Biomechanics of Traumatic Brain Injury: Influences of the Morphologic

Heterogeneities of the Cerebral Cortex

R.J.H. CLOOTS, H.M.T. GERVAISE, J.A.W. VAN DOMMELEN, and M.G.D. GEERS

Materials Technology Institute, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

(Received 6 September 2007; accepted 28 April 2008; published online 9 May 2008)

Abstract—Traumatic brain injury (TBI) can be caused byaccidents and often leads to permanent health issues or evendeath. Brain injury criteria are used for assessing theprobability of TBI, if a certain mechanical load is applied.The currently used injury criteria in the automotive industryare based on global head kinematics. New methods, based onfinite element modeling, use brain injury criteria at lowerscale levels, e.g., tissue-based injury criteria. However, mostcurrent computational head models lack the anatomicaldetails of the cerebrum. To investigate the influence of themorphologic heterogeneities of the cerebral cortex, a numer-ical model of a representative part of the cerebral cortex witha detailed geometry has been developed. Several differentgeometries containing gyri and sulci have been developed forthis model. Also, a homogeneous geometry has been made toanalyze the relative importance of the heterogeneities. Theloading conditions are based on a computational head modelsimulation. The results of this model indicate that theheterogeneities have an influence on the equivalent stress.The maximum equivalent stress in the heterogeneous modelsis increased by a factor of about 1.3–1.9 with respect to thehomogeneous model, whereas the mean equivalent stress isincreased by at most 10%. This implies that tissue-basedinjury criteria may not be accurately applied to mostcomputational head models used nowadays, which do notaccount for sulci and gyri.

Keywords—Traumatic brain injury, Brain tissue, Cerebrum,

Cerebral cortex, Inhomogeneities, Finite element model.

INTRODUCTION

The brain is often one of the most seriously injuredparts of the human body in case of a road traffic crashsituation.1,21,41 The incidence rate and mortality rate inEurope are estimated to be 235 and 15.4 per 100,000 ofthe population per year, respectively.41 Traumaticbrain injury (TBI) is therefore considered as a wide-spread problem. Understanding the mechanisms

inducing TBI is necessary for reducing the number ofoccurrences, e.g., by developing more appropriateprotective systems and diagnostic tools.

Brain injury criteria are used for the assessment ofthe probability of TBI for certain mechanical loadingconditions. The most commonly used injury criterionin the automotive industry is the Head Injury Criterion(HIC).16,43 It is developed to predict TBI resultingfrom a translational acceleration of the head. One ofthe drawbacks of the HIC is that it is based on globalkinematic data to predict TBI, whereas actual braindamage is caused at the cellular level as a consequenceof tissue strains and stresses.35 Furthermore, it is basedon experimental data, in which only anterior–posteriorcontact loading has been applied to human cadavers,not accounting for angular accelerations of the head.For a better approximation of the relation betweenTBI and a mechanical load, more advanced methodshave been developed. For instance, three-dimensionalfinite element (FE) head models have been developedto predict brain injury.2,6,7,9,17,22,38,42,45,47 With thesenumerical head models, different injury mechanismsand loading conditions can be distinguished. However,in these models, the heterogeneous anatomy of thecerebrum is usually represented by a relatively homo-geneous geometry. A comparison between the homo-geneous geometry of a typical finite element headmodel and the complex structure of a real brain is gi-ven in Fig. 1. The main function of the heterogeneousmorphology is to increase the cortical surface in orderto obtain a more complex level of the brain func-tions.34 The most recent numerical head models in-clude ventricles and the invaginations of the duramater, but none include the convolutions of the cere-bral cortex. Consequently, the stresses and strains thatare predicted from these models likely do not representactual tissue stresses and strains, at least in the cortex.Therefore, although tissue-based injury criteria may beused, their accuracy is expected to be limited. Thismight prohibit the direct use of tissue-based injury

Address correspondence to R.J.H. Cloots, Materials Technology

Institute, Eindhoven University of Technology, P.O. Box 513,

5600 MBEindhoven, The Netherlands. Electronic mail: r.j.h.cloots@

tue.nl

Annals of Biomedical Engineering, Vol. 36, No. 7, July 2008 (� 2008) pp. 1203–1215

DOI: 10.1007/s10439-008-9510-3

0090-6964/08/0700-1203/0 � 2008 The Author(s)

