Multi-Directional Dynamic Model for Traumatic Brain Injury Detection

Abstract

Given the worldwide adverse impact of traumatic brain injury (TBI) on the human population, its diagnosis and prediction are of utmost importance. Historically, many studies have focused on associating head kinematics to brain injury risk. Recently, there has been a push toward using computationally expensive finite element (FE) models of the brain to create tissue deformation metrics of brain injury. Here, we develop a new brain injury metric, the brain angle metric (BAM), based on the dynamics of a 3 degree-of-freedom lumped parameter brain model. The brain model is built based on the measured natural frequencies of an FE brain model simulated with live human impact data. We show that it can be used to rapidly estimate peak brain strains experienced during head rotational accelerations that cause mild TBI. In our data set, the simplified model correlates with peak principal FE strain (R² = 0.82). Further, coronal and axial brain model displacement correlated with fiber-oriented peak strain in the corpus callosum (R² = 0.77). Our proposed injury metric BAM uses the maximum angle predicted by our brain model and is compared against a number of existing rotational and translational kinematic injury metrics on a data set of head kinematics from 27 clinically diagnosed injuries and 887 non-injuries. We found that BAM performed comparably to peak angular acceleration, translational acceleration, and angular velocity in classifying injury and non-injury events. Metrics that separated time traces into their directional components had improved model deviance compare with those that combined components into a single time trace magnitude. Our brain model can be used in future work to rapidly approximate the peak strain resulting from mild to moderate head impacts and to quickly assess brain injury risk.

Introduction

Mild traumatic brain injury (mTBI), or concussion, has received heightened awareness because of its adverse effects not only on professional athletes and military personnel, but more broadly, on the general public. Aside from acute neurocognitive deficits, mounting evidence suggests increased risk of chronic neurodegeneration with repeated mTBI.¹ There have been multiple reports of contact athletes and service veterans experiencing memory loss, behavioral changes, and motor function abnormalities later in life.² In severe cases, retired professional football players in their middle years have shown extreme changes in personality and suicidal tendencies.³ Return to play guidelines and legislations have been introduced to protect athletes from repeat trauma and to reduce the risk of long-term brain damage.⁴ However, there is low compliance because of under-reporting of injuries.^5
–7 Despite increased awareness of mTBI, timely diagnosis and prevention of this injury is difficult because of a lack of understanding of injury mechanisms.

In the mid-twentieth century, rising motor vehicle and sporting deaths led to the establishment of safety standards that targeted reduction of forces that deform or fracture the skull. In the 1960s, the Wayne State University tolerance threshold⁸ was developed, motivating two translational acceleration standards: the Gadd Severity Index (SI)⁹ and Head Injury Criterion (HIC).¹⁰ The two biggest regulating bodies for enforcing safety standards, the National Highway Traffic Safety Administration (NHTSA) and the National Operating Committee on Standards for Athletic Equipment (NOCSAE), still evaluate injury risk based on translational acceleration. NHTSA uses a metric based on the time history of translational head acceleration as the only federally mandated brain injury metric in automobile safety regulation,¹⁰ whereas NOCSAE uses a maximum resultant translational acceleration criterion to evaluate helmet design.¹¹ Despite widespread use of translation-based metrics to predict brain injury, some question whether they are suitable for assessing all types of brain injury.¹²

Diffuse brain injury, which can occur through purely inertial head acceleration even in the absence of skull deformation, has become better understood since the development of HIC and SI. Rapid head rotations can shear and deform the white matter of the brain causing diffuse axonal injury (DAI),¹³ a fundamentally different injury than focal injuries caused by skull deformation, which are primarily caused by translational motions,^12,14 Indeed, although head translational acceleration is an important factor for focal trauma such as skull fracture, rotational acceleration causes time-dependent inertial loading of the brain and better correlates with brain trauma severity in animal experiments.^15
–17 More recently, NHTSA developed the brain injury criterion (BrIC) to predict TBI risk by relating head rotational velocity to critical brain strains, and this criterion has been proposed to be used in the New Car Assessment Programs rating.¹⁸ Realizing that both translational and rotational head kinematics may factor into injury risk, some 6 degree-of-freedom (DOF) criteria, such as head impact power (HIP)¹⁹ and generalized acceleration model for brain injury threshold (GAMBIT),²⁰ have also been developed, which include both translational and rotational components of acceleration. Brain tissue deformation metrics calculated from finite element (FE) simulations are another category of frequently investigated injury criteria. Using these FE models, researchers simulate metrics such as tissue strain and strain rate to predict injury risk. Morphologically based metrics such as fiber tract-oriented strain, which better accounts for anisotropy in brain tissue, have also been found to correlate with injury.^21,22 Although FE model-derived criteria provide more physical intuition behind injury risk prediction than skull-kinematics-based criteria, they are severely limited by long computational running time and may not be practical options as federal standards (Table 1). ^23

–29

Table 1.

