A Novel Finite Element Model for Annulus Fibrosus Tissue Engineering Using Homogenization Techniques

8/3/2019 A Novel Finite Element Model for Annulus Fibrosus Tissue Engineering Using Homogenization Techniques

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*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mailaddresses: [email protected]. (Z.Hu). Tel/Fax: +1-605-688-4817/+1-605-688-5878.E-mail address: [email protected]. 2012. American Transactions on Engineering& Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf

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American Transactions on Engineering& Applied Sciences

http://TuEngr.com/ATEAS, http://Get.to/Research

A Novel Finite Element Model for Annulus Fibrosus

Tissue Engineering Using Homogenization Techniques

Tyler S. Remunda, Trevor J. Layh

b, Todd M. Rosenboom

b,

Laura A. Koepsella, Ying Deng

a*, and Zhong Hu

b*

aDepartment of Biomedical Engineering Faculty of Engineering, University of South Dakota, USA

bDepartment of Mechanical Engineering Faculty of Engineering, South Dakota State University, USA

A R T I C L E I N F O A B S T RA C TArticle history :Received September 06, 2011Received in revised form -

Accepted September 24, 2011Available online: September 25,

2011

Keywords:

Finite Element Method

Annulus Fibrosus

Tissue Engineering

Homogenization

In this work, a novel finite element model using themechanical homogenization techniques of the human annulus

fibrosus (AF) is proposed to accurately predict relevant moduli of

the AF lamella for tissue engineering application. A generalformulation for AF homogenization was laid out with appropriate

boundary conditions. The geometry of the fibre and matrix were

laid out in such a way as to properly mimic the native annulusfibrosus tissues various, location-dependent geometrical and

histological states. The mechanical properties of the annulus

fibrosus calculated with this model were then compared with theresults obtained from the literature for native tissue.

Circumferential, axial, radial, and shear moduli were all in

agreement with the values found in literature. This study helps to

better understand the anisotropic nature of the annulus fibrosus

tissue, and possibly could be used to predict the structure-functionrelationship of a tissue-engineered AF.

2012 American Transactions on Engineering and Applied Sciences.

2012 American Transactions on Engineering & Applied Sciences


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2 Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu

1. IntroductionThe annulus fibrosus (AF) is an annular cartilage in the intervertebral disc (IVD) that aids in

supporting the structure of the spinal column. It experiences complex, multi-directional loads

during normal physiological functioning. To compensate for the complex loading experienced,

the AF exhibits anisotropic behavior, in which fibrous collagen bundles that are strong in tension,

run in various angles in an intersecting, crossing pattern which helps to absorb the loadings. (Wu

and Yao 1976) The layers of the AF are composed of fibrous collagen fibrils that are oriented in

such a way that the angles rotate from 28 degrees relative to the transverse axis of the spine in

the outer AF (OAF) to 44 degrees relative to the transverse axis of the spine in the inner AF

(IAF). (Hickey and Hukins 1980; Cassidy, Hiltner et al. 1989; Marchand and Ahmed 1990).

The approach that homogenization offers to deal with anisotropic materials includes

averaging the directionally-dependent mechanical properties in what is called a representative

volume elements (RVE). These RVE are averages of the directionally- and spatially-dependent

material properties. When summed over the volume of the material, they can be very useful in

describing the macroscopic mechanical properties of materials with complex microstructures.

(Bensoussan A 1978; Sanchez-Palencia E 1987; Jones RM 1999) Homogenization has been

applied to address some of the shortcomings of structural finite element analysis (FEA) models

that utilized truss and cable elements (Shirazi-Adl 1989; Shirazi-Adl 1994; Gilbertson, Goel et al.1995; Goel, Monroe et al. 1995; Lu, Hutton et al. 1998; Lee, Kim et al. 2000; Natarajan,

Andersson et al. 2002) and fiber-reinforced strain energy models (Wu and Yao 1976; Klisch and

Lotz 1999; Eberlein R 2000; Elliott and Setton 2000; Elliott and Setton 2001) for modeling the

AF. Homogenization has also been used to describe biological tissues such as trabecular bone

(Hollister, Fyhrie et al. 1991), articular cartilage (Schwartz, Leo et al. 1994; Wu and Herzog

2002) and AF. (Yin and Elliott 2005).

