Introduction 1 21 41 41 16 43 35 2 6 7 9 17 22 38 42 45 47 1 34 in vitro in vivo 3 4 10 11 13 31 32 et al 30 33 Figure 1 6 7 et al 44  14 et al 5 −2 et al 6 Methods In this study, plane strain models of small sections of the cerebrum are made using the FE code Abaqus 6.6-1 (HKS, Providence, USA). An explicit time integration is used, anticipating a dynamic load with a high magnitude and a short duration. The time increments are limited by the stability condition, which is determined in the global estimator function in Abaqus. Geometries 2 2 2 29 34 29 34 34 Figure 2 (a) Heterogeneous geometry 1 and (b) its spatial discretization. (c) Heterogeneous geometry 2. (d) Heterogeneous geometry 3. (e) Homogeneous geometry  x y et al 27 2 Material Properties 4 1 \documentclass[12pt]{minimal}
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\begin{document}$$\dot{\gamma}$$\end{document} 2 47 Table 1 Linear material parameters   Bulk modulus (GPa) Shear modulus (Pa) Time constant (s) CSF 2.2 a ∞ b ∞ Brain tissue 2.5 182.9 ∞ 9884 0.00013 835.5 0.012 231.2 0.35 67.1 4.62 3.61 12.1 2.79 54.3 a b et al 19 N \documentclass[12pt]{minimal}
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\begin{document}$$ \varvec{\sigma}_{\rm e}^{\rm d} = \frac{G_{\infty}}{J} \left[ (1-A)\hbox{exp}\left(-C\sqrt{b\tilde{I}_{1} + (1-b)\tilde{I}_{2} - 3}\right) + A \right]\left[b\tilde{\user2{B}}^{\rm d} - (1-b)(\tilde{\user2{B}}^{-1})^{\rm d} \right] $$\end{document} G ∞ \documentclass[12pt]{minimal}
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\begin{document}$$ \eta(\tau) = \eta_{\infty} + \frac{\eta_{0} - \eta_{\infty}} {1 + \left(\frac {\tau} {\tau_{0}}\right)^{(n-1)}} $$\end{document} 0 G ∞ k 0 et al 19 18 et al 39 1 2 Table 2 Non-linear material parameters for brain tissue Elastic Viscous A 0 C n a k b  Boundary Conditions 3 x y \documentclass[12pt]{minimal}
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\begin{document}$$ 0.010\,\hbox{s} < t \leq 0.030\,\hbox{s} : \dot{\omega}(t) = -125 \pi\sin\;(50\pi (t-0.010)) $$\end{document} Figure 4 The loading conditions of the cerebral cortex model (micro-level) are derived from the region of interest in a parasagittal cross-section (15 mm offset from the midsagittal plane) of the head model (macro-level). Shown at the macro-level is the equivalent stress field of the head model at 10 ms  12 13 \documentclass[12pt]{minimal}
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\begin{document}$$\ddot{u}(y,t),$$\end{document} 4 In the first approach, the input accelerations of the head model are used to define the loading condition of the cerebral cortex model. This approach will be referred to as loading condition A. \documentclass[12pt]{minimal}
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\begin{document}$$\ddot{u}$$\end{document} 1 3 5 5 y The disadvantage of this loading condition is that a spatially constant acceleration gradient is assumed and therefore it does not account for the influence of the geometry of the cranium. To account for the geometry of the head, another loading condition has been developed that is described next. et al 6 et al 19 4 x x 6 y y Figure 5 Loading condition A. (a) Acceleration at the upper and lower boundary of the cerebral cortex model. (b) Acceleration profiles at different times  Figure 6 Loading condition B: displacement (top) and acceleration (bottom) profiles derived from the output of the head model  \documentclass[12pt]{minimal}
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\begin{document}$$\varvec{\sigma}.$$\end{document} 30 3 30 Results 7 1 Figure 7 The equivalent stress fields as a result of loading condition A  8 Figure 8 Maximum and mean equivalent stress for the heterogeneous and homogeneous models as a result of loading condition A  To investigate the influence of the heterogeneities, the equivalent stress of the cerebral cortex in the heterogeneous models is taken relative to that of the homogeneous model. For the maximum equivalent stress, this will be done by taking the maximum values, whereas for the mean equivalent stress, this will be done by taking the time averaged values. The maximum equivalent stress of the heterogeneous models 1, 2, and 3 is 1.31, 1.84, and 1.83 times higher than the homogeneous model, respectively. The mean equivalent stress of the heterogeneous models 1, 2, and 3 with respect to the homogeneous model is 1.09, 1.08, and 1.10, respectively. 9 Figure 9 The equivalent stress fields as a result of loading condition B  10 Figure 10 Maximum and mean equivalent stress for the heterogeneous and homogeneous models as a result of loading condition B  To quantify the influence of the heterogeneities, the equivalent stress of the brain tissue of the heterogeneous models is taken relative to the homogeneous model in the same manner as described previously for loading condition A. The maximum equivalent stress of the heterogeneous models 1, 2, and 3 has increased by 1.44, 1.74, and 1.92 with respect the homogeneous model, respectively. The mean equivalent stress of the heterogeneous models 1, 2, and 3 is 0.97, 0.99, and again 0.99 relative to the homogeneous model, respectively. 11 9 Figure 11 The maximum principal logarithmic strain field as a result of loading condition B at 10 ms  Discussion and Conclusions In this study, the influences of the heterogeneities in the cerebral cortex were investigated. This was done with FE models of several different geometries from small detailed parts of the cortex. In a preliminary study, the boundary constraints were tested. The loading conditions were derived from a numerical head model. 1 3 3 x 1 1 2 4 6 9 6 12 7 4 9 4 19 12 26 et al 37 et al 37 2 28 30 47 et al 20 17 24 8 15 40 The two loading conditions and the different geometries resulted in different equivalent stress fields. The simulations with loading condition A resulted in a lower mean and maximum equivalent stress compared to the simulations with loading condition B. However, relative to the homogeneous model, it was observed that the equivalent stress was almost independent of the different loading conditions used in this study. The differences between the several heterogeneous geometries had more influence on the relative mean and maximum equivalent stress. The morphologic heterogeneities of the cerebral cortex led to an increase of the maximum equivalent stress by a factor of about 1.3–1.9, depending mostly on the geometry, whereas the relative mean equivalent stress values of all the geometries were 1.1 and 1.0 for loading condition A and B, respectively. Furthermore, the peak maximum principal logarithmic strain was increased by a factor of about 1.2–1.9 due to the morphologic heterogeneities of the cerebral cortex. This is a strong indication that predictions of brain injury obtained from head models with a homogeneous cerebrum should be interpreted with care. To obtain a more accurate assessment of injury, the influence of the morphologic heterogeneities in the cerebral cortex should be accounted for.