Application of Optimization Methodology and Specimen-Specific Finite Element Models for Investigating Material Properties of Rat Skull
Finite element (FE) models of rat skull bone samples were developed by reconstructing the three-dimensional geometry of microCT images and voxel-based hexahedral meshes. An optimization-based material identification method was developed to obtain the most favorable material property parameters by minimizing differences in three-point bending test responses between experimental and simulation results. An anisotropic Kriging model and sequential quadratic programming, in conjunction with Latin Hypercube Sampling (LHS), are utilized to minimize the disparity between the experimental and FE model predicted force-deflection curves. A selected number of material parameters, namely Young's modulus, yield stress, tangent modulus, and failure strain, are varied iteratively using the proposed optimization scheme until the assessment index 'F', the objective function comparing simulation and experimental force-deflection curves through least squares, is minimized. Results show that through the application of this method, the optimized models' force-deflection curves are closely in accordance with the measured data. The average differences between the experimental and simulation data are around 0.378 N (which was 3.3% of the force peak value) and 0.227 N (which was 2.7% of the force peak value) for two different test modes, respectively. The proposed optimization methodology is a potentially useful tool to effectively help establish material parameters. This study represents a preliminary effort in the development and validation of FE models for the rat skull, which may ultimately serve to develop a more biofidelic rat head FE model.
Application of Optimization Methodology and Specimen-Speciﬁc Finite
Element Models for Investigating Material Properties of Rat Skull
YENER N. YENI,
and KING H. YANG
State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Hunan, China;
Bioengineering Center, Wayne State University, 818 W. Hancock, Detroit, MI 48201, USA; and
Henry Ford Hospital, Detroit,
(Received 19 April 2010; accepted 5 July 2010; published online 23 July 2010)
Associate Editor Kyriacos A. Athanasiou oversaw the review of this article.
Abstract—Finiteelement(FE) models of rat skull bone samples
were developed by reconstructing the three-dimensional geom-
etry of microCT images and voxel-based hexahedral meshes.
An optimization-based material identiﬁcation method was
developed to obtain the most favorable material property
parameters by minimizing differences in three-point bending
test responses between experimental and simulation results. An
anisotropic Kriging model and sequential quadratic program-
ming, in conjunction with Latin Hypercube Sampling (LHS),
are utilized to minimize the disparity between the experimental
and FE model predicted force–deﬂection curves. A selected
number of material parameters, namely Young’s modulus,
yield stress, tangent modulus, and failure strain, are varied
iteratively using the proposed optimization scheme until the
assessment index ‘F’, the objective function comparing simu-
lation and experimental force–deﬂection curves through least
squares, is minimized. Results show that through the applica-
tion of this method, the optimized models’ force–deﬂection
curves are closely in accordance with the measured data. The
average differences between the experimental and simulation
data are around 0.378 N (which was 3.3% of the force peak
value) and 0.227 N (which was 2.7% of the force peak value)
for two different test modes, respectively. The proposed
optimization methodology is a potentially useful tool to
effectively help establish material parameters. This study
represents a preliminary effort in the development and valida-
tion of FE models for the rat skull, which may ultimately serve
to develop a more bioﬁdelic rat head FE model.
Keywords—Rat skull, Specimen-speciﬁc ﬁnite element models,
Anisotropic Kriging, Optimization, Material identiﬁcation.
Numerous in vivo rodent experimental models
have been developed to investigate different types of
traumatic brain injury (TBI), associated behavioral
changes, and efﬁcacy of therapeutic methods. Although
the extern al parameters used in these experimental
models (such as the impact speed, impact depth, and
weight of the impactor) can be precisely controlled,
there are no direct methods available for investigating
intracranial responses that are directly related to tissue
damage. At present, ﬁnite elemen t (FE) models are
probably the best means to acq uire such responses. In
recent years, several FE models of the rat head have
been developed to predict internal respon ses of the
brain under injury scenarios to complement experi-
In order to ensure accurately
predicted brain internal responses using FE modeling
techniques, accurate material properties are necessary.
Several studies have been conducted to determine
human crania l bone propert ies in diﬀerent loading
modes and directions.
