Page 1

Application of Optimization Methodology and Specimen-Specific Finite

Element Models for Investigating Material Properties of Rat Skull

FENGJIAO GUAN,1,2XU HAN,1HAOJIE MAO,2CHRISTINA WAGNER,2YENER N. YENI,3and KING H. YANG1,2

1State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Hunan, China;

2Bioengineering Center, Wayne State University, 818 W. Hancock, Detroit, MI 48201, USA; and3Henry Ford Hospital, Detroit,

MI, USA

(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)modelsofratskullbonesamples

were developedbyreconstructingthethree-dimensionalgeom-

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

testresponsesbetweenexperimentalandsimulationresults.An

anisotropic Kriging model and sequential quadratic program-

ming, in conjunction with Latin Hypercube Sampling (LHS),

are utilizedto minimize the disparitybetween 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—Ratskull,Specimen-specificfiniteelementmodels,

Anisotropic Kriging, Optimization, Material identification.

INTRODUCTION

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 external 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 element (FE) models are

probably the best means to acquire such responses. In

recent years, several FE models of the rat head have

been developed to predict internal responses of the

brain under injury scenarios to complement experi-

mental studies.17–20,24In 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 cranial bone properties in different loading

modes and directions.21,23In 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

et al.23found that the compressive modulus of the

human diploe ¨ 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

(±2.90) GPa,21with 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,21but other

studies have reported values of half that magnitude.1,28

To the best of our knowledge, only one study has

reported the material properties of rat skull bone.8

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: king.yang@wayne.edu

Annals of Biomedical Engineering, Vol. 39, No. 1, January 2011 (? 2010) pp. 85–95

DOI: 10.1007/s10439-010-0125-0

0090-6964/11/0100-0085/0 ? 2010 Biomedical Engineering Society

85

Page 2

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.8to determine the skull bone properties.

Experimentally, machining a small, curved rat skull

into rectangular specimens for direct material testing

maypresentsometechnicalchallenges.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

identification problems,4,14,32,33little information was

providedregardingwhichoptimizationprocedureswere

applied.

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. Considering 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.,26was used

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 microCT 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-specific 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.

one skull sample. The size bar indicates 200 lm.

A typical microCT image showing the porosity of

FIGURE 2.

using 35,613 hexahedral elements.

A sample-specific FE mesh of one skull sample

Steel shaft to Instron ram

Impactor diameter 2mm

Support diameter 2mm

Skull sample

Steel support

Load cell to fixed plate

FIGURE 3.

test setup.

Schematic diagram of the three-point bending

GUAN et al.86

Page 3

mid-pointbetween thebregma andlambda 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 channel 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:

E ¼L3m

4bd3

ð1Þ

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.

FE Simulations

The boundary and loading conditions were precisely

defined for each FE model using Hypermesh (Altair

Engineering, Troy, MI) according to the specimen-

specific experimental setup. The implicit 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.10,36

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

explicitsolvertoensurethatthekineticenergywasclose

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

60 ms21for one skull sample. To check its validity, the

force–deflection curve predicted by the combined

explicit solver with the addition of 60 ms21damping

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 quasi-static simula-

tions to calculate the optimal material properties. In

solvinghigh-speedtransientdynamicproblemsusingan

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.

Theconstitutivematerial

*MAT_PIECEWISE_LINEAR_PLASTICITY

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.

lawassumed was

from

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

Eq. (2).

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

i¼1

F ¼

X

n

fmi? fci

ðÞ2

,

n

v

u

u

t

ð2Þ

Here fmiindicates those values measured from tests, fci

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.

and explicit solution method.

Comparison between implicit solution method

Application of Optimization Methodology and Specimen-Specific FE Models 87

Page 4

require different n values, which are determined by

total loading time and sampling frequency. In this

study (fmi2 fci)2, in correspondence with the absolute

difference, was adapted in the assessment index (Eq. 2)

instead of (fmi2 fci)2/fmi, 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,35the

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 specimen are listed in

Tables 1and2.Theseranges defineafeasibledomain 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

task.

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

ofDesign ofComputer

response surface method, and optimization technique

is ideal for reverse engineering, taking into account

a certain degree of uncertainty in the physical

Experiments(DOCE),

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-

mization study.

