grytz, r. & meschke, g., constitutive modeling of crimped collagen

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J O U R N A L O F T H E M E C H A N I C A L B E H AV I O R O F B I O M E D I C A L M A T E R I A L S 2 ( 2 0 0 9 ) 5 2 2 – 5 3 3

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Research paper

Constitutive modeling of crimped collagen fibrils insoft tissues

Rafael Grytz∗, Günther Meschke

Institute for Structural Mechanics, Ruhr-University Bochum, 44801 Bochum, Germany

A R T I C L E I N F O

Article history:

Received 12 June 2008

Received in revised form

12 December 2008

Accepted 22 December 2008

Published online 6 January 2009

A B S T R A C T

A microstructurally oriented constitutive formulation for the hyperelastic response of

crimped collagen fibrils existing in soft connective tissues is proposed. The model is based

on observations that collagen fibrils embedded in a soft matrix crimp into a smooth three-

dimensional pattern when unloaded. Following ideas presented by Beskos and Jenkins

[Beskos, D., Jenkins, J., 1975. A mechanical model for mammalian tendon. ASME Journal

of Applied Mechanics 42, 755–758] and Freed and Doehring [Freed, A., Doehring, T., 2005.

Elastic model for crimped collagen fibrils. Journal of Biomechanical Engineering 127,

587–593] the collagen fibril crimp is approximated by a cylindrical helix to represent the

constitutive behavior of the hierarchical organized substructure of biological tissues at the

fibrillar level. The model is derived from the nonlinear axial force–stretch relationship of

an extensible helical spring, including the full extension of the spring as a limit case.

The geometrically nonlinear solution of the extensible helical spring is carried out by an

iterative procedure. The model only requires one material parameter and two geometrical

parameters to be determined from experiments. The ability of the proposed model to

reproduce the biomechanical response of fibrous tissues is demonstrated for fascicles from

rat tail tendons, for porcine cornea strips, and for bovine Achilles tendons.c© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The constitutive response of biological tissues existing

in the human eye, heart, veins, arteries or tendons is

mainly determined by the elastic behavior of collagen

fibrils. Organized collagen fibrils form fibrous networks that

introduce strong anisotropic and highly nonlinear attributes

into the constitutive response of soft tissues (Fung, 1993). The

main goal of this contribution is to derive a microstructurally

based constitutive formulation for crimped collagen fibrils

∗ Corresponding author. Tel.: +49 234 3229051; fax: +49 234 3214149.E-mail addresses: [email protected] (R. Grytz), [email protected] (G. Meschke).URL: http://www.sd.rub.de (G. Meschke).

embedded in soft tissues to be used in numerical analysissuch as the finite element method.

Due to their huge aspect ratio (Trotter et al., 1994),collagen fibrils embedded in a soft matrix buckle or crimpwhen the tissue is unloaded. Crimp usually occurs at thelevel of aggregated fibrils, e.g. at the level of fascicles intendons (Kastelic et al., 1980) or at the level of lamellaein the corneal stroma (see Fig. 1). The wavy structure ofundulated fibrils in unloaded tissues has a typical crimpangle of between 5◦ and 30◦ depending on the tissue,where different wave shapes of crimped fibrils have been

1751-6161/$ - see front matter c© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.jmbbm.2008.12.009


J O U R N A L O F T H E M E C H A N I C A L B E H AV I O R O F B I O M E D I C A L M A T E R I A L S 2 ( 2 0 0 9 ) 5 2 2 – 5 3 3 523

Fig. 1 – Electron micrograph of wavy lamellae in thecentral rabbit stroma fixed at 0 mmHg intraocularpressure (reproduced from Andreo and Farrell (1982)).

observed experimentally. By rotating tendon specimensbetween crossed polaroids, Diamant et al. (1972) and Daleet al. (1972) showed that the wave shape is planar. Incontrast, scanning electron microscope (SEM) photographstaken by Liao (2003) show a helical crimp of fibrils in mitralvalve chordae tendineae. Evans and Barbenel (1975) as wellas Yahia and Drouin (1989) have reported that both planar andhelical crimp forms exist.

Collagen fibrils have a hierarchically organized substruc-ture starting from collagen molecules, which are built frompolypeptide chains. These molecules are synthesized astropocollagen polymers with a triple-helix geometry. Fivetropocollagens form a microfibril. Microfibrils aggregate toform subfibrils, which themselves aggregate to form fibrils.These structures are held together by lateral cross connec-tions of different kinds. The typical “J-type” shape of thestress–stretch curve of collagen fibrils embedded in a softtissue can be traced back to miscellaneous phenomena oc-curring at different levels of the hierarchical structure ofcollagen. The region of small strains (the “toe” region) cor-responds to the removal of the crimp at the fibrillar level(Diamant et al., 1972). At larger strains (the “heel” region) thenonlinear material response may correspond to straighten-ing of molecular kinks within the gap regions of microfib-rils (Misof et al., 1997). If collagen fibrils are stretched beyondthe “heel” region, the elastic response is assumed to resultfrom stretching of the collagen triple-helices and from glid-ing of neighboring molecules (Fratzl et al., 1997; Sasaki andOdajima, 1996; Folkhard et al., 1987). This last region of thematerial response of collagen fibrils is characterized by a lin-ear relation between stretch and stress. It may be concludedthat the nonlinear elastic response (“toe” and “heel” region)of soft tissues is mainly caused by straightening of crimps orkinks of the collagen fibril structure, first at the fibrillar levelthan at the molecular level. A bottom-up approach consider-ing the various length scales of the underlying mechanisms,although it would be helpful in understanding the physics offibrillar collagen, would not be advantageous for the purposeof developing a hyperelastic model to be incorporated withina numerical model of soft biological tissues at the macro-scale. Consequently, it is not the wish of the authors to modelall the phenomena responsible for the hyperelastic constitu-tive behavior of collagen fibrils at the various length scales,but to capture the basic phenomena with a minimum of mi-crostructurally motivated parameters on the fibrillar level. Animportant criterion for the design of the constitutive model

is the convexity of the stress–stretch curve to ensure mate-rial and numerical stability for algorithms used in the finiteelement method.

