



ORIGINAL ARTICLE 

Year : 2017  Volume
: 6
 Issue : 3  Page : 4148 

A Viscohyperelastic model for prediction of the brain tissue response and the traumatic brain injuries
Hossein Ashrafi^{1}, M Shariyat^{2}
^{1} Department of Applied Design, Faculty of Mechanical Engineering, University of Kashan, Iran ^{2} Department of Applied Design, Faculty of Mechanical Engineering, K. N. Toosi University of Technology, Tehran, Iran
Date of Web Publication  29Nov2017 
Correspondence Address: Dr. Hossein Ashrafi Faculty of Mechanical Engineering, University of Kashan, Kashan Iran
Source of Support: None, Conflict of Interest: None
DOI: 10.4103/atr.atr_9_17
Introduction: Numerous geometrically simplified models may be found in the literature on simulation of the traumatic brain injuries due to the increased intracranial pressure induced by severe translational accelerations of the brain inside the cranium following the impact waves. While numerous researchers have utilized viscoelastic models, some have employed specific hyperelastic models for behavior analysis of the brain tissue. No research has been presented so far based on the more realistic viscohyperelastic model. Materials and Methods: In the present research, a realistic finite element model and four viscohyperelastic constitutive models (viscoelastic models on the basis of the polynomial, Yeoh, ArrudaBoyce, and Ogden hyperelastic models) are employed to accomplish the outlined task. Therefore, the main motivation of the present research is checking the accuracy of the modeling procedure rather than presenting clinical results. In this regard, a realistic skullbrain model is constructed in CATIA computer code based on the magnetic resonance imaging scans and optimized in the HYPERMESH finite element software. Results: Influence of the contact and nonlinear characteristics of the brain tissue are considered in the simulation of the relative motions in LSDYNA software to predict time histories of the acceleration and the coup and countercoup pressures by means of ANSYS finite element analysis software. Discussion: Comparing results of the four proposed viscohyperelastic constitutive models with the available experimental reveals that employing Arruda–Boyce or Ogdentype viscoelastic models may lead to inaccurate or even erroneous results. Keywords: Brain tissue, intracranial pressure, traumatic brain injuries, viscohyperelastic finite element model
How to cite this article: Ashrafi H, Shariyat M. A Viscohyperelastic model for prediction of the brain tissue response and the traumatic brain injuries. Arch Trauma Res 2017;6:418 
How to cite this URL: Ashrafi H, Shariyat M. A Viscohyperelastic model for prediction of the brain tissue response and the traumatic brain injuries. Arch Trauma Res [serial online] 2017 [cited 2018 Aug 21];6:418. Available from: http://www.archtrauma.com/text.asp?2017/6/3/41/219407 
Introduction   
The relative translational motions of the brain due to severe accelerations and decelerations, lead to increased intracranial pressure gradients on the brain tissue, cerebrospinal fluid (CSF), and the brain's blood supply vessels. The resulting pressures gradients can lead to traumatic brain injuries, blood vessels damages, and restricting blood flow to the brain and brain trauma, and consequently, to fatal conditions if the pressure exceeds 40 mmHg in adult persons.^{[1]} The relative motions of the brain with respect to the skull, striking and bouncing of the parenchyma against the inner skull protrusions, cavitation phenomena induced by negative pressures, and rupture of the bridging veins, axonal fibers, and vascular tissue are among the frequent origins of the damages.^{[2],[3]} Yue et al.^{[4],[5]} studied dynamic characteristics of the human skulldura mater system and deformations of the human skull due to variations in the intracranial pressure, using a hollow sphere finite element model. The skull was simulated by a thinwalled composite shell, and Maxwell's viscoelasticity model was employed for the human skulldura mater. The important mechanical characteristics of the cancellous bone and dura mater is viscoelasticity.^{[6],[7]} Willinger et al.^{[8]} and Ding et al.^{[9]} determined mechanical properties and characteristics of the compact bone, cancellous bone, and dura mater.
