2012 American Transactions on Engineering & Applied Sciences                           American Transactions on Engineerin...
1. Introduction      The annulus fibrosus (AF) is an annular cartilage in the intervertebral disc (IVD) that aids insuppor...
modulus can be explained by the fact that the models assume the tissue to be firmly anchored insurrounding tissue, whereas...
Cα , β : overall average material properties                   ci , j ,k ,l : non-homogeneous material properties         ...
C11 =                 ρE f                             +                               (1 − ρ )Em − ρE fν f 2 − (1 − ρ )Em...
2.1.1 Fiber angle and fiber volume fraction      The first two important geometric considerations are the volumetric ratio...
the crossing collagen fibers are separated by a section of proteoglycan matrix, whereas in theoriginal model they were wel...
Figure 1: Meshed 3D geometric representation of matrix and fiber orientation along with            coordinate system, dime...
Figure 2: Meshed 3D geometric representation of composite RVE along with corresponding                             axes, d...
are four fibers within the RVE, that there are five equal divisions of width.              2⋅n⋅c⋅r         d=           +r...
tissues. An elastic modulus of 500 MPa and a Poisson’s Ratio of 0.35 were adopted for thecollagen fibers (Goel, Monroe et ...
Figure 3 looks at how the circumferential modulus varies with varying FVF and fiber angle.At a fiber angle of 20 degrees t...
Figure 4: Axial modulus vs. fiber volume fraction at various fiber angles.           Figure 5: Shear modulus vs. fiber vol...
In Figure 5 the shear modulus is evaluated against fiber volume fraction at various fiberangles. The shear modulus, at a f...
Figure 7: Circumferential modulus vs. fiber angle at various fiber volume fractions.           Figure 8: Axial modulus vs....
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 ...
The changes to the moduli are mostly linear. But while the axial- and shear- moduli (Figures8-9) increase with increasing ...
The existing model has a circumferential modulus in the 11 MPa range, an axial modulus ofaround 2 MPa, and a shear modulus...
fibers and matrix within the RVE. In both this model and the original, there are four fibers. Inthe original model there a...
It should be noted that this model, like those proposed in the past, does not take interlamellarinteractions into account....
Eberlein R, H. G., Schulze-Bauer CAJ (2000). "An anisotropic model for annulus tissue and       enhanced finite element an...
Ohshima, H., H. Tsuji, et al. (1989). "Water diffusion pathway, swelling pressure, and      biomechanical properties of th...
Todd M. Rosenboom holds a BS in Mechanical Engineering from South Dakota State             University. He currently works ...
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A Novel Finite Element Model for Annulus Fibrosus Tissue Engineering Using Homogenization Techniques

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In this work, a novel finite element model using the mechanical homogenization techniques of the human annulus fibrosus (AF) is proposed to accurately predict relevant moduli of the AF lamella for tissue engineering application. A general formulation 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 annulus fibrosus tissue’s various, location-dependent geometrical and 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.