Elastic constitutive models available for collagen fibrilsembedded in soft tissues can be essentially classified into twogroups: phenomenological and microstructural models. Dueto the good agreement of the nonlinear stiffening behaviorand the straightforward implementation into numericalalgorithms, exponential functions are the most commonlyused phenomenological models for collagen fibrils (Fung,1967; Holzapfel et al., 2000). However, model parameters ofphenomenological formulations have no direct physical ormicrostructural relevance.

Microstructural models are characterized by physiologicalparameters, which can be experimentally measured. Themain difference between existing microstructural models forcrimped collagen fibrils is the assumption of the wave shapeat the fibrillar level or at the molecular level. One groupof physiological models is based on statistical distributionsof the crimp shape or crimp behavior of collagen fibrils.Pioneering work in this class of model was performed by Lanir(1979, 1983). The idea of Lanir was that individual crimpedcollagen fibrils within a soft tissue have different lengthsso that for a given macroscopic stretch individual fibrils areundulated and stretched differently. Lanir used a statisticaldistribution in either the stretch of the fibrils or their length.Recently, Cacho et al. (2007) presented a model, where arandomly crimped morphology of individual collagen fibrilsis assumed. Traditionally applied to model the DNA doublehelix, the wormlike chain model (Kratky and Porod, 1949;Flory, 1969) was also successfully applied to describe theconstitutive behavior of crimped collagen fibrils (Bischoffet al., 2002; Garikipati et al., 2004; Kuhl et al., 2005). Incontrast to the majority of available constitutive models,the wormlike chain model is based on the entropy of thesubstructure of collagen fibrils at the molecular level. Thewave shape of a wormlike chain is characterized through asmooth curvature, the direction of which changes randomlybut in a continuous manner. However, Garikipati et al. (2008)pointed out that there are strong physical and physiologicalreasons to consider entropic elasticity as the wrong approachto adopt.

Another class of models assume a pre-defined planarshape of crimped collagen fibrils like sinusoidal or circularwave forms (Comninou and Yannas, 1976; Garikipati et al.,2008) or zig-zag shapes (Diamant et al., 1972; Kastelic et al.,1980; Stouffer et al., 1985) at the fibrillar level. Zig-zag modelsare based on kinematic linkages with rigid or flexible hingesand links.

In the presented work a pre-defined smooth three-dimensional wave form for the description of the crimpat the fibrillar level is assumed. Following ideas presentedby Beskos and Jenkins (1975) and Freed and Doehring (2005)the crimp is approximated by a cylindrical helix. Beskos andJenkins (1975) were the first to derive a constitutive modelfor crimped collagen fibrils based on a helical configurationof the wave shape. Their model predicts an infinite stiffnessat full extension due to the assumption of inextensiblefibrils, which means that the admissible stretch rangeis bounded. This unrealistic deficiency was eliminated in


524 J O U R N A L O F T H E M E C H A N I C A L B E H AV I O R O F B I O M E D I C A L M A T E R I A L S 2 ( 2 0 0 9 ) 5 2 2 – 5 3 3

the model proposed by Freed and Doehring, who definedthe stress–stretch relationship piecewise decomposed intoa nonlinear and a linear function. The function for thenonlinear region is basically identical to the model proposedby Beskos and Jenkins assuming fiber inextensibility fora helical spring under tension. The basic idea of Freedand Doehring was to force the stress function to becontinuous and differentiable at the transition point betweenthe nonlinear and linear region. Unfortunately, thesemathematical constraints change the original shape ofthe microstructurally motivated stress function, which, incontrast to the geometrically exact solution proposed in thepresent paper, affects the physical interpretation of theirmodel parameters. This is especially true in the case ofvery large deformations as assumed in the model of Freedand Doehring, who consider helical springs up to theirfull extension. Furthermore, higher derivatives of the stressfunction proposed by Freed and Doehring are not continuous,which might cause numerical instabilities in applications likefinite elements.

The present model is derived from the geometricalnonlinear solution of an extensible helical spring under anaxial force based on the Kirchhoff–Love hypotheses. Thephysical relevance of the model assumption is investigatedby comparing the microstructural parameters with directlymeasured values from experimental observations. Theproposed stress function is strictly convex and differentiablemultiple times. Accordingly, the respective stiffness responseis a priori a continuous function of the axial stretch includingcompressive states and the limit case of fully extendedfibrils. Considering the continuity of higher derivatives ofthe fiber stress function will likely add stability to numericalalgorithms.

Interactions between neighboring fibrils or between a fibriland the ground substance matrix are not considered in themodel.

The outline of the present paper is as follows: in Section 2 amathematical model for extensible helical springs subjectedto an axial force based on the classical Kirchhoff–Love rod-theory is presented. The geometrical nonlinear solution ofthe extensible helical spring problem is used in Section 3to formulate the constitutive response of crimped collagenfibrils, including the transition to a macroscopic strainenergy density function from which the tissue stress–stretchresponse can be obtained. The ability of the presented modelto easily fit and reproduce the stress–stretch response offibrous tissues under uniaxial tension is demonstrated inSection 4. Also the sensitivity and physical relevance ofthe respective model parameters is investigated. Section 5closes the paper with an extensive discussion on the modelframework and obtained results.

