Topological features dictate the mechanics of the mammalian brains

Topological features dictate the mechanics of the mammalian brains1 P. S´aeza,b,∗, C. Du˜n´oa, L.Y. Sunc, N. Antonovaited, M. Malv`ee,f, D. Tostc,g, A. Gorielyh 2 aLaboratori de Calcul Numeric (LaCaN), Universitat Polit`ecnica de Catalunya, Barcelona, Spain.3 bBarcelona Graduate School of Mathematics–BGSMath, Barcelona, Spain4 cCentre de Recerca en Enginyeria Biom`edica (CREB),Universitat Polit`ecnica de Catalunya, Barcelona, Spain5 dDepartment of Physics and Astronomy and LaserLab Amsterdam, Vrije Universiteit Amsterdam, De Boelelaan6 1085, 1081 HV Amsterdam, Netherlands7 eCentro de Investigaci´on Biom´edica en Red en Bioingenier´ıa Biomateriales y Nanomedicina (CIBER-BBN),8 Arag´on Institute of Engineering Research (I3A), Universidad de Zaragoza9 fDepartment of Engineering, Public University of Navarra, Pamplona, Spain10 gInstitut de Recerca Sant Joan de D´eu, Barcelona, Spain.11 hMathematical Institute, University of Oxford, Oxford, UK.12 Abstract13 Understanding brain mechanics is crucial in the study of pathologies involving brain deformations such as tumor, strokes, or in traumatic brain injury. Apart from the intrinsic mechanical properties of the brain tissue, the topology and geometry of the mammalian brains are particularly important for its mechanical response. We use computational methods in combination with geometric models to understand the role of these features. We find that the geometric quantifiers such as the gyrification index play a fundamental role in the overall mechanical response of the brain. We further demonstrate that topological diversity in brain models is more important than differences in mechanical properties: Topological differences modify not only the stresses and strains in the brain but also its spatial distribution. Therefore, computational brain models should always include detailed geometric information to generate accurate mechanical predictions. These results suggest that mammalian brain gyrification acts as a damping system to reduce mechanical damage in large-mass brain mammals. Our results are relevant in several areas of science and engineering related to brain mechanics, including the study of tumor growth, the understanding of brain folding, and the analysis of traumatic brain injuries. Keywords: Brain shape, animal-scale laws, Brain Mechanics, Finite Element Method.14 1. Introduction15 The mammalian brain is, arguably, the most intriguing and unexplored organ. It acts as a16 command center for the nervous system where distinct regions are mostly responsible for controlling17 specific cognitive functions such as perception, emotion, behavior, or motor function. The majority18 of brain research is focused on understanding the ways in which the brain cognition works. However,19 brain mechanics, the way in which the brain deforms under external or internal mechanical loads,20 is fundamental for the analysis of a large number of brain pathologies that are strain-dependent21 and that modify brain function [11, 67, 25, 24]. For example, tumor growth has been demonstrated22 to be highly dependent on the mechanical environment [29, 8]. Computational models have been23 used to quantify the deformation that a brain experiences during tumor progression [57, 62, 2].24 Swelling process in the brain induce large deformation in the brain tissue [21, 68, 43] and different25 mathematical and computational models have been developed to analyze the mechanical response26 of the brain under such internal loads [26, 15, 44]. Computational models have also been used27 ∗Corresponding author: P. S´aez Email address: [email protected] (P. S´aez ) Preprint submitted to International Journal of Mechanical Sciences August 31, 2020 to evaluate mechanical deformations during decompressive craniotomies [16, 19, 70]. Traumatic28 brain injuries (TBI) depend on the mechanical injury that the brain suffers [22, 54, 72] and many29 experimental [10, 45, 39, 58] and computational models [38, 17, 41, 71, 42] have been used to30 evaluate the impact of mechanical loads in brain damage mechanics.31 At the tissue level, experimental tests have been conducted to determine the mechanical prop-32 erties of the brain [51, 18, 12, 6] where a wide range of stiffness values of mammalian brains have33 been reported. Today, it is accepted that the stiffness of brain tissue is in the range 0.1-4 kPa34 [37, 6], depending on the brain region probed [40, 3]. Measured through Magnetic Resonance35 Elastography (MRE), the stiffness of the brain tissue has been measured to be within a range of36 1 to 20 kPa [50, 35, 69, 33], but most studies place brain tissues within the 3-6kPa range, close37 to the values reported by mechanical testing. By means of MRE it was also suggested that there38 are not variations in tissue stiffness between species [69] and that stiffness differences mainly arise39 due to