The Strain Rates in the Brain, Brainstem, Dura, and Skull Under Dynamic Loadings

Mathematical and Computational Applications

Article The Strain Rates in the Brain, Brainstem, Dura, and Skull under Dynamic Loadings

Mohammad Hosseini-Farid 1,2,* , MaryamSadat Amiri-Tehrani-Zadeh 3, Mohammadreza Ramzanpour 1, Mariusz Ziejewski 1 and Ghodrat Karami 1

1 Department of Mechanical Engineering, North Dakota State University, Fargo, ND 58104, USA; [email protected] (M.R.); [email protected] (M.Z.); [email protected] (G.K.) 2 Department of Orthopedic Surgery, Mayo Clinic, Rochester, MN 55905, USA 3 Department of Computer Science, North Dakota State University, Fargo, ND 58104, USA; [email protected] * Correspondence: [email protected]; Tel.: +1-7012315859

Received: 7 March 2020; Accepted: 5 April 2020; Published: 7 April 2020

Abstract: Knowing the precise material properties of intracranial head organs is crucial for studying the biomechanics of head injury. It has been shown that these biological tissues are significantly rate-dependent; hence, their material properties should be determined with respect to the range of deformation rate they experience. In this paper, a validated finite element human head model is used to investigate the biomechanics of the head in impact and blast, leading to traumatic brain injuries (TBI). We simulate the head under various directions and velocities of impacts, as well as helmeted and unhelmeted head under blast shock waves. It is demonstrated that the strain rates for the brain 1 are in the range of 36 to 241 s− , approximately 1.9 and 0.86 times the resulting head acceleration under impacts and blast scenarios, respectively. The skull was found to experience a rate in the range 1 of 14 to 182 s− , approximately 0.7 and 0.43 times the head acceleration corresponding to impact and blast cases. The results of these incident simulations indicate that the strain rates for brainstem and 1 dura mater are respectively in the range of 15 to 338 and 8 to 149 s− . These findings provide a good insight into characterizing the brain tissue, cranial bone, brainstem and dura mater, and also selecting material properties in advance for computational dynamical studies of the human head.

Keywords: ﬁnite element; head model; brain tissue; strain rate; material characterization

1. Introduction Intracranial head organs have been known to be the most sensitive organs involved in life- threatening injuries caused by impact incidents and blast waves [1–3]. Due to the complex mechanical and physical reactions of the head and brain under impact and explosion waves, the mechanism of traumatic brain injuries (TBI) is not well understood [4]. Finite element (FE) simulations provide a convincing framework for determining biomechanical responses of the head exposed to those high rated loads [5]. These numerical models have been introduced to predict brain deformation for diﬀerent applied loads and study intracranial organ behavior under TBI conditions [6–13]. The bioﬁdelity of such computational simulations is strictly related to the accuracy of the material properties used to model these tissues. Mechanical properties of brain tissue are a fundamental subject of biomechanics and have been extensively studied in the last few decades [3,14,15]. It has been shown that the mechanical behavior of the brain is dominated by the loading rate and varies nonlinearly with any change in strain, as well as with its strain rate [16–20]. Typically, soft biological materials such as brain and brainstem present complex mechanical responses characterized by large strains, load history, and rate sensitivity [21–26].

Math. Comput. Appl. 2020, 25, 21; doi:10.3390/mca25020021 www.mdpi.com/journal/mca Math. Comput. Appl. 2020, 25, 21 2 of 15

Since brain and brainstem tissues are rate-dependent materials, great care should be taken in selecting proper material properties from corresponding strain rates in different scenarios. Dura mater (a surrounding membrane of brain) and skull, which hold and protect the brain, have significant roles in the analysis of the TBI incidents. Therefore, determining the material properties of the dura and cranial bone was the subject of earlier research in biomechanics [27–29]. Both theoretical and experimental studies have shown that the dura and cranial bone present rate-dependent material properties [30–34]. Although linear and hyperelastic models can approximate the behavior of dura and skull very well at each rate [31,35,36], they behave differently under various speeds of loading. For instance, Persson et al. [31] tested dura mater under uniaxial tension at three various strain rates of 1 0.01, 0.1, and 1.0 s− . They postulated the mechanical responses of dura is rate-dependent; however, they fitted their results to the rate-independent hyperelastic Ogden model. Additionally, experimental tensile results indicated the Young’s modulus of the cranial bone varies from 8 to 19.5 GPa for quasistatic to dynamic rates [37]. Therefore, to have a correct result at each FE simulation, the right material properties corresponding to the rate of that application should be selected. The majority of biomechanical characterizations for human head organs have been performed within experiments in relatively low strain rates. Recently, De Kegel et al. [35] conducted some 1 experiments for human dura specimens within a strain rate of 1.0 s− . They have obtained a highly nonlinear behavior for dura and characterized its mechanical response using three different hyperelastic material models. Franceschini et al. [38] have experimentally studied the mechanical behavior of 1 human brain tissue at strain rates ranging between 0.0055 and 0.0093 s− . Although their results are valid to be employed for studies in quasistatic loading, those material properties have been used in many computational simulations of TBI at dynamic loads [39,40]. It was confirmed [16,41] that, in addition to selecting an appropriate constitutive model, using the material constants derived from a mismatched strain rate may considerably affect the validity of the results. Farid et al. [16] showed that hyper-viscoelastic material models, which are optimized for various low strain rates, will result in considerable errors when predicting brain behavior at higher strain rates. Therefore, it is crucial to know the range of strain rates at dynamic loading conditions and characterize the material properties of head organs at those rates. Finding the ranges of strain rate for brain, brainstem, dura, and skull in TBI through simulations can help in determining and using the material properties at the right rate. The strain rates of different incidents cannot be measured in vivo, and therefore, a computational finite element analysis of the human head was conducted to predict the strain rates for loading scenarios associated with brain injury [12]. In this paper, impact and blast assault simulations are conducted on a developed FE human head model to predict the range of strain rates. In this paper, it was assumed that the brain will experience minor injuries by blunt impact [1] and an effective (mild to severe) brain injury caused by blast [42]. This approach came from the fact that impact and blast load types are inherently different and cannot be directly compared to each other. The findings provide a ground basis for determining more relevant material properties for the strain rates corresponding to impact and blast cases.

