Introduction

Traumatic brain injuries (TBI) are a significant public health burden, and account for approximately one-third of the injury-related deaths in the United States.10 The ability to predict a TBI from environmental information about an impact is crucial for designing better protective head equipment and safety countermeasures. Recent TBI studies18,19 have focused on the development of kinematics-based injury metrics that predict brain injury based on impact conditions (translational and rotational head kinematics). While these injury metrics are essential to injury prevention, they cannot predict tissue-level, local damage to the brain. Computational models are crucial to understanding the mechanisms of TBI at the tissue level, because of their ability to link external head impact conditions to the mesoscopic (tissue level ~ 1 mm) and even microscopic (cellular level ~ 10 μm) responses of brain tissue that leads to injury.

While the microstructure of the brain white matter (WM) is heterogeneous and anisotropic, most current computational brain models44,47,60 have adopted an isotropic representation of the material. More importantly, the lack of mesoscopic WM structures may limit their capability in predicting TBI as the damage to axons is believed to be one of the critical mechanisms of TBI.45 The significance of WM anisotropy on brain tissue responses has been recently studied54,63,66 and it is believed that the incorporation of WM anisotropy is necessary for the development of more precise injury metrics.28,55

The axonal fiber architecture of human WM can be characterized in vivo through diffusion-weighted magnetic resonance imaging (DWI) and subsequent deterministic tractography analyses by exploiting the Brownian motion of water molecules in tissues.39 Recent attempts to incorporate WM anisotropy in FE brain models (Table 1) have been made based on tractography information. The majority of these studies have implicitly incorporated fiber tractography to inform anisotropic, fiber-reinforced constitutive models.12,21,27,63,66 However, this approach over-simplifies the brain parenchyma heterogeneity and requires the use of weighted-average fiber orientation for each element, which may not be aligned with the actual orientation of the axonal fiber bundles.66 Garimella and Kraft22 discussed the limitations associated with this technique in detail, and instead suggested that axonal fibers be explicitly modeled as embedded elements. This method allows for the incorporation of multiple fiber orientations for a single element and takes advantage of the full axonal fiber tractography. However, the embedding method introduces new challenges associated with, but not limited to, mechanical characterization of axonal fibers and interaction between fibers and the ground substance. Also, deterministic tractography can be an erroneous process due to differences in reconstruction methods and tracking algorithms.43 While most studies (Table 1) obtain WM tracts from a single subject, a population-based atlas is preferred for increasing the validity of the fiber tractography and modeling representative topological interconnectivity in the general population.64

Table 1 3D anisotropic finite element brain models.
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The first objective of this study was to develop a methodology to incorporate axonal fibers into an existing isotropic FE human brain model. The Global Human Body Models Consortium (GHBMC) owned 50th percentile male (M50) brain model44 was used to demonstrate the applicability of this methodology. The second objective was to improve the biofidelity and prediction capability of the original model. From a broad perspective, this study provides a novel and generalized framework for incorporating mesoscopic anatomical details in multi-scale FE models.

Materials and Methods

Baseline Model

The GHBMC M50 v4.3 brain model44 has been used extensively in TBI research.18,20,56,57 The geometry of the model was based on computed tomography (CT), and magnetic resonance imaging (MRI) scans of an adult male representing the 50th percentile height and weight of the population. The brain model has 82,083 hexahedral elements in total and includes anatomical representation of the cerebrum, cerebellum, brainstem, corpus callosum, ventricles, thalamus, bridging veins, cerebrospinal fluid (CSF), and membranes (falx, tentorium, pia, arachnoid, dura). Brain model responses, including pressure and relative brain-skull motion, were previously validated.44 In this study, the GHBMC baseline model was modified to include axonal WM fiber 1-D elements, and new constitutive material models were applied to the ground substance and fiber components. Finally, the brain deformation response was validated under a battery of impact cases. The updated model is hereafter referred to as the “axon-based” model.

