# Anisotropic constitutive equations and experimental tensile behavior of brain tissue.

**ABSTRACT** The present study deals with the experimental analysis and mechanical modeling of tensile behavior of brain soft tissue. A transversely isotropic hyperelastic model recently proposed by Meaney (2003) is adopted and mathematically studied under uniaxial loading conditions. Material parameter estimates are obtained through tensile tests on porcine brain materials accounting for regional and directional differences. Attention is focused on the short-term response. An extrapolation of tensile test data to the compression range is performed theoretically, to study the effect of the heterogeneity in the tensile/compressive response on the material parameters. Experimental and numerical results highlight the sensitivity of the adopted model to the test direction.

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**ABSTRACT:**Characterizing the dynamic mechanical properties of brain tissue is deemed important for developing a comprehensive knowledge of the mechanisms underlying brain injury. The results gathered to date on the tissue properties have been mostly obtained in vitro. Learning how these results might differ quantitatively from those encountered in vivo is a critical step towards the development of biofidelic brain models. The present study provides novel and unique experimental results on, and insights into, brain biorheology in vivo, in situ and in vitro, at large deformations, in the quasi-static and dynamic regimes. The nonlinear dynamic response of the cerebral cortex was measured in indentation on the exposed frontal and parietal lobes of anesthetized porcine subjects. Load-unload cycles were applied to the tissue surface at sinusoidal frequencies of 10, 1, 0.1 and 0.01 Hz. Ramp-relaxation tests were also conducted to assess the tissue viscoelastic behavior at longer times. After euthanasia, the indentation test sequences were repeated in situ on the exposed cortex maintained in its native configuration within the cranium. Mixed gray and white matter samples were subsequently excised from the superior cortex to be subjected to identical indentation test segments in vitro within 6-7 h post mortem. The main response features (e.g. nonlinearities, rate dependencies, hysteresis and conditioning) were measured and contrasted in vivo, in situ and in vitro. The indentation response was found to be significantly stiffer in situ than in vivo. The consistent, quantitative set of mechanical measurements thereby collected provides a preliminary experimental database, which may be used to support the development of constitutive models for the study of mechanically mediated pathways leading to traumatic brain injury.Acta biomaterialia 06/2011; 7(12):4090-101. · 5.09 Impact Factor - SourceAvailable from: espace.uq.edu.au[Show abstract] [Hide abstract]

**ABSTRACT:**Past research into brain injury biomechanics has focussed on short duration impulsive events as opposed to the oscillatory loadings associated with Shaken Baby Syndrome (SBS). A series of 2D finite element models of an axial slice of the infant head were created to provide qualitative information on the behaviour of the brain during shaking. The test series explored variations in subarachnoid cerebrospinal fluid (CSF) thickness and geometry. A new method of CSF modeling based on Reynolds lubrication theory was included to provide a more realistic brain-CSF interaction. The results indicate that the volume of subarachnoid CSF, and inclusion of thickness variations due to gyri, are important to the resultant behaviour. Stress concentrations in the deep brain are reduced by fluid redistribution and gyral contact. These results provide direction for future 3D modeling of SBS. - SourceAvailable from: James Stuhmiller[Show abstract] [Hide abstract]

**ABSTRACT:**The purpose of this publication is to provide an overview of the USAMRMC blast research program, to show how quantitative physiology has provided useful solutions to operational medicine, and to indicate future directions of research.05/2008;

Page 1

Biomech Model Mechanbiol (2006) 5: 53–61

DOI 10.1007/s10237-005-0007-9

SHORT COMMUNICATION

F. Velardi · F. Fraternali · M.Angelillo

Anisotropic constitutive equations and experimental tensile behavior

of brain tissue

Received: 4 May 2005 /Accepted: 7 October 2005 / Published online: 29 November 2005

© Springer-Verlag 2005

Abstract Thepresentstudydealswiththeexperimentalanal-

ysisandmechanicalmodelingoftensilebehaviorofbrainsoft

tissue. A transversely isotropic hyperelastic model recently

proposed by Meaney (2003) is adopted and mathematically

studied under uniaxial loading conditions. Material param-

eter estimates are obtained through tensile tests on porcine

brainmaterialsaccountingforregionalanddirectionaldiffer-

ences. Attention is focused on the short-term response. An

extrapolation of tensile test data to the compression range

is performed theoretically, to study the effect of the hetero-

geneity in the tensile/compressive response on the material

parameters.Experimentalandnumericalresultshighlightthe

sensitivity of the adopted model to the test direction.

