# Application of Optimization Methodology and Specimen-Specific Finite Element Models for Investigating Material Properties of Rat Skull

**Abstract**

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

FENGJIAO GUAN,

1,2

XU HAN,

1

HAOJIE MAO,

2

CHRISTINA WAGNER,

2

YENER N. YENI,

3

and KING H. YANG

1,2

1

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

2

Bioengineering Center, Wayne State University, 818 W. Hancock, Detroit, MI 48201, USA; and

3

Henry 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) 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.

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 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-

mental studies.

17–20,24

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.

21,23

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

et al.

23

found 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,

21

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,

21

but 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 Engin eering, Vol. 39, No. 1, January 2011 (Ó 2010) pp. 85–95

DOI: 10.1007/s10439-010-0125-0

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

85

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.

8

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

identiﬁcation problems,

4,14,32,33

little information was

provided regarding which optimization procedures were

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. 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.,

26

was 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 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

Skull sample

Steel support

Load cell to fixed plate

FIGURE 3. Schematic diagram of the three-point bending

test setup.

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:

E ¼

L

3

m

4bd

3

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) 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.

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

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

60 ms

21

for one skull sample. To check its validity, the

force–deﬂection curve predicted by the combined

explicit solver with the addition of 60 ms

21

damping

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

*MAT_PIECEWISE_LINEAR_PLASTICITY from

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

Eq. (2).

F ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

X

n

i¼1

f

mi

f

ci

ðÞ

2

,

n

v

u

u

t

ð2Þ

Here f

mi

indicates those values measured from tests, f

ci

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

study (f

mi

2 f

ci

)

2

, in correspondence with the absolute

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

instead of (f

mi

2 f

ci

)

2

/f

mi

, 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,

35

the

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

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

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-

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.

7,29–31

Many

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.

22

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

m

grid on the experimental space. Independent

TABLE 1. Range of design parameters for loading velocity of

200 mm/s.

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

0.02 mm/s.

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

9

recommended a

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.

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,

yxðÞ¼fxðÞþZxðÞ ð3Þ

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 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

2

, 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:

cov Zx

i

; Zx

j

¼ r

2

R Rx

i

; x

j

; i; j ¼ 1; 2; ...n

ð4Þ

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

consisting of a spatial correlation function, and

R(x

i

, x

j

) is the correlation function between any two

sampling points x

i

and x

j

, r

2

is the variance which

depicts the scalar of the spatial correlatio n function

quantifying the correlation between x

i

and x

j

, 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

i

ðÞ; Zx

j

ðÞ½is computed considering

different variogram ranges for each single input vari-

able, with a common sill and nugget.

34

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

parameter, R

2

, used as an error indicator to gauge the

accuracy of the anisot ropic Kriging model, is repre-

sented as follows:

R

2

¼ 1

P

n

i¼1

y

i

~

yðÞ

2

P

n

i¼1

y

i

yðÞ

2

ð5Þ

In Eq. (5), y

i

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

2

calculated 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 pa rameter R

2

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

2

was

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.

3,13

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.

3,13

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

2

) 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.

5

The 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 . 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

25

mm with P as 0.5 N, L as 3.5 mm, E

1

as

210 GPa, v

1

as 0.30, E

2

as 5.917 GPa, and v

2

as 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 ¼

4P

pL

1 v

2

1

E

1

þ

1 v

2

2

E

2

ð6Þ

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

1

and v

1

are the modulus and Poisson’s ratio for steel, E

2

and v

2

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,

8

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

surface.

21,35

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

anisotropic elasticity.

11,15

These anisotropic character-

istics were also conﬁrmed through microindentation

experiments by Daegling et al.

6

In the only published

study to investigate rat skull material properties exper-

imentally, Gefen et al.

8

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.

21

and

Wood,

35

as 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 experi mentally along three

mutually perpendicular axes, CT imaging with elastic

trajectories,

11

and FE simulations with elasticity tensors

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-

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.

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–

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

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 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

a

0.52

a

0.067

b

0.378

S.D. 2.00 0.013

a

0.46

a

0.019

b

0.181

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

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

a

Exclude sample #5.

b

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.

GUAN et al.94

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- CitationsCitations11
- ReferencesReferences41

- "Compared with the traditional curve fitting methods, this method performed specimen-specific optimization therefore can more accurately obtain complicated nonlinear hyperelastic parameters and viscoelastic parameters at the same time. NSGA-II optimization method was widely adopted by researchers in the field of FEM optimization [49] and has made significant contributions in improving the calculation accuracy in biomechanical field [50]. By applying this method to the establishment of bovine eye FEM, we not only obtained more accurate material parameters, as shown inTable 2, but also established a more realistic eyeball model and provided a reliable basis to truly simulate the biomechanical responses when IOP was elevated. "