The mechanical complexity of the AF has posed substantial problems for engineers

attempting to model the system. To date, the circumferential modulus and axial modulus have

been predicted accurately, but the predicted shear modulus has been consistently two orders of

magnitude high. An explanation proposed in a recent paper (Yin and Elliott 2005), which offered

a novel homogenization model for the AF, is that the high magnitude prediction for shear


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3

modulus can be explained by the fact that the models assume the tissue to be firmly anchored in

surrounding tissue, whereas the experimentally measured tissue is removed from its surrounding

tissue. This removal of the sample from surrounding tissue releases the fibers near the edge,

which prevents a portion of the fiber stretch component from being included as a part of the

overall shear measurement.

The purpose of this paper was to establish a novel method for modeling the AF using FEA

and homogenization theory that predicts the circumferential-, axial-, and radialmodulus

accurately while also predicting a shear modulus that accurately represents that of the

experimentally measured tissue. A general formulation for annulus fibrosus lamellar

homogenization was laid out. Appropriate changes to the boundary conditions as well as the

geometry of the structural fibres was made to accommodate the measurements of the mechanical

properties under various annulus fibrosus volume fractions and orientations. The specific

changes in the three dimensional location and orientation of the cylindrical, crossing fibers within

the matrix was taken into account. And the mechanical properties of the human AF by modeling

were compared with the results obtained in the literatures for the native tissues.

2. MathematicalModelThe general homogenization formulation used here was applied to the AF before. (Yin and

Elliott 2005) In the homogenization approach volumetric averaging is used to arrive at the

general formulation. (Sanchez-Palencia 1987; Bendsoe 1995; Jones RM 1999) The

homogenization formula is created by averaging material properties for a material that is assumed

to be linear elastic over discrete, volumetric segments. The overall material is assumed to have

inhomogeneous properties throughout the entire volume. So, the average material properties can

be calculated by multiplying the inhomogeneous, localized material properties c by the

independent strain rates u, in independent strain states , , over the volume of the tissue like

in Eq. (1).

= duuC lkji

,,,

1(1)


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,C : overall average material properties

lkjic ,,, : non-homogeneous material properties

jiu , : independent strain rates

, : independent strain rates

: volume

The stiffness tensor Eq. (2) rotates around a certain angle, , in both the positive and

negative direction. This tensor thus rotates the average material properties to simulate the

direction of the AF collagenous fibers. This angle, , is measured from the midline, , and it

changes with spatial location.

RCRCT

=

(2)

C : average elasticity tensor for two lamellae

R: rotation tensor

The elasticity tensor of two, combined lamella Eq. (3) rotated at the same angle, , in

opposite directions .

2

/

+

+ +=CC

C (3)

There are four in-plane material properties: 11C , 22C , 12C , and 66C that are calculated for a

single lamella. They are arranged in matrix notation, like in Eq. (4).

C

=

66

2212

1211

00

0

0

C

CC

CC

(4)

And the values for 11C , 22C , 12C , and 66C can be calculated from the system of equations

shown in Eq. (5) using the height of the fiber portion of the segment , the elastic modulus of

the fiber and matrix mf EE , respectively and the Poisson ratio of the fiber and matrix mf ,

respectively:


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5

( ) ( ) ( )( )( ) ( )( ) fmfm

fmmf

m

m

f

ff

m

m

f

f

EE

EEEEEEC

22

2

2

2

2

2

2211111

1

1

1

11

1

1

+

++

+

=

( )

( ) ( )( ) fmfmfmmf

EE

EEC

2212111

1

+

+=

( ) ( )( ) fmfmfm

EE

EEC

2222111 +

=

( ) ( )( ) fmfmfm

EE

EEC

+++=

1112

166

(5)

:height of the fiber

fE : elastic modulus of the fiber

mE : elastic modulus of the matrix

fv : Poisson ratio of the fiber

mv : Poisson ratio of the matrix

Taken together, this system of equations accurately modeled the AF in the existing model.