1 Introduction

Biomechanicalmodelingofthehumanheadisataskofgreat

interest for both medical and engineering reasons, which are

mainlyrelatedtothedevelopmentofcomputersimulationsof

traumaticbraininjuriesunderimpactloads(focalanddiffuse

injuries); virtual reality and robotic techniques in neurosur-

gery; design and efficiency assessment of helmets and other

protective tools.

This topic involves several research aspects, including:

formulationofconstitutiveequationsforbiologicalbrainmate-

rials and particularly for soft brain tissue, accounting for

directional properties, age effects, time-dependent behavior,

F. Velardi

Section of Pediatric Neurosurgery, Institute of Neurosurgery, Catholic

University Medical Centre, LargoA. Gemelli 1, 00168 Rome, Italy

F. Velardi

Pediatric Research Hospital “Bambino Ges` u”, Piazza S. Onofrio 2,

00165 Rome, Italy

F. Fraternali (B )· M.Angelillo

Department of Civil Engineering, University of Salerno,

84084 Fisciano (SA), Italy

E-mail: f.fraternali@unisa.it

Fax: +39-089-964045

and regional heterogeneities (see, e.g., Miller and Chinzei

1997, 2002; Arbogast and Margulies 1998, 1999; Miller et

al. 2000; Miller 2001; Bilston et al. 2001;Prange and Mar-

gulies 2002; Gefen and Margulies 2004); definition of auto-

matic procedures for brain topology reconstruction from im-

agedata(cf.,BartesaghiandSapiro2001;Ramonetal.2004);

and formulation of detailed finite element models of the hu-

man head (see Huang et al. 1999, 2000; Zhang et al. 2001;

Kleiven 2002; Mota et al. 2003).

Brainmatterconsistsofabasematrix(neuronsandextra-

cellular components: gray matter) crossed by a network of

neural tracts (or axonal fibers) in the so-called white matter.

The fibers are highly uniaxially oriented in the corpus callo-

sum (where they pass from one to the opposite brain hemi-

sphere), and arranged in a more disordered pattern, having

always a preferential axis, in the corona radiata.

Mechanical properties of human brain tissue have been

measured by several authors, both in vitro and in vivo. Con-

cerninginvitroexperiments,inrecentyearsresearchershave

focused their attention on uniaxial and shear testing (Miller

2001; Miller and Chinzei 1997, 2002; Arbogast et al. 1997;

Arbogast and Margulies 1998; Bilston et al. 2001; Prange

andMargulies2002).Invivoindentationtestshavealsobeen

carried out (Miller et al. 2000; Gefen and Margulies 2004)

to study the effects of blood pressure in vasculature on the

mechanical response of brain. Commonly, experiments are

conducted on porcine brain tissues, which have been found

to have some similarities with human brain material.

A wide dispersion of results between different authors

has been found, with material properties varying up to an

orderofmagnitude,inrelationtotestingconditions,prepara-

tion of samples, and differences in regional, directional, age,

and post-mortem conditions of brain tissues. In most cases,

experimental results have been correlated with rubber-like

hyperelastic constitutive models, obtaining material parame-

terestimatesforelasticformulationsofisotropicOgden-type

models (Ogden 1984).

Thepresentpaperdealswiththeexperimentalverification

of a transversely isotropic model, which appeared recently

in the literature for brain tissue (Merodio and Ogden 2003).

Page 2

54F. Velardi et al.

Several tensile tests on porcine samples are presented, con-

sidering tissues coming from various brain regions and with

different axonal fiber orientations. Attention is focused on

short- or middle-term tissue response under impact/accel-

eration loading, disregarding viscous effects. Indeed, such

effects have been found to have a limited influence on the

short-term response of brain tissue under impact actions (cf.,

e.g.,Aida2000).Atwo-levelprocedureforfittingexperimen-

tal data is presented and used in order to obtain Meaney’s

model parameter estimates. The given results are in good

agreement with those obtained by Miller and Chinzei (2002)

through uniaxial tests on cylindrical samples (isotropic mod-

eling).A central topic of the paper is the discussion about the

difference of the tissue behavior in tension and in compres-

sion (cf., e.g., Miller and Chinzei 2002). We found remark-

ablydifferentparameterestimatesbyconsideringonlytensile

and combined tensile-compressive behaviors, with different

signs of stretch exponents in the two cases.

