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

2012 American Transactions on Engineering & Applied Sciences

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
a b b
Tyler S. Remund , Trevor J. Layh , Todd M. Rosenboom ,
a a* b*
Laura A. Koepsell , Ying Deng , and Zhong Hu

a
Department of Biomedical Engineering Faculty of Engineering, University of South Dakota, USA
b
Department of Mechanical Engineering Faculty of Engineering, South Dakota State University, USA

ARTICLEINFO A B S T RA C T
Article history: In this work, a novel finite element model using the
Received September 06, 2011
Received in revised form - mechanical homogenization techniques of the human annulus
Accepted September 24, 2011 fibrosus (AF) is proposed to accurately predict relevant moduli of
Available online: September 25, the AF lamella for tissue engineering application. A general
2011
formulation for AF homogenization was laid out with appropriate
Keywords:
boundary conditions. The geometry of the fibre and matrix were
Finite Element Method
Annulus Fibrosus
laid out in such a way as to properly mimic the native annulus
Tissue Engineering fibrosus tissue’s various, location-dependent geometrical and
Homogenization histological states. The mechanical properties of the annulus
fibrosus calculated with this model were then compared with the
results 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-function
relationship of a tissue-engineered AF.

2012 American Transactions on Engineering and Applied Sciences.

*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mail
addresses: ying.deng@usd.edu. (Z.Hu). Tel/Fax: +1-605-688-4817/+1-605-688-5878.
E-mail address: Zhong.hu@sdstate.edu. 2012. American Transactions on Engineering 1
& Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660
Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf

1. Introduction
The 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

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

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 radial- modulus
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. Mathematical Model
The 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 α , β , over the volume of the tissue Ω like
in Eq. (1).
1 α β
Cα , β = ∫ ui, juk ,l dΩ
ΩΩ
(1)


Cα , β : overall average material properties

ci , j ,k ,l : non-homogeneous material properties

ui, j : independent strain rates

α , β : independent strain rates
Ω: volume

The stiffness tensor Eq. (2) rotates around a certain angle, α , 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, α , is measured from the midline, θ , and it
changes with spatial location.

C α = RT C ⋅ R (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, α , in
opposite directions .

C + α + C −α
C + / −α = (3)
2

There are four in-plane material properties: C11 , C22 , C12 , and C66 that are calculated for a
single lamella. They are arranged in matrix notation, like in Eq. (4).

⎡C11 C12 0 ⎤
C = ⎢C12 C22
⎢ 0 ⎥⎥ (4)
⎢0
⎣ 0 C66 ⎥
⎦

And the values for C11 , C22 , C12 , and C66 can be calculated from the system of equations

shown in Eq. (5) using the height of the fiber portion of the segment ρ , the elastic modulus of
the fiber and matrix E f , E m respectively and the Poisson ratio of the fiber and matrix υ f , υ m

respectively:


C11 =
ρE f
+
(1 − ρ )Em − ρE fν f 2 − (1 − ρ )Emν 2 + (ρν + (1 − ρ ) m ) Em E f
ν 2

( ) ( )
f

1 −ν f
2
1 −ν m
2
1 −ν f
2
1 −ν m
2
ρEm 1 − ν f 2 + (1 − ρ ) 1 − ν m 2 E f

(ρν + (1 − ρ ) m )E m E f
ν
C12 =
f

(
ρE m 1 − ν f
2
)+ (1 − ρ )(1 −ν )E
2
m f

Em E f
C 22 =
(
ρE m 1 − ν f 2
)+ (1 − ρ )(1 −ν )E
m
2
f

1 Em E f
C66 =
2 ρEm (1 + ν f ) + (1 − ρ )(1 + ν m )E f
(5)
ρ: height of the fiber
Ef : elastic modulus of the fiber

Em : elastic modulus of the matrix

vf : Poisson ratio of the fiber

vm : 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 Model from the literature
The 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.


2.1.1 Fiber angle and fiber volume fraction
The 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 Fiber configuration
The 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 Boundary conditions
The 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 when they set the boundary conditions
for the shear case, they fixed the edges along both the θ - and z- axis when they applied a shear
along z = 1 and θ = 1 . (Sanchez-Palencia 1987) The proposed model has adopted a boundary
condition from (K. Sivaji Babu 2008), It constrains the rz-surface at θ = 0 and applies a shear to
the rz surface at θ = 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
tissue’s mechanical behavior to be accurate.

