#### 3.1. Simulation of Impact

The recorded impact force from the frontal cadaveric impact experiment conducted by Nahum, as discussed in the preceding section, is directly applied to the model in the form of a distributed load with the peak pressure input of 4.37 MPa on the frontal region of the skull. The impact pulse lasts about 9 ms and the simulation is run for 15 ms. Rotation of the model is permitted as the base of the skull is not constrained. We find that this free boundary condition gives better correlation of coup pressure to experimental results [

17] than the fixed boundary condition. The FE simulations are carried out using Abaqus/Explicit.

The pressure–time history for the Nahum loading is presented in

Figure 4 at the coup impact site. We plot the results for both the homogeneous and heterogeneous models. The model predicts tensile pressure for nearly the whole duration of impact. The peak pressure predicted by the heterogeneous material model more closely matches that from Nahum’s experiments than the homogeneous model. In both cases the peak pressure occurs roughly at the same time, indicating very similar wave speeds.

We next compare the displacement response to the C383-T1 impact from Hardy et al. experiments [

32]. This is a frontal impact experiment lasting 118 s with displacements captured by two columns of six NDTs. We plot the response for relative displacements for the

$x\u2013$ and

$y\u2013$directions for two NDTs each in the anterior (A) and posterior (P) positions (labeled A1, A2 and P1, P2, respectively); see

Figure 5 and

Figure 6. To quantitatively compare the response of the two models, we follow a similar argument as in [

26] and compute the displacement magnitude (excursion) at the NDTs. Since injury criteria are based on the magnitude of tissue strain, it is argued that the extension rather than the entire NDT trajectory is more important. We find that the heterogeneous model more consistently and closely predicts the excursion determined experimentally, as presented in

Table 4. Additionally, we have previously validated our homogeneous model using tagged MRI and HARP imaging analysis techniques in [

18], which allows comparisons of displacement fields for the entire cerebrum in vivo.

We plot the contours of the von Mises stress distribution on the sagittal plane due to the frontal impact in

Figure 7 (also see

supplementary simulation video available online). The spherically convergent structure of the shear wave, just like the one found using the linear elastic model of brain’s gray and white matters [

17], is clearly observed. The wave attenuates as it travels inwards and eventually dissipates. Reflections of wave due to scattering from heterogeneous white matter structures can also be observed at later times.

Next, we consider three distinct points along the sagittal plane, as depicted in

Figure 8. The points are chosen within regions of strong heterogenities due to the presence of highly aligned axon tracts, such as the corpus callosum and corona radiata. The differences in mechanical properties of these regions are given in

Table 5. We see that the material phases at these points are relatively stiffer than the corresponding points in the homogeneous model. As a result, the response in

Figure 8 is affected accordingly. We find that the difference in peak pressure response is proportional to the difference in shear stiffness between the homogeneous and heterogeneous models. However, the time at which these events occur is not significantly affected. This indicates that the pressures in regions of high stiffness within the brain are over-estimated in the homogeneous models. In summary, relative to the MRI-based model, the new MRE-based heterogeneous model more accurately predicts the local response within the white matter by taking into account the differences in tissue stiffness of local white matter structures.

A few points are in order regarding the qualitative differences between the shear modulus of different regions in our model. Globally, the white matter is found to be approximately

$32\%$ stiffer than the gray matter. In general, the white matter properties in local regions differ significantly from the average ones. For instance, the storage modulus

${G}^{\prime}$ is significantly lower in the rest of the white matter than within the corpus callosum and the corona radiata [

10]. This is quite logical given the fact that the corpus callosum contains highly oriented, tightly packed axon tracts. The corpus callosum, in fact, is stiffer than the corona radiata, again evident from the composition of the corona radiata, which contains axon fibers that fan out and are not as highly aligned as the corpus callosum. Indeed, experiments have found that the fractional anisotropty (FA) values for the corpus callosum and corona radiata are in the range of 0.6–1.0 and 0.4–0.6, respectively [

34,

35].