2. Material and Methods

2.1. Finite Element of Head Model Using MRI and CT scan techniques, the geometry of the 50th percentile human head–neck model was developed by Horgan and Gilchrest [43], and the detailed explanation of this head model has been provided in earlier studies [44,45]. As shown in Figure1, the model contains all the major parts of the head and the neck, including the brain, falx, tentorium, cerebrospinal fluid (CSF), dura mater, pia mater, neck muscle, neck bone, facial bone, skull, and scalp. In this paper, linear elastic material behavior is used for all head components (Table1) except the brain and brainstem, which are modeled as hyper-viscoelastic (Table2) and viscoelastic (Table3) material [ 1], respectively. The Mooney–Rivlin strain energy coupled with the linear viscoelastic function was used as the constitutive relation of Math. Comput. Appl. 2020, 25, 21 3 of 15 the brain given by Equation (1). Furthermore, the linear viscoelastic material model expressed by Equation (2) was employed to represent the shear characteristics of the brainstem. A linear material model with 10 GPa of Young’s modulus was considered for the skull. The developed head model was earlier examined and validated under different blast scenarios [42,46,47]. Additionally, Horgan and Gilchrist [43,48] verified this head model against various mostly referenced cadaveric experiments of Nahum et al. [49], Hardy et al. [50], and Trosseille et al. [51].     Z t M  N   X   d  X   (t τ)/τk    i j W = 1 gk 1 e− −    Cij(I1 3) (I2 3) dτ (1)  − −  ∗ dτ − −  0  k = 1 i+j = 1 

βt Math. Comput. Appl. 2020, 25, x FOR PEER REVIEWG(t) = G + (G0 G )e− 4 of 16 (2) ∞ − ∞

Figure 1. The ﬁnite element (FE) head model used in this study includes the main components of the humanFigure head 1. [The44]. finite element (FE) head model used in this study includes the main components of the human head [44]. Table 1. Head model component properties and mechanical properties [1,52]. 2.2. Impact Case Segment Constitutive Density Poisson’s FE Model Young’s Modulus (MPa) NameThe injury causedModel by a golf ball impact is one of the(kg /mostm3) leadingRatio causes of head injury in sports- relatedTentorium impacts Linear[53]. The Elastic possibility Shell of being Element hit by an 1133 occasional 0.45 golf ball is common 31.5 for both the players and the people who watch the game [54]. This case of impact was simulated to evaluate the Dura Mater Linear Elastic Shell Element 1133 0.45 31.5 strain rate as an example of the general blunt impact applications. The simulations of a golf ball impactPia Mater were done Linear for a Elastic ball flying Shell at speeds Element of 10, 15, 1130 and 20 m/s. 0.45 As suggested by Lee 11.5 et al. [52], theseFalx ball speeds Linear were Elasticselected in Shellsuch Elementa way to avoid 1133 severe brain 0.45 injury. Since directionality 31.5 has notable influences [11], the simulation was performed for three directions of back, frontal, and lateral Skull Linear Elastic Solid Element 1300 0.24 10000 impact. Figure 2 presents the impact setup for two directions of frontal and lateral. Table 4 provides Neck-Bone Linear Elastic Solid Element 1300 0.24 10000 the summary of the simulated impact scenarios in this study. The neck is assumed to be fixed in the translationalNeck-Flesh displacements Linear Elastic [52]. Solid Element 1130 0.45 0.1 Face-Bone Linear Elastic Solid Element 2100 0.42 5540 Face-Skin Linear Elastic Solid Element 1200 0.42 16.7 Scalp Linear Elastic Solid Element 1200 0.42 16.7 Golf Ball Linear Elastic Solid Element 1100 0.49 300 Poisson’s Bulk Viscosity Linear Elastic CSF Solid Element 1040 Ratio Modulus Coeﬃcient (Fluid Opt.) 0.49 2190 0.2

Figure 2. Setup of frontal and lateral impact by golf ball in the FE head model.

