in the brain. Three regimes are investigated with shock wave formation distances of 2.54m,0.018m, and 0.0064m. We demonstrate that under realistic loading scenarios, with nonlinear properties consistent with measurements in the brain, and when the shock wave propagation distance and focal distance coincide, nonlinear propagation can easily overcome attenuation to generate shear shocks deep inside the brain. Due to these effects, the accelerations in the focal are larger by a factor of 15 compared to acceleration at the skull surface. These results suggest that shock wave focusing could be responsible for diffuse axonal injuries.

FIGURES IN THIS ARTICLE
<>
Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.

References

Catheline, S. , Gennisson, J.-L. , Tanter, M. , and Fink, M. , 2003, “ Observation of Shock Transverse Waves in Elastic Media,” Phys. Rev. Lett., 91(16), p. 164301. [CrossRef] [PubMed]
Whitham, G. , 1979, Lectures on Wave Propagation, Springer-Verlag, Berlin.
Catheline, S. , Gennisson, J.-L. , Delon, G. , Fink, M. , Sinkus, R. , Abouelkaram, S. , and Culioli, J. , 2004, “ Measurement of Viscoelastic Properties of Homogeneous Soft Solid Using Transient Elastography: An Inverse Problem Approach,” J. Acoust. Soc. Am., 116(6), pp. 3734–3741. [CrossRef] [PubMed]
Duck, F. A. , 2013, Physical Properties of Tissues: A Comprehensive Reference Book, Academic Press, London.
Sandrin, L. , Tanter, M. , Catheline, S. , and Fink, M. , 2002, “ Shear Modulus Imaging With 2-D Transient Elastography,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 49(4), pp. 426–435. [CrossRef]
Hamilton, M. , Ilinskii, Y. , and Zabolotskaya, E. , 2004, “ Separation of Compressibility and Shear Deformation in the Elastic Energy Density (L),” J. Acoust. Soc. Am., 116(1), pp. 41–44. [CrossRef]
Destrade, M. , and Ogden, R. W. , 2010, “ On the Third-and Fourth-Order Constants of Incompressible Isotropic Elasticity,” J. Acoust. Soc. Am., 128(6), pp. 3334–3343. [CrossRef] [PubMed]
Zabolotskaya, E. , Hamilton, M. , Ilinskii, Y. , and Meegan, G. D. , 2004, “ Modeling of Nonlinear Shear Waves in Soft Solids,” J. Acoust. Soc. Am., 116(5), pp. 2807–2813. [CrossRef]
Wochner, M. , Hamilton, M. , Ilinskii, Y. , and Zabolotskaya, E. , 2008, “ Cubic Nonlinearity in Shear Wave Beams With Different Polarizations,” J. Acoust. Soc. Am., 123(5), pp. 2488–2495. [CrossRef] [PubMed]
Kuznetsov, V. , 1971, “ Equations of Nonlinear Acoustics,” Sov. Phys. Acoust., 16(4), pp. 467–470.
Jacob, X. , Catheline, S. , Genisson, J.-L. , Barrière, C. , Royer, D. , and Fink, M. , 2007, “ Nonlinear Shear Wave Interaction in Soft Solids,” J. Acoust. Soc. Am., 122(4), pp. 1917–1926. [CrossRef] [PubMed]
Gennisson, J.-L. , Rénier, M. , Catheline, S. , Barrière, C. , Bercoff, J. , Tanter, M. , and Fink, M. , 2007, “ Acoustoelasticity in Soft Solids: Assessment of the Nonlinear Shear Modulus With the Acoustic Radiation Force,” J. Acoust. Soc. Am., 122(6), pp. 3211–3219. [CrossRef] [PubMed]
Rénier, M. , Gennisson, J.-L. , Barrière, C. , Royer, D. , and Fink, M. , 2008, “ Fourth-Order Shear Elastic Constant Assessment in Quasi-Incompressible Soft Solids,” Appl. Phys. Lett., 93(10), p. 101912. [CrossRef]
Pinton, G. , Coulouvrat, F. , Gennisson, J.-L. , and Tanter, M. , 2010, “ Nonlinear Reflection of Shock Shear Waves in Soft Elastic Media,” J. Acoust. Soc. Am., 127(2), pp. 683–691. [CrossRef] [PubMed]