Kinematics Expressed in Mean and Standard Deviation Values from Different Data Sources

Authors	Year	Description	Sample size		Lin. Accel. (g)			Rot. Accel. (rad/s²)			Rot. Vel (rad/s)
			(injured)	Lateral	Ant-Post	Inf-Sup	Coronal	Sagittal	Axial	Coronal	Sagittal	Axial
Wu et al.²³	2016	Collegiate Football	139 (0)	15.4 ± 9.6	17.0 ± 14.3	11.1 ± 9.3	1464.8 ± 1940.5	830.9 ± 1015.7	664.3 ± 483.2	7.5 ± 5.4	7.6 ± 4.1	6.5 ± 4.0
Hernandez et al.²⁴	2014	Collegiate Football	537 (2)	16.1 ± 13.5	12.3 ± 10.6	13.8 ± 15.4	670.7 ± 872.5	1091.9 ± 1403.1	543.6 ± 487.1	6.4 ± 4.5	9.3 ± 6.8	6.5 ± 4.4
Sanchez et al.²⁵	2003	NFL Football	53 (20)	53.9 ± 33.2	35.2 ± 18.9	29.4 ± 13.4	2593.5 ± 1734.0	1890.4 ± 1088.1	2202.9 ± 1643.1	24.8 ± 11.3	12.6 ± 6.7	16.6 ± 12.7
Supplementary Materials	2018	High school football	1 (1)	12.2	5.9	31.2	2189	4922	1651	48.9	96.2	66.5
Ewing et al.²⁶	1976	Navy sled tests	51 (0)	18.2 ± 1.3	17.9 ± 1.2	17.9 ± 1.2	827.8 ± 365.6	199.9 ± 0.1	199.9 ± 0.1	16.5 ± 8.0	1.2 ± .04	1.3 ± 0.6
Ewing et al.²⁷	1975	Navy sled tests	100 (0)	18.3 ± 1.8	18.1 ± 1.3	17.9 ± 0.8	199.9 ± 0.1	773.6 ± 349.6	199.9 ± 0.1	1.1 ± 0.5	15.5 ± 7.6	1.0 ± 0.4
Hernandez and Camarillo²⁸	2019	Voluntary motion	29 (0)	25.0 ± 10.5	32.3 ± 9.2	21.6 ± 9.7	103.1 ± 51.4	54.9 ± 23.1	211.4 ± 121.2	4.8 ± 3.9	2.0 ± 1.0	11.3 ± 7.1
OKeeffe et al.²⁹	2019	Mixed martial arts	4 (4)	106.4 ± 82	50.2 ± 44	116.7 ± 94	5381 ± 2973	11522 ± 8030	9969 ± 9153	15.0 ± 3.6	36.0 ± 26	19.8 ± 14

Simplified mechanical models of the brain have been developed and used to gain insight into brain tissue deformation since the 1950s, when Kornhauser first investigated the sensitivity of mass-spring systems to transient accelerations in the context of brain injury.^30,31 Since then, both rotational and translational lumped-parameter brain models have been developed to enable better understanding of how the brain deforms under head acceleration; however, many of these models were not effectively validated because of lack of experimental data.^32
–34 More recently, with the development of validated FE brain models, there have been efforts toward developing simplified, linear mechanical analogs of these brain FE models. Gabler and coworkers developed a mass-spring-damper lumped model of the brain, fitting parameters to match mass displacement to maximum principal strain from brain FE models over a range of idealized force profiles applied to the skull.³⁵ Gabler and coworkers found that the peak principal strain from FE models can be adequately reproduced using simple mechanical systems.

To identify promising injury criteria for predicting human injury risks, the ideal approach is to compare the performance of all candidate injury criteria with a large human injury and non-injury data set. However, only a small number of studies have compared different injury criteria using a common data set. In our previous study,²⁴ injury criteria were evaluated using a 6 DOF human injury data set containing two injuries. These results helped provide insight into promising injury criteria, showing the importance of rotational measurements for predicting injury. However, the scarcity of full 6 DOF human injury data has hindered the ability to make statistically significant comparisons of all injury criteria. In addition, because of the relative infrequency of concussions compared with non-concussive impacts, existing data sets are typically biased and injury functions are developed using similar numbers of injuries and non-injuries,³⁶ with few studies considering injury risk variations with sampling variability and sampling bias.³⁷

In this article, we developed a 3 DOF, mass-spring-damper model of the brain, with parameters based on modal analysis of brain displacements caused by real-world impacts,³⁸ contrary to previous approaches of fitting lumped model parameters to FE simulation results using idealized pulses.^35,39 Using our lumped parameter model, we present the brain angle metric (BAM), a metric for classifying between injurious and non-injurious impacts. We compare BAM to several existing injury criteria, using a combined 6 DOF human head kinematics data set from multiple loading regimes.