The early finite element models for the skull/brain complex were presented based on a simple fluidfilled spherical shell. Zhang et al.,^{[10]} Willinger et al.,^{[11]} Kleiven and von Holst,^{[12]} and Horgan and Gilchrist ^{[13]} have recently developed more complicated threedimensional (3D) models based on anatomical drawings and medical images, such as computed tomography (CT) and magnetic resonance imaging (MRI) scans. Liu et al.^{[14]} employed a transparent physical head model with air bubbles to study the brain cavitation phenomenon in a head deceleration impact. The transparent skull model was generated based on a real human skull through a turnover formwork technique, and a transparent gel was used to represent the brain tissue. Recently, Chen and OstojaStarzewski ^{[15]} presented a 3D finite element model for the human head that accounts for the important geometric characteristics of the various components within the human head, through using an efficient MRI voxelbased mesh generation method. El Sayed et al.^{[2]} presented a biomechanical model for simulation of the traumatic brain injury and damages of the soft tissues to simulate the axonal damage and cavitation injury through inelastic deformations due to frontal and oblique head impacts with external objects. The material response was divided into elastoplastic and viscoelastic components, including rate effects, shear and porous plasticity, and finite viscoelasticity.
Some researches have considered simple hyperelastic or finite viscoelastic models.^{[16],[17],[18],[19],[20],[21],[22]} Elastic and hyperelastic properties of the brain gray and white matters were characterized by Kaster et al.,^{[23]} based on the forcedisplacement data of the tissues for 25 different brain samples, using an indentation apparatus, and the material properties of the Polynomial, Yeoh, Arruda–Boyce, and Ogden hyperelastic models were obtained. Recently, Post et al.^{[24]} studied effects of loadings with an identical area beneath the curves, on the time history of the resulting von Mises stress and maximum principal strain as measures of the brain tissue damage.
In the present paper, several viscohyperelastic finite element models are presented for simulation of the traumatic brain injuries due to the increased intracranial pressure caused by severe translational accelerations and decelerations. In this regard, a realistic skull–brain model based on the MRI is created in CATIA modeling software and optimized in HYPERMESH finite element computer code. Influence of the contact and nonlinear characteristics of the brain tissues are considered in the simulation of the relative motions in LSDYNA software. Finally, time histories of the acceleration and the pressures are determined based on ANSYS finite element analysis software. In this regard, results of various viscohyperelastic constitutive models are studied and compared with each other and with the available experimental results.
Geometric and FiniteElement Modeling   
The skull is composed of a braincase (neurocranium) and a facial skeleton (splanchnocranium, viscerocranium), both sharing the base of the skull, which descends obliquely backward [Figure 1].
The cerebrum is the highest and biggest region of the brain that covers the majority of the portions of the brain. Almost all of the skull space is filled by cerebrum that is composed of the right and left cerebral hemispheres, and several bilateral gray nuclei (basal ganglia). These two parts are separated at all sections, but in some inside regions, they have little connections by some white fibers. The cerebral hemisphere consists of an outer gray cortical layer and an inner white core composed mainly of nerve fibers. The CSF is a clear fluid, which fills the whole subarachnoid space and acts as a protective fluid cushion around the brain and the spinal cord and damps the externally imposed shocks.
The realistic brain model is constructed based on the MRI of the brain and cranium which some of them are shown in [Figure 2].^{[25]} The whole scan set consists of 55 parallel sections. The 3D geometric and finite element models were reconstructed based on the axial MRI available in “The Whole Brain Atlas ^{More Details}” of Harvard Medical School.^{[26]} In the present research, each section is defined by many key points; so that, the assembly of the whole sections of the brain have constituted clouds of key points. The geometric model of the brain tissue is constructed in CATIA computer code, through passing surfaces through the key points of the clouds associated with each individual matter of the brain to form the whole brain model. It is known that each closed surface is identified as a volume in the computeraided design software, for example, CATIA computer code. The reader may also refer to a book by Saba ^{[27]} on image principles of the brain, for more details. The finite element model [Figure 3] is constructed in HYPERMESH finite element software to obtain elements with optimized topologies. The resulting finite element model includes the: (1) CSF in the form of a 3mm thick layer, (2) gray matter, (3) white matter, (4) cerebellum, (5) corpus callosum, (6) telencephalic nuclei, (7) brain stem, and (8) ventricles. The whole assembly is illustrated in [Figure 4].  Figure 2: Magnetic resonance imaging scans of the brain and cranium.^{[2]}
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 Figure 4: The skull–brain finite element assembly constructed in HYPERMESH software
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10noded secondorder tetrahedral composite elements whose material properties are similar to those of the Wayne State Brain Injury Model,^{[26]} are employed to discretize the model. The 3 (mm) gap between the inner surface of the cranium and the outer surface of the brain is filled with the CSF. Due to the complexity of the geometries, the general contact constraint is defined between these three media and the dynamic simulation is accomplished in LSDYNA software.