In the adult, cranial bone
can be divided into compact bone (outer and inner
tables) and trabecular bone (diploe
), which have been
tested both as a composite and separately. Melvin
found that the compressive modulus of the
layer ranged from 0.39 to 2.75 GPa. For
the intact three-layered structure, McElhaney’s study
of dog-bone shaped human skull samples in in-plane
tension indicated an average elastic modulus of 5.38
with no signiﬁcant differences found in
compressive modulus when skull bone samples were
tested in different tangential directions. Testing of
human skull bone perpendicular to the surface has
resulted in an elastic modulus of 2.4 GPa,
studies have reported values of half that magnitude.
To the best of our knowledge, only one study has
reported the material properties of rat skull bone.
That study utilized several two-dimensional FE models
to simulate indentation tests conducted on rat skulls to
Address correspondence to King H. Yang, Bioengineering Cen-
ter, Wayne State University, 818 W. Hancock, Detroit, MI 48201,
USA. Electronic mail: firstname.lastname@example.org
Annals of Biomedical Engin eering, Vol. 39, No. 1, January 2011 (Ó 2010) pp. 85–95
0090-6964/11/0100-0085/0 Ó 2010 Biomedical Engineering Society
calculate material property parameters. The authors
reported an average elastic modulus of 6.01 MPa for
the skull of 43-day-old adult rat, which was at least 60
times lower than that of human. Some of the difference
in moduli between rat and human skulls may be due to
variance in porosity or trabecular arrangement of these
bones, in addition to the indirect method used by
Gefen et al.
to determine the skull bone properties.
Experimentally, machining a small, curved rat skull
into rectangular specimens for direct material testing
may present some technical challenges. Additionally, no
literature has been published to date reporting rat skull
properties at diﬀerent loading rates. Finally, although
optimization methods have been used in biomaterial
little information was
provided regarding which optimization procedures were
The primary objective of this study was, therefore,
to develop a methodology for investigating rate-
dependent elastic–plastic properties (Young’s modu-
lus, yield stress, tangent modulus, and failure strain) of
the rat skull. Experimental three-point bending tests at
two diﬀerent rates were conducted to obtain force–
deﬂection curves. Consid ering that each skull sample
had diﬀerent dimensions and varying porosities, spec-
imen-speciﬁc FE models were developed and used in
conjunction with optimization-based material identiﬁ-
cation method to match experimentally measured and
model predicted force–deﬂection curves to identify
aforementioned material property parameters.
MATERIALS AND METHODS
Geometry and Hexahedral Mesh Generation
Two skull samples, each approximately 14 by 3 mm,
were dissected using a Dremel rotary tool (Model 750,
Robert Bosch Tool Corporation, IL) from each of 10
young adult Sprague-Dawley rats with an average
mass of 266 (±7) g. The microcomputed tomography
(microCT) scanner built in-house at Henry Ford
Hospital (Detroit, MI), similar to the original system
previously described by Reimann et al.,
to scan all 20 samples at a spatial resolution of
16 9 16 9 16 lm to depict detailed geometric proﬁles
and internal porosities. Figure 1 shows a coronal sec-
tion of a typical sample with detailed geometric
structure obtained from micr oCT scanning, in which
the porous characteristics are shown. The average
thickness of each skull sample was measured from the
microCT images. An imaging post-processing soft-
ware, Mimics (version 12, Materialise Inc., Leuven,
Belgium), was used to segment the bony portion from
each microCT dataset before a voxel-based hexahedral
mesh was generated to create sample-spe ciﬁc FE
meshes with an element resolution of 64 9 64 9
128 lm (Fig. 2). The sample-speciﬁc FE models served
to eliminate any geometric effect due to intersample
variations, such as varying thickness and locations
without bony tissues (i.e., voids or holes), on model
predicted force–deﬂection curves.
Three-point Bending Test Setup
The two samples taken from each rat were tested in
three-point bending on an Instron material testing sys-
tem (Model 1321 frame with Model 8500 controller,
Canton, MA) at a randomly selected loading velocity of
either 0.02 or 200 mm/s using the test setup shown in
Fig. 3. The center of loading was aligned with the
FIGURE 1. A typical microCT image showing the porosity of
one skull sample. The size bar indicates 200 lm.
