## Abstract

Head rotational kinematics and tissue deformation metrics obtained from finite element models (FEM) have the potential to be used as traumatic axonal injury (TAI) assessment criteria and headgear evaluation standards. These metrics have been used to predict the likelihood of TAI occurrence; however, their ability in the assessment of the extent of TAI has not been explored. In this study, a pig model of TAI was used to examine a wide range of head loading conditions in two directions. The extent of TAI was quantified through histopathology and correlated to the FEM-derived tissue deformations and the head rotational kinematics. Peak angular acceleration and maximum strain rate of axonal fiber and brain tissue showed relatively good correlation to the volume of axonal injury, with similar correlation trends for both directions separately or combined. These rotational kinematics and tissue deformations can estimate the extent of acute TAI. The relationships between the head kinematics and the tissue strain, strain rate, and strain times strain rate were determined over the experimental range examined herein, and beyond that through parametric simulations. These relationships demonstrate that peak angular velocity and acceleration affect the underlying tissue deformations and the knowledge of both help to predict TAI risk. These relationships were combined with the injury thresholds, extracted from the TAI risk curves, and the kinematic-based risk curves representing overall axonal and brain tissue strain and strain rate were determined for predicting TAI. After scaling to humans, these curves can be used for real-time TAI assessment.

## 1 Introduction

Traumatic brain injury (TBI) is a major cause of cognitive and behavioral deficits in the U.S. and worldwide and occurs in a variety of sports incidents, falls, or automotive accidents. The main driving cause of TBI is recognized to be the brain tissue deformations and axonal stretch caused by rapid head rotations and head biomechanical loadings [1]. The axonal and brain tissue deformation responses due to head rotational kinematic loadings can be quantified by reconstruction simulations using biofidelic brain finite element models (FEM). Therefore, rotational kinematics and FEM-derived tissue deformation metrics have the potential to predict and/or assess the risk of occurrence and extent of TBI at different accidental events and thus to be used as design criteria and evaluation standards for protective headgears. An important question is that whether and how accurately these metrics can estimate the risk and extent of TBI especially in case of diffuse microscopic damage such as traumatic axonal injury (TAI). TAI is one of the common pathological features of TBI, but it is a diffuse microscopic damage and the definite determination of its location and extension is still challenging in the clinical setting. Therefore, a practical approach to answer this question is to use animal models of TBI, in which the precise location and extent of TAI can be quantified through microscopic histopathology analysis after sacrifice.

Fig. 1
Fig. 1
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In addition, these generalized relationships can be coupled with tissue injury thresholds determined from risk curves, commonly generated from animal experimental studies resulting in injury occurrence, to develop kinematic-based TBI criteria that are inspired by axonal/brain tissue deformation responses. There have been previous efforts to develop such injury tolerance criteria for severe diffuse axonal injury by linking head rotational loading conditions and brain tissue strain using analytical and physical skull models [1]. With advancements in imaging and computational modeling techniques, the biofidelity of the brain FEMs has improved, which makes the calculation of deformation response along the axonal tracts possible. In addition, many recent in vitro and in vivo studies, that examined the effect of strain and strain-rate on neuro-axonal damage, suggested that the extent of TBI is dependent not only on strain but also on strain rate. Therefore, there is a need to investigate and develop kinematic-based risk curves inspired by tissue deformation for mild TAI based on both strain and strain rate of brain tissue and axonal fibers by taking advantage of axonal tract embedding modeling technique. Such kinematic-based curves can be used for real-time TBI assessment using wearable sensor measurements as inputs.

## 2 Methods

### 2.1 Pig Traumatic Brain Injury Experiments, Head Kinematic Measurements, and Traumatic Axonal Injury Pathology Assessment.

Fig. 2
Fig. 2
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All protocols for these experiments were approved by the Institutional Animal Care and Use Committee of the University of Pennsylvania, where these experiments were previously conducted. Following the TBI experiments, pigs were sacrificed at 6 h postinjury, brains were perfusion-fixed and sectioned in coronal slices at every 3 mm. Each brain section was cut into 6 μm thick slices and stained for beta-amyloid-precursor-protein, and areas with positive axonal damage profiles were identified. To quantify the extent of acute TAI, the cumulative sum of marked axonal damage areas over all the brain sections throughout the whole brain was calculated as the axonal injury volume (AIV) for each animal expressed as a percentage of cerebral volume. The animals sustained different levels of TAI with AIV ranging from 0.02% to 1.65%, which represents levels of TBI from no/very minor with no significant behavioral or cognitive deficits, to mild TBI. In summary, peak angular velocity, peak angular acceleration, angular velocity time history, and AIV were extracted from each pig TBI experiment for further analysis.