Fiber Tractography Model and Embedded Element Method

A combination of open-source medical imaging tools and in-house scripts were utilized to embed the baseline model with axonal fiber tract elements. Figure 1 shows a schematic of the process, which involves five steps: (1) a FE mesh of axonal fiber networks is created based on a population-based tractography template; (2) the axon tract mesh is morphed from the geometry of the tractography template to the geometry of the baseline FE brain model; (3) the morphed fiber mesh is mathematically embedded into solid elements of the baseline model; (4) the brain elements (cable and solid) are categorized based on the fractional anisotropy (FA) values of the tracts; and (5) mechanical properties of the both the axon tract elements and the isotropic solid element are assigned based on multi-modal tissue data in the literature. Each step of this process is explained in detail below.

Figure 1
figure 1

Procedural flowchart adopted to embed axonal fibers in the baseline brain finite element model. (a) Diffusion magnetic resonance imaging template; (b) Reconstructed axonal fiber tractography (c) Fiber tractography finite element model; (d) Morphing process; (e) Embedding fibers into the baseline model.

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The deterministic whole-brain fiber tractography process was performed using a freely available, pre-processed, group-averaged (N = 842; M: 372 F; 470; Ages: 20–40 years) tractography dataset (HCP-842 tractography template) consisting of DWI data from the Human Connectome Project.64 Data were accessed under the WU-Minn HCP open access agreement and were initially acquired using a multi-shell diffusion scheme (b-values: 1000, 2000, and 3000 s/mm2; diffusion sampling directions: 90, 90, and 90; in-plane resolution and the slice thickness: 1.25 mm). The tractography reconstruction was conducted using DSI Studio (http://dsi-studio.labsolver.org) in the standard MNI atlas space. After 5000 randomized seeding attempts, the resulting tractography included in the model had 3446 fiber tracts with a maximum length of 297.0 mm, a minimum length of 29.3 mm and a mean length of 78.6 ± 38.24 mm. An in-house script was used to convert the axonal tractography data obtained from DSI Studio into a FE mesh using a network of 1-D cable (tension-only) elements. The axonal fiber tractography FE mesh had a total of 104,866 cable elements with an element size of around 2.5 mm. The elements were then categorized into ten groups based on their FA values (Fig. 2). FA is a widely used metric of diffusion anisotropy and ranges from 0, representing an isotropic movement of water molecules (e.g., CSF), to 1, highly anisotropic movement of water molecules (e.g., fiber bundles).

Figure 2
figure 2

Fractional anisotropy distribution for the cable fiber elements.

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Matching the tractography mesh with FE brain model is a challenging procedure because usually they are originally in different spatial orientations and are associated with brains that have a different shape and size. Here, we adopted a morphing technique to align the fiber FE model with the baseline model, based on a technique by Park et al.50 First, the geometry of the template MNI brain and CSF surface was aligned and scaled to the target geometry of baseline model using iterative closest point approximation.4 Next, the MNI surface nodes served as landmarks and were mapped to the baseline model brain surface using an iterative registration method (Burr’s elastic registration7) to match the external geometry of the two surfaces. The same transformation in this step is then applied to the tractography mesh using thin-plate spline method with radial basis function56 to interpolate and smooth to match the axonal tracts to the internal geometry of the baseline model brain. The results of the morphing can be visually checked in Fig. 3. The mean distance between the MNI surface nodes and the baseline model surface after registration was less than 0.1 mm.

Figure 3
figure 3

Morphing results demonstrating the morphed MNI surface and morphed fiber tractography model. The baseline model dura surface is shown in black.

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Once the tractography mesh was in the same anatomical space as the volumetric baseline model brain, they were constrained as embedded elements using the *CONSTRAINED_BEAM_IN_SOLID Keyword in LS-DYNA (v971 R9.2.0, LSTC, Livermore, CA). This technique has been applied previously to model rebar-reinforced concrete composites3 and ensures that the axonal fibers and volumetric ground substance are continuous and have the same accelerations and velocities. Steps were taken to ensure this method would satisfy the structural conditions of equilibrium, energy balance, and compatibility.