Methods

Head kinematics data sets

In this study, we included human male injury and non-injury data sets from multiple loading regimes, as listed in Table 1. We adopted an inclusion criterion in which for each data point either (1) all 6 DOF head kinematics were available, or (2) the head motion protocol was mainly constrained to a single plane and the kinematical measurements in that plane were available. For case 2, all head motion outside of the single recorded plane was set to random noise with magnitude <10% of the peak value in the recorded plane of motion to account for small out of plane movements. Using these criteria, we included data from the following experiments. Wu and coworkers measured head impact kinematics during football practice and game events with instrumented mouthguards (n_injury = 0, n_noninjury = 139).²³ Hernandez and coworkers measured head impact kinematics during athletic events using instrumented mouthguards (n_injury = 2, n_noninjury = 535).²⁴ More recently, a single concussive impact was recorded using the Stanford Mouthguard (MG)in high school football using the same measurement protocols as described by Hernandez and coworkers²⁴ (n_injury = 1, n_noninjury = 0). Together, these three data sets constitute the Stanford MG. Hernandez and coworkers measured head kinematics during rapid voluntary head rotations using instrumented mouthguards (n_injury = 0, n_noninjury = 29).²⁸ O'Keeffe and coworkers measured head kinematics from four mixed martial arts (MMA) fighters who received clinically diagnosed concussions (n_injury = 4, n_noninjury = 0).²⁹ Pellman and coworkers reconstructed injury and non-injury National Football League (NFL) head impacts by video analysis and dummy models.³⁶ However, the data from this original publication were found to be erroneous because of a faulty accelerometer, and have since been reanalyzed and corrected by Sanchez and coworkers (n_injury = 20, n_noninjury = 33).²⁵ All the 6 DOF measurements for the head kinematic data mentioned so far were available. In addition, we included the following three studies, in which the head motion was mainly constrained to a single plane. Ewing and coworkers performed non-injury sled tests on Navy volunteers in the coronal and sagittal directions (n_injury = 0, n_noninjury = 151).^26,27,40 In total, we have 27 injury data points and 887 non-injury data points. Injury cases were defined to be cases in which there was a clinical diagnosis of concussion from a physician. For more information on head kinematics measurements, refer to Supplementary Materials.

All the previously described data sets contain measurements of translational acceleration and rotational velocity. The Wu and Hernandez athletic data were recorded at 1000 Hz for a duration of 100 ms. The Hernandez voluntary motion data were recorded at 1000 Hz for 500 ms. The O'Keefe data were recorded at 1000 Hz for a duration of 200 ms. The sampling frequency of the Navy volunteer data and NFL reconstruction data were not reported, and the duration varied between 50 and 300 ms. All the data were projected to the center of gravity of the head and rotated to anatomical axes (x - anterior/posterior translation and coronal rotation, y - left/right translation and sagittal rotation, z - inferior/superior translation and axial rotation). The translational acceleration data were filtered at the CFC180 filter of 300 Hz, and the angular velocity data were filtered at the lowest sensor bandwidth of 184 Hz.⁴¹

Brain FE modeling

To calculate local brain tissue deformations resulting from head impacts, we simulated head impacts using a validated FE head model developed at the KTH Royal Institute of Technology (Stockholm, Sweden), which represents an average adult male human head⁴² (Fig. 1A). Because of the computational cost of running FE simulations, we only simulated a subset of the American football head impacts with higher kinematics. All impacts from the NFL data set were simulated. Within the Stanford MG data set, we simulated a total of 188 cases using our FE model, including all the impacts resulting in a clinically diagnosed concussion. Further, all impacts in which the peak value of at least one translational or rotational component exceeded that of any of the three clinically diagnosed concussions recorded by the Stanford MG were simulated, along with a random sample of 10% of the remaining impacts. The FE-simulated impacts were thus biased toward higher severity impacts that would be most difficult to classify for a machine learning classifier. For these simulations, we used the measurements of skull translational accelerations and rotational velocities as input to the model and simulated the entire duration of the impact. From the simulations, we computed two commonly used deformation metrics: peak principal strain in the brain and 15% cumulative strain damage metric (CSDM). Peak principal strain is the maximum strain among any element over the entire time trace. CSDM represents the cumulative volume of the brain matter experiencing strains over a critical level of 15%. Lastly, we calculated the peak axonal strain in the corpus callosum, as it has been suggested that the strain along the axonal fiber tracts may correlate well with injury risk (Fig. 1B).⁴³ To do this, we projected the tissue strain in the brain along the fiber tract directions,⁴³ and took the maximum value experienced within the corpus callosum.

FIG. 1.

(A) Schematic view of the isotropic KTH finite element (FE) model, (B) FE strain can be projected along axonal fiber tracts taken from diffusion tensor imaging (DTI), (C) Schematic of lumped model with discrete elements connecting the brain to the skull. Color image is available online.

Lumped parameter brain model

Although brain tissue and the brain–skull interface exhibit non-linear viscoelastic behavior,^{21,42,44
–46} many studies have simplified this complex relationship through linear mechanical elements.^{31

–35,39,47,48} Here, we developed a new lumped parameter brain model by creating a 3 DOF mechanical analog of the brain as follows: we assumed a rigid-body behavior for the brain's motion^47,48 in three anatomical directions and therefore used three separate spring-damper systems attached to the mass of the brain (Fig. 1C). This mechanical mass-spring-damper system models the rotational deformation of the brain from skull loading. To model accelerations of the skull, an input was applied as an excitation to the base of the system. The equations of motion for this system are as follows: $I ({\ddot{θ}}_{b r a i n} + {\ddot{θ}}_{s k u l l}) = - k θ_{b r a i n} - c {\dot{θ}}_{b r a i n}$

where I is the moment of inertia of the mass, k and c are the stiffness and damping values of the system, and $θ_{b r a i n}$ and $θ_{s k u l l}$ represent the angles of the brain (the mass) and the skull (the base). We assigned previously reported moments of inertia and dynamics parameters (dominant frequency [ω_x , ω_y , ω_z ] and decay rate [λ_x , λ_y , λ_z ]) in each anatomical direction.³⁸ We used inertia values for a 50th percentile male brain of I = (0.016, 0.024, 0.022) kg² for coronal, sagittal, and axial directions, respectively,²⁴ and derived the corresponding spring and damper coefficients using the following relationships: $ω_{n} = \sqrt{\frac{k}{I}}, λ_{x, y, z} = - \frac{c_{x, y, z} ω_{n}}{2 \sqrt{k_{x, y, z} I_{x, y, z}}}, ω_{x, y, z} = ω_{n} \sqrt{1 - {(\frac{λ_{x, y, z}}{ω_{n}})}^{2}}$

Table 2 lists model parameters in each anatomical direction. We simulated each impact using the lumped parameter model by applying the angular skull kinematics to the base of the mass spring damper system. For each impact, measured time traces in each anatomical direction were applied to the corresponding mass-spring-damper model. This resulted in a vector of the three peak relative brain angle values in each direction ( ${\vec{θ}}_{b r a i n}$ ).

