Theoretical and numerical analysis for angular acceleration being determinant of brain strain in mTBI

Mild traumatic brain injury (mTBI, also known as concussion) caused by head impact is a crucial global public health problem, but the physics of mTBI is still unclear. During the impact, the rapid movement of the head injures the brain, so researchers have been endeavoring to investigate the relationship between head kinematic parameters (e.g., linear acceleration, angular velocity, angular acceleration) and brain strain, which is associated with the injury of the brain tissue. Although previous studies have shown that linear acceleration had limited contribution to brain strain, whether angular velocity or angular acceleration causes brain strain is still unclear because of their interdependency (acceleration being the velocity’s time-derivative). By reframing the problem through the lens of inertial forces, we propose to use the skull frame of reference instead of the ground frame of reference to describe brain deformation during head impact. Based on the rigid-body rotation of the brain, we present a theoretical framework of mechanical analysis about how the inertial forces cause brain strain. In this way, we theoretically show that angular acceleration determines brain strain, and we validate this by numerical simulations using a finite element head model. We also provide an explanation of why previous studies based on peak values found the opposite: that angular velocity determined brain strain in certain situations. Furthermore, we use the same framework to show that linear acceleration causes brain strain in a different mechanism from angular acceleration. However, because of the brain’s different resistances to compressing and shearing, the brain strain caused by linear acceleration is small compared with angular acceleration. Mild traumatic brain injury (mTBI, Table 1 gives abbreviation and symbols) is a crucial global public health problem because it significantly impairs patients’ quality of life [1,2]. Because of the relatively thick skull of humans, the stress wave caused by a head impact will not pass through the skull [3]. Instead, the head impact will cause a rapid movement of the head, which in turn deforms the brain because of its inertia. Therefore, head kinematics are the key to describing the severity of the loading on the brain. The traumatic brain injury community has endeavored to learn how head kinematics predict brain strain, a mechanical parameter describing deformation and closely relevant to brain injury [4,5]. In 1943, Holbourn hypothesized that linear acceleration would not deform the brain significantly and was unlikely to cause brain injury because of the incompressibility of the brain [6]. After that, researchers still debated whether translation or rotation of the head accounted for brain injury [7-9] until the recent development of finite element (FE) head models [3,10-12] which could calculate brain strain according to head kinematics. The simulation using a FE head model helped confirm the Houlbourn hypothesis that brain strain was most dependent on head rotation instead of translation [3], but the mechanism behind this has not been formally discussed. Angular acceleration and angular velocity of the head have been studied heavily over the last two decades to better understand how the brain is deformed [13-17]. Based on these theories, reduced-order models were developed to predict brain strain. The effects of linear and angular acceleration have been combined to predict brain strain [18-22]. In contrast, angular velocity was used [12,23-25] and found to have better predictability for the peak brain strain [12,26]. According to the studies comparing reduced-order models [12,24,27], the angular velocity-based models have achieved a higher predictability than angular acceleration, indicating that the angular velocity decides brain strain. However, the better predictability of angular velocity contradicted both experimental and numerical results [28], where volunteers experienced angular velocity higher than the injury threshold [13], but had no brain injury and low brain strains. This finding [28] could be explained by the classification of the impacts according to the impulse duration [16]: the peak brain strain depends on angular velocity peak in short-duration impacts, on angular acceleration peak in long-duration impacts and on both in moderate-duration impacts. However, the mechanism behind this classification [16] is still unclear. Furthermore, it should be noted that only peak values of kinematic parameters and brain strain were investigated [11-13,23-25,27], and the angular velocity with different profiles but the same peak were found to yield different brain strains [29-31]. Recently, combining the effects of both angular velocity and acceleration, researchers developed mass-spring-damper models [16,17,32,33] and machine learning models [26,34,35] to predict brain strain and have achieved promising accuracy. However, the