Constitutive Model

The brain tissue response was decomposed into an isotropic ground substance and an anisotropic component governed by the myelinated axons. The isotropic ground substance was assumed to have material properties the same as gray matter (GM). Therefore, the distinguishing feature between WM and GM was the presence of the myelinated axons. Both the ground substance and fiber materials were modeled as hyper-viscoelastic and implemented in LS-DYNA as user-defined materials. For the ground substance material, the isotropic hyper-elastic strain energy density function is based on the Holzapfel-Gasser-Ogden (HGO) model24:

$$W = \frac{G}{2} ( {\tilde{I}_{1} - 3} ) + K\left( {\frac{{J^{2} - 1}}{4} - \frac{1}{2}\ln (J)} \right) + \frac{{k_{1} }}{{2k_{2} }}(e^{{k_{2} \tilde{E}_{a}^{2} }} - 1)$$
(1)
$$\tilde{E}_{a} = \frac{1}{3}\left( {\tilde{I}_{1} - 3} \right)$$
(2)

\(\tilde{I}_{1}\) is the first invariant of the isochoric right Cauchy-Green deformation tensor and \(J = \det F\) is the volume change ratio. \(G\) is the shear modulus, K is the bulk modulus, \(k_{1}\) is a stress-like parameter, and \(k_{2}\) is a dimensionless parameter.

While the strain energy density function for the axonal fiber was formulated as Eq. (3), which is also based on the HGO model.24

$$W = \frac{{k_{1} }}{{2k_{2} }}(e^{{k_{2} E_{a}^{2} }} - 1)$$
(3)
$$E_{a} = \kappa \left( {I_{1} - 3} \right) + (1 - 3\kappa )(I_{4a} - 1)$$
(4)

The Green–Lagrange strain-like quantity \(E_{a}\) is a function of \(I_{4a} = \tilde{C}:n_{0a} \otimes n_{0a}\) (where \(\tilde{C}\) is the isochoric part of the right Cauchy–Green strain tensor and \(n_{0a}\) is the unit vector of fiber direction in the undeformed configuration) and \(\kappa\). The dimensionless structure parameter \(\kappa\) accounts for the orientation distribution of the axons in a voxel-scale fiber bundle and can be related with FA through Eq. (5) by assuming similarity between mechanical and diffusion anisotropy.27,63 At the lower limit, \(\kappa\) = 0 (FA = 1), axons are perfectly aligned and at the upper limit, \(\kappa\) = 1/3 (FA = 0), axons are randomly oriented and isotopically distributed.

$$\kappa = \frac{1}{2}\frac{{ - 6 + 4{\text{FA}}^{2} + 2\sqrt {3{\text{FA}}^{2} - 2{\text{FA}}^{4} } }}{{ - 9 + 6{\text{FA}}^{2} }}$$
(5)

The temporal response of the deviatoric stress component was modeled using a quasi-linear viscoelastic (QLV) mathematical framework,17 and the volumetric behavior was assumed to be independent of time.

$$\sigma^{d} \left( t \right) = \mathop \int \limits_{0}^{t} \left[ {G_{\infty } + \mathop \sum \limits_{i = 1}^{4} G_{i} e^{{ - \beta_{i} (t - \tau )}} } \right]\frac{{\partial \sigma_{e}^{d} }}{\partial \tau }d\tau$$
(6)

where \(\sigma_{e}^{d}\) is the instantaneous, deviatoric elastic response. A Prony series with four time-constants was chosen to model the relaxation behavior. \(G_{\infty }\) and \(G_{i}\) are the linear coefficients of the reduced relaxation coefficients, and \(\beta_{i}\) are the relaxation time constants.