LHS Method

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.7,29–31Many

methods allow only two to three levels for each input

variable to avoid a rapid increase in the number of

computer experiments.

The LHS technique, which has been used extensively

in many DOCE, was adopted in this study. LHS was

first proposed by McKay et al.22and 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

formannmgridontheexperimentalspace. Independent

TABLE 1. Range of design parameters for loading velocity of

200 mm/s.

No. Parameters Lower boundUpper bound

1

2

3

4

Young’s modulus (GPa)

Yield stress (GPa)

Tangent modulus (GPa)

Failure strain

10.00

0.050

0.01

0.010

20.00

0.180

1.80

0.150

TABLE 2.Range of design parameters for loading velocity of

0.02 mm/s.

No. ParametersLower bound Upper bound

1

2

3

4

Young’s modulus (GPa)

Yield stress (GPa)

Tangent modulus (GPa)

Failure strain

6.00

0.050

0.005

0.010

15.00

0.180

0.150

0.150

GUAN et al.88

Page 5

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.Inthisway,theexactinputvaluesarerelatively

uniformlysampledoverthedesign space. Alsonotethat

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 small number of

computer experiments with multiple levels will be suf-

ficient to investigate the potentially nonlinear relation-

ships between input variables and output response. As

illustrated in Fig. 6 for a two-dimensional problem,

each variable in an LHS scheme is divided into five

equal subintervals and then organized simultaneously

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 Yang9recommended a

minimum of three times the number of design variables

to be used for initial simulations. Based on this

assumption, for each test sample, 12 FE model simu-

lations were conducted initially within the design space

to construct the first response surface.

Anisotropic Kriging

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.2,12,16,29,30

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.25

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.

Mathematically,

y x

ð Þ ¼ f x

ð Þ þ Z x

ð Þð3Þ

Define reverse analysis :

Objectives, constraints

Select Latin hypercube sampling

(experimental design)

Construct anisotropic

Kriging model

(response surface )

Accuracy?

Yes

Stop

No

Add

new point

to

reconstruct

response

surface

Use sequential quadratic

programming optimization

base on anisotropic Kriging model

Obtain the optimum parameters

Fulfil the design purpose?

Yes

No

Decide the domain of

assessment variables

LS-DYNA solver

Solver script file

update

FE model of

rat skull sample

Force-deflection

curve

Assessment index

Comparison with

experimental curve

FIGURE 5. Optimization procedures used to determine material parameters.

Application of Optimization Methodology and Specimen-Specific FE Models89

Page 6

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, variance r2, 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:

? ?;Z xj

cov Z xi

? ???¼ r2R R xi;xj

????;

i;j ¼ 1;2;...n

ð4Þ

In Eq. (4), R stands for the correlation matrix

consisting of a spatial correlation function, and

R(xi, xj) is the correlation function between any two

sampling points xiand xj, r2is the variance which

depicts the scalar of the spatial correlation function

quantifying the correlation between xiand xj, and it can

control the smoothness of the Kriging model, the effect

of thenearby points, anddifferentiability 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 Z xi

ð Þ;Z xj

different variogram ranges for each single input vari-

able, with a common sill and nugget.34In 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 surface, so it is a preferable correlation

function when a gradient-based optimization algo-

rithm is to be adopted next stage. The regression

parameter, R2, used as an error indicator to gauge the

ð Þ½? is computed considering

accuracy of the anisotropic Kriging model, is repre-

sented as follows:

R2¼ 1 ?

P

P

n

i¼1

n

yi? ~ y

ðÞ2

i¼1

yi? ? y

ðÞ2

ð5Þ

In Eq. (5), yiis the actual value, ~ y is the value pre-

dicted by the anisotropic Kriging model, and ? y is the

average of all actual values. When R2calculated from

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 parameter R2was 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 R2was

greater than 0.95, which indicated that the resulting

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 programming method is one of the most

powerful nonlinear programming algorithms for solv-

ing differentiable nonlinear programming problems in

an efficient and reliable way.3,13Many nonlinear pro-

gramming problems, such as least squares or min–max

optimization, can be solved using this method. Within

each numerical iteration, 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.3,13

FIGURE 6.

LHS scheme of DOCE.