2.2 Proposed model changes
Changes 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

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 Fiber angle and fiber volume fraction
The ranges for this study were based loosely on the values used for the original study. In this
simulation graphs of circumferential-, axial-, and radial- modulus 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 Fiber configuration
In 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 case’s 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 of b is dependent on the fiber angle α
and the length of a. Eq. (7) The length of a 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 of a
Eq. (8). So a can be derived from Eq. (9) by substitution of Eq. (10) and then rearranging.

c = 13 ⋅ r (6)
b = a ⋅ tan(α ) (7)

4π ⋅ r 2
a= (8)
ρ ⋅ c ⋅ sin (α )

Figure 1: Meshed 3D geometric representation of matrix and fiber orientation along with
coordinate system, dimensions, and tensile boundary conditions.


Figure 2: Meshed 3D geometric representation of composite RVE along with corresponding
axes, dimensions, and shear boundary conditions.

V fiber π ⋅n⋅lf ⋅r2
ρ= = (9)
VRVE a ⋅b⋅c

l f = a 1 + tan 2 (α ) (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 d was derived as given by Eq. (11). It makes use of the idea that when there

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

2⋅n⋅c⋅r
d= +r (11)
5
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
α : 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 Boundary conditions
The 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. Material Properties
It 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

tissues. An elastic modulus of 500 MPa and a Poisson’s 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 Poisson’s 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. Results
The 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.
Lamina Layer Inner 2nd 3rd 4th 5th Outer References
Annulus fibrosus (Lu, Hutton et al.
0.06 0.11 0.163 0.22 0.2662 0.195
cross sectional area 1998)
Collagen fiber (Yin and Elliott
0.05 0.09 0.13 0.17 0.2 0.23
volume fraction 2005)
(Lu, Hutton et al.
Fiber angle Annulus Fiber orientation average: 29.6 (range 24.5‐36.3)
1998)


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.


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

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


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.


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

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 shear- modulus
measured experimentally as found in the literature.
Modeling Ranges
Real
Modulus (Mpa) Fα[20‐30] FVF Experimental
Case
[0.05‐0.30]
Circumferential 18±14
1.92≤E≤25.35 7.09
Modulus (Elliott and Setton 2001)
0.7±0.8
(Acaroglu, Iatridis et al. 1995)
Axial Modulus 0.91≤E≤9.09 2.12
(Ebara, Iatridis et al. 1996)
(Elliott and Setton 2001)
Radial Modulus 1.10≤E≤1.57 1.34
0.1
Shear Modulus 0.08≤G≤1.20 0.16
(Iatridis, Kumar et al. 1999)


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 don’t 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. Discussion
Here 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 moduli
were 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.

The existing model has a circumferential modulus in the 11 MPa range, an axial modulus of
around 2 MPa, and a shear modulus of around 18 MPa. Conversely, the proposed model had a
circumferential modulus of about 7 MPa, an axial modulus of about 2 MPa, and a shear modulus
of around 0.5 MPa. The experimentally measured values for these parameters are a
circumferential modulus in the range of 4-32 MPa, an axial modulus in the range of 0.1-1.5 MPa,
and a shear modulus of 0.1 MPa. (Table 2).

While there is agreement between the various models and the experimentally-measured
values from literature when it comes to tensile moduli, the models uniformly disagree with the
experimentally measured data from the literature when it comes to the shear modulus. The shear
modulus is over two orders of magnitude higher in the models than in the experimentally
measured data from the literature. The author suggested that this is because the tissue has to be
removed from its surroundings to be measured experimentally. (Yin and Elliott 2005) This frees
up the ends of the fibers so there is fiber sliding but not fiber stretching contributing to overall
shear measurements. Whereas the nature of the models can have more realistic in vivo boundary
conditions, so the tissue can experience both fiber stretch and fiber sliding in its shear
measurement. Conversely, the proposed model will more accurately emulate the former.