Parameters Calibration

A single set of coefficients for the ground substance material model were first calibrated using available human GM tissue material testing data.37 These data were obtained from experiments conducted at a set of constant strain rates (0.5/s, 10/s, 30/s) under various loading modes including simple shear, compression, tension in terms of engineering stress and engineering strain. The explicit constitutive relations were derived analytically and formulated in terms of Cauchy stress and deformation gradient (F) during calibration. The calibration process was performed through a generalized reduced gradient nonlinear optimization to minimize the sum of squared error (SSE) between the experimental data and model predicted stress. The instantaneous and viscoelastic coefficients were optimized simultaneously.

The fiber properties were calibrated based on the composite response of the axonal fibers and the ground substance material, i.e., the mechanical properties of white matter. Since the stiffness of white mater in the model will be region dependent due to the heterogeneity of fiber architecture, the effective shear stiffness of the model at the corona radiata region was used as a benchmark. The calibration of fiber parameters was conducted using a single element inverse FE approach (details are reported in the supplementary materials). The same set of coefficients identified for the ground substance were used to model the relaxation behavior of the fibers.

Model Validation Data

Experiments from two separate studies were simulated to assess the biofidelity of the axon-based and baseline model brain deformations. The first study was a series of cadaveric impact tests conducted by Hardy et al.31,32 to measure relative brain-skull displacements under high-rate impacts using embedded radiopaque, neutral density targets (NDT). Although not designed for model validation, these experiments have been widely used to validate brain FE models.44,46,66 Recently, Alshareef et al.1 introduced a novel method for quantifying 3-D human brain deformation using sonomicrometry (SONO). Sonomicrometry provides significant advantages over previous experiments because of its capability in measuring displacement in three dimensions under multiple loading severities in all three directions of head rotation for the same specimen. These tests were conducted in a well-controlled pure rotational boundary condition with high repeatability and specifically to obtain validation targets for brain FE models. The axon-based and baseline models were validated using a subset of 12 loading cases from a single cadaver (male; 53 years old; height of 173 cm). The validation cases (n = 17) are summarized in Table 2 and represent an array of loading severities, impact durations, and impact directions.

Table 2 Summary of experimental test conditions.
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Data Analysis

All simulations were performed using LS-DYNA (v971 R9.2.0, double precision; LSTC, Livermore, CA) with 24 CPUs. Six-degree-of-freedom (DOF) head kinematics were applied to the rigid dura through the center of gravity in both brain models. For each validation case, the predicted displacement–time histories of the relevant nodes were compared with the experimental measurements. Model biofidelity was quantified using the CORrelation and Analysis objective rating system (CORA25). A weighted averaging was applied to obtain a single representative objective rating for each 3-D signal.14 The weighted CORA score was calculated by weighting the component CORA scores (\({\text{CORA}}_{x,y,z}\)) by the peak-to-peak displacement values of motion in the three axes (\(d_{x}\), \(d_{y}\), \(d_{z}\)) from the same signal as per Eq. (7). For each validation case, an overall score was computed by averaging the individual signal scores.

$${\text{Weighted CORA}} = \frac{{d_{x} \times {\text{CORA}}_{x} + d_{y} \times {\text{CORA}}_{y} + d_{z} \times {\text{CORA}}_{z} }}{{d_{x} + d_{y} + d_{z} }}$$
(7)

In this study, further investigation on strain-based metrics was performed using the SONO loading cases. The element-wise maximum principal strain (MPS) for the solid brain tissue and the element-wise maximum of axial strain for the axonal fiber tracts (MAS) were calculated.

Finally, to study the effect of the anisotropy on the tissue response, the element-wise MPS response of the baseline model was compared with those from its isotropic derivative model that only included the ground substance material (referred as GS-based model).