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Table 2. Mooney–Rivlin hyper-viscoelastic constants of the brain used in FE modeling [1].

3 1 1 Density (kg/m ) C10 (Pa) C01 (Pa) g1 (Pa) g2 (Pa) β1 (s− ) β2 (s− ) 1040 31 35 407 233 125 6.7

Table 3. Viscoelastic material parameters of the brainstem used in FE modeling [1].

3 1 Density (kg/m ) Bulk Modulus (GPa) G0 (kPa) G (kPa) β (s− ) ∞ 1060 2.19 22.5 4.5 80 Figure 1. The finite element (FE) head model used in this study includes the main components of the 2.2. Impacthuman Case head [44].

2.2.The Impact injury Case caused by a golf ball impact is one of the most leading causes of head injury in sports-relatedThe injury impacts caused [53 by]. a The golf possibility ball impact ofis one being of the hit most by an leading occasional causes golf of head ball injury is common in sports- for both the playersrelated impacts and the [53]. people The whopossibility watch of the being game hit by [54 an]. Thisoccasional case ofgolf impact ball is was common simulated for both to the evaluate the strainplayers rate and asthe an people example who ofwatch the generalthe game blunt [54]. This impact case applications. of impact was Thesimulated simulations to evaluate of a the golf ball impactstrain were rate doneas an forexample a ball of flying the general at speeds blunt of impact 10, 15, applications. and 20 m/s. The As simulations suggested of by a Leegolf etball al. [52], theseimpact ball speedswere done were for selected a ball flying in such at speeds a way of to 10 avoid, 15, and severe 20 m/s. brain As sugg injury.ested Since by Lee directionality et al. [52], has notablethese influences ball speeds [11 were], the selected simulation in such was a way performed to avoid forsevere three brain directions injury. Since of back, directionality frontal, and has lateral impact.notable Figure influences2 presents [11], the the simulation impact setup was performed for two directions for three directions of frontal of and back, lateral. frontal, Table and 4lateral provides impact. Figure 2 presents the impact setup for two directions of frontal and lateral. Table 4 provides the summary of the simulated impact scenarios in this study. The neck is assumed to be fixed in the the summary of the simulated impact scenarios in this study. The neck is assumed to be fixed in the translationaltranslational displacements displacements [52 [52].].

FigureFigure 2. 2.Setup Setup of of frontal frontal and lateral lateral impact impact by by golf golf ball ball in the in theFE head FE head model. model. Table 4. The summary of impact cases by golf ball applied in FE modeling for estimating the ranges of strain rates.

Case Code Impactor Speed (m/s) Direction Impact# 1 Golf ball 10 Lateral Impact# 2 Golf ball 15 Lateral Impact# 3 Golf ball 20 Lateral Impact# 4 Golf ball 10 Back Impact# 5 Golf ball 15 Back Impact# 6 Golf ball 20 Back Impact# 7 Golf ball 10 Frontal Impact# 8 Golf ball 15 Frontal Impact# 9 Golf ball 20 Frontal Math. Comput. Appl. 2020, 25, x FOR PEER REVIEW 5 of 16

Table 4. The summary of impact cases by golf ball applied in FE modeling for estimating the ranges of strain rates.

Case Code Impactor Speed (m/s) Direction Impact# 1 Golf ball 10 Lateral Impact# 2 Golf ball 15 Lateral Impact# 3 Golf ball 20 Lateral Impact# 4 Golf ball 10 Back Impact# 5 Golf ball 15 Back Impact# 6 Golf ball 20 Back Impact# 7 Golf ball 10 Frontal Impact# 8 Golf ball 15 Frontal Math. Comput. Appl.Impact#2020, 25, 9 21 Golf ball 20 Frontal 5 of 15