### 2.2 Finite Element Simulation of Pig Traumatic Brain Injury Experiments and Finite Element Model-Derived Tissue Deformations.

To calculate the brain tissue and axonal fiber deformations experienced by the pig brains during these experiments, a newly developed and evaluated anisotropic multiscale axonal tract embedded pig brain FEM [17] was used to reconstruct those experiments using their measured angular velocity traces as input loading conditions. The brain deformation response obtained from this FEM was compared with the brain deformation measured in ex vivo hemisection experiment in a high strain and strain rate condition similar to the experiments used for reconstruction in this study and relatively good statistical correlation (p-value < 0.1) was observed between them [17]. This model can predict the overall presence/absence of TAI with 73–90% accuracy rate [17]. For each reconstruction simulation, the base pig brain FEM was scaled accordingly using the mass scaling approach ($λx= λy= λz=(manimalmbase)1/3)$[21]. All simulations were performed in LS-DYNA with temporal resolution of 0.1 ms. Axial strain of every axonal element and maximum principal strain (MPS) of every brain element at each time-step (0.1 ms) were extracted from each simulation. The strain rate was then calculated as the discrete derivative of the strain time history between time points for each element. Examples of spatial distribution of axonal fiber strains for an axial and a sagittal pig experiments with similar peak angular velocity conditions at six time frames throughout the whole-time window of simulations were shown in Fig. 3.

Fig. 3
Fig. 3
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For each simulation, the maximum values of stain, strain rate, and the product of strain and strain rate over the entire simulation period were calculated for each brain element and axonal fiber element. The 95th percentile maximum strain, strain rate, and strain times strain rate values, at or below which were experienced by 95% of the brain elements and axonal elements, were calculated as the overall MPS, maximum axonal strain (MAS), maximum principal strain rate (MPSR), maximum axonal strain rate (MASR), maximum principal strain times strain rate (MPSxSR), and maximum axonal strain times strain rate (MASxSR) extracted for each animal and used for further analysis. The 95th percentile maximum values were used instead of the largest (100th percentile maximum) values to eliminate any possible numerical noise.

### 2.3 Correlation Analysis.

Correlation analyses were performed between the kinematic parameters including peak angular velocity or peak angular acceleration extracted from the pig TBI experiments and the AIV from histopathology from each subject for each rotational direction separately and for both directions combined. Similar analyses were performed between the FEM-derived tissue deformation parameters including MAS, MPS, MASR, MPSR, MASxSR, and MPSxSR extracted from pig FEM simulations and AIV for this animal injury dataset. Correlation between AIV and each parameter was assessed using the correlation coefficient (R2). A power function ($y=a xb$) was employed in these analyses because this function showed higher correlation coefficients for the parameters examined than Gaussian, linear, and exponential functions. In addition, a linear surface contour was fit to the three-dimensional AIV, peak angular velocity, and peak angular acceleration results for all animals and the goodness of fit (R2) were reported.

### 2.4 Finite Element Model Parametric Simulations.

In order to quantify the possible relationships between the head angular kinematics and the axonal/brain tissue deformations beyond the experimental loadings studied herein, a series of 104 simulations per direction were performed for axial and sagittal rotations for a wide range of angular loading conditions with different values of peak angular velocity and peak angular acceleration varied from 25–400 rad/s and 25–250 krad/s2, respectively. This range was selected to cover the current and previous pig TBI experiments performed in our laboratory [1416,19], sport- or fall-related human head impact kinematics measured on-field [6,7,9] or reconstructed in laboratory [8,1013], and primate TBI experiments that were previously performed for severe diffuse axonal injury [1], all mass scaled to pig. For these parametric simulations, idealized full cycle sinusoidal angular acceleration signals were used as the loading traces. An example of these idealized traces is shown in Fig. 4(a). The head kinematic pulse durations (τ) for these parametric angular motion traces ranged from 0.6 to 100 ms. Because the parametric rotational traces were idealized full cycle sinusoidal, in which the pulse duration directly related to the peak angular velocity and acceleration values $(2πτ=peak angular accelerationpeak angular velocity)$, only peak angular velocity and acceleration were reported from these traces. The range and distribution of loading matrix selected for the parametric simulations in this study was shown in Fig. 4(b).

