Fatigue of soft fibrous tissues: Multi-scale mechanics and constitutive modeling

Acta Biomaterialia xxx (2018) xxx–xxx

Contents lists available at ScienceDirect

Acta Biomaterialia journal homepage: www.elsevier.com/locate/actabiomat

Full length article

Fatigue of soft fibrous tissues: Multi-scale mechanics and constitutive modeling Kevin Linka ⇑, Markus Hillgärtner, Mikhail Itskov Department of Continuum Mechanics, RWTH Aachen University, Kackertstr. 9, 52072 Aachen, Germany

a r t i c l e

i n f o

Article history: Received 21 November 2017 Received in revised form 21 February 2018 Accepted 5 March 2018 Available online xxxx Keywords: Multi-scale modeling Soft fibrous tissues Fatigue

a b s t r a c t In recent experimental studies a possible damage mechanism of collagenous tissues mainly caused by fatigue was disclosed. In this contribution, a multi-scale constitutive model ranging from the tropocollagen (TC) molecule level up to bundles of collagen fibers is proposed and utilized to predict the elastic and inelastic long-term tissue response. Material failure of collagen fibrils is elucidated by a permanent opening of the triple helical collagen molecule conformation, triggered either by overstretching or reaction kinetics of non-covalent bonds. This kinetics is described within a probabilistic framework of adhesive detachments of molecular linkages providing collagen fiber integrity. Both intramolecular and interfibrillar linkages are considered. The final constitutive equations are validated against recent experimental data available in literature for both uniaxial tension to failure and the evolution of fatigue in subsequent loading cycles. All material parameters of the proposed model have a clear physical interpretation. Statement of significance Irreversible changes take place at different length scales of soft fibrous tissues under supra-physiological loading and alter their macroscopic mechanical properties. Understanding the evolution of those histologic pathologies under loading and incorporating them into a continuum mechanical framework appears to be crucial in order to predict long-term evolution of various diseases and to support the development of tissue engineering. Ó 2018 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

1. Introduction The last two decades came along with significantly increasing attention to the continuum biomechanics of growth and remodeling of soft tissues, necessary e.g. to describe the continuing enlargement of aneurysms [13]. Along with growth and remodeling effects, such modeling techniques require the proper description of cell turnover, in turn, to account for the description of e.g. dilatation due to aging or hypertension. It has been observed that the collagen half-life decreases from 60–70 days down to 16 days due to hypertension [62]. So far, the research has only focused on phenomenological descriptions of collagen turnover (e.g. [25,13]) without any microstructural reasoning. Only few studies have addressed the

Abbrevations: CP, Core protein; ECM, Extracellular matrix; (e)WLC, (extensible) worm-like chain; GAG, Glycosaminoglycan; PG, Proteoglycan; IFM, Interfibrillar matrix; SDF, Small diameter fibril; TC, Tropocollagen. ⇑ Corresponding author. E-mail address: [email protected] (K. Linka).

issue of long-term fatigue modeling of soft fibrous tissues [56,57]. Despite the ability of the proposed models to capture the fatigue enforced stress reduction, the link between damage factors and the histology has not been revealed. Furthermore, tendon injuries are a common issue in modern medicine [53,80] and their purposive treatment is of high importance especially for athletes and the elder generation. It is well known that tendons without pathological findings are less prone to rupture and tendon injuries are mostly accompanied by a changed histology [53,43]. Hence, predictions by a histologically based material model can further strengthen the scientific basis towards better understanding of the long term behavior of soft fibrous tissues as well as the etiology of tendon diseases. Moreover, patient-specific biomechanics has attracted considerable interest, and enormous effort has been dedicated towards understanding of fibrous tissues mechanics. Experimental procedures reported in literature are ranging from macromolecular tests by applying e.g optical/magnetical tweezers or Atomic force microscopy (AFM) to estimate the force-extension behavior [70,29] up to the measurement of collagen fiber dispersion

https://doi.org/10.1016/j.actbio.2018.03.010 1742-7061/Ó 2018 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Please cite this article in press as: K. Linka et al., Fatigue of soft fibrous tissues: Multi-scale mechanics and constitutive modeling, Acta Biomater. (2018), https://doi.org/10.1016/j.actbio.2018.03.010

2

K. Linka et al. / Acta Biomaterialia xxx (2018) xxx–xxx

by (polarized) light microscopy in whole tissues/organs (e.g. [26,18]). These procedures deliver valuable input for a multi-scale based constitutive model. Indeed, biological tissues are very patient-specific in their mechanical properties due to their varying micro-structural parameters. For example, their stress response can vary between different individuals by an order of magnitude [38]. Hence, the material model proposed in the present paper aims to incorporate structural and mechanical information at different length-scales accessible by minimal-invasive testing. Due to the complex hierarchical arrangement of tendon tissue and its main constituent, collagen, the damage progression of the biofilament network represents a versatile interplay between different length scales. Histological changes may be triggered by either biochemical or mechanobiological sources. The latter ones are often related to a dynamic loading of the tissue and mostly involve a high number of loading cycles. Recent experimental investigations under cyclic loading protocols indicate that tendon fatigue is a local damage accumulation on the micro-level, which starts even under physiological loading conditions [23,22,90]. The main underlying damage mechanism is considered to be a permanent opening of the triple helical collagen molecules, which leads to irreversible kinking of collagen fibrils and widening of the interfiber space under long-term loading [21,87]. This mechanism was further experimentally confirmed by utilizing hybridizing peptide, which binds to unfolded TCs, while applying a mechanical overload on rat tail tendon [93]. While the fibrillar damage mainly takes places at high fatigue stages, early fatigue was observed to correlate with an irreversible damage of the interfibrillar matrix (IFM) [83,47]. This was detected by a reduction of interfibrillar sliding, which can be quantified by a debonding between small diameter fibrils (SDFs) and proteoglycans (PGs) surrounding collagen fibrils [83,47,84]. In summary, tendon evolves under cyclic loading as follows. At the low fatigue phase, uncrimping of the collagen fiber network starts due to interfibrillar damage and results in a stiffer loading response, while during the moderate fatigue phase local fiber angulations initiate. Finally, the high fatigue phase is characterized by rupturing of whole collagen fiber layers and ends up with the complete tendon failure [21–23,36,47]. Although, many aspects of tendon fatigue have already been discussed from the experimental point of view [65,67,92,77,76,79], there is still a lack of an integrated material model capturing the

alteration from the physiological to the pathological state, while linking them to the tendon histology. To the best of our knowledge, such a fatigue model for soft fibrous tissues has not so far been addressed. Thus, the present work aims to close this gap by introducing a multi-scale constitutive model including damage formulation in accordance with recent experimental findings [87,21,36]. In particular, the model captures irreversible damage of the IFM and the fibrillar structures by a force-time coupled criterion. Finally, this allows to predict long term damage accumulation even under physiological loads. The paper is organized as follows. In Section 2 we discuss mechanics of soft fibrous tissues and present the fatigue model. Predictions of the model are compared against various experimental data and possibilities of the parameter identification are discussed in detail in Section 3. 2. Mechanics of collagen fibers As mentioned above mechanical properties of soft fibrous tissues result mainly from their complex hierarchical organization. Accordingly, in the following we briefly summarize the hierarchical structure of collagen. With this in mind, the constitutive equations are formulated first for the elastic case in Section 2.3. In Section 2.4 the constitutive equations are further enhanced to inelasticity to take into account the damage evolution due to fatigue mentioned above. 2.1. Hierarchical structure and mechanics of collagen Tropocollagen is a triple helical arrangement of three coiled collagen-protein chains (see Fig. 1). The chains are bonded together by weak hydrogen bonds. With a D-banding pattern TC molecules are assembled to their higher order-structure referred to as collagen fibrils [42]. The TCs are attached to each other at their tips by covalent cross-linkers which provide intermolecular integrity [7]. In the periodical packing of collagen fibrils there are molecular gaps with the highest molecular disorder [19]. A single collagen fiber is organized by a number of collagen fibrils and embedded into the IFM. The IFM, in turn, consists of small diameter fibrils (SDFs) wrapping bigger diameter fibrils and a large amount of proteoglycans (PGs) [84]. Due to the high negative fixed charge density, the PGs osmot-

Fig. 1. Multi-scale representation of soft fibrous tissues starting on the meso level with the ground substance reinforced by collagen fibers. A collagen fiber is seen on the micro-level as a composition of collagen fibrils embedded in the interfibrillar materix, in turn, consisting of proteoglycans (PGs) and small diameter fibrils (SDFs). The PGs consist of a core protein (CP) to which one or more GAG side chains are covalently bonded. The fibrils theirself are D-periodically assembled by tropocollagen molecules.