Table 2.

Parameters Used in Mass-Spring-Damper Brain Model

Anatomical direction	x	y	z
Moment of inertia (kg²)	0.016	0.024	0.022
Decay rate (1/s)	−32	−38	−30
Natural frequency (Hz)	22	22	25
Spring stiffness (Nm/rad)	322.1	493.2	562.6
Damper viscosity (Nms/rad)	1.024	1.824	1.320

With the tissue deformation metrics calculated from the FE simulations of the KTH model brain, we ran a linear regression between the maximum resultant brain angle ( $θ_{b r a i n}^{r}$ ) and peak principal strain, and between $θ_{b r a i n}^{r}$ and CSDM. Additionally, we ran a multi-dimensional linear regression of maximum ${\vec{θ}}_{b r a i n}$ in each direction with tract-oriented corpus callosum strain.

Statistical analyses

Having shown the correlation of the brain angle measures with local tissue and axon deformations, we used the maximum brain angle ( ${\vec{θ}}_{b r a i n}$ ) in each anatomical direction as a new brain injury metric, the BAM. Further, we compared our BAM metric's performance against other existing injury criteria. Using the kinematics data, we computed the following rotational kinematics-based injury criteria: peak rotational acceleration in each direction ( $\vec{α}$ ), peak resultant rotational acceleration ( $α_{r}$ ), peak change in rotational velocity in each direction ( $Δ \vec{ω}$ ), peak resultant change in rotational velocity ( $Δ ω_{r}$ ), brain injury criterion (BrIC), and rotational injury criterion (RIC). We also computed translational kinematics-based injury criteria: peak translational acceleration in each direction ( $\vec{a}$ ), peak resultant translational acceleration (a_r ), HIC₁₅ , HIC₃₆ , and Gadd SI. Lastly, we included injury criteria that take into account both rotation and translation: the 6 DOF HIP, HIP separated into each direction (HIP_3D ), GAMBIT, and the Virginia Tech combined probability metric (VTCP). For more detailed information about each criterion, refer to the Supplementary Materials.

Because the injury and non-injury data likely fall in a binomial distribution, we fit a logistical regression model for each kinematic injury criteria on our full data set of injuries and non-injuries. In order to understand the ability of the BAM metric to predict injuries compared with the FE results, we also fit a logistical regression model on a smaller subset of just the football head impacts for strain-based criteria. We fit the logistical model to each injury criterion using the following equation, $p_{i n j u r y} = {(1 + e^{- β_{0} - \sum β_{i} x_{i}})}^{- 1}$ , where p_injury is the probability of injury, x_i are the components of the injury criterion, and β_i are the fitted coefficients, with i = 1, 2, 3 representing the anatomical directions. The following performance measures were computed and compared to assess the predictive value of each injury criterion. The deviance (D) statistic given by $D = - 2 ln (\frac{l i k e l i h o o d o f t h e f i t t e d m o d e l}{l i k e l i h o o d o f s a t u r a t e d m o d e l})$ assesses the quality of fit of a logistical regression (analogous to R² in linear regression) and has been used to assess mTBI prediction (also known as −2LLR).^19,36,49 For each model, we calculated the difference in deviance between the model and the null model (prediction using only the intercept term). The receiver operating curve (ROC) is created by plotting the true positive rate (TPR, sensitivity) against the false positive rate (FPR, 1 precision) at various threshold settings. The area under the ROC curve (AUC_ROC) is a measure of how well a binary classifier, based on the logistical fit, separates the two classes of events, with an AUC_ROC of <0.5 being worse than random guessing. In addition, we computed the area under the precision recall (PR) curve (AUC_PR), which plots precision over recall (sensitivity). Precision is the percentage of true positive detections in all positive detections and is a good measure of the classifier's performance in highly imbalanced data sets, as it is not affected by the imbalanced sample proportions. Confidence intervals on AUC metrics were computed empirically with 1000 bootstrap replicas. Using the method outlined by Delong and coworkers,⁵⁰ we compared the AUC_ROC for each metric against the ${\vec{θ}}_{b r a i n}$ criteria to test for statistical significance. In all statistical analyses, we accounted for multiple comparisons using the Bonferroni⁵¹ correction method. In this method, the standard significance value of 0.05 is divided by the total number of comparisons, setting our cutoff value to be p < 0.004.