Results

Calibration Results

The optimized constitutive model for the ground substance material compared with experimental corridors are shown in Fig. 4a. To verify the response of the calibrated constitutive model, hypothetical shear oscillation tests were analytically computed and the stress output from the constitutive model was compared to frequency sweep data available in the literature.2,5,6,23,35,42,47,52,58,61 These results are shown in Fig. 4b. This figure also demonstrates the differences between the calibrated model and the baseline model materials. The calibration results for the axonal fiber tracts are reported in the supplementary materials.

Figure 4
figure 4

Constitutive model and experimental tissue tests results. a Constant strain rate mechanical tests; b Complex shear modulus and tan delta of brain tissue from shear oscillation tests in the literature between 0.01 and 10,000 Hz.

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Table 3 summarizes the detailed material properties employed in the axon-based model. The thickness of the falx, tentorium, and pia was modified based on recently published experimental data.30,36 Material properties of other brain regions remained identical to the unmodified baseline model.

Table 3 Material properties used in the axon-based model.
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Nodal Displacements and CORA Score

All the FE models in this study were stable and terminate normally. The simulation with the Axon-based model approximately requires 2.4 times more computational cost than that of the baseline model. Table 4 summarizes the validation performances of the axon-based and baseline models in the selected NDT and SONO cases. For the NDT impacts (Table 4, Study I), the axon-based and baseline models had overall CORA scores of 0.450 and 0.430, respectively, based on the average of the five tests (individual plots reported in the supplementary materials). These performed as well as other state-of-the-art FE models for this particular experimental dataset (details reported in supplementary materials). For the SONO cases, the axon-based and baseline models exhibited overall weighted CORA scores of 0.569 and 0.535 respectively. Exemplar brain deformation trajectories for the axon-based model are shown in Figs. A-10 (in the supplementary materials) for SONO coronal rotation cases. Overall, the axon-based model reported a higher CORA score for 16 out of 17 total NDT and SONO cases (the only exception is the C383-T4 case). However, CORA score differences between the two models were not statistically significant.

Table 4 Model validation results.
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Strain Results

Although the axon-based and baseline models demonstrated similar CORA scores, the strain responses in the two models were found to be significantly different, especially in more severe loading conditions. For the SONO simulations, the MPS predicted by the baseline model, MPS predicted by the axon-based model, and MAS predicted by the axon-based model are illustrated in Fig. 5 as cumulative distributions across all brain tissue elements. In the most severe loading case (SONO 846 M-Z4, \(\omega_{p} = 40 {\text{rad}}/{\text{s}},\alpha_{p} = 5.1 {\text{Krad}}/{\text{s}}\)), the maximum MPS of all elements predicted by the baseline model is more than 90%, while the axon-based model reported a maximum MPS of 56%.

Figure 5
figure 5

Comparison of strain results of the FE models in sonomicrometry simulations. The cumulative distributions show the percentage of elements above specific peak strain values.

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Effects of Anisotropy

The element-wise MPS response of the axon-based model was compared with those from the isotropic GS-based model using linear regression. This result is illustrated in Fig. 6a showing the element-wise MPS under the most severe SONO loading case (SONO 846 M-Z4, \(\omega_{p} = 40 {\text{rad}}/{\text{s}},\alpha_{p} = 5.1 {\text{Krad}}/{\text{s}}\)). Globally, the effects of anisotropy on strain responses were not significant. However, the inclusion of anisotropy does lead to some local differences in the strain pattern for the inner WM region (Fig. 6b), which were mainly composed of highly aligned axonal fibers.

Figure 6
figure 6

Effect of anisotropy on strain results for the SONO 846 M-Z case. a Comparison of strain results of solid elements for the whole brain; b Comparison of strain distributions.

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Discussion

Advanced brain FE models are fundamental for investigating and understanding TBI. With increasing interest of understanding injury mechanisms at the mesoscale, the biofidelity of the brain models needs to be improved in both anatomical representation and predicting biomechanical responses. In this study, we developed a novel framework based on an embedded element method for incorporating axonal fiber tracts into the existing isotropic brain FE models. We demonstrated the applicability of this framework on an existing brain model without extra efforts on mesh refinements. The responses of the newly developed axon-based model correlated well with experimental studies under various loading cases relevant to concussion (Tables 2 and 4).

