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Submitted to the ASME Journal of Biomechanical Engineering

DETERMINISTIC MATERIAL-BASED AVERAGING THEORY MODEL

OF COLLAGEN GEL MICROMECHANICS

Preethi L. Chandran and Victor H. Barocas*

Department of Biomedical Engineering

University of Minnesota

312 Church St SE

Minneapolis, MN 55455

*Corresponding author,

PHONE: 612-626-5572

FAX: 612-626-6583

E-MAIL: baroc001@umn.edu

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INTRODUCTION

Type I Collagen extracted from tissues can be reconstituted in vitro, forming a highly

hydrated fibril network with a gel-like appearance. Scanning electron microscopy [1-3]

and confocal microscopy [4,5] images show nearly straight fibrils, 50 – 200 nm in

diameter, and a resulting pore size on the order of 1μm [6].

A pronounced in vivo character is retained in the gel’s mechanical response and the

interaction with cells [7,8]. Contractile cells compact and remodel the gel microstructure,

a process reminiscent of in vivo wound contraction and morphogenesis [9] [7]. Cellular

remodeling [10,11] and mechanical studies [5,12] suggest that fibril interconnections

behave as crosslinks rather than entanglements over the time scale of observation,

sustaining force transmission and enabling strain recovery. Similar crosslink conditions

have been invoked to explain the swelling nature of cartilage [13] and the mechanical

response of fibrillar tissues [14]. Collagen gel mechanics in compression are modulated

by a biphasic fluid-solid coupling [2,5,15], not unlike that long discussed in the cartilage

community [16-18]. Finally, the collagen gel provides a simple model for correlating

fibril arrangement and mechanical behavior, and thereby the pliability of skin versus the

rigidity of tendon, both type-I-collagen-based tissues [19].

Modeling collagen gel mechanics is an important step towards the systematic analysis of

complex tissue behavior. Since the mechanics are determined at the level of the fibril

network, a comprehensive model should work at the network scale (microns). In this

paper the tensile loading of the gel is considered. The important micro-scale features

determining equilibrium mechanics are fibril constitutive behavior, inter-fibril force

transfer, and the fibril arrangement. Fibril dimensions being on the order tens to hundreds

of nm, Brownian motion and configurational entropy are not applicable. With fibril

response considered elastic [20], and the interconnections as crosslinks [9], the relevant

microstructure of a collagen gel can be likened to a random network of nonlinear

rigid/semirigid springs. A number of schemes exist to model such microstructures, most

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of them approximations to the realistic but computationally demanding problem of

balancing forces in a deforming network.

A common approach to modeling fibrous and networked structures is to describe the

microstructure as a statistical collection of independently acting strain-energy springs.

The assumption of affine kinematics is used to model the fibril response to macroscopic

strain. The affine assumption is routinely used to describe inclusion kinematics in dilute

non-interacting systems where the primary load transfer is between the inclusion and

matrix1. The inclusion deforms homogeneously with the matrix, and its deformation is

thereby obtained as a direct projection of the macroscopic deformation [21].

Microstructure-based models of planar fibrous tissues [22-26] use the statistical approach

and assume affine kinematics.

The conception of the gel microstructure as a lattice of springs can also be treated

deterministically - fibril response to macroscopic strain governed by an explicit force

balance at crosslinks. Unlike the statistical approach, the deterministic approach requires

no assumption on crosslink displacement, but there is a significant increase in numerical

complexity. Such deterministic network approaches have been used to model granular

material [27], erythrocyte membrane [28], and lung microstructure [29].

Given two possible approaches to modeling network fibril kinematics – solving the full

force balance problem (network model) or using an affine assumption (affine model), we

previously [5] compared the two for a two-dimensional uniaxial-strain problem for a

network topology based on collagen gel microstructure. Importantly,

- A negative correlation between fibril length and stretch was seen in the network

model, but absent in the affine model.

- A strong correlation between fibril orientation and stretch, seen in the affine

model, was weak or absent in the network model.

1 Throughout this paper, we use “matrix”, in the composite-materials sense of non-fibrillar material, rather than in the biological sense of extracellular matrix.

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- Stresses were significantly smaller in network model.

- A toe region of very low stress was observed at small strains in the network case,

but not in the affine model.

All qualitative observations were independent of the fibril constitutive equation. The

network kinematics suggested fibril using the extensive reorientation capacity to lower

strain energy, by containing fibril strains and utilizing length heterogeneities to store the

larger strains in shorter fibrils. As a result, the network model gave consistently lower

stresses than the affine model.

Similar differences in affine and network stresses have been noted in other studies. For

instance, models of polymer networks use a statistical approach and describe polymers

chains as entropic springs, unlike the internal energy springs described above. The stress

response is a function of the configuration space available for a chain to explore and

crosslinks are described as constraints decreasing the configuration space of the chain

between [30]. Models that assume crosslink displacement to be affine [31] give larger

stresses than ‘Phantom’ models, which assume crosslink displacement to fluctuate

(Gaussian) about the mean affine [30]. Studies of the deterministic behavior of Hookean

spring networks [27,28] showed overall network stiffness to decrease with increase in

network randomness. Since in the absence of any randomness, a network of springs is

naturally affine, the decrease in stiffness corroborates our earlier observation that the

network uses heterogeneities in the system to minimize internal energy. In documenting

the effect of randomness in fibril length, fibril connectivity and spring constant, all

parameters defining the topology of a network, the Hansen study [28] raises the

importance of topology and arrangement in network mechanics.

Since network kinematics is not affine, and a statistical approach does not account for

network topology as a determinant of overall mechanics, a deterministic network

approach is attractive. The large difference, however, between the scale of observation

(domain boundary in mm) and mechanics (fibril scale in um), precludes the direct

simulation of the entire network, and a multi-scale scheme is required [32-34] . A multi-

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scale network scheme for collagen gels was first outlined by Agoram and Barocas [35].

The micro-scale network problem was coupled to a macro-scale finite element analysis.

The original model has undergone a number of modifications to improve the theoretical

and numerical framework. In this paper we present the current version, which we call the

Representative Microstructure Finite Element (RMFE) model, and demonstrate its

application to a simplified model of scar mechanics, in which an isotropic scar tissue

interacts with anisotropic tissue surrounding it.

METHODS

Theoretical Development for a material averaging volume

All summation and differentiation operations are represented using the indicial scheme of

Einstein. Uppercase letters (e.g. S, X) denote macro-scale variables, while lowercase

letters (s, x) denote micro-scale variables. Directional indices are always lowercase

subscripts, since the cardinal directions are scale-independent in our model. In the

differentiation notation, the case convention is applied to indicate the scale of the

differentiating variable (eq.1).

iI X

UU∂∂

≡, (1)

Summation over repeated indices is not case sensitive, the directions being scale

independent. For instance,

2

2

1

1, X

SXS

S jjIij ∂

∂+

∂

∂≡ (2)

is the divergence of the macro-scale stress Sij with respect to the macro-scale coordinates

Xi. All variables are in the current state. Finally, the variable E (and therefore e) is used

for deformation-gradient instead of the standard F, to eliminate conflict with the natural

use of f for force in a fibril.

For a hyper-elastic microstructured material, the quasi-static momentum balance (eq.3) is

required to be solved at the scale of the microstructural elements,

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0, =iijs (3)

where s is the micro-scale stress and the applied case convention is as described above. It

is practical, however, to work at a higher scale, using averages of the micro-scale

quantities. In doing so, not only are short-range fluctuations filtered out, but resulting

fields are more relevant to macro-scale observation and measuring tools. In order that the

average be a meaningful representation of local material behavior, the averaging region

should be larger than the scale of microscopic gradients but smaller than that of the

macroscopic [36].