1203

criteria. Such criteria predict injury at the tissue leveland are based on in vitro and in vivo experi-ments.3,4,10,11,13,31,32 For a direct application of tissue-based injury criteria in a computational head model, amore detailed description of the biomechanicalbehavior of the cerebrum may be required, which canbe achieved by including its morphologic heterogene-ities in these models. A few two-dimensional FEmodels of the brain containing the convolutions of thecerebral cortex have been described in literature. Milleret al.30 compared different modeling techniques for therelative motion between the brain and the cranium.Nishimoto and Murakami33 developed a model toinvestigate the relation between brain injury and theHIC. However, these models have not been developedwith the purpose of investigating the local biome-chanics at the level of these convolutions. No conclu-sions have been drawn from these studies on thebiomechanical influence of the heterogeneities of thecerebral cortex, due to the limited spatial resolution ofthe mesh.

Physical experiments have been conducted in severalstudies to investigate the biomechanical consequencesof the heterogeneities of the cerebrum.14 In a study byBradshaw et al.,5 a gel-filled chamber that representedthe brain and skull in a coronal plane including the falxcerebri and the sulci of the cerebral cortex was sub-jected to a rotation with a peak acceleration ofapproximately 7800 rad s-2. An increase of the maxi-mum principle strain in the cerebral cortex due to thesulci was found.

The aim of this study is to investigate the biome-chanical influences of the morphologic heterogeneitiesin the cerebral cortex. To achieve this, several two-dimensional FE models with detailed geometries of apart of the cerebral cortex have been developed. Also,an FE model with a homogeneous morphology of thecortex has been made. The loading conditions arebased on simulations with a computational head modelas used by Brands et al.6 The results of the simulationsof the heterogeneous models will be compared to thoseof the homogeneous model.

METHODS

In this study, plane strain models of small sectionsof the cerebrum are made using the FE code Abaqus6.6-1 (HKS, Providence, USA). An explicit time inte-gration is used, anticipating a dynamic load with ahigh magnitude and a short duration. The time incre-ments are limited by the stability condition, which isdetermined in the global estimator function in Abaqus.

Geometries

To investigate the influence of the heterogeneities ofthe cerebral cortex, a homogeneous model and threeheterogeneous models have been developed. The heter-ogeneous models, which are shown in Figs. 2a, 2c, and2d have detailed geometries of a small part of the cere-brum, including also a part of the cerebrospinal fluid

FIGURE 1. (a) Numerical head model developed by Claessens.6,7 (b) Lateral view of the human brain. Adapted from Welkeret al.44

CLOOTS et al.1204

(CSF). The cranium is modeled by a boundary con-straint, as will be detailed further on. Since the duramater and the arachnoid are connected to the insideof the cranium in the region that is modeled,29,34 it isassumed that they can be ignored for this situation. Thepia mater, which is a thin and delicate membranecovering the brain,29,34 is also not included, since it isexpected to have no mechanical influence for the usedloading conditions. The same assumption is used for thearachnoid trabeculae, which extend from the arachnoidto the pia mater and are less existent inside the sulci.34

The first geometry has one narrow sulcus on theright hand side and a small part of a sulcus on the lefthand side. The second geometry contains two deeperand wider sulci than the other two geometries. Thethird geometry consists of one vertical sulcus and onepartly horizontal sulcus, where horizontal and verticalrefer to the x- and y-direction, respectively. Thesegeometries, which represent typical stylized shapes ofthe cerebral cortex, are based on the topologicalstudies by Mai et al.27 The left and right boundaries ofthe models are chosen to be periodic, i.e., the internalgeometries near the opposite boundaries match. Theperiodicity of the boundary conditions will be ex-plained further on. The models do not distinguish be-tween gray (cerebral cortex) and white matter. InFig. 2e, the homogeneous model is shown. Similar tothe heterogeneous models, it also consists of CSF andbrain tissue, but it does not contain any gyri and sulci.The outer dimensions of each model are 32 mm by24 mm. The meshes consist of bi-linear, quadrilateral,reduced integration elements with hourglass control.

The heterogeneous models also contain a small num-ber of triangular elements. The total number of ele-ments of the heterogeneous models ranges from 4243to 4533 elements. The homogeneous geometry consistsof 3072 elements.