FIGURE 2. A sample-speciﬁc FE mesh of one skull sample
using 35,613 hexahedral elements.
Steel shaft to Instron ram
Impactor diameter 2mm
Support diameter 2mm
Load cell to fixed plate
FIGURE 3. Schematic diagram of the three-point bending
GUAN et al.86
mid-point between the bregma and lambda sutures. The
long axis of a cylindrical steel rod with a diameter of
2 mm served as the impactor surface, held in place by a
solid steel shaft attached to the Instron ram with an
aluminum plate. Two other rods of the same diameter
were used to support the specimen during bending. A
22.24 N capacity load cell (Model MDB-5, Transducer
Techniques, CA), screwed ﬁrmly to the Instron frame
underneath the sample supports, was used to measure
the force–time histories. SAE ch annel frequency class
(CFC) 600 ﬁlter at a corner frequency of 1000 Hz was
used to ﬁlter all force–time curves. Equation (1), taken
from classical beam theory, was used also to calculate
the Young’s modulus from three-point bending test
results for comparison with optimization results:
where E stands for the elastic modulus in MPa;
L stands for support span in mm; m stands for slope of
the initial straight line portion of the force–deﬂection
curve in N/mm; b stands for width of test sample in
mm; and d stands for thickness of test sample in mm.
The boundary and loading conditions were precisely
deﬁned for each FE model using Hypermesh (Altair
Engineering, Troy, MI) accordi ng to the specimen-
speciﬁc experimental setup. The impl icit FE method is
generally the preferred method when solving quasi-
static problems. When the entire system’s degrees-
of-freedom are very large, this method requires an
enormous amount of random access memory to store
the structural stiﬀness matrix before it is inversed to
calculate nodal displacements. It may become very
computationally expensive when acquiring time histo-
ries at small time steps as compared to using the
explicit FE method. Additionally, the current optimi-
zation study called for a large number of simulations
to determine optimal material parameters. Conse-
quently, shortening the time needed for each simula-
tion was greatly desired. One way to overcome these
problems is through the application of a damping
factor using the explicit solver, so long as the kinetic
energy is controlled near zero. This method is recom-
mended by the LS-DYNA theory manual and has been
used successfully by Zhang.
To simulate quasi-static loading at 0.02 mm/s with
minimal computational time and random access mem-
ory, a damping factor was applied in the LS-DYNA
explicit solver to ensure that the kinetic energy was close
to zero. An iterative process was set up to choose the
best damping factor until the kinetic energy nearly
vanished. This resulted in a damping factor (n)of
for one skull sample. To check its validity, the
force–deﬂection curve predicted by the combined
explicit solver with the addition of 60 ms
factor was then compared to that predicted using the
implicit method. Figure 4 demonstrates that the dif-
ferences between implicit and explicit predictions were
minimal (less than 1%). Subsequently, the same
damping factor was applied to all qua si-static simula-
tions to calculate the optimal material properties. In
solving high-speed transient dynamic problems using an
explicit FE solver, no damping factor is needed as long
as the time step is smaller than that needed for the wave
to pass through each element.
The constitutive material law assumed was
the LS-DYNA material library. In order to reduce the
simulation time, a Massively Parallel Processing (MPP)
version of the solver was used on an eight-node cluster,
which has two AMD Opteron (tm) processors with a
clock speed of 2.4 GHz for each node.
Optimization-based Material Identiﬁcation
The objective function selected for the optimization
scheme was based on an assessment index F, which
aimed to minimize the average error between the
experimentally obtained and model calculated force–
deﬂection curves on a least squares basis, as shown in
indicates those values measured from tests, f
indicates the FE model calculated corresponding force
values, m stands for the ‘‘measured’’ value, c stands for
the ‘‘calculated’’ value, and n stands for the number
of points of measured data. Different loading speeds
FIGURE 4. Comparison between implicit solution method
and explicit solution method.
Application of Optimization Methodology and Specimen-Speciﬁc FE Models 87
require different n values, which are de termined by
total loading time and sampling frequency. In this
, in correspondence with the absolute
difference, was adapted in the assessment index (Eq. 2)
instead of (f
, related to relative difference,
to avoid over-ﬁtting of the initial phases of elastic
deformation. The sampling interval in each simulation
was 0.02 ms. As evident from Eq. (2), an F value of
zero (0) represents the best possible match.
