ORIGINAL ARTICLE

An anisotropic viscous hyperelastic constitutive law for brainmaterial finite-element modeling

Simon Chatelin • Caroline Deck • Remy Willinger

Received: 13 January 2012 / Accepted: 3 October 2012

� Japanese Society of Biorheology 2012

Abstract Recent experimental studies have highlighted

the significant influence of axonal fibers on the nonlinear

and anisotropic behavior of brain tissue. This study aims to

implement these properties in a transversely isotropic

visco-hyperelastic constitutive law for brain tissue. The

second step consists in implementation of this law under

finite-element (FE) code to improve brain FE modeling by

including brain tissue specificities proposed in the recent

literature. Validation of the model is shown by comparison

of numerical simulation of unconfined compression with

experimental tests from the literature. This study represents

a step toward more realistic brain FE modeling.

Keywords Brain finite-element model �Brain matter constitutive law � Visco-hyperelasticity �Anisotropy � Transverse isotropy � Mooney–Rivlin

Abbreviations

F Deformation gradient tensor

J Jacobian of the deformation (determinant of F)

C Right Cauchy–Green deformation tensor

W Strain energy function

Ii (i = 1, 2, 3, 4, 5) invariants of C

K Bulk modulus

G Shear modulus

ki (i = 1, 2, 3) principle stretches of F

S Second Piola–Kirchhoff stress tensor

r Cauchy stress

N Nominal stress tensor

Introduction

Brain injury remains one of the main causes of death and

disability after head trauma. In France, brain injury results in

death in 22 % of cases of head trauma [41]. Since the 1970s,

finite-element human head models (FEHM) have been

developed as tools to assess head injury risk and eventually

injury location. In the last decades, about twenty FEHM

have been reported in the literature [1, 5, 6, 10, 14, 15, 17–

20, 23, 34, 36, 38, 39, 43, 45, 47]. None of the previously

presented mechanical laws for brain tissue take axons and

resulting anisotropy into account, and only a few of them

include nonlinear viscoelastic brain material behavior. At

the same time, experimental studies have shown a huge

number of detailed mechanical properties for brain tissue

[8], such as strain rate sensitivity [30] or strong nonlinear-

ities resulting in stiffening at high strain level [12]. In 1998,

experimental studies on porcine brain stem samples were

performed along three directions using in vivo shear

dynamic mechanical analysis by Arbogast and Margulies,

showing significant mechanical anisotropy of the tissue. A

first attempt to include axonal anisotropy in brain finite-

element simulations was proposed in 2011 by Chatelin et al.

[7] by looking at local strain values along the main direc-

tions of axonal fibers during head injury simulations. The

objective of this study is to develop a new constitutive law

for brain tissue based on experiments from the literature [11]

and improved by brain tissue properties recently highlighted

in the experimental literature. Thus, a transversely isotropic

visco-hyperelastic law is proposed for human brain tissue.

Evaluation of this law is performed by finite-element

Electronic supplementary material The online version of thisarticle (doi:10.1007/s12573-012-0055-6) contains supplementarymaterial, which is available to authorized users.

S. Chatelin � C. Deck � R. Willinger (&)

University of Strasbourg, IMFS-CNRS,

2 rue Boussingault, 67000 Strasbourg, France

e-mail: remy.willinger@unistra.fr

123

J Biorheol

DOI 10.1007/s12573-012-0055-6

simulation of unconfined compression tests from the liter-

ature by superimposing experimental and numerical brain

sample responses. This study constitutes a step towards

more realistic brain finite-element (FE) modeling.

Transversely isotropic viscous hyperelastic material

model

General format for the constitutive model

The model developed in this study is based on research

published by Weiss et al. [44] and Puso and Weiss [31], who

aimed to include collagen fibers in human tendon models. In

the present approach, it is proposed to apply a similar model

by considering that axonal fibers have the same kind of

influence on brain tissue as collagen fibers on ligament

mechanical behavior (Fig. 1). The white matter tracts con-

sist of densely packed axons in addition to various types of

neuroglia and other small populations of cells. Despite this

complex environment, this study was based on one main

assumption: as opposed to axon fibers, the multiple com-

ponents of the neuroglia have no significant influence on

brain tissue mechanical anisotropy. The influence of the

neuroglia and other small populations of cells is supposed to

underlie the ‘‘matrix’’ mechanical properties and, due to

their shape and size, to have no significant influence on the

global anisotropic mechanical behavior of brain matter. It is

therefore suggested that axonal fibers are a source of

mechanical brain anisotropy along one main direction.

Throughout this study, the deformation gradient is

denoted by F and the Jacobian J of the deformation is

defined by Eq. 1.

J :¼ detðFÞ: ð1Þ

The right Cauchy–Green deformation C is given by Eq. 2.

C :¼ FTF: ð2Þ

Brain tissue is considered to be made up of axonal fibers

embedded in an isotropic matrix that corresponds to

neuroglia (Fig. 1). For a hyperelastic fiber-reinforced

anisotropic material, strain energy can be fully described

by the five invariants of C as expressed in equations 3–7

[37].

I1 ¼ trðCÞ; ð3Þ

I2 ¼1

2ðtrðCÞ2 � trðC2ÞÞ; ð4Þ

I3 ¼ detðCÞ ¼ ðJÞ2; ð5ÞI4 ¼ a0 � C � a0; ð6Þ

I5 ¼ a0 � C2 � a0: ð7Þ

Classically, the strain energy W ¼ Wð~I1; ~I2; ~I5; ~I4; ~I5Þ is

defined as a combination of these five invariants [37].