2 Tensile tests on porcine brain tissue

In order to obtain quantitative and qualitative information

aboutregionalanddirectionalpropertiesofbraintissuemate-

rial, several tensile tests were carried out on tissue samples,

using porcine brain matter.

One of the practical difficulties in conducting uniaxial

tension tests on brain tissue is placing the samples in the

testing machine in a reliable and repeatable way. Miller and

Chinzei (1997, 2002) solved the problem by extracting short

cylindrical samples and gluing them to the plates of the ten-

sile testing machine. The main shortcoming of their method

is the identification of material parameters by relating exper-

imental data to an analytic solution of finite elasticity that

refers to the extension of a short cylinder (Miller 2001).

Instead we dealt with standard tensile tests on prismatic

samples that were accurately excised from porcine brains

through surgical techniques. In this section, we describe the

proceduresweadoptedtoovercomeinherentdifficultiescon-

nectedwithtensiletestingofsoftbiologicaltissues.Wechose

quitelongsamples(4–6cm)inordertorealizeuniaxialload-

ing conditions in the specimen central region, and also to

accurately locate the direction of fibers within the sample.

An analogous result cannot be obtained through cylindrical

samples, in which gray and white matter are mixed.

2.1 Specimen preparation

Brain characteristics A total of six swine brains (Fig.1a),

extracted from adult animals (age between 1 and 2 years),

were collected from a slaughter house in three different lots

(2–2–2).Apigheadwasalsotakenandsubjectedtomagnetic

resonance (Fig. 1c, d).

Storage The brains were stored in a physiological solution

and kept at a temperature of 3–7◦C. Transportation to the

laboratory took half an hour. Experiments were completed

within 5–6h post-mortem.

Shapeofsamples Samplesweretakenfromdifferentregions

of the brain to assess regional and directional properties of

thebraintissue(seebelow).Thesampleswerecutwithalan-

cet into strip shapes, approximately 4–6cm long, 1cm wide,

and 0.2–0.5cm thick. Obtaining an exact strip shape is diffi-

cult since the brain material is very soft and adheres, upon

contacting, to any body. Therefore, the areas of the sample

cross sections we used to convert load into stress must be

understood as averages.

Nature of tissues To assess regional and directional proper-

tiesofthebraintissue,thefollowingdifferentbrainmaterials

were tested:

(1) pure gray matter from motor strip (number of samples:

ns= 12);

(2) whitematterfromthecorpuscallosumwithaxonalfibers

along the longitudinal direction (aligned with load; ns=

6);

(3) white matter from the corona radiata with fibers in the

longitudinal direction (ns= 12);

(4) white matter from the corona radiata with fibers in the

transverse direction (ns= 12).

Massdensitywasslightlygreaterinwhitematterthaningray

matter (1,039g/cm3in white matter and 1,036g/cm3in gray

matter).

2.2 Experimental setup

Testing machine The machine employed for testing was an

INSTRON 4301. The mounted load cell allowed measure-

ment of axial force in the range 0.02–5N, with an error of

less than 0.1% of the maximum load.

Recording TheexperimentsweredocumentedbytakingCCD

camera images to ensure that during loading samples did not

slip between the platens. Tests were conducted in displace-

ment control, and load–displacement plots were automati-

cally produced.

Placing of the samples To prevent slip of samples from the

grips (cf., Fig.1b) and to preserve integrity of brain material,

we operated as follows:

– strips were continuously moistened with a physiological

solution before the placing in the testing machine, and

during the whole test;

– the strips were wrapped in tissue paper at the ends before

insertion into the grips;

– grips were tightened manually;

– the no-slip condition was checked by visual inspection

during testing and also on inspecting the CCD recording.