Brain Material Properties

Most of the current brain models21,22,27,63,66 utilize material properties that have been calibrated using experimental brain tissue mechanical data obtained from a single loading mode, which might not fully capture the various aspects of the complex response of human brain tissue. Moreover, large disparities in testing protocols and results reported in the literature complicate the selection of a single experimental data set that accurately represents the mechanical behavior of the brain.11 It has been shown, because of the vast variations in material properties, the resulting constitutive models will lead to large disparities in strain-based injury metrics.65 These disparities could be partly attributed to the viscoelasticity and frequency dependence of brain tissue as different studies have characterized brain tissue at different rates of deformation. To address these limitations, we have simultaneously identified a single parameter set for shear, compression, tension at a series of constant strain rates, and verified the same parameter set with shear oscillation tests across a broad range of frequencies (0.01–10,000 Hz).

Constitutive Model for Axonal Fibers

One challenge inherent to explicitly modeling axonal fiber tracts is isolating the material properties of axonal fibers since the mechanical properties of axons, and WM directional-dependence are ambiguous in the literature.49 Even if reliable axon material property data were available, these would be challenging to incorporate into the model because the stiffness contribution from the axonal fibers depends on structural features (such as cross-sectional area, element size, etc.), as well as the underlying ground substance constitutive model. Therefore, instead of calibrating the axonal fiber constitutive material model directly, the fiber properties were calibrated based on the stiffness differences between the WM and GM in this study. Because of the lack of well-characterized experimental data, it was worth noting that the constitutive model parameters obtained through the inverse FE approach might not be unique.

Individual axons in the axonal fiber tracts are not perfectly aligned but dispersed around some referential, preferred direction. In the axon-based model, these fibers were represented as ‘cable’ elements. Depending on the dispersion, the cables should have different mechanical properties. To account for the dispersion of the axons, most studies assume a probability distribution of the axons and perform a pre-integration of the distribution to achieve improved computational efficiency. The best-known model of this kind is the Holzapfel-Gasser-Ogden (HGO) model, which has been widely used for modeling anisotropic brain tissue.21,27,66 As recognized by Holzapfel and Ogden,33 the limiting issue of using the HGO model in fiber-reinforced anisotropic materials is the tension–compression switch criterion. This switch criterion is required because fibers do not support compression. Also, compressive axonal strains should not be included when using strain-based metrics to relate brain deformation to injury. The switch used by previous studies21,27 was based on an averaged structure invariant: \(E_{a}\) (denoted as ‘axonal strain’ in those studies). That is to say:

$$\left\{ {\begin{array}{*{20}c} {w_{\text{fiber}} = \frac{{k_{1} }}{{2k_{2} }}(e^{{k_{2} E_{a}^{2} }} - 1)} & {E_{a} > 0} \\ {w_{\text{fiber}} = 0} & {E_{a} \le 0} \\ \end{array} } \right.$$
(8)
$$E_{a} = \kappa \left( {I_{1} - 3} \right) + (1 - 3{\kappa })(I_{4a} - 1)$$
(9)

However, this switch can give erroneous results, as deformation states may exist for which the axon family is extended according to the averaged structure invariant (\(E_{a} > 0\)), but the fiber in the corresponding preferred direction is under compression (λ < 0). This is apparent in Fig. 7, which illustrates the dependence of \(E_{a}\) on λ for several values of κ. As shown, \(E_{a}\) > 0 does not necessarily require stretch λ > 1. In the current application, the effects of using different switches on the mechanical responses of brain tissue might not be significant (because the effect of anisotropy is small), but using \(E_{a}\), instead of the strain in the main fiber direction, as an injury metric27 might lead to erroneous results, since \(E_{a}\) cannot differentiate compressive and tensile strains.