However, in practice, strain energy can be restricted a priori

so that the parametric coefficients can be identified from

material tests [9]. In the present study, we assume that W is a

function of the first four invariants to describe the brain

tissue material from biaxial stretch experimental data. As

expressed by Eq. 8, the strain energy function for the soft

tissue material has three terms:

W ¼ WdMatrixð~I1; ~I2Þ þWd

Fibersð~I4Þ þWmðI3Þ: ð8Þ

The functions WdMatrix and Wd

Fibers are the matrix and fiber

distortional strain energies, respectively. ~I1 ¼ I�1=33 I1; ~I2 ¼

Fig. 1 Illustration of the

assumption made when

modeling neuronal structure

mainly composed of neuroglia

and axonal fibers by the fiber-

reinforced composite model

proposed for brain tissue. A

‘‘mean’’ a0 direction results

from the axonal fiber orientation

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123

I�2=33 I2; and ~I4 are the first, second, and fourth distortional

invariants of C, respectively. Wm represents volumetric

changes and depends on the compressibility of the material,

in accordance with Eq. 9. The effective bulk modulus for

the material is denoted by K. In the case of the perfect

incompressibility assumption, this volumetric energy is

considered to be negligible:

Wvð~I3Þ ¼1

2K ln ðJÞ2: ð9Þ

Material model for matrix

According to most of the recent experimental data, brain

tissue exhibits nonlinear mechanical behavior with a sig-

nificant increase of stiffness at large strain, and should be

modeled as a nonlinear solid with very small volumetric

compressibility [12]. This remains valid not only for white

but also for gray matter, which includes almost no fibers.

To incorporate nonlinear material, a Mooney–Rivlin

material model is introduced for the matrix. The Mooney–

Rivlin model is defined by Eq. 10, where C10 and C01 are

two model coefficients [33].

WdMatrixð~I1; ~I2Þ ¼ C10ð~I1 � 3Þ þ C01ð~I2 � 3Þ: ð10Þ

It is assumed that brain tissue stiffening at high strain is

due to matrix strain energy. The near-incompressibility is

ensured by defining a bulk modulus K at least three

times larger in magnitude the parameter ðC10 þ C01Þ=2,

which corresponds to the initial isotropic shear

modulus.

Material model for fibers

Previous experimental studies have shown different stiff-

nesses when testing brain white matter in compression and

tension [24]. Several assumptions have to be made in the

fiber strain energy model. First, we consider that the

influence of fibers is negligible in compression. Then, the

mechanical contribution of axonal fibers in tension is

considered as stiffening, exponentially increasing with

fiber stretch. Finally, it is assumed that the deviatoric local

fiber stretch ~k depends on the main fiber orientation a0, as

defined in Eq. 11, where ~C is the deviatoric right Cauchy–

Green deformation tensor.

~k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a0 � ~C � a0p

¼ffiffiffiffi

~I4

q

: ð11Þ

These observations lead to the strain energy defined by

Eq. 12, where C3 and C4 are constant parameters of the

model, depending on the mechanical properties and density

of the axonal fibers, as proposed in 1998 by Puso and

Weiss [31].

~koWd

Fibers

okð~kÞ ¼

0 0� ~k\1

C3 eC4ð~k�1Þ � 1� �

~k� 1

(

ð12Þ

Consequently, fibers have no mechanical influence either in

compression or when stretched perpendicularly to the fiber

orientation ð~k ¼ 0Þ. Due to the influence of the fibers,

differences in stiffness between tensile and compressive

behavior are in accordance with experimental results from

the literature [12, 24].

Viscoelastic component introduction

Due to the high strain rate sensitivity, viscosity plays a

significant role in the response of brain matter to quasistatic

loading as well as impact scenarios, as shown experimen-

tally, notably by Prevost et al. [30]. The most general

viscoelastic behavior is described by Eq. 14 by considering

the time-dependent second Piola–Kirchhoff stress S(C, t),

as proposed in 1981 by Fung [13].

r ¼ J�1F � S � FT; ð13ÞSðC; tÞ ¼ SeðCÞ þ SvðC; tÞ: ð14Þ

SeðCÞ is the equilibrium stress representing the long-time

elastic material behavior. S is related to the Cauchy stress rby Eq. 13. The hyperelastic strain energy represents this

elastic long-time response of the material. Rate effects are

taken into account through linear viscoelasticity via a

convolution integral representation, as indicated in Eq. 15.

SvðC; tÞ ¼Z

t

0

2Gðt � sÞ oW

oCðsÞ ds: ð15Þ

Gðt � sÞ is the reduced relaxation function. Most of the

experimental studies have pointed out the importance

of viscosity in the mechanical behavior of brain tissue.

Many authors have described relaxation as a decreasing

exponential function. However, it is important for the

simulation of head impacts to ensure the accuracy of the

law extracted from experimental data, especially at short

times. To accurately model the viscoelasticity of the brain

and to be able to add it to the hyperelastic term, we propose

to rely on n-order Prony series, which has already been

shown to be a powerful method for numerical modeling of

soft tissues [40]. In this method, time-dependent stresses

are added to the hyperelastic stress tensor and the reduced

relaxation modulus is expressed as in Eq. 16.