Page 3

Anisotropic constitutive equations and experimental tensile behavior of brain tissue55

Fig. 1 View of one of the porcine brains tested, where the removal of a strip of gray matter from the motor strip is visible (a); image of a strip

under testing (b); magnetic resonance images (3T) of a swine head (c, d): c coronal; d sagittal. Regional flags: 1=motor strip; 2=corpus callosum;

3=corona radiata

Temperature Tests were conducted at room temperature

(20–25◦C).

Loadinghistory Adisplacementrateof0.5mm/s,correspond-

ingtoastrainrateofabout0.01/s,wasfixedforalltests.This

rate was low enough to minimize inertia effects. Tests were

continued until failure or slipping of the samples from the

grips. Only one load cycle was allowed. No preconditioning

was performed due to the extreme delicacy and adhesiveness

of brain tissue (cf., Miller and Chinzei 2002).

2.3 Test results

The load–displacement plots were converted into nominal

stress S versus longitudinal stretch λ curves, by dividing the

applied force by the (averaged) initial cross-sectional area,

and the relative displacement between the platens by the ini-

tial length of the samples, respectively.

The initial cross-sectional area was determined by aver-

aging two measurements carried out in correspondence with

the central region of the specimen.

Figure2a–d shows the S–λ curves obtained by averag-

ing experimental data for each of the conducted test (see

above).Thegraphsincludestandarddeviationbars.Theratio

between standard deviation and mean value of S (coefficient

ofvariation),foranyfixedvalueofλ,rangedbetween0.2and

0.4 between the different tests, with lower values for small

stretches. The ranges of λ in Fig.2 correspond to the stretch

intervals for which all the tests for a given material were car-

ried out successfully until material failure or sample slipping

from machine grips occurred.

3 Meaney’s model for brain tissue

Due to its peculiar nature, the mechanical behavior of brain

tissue is expected to be sufficiently well described through

an unidirectional fiber reinforced composite model, and, in

particular, by means of a transversely isotropic hyperelastic

model.

Recently, Merodio and Ogden (2003) has proposed a

transversely isotropic model which consists of a first-order

Ogden model augmented by a I4-type reinforcing term (cf.,

Spencer 1984; Holzapfel 2000; Ogden 2003). It deals with

the following strain-energy function

W =2µ

α2

+2kµ

β2

?λα

1+ λα

?

2+ λα

+ 2I−β/4

3− 3?

4

Iβ/2

4

− 3

?

,λ1λ2λ3= 1,

(1)

Page 4

56F. Velardi et al.

Fig.2 Averagednominalstressagainstlongitudinalstretchinsimpletensiontestsondifferentbrainporcinesamples.Errorbarsindicatestandard

deviation

where λ1,λ2,λ3are the principal stretches of the (incom-

pressible)material,andI4coincideswiththesquareofmate-

rial stretch in the fiber direction.

In Eq.1, µ is the infinitesimal shear modulus of the unre-

inforcedmaterial(nofibers);α andβ areparameters;k(> 0)

is a coefficient which measures the increase of stiffness of

the material in the fiber direction. The case with k = 0 cor-

responds to the gray matter tissue.

Meaney suggests to set β = α, which is motivated by the

fact that experimental results on white matter tissues show

small changes of α with the test direction (cf., Prange and

Margulies 2002).

Subsequently,webrieflyexaminethemathematicalprop-

ertiesoftheMeaneymodelunderuniaxialloading,following

theapproachadoptedin(MerodioandOgden2005).Wedeal

with the strain-energy function (α = β)

W =2µ

α2

+2kµ

α2

?λα

1+ λα

?

2+ λα

+ 2I−α/4

3− 3?

4

Iα/2

4

− 3

?

,λ1λ2λ3= 1,

(2)

assumingthattheloadisappliedalongtheX2-axisofagiven

Cartesian frame X1,X2,X3and that fibers can be aligned

along either X2or X1.

We use the short-hand notation λ for the stretch in the

loading direction (λ ≡ λ2), and the symbols S and T for

the first Piola-Kirchoff stress tensor and the Cauchy stress

tensor, respectively. Such tensor fields are derived from the

strain-energy function through (see, e.g.,Ogden 1984)

S =

3

?

3

?

i=1

?∂W

?

∂λi

− pλ−1

i

?

v(i)⊗ u(i);

T =

i=1

λi∂W

∂λi

− p

?

v(i)⊗ v(i),

(3)

u(i)and v(i)being the eigenvalues of the right and left stretch

tensors U and V, respectively.