In classifying rare events, in which the rate of incidence in a certain class is disproportionally smaller than the other class or classes, the maximum likelihood estimation of the logistical model suffers from small sample bias. The degree of bias is strongly dependent on the number of cases in the less frequent of the two categories. Reported concussions in sports occur at a rate of close to 5.5 cases per 1000 head impacts,⁵² which is by definition a rare event. This needs to be taken into account when performing statistical analysis such as logistical regression. In addition, a majority of previously published injury criteria from injury and non-injury events have been based on severely skewed data sources, meaning that in such analysis, the percentage of injury-inducing head impacts is much greater than the actual incidence rate.⁵² We used a formal approach to address these two challenges. We applied prior correction and bias correction methods proposed by King and Zeng for logistical regression analysis, using the ReLogit package in R.⁵³

Results

We included a total of 914 head kinematics including 27 clinically diagnosed brain injuries, from American football, boxing, MMA, sled tests, and rapid voluntary head motions. The translational and rotational accelerations as well as rotational velocities are shown in Figure 2A–C. We also present the incidence of each kinematic measure in the histogram plots in Figure 3.

FIG. 2.

Loading regimes: distribution linear and rotational kinematics in each anatomical plane. The filled circles represent non-concussive events whereas open circles represent concussive events. Color image is available online.

FIG. 3.

Loading regimes: histogram distribution of head kinematics and comparison of directional kinematics in (A) linear acceleration, (B) rotational acceleration, and (C) rotational velocity. Color image is available online.

We used three strain-based FE metrics to compare against the output of our lumped-parameter model, ${\vec{θ}}_{b r a i n}$ : peak principal strain ( $θ_{m a x}$ ) and CSDM15, which are common metrics in the injury biomechanics field (Fig. 4A,B).^42,54 We also used strain in the axonal fiber directions, a relatively new metric that has been shown to be a better predictor of microstructural damage to the brain tissue and may be appropriate for the analysis of mild TBI (Fig. 4C).^21,55 ${\vec{θ}}_{b r a i n}$ follows the $θ_{m a x}$ trends closely. The linear fit begins to deviate at higher levels of strain (> 30%), indicating less accurate approximations by the lumped parameter model in more severe head motions. When analyzing peak axonal strain in the corpus callosum, we first ran a three-dimensional (3D) linear regression against the peak brain displacement in each plane of motion and found that the sagittal brain displacement had no significant correlation with strain in the corpus callosum. Rerunning the linear regression while excluding the sagittal brain angle direction, we found that our mechanical model was well correlated with axonal strains in the corpus callosum. For all further analyses, we used the results from the 3D lumped-parameter model because of the computational cost of FE simulation of all the events.

FIG. 4.

Performance of the rigid brain model: comparison of the maximum relative brain displacements ( ${\vec{θ}}_{b r a i n}$ ) predicted from the lumped model against finite element results. (A) Peak principal strain, (B) 15% cumulative strain damage measure (CSDM), (C) peak axonal strain in the corpus callosum. For peak principal strain and CSDM, the maximum resultant brain angle was used in linear regression fit. For peak axonal strain in the corpus callosum, a three dimensional linear regression was run against the peak brain displacement in each plane of motion. It was found that sagittal plane had no significant correlation with peak axonal strain (p = 0.6), and therefore was removed from linear regression, such that only coronal and axial brain angles were used. Color image is available online.

Next, we compared the performance of the kinematic injury metrics with that of BAM, our metric based on the ${\vec{θ}}_{b r a i n}$ output of our lumped-parameter model, in classifying the clinical injury diagnosis of each event. We hypothesized that because brain-skull is a dynamic system, a good predictor needs to take into account not only the peak kinematics of the head motion, but also the internal dynamics of skull and brain to account for any lag of the brain's mass. The results are shown in Figures 5 and 6, as well as Figure S1 in the Supplementary Materials. In Figure 5, the deviance of each model fit is plotted, with a lower deviance indicating a better model fit. All metrics performed significantly better than the null model (a fit using a single intercept value), as denoted with asterisks. BAM had among the lowest deviances, performing comparably to $\vec{α}$ , $\vec{a}$ , BRIC, and $Δ \vec{ω}$ metrics. In Figure 6, we compare the precision recall curve of BAM with three commonly-used kinematic metrics: BrIC, HIC, and SI. Further, we show the four logistical regression functions with injuries and non-injuries from each data source. The rotational metrics (BrIC and BAM) had much higher sensitivity, precision and AUC_PR than metrics that rely on translational acceleration magnitude (HIC₁₅ and SI). The AUC_ROC and AUC_PR for all models are also shown in Figure S1A and S1B in the Supplementary Materials. BAM was among the metrics with AUC_PR > 0.70, with the others being $\vec{α}, Δ \vec{ω}$ , and BrIC metrics. Further, BAM shows statistically significantly larger AUC_ROC over RIC, SI, and HIP, with comparable AUC_ROC to $\vec{α}, Δ \vec{ω}$ , and BrIC metrics. Vector metrics $\vec{α}$ , $Δ \vec{ω}$ , and $\vec{a}$ had lower deviance and higher AUC_ROC and AUC_PR than the kinematic metrics based on resultant traces: $α_{r}$ , $Δ ω_{r}$ , and $a_{r} .$

FIG. 5.

Deviance is a statistical measure that quantifies the “goodness-of-fit” of each logistical regression, with lower values indicating a better fit. Brain angle metric (BAM) is among the metrics with the lowest deviance. All models were statistically significant when compared against the null model, as denoted by an asterisk. Logistical regression coefficients for each metric are listed in Table S1 in Supplementary Materials. Color image is available online.

FIG. 6.