As in any material testing, the force–deﬂection curve
drops sharply after failure occurs. To minimize over-
exaggeration of this eﬀect on curving ﬁtting, the
magnitude of failure strain was matched separately.
Therefore, the material identiﬁcation of each skull
sample was divided into two stages. In the ﬁrst stage,
the force–deﬂection curves before failure were used to
determine the most favorable Young’s modulus, yield
stress, and tangent modulus based on the optimization
procedures. In the second stage, an FE analysis based
on the Young’s modulus, yield stress, and tangent
modulus obtained in the ﬁrst stage was performed to ﬁt
the time of failure with an appropriate failure strain.
Thus, over-ﬁtting of the curve based on post-failure
behavior is avoided. In this manner, precise failure
strain can be revealed more clearly.
A ‘‘design domain’’ which covered the possible range
of the Young’s modulus, yield stress, and tangent
modulus was deﬁned. Because the Young’s modulus in
the case of skull bone is loading rate-dependent,
ranges of Young’s moduli under the two different
loading conditions could be dissimilar, with the lower
loading rate resulting in a lower value of Young’s
modulus. The ranges selected for identifying the opti-
mized Young’s modulus, yield stress, tangent modulus,
and failure strain for each specim en are listed in
Tables 1 and 2. These ranges deﬁne a feasible domain of
the parameters to be identiﬁed. An optimization soft-
ware modeFRONTIER (Esteco, Srl, Itlay) was used to
automatically update these input parameters and sub-
mit the new keyword ﬁle to LS-DYNA, which ran in
MPP mode, to reduce the time needed to complete the
The Latin Hypercube Sampling (LHS) method
(further described in ‘‘LHS Method’’ section) was then
used to obtain a uniform allocation inside the design
domain. A sequential quadratic program (further
described in ‘‘Sequential Quadratic Programming
Algorithm’’ section) was then used to ﬁnd the minimal
F from the response surface formed by an anisotropic
Kriging model (further described in ‘‘Anisotropic
Kriging’’ section). It is proposed that the combination
of Design of Computer Experiments (DOCE),
response surface method, and optimization technique
is ideal for reverse engineering, taking into account
a certain degree of uncertainty in the physical
experiments. At the end of the optimization, the opti-
mal material parameters obtained from this procedure
are used as input to specimen-speciﬁc FE models
undergoing validation. This process is desired to judge
the validity of the response surface.
Figure 5 shows the ﬂowchart used to identify the
material property parameters of each sample using
modeFRONTIER version 4.1 with the Young’s mod-
ulus, yield stress, and tangent modulus as design
variables and F as the objective function of the opti-
A response surface identiﬁed through an anisotropic
Kriging model was proposed here to ﬁnd the optimal
material parameters that minimize F. Considering the
large range of material properties to be investigated,
there is a need to reduce the number of FE simulations
before utilizing the anisotropic Kriging model. The
DOCE method is aimed at minimizing the number of
runs while simultaneously acquiring as much infor-
mation as possible. There are several DOCE methods
that have been proposed in the literature.
methods allow only two to three levels for each input
variable to avoid a rapid increase in the number of
The LHS technique, which has been used extensively
in many DOCE, was adopted in this study. LHS was
ﬁrst proposed by McKay et al.