Material Properties

For the material properties of the CSF, a nearlyincompressible, low shear modulus elastic solid hasbeen assumed, since the shear stress in the brain tissuedue to the applied loading conditions is estimated to beabout a factor 104 higher than that in the CSF. Thematerial properties are listed in Table 1. The shearmodulus of CSF is estimated from the loading condi-tions that are described further on by using G ¼ g _c=c;in which G is the elastic shear modulus, g is the vis-cosity, c is the estimated shear strain, and _c is theestimated shear rate. Because two different loadingconditions have been used, also two different estimatesfor the CSF shear modulus have been used. However,with these shear moduli being much lower than that ofthe brain tissue, the exact values of these estimates donot affect the outcome of this study. The bulk modulusis obtained from literature.2,47

The material properties of the brain tissue aredescribed by a non-linear viscoelastic constitutivemodel that has been developed by Hrapko et al.19 Thismodel was found to accurately describe the response ofbrain tissue for large deformations in both shear andcompression. This model is extended here to accountfor compressibility.

(a) (b)

(c) (e)(d)

CSF

Brain tissue

y

x

FIGURE 2. (a) Heterogeneous geometry 1 and (b) its spatial discretization. (c) Heterogeneous geometry 2. (d) Heterogeneousgeometry 3. (e) Homogeneous geometry.

Biomechanics of the Cerebral Cortex 1205

The constitutive model consists of an elastic part, de-noted by the subscript ‘e’, and a (deviatoric) viscoelasticpart, denoted by the subscript ‘ve’, with N viscoelasticmodes. The total Cauchy stress tensor r is written as

r ¼ rhe þ rd

e þXN

i¼1rdvei

ð1Þ

in which the superscripts ‘h’ and ‘d’ denote thehydrostatic and the deviatoric part, respectively. Forsimplicity, the subscript i indicating the number of theviscoelastic mode will be omitted from this point on.The hydrostatic part of Eq. (1) is defined as

rhe ¼ KðJ� 1ÞI ð2Þ

where K is the bulk modulus and J ¼ detðFÞ is thevolume change ratio.

The deviatoric elastic mode describes a non-linearresponse to the deformation gradient tensor F, which isgiven by

rde ¼

G1Jð1� AÞexp �C

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib~I1 þ ð1� bÞ~I2 � 3

q� �þ A

� �

b~Bd � ð1� bÞð~B�1Þdh i

ð3Þ

where G¥ is the elastic shear modulus, ~B ¼ J�23B is the

isochoric part of the Finger tensor B, and ~I1 and ~I2 arethe first and second invariant of the isochoric Fingertensor ~B; respectively. A, C, and b are fitting parame-ters describing the non-linearity of the elastic response.

The third term on the right hand side of Eq. (1)consists of the summation of the viscoelastic modes.The deformation gradient tensor F is partitioned intoan elastic deformation gradient tensor Fe and a viscousdeformation gradient tensor Fv by assuming multipli-cative decomposition25,36:

F ¼ Fe � Fv ð4Þ

The decomposition involves a fictitious intermediatestate, which could exist after application of merely the

viscous deformation gradient tensorFv:This is the stress-free state, which after application of the elastic defor-mation tensor Fe transforms into the final state. Thethird term on the right hand side of Eq. (1) describes theviscoelastic contribution to the stress as follows:

rdve ¼

G

Ja~Bd

e � ð1� aÞð~B�1e Þd

h ið5Þ

with G the shear modulus, ~Be ¼ J�23Be the isochoric

part of the elastic Finger tensor Be; and a a fittingparameter.

The viscous deformation Fv is assumed to be vol-ume-invariant, i.e., detðFvÞ ¼ 1 and Je ¼ detðFeÞ ¼ J:The viscous rate of deformation tensor is calculatedfrom the flow rule as

Dv ¼rdve

2gðsÞ ð6Þ

where the dynamic viscosity g is a function of the

scalar equivalent stress measure s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 rd : rd

q; for

which the Ellis model is adopted:

gðsÞ ¼ g1 þg0 � g1

1þ ss0

� �ðn�1Þ ð7Þ

with subscripts 0 and ¥ denoting the initial and infinitevalues, respectively. The initial value for the viscosity isdefined as g0 = Gk, whereas the infinite viscosity isdefined as g¥ = kg0. Here, k refers to the time constant.

Although differences between the material proper-ties of the gray and white matter may exist, these dif-ferences are not well characterized. Therefore, nodistinction between gray and white matter has beenmade in this study, except for the investigation of theinfluence of varying the material properties of graymatter with respect to those of white matter (see theDiscussion and Conclusions). For simulating a headimpact situation representative of road traffic acci-dents, an extra viscoelastic mode with a smaller timeconstant has been added to the behavior as charac-terized by Hrapko et al.19 The extra mode18 is basedon the experimental data from Hrapko and co-workersin combination with the data by Shen et al.39 Thelinear material properties are listed in Table 1. Thevalues of the non-linear viscoelastic parameters areshown in Table 2.