Obviously, under uniaxial loading along the X2-axis, the

unique nonzero components of S and T are S(≡ S22) and

T(≡ T22), respectively.

Page 5

Anisotropic constitutive equations and experimental tensile behavior of brain tissue57

Fig. 3 Plots S/µ against stretch λ in the fiber direction for α = −5 and k = 0,0.5,5,20 (a); k = 2 and α = −5,2,5 (b)

Fig. 4 Plots of S/µ against stretch λ in the direction orthogonal to the fibers for k = 0,0.5,5,20 and α = 5 (a); α = −5 (b)

For uniaxial load in the fiber direction, upon imposing

Sij= 0 for (i,j) ?= (2,2) and enforcing the incompressibil-

ity constraint (J = 1), it is easy to obtain

λ1= λ3= λ−1/2,

α

p =2µλ−α/2

,I4= λ2;

(4)

W =2µ(1 + k)

S =2µ(1 + k)λ−1−α/2?λ3α/2− 1?

T = λS =2µ(1 + k)λ−α/2?λ3α/2− 1?

It is not difficult to verify that S (as well as T) is a monotonic

functionofλ,foranyvalueofk,approaching−∞forλ → 0

and +∞ for λ → ∞.

TheresponseofthematerialisillustratedinFig.3,which

shows plots of S/µ against λ for several values of k and α.

Let us now consider the case of uniaxial load orthogonal

to the fibers (fibers aligned along X1). Differently from the

previous case, here we have λ1= λ3= λ−1/2if and only if

k = 0 (excluding the meaningless case α=0).

α2

?λα+ 2λ−α/2− 3?,

α

,

α

.

(5)

By writing λ3 = 1/λ1λ (due to incompressibility) and

imposing Sij = 0 for (i,j) ?= (2,2), we get the following

relation between λ1and λ

λ1

?

λα

1+ k

?

λα

1− λ−α/2

1

??1/α

= λ−1

(6)

which cannot be solved analytically for λ1except for special

valuesofα.Itisinsteadpossibletonumericallydetermineλ1

as a function of λ, once α and k are given and inadmissible

rootsofEq.6(λ1< 0)arediscarded.Underuniaxialloading

orthogonal to the fibers, it can then be shown that the stretch

in the fiber direction exhibits a limiting nonzero minimum

value (limiting contractive stretch) for α > 0 and k ?= 0.

Differently,thesamestretchexhibitsafinitemaximumvalue

(limiting extensional stretch) for α < 0 and k ?= 0.

Once λ1= λ1(λ) has been numerically determined for

given α and k, it is possible to express p,I4,W,S, and T

as functions of λ. The dependence of S/µ on λ is shown in

Fig.4forseveralvaluesofα andk.AscanbeseenS israther

insensitive to k in the tensile range for α > 0 (cf., Fig.4a), or

in the compressive range for α < 0 (Fig.4b). In each case,

the effects of the transverse reinforcement are very weak for

Page 6

58F. Velardi et al.

Table 1 Uniaxial response in the direction of the fibers (a) and orthogonal to the fibers (b) using the model by Meaney (2003), i.e. eq. (2).

Different values for α,k and λ are displayed, while ?% denotes the normalized difference in percent (see the text)

S/µ (load?fibers)

k = 0

α = −5

−1.239

−0.411

+0.392

+0.510

S/µ (load⊥fibers)

k = 2

α = −5

−1.333

−0.465

+0.527

+0.718

(a)

k = 2

α = −5

−7.438

−2.467

+2.351

+3.060

k = 10

α = −5

−26.033

−8.634

+8.229

+10.710

?%

λ

0.80

0.90

1.20

1.30

α = 5

−0.709

−0.316

+0.618

+0.983

α = 5

−4.258

−1.896

+3.709

+5.897

α = 5

−14.902

−6.635

+12.981

+20.638

+74.7

+30.1

−36.6

−48.1

(b)

k = 10

α = −5

−1.355

−0.484

+0.632

+0.915

λ

0.80

0.90

1.20

1.30

α = 5

−0.977

−0.405

+0.675

+1.043

?%

+36.4

+14.8

−21.9

−31.2

α = 5

−1.209

−0.462

+0.690

+1.056

?%

+12.1

+4.8

−8.4

−13.3

low values of k(0 < k ? 5) and moderately large values of

λ(0.8 ? λ ? 2).