and has become one of
the most popular design types for Kriging models. LHS
is a space-ﬁlling design with constrainedly stratiﬁed
sampling method. Once the number of computer
experiments (n) is determined, each input range of input
variables (m) is split into n intervals of equal length and
form an n
grid on the experimental space. Independent
TABLE 1. Range of design parameters for loading velocity of
No. Parameters Lower bound Upper bound
1 Young’s modulus (GPa) 10.00 20.00
2 Yield stress (GPa) 0.050 0.180
3 Tangent modulus (GPa) 0.01 1.80
4 Failure strain 0.010 0.150
TABLE 2. Range of design parameters for loading velocity of
No. Parameters Lower bound Upper bound
1 Young’s modulus (GPa) 6.00 15.00
2 Yield stress (GPa) 0.050 0.180
3 Tangent modulus (GPa) 0.005 0.150
4 Failure strain 0.010 0.150
GUAN et al.88
sampling with the same probability is then performed
within each subinterval. Random samples from subin-
tervals can be taken one at a time, while this technique
remembers which samples were taken so far. Thus,
n points are selected among the grid so that n levels of
each variable are represented only once in the design
domain. In this way, the exact input values are relatively
uniformly sampled over the design space. Also note that
this sampling scheme does not require more samples
for more variables; therefore, the key advantage of
this technique is that the number of samples does not
increase exponentially with the number of variables,
and at the same time it ensures that a smal l number of
computer experiments with multiple levels will be suf-
ﬁcient to investigate the potentially nonlinear relation-
ships between input variables and outp ut response. As
illustrated in Fi g. 6 for a two-dimensional problem,
each variable in an LHS scheme is divided into ﬁve
equal subintervals and then organized simulta neously
to form a matrix of random sampling points to ensure
that all portions of the design space are captured.
The number of simulations in LHS is determined by
the total number of design variables involved. To
construct a reasonably accurate approximated model
for optimization, Gu and Yang
minimum of three times the number of design variables
to be used for initial simulation s. Based on this
assumption, for each test sampl e, 12 FE model simu-
lations were conducted initially within the design space
to construct the ﬁrst response surface.
After generating the LHS sample points, FE simu-
lations were completed using LS-DYNA. The corre-
sponding assessment indices (F), calculated from these
simulations, were used as the response to construct the
approximated model. Some literature sources indicate
that a simple polynomial model may not be sufﬁcient
for modeling complex nonlinear responses.
For example, second-order response surfaces are
incapable of modeling surfaces with multiple extrema
because they do not have a very ﬂexible shape.
An anisotropic Kriging model was adapted here to
construct the approximate model. Anisotropic Kriging
is a reﬁned version of the Kriging model which oﬀers
the possibility of controlling the relative importance
between input variables.
The basis of the Kriging response surface method is
the estimation of the response as a combination of two
components, a global model plus a localized departure.
Define reverse analysis :
Select Latin hypercube sampling
(response surface )
Use sequential quadratic
base on anisotropic Kriging model
Obtain the optimum parameters
Fulfil the design purpose?
Decide the domain of
Solver script file
FE model of
rat skull sample
FIGURE 5. Optimization procedures used to determine material parameters.
Application of Optimization Methodology and Specimen-Speciﬁc FE Models 89
In Eq. (3), y(x) indicates the unknown function to
be estimated and f(x) is a polynomial-function based
model designed to indicate the general trend over the
design space, and Z(x) represents a stochastic process
with a zero mean value, varia nce r
, and non-zero
covariance. The purpose of Z(x) is to create a localized
deviation by quantifying the interpolation values
between sampling points with a correlation function,
and the covariance matrix of Z(x) is represented by:
; i; j ¼ 1; 2; ...n
In Eq. (4), R stands for the correl ation matrix
consisting of a spatial correlation function, and
) is the correlation function between any two
sampling points x
is the variance which
depicts the scalar of the spatial correlatio n function
quantifying the correlation between x
, and it can
control the smoothness of the Kriging model, the effect
of the nearby points, and diff erentiability of the surface.
Generally, variations in diﬀerent variables result in
changes of diﬀerent magnitudes in the responses. For
example, if the oscillations in the response are twice as
dense in one direction as opposed to others, the vari-
ogram range should be adjusted accordingly, i.e., twice
as small. For anisotropic Kriging, the covariance
function cov Zx
ðÞ½is computed considering
different variogram ranges for each single input vari-
able, with a common sill and nugget.
In this study, a
popular correlation function known as the Gaussian
correlation function was utilized. The Gaussian func-
tion can provide a relatively smooth and inﬁnitely
differentiable surfa ce, so it is a preferable correlation
function when a gradient-based optim ization algo-
rithm is to be adopted next stage. The regression
, used as an error indicator to gauge the
accuracy of the anisot ropic Kriging model, is repre-
sented as follows:
In Eq. (5), y
is the actual value,
y is the value pre-
dicted by the anisotropic Kriging model, and
y is the
average of all actual values. When R
the anisotropic Kriging model is sufﬁciently close to
one, the process of constructing the approximated
model stops. Otherwise, new sampling points are
added to update the approximated model.