(Yin and Elliott 2005) It addressed many of the shortcomings of structural truss and cable

models and of strain energy models. However it did predict a shear modulus that was two orders

of magnitude higher than native tissue.

2.1 ModelfromtheliteratureThe homogenization model for the AF created by Yin et al. accurately predicted most of the

important mechanical properties of the AF tissue. But it did not make accurate shear modulus

predictions. As a matter of fact, the predictions from this model were two orders of magnitude

higher than the measurements reported in the literature. In this section we will detail some

aspects of the published model that may contribute to the unnaturally high modulus prediction.


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2.1.1 FiberangleandfibervolumefractionThe first two important geometric considerations are the volumetric ratio of fiber to matrix

fiber volume fraction (FVF) within the RVE and the fiber angle. (Table 1) (Ohshima, Tsuji et al.

1989; Lu, Hutton et al. 1998) These ratios are used extensively in the calculations. Both the

FVF and the fiber angle vary by which lamina they are located in. But the finite element method

is a great tool for taking these variabilities into account. The original model used fiber angles in

the range of 15 to 45 degrees. It also used FVFs in the range of 0 to 0.3. These ranges were used

first in parametric studies in order to better understand how the fiber angle and FVF affect the

various relevant moduli. Also, beings fiber angle, and to a lesser extent FVF, can be determined

experimentally, the parametric studies helped in determining some of the more difficult to

elucidate material properties of the collagen fibers and the proteoglycan matrix.

2.1.2 FiberconfigurationThe second important geometric consideration is the 3D arrangement of the fibers and matrix

within the composite RVE. In the original formulation, (Yin and Elliott 2005) they assumed the

two fiber populations to be within a single continuous material and not layered as in native tissue

structure. (Sanchez-Palencia 1987)

2.1.3 BoundaryconditionsThe final important consideration is the boundary conditions applied to the RVE. The

boundary condition for the tensile case can be seen in Figure 1. A similar boundary condition for

the tensile case was applied to the proposed model. But whenthey set the boundary conditions

for the shear case, they fixed the edges along both the - and z- axis when they applied a shear

along 1=z and 1= . (Sanchez-Palencia 1987) The proposed model has adopted a boundary

condition from (K. Sivaji Babu 2008), It constrains the rz-surface at 0= and applies a shear to

the rz surface at 1= . (K. Sivaji Babu 2008) This boundary condition can be visualized in

Figure 2. Taken together, these geometric considerations allow the proposed model of the AF

tissues mechanical behavior to be accurate.

2.2 ProposedmodelchangesChanges to the original model are proposed here. They include changes to the fiber angle

and FVF in order to bring them closer to the physiological range. Changes in the fiber

configuration were proposed in order to more closely mimic the native state of the tissue where


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7

the crossing collagen fibers are separated by a section of proteoglycan matrix, whereas in the

original model they were welded together in the shape of an X. The final change made to the

original model was in the applied boundary conditions.

2.2.1 FiberangleandfibervolumefractionThe ranges for this study were based loosely on the values used for the original study. In this

simulation graphs of circumferential-, axial-, and radialmodulus as well as shear modulus

against fiber volume fraction at fiber angles of 20, 25, 30, and 35 degrees were generated.

Graphs were also generated for axial- and circumferential- modulus as well as shear modulus

against varying fiber angle at fiber volume fractions of 0.05, 0.1, 0.15, 0.2, 0.25, and 0.3. The

angles of collagen in native tissue range from 24.5-36.3 degrees to the transverse plane with an

average of 29.6 degrees.

