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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

## Abstract

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 $2.54 m, 0.018 m$, and $0.0064 m$. 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.

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## Figures

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].

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.

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.

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

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.

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.

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.

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.

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.

Fig. 10

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

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