Additionally, while both white and gray matter have similar values of damping ratio,

$\mathsf{\xi}$ (which reflects the amount of motion attenuation within the tissue), the corpus callosum has a lower value while the corona radiata has a higher value. This can be explained by examining the microstructure of each of these regions. Experiments by Guo et al. [

36] demonstrated that the damping ratio (and thus, the attenuation) in soft tissue composites increases as the number of cross-links between fibers increases. The corona radiata consists of fibers arranged in a grid-like pattern [

37] with a large number of cross-links. These crossings do not exist in the corpus callosum, offering a possible explanation for the distribution of

$\mathsf{\xi}$.

Since mechanical measures from MRE and diffusivity measures from diffuse tensor imaging (DTI) both provide insight into the heterogeneity within the white matter, a natural question arises here: What is the difference between our heterogeneous (yet isotropic) model and the more common anisotropic FE models where fiber anisotropy is determined from DTI scans? Many such examples of the latter exist in the literature: for instance in [

38,

39,

40,

41]. Johnson et al. [

10] performed both MRE and DTI measurements on a group of seven volunteers to determine the correlation of mechanical and diffusivity measures within the corpus callosum and corona radiata. They determined that MRE and DTI measures correlate well with each other within the corpus callosum—not surprising since they are both sensitive to the underlying tissue microstructure. They hypothesize that these measures are highly dependent on axon diameter since larger axons provide greater structural rigidity to the tissue [

42]. Within the corona radiata, however, the correlation is not as significant. The corona radiata comprises fiber tracts that fan towards the cortex and contain numerous crossings [

37] which are not captured well by DTI [

43]. This has been hypothesized as the reason for the poor correlation within the corona radiata. More work is needed to determine the differences between these two methods when used within FE models.

#### 3.2. Stochastic Wave Propagation

The highly heterogeneous structure of the brain tissue introduces wave scattering that competes with wave amplification due to spherically convergent implosion. Following the development in [

44], we investigate this effect by considering the theory of wavefronts. For the case of one-dimensional wave motion, we assume that a compressive load produces a shock wavefront that propagates from a disturbed domain to an undisturbed one with a speed

${c}_{T}$. The initial conditions can be given as

$u(x,0)={u}_{,t}(x,0)=0;\tau (0,t)=-{\tau}_{0}H\left(t\right)$ where

H is the Heaviside function.

Assuming a plane wave in a homogeneous medium, we have the dynamic compatibility condition

$\left[\tau \right]=-\rho {c}_{T}\left[u{,}_{t}\right]$ in the

$(x,t)$-plane, where

$[\xb7]$ denotes the discontinuity in a function across the boundary of two materials,

$\sigma $ is the shear stress,

$\rho $ is the mass density,

u is the displacement normal to the direction of wave motion, and

${c}_{T}$ is the transverse wave speed. The linear viscoelastic stress–strain relation for a process that started at time

$t={t}_{0}^{+}$ is

where

$\epsilon $ is the shear strain. We can derive the relationship for the wave speed as

${c}_{T}=\sqrt{G\left(0\right)/\rho}$, where

$G\left(0\right)$ is the glassy modulus. Following the derivation in [

44], we obtain the equation governing the evolution of the discontinuity of

$\tau $ at the wavefront as:

On account of the initial conditions above, the solution of Equation (

10) is

Given that

${G}_{,t}\left(0\right)\le 0,$ and

$G\left(0\right)>0$, the stress jump exhibits exponential attenuation and has a tendency for blow-up as

$r\to 0$. As our simulations here and in [

19] demonstrate, the attenuation is sufficiently strong so that the imploding waves generated from transient impacts do not blow up into a singularity at the head center.

The impact results not only in a fast pressure wave, but also in a slower shear wave. Due to its relatively low shear modulus, brain tissue deforms much more easily in shear than in dilatation mode. Thus, the shear wave is potentially more damaging. Recall that the spherically convergent shear wave patterns are observed even in the case of homogeneous material description, c.f. [

17]. The attenuations of pressure along the sagittal plane for both the homogeneous and heterogeneous models are presented in

Figure 9. It is clear that the attenuation is greater in the heterogeneous model as predicted. This is consistent with studies of transient wave propagation in elastodynamics of random media [

45,

46].