2.2.2 FiberconfigurationIn this paper it is assumed that the fiber populations are layered and separated by matrix

material. The three dimensional geometric arrangement for this fiber and matrix composite is

shown in Figure 1 as a RVE along with the tensile cases boundary conditions. The

corresponding RVE for the shear case is shown in Figure 2. With the material being a

composite, it is important to assign dimensions to repeating components within the RVE. The

width of the segment, which is denoted by c in Eq. (6) was set to be equal to 13 times the radius,

r, of the fiber when the number of fibers, n, within the RVE is 4. This means that the distance

between fibers is the equivalent of one radius. The length ofb is dependent on the fiber angle

and the length ofa. Eq. (7) The length ofa was derived from looking at the ratio of total fiber

volume to total segment volume. A number of new variables are introduced in the derivation ofa

Eq. (8). So a can be derived from Eq. (9) by substitution of Eq. (10) and then rearranging.

rc =13 (6)

( )tan= ab (7)

( )

sin

4 2

=

c

ra (8)


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Figure 1: Meshed 3D geometric representation of matrix and fiber orientation along with

coordinate system, dimensions, and tensile boundary conditions.


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9

Figure 2: Meshed 3D geometric representation of composite RVE along with corresponding

axes, dimensions, and shear boundary conditions.

cba

rln

V

V f

RVE

fiber

==

2 (9)

( )2tan1+= alf (10)

After substituting, making use of a trigonometric identity, and rearranging, the simplified

formula for a,becomes clear.

So to equally space the four fibers along the c edge from each other and also the edge of the

matrix, the length dwas derived as given by Eq. (11). It makes use of the idea that when there


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are four fibers within the RVE, that there are five equal divisions of width.

rrcn

d +

=5

2(11)

a : width of the representative volume element

b : height of the representative volume element

c : length of the representative volume element

d: distance between fibers

n : number of fibers in the representative volume element

r: radius of the fibers

: angle between fibers.

So by putting the above equations into the prototype code, a master program code was

developed that is useful for predicting the various moduli at each variation of fiber angle and

FVF.

2.2.3 BoundaryconditionsThe original paper had fixed boundary conditions along two adjoining faces of the RVE and

applied shear on the two opposite faces of the RVE. In the proposed model one face has fixed

boundary conditions, and the opposite face has an applied shear. These changes taken together

make for a model that predicts all moduli, including the shear modulus, accurately.

3. MaterialPropertiesIt is also important to assign material properties to the parameters that remain constant

regardless of where they are measured throughout the AF. The elastic modulus and Poisson ratio

for the collagen fibers and proteoglycan matrix can be assigned specific values. For modeling the

varying conditions of the AF tissue, laminae, and IVD, the parameters were chosen based on the

literature of past numerical models of the AF, and in some cases, direct measurements of the


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11

tissues. An elastic modulus of 500 MPa and a Poissons Ratio of 0.35 were adopted for the

collagen fibers (Goel, Monroe et al. 1995; Lu, Hutton et al. 1998), while an elastic modulus of

0.8 Mpa (Lee, Kim et al. 2000; Elliott and Setton 2001) and a Poissons Ratio of 0.45 (Shirazi-

Adl, Shrivastava et al. 1984; Goel, Monroe et al. 1995; Tohgo and Kawaguchi 2005) were

assigned to the proteoglycan matrix. Fiber volume fractions and fiber angles were varied over

ranges found in previous homogenization.

4. ResultsThe first input parameter from the lamina that is varied in order to investigate the effect on

the various moduli is the FVF. The FVF is varied from 0.05 to 0.3, which are normal

physiological ranges. (Table 1) Table 1 gives estimates for the cross-sectional area of the AF,

FVF of the AF, and fiber angle. Each are estimated for the corresponding lamella. Of course

these parameters are variable throughout the AF. But this list was compiled for the original

model, so it was used here for ease of comparison. There are also more than six lamellar layers

in the AF, but six is a reasonable approximation.

Table 1: Annulus fibrosus cross-sectional area for each of the lamina layers, collagen fiber

volume fraction for each of the lamina layers, and fiber orientation angle as reported in the

literatures. These values were inserted into the proposed formulation.