Hamhaber, U. , Klatt, D. , Papazoglou, S. , Hollmann, M. , Stadler, J. , Sack, I. , Bernarding, J. , and Braun, J. , 2010, “ In Vivo Magnetic Resonance Elastography of Human Brain at 7 T and 1.5 T,” J. Magn. Reson. Imaging, 32(3), pp. 577–583. [CrossRef] [PubMed]
Johnson, C. L. , McGarry, M. D. , Houten, E. E. , Weaver, J. B. , Paulsen, K. D. , Sutton, B. P. , and Georgiadis, J. G. , 2013, “ Magnetic Resonance Elastography of the Brain Using Multishot Spiral Readouts With Self-Navigated Motion Correction,” Magn. Reson. Med., 70(2), pp. 404–412. [CrossRef] [PubMed]
Fu, M. , Barlaz, M. S. , Shosted, R. K. , Liang, Z.-P. , and Sutton, B. P. , 2015, “ High-Resolution Dynamic Speech Imaging With Deformation Estimation,” 37th Annual International Conference of the IEEE on Engineering in Medicine and Biology Society (EMBC), Milan, Italy, Aug. 25–29, pp. 1568–1571.
Walker, W. F. , and Trahey, G. E. , 1995, “ A Fundamental Limit on Delay Estimation Using Partially Correlated Speckle Signals,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 42(2), pp. 301–308. [CrossRef]
Pinton, G. F. , Dahl, J. J. , and Trahey, G. E. , 2006, “ Rapid Tracking of Small Displacements With Ultrasound,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 53(6), pp. 1103–1117. [CrossRef]
Pinton, G. , Gennisson, J.-L. , Tanter, M. , and Coulouvrat, F. , 2014, “ Adaptive Motion Estimation of Shear Shock Waves in Soft Solids and Tissue With Ultrasound,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 61(9), pp. 1489–1503. [CrossRef]
Hart, T. , and Hamilton, M. , 1988, “ Nonlinear Effects in Focused Sound Beams,” J. Acoust. Soc. Am., 84(4), pp. 1488–1496. [CrossRef]
Averkiou, M. , and Hamilton, M. , 1997, “ Nonlinear Distortion of Short Pulses Radiated by Plane and Focused Circular Pistons,” J. Acoust. Soc. Am., 102(5), pp. 2539–2548. [CrossRef] [PubMed]
Marchiano, R. , Coulouvrat, F. , and Grenon, R. , 2003, “ Numerical Simulation of Shock Wave Focusing at Fold Caustics, With Application to Sonic Boom,” J. Acoust. Soc. Am., 114(4), pp. 1758–1771. [CrossRef] [PubMed]
Marchiano, R. , Thomas, J.-L. , and Coulouvrat, F. , 2003, “ Experimental Simulation of Supersonic Superboom in a Water Tank: Nonlinear Focusing of Weak Shock Waves at a Fold Caustic,” Phys. Rev. Lett., 91(18), p. 1843901. [CrossRef]
Chen, Y. , and Ostoja-Starzewski, M. , 2010, “ MRI-Based Finite Element Modeling of Head Trauma: Spherically Focusing Shear Waves,” Acta Mech., 213(1–2), pp. 155–167. [CrossRef]
Shugar, T. , 1977, “ A Finite Element Head Injury Model,” Department of Transportation, National Highway Traffic Safety Administration, Report No. DOT HS-289-3-550.
Hosey, R. , and Liu, Y. , 1982, “ A Homeomorphic Finite Element Model of the Human Head and Neck,” Finite Elements in Biomechanics, Wiley, New York, pp. 379–401.
Zhang, L. , Yang, K. , and King, A. , 2001, “ Comparison of Brain Responses Between Frontal and Lateral Impacts by Finite Element Modeling,” J. Neurotrauma, 18(1), pp. 21–30. [CrossRef] [PubMed]
Taylor, P. A. , and Ford, C. , 2009, “ Simulation of Blast-Induced Early-Time Intracranial Wave Physics Leading to Traumatic Brain Injury,” ASME J. Biomech. Eng., 131(6), p. 61007. [CrossRef]