Fig. 4
Fig. 4
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Similar to the pig injury reconstruction simulations (described in Sec. 2.2), the rotational traces were applied to the rigid skull with the same center of rotation at a point in the neck, simulations were run longer than the rotational signals to let the brain to return to its initial undeformed state, and six deformation parameters including MAS, MASR, MASxSR, MPS, MPSR, and MPSxSR were extracted from each simulation. These FEM-derived deformation parameters were then related to the rotational kinematic parameters to establish kinematic-based tissue deformation response surface contour plots for both axial and sagittal rotational directions over wide range of loading conditions that were parametrically investigated in this study.

## 3 Results

Correlations between the kinematic parameters and sustained AIV from histopathology analysis were determined (Fig. 5). When data of the axial and sagittal directions were combined, poor correlation (R2 = 0.29) was observed between peak angular velocity and AIV (Fig. 5(a)). However, data from each of these directions separately showed fair correlation with AIV (R2 = 0.47 for sagittal and R2 = 048 for axial) and the correlation trends were dependent on the rotational direction (Fig. 5(a)). In contrary, peak angular acceleration from sagittal and axial rotations showed relatively good correlation to AIV (R2 = 0.53–0.63) with a similar trend for both directions (Fig. 5(b)), suggesting that correlation of peak angular acceleration experienced by the head and the resulting TAI may be less sensitive to the rotational directions. The multiple regression analysis for correlating the two kinematic parameters and the AIV was also performed for data from both directions combined (Fig. 5(c)). The coefficient of the peak angular acceleration parameter (0.02128) in the AIV correlation function was twenty times larger than the coefficient of the peak angular velocity (-0.00117). Power correlation analysis was also performed between FE-derived tissue deformation parameters including MAS, MPS, MASR, MPSR, MASxSR, MPSxSR, and AIV (Fig. 6). The analysis revealed relatively good correlations between MPSR and MASR with AIV (R2 = 0.48–0.56), which also were insensitive to rotational directions (Figs. 6(c) and 6(d)). On the other hand, the correlations of MPS and MAS with AIV were shown to be very different for the two rotational directions; both MPS and MAS showed high correlation with AIV for axial direction (R2 = 0.60–0.64) while no correlation was observed for sagittal direction (R2 ≤ 0.05). The correlations between MPSxSR and MASxSR with AIV (R2 = 0.36–0.57) were not as directional dependent as strain parameters and were not as directional independent as strain rate parameters.

Fig. 5
Fig. 5
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Fig. 6
Fig. 6
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The relationships between kinematic metrics and the axonal fiber and brain tissue deformation metrics in the pig TBI experiment dataset were also examined using linear regression analysis (Fig. 7). The results showed that, in the loading regimes and characteristics applied in the animal TBI experiments, the peak angular acceleration was highly correlated to MASR and MPSR (Figs. 7(e) and 7(k)) for both axial (R2 = 0.67–0.73) and sagittal (R2 = 0.92–0.96) rotations, and the trend of the correlations were similar for both rotational directions. The linear correlation between the peak angular velocity and MPS (R2 = 0.57–0.75) and MAS (R2 = 0.43–0.67) were also relatively good and the correlation trends were similar for both directions (Figs. 7(a) and 7(g)). However, peak angular acceleration showed correlation to MPS (R2 = 0.82) and MAS (R2 = 0.78) only for axial direction, but not for sagittal direction.

Fig. 7
Fig. 7
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To explore the relationships between head rotational kinematic metrics and the axonal/brain tissue metrics beyond the loading range and characteristics applied in the animal experiments, a matrix of head rotational movements over a wide range of possible loading conditions and characteristics observed in different sports and head impact accidental events were parametrically simulated, as described in Sec. 2.4, for axial and sagittal directions. It should be noted that same loading traces were used for both axial and sagittal simulations. The relationships between axonal fiber and brain tissue deformation metrics including MAS, MASR, MASxSR, MPS, MPSR, and MPSXSR and rotational kinematic metrics including peak angular acceleration and peak angular velocity were determined through surface fitting and resulted in high goodness of fit (R2 ≥ 0.99). The surface contour curves were given in Fig. 8. Each contour curve represents a constant level of a tissue deformation metric as a function of peak angular acceleration and peak angular velocity applied to the head.