LaminaLayer Inner 2nd 3rd 4th 5th Outer References

Annulusfibrosus

crosssectionalarea0.06 0.11 0.163 0.22 0.2662 0.195

(Lu,Huttonetal.

1998)

Collagenfiber

volumefraction0.05

0.09

0.13

0.17

0.2

0.23

(Yin

and

Elliott

2005)

Fiberangle AnnulusFiberorientationaverage:29.6(range24.536.3)(Lu,Huttonetal.

1998)


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Figure 3 looks at how the circumferential modulus varies with varying FVF and fiber angle.

At a fiber angle of 20 degrees the circumferential modulus varies from 7 Mpa at a FVF of 0.05 to

26 Mpa at a FVF of 0.3. At a fiber angle of 35 degrees the circumferential modulus varies from 2

Mpa at a FVF of 0.05 to 17 Mpa at a FVF of 0.3.

Figure 3: Circumferential modulus vs. fiber volume fraction at various fiber angles.

Figure 4 takes a look at how the axial modulus varies with FVF and fiber angle. The axial

modulus at a fiber angle of 20 degrees varies from 1 Mpa at a FVF of 0.05 to 4 Mpa at a FVF of

0.3. It also varies from 1 Mpa at a FVF of 0.05 to 9 Mpa at a FVF of 0.3 when the fiber angle is

35 degrees.


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13

Figure 4: Axial modulus vs. fiber volume fraction at various fiber angles.

Figure 5: Shear modulus vs. fiber volume fraction at various fiber angles.


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In Figure 5 the shear modulus is evaluated against fiber volume fraction at various fiber

angles. The shear modulus, at a fiber angle of 20 degrees, was 0.1 Mpa at a FVF of 0.05 and was

0.6 Mpa at a FVF of 0.3. The shear modulus, at a fiber angle of 35 degrees, was 0.3 Mpa at a

FVF of 0.05 and was 1.2 Mpa at a FVF of 0.3.

Figure 6 shows that the radial modulus seemed to depend very little on fiber angle. But it

also shows that radial modulus increases linearly with increasing FVF from 0 Mpa at a FVF of

0.05 to 1.6 Mpa at a FVF of 0.3.

Figure 6: Radial modulus vs. fiber volume fraction at various fiber angles.

The next input parameter from the lamina that is varied in order to investigate the effect on

the various moduli is the fiber angle. The physiologically-relevant range of fiber angles is

roughly 20 to 35 degrees(Table 1).

In Figure 7 the circumferential modulus at a FVF of 0.05 varies from 7 Mpa at a fiber angle

of 20 degrees to 2 Mpa at a fiber angle of 35 degrees, and at a FVF of 0.3 it varies from 25 Mpa

at a fiber angle of 20 degrees to 16 Mpa at a fiber angle of 35 degrees.


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15

Figure 7: Circumferential modulus vs. fiber angle at various fiber volume fractions.

Figure 8: Axial modulus vs. fiber angle at various fiber volume fractions.

In Figure 8 the axial modulus at a FVF of 0.05 is 1 Mpa, and at a FVF of 0.3 it varies from


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3.5 Mpa at a fiber angle of 20 degrees to 9 Mpa at a fiber angle of 35 degrees.

In Figure 9 the shear modulus at a FVF of 0.05 varies from 0.6 Mpa at a fiber angle of 20

degrees to 1.2 Mpa at a fiber angle of 35 degrees, and at a FVF of 0.3 it varies from 0.1 Mpa at a

fiber angle of 20 degrees to 0.2 Mpa at a fiber angle of 35 degrees.

Figure 9: Shear modulus vs. fiber angle at various fiber volume fractions.

Table 2: Values predicted by the model in both range form and real case calculations as

compared to the corresponding values of circumferential-, axial-, radial-, and shearmodulus

measured experimentally as found in the literature.