Description of the Material Properties   
Mechanical properties of the human skull and the CSF are listed in [Table 1]. Based on the studies performed on deformations of the brain tissue under impact loading, it has been observed that these tissues exhibit hyperelastic behaviors.^{[16],[17],[18],[19],[20],[21],[22]} Hyperelasticity refers to a constitutive response that is derivable from an elastic potential function W^{ (e)} and is typically used for nearly incompressible materials which experience nonlinear large elastic deformations such as rubber and some biological materials. When the strain energy density function per unit
undeformed volume is defined, one may determine the stressstrain expressions from:
Where S_{ij}, E_{ij}, C_{ij}, and (I_{1}, I_{2}, I_{3}) are components of the second Piola–Kirchhoff stress tensor, components of the Lagrangian strain tensor, components of the right Cauchy–Green deformation tensor, and the stress invariants, respectively. Several forms of the strain energy potential have been proposed for simulation of the incompressible or nearly incompressible hyperelastic materials, among them: Arruda–Boyce, BlatzKo, extended tube, Gent, Mooney–Rivlin, neoHookean, Ogden potential, polynomial form, and Yeoh models.
Evaluation of efficiency and accuracy of the hyperelastic models for the behavior and injury simulation of the skullbrain assembly is an important issue. In the present research, four wellknown hyperelasticity models are adopted and modified to include the viscoelastic behaviors of the material, and their predictions are compared with the available experimental results.
Polynomial model
In this hyperelastic model, the strain energy potential is expressed by:
Where C_{ij}, I_{1}, I_{2}, I_{3}, and D_{k} are material constants, the first and second invariants of the strain tensor, the elastic volume strain, and the material incompressibility parameter, respectively. The neoHookean model can be obtained by setting n = 1 and C_{01 }= 0. In our comparative analyses, the secondorder polynomial strain energy density function which is equivalent to the fiveparameter Mooney–Rivlin model (n = 2) is adopted. In this model, the initial shear and bulk moduli are defined as:
Yeoh model
The Yeoh model ^{[28]} is a reduced and revised form of the thirdorder polynomial model, wherein the strain energy potential depends on the first strain invariant only:
The notations are identical to those of the previous model. In this model, the initial shear and bulk moduli are defined as:
ArrudaBoyce model
Arruda–Boyce model depends on the first strain invariant, only and the relevant strain energy density function is:
Where, , , , , λ is the locking stretch, and μ is the initial shear modulus.^{[29]}λ and μ are measurable parameters and:
Therefore, as the parameter λ tends to infinity, the model tends to the neoHookean model.
Ogden model
The Ogden form of the strainenergy potential density is dependent on the principal stretches of leftCauchy strain tensor:
Where α_{i} are nondimensional exponents and λ_{i} are the deviatoric stretches. The firstorder Ogden model depends on the two parameters that are the initial shear modulus μ and α_{i}. In this model, the initial shear and bulk moduli may be found from:
The constitutive equations of the viscohyperelastic models
The total strain tensor of the viscohyperelastic material may be considered to be composed of hyperelastic and viscoelastic components. Using Volterra–Boltzmann representation of the elastic behavior, one may write ^{[2],[30],[31],[32],[33],[34]}:
Where N, , a_{r} , K , e_{j} , and q are the number of the required Ogden's functions to model the shear deformations, shear moduli, dimensionless stretch component, bulk modulus, and eigenvalues of the elastic logarithmic shear strains associated with the principal stretches and logarithmic volumetric strain,^{[2]} respectively. , that is, the strain energy density function due to the hyperelastic nature of the material per unit volume were given in equations 2, 4, 6, 8 for the considered four hyperelastic models.