GðtÞ ¼X

n

i¼1

Sie�t=Ti : ð16Þ

Si and Ti are shearing relaxation moduli and decay

constants, respectively, which characterize the strain rate

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123

sensitivity of the model. The resulting series capture strain

rate loading and unloading effects and also produce an

excellent fit to the complex loading sequence. At least two

orders are required to ensure correct both short-time and

long-term relaxation behaviors.

Identification of material constants

Identifying brain matrix material constants

The parameters for the anisotropic visco-hyperelastic law

were identified from experimental data obtained in vivo by

magnetic resonance elastography (MRE) in 2007 by Kruse

et al. [21], and were completed for nonlinear effects at high

strain by using in vitro experimental results from Ning et al.

[28], Pervin and Chen [29], and Prevost et al. [30] at high

strain rate. To perform this identification, the theoretical

model was expressed in terms of stress/stretch for uniaxial

compression and tensile loading along the fiber direction a0.

For this particular configuration, the hyperelastic first

principal Cauchy stress is expressed by Eq. 17 under the

incompressibility assumption. k is the stretch ratio, such as

k1 ¼ k; k2 ¼ k3 ¼ 1=ffiffiffi

kp

, and p is the hydrostatic pressure.

Viscous parameters were then identified from experimental

stress relaxation curves from the literature.

reðkÞ ¼ koWd

Matrix

okþ k

oWdFibers

ok� p: ð17Þ

In compression (ð0\k\1Þ), Eq. 17 becomes Eq. 18,

where fibers have no influence.

reðkÞ ¼ 2 k2 � 1

k

� �

C10 þC01

k

� �

: ð18Þ

The Mooney–Rivlin model parameters C10 and C01 (brain

matrix) were identified from in vivo MRE tests performed

in 2007 by Kruse et al. [21] on healthy human subjects. In

addition, more recent results from the literature have shown

an important stiffening of brain tissue between 50 and

60 % stretch ð0:4� k\0:5Þ in compression at low [30] as

well as high strain rate values [29]. This observation is also

taken into account in the present study to identify hyper-

elastic parameters. Identification of parameters was per-

formed using optimization by the downhill simplex method

[26]. The curve shown in Fig. 2 was obtained with C10 and

C01 identified at -1.03 and 7.81 kPa, respectively. Under

10 % stretch ð0� k\0:1Þ, brain tissue behavior is con-

sidered as linear [27], and a small-strain tangent modulus E

(slope at small strains) of 40.7 kPa is obtained from the

identified curve. Under the incompressibility assumption,

this corresponds to a shear modulus of G ¼ E=3 of

13.6 kPa. At small stretch, results are similar to experi-

mental data obtained in compression by Estes and

McElhaney [11] at 40 s-1 strain rate. Considering the size

of the samples tested by the authors, this strain rate is

equivalent to a compression speed of about 1 m s-1, which

corresponds to the accidentology speed range.

Identifying axon-fiber-reinforced material constants

In tension ðk[ 1Þ, Eq. 16 becomes Eq. 19 with the com-

bined influence of brain matrix and fibers. While stretching

along the fiber direction involves ~k ¼ k, stretching

orthogonally to the fiber direction does not include the

exponential fiber stiffening.

reðkÞ ¼ 2 k2 � 1

k

� �

C10 þC01

k

� �

þ C3ðeC4ð~k�1Þ � 1Þ:

ð19Þ

The previously identified C10 and C01 parameter values

enabled interpolation of brain matrix behavior in tension

ðk[ 1Þ. This stress/stretch curve is shown in Fig. 3 and

corresponds to the response in tension of brain tissue

stretched perpendicularly to the fibers. The tangent moduli

in the initial state ð0� k\0:1Þ, at 10 % stretch ðk ¼ 0:1Þ,and at 50 % stretch ðk ¼ 0:5Þ are 40.7, 40.5, and 22.1 kPa,

respectively.

At small strain, the shear modulus of fibers is 1.8 [42] to

2 [3] times higher than the matrix shear modulus. Brain

tissue behavior is considered as linear up to 10 % stretch

ð0� k\0:1Þ [27]. In the same manner, the stiffness of

fibers is about 10 times higher than the brain matrix stiff-

ness at 50 % stretch ðk ¼ 0:5Þ [28]. For brain matrix

stiffness of 40.5 and 22.1 kPa, the exponential fiber

behavior (parameters C3 and C4 in Eq. 12) was fitted to

obtain tangent moduli of about 81.0 and 221.0 kPa at 10

and 50 % stretch, respectively, as shown in Fig. 3. This

calculation was performed using the downhill simplex

method [26] in MATLAB� software (Mathworks Inc.,

Natick, MA, USA). It was ensured that the curve was C0-

and C1-continuous. The parameters C3 and C4 related to

fibers are identified as 13.6 and 4.6 kPa. Using this algo-

rithm, the tangent moduli from the stress/stretch curves are

81 and 219.3 kPa at 10 and 50 % tensile stretch, respec-

tively, as shown in Fig. 3. All these values as well as the

identified curve are reported in Fig. 3. The maximal stiff-

ening of fibers is limited by the vertical asymptote at 60 %

tensile stretch ðk ¼ 0:6Þ, which is in accordance with

experimental results from the literature [30]. A first-order

Taylor series of Eq. 19 for k! 1 leads to Eq. 20.

reðk!1;k [ 1ÞðkÞ ¼ 6 C10 þ C01ð Þkþ C3C4k: ð20Þ

Now, the tangent modulus E0,Fibers at small strains in

tension for axonal fibers can be calculated for our model

using Eq. 21.