Please cite this article in press as: K. Linka et al., Fatigue of soft fibrous tissues: Multi-scale mechanics and constitutive modeling, Acta Biomater. (2018), https://doi.org/10.1016/j.actbio.2018.03.010

3

K. Linka et al. / Acta Biomaterialia xxx (2018) xxx–xxx

ically attract water inside the matrix, which in turn keeps the matrix hydrated [71,3]. Recent studies suggest that the inferfibrillar matrix is crucial for the shear load transfer between the discontinuous collagen fibrils [84,47]. Accordingly, SDFs wrapping larger diameter fibrils are mainly responsible for the load transfer. Indeed, after a selective elimination of glycosaminoglycans (GAGs) via enzymatic removal, only a little alteration on the interfibrillar sliding was observed [72,71,84,47]. 2.2. Kinematics In order to take slight compressibility of soft tissues into account, we apply the multiplicative decomposition of the defor
into a volumetric part J 1=3 and an isomation gradient F ¼ ðJ 1=3 ÞF
[17,37], where J ¼ det F is the choric (distortional) part F T

determinant of F. In the following C ¼ F F denotes the right
¼F
represents its distortional part.
T F Cauchy-Green tensor and C Furthermore, we assume affine deformation for the bundles of collagen fibers with respect to the macroscopic deformation. Let us consider a mean fiber direction i ði ¼ 1; 2; . . . ; nÞ specified by a unit vector mi and the corresponding structural tensor Li ¼ mi  mi . Now, assuming a rotationally symmetric dispersion of fibers ~i and the isochoric fiber around mi , a generalized structural tensor L

stretch vi can be expressed in terms of the isochoric structural invariant
I4i by [28,14]

wi L~i ¼ I þ ð1  wi ÞLi ; 3

vi ¼

qffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi qffiffiffiffiffi
: L~i ¼ tr C
L~i ¼
I4i ; C

ð1Þ

where wi is referred to as a weighting dispersion parameter. The upper limit wi ¼ 1 describes an isotropic collagen fiber distribution, while the lower limit wi ¼ 0 corresponds to a perfect collagen fiber alignment in the direction mi (cf. [27]). I denotes here the identity tensor. Current investigations reveal that the IFM rather promotes collagen fibril sliding than a direct force transmission between the fibrils under tension [78,83,84]. Indeed, X-ray diffraction and AFM interventions on soft fibrous tissues confirm that the fibril stretch under tensile loads is smaller than the tissue stretch [69,72,83]. By this means, a higher IFM volume fraction results in larger possible tissue deformations without causing fibril damage due to relative fibril sliding. In particular, this is important e.g. for energy storing tendons [85]. In order to take this strain relaxing role of the IFM

into account we apply a linear relationship between the fibril stretch kifb and the tissue stretch Fig. 2) [51]

vi in the direction i as (see

  kifb ¼ max ð1  /ifm Þðvi  1Þ þ 1; 1 ;

ð2Þ

where /ifm 2 ½0; 1 denotes the IFM volume fraction around collagen

fibrils. Note that according to (2) kifb ¼ 1 when vi 6 1, which implies that fibrils do not carry load under compression of the tissue. Moreover, there is a considerable structural inhomogeneity of the collagen fibrils due to their assembly by covalent crosslinkage of collagen molecules. This is modeled in the present study by a strain amplification in the uncross-linked part with the length (R) of the collagen molecules [49,51]. Accordingly, we consider the cross-linked protein part ðLcl Þ with the relative length c 2 ½0; 1 to be very stiff in comparison to the uncross-linked one, resulting in the molecular stretch ki which can be represented as [49]

ki ¼

kifb  c ; 1c

c¼

Lcl : R þ Lcl

ð3Þ

This form of the strain amplification is very similar to that one applied for filler reinforced rubbers (see e.g [6,35]). 2.3. Elasticity of collagen fibers In the following, constitutive relations for the collagen molecular-, the collagen fibril- and collagen fiber-level are proposed. The relations are histologically based and interrelated by the kinematic assumptions presented in Section 2.2. Accordingly, we assume an additive decomposition of the fibril and IFM contribution to the stress response. Thus, the resulting macroscopic isochoric nominal (1st Piola-Kirchhoff) stress can be given as


iso ¼ P

n X
i þ P
ifm ; P fn

ð4Þ

i


i represent the fibril stresses in the preferred direction mi where P fn
ifm is the IFM stress contribution. Within this (i ¼ 1; 2; . . . ; n) and P framework, the elasticity description results from the response of single collagen molecules. They are described by the extensible worm-like chain (eWLC) model based on statistical mechanics. 2.3.1. Tropocollagen mechanics The Helmholtz free energy of a tropocollagen molecule can be expressed as

Wtc ¼ We  Ts;

ð5Þ

where We is the internal energy per unit current volume. T is absolute temperature, s denotes the entropy, while Ts ¼ Ws represents thus the entropic energy. Due to strain induced structural changes of the biopolymer, elastic deformations of collagen molecules can be divided into two different regimes. Under tension the molecules become first aligned and straightening of molecular kinks initiates [59]. This kinematic alteration is accompanied by a reduction of the possible conformations of the molecules until the extended helix length L is reached, see also Fig. 3. This process is referred to as the entropic deformation regime [4,9]. The pure entropic force F in this regime can well be captured by the worm-like chain (WLC) model (see e.g. [4,7,52]) and is often interpolated as a function of the molecular stretch k by [55]

kB T F¼ lp Fig. 2. Tissue stretch vi and the resulting fibril stretch kifb according to (2) versus time for different IFM volume fractions /ifm under a specific test protocol.

! Rk 1 1 þ ;  L 4ð1  Rk=LÞ2 4

ð6Þ

where the persistence length lp characterizes the bending stiffness of the filament, L denotes the extended molecular length (also

Please cite this article in press as: K. Linka et al., Fatigue of soft fibrous tissues: Multi-scale mechanics and constitutive modeling, Acta Biomater. (2018), https://doi.org/10.1016/j.actbio.2018.03.010

4

K. Linka et al. / Acta Biomaterialia xxx (2018) xxx–xxx Table 1 Material parameters utilized in this study for the prediction of the TC molecule force response in Fig. 4. Best fit Proposed model eWLC WLC

Fig. 3. Deformation of a single TC molecule from the reference end-to-end distance R and to the extended molecular length L.

referred to as contour length) and kB denotes the Boltzmann constant. Furthermore, R represents the reference end-to-end length of the filament. The WLC model [46] is based on the total bending energy due to thermal fluctuations. However, under higher tension forces the molecules unwind and initiate bond stretching. This deformation regime is often referred to as energetic or intrinsic [9] and cannot more be described by the WLC model (see Fig. 4). Against this background, Maceri et al. [52] proposed a piecewise defined function, which consists of an interpolation of the WLC [54] for the entropic range and a linear elastic model for the energetic regime. However, this approach does not define a physically motivated transition point between these two regimes. In this context, the extensible worm-like chain model [63] seems to be reasonable, since it predicts this transition smoothly. In contrast to the WLC model (6), the eWLC model is formulated additionally in terms of the elastic modulus of the filament K and can be given as [63]

k¼

" #  1 L 1 kB T 2 F 1 þ : R 2 F lp K

ð7Þ

The eWLC model demonstrates good agreement with experimental data from TC molecule in the whole range of deformations, see Fig. 4. However, for the constitutive modeling the inverse form FðkÞ of the relation (7) is necessary, which has a very complex expression and is thus inconvenient to use. Taking advantage of the fact that for the numerically obtained set of material parameters (given in Table 1) the entropic and energetic contributions of (7) differ by several orders of the magnitude, we propose a piecewise defined formulation of the eWLC. Accordingly, the energetic and entropic parts are considered separately, where bond stretching and helix untwisting are neglected in the entropically dominated

p

R

L

K

4.71 nm 2.65 nm 4.71 nm

214 nm 214 nm 214 nm

315 nm 315 nm 315 nm

1592.4 pN 4738 pN –

regime ðK ¼ 1Þ up to the transition stretch ktr . Beyond this stretch a linear energetic contribution can be considered as the only relevant source of deformation resistance, leading to