TABLE 1. Linear material parameters.

Bulk modulus

(GPa)

Shear modulus

(Pa)

Time

constant (s)

CSF 2.2 0.036a ¥0.12b ¥

Brain tissue 2.5 182.9 ¥9884 0.00013

835.5 0.012

231.2 0.35

67.1 4.62

3.61 12.1

2.79 54.3

aShear modulus in case of loading condition A.bShear modulus in case of loading condition B.

TABLE 2. Non-linear material parameters for brain tissue.

Elastic Viscous

A = 0.73 s0 = 9.7 Pa

C = 15.6 n = 1.65

a = 1 k = 0.39

b = 1

CLOOTS et al.1206

Boundary Conditions

The boundary conditions have been chosen such thatthey represent the biomechanical influences of the sur-roundings on the cerebral cortex model. Figure 3 showsthe labeling of the corner nodes and the boundaries. Thesymbols x and y denote the components of positionvector~x with respect to a Cartesian vector basis ð~ex;~eyÞ;whereas u and v are the components of the displacementvector ~u with respect to this basis.

The Young’s modulus of the cranium is muchhigher than that of brain tissue.19,46 Still, in a contactloading situation of the head the deformation of theskull is important, because it initiates strain waves inthe brain tissue. In this study, however, only inertialloading of the head is considered and therefore thecranium is assumed to be rigid. The cranium is incor-porated in the boundary condition at C3. Because ofthe low shear modulus of the CSF, the influence of therigid constraint associated with the cranium atboundary C3 in the x-direction can be neglected. Pro-vided no rotation of the model occurs, the constraintequation for all nodes on boundary C3 is

vjC3¼ vs ð8Þ

with vs the vertical displacement of the skull.

The boundaries C2 and C4 are subjected to periodicboundary conditions23:

~ujC2�~ujC4

¼ ~ujC2�~ujC1

ð9Þ

These constraints imply that throughout the defor-mation process the shapes of the opposite boundaries,C2 and C4, remain identical to each other, while thetractions on opposite boundaries are opposite to sat-isfy stress continuity, which can be written as

r �~n2 ¼ �r �~n4 ð10Þ

with r the Cauchy stress tensor and~ni the unit outwardnormal vector of boundary Ci.

The lower boundary, C1, of the brain tissue in themodel lies adjacent to brain tissue in neighboringregions. Therefore, boundary C1 has to be constrainedaccordingly. The applied constraint on C1 is obtainedby tying all nodal displacements on C1 to a linearinterpolation between the displacements of cornernodes C1 and C2. For any node on boundary C1, thisresults in

~ujC1¼ ~ujC1

þjj~x0jC1

�~x0jC1jj

jj~x0jC2�~x0jC1

jj ~ujC2�~ujC1

� ð11Þ

with the subscript 0 denoting the initial configuration.The displacements of corner nodes C1 and C2 areprescribed and calculated from the applied loadingconditions.

The loading conditions of the cerebral cortex model(micro-level in Fig. 4) are based on the loading con-ditions that have been used by Brands et al.6 for athree-dimensional numerical head model (macro-levelin Fig. 4). In that model, an eccentric rotation has beenapplied to the skull to simulate an angular headacceleration around the neck-shoulder joint in the

Γ1

Γ2

Γ3

Γ4

C1 C2

x

y

FIGURE 3. Labeling of corner nodes and boundaries.

x

y

r

y=0 mm

y=24 mm

level-orciMlevel-orcaMAxis of rotation

σ̄ (Pa)

40036032028024020016012080400

FIGURE 4. The loading conditions of the cerebral cortex model (micro-level) are derived from the region of interest in a para-sagittal cross-section (15 mm offset from the midsagittal plane) of the head model (macro-level). Shown at the macro-level is theequivalent stress field of the head model at 10 ms.

Biomechanics of the Cerebral Cortex 1207

sagittal plane in the anterior–posterior direction. Theeccentricity has been chosen to represent a typical necklength. The axis of rotation has been positioned at155 mm below the anatomical origin, i.e., the ear holeprojected to the sagittal plane. The rotation of the headmodel consists of two successive sine functions thatdescribe the angular acceleration:

0 s<t � 0:010 s : _xðtÞ ¼ 250p sin ð100ptÞ ð12Þ

0:010 s<t � 0:030 s : _xðtÞ ¼ �125p sin ð50pðt� 0:010ÞÞð13Þ

In Eqs. (12) and (13), the angular acceleration _x isgiven in rad s-2.