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123

E0;Fibers ¼ C3 � C4: ð21Þ

E0,Fibers is 63.3 kPa in tension for the model. Adopting

the near-incompressibility assumption, this corresponds to

shear modulus of about 21.1 kPa for fibers in tension.

Figure 4 synthesizes the stress/stretch curves in both

compression and tension for brain matrix, fiber, and total

homogeneous brain material using the previously identified

model. While the model depends only on the neuroglial

matter in compression, stiffening by axonal fibers is sig-

nificant in tension. This stiffening is absent and maximum

when tension is applied perpendicularly and parallel to the

main axonal fibers, respectively.

Identifying brain material viscosity constants

The viscosity parameters Si and Ti were identified from

experimental relaxation data in shearing dynamic

mechanical analysis by Shuck and Advani [35], as pre-

sented in Fig. 5, which does not take fibers into account.

The resulting relaxation modulus versus time curve is

Fig. 2 Identification of

hyperelastic parameters C01 and

C10 (matrix) from experimental

in vivo MRE tests in 2007 by

Kruse et al. [21]. Results are

presented in terms of stress/

stretch curve [22]. At small

strain (linear domain), results

are similar to those from

compression tests at constant

(40 s-1) strain rate by Estes and

McElhaney [11]. According to

more recent experimental

studies, significant stiffening of

brain matter is observed at high

strain

Fig. 3 Matrix and fibers stress/stretch curves proposed for the fiber-reinforced composite model. In accordance with experimental results from

the literature [5, 28, 41], stiffness values are imposed at 0 % (k = 1), 10 % (k = 1.1), and 50 % (k = 1.5) stretch

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123

scaled to 13.6 kPa [21] to ensure continuity between the

viscoelastic (linear) and hyperelastic (nonlinear) models, in

accordance with Eq. 22. The viscoelastic part of our model

is presented in Fig. 5 by identifying the following second-

order Prony series parameters: S1 = 4.50 kPa, S2 = 9.11 kPa,

T1 = 1 9 109 s-1 and T2 = 6.90 s-1.

Gðt ¼ 0Þ � 2ðC10 þ C01Þ: ð22Þ

Finite-element simulation of unconfined compression

experiments

The new fiber-reinforced brain model was evaluated using

FE simulations of unconfined compression, aiming to

reproduce rheological experimental test results on brain

tissue performed in compression at four different strain rates

(0.08, 0.8, 8, and 40 s-1) by Estes and McElhaney [11].

Fig. 4 Stress/stretch summary curves for fiber, matrix, and anisotropic brain model in compression and tension

Fig. 5 Relaxation curve implemented for the viscoelastic part of the brain constitutive law. Data are identified from both in vitro experimental

data by Shuck and Advani [35] and in vivo magnetic resonance elastography values for human white matter by Kruse et al. [21, 22, 35]

J Biorheol

123

To verify the response at high strain rate, experimental

compression tests of Pervin and Chen [29] at strain rate of

1000 s-1 were also simulated.

Finite-element model of the tested sample

The sample to be modeled is a cylinder (24 mm high,

32 mm diameter) meshed by 17472 brick elements dis-

tributed in 24 layers, as presented in Fig. 6. The shape and

dimensions are close to those of the samples used by Estes

and McElhaney [11] in the experimental rheological tests.

The FEM sample was meshed using LS-DYNA� software

(Dynamore GmbH, Stuttgart, Germany) with 1 mm char-

acteristic size. The cylinder symmetry axis is denoted by

Z. Two plates (upper and lower) were meshed with shell

elements and 1 mm characteristic size to reproduce the

experimental boundary conditions. For a given sample

model, all elements have fibers oriented along the same

direction. Three configurations are studied:

– Fibers along compression axis (a0 = Z axis)

– Fibers orthogonal to compression axis (a0 = X axis)

– Fibers along ‘‘oblique’’ direction (a0 = (X ? Y ? Z)/

|X ? Y ? Z| axis), which corresponds to the diagonal

of the cube encompassing the cylindrical FEM sample

Material properties

The anisotropic visco-hyperelastic law identified in the first

part of this study was implemented in a brain sample model

under LS-DYNA� software using *MAT_092_SOFT_

TISSUE_VISCO material. In the present section, the

parameters for this law are detailed in Table 1, where a

correspondence is proposed with the theoretical constitutive

law parameters identified in the first part of this study. The

anisotropy vector a0 is defined in a local frame linked with

each of the element geometries. Considering its high

water content, the brain tissue density is assumed to be

1040 kg m-3. The bulk modulus value of 1125 GPa ensures

the incompressibility of the material insofar as this value has

to be at least three order of magnitude of the ðC10 þ C01Þ=2

parameter, which corresponds to the initial isotropic

shear modulus. All the parameters implemented under

LS-DYNA� for brain material are collated in Table 2.