A macro-scale field of averaged stress S(X) and deformation E(X), can be obtained by

averaging the micro-scale stress (s) and deformation gradient (e) fields in the region a

surrounding every point X.

(4)

∫=a

ijij dxdxea

XE 211)( (5)

where s and e are functions of the local x surrounding X.

While a number of theories [37] exist that relate micro and macro-scale variables, the

ideas of averaging theory, that macro-scale fields are locally averaged micro-scale fields

are used here. Unlike homogenization [37,38], averaging theory permits a greater

freedom in the description of microstructural mechanics.

In eqs.4 and 5, though x and X characterize changes on the micro and macro-scale

respectively, they themselves vary on the same scale; that is, dx = dX.

Using Gauss’ theorem, an average deformation gradient (eq.4) can be shown to be

determined by boundary displacement alone, for compatible deformation fields [36] .

∫=a

ijij dxdxsa

XS 211)(

7

∫∫∫ ==a

ija

jia

ij dluna

dxdxua

dxdxea δ

11121,21 (6)

where δa is the boundary of the micro-scale area a, and u is the displacement.

Unlike previous averaging schemes [39-41] where the averaging volume is constant, we

define the averaging volume to be material in the macroscopic deformation field. A

material averaging volume offers three advantages. First, it facilitates a consistency in the

representation of the discrete microstructure responding to a macro-scale deformation.

Second, with only a single phase contributing to the momentum balance, an averaging

volume material in that phase implicitly assures mass conservation in an integral sense.

Third, macroscopic gradients the size of the finite averaging volume may modulate

network rearrangement, and in this scheme they are naturally imposed. The averaging

volume is sufficiently small, however, that the resulting long-range gradients in

microstructural behavior do not affect the average description.

The differential micro-scale momentum balance (eq.3) must be expressed in terms of the

averaged variables. Since the averaging volume is material, Leibnitz’s rule (eq.7) is used

whenever the operations of integration and differentiation are to be interchanged:

∫∫∫ +=⎟⎟⎠

⎞⎜⎜⎝

⎛

)(,

)(,

,)(

),(Xa

ktkta

t

tta

dlnhudatxhhdaδ

(7)

where t is a parameter and u is the displacement at the surface δa. The version of Leibnitz

rule with t representing time is the ubiquitous Reynolds Transport theorem. The micro-

scale variables act like h of eq.7, whereas the macroscopic displacement X, like t of eq.7,

parameterizes the micro-scale variables by fixing the center of the averaging area.

Differentiating the averaged stress, applying chain rule and noting that sij is a function of

(X+x) and the averaging volume a is a function of X, we get

∫∫∫ ⎟⎠⎞

⎜⎝⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

aij

IIaij

Iaij das

adas

adas

a ,,,

111 (8)

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The integration surface changes with macro-scale displacement. Applying eq.7

∫∫∫ +=⎟⎟⎠

⎞⎜⎜⎝

⎛

akIkij

aIij

Iaij dlnusdasdas

δ,,

, (9)

∫∫ −=⎟⎟⎠

⎞⎜⎜⎝

⎛−=−=⎟

⎠⎞

⎜⎝⎛

)(,2

,2,2

,

1111

XakIk

IaI

I

danua

daa

aaa δ

(10)

The integrand in the first term of the RHS of eq.9 is zero by the micro-scale momentum

balance (eq.3). Combining eq.4 and 8 – 10, and reordering gives

∫ −=a

kIkijijIij dlnuSsa

Sδ

,, )(1 (11)

The LHS is divergence of the averaged stress in the macroscopic scale. A new surface

integral term (RHS) appears, working like a body force that compensates for the changes

in averaging volume due to macro-scale displacement gradient. This term is the product

of two boundary fluctuations, going to zero when either the deformation or stress field is

homogenous on the micro-scale. It is a measure of the correlation between the two

fluctuation fields, similar to the Reynolds stresses that appear in turbulence averaging as

a correlation of the velocity fluctuations in the different directions.

Thus the final macro-scale momentum conservation equation is

(12)

where,

( )∫ −=a

kIkijijj dlnuSsa

Qδ

,,1

(13)

The macroscopic momentum balance can be solved in the domain A using Finite Element

analysis. Instead of constitutive equations, the average stress and body force are supplied

0, =− jIij QS

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by a discrete micro-scale analysis, using the concept of a representative element to

replace the microstructure in the averaging region.

Creation and analysis of Representative Volume Element A representative volume element (RVE) is defined as the smallest possible unit that is

representative of a material [37]. For instance, given a large statistically homogenous

fibril network, under a homogenous macroscopic deformation, an RVE is the minimal

region behaving similarly to the entire sample. Beyond this size, any intensive response

(or appropriately scaled extensive response) is independent of fibril number and the

precise fibril arrangement. The representativeness of an RVE permits material behavior

to be derived from a smaller fibril set and a single network realization, significantly

reducing computational time and complexity. The size of the RVE must be larger than

the correlation lengths up to which fibril interactions are felt for a given macro-scale

deformation. In this formulation, we represent the averaging volume by an RVE in the

computational domain. The use of a materially deforming RVE implies that the

correlation lengths for the independent response of an interacting microstructure can

become large enough for a point-wise homogenous macroscopic deformation to be

irrelevant.

The discrete representation within the RVE incorporates the important microstructural

features assumed to affect the material’s mechanical response – fibrils, interconnections

and the relative arrangement. The microstructure is represented by a network generated in

a box of size unity [5] using a method based on collagen fibrillogenesis and gel formation

[42]. The network is referred to as a ‘micromesh’. Segments grow from randomly

generated nucleation sites at a fixed rate, the direction determined by a Von Mises

distribution. The growing tip stops on hitting another segment or RVE edge, and an

interconnection referred to as a ‘micronode’ is formed. A typical micronode joins three

segments.

The collagen fibrils in collagen gels have diameters between 50 nm and 500 nm [6] and

pore size of 1 µm. A wide range of fibril lengths exists [5], and aspect ratios at typical

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fibril lengths are about 20 [2]. Reported values of linear Young modulus vary between

200 and 500 MPa for a collagen fibril [43,44] and 200 to 480 MPa for tendons [45].

Though a collagen fibril assembly is known to exhibit long time viscous retardation [46],

only the elastic response at relatively short time scales is studied here [20].

While tissue behavior is known to be non-linear, the constitutive behavior of a collagen

fibril has been found to be nearly linear [47,48]. In the model, a fibril is represented by its

linear distance between interconnections, in the linear segment between micronodes. The

constitutive behavior is expressed as a function of segment strain, and is non-linear to

account for two phenomena. First, there are near-zero negative stresses in segment

compression (fibril buckling). Second, the stress-strain curve should exhibit an initial

region of low stresses during segment extension (release of fibril rest-state slack),

followed by a stiff response corresponding to actual fibril strain. A constitutive equation

(eq.14) describing fibril force (f) as an exponential function of segment Green strain εs is

used (cf. [22]).

]1)[exp( −= sf βεα (14)

)1(5.0 2 −= ss λε (15)

The parameters α and β affect the magnitude and nonlinearity of the force response

respectively. Our fibril constitutive eq.4.14 and α = 40μN and β= 6 corresponds roughly

to a Young’s modulus E = 1.3 MPa in the small strain region and E = 250 MPa in the stiff

region (30% segment strain) for a fibril diameter of 200 nm.