2. Mathematical model for extensible helicalsprings

2.1. Geometry of helical springs

The undeformed body B0 of the helical spring consideredhere is a curved rod with circular cross section A0 = πr

20. The

Fig. 2 – Reference and current configuration of the helicalfilament; kinematic variables.

position vector of a helical space curve C0 representing themid-line of B0 can be expressed by means of one curvilinearcoordinate s

0X (φ(s)) = R0 [cos(φ)i1 + sin(φ)i2 + cot(θ0)φi3] (1)

with φ(s) = s sin(θ0)/R0, where R0 is the radius (or amplitude)and θ0 is the crimp angle of C0 (see Fig. 2). The Cartesiandirection i3 represents the centerline of the helical curve. Forthe sake of simplicity, we consider only one helical revolutions ∈ [0, L0] in the reference configuration, where L0 is the arclength of the filament and H0 is the height (or wavelength)of the helix. Note that, in contrast to the force–displacementrelationship, the stress–stretch relationship of a elastic springdoes not depend on the number of helical revolutions or theabsolute length of the filament. For the geometrical variablesof the undeformed state the relation(2πR0L0

)2+

(H0L0

)2= sin2(θ0)+ cos2(θ0) = 1 (2)

holds. Let A3 = ∂0X /∂s be the unit tangent vector of the

undeformed curve and Ai an orthonormal base vector setdefined at each point of C0

A1 = −cos(φ)i1 − sin(φ)i2

A2 = cos(θ0) [sin(φ)i1 − cos(φ)i2]+ sin(θ0)i3

A3 = sin(θ0) [−sin(φ)i1 + cos(φ)i2]+ cos(θ0)i3

(3)

with φ(s) = s sin(θ0)/R0, where Aα are the principal axes ofthe cross section. Since the base vectors (3) of C0 form anorthonormal triad, the rate of change along the helical curveis given by

Ai,3 =∂Ai∂s= κ0 ×Ai, (4)

where κ0(s) = κ0i Ai(s) is a vector valued function of s. The

components of the initial curvature κ0α and twist κ03 of thehelical curve can be computed from (4) as

κ01 = −A2,3 ·A3 = A2 ·A3,3 = 0

κ02 = κ0 = −A1,3 ·A3 = A1 ·A3,3 =sin(θ0)

2

R0

κ03 = τ0 = −A2,3 ·A1 = A1,3 ·A2 =sin(θ0) cos(θ0)

R0.

(5)



For the present case the two non-zero curvature componentsare constant and the vector κ0 can be reduced by means of (5)to

κ0 = κ0A2+ τ0A

3=

sin(θ0)2

R0A2+

sin(θ0) cos(θ0)R0

A3. (6)

The basic assumption used in the present derivation isthat after deformation the crimp of the spring remains acylindrical helix with deformed values for the radius R, thecrimp angle θ, the height H and the length of the filamentL. Considering the curvilinear coordinate s to be convective,the geometry of the mid-line of the spring in the currentconfiguration C (Fig. 2) is defined by

0x (φ̄(s)) = R

[cos(φ̄)i1 + sin(φ̄)i2 + cot(θ)φ̄i3

](7)

with φ̄ = sλL sin(θ)/R, where λL is the stretch of the filament.Beside λL, the axial stretch λH and the torsional stretch λφ

around the helical axis i3 are introduced:

λL =LL0, λH =

HH0

, λφ = 1+∆φ

2π. (8)

∆φ is the twist angle defined in Fig. 2. Consideration of thetorsional stretch λφ is of importance since the extension orcompression of helical springs is always accompanied bytorsional deformations and vice versa. The axial stretch λH isused in Section 3 to accomplish the scale transition from themicromechanical model of crimped collagen fibrils as helicalsprings to a one-dimensional constituent at the tissue level.At the deformed configuration the geometrical relation (2)takes the form(2πRLλφ

)2+

(HL

)2= sin2(θ)+ cos2(θ) = 1. (9)

Since s is the arc length of the undeformed helical curve the

unit normal tangent vector of C can be given by a3 =0x,3/λL.

In accordance with (3) let ai be the orthonormal base vectorsof the deformed helical curve

a1 = −cos(φ̄)i1 − sin(φ̄)i2

a2 = cos(θ)[sin(φ̄)i1 − cos(φ̄)i2

]+ sin(θ)i3

a3 = sin(θ)[−sin(φ̄)i1 + cos(φ̄)i2

]+ cos(θ)i3

(10)

with φ̄ = sλL sin(θ)/R. Then the definition of κ

ai,3 =∂ai∂s= κ× ai (11)

can be used to identify the components of the currentcurvature and twist of C as

κ = κa2 + τa3 =sin(θ)2

Ra2 +

sin(θ) cos(θ)R

a3. (12)

2.2. Geometrically nonlinear solution of extensible helicalsprings

In this subsection the nonlinear relationship between theaxial force Paxial = Paxiali3 acting along the centerline of thehelix and the axial stretch λH is derived. This relation willbe used in the following section to define the constitutiveformulation of crimped collagen fibrils. The presentedderivation is based on the classical theory of thin rods at finitedisplacements as developed by Kirchhoff (1859) and Clebsch

Fig. 3 – Resultant forces and moments acting along thehelical filament and along the centerline of the helix.

(1862) and presented by Love (1892). A helical spring can becalled a thin rod when α =max{r0/L0, r0/κ0, r0/τ0} � 1, whichholds for collagen fibrils. The Kirchhoff–Love hypothesesstates that (i) the cross section remains plane and normal tothe mid-line of the rod, (ii) the transverse stresses are zeroand (iii) the bending and twisting moments are proportionalto the components of curvature and twist of the mid-lineC . Dill (1992) showed that Kirchhoff’s theory of rods canbe derived from the framework of continuum mechanics byneglecting terms of order O(α2).

Let F = Q1a1 + Q2a2 + Na3 be the resultant force andM = Miai the resultant moment vectors of the helical spring.Due to the symmetry of the helical spring the bendingmoment M1 = M1a1 does not exist. Furthermore, if externaldistributed forces andmoments are not applied on the spring,the shear force Q1 = Q1a1 is also zero and F, M are constant.

The constitutive equations for the generalized Bernoullitheory are

N = EA0(λL − 1) = Eπ r20(λL − 1)

M2= EI2(κ− κ0) =

Eπ r404

(κ− κ0)

M3= GI3(τ − τ0) =

Eπ r406

(τ − τ0),

(13)

where incompressibility (ν = 0.5) is imposed for the materialbehavior of the spring. Note, that there is no constitutiverelation for the shear force Q2 = Q2a2, which can bedetermined by the balance of momentum as in the lineartheory of bending rods.