Boundary conditions
In the present research, the possibility of the brain injury is investigated through simulation of the impact of the frontal region of the skull with a rigid plate. The initial velocity of the skull is assumed to be 7.5 m/s (27 km/m). Value of the initial velocity is chosen identical to that of an experiment performed by Nahum et al.^{[35]}
Results and Discussions   
In the present section, a comparative study is carried out for a better judgment about the usefulness of the viscohyperelastic models introduced in section 3, for behavior simulation of the skullbrain system. Such comparative study has not been performed so far. The imposed boundary conditions are defined in section 4. Material properties of the adopted hyperelastic models are listed in [Table 2]. The viscoelastic parameters of the materials are given in [Table 3].  Table 2: Material properties of the considered hyperelasticity models^{[26]}
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Results of the present finite element model are compared with the experimental ones. Quadratic pyramid elements are used to discretize the viscohyperelastic model. Convergence of the resulting finite element mesh is checked through comparing results of successive refinement of the mesh; so that finally, ignorable changes in the results were noticed by further refinement of the mesh. The performed convergence study whose results are not included here to save space has revealed that choosing about 20000 elements is vital for achieving convergent results.
The experimental results include time variations of pressures of the coup and countercoup regions [Figure 5] reported by Nahum et al.^{[35]} and time history of acceleration of the center of gravity of the brain measured by Trosseille et al.^{[36]} The time histories of pressures of the coup and countercoup regions are extracted from an article by Chen and OstojaStarzewsky.^{[15]} Timedependency of the shear modulus, due to the viscoelastic nature of the material properties, may be defined as:  Figure 5: Different views of the coup and countercoup regions of the brain^{[37]}
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Where and are the shortterm and longterm shear moduli, respectively.
Results of the viscohyperelastic version of the polynomial, Yeoh, Arruda–Boyce, and Ogden hyperelasticity models are illustrated in [Figure 6] and [Figure 7] for the coup and countercoup regions, respectively, and compared with the experimental results. Deviations of the predicted peak pressures from those of the experimental ones are given in [Table 4], for various viscohyperelastic models.  Figure 6: A comparison among time histories of the coup pressure predicted by various visco.hyperelastic models and Nahum et al. experimental results
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 Figure 7: A comparison among time histories of the countercoup pressure predicted by various viscohyperelastic models and Nahum et al. experimental results
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 Table 4: Deviations of the peak pressures prediced by various models from Nahum et al. experimental results
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Results illustrated in [Figure 6] and [Figure 7] reveal that the maximum magnitudes of the coup and countercoup pressures predicted by the polynomial viscohyperelastic model are less than those of the experimental results. However, based on results listed in [Table 4], the discrepancies between the finite element and the experimental results are lowest for the polynomial constitutive model. Moreover, curvatures of the time history graphs of the finite element and experimental results show the highest concordance; so that effects of the vacuums occurred in the coup and countercoup regions in the CSF have been simulated more accurately. Therefore, the accuracy of predictions of these models is more reliable.
On the other hand, since the polynomial viscohyperelastic model has underestimated the coup and countercoup peak pressures, its results are not on the safe side. The peaks are slightly postponed in comparison with the experimental results. Results of Yeoh viscohyperelastic model [Figure 7], [Figure 8], [Figure 9], [Figure 10] are slightly greater than the experimental results. Thus, they are on the safe side. On the other hand, accuracy on these results is located in the second rank (after the polynomial viscohyperelastic model). The predicted behaviors for the vacuum regions are not reliable. The peak pressures have occurred slightly before the real ones.  Figure 8: Distribution of the coup pressure predicted by the polynomial viscohyperelastic model
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 Figure 9: Distribution of the coup pressure predicted by Yeoh viscohyperelastic model
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 Figure 10: Distribution of the coup pressure predicted by Arruda.Boyce viscohyperelastic model
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In Arruda–Boyce viscohyperelastic model, occurrence times of the peak pressures are close, to a great extent, to those of the experimental ones. But as results of [Table 4] confirm, discrepancies between predictions of this model and the experimental results are significant. Magnitudes of the peak pressures predicted by this model are remarkably higher than those of the experimental ones. In comparison to the previous viscohyperelastic models, a much oscillatory response is observed in times following the peak pressures. However, responses of this region of the time history are more consistent with the experimental results, in comparison to Yeoh model. Therefore, it may be concluded that this model does not simulate the viscohyperelastic nature of the brain tissue accurately.