tissue composition [40, 3]. The brain tissue also exhibits viscoelastic behavior [53, 12, 6],40 both at small and large strain although it is still unclear which components of the brain tissue41 determine the viscoelastic response at the tissue scale. Along with experimental characterizations42 and modeling efforts, different computational models have been proposed to study the mechanical43 response of the mammalian brain under static and dynamics loads [49, 63, 34]. At the organ level,44 computational models are used to predict mechanical states under internal or external mechanical45 loads, bypassing costly and potentially unethical experimental tests. However, many computational46 models are not fully reliable because the complexity of the brain geometry under study is not well47 represented, which is central to many disciplines that involve brain mechanics. In particular, the48 influence of brain geometry has not been analyzed in computational models and the vast majority of49 models rely on coarse geometrical representations [55, 36, 52, 66]. Indeed, only a few models have50 been able to present accurate geometrical reconstruction and perform computational mechanics51 simulations [41, 64]. Moreover, many experimental and computational models use results from one52 specific mammalian species to extrapolate mechanical information to another mammalian brain,53 usually humans. It is therefore key to understand whether the mechanical response of the brains54 in different animal species under a given mechanical load is equivalent and what are the key brain55 features that make brain mechanics different from one species to another.56 A standard quantifier for brain morphology is the gyrification index (GI) defined as the ratio57 of total cortical surface to the area of an outer surface (the convex hull) that smoothly encloses58 the cortex. The GI varies dramatically between species [46] with values close to one for mammals59 with smooth brains such as manatees and rodents, and as high as 2.5 for the human brain and 2.760 for some odontocetes [59, 48]. Several physical models have demonstrated that mechanics forces61 control folding during development [25, 61, 32]. However, only few studies have accounted for62 the effect of small topological features, e.g. sulci, in the mechanics of the brain [14, 30]. Here,63 we argue that brains with very different folding patterns present very different overall mechanical64 responses. Specifically, we address two important questions in brain mechanics: What is the role of65 mammalian brain folding in the mechanics of the brain? How important are topological variations66 in comparison to the elastic and viscoelastic responses for brain mechanics?67 2. Material and Methods68 2.1. Image-based geometrical reconstruction69 We reconstructed the external surface of three mammalian brains: a human, a macaque and a70 mouse brain, which represent a GI ≈2.5, 1.75 and 1, respectively. DICOM data (Digital Imaging71 and Communications in Medicine) were collected from the Center For in Vivo Microscopy at the72 Duke University Medical Center for the mouse and the manatee models. The mouse brain was73 2 obtained using a wild-type adult male C57BL/6 mouse. The human brain were obtained by means74 of MRI data collected from the Human Connectome Project (Subject Id: Id 100307). The medical75 images were manually segmented using the software package 3DSlicer (BSD-style open source, The76 Open Source Initiative). Each STL file coming from the geometrical segmentation was imported77 in the commercial software Rhinoceros for a further smoothing and merging of the different parts78 composing every single brain model to obtain surfaces that compose the 3D geometry, an STL79 (Stereo Lithography) file.80 2.2. Generation of the computational grid81 The reconstructed STL files containing the various parts of each brain model (cerebellum, left82 and right cerebrum, gray matter, cranium etc....) were imported into the commercial computer83 aided design software (CAD) Rhinoceros (McNeel and Associates, Indianapolis, IN, USA). With84 this software, all the brain structures were merged in one single model for each considered case.85 Each complete model was finally exported and used to create 3D computational grids. Fig. 1 shows86 the resulting geometry models of the three mammal brains. The meshes were generated by means of87 the commercial software package Ansys IcemCFD (Ansys Inc., Canonsburg, Pennsylvania, USA).88 The STL files containing the compete model of the rat, the macaque and the human brain were89 separately imported into Ansys IcemCFD. Due to the intrinsic complexity of the brain geometry90 that is composed by several tortuous regions, an unstructured topology was selected. The semi-91 automated octree algorithm of Ansys IcemCFD was used to generate the unstructured tetrahedral92 grids by assigning separate surface cell sizes to different