The two plates which define the sample’s boundary

conditions are supposed to be highly stiff and compress-

ible, with Young’s modulus and Poisson’s ratio of 210 GPa

and 0.285, respectively, corresponding to structural steel

[4]. Two interfaces (AUTOMATIC_NODES_TO_SURFACE

under LS-DYNA�) were defined between the sample and

each of the plates. In accordance with the literature related

to friction properties between human skin and steel [46],

a friction coefficient of 0.3 was allocated to the interfaces.

This coefficient represents the friction induced by sand-

paper between the steel plates and the brain tissue sample

during the experimental study [16].

Numerical simulation of experimental unconfined

compression tests

The aim of this numerical simulation is to validate our

model against experimental tests carried out in 1970 by

Estes and McElhaney [11]. In accordance with protocols

reported in the literature by Estes and McElhaney for

rheometric compression tests, displacement was imposed on

the upper plate nodes along the Z direction at constant strain

rate. Compression was applied until 50 % strain. The lower

plate was fixed. Stress relaxation was then observed during

Fig. 6 Finite-element model used to simulate unconfined compres-

sion experimental tests reported in the literature [11]. While the lower

plate is fixed, the upper plate is under controlled displacement. Both

plates are supposed to be highly stiff and have friction coefficient of

0.3, as for steel/biological soft tissue contacts

Table 1 Correspondence of brain constitutive law parameters and

parameters implemented under LS-DYNA� finite-element code

LS-DYNA

parameters

Constitutive

parameters

Units Meaning

q q kg m-3 Density

K K Pa Bulk modulus

C1 C10 Pa Mooney–Rivlin parameters

C2 C01 Pa

C3 C3 Pa Fiber reinforcement

parametersC4 C4 –

Si Si Pa Long-term shear moduli

Ti Ti s Time constants

AX, AY, AZ – – Local element frame

R(element) definitionBX, BY, BZ

LAX, LAY,

LAZ

a0X, a0Y, a0Z – Main anisotropy vector a0

defined in the local frame

The mechanical law is refered by *MAT_092_SOFT_TISSUE_

VISCO (LS-DYNA). Fiber orientation is expressed in the local ele-

ment frame

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123

10 ms, maintaining the strain at 50 %. The curve displace-

ment along the Z axis is shown in Fig. 7 for a 40 s-1

(*1 m s-1) strain rate. In order to simulate the experi-

mental loadings of the tests carried out by Estes and

McElhaney [11], the model was evaluated at three strain

rates: 0.8 s-1 (*24 mm s-1 speed), 8 s-1 (*0.24 m s-1

speed), and 40 s-1 (*1 m s-1 speed); also, to extend the

strain rate range and to simulate the high-speed tests of

Pervin and Shen [29], the simulation was also performed at a

fourth strain rate of 1000 s-1 (*24 m s-1 speed).

Results

To express the isotropic stress and observe the influence of

fibers in the sample, the von Mises stress was supplied

by LS-DYNA� software, as defined by Eq. 23, where the

~rij are the components of the deviatoric stress tensor.

rVM ¼2

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3

2~r2

xx þ ~r2yy þ ~r2

zz

� �

þ 3

4~r2

xy þ ~r2yz þ ~r2

xz

� �

r

:

ð23Þ

Results are illustrated in Fig. 8 in terms of the von Mises

stress for the three fiber orientations. Samples in the final

state are shown. The influence of the fibers is clearly

significant, involving higher stress values, anisotropic

stress distribution in the transverse (X, Y) plane, and

shear effects, for fibers along the Z, X, and oblique

direction, respectively. Results were also expressed using

stress/stretch curves. While the parametric coefficients of

the mechanical law were implemented using Cauchy

stresses, the results are expressed in terms of the nominal

stress N, which is linked to the Cauchy stress r by Eq. 24.

N ¼ JF�1 � r: ð24Þ

The maximal principal nominal stress n(t) and stretch ratio

k(t) were calculated from the upper plate displacement UZ

and force FZ at the lower interface using Eqs. 25 and 26,

where L0 and S0 are the initial height and upper plate

surface of the sample.

nðtÞ ¼ FZðtÞS0

; ð25Þ

kðtÞ ¼ UZðtÞL0

þ 1: ð26Þ

Assuming that this brain tissue characterization involves

low-frequency phenomena, a 100-Hz low-pass filter was

applied to all the resulting curves, in accordance with

Society of Automotive Engineer (SAE) norms. Stress/

stretch curves are presented in Fig. 9 for the three fiber

orientations. Results are compared with experimental

curves of Estes and McElhaney [11] and Pervin and Chen

[29] obtained from similar in vitro compression tests

[11, 29]. In Fig. 9, results are also compared with main

results proposed in the literature from rheometric com-

pression tests [11, 25, 29, 32].

The tangent modulus was calculated by considering the

slopes at small strain in Fig. 9. Results are shown in Fig. 10.

This figure confirms that, while in tension the fibers have a

significant influence when oriented along the tension

direction, in compression the fibers only have an influence

when oriented in the plane transverse to the compression

direction.

Discussion

According to the theory, fibers have no influence on the

model response under unconfined compression with fibers

oriented along the Z compression axis. This can be

observed in Fig. 8. However, due to the near-incompress-

ibility, stretches appear in the (X, Y) transverse plane. The

more the fibers are oriented in this transverse plane, the

more anisotropic the stress distribution is. Due to the tissue

incompressibility, the influence of the fibers appears in this

case not only in tension but also in compression as an

increase of the shear stress in the model.