Changes in relative angles between adjacent fibrils intersecting at a node (Fig.4.1) are

resisted by a restoring force proportional to the angle change and acting on the free fibril

ends, along the chord of the subtended angle,

mnomonmonmon gKB )( θθ −= (16)

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where Bmon is the restoring force at the free ends m and n of two neighboring fibrils

connected at node o, with angle θmon subtended in between. K is the segment rotational

stiffness (related to the bending stiffness of a fibril), and g is the determinant of the angle

θmon, positive if acute and negative otherwise. The stiffness to fibril rotation is derived

assuming an inter-fibril angle change as equivalent to a localized bending of the fibril at

the crosslink (cf. [49]). This region of bending is much smaller than the fibril length,

therefore giving restoring forces independent of the fibril length. In fig.1a, the rotational

forces acting on a fibril group at a cross-link are shown. The typical effect of small and

large rotational stiffness on the fibril kinematics (fig.1b and c) can be seen in the response

of a single micromesh under 4% macro-scale, homogenous uniaxial extension (cf. [50]).

At these small boundary strains, fibrils with low rotational stiffness (K=0.01)

accommodate the macro-scale strain by reorienting and few fibrils undergo stretch (fibril

stretch ratio > 1). With large rotational stiffness, more fibrils are recruited into stretching

to accommodate the boundary strain. In fig.1b and c the fibril stretch ratios at 4% strain

are shown as a function of initial fibril orientation. The predictions of the affine model

are shown alongside in gray.

All angles are measured counterclockwise from a designated reference segment at the

common node. This permits a rotational invariance and easy detection of segments

jumping past their neighboring at the common node. A large potential well when fibrils

approached θ<0.01 radians was introduced to prevent the segments from crossing over

each other.

In this scheme, there are three inherent scales:

1. The macroscopic length scale (L) corresponding to the size of the boundary value

problem.

2. The averaging length scale (l) corresponding to the size of the averaging volume.

3. The RVE length scale (r) which is of order 1, since the RVE is a unit square.

The scale ratios μ and η relate the averaging scale to that of the macroscopic domain and

the RVE respectively.

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Ll

=μ (19)

μ is a measure of the scale separation in the multi-scale problem, and the extent of

macroscopic deformation contained within the averaging volume.

rl

=η (20)

η recognizes the RVE to be a representation of a sub-region of the microstructure,

equivalent to a material unit, and on a different length scale (since the area of the RVE is

always unity). The value of η can be obtained by matching the fibril length density (ρ) in

the RVE and the averaging volume.

( ) ( )R

r

Rr

Rv

Rr

Vv

Vv

a

l

a

l

a

l ∑∑∑===ηη

ηρ 1

2 (21)

( )ρ

η∑

= r

Rrl

(22)

where the superscript V represents the averaging volume, and R represents the RVE. The

symbols a and l denote area and fibril length respectively. The summation is over fibrils v

and r in the averaging and representative volume. In eq.22, the fibril length density (ρ)

can be obtained by regular microscopy or other experimental techniques, while the

numerator is readily available from computer-generated realizations.

Computational Implementation

The concept of averaged momentum balance and localized microstructure analysis is

integrated in a multi-scale scheme, which uses finite elements at the macro-scale and

RVE on the micro-scale. We call the formulation, the Representative Microstructure

Finite Element (RMFE) method, and it can be seen as a continuous cycling between four

events (cf.[3,35])

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1. The Macroscopic Problem, in which the finite element equations are assembled

and used to generate a guess of macro-scale deformation (FE node displacement).

2. Downscaling, in which the RVE boundary deformation is determined from the

calculated FE deformation field.

3. The Microscopic Problem, in which the micro-scale network equations are

assembled and solved for the specified boundary displacement. Equilibrium

cross-link displacements are determined and fibril forces arise at RVE boundary.

4. Upscaling, in which the macro-scale quantities - average stress and body force -

are determined the equilibrium micro-scale state.

In the following section the numerical details in each stage are elaborated, for an

illustrative boundary value problem of uniaxial extension (Fig.2).

Macroscopic Problem

With each entry into this stage, the FE residual and Jacobian matrices are assembled and

one Newton-Raphson iteration is performed using direct Gauss elimination. A guess of

FE node displacement is generated.

The Galerkin FE form of Eq.10 is obtained by multiplying by test function φ, integrating

over the domain, and integrating by parts to get the weak form. The j-th component of

the residual associated with a given test function φ is

dAQdLSNdASRA

jA

ijiA

Iijj ∫∫∫ −−= ϕϕϕδ

, (4.23)

where N is the outward normal to the boundary δA of the macroscopic domain A. Bilinear

basis functions are used to interpolate the macro-scale displacement. The integrands are

evaluated using four-point Gauss quadrature.

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Downscaling

In specifying the deformation of the averaging volume, two requirements must be

satisfied. The averaging volume must be material in the macroscopic deformation, and

the average micro-scale deformation gradient must be equal the macro-scale deformation

gradient at its centering X (eq.3). The first is satisfied by moving the RVE boundary

according to the bilinear FE displacement field. The second is satisfied by noting that a

micro-scale deformation field, bilinear in position and displacement, would satisfy eq.5

for all boundary displacements. Eq.6 states that the average deformation gradient can be

fixed by specifying the boundary micro-scale deformations alone. Therefore an average

gradient as defined by eq.3 can be assured by requiring the boundary RVE crosslinks

alone to move affine with the macro-scale bilinear displacement field. Any artifacts

introduced by specifying boundary crosslink displacement are expected to die out rapidly.

For large RVEs where the scaled response is independent of RVE size, boundary effects

are essentially negligible.

Microscopic Problem

The microscopic problem (cf. [5]) involves finding the crosslink displacements such that

the fibril stretching and bending forces balance. The residual r solved at each internal

micronode c is

∑∑ +=)(

)(,)( cp

pqpicp

pici bfr (24)

where f is the force due to the fibril p acting at crosslink c, and b is the rotational force

between the fibril p and its neighbor q interacting at the other end. Displacement of

boundary micronodes is specified.

The nonlinear micro-scale network problem was solved using damped Newton-Raphson

iteration. The disruption of the iteration process by the highly nonlinear force function

(nearly flat at compressive and small tensile strains and increasing exponentially after),

was prevented by approximating the force to increase linearly for fibril strains that were

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larger than the macro-scale strain by 5%. Observations in the earlier study of network

behavior [50] that fibril strains tend to be smaller than the overall macro-scale, suggested

λ +0.05 to be a safe upper limit to use, and this was confirmed by inspection of the

problem solution.

Averaging

The average stress S is calculated from the discrete forces f at the RVE surface using

Gauss’s theorem.

∫∫∫∫ +−==RRRR a

kikja

kika

ikkja

ij dlnsxdansdasxdasδ

,

∑∫ == ijRa

kikj fxa

dlnsxR

1

δ (25)

where scaling the area and length to dimensions of the AVE gives

∑= ijRij fxa

S η (26)

Fibril forces depend only on dimensionless stretch and are unaffected by the scaling.

Similarly,

∫∫ −=RR a

kIkijRa

kIkRij

j dlnusa

dlnuaS

Qδδ

ηη,, (27)

The source term Q is the combined effect of inhomogeneities in the micro-scale stress

and strain fields. The calculation of the RVE boundary deformation as a function of

macroscopic position is straightforward. The second term requires knowledge of the

micro-scale stress distribution along the boundary. The relation between micro-scale

stress and boundary fibril force f assumed in the derivation of average stress (eq.2.22) is

Bjij

fi nsnf = (28)

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nf and nB are the unit vectors in the direction of the fibril and the boundary normal, and f

is the force in a fibril intersecting the boundary.

Therefore,

fnns Bj

fiij = (29)

The average stress and source term calculated in this stage are available for the FE

equations.

The calculation of the FE Jacobian proceeds along similar lines as outlined previously

[35]. The initial guess for the micronode displacement at each FE iteration step was

calculated using a first-order continuation method.