Two variables with five levels sampling using

GUAN et al.90

Page 7

RESULTS AND DISCUSSION

Identicaloptimizationprocedureswereappliedfor20

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 (R2) 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 acceptable 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

beginningin15outofthe20specimenstested,similarto

those depicted by Currey and Butler.5The maximum

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. Firstly, only the slope of initial straight

line portion of the force–deflection curve was needed to

calculate the Young’s modulus. Therefore, 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

whentheywereplacedontopofthethree-pointbending

fixture.Asaresult,onlypartofthespecimenwasloaded

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 1025mm with P as 0.5 N, L as 3.5 mm, E1as

210 GPa, v1as 0.30, E2as 5.917 GPa, and v2as 0.22.

However, for tests within the nano range, the Hertzian

contact could affect results significantly, and an

unloading protocol would be needed.27

d ¼4P

pL

1 ? v2

E1

1

??

þ

1 ? v2

E2

2

??

??

ð6Þ

where d is indentation depth, P is the loading force, E1

and v1are the modulus and Poisson’s ratio for steel, E2

and v2are 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 standarddeviation 1.86 GPa. The Young’smodulus

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 2003,8the values obtained 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 reported to be

transversely isotropic in directions tangent to the skull

surface.21,35In contrast, facial bones are subjected to

Application of Optimization Methodology and Specimen-Specific FE Models91

Page 8

FIGURE 7.

lines simulation curves. Horizontal axis unit: deflection (mm), vertical axis unit: force (N).

Comparison between experimental curves and simulation results (200 mm/s). Solid lines experimental curves, dash

GUAN et al.92

Page 9

FIGURE 8.

lines simulation curves. Horizontal axis unit: deflection (mm), vertical axis unit: force (N).

Comparison between experimental curves and simulation results (0.02 mm/s). Solid lines experimental curves, dash

Application of Optimization Methodology and Specimen-Specific FE Models 93

Page 10

masticatory loads and may exhibit directionally

dependentmaterialproperties.StudiesonanFEhuman

mandible model demonstrated that the FE-predicted

peak volumetric strain decreased when considering

anisotropic elasticity.11,15These anisotropic character-

istics were also confirmed through microindentation

experiments by Daegling et al.6In the only published

study to investigate rat skull material properties exper-

imentally, Gefen et al.8analyzed 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.21and

Wood,35as well as the rat data from the Gefen et al.8

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 experimentally along three

mutually perpendicular axes, CT imaging with elastic

trajectories,11andFEsimulationswithelasticitytensors

defined accordingly.

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-

tiesasshownfrommicroCTimages,specimen-specificFE

modelsservedtoeliminatesuchgeometriceffect.Whilein

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.

CONCLUSIONS

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–

deflectioncurvescanbemappedaccurately, reducingthe

total simulation time and improving accuracy of the

materialparameters.Themethodwasappliedtoatypical

rat skull material identification problem. Through this

study,theratskullmaterialpropertyparameters(suchas

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

procedurestoobtainaccuratematerialparametersforFE

models development.

ACKNOWLEDGMENTS

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 Republic of China and in part by

the Bioengineering Center at Wayne State University.

TABLE 3.Optimal material parameters for loading velocity

of 200 mm/s.

SampleE (GPa)Y (GPa)Et (GPa) FSF (N)

1

3

5

7

9

11

13

15

17

19

Average

S.D.

14.47

19.50

17.50

16.08

17.73

19.09

15.80

14.00

15.51

14.10

16.38

2.00

0.121

0.148

None

0.105

0.133

0.116

0.127

0.125

0.131

0.113

0.124a

0.013a

1.25

0.01

None

0.10

0.05

0.71

1.00

0.70

0.74

0.10

0.52a

0.46a

0.059

0.105

None

0.048

0.075

0.052

0.060

0.080

0.058

None

0.067b

0.019b

0.294

0.771

0.639

0.242

0.389

0.269

0.244

0.269

0.312

0.349

0.378

0.181

S.D. standard deviation, E Young’s modulus, Y yield stress,

Et tangent modulus, FS failure strain, F assessment index.

aExclude sample #5.

bExclude sample #5 and #19.

TABLE 4.Optimal material parameters for loading velocity

of 0.02 mm/s.