[Show abstract] [Hide abstract]**ABSTRACT:**Glaucoma mainly induced by increased intraocular pressure (IOP), it was believed that the pressure that wall of eyeball withstands were determined by material properties of the tissue and stereoscopic geometry of the eyeball. In order to study the pressure changes in different parts of interior eyeball wall, it is necessary to develop a novel eye ball FEM with more accurate geometry and material properties. Use this model to study the stress changes in different parts of eyeball, especially the lamina cribrosa (LC) under normal physiological and pathological IOP, and provide a mathematical model for biomechanical studies of selected retinal ganglion cells (RGCs) death. (1) Sclera was cut into 3.8-mm wide, 14.5-mm long strips, and cornea was cut into 9.5-mm-wide and 10-mm-long strips; (2) 858 Mini BionixII biomechanical loading instrument was used to stretch sclera and cornea. The stretching rate for sclera was 0.3 mm/s, 3 mm/s, 30 mm/s, 300 mm/s; and for cornea were 0.3 mm/s and 30 mm/s. The deformation-stress curve was recorded; (3) Naso-temporal and longitudinal distance of LC were measured; (4) Micro-CT was used to accurately scan fresh bovine eyes and obtain the geometrical image and data to establish bovine eye model. 3-D reconstruction was performed using these images and data to work out the geometric shape of bovine eye; (5) IOP levels for eyeball FEM was set and the inner wall of eyeball was used taken as load-bearing part. Simulated eyeball FE modeling was run under the IOP level of 10 mmHg, 30 mmHg, 60 mmHg and 100 mmHg, and the force condition of different parts of eyeball was recorded under different IOP levels. (1) We obtained the material parameters more in line with physiological conditions and established a more realistic eyeball model using reversed engineering of parameters optimization method to calculate the complex nonlinear super-elastic and viscoelastic parameters more accurately; (2) We observed the following phenomenon by simulating increased pressure using FEM: as simulative IOP increased, the stress concentration scope on the posterior half of sclera became narrower; in the meantime, the stress-concentration scope on the anterior half of scleral gradually expanded, and the stress on the central part of LC is highest. As simulative IOP increased, stress-concentration scope on the posterior half of sclera gradually narrowed; in the meantime, the stress-concentration scope on the anterior half of sclera gradually expanded, and the stress on the LC is mainly concentrated in the central part, suggesting that IOP is mainly concentrated in the anterior part of the eyeball as it increases. This might provide a biomechanical evidence to explain why RGCs in peripheral part die earlier than RGCs in central part under HIOP.- "The material properties of impact surfaces can be determined by FE simulation and optimisation method through matching the simulation results with those from the drop tests. This method has been used in the literature [10,12,34].Figure 3 shows the processes of determining the material properties of impact surfaces using this method. First of all, the parameters in the material model were set as variables and input into the FE model of the impactor. "

[Show abstract] [Hide abstract]**ABSTRACT:**The mechanism and tolerance of paediatric head injuries are not well established mainly due to the limited cadaveric tests available. Weber's studies [Experimental studies of skull fractures in infants, Z Rechtsmed. 92 (1984), pp. 87–94; Biomechanical fragility of the infant skull, Z Rechtsmed. 94 (1985), pp. 93–101] in mid 1980s contained the most extensive paediatric cadaver test data under various fall conditions in the literature. However, the limited injury measurements and the unknown material properties of the impact surfaces in Weber's tests limited their application in paediatric fall protection. In the present study, drop tests, inverse finite element modelling and optimisation were first conducted to quantify the material characteristics of four impact surfaces (carpet, vinyl, foam and blanket) used in the Weber's tests. With the impact surface material determined, five cadaver tests from Weber's studies were reconstructed using a parametric paediatric head finite element model. Results showed that the simulated strain and stress distributions on the skulls correlated well with the fracture patterns reported in the cadaver tests. The impact surface material properties developed in this study and the methods of using the parametric paediatric head model to reconstruct the cadaver tests provided valuable information for future ground surface designs for child fall protection and development of paediatric head injury criteria.- "Due to its importance in clinical and research applications, subject-specific finite element (FE) modelling of bone is a fast growing domain. A number of specimen-specific modelling procedures were proposed in the past decade (Weinans et al. 2000; Anderson et al. 2005; Wullschleger et al. 2010; Guan et al. 2011). However, the assessment of the reliability of the mechanical predictions obtained using such models remains a challenging issue as it depends on both the methods applied and the research discipline. "

[Show abstract] [Hide abstract]**ABSTRACT:**The authors propose a protocol to derive finite element (FE) models from micro computer tomography scans of implanted rat bone. A semi-automatic procedure allows segmenting the images using specimen-specific bone mineral density (BMD) thresholds. An open-source FE model generator processes the segmented images to a quality tetrahedral mesh. The material properties assigned to each element are integrated from the BMD field. Piecewise, threshold-dependent density-elasticity relationships are implemented to limit the effects of metal artefacts. A detailed sensitivity study highlights the coherence of the generated models and quantifies the influence of the modelling parameters on the results. Two applications of the protocol are proposed. The stiffness of bare and implanted rat tibiae specimens is predicted by simulating three-point bending and inter-implant displacement, respectively. Results are compared with experimental tests. The mean value and the variability between the specimens are well captured in both tests.

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