( Fðk; LÞ ¼



2 F s ðk; LÞ ¼ k4BlTp 1  Rk  F0; L

if k 6 ktr ;

F e ðk; LÞ ¼ K ðk  ktr Þ þ F s ðktr ; LÞ; if k > ktr :

ð8Þ



2 Herein F 0 ¼ k4BlTp 1  RL provides the stress-free reference configuration, while F s ðktr ; LÞ ensures continuity between the entropic and energetic regime. The transition point ktr between these regimes was obtained in [63] as the inflection point of the curve kðln FÞ resulting from (7). Accordingly, the force F  developed by the molecule at this point can be expressed by [63]

F ¼

 1 1 kB T K 3 : 4 lp

ð9Þ

Inserting this result into (8), as F  ¼ F s ðktr ; LÞ þ F 0 , we further get

"  1 # L 1 kB T 2 : 1 ktr ðLÞ ¼ R 2 F  lp

ð10Þ

For validation purposes (8) is fitted against the force response of a single TC molecule (see Fig. 4). The corresponding material parameters are given in Table 1. The elastic modulus in (7) and (8) can be obtained from a helical potential accounting for the triple helical TC structure [44,54,29,31,49]. Accordingly, K depends on the twisting rigidity C of a helix, the stretch modulus S and the parameter g quantifying the twist-stretch coupling relation as

K¼

S C  g2 : C

ð11Þ

The expression (11) was successfully applied to predict the forceextension behavior for double-stranded DNA [31], whose structural configuration is similar to that one of TC molecules. While we proposed an entropic-energetic approach in order to capture the TC molecule response, an alternative approach in the context of implicit elasticity was recently reported [20]. There, a one-dimensional energy function capable of capturing the typical J shape response of biological fibers was proposed. Accordingly, an elastic part for the toe region was utilized, while the implicit part corresponds to the stiffening behavior due to straightening of biological fibers. 2.3.2. Collagen fibril mechanics Due to a strong periodical assembly of a fibril [42], we consider an averaged end-to-end reference length R of TCs. In contrast, every TC is considered with a different extended helix length L experimentally measured in [4]. These extended helix lengths can be described by the Beta probability density function (PDF) as [49]

.ðLÞ ¼ Fig. 4. Force developed by a TC molecule versus molecular stretch: comparison of different TC models to experimental data [4]. The best fits for the proposed model and the eWLC model were obtained for the same initial elongation ratio RL.

ðL  Lmin Þa1 Bða; bÞðLmax  Lmin Þaþb1

ðLmax  LÞb1 ;

ð12Þ

where Bða; bÞ denotes the beta function. .ðLÞ (12) is determined by two shape parameters a and b, the minimal and maximal extended

Please cite this article in press as: K. Linka et al., Fatigue of soft fibrous tissues: Multi-scale mechanics and constitutive modeling, Acta Biomater. (2018), https://doi.org/10.1016/j.actbio.2018.03.010

5

K. Linka et al. / Acta Biomaterialia xxx (2018) xxx–xxx

helix length Lmin amd Lmax , respectively. It is assumed that the shortest molecule of the considered interval L 2 ½Lmin ; Lmax  is in the energetic regime, such that ktr ¼ 1 is enforced at L ¼ Lmin (cf. [49]). The molecular force F, given in (8), can be expressed in terms of the molecular strain energy w as F ¼ @w=@ðRkÞ. The fibril strain-energy can further be given in terms of w and the average amount of collagen molecules inside a single fibril N as (cf. [49,51])

Z wfb ¼ N

Lmax

Lde

.ðLÞ wðk; LÞ dL:

ð13Þ

Thus, the force developed by a single fibril can further be obtained by the chain rule and in view of (8) as

F fb ¼

@wfb 1 @wfb @k N @k ¼ ¼ @ðRfb kfb Þ Rfb @k @kfb Rfb @kfb ¼

N @k Rfb @kfb

Z Z

Lmax Lde Lmax Lde

.ðLÞ

@wðk; LÞ dL; @k

.ðLÞ FR dL;

ð14Þ

where Rfb is the reference end-to-end fibril length, while @k=@kfb ¼ 1=ð1  cÞ results from (3). In the elastic case the lower limit of the integral in (14) coincides with Lmin (Lde ¼ Lmin ), but will increase upon breakage of short TC molecules as described in Section 2.4.3. This increase of Lde is of dissipative nature and thus does not contribute to the fibril force (14). 2.3.3. Constitutive modeling of collagen fibers The IFM surrounds the much stiffer discontinuously dispersed collagen fibrils. This assembly represents the main building block of collagen fibers, whose bundles are spatially arranged in soft fibrous tissues [84]. The IFM is considered to be isotropic and will be modeled by the neo-Hookean strain energy function Wifm ¼ ,ð
IC  1Þ, expressed in terms of the first principal invariant
and the constant ,. Assuming independent contributions
IC ¼ trC
fib we of fibrils and IFM to the isochoric part of the fiber stress P can write by virtue of (1), (2) and (14)


i ¼ P
i þ P
ifm ¼N0 @wfb þ / @ Wifm ; P ifm fib fn

@F @F i @wfb @kfb @ vi @ Wifm þ /ifm ¼N0 i

; @ v @ F @F @k i fb

ð1  /ifm Þ @wfb
~
¼N0 FLi þ 2/ifm ,F; vi @kifb

ð15Þ

where N 0 denotes the average amount of collagen fibrils within one preferred fiber family i per unit reference volume. 2.4. Inelasticity of collagen fibers 2.4.1. Dissociation of non-covalent interactions Staggered arrays of intercrosslinked collagen molecules form fibrils, which represent discontinuous rope-like structures embedded into an IFM in order to form collagen fibers. Along with the covalent cross-links, primarily developed at the ends of TC molecules [8], interaction forces such as intermolecular and interfibrillar junctions ensure integrity and strength of collagen fibers [42,75]. The intermolecular and interfibrillar bonds establish weaker non-covalent molecular linkages. Hence, those bonds are prone to rupture first compared to the covalent cross-links [87]. In the following, we will discuss the intermolecular and intrafibrillar interactions and propose a model describing breakage of their non-covalent adhesive bonds. Under a mechanical force F applied to non-covalent chemical bonds their energy state decreases, which can finally lead to a separation of adjacent cell interfaces. Indeed, molecular adhesive bonds fail under any external force, applied sufficiently long enough [2,15,16]. This will be modeled in the following by a force depending bond failure rate XðFÞ (also referred to as off-rate), where F ¼ f ðkÞ. By this means, one obtains a likelihood of the bond survival under the applied force F up to time t as [16]

 Z t pðF; tÞ ¼ XðFÞ exp  XðF Þ ds ;

ð16Þ

0

which is often referred in the literature to as the Bell-Evans model. Accordingly, p is an exponentially decaying function of time at a constant force. In particular, Evans and Ritchie (1997) [16] proposed a formulation of a force depending off-rate XðFÞ on the basis of a molecular interaction potential described by an inverse power law as E ¼ Eb =xn , where the lower limit is fixed by a repulsive wall at the normalized-distance xm ( 1), defined by the energy barrier Eb . Under consideration of an external force F, the combined energy ~ xm Þ ¼ E  F ^x, where ^x ¼ x  xm (see Fig. 5). potential reads as Eðx; The detailed derivation for the off-rate XðFÞ can be found in the Appendix A. Now, assuming that interaction forces F depend linearly on ~ ðkÞ. For the sake stretch k [16,8], the off-rate can be expressed as X of simplification and a better numerical performance in the context

Fig. 5. (a) The attractive potential E  Eb =xn in a force free state and the energy landscapes which result from the application of different forces ðF 1 < F 2 < F 3 Þ. The maximums of the combined potentials are reached at the transition states xts (marked by a cross). (b) Illustration of two adhesive bonds with a distance x, subjected to a force F.