SampleE (GPa)Y (GPa)Et (GPa) FSF (N)

2

4

6

8

10

12

14

16

18

20

Average

S.D.

9.36

8.50

8.98

10.70

10.50

7.80

9.77

7.11

7.64

10.63

9.10

1.31

0.132

0.094

0.087

0.082

0.132

0.120

0.090

0.120

0.094

0.105

0.106

0.019

0.050

0.050

0.005

0.010

0.090

0.005

0.031

0.005

0.015

0.012

0.027

0.028

0.052

0.097

0.125

0.060

0.047

0.060

0.115

0.095

0.135

0.065

0.085

0.032

0.320

0.188

0.094

0.133

0.120

0.161

0.160

0.638

0.321

0.138

0.227

0.164

S.D. standard deviation, E Young’s modulus, Y yield stress,

Et tangent modulus, FS failure strain, F assessment index.

GUAN et al.94

Page 11

REFERENCES

1Barber, T. W. Static compression testing of specimens

from an embalmed human skull. Tex. Rep. Biol. Med.

28(4):497–508, 1970.

2Bates, R. A., R. J. Buck, E. Riccomagno, and H. P. Wynn.

Experimental design and observation for large systems.

J. R. Stat. Soc. B 58:77–94, 1996.

3Boggs, P. T., and J. W. Tolle. Sequential quadratic pro-

gramming for large-scale nonlinear optimization. J. Com-

put. Appl. Math. 124:123–137, 2000.

4Chawla, A., S. Mukherjee, and B. Karthikeyan. Charac-

terization of human passive muscles for impact loads using

genetic algorithm and inverse finite element methods.

Biomech. Model. Mechanobiol. 8:67–76, 2009.

5Currey, J. D., and G. Butler. The mechanical properties of

bonetissueinchildren.J.BoneJointSurg.57:810–814,1975.

6Daegling, D. J., J. L. Hotzman, W. S. McGraw, and

A. J. Rapoff. Material property variation of mandibular

symphyseal bone in colobine monkeys. J. Morphol.

270:194–204, 2009.

7Fang, K.-T., R. Li, and A. Sudjianto. Design and modeling

for computer experiments. Boca Raton: Taylor & Francis

Group, 2006.

8Gefen, A., N. Gefen, Q. Zhu, R. Raghupathi, and

S. S. Margulies. Age-dependent changes in material prop-

erties of the brain and braincase of the rat. J. Neurotrauma

20:1163–1177, 2003.

9Gu, L., and R. J. Yang. Recent applications on reliability-

based optimization of automotive structures. SAE Tech-

nical Paper Series, 2003-01-0152, 2003.

10Hallquist, J. O. LS-DYNA Theoretical Manual. Liver-

more, CA: Livermore Software Technology Co, 2005.

11Hellmich, C., C. Kober, and B. Erdmann. Micromechan-

ics-based conversion of CT data into anisotropic elasticity

tensors, applied to FE simulations of a mandible. Ann.

Biomed. Eng. 36(1):108–122, 2008.

12Jourdan, A. How to repair a second-order surface for

computer experiments by Kriging. Chemom. Intell. Lab.

Syst. 96(2):108–116, 2009.

13Kim, K.-Y., and D.-Y. Shin. Optimization of a staggered

dimpled surface in a cooling channel using Kriging model.

Int. J. Therm. Sci. 47(11):1464–1472, 2008.

14Kim, J. E., Z. P. Li, Y. Ito, C. D. Huber, A. M. Shih,

A. W. Eberhardt, K. H. Yang, A. I. King, and B. K. Soni.

Finite element model development of a child pelvis with

optimization-based material identification. J. Biomech.

42:2191–2195, 2009.

15Kober, C., B. Erdmann, C. Hellmich, R. Sader, and

H. F. Zeilhofer. Consideration of anisotropic elasticity

minimizes volumetric rather than shear deformation in

human mandible. Comput. Methods Biomech. Biomed. Eng.

9(2):91–101, 2006.

16Koehler, J. R., and A. B. Owen. Computer experiments. In:

Handbook of Statistics, 13: Designs and Analysis of

Experiments, edited by S. Ghosh, and C.R. Rao. North-

Holland: Amsterdam, 1996, pp. 261–308.