As the ﬁrst step to test the accuracy of the aniso-
tropic Kriging model, four additional FE model sim-
ulations were performed based on random sampling. If
the regression pa rameter R
was below 0.95, which
indicated that the initial response surface based on the
ﬁrst 12 simulations was not ideal; results from all 16
simulations were used to construct a second response
surface. The accuracy of the second response surface
was then tested by four additional FE simulations, also
based on random sampling. This procedure was con-
tinued until the resulting regression parameter R
greater than 0.95, which indicated that the resul ting
response surface was accurate enough to identify the
design parameters. This step also demonstrated that
the anisotropic Kriging model approach maps the
relationship between the material parameters and the
assessment index F in an acceptably accurate manner.
Sequential Quadratic Programming Algorithm
A sequential quadratic programming algorithm was
used as the optimization strategy to minimize the
objective function in this material identiﬁcation prob-
lem. This program ming method is one of the most
powerful nonlinear programming algorithms for solv-
ing diﬀerentiable nonlinear programming problems in
an eﬃcient and reliable way.
Many nonlinear pro-
gramming problems, such as least squares or min–max
optimization, can be solved using this method. Within
each numerical iterati on, the basic idea is to solve a
quadratic programming sub-problem that is formu-
lated by replacing the objective function with a qua-
dratic approximation and replacing the constraint
function by linear approximation. The three main steps
included in the sequential quadratic programming
implementation are: updating the Hessian matrix of
the Lagrangian function, solving the quadratic pro-
gramming sub-problem, and the formation of a new
iteration using a line search.
FIGURE 6. Two variables with ﬁve levels sampling using
LHS scheme of DOCE.
GUAN et al.90
RESULTS AND DISCUSSION
Identical optimization procedures were applied for 20
skull samples and FE models. Through this method,
each experimental force–deﬂection curve was matched
to the simulation curve using the most favorable mate-
rial parameters obtained from optimization-based
material identiﬁcation calculation (Figs. 7, 8). Further-
more, the optimized material parameters and assess-
ment index F are listed in Tables 3 and 4. Different
samples may have varying numbers of sampling points,
response surface accuracies, and assessment index val-
ues. The numbers of LHS sampling points were all 12,
and the numbers of additional sampling points ranged
from 0 to 8. The quantitative accuracies of the aniso-
tropic Kriging model (R
) ranged from 0.966 to 0.998.
The average diﬀerences between experimental data
and simulation results were 0.378 N (which was 3.3%
of the force peak value) at 200 mm/s and 0.227 N
(which was 2.7% of the force peak value) at 0.02 mm/
s. From Fig. 7 and Table 3, it can be seen that sample
7 had the best curve ﬁtting result with an assessment
index value of 0.242 N, which is less than 2.6% of the
peak force measured, while the worst ﬁt, sample 3, was
still accepta ble with an assessment index value of
0.771 N, less than 5.1% of the peak force measured.
Similarly, as Fig. 8 and Table 4 show, sample 6 had
the best curve ﬁtting result with the assessment index
value of 0.094 N, less than 1.3% of the peak force
measured, and sample 16 had the poorest curve ﬁtting
result with the assessment index value of 0.638 N, less
than 5.8% of the peak force measured. The average
most favorable material parameters for the 200 mm/s
test group were a Young’s modulus of 16.38 (±2.00)
GPa, yield stress of 0.124 (±0.013) GPa, tangent
modulus of 0.52 (±0.46) GPa, and failure strain of
0.067 (±0.019) as shown in Table 3. Simultaneously,
for a quasi-static velocity of 0.02 mm/s, the average
optimized parameters were a Young’s modulus of 9.10
(±1.31) GPa, yield stress of 0.106 (±0.019) GPa, tan-
gent modulus of 0.027 (±0.028) GPa, and failure strain
of 0.085 (±0.032) as shown in Table 4.