Please cite this article in press as: K. Linka et al., Fatigue of soft fibrous tissues: Multi-scale mechanics and constitutive modeling, Acta Biomater. (2018), https://doi.org/10.1016/j.actbio.2018.03.010

6

K. Linka et al. / Acta Biomaterialia xxx (2018) xxx–xxx

Fig. 6. (a) Comparison of the off-rate (25) and its piecewise defined approximation (17) plotted vs. stretch (b) their resulting likelihoods of bond survival according to the Bell-Evans model (16) vs. stretch. k ¼ 1:19 denotes the transition stretch.

Fig. 7. Sensitivity analysis of the bond density qðk; tÞ (18) and the likelihoods of bond survival p½f ðkÞ; t  (16) with respect to different strain rates e_ plotted against the stretch (a & c) and time (b & d) for a ¼ 0:017s-1, b ¼ 0:37 s-1 and the transition stretch k ¼ 1:19.

Please cite this article in press as: K. Linka et al., Fatigue of soft fibrous tissues: Multi-scale mechanics and constitutive modeling, Acta Biomater. (2018), https://doi.org/10.1016/j.actbio.2018.03.010

7

K. Linka et al. / Acta Biomaterialia xxx (2018) xxx–xxx

of a finite element (FE) implementation, we propose a piecewise ~ ðkÞ as linear approximation of X

~ ðkÞ ¼ X

aðk  1Þ

if k 6 k ;

bðk  k Þ þ aðk  1Þ if k > k ; 



ð17Þ

while a and b denote two shape parameters. The approximated off-

~ ðkÞ is illustrated in Fig. 6a in comparison to the prediction of rate X the Evans-Ritchie model (25). The behavior of the likelihood of bond survival resulting from (16) is also shown in Fig. 6b. Accordingly, good agreement between the Evans-Ritchie model and the pro~ ðkÞ and posed approximation is obtained both with respect to X

p½f ðkÞ; t . The transition between the low and high rupture probability range is described by the transition stretch k . By this value, the physiological loading range, where molecular degeneration is compensated by cell remodeling is separated from the supraphysiological loading range. Now, integrating (16) over time leads to an expression for the percentage of broken adhesive bonds as

qðk; tja; b; k Þ ¼ 1 

Z

t 0

p½f ðkÞ; s ds;

ð18Þ

referred to as bonding density q 2 ½0; 1. It is the function of k and t while a, b and k (appearing after the vertical bar) represent its parameters. With respect to stretch, q is monotonically decreasing, which implies irreversible debonding. The numerical implementation of the procedure calculating the bond density q is presented in Algorithm 1, while in Fig. 7 the influence of the strain rate e_ is studied. As expected, the mean dissociation stretch increases with the strain rate (e.g [16,34]). Algorithm 1. Numerical implementation of the bond density q calculation, with ðÞ0 denoting variables at the previous time t0 and ðÞ1 denoting variables at the current time t1 . x represents an internal variable used to store the integrated ~. history of the off-rate X ~ ðk0 ; k1 ; t0 ; t1 ; x0 ; q0 ; a; b; k 1: function q 2: Calculate kðtÞ by linearisation between ðk0 ; t0 Þ and ðk1 ; t 1 Þ Rt ~ ðkÞ dt 3: Numerically integrate Dx ¼ 1 X 

t0

4: q1 ¼ q0  expðx0 Þ þ expðDx  x0 Þ 5: x1 ¼ x0 þ Dx 6: return fq1 ; x1 g 7: end function

2.4.2. Long term behavior of the interfibrillar matrix Collagen fibrils are surrounded by PGs and SDFs which enable interfibrillar sliding (see Fig. 1) [71,47,84]. In the IFM, the PGs consist of a core protein (CP) to which one or more GAG side chains are covalently bonded. In turn, the core protein attaches to collagen fibrils, while the GAGs are adhesively connected to each other by hyalurona-bindings, PG cores or collagen fibrils (see Fig. 1) (e.g [61,32,66,25]). The adhesive bonds are much weaker in comparison to the links of the core protein to collagen fibrils and rupture first [89]. Furthermore, the IFM is enriched by small diameter interweaving collagen fibrils fused to large diameter fibrils, which have not formed fibril bundles [68]. Those SDFs and the presence of the PGs in the IFM are reported to establish functionally continuous collagen fibrils [68,84]. In multi-scale structural investigations it was observed that tendon damage starts in the interfibrillar structure due to non-recoverable interfibrillar sliding [47]. This damage mechanism implies a breakage of IFM linkages to the collagen fibrils and might be described in view of (18) in terms of the IFM bonding density as qifm ¼ q ðvi ; tjaifm ; bifm ; kifm Þ. qifm describes the current fraction of IFM linkages to a fibril.

Moreover, due to a reduction of interfibrillar sliding by a successive IFM turnover, the toe-region of the tissue stress response shifts to higher stretches [47], which leads to the typical nonlinear fibril response [19]. By this means, the IFM contribution reduces due to interfibrillar damage and the force between fibrils becomes transmitted by long spanning fibrils themselves [68,82]. Consequently, the stress contribution of the IFM can be described in view of (15) as


ifm ¼ 2q / ,F;
P ifm ifm

ð19Þ

where the qifm acts as damage-like variable. 2.4.3. Long term behavior of collagen fibrils So far, we have only considered the elastic behavior of TC molecules by neglecting inelastic effects due to overstretching or long term loading. Current experimental investigations show that cyclic loading with a moderate (still physiological) force amplitude can cause the same damage pattern at the nanoscale as a mechanical overloading [36,87]. This damage pattern is referred to as local plasticity and displays discrete kinks along the collagen fibril. These kinks are associated with a permanent opening of collagen molecules [40,88]. This failure mechanism was further experimentally confirmed as an origin of a subsequent tissue failure [93]. The triple-helical arrangement of the TC molecules is mainly sustained by intramolecular adhesion due to hydrated molecules [64,48]. In [8] the Lennard-Jones potential was successfully applied in order to capture the behavior of single intermolecular interactions. Accordingly, adhesive debonding results in local fibril defects and decreases the fibril stress. The debonding is considered here to initiate with the shortest TC molecules and evolves to longer ones (see cf. [30] and Fig. 8). Hence, the minimal length Lde of the integration limits in (15) changes due to damage to

Lde ðt max Þ ¼ ðLmax  Lmin Þ ð1  qint Þ þ Lmin ;

with Lmin 6 Lde 6 Lmax ; ð20Þ

where the intermolecular bond density qint ¼ q ðkint ; tjaint ; bint ; kint Þ is given by (18). Furthermore, the energetic regime increases linearly with molecular stretch Les ¼ Lmin k within the interval Lde 6 Les 6 Lmax , since the shortest molecule in the considered interval is assumed to belong to the energetic regime as discussed in

Fig. 8. Illustration of the beta distribution .ðLÞ for the TC extended helix length and the corresponding deformation based intervals. Accordingly, Lde determines the boundary between the damaged and energetic regime, while Les separates the energetic from the entropic regime.