Modulus(Mpa)

ModelingRanges

F[2030]FVF

[0.050.30]

Real

CaseExperimental

Circumferential

Modulus1.92E25.35 7.09

1814

(ElliottandSetton2001)

AxialModulus 0.91E9.09 2.12

0.70.8

(Acaroglu,Iatridisetal.1995)

(Ebara,Iatridisetal.1996)

(ElliottandSetton2001)

RadialModulus 1.10E1.57 1.34

ShearModulus 0.08G1.20 0.160.1

(Iatridis,Kumaretal.1999)


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17

The changes to the moduli are mostly linear. But while the axial- and shear- moduli (Figures

8-9) increase with increasing fiber angle, the circumferential modulus (Figure 7) decreases with

increasing fiber angle (Table 2).

While modeling ranges allow us to evaluate the effect of changing the input parameters such

as fiber angle and fiber volume fraction on the various mechanical characteristics of the tissue,

they dont allow us to compare our model to the real case. Table 2 shows the ranges of the

moduli predicted by the model accompanied by the modulus predicted when the input parameters

used were what was assumed to be found in the human body. These values were then compared

to experimentally measured values found in literature.

5. DiscussionHere comparisons between the proposed model and existing homogenization model, as well

as the experimentally measured data from the literature, will be made. It is worth repeating that

in the 3D homogenization models, the fibres of the AF are modelled as truss or cable elements

that are strong in tension but not capable of resisting compression or bending moment. This

holds true for both the proposed as well as the existing homogenization model. Also, the surfaces

of the fiber and matrix that come into contact with each other are glued as if the surfaces that

those two features share are actually one in the same. So the interface is a blend and there is no

slippage between the components at their respective interfaces.

An explanation would be in order for how the real case moduli (Table 2) were calculated.

The fiber angle in the native tissue varies not only from lamella-to-lamella, but also within each

lamella. So an average fiber angle of 29.6 degrees was taken from the literature (Lu, Hutton et al.

1998). Fiber volume fraction is also variable, so a weighted FVF was used. To arrive at this

weighted FVF, an approximate FVF from each lamella was considered (Yin and Elliott 2005)

along with the cross sectional area of the corresponding lamella (Lu, Hutton et al. 1998). Using

these parameters, calculations were made for the moduli for each of the lamella. Then the moduliwere weighted based on the cross-sectional areas (Table 1) of the various lamellas relative to the

overall cross sectional area. Once the weighting factors were multiplied by the modulus for that

specific lamella, the various weighted moduli were summed to come to an actual modulus.


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It should be noted that this model, like those proposed in the past, does not take interlamellar

interactions into account. To this point, it has not been determined if the interlamellar

interactions and interweaving, that have been observed in the literature, are of mechanical

significance.

6. ConclusionIn summary, this study established a novel approach to an existing homogenization model. It

more closely models the anisotropic AF tissues in-plane shear modulus as if it were excised

from the IVD. It did this while still making accurate predictions of circumferential-, axial-, and

radial- moduli. The lower shear stress predictions were more in line with experimental

measurements than past models. The model also elucidates the relationship between FVF, fiber

angle, and composite mechanical properties. The proposed model will also help to better

understand the structure-function relationship for future work with disc degeneration and

functional tissue engineering.

7. AcknowledgementsThis research was partially supported by the joint Biomedical Engineering (BME) Program

between the University of South Dakota and the South Dakota School of Mines and Technology.

The authors would also acknowledge the South Dakota Board of Regents Competitive Research

Grant Award (No. SDBOR/USD 2011-10-07) for the financial support.

8. ReferencesAcaroglu, E. R., J. C. Iatridis, et al. (1995). "Degeneration and aging affect the tensile behavior of

human lumbar anulus fibrosus." Spine (Phila Pa 1976) 20(24): 2690-2701.

Bendsoe (1995). "Optimization of structural topology, shape, and material." Berlin.

Bensoussan A, L. J., Papanicolaou G. (1978). Asymptomatic Analysis for Periodic Structures.North Holland, Amsterdam.

Cassidy, J. J., A. Hiltner, et al. (1989). "Hierarchical structure of the intervertebral disc." ConnectTissue Res 23(1): 75-88.

Ebara, S., J. C. Iatridis, et al. (1996). "Tensile properties of nondegenerate human lumbar anulus

fibrosus." Spine 21(4): 452-461.