Logistical regression results comparing brain angle metric (BAM) with three different commonly used metrics: brain injury criterion (BrIC), head injury criterion (HIC)₁₅, and severity index (SI). (A) Precision-recall curves for each metric quantitatively measure the performance of each classifier in the trade-off between precision and recall at various thresholds in the logistical function. The area under the curve (AUC)_PR summarizes performance, with higher values indicating increasing model accuracy. (B–E) The logistical function of each classifier is plotted against $β^{t} x$ where $β$ are the coefficients of the logistical regression fit and x is the injury criteria vector. Injuries and non-injuries are superimposed on each curve. Color image is available online.

To understand how our proposed metric compared with FE metrics, we compared its performance to $θ_{m a x}$ on the subset of the 169 simulated football head impacts. Specifically, we fit logistical regression models to $θ_{m a x}$ , $θ_{b r a i n}^{r}$ (max resultant brain angle), and BAM ( ${\vec{θ}}_{b r a i n}$ , the maximum brain angle in each direction). On this smaller data set, $θ_{m a x}$ had a deviance of 98.64, AUC_ROC of 0.91, and AUC_PR of 0.65, whereas $θ_{b r a i n}^{r}$ had a deviance of 93.10, AUC_ROC of 0.92, and AUC_PR of 0.74. BAM had a deviance of 64.93, AUC_ROC of 0.96, and AUC_PR of 0.88.

Note that prior correction and bias correction have been applied to adjust for the sample proportion bias in consideration of real-world concussion incidence rates. We found that these analyses made a difference in the fitted logistical parameters. For example, in the case of BAM, the beta coefficients in the logistical regression change from β_original = (8.288, −29.59, −10.35, −39.38) to β_corrected = (9.845, −28.86, −10.23, −37.94). Beta coefficients for all model logistic regression fits are shown in the Supplementary Materials in Table S1.

Using BAM, we developed a risk curve to classify the injury and non-injury events. The results are given in Figure 7, where 50% risk of concussion is signified by the green plane. Critical brain angle values (axes intercept) for 50% injury risk correspond to 0.34 rad in the coronal direction, 0.26 rad in the axial direction, and 0.96 rad in the sagittal direction, although there were not enough injury data in the sagittal direction for this critical value to be statistically significant. In the case of classifying concussions, a high sensitivity classifier is desirable to minimize the number of false negatives. The classifier hyper-plane can be tuned to different sensitivity levels by adjusting the classification threshold. To obtain 50% sensitivity with the BAM, coronal, sagittal, and axial critical values are 0.29, 0.83, and 0.22; to obtain 90% sensitivity, the critical values are 0.17, 0.49, and 0.13. Risk curves for $\vec{a}$ and $Δ \vec{ω}$ are shown in the Supplementary Materials Section in Figure S2.

FIG. 7.

Logistical regression results: classifier planes based on brain angle metric (BAM), separating injured from non-injured data points. For illustration purposes, we show a 50% risk injury plane. Critical brain angle values for 50% injury risk correspond to 0.34 rad in the coronal direction, 0.96 rad in the sagittal direction, and 0.26 rad in the axial direction. It is of note that there were not enough injury data in the sagittal direction to determine a statistically significant critical brain angle in the sagittal direction, and with further data this critical value would be expected to be lower. The 50% injury risk plane, if used as a classifier, would have a sensitivity of 29.6%, a specificity of 99.9%, an accuracy of 97.8%, and precision of 88.9%. The classifier hyper-plane can be tuned to different sensitivity levels by adjusting the classification threshold. The coronal, sagittal, and axial critical values that correspond to a 50% sensitivity value are 0.29, 0.83, and 0.22; those that correspond to a 90% sensitivity value are 0.17, 0.49, and 0.13. Color image is available online.

Discussion

In pursuit of brain injury mitigation technology, national safety standards are being researched and evaluated to obtain new insights into the mechanisms of brain injury. However, standards have yet to converge to an injury criterion that encompasses understanding of both the kinematics of head motion and the dynamics of the brain. We hypothesized that there is a need to understand and incorporate the dynamic response of the skull-brain system into a chosen injury predictor. Brain deformation-based criteria calculated from finite element models have been found to provide more physical insight into brain injury than do kinematics criteria, but they are not practical for rapid injury prediction because of the computational complexity. Further, promising injury criterion may necessitate sensitivities to different anatomical planes.

In this article, we tested a number of different injury predictors on a large data set of 6 DOF human injury and non-injury head kinematic data from different loading regimes. We presented an injury predictor based on a multi-directional, mechanical model of brain deformation, taking into account the dynamics of the brain (time history of loading). We found that our proposed metric, BAM, performed similarly to peak angular acceleration ( $\vec{α}$ ), translational acceleration ( $\vec{a}$ ), BrIC, and peak change in rotational velocity ( $Δ \vec{ω}$ ) in its logistical regression deviance, AUC_PR, and AUC_ROC, but outperformed the other existing metrics based on both rotational and translational kinematics. Injury predictors that considered each anatomical direction separately performed better than metrics that did not. For example, vector metrics $\vec{α}$ , $Δ \vec{ω}$ , and $\vec{a}$ had better performance than the kinematic metrics based on resultant traces: $α_{r}$ , $Δ ω_{r}$ , and $a_{r} .$ Further, metrics such as HIC and SI, which analyze translational acceleration magnitudes, performed with lower sensitivity than those that treated each direction separately. The VTCP takes into account both peak rotational and translational accelerations, and has been used extensively to predict brain injury risk and evaluate bicycle, hockey, and football helmets.^56
–58 However, this metric had higher model deviance and lower AUC_PR and AUC_ROC than metrics that treated each anatomical direction separately, suggesting that its value in predicting injury would be substantially improved if it accounted for different anatomical directions rather than using the acceleration time trace magnitudes. In addition, although BrIC takes into account the peak angular velocity in three directions of motion, its logistical regression had higher deviance than the peak $Δ \vec{ω}$ metric. This suggests that the critical values used in calculating BrIC could be further tuned to better predict the injuries in American football and MMA used in our data set.