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

  1. 1. 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 Hua Department of Biomedical Engineering Faculty of Engineering, University of South Dakota, USAb Department of Mechanical Engineering Faculty of Engineering, South Dakota State University, USAARTICLEINFO A B S T RA C TArticle history: In this work, a novel finite element model using theReceived September 06, 2011Received in revised form - mechanical homogenization techniques of the human annulusAccepted September 24, 2011 fibrosus (AF) is proposed to accurately predict relevant moduli ofAvailable online: September 25, the AF lamella for tissue engineering application. A general2011 formulation for AF homogenization was laid out with appropriateKeywords: boundary conditions. The geometry of the fibre and matrix wereFinite Element MethodAnnulus Fibrosus laid out in such a way as to properly mimic the native annulusTissue Engineering fibrosus tissue’s various, location-dependent geometrical andHomogenization 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-mailaddresses: 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 2229-1660Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf
  2. 2. 1. Introduction   The annulus fibrosus (AF) is an annular cartilage in the intervertebral disc (IVD) that aids insupporting the structure of the spinal column. It experiences complex, multi-directional loadsduring 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. (Wuand Yao 1976) The layers of the AF are composed of fibrous collagen fibrils that are oriented insuch a way that the angles rotate from ± 28 degrees relative to the transverse axis of the spine inthe 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 includesaveraging the directionally-dependent mechanical properties in what is called a representativevolume elements (RVE). These RVE are averages of the directionally- and spatially-dependentmaterial properties. When summed over the volume of the material, they can be very useful indescribing the macroscopic mechanical properties of materials with complex microstructures.(Bensoussan A 1978; Sanchez-Palencia E 1987; Jones RM 1999) Homogenization has beenapplied to address some of the shortcomings of structural finite element analysis (FEA) modelsthat 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 andLotz 1999; Eberlein R 2000; Elliott and Setton 2000; Elliott and Setton 2001) for modeling theAF. 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 Herzog2002) and AF. (Yin and Elliott 2005). The mechanical complexity of the AF has posed substantial problems for engineersattempting to model the system. To date, the circumferential modulus and axial modulus havebeen predicted accurately, but the predicted shear modulus has been consistently two orders ofmagnitude high. An explanation proposed in a recent paper (Yin and Elliott 2005), which offereda 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
  3. 3. modulus can be explained by the fact that the models assume the tissue to be firmly anchored insurrounding tissue, whereas the experimentally measured tissue is removed from its surroundingtissue. 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 theoverall shear measurement. The purpose of this paper was to establish a novel method for modeling the AF using FEAand homogenization theory that predicts the circumferential-, axial-, and radial- modulusaccurately while also predicting a shear modulus that accurately represents that of theexperimentally measured tissue. A general formulation for annulus fibrosus lamellarhomogenization was laid out. Appropriate changes to the boundary conditions as well as thegeometry of the structural fibres was made to accommodate the measurements of the mechanicalproperties under various annulus fibrosus volume fractions and orientations. The specificchanges in the three dimensional location and orientation of the cylindrical, crossing fibers withinthe matrix was taken into account. And the mechanical properties of the human AF by modelingwere 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 andElliott 2005) In the homogenization approach volumetric averaging is used to arrive at thegeneral formulation. (Sanchez-Palencia 1987; Bendsoe 1995; Jones RM 1999) Thehomogenization formula is created by averaging material properties for a material that is assumedto be linear elastic over discrete, volumetric segments. The overall material is assumed to haveinhomogeneous properties throughout the entire volume. So, the average material properties canbe calculated by multiplying the inhomogeneous, localized material properties c by theindependent strain rates u, in independent strain states α , β , over the volume of the tissue Ω likein Eq. (1). 1 α β Cα , β = ∫ ui, juk ,l dΩ ΩΩ (1)*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mailaddresses: 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 3& Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf
  4. 4. 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 andnegative direction. This tensor thus rotates the average material properties to simulate thedirection of the AF collagenous fibers. This angle, α , is measured from the midline, θ , and itchanges 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 , inopposite 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 asingle 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 equationsshown in Eq. (5) using the height of the fiber portion of the segment ρ , the elastic modulus ofthe fiber and matrix E f , E m respectively and the Poisson ratio of the fiber and matrix υ f , υ mrespectively: 4 Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu
  5. 5. 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 cablemodels and of strain energy models. However it did predict a shear modulus that was two ordersof 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 theimportant mechanical properties of the AF tissue. But it did not make accurate shear moduluspredictions. As a matter of fact, the predictions from this model were two orders of magnitudehigher than the measurements reported in the literature. In this section we will detail someaspects of the published model that may contribute to the unnaturally high modulus prediction.*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mailaddresses: 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 5& Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf
  6. 6. 2.1.1 Fiber angle and fiber volume fraction   The first two important geometric considerations are the volumetric ratio of fiber to matrixfiber 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 theFVF and the fiber angle vary by which lamina they are located in. But the finite element methodis a great tool for taking these variabilities into account. The original model used fiber angles inthe range of 15 to 45 degrees. It also used FVFs in the range of 0 to 0.3. These ranges were usedfirst in parametric studies in order to better understand how the fiber angle and FVF affect thevarious relevant moduli. Also, beings fiber angle, and to a lesser extent FVF, can be determinedexperimentally, the parametric studies helped in determining some of the more difficult toelucidate 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 matrixwithin the composite RVE. In the original formulation, (Yin and Elliott 2005) they assumed thetwo fiber populations to be within a single continuous material and not layered as in native tissuestructure. (Sanchez-Palencia 1987)2.1.3 Boundary conditions  The final important consideration is the boundary conditions applied to the RVE. Theboundary condition for the tensile case can be seen in Figure 1. A similar boundary condition forthe tensile case was applied to the proposed model. But when they set the boundary conditionsfor the shear case, they fixed the edges along both the θ - and z- axis when they applied a shearalong z = 1 and θ = 1 . (Sanchez-Palencia 1987) The proposed model has adopted a boundarycondition from (K. Sivaji Babu 2008), It constrains the rz-surface at θ = 0 and applies a shear tothe rz surface at θ = 1 . (K. Sivaji Babu 2008) This boundary condition can be visualized inFigure 2. Taken together, these geometric considerations allow the proposed model of the AFtissue’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 angleand FVF in order to bring them closer to the physiological range. Changes in the fiberconfiguration were proposed in order to more closely mimic the native state of the tissue where 6 Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu
  7. 7. the crossing collagen fibers are separated by a section of proteoglycan matrix, whereas in theoriginal model they were welded together in the shape of an ‘X’. The final change made to theoriginal 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 thissimulation graphs of circumferential-, axial-, and radial- modulus as well as shear modulusagainst 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 modulusagainst varying fiber angle at fiber volume fractions of 0.05, 0.1, 0.15, 0.2, 0.25, and 0.3. Theangles of collagen in native tissue range from 24.5-36.3 degrees to the transverse plane with anaverage of 29.6 degrees.2.2.2 Fiber configuration  In this paper it is assumed that the fiber populations are layered and separated by matrixmaterial. The three dimensional geometric arrangement for this fiber and matrix composite isshown in Figure 1 as a RVE along with the tensile case’s boundary conditions. Thecorresponding RVE for the shear case is shown in Figure 2. With the material being acomposite, it is important to assign dimensions to repeating components within the RVE. Thewidth 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 distancebetween fibers is the equivalent of one radius. The length of b is dependent on the fiber angle αand the length of a. Eq. (7) The length of a was derived from looking at the ratio of total fibervolume to total segment volume. A number of new variables are introduced in the derivation of aEq. (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 )*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mailaddresses: 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 7& Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf
  8. 8. Figure 1: Meshed 3D geometric representation of matrix and fiber orientation along with coordinate system, dimensions, and tensile boundary conditions.8 Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu
  9. 9. 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 simplifiedformula for a, becomes clear. So to equally space the four fibers along the c edge from each other and also the edge of thematrix, the length d was derived as given by Eq. (11). It makes use of the idea that when there*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mailaddresses: 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 9& Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf
  10. 10. 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 wasdeveloped that is useful for predicting the various moduli at each variation of fiber angle andFVF.2.2.3 Boundary conditions  The original paper had fixed boundary conditions along two adjoining faces of the RVE andapplied shear on the two opposite faces of the RVE. In the proposed model one face has fixedboundary conditions, and the opposite face has an applied shear. These changes taken togethermake 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 constantregardless of where they are measured throughout the AF. The elastic modulus and Poisson ratiofor the collagen fibers and proteoglycan matrix can be assigned specific values. For modeling thevarying conditions of the AF tissue, laminae, and IVD, the parameters were chosen based on theliterature of past numerical models of the AF, and in some cases, direct measurements of the 10 Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu
  11. 11. tissues. An elastic modulus of 500 MPa and a Poisson’s Ratio of 0.35 were adopted for thecollagen fibers (Goel, Monroe et al. 1995; Lu, Hutton et al. 1998), while an elastic modulus of0.8 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) wereassigned to the proteoglycan matrix. Fiber volume fractions and fiber angles were varied overranges found in previous homogenization.4. Results  The first input parameter from the lamina that is varied in order to investigate the effect onthe various moduli is the FVF. The FVF is varied from 0.05 to 0.3, which are normalphysiological 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 coursethese parameters are variable throughout the AF. But this list was compiled for the originalmodel, so it was used here for ease of comparison. There are also more than six lamellar layersin 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) *Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mailaddresses: 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 11& Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf
  12. 12. 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 to26 Mpa at a FVF of 0.3. At a fiber angle of 35 degrees the circumferential modulus varies from 2Mpa 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 axialmodulus at a fiber angle of 20 degrees varies from 1 Mpa at a FVF of 0.05 to 4 Mpa at a FVF of0.3. 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 is35 degrees. 12 Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu
  13. 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.*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mailaddresses: 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 13& Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf
  14. 14. In Figure 5 the shear modulus is evaluated against fiber volume fraction at various fiberangles. The shear modulus, at a fiber angle of 20 degrees, was 0.1 Mpa at a FVF of 0.05 and was0.6 Mpa at a FVF of 0.3. The shear modulus, at a fiber angle of 35 degrees, was 0.3 Mpa at aFVF 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 italso shows that radial modulus increases linearly with increasing FVF from 0 Mpa at a FVF of0.05 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 onthe various moduli is the fiber angle. The physiologically-relevant range of fiber angles isroughly 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 angleof 20 degrees to 2 Mpa at a fiber angle of 35 degrees, and at a FVF of 0.3 it varies from 25 Mpaat a fiber angle of 20 degrees to 16 Mpa at a fiber angle of 35 degrees. 14 Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu
  15. 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*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mailaddresses: 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 15& Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf
  16. 16. 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 20degrees 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 afiber 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)  16 Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu
  17. 17. The changes to the moduli are mostly linear. But while the axial- and shear- moduli (Figures8-9) increase with increasing fiber angle, the circumferential modulus (Figure 7) decreases withincreasing fiber angle (Table 2). While modeling ranges allow us to evaluate the effect of changing the input parameters suchas 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 themoduli predicted by the model accompanied by the modulus predicted when the input parametersused were what was assumed to be found in the human body. These values were then comparedto experimentally measured values found in literature.5. Discussion  Here comparisons between the proposed model and existing homogenization model, as wellas the experimentally measured data from the literature, will be made. It is worth repeating thatin the 3D homogenization models, the fibres of the AF are modelled as truss or cable elementsthat are strong in tension but not capable of resisting compression or bending moment. Thisholds true for both the proposed as well as the existing homogenization model. Also, the surfacesof the fiber and matrix that come into contact with each other are ‘glued’ as if the surfaces thatthose two features share are actually one in the same. So the interface is a blend and there is noslippage 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 eachlamella. 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 thisweighted 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). Usingthese 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 theoverall cross sectional area. Once the weighting factors were multiplied by the modulus for thatspecific lamella, the various weighted moduli were summed to come to an actual modulus.*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mailaddresses: 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 17& Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf
  18. 18. The existing model has a circumferential modulus in the 11 MPa range, an axial modulus ofaround 2 MPa, and a shear modulus of around 18 MPa. Conversely, the proposed model had acircumferential modulus of about 7 MPa, an axial modulus of about 2 MPa, and a shear modulusof around 0.5 MPa. The experimentally measured values for these parameters are acircumferential 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-measuredvalues from literature when it comes to tensile moduli, the models uniformly disagree with theexperimentally measured data from the literature when it comes to the shear modulus. The shearmodulus is over two orders of magnitude higher in the models than in the experimentallymeasured data from the literature. The author suggested that this is because the tissue has to beremoved from its surroundings to be measured experimentally. (Yin and Elliott 2005) This freesup the ends of the fibers so there is fiber sliding but not fiber stretching contributing to overallshear measurements. Whereas the nature of the models can have more realistic in vivo boundaryconditions, so the tissue can experience both fiber stretch and fiber sliding in its shearmeasurement. Conversely, the proposed model will more accurately emulate the former. In this study, a homogenization model of the AF was revised to address the discrepancybetween the shear modulus prediction in the previously proposed model and the experimentaldata of human AF tissue. The original model had a shear modulus two orders of magnitudehigher than that of the experimental values for native AF tissue. It was suggested that the shearwas lower in the experimental values, because the pieces of AF tissue were removed from theirnative surroundings. This causes the fibers of the tissue near the edges to not be anchored intothe surrounding tissue. So the stretch of the tissue’s fibers may not have been contributing toshear measurements. Here is suggested a model that gives accurate accounts of the shearmodulus 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) toaddress the discrepancy between the shear modulus in the model and that experimentallymeasured in the native tissue. The first change made to the model was the arrangement of the 18 Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu
  19. 19. fibers and matrix within the RVE. In both this model and the original, there are four fibers. Inthe original model there are two fibers on each opposing face. The two crossing fibers are in thesame plane, so they are in effect welded together. One of the changes made to this model is inthe geometrical layout of the fibers. The alternating fibers are separated in space and by matrixmaterial. This separation of the fibers allows them to slide against each other. Once thearrangement of the fibers and the matrix were changed, the shear modulus prediction wasdecreased. But it had decreased to a level much smaller than that of the native tissue value. Thevalue the model had predicted was actually 10 −12 MPa. This is much, much smaller than thevalue tested in native tissue of roughly 0.1 MPa. So a literature search was performed to try tofind alternative approaches to improving shear predictions in homogenization models. The paperthat was found called for changing the boundary conditions. In the original model, two adjoiningsides of the RVE are constrained, and the opposing two sides of the RVE have the shear loadingsapplied. This model has one side constrained at a time. The opposing side of the RVE has theshear loading applied. This has brought the shear modulus prediction much closer to that testedin the native AF tissue. And while the original model is likely more accurate for 3D predictionsas the tissue is in the IVD in vivo, if the aim is to develop a model that more accurately predictsthe mechanical properties of a resected piece of AF tissue as is measured in the literature, thenboundary conditions used in the proposed model are more applicable. This is because theboundary conditions in the proposed model allow for the fibres to slide more freely, avoidingincorporating fiber stretch, and resulting in significantly lower shear measurements. This model is important in understanding the mechanics of the AF, especially when tissuesamples are resected from the greater IVD. It can be useful for better understanding discdegeneration and for improving approaches to designing functional tissue engineered constructs.It can help in understanding disc degeneration as the process is usually characterized by adegradation of the proteoglycan matrix. Through the alteration of the matrix, disc degradationcan be modeled accurately. Also, more appropriate benchmarks for the design of functionaltissue engineered constructs can be set through the better understanding of the interaction of theAF subcomponents that this model provides.*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mailaddresses: 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 19& Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf
  20. 20. It should be noted that this model, like those proposed in the past, does not take interlamellarinteractions into account. To this point, it has not been determined if the interlamellarinteractions and interweaving, that have been observed in the literature, are of mechanicalsignificance.6. Conclusion  In summary, this study established a novel approach to an existing homogenization model. Itmore closely models the anisotropic AF tissue’s in-plane shear modulus as if it were excisedfrom the IVD. It did this while still making accurate predictions of circumferential-, axial-, andradial- moduli. The lower shear stress predictions were more in line with experimentalmeasurements than past models. The model also elucidates the relationship between FVF, fiberangle, and composite mechanical properties. The proposed model will also help to betterunderstand the structure-function relationship for future work with disc degeneration andfunctional tissue engineering.7. Acknowledgements  This research was partially supported by the joint Biomedical Engineering (BME) Programbetween 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 ResearchGrant Award (No. SDBOR/USD 2011-10-07) for the financial support.8. References  Acaroglu, 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." 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. 20 Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu
  21. 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 in Biomechanics 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." J Biomech 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 a disc. 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 of collagenous 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 annulus fibrosus." 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-element model." 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 in biomechanical properties in a herniated lumbar intervertebral disc." J Biomech Eng 124(2): 229-236.*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mailaddresses: 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 21& Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf
  22. 22. 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 unit in 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." J Biomech 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." Journal of Biomechanics 38(8): 1674-1684. 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. 22 Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu
  23. 23. 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.*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mailaddresses: 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 23& 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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