The last viscohyperelastic model, that is, Ogden model significantly overestimates the peak pressures of both the coup and countercoup regions of the brain. Results presented in [Figure 6] and [Figure 7] and results reported in [Table 4] reveal that in spite of using this model by some researchers,^{[2]} results of this model are not reliable at all. Results of Ogden model show the greatest discrepancies with respect to the experimental results. Furthermore, redundant oscillations have appeared in the predicted responses shown in [Figure 6] and [Figure 7].
However, the absolute peak of each response is the most important parameter for injury assessment of the brain tissue and the blood vessels that are in the neighborhood of the coup and countercoup regions. Therefore, the minor oscillations can be ignored in the responses. Comparing results shown in [Figure 6] and [Figure 7] with experimental results reported in ^{[36],[38],[39]} reveals that predictions of the polynomial model are even closer to those of the real ones.
[Figure 6] and [Figure 7] reveal that the coup peak pressures are greater than those of the countercoup region. The distributions predicted for the coup pressure by the four viscohyperelastic models are shown in [Figure 8], [Figure 9], [Figure 10], [Figure 11].  Figure 11: Distribution of the coup pressure predicted by Ogden viscohyperelastic model
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One of the discrepancy sources is the shortcoming in accurately modeling the inertial and volume characteristics. For this reason, time histories of the acceleration of the center of gravity of the brain are determined by LSDYNA software for the various viscohyperelastic models and plotted in [Figure 12]. Results of the polynomial viscohyperelastic model show a better agreement with the time history reported by Trosseille et al.^{[36]} regarding both the peak acceleration and the peak occurrence time. Again, Ogden model has led to the worst results. Results of Yeoh model are still in the second rank. Therefore, both acceleration of the center of gravity of the brain and the pressure results have led to identical conclusions that the models may be ordered with respect to the accuracy as the polynomial, Yeoh, Arruda–Boyce, and Ogden models.
Zhou et al.^{[40]} reported that the mild traumatic brain injuries occur at a brain von Mises stress of 20kPa. Baumgartner and Willinger ^{[41]} observed that a brain von Mises stress of 18 kPa generates moderate neurological lesions which become severe from 38 kPa. Magnitudes of all the obtained coup and countercoup stresses are above 38 kPa. Therefore, all of the considered viscohyperelastic models predict severe neurological lesions.
Conclusions   
In the present research, a comparison is made among results of various (polynomial, Yeoh, Arruda–Boyce, and Ogden) viscohyperelastic models that may be employed for simulation of the traumatic brain injuries, for the first time. CATIA geometry modeling, HYPERMESH and ANSYS finite element modeling, and LSDYNA nonlinear dynamic finite element analysis computer codes are employed in the present research. A pair of experimental results is employed to evaluate the accuracy of the resulting peak pressures and accelerations of the center of gravity of the brain predicted by the mentioned four viscohyperelastic models. Comparing with both experimental results confirm that employing Arruda–Boyce or Ogden models may lead to inaccurate or even erroneous results. Comparisons made with the experimental results of accelerations of the center of gravity of the brain and pressure results of the coup and countercoup regions have led to an identical conclusion that the models may be ordered with respect to the accuracy as the polynomial, Yeoh, Arruda–Boyce, and Ogden models.
Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
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[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7], [Figure 8], [Figure 9], [Figure 10], [Figure 11], [Figure 12]
[Table 1], [Table 2], [Table 3], [Table 4]
