Biomech Model Mechanbiol (2006) 5: 53–61
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
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
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
sensitivity of the adopted model to the test direction.
interest for both medical and engineering reasons, which are
injuries); virtual reality and robotic techniques in neurosur-
gery; design and efficiency assessment of helmets and other
This topic involves several research aspects, including:
rials and particularly for soft brain tissue, accounting for
directional properties, age effects, time-dependent behavior,
Section of Pediatric Neurosurgery, Institute of Neurosurgery, Catholic
University Medical Centre, LargoA. Gemelli 1, 00168 Rome, Italy
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
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-
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).
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-
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
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
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-
models (Ogden 1984).
of a transversely isotropic model, which appeared recently
in the literature for brain tissue (Merodio and Ogden 2003).
54 F. 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.,
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-
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
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
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
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.
of the brain to assess regional and directional properties of
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-
(1) pure gray matter from motor strip (number of samples:
along the longitudinal direction (aligned with load; ns=
(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).
matter (1,039g/cm3in white matter and 1,036g/cm3in gray
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.
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-
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.
Anisotropic constitutive equations and experimental tensile behavior of brain tissue 55
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;
Temperature Tests were conducted at room temperature
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
between standard deviation and mean value of S (coefficient
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
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
56 F. Velardi et al.
where λ1,λ2,λ3are the principal stretches of the (incom-
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
with the strain-energy function (α = β)
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)
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.
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,
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 → ∞.
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).
?λα+ 2λ−α/2− 3?,
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
which cannot be solved analytically for λ1except for special
as a function of λ, once α and k are given and inadmissible
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.
(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
58 F. 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)
k = 0
α = −5
k = 2
α = −5
k = 2
α = −5
k = 10
α = −5
α = 5
α = 5
α = 5
k = 10
α = −5
α = 5
α = 5
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
about +75% for λ = 0.80 to about −48% for λ = 1.30 (see
? 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.
mates for µ0and α through shear tests on swine tissues ob-
they found µ0-values varying between 180 and 290Pa and
α-values varying between 0.03 and 0.075, in relation to the
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
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
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
corona radiata, we employed the two-level fitting procedure
as described above, to estimate the complete set of material
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
α = 3.50 (cf., Fig.5a). For white
α = 2.38 (cf., Fig.5b).
α = 6.84
Anisotropic constitutive equations and experimental tensile behavior of brain tissue 59
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
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.
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
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
5 Concluding remarks
In this work we have discussed the mechanical behavior of
soft brain tissues. We have focused our attention on trans-
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
60 F. 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
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
4− νr− ξr
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
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
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-
le, from the Institute of Radiology, University “Federico II” of Naples,
Sollievo della Sofferenza” of S. Giovanni Rotondo (Foggia).
Aida T (2000) Study of human head impact: brain tissue constitutive
models. PhD Thesis, West Virginia University
Arbogast KB, Margulies SS (1998) Material characterization of the
brainstem from oscillatory shear tests. J Biomech 31:801–807
of the viscoelastic behavior of the brainstem in shear. J Biomech
A high-frequency shear device for testing soft biological tissues.
J Biomech 30:757–759
Ball JM (1977) Convexity conditions and existence theorems in non-
linear elasticity.Arch Ration MechAnal 63:337–403
Bartesaghi A, Sapiro G (2001) A system for the generation of curves
on 3D brain images. Hum Brain Mapp 14:1–15
Anisotropic constitutive equations and experimental tensile behavior of brain tissue61 Download full-text
Bilston LE, Liu Z, Phan-Thien N (2001) Nonlinear viscoelastic be-
haviour of brain tissue in shear: some new experimental data and a
differential constitutive model. Biorheol 38:335–345
Ciarlet PG (1988) Mathematical elasticity. Three-dimensional elastic-
ity, vol I. Elsevier,Amsterdam, The Netherlands
Gefen A, Margulies S (2004) Are in vivo and in situ brain tissues
mechanically similar? J Biomech 37:1339–1352
Holzapfel GA (2000) Nonlinear solid mechanics. Wiley, Chichester
sional finite element analysis of subdural hematoma. J Trauma Inj
Infect Crit Care 47:538–544
analysis of brain contusion: an indirect impact study. Med Biol Eng
Kleiven S (2002) Finite element modeling of the human head. PhD
Dissertation, Kungl Tekniska H¨ ogskolan (Royal Institute of Tech-
nology), Stockholm, Sweden
Meaney DF (2003) Relationship between structural modeling and
hyperelastic material behavior: application to CNS white matter.
Biomech Model Mechanobiol 1:279–293
Merodio J, Ogden RW (2002) Material instabilities in fiber-reinforced
nonlinearly elastic solids under plane deformation. Arch Mech
Merodio J, Ogden RW (2003) Instabilities and loss of ellipticity in
deformation. Int J Solids Struct 40:4707–4727
Merodio J, Ogden RW (2005) Mechanical response of fiber-reinforced
incompressible non-linearly elastic solids. Int J Non-linear Mech
40:213–227 (available online at www.sciencedirect.com)
Miller K (2001) How to test very soft biological tissues in extension?
J Biomech 34:651–657
iment and theory. J Biomech 30:1115–1121
Miller K, Chinzei K (2002) Mechanical properties of brain tissue in
tension. J Biomech 35:483–490
Miller K, Chinzei K, Orssengo G, Bednarz P (2000) Mechanical prop-
erties of brain tissue in-vivo: experiment and computer simulation.
J Biomech 33:1369–1376
MotaA, Klug WS, Ortiz M, West M, PandolfiA (2003) Finite element
simulation of firearm injury to the human cranium. Comput Mech
Ogden RW (1984) Non-linear elastic deformations. Ellis Horwood,
Ogden RW (2003) Nonlinear elasticity, anisotropy, material stability
and residual stresses in soft tissue. In: Holzapfel GA, Ogden RW
(eds) Biomechanics of soft tissue in cardiovascular systems, CISM
courses and lecture series 441. Springer-Verlag,Wien NewYork, pp
Ogden RW, Saccomandi G, Sgura I (2004) Fitting hyperelastic model
to experimental data. Comput Mech 34:484–502
Prange MT, Margulies SS (2002) Regional, directional, and age depen-
Ramon C, Schimpf P, Haueisen J, Holmes M, Ishimaru A (2004) Role
Spencer AJM (1984) Constitutive theory for strongly anisotropic sol-
ids. In: Spencer AJM (ed) Continuum theory of the mechanics of
fibre-reinforced composites, CISM courses and lecture series 282.
Springer-Verlag, Wien, pp 1–32
Twizell EH, Ogden RW (1983) Non-linear optimization of the material
constants in Ogden’s stress–deformation relation for incompressible
isotropic elastic materials. JAust Math Soc B 24:424–434
Wolfram S (1999) The mathematica book, 4th edn. Wolfram
Media/Cambridge University Press, Champaign
Zhang L, Yang KH, King AI (2001) Comparison of brain responses
between frontal and lateral impacts by finite element modeling.
J Neurotrauma 18:21–30