21/23


21

Eberlein R, H. G., Schulze-Bauer CAJ (2000). "An anisotropic model for annulus tissue and

enhanced finite element analyses of intact lumbar bodies." Computational Methods inBiomechanics and Biomedical Engineering: 1-20.

Elliott, D. M. and L. A. Setton (2000). "A linear material model for fiber-induced anisotropy of

the anulus fibrosus." J Biomech Eng 122(2): 173-179.

Elliott, D. M. and L. A. Setton (2001). "Anisotropic and inhomogeneous tensile behavior of the

human anulus fibrosus: experimental measurement and material model predictions." JBiomech Eng 123(3): 256-263.

Gilbertson, L. G., V. K. Goel, et al. (1995). "Finite element methods in spine biomechanics

research." Crit Rev Biomed Eng 23(5-6): 411-473.

Goel, V. K., B. T. Monroe, et al. (1995). "Interlaminar shear stresses and laminae separation in adisc. Finite element analysis of the L3-L4 motion segment subjected to axial compressive

loads." Spine (Phila Pa 1976) 20(6): 689-698.

Hickey, D. S. and D. W. Hukins (1980). "X-ray diffraction studies of the arrangement ofcollagenous fibres in human fetal intervertebral disc." J Anat 131(Pt 1): 81-90.

Hollister, S. J., D. P. Fyhrie, et al. (1991). "Application of homogenization theory to the study of

trabecular bone mechanics." J Biomech 24(9): 825-839.

Iatridis, J. C., S. Kumar, et al. (1999). "Shear mechanical properties of human lumbar annulusfibrosus." J Orthop Res 17(5): 732-737.

Jones RM (1999). Mechanics of Composite Materials. London, England, Taylor and Francis.

K. Sivaji Babu, K. M. R., V. Rama Chandra Raju, V. Bala Krishna Murthy, and MSR Niranjan

Kumar (2008). "Prediction of Shear Moduli of Hybrid FRP Composite with Fiber-Matrix

Interface Debond." International Journal of Mechanics and Solids 3(2): 147-156.

Klisch, S. M. and J. C. Lotz (1999). "Application of a fiber-reinforced continuum theory to

multiple deformations of the annulus fibrosus." J Biomech 32(10): 1027-1036.

Lee, C. K., Y. E. Kim, et al. (2000). "Impact response of the intervertebral disc in a finite-elementmodel." Spine (Phila Pa 1976) 25(19): 2431-2439.

Lu, Y. M., W. C. Hutton, et al. (1998). "The effect of fluid loss on the viscoelastic behavior of the

lumbar intervertebral disc in compression." J Biomech Eng 120(1): 48-54.

Marchand, F. and A. M. Ahmed (1990). "Investigation of the laminate structure of lumbar disc

anulus fibrosus." Spine (Phila Pa 1976) 15(5): 402-410.

Natarajan, R. N., G. B. Andersson, et al. (2002). "Effect of annular incision type on the change inbiomechanical properties in a herniated lumbar intervertebral disc." J Biomech Eng

124(2): 229-236.


22/23


Ohshima, H., H. Tsuji, et al. (1989). "Water diffusion pathway, swelling pressure, and

biomechanical properties of the intervertebral disc during compression load." Spine (Phila

Pa 1976) 14(11): 1234-1244.

Sanchez-Palencia E, Z. A. (1987). Homogenization Techniques for Composite Media. Verlag,Berlin, Springer.

Sanchez-Palencia, E. Z. A. (1987). Homogenization techniques for composite media. Berlin,

Springer Verlag.

Schwartz, M. H., P. H. Leo, et al. (1994). "A microstructural model for the elastic response of

articular cartilage." J Biomech 27(7): 865-873.

Shirazi-Adl, A. (1989). "On the fibre composite material models of disc annulus--comparison of

predicted stresses." J Biomech 22(4): 357-365.