2.3. Blast Case 2.3. Blast Case High strength explosion waves are a possible source of TBI. In this paper, the same head model was alsoHigh being strength used explosion for blast wavessimulations. are a possibleProtecting source the head of TBI. with In thisa helmet paper, can the reduce same headthe severity model wasof brain also injury, being used but injury for blast can simulations. still occur when Protecting the person the head is exposed with a helmet to blas cant [55]. reduce As illustrated the severity in ofFigure brain 3, injury, a three-dimensional but injury can stillmodel occur of an when adva thenced person combat is exposed helmet to(ACH) blast is [55 integrated]. As illustrated with the in Figurehead–neck3, a three-dimensionalmodel. The detailed model explanation of an advanced of helmet combat and its helmet padding (ACH) system is integrated modeling withand their the head–neckmechanical model.properties The can detailed be found explanation in earlier ofstudy helmet conducted and its by padding Salimi systemJazi et al. modeling [55]. In this and study, their mechanicalthe blast simulation properties was can conducted be found infor earlier the protect studyed conducted and unprotected by Salimi head Jazi in et al.two [55 back]. In and this lateral study, thedirection. blast simulation The standoff was distances conducted are forset to the be protected one and two and meters unprotected from the head explosion in two backsite. In and LS-Dyna lateral direction.(LS-Dyna TheTheory stando Manual,ﬀ distances 2007), are the set standard to be one trinitrotoluols and two meters (TNT) from is theselected explosion as the site. high In explosive LS-Dyna (LS-Dyna(HE) material. Theory Because Manual, of a 2007), very theshort standard effective trinitrotoluols time on the (TNT)blast (less is selected than 1 asms), the a highfree boundary explosive (HE)condition material. was applied Because to of the a very end shortof neck. eﬀ ectiveRezaei time et al. on [56] the have blast shown (less than the blast 1 ms), waves a free of boundary 95 g HE conditionplaced 750 was mm applied far from to the the human end of head neck. can Rezaei cause et a al.severe [56] and have fatal shown brain the injury. blast wavesIn addition of 95 to g this HE placedcase (95 750 gr mmTNT), far which from theis referred human headto as can“heavy cause blast” a severe in this and paper, fatal eight brain other injury. blast In addition scenarios to with this case75 gr (95 of HE gr TNT), mass were which chosen is referred and are to aslisted “heavy in Table blast” 5. in this paper, eight other blast scenarios with 75 gr of HE mass were chosen and are listed in Table5.

Figure 3. The position of the detonation at side and back of the protected head model. Figure 3. The position of the detonation at side and back of the protected head model. Table 5. The summary of blast scenarios implemented in FE simulations to predict the ranges of strain rate associated to traumatic brain injuries (TBI). Case Code Head Condition Explosive Mass (gr) Distance (m) Direction Blast# 1 Protected 75 1 Back

Blast# 2 Protected 75 2 Back Blast# 3 Protected 75 1 Lateral Blast# 4 Protected 75 2 Lateral Blast# 5 Protected 95 0.75 Back Blast# 6 Unprotected 75 1 Back Blast# 7 Unprotected 75 2 Back Blast# 8 Unprotected 75 1 Lateral Blast# 9 Unprotected 75 2 Lateral Blast# 10 Unprotected 95 0.75 Back Math. Comput. Appl. 2020, 25, 21 6 of 15

Upon a burst in the air, a shock wave is generated moving away from the center of detonation mass. In this study, the multimaterial arbitrary Lagrangian–Eulerian (ALE) formulation method was used for simulating the blast scenarios in LS-Dyna. This method is established on explicit modelling of the air and exploding mass with multimaterial ALE equation for simulating the blast wave propagation. It solves the movement of the mesh using mesh smoothing algorithms [42]. To determine the behavior of the explosive gas, the equation of state (EOS) must be defined. This equation of state (EOS) explains pressure–volume–energy relationships of the gaseous form of detonation. The EOS regarding blast pressure wave generated by HE detonation can be expressed using the Jones–Wilkins–Lee (JWL) equation as [45]: ! ω ω ωE p = A 1 eR1V + B 1 eR2V + (3) − R1V − R2V V where ω, A, B, R1, and R2 are constants that are defined by the user, E is the internal energy per initial volume, and V is the relative volume. The numerical data for these parameters are derived from the explosive handbook by Dobratz [57]. The discretized blast domain depicted in Figure3 shows the location and direction of HE material. To model the blast wave–head interaction, a penalty-based fluid–structure interaction (FSI) algorithm was also used. FSI was employed through a coupling algorithm modeling the air as a moving fluid along with a moving mesh. For a more detailed explanation of blast modeling procedures, the readers are referred to earlier studies of Salimi Jazi et al. [55] and Rezaei et al. [42].

3. Results FE simulations of the head impacted by a golf ball are performed for diﬀerent impact directions and ball speeds as stated in Table4. Moreover, the mentioned blast scenarios presented in Table5 were implemented in the head model. In this study, the resultant linear acceleration of the skull was assumed to be the same as human head acceleration [58]. Therefore, this acceleration for three cases of impact and four scenarios of the blast are measured and shown in Figure4a,b. Instead of measuring the acceleration at one point or element, the whole skull geometry is considered as one body, and the averagedMath. Comput. acceleration Appl. 2020, 25 of, x this FOR part PEER was REVIEW evaluated by LS-Dyna. 7 of 16

a) b)

FigureFigure 4. 4.Resultant Resultant linear linear acceleration acceleration for for the the head head skull skull (head) (head) at: at: ( a(a)) impact impact in in three three directions, directions, and and (b()b four) four cases cases of of blast. blast.