geometric parts and an overall grid expan-93 sion ratio. The first preliminary step for grid generation was to split-up the geometry into multiple94 surface parts, which permitted regional specification of grid resolution. With this aim, a multiple95 external and internal surfaces were defined, permitting variable cerebrum and cerebrospinal fluid96 grid refinement. The element size, that is the minimum length of any side of all tetrahedral el-97 ements in the model, should be enough small to acceptably discretize the model volume. In the98 other side, it is well known that an excessive number of elements may results in an increase of99 computational costs. For this reason, for establishing the adequate element size for each model,100 a mesh independence analysis was carried out. Different resolutions were considered progressively101 increasing the element size and a structural analysis was performed in each case. The computed102 strains were compared among different mesh resolutions. After this analysis, the element size se-103 lected for the mouse resulted in 0.1mm, for the macaque was 0.7mm and for the human brain104 resulted in 1.2mm. This means that meshes using smaller element sizes resulted in an increase of105 computational time without adding precision to the numerical solution while coarser grids resulted106 inadequate. The mouse brain was composed of 67820 nodes and 33914 elements. The brain tissue107 was composed of 27298 elements and 54592 nodes and the CSF section was made of 6616 elements108 and 13228 nodes. The macaque model was composed by 295302 elements and 590640 nodes. The109 brain tissue was composed by 528472 nodes and 264216 elements and the CSF was made of 31086110 nodes and 62168 elements. Finally, the human model was composed of 2,062,416 elements for the111 CSF. The grey and white matter are made of 1,678,496 and 5,301,024 elements, respectively. The112 right and left hemisphere of the gray matter have 844,640 and 833,856 elements respectively. The113 white matter is made of 777,952 and 782,160 for the right and left side respectively. Multiple114 smoothing iterations were carried out for improving the grid quality.115 2.3. Mechanical testing of the brain tissue116 We provide here a basic explanation of how the mechanical tests were performed (see [3] for117 more details). Horizontal brain tissue slices of 300 um thickness were extracted from 6 months118 old C57 / BL6(Harlan) mouse. All experiments were performed in accordance with protocols and119 3 Figure 1: Coronal sections of a human (a), rhesus macaque (b) and common mouse (c) brain (adapted from http://brainmuseum.org). Reconstruction of a finite element mesh for the three species (d-f). Fluorescence image of hippocampus stained for nuclei (Hoechst). Storage modulus map over regions with low density of axons and cells obtained at 6.6% strain, 5.62Hz frequency with equilibrium frequency sweep (f). Storage modulus map of other hippocampal regions at 6.6% strain, 5.62Hz frequency previously obtained with oscillatory ramp (g)[3]. guidelines approved by the Institutional Animal Care and Use Committee (UvA-DEC) operating120 under standards set by EU Directive 2010/63/EU. Sample was placed in a glass bottom chamber121 for imaging with inverted microscope, supplied with carbonated artificial cerebrospinal fluid to122 maintain the viability and gently pressed down with harp for stabilization. Measurements were123 performed within 6 hours after extraction.124 Indentation setup consist of a cantilever-based Ferrule-top probe (k=0.2 N/m, R=95 um)125 equipped with interferometric readout mounted on a piezo transducer and XYZ manipulator (more126 details [3]). Indentation was performed in an indentation-depth controlled mode at 5 um/s inden-127 tation speed, up to 10 um depth, followed by 30 s load relaxation and frequency sweep between128 1Hz and 10 Hz. Stratum lacunosum-moleculare region situated in hippocampus was selected for129 the measurements due to relatively low density of cells and axons. In comparison to white matter130 we can see that density of axons is low here. The fluorescence image of the region of interest and131 the storage modulus map obtained from a Hertz model is shown in Fig. 1.132 2.4. Kinematics of the brain dynamics problem133 We use the theory of nonlinear elasticity [27]. We characterize the spatial motion problem is134 through the spatial motion map,135 x=χ(X, t ) : Ω0→Ωt,(1) between the material placement Xof a particle in Ω0, to the spatial placement xof the same136 particle in the spatial configuration Ωt. The deformation gradient Fand its Jacobian Jare defined137 then as138 F=∇Xχ(X, t) : TΩ0→TΩtJ= det F>0,(2) 4 representing the linear tangent map from the tangent space TΩ0to the time–dependent tangent139 space TΩt. We also define the right spatial Cauchy– Green strain tensor C=Ft·F. Finally, we140 define the material time derivative Dtof the spatial velocity