In this study, a homogenization model of the AF was revised to address the discrepancy
between the shear modulus prediction in the previously proposed model and the experimental
data of human AF tissue. The original model had a shear modulus two orders of magnitude
higher than that of the experimental values for native AF tissue. It was suggested that the shear
was lower in the experimental values, because the pieces of AF tissue were removed from their
native surroundings. This causes the fibers of the tissue near the edges to not be anchored into
the surrounding tissue. So the stretch of the tissue’s fibers may not have been contributing to
shear measurements. Here is suggested a model that gives accurate accounts of the shear
modulus in the AF tissue while not sacrificing modulus predictions in the circumferential-, axial-,
and radial-directions.

Several significant changes have been made to the reported model (Yin and Elliott 2005) to
address the discrepancy between the shear modulus in the model and that experimentally
measured in the native tissue. The first change made to the model was the arrangement of the


fibers and matrix within the RVE. In both this model and the original, there are four fibers. In
the original model there are two fibers on each opposing face. The two crossing fibers are in the
same plane, so they are in effect welded together. One of the changes made to this model is in
the geometrical layout of the fibers. The alternating fibers are separated in space and by matrix
material. This separation of the fibers allows them to slide against each other. Once the
arrangement of the fibers and the matrix were changed, the shear modulus prediction was
decreased. But it had decreased to a level much smaller than that of the native tissue value. The
value the model had predicted was actually 10 −12 MPa. This is much, much smaller than the
value tested in native tissue of roughly 0.1 MPa. So a literature search was performed to try to
find alternative approaches to improving shear predictions in homogenization models. The paper
that was found called for changing the boundary conditions. In the original model, two adjoining
sides of the RVE are constrained, and the opposing two sides of the RVE have the shear loadings
applied. This model has one side constrained at a time. The opposing side of the RVE has the
shear loading applied. This has brought the shear modulus prediction much closer to that tested
in the native AF tissue. And while the original model is likely more accurate for 3D predictions
as the tissue is in the IVD in vivo, if the aim is to develop a model that more accurately predicts
the mechanical properties of a resected piece of AF tissue as is measured in the literature, then
boundary conditions used in the proposed model are more applicable. This is because the
boundary conditions in the proposed model allow for the fibres to slide more freely, avoiding
incorporating fiber stretch, and resulting in significantly lower shear measurements.

This model is important in understanding the mechanics of the AF, especially when tissue
samples are resected from the greater IVD. It can be useful for better understanding disc
degeneration and for improving approaches to designing functional tissue engineered constructs.
It can help in understanding disc degeneration as the process is usually characterized by a
degradation of the proteoglycan matrix. Through the alteration of the matrix, disc degradation
can be modeled accurately. Also, more appropriate benchmarks for the design of functional
tissue engineered constructs can be set through the better understanding of the interaction of the
AF subcomponents that this model provides.


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. Conclusion
In summary, this study established a novel approach to an existing homogenization model. It
more closely models the anisotropic AF tissue’s 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. Acknowledgements
This 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. References
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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." Connect
Tissue 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.


Eberlein R, H. G., Schulze-Bauer CAJ (2000). "An anisotropic model for annulus tissue and
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Elliott, D. M. and L. A. Setton (2000). "A linear material model for fiber-induced anisotropy of
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Tyler S. Remund is a PhD candidate in the Biomedical Engineering Department at the
University of South Dakota. He holds a BS in Mechanical Engineering from South Dakota State
University. He is interested in tissue engineering of the annulus fibrosus.

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


Todd M. Rosenboom holds a BS in Mechanical Engineering from South Dakota State
University. 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 the
University of South Dakota. She is a Postdoctoral Research Associate at the University of
Nebraska Medical Center Department of Orthopedics and Nano-Biotechnology. She is
interested in cellular adhesion, growth, and differentiation of mesenchymal stem cells on
titanium dioxide nanocrystalline surfaces. She is trying to better understand any inflammatory
responses evoked by these surfaces and to evaluate the expression patterns and levels of
adhesion 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 of
South Dakota at Sioux Falls where she is currently assistant Professor of Biomedical
Engineering. 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 State
University, Brookings, South Dakota, USA. He has about 70 publications in the journals and
conferences in the areas of Nanotechnology and nanoscale modeling by quantum
mechanical/molecular dynamics (QM/MD); Development of renewable energy (including
photovoltaics, wind energy and energy storage material); Mechanical strength evaluation and
failure prediction by finite element analysis (FEA) and nondestructive engineering (NDE);
Design and optimization of advanced materials (such as biomaterials, carbon nanotube, polymer
and 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 journal’s website.


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

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