Table 2 Parameter values for *MAT_092_SOFT_TISSUE_VISCO

material implemented for the brain tissue sample under LS-DYNA�

software

q (kg m-3) 1040 K (MPa) 1125

Matrix C1 (kPa) -1.034 C2 (kPa) 7.809

Fibers C3 (kPa) 13.646 C4 4.64

Matrix viscosity S1 (kPa) 4.5 T1 (s) 1.10-9

S2 (kPa) 9.11 T2 (s) 0.1450

Hyperelastic matrix, fiber, and viscoelastic parameters are defined

individually

Fig. 7 Displacement curve imposed in the Z direction on the upper

plate for unconfined compression simulations. Constant strain rate is

followed by relaxation in the simulations

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123

Compression tests were simulated successively at four

strain rates (0.8, 8, 40, and 1000 s-1) for the three fiber

orientations (along Z axis, X axis, and X ? Y ? Z direc-

tion), in accordance with the experimental protocols of Estes

and McElhaney [11] and Pervin and Chen [29]. At small

strain, significant stiffening with strain rate was observed, in

accordance with experimental data from the literature at low

speeds from Estes and McElhaney [11], as well as at high

speeds for the curves obtained by Pervin and Chen [29]. This

stiffening depends on the fiber orientation for strain rates up

to 40 s-1. This aspect is in accordance with literature data

on experimental brain tissue mechanics that show signifi-

cant anisotropy at low speed [2] but isotropy at high speed

[29]. Beyond this limit, the model response is independent

of strain rate. At low strain rates, the model with fibers

orthogonal to the compression (X) axis has 1.5 times higher

stiffness than the model with fibers along the compression

axis (Z). This can be explained by considering that, while in

the first case brain matrix and fibers are tested, the second

configuration corresponds to simulation of brain matrix

only. While fibers have more influence when oriented

orthogonally to the main testing direction in tension,

anisotropy appears in compression essentially with fibers

orthogonal to the main displacement axis.

Moreover, when considering fibers basically oriented

along the main compression axes, deformation could

indeed induce bending of the fibers (and also of the

‘‘equivalent fiber’’ of each of the FE elements), which, in

such a case, could have a significant influence on the

mechanical behavior of brain tissue (the influence of fibers

also no longer being negligible). When considering fibers

basically oriented along another axis, this ‘‘bending effect’’

is drastically reduced (up to the extreme case of fibers

perpendicular to the compression axis, in which case the

bending effect is completely removed).

Independent of the strain rate and main fiber orientation,

nonlinear behavior is observed by significant stiffening at

high strain. However, the maximal acceptable strain for

this model (vertical asymptotes on Fig. 9) depends on the

strain rate and main fiber orientation. The model overload

limit is lower for fibers oriented orthogonally to the com-

pression axis and for high load speed. At least, overloading

occurs at about 50 % strain at low speed and at 20 % at

high speed, in accordance with experimental data from the

literature [29, 30]. It can be noted that the stiffness increase

due to the hyperelastic part in compression is much greater

in the simulation results than in the experimental data from

the literature. This aspect could be improved by separating

Fig. 8 Illustration of von Mises stress distribution at 50 % stretch with fibers oriented along Z, X, or oblique direction (X ? Y ? Z) for the FEM

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123

the hyperelastic parameters implemented for compressive

and tensile behavior.

From Fig. 10, the tangent modulus at small stretch

ð0� k\0:1Þ confirms the significant stiffening with strain

rate. Whatever the strain rate, the highest stiffness is

always obtained when the fibers are oriented in the trans-

verse plane (X, Y). The original tangent modulus of about

40 kPa is obtained at 42 s-1, in accordance with experi-

mental values obtained by Estes and McElhaney at 40 s-1

[11]. The corresponding speed (1 m s-1) is close to those

used for road accident simulations.

One of the limitations of this work is due to the neg-

ative value identified for one of the hyperelastic Mooney–

Rivlin coefficients, which could result in numerical

instabilities during FEM simulations. However, this effect

has not been observed in LS-DYNA� software for the

numerical conditions used in this study. Consequently, at

high stretch level, the stress/stretch curve will have an

asymptote close to 0 or even slightly negative. However,

this is also balanced by either the viscoelastic part or the

fiber reinforcement of the material. This aspect could lead

one to reconsider the choice of the Mooney–Rivlin model

Fig. 9 Comparison at 4 different constant strain rates in compression

in terms of stress/stretch ratio for simulations of the cylindrical model

using the new mechanical law. Fibers are oriented along Z, X, or

‘‘oblique’’ direction (X ? Y ? Z). Curves are compared for validation

with data from the literature provided from experimental tests at

similar strain rates by Estes and McElhaney [11] as well as Pervin and

Chen [29]

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for the hyperelastic behavior, which could also be

substituted by an Ogden model as a next perspective for

this study.

Overall, a new dedicated brain tissue constitutive law

that can be implemented in brain FEM has been developed

in a realistic way and in accordance with experimental

literature results. Development of a complete brain FEM

would necessitate the definition of a particular fiber ori-

entation vector a0 as well as a definition of the C3 and C4

parameters taking the density of axonal fibers into account.

Indeed, while the fiber parameters have to be significant for

white matter, they could be considered as negligible for

gray matter, which is mainly composed of brain matrix.