v= Dtχ(X, t).141 2.5. Balance of linear momentum142 The mass specific version of the balance of linear momentum is based on the kinetic energy143 density, K=1 2v·v. The motion momentum density pis defined by the partial derivative with144 respect to the spatial velocity vas145 p=∂vK=v(3) The rate of change of the mass specific spatial motion momentum density pis balanced with146 the momentum fluxes σand the reduced momentum sources bas147 ρdtp=∇ · σ+b(4) where the Neumann boundary are148 σ·n=t(5) 2.6. Constitutive equations in brain dynamics149 We model the tissue as a quasi-incompressible hyperelastic material with a viscoelastic compo-150 nent and use the framework of large strains theory, that relies on the definition of a strain-energy151 density function (energy for short) Ψ( ¯ C). The quasi-incompressible behavior of the tissue is repro-152 duced through a volumetric-isochoric decomposition of the deformation gradient, which was first153 proposed by [20]. The deformation gradient Fis decoupled into dilatational and volume-preserving154 part as F= J1/3¯ F, where ¯ Fis the isochoric deformation gradient. Consequently, we split the155 energy as156 Ψ(J, ¯ C) = Ψvol(J)+Ψich(¯ C),(6) where Ψvol is related with the water content in the brain. The second term Ψich(¯ C) is associated157 with the isochoric contribution of the deformation gradient, which is associated with the solid158 components of the tissue and ¯ Ccorresponds with the right Cauchy Green tensor. The isochoric159 contribution can again be split up into different parts to model the behavior of the different com-160 ponents. In terms of the mechanical behavior of the brain tissue, we consider the gray matter161 to be mechanically isotropic. On the other hand, white matter can be described as anisotropic162 due to axonal structure lining in a preferential direction as it has been described in [41]. In this163 contribution and for sake of analyzing exclusively the mechanics of the folding pattern, we consider164 also the white matter as isotropic. The elastic response of the both the gray and white matter is165 fully characterized by a Mooney-Rivlin energy function166 Ψiso(I1, I2) = C10[I1−3] + C01[I2−3] (7) where C10 kPa and C01 kPa are material parameters and I1=tr(¯ C) and I2= 0.5I2 1−(¯ C·¯ C)] are167 the first and second invariant of the isochoric part of the deformation. The elastic Piola-Kirchhoff168 stress tensor is then given by169 S= 2∂CΨvol + 2∂CΨich(¯ C) = Svol +Sich,with (8) 170 Svol = 2∂CΨvol and Sich = 2∂¯ CΨich(¯ C) : ∂C¯ C= J−2/3P:¯ S(9) the volumetric and isochoric Piola-Kirchhoff stress tensor. ¯ S=∂¯ CΨich(¯ C) is the fictitious second171 Piola-Kirchhoff stress and Pis the fourth order projection tensor in the material reference defined172 as P=I−1/3C−1⊗C. Then, we recover the Cauchy stress tensor as σ=F·S·FT 173 5 Average 14 samples C10 [Pa] C01 [Pa] g1[-] τ1[s] Mean 668 431 0,90 10,5 Standard deviation 175 207 0,02 3,5 Table 1: Averaged values of the elastic and viscoelastic parameters. The equivalent shear modulus of the tissue based on the values of the Mooney-Rivlin model (C10=668Pa C01=441Pa) is µ= 2(C01 + C01) ≈2.2kPa and the Young’s modulus is E=6.6kPa. These two values of shear and Young modulus are within the range of values reported in literature (see the Introduction section). To include the viscoelastic behavior, we use a one-term Prony formulation which is a powerful174 method for modeling of soft tissues [6]. The evolution of stiffness moduli in time is then given by175 Ci(t) = Ci0h1−g1[1 −e−(t/τ1)]i,(10) where g1is the characteristic time constants of the material and τ1is the the stiffness weight176 associated with the time constant. At long enough tomes, we obtain the steady-state moduli177 Ci∞= Ci0[1 −g1].178 We use the convolution integral to include the Prony series in the Mooney-Rivlin material model179 as180 Ψiso(I1, I2) = Zt 0C10(t−τ)[I1−3] ∂τ +C01(t−τ)[I2−3] ∂τ dτ (11) Note that we work with the deviatoric part of the energy as the volumetric behavior is assumed181 to be time independent. τare the time decay constant.182 We make use of a genetic algorithm strategy to find the best set of material parameters that183 reproduce each of the 14 indentation tests (see more details in [3]). The result of the material184 parameters were later spatially averaged to obtain a single value characterizing the brain that was185 used over the entire domain of the finite element models. Table 1 summarize the averaged quantities186 over the 14 samples.187 2.7. Finite element simulations188 The initial condition is a uniform velocity v0= 0.1m/s at time t0= 0s in all the nodes of the189 model, making a straight trajectory with a predefined displacement. Then, a linear deceleration190 beginning at t>0 up to t1= 25ms is followed by an imposed zero velocity at the nodes that belong191 to the outer surface of the model, corresponding with the skull, until the final time of