Please cite this article in press as: K. Linka et al., Fatigue of soft fibrous tissues: Multi-scale mechanics and constitutive modeling, Acta Biomater. (2018), https://doi.org/10.1016/j.actbio.2018.03.010

8

K. Linka et al. / Acta Biomaterialia xxx (2018) xxx–xxx

Section 2.3.2. The intermolecular stretch kint develops only during energetic deformations (due to bond stretching and twisting effects, see e.g. [8,40]) and can be given in view of (8) and (20) as

Z kint ¼

Les

Lde

.ðLÞðk  ktr Þ dL þ 1; with Lde 6 Les 6 Lmax :

ð21Þ

Note that qint (18) is a monotonically decreasing function of stretch due to the fact that the TC bonding density always reduces under loading. A certain level of TC damage in soft tissues during physiological loadings belongs to the normal tissue functionality [93]. This protein loss is compensated by remodeling and growth processes of the extracellular matrix (ECM). However, the remodeling effects take place at time scales (of several days to month) much longer then that of fatigue (hours) considered in the present model and will not be addressed here. 2.5. Constitutive relations The final constitutive model includes mechanical contributions of the IFM and the dispersed collagen fibers. Both contributions degrade under mechanical load due to dissolving linkages. For the numerical implementation the tissue is considered to be nearly incompressible. It is enforced by a penalty term UðJÞ ¼ j ðJ  1Þ2 , where j denotes the bulk modulus. Accordingly, the first PiolaKirchhoff stress tensor P can be represented in view of (4), (15) and (19) by

P ¼ Piso þ Pvol ;

ð22Þ

where Pvol ¼ U 0 JFT is the volumetric stress contribution. The isochoric stress contribution can further be expressed by (see [41])


iso  1 ðP
iso : FÞFT : Piso ¼ J 2=3 P 3

ð23Þ

The numerical implementation of (22) for a single preferred fiber orientation is presented in Algorithm 2. Algorithm 2. Calculate tissue stress Pi for a single fiber direction i for m time steps, where ðÞ½j denotes a variable at the time t½j (j ¼ 1; 2; . . . ; m).

3. Results and discussion In this section, we evaluate and discuss the predictive capability of the proposed constitutive model in comparison to different experimental data reported in the literature. 3.1. Model validation in comparison to experimental data on bovine tendon Experimental data on bovine tendon were reported in [36]. There, in a first characterization step uniaxial tensions (UT) to rupture as well as 1000 load cycles at a force magnitude of 30% of the ultimate failure force were applied. Then, by a scanning electron microscopy analysis, nanoscale structural alterations referred to as discrete plasticity were identified for both loading steps (see [36] for details). The model predictions have been fitted to both loading protocols simultaneously. The so resulting set of material parameters for this particular testing is presented in Table 2. Note that the material constants related to TC molecules were validated in a previous study [49] against another experimental data and are only shown in Table 2 for the sake of completeness. The fitting results for the rupture and the fatigue are shown in Figs. 9a and 10a, respectively. In general, both predictions are in good agreement to the experimental data. However, in our opinion, fitting to macroscopic experimental stress-responses is not a proper approach to reveal the associated values and can only serve as a gross estimation. Hence, force spectroscopy or molecular dynamics (MD) simulations are required to identify values of the interaction parameters (a; b; k ) for both, the IFM and intramolecular behavior (see, e.g., [1,34]). Note also that the availability of experimentally measured bond interactions might require to redefine the approximation (18). In the reference configuration the model predicts all TC molecules to be entirely in the entropic regime. Subjected to a sufficiently large force applied during sufficiently long time period the molecules pass over to the energetic state before failing (see e.g. Fig. 9b). As expected, the damage progression under the long-term loading is initially strongly correlated to the IFM density decrease. Within the progressive damage of the IFM, the fibril damage develops (see Figs. 10b and 11b). This is due to the increasing likelihood of bond detachment predicted by (18) for both intramolecular and interfibrillar linkages. In contrast, the damage due to overstretching results from subsequently overpassing the critical TC stretch of individual collagen molecules. Thus, the IFM bonding density remains nearly unaltered while the amount of damaged fibrils continuously increases (see Fig. 11a). 3.2. Triphasically tested rat tendon The second set of experimental data [23] serves to verify the model over an entire lifetime of the tissue, in particular tendon. In these experiments, samples from the rat flexor digitorum longus (FDL) tendon were loaded cyclically up to 16 N, which corresponds approximately to 33% of the failure strength under monotonic loading. Additionally, light microscopy interventions directly after the loading revealed a continuous damage accumulation, starting from local kinked fibers at low fatigue up to isolated fibers rupturing at high fatigue levels. This was hypothesized to be caused by collagen molecule degeneration on the molecular level [21]. Note that this subsequent damage evolution on the basis of individual TC failing is one of the key-features of the proposed model (22) described by physically based internal variables (e.g. in (18)). In addition, the fatigue experiments were enriched by postmechanical tests, which determined a residual degeneration of

Please cite this article in press as: K. Linka et al., Fatigue of soft fibrous tissues: Multi-scale mechanics and constitutive modeling, Acta Biomater. (2018), https://doi.org/10.1016/j.actbio.2018.03.010

9

K. Linka et al. / Acta Biomaterialia xxx (2018) xxx–xxx Table 2 Material parameters utilized in this study for the prediction of the rupture and fatigue tests of bovine tendon in Figs. 9 and 10.

TC

Material parameter

Notation

Numerical value

Source

Modulus Shape parameters Min./Max. length Persistence length Off-rate parameters

K a; b Lmin ; Lmax lp aint ; bint

1592.4 pN 4.3 185 nm, 315 nm 4.71 nm

Table 1 [49] Fitting, [49] Table 1 Fitting

Transition parameter IFM

FB

kint

,

4:5 108 s1 , 1.1 s-1 1.21

/ifm

3.03 MPa 3%

Transition parameter

kifm

5:6 103 s1 ; 2:2 102 s1 1.13

Amount of TCs

N0 N

Modulus IFM fraction Off-rate parameters

Cross-link density Fibril dispersion

aifm ; bifm

c w

8:6 109 mm3 0.46 0.03

Fitting Fitting Fitting Fitting Fitting Fitting [73] Fitting

Fig. 9. Uniaxial tension to failure of bovine tendon [36]: (a) model predictions vs. experimental data (b) distribution of TC molecules between the entropic, energetic or damaged range during loading. In addition, the PDF .ðLÞ is depicted for example for the last time step.

Fig. 10. Cyclic fatigue loading of bovine tendon [36]: (a) model predictions vs. experimental data of 1000 loading cycles (b) distribution of TC molecules between the entropic, energetic or damaged range during loading. In addition, the PDF .ðLÞ is depicted for example for the last time step.

Please cite this article in press as: K. Linka et al., Fatigue of soft fibrous tissues: Multi-scale mechanics and constitutive modeling, Acta Biomater. (2018), https://doi.org/10.1016/j.actbio.2018.03.010

10

K. Linka et al. / Acta Biomaterialia xxx (2018) xxx–xxx

Fig. 11. Model prediction of the decreasing bonding densities ðqint ; qifm Þ under (a) uniaxial tension until rupture and (b) cyclic loading of bovine tendon.

Fig. 12. (a) Model prediction of a triphasic fatigue evolution vs. experimental data of the rat FDL tendon [23] (b) distribution of TC molecules between the entropic, energetic or damaged range during loading. In addition, the PDF .ðLÞ is depicted for example for the last time step.

mechanical properties after certain recovery time. This is also covered by the proposed model, where damage variables are described by monotonically decreasing functions. For more details about the experimental method an interested reader is referred to [23]. In Fig. 12a the peak strain predicted by the model is plotted vs. the number of cycles and compared with the corresponding experimental data. The corresponding material parameters can be found in Table 3. Despite the overall good agreement between the model prediction and the measured peak strain, a slight disagreement especially in the very early loading range is observable. This might be due to viscoelastic effects which are not considered in the present approach. Indeed, already after a small amount of loading cycles viscous damping has a considerable impact to the delayed damage progression [32,81]. 3.3. Experimental identification of material parameters Although the majority of model parameters were identified by mathematical fitting, promising experimental methods have been

reported in order to capture the histological information needed for the proposed material model. On the molecular level, the persistence length lp of TC molecules was evaluated by a molecular dynamics simulation, optical tweezers and atomic force microscopy [4,9,70]. lp reflects the bending stiffness of macromolecules, necessary to describe the molecular reaction force due to thermal fluctuations. Applying rotor bead tracking simultaneously with force measurements by magnetic tweezers, the twist-stretch coupling of DNA molecules was evaluated [29,31]. Accordingly, from the forces sufficient to suppress bending fluctuations the three necessary parameters g, S and C in (11) can be identified within a linear theory. However, this method was so far applied only for DNA while similar results for collagen are still missing. The size distribution including maximal, minimal and means molecular lengths was extensively studied by atomic force microscopy, both topologically and force spectroscopy [4]. In turn, such results can deliver the shape parameters a and b for the beta distribution and the corresponding minimal and maximal molecular length ðLmin ; Lmax Þ within one fibril. X-ray diffractometry

Please cite this article in press as: K. Linka et al., Fatigue of soft fibrous tissues: Multi-scale mechanics and constitutive modeling, Acta Biomater. (2018), https://doi.org/10.1016/j.actbio.2018.03.010

11

K. Linka et al. / Acta Biomaterialia xxx (2018) xxx–xxx Table 3 Material parameters utilized in this study for the prediction of fatigue tests of rat FDL tendon in Fig. 12a.