InTables1a and b, we examined the effects of the sign of

the stretch exponent α on the material response, both in the

direction of the fibers (Table1) and in the direction orthog-

onal to the fibers (Table2). We considered two values of α

(−5 and 5), and recorded S/µ for selected values of λ, in

compression (λ = 0.8,0.9) and in tension (λ = 1.2,1.3).

For loading parallel to the fibers, the normalized differ-

ence ? = (S|α=−5− S|α=+5)/ S|α=+5between the two

examinedresponsesisindependentofthevalueofthestrength-

eningparameterk (cf.,Eq.52).Suchadifferencerangesfrom

about +75% for λ = 0.80 to about −48% for λ = 1.30 (see

alsoFig.3b).Differently,forloadingorthogonaltothefibers,

? depends on k and decreases as k increases (cf., Fig.4).

Miller and Chinzei (2002) and Prange and Margulies

(2002) have experimentally validated a first order, isotropic

(k = 0),andviscoelasticformulationofmodel2.Itisknown

that policonvexity of the isotropic model holds if |α| > 1

(cf., Ball 1977; Ciarlet, 1998). Miller and Chinzei estimated

µ0= 842Pa (instantaneous value of µ) and α ≈ −4.7 (con-

stant in time) for porcine brain tissue using uniaxial tests.

InsteadPrangeandMarguliesobtainedseveraldifferentesti-

mates for µ0and α through shear tests on swine tissues ob-

tainedfromvariousbrainregions.Foradultporcinesamples,

they found µ0-values varying between 180 and 290Pa and

α-values varying between 0.03 and 0.075, in relation to the

differentnatureofthetissues(grayorwhitematter),regional

origin (thalamus, corona radiata, corpus callosum) and rela-

tive position of axonal fibers with respect to test direction (in

white matter and mixed white/gray matter samples).

4 Material parameter estimation

In the case of uniaxial (tensile and/or compressive) loading

parallel to fibers, it is possible to fit test data (nominal stress

vs. fiber stretch) to model52and estimate (1 + k)µ and α.

Successively, it is possible to estimate µ, and thus also k,

through uniaxial tests transverse to the fiber direction, tak-

ing (1 + k)µ and α as fixed. In this second phase, due to the

impossibilityofobtainingananalyticexpressionfortheMea-

ney model, one could fit data to the Ogden model. This can

be acceptable for moderately large stretches (see Section 3).

Alternatively, one could use the fitting procedures proposed

by Ogden et al. (2004) for multiple data sets.

For gray matter (from motor strip), we fitted the isotro-

pic formulation of model52(k = 0) to the tensile test data

inFig.2a(meanS–λcurve),employingtheLevenberg–Mar-

quardt optimization algorithm (see, e.g., Twizell and Ogden

1983), which is available under the add-on package <Sta-

tistics—“NonlinearFit”> of Mathematica

We obtained the following estimates for the material param-

eters: µ = 319.28Pa,

matter from the corpus callosum under uniaxial load aligned

with fibers, we fitted the Meaney model to data in Fig.2b,

obtaining (1 + k)µ = 502.12Pa,

Since we did not test brain material from this region under

load transverse to the fibers, we were not able to estimate µ

andkseparatelyinthiscase.Finally,forwhitematterfromthe

corona radiata, we employed the two-level fitting procedure

as described above, to estimate the complete set of material

parametersµ,α,andk.Indetail,firstwefittedmodel52tothe

data in Fig.2c, obtaining (1+k)µ = 378.55Pa,

(Fig.5c).Then,wefittedtheisotropicmodelwithα = 6.84to

the data in Fig.2d, obtaining µ = 136.82Pa, which implies

k = 1.77 (Fig.5d).

In order to evaluate the influence of the different brain

material responses under tensile and compressive loadings

on parameter estimation, we also addressed an extrapola-

tion of the data presented in Fig.2 in the compression range

λ ∈ (0.8,1.0). To this end, we adopted the model in Table2

for the response of the brain material in compression, which

was deduced from Fig.4 of Miller and Chinzei (2002). In

?

R(Wolfram 1999).