Chatelin, S. , Deck, C. , Renard, F. , Kremer, S. , Heinrich, C. , Armspach, J.-P. , and Willinger, R. , 2011, “ Computation of Axonal Elongation in Head Trauma Finite Element Simulation,” J. Mech. Behav. Biomed. Mater., 4(8), pp. 1905–1919. [CrossRef] [PubMed]
Nyein, M. , Jerusalem, A. , Radovitzky, R. , Moore, D. , and Noels, L. , 2008, “ Modeling Blast-Related Brain Injury,” MIT, Cambridge, MA, DTIC Document No. ADA504193.
Moore, D. , Jérusalem, A. , Nyein, M. , Noels, L. , Jaffee, M. , and Radovitzky, R. , 2009, “ Computational Biology—Modeling of Primary Blast Effects on the Central Nervous System,” Neuroimage, 47, pp. T10–T20. [CrossRef] [PubMed]
Mendis, K. , 1992, “ Finite Element Modeling of the Brain to Establish Diffuse Axonal Injury Criteria,” Ph.D. thesis, The Ohio State University, Columbus, OH.
Brands, D. , Peters, G. , and Bovendeerd, P. , 2004, “ Design and Numerical Implementation of a 3-D Non Linear Viscoelastic Constitutive Model for Brain Tissue During Impact,” J. Biomech., 37(1), pp. 127–134. [CrossRef] [PubMed]
Landau, L. , and Lifshitz, E. , 1959, “ Course of Theoretical Physics,” Theory and Elasticity, Vol. 7, Pergamon Press, Oxford, UK.
Khokhlova, V. , Ponomarev, A. , Averkiou, M. , and Crum, L. , 2006, “ Nonlinear Pulsed Ultrasound Beams Radiated by Rectangular Focused Diagnostic Transducers,” Acoust. Phys., 52(4), pp. 481–489. [CrossRef]
Yang, L. , Chen, Y.-Y. , and Yu, S.-T. J. , 2013, “ Viscoelasticity Determined by Measured Wave Absorption Coefficient for Modeling Waves in Soft Tissues,” Wave Motion, 50(2), pp. 334–346. [CrossRef]
Lee, Y.-S. , and Hamilton, M. , 1995, “ Time-Domain Modeling of Pulsed Finite-Amplitude Sound Beams,” J. Acoust. Soc. Am., 97(2), pp. 906–917. [CrossRef]
Coulouvrat, F. , 2000, “ Focusing of Weak Acoustic Shock Waves at a Caustic Cusp,” Wave Motion, 32(3), pp. 233–245. [CrossRef]
Strang, G. , 1968, “ On the Construction and Comparison of Difference Schemes,” SIAM J. Numer. Anal., 5(3), pp. 506–517. [CrossRef]
Loubeau, A. , and Coulouvrat, F. , 2009, “ Effects of Meteorological Variability on Sonic Boom Propagation From Hypersonic Aircraft,” AIAA J., 47(11), pp. 2632–2641. [CrossRef]
Dagrau, F. , Rénier, M. , Marchiano, R. , and Coulouvrat, F. , 2011, “ Acoustic Shock Wave Propagation in a Heterogeneous Medium: A Numerical Simulation Beyond the Parabolic Equation,” J. Acoust. Soc. Am., 130(1), pp. 20–32. [CrossRef] [PubMed]
Yang, X. , and Cleveland, R. , 2005, “ Time Domain Simulation of Nonlinear Acoustic Beams Generated by Rectangular Pistons With Application to Harmonic Imaging,” J. Acoust. Soc. Am., 117(1), pp. 113–123. [CrossRef] [PubMed]
Coulouvrat, F. , 2009, “ A Quasi-Analytical Shock Solution for General Nonlinear Progressive Waves,” Wave Motion, 46(2), pp. 97–107. [CrossRef]
McDonald, B. , and Ambrosiano, J. , 1984, “ High-Order Upwind Flux Correction Methods for Hyperbolic Conservation Laws,” J. Comput. Phys., 56(3), pp. 448–460. [CrossRef]
Boris, J. , and Book, D. , 1973, “ Flux-Corrected Transport. I. SHASTA, A Fluid Transport Algorithm That Works,” J. Comput. Phys., 11(1), pp. 38–69. [CrossRef]
LeVeque, R. J. , 2002, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge.
Marston, P. , 1992, “ Geometrical and Catastrophe Optics Methods in Scattering,” Physical Acoustics, Vol. XXI, Academic Press, San Diego, CA, pp. 1–234.