Peak translational acceleration ( $\vec{a}$ ) had lower deviance, AUC_PR, and AUC_ROC than peak angular kinematics and BAM, but still outperformed many other metrics. Although many previous studies suggest that rotation is a primary cause of brain injury,^16,17,59 the results shown here indicate that translational acceleration still has predictive value in classifying brain injuries. This could be because in our data set, with the majority of injuries taken from laboratory reconstruction data, the translational and angular acceleration values may be coupled more than in vivo data. However, single variate injury criteria based on resultant translational acceleration had extremely low sensitivity in our data set. We saw that both HIC₁₅ and SI predicted only a single event with a >50% risk of injury on our data set because of a few non-injury events with high HIC₁₅ and SI values.

The ${\vec{θ}}_{b r a i n}$ output of our brain model was compared against peak principal strain, CSDM, and peak axonal strain extracted from finite element simulations of head impacts (Fig. 4). We showed that our simplified lumped-parameter model correlated with peak principal strain with an R² value of 0.82, suggesting that our mechanical model can capture the complex dynamics of an FE model and supporting the results found by Gabler and coworkers.³⁵ Further, we found that peak tract-oriented strain in the corpus callosum was well correlated with coronal and axial brain displacement from our mechanical model, but not from sagittal brain rotation. Intuitively this makes sense, as the axonal tracts within the corpus callosum run laterally,⁵⁵ and should only be strained from coronal or axial rotations. In fact, these results suggest that more complex tissue-deformation metrics, such as axonal tract-oriented strain, could correlate to simplified mechanical models if the tract direction is known for specific regions of interest. Whereas other kinematic metrics provide only a single non-dimensional output, BAM is correlated to physical deformations in the brain at lower strain levels. However, comparing the predictive value of BAM and $θ_{m a x}$ , we found that the BAM logistical fit had lower deviance, and larger AUC_ROC and AUC_PR, suggesting that it can more accurately classify injury than using maximum principal strain alone for the data set evaluated here. Although the correlation between $θ_{m a x}$ and ${\vec{θ}}_{b r a i n}$ may deviate at severe impacts (> 30% strain), we showed that the primary purpose of our brain model, to classify injury through BAM, remains valid and perhaps even more accurate than traditional finite element metrics, suggesting that this deviation in correlation does not negatively affect injury classification. However, it should be noted that FE models provide more complete and detailed information about tissue response of the brain to impact, giving temporal and spatial information about stresses, strains, and strain rates, whereas BAM reduces this information into a single direction-dependent value. The presented lumped-parameter model is not meant to be a replacement of an in-depth analysis of individual impact cases, but rather a tool to rapidly estimate brain injury risk.

We hypothesized that simulating the dynamics of the brain would improve brain injury classification. However, on our data set, BAM showed no statistically significant differences from peak angular acceleration ( $\vec{α}$ ), BrIC, and peak change in rotational velocity ( $Δ \vec{ω}$ ) in AUC_ROC or model deviance. This is likely because the majority of our injuries within our data set are from American football, where both angular acceleration and angular velocity play a significant role in determining tissue strain. To visualize this, we plotted the injury risk predicted by three metrics – BAM, peak angular acceleration ( $\vec{α}$ ), and BrIC – from idealized half-sinusoidal angular acceleration pulses over a wide range of amplitudes and durations (Fig. 8). For the purpose of creating this illustration, the half-sinusoidal pulses were applied equally in each anatomical plane. Injuries and non-injuries from our data set were superimposed, and a past injury curve presented by Margulies and Thibault was superimposed on BAM.⁶⁰ BAM is sensitive to both angular velocity and angular acceleration, and injury and non-injury regimes are separated by an L-shaped curve. The shape of our BAM injury risk curves is similar to the shape of injury curves from past primate and human studies,^59
–61 supporting the notion that BAM may be a better approximation of reality than metrics based solely on peak angular acceleration or velocity. As shown in Figure 8, injuries in our data set lie near the corner of this L-shaped curve where both angular acceleration and velocity determine injury risk. However, in other loading regimes away from contact sports head impacts, such as voluntary head motion in which angular velocities are high but accelerations are low,²⁸ BrIC and $Δ \vec{ω}$ injury metrics would predict high injury risk, whereas BAM could capture the dynamics of the brain and predict a more realistic, low injury risk. Similarly, in an extremely short duration impact with very high acceleration but low velocity, one would expect low tissue strains and injury risk,³⁵ which would be captured by BAM but not by peak $\vec{α}$ or $Δ \vec{ω}$ . In Figure S3 (see Supplementary Materials Section), we split the kinematics data into different loading regimes, and show that peak angular acceleration ( $\vec{α}$ ) performs best in the high acceleration, low velocity regime, whereas BrIC performs best in the low acceleration, high velocity regime; BAM performs similarly well across all regimes, indicating that BAM captures the dynamics of the skull-brain system. In summary, we would expect BAM to be more versatile to different loading regimes than metrics based only on angular acceleration or velocity, which we hypothesize would have been reflected more substantially if our data set had had injuries from other sports (e.g. unhelmeted impacts) or impact scenarios. However, in the case of predicting injury in American football and other contact sports, a classifier based on peak rotational velocity or acceleration alone is likely sufficient.