α = 3.50 (cf., Fig.5a). For white

α = 2.38 (cf., Fig.5b).

α = 6.84

Page 7

Anisotropic constitutive equations and experimental tensile behavior of brain tissue59

Fig. 5 Fits of the augmented Ogden model to simple tension tests (Levenberg–Marquardt nonlinear fit method)

Table 2 Theoretical model used to extrapolate tensile data to the com-

pression range?¯S = S|λ=1.2

λS22/¯S22

0.80

−4.20

0.85

−2.60

0.90

−1.30

0.95

−0.67

?

Table2,¯S denotes the value of S at λ = 1.2. The fitting of

the extended test data to the Meaney model, conducted as

described above, leads us to new estimates of the material

parameters, which are displayed in Fig.6.

Ascanbeseenquitedifferentestimatesofµ,α,andkmay

be obtained with respect to pure tension data (cf., Figs.5,6),

and negative values of α may be found. The results in Fig.6

agree well with those given by Miller and Chinzei (2002).

The values of α obtained in pure tension are not very far

from +5.0, while those corresponding to the complete uni-

axial response are not far from −5.0. Hence, data given in

Tables1and2areusefultopredictthe(remarkable)errorthat

would occur in a finite element model by using tensile mate-

rial constants for the computation of the complete (tensile

and compressive) brain response due to impact/acceleration

loading.

5 Concluding remarks

In this work we have discussed the mechanical behavior of

soft brain tissues. We have focused our attention on trans-

verselyisotropicconstitutiveequations,regionaldifferences,

directional properties, and tensile testing. Fitting procedures

for material parameter estimation have been proposed and

employed in practice, obtaining estimates for porcine brain

materials under pure uniaxial tension and combined uniax-

ial compression-tension. In the latter case, tensile test data

were associated to a theoretical model of the compressive

response, deduced by other available experimental studies

(Miller and Chinzei 2002).

The results obtained highlight the sensitivity of mate-

rial parameters to test conditions. In particular, the exponent

α of the principal stretches appearing in the strain-energy

function changed the sign passing from simple tension (α >

0) to compression-tension loads (α < 0). All the estimates

obtainedforα fallwithintherange|α| > 1.Whitematterwas

Page 8

60F. Velardi et al.

Fig. 6 Fits of the augmented Ogden model to complete uniaxial load data (*: tensile experimental data extrapolated in compression, see Table2)

found to be stiffer than gray matter, and, within the former,

the corpus callosum showed higher shear modulus than the

corona radiata.

Itisevidentlynecessarytoadoptmodelswithatleasttwo

terms for the isotropic part, one describing the response of

the brain matrix tissue in tension and the other the response

in compression, and at least one term for the fiber reinforc-

ing part. Refinements of the Meaney model are obtained by

dealing with strain energies of the form

W =

N

?

n=1

µn

αn

?λαn

νrIβr

1+ λαn

2+ λαn

3− 3?

+

R

?

r=1

?

4+ ξrIγr

4− νr− ξr

?

,λ1λ2λ3= 1,

(7)

whereN andR arepositiveintegersandµn,αn,νr,ξr,βr,γr

are material parameters.

Significant developments are expected in future studies,

with reference to a mathematical analysis of model7 under

general loading conditions (cf., Merodio and Ogden 2002,

2003, 2005); fitting of multiple experimental data relative to

indentation,uniaxial,andsheartests;time-dependentbehav-

ior; tissue damage; and large-scale assessment of material

models through finite element modeling of the human head.

Acknowledgements The authors wish to express their sincere thanks

to Professor Ray W. Ogden for suggesting and discussing reinforcing

modelsforbraintissue,andforhisassistancewiththemathematicaland

mechanical aspects of the present work. They also wish to gratefully

acknowledge the very kind assistance with the experimental aspects of

the present research offered by Professors Loredana Incarnato andVit-

toriaVittoriaandDr.GiulianaGorrasifromtheDepartmentofChemical

Engineering,UniversityofSalerno,andbyProfessorFrancescoDeSal-

le, from the Institute of Radiology, University “Federico II” of Naples,

andDr.TommasoScarabino,fromtheDepartmentofRadiology,“Casa

Sollievo della Sofferenza” of S. Giovanni Rotondo (Foggia).

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