The Modeling of Shock-Wave Pressures, Energies, and Temperatures Within the Human Brain Due to Improvised Explosive Devices (IEDs) Using the Transport and Burgers' Equations

Abstract

This second paper adopts a more rigorous, in-depth approach to modeling the resulting dynamic-pressures in the human brain, following a transitory improvised explosive device (IED) shock-wave entering the head. Determining more complicated boundary conditions, a set of particular-solutions for both Burgers' and the Transport equations has been obtained to describe the highly damped neurological pressures, complete with respective graphical plots. Many of these two-dimensional solution-curves closely resemble the Friedlander curve [1][2][3][4], not only illustrating enormous over-pressures that result almost immediately after the initial impact, but under-pressures experimentally depicted in all cases, due to oscillatory motion. It appears, given experimental evidence, that most-if not all-of these models can be aptly described by damped sinusoidal functions, these facts being further corroborated by existing literature, referencing models expounded by Friedlander's seminal work [1][2][3][4]. Using other advanced mathematical techniques, such as the Hopf-Cole Transformation, application of the Dirac-delta function and the Heat-Diffusion equation, expressions have been determined to model and predict the associated energies and temperatures within this paper.Keywords: traumatic brain injury, PDEs, IED-blast, pressure, under-pressure, energy, temperature, Heat-Diffusion equation INTRODUCTIONIn Paper 1, a one-dimensional model of the human head, through which a shock-wave due to an improvised explosive device (IED) blast traveled, was developed. Within this scenario, the initial shock-wave was modeled with a partial differential equation (PDE) known as Burgers' equation and, from the assumed initial boundary-condition, several shock-wave solution-curves were graphically plotted using experimental data derived from existing literature. It was hoped that, given this primary data, the theoretical calculations would match the experimental observations. Thus, future predictions about as-yet unknown IED neurological shock-waves should be possible. In setting up the preparatory conditions to tackle the challenges of this paper, it was indicated that the sheer violence of the shock-wave may have significantly harmful effects upon the human brain during and after battle, and a physics-based discussion about some of these issues followed. Mason IED Neurological Shock-Waves and EnergiesIn addition to high-velocity longitudinal pressure-waves, observed to cause significant brain trauma, it is mentioned that shear-or transverse-waves can be explained by similar physics-based mathematics, in terms of the way in which waves behave during their propagation through the brain.In continuing this mathematical discourse, we turn to further worked solutions of Burgers' equation by first considering damped sinusoidal motion of the resulting, dynamic, variation in pressure following initial transitory shock-wave propagation through the human head. MODELING OF VARIATION OF PRESSURE IN TERMS OF U (T, X)From each of the Equations (19) and (20...

Acoustic light-sheet microscopy

“…This single-shot image acquisition can be repeated with the lasers' repetition frequency or camera's frame rate to obtain 3D data by sliding the sample in an x-axis direction. The acoustic pulses used in our experiments resemble smoothed Friedlander curves (24), displaying a sharp rise of pressure (i.e. high refractive index contrast) at the leading wavefront with a following elongated negativepressure area (fig.…”