FEKKinewi Uxx ,= (30)

where xi,K is solution to

KpKiip rxr ,,, = (31)

and rp is the micro-scale residual and rp,i is the micro-scale Jacobian. The Jacobian of the

micro-scale residual with respect to the macro-scale displacements, rp,K, is obtained from

the deformation field and position of the boundary micronodes.

Forces at the boundary of the macro-scale domain are calculated by constructing RVEs at

the corresponding FE boundary and performing 2-point Gauss integration of the

calculated average stresses.

Cases Studied

Case 1: Characterization of RVE

A micromesh belonging to a parameter set is considered an RVE if all micromeshes

within the set behave similar. The response being independent of fibril number and

arrangement satisfies the RVE requirement of statistical homogeneity and equivalence.

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In this section the micromeshes were parameterized according to their geometric and

mechanical character and the criterion for acceptability as RVE was determined. It was

hypothesized that two parameters – fibril orientation and density- sufficiently

characterize the system.

While the density parameter arises from the scaling of mechanical response (eqs.2.21-22

and 26), the orientation parameter arises naturally in the description of fibrous

microstructure. The -xx component of the orientation tensor (Ω) is shown in eq.32.

∑∑

=Ω

ff

fff

xx l

l θ2cos

(32)

where lf cosθf is the segment projection along the horizontal axis, and summation is over

all fibrils. For a two-dimensional system with the coordinate axes coincident with the

alignment axes, Ωxx contains all the orientation information. The sensitivity of mesh

response to the density and orientation parameter was examined. Single micromeshes of

different realizations and sizes were subjected to a strip biaxial extension test (cf.

[5,51,52]). The mechanical and geometric state at 30% strain was plotted as a function of

the initial Ωxx.

Case2: Homogenous cross-aligned TE in uniaxial extension

In this section, the multi-scale model was applied to the study of a homogenous and

aligned TE, subjected to uniaxial extension perpendicular to the direction of alignment.

An experimental study by Tower et al. [12] reported the simultaneous mechanical and

birefringent response of a cross-loaded TE. It was hypothesized that the mechanical and

kinematic observations in Tower et al. [12] are due to network microstructural effects and

would be predicted by the RMFE model.

Tower [12] reported a pre-compaction collagen density of 2 mg/ml and suggested fibril

diameter of 50-500 nm. The sample thickness was 0.4 mm. Based on the reported area

dimensions, we specified a 3 mm long Χ 5 mm wide sample with extension of up to 1.5

18

mm (50% engineering strain). A 9 Χ 15 FE mesh was used with 500 fibrils in each RVE.

Mesh convergence was confirmed by comparison with a 6 Χ 10 mesh. Since the

uncrosslinked sample [12] showed large and free fibril reorientations with negligible

accompanying stress, we assumed that the rotational resistance was small and specified a

stiffness of 0.005 MPa/rad for the fibril reorientation force. The exact fibril distribution in

the TE was not known, and was approximated by a Von Mises distribution (κ = 0.3) with

an average orientation of 88° during the seeding phase of the mesh generation. The same

micromesh was used at all Gauss points of all FEs.

Microstructural data from the central finite element were used to generate quantities for

comparison to the birefringence parameters reported by Tower [12]. A material

birefringence is a measure of the anisotropy in its refractive index [53]. The optical

anisotropy is usually induced by a structural anisotropy which can now be estimated from

the birefringence measurements. A material birefringence is described by two parameters,

the retardation and the extinction angle. The retardation is a measure of the extent of

optical and therefore structural anisotropy. The extinction angle is the direction of the

optical anisotropy. In a collagenous system, the collagen fibrils show different refractive

indices along and across the fibril direction [54]. The micro-scale anisotropies manifest

on the macro-scale as birefringence, from which in turn features of the fibril arrangement

can be estimated. Tower et al [12] showed the evolution of the TE retardation and its

components with strain alongside the TE force. To compare the simulation response with

these experimental data, quantities comparable to retardation and extinction angle were

estimated from the microstructural deformation within each RVE. The extinction angle φ

is the direction of the net fiber orientation and was estimated from the angle between the

eigenvectors of the fiber orientation tensor Ω and the coordinate axes. A simulation

equivalent for the retardation, however, is not readily apparent. Assuming a collagen

monomer as a transversely isotropic rod, and a collagenous system such as the TE to be

an assembly of such rods, the retardation can be shown proportional to the fibril

concentration and orientation anisotropy [5,55]. The proportionality constant is, however,

complex and may be nonlinear with fibril strain and rearrangement. We assumed the

19

proportionality constant to be constant and estimated the simulated retardation as the

product of fibril concentration and orientation anisotropy, given by γ in eq.34.

a

lf

f∑Ω−Ω= )( 21γ (34)

The term within parenthesis is the difference between the principal directions of the

orientation tensor (eq.32). The horizontal and vertical components of the simulated

retardation are defined similar to those in [12].

ϕγγ 2cos=xx (35a)

ϕγγ 2sin=yy (35b)

where φ is the mean orientation or extinction angle.

The simulated retardation estimated using eq.34 makes assumptions on the relation

between the micro-scale arrangement and the macro-scale birefringence. The simulated

extinction angle, given by the net fiber orientation, is however a more reliable estimate of

the experimental extinction angle and a comparison of the two was also made. The

components of the extinction angle are not documented in Tower et al [12], and were

extracted from the plots of retardation and its components in [12] using eq.35a and b.

Case3: Non-homogenous wound tissue in uniaxial extension

The final case study involved an idealized wound mechanics problem (cf. [56]). The

microstructural specification was based on the small-angle-light-scattering data of Bowes

et al. [57], who studied normal vs. wounded porcine skin (specifically, the inputs to the

microstructural model were based on the TGF-treated wound case of [57]). The

mechanical response of a homogenous normal and wounded tissue was compared to that

of a non-homogenous tissue where a wounded region was present in the middle of the

normal tissue. The non-homogenous tissue case was a simplified representation of the

mechanical situation encountered with a wound region in vivo. The aims of this study

were to understand how the presence of macro-scale inhomogeneities affected the

20

deformation of a networked tissue and to assess the effect of fibril orientation on network

mechanics for samples of similar fibril density.

All tissues (homogenous normal, homogenous wounded, and nonhomogenous) were

simulated by a 11 Χ 11 square finite element mesh. In the non-homogenous case, the

central 5 Χ 5 finite elements, about 20% of total tissue, had RVEs with wound

microstructure. The RVE microstructure was generated from a Von Mises distribution

having the same features as the documented fibril data in [57]. The micromesh

representing normal skin was generated from a Von Mises distribution having a mean

orientation of 52.1° and a concentration parameter κ = 0.832, which describes the

concentration of the distribution about the mean. The micromesh representing the

wounded skin was generated from a distribution with a mean of 0° and concentration

parameter κ = 1.51. The wound tissue was therefore more strongly aligned than the

normal tissue. The concentration parameters for the normal and wounded tissue

correspond to an Orientation Index (OI) of 36.5 and 18.2 respectively in [57]. The

uniaxial tensile test was performed by pulling the tissue up to 30% strain along the 0°

axis. Thus, the wounded region had its axis of anisotropy along the macroscopic direction

of stretch. Only two RVEs were used, one for all wound region and one for all normal

region. Both had similar fibril densities of about 550 fibrils.

RESULTS

Case1: Characterization of RVE

The scatter plots of fig.3-4 show Ωxx and Sxx at 30% strain as a function of initial

orientation Ωoxx. The different RVE sizes are shown by different symbols, and multiple

realizations were used for each RVE size.

