A three-dimensional (3D) geometric representation was derived from an idealized human brain axial section as depicted in Figure 1. This axial section was constructed via computer aided drafting (CAD) software (SolidWorks, 2018) and then extruded to match the overall height of a human brain. The values of height, width, thickness as well as the ratios and dimensions of sulcal depth to gyral width, sulcal width, CSF layer thickness, gray / white matter, ventricle size were compared to principal investigator's (35-year-old-male, healthy) brain magnetic resonance image (MRI) imaging. The aim of this geometry was to model the key phenomenology of intracranial interface dynamics, yet was simplified enough to provide mechanistic insight into blast traumatic brain injury. The derived anatomy is a generalized representation / notional model of a human brain to study phenomenology and does not reflect any personalized attributes. Although geometrically simple, the uniqueness of this model is the high spatial resolution in which we meshed this simplified anatomy, in order to ensure that simulations resolve the appropriate interface mechanics. Additionally, this design can be fabricated into experimental phantoms for future efforts to support the results of these computational simulations. The geometry was reviewed and verified by a board-certified neurologist who confirmed the relative size and shape of the human head was maintained in the brain phantoms while maintaining the dominant length scales of the gyri/sulci. See Figure 1.
Using the developed CAD model, Cubit software was used to create all hexahedral finite element meshes of the surrogate head models. The mesh design was optimized to resolve the thin gray matter layer and CSF layer, each employing a minimum of 5 elements through the thickness. Without vasculature, the model consists of 2.3 million elements, with a more detailed breakdown of mesh statistics given in Table 1. The vascular model also included small diameter channels (0.6, 0.8, 1.2, and 1.6 mm) which were surrounded by a layer of five elements to model a vascular wall 10% of the entire vessel diameter. This highly refined vasculature model contains 11.8 million elements and details are provided in Table 2. The element "size" provided is a measure of the element diameter and is directly related to the fidelity of the computational results. Typical linear finite element methods (as used here) converge quadratically as the element size is decreased.
Blast loading onto the head model is accomplished with the shock wave physics code CTH CTH is a multidimensional finite volume code capable of accurately resolving shock wave propagation. Air is modeled via a SESAME tabulated equation of state in CTH. A uniform grid of 0.25 mm cubic cells is used to resolve the shock wave propagation in air. Shock loading is generated from an energized slab of air tuned to give a specific overpressure and Friedlander shaped shock profile with overpressure durations of 1 ms, see Table 3. Described in Table 3 are the thermodynamically consistent CTH conditions and the required stand-off distances to generate the Friedlander profiles of the specified duration (1 ms) and overpressures (150, 250, and 500kPa). The blast characteristics were not aimed to directly model any particular blast exposure. The choice of the 1 ms overpressure duration is convenient but representative of smaller blasts at close range:pipebombs or rocket propelled grenade blasts. This shorter duration also improved computational efficiency secondary to shorter simulations and smaller computational domains, furthermore it was chosen to match the overpressure durations in upcoming experimental validation studies. Amplitudes where chosen such that reflected waves overpressures would approximately achieve the levels in which NHP studies began demonstrating changed in cerebrospinal fluid injury biomarkers. A reflective boundary condition must be used adjacent to this energized air to direct the energy toward the test object. All other boundaries utilize an outflow boundary condition. Simulations were run to determine a sufficiently large domain size to ensure that boundary conditions do not affect the mechanical response of the head in the time periods considered herein. Three blast scenarios were simulated (see Figure 3), front blast, side blast, and wall blast. In order to investigate if a more complex blast wave (which included reflections / mach stem) altered the phenomenology of blast TBI, we included a wall at an arbitrary distance / angle to the incident blast. The wall blast was identical to the front blast, except for the inclusion of a reflecting surface 6.3 cm from side of head at angle of 26.5  off from direction of blast.
Finite element simulations of the surrogate head model (and all intracranial contents) were conducted using the Sierra Solid Mechanics software from Sandia National Laboratories. We use this Lagrangian finite element software to explicitly integrate the dynamical equations of motion for a 3-dimensional solid. Material models were used for the constituent parts of skull, gray matter, white matter, and CSF. For the skull, a hyperelastic solid (see Table 4) was used. There is significant variation in skull properties reported and our parameters fit within experimentally derived values. Our gray matter and white matter models were modeled via a Tillotson-Brundage equation of state (EOS) for dilatational response (Table 5) while using viscoelastic model for deviatoric response (Table 6). Reported values of the parameters to model white and gray matter also vary significantly between study, species, and age of specimen, thus we used both the meticulously derived values of Taylor and simulated with values reduced by an order of magnitude to ensure stability of results to material parameters. CSF and blood are indistinguishable in our simulations. They are both modeled using the same Tillotson-Brundage equation of state (EOS) for dilatational response (Table 5) which had no deviatoric strength. We did not include separate tissue models for vessel walls, thus our model of vascular structures contained only a fluid filled cavity embedded within the brain model, see Figures 2 and 5. Thus, our vascular models were fluid channels embedded in the brain without any surrounding endothelium. Although this model will not resolve the important mechanical interactions of the mechanics within vascular tissue (which could provide mechanical insight into the development of pseudoaneurysms or vasospasm), it will capture the fluid-structure interaction and any possible cavitation events within the vasculature.
Blast-on-structure simulations are challenging due to the need to couple two different physical domains, air with a spatial description and the head with a material description. Adding to this complexity is the technical challenge of coupling the different software pieces. The Sierra Zapotec tool couples the air blast in CTH to the finite element simulation in Sierra Solid Mechanics yielding a powerful yet scalable computational framework capable of simulating highly resolved computational meshes Figure 2.
Cavitation events are modeled using a minimum pressure approach, similar to other TBI studies. Thus, once a defined minimum pressure / tension is achieved, local material stress will be fixed at that prescribed level:essentially providing a yield point of the material:where further deformation will not further increase material tension. Cavitation rebound occurs when an element's pressure increases above the cavitation threshold, at which point the element is no longer cavitating. No damage modeling mechanisms are being modeled here, and material response after cavitation rebound is unchanged by the cavitation event itself. That is, an element's cavitation history does not impact its instantaneous response.
Cavitation inception at the scale of element size is captured, but the fine details of cavitation bubble growth and subsequent collapse are not resolved. Therefore, cavitated elements can be viewed as markers of where cavitation may occur, but would require more localized studies of cavitation events to understand the effects on surrounding tissue. Thresholds were set at gauge pressures of -200 kPa for CSF and -300 kPa for intracranial tissue.