The loading conditions are applied to the cerebralcortex model by means of body forces. In all integra-tion points of the elements in the model, a non-uniformbody force is imposed that reversely simulates theinertial forces:

~qð~x; tÞ ¼ qð~xÞ€uðy; tÞ~ex ð14Þ

in which ~q represents the distributed load per unit ofvolume, q is mass density, t is time, and €u refers to theacceleration in the x-direction that is represented bythese body forces. Note that for the head model theloading conditions contain an angular component,whereas the cerebral cortex model uses translationalloading conditions. Because only a small part of thehead is modeled and because of the small rotation ofthe head model with a maximum of 4�, the loading ofthe cerebral cortex model is assumed to be transla-tional in x-direction only.

The loading conditions of the cerebral cortex model,i.e., the representative accelerations €uðy; tÞ; are calcu-lated from the head model (from the region indicatedin Fig. 4) in two different approaches:

A. In the first approach, the input accelerations of thehead model are used to define the loading conditionof the cerebral cortex model. This approach will bereferred to as loading condition A.

The translational acceleration €u can be calcu-lated using

€uðy; tÞ ¼ _xðtÞ rðyÞ ð15Þ

with _x the angular acceleration, which is defined byEqs. (12) and (13), and r the radius from the axis ofrotation (neck-shoulder) in the head model to apoint in the region of interest. The radius r is afunction of the y-position in the cerebral cortexmodel. It varies between r(0) = 0.251 m atboundary C1 to r(0.024) = 0.275 m at C3. Theaccelerations €u at C1 and C3 are depicted in Fig. 5a.All other accelerations are interpolated linearlybetween these two boundaries, thereby creating agradient across the height of the model. Theacceleration gradient is important for the resultingshear stresses. Figure 5b shows the accelerationprofile of the cerebral cortex model. The accelera-tions are used to calculate the body forces as afunction of both time and y-position.

The disadvantage of this loading condition is thata spatially constant acceleration gradient is assumedand therefore it does not account for the influence ofthe geometry of the cranium. To account for thegeometry of the head, another loading condition hasbeen developed that is described next.

B. The second approach, loading condition B, usesoutput accelerations from a global head modelsimulation as the input of the cerebral cortex model.For this, a modified version of the head model, asused by Brands et al.,6 has been employed in thesimulation code Madymo, in which the constitutivemodel for brain tissue by Hrapko et al.19 has beenimplemented. The accelerations obtained from theregion inside the box in Fig. 4 from the head modelare imposed on the cerebral cortex model. Hence,the influence of the geometry of the head is modeledindirectly by means of an acceleration profile that isobtained from the head model.

The displacements of the brain tissue in the headmodel in the field of interest are almost entirely in

(a) (b)

acceleration (m s−2)acceleration (m s−2)

sm(

noitarelecca−2

)

y)

mm(

y)

mm(

time (ms)

y=24 mmy= 0 mm

5ms 20ms

24 24

20 20

16 16

12 12

8 8

4 4

0 00 0

0

0-150

-100 -100-100 -50 -50

-50

50 50

50

100 100

100

150 150

150

200 200

200

250

5 10 15 20 25 30

FIGURE 5. Loading condition A. (a) Acceleration at the upper and lower boundary of the cerebral cortex model. (b) Accelerationprofiles at different times.

CLOOTS et al.1208

the x-direction justifying the assumption of inertialloading (of the cerebral cortex model) in thex-direction only. In Fig. 6, the acceleration profilesas a function of the y-position are shown at 5, 10,and 20 ms. Similar to loading condition A, theaccelerations are used to calculate the body forcesas a function of both time and y-position.

In order to quantify the influence of the morphologic

heterogeneities, the equivalent stress �r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32 rd : rd

qis

used, in which rd is the deviatoric part of the Cauchystress tensor r: The equivalent stress is chosen, becausethe simulations are based on an angular acceleration ofthe head, in which deviatoric stresses are considered tobe the most important.30 The maximum principal strainis considered important as well with respect to diffuseaxonal brain injury.3,30 Therefore, also the maximumprincipal logarithmic strain is used to quantify theinfluence of the morphologic heterogeneities.

RESULTS

Figure 7 depicts the development in time of theequivalent stress fields for the homogeneous model(top row) and the heterogeneous models from thesimulation with loading condition A. Stress concen-trations are present in the heterogeneous models at thesurface of the brain tissue between two gyri at 5, 10,and 20 ms. Near boundary C1, all heterogeneousmodels have lower equivalent stresses compared to thehomogeneous model at 20 ms.