FIG. 8.

Comparing injury and non-injury regimes for different metrics: probability of injury is plotted for (A) brain angle metric (BAM), (B) peak angular acceleration, and (C) brain injury criterion (BrIC) to show how injury and non-injury regimes differ using different metrics. BAM is sensitive to both peak angular acceleration and angular velocity, thus separating the injury and non-injury regimes by L-shaped curves. Our current data set of injuries and non-injuries is superimposed on top of this surface plot, and a past diffuse axonal injury tolerance curve presented by Margulies and coworkers is superimposed on BAM with a dashed line.⁶⁰ Most of the injuries and non-injuries lie near the corner of the L-curves, where peak angular velocity and angular acceleration have equal effect on injury probability, explaining why our metric performs comparably to metrics based on peak kinematics. Color image is available online.

In developing our mechanical brain model, we have modeled the complex dynamics of brain-skull motion into an injury predictor, which correlates to peak principal tissue strain as well as axonal strains in the corpus callosum in mild to moderate impacts. Although FE models are computationally expensive, we showed our metric to be an effective and easy-to-implement injury criterion that requires few inputs and low computational power. In future work, this classifier has potential to be used in real time in conjunction with wearable sensors to help inform players and coaches of the injury risk resulting from an on-field head impact. The brain model could also be used with simplified rigid-body models of the head and cervical spine⁶² to give rapid mapping between input force to output brain displacement.

Recent previous work has also focused on the development of similar reduced order spring-damper models of the brain to estimate brain deformation from rotational skull kinematics.³⁵ In this article, the authors fit stiffness and viscosity parameters to match the maximum principal strain resulting from applying a series of idealized rotational pulses to the Global Human Body Models Consortium (GHBMC) brain finite element model. Conversely, we obtained model parameters by characterizing the dominant frequencies and decay rate of a brain finite element model excited by real-world injury and non-injury pulses.³⁸ The model presented by Gabler and coworkers correlates with the peak principal strain, simulated on the GHBMC FE brain model, with an R² value of 0.95. There may be differences in the ability of the KTH model and the GHBMC model to be correlated with simple mechanical systems. Further, whereas the Gabler and coworkers model shows high correlation to the GHBMC peak principal strain, we show that our mechanical model can classify injuries from real-world data more effectively than many existing kinematic criteria, which have not been previously tested using simple mechanical brain models.

One limitation to this study is the scope of the data set. As mentioned previously, the ideal approach is to compare the performance of all candidate injury criteria with a large human injury and non-injury data set. Although head impact monitoring technology has advanced over the past years, there is still a lack of available, quality 6 DOF injury data in literature. Further, because of a lack of injury data in which head rotation was primarily in the sagittal plane, logistical regression coefficients for many of the metrics in the sagittal plane were not statistically significant. More injury and non-injury data, within different loading regimes, would allow for more extensive validation of the correlation between ${\vec{θ}}_{b r a i n}$ and maximum principal strain, as well as the accuracy of our proposed metric BAM. For example, there were no injuries in the voluntary or navy data sets, which had loading regimes different than those for contact sport head impacts. Therefore, these data points had a lesser effect on our logistical regression analyses, with all predictors suggesting low injury risk; had these data sets contained injury points, we would expect the implications on metric performance to be greater. Another limitation of ${\vec{θ}}_{b r a i n}$ is that the mass-spring model includes only decoupled motion in each anatomical direction and ignores the interactions. In reality, the motion and deformation of brain tissue inside the skull is non-linear, and may have coupling within different anatomical planes of motion. However, in a related study, it was found that off-axis elements in simplified mass-spring-damper brain models have minimal contribution to predicting total brain strain.³⁹ This indicates that coupling elements would only marginally improve model accuracy in recreating FE results. Lastly, the data and anatomical values used in this work represent only male subjects, and, therefore, further work would need to be performed toward developing a female-specific brain model and injury metric, as susceptibility to concussion and anatomical skull and brain properties vary significantly across sexes. Moving forward, efforts should be focused on collecting more concussive head impact data from different loading regimes. Doing so will allow for further tuning of the presented logistical regression analyses to obtain greater statistical significance toward creating more sensitive and accurate injury criteria. In conclusion, we investigated a new injury criterion that simulates the dynamics of the brain to efficiently estimate deformation-based criteria from FE model simulations and predict risk of brain injury.

Footnotes

Funding Information

This study was supported by the Child Health Research Institute, Lucile Packard Foundation for Children's Health, and Stanford Clinical and Translational Science Award Program (CTSA) (UL1 TR001085). The study was also partially supported by the National Institutes of Health (NIH) National Institute of Biomedical Imaging and Bioengineering (NIBIB) 3R21EB01761101S1; the David and Lucile Packard Foundation 38454; Child Health Research Institute – Transdisciplinary Initiatives Program, National Science Foundation (NSF) Graduate Research Fellowship Program, and NIH UL1 TR000093 for biostatistics consultation. We thank the Ford Motor Company for supporting this study through the Ford–Stanford research alliance.

Author Disclosure Statement

No competing financial interests exist.

Supplementary Material

Supplementary Materials

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