Fig. 8
Fig. 8
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These kinematic-based tissue deformation contour curves were also combined with the results of the TAI risk curves that we previously developed [17] and contour curves corresponding to 10%, 25%, 50%, 75%, and 90% likelihood of sustaining TAI for both axial and sagittal directions (Fig. 9) were derived for all of the six tissue deformation metrics used in this study. The 10–90% tissue injury threshold values, extracted from the previously developed binary logistic regression TAI risk curves [17], are given in Table 1. In addition, the kinematic-based TAI risk curves at the 50% likelihood were compared between strain and strain-rate related parameters for axial and sagittal as shown in Fig. 10 (top row). These results were scaled to the human head kinematics using mass scaling approach . In some loading conditions, strain-related TAI risk curves were more conservative while in many other loading conditions the strain-rate-related TAI risk curves were more conservative. For example, at the same peak angular acceleration, strain rate-related curves were predicted TAI at lower peak angular velocity than strain-related curves.

Fig. 9
Fig. 9
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Fig. 10
Fig. 10
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Table 1

Averages of the FE-derived tissue deformation TAI thresholds derived from the 50-repetitions fivefold binary logistic regression risk curves [17]

Pig dataset10% likelihood threshold25% likelihood threshold50% likelihood threshold75% likelihood threshold90% likelihood threshold
MAS0.0890.10480.12070.13650.1524
MASR (s−1)47.0356.766.476.185.8
MASxSR (s−1)3.54.24.95.46.3
MPS0.22980.25750.28520.31290.3405
MPSR (s−1)103.49122.12140.76156.39178.03
MPSxSR (s−1)17.121.024.928.832.7
Pig dataset10% likelihood threshold25% likelihood threshold50% likelihood threshold75% likelihood threshold90% likelihood threshold
MAS0.0890.10480.12070.13650.1524
MASR (s−1)47.0356.766.476.185.8
MASxSR (s−1)3.54.24.95.46.3
MPS0.22980.25750.28520.31290.3405
MPSR (s−1)103.49122.12140.76156.39178.03
MPSxSR (s−1)17.121.024.928.832.7

## 4 Discussion

### 4.1 Correlation of Tissue Deformation Responses and Head Rotational Kinematics With the Extent of Traumatic Axonal Injury.

The TBI metrics are commonly developed based on binary injury data to assess absence or presence of TBI. However, the relatively good correlations of peak angular acceleration, MASR and MPSR with AIV observed in this study suggest that these metrics are good candidates for estimating the extent of TBI. Moreover, the correlations of the FEM-derived axonal fiber and brain tissue strain-rates and peak angular acceleration with the extent of TAI were shown to be less sensitive to rotational direction, at least in the head rotational kinematic ranges and characteristics applied in this study which, when scaled to humans, are similar to the head impact conditions measured in sports such as football and hockey. The similar trends of the correlation of these metrics to AIV for different rotational directions make them good candidates for assessment of TBI in the real-world head trauma where the head impact incidents are mostly multidirectional.

### 4.2 Generalized Relationships Between Tissue Deformation Responses and Head Rotational Kinematics.

There have been studies investigating the relationships between the brain tissue strain and head impact kinematics [14,17,2224,29] but the effect of the head kinematic characteristics on the underlying tissue strain-rate, which has been shown by many in vitro and in vivo studies [25] to highly affect the extent of neuro-axonal injury, has not been yet determined. In this study, we experimentally and parametrically investigated and demonstrated the relationships between head angular velocity and acceleration to strain, strain rate, and product of strain and strain rate of axonal fiber and brain tissue over a wide range of possible head kinematic conditions. The results, shown in Figs. 7 and 8, illustrated that both head angular velocity and acceleration affect the underlying deformation responses (strain and strain-rate related parameters) of the axonal fiber bundles and brain tissue and thus the knowledge of both of these kinematic metrics can help to better predict the risk of brain injury at different head loading conditions.

### 4.3 Tissue Deformation Inspired Head Kinematic-Based Traumatic Axonal Injury Risk Curves.

Although the magnitude and rate of axonal and brain tissue deformations have been shown to be the leading cause of TBI, the TBI risk metrics are commonly based on head kinematics, as these metrics have the advantage of low computational cost and capability of real-time assessment of potential injury. In this study, the generalized tissue deformation surface contours were combined with the tissue injury thresholds extracted from the traditional risk curves, recently developed using the same dataset and the anisotropic axonal tract embedded brain FEM for predicting likelihood of TAI occurrence [17]. Then the kinematic-based risk curves representing overall axonal and brain tissue strain, strain rate, and strain times strain rate were determined for 10% to 90% likelihood of sustaining TAI for pigs. These curves were then scaled to human kinematics.

## Acknowledgment

The views expressed are solely those of the authors and do not represent those of any funding sources or their affiliates.

## Funding Data

• Biomechanics Consulting and Research, LLC (BioCore, LLC).

• National Institutes of Health (R01NS097549 and R56NS055951; Funder ID: 10.13039/100000002).

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