FigureFigure5 5presents presents the the blast blast wavewave propagationpropagation and intracranial intracranial pressure pressure (ICP) (ICP) distribution distribution in in the thebrain brain which which is necessary is necessary to predict to predict brain brain behavior behavior during during the theblast. blast. As can As canbe seen, be seen, for this for thiscase case(backward (backward heavyheavy blast blast), the), the ICP ICP in inbrain brain exceeds exceeds 300 300 kPa, kPa, and and therefore, therefore, a severesevere TBITBI shouldshould be be expectedexpected [1 [1].]. Furthermore, Furthermore, in in comparison comparison to Rezaeito Rezaei et al.’set al.’s study study [56 ],[56], it proves it proves that that the selectionthe selection of HE of massesHE masses and itsand distance its distance from from the head the head was adequatewas adequate to create to create such asuch condition. a condition.

Figure 5. Intracranial pressure (ICP) (kPa) contours in the brain for the protected head in heavy blast for whole brain and also cross-sectional view.

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a) b)

Figure 4. Resultant linear acceleration for the head skull (head) at: (a) impact in three directions, and (b) four cases of blast.

Figure 5 presents the blast wave propagation and intracranial pressure (ICP) distribution in the brain which is necessary to predict brain behavior during the blast. As can be seen, for this case (backward heavy blast), the ICP in brain exceeds 300 kPa, and therefore, a severe TBI should be expected [1]. Furthermore, in comparison to Rezaei et al.’s study [56], it proves that the selection of Math.HE masses Comput. and Appl. its2020 distance, 25, 21 from the head was adequate to create such a condition. 7 of 15

FigureFigure 5.5. IntracranialIntracranial pressurepressure (ICP)(ICP) (kPa)(kPa) contourscontours inin thethe brainbrain forfor thethe protectedprotected headhead inin heavyheavy blastblast forfor wholewhole brainbrain andand alsoalso cross-sectionalcross-sectional view. view.

The strain rate distributions for all elements in the brain, brainstem, skull, and dura were examined. Similar to other mechanical parameters, such as ICP, strain or stress, the strain rate has a non-uniform distribution. For instance, in the brain, the maximum strain rate occurs only in a relatively small zone (some elements) which takes place in different locations for each case, and the rest of the brain observes a moderate regime of rates. Figure6a,b illustrates the history of 1st principal strain rate in three different locations of the brain for the head impacted by a golf ball. For the case where the ball hit the front of the head with 15 m/s of speed, the maximum strain rate does not happen in coup nor contrecoup. However, it was found that for lateral impacts, the highest rate happens at the coup zone. This shows the location of the maximum rate can occur in every area of the brain and varies for several impact scenarios. Similarly, Figure6c,d depicts the brain strain rates by the same measure (1st principal) for the protected and unprotected heads under heavy blast conditions. As shown in Figure6c,d, for the heavy-blast scenario, the highest estimated strain rates of the protected and unprotected head are close to each other, although due to inherent high nonlinearity of blast analysis, it is difficult to find a rational explanation for this, and it seems the helmet in this scenario has a minor effect on the strain-rate measure. One probable reason for this can be the severe explosion happening in a very short distance from the head, which leads to the helmet not being effective in serving as a deep protective tool here. For blunt impact cases, the elements of the skull, which are located at the impact side, are observed to have the highest strain rates. Additionally, for all blast simulations (lateral and backside), the maximum rates occur at the temporal bone, which has the lowest cranial thickness. Math. Comput. Appl. 2020, 25, x FOR PEER REVIEW 8 of 16

The strain rate distributions for all elements in the brain, brainstem, skull, and dura were examined. Similar to other mechanical parameters, such as ICP, strain or stress, the strain rate has a non-uniform distribution. For instance, in the brain, the maximum strain rate occurs only in a relatively small zone (some elements) which takes place in different locations for each case, and the rest of the brain observes a moderate regime of rates. Figure 6a,b illustrates the history of 1st principal strain rate in three different locations of the brain for the head impacted by a golf ball. For the case where the ball hit the front of the head with 15 m/s of speed, the maximum strain rate does not happen in coup nor contrecoup. However, it was found that for lateral impacts, the highest rate happens at the coup zone. This shows the location of the maximum rate can occur in every area of the brain and varies for several impact scenarios. Similarly, Figure 6c,d depicts the brain strain rates by the same measure (1st principal) for the protected and unprotected heads under heavy blast conditions. As shown in Figure 6c,d, for the heavy-blast scenario, the highest estimated strain rates of the protected and unprotected head are close to each other, although due to inherent high nonlinearity of blast analysis, it is difficult to find a rational explanation for this, and it seems the helmet in this scenario has a minor effect on the strain-rate measure. One probable reason for this can be the severe explosion happening in a very short distance from the head, which leads to the helmet not being effective in serving as a deep protective tool here. For blunt impact cases, the elements of the skull, which are located at the impact side, are observed to have the highest strain rates. Additionally, for all blast simulations (lateral and backside), the maximum rates occur at the temporal bone, which has the Math. Comput. Appl. 2020, 25, 21 8 of 15 lowest cranial thickness.