TC

Material parameter

Notation

Numerical value

Source

Modulus Shape parameters Min./Max. length Persistence length Off-rate parameters

K a; b Lmin ; Lmax lp aint ; bint

1592.4 pN 4.3 193 nm, 315 nm 4.71 nm

Table 1 [49] Fitting, [49] Table 1 Fitting

Transition parameter IFM

FB

Modulus IFM fraction Off-rate parameters

,

4.65 MPa 2:35%

/ifm aifm ; bifm

Transition parameter

kifm

Amount of TCs

N0 N

Cross-link density Fibril dispersion

3:7 105 s1 , 1.18 s-1 1.15

kint

5:6 105 s1 , 0.17 s-1 1.08 2:3 1010 mm3 0.46 0.03

c w

was applied in order to study the impact of macroscopical loading on the straining mechanism of collagen fibrils [74]. There, average amounts of TC molecules within a fibril N as well as the average cross-link density c were measured. TC molecular failure mechanics has well been disclosed by means of MD simulations. For instance, in [1] the unfolding and sequential rupture of individual protein building blocks as a function of the density of their hydrogen bonds have been studied and related to the Bell-model. This study was extended in [86], where in further MD simulations TCpairs were subjected to a force until the strand separation took place. The purpose of the study was mainly to study TC-TC crosslinking. In this regards, MD studies could be beneficial to characterize the Bell-Evans parameters of the TCs ðaint ; bint ; kint Þ used in (20). The inverted confocal microscope image analysis for example during uniaxial tension tests of rat tail fascicles [83] delivered information on the interfibrillar sliding and thus, enabled to quantify the difference between the tissue and fibril stretch, represented in the proposed model by /ifm . These measurements were later on enriched by a fatigue cyclic loading protocol, which disclosed the intefibrillar sliding evolution as a function of IFM damage [47]. Those data in combination with dynamic force spectroscopy of the IFM (e.g performed to reveal self-adhesive properties of PGs [24,34]) could build a solid scientific basis to describe the failure parameters of the Bell-Evans model for the IFM ðaifm ; bifm ; kifm Þ. A non-invasive method to reveal the histological composition of soft fibrous tissues could be the multi-parametric functional magnetic resonance imaging. By a proper evaluation of relaxation times of magnetic fields and spins this method delivers information about the protein contents, fluid fractions and fibril orientations [60,50]. 3.4. Limitations of the model and future research directions The model proposed in the paper describes the histologically observed fatigue evolution of soft fibrous tissues (e.g. tendon) at relatively short time scales. Under in vivo conditions growth and remodeling effects which are not considered here, compensate turnover effects up to a certain limit. In this context, concepts like the recently proposed machanobiological free energy, motivated by homeostatic effects and the presence of a reservoir of remodeling energy, appear very promising [11]. Thus, further research should address a combination of mechanobiological models of growth and remodeling in the ECM (e.g. by constrained mixture theory [39,12,5]) and histologically motivated long-term damage (e.g by the model proposed here). This may shed more light to long-term damage of soft fibrous tissues and serve as a possible

Fitting Fitting Fitting Fitting Fitting Fitting [73] Fitting

mirco-mechanical elucidation of the collagen half-life reduction under e.g. growing aneurysm or aging [10,91]. By this means, simulations of long time effects like wrinkling of skin or the evolution and growth of aneurysms might become reliable in future. Further research should especially address those simulations on the patient-specific basis. 4. Conclusions We presented a model which encounters the entropic and energetic contributions of single collagen molecules, whose lengths within a fibril are treated statistically. A force-time depending exponential decay function based on Kramers’s theory of reaction kinetics is utilized to predict the decreasing bonding densities of intramolecular and interfibrillar linkages [45,2,16]. The decrease of the IFM bonding density leads to a reduction of the IFM stress contribution due to less interfibrillar friction [71,47], while intramolecular interaction turnover results in a degeneration of TC molecules [93]. Good agreement between model predictions and experiment data from different kinds of loading protocols were obtained. Thus, the mechanical assumptions underlying the model appear to be adequate for describing fatigue of soft fibrous tissues. In this context, application of the proposed material model in numerical simulations on the organ level or in order to improve the design of engineered tissues seems to be promising. Appendix A A.1. Derivation of the Evans-Ritchie model As discussed in Section 2.4.1, in the Evans-Ritchie model [16] the molecular interaction is governed by the attractive potential described by a power law as E ¼ Eb =xn , where the lower limit is fixed by a repulsive wall at the normalized-distance xm ( 1), defined by the energy barrier Eb [16]. The case n ¼ 6 corresponds to the van der Waals type interaction. The combined energy poten~ xm Þ ¼ E  F x ^, where ^ tial reads as Eðx; x ¼ x  xm (see Fig. 5a) [16]. The transition point xts between the bonded and the free state is @ ~ given by the maximum of the potential derived from @x E ¼ 0 as 1 F 1 nþ1 , where F 1 denotes the total bond strength at the maxxts ¼ F imal attraction nEb [16]. The energy barrier DEb represents the value of the potential e E at the transition point and can be calculated by [16]

" # n  nþ1 F F ~ : DEb ðFÞ ¼ EðxÞj ¼ E ðn þ 1Þ  n b xts F1 F1

ð24Þ

Please cite this article in press as: K. Linka et al., Fatigue of soft fibrous tissues: Multi-scale mechanics and constitutive modeling, Acta Biomater. (2018), https://doi.org/10.1016/j.actbio.2018.03.010

12

K. Linka et al. / Acta Biomaterialia xxx (2018) xxx–xxx

The bond failure rate XðFÞ can be given in a generic form based on Kramers’ theory of reaction kinetics as [45,15,16]

XðFÞ ¼

X0

b ðFÞ

exp ½DEb ðFÞ;

ð25Þ

where X0 ¼ ð1=t D Þ expðEb Þ denotes a scaling factor depending on a characteristic time constant tD . The time constant t D reflects bond detachment slowed due to viscous damping effects (see [16,45,33]). The energy barrier width b in (25) expresses the statistical weighted distance between the bonded and the free state within the energy landscape. It can be given by an improper expo~ xts Þ  Ets nential integration of the combined energy difference Eðx; over the distance x [45,33]. Approximating the combined energy ~ xts Þ by second order Taylor series developed around xts , one Eðx; obtains the following Gaussian-type integral [16]