mechanisms underlying the relationship between brain strain and head kinematics are still unknown. The difficulty in separating the effects of the head angular velocity and acceleration is the interdependency between them, that is, angular acceleration is the derivative of angular velocity. The movement of the head in the ground frame of reference (FoR) will be determined when each of them is given (Fig.1A, the FoR is fixed to the ground). Therefore, the brain strain caused by angular acceleration and angular velocity can not be investigated independently. To address this, we studied brain deformation in the skull FoR (Fig.1B, the FoR moves along with the skull). In the skull FoR, the skull is fixed, and the brain tissue is deformed by the inertial forces caused by each kinematic parameter as shown in Fig.1B. In this manner, the effect of each kinematic parameter can be investigated independently. To explain how each inertial force causes brain strain, the rigid-body movement of the brain [36] found in low-severity cadaver head impacts [37] was used as a bridge. The rigid-body movement of the brain means that most of the brain tissue is displaced similarly as a rigid body. However, because of the constraint of the skull and the spatial distribution of the inertial force, brain strain will still be caused. In this study, we assumed that brain strain is linearly correlated with the magnitude of the rigid-body rotation of the brain (RRB) and validated this assumption by FE simulations. In this letter, we show that only the inertial force by angular acceleration causes non-zero torque on the brain (Fig.1C), therefore only angular acceleration leads to the RRB and the brain strain. Then, to validate this finding, we performed FE simulations using the KTH head model [11] and 118 on-field football head impacts (Fig.2) collected by instrumented mouthguards [38-40]. Considering the difference between our findings and previous studies [12,16,24,27], we provide a further explanation based on the brain strain by constant angular acceleration. Moreover, we discuss the rigid-body translation of the brain (RTB) caused by linear acceleration, which provides further explanation to the Holbourn hypothesis [6]. The skull FoR is non-inertial, and the inertial force (Eq.1) caused by the movement of the FoR is [41], F(r) = ρ(r)a − ρ(r)ω × (ω × r) − β × r − 2ρ(r)ω × vr(r) (Eq.1) Where F(r) is the inertial force per unit mass at the location r; a, ω and β are the linear acceleration at coordinate origin, the angular velocity, and the angular acceleration of the head, respectively; vr(r) is the relative velocity of location r at the skull FoR; r is the position vector, and the coordinate origin of r is set at the center of gravity (CoG) of the brain (Fig.1A), therefore, ∫∫∫rρ(r)dv = 0 (Eq.2) Because the relative movement of the brain tissue in the skull was small [36,37], the Coriolis force item (2ρ(r)ω × vr(r)) can be neglected, and this is validated by FE simulation later. Therefore, Eq.1 is simplified as Eq.3. It should be noted that the items in Eq.3 with a, ω and β are separated, so the effect of each kinematic parameter can be studied independently. We defined the inertial forces by each kinematic parameter as, F(r) = ρ(r)a − ρ(r)ω × (ω × r) − β × r = FLinAcc(r) + FAngVel(r)+FAngAcc(r) (Eq.3) Where FLinAcc, FAngVel and FAngAcc are the inertial forces by linear acceleration, angular velocity, angular acceleration, respectively (Fig.1B), and are defined as, FLinAcc(r) = ρ(r)a (Eq.4) FAngVel(r) = −ρ(r)ω × (ω × r) (Eq.5) FAngAcc(r) = −ρ(r)β × r (Eq.6) Integrating over the brain, the inertial torque on the brain caused by the head movement is calculated at the CoG of the brain, T = ∫∫∫r × F(r)dv (Eq.7) Replacing F(r) by Eq.3, the inertial torque can be written as the sum of the inertial torques caused by each kinematic parameter, T = TLinAcc + TAngAcc + TAngAcc (Eq.8) Here, the inertial torque by linear acceleration is, TLinAcc = ∫∫∫r × FLinAcc(r)dv (Eq.9) Replacing Eq.4 into Eq.9 and considering a is constant over the brain, TLinAcc = ∫∫∫ρ(r)r × adv = ∫∫∫ρ(r)rdv × a = 0 (Eq.10) Then, the inertial torque by angular velocity is, TAngVel = ∫∫∫r × FAngVel(r)dv (Eq.11) Replacing Eq.5 into Eq.11, TAngVel = ∫∫∫ρ(r)(ω ∙ r)ω × rdv (Eq.12) Further, the inertial torque by angular acceleration is TAngAcc = ∫∫∫r × FAngAcc(r)dv (Eq.13) Replacing Eq.6 into Eq.13, TAngAcc = ∫∫∫ρ(r)(r β − (r ∙ β)r)dv (Eq.14) We further assume that the shape of the human brain is close to a sphere (radius r0) and has a homogeneous density ρ0. Because of the symmetry in Eq.12, the inertial torque by angular velocity is, TAngVel sphere = 0 (Eq.15) Applying the sphere assumption to Eq.14, we get the inertial torque by angular velocity for a spherical brain as (see deduction of Eqs. 15 and 16 in Supplementary Information S1), TAngAcc sphere = 8 15 ρ0πr0 β (Eq.16) Therefore, for the actual brain, TAngVel ≪ TAngAcc (Eq.17) The accuracy of Eq.17 depends on the closeness of the human brain to the sphere. To valid