FigureFigure 6.6. Histories of strain rate at three diﬀfferenterent regionsregions ofof thethe brainbrain under:under: ((aa)) frontalfrontal andand ((bb)) laterallateral impactimpact byby golfgolf ball,ball, andand heavyheavy blastblast scenariosscenarios forfor ((cc)) protectedprotected andand ((dd)) unprotectedunprotected head. head.

InIn thisthis study,study, thethe maximummaximum shearshear strainstrain raterate waswas determineddetermined forfor brainstembrainstem sincesince itit hashas beenbeen shownshown toto bebe thethe dominantdominant cause cause of of injury injury at at brainstem brainstem occurring occurring due due to to shear shear strain strain [55 [55].]. Figure Figure7a–d 7a– demonstratesd demonstrates the the histories histories of the of strain the strain rate at rate regions at regions with highest with valueshighest of valu ratees in brain,of rate brainstem, in brain, skull, and dura under three directions of impact. In addition, the computed strain rates at points with maximum rates in these head parts under four scenarios of blast are plotted and shown in Figure8a–d. Finally, the maximum linear acceleration of the head and highest calculated strain rate in the brain, brainstem, skull, and dura for all impact and blast cases are demonstrated in Tables6 and7, respectively. Since the values of the acceleration in all impact scenarios are less than 90 g [1], the damage level can be considered minor or close to the mild TBI. Head accelerations in some blast cases exceed 90 g, so severe brain injuries are expected under those circumstances.

Table 6. The estimated maximum linear acceleration of skull, and strain rate of the brain, brainstem, skull, and dura at diﬀerent impact scenarios.

Max. Skull Max. Strain Rate Max. Strain Rate Max. Strain Rate Max. Strain Rate Case Code 1 1 1 1 Acceleration (g) in Brain (s− ) in Brainstem (s− ) in Dura (s− ) in Skull (s− ) Impact# 1 33 85 24 49 35 Impact# 2 47 105 35 68 45 Impact# 3 82 180 68 103 48 Impact# 4 23 42 21 38 22 Impact# 5 61 67 43 62 37 Impact# 6 68 132 60 75 61 Impact# 7 34 56 15 43 16 Impact# 8 52 81 24 54 31 Impact# 9 70 142 35 76 42 Math. Comput. Appl. 2020, 25, x FOR PEER REVIEW 9 of 16

brainstem, skull, and dura under three directions of impact. In addition, the computed strain rates at points with maximum rates in these head parts under four scenarios of blast are plotted and shown in Figure 8a–d. Finally, the maximum linear acceleration of the head and highest calculated strain rate in the brain, brainstem, skull, and dura for all impact and blast cases are demonstrated in Tables 6 and 7, respectively. Since the values of the acceleration in all impact scenarios are less than 90 g [1], the damage level can be considered minor or close to the mild TBI. Head accelerations in some blast Math.cases Comput.exceed Appl. 90 g,2020 so, 25severe, 21 brain injuries are expected under those circumstances. 9 of 15

Brain@ Impact Brainstem@ Impact a)150 b) 80 Back Impact 20m/s Back Impact 20m/s Front Impact 15m/s Front Impact 15m/s Side Impact 15m/s 60 Side Impact 15m/s 100

50 20

0 0 0 5 10 15 0 5 10 15 Time (ms) Time (ms) c)Dura@ Impact d) Skull@ Impact 80 80 Back Impact 20m/s Back Impact 20m/s Front Impact 15m/s Front Impact 15m/s 60 Side Impact 15m/s 60 Side Impact 15m/s

40 40

20 20

0 0 0 5 10 15 0 5 10 15 Time (ms) Time (ms) Figure 7. History of highest calculated strain rate under three blunt impact cases for: (a) brain, Math. Comput.Figure Appl.7. History 2020, 25 of, xhighest FOR PEER calculated REVIEW strain rate under three blunt impact cases for: (a) brain, (b 10) of 16 (b) brainstem, (c) dura, and (d) skull. brainstem, (c) dura, and (d) skull. a) b) ) -1 Strain Rate(s

c) d) ) -1 Strain Rate(s Strain

FigureFigure 8. HistoryHistory of of highest highest calculated calculated strain strain rate duringrate during four blast four scenarios blast scenarios for: (a) brain,for: ( (ab)) brain, brainstem, (b) brainstem,(c) dura, and (c) (dura,d) skull. and (d) skull.

Table 6. The estimated maximum linear acceleration of skull, and strain rate of the brain, brainstem, skull, and dura at different impact scenarios.

Case Max. Skull Max. Strain Rate Max. Strain Rate in Max. Strain Rate Max. Strain Rate Code Acceleration (g) in Brain (s−1) Brainstem (s−1) in Dura (s−1) in Skull (s−1) Impact# 33 85 24 49 35 1 Impact# 47 105 35 68 45 2 Impact# 82 180 68 103 48 3 Impact# 23 42 21 38 22 4 Impact# 61 67 43 62 37 5 Impact# 68 132 60 75 61 6 Impact# 34 56 15 43 16 7 Impact# 52 81 24 54 31 8 Impact# 70 142 35 76 42 9

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Table 7. The predicted maximum linear acceleration of skull, and strain rate of the brain, brainstem, skull, and dura at diﬀerent blast scenarios.