In fig.3 the geometric response (Ωxx) was found to be highly sensitive to the initial

orientation state Ωoxx. While the orientation parameter (Ωo

xx), an intensive quantity,

sufficiently parameterized the geometric behavior, the mechanical behavior (Fig.4a)

21

required additional scaling by the fibril density parameter (i.e. dividing stress by the

initial total segment length) to give a consistent response (Fig.4b). The plots of fig.3 and

4b were fitted with a linear and power function respectively for each mesh size and the

regression coefficients tabulated (Table 1). For fibril numbers greater than 500, the

micromesh was found to be sufficiently consistent in its geometric and scaled mechanical

response (regression coefficient r2 > 0.9). Micromeshes with fibril numbers 500 or

greater were therefore considered suitable RVEs.

Case 2: Homogenous cross-aligned TE in uniaxial extension

Fig.5 shows the macroscopic deformation state of a cross-loaded TE. As expected, the

sample compacts laterally. In fig.6, the mechanical and birefringent data from the

simulation results are plotted along with the experimental observations. The experimental

data were extracted from one of the stabilized preconditioning cycle plots documented in

fig.7 of [12]. Fig. 6a and b show the simulated and experimental curves for the force,

scaled to fit the plot, alongside the retardation and its components. Fig. 6c and d show

the simulated and experimental curves for the force, scaled to fit the plot, alongside the

extinction angle and its components. The simulation results are detailed below.

The simulated TE sample shows an initial strong vertical alignment (γyy >> γxx) and is not

perfectly cross-loaded ( 0cos ≠ϕ ) (fig.6a). As the sample gets stretched, the stress and

fibril kinematic behavior can be summarized as occurring in two phases.

Initially, little stress develops, indicating that fibril reorientation and not fibril stretch is

the dominant deformation mode. γ decreases indicating that the strength of vertical

alignment is decreasing as fibrils reorient. The direction of net fibril alignment changes

little, however, as seen in the near constant values of cosφ and sinφ, and the proportional

changes in γ, γxx, and γyy. We refer to this region of up to 10% strain, where stresses are

low and fibrils reorient to accommodate macro-scale strain but without changing the

average orientation, as the rearrangement phase.

22

Beyond the rearrangement phase, the stress increases steadily, indicating fibril stretch to

be a dominant mode. The horizontal component of retardation increases, indicating that

the direction of net fiber orientation rotates towards the horizontal stretch axis. The

retardation continues decreasing, but reaches a minimal value where its components cross

each other rapidly. The components of the extinction angle also show a crossover,

indicating that average orientation rotates rapidly, over the finite strain period, from the

vertical to the horizontal direction. We refer to this region of increasing stress, fibril

stretch, and rotation of the net fiber direction as the realignment phase.

Case 3:Non-homogenous wound tissue in uniaxial extension

Figure 9 shows the mesh deformation at 10%, 20%, and 30% strain for the normal,

wounded, and non-homogenous tissue. The wounded tissue, with its axis of anisotropy

along the direction of stretch, shows a typical deformation expected from a homogenous

isotropic material. The normal tissue, aligned at 52.1°, shows a distinct line of stress,

running from grip to grip, nearly in the same direction as the structural anisotropy. The

wounded tissue also gives a much higher stress response than the normal tissue, about

11Χ greater at 30% strain (fig.8). In the non-homogenous tissue, the normal skin region

surrounds the wound region, and being significantly more compliant absorbs most of the

grip-to-grip uniaxial strain. As a result the non-homogenous stress response, though

greater than the homogenous normal tissue, is still closer to it than to the homogenous

wound tissue. With the normal region determining the strain field in the non-homogenous

tissue, and the stiffness along one diagonal direction greater than the other, the central

wound gets rotated like a rigid body (Fig.7).

The initial and deformed state of the wound and skin micromeshes from specific regions

of the three tissues is shown in fig.9. The wound micromesh in the middle of the non-

homogenous tissue undergoes less strain than the normal or wound micromesh at the

same location in the homogenous tissue. The wound micromesh located near the free

surface in the homogenous wound tissue shows significant lateral compaction. The

deformation of the normal skin micromeshes in the corner and surface locations of the

23

homogenous and non-homogenous tissues is nearly equivalent, indicating that the

disturbance from the central non-homogeneity decays before reaching the boundaries.

DISCUSSION

The important micro-scale feature affecting TE mechanics is its networked architecture

where primary load transfer occurs from fibril to fibril at the interconnections. Traditional

statistical models of networked material do not permit mechanical equilibrium at

interconnections, and the fibril kinematics is determined solely by the macro-scale

boundary. In this paper, a multi-scale protocol was developed to permit a deterministic

analysis of networked materials.

Ideas of averaging theory were used to frame the macro-scale balance equations. Unlike

most averaging formulations, the averaging volume was assumed material in the

macroscopic deformation. While a material averaging volume naturally satisfies mass

conservation in an integral sense, an additional source term appears in the momentum

balance. The source term corrects for correlations between mass and micro-scale

momentum gradients. The macro-scale field equations were solved numerically by the

finite element (FE) technique. Averaged variables were evaluated by discrete analysis of

the local microstructure at each FE Gauss point.

The microstructure within an averaging volume was replaced by simulated networks or

micromeshes using the concept of Representative Volume. The micromeshes were

generated by a 2D simplification of the collagen fibrillogenesis. It should be noted that

the micromesh only bears information on the fibril segment length and all accounting of

the fibril contour length, diameter, and Poisson’s ratio is via the constitutive equation that

gives fibril force as a function of segment length. Since the averaging area is material in

the solid phase, knowledge of the true fibril state at any boundary displacement is not

required. Also, if the scaling parameter η is based on micromesh segment length density,

the corresponding fibril length density should be determined from micrographs of the

24

averaging region by using the straight line distance between fibril nodes and not the

contour length.

Micromesh response was found sensitive to the fibril orientation and fibril density

parameter. For fibril numbers greater than 500, micromeshes within a parameter set were

found to be statistically equivalent and were classified as suitable RVEs. From scaling

arguments (eq.2.21 and 22) a linear dependence between macro-scale stress and fibril

density is predicted. This is also seen in composite [43] and mixture type models [58].

Roeder et al. [59] showed an increase in stiffness with increased collagen density, but the

plot of stiffness vs. density contained only four data points and was not decisively linear.

The model was used to simulate the uniaxial extension of an anisotropic TE loaded

perpendicular to its direction of net fibril alignment. The domain dimensions were based

on Tower et al. [12]. The simulation results for force and microstructural rearrangements

can be discussed as a two-phase response. At low strains, a rearrangement phase occurs

where stresses are low as fibril reorient predominantly to accommodate boundary strain

but maintain the average direction of fibril orientation. At large strains, an alignment

phase occurs where stress increases, fibrils stretch predominantly and the direction of

fibril orientation rotates. The two phases can be distinguished in the experimental results.

First, a long ‘toe’ region is seen in the stress-strain curve (fig.6b). In this region the

retardation decreases (fig.6b) and the average orientation axis is nearly constant (fig.6d).

Secondly, at around 1.5mm displacement (30% strain), the stress increases along with the

horizontal component of retardation. The retardation and extinction angle component

cross over rapidly while total retardation is nearly constant, suggestive of a rotation of the

fibril orientation axis over a finite strain period. The two experimental observations are

captured by the rearrangement and realignment phase behavior in the simulations.

The observation that the toe region strain could be accommodated by fibril reorientation

is striking because in an affine model, fibrils reorient and stretch occur simultaneously

and are correlated. This observation was also predicted in our earlier study [5] where the

25

minimal fibril stretch at small strains gave a ‘toe’ region in the macro-scale stress

response.

In both the experimental and simulation study, the TE was not perfectly cross-loaded.