In order to obtain a good comparison of the resultsfor all geometries during the complete simulation time,

the maximum and mean equivalent stress from thesimulations with loading condition A are shown inFig. 8 as a function of time. It shows the stresses in thebrain tissue only. It can be noticed that the heteroge-neous models have a higher maximum equivalent stressthan the homogeneous model. Among the heteroge-neous configurations, geometry 1 causes a noticeablylower maximum equivalent stress of 112 Pa comparedto geometries 2 and 3, with a maximum equivalentstress of approximately 156 Pa. The large maximumequivalent stress in heterogeneous geometry 2 lastslonger than the stresses of the other geometries. Themaximum equivalent stress of the homogeneous modelreaches a value of 80 Pa. The mean equivalent stressesare nearly the same for all geometries.

To investigate the influence of the heterogeneities,the equivalent stress of the cerebral cortex in the het-erogeneous models is taken relative to that of thehomogeneous model. For the maximum equivalentstress, this will be done by taking the maximum values,whereas for the mean equivalent stress, this will bedone by taking the time averaged values. The maxi-mum equivalent stress of the heterogeneous models 1,2, and 3 is 1.31, 1.84, and 1.83 times higher than thehomogeneous model, respectively. The mean equiva-lent stress of the heterogeneous models 1, 2, and 3 withrespect to the homogeneous model is 1.09, 1.08, and1.10, respectively.

The equivalent stress fields obtained with loadingcondition B are displayed in Fig. 9. During thebeginning of the simulation, the equivalent stress fieldsin the brain tissue are comparable for all models. Whenthe field of higher equivalent stress moves downwards,the heterogeneities result in local peak stress concen-

acceleration m s−2 acceleration m s−2 acceleration m s−2

)m

m(y

)m

m(y

)m

m(y

)m

m(y

displacement (mm)

5 ms

5 ms 10 ms

10 ms 15 ms

20 ms

20 ms

2424

24

24

2020

20

20

1816

1616

16

16

1412

1212

12

12

108

88

8

8

64

44

4

4

2

00000

0

0-100-100-100 -50-50-50 505050 100100100 150150150 200200200

FIGURE 6. Loading condition B: displacement (top) and acceleration (bottom) profiles derived from the output of the head model.

Biomechanics of the Cerebral Cortex 1209

trations, which can be seen at 10 ms for geometries 1and 3. Later on, at 20 ms, the heterogeneous geome-tries 1 and 3 have less influence on the equivalent stressfields. The differences of geometry 2 with respect togeometries 1 and 3 are a consequence of the deepersulci in geometry 2.

The maximum and mean equivalent stress of thecerebral cortex as a function of time obtained withloading condition B is shown in Fig. 10. The maximumequivalent stress is higher for the heterogeneousmodels than for the homogeneous models, but not forthe complete duration of the simulation. After about10–15 ms, the maximum equivalent stress of the het-erogeneous models drops to approximately the samemagnitude as the one obtained for the homogeneousmodel. For the heterogeneous models, the maximumequivalent stress reaches values of approximately470, 565, and 624 Pa for geometries 1, 2, and 3,respectively. The homogeneous model has a maximumequivalent stress reaching 325 Pa. Also, the moment in

5 ms 10 ms 20 ms

σ̄ (Pa)

1501009080706050403020100

Homogeneous model

Heterogeneous geometry 1

Heterogeneous geometry 2

Heterogeneous geometry 3

FIGURE 7. The equivalent stress fields as a result of loading condition A.

time (ms)

σ̄)a

P(

02 4 6 8 10 12 14 16 18 20

20

40

60

80

100

120

140

160

Homogeneous, maximumGeometry 1, maximumGeometry 2, maximumGeometry 3, maximum

Homogeneous, meanGeometry 1, meanGeometry 2, meanGeometry 3, mean

FIGURE 8. Maximum and mean equivalent stress for theheterogeneous and homogeneous models as a result ofloading condition A.

CLOOTS et al.1210

time at which the maximum occurs differs from onegeometry to the other. The mean equivalent stressvalues of all the geometries are similar.

To quantify the influence of the heterogeneities, theequivalent stress of the brain tissue of the heterogeneousmodels is taken relative to the homogeneous model inthe same manner as described previously for loadingcondition A. The maximum equivalent stress of theheterogeneous models 1, 2, and 3 has increased by 1.44,1.74, and 1.92 with respect the homogeneous model,respectively. The mean equivalent stress of the hetero-geneous models 1, 2, and 3 is 0.97, 0.99, and again 0.99relative to the homogeneous model, respectively.

