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.

References

References
1.
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
.
2.
Whitham
,
G.
,
1979
,
Lectures on Wave Propagation
,
Springer-Verlag
,
Berlin
.
3.
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
.
4.
Duck
,
F. A.
,
2013
,
Physical Properties of Tissues: A Comprehensive Reference Book
,
Academic Press
, London.
5.
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
.
6.
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
.
7.
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
.
8.
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
.
9.
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
.
10.
Kuznetsov
,
V.
,
1971
, “
Equations of Nonlinear Acoustics
,”
Sov. Phys. Acoust.
,
16
(
4
), pp.
467
470
.
11.
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
.
12.
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
.
13.
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
.
14.
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
.
15.
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
.
16.
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
.
17.
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
.
18.
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
.
19.
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
.
20.
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
.
21.
Hart
,
T.
, and
Hamilton
,
M.
,
1988
, “
Nonlinear Effects in Focused Sound Beams
,”
J. Acoust. Soc. Am.
,
84
(
4
), pp.
1488
1496
.
22.
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
.
23.
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
.
24.
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
.
25.
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
.
26.
Shugar
,
T.
,
1977
, “
A Finite Element Head Injury Model
,” Department of Transportation, National Highway Traffic Safety Administration, Report No. DOT HS-289-3-550.
27.
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
.
28.
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
.
29.
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
.
30.
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
.
31.
Nyein
,
M.
,
Jerusalem
,
A.
,
Radovitzky
,
R.
,
Moore
,
D.
, and
Noels
,
L.
,
2008
, “
Modeling Blast-Related Brain Injury
,” MIT, Cambridge, MA,
DTIC Document No. ADA504193
.
32.
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
.
33.
Mendis
,
K.
,
1992
, “
Finite Element Modeling of the Brain to Establish Diffuse Axonal Injury Criteria
,”
Ph.D. thesis
, The Ohio State University, Columbus, OH.
34.
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
.
35.
Landau
,
L.
, and
Lifshitz
,
E.
,
1959
, “
Course of Theoretical Physics
,”
Theory and Elasticity
, Vol.
7
,
Pergamon Press
, Oxford, UK.
36.
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
.
37.
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
.
38.
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
.
39.
Coulouvrat
,
F.
,
2000
, “
Focusing of Weak Acoustic Shock Waves at a Caustic Cusp
,”
Wave Motion
,
32
(
3
), pp.
233
245
.
40.
Strang
,
G.
,
1968
, “
On the Construction and Comparison of Difference Schemes
,”
SIAM J. Numer. Anal.
,
5
(
3
), pp.
506
517
.
41.
Loubeau
,
A.
, and
Coulouvrat
,
F.
,
2009
, “
Effects of Meteorological Variability on Sonic Boom Propagation From Hypersonic Aircraft
,”
AIAA J.
,
47
(
11
), pp.
2632
2641
.
42.
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
.
43.
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
.
44.
Coulouvrat
,
F.
,
2009
, “
A Quasi-Analytical Shock Solution for General Nonlinear Progressive Waves
,”
Wave Motion
,
46
(
2
), pp.
97
107
.
45.
McDonald
,
B.
, and
Ambrosiano
,
J.
,
1984
, “
High-Order Upwind Flux Correction Methods for Hyperbolic Conservation Laws
,”
J. Comput. Phys.
,
56
(
3
), pp.
448
460
.
46.
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
.
47.
LeVeque
,
R. J.
,
2002
,
Finite Volume Methods for Hyperbolic Problems
,
Cambridge University Press
, Cambridge.
48.
Marston
,
P.
,
1992
, “
Geometrical and Catastrophe Optics Methods in Scattering
,”
Physical Acoustics
, Vol.
XXI
,
Academic Press
,
San Diego, CA
, pp.
1
234
.
49.
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
.
50.
Guiraud
,
J.-P.
,
1965
, “
Acoustique Géométrique, Bruit Balistique des Avions Supersoniques et Focalisation
,”
J. Mécanique
,
4
, pp.
215
267
.
51.
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
.
52.
Arbogast
,
K.
,
Meaney
,
D.
, and
Thibault
,
L.
,
1997
, “
Biomechanical Characterization of the Constitutive Relationship for the Brainstem
,” SAE Technical Paper No. 952716.
53.
Bilston
,
L.
,
Liu
,
Z.
, and
Phan-Thien
,
N.
,
1997
, “
Linear Viscoelastic Properties of Bovine Brain Tissue in Shear
,”
Biorheology
,
34
(
6
), pp.
377
395
.
54.
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
.
55.
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
.
56.
Shuck
,
L.
, and
Advani
,
S.
,
1972
, “
Rheological Response of Human Brain Tissue in Shear
,”
ASME J. Basic Eng.
,
94
(
4
), pp.
905
911
.
57.
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
.
58.
Gadd
,
C.
,
1966
, “
Use of a Weighted-Impulse Criterion for Estimating Injury Hazard
,” SAE Technical Paper No. 660793.
59.
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
.
60.
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
.
61.
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
.
62.
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.
63.
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
.
You do not currently have access to this content.