The net fibril orientation in the experimental study was around 72°, and in the simulation

study it was 88°. The experimental observation that the angle of orientation did not rotate

for a large part of the initial sample deformation, but then did rapidly over a small strain

period, is interesting. The initial resistance to changing the net fibril direction until fibril

stretch is extensively recruited could arise as a result of the fibril interconnections which

favor a semblance to the initial relative fibril arrangement. This tendency is clearly

demonstrated in the study of the normal and wounded tissue. In the normal tissue, the

fibrils were aligned 52.1° to the direction of stretch. In a composite or affine model where

fibril interaction does not affect fibril kinematics, the net direction of stretch would

progressively turn towards the horizontal stretch axis. However in the network model,

presumably because of the interconnected nature, it appears more favorable to retain the

initial orientation. This results in a line of stress developing along that direction and a

Poisson’s ratio that seems significantly greater in the direction perpendicular to it. When

a stiffer wound region with a different direction of alignment is introduced in the normal

skin tissue, it not only gets spared the full impact of the boundary uniaxial strain, but

because of the intrinsic resistance to change in alignment direction, it gets rotated rather

than distorted by the anisotropic compliance field surrounding it.

This study gives a modeling framework to analyze networked materials. The model was

able to capture important qualitative trends in the mechanical and fibril kinematic

response of tissue equivalents in uniaxial extension. To perform more quantitative

studies, however, more rigorous descriptions are required both at the microstructural and

numerical level. The fibril constitutive equation as a function of segment length needs to

be more carefully defined. Since the formulation used centrally acting springs, the

constitutive equation must capture many features (e.g. the fibril stress-strain behavior,

bending and buckling properties, area and Poisson effects, curvature changes during

inter-fibril rotation) as a function of only the segment, the straight-line distance between

26

fibril cross-links. Since this would be an important part of any quantitative analysis, more

rigorous work is required in defining the fibril constitutive equation. Also, the model

assumes uniform fibril diameters and a Von Mises distribution of fibril orientations. In a

TE, evident from SEM micrographs, there are fibrils of different diameters.

On the numerical side, a three-dimensional micromesh would permit a larger degree of

fibril arrangement and inter-fibril contact effects. The resulting network kinematics

would be a better indicator of real-life response. Also, a higher-order function than the

bilinear used to interpolate displacement in this study, would be an improved

representation of the complex macro-scale deformation field in a network material. In

spite of these limitations, the model was able to predict qualitatively the experimental

observations of TE extensile behavior and give insight into wound mechanics, both not

entirely intuitive from other models of fibril kinematics. The results highlight the need to

identify the situations where the networked or affine character plays the dominant

kinematic role in interconnected fibril systems.

ACKNOWLEDGEMENTS

This work was supported primarily by the MRSEC Program of the National Science

Foundation under Award Number DMR-0212302. Computations were made possible by

a grant from the Minnesota Supercomputing Institute.

27

Tables

RVE size (No of fibrils) Geometric response Mechanical response

130 ±10 .9766 .6352

270±10 .9919 .9286

555±10 .9964 .954

850±10 .9968 .9761

1130±10 .997 .9766

Table.1. Micromesh as Representative Volume Element (RVE)

Correlation coefficients describing RVE geometric (net fibril orientation) and mechanical

response (stress) at 30% strain as a function of the initial geometric state. The geometric response

is fitted with a linear function and the mechanical response with a 2nd order polynomial. The

correlation increases with increasing micromesh size (increasing fibril number within

micromesh), indicating better suitability as RVE.

28

Table.1. Micromesh as Representative Volume Element (RVE)

Correlation coefficients describing RVE geometric (net fibril orientation) and mechanical

response (stress) at 30% strain as a function of the initial geometric state. The geometric

response is fitted with a linear function and the mechanical response with a 2nd order

polynomial. The correlation increases with increasing micromesh size (increasing fibril

number within micromesh), indicating better suitability as RVE.

29

Fig.1 Inter-fibril rotational force

a) The rotational force due to changes in angle between neighboring fibrils

connected at a node is assumed to arise due to localized bending off the fibril

close to the node. The force acts along the chord of the angle subtended by the

connected fibrils.

b) Segment stretch ratio vs. initial orientation at 4% strain for K=0.01. Few fibrils

are recruited into stretch. The corresponding affine prediction for 4% strain is

also shown in grey.

c) Segment stretch ratio vs. initial orientation at K=10 at 4% strain for K=10. A

large number of fibrils undergo stretch. The affine prediction at 4% strain is

shown in grey. The network model fibril strains are comparable to the affine.

d) Network stresses for small (K=0.01) and large (K=10) values of the rotational

stiffness. The toe region is lost and stresses are high for the large rotational

stiffness case.

Fig.2 Macroscopic boundary value problem - Uniaxial Tensile Test

Fig.3 Parameterization of micromesh geometric response

Net orientation at 30% strain as a function of initial orientation. Orientation state of

deformed micromesh is to large extent completely defined by initial orientation state.

Fig.4 Parameterization of micromesh mechanical response

a) Stress response at 30% as function of initial orientation state. Stress response

is dependent on initial orientation, but differs between each size group.

b) Scaling stress response by fibril density gives consistent response between the

different size groups.

30

Fig.5. The undeformed (a) and deformed state at 27% strain (b) of a cross-loaded

TE with initial alignment 880 to the stretch axis.

The RVEs within each finite element are shown at 10X magnification.

Fig.6 Comparison of simulated and experimental response of cross-loaded TE in

uniaxial extension

a) Simulation result: Force (scaled to fit the plot), retardation and its components

as function of sample strain.

b) Experimental result: Force (scaled to fit the plot), retardation and its

components as function of sample strain.

c) Simulation result: Force (scaled to fit the plot), components of the orientation

angle as a function of sample strain

d) Experimental result: Force (scaled to fit the plot), components of the

orientation angle as a function of sample strain

Fig.7. The deformed state of the three tissue types – normal skin, wounded skin,

nonhomogenous scar (normal skin with central wounded region) at 10%, 20 %

and 30% strain.

The wounded region is shown in grey. Each column is one tissue type and each row is

one strain level.

Fig.8 Stress response of homogenous normal and wounded skin tissue, and non-

homogenous normal tissue with central wounded region.

The wounded skin tissue, highly aligned and oriented along the direction of stretch

gives a stiffer response than the normal skin tissue. The force at 30% strain is 11Χ that

of the normal tissue. However if the wounded tissue is within a surrounding normal

tissue, the stiffness of the resulting non-homogenous tissue is much lower. The force

at 30% strain is less than 2Χ that of the normal tissue.

Fig.9 The undeformed micromeshes used to model normal and wounded skin,

and their deformed state at selected regions of the tissue at 30% strain.

31

a. Input micromeshes for the normal and wounded tissue. The normal skin

micromesh has a net orientation of 52.20. The wounded tissue micromesh has a

net orientation of 00 and is more aligned than the normal one.

b. Reference positions of the micromeshes shown in 9b.

c. Typical deformation state of micromesh at the locations marked in 9b. Each

column corresponds to one tissue type. Each row corresponds to one reference

location.

All micromeshes are of the same scale.

32

a)

b)

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

0 0.2 0.4 0.6 0.8 1

networkaff ine

segm

ent s

tretc

h ra

tio

cosθs

c)

0.98

1

1.02

1.04

1.06

1.08

0 0.2 0.4 0.6 0.8 1

networkaffine

segm

ent s

tretc

h ra

tio

cosθs

Fig.1

o

n

m

Rotational force

mnomonmonmon gKB )( θθ −=

33

Fig.2

0)( =+ XQSdXd

iji

0=kjk SN

0==

Y

XX

UUU

0=kjk SN

00

==

Y

X

UU

34

0.3

0.4

0.5

0.6

0.7

0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8

129 fibers

270 fibers

548 fibers

848 fibers1130 fibers

final

orie

ntat

ion

para

met

er Ω

xx

inital orientation parameter Ωoxx

Fig.3

35

0

5 102

1 103

2 103

2 103

0.2 0.3 0.4 0.5 0.6 0.7 0.8

129 fibers

270 fibers

548 fibers

848 fibers

1130 fibers

final

stre

ss σ

xx

initial orientation parameter Ω0xx

0

1 101

2 101

3 101

4 101

5 101

0.2 0.3 0.4 0.5 0.6 0.7 0.8

129 fibers

270 fibers

548 fibers

848 fibers

1130 fibers

final

sca

led

stre

ss σ

xx

initial orientation parameter Ωxx

Fig.4

b)

a)

36

a)

b)

Fig.5

37

a)

0

2

4

6

8

10

0

200

400

600

800

1000

0 0.05 0.1 0.15 0.2 0.25 0.3

γxx

γyy

γ

force

reta

rdat

ion

(γ)

strain

forc

e

b)

0

5

10

15

20

0246810121416

0 0.5 1 1.5 2 2.5

γxx

γxx

γ

force

reta

rdat

ion

(γ)

force

displacement (mm)

c)

0

0.5

1

1.5

2

0

200

400

600

800

1000

0 0.05 0.1 0.15 0.2 0.25 0.3

cosϕsinϕ

force

strain

forc

e

d)

-0.20

0.20.40.60.8

11.2

0246810121416

0 0.5 1 1.5 2 2.5

cosϕsinϕ force

force

displacement (mm)

Fig.6

b)

38

Normal Skin Tissue Wounded Skin Tissue Non-homogenous Scar Tissue

10% strain

20% strain

30% strain

Fig.7

39

0

1000

2000

3000

4000

5000

6000

0 0.1 0.2 0.3 0.4

displacement

forc

e

skin tissuewound tissueskin+wound tissue

Fig.8

40

Normal Wounded Non-homogenous

Mid

Surface

Corner

Fig.9

Mid

Surface Corner a) b)

c)

Normal Wound

41

References

[1]Allen, T. D., Schor, S. L., Schor, A. M., 1984, "An Ultrastructural Review of

Collagen Gels, a Model System for Cell-Matrix, Cell-Basement Membrane and Cell-Cell Interactions," Scanning Electron Microscopy., pp. 375-390.

[2]Knapp, D. M., Barocas, V. H., Moon, A. G., Yoo, K., Petzold, L. R., Tranquillo, R. T., 1997, "Rheology of Reconstituted Type I Collagen Gel in Confined Compression," Journal of Rheology, 41, pp. 971-993.

[3]Agoram, B., Barocas, V. H., 2001, "Coupled Macroscopic and Microscopic Scale Modeling of Fibrillar Tissues and Tissue Equivalents," Journal of Biomechanical Engineering, 123, pp. 362-369.

[4]Voytik-Harbin, S. L., Roeder, B. A., Sturgis, J. E., Kokini, K., Robinson, J. P., 2003, "Simultaneous Mechanical Loading and Confocal Reflection Microscopy for Three-Dimensional Microbiomechanical Analysis of Biomaterials and Tissue Constructs," Microscopy & Microanalysis., 9, pp. 74-85.

[5]Chandran, P. L., Barocas, V. H., 2004, "Microstructural Mechanics of Collagen Gels in Confined Compression: Poroelasticity, Viscoelasticity, and Collapse.," Journal of Biomechanical Engineering, 126, pp. 152-166,.

[6]Tranquillo, R. T., 1999, "Self-Organization of Tissue-Equivalents: The Nature and Role of Contact Guidance," Biochemical Society Symposia., 65, pp. 27-42.

[7]Tranquillo, R. T., Durrani, M. A., Moon, A. G., 1992, "Tissue Engineering Science - Consequences of Cell Traction Force," Cytotechnology, 10, pp. 225-250.

[8]Elsdale, T., Bard, J., 1972, "Collagen Substrata for Studies on Cell Behavior," Journal of Cell Biology., 54, pp. 626-637.

[9]Grinnell, F., Lamke, C. R., 1984, "Reorganization of Hydrated Collagen Lattices by Human Skin Fibroblasts," Journal of Cell Science, 66, pp. 51-63.

[10]Guidry, C., Grinnell, F., 1986, "Contraction of Hydrated Collagen Gels by Fibroblasts: Evidence of Two Mechanisms by Which Collagen Fibrils Are Stabilized," Collagen and Related Research, 6, pp. 515-529.

[11]Sawhney, R. K., Howard, J., 2002, "Slow Local Movements of Collagen Fibers by Fibroblasts Drive the Rapid Global Self-Organization of Collagen Gels," Journal of Cell Biology., 157, pp. 1083-1091.

[12]Tower, T. T., Neidert, M. R., Tranquillo, R. T., 2002, "Fiber Alignment Imaging During Mechanical Testing of Soft Tissues," Annals of Biomedical Engineering., 30, pp. 1221-1233.

[13]Maroudas, A., Ziv, I., Weisman, N., Venn, M., 1985, "Studies of Hydration and Swelling Pressure in Normal and Osteoarthritic Cartilage," Biorheology., 22, pp. 159-169.

[14]Harkness, M. L. R., R.D., H., 1959, "Effects of Enzymes on Mechanical Properties of Tissue," Nature,Lond, 183, pp. 1821-1822.

[15]Girton, T. S., Barocas, V. H., Tranquillo, R. T., 2002, "Confined Compression of a Tissue-Equivalent: Collagen Fibril and Cell Alignment in Response to Anisotropic Strain," Journal of Biomechanical Engineering., 124, pp. 568-575.

42

[16]Lai, W. M., Mow, V. C., 1980, "Drag-Induced Compression of Articular Cartilage During a Permeation Experiment," Biorheology., 17, pp. 111-123.

[17]McCutchen, C. W., 1982, "Cartilages Is Poroelastic, Not Viscoelastic (Including an Exact Theorem About Strain Energy and Viscous Loss, and an Order of Magnitude Relation for Equilibration Time)," Journal of Biomechanics., 15, pp. 325-327.

[18]Gu, W. Y., Lai, W. M., Mow, V. C., 1993, "Transport of Fluid and Ions through a Porous-Permeable Charged-Hydrated Tissue, and Streaming Potential Data on Normal Bovine Articular Cartilage," Journal of Biomechanics, 26, pp. 709-723.

[19]Silver, F., Biological Materials: Structure, Mechanical Properties and Modeling of Soft Tissues, New York University Press, New York.

[20]Misof, K., Rapp, G., Fratzl, P., 1997, "A New Molecular Model for Collagen Elasticity Based on Synchrotron X-Ray Scattering Evidence," Biophysical Journal., 72, pp. 1376-1381.

[21]Sokolnikoff, Ivan, S., 1956, Mathematical Theory of Elasticity., New York, McGraw-Hill,

[22]Billiar, K. L., Sacks, M. S., 2000, "Biaxial Mechanical Properties of the Native and Glutaraldehyde-Treated Aortic Valve Cusp: Part II--a Structural Constitutive Model," Journal of Biomechanical Engineering, 122, pp. 327-335.

[23]Pinsky, P. M., Datye, D. V., "A Microstructurally-Based Finite Element Model of the Incised Human Cornea," pp. 907-922, 1991.

[24]Barocas, V. H., Tranquillo, R. T., 1997, "An Anisotropic Biphasic Theory of Tissue-Equivalent Mechanics: The Interplay among Cell Traction, Fibrillar Network Deformation, Fibril Alignment, and Cell Contact Guidance," Journal of Biomechanical Engineering, 119, pp. 137-145.

[25]Lanir, Y., 1983, "Constitutive Equations for Fibrous Connective Tissues," Journal of Biomechanics., 16, pp. 1-12.

[26]Lanir, Y., 1979, "A Structural Theory for the Homogeneous Biaxial Stress-Strain Relationships in Flat Collagenous Tissues," Journal of Biomechanics., 12, pp. 423-436.

[27]Ostoja-Starzewski, M., Wang, C., 1989, "Linear Elasticity of Planar Delaunay Networks: Random Field Characterization of Effective Moduli," Acta Mechanica, 80, pp. 68-80.

[28]Hansen, J. C., Skalak, R., Chien, S., Hoger, A., 1996, "An Elastic Network Model Based on the Structure of the Red Blood Cell Membrane Skeleton," Biophysical Journal, 70, pp. 146 - 166.

[29]Geoffrey, N. M., Jeffrey, J. F., Jason, H. T. B., 1998, "Force Heterogeneity in a Two-Dimensional Network Model of Lung Tissue Elasticity," J. Appl. Physiol., 85, pp. 1223-1229.

[30]Edwards, F. S., 1986, "The Theory of Macromolecular Networks," Biorheology, 23, pp. 589 - 603.

[31]Flory, P. J., 1977, "Theory of Elasticity of Polymers Networks. The Effect of Local Constraints on Junctions," The Journal of Chemical Physics, 66, pp. 5720-5727.

[32]Stoneham, M. A., Harding, J. H., 2003, "Not Too Big, Not Too Small: The Appropriate Scale," Nature Materials, 2, pp. 77-83.

43

[33]Guilak, F., Mow, V. C., 2000, "The Mechanical Environment of the Chondrocyte: A Biphasic Finite Element Model of Cell-Matrix Interactions in Articular Cartilage," Journal of Biomechanics, 33, pp. 1663-1673.

[34]Breuls, R. G., Sengers, B. G., Oomens, C. W., Bouten, C. V., Baaijens, F. P., 2002, "Predicting Local Cell Deformations in Engineered Tissue Constructs: A Multilevel Finite Element Approach," Journal of Biomechanical Engineering, 124, pp. 198-207.

[35]Agoram, B. 2000. A Macroscopic-Microscopic Model of Fibrillar Materials. Ph.D. University of Colorado, Boulder, CO.

[36]Oda, M., Iwashita, K., 1999, Mechanics of Granular Materials : An Introduction, Balkema, Rotterdam.

[37]Hori, M., Nemat-Nasser, S., 1999, "On Two Micromechanics Theories for Determining Micro-Macro Relations in Heterogeneous Solids," Mechanics of Materials, 31, pp. 667-682.

[38]Hollister, S. J., Brennan, J. M., Kikuchi, N., 1994, "A Homogenization Sampling Procedure for Calculating Trabecular Bone Effective Stiffness and Tissue Level Stress," Journal of Biomechanics., 27, pp. 433-444.

[39]Drew, D., 1971, "Averaged Field Equations for Two-Phase Media," Studies in Applied Mathematics, 50, pp. 133-166.

[40]Whitaker, S., 1986, "Flow in Porous Media III: Deformable Media," Transport in Porous Media, 1, pp. 127-154.

[41]Armstrong, C. G., Lai, W. M., Mow, V. C., 1984, "An Analysis of the Unconfined Compression of Articular Cartilage," Journal of Biomechanical Engineering, 106, pp. 165-173.

[42]Veis, A., 1982, "Collagen Fibrillogenesis," Connective Tissue Research., 10, pp. 11-24.

[43]Jain, M. K., Chernomorsky, A., Silver, F. H., Berg, R. A., 1988, "Material Properties of Living Soft-Tissue Composites," Journal of Biomedical Materials Research-Applied Biomaterials, 22, pp. 311-326.

[44]Sasaki, N., Shukunami, N., Matsushima, N., Izumi, Y., 1999, "Time-Resolved X-Ray Diffraction from Tendon Collagen During Creep Using Synchrotron Radiation," Journal of Biomechanics., 32, pp. 285-292.

[45]Silver, F., Biological Materials, 1987, Structure, Mechanical Properties and Modeling of Soft Tissues, New York University Press, New York.

[46]Mosler, E., Folkhard, W., Knorzer, E., Nemetschek-Gansler, H., Nemetschek, T., Koch, M. H., 1985, "Stress-Induced Molecular Rearrangement in Tendon Collagen," Journal of Molecular Biology., 182, pp. 589-596.

[47]Sasaki, N., Odajima, S., 1996, "Elongation Mechanism of Collagen Fibrils and Force-Strain Relations of Tendon at Each Level of Structural Hierarchy," Journal of Biomechanics., 29 (9),pp. 1131-1136.

[48]Fratzl, P., Misof, K., Zizak, I., Rapp, G., Amenitsch, H., Bernstorff, S., 1998, "Fibrillar Structure and Mechanical Properties of Collagen," Journal of Structural Biology, 122, pp. 119-122.

[49]Wang, C. W., Berhan, L., Sastry, A. M., 2000, "Structure, Mechanics and Failure of Stochastic Fibrous Networks: Part1-Microscale Consideration," ASME Journal of Engineering Materials and Technology, 122, pp. 450-459.

44

[50]Chandran, P. L., Barocas, V. H., 2005, "Affine Vs. Non-Affine Fibril Kinematics in Collagen Networks: Theoretical Studies of Network Behavior," Journal of Biomechanical Engineering., submitted,

[51]Kang, A. H., Gross, J., 1970, "Relationship between the Intra and Intermolecular Cross-Links of Collagen," Proceedings of the National Academy of Sciences of the United States of America., 67, pp. 1307-1314.

[52]Sun, W., Sacks, M. S., Sellaro, T. L., Slaughter, W. S., Scott, M. J., 2003, "Biaxial Mechanical Response of Bioprosthetic Heart Valve Biomaterials to High in-Plane Shear," Journal of Biomechanical Engineering, 125, pp. 372-380.

[53]Collett, E., 1993, Polarized Light : Fundamentals and Applications, Marcel Dekker, New York.

[54]Wolman, M., Kasten, F. H., 1986, "Polarized Light Microscopy in the Study of the Molecular Structure of Collagen and Reticulin," Histochemistry., 85, pp. 41-49.

[55]Fuller, G. G., 1995, Optical Rheometry of Complex Fluids, Oxford University Press, New York.

[56]Dallon, J., Sherratt, J., Maini, P., Ferguson, M., "Biological Implications of a Discrete Mathematical Model for Collagen Deposition and Alignment in Dermal Wound Repair," pp. 379-393, 2000 Dec.

[57]Bowes, L. E., Jimenez, M. C., Hiester, E. D., Sacks, M. S., Brahmatewari, J., Mertz, P., Eaglstein, W. H., 1999, "Collagen Fiber Orientation as Quantified by Small Angle Light Scattering in Wounds Treated with Transforming Growth Factor-Beta2 and Its Neutalizing Antibody," Wound Repair and Regeneration, 7, pp. 179-186.

[58]Gu, W. Y., Lai, W. M., Mow, V. C., 1997, "A Triphasic Analysis of Negative Osmotic Flows through Charged Hydrated Soft Tissues," Journal of Biomechanics, 30, pp. 71-78.

[59]Roeder, B. A., Kokini, K., Surgis, J. E., Robinson, J. P., Voytik-Harbin, S. L., 2002, "Tensile Mechanical Properties of Three-Dimensional Type I Collagen Extracellular Matrices with Varied Microstructure," Journal of Biomechanical Engineering, 124, pp. 214-223.