Shirazi-Adl, A. (1994). "Nonlinear stress analysis of the whole lumbar spine in torsion--

mechanics of facet articulation." J Biomech 27(3): 289-299.

Shirazi-Adl, S. A., S. C. Shrivastava, et al. (1984). "Stress analysis of the lumbar disc-body unitin compression. A three-dimensional nonlinear finite element study." Spine (Phila Pa

1976) 9(2): 120-134.

Tohgo, K. and T. Kawaguchi (2005). "Influence of material composition on mechanical

properties and fracture behavior of ceramic-metal composites." Advances in Fracture and

Strength, Pts 1- 4 297-300: 1516-1521.

Wu, H. C. and R. F. Yao (1976). "Mechanical behavior of the human annulus fibrosus." JBiomech 9(1): 1-7.

Wu, J. Z. and W. Herzog (2002). "Elastic anisotropy of articular cartilage is associated with the

microstructures of collagen fibers and chondrocytes." Journal of Biomechanics 35(7):931-942.

Yin, L. Z. and D. M. Elliott (2005). "A homogenization model of the annulus fibrosus." Journalof Biomechanics 38(8): 1674-1684.

Tyler S. Remund is a PhD candidate in the Biomedical Engineering Department at theUniversity of South Dakota. He holds a BS in Mechanical Engineering from South Dakota StateUniversity. He is interested in tissue engineering of the annulus fibrosus.

Trevor J. Layh holds a BS in Mechanical Engineering from South Dakota State University. Aftergraduation he was accepted into the Department of Defense SMART Scholarship for ServiceProgram in August 2010, Trevor is now employed by the Naval Surface Warfare CenterDahlgren Division in Dahlgren, VA as a Test Engineer.


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*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mailaddresses: [email protected]. (Z.Hu). Tel/Fax: +1-605-688-4817/+1-605-688-5878.E-mail address: [email protected]. 2012. American Transactions on Engineering& Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660Online Available at http://TUENGR COM/ATEAS/V01/01-23 pdf

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Todd M. Rosenboom holds a BS in Mechanical Engineering from South Dakota StateUniversity. He currently works as an application engineer for Malloy Electric in Sioux Falls,SD.

Laura A. Koepsell holds a PhD in Biomedical Engineering and a BS in Chemistry, both from theUniversity of South Dakota. She is a Postdoctoral Research Associate at the University ofNebraska Medical Center Department of Orthopedics and Nano-Biotechnology. She isinterested in cellular adhesion, growth, and differentiation of mesenchymal stem cells ontitanium dioxide nanocrystalline surfaces. She is trying to better understand any inflammatoryresponses evoked by these surfaces and to evaluate the expression patterns and levels ofadhesion and extracellular matrix-related molecules present (particularly fibronectin).

Dr. Ying Deng received her Ph.D. from Huazhong University of Science and Technology in 2001.She then completed a post-doctoral fellowship at Tsinghua University and a second post-doctoral fellowship at Rice University. In 2008, Dr. Deng joined the faculty of the University ofSouth Dakota at Sioux Falls where she is currently assistant Professor of BiomedicalEngineering. She has authored over 15 scientific publications in the biomedical engineering area.

Dr. Zhong Hu is an Associate Professor of Mechanical Engineering at South Dakota StateUniversity, Brookings, South Dakota, USA. He has about 70 publications in the journals andconferences in the areas of Nanotechnology and nanoscale modeling by quantummechanical/molecular dynamics (QM/MD); Development of renewable energy (includingphotovoltaics, wind energy and energy storage material); Mechanical strength evaluation andfailure prediction by finite element analysis (FEA) and nondestructive engineering (NDE);Design and optimization of advanced materials (such as biomaterials, carbon nanotube, polymerand composites). He has been worked on many projects funded by DoD, NSF RII/EPSCoR,NSF/IGERT, NASA EPSCoR, etc.

Peer Review: This article has been internationally peer-reviewed and accepted for publication

according to the guidelines given at the journals website.

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A Novel Finite Element Model for Annulus Fibrosus Tissue Engineering Using Homogenization Techniques