Figure 7
figure 7

The plot of the function Ea for different dispersion values. Note that \(\varvec{E}_{\varvec{a}} > 0\) exists under compressive loading.

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We further noticed this criterion was not the original proposal of Gasser et al.24 as per Equation

$$w_{\text{fiber}} = \frac{{k_{1} }}{{2k_{2} }}\left( {e^{{k_{2} E_{a}^{2} }} - 1} \right)$$
(10)
$$\left\{ {\begin{array}{*{20}c} {E_{a} = \kappa (I_{1} - 3) + (1 - 3\kappa )(I_{4a} - 1)} & {I_{4a} > 1} \\ {E_{a} = \kappa (I_{1} - 3\kappa )} & {I_{4a} \le 1} \\ \end{array} } \right.$$
(11)

However, this original criterion was also criticized for resulting in non-physical stress discontinuities.41 To our knowledge, there is no simple correction for this issue from a constitutive perspective.34 In this study, since the fibers and ground substance were explicitly modeled, and the nonlinear behavior of the fibers was decoupled from the ground substance, the exclusion of the compressive strain and stress for these 1-D elements was straightforward and did not cause any stress discontinuities.

Embedded Elements Method

The embedding between the fibers and the ground substance was assumed to be no-slip, and the initial structure of the fiber is non-undulated. The no-slip assumption may be appropriate, as brain tissues can remain intact under large deformation (up to 50% strain53). Axonal undulation is present in some intracranial nervous tissue as a physiological adaptation, such as the optic nerve, the root of the trigeminal nerve and cranial nerves VI–XII.48 Axon tracts in most other WM regions were found to be fully coupled to the surrounding tissue, at least in porcine brain tissue (Dave Meaney, personal communication, October 13, 2017).

The embedded elements method was developed before its application in modeling soft tissue.16 In general, two issues should be considered when implementing embedded elements on a fiber-reinforced composite: the interpenetration of the contacting fibers and volume redundancy.59 In this study, the first issue was irrelevant, because physically the fibers cannot come into contact with each other unless the surrounding tissues were damaged. TBI injuries typically present without visible physical damage or gross tissue disruption.26 However, the volume redundancy issue needed special consideration to correct the resulting mass and stiffness redundancy. In the embedded element method, the ground substance occupies the full volume of the brain, including the volume under the fiber reinforcement. The addition of reinforcing axonal fibers that have finite cross-sectional area results in mass and stiffness redundancies. To resolve this volume redundancy, we artificially decreased the density of the fibers to be negligible. To address the stiffness redundancy, the constitutive contribution of the ground substance material was subtracted from the constitutive contribution of the axonal fibers.59

Effects of Anisotropy

In this study, we found the effects of anisotropy on strain responses were not substantial. Previous studies have also noted minimal effects of anisotropy on strain outcomes.63,66 However, varying conclusions exist depending on how the fiber-reinforcement term was defined. In fact, the relative stiffness contribution of the fibers defined in constitutive model approaches was very different in the literature. For example, Sahoo et al.54 concluded that the inclusion of DTI parameters (anisotropy) to the brain FE model had a significant influence on the local brain deformation. This was expected since the contribution ratio between the fiber and ground substance terms in their constitutive model was relatively large, the fiber term contributes up to 70% of the overall stiffness under 1.5 stretch at the corona radiata region. Cloots et al.,13 Giordano et al.,27 and Zhao and Ji66 reported conflicting findings on the significance of anisotropy despite using fiber material properties based on the same experimental study.49 Nevertheless, a direct comparison with these previous studies was not feasible, because they modeled the axon contribution through a fiber reinforcement term in the strain energy function (not a physical fiber in the model). In this study, the fiber contribution was not solely determined by the constitutive model but also related to the physical fiber architecture (e.g., distribution, cross-section area, numbers of cable elements).