The distribution of maximum principal strains forloading condition B at 10 ms is shown in Fig. 11. Onecan notice that a concentration of maximum principalstrains at 10 ms occurs in the same location as theequivalent stress concentration at 10 ms (Fig. 9), bothin case of loading condition B. The same method forthe quantification of the influence of the heterogene-ities is used, but with the maximum principal strain

5 ms 10 ms 20 ms

σ̄ (Pa)

Homogeneous model

Heterogeneous geometry 1

Heterogeneous geometry 2

Heterogeneous geometry 3

40036032028024020016012080400

FIGURE 9. The equivalent stress fields as a result of loading condition B.

time (ms)

σ̄)a

P(

02 4 6 8 10 12 14 16 18 20

100

200

300

400

500

600

700

Homogeneous, maximumGeometry 1, maximumGeometry 2, maximumGeometry 3, maximum

Homogeneous, meanGeometry 1, meanGeometry 2, meanGeometry 3, mean

FIGURE 10. Maximum and mean equivalent stress for theheterogeneous and homogeneous models as a result ofloading condition B.

Biomechanics of the Cerebral Cortex 1211

instead of the equivalent stress. For the simulationswith loading condition A, the peak maximum principalstrain in the brain tissue of the heterogeneous models1, 2, and 3 has increased with respect to the homoge-neous model by 1.22, 1.92, and 1.80, respectively. Ifloading condition B is used, the increases are 1.43,1.84, and 1.90, respectively.

DISCUSSION AND CONCLUSIONS

In this study, the influences of the heterogeneities inthe cerebral cortex were investigated. This was donewith FE models of several different geometries fromsmall detailed parts of the cortex. In a preliminarystudy, the boundary constraints were tested. Theloading conditions were derived from a numerical headmodel.

In order to determine which constraints on theboundaries would represent the surroundings best, apreliminary study was conducted in which severaldifferent constraints were applied to boundaries C1 andC3. The different conditions on boundary C3, i.e., rigidconstraint or slip-condition in the x-direction, did notcontribute to differences in the results of the braintissue. This was probably caused by the low shearmodulus of the CSF. For boundary C1, several dif-ferent boundary conditions were tested and comparedto the results of models with the same width, but withtwice the height of the models in this study. By com-paring the ‘normal’ and the ‘high’ models, theboundary condition at boundary C1 that representedadjacent brain tissue could be determined. The peri-odic boundary condition that was applied to bound-aries C2 and C4, was used because of the assumedperiodicity in the cerebral cortex.

Since the loading conditions were dynamic theycould not be applied to the model by directly imposing

a deformation, which would induce boundary effects.For this reason, an indirect deformation was imposedby means of body forces. This approach worked wellfor the first 20 ms of the simulation, which had a totalduration of 30 ms. After 20 ms the deformations of thebrain tissue differed from the deformations derivedfrom the head model that were indirectly applied to themodel (Figs. 6 and 9). Therefore, only the first 20 msof the simulation are considered to be realistic.

Loading condition A was derived from the acceler-ation pulse that has been applied to a head model.6,12

The equivalent stress fields of the homogeneous modelscaused by loading condition A (Fig. 7) showed nosimilarities to the equivalent stress field of the headmodel (Fig. 4). Provided that the equivalent stress fieldof the head model is realistic, loading condition A canbe considered unrealistic. Loading condition B wasobtained from the resulting accelerations in the regionof interest of the head model. The equivalent stressfields of the homogeneous models from simulationswith loading condition B (Fig. 9) were approximatelysimilar to the stress fields in the corresponding regionof the head model (Fig. 4) during the first 20 ms.Nevertheless, differences in the equivalent stress fieldsexisted at the surface of the cortex. This is due to theCSF layer in the cerebral cortex model being describedelastically with a low shear modulus, as opposed to thehead model, which contains a relatively stiff CSF/duralayer. In spite of all these obvious differences, bycomparing these two loading conditions, it was shownthat the different loading conditions have hardly anyeffect on the relative mean and maximum equivalentstress and the relative peak maximum principal loga-rithmic strain. Hence, the mechanical influences of theheterogeneities of the cerebral cortex seem to be inde-pendent of the loading conditions.