Max. Skull Max. Strain Rate Max. Strain Rate Max. Strain Rate Max. Strain Rate Case Code 1 1 1 1 Acceleration (g) in Brain (s− ) in Brainstem (s− ) in Dura (s− ) in Skull (s− ) Blast# 1 93 100 165 39 55 Blast# 2 30 40 70 8 14 Blast# 3 140 120 228 55 51 Blast# 4 41 36 71 13 26 Blast# 5 226 205 338 144 122 Blast# 6 180 115 214 62 64 Blast# 7 38 48 62 15 20 Blast# 8 205 180 151 122 66 Blast# 9 63 73 47 32 28 Blast# 10 432 241 412 149 182

4. Discussion Evaluating mechanical properties of the human head organs is fundamental in understanding intracranial brain deformation under different loading conditions. It is known that the brain has a rate-dependent behavior and gets stiffer by increasing the loading rate [16,59]. That is why, in literature, it is common to conduct in vitro tests with certain strain rates to capture the wider properties of the brain [18,19,60]. Similar approaches have been applied to identify the material parameters of brainstem, dura, and other rate-dependent biological tissue [31,61]. Cranial bone was found to behave differently 1 at various rates [30]. It was reported that the bone tissue for the skull at high rates (150 s− ) can become up to 200% stiffer as skull in quasistatic rates [37]. Therefore, to simulate these rate-dependent materials in dynamical studies such as TBI cases, the validated properties over the desired range of strain rates must be employed. In this regard, the findings of this paper propose a measure for the range of the rates needed for conducting the in vitro test and also selecting material properties from the right strain rates. The time duration of a general head impacts is in the order of milliseconds, while in the case of a high-speed load such as a blast wave less than one millisecond. Regardless of the strain value, it was 1 found that brain injury caused by impact happens with a brain strain rate of 23 to 140 s− with an 1 average of 84 s− [62,63]. Using a validated FE head model is a very feasible approach for determining the strain rates of the brain and skull under loading. Clinical studies of reconstructed head impact 1 simulations which led to mild TBI for 58 football players proposed a range of 35–97 s− for the brain strain rate [64]. Viano et al. [5] analyzed 28 cases of NFL impacts involving 22 concussions by FEM, 1 and they found the brain strain rates for those incidents were in the range of 19–162 s− . To the best of the authors’ knowledge, there is no reported study in the literature pointing out the range of the strain rate from experimental measurement of an in vivo or cadaver test. In this paper, some impact and blast scenarios were modeled on a validated head model. Results (Tables6 and7) showed that under these dynamic loads, the response of brain and brainstem were 1 in the range of 36 to 241 and 15 to 412 s− , respectively. Meanwhile, the estimated strain rates of the dura and skull during such impact and blast loads were respectively in the range of 8 to 149 and 14 to 1 182 s− . Significant correlations of strain rates with linear head acceleration for brain, brainstem, dura, and skull were found and are presented in Figure9a–d. These linear regressions were independent of the effects of direction and speed of impactor, as well as the HE mass, distance, direction, and head condition (protection) under blast [65]. The proposed strain rates of brain, brainstem, dura, and skull are, respectively, 1.9 (R = 0.8, p < 0.01), 0.7 (R = 0.9, p < 0.01), 1.18 (R = 0.9, p < 0.001) and 0.7 (R = 0.6, p < 0.01) times of resulting head acceleration by impact. In addition, under blast incidents, the brain, brainstem, dura, and skull behaved with a strain rate of 0.86 (R = 0.96, p < 0.001), 1.25 (R = 0.9, Math. Comput. Appl. 2020, 25, 21 11 of 15 p < 0.001), 0.52 (R = 0.95, p < 0.001) and 0.43 (R = 0.9, p < 0.001) times of acceleration, respectively. Figure9 shows that for the equivalent linear acceleration of head, the strain rates induced by impact Math. Comput. Appl. 2020, 25, x FOR PEER REVIEW 12 of 16 for the brain, dura, and skull are higher in comparison to the blast one.