Force–deﬂection curves measured from rat skull
samples showed a very small nonlinearity at the very
beginning in 15 out of the 20 specimens tested, similar to
those depicted by Currey and Butler.
forces in the nonlinear region were all below 0.46 N.
These slight nonlinear regions were removed by
extrapolating the force–deﬂection curve in the linear
region back to zero force. We performed such analysis
for two reasons . First ly, only the slope of initial straight
line portion of the force–deﬂection curve was needed to
calculate the Young’s modulus. Ther efore, the effect of
the initial nonlinearity was neglected in order to com-
pare our analysis with that obtained from traditional
method. Secondly, skull samples were not perfectly ﬂat
when they were placed on top of the three-point bending
ﬁxture. As a result, only part of the specimen was loaded
initially and hence the nonlinear response was mainly
from stabilization of the specimen in the beginning of
three-point bending tests. The potential Hertzian con-
tact between the steel rollers and the rat skull samples
should be small. Using Eq. (6) below, the calculated
indentation depth due to Hertzian contact was only
3.01 9 10
mm with P as 0.5 N, L as 3.5 mm, E
210 GPa, v
as 0.30, E
as 5.917 GPa, and v
However, for tests within the nano range, the Hertzian
contact could affect results signiﬁcantly, and an
unloading protocol would be needed.
where d is indentation depth, P is the loading force, E
are the modulus and Poisson’s ratio for steel, E
are the modulus and Poisson’ s ratio for skull.
L is length of the bone sample.
Based on classical beam theory (Eq. 1), the Young’s
modulus for the 200 mm/s group was 9.49 GPa with
the standard deviation 1.86 GPa. The Young’s modulus
for 0.02 mm/s group was 5.92 GPa with the standard
deviation 0.763 GPa. Comparing with Gefen’s data
(6.01 MPa) reported in 2003,
the values obtaine d from
this optimization study are much larger than Gefen’s
results, and are in the same order of magnitude as the
analytical solution based on classical beam theory. The
analytical method tends to underestimate the elastic
moduli because the samples are simpliﬁed as perfect
beam without considering internal porosity, curvature,
and change of thickness.
Mechanical properties of the rat skull may aﬀect
intracranial responses during in vivo TBI experiments
in which the skull is either open (such as in a controlled
cortical impact) or close (such as in a Marmarou
weight drop). Initially, it was hoped that the effect of
structural inhomogeneity could be eliminated through
the use of Micro CT and specimen-speciﬁc FE models
to yield a narrower range in mechanical properties.
However, the optimized Young’s modulus still varied
greatly ranging from 14.0 to 19.5 GPa at a loading
velocity of 200 mm/s and 7.1 to 10.7 GPa at a loading
velocity of 0.02 mm/s. Future work should consider
the inclusion of material property assignment based on
gray scale values measured using Micro CT into opti-
mization processes to better quantify the mechanical
properties of rat skull so that intracranial responses
can be more accurately predicted.
Human cranial bones have been report ed to be
transversely isotropic in directions tangent to the skull
In contrast, facial bones are subjected to
Application of Optimization Methodology and Specimen-Speciﬁc FE Models 91
FIGURE 7. Comparison between experimental curves and simulation results (200 mm/s). Solid lines experimental curves, dash
lines simulation curves. Horizontal axis unit: deﬂection (mm), vertical axis unit: force (N).
GUAN et al.92
FIGURE 8. Comparison between experimental curves and simulation results (0.02 mm/s). Solid lines experimental curves, dash
lines simulation curves. Horizontal axis unit: deﬂection (mm), vertical axis unit: force (N).
Application of Optimization Methodology and Specimen-Speciﬁc FE Models 93
masticatory loads and may exhibit directionally
dependent material properties. Studies on an FE human
mandible model demonstrated that the FE-predicted
peak volumetric strain de creased when considering
These anisotropic character-
istics were also conﬁrmed through microindentation
experiments by Daegling et al.
In the only published
study to investigate rat skull material properties exper-
imentally, Gefen et al.
analyzed the tissue as an iso-
tropic elastic material. Therefore, for the current study,
isotropic elastic–plastic behavior was assumed based on
human data published by McElhaney et al.
as well as the rat data from the Gefen et al.
study. This is believed to be appropriate because the rat
skull is thin and shell-like, which makes it very difﬁcult
to study material behavior in the axis perpendicular
to the skull surface and to date no rat skull properties
have been reported for this direction in the literature.
In future studies, the potential anisotropic property of
rat skull could be studied experi mentally along three
mutually perpendicular axes, CT imaging with elastic
and FE simulations with elasticity tensors
The proposed optimization-based material identiﬁca-
tion methodology is a helpful tool to eﬃciently and
accurately reveal material parameters through reverse
engineering, providing scientiﬁc basis for FE model
development. Considering that each skull sample was
naturally curved, diﬀerent thickness and varying porosi-
ties as shown from micro CT images, specimen-speciﬁc FE
models served to eliminate such geometric eﬀect. While in
the conventional beam theory, the skull sample was ide-
ally assumed as perfect rectangular, ﬂat, with uniform
thickness, and without porosity. Such simpliﬁcation and
idealization when applying classical beam theory could
induce errors in calculating material parameters.
This paper proposed an application of optimization
methodology to biomaterial parameter identiﬁcation
through the combination of the sample-speciﬁc FE
models, anisotropic Kriging modeling, and sequential
quadratic programming. The anisotropic Kriging model
predicted results show excellent accuracy to the experi-
ments. Using anisotropic Kriging to construct the
response surface model and optimize the objective func-
tion through sequential quadratic programming, the
relationships among the material parameters and force–
deﬂection curves can be mapped accurately, reducing the
total simulation time and improving accuracy of the
material parameters. The method was applied to a typical
rat skull material identiﬁcation problem. Through this
study, the rat skull material property parameters (such as
the Young’s modulus, yield stress, tangent modulus, and
failure strain as described in Tables 3 and 4)were
obtained for two different loading speeds. The same
method can be adapted to other reversing engineering
procedures to obtain accurate material parameters for FE
This study is supported by the National 973
Program under Grant number 2010CB832705 and the
National Science Fund for Distinguished Young
Scholars (10725208), both funded through the Chinese
government. The primary author of this manuscript is
supported by a fellowship provided by the China
Scholarship Council funded by the Ministry of Edu-
cation of the People’s Republ ic of China and in part by
the Bioen gineering Center at Wayne State University.
TABLE 3. Optimal material parameters for loading velocity
of 200 mm/s.
Sample E (GPa) Y (GPa) Et (GPa) FS F (N)
1 14.47 0.121 1.25 0.059 0.294
3 19.50 0.148 0.01 0.105 0.771
5 17.50 None None None 0.639
7 16.08 0.105 0.10 0.048 0.242
9 17.73 0.133 0.05 0.075 0.389
11 19.09 0.116 0.71 0.052 0.269
13 15.80 0.127 1.00 0.060 0.244
15 14.00 0.125 0.70 0.080 0.269
17 15.51 0.131 0.74 0.058 0.312
19 14.10 0.113 0.10 None 0.349
Average 16.38 0.124
S.D. 2.00 0.013
S.D. standard deviation, E Young’s modulus, Y yield stress,
Et tangent modulus, FS failure strain, F assessment index.
Exclude sample #5.
Exclude sample #5 and #19.
TABLE 4. Optimal material parameters for loading velocity
of 0.02 mm/s.
Sample E (GPa) Y (GPa) Et (GPa) FS F (N)
2 9.36 0.132 0.050 0.052 0.320
4 8.50 0.094 0.050 0.097 0.188
6 8.98 0.087 0.005 0.125 0.094
8 10.70 0.082 0.010 0.060 0.133
10 10.50 0.132 0.090 0.047 0.120
12 7.80 0.120 0.005 0.060 0.161
14 9.77 0.090 0.031 0.115 0.160
16 7.11 0.120 0.005 0.095 0.638
18 7.64 0.094 0.015 0.135 0.321
20 10.63 0.105 0.012 0.065 0.138
Average 9.10 0.106 0.027 0.085 0.227
S.D. 1.31 0.019 0.028 0.032 0.164
S.D. standard deviation, E Young’s modulus, Y yield stress,
Et tangent modulus, FS failure strain, F assessment index.
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