simulation192 tf= 50ms. An adaptive time increment scheme was used to speed-up the simulations. We use193 the finite element software Abaqus [60] to solve the dynamic simulation. The transient dynamic194 problem is solved explicitly in time ensuring stability criteria. The material models are implemented195 in user subroutines. The problem is parallelized and solved in a computer cluster with 5 nodes with196 24 cores Intel Xeon E5-2650L v3 (1,8GHz, 12N/24S, 9.60GT/s i 65W) and 256 GB RAM memory.197 3. Results198 3.1. Mechanical response of mouse, macaque and human brains199 We use the baseline elastic and viscoelastic parameters obtained by performing standard in-200 dentation tests on mouse brain slices, following the procedure described in Materials and Methods201 to compute the mechanical response of the brain models. We used the geometric model of the202 brain of three different mammalian species: a model of human brain (GI ≈2.5), a rhesus macaque203 6 (GI≈1.75), and a common mouse (GI≈1) using 3D MRI images from which we extracted polygonal204 surfaces after filtering and segmentation (see Material and Methods for details). In our simulation,205 an impact load was applied laterally to the brains and results of the maximum principal strain,206 maximum principal stress, maximum shear stress and pressure were computed. These mechanical207 variables are widely used in brain damage mechanics [17, 71]. As expected, the mechanical response208 varied dramatically between species due to the large differences in brain mass (data not shown).209 For instance, the mass of the human brain is about 120 times larger than the mass of the mouse210 brain. Hence the same impact induces much higher inertial forces within the brain and no sensible211 comparison can be obtained.212 3.2. Analysis of mass-scaled brain models213 To take into account the variation in mass and isolate the role of the topological differences214 among species, we scaled the volumes of the mouse and macaque brains so that they have the same215 mass as the human brain and computed the mechanical response of these three mass-scaled brains.216 We create a probability distribution function (PDF) by fitting a kernel-smoothing distribution of217 the maximum accumulated value over time of the variables of interest in all nodes of the finite218 element mesh. We choose a kernel distribution in order to avoid making assumption about the219 distribution of the data. The estimation of the PDF is given by220 ˆ fh(x) = 1 nh n X i=1 Kx−xi h,(12) where n is the sample size, K is the kernel function, h>0 is the bandwidth and x represent the221 sample data.222 The PDF provides a measure of the relative volume of brain tissue, subjected to a specific value223 of the mechanical variables. Doing so we can analyze quantitatively the differences in the mechanical224 variables, compare the maximum values of the variables and the distribution of the variable values225 for the brain models. Fig. 2 shows the results of the PDF and the map of mechanical variables in226 the brain models.227 The PDF along the three brain models, shown in Fig. 2(a-d), indicate that for every value228 of the mechanical variables the same relative amount of tissue subjected to that mechanical state229 would be obtained for the human and macaque models. However, we see that the mouse-scaled230 brain has a different response for all mechanical variables, suggesting that the GI of mammalian231 brains modify the amount of tissue that suffers the range of maximum mechanical stimuli under232 external loads.233 Fig. 2(e-m) shows how these mechanical variables are distributed in space by only showing234 elements subjected to values higher than a given critical value. This figure demonstrate that the235 spatial distribution of the mechanical variables are significantly different along the three models.236 In the human brain the locations reaching a high value are highly dispersed throughout the brain.237 In the mouse model, the maximum principal stress and pressure are localized in the lateral part238 of the brain along the zone of impact and both the maximum principal strain and shear stress239 are localized in mid regions of both hemispheres. These results clearly suggest that topological240 differences modify not only the values of stresses and strains in the brain but also their spatial241 distribution. In addition, the fact that high values of stresses are disperse through the brain as242 the GI increases suggests that the folding pattern of the mammalian brain could act as a damping243 system to reduce mechanical damage for high-mass brain mammals.244 7 Figure 2: PDF distribution for the maximum shear stress (a), maximum principal stress (b), maximum principal strain (c) and maximum pressure (d) for the mouse (pink), macaque (red) and human (blue) models. The mechanical variables are plotted over the human (e-h), macaque (i-l) and mouse brain models (m-p). Only elements with a maximum shear stress (e,i,m), maximum principal stress (f,j,n), maximum principal strain (g,k,o) and pressure (h,l,p) with values higher than 0.12kPa, 22kPa, 4.10−3and 0.1kPa, respectively, are shown. The human and macaque-scaled models behave similarly, with the curves for all mechanical variables overlapping very closely. For the maximum principal stress and pressure, we observed that the mouse model increases the PDF (i.e. the relative amount of tissue) subjected to the highest values, 0.2-0.3 kPa and 0.1-0.15 kPa, respectively. The maximum shear stress also presents a significant increase of the amount of tissue subjected to a shear stress of ≈0.1 kPa. 8 3.3. Analysis of elastic and viscous response of the brain245 To investigate the relative effect of topological differences compared to the effect of material246 properties we vary the elastic and viscoelastic properties of the brain tissues in our simulations.247 We took the macaque model as control as an intermediate folding case between the human and248 the mouse model. First, we increased the Young’s modulus to simulate a tissue that is either 2.5249 times softer or 2.5 times stiffer. This range of stiffness covers previously reported values [51, 18,250 12, 6, 40, 3]. Second, we modified the viscoelastic behavior of the brain. We took the results of our251 mechanical tests as control and compared the results with an elastic model, without viscoelastic252 behavior. Then, we considered a more viscous tissue by using the values reported in [6] with two253 Prony terms (g1= 0.599kPa, g2= 0.241kPa, τ1= 3.49s, τ2= 298.55s). The PDF for the four254 mechanical variables of interest are shown in Fig. 3.255 Despite large variation of the tissue elastic properties, the simulations showed similar response256 for the maximum principal stress for the three values of the elastic properties. As expected, the257 distribution of principal strain and pressure of the softer material moved to the higher values of258 the variables. In addition, the amount of tissue subjected to the higher value of the variable259 visibly increased. The stiffer material moved the PDF of maximum shear stress to the right,260 i.e to higher values of the variables. However, we did not observed a non-uniform increase in261 the amount of tissue subjected to the higher values of the mechanical variables, as found in the262 mass-scaled models. Qualitatively, these changes are similar to the variations in the viscoelastic263 properties. Quantitatively, the variations in the amount of tissue for the maximum principal strain264 and pressure of the non-viscoelastic model clearly show a large increase of the amount of tissue265 subjected to large strain and pressure values. Note, however, that this is expected. In the absence of266 viscosity, there is no damping in the system and large stresses are expected. These results indicate267 that variations of the brain material response, such as elastic and viscoelastic properties, modify268 the state of stresses and strains in a uniform way. Our results also indicates that no changes for269 the amount of tissue subjected to specific ranges of the mechanical variables are observed for those270 realistic combinations of material parameters.271 3.4. Topological differences dictate the mechanics of mammalian brains272 To investigate the differences between mechanical properties and topological features in brain273 mechanics, we considered variations in the stiffness and viscoelastic parameters as well as the three274 mammal models. We were interested in investigating whether variations of mechanical properties275 are more or less relevant than differences in the folding pattern of the brain. We normalized the276 PDF of all models with respect to the macaque brain model, scaled to the human brain mass, with277 the mechanical values of our experiments which we took as the control case. Fig. 4 shows the278 normalization for the 4 variables of interests. We are interested in regions of higher values of the279 mechanical variables, identified as those lying on the right-hand side of the line marked as low-stress280 and low-strain regime in Fig. 4. We assume that variations in the lower regimens have no effect on281 possible damages. We are also interested in normalized values larger than one, meaning an increase282 in the value of the mechanical variable of interest.283 The stiffer material and the mouse-scaled model show a higher increase of brain tissue subjected284 to the high shear stress. The mouse-scaled brain is the only model that shows an increase of amount285 of tissue under the highest values of maximum principal stress. The mouse-scaled model also shows286 a remarkable increase of strain for the larger strain values. However, this increase was higher for287 the softer material and even higher for the non viscoelastic model. 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