Figure 9. Linear regression of highest evaluated strain rate versus maximum acceleration of head Figure 9. Linear regression of highest evaluated strain rate versus maximum acceleration of head under impact and blast cases for: (a) brain, (b) brainstem, (c) dura, and (d) skull. under impact and blast cases for: (a) brain, (b) brainstem, (c) dura, and (d) skull. As the first limitation of this study, the result was not based on a general impact or blast conditions. NumerousAs the cases first oflimitation other simulations of this study, forvarious the result scenarios was not of bluntbased impacton a general and blast impact arerequired or blast insteadconditions. of the Numerous chosen limited cases representativeof other simulations cases. Therefore,for various it shouldscenarios be of mentioned blunt impact that theand estimated blast are rangerequired of ratesinstead in thisof the study chosen is more limited expected representative for a blunt cases. impact Therefore, causing it should mild TBI, be andmentioned blast cases that generatingthe estimated concussion range of or rates brain in injury this study [1,63 ].is Inmore this expected research, wefor dida blunt not studyimpact the causing effect of mild element TBI, sizeand onblast the cases strain generating rate. We haveconcussion used a or validated brain injury head [1,63]. model In that this has research, been used we indid many not study computational the effect studiesof element of human size on head the strain under rate. similar We have dynamical used a loads. validated Therefore, head model we did that not has change been theused mesh in many size, andcomputational of course, oncestudies we of had human changed head the under element similar sizes, dynamical we needed loads. to validateTherefore, the we new did head not change model again.the mesh As anothersize, and limitation, of course, the once FE we simulations had changed were the done element with onesizes, set we of materialneeded to properties, validate the and new we usedhead amodel simple again. elastic As material another model limitation, for the the cranial FE bonesimulations and dura were mater. done The with most one accurate set of materialmaterial propertiesproperties, for and this we kind used of a simulation simple elastic would material be the model nonhomogeneous for the cranial anisotropic bone and visco-hyperelastic dura mater. The materials.most accurate However, material first, properties we could notfor findthis thesekind propertiesof simulation for humanwould headbe the tissues nonhomogeneous in the current literature,anisotropic and visco-hyperelastic second, our head mate modelrials. was However, not developed first, for we utilizing could suchnot find material these models. properties Among for materialhuman head properties tissues available in the curre in thent literature,literature, thoseand second, which wereour head determined model atwas the not highest developed dynamic for ratesutilizing were such selected. material However, models. theAmong findings material of this properties study still available suggest in thatthe literature, there is a necessitythose which to characterizewere determined those at material the highest properties dynamic at rates even were higher selected. ranges. However, Thus, the the results findings of this of this paper study can still be consideredsuggest that as there the firstis a necessity step in having to characterize the true estimation those material of selecting properties the at right even material higher ranges. properties Thus, in dynamicalthe results simulations.of this paper can be considered as the first step in having the true estimation of selecting the right material properties in dynamical simulations. 5. Conclusions 5. Conclusions Cranial bone, and particularly brain, brainstem, and dura are highly rate-dependent materials and willCranial behave bone, di ffanderently particularly in various brain, strain brainstem, rates. In and order dura to study are highly the biomechanics rate-dependent oftraumatic materials brainand will injury behave (TBI) differently through FE in methods, various itstrain is necessary rates. In to order implement to study correct the bi propertiesomechanics associated of traumatic with brain injury (TBI) through FE methods, it is necessary to implement correct properties associated with anticipated application rates. However, to calculate the material properties of the brain, brainstem, dura, and skull related to TBI cases, we need to know for which range of strain rates they are required to be examined. In this study, FE simulation was performed to estimate the strain rate ranges in impact and blast. It was found that the brain and skull experience strain rates in the range of 36 to

Math. Comput. Appl. 2020, 25, 21 12 of 15 anticipated application rates. However, to calculate the material properties of the brain, brainstem, dura, and skull related to TBI cases, we need to know for which range of strain rates they are required to be examined. In this study, FE simulation was performed to estimate the strain rate ranges in impact and blast. It was found that the brain and skull experience strain rates in the range of 36 to 1 241 and 14 to 182 s− for impact and blast incidents, respectively. Additionally, in such dynamical 1 simulations, the brainstem and dura behave at strain rates of 15 to 412 and 8 to 149 s− , respectively. Knowing these ranges provides a good measurement for the more accurate characterization of these materials. Additionally, representing the rates based on head acceleration helps to get a better idea of selecting material properties in advance. It was shown in the impact incidents that the strain rates of the brain, brainstem, dura, and skull were approximately 1.9, 0.7, 1.18, and 0.7 times the resulting head acceleration. Furthermore, under blast loadings, the strain rates for those head organs were around 0.86, 1.25, 0.52, and 0.43 times head acceleration.

Author Contributions: Conceptualization, M.H.-F. and G.K.; methodology, M.H.-F.; software, M.H.-F. and M.A.-T.-Z.; validation, M.H.-F. and M.R.; formal analysis, M.H.-F.; investigation, M.H.-F.; resources, M.H.-F., M.Z., G.K.; data curation, M.H.-F. and M.A.-T.-Z.; writing—original draft preparation, M.H.-F.; writing—review and editing, M.H.-F., M.A.-T.-Z., M.R., M.Z., G.K.; visualization, M.H.-F.; supervision, M.Z. and G.K.; All authors have read and agreed to the published version of the manuscript. Acknowledgments: The Authors would like to thank Mehdi Salimi Jazi, Hesam Sarvghad-Moghaddam, Asghar Rezaei and Ashkan Eslaminejad for their support and providing us with the validated FE head model. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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