The constitutive model for brain tissue was based onexperiments on porcine white matter.19 This was usedas a substitute for human brain tissue, as it was readilyavailable and it allowed to conduct experiments with ashorter post-mortem time.12,26 Experiments conductedby Prange et al.37 on fresh human brain tissue indi-cated that its mechanical properties are approximately30% stiffer than those of fresh porcine brain tissue.Although the constitutive model for brain tissue in thecerebral cortex model was not based on human braintissue, the geometries of the models were based on thehuman cerebrum. This assumption is expected to havemore effect on the absolute equivalent stress than onthe relative equivalent stress, which is dominated bythe heterogeneities of the model. Furthermore, nodistinction between the mechanical properties of whiteand gray matter was made, so that the materialproperties of the cerebral cortex, which consists of graymatter, were based on experiments on white matter, as

max (-)

0.0000.0150.0300.0450.0600.0750.0900.1050.1200.1350.1500.997

FIGURE 11. The maximum principal logarithmic strain fieldas a result of loading condition B at 10 ms.

CLOOTS et al.1212

well. In their results, Prange et al.37 found differencesbetween the material properties of the corpus callo-sum, i.e., the white matter that connects the twocerebral hemispheres, and the cerebral cortex, but notbetween the corona radiata, i.e., the white matter thatlies inferior to the cerebral cortex, and the cerebralcortex. Nevertheless, in several other studies of headand brain models, different material properties wereused for gray and white matter based on the assump-tion that white matter was more fibrous than graymatter.2,28,30,47 Therefore, simulations of the cerebralcortex model with shear moduli for gray matter rang-ing from 75% to 125% with respect to the shearmoduli white matter were conducted as well. Theresults of these simulations indicated that the materialproperties had an effect on the magnitude of theequivalent stress, but not on the regions in which thepeak equivalent stress was observed. The equivalentstress of the heterogeneous model relative to thehomogeneous model was hardly affected by the dif-ferent material properties.

In this model, the meninges and the blood vesselswere considered to have no mechanical influence onthe cerebral cortex for the used loading conditions.Although Jin et al.20 suggested that the pia-arachnoidcomplex can have a mechanical influence on the braintissue during an impact, it is still not clear if it wouldaffect the influence due to the gyrification of the cere-bral cortex. In a study performed by Ho and Kleiven,17

it was found that the vasculature of the brain can beneglected, as far as the mechanical influences areconcerned in a head model not containing the gyrifi-cation of the cortex.

In order to validate this model, a comparison withphysical experiments is required. However, data ofphysical experiments at a typical length scale of thislevel is rather limited. Parallel to this study, the resultsof physical experiments, in which brain slices havebeen accelerated, showed increased equivalent strainsnear the sulci.24 Furthermore, some studies showedthat angular accelerations of the head induce highstress concentrations in and near the cerebral cor-tex.8,15 Another method of validation can be per-formed by comparing the results of the model toclinically observed injury. It has been shown that smallcortical infarcts exist in diffuse brain injury at thebottom of the sulci.40 This is in accordance with thelocations of the high stress and strain regions inthe cerebral cortex model.

The two loading conditions and the different geom-etries resulted in different equivalent stress fields. Thesimulations with loading condition A resulted in a lowermean and maximum equivalent stress compared to thesimulations with loading condition B. However, relativeto the homogeneous model, it was observed that the

equivalent stress was almost independent of the differentloading conditions used in this study. The differencesbetween the several heterogeneous geometries hadmoreinfluence on the relative mean andmaximum equivalentstress. The morphologic heterogeneities of the cerebralcortex led to an increase of the maximum equivalentstress by a factor of about 1.3–1.9, depending mostly onthe geometry, whereas the relative mean equivalentstress values of all the geometries were 1.1 and 1.0 forloading condition A and B, respectively. Furthermore,the peak maximum principal logarithmic strain wasincreased by a factor of about 1.2–1.9 due to themorphologic heterogeneities of the cerebral cortex. Thisis a strong indication that predictions of brain injuryobtained from head models with a homogeneous cere-brum should be interpreted with care. To obtain a moreaccurate assessment of injury, the influence of themorphologic heterogeneities in the cerebral cortexshould be accounted for.

ACKNOWLEDGMENTS

This work has been supported by the EuropeanIntegrated Project APROSYS and the Dutch Tech-nology Foundation STW, applied science division ofNWO and the Technology Program of the Ministry ofEconomic Affairs.

OPEN ACCESS

This article is distributed under the terms of theCreative Commons Attribution NoncommercialLicense which permits any noncommercial use, distri-bution, and reproduction in any medium, provided theoriginal author(s) and source are credited.

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