The parameters C3 and C4 contain information relating

directly to the local brain axonal fiber density. For about

10 years, a new imaging technique, called diffusion tensor

imaging (DTI), has enabled observation of the Brownian

movement of water molecules constrained by axonal fibers

in the brain [22] and could also provide information on

fiber density and, indeed, local values of the C3 and C4

parameters in the brain.

Our first aim was to simulate and compare this law with

some experiments on brain tissue from the literature.

Nevertheless, it may be possible to extend some observa-

tions from these FE simulations of rheological tests to

further head impact FE simulations. In this regard, if the

fibers are not strictly oriented along the main direction of

brain compression, axon bending could result in a combi-

nation of multiple compressions along different directions

and also tissue shearing, which significantly increases

stress values and could result in tissue injury.

Conclusions

A transversely isotropic visco-hyperelastic brain constitu-

tive law has been developed and implemented in a FE code

to simulate a large range of experimental brain sample

compression tests reported in the literature. The aim is to

evaluate the sensitivity of the model to the fiber orientation

and applied strain rate. Validation of this law has been

presented based on experimental tests performed in 1970 by

Estes and McElhaney [11] as well as high-strain-rate tests

published by Pervin and Chen [29]. Significant stiffening at

high strain with dependence on the axonal fiber orientation

as well as on the strain rate has also been highlighted. The

results of this study can be considered as a step towards

reliable finite-element modeling of human brain.

References

1. Al-Bsharat A, Hardy W, Yang K, Kahlil T, Tashman S, King A.

Brain/skull relative displacement magnitude due to blunt head

impact: new experimental data and model. In: Proc. 43th stapp

car crash Conf., 1999. Warrendale: Society of Automotive

Engineers; 1999. p. 321–32.

2. Arbogast KB, Margulies SS. Material characterization of the

brainstem from oscillatory shear tests. J Biomech. 1998;31:801–7.

3. Arbogast KB, Meaney DF. Biomechanical characterization of the

constitutive relationship of the brainstem. In: Proceedings of the

Society of Automotive Engineers, 1995. p. 153–9.

4. Ashby MF, Jones DRH. Materiaux 1. Proprietes et applications.

3rd ed. Dunod, 2008.

5. Bandak FA, Van Der Vorst MJ, Stuhmiller LM, Mlakar PF,

Chilton WE, Stuhmiller JH. An imaging based computational and

experimental study of skull fracture: finite element model

development. In: Proc. of the head injury symposium, Wash-

ington DC, 1994.

6. Brands DWA. Predicting brain mechanics during closed head

imapect—numerical and constitutive aspects. Ph.D. Dissertation

Thesis, 2002. University of Eindhoven, Eindhoven.

7. Chatelin S, Deck C, Renard F, Kremer S, Heinrich C, Armspach

J-P, Willinger R. Computation of axonal elongation in head

trauma finite element simulation. J Mech Behav Biomed Mater

2011. ISSN 1751-6161, 10.1016.

8. Chatelin S, Constantinesco A, Willinger R. Fifty years brain

tissue mechanical testing: from in vitro to in vivo investigations.

Biorheology. 2010;47:255–76.

9. Criscione JC, Douglas AS, Hunter WC. Physically based strain

invariant set for materials exhibiting transversely isotropic

behavior. J Mech Phys Sol. 2001;49:871–97.

10. DiMasi F, Marcus J, Eppinger R. 3D anatomic brain model for

relating cortical strains to automobile crash loading. In: Proc. of

the international technical conference on experimental safety

vehicles, NHTSA, vol. 2, 1991. p. 916–23.

11. Estes MS, McElhaney JH. Response of brain tissue to compres-

sive loading. In: Proc. of the 4th ASME Biomechanics Conf.,

1970, 70-BHF-13.

12. Franceschini G, Bigoni D, Regitnig P, Holzapfel GA. Brain tissue

deforms similarly to filled elastomers and follows consolidation

theory. J Mech Phys Solids. 2006;54:2592–620.

Fig. 10 Tangent moduli calculated at small stretch (0.9 \ k\ 1)

from the curves presented in Fig. 9, for four different strain rates and

in the three applied fiber orientations

J Biorheol

123

13. Fung YC. Biomechanics: mechanical properties of living tissues.

NY: Springer; 1981.

14. Horgan TJ. A finite element model of the human head for use in

the study of pedestrian accidents. PhD thesis, 2005. University of

Dublin, Ireland.

15. Hosey RR, Liu YK. A homeomorphic finite element model of

impact head and neck injury. I.C.P. Finite Elements Biomech.

1980;2:379–401.

16. Hrapko M, van Dommelen JAW, Peters GWM, Wismans JSHM.

The influence of test conditions on characterization of the

mechanical properties of brain tissue. J Biomech Eng. 2008;

130(3):031003–10.

17. Kang HS, Willinger R, Diaw BM, Chinn B. Validation of a 3D

human head model and replication of head impact in motorcycle

accident by finite element modeling. In: Proc. 41th stapp car

crash conf. Lake Buena Vista: Society of Automotive Engineers;

1997. p. 329–38.

18. King A, Yang K, Zhang L, Hardy W. Is head injury caused by

linear or angular acceleration? In: IRCOBI conference, 2003.

p. 1–12.

19. Kleiven S. Predictors for traumatic brain injuries evaluated

through accident reconstruction. In: Proc. 51th stapp car crash

conf. Society of Automotive Engineers paper 2007-22-0003.

p. 81–114.