The six equations of local equilibrium presented in Love(1892) reduce to the two expressions shown below at thefilament level:

N κ− Q2 τ = 0

M3 κ−M2 τ − Q2= 0.

(14)

After transformation to the centerline of the helical curve, thenon-zero resultant forces and moments of F and M can beexpressed by means of the axial force Paxial = Paxiali3 and theaxial moment Maxial = Maxiali3. From Fig. 3 and considering(14)1 the resulting forces and moments at the centerline ofthe helical curve reduce to

Paxial = (N cos(θ)+ Q2 sin(θ))i3 =N

cos(θ)i3

Maxial = (M3 cos(θ)+M2 sin(θ)+NR sin(θ)− Q2R cos(θ))i3

= (M3 cos(θ)+M2 sin(θ))i3.

(15)



To derive an exclusive relationship between the axial stretchλH and Paxial a further assumption for either λφ or Maxialis needed. In several fibrous tissues the matrix material isessentially an aqueous, highly compliant gel-like substance(also called ground substance). We assume that crimpedcollagen fibrils embedded in such a matrix can rotate freelyaround the centerline of the helical curve, which impliesMaxial = 0. This assumption conforms with the hypothesisthat the axial force is the only load carried by individualcollagen fibrils. Consequently, extension or compression ofsuch fibrils is accompanied by torsional deformations aroundthe axis i3 (λφ 6= 1). However, for helical springs under tensionwith crimp angles θ0 < 30◦ the torsional deformations λφare small and the assumption Maxial = 0 has a negligibleinfluence on the axial stretch–force relationship. ConsideringMaxial = 0 in (15)2 the relation between M3 and M2 in theequilibrium state can be found

M3 cos(θ)+M2 sin(θ) = 0. (16)

Inspired by the work of Jiang et al. (1989), who introducedtwo auxiliary strain variables for their solution of thehelical spring boundary problem, two auxiliary stretches areintroduced here

λ̂H =λHλL, λ̂φ =

λφ

λL. (17)

The advantage of (17) is that all geometrical variables in (14)and (16) can be directly substituted:

cos(θ) = λ̂H cos(θ0)

sin(θ) =√1− λ̂H2 cos2(θ0)

τ = τ0λ̂Hλ̂φ

κ = κ0λ̂φsin(θ)sin(θ0)

.

(18)

Inserting the constitutive equation (13)2,3 together with (18)into (16) the auxiliary stretch λ̂φ can be written as a functionof λ̂H

λ̂φ(λ̂H) =cot(θ)

λ̂H cot(θ0)

3+ 2 cot(θ) cot(θ0)

3+ 2 cot2(θ)(19)

with cot(θ) being a nonlinear function of λ̂H according to(18)1,2. The equilibrium equation (14) is used together withthe constitutive equation (13) and relations (18) to identify thestretch λL as a function of both auxiliary stretches λ̂H and λ̂φ:

λL(λ̂H, λ̂φ) = 1+14

(r0R0

)2sin2(θ0) cos

2(θ0)λ̂Hλ̂φ

×

(cot(θ)cot(θ0)

−2+ λ̂Hλ̂φ

3

). (20)

By repeating the definition of the auxiliary stretches (17) andconsidering (19) and (20) the stretch λH can be expressed as afunction solely of λ̂H

λH(λ̂H) = λ̂H λL

(λ̂H, λ̂φ(λ̂H)

). (21)

Note that (21) represents a multiplicative decompositionof the axial stretch λH into uncrimping λ̂H and filamentstretching λL components. Unfortunately, Eq. (21) is highlynonlinear with respect to λ̂H such that the inverse relationλ̂H(λH) can only be solved by an iterative procedure. A

Table 1 – Newtonian algorithm for iterative solution ofλ̂H(λH).

If λH = 1 then λ̂H = 1 elseλ̂0H = 1Do while µ > tol

cot(θn) =λ̂nH cos(θ0)√

1−(λ̂nH)2 cos2(θ0)

λ̂nφ=

cot(θn)λ̂nH cot(θ0)

·3+2 cot(θn) cot(θ0)

3+2 cot2(θn)

λnL = 1+ 14

(r0R0

)2sin2(θ0) cos

2(θ0)λ̂nHλ̂

nφ

(cot(θn)cot(θ0)

−

2+λ̂nH λ̂nφ

3

)

(λnH)′= 1+ 1

4

(r0R0

)2sin2(θ0) cos

2(θ0)λ̂nHλ̂

nφ(

4 cot(θn)cot(θ0)

+cot3(θn)cot(θ0)

−

6+5λ̂nH λ̂nφ

3

)λ̂n+1H = λ̂nH −

λ̂nHλnL−λH

(λnH)′

µ =|λ̂n+1H −λ̂nH|

|λ̂n+1H −λ̂0H|

If λ̂n+1H > λlock then λ̂n+1H = λlock − (λlock − 1)10−n

with λlock =1

cos(θ0)

n++End do

End if

Newtonian algorithm is presented in Table 1 for the iterativesolution of λ̂H(λH).

To briefly discuss the nonlinear shape of the functionλ̂H(λH) let us introduce the locking stretch λlock at which aninextensible spring (λL = 1) would be fully extended

λlock =L0H0=

1cos(θ0)

. (22)

For models assuming the inextensibility of fibrils the lockingstretch represents the limiting value of the stretch in the axialdirection λH (Beskos and Jenkins, 1975). In the present caseconsidering extensible fibrils the locking stretch representsthe limiting horizontal asymptote of the auxiliary stretchfunction λ̂H(λH) (see the diagram in Table 1)

limλH→∞

λ̂H(λH) = λlock. (23)

A more detailed interpretation of the locking stretch and itsimportant implication for the present model is presented inSection 3.

To provide a generally valid constitutive framework theiterative solution of λ̂H(λH) in Table 1 starts for any λH withthe initial value λ̂0H = 1. Considering the convexity of thefunction λ̂H(λH) together with (23) the solution of the iterativeprocedure in Table 1 is guaranteed even for large values of λHfor any helical curve geometry. In case of a value λ̂n+1H > λlockin iteration step n + 1 this value is reset to a recovery valuewithin the admissible range 0 < λ̂H < λlock. The recoveryterm λ̂n+1H = λlock−(λlock−1)10

−n tends to λlock for increasingiteration steps.

After a successful iterative identification of λ̂H(λH) theaxial force can finally be computed by considering the identity(15)1 together with (18)1 and the constitutive equation (13)1 as

Paxial(λH) =Eπr20(λH − λ̂H)

λ̂2H cos(θ0)i3 (24)

with λ̂H = λ̂H(λH).



Fig. 4 – Reference configuration of a crimped collagen fibrilwith fiber direction e0.

3. Constitutive formulation of crimped colla-gen fibrils as helical springs

At the tissue level a single collagen fibril is considered as aone-dimensional constituent with a distinct fiber directionin the reference configuration defined through a unit normalvector e0 (see Fig. 4). For the transition from the helical springproblem presented in the previous section to the tissue levelthe fiber direction e0 is assumed to coincide with the axialdirection i3 in Fig. 2. We assume that collagen fibrils andthe surrounding matrix undergo affine deformations in fiberdirection. Consequently, the fiber stretch at the tissue level

λf =√IVC =

√

C : A =√e0Ce0 (25)

corresponds to the axial stretch of the cylindrical helix λH,where C = FTF is the right Cauchy–Green strain tensor, A =e0⊗e0 a structure tensor and IVC the fourth pseudo-invariantof C and A. The first Piola–Kirchhoff fiber stress of individualcollagen fibrils is defined through (24) as

Pf(λf) =i3 · Paxial(λH)|λH=λf

πr20 cos(θ0)= E

λf − λ̂H

λ̂H2 cos2(θ0)(26)

with λ̂H = λ̂H(λH)|λH=λf according to Table 1, where πr20 cos(θ0)is the projection of the undeformed area A0 of the filamentto the plane perpendicular to the fiber direction e0. In viewof (26) and the terms in Table 1 the presented constitutiveformulation contains three independent model parameters:one material parameter, the elastic modulus of the filamentE; and two microstructural parameters, the crimp angle θ0and the ratio of the amplitude of the helix to radius of thefilament cross-section R0/r0. The microstructural parametersθ0 and R0/r0 have physical meanings and can be measuredexperimentally, e.g. from SEM photographs.

Let the strain energy density W stored in a fibrous softtissue be composed of an isotropic and anisotropic partrepresenting the energy contribution of the ground substance(g) and n collagen fibril families (f)

W(C,Ai) =Wg(C)+n∑

i=1

NiNtot

Wfi(C,Ai), (27)

where Ntot is the total number of fibrils per referencevolume and Ni is the number of fibrils of family i witha given orientation and identical constitutive properties.Several hyperelastic models are suitable for describing theisotropic part Wg of the additive split in (27) (Holzapfel,2000), where due to its simplicity the neo-Hookean modelWg(IC) = c(IC − 3) with IC = tr(C) is often used inbiomechanics of soft tissues (Holzapfel et al., 2000). In the

present paper the anisotropic part Wf representing the strainenergy contribution of a single crimped collagen fibril isdefined as the integral over the fiber stress–stretch curve

Wf(IVC) =

∫ λf=√IVC

1Pf(λ)dλ (28)

with Pf(λ) according to (26). The definition (28) of Wf, whichhas also been used in Einstein et al. (2006), has the advantagethat Pf can be an arbitrary function of λ, where no closedform representation of Pf(λ) is needed. Note that a closedform representation of the presented fiber stress function (26)can only be provided by introducing further assumptions intothe helical spring model such as the inextensibility of thefilament λL = 1 (Beskos and Jenkins, 1975).

In order to put the focus on the presented helical springmodel representing the crimped collagen fibrils, the strainenergy contribution Wg associated with the surroundingextrafibrillar matrix material in (27) is omitted and onlyone family of parallel organized collagen fibrils (n = 1)with identical material and microstructural properties isconsidered for the numerical examples in the subsequentsection. In general, the ground substance in fibrous tissueshas a negligible tensile stiffness in comparison with collagenfibrils, which justifies the above assumption. Uncoiledcollagen fibrils have a Young’s modulus of the order of 1.0GPa along the fibril direction, whereas the Young’s modulusand shear modulus of the ground substance is of the orderof 10 kPa (Pinsky and Datye, 1991). Considering only onefamily of collagen fibrils in (27) restricts the numericalinvestigations presented in Section 4 to tissues with almostparallel organized collagen fibrils. In this case Wf willrepresent the strain energy density of an average collagenfibril representing the whole collagen network of the tissue.

For a successful implementation of the presented modelinto a finite element environment the first and secondvariation of (28) is needed. Considering the principle of virtualwork at the reference configuration, the first variation of (28)can be written as

δWf =12Sf : δC with Sf = 2

∂Wf∂C=

Pf(λf)λf

A, (29)

where Sf is the second Piola–Kirchhoff stress tensorof individual collagen fibrils. Accordingly, the directionalderivative of (29) reads

∆δWf =14∆C : Cf : δC+

12Sf : ∆δC (30)

with

Cf = 4∂2Wf∂C∂C

=1

λ2f

(Cf(λf)−

Pf(λf)λf

)A⊗A, (31)

where Cf(λf) = ∂Pf/∂λf is the fiber stiffness.For the derivation of Cf let us consider Pf and Mf =

|Maxial|/(πr20 cos(θ0)) to be functions of the axial stretch λH

and the torsional stretch λφ. For Mf = Mf(λH, λφ) = 0 andPf = Pf(λH, λφ) the fiber stiffness after static condensationreads

Cf =∂Pf∂λH−∂Mf∂λH

(∂Mf∂λφ

)−1 ∂Pf∂λφ

. (32)



Table 2 – Calculation of fiber stiffness Cf(λf).

λ̂H = λ̂H(λH)|λH=λfaccording to Table 1

cot(θ) =λ̂H cos(θ0)√

1−λ̂2H cos2(θ0)

λ̂φ =cot(θ)

λ̂H cot(θ0)3+2 cot(θ) cot(θ0)

3+2 cot2(θ)

∂λH∂λ̂H= 1+ 1

4

(r0R0

)2sin2(θ0) cos

2(θ0)λ̂Hλ̂φ[cot(θ)cot(θ0)

(cot2(θ)

λ̂2H cos2(θ0)+ 2

)−

4+3λ̂H λ̂φ3

]∂λH∂λ̂φ=

14

(r0R0

)2sin2(θ0) cos

2(θ0)λ̂2H

[cot(θ)cot(θ0)

−2+2λ̂H λ̂φ

3

]∂λφ

∂λ̂H=

14

(r0R0

)2sin2(θ0) cos

2(θ0)λ̂2φ[

cot(θ)cot(θ0)

(cot2(θ)

λ̂2H cos2(θ0)+ 1

)−

2+2λ̂H λ̂φ3

]∂λφ

∂λ̂φ= 1+ 1

4

(r0R0

)2sin2(θ0) cos

2(θ0)λ̂Hλ̂φ

[2 cot(θ)cot(θ0)

−4+3λ̂H λ̂φ

3

]det(D) =

∂λH∂λ̂H

∂λφ

∂λ̂φ−∂λH∂λ̂φ

∂λφ

∂λ̂H

∂Pf∂λ̂H=

14

(r0R0

)2Eλ̂φ

[cot3(θ)

cot3(θ0)λ̂3H− sin2(θ0)

λ̂φ3

]∂Pf∂λ̂φ=

14

(r0R0

)2E sin2(θ0)

[cot(θ)cot(θ0)

−2+2λ̂H λ̂φ

3

]Cf =

1det(D)

[∂Pf∂λ̂H

∂λφ

∂λ̂φ−

∂Pf∂λ̂φ

∂λφ

∂λ̂H

]

Neglecting the second term in (32), which is much smallerthan the first one, and applying the chain rule Eq. (32) is recastto

Cf =∂Pf∂λ̂H

∂λ̂H∂λH+∂Pf∂λ̂φ

∂λ̂φ

∂λH

=1

det(D)

[∂Pf∂λ̂H

∂λφ

∂λ̂φ−∂Pf∂λ̂φ

∂λφ

∂λ̂H

](33)

with

det(D) =∂λH

∂λ̂H

∂λφ

∂λ̂φ−∂λH

∂λ̂φ

∂λφ

∂λ̂H. (34)

All required terms to compute Cf from (33) for a given valueof λf are summarized in Table 2. Thus, the fiber stress Pf andthe fiber stiffness Cf can be determined as functions of thefiber stretch λf = λH via (26) and Table 2, respectively, aftercomputing λ̂H from Table 1.

According to the limiting value of the auxiliary stretch λ̂H(23) the fiber stress function (26) converts to a linear functionat the limit state of fully extended fibrils

Pf(λf)|λf→∞ = E(λf − λlock) (35)

while the fiber stiffness (33) tends towards the constantelastic modulus of the filament

Cf(λf)|λf→∞ = E. (36)

The linear function (35) represents the inclined asymptoteto which the stress–stretch curve converges at the higherstretch level. The locking stretch (22) represents theintersection point of the asymptote with the abscissa in astress–stretch diagram. Parameter studies presented in thesubsequent section demonstrate that the locking stretch alsocharacterizes the axial stretch level λH = λlock at which

the curvature of the fiber stiffness–stretch curve changesthe sign. Accordingly, the locking stretch λlock represents acharacteristic quantity of the present constitutive framework.

It should be noted that for model parameters E > 0,R0/r0 > 1 and 0◦ < θ0 < 90◦ the strain energy function of thehelical spring model (28) represents a strictly convex functionincluding compressive states and the almost linear regionof fully extended fibrils, which is important to ensure bothmaterial stability and numerical stability for algorithms usedin the finite element method. In the undeformed state λf = 1the fiber stiffness (33) simplifies to

Cf|λf=1 = E2+ cos2(θ0)

12(R0/r0)2 + cos2(θ0)+ 2 cos4(θ0), (37)

where Cf|λf=1 ≤ 0.2E for any admissible set of modelparameters. Under compression, λf < 1, the fiber stiffnessCf decreases further to the mathematical limit case of fullcompression

Cf|λf→0 = E sin4(θ0)6(R0/r0)

2+ sin2(θ0) cos

2(θ0)

36(R0/r0)4. (38)

It is unclear if collagen fibrils embedded in soft tissues canbear load under compression. However, the stiffness responseof the presentedmodel is very soft in these states without lossof convexity.

4. Numerical examples

In this section the dependence of the presented helical springmodel for crimped collagen fibrils on the respective modelparameters is investigated. The ability of the proposed modelto reproduce the mechanical response of rat tail tendons,porcine corneas and bovine Achilles tendons under uniaxialextension is demonstrated.

There are two geometrical parameters involved in theconstitutive formulation presented in (26), Table 1 and Table 2:the crimp angle θ0 and the ratio R0/r0. In Figs. 5 and 6,respectively, the dependence of the fiber stress Pf and fiberstiffness Cf on the geometric model parameters θ0 and R0/r0are illustrated. From (22) the direct influence of the crimpangle on the locking stretch λlock = 1/ cos(θ0) is evident. Thus,for increasing values of θ0 the locking stretch increases andthe stress–stretch curve as well as the stiffness–stretch curveare stretched along the abscissa (see left diagrams in Fig. 5).However, the normalized diagrams on the right-hand side ofFig. 5 illustrate that the parameter θ0 has almost no influenceon the overall shape of the stress and stiffness function. Forthe limit case θ0 → 0◦ the constitutive model predicts an apriori fully extended spring (λlock = 1) with linear stiffnessE, while for θ0 → 90◦ the locking stretch moves to infinity(λlock = ∞).

In contrast to θ0, the parameter R0/r0 changes the shapeof the function Pf(λf) and Cf(λf) as can clearly be seen inthe normalized diagrams on the right-hand side of Fig. 6.For increasing values of R0/r0 the belly of the stress functionPf takes a more pronounced shape at the locking stretch(see upper diagrams in Fig. 6). Accordingly, the shape of thestiffness function Cf is rotated clockwise around Cf(λlock) forincreasing values of R0/r0 (see lower right diagram in Fig. 6).



Fig. 5 – Dependences of fiber stress Pf and fiber stiffness Cf on the crimp parameter θ0 with E = 400 MPa and R0/r0 = 2 fixed.

For the limiting case of R0/r0 → ∞ the stiffness functiontends to the shape of a jump function with stiffness valuesCf(λf) = Cf(1) for λf < λlock and Cf(λf) = E for λf > λlock.According to the lower right diagrams in Figs. 5 and 6 thelocking stretch characterizes the axial stretch level λH = λlockat which the curvature of the stiffness–stretch curve changesthe sign.

The ability of the proposed model to reproduce thebiomechanical response of fibrous tissue is demonstratedfor uniaxial extension tests of fascicles from rat tailtendons performed by Hansen et al. (2002). The predictedstress–stretch response of the helical spring model is inexcellent agreement with the experimental data presented inFig. 7, where the model parameters have been identified bymeans of the nonlinear Levenberg–Marquardt algorithm asE = 476 MPa, θ0 = 13.88◦, and R0/r0 = 3.05. This algorithmoptimizes the model parameters such that the sum of squareresiduals is minimized

χ2 =nd∑i=1

[Pexp,i − Pf(λexp,i)

]2, (39)

where nd is the number of data points and Pexp,i and λexp,iare the experimentally obtained stress and stretch values forthe ith data record, respectively. The quality of the optimizedparameters can be measured by means of the root meansquare error

ε =

√χ2

nd−np

Prefwith Pref =

1nd

nd∑i=1

Pexp,i, (40)

where np is the number of fitted parameters. An error ofε = 0.03 has been obtained for the data fit in Fig. 7. InFig. 8 the objective function (39) is plotted for varying valuesof the geometrical parameters θ0 and R0/r0, illustrating thatχ2 is considerably more sensitive with respect to the crimpangle θ0 compared to R0/r0. It should be noted that forthe three independent model parameters E, θ0 and R0/r0the objective function (39) is characterized by only one(global) minimum (see Fig. 8), which will most likely befound by a gradient based optimization algorithm like thenonlinear Levenberg–Marquardt algorithm. Accordingly, forexperimental stress–stretch curves which include the linearregion, a unique optimal parameter set can be easily foundsince the residual function does not exhibit local minima. Ifthe experimental data do not include the linear region, thesensitivity of the function (39) decreases with respect to themodel parameters E and θ0. In this case, it might be advisableto estimate the crimp angle parameter directly from the fibrilmicrostructure of the investigated tissue.

Note that the elastic modulus E as well as the crimpangle θ0 can be directly extracted from the shape of theexperimentally obtained stress–stretch curve. To this end, theasymptotic line to which the experimental data converge athigher stretch levels of (almost) fully extended fibrils has tobe specified (see the doted line in Fig. 7). Then, the elasticmodulus E of the helical spring model can be obtained fromthe slope of the asymptote. The intersection point of theasymptote with the abscissa represents the locking stretchλlock of the presented model, which can be used to identifythe crimp angle θ0 through identity (22).



Fig. 6 – Dependences of fiber stress Pf and fiber stiffness Cf on the parameter R0/r0 with E = 400 MPa and θ0 = 10◦ fixed.

Fig. 7 – A fit of the helical spring model to data taken fromuniaxial extension experiments on fascicles from rat tailtendons (Hansen et al., 2002).

As a second example Fig. 9 contains a comparisonbetween results from strip extensiometry experiments onporcine corneas performed by Anderson et al. (2003) andmodel results. The respective model parameters have beenidentified as E = 27 MPa, θ0 = 27.52◦ and R0/r0 = 1.94with an error value of ε = 0.02. Also for this example,the proposed model replicates the typical “J-type” shape ofthe stress–stretch curve in excellent agreement with theexperimental observations.

Sasaki and Odajima (1996) investigated the elongationmechanism of collagen fibrils in bovine Achilles tendons at

Fig. 8 – The sum of square residuals χ2 for varyingparameters θ0 and R0/r0 with E = 476 MPa fixed.

different levels of the structural hierarchy. In Fig. 10 theexperimentally obtained stretch values of a tendon, of anembedded fibril and of a collagen molecule are plotted forvarying values of the first Piola–Kirchhoff stressexternallyapplied to the specimen. The solid line in Fig. 10 representsthe fit of the helical crimp model to the experimentaldata obtained for the tendon, which is again in excellentagreement with the experimental data. The respective modelparameters have been optimized to E = 426 MPa, θ0 = 12.82◦,and R0/r0 = 2.15 with an error of ε = 0.03. To investigatethe physical relevance of the proposed model the stretch ofthe filament λL defined in (8) using the above parametersis plotted for varying values of the fiber stress Pf (brokenline in Fig. 10). Obviously, the stress–stretch curve Pf –λL at



Fig. 9 – A fit of the helical spring model to data taken fromstrip extensiometry experiments on porcinecorneas (Anderson et al., 2003).