Pearcey, T. , 1946, “ The Structure of an Electromagnetic Field in the Neighbourhood of a Cusp of a Caustic,” Philos. Mag., 37(268), pp. 311–317. [CrossRef]
Guiraud, J.-P. , 1965, “ Acoustique Géométrique, Bruit Balistique des Avions Supersoniques et Focalisation,” J. Mécanique, 4, pp. 215–267.
Chatelin, S. , Constantinesco, A. , and Willinger, R. , 2010, “ Fifty Years of Brain Tissue Mechanical Testing: From In Vitro to In Vivo Investigations,” Biorheology, 47(5), pp. 255–276. [PubMed]
Arbogast, K. , Meaney, D. , and Thibault, L. , 1997, “ Biomechanical Characterization of the Constitutive Relationship for the Brainstem,” SAE Technical Paper No. 952716.
Bilston, L. , Liu, Z. , and Phan-Thien, N. , 1997, “ Linear Viscoelastic Properties of Bovine Brain Tissue in Shear,” Biorheology, 34(6), pp. 377–395. [CrossRef] [PubMed]
Ning, X. , Zhu, Q. , Lanir, Y. , and Margulies, S. , 2006, “ A Transversely Isotropic Viscoelastic Constitutive Equation for Brainstem Undergoing Finite Deformation,” ASME J. Biomech. Eng., 128(6), pp. 925–933. [CrossRef]
Prange, M. , Meaney, D. , and Margulies, S. , 2000, “ Defining Brain Mechanical Properties: Effects of Region, Direction, and Species,” Stapp Car Crash J., 44, pp. 205–213. [PubMed]
Shuck, L. , and Advani, S. , 1972, “ Rheological Response of Human Brain Tissue in Shear,” ASME J. Basic Eng., 94(4), pp. 905–911. [CrossRef]
Jiang, Y. , Li, G. , Qian, L.-X. , Liang, S. , Destrade, M. , and Cao, Y. , 2015, “ Measuring the Linear and Nonlinear Elastic Properties of Brain Tissue With Shear Waves and Inverse Analysis,” Biomech. Model. Mechanobiol., 14(5), pp. 1–10. [PubMed]
Gadd, C. , 1966, “ Use of a Weighted-Impulse Criterion for Estimating Injury Hazard,” SAE Technical Paper No. 660793.
Greenwald, R. , Gwin, J. , Chu, J. , and Crisco, J. , 2008, “ Head Impact Severity Measures for Evaluating Mild Traumatic Brain Injury Risk Exposure,” Neurosurgery, 62(4), pp. 789–798. [CrossRef] [PubMed]
Chinn, B. , Doyle, D. , Otte, D. , and Schuller, E. , 1999, “ Motorcyclists Head Injuries: Mechanisms Identified From Accident Reconstruction and Helmet Damage Replication,” International Research Council on the Biomechanics of Injury Conference (IRCOBI), Vol. 27, pp. 53–71.
Raymond, D. , Van Ee, C. , Crawford, G. , and Bir, C. , 2009, “ Tolerance of the Skull to Blunt Ballistic Temporo-Parietal Impact,” J. Biomech., 42(15), pp. 2479–2485. [CrossRef] [PubMed]
Shridharani, J. K. , Wood, G. W. , Panzer, M. B. , Capehart, B. , Nyein, M. K. , Radovitzky, R. A. , and Bass, C. R. , 2012, “ Porcine Head Response to Blast,” Front. Neurol., 3, p. 70.
Zhang, L. , Yang, K. , and King, A. , 2004, “ A Proposed Injury Threshold for Mild Traumatic Brain Injury,” ASME J. Biomech. Eng., 126(2), pp. 226–236. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Left: Solutions of Eq. (15) calculated over one period at position x = 30, for an initial sinusoidal plane wave with an angular frequency ω0. Dotted line: implicit analytical Poisson's solution. Dark-dashed line: numerical solution with minmod limiter. Light-dashed line: numerical solution with Boris and Book limiter. Right: energy as function of distance for x∈[0 ,30] without dissipation (dotted line) and with dissipation for a Gol'dberg number equal to 60 (dashed line). Inset figure: zoom for x∈[0 ,1.5].