20. Kleiven S, Hardy WN. Correlation of an FE model of the human

head with experiments on localized motion of the brain-conse-

quences for injury prediction. Stapp Car Crash J. 2002;46:123–44.

21. Kruse SA, Rose GH, Glaser KJ, Manduca A, Felmlee JP, Jack CR

Jr, Ehman R. Magnetic resonance elastography of the brain.

NeuroImage. 2007;39:231–7.

22. LeBihan D, Mangin J-F, Poupon C, Clark CA, Pappata S, Molko

N, Chabriat H. Diffusion tensor imaging: concepts and applica-

tions. J Magn Reson Imaging. 2001;13:534–46.

23. Mendis K. Finite element modelling of the brain to establish

diffuse axonal injury criteria. PhD dissert, 1992, Ohio State

University.

24. Miller K, Chinzei K. Mechanical properties of brain tissue in

tension. J Biomech. 2002;35(4):483–90.

25. Miller K, Chinzei K. Constitutive modeling of brain tissue:

experiment and theory. J Biomech. 1997;30:1115–21.

26. Nelder JA, Mead R. A simplex method for function minimiza-

tion. Comput J. 1965;7:308–13.

27. Nicolle S, Lounis M, Willinger R. Shear properties of brain tissue

over a frequency range relevant for automotive impact situations:

new experimental results. Stapp Car Crash J. 2004;48:239–58.

28. Ning X, Zhu Q, Lanir Y, Margulies SS. A transversely isotropic

viscoelastic constitutive equation for brainstem undergoing finite

deformation. J Biomech Eng T ASME. 128;2006:925–33.

29. Pervin F, Chen WW. Dynamic mechanical response of bovine

gray matter and white matter brain tissues under compression.

J Biomechanics. 2009;42:731–5.

30. Prevost TP, Balakrishnan A, Suresh S, Socrate S. Constitutive

response of brain tissue. Acta Biomateriala. 2010;7:83–95.

31. Puso MA, Weiss JA. Finite element implementation of anisotropic

quasilinear viscoelasticity. ASME J Biomech Eng. 1998;120(1):

62–70.

32. Rashid B, Destrade M, Gilchrist MD. Modelling and simulation

of brain tissue in tension and compression at high strain rates. In:

Workshop on computational mechanics of materials, Limerick,

2011.

33. Rivlin RS, Saunders DW. Large elastic deformations of isotropic

materials VII. Experiments on the deformation of rubber. Phil.

Trans. Roy. Soc. London A. 1951;243:251–88.

34. Ruan JS, Kahlil T, King AI. Human head dynamic response to

side impact by finite element modeling. J Biomech Eng.

1991;113:276–83.

35. Shuck LZ, Advani SH. Rheological response of human brain

tissue in shear. J Basic Eng. 1972;94:905–11.

36. Shugar TA. A finite element head injury model. Report No DOT

HS 289-3-550-TA, vol. 1, 1977.

37. Spencer AJM. Continuum theory of the mechanics of fiber-

reinforced composites. New York: Springer; 1984.

38. Takhounts EG, Eppinger RH, Campbell JQ, Tannous RE, Power

ED, Shook SS. On the development of the SIMon finite element

head model. Stapp Car Crash J. 2003;47:107–33.

39. Takhounts EG, Hasija V, Ridella SA, Tannous RE, Campbell JQ,

Malone D, Danelson K, Stitzel J, Rowson S, Duma S. Investi-

gation of traumatic brain injuries using the next generation of

simulated injury monitor (SIMon) finite element head model.

Stapp Car Crash J. 2008;52:1–32.

40. Taylor Z, Comas O, Cheng M, Passenger J, Hawkes D, Atkinson

D, Ourselin S. On modelling of anisotropic viscoelasticity for soft

tissue simulation: numerical solution and gpu execution. Med

Image Anal. 2009;13:234–44.

41. Tiret L, Hausherr E, Thicoipe E, Garros B, Maurette P, Castel JP,

Hatton F. The epidemiology of head trauma in Aquitaine

(France), a community-based study of hospital admissions and

deaths. Int J Epidemiol. 1986;19(1):133–40.

42. Velardi F, Fraternali F, Angelillo M. Anisotropic constitutive

equations and experimental tensile behavior of brain tissue.

Biomech Model Mechanobiol. 2006;5(1):53–61.

43. Ward CC, Chan M, Nahum AM. Intracranial pressure: a brain

injury criterion. In: Proc. 24th stapp car crash Conf., SAE paper,

1980;80:1304.

44. Weiss J, Maker B, Govindjee S. Finite element implementation of

incompressible, transversely isotropic hyperelasticity. Comput

Methods Appl Mech Eng. 1996;135:107–28.

45. Zhang L, Yang K, Dwarampudi R, Omori K, Li T, Chang K,

Hardy W, Kahlil T, King A. Recent advances in brain injury

research: a new human head model development and validation.

Stapp Car Crash J. 2001;45:396–93.

46. Zhang M, Mak AFT. In vivo friction properties of human skin.

Prosthet Orthot Int. 1999;23:135–41.

47. Zhou C, Khalil TB, King AI. A 3D human finite element head for

impact injury analyses. In: Symposium proc. of prevention

through biomechanics, 1995. p. 137–48.

J Biorheol

123