17Levchakov, A., E. Linder-Ganz, R. Raghupathi, S. S.

Margulies, and A. Gefen. Computational studies of strain

exposures in neonate and mature rat brains during closed

head impact. J. Neurotrauma 23:1570–1580, 2006.

18Mao, H., X. Jin, L. Zhang, K. H. Yang, T. Igarashi,

L.Noble-Haeusslein,andA.I.King.Finiteelementanalysis

of controlled cortical impact induced cell loss. J. Neuro-

trauma 27:877–888, 2010.

19Mao, H., K. H. Yang, A. I. King, and K. Yang. Compu-

tational neurotrauma—design, simulation, and analysis of

controlled corticalimpact

Mechanobiol., 2010. doi:10.1007/s10237-010-0212-z.

20Mao, H., L. Zhang, K. H. Yang, and A. I. King. Appli-

cation of a finite element model of the brain to study

traumatic brain injury mechanisms in the rat. Stapp Car

Crash J. 50:583–600, 2006.

21McElhaney, J. H., J. L. Fogle, J. W. Melvin, R. R. Haynes,

V. L. Roberts, and N. M. Alem. Mechanical properties on

cranial bone. J. Biomech. 3:495–511, 1970.

22McKay, M. D., R. J. Beckman, and W. J. Conover. A

comparison of three methods for selecting values of input

variables in the analysis of output from a computer code

(JSTOR Abstract). Technometrics 21(2):239–245, 1979.

23Melvin, J. W., D. H. Robbins, and V. L. Roberts. The

mechanical properties of the diploe ¨ layer in the human

skullincompression.Developments

5:811–818, 1969. Paper No. 05-0250.

24Pena, A., J. D. Pickard, D. Stiller, N. G. Harris, and

M. U. Schuhmann. Brain tissue biomechanics in cortical

contusion injury: a finite element analysis. Acta Neurochir.

Suppl. 95:333–336, 2005.

25Rasmussen, C. E., and C. K. I. Williams. Gaussian Pro-

cesses for Machine Learning. Cambridge: MIT Press, 2006.

26Reimann, D. A., S. M. Hames, M. J. Flynn, and D. P.

Fyhrie. A cone beam computed tomography system for

true 3d imaging of specimens. Appl. Radiat. Isot. 48(10–12):

1433–1436, 1997.

27Rho, J. Y., J. D. Currey, P. Zioupos, and G. M. Pharr. The

anisotropic Young’s modulus of equine secondary osteones

and interstitial bone determined by nanoindentation.

J. Exp. Biol. 204:1775–1781, 2001.

28Robbins, D. H., and J. L. Wood. Determination of

mechanical properties of the bones of the skull. Exp. Mech.

9(5):236–240, 1969.

29Sacks, J., S. B. Schiller, and W. J. Welch. Designs for

computer experiments. Technometrics 31:41–47, 1989.

30Sacks, J., W. J. Welch, T. J. Mitchell, and H. P. Wynn.

Design and analysis of computer experiments. Stat. Sci.

4(4):409–423, 1989.

31Santner, T. J., B. J. Williams, and W. Notz. The design and

analysis of computer experiments. New York: Springer

Series in Statistics, Springer, 2003.

32Untaroiu, C., K. Darvish, J. Crandall, B. Deng, and

J. T. Wang. Characterization of the lower limb soft tissues

in pedestrian finite element models. The 19th International

Technical Conference on the Enhanced Safety of Vehicles.

US Department of Transportation, National Highway

Traffic Safety Administration, Washington, DC, 2005.

33Untaroiu, C., J. Kerrigan, and J. Crandall. Material iden-

tification using successive response surface methodology,

with application to a human femur subjected to three-point

bending loading, SAE Technical Paper Number 2006-01-

0063. Warrendale, PA, 2006.

34Wackernagel, H. Multivariate Geostatistics: An Introduc-

tion with Applications. New York: Springer, 2003.

35Wood, J. L. Dynamic response of human cranial bone.

J. Biomech. 4:1–12, 1971.

36Zhang J. A finite element modeling of anterior lumbar

spinal fusion, M.S. Thesis, Wayne State University, 1992.

model.Biomech. Model.

inMechanics

Application of Optimization Methodology and Specimen-Specific FE Models95