Grahic Jump Location
Fig. 2

Left: Schematic diagram of the domain of calculation used for the caustic cusp geometry. O is the geometrical caustic point, at the origin. Right: maximum particle velocity determined by the numerical solution for a focused N-wave for the case Nτ= 70,001. The geometrical caustic cusp is shown as a line. The solution is zoomed around the caustic cusp.

Grahic Jump Location
Fig. 3

Top left: comparison between the analytical solution (solid line) and numerical solutions with time step Δτ=10−3 (circles), Δτ=5×10−4 (crosses), and Δτ=10−4 (black squares) for an N-wave at the theoretical caustic point O. Top right: zoom of first plot centered around the first shock. Down: comparison between the analytical solution (solid line) and numerical solutions for a time step of Δτ=10−4 for first-order (circles) and second-order (crosses) schemes. Only the zoom centered on the first shock is shown. (a) Particle velocity at the caustic point, (b) first shock of the solution at the caustic point, and (c) first shock of the solution at the caustic point: comparison between the first- and second-order schemes.

Grahic Jump Location
Fig. 4

Velocity field for different values of γ for the self-similar case of a focused step shock with boundary condition given by Eq. (26). Left column: direct simulations calculated on a fixed domain. The solid line is the geometrical cusp caustic. Right column: the same simulations rescaled by the self-similar variables. The rectangles on the left column show the rescaled region in the right column: (a) γ = 0.05, (b) γ = 0.5, (c) γ = 1, (d) γ = 0.05, (e) γ = 0.5, and (f) γ = 1.

Grahic Jump Location
Fig. 5

Values of self-similar variables x¯ (circle), y¯ (triangle), v¯ (disk), and −t¯ (square) of the point of maximum amplitude for different values of γ. In the ideal case, each value would follow a horizontal line.

Grahic Jump Location
Fig. 6

Velocity as a function of retarded time (self-similar variables) at the point of maximum amplitude for γ varying from 0.05 to 1: (a) full time signal and (b) zoom centered on the shock

Grahic Jump Location
Fig. 7

CT of a human head. Dark gray: bone. Light gray: soft tissues. Left: vertical central cut. The horizontal line indicates the horizontal cut. Middle: horizontal cut. The arrow indicates the selected parietal area. Right: the selected parietal area. The doted line indicates the input plane for numerical simulations.

Grahic Jump Location
Fig. 8

Left: Initial waveform. Right: velocities calculated at the focal spot for different initial amplitudes. Solid line: 0.05 m/s. Dashed line: 0.6 m/s. Dotted line: 1 m/s.

Grahic Jump Location
Fig. 9

Maximum of energy (top), acceleration (middle), and shear stress σ31 (bottom) on the calculation domain for initial amplitudes of 0.05 m/s ((a), (d), and (g)), 0.6 m/s ((b), (e), and (h)), and 1 m/s ((c), (f), and (i)). These values correspond to shock formation distances that are, respectively, well past the focus, near the focus, and well before the focus.

Grahic Jump Location
Fig. 10

Maximum particle velocity (a), acceleration (b), and shear stress (c) at the focus point for different initial particle velocities

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In