Although the effects of anisotropy on mechanical responses were subtle, the fact that the MAS was significantly different from the MPS in values and distribution revealed the potential importance of incorporating anisotropic axonal fibers into brain FE models. For the SONO simulations using the axon-based model, MAS was significantly lower than MPS, and large strains were occurring either in non-fiber directions or non-fiber regions (e.g., cerebral cortex). If axonal damage were indeed an injury mechanism of TBI, the differences in MPS and MAS could result in different injury risk outcomes. Several studies have explored the correlation between axonal strain and TBI. Sahoo et al.55 showed that axonal strain was the most appropriate parameter for predicting DAI, based on 109 reconstructed pedestrian accident cases. Giordano et al.28 found that strain in the axonal direction was a better injury predictor than MPS for a data set of 58 mild TBI reconstructions. Nevertheless, whether the axonal strain in an FE model is a better injury predictor requires further investigation.

Limitations

This study assumed correlations between the mechanical properties of nervous tissue and its underlying microstructure. The regional dependence or mechanical heterogeneity was typically found in biomechanical tests,37 indentation tests,8 and in vivo magnetic resonance elastography.38 Fiber-rich regions like the brainstem and corona radiata were generally found to be stiffer than fiber-deficient regions such as the cortex and thalamus. However, the authors acknowledge the contrary evidence in the literature on the mechanical anisotropy of WM. Velardi et al.62 found significantly stiffer response in the fiber direction than perpendicular to it under uniaxial tests. Prange and Margulies,53 Arbogast et al.,2 Feng et al.15 and Jin et al.37 found significant anisotropy in shear but their conclusions were contradictory, and the directional dependence did not correlate well with expected fiber orientation. Pervin and Chen,51 Nicolle et al.47 and Budday et al.9 revealed no statistically significant dependencies on fiber orientation. The contradictory experimental results could be potentially due to the complexity of the microstructure in the brain, more than 80% of the white matter voxels in the HCP-842 template had more than one fiber orientation even at 2.5-mm3 resolution,64 extracting specimens that exhibit distinct fiber direction would be difficult. Understanding the directional and regional dependence of brain mechanical properties both in vitro and in vivo is still a topic of intense ongoing research and inconclusive.

Another limitation is the fundamental ambiguities inherent in tract reconstruction techniques. The group-averaged whole-brain fiber tractography was obtained from DWI and subsequent deterministic tractography analyses. Group-averaged tractography would potentially reduce random errors associated with individual fiber tracking process, but there would still be error between the tractography and the true fiber architecture. It has been shown recently that invalid bundles occur systematically across different research groups using different tract reconstruction methods when evaluated with ground truth bundles.43 The encouraging finding reported in the evaluation study43 is that the deterministic fiber tracking method used to obtain the current tractography template (HCP-842) has achieved the highest 92% valid connection among 96 methods. There were also limitations associated with the CORA objective rating system and its widespread use for validating brain FE models, particularly for brain deformation with nodal displacement–time histories.66 For example, the axon-based and baseline models yielded very similar responses for the cadaveric impacts, and these were reflected in the CORA scores. However, they differed significantly in strain outputs, and these differences were not reflected in the CORA score. This discrepancy is important as the majority of injury metrics (e.g., MPS and CSDM) are strain-based metrics.18 However, considering the low data quality of experimental strain measurement40 and mesh-sensitivity of strain output in current FE brain models,29 validation using nodal displacement–time histories instead of using experimental strain measurement is still a more robust approach for current FE models.

Overall, this study developed an anisotropic and heterogeneous brain model by explicitly incorporating axonal fibers as embedded cable elements into the previously validated brain model. The updated model demonstrated good biofidelity when simulating the latest human brain deformation data (SONO data1). The improved model will help advance our understanding of injury mechanisms and facilitate research in predicting and mitigating TBI. The framework presented here can also be generalized to include other mesoscopic anatomical details in finite element models without additional mesh generation.