b ðFÞ ¼

Z

1

1

!  1 @2E 2p xts 2 2 exp jxts ðx  xts Þ dx ¼ : 2 2 @x ðn þ 1ÞF

ð26Þ

Inserting (24) and (26) into (25) yields the fully defined off-rate

XðFÞ. The off-rate (25) depends on two physically based material parameters t D and Eb , which can be determined by advanced experimental methods e.g. dynamic force spectroscopy [58,34]. References [1] Theodor Ackbarow, Xuefeng Chen, Sinan Keten, Markus J. Buehler, Hierarchies, multiple energy barriers, and robustness govern the fracture mechanics of ahelical and b-sheet protein domains, Proc. Nat. Acad. Sci. 104 (42) (2007) 16410–16415. [2] George I. Bell, Models for the specific adhesion of cells to cells, Science 200 (4342) (1978) 618–627. [3] Markus Böl, Alexander E. Ehret, Kay Leichsenring, Michael Ernst, Tissue-scale anisotropy and compressibility of tendon in semi-confined compression tests, J. Biomech. 48 (6) (2015) 1092–1098. [4] Laurent Bozec, Michael Horton, Topography and mechanical properties of single molecules of type I collagen using atomic force microscopy, Biophys. J. 88 (6) (2005) 4223–4231. [5] F.A. Braeu, A. Seitz, R.C. Aydin, C.J. Cyron, Homogenized constrained mixture models for anisotropic volumetric growth and remodeling, Biomech. Model. Mechanobiol. 16 (3) (2017) 889–906. [6] F. Bueche, Molecular basis for The Mullins effect, Rubber Chem. Technol. 34 (10) (1961) 493–505. [7] Markus J. Buehler, Nature designs tough collagen: explaining the nanostructure of collagen fibrils, Proc. Nat. Acad. Sci. USA 103 (33) (2006) 12285–12290. [8] Markus J. Buehler, Nanomechanics of collagen fibrils under varying cross-link densities: atomistic and continuum studies, J. Mech. Behav. Biomed. Mater. 1 (1) (2008) 59–67. [9] Markus J. Buehler, Sophie Y. Wong, Entropic elasticity controls nanomechanics of single tropocollagen molecules, Biophys. J. 93 (1) (2007) 37–43. [10] M. Carmo, L. Colombo, A. Bruno, F.R.M. Corsi, L. Roncoroni, M.S. Cuttin, F. Radice, E. Mussini, P.G. Settembrini, Alteration of elastin, collagen and their cross-links in abdominal aortic aneurysms, Eur. J. Vascular Endovascular Surg. 23 (6) (2002) 543–549. [11] C.J. Cyron, R.C. Aydin, Mechanobiological free energy: a variational approach to tensional homeostasis in tissue equivalents, ZAMM-J. Appl Math Mech-Uss. 97 (9) (2017) 1011–1019. [12] C.J. Cyron, R.C. Aydin, J.D. Humphrey, A homogenized constrained mixture (and mechanical analog) model for growth and remodeling of soft tissue, Biomech. Model. Mechanobiol. 15 (6) (2016) 1389–1403. [13] C.J. Cyron, J.D. Humphrey, Growth and remodeling of load-bearing biological soft tissues, Meccanica 52 (3) (2017) 645–664. [14] Alexander E. Ehret, Mikhail Itskov, A polyconvex hyperelastic model for fiberreinforced materials in application to soft tissues, J. Mater. Sci. 42 (21) (2007) 8853–8863. [15] E. Evans, D. Berk, A. Leung, Detachment of agglutinin-bonded red blood cells. I. Forces to rupture molecular-point attachments, Biophys. J. 59 (4) (1991) 838– 848. [16] E. Evans, K. Ritchie, Dynamic strength of molecular adhesion bonds, Biophys. J. 72 (4) (1997) 1541–1555. [17] P.J. Flory, Thermodynamic relations for high elastic materials, Trans. Faraday Soc. 57 (1961) 829–838. [18] Caroline Forsell, Jesper Swedenborg, Joy Roy, T. Christian Gasser, The quasistatic failure properties of the abdominal aortic aneurysm wall estimated by a mixed experimental-numerical approach, Ann. Biomed. Eng. 41 (7) (2013) 1554–1566.

[19] P. Fratzl, K. Misof, I. Zizak, G. Rapp, H. Amenitsch, S. Bernstorff, Fibrillar structure and mechanical properties of collagen, J. Struct. Biol. 122 (1–2) (1998) 119–122. [20] Alan D. Freed, K.R. Rajagopal, A promising approach for modeling biological fibers, Acta Mech. 227 (6) (2016) 609–1619. [21] Benjamin R. Freedman, Joseph J. Sarver, Mark R. Buckley, Pramod B. Voleti, Louis J. Soslowsky, Biomechanical and structural response of healing Achilles tendon to fatigue loading following acute injury, J. Biomech. 47 (9) (2014) 2028–2034. [22] David T. Fung, Vincent M. Wang, Nelly Andarawis-Puri, Jelena Basta-Pljakic, Yonghui Li, Damien M. Laudier, Hui B. Sun, Karl J. Jepsen, Mitchell B. Schaffler, Evan L. Flatow, Early response to tendon fatigue damage accumulation in a novel in vivo model, J. Biomech. 43 (2) (2010) 274–279. [23] David T. Fung, Vincent M. Wang, Damien M. Laudier, Jean H. Shine, Jelena Basta-Pljakic, Karl J. Jepsen, Mitchell B. Schaffler, Evan L. Flatow, Subrupture tendon fatigue damage, J. Orthopaedic Res. 27 (2007) 264–273. [24] Sergi Garcia-Manyes, Iwona Bucior, Robert Ros, Dario Anselmetti, Fausto Sanz, Max M. Burger, Xavier Fernandez-Busquets, Proteoglycan mechanics studied by single-molecule force spectroscopy of allotypic cell adhesion glycans, J. Biol. Chem. 281 (9) (2006) 5992–5999. [25] T.C. Gasser, An irreversible constitutive model for fibrous soft biological tissue: a 3-D microfiber approach with demonstrative application to abdominal aortic aneurysms, Acta Biomater. 7 (6) (2011) 2457–2466. [26] T.C. Gasser, Bringing vascular biomechanics into clinical practice. Simulationbased decisions for elective abdominal aortic aneurysms repair, PatientSpecific Comput. Model. 5 (2012) 1–37. [27] T.C. Gasser, A. Gerhard Holzapfel, Modeling the propagation of arterial dissection, Eur. J. Mech. A. Solids 25 (4) (2006) 617–633. [28] T.C. Gasser, Ray W. Ogden, Gerhard A. Holzapfel, Hyperelastic modelling of arterial layers with distributed collagen fibre orientations, J. R. Soc., Interface/ the R. Soc. 3 (6) (2006) 15–35. [29] Jeff Gore, Zev Bryant, Marcelo Nöllmann, Mai U. Le, Nicholas R. Cozzarelli, Carlos Bustamante, DNA overwinds when stretched, Nature 442 (1) (2006) 836–839. [30] Sanjay Govindjee, Juan Simo, A micro-mechanically based continuum damage model for carbon black-filled rubbers incorporating Mullins’ effect, J. Mech. Phys. Solids 39 (1) (1991) 87–112. [31] Peter Gross, Niels Laurens, Lene B. Oddershede, Ulrich Bockelmann, Erwin J.G. Peterman, Gijs J.L. Wuite, Quantifying how DNA stretches, melts and changes twist under tension, Nat. Phys. 7 (9) (2011) 731–736. [32] H.S. Gupta, J. Seto, S. Krauss, P. Boesecke, H.R.C. Screen, In situ multi-level analysis of viscoelastic deformation mechanisms in tendon collagen, J. Struct. Biol. 169 (2) (2010) 183–191. [33] Peter Hänggi, Peter Talkner, Michal Borkovec, Reaction-rate theory: fifty years after Kramers, Rev. Mod. Phys. 62 (2) (1990) 251–341. [34] Alexander Harder, Volker Walhorn, Thomas Dierks, Xavier FernàndezBusquets, Dario Anselmetti, Single-molecule force spectroscopy of cartilage aggrecan self-adhesion, Biophys. J. 99 (November) (2010) 3498– 3504. [35] J.A.C. Harwood, L. Mullins, A.R. Payne, Stress softening in natural rubber vulcanizates. Part II. Stress softening effects in pure gum and filler loaded rubbers, Rubber Chem. Technol. 39 (1966) 814–822. [36] Tyler W. Herod, Neil C. Chambers, Samuel P. Veres, Collagen fibrils in functionally distinct tendons have differing structural responses to tendon rupture and fatigue loading, Acta Biomater. 42 (2016) 296–307. [37] G.A. Holzapfel, Nonlinear Solid Mechanics. A Continuum Approach for Engineering, J. Wiley and Sons, 2000. [38] Holzapfel, Gerhard A., et al. Determination of layer-specific mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling. Am. J. Physiol-Heart. C. 289 (5) (2005) H2048–H2058. [39] J.D. Humphrey, K.R. Rajagopal, A constrained mixture model for arterial adaptations to a sustained step change in blood flow, Biomech. Model. Mechanobiol. 2 (2) (2003) 109–126. [40] Pieter J. in’t Veld, Mark J. Stevens, Simulation of the mechanical strength of a single collagen molecule, Biophys. J. 95 (1) (2008) 33–39. [41] Mikhail Itskov, Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics, fourth ed., Springer Publishing Company, Incorporated, 2015. [42] K.E. Kadler, D.F. Holmes, J.A. Trotter, J.A. Chapman, Collagen fibril formation, The Biochem. J. 316 (Pt 1) (1996) 1–11. [43] Christopher Kaeding, Thomas M. Best, Tendinosis: pathophysiology and nonoperative treatment, Sports Health: A Multidisciplinary Approach 1 (4) (2009) 284–292. [44] R.D. Kamien, T.C. Lubensky, Direct determination of DNA twist-stretch coupling, EPL (Europhys. Lett.) 271 (1997). [45] Hendrik Anthony Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7 (4) (1940) 284–304. [46] O. Kratky, Das Studium gelöster Fadenmoleküle mittels der Röntgenkleinwinkelmethode 1 (1939) 1955. [47] Andrea H. Lee, Spencer E. Szczesny, Michael H. Santare, Dawn M. Elliott, Investigating mechanisms of tendon damage by measuring multi-scale recovery following tensile loading, Acta Biomater. 57 (2017) 363–372. [48] Yang Li, S. Michael Yu, Targeting and mimicking collagens via triple helical peptide assembly, Curr. Opinion Chem. Biol. 17 (6) (2013) 968–975.