Fig. 10 – Comparison of stress–stretch curves of the helicalspring model for the crimped fibril (solid line) and thefilament (broken line) to experimental data obtained for atendon (©), a collagen fibril (�) and a collagen molecule(•) (from Sasaki and Odajima (1996)).

the filament level of the helical crimp model correspondsvery well to the stress–stretch data experimentally measuredby Sasaki and Odajima for a collagen fibril. In accordance withthe experimental observations, the slope of the broken lineEL = 437 MPa is slightly higher than the optimized elasticmodulus parameter E = 426 MPa representing the slope of thesolid line in the linear region. Note that according to (17)1, (22)and (26) the ratio between the predicted elastic moduli EL/Erelates to the locking stretch at the limit state of full extendedfibrils

ELE

∣∣∣∣λH→∞

=λHλL

∣∣∣∣λH→∞

=1

cos(θ0)= λlock. (41)

5. Discussion

A physiologically motivated constitutive formulation hasbeen proposed for the hyperelastic response of crimpedcollagen fibrils. Assuming a helical form of the fibrils,the model has been derived from the nonlinear axialforce–stretch relationship of an extensible helical spring. Only

one material parameter (the elastic modulus of the fibrils)and two microstructural parameters (the crimp angle θ0 andthe ratio R0/r0 between the radii of the helical wave formand of the fibril cross section) are required. Consideringlarge deformations, an iterative algorithm has been presentedfor the straightforward implementation of the nonlinearrelations between the fiber stress and stiffness and the fiberstretch into a numerical code. The proposed strain energyfunction is strictly convex, includes compressive states andallows for a smooth transition from a crimped to a fullyelongated state of collagen fibrils. Themodel prediction of thestress–stretch response for rat tail tendon fascicles, porcinecornea strips and bovine Achilles tendons is in excellentagreement with experimental observations.

To examine the physiological relevance of the proposedmodel the predicted model parameters are compared todirectly measured values of the microstructure of rat tailtendons. Diamant et al. (1972) measured crimp angles of12.5◦ and 15.4◦ for undulated fibrils of tendons taken from 29and 13 month old rats, respectively. The optimal crimp angleparameter of 13.88◦ predicted by the helical spring model forthe experimental data presented in Fig. 7 lies well within therange of those directly measured values.

If the assumption holds that each collagen fibril of afascicle acts separately without interaction with other fibrilsor other constituents, the helical spring filament of thepresented model would represent an average collagen fibril.An electron-microscopic study performed by Gotoh and Sugi(1985) revealed an average diameter of 318 ± 12 nm ofcollagen fibrils in rat tail tendons. From the experimentalobservationsmade by Diamant et al. (1972) a crimp amplitudeof approximately 12.5 µm can be deduced. Thus, for anaverage collagen fibril in rat tail tendons the experimentallymeasured ratio R0/r0 has an approximate value of 80.However, the helical spring model predicted a much smallervalue of 3.05 for the respective model parameter. Note thatthe parameter R0/r0 has nothing to do with the fibril aspectratio, which is a different geometric property of collagen thatthis model does not take into account.

As the parameter R0/r0 is responsible for the nonlinearshape of the stress–stretch curve (see the interpretation ofFig. 6) the physical relevance of this parameter is stronglyinfluenced by the assumptions used in the present derivation.Due to their hierarchical substructure, collagen fibrils are notisotropic continua and might not exhibit a rod-like bendingor torsion stiffness as assumed for the derivation of thepresented model. Buehler and Wong (2007) estimated bymeans ofmolecular dynamics simulations a bending stiffnessof EI2 = 1.247×10−29 Nm2 for single tropocollagenmolecules.It was noted that the tensile response of collagen moleculesis first dominated by entropic elasticity followed by energeticelasticity. Buehler and Wong also predicted that the bendingstiffness increases significantly after assembly into a fibril,thus reducing the significance of entropic elasticity, whilethe tensile stiffness decreases continuously with increasingnumber of molecules. In this case, collagen fibrils mostlikely exhibit bending and torsional stiffnesses as assumedhere. However, the ratio between the bending or torsionalstiffnesses and the tensile stiffness might be very differentfrom what we know for elastic rods.



Furthermore, the nonlinear elastic response of crimpedcollagen fibrils in the “heel” region of the stress–stretchcurves is more likely to be dominated by the reduction ofentropy at the molecular gap level (Sasaki and Odajima, 1996;Fratzl et al., 1997) and not by a rod-like behavior of collagenfibrils. This implies that the direct incorporation of measuredvalues R0/r0 into the model would require an enhancementto consider the hierarchical substructure of collagen fibrilsincluding lower scale phenomena such as the mentionedreduction of kinks in the molecular gap region. According tothemacroscopic mode of description adopted in the proposedmodel, the parameter R0/r0 has to be determined indirectlyas described before, sacrificing, however, its direct physicalinterpretation.

The assumption that the stress behavior of a fascicle canbe determined by the geometric properties of an averagefibril might also not hold. Phenomena like fiber recruitmentor other interactions between neighboring fibrils might playan important role in understanding the microstructuralrelevance of R0/r0 at the tissue level. In fact, collagen fibrilscan interact with neighboring fibrils due to proteoglycans,which may link fibrils together (Liao and Vesely, 2003). Iflinked fibrils “act” together when the tissue is stretched, aneffective radius reff0 , which might be much larger than thecross section radius r0 of a single fibril, should be used forthe microstructural model parameter R0/r

eff0 .

Note that the physical relevance of the parameters θ0 and Eis not influenced by assuming a rod-like behavior for collagenfibrils. The crimp angle parameter solely describes the stretchvalue at which fibrils lock (see the interpretation of Fig. 5),which might result from the wave form of the fibril itselfas well as from the entropy at the molecular gap region.If the contribution to the locking stretch from straighteningmolecular kinks is small compared to the uncrimping ofthe fibril itself, the model parameter θ0 can be estimated bythe physiological crimp angle of the fibrils as shown herefor the rat tail tendon. The elastic modulus parameter Edescribes the stiffness of the fibril in the linear region of thestress–stretch curve, which might result from stretching thecollagen molecule or from gliding of neighboring molecules.

The excellent fit of the helical spring model to theexperimental stress–stretch data throughout the wholestretch range and the good agreement of the optimal crimpangle parameter θ0 with experimentally observed valuessupport the physical relevance of the model. Also, the goodagreement of the model elongation response at the filamentlevel with experimental data from bovine Achilles tendonscorroborates the physical relevance of two of the modelparameters (θ0,E) of the proposed model.

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