Please cite this article in press as: K. Linka et al., Fatigue of soft fibrous tissues: Multi-scale mechanics and constitutive modeling, Acta Biomater. (2018), https://doi.org/10.1016/j.actbio.2018.03.010

K. Linka et al. / Acta Biomaterialia xxx (2018) xxx–xxx [49] Kevin Linka, Mikhail Itskov, Mechanics of collagen fibrils: a two-scale discrete damage model, J. Mech. Behav. Biomed. Mater. 58 (2016) 163–172. Special issue: Mechanics of biological membranes. [50] Kevin Linka, Mikhail Itskov, Daniel Truhn, Sven Nebelung, Johannes Thüring, T2 MR imaging vs. computational modeling of human articular cartilage tissue functionality, J. Mech. Behav. Biomed. Mater. 74 (2017) 477–487. [51] Kevin Linka, Vu Ngoc Khiêm, Mikhail Itskov, Multi-scale modeling of soft fibrous tissues based on proteoglycan mechanics, J. Biomech. 49 (12) (2016) 2349–2357. Cardiovascular Biomechanics in Health and Disease. [52] Franco Maceri, Michele Marino, Giuseppe Vairo, A unified multiscale mechanical model for soft collagenous tissues with regular fiber arrangement, J. Biomech. 43 (2) (2010) 355–363. [53] Nicola Maffulli, Per Renström, Wayne B. Leadbetter, Tendon Injuries, Springer, 2005. [54] J.F. Marko, Stretching must twist DNA, EPL (Europhys. Lett.) 183 (1997). [55] John F. Marko, Eric D. Siggia, Stretching DNA, Macromolecules 28 (26) (1995) 8759–8770. [56] Caitlin Martin, Wei Sun, Simulation of long-term fatigue damage in bioprosthetic heart valves: effects of leaflet and stent elastic properties, Biomech. Model. Mechanobiol. 13 (4) (2014) 759–770. [57] Caitlin Martin, Wei Sun, Fatigue damage of collagenous tissues: experiment, modeling and simulation studies, J. Long-term Effects Med. Implants 25 (1–2) (2015). [58] R. Merkel, P. Nassoy, A. Leung, K. Ritchie, E. Evans, Energy landscapes of receptor-ligand bonds explored with dynamic force spectroscopy, Nature 397 (6714) (1999) 50–53. [59] K. Misof, G. Rapp, P. Fratzl, A new molecular model for collagen elasticity based on synchrotron X-ray scattering evidence, Biophys. J. 72 (3) (1997) 1376– 1381. [60] S. Nebelung, B. Sondern, S. Oehrl, M. Tingart, B. Rath, T. Pufe, S. Raith, H. Fischer, C. Kuhl, H. Jahr, D. Truhn, Functional mr imaging mapping of human articular cartilage response to loading, Radiology 282 (2) (2017) 464–474. [61] Laurel Ng, Alan J. Grodzinsky, Parth Patwari, John Sandy, Anna Plaas, Christine Ortiz, Individual cartilage aggrecan macromolecules and their constituent glycosaminoglycans visualized via atomic force microscopy, J. Struct. Biol. 143 (2003) 242–257. [62] Reinhart Nissen, George J. Cardinale, Sidney Udenfriend, Increased turnover of arterial collagen in hypertensive rats, Proc. Nat. Acad. Sci. 75 (1) (1978) 451– 453. [63] Theo Odijk, Stiff chains and filaments under tension, Macromolecules 28 (20) (1995) 7016–7018. [64] Joseph P.R.O. Orgel, Thomas C. Irving, Andrew Miller, Tim J. Wess, Microfibrillar structure of type i collagen in situ, Proc. Nat. Acad. Sci. 103 (24) (2006) 9001–9005. [65] M.M Panjabi, E. Yoldas, T.R. Oxland, 3Rd Crisco J.J. Subfailure injury of the rabbit anterior cruciate ligament, J. Orthop. Res. 14 (23) (1996) 216–222. [66] Geraint J. Parfitt, Christian Pinali, Robert D. Young, Andrew J. Quantock, Carlo Knupp, Three-dimensional reconstruction of collagen–proteoglycan interactions in the mouse corneal stroma by electron tomography, J. Struct. Biol. 170 (2) (2010) 392–397. [67] A.V. Pike, R.F. Ker, R.M. Alexander, The development of fatigue quality in highand low-stressed tendons of sheep (Ovis aries), J. Exp. Biol. 203 (2000) 2187– 2193. [68] Paolo P. Provenzano, Ray Vanderby, Collagen fibril morphology and organization: implications for force transmission in ligament and tendon, Matrix Biol. 25 (2) (2006) 71–84. [69] R. Puxkandl, I. Zizak, O. Paris, J. Keckes, W. Tesch, S. Bernstorff, P. Purslow, P. Fratzl, Viscoelastic properties of collagen: synchrotron radiation investigations and structural model, Philos. Trans. R. Soc. London. Series B, Biol. Sci. 357 (1418) (2002) 191–197. [70] Naghmeh Rezaei, Benjamin P.B. Downing, Andrew Wieczorek, Clara K.Y. Chan, Robert Lindsay Welch, Nancy R. Forde, Using optical tweezers to study mechanical properties of collagen, Spie 8007 (2011). 80070K–80070K–10. [71] S. Rigozzi, R. Müller, A. Stemmer, J.G. Snedeker, Tendon glycosaminoglycan proteoglycan sidechains promote collagen fibril sliding-AFM observations at the nanoscale, J. Biomech. 46 (4) (2013) 813–818.

13

[72] S. Rigozzi, A. Stemmer, R. Müller, J.G. Snedeker, Mechanical response of individual collagen fibrils in loaded tendon as measured by atomic force microscopy, J. Struct. Biol. 176 (1) (2011) 9–15. [73] Naoki Sasaki, Shingo Odajima, Elongation mechanism of collagen fibrils and force-strain relations of tendon at each level of structural hierarchy, J. Biomech. 29 (9) (1996) 1131–1136. [74] Naoki Sasaki, Singo Odajima, Stress-strain curve and young’s modulus of a collagen molecule as determined by the x-ray diffraction technique, J. Biomech. 29 (5) (1996) 655–658. [75] J.E. Scott, Elasticity in extracellular matrix ‘shape modules’ of tendon, cartilage, etc. A sliding proteoglycan-filament model, J. Physiol. 553 (Pt 2) (2003) 335– 343. [76] Jedd B. Sereysky, Nelly Andarawis-Puri, Karl J. Jepsen, Evan L. Flatow, Structural and mechanical effects of in vivo fatigue damage induction on murine tendon, J. Orthop. Res. 30 (June) (2012) 965–972. [77] Jedd B. Sereysky, Nelly Andarawis-Puri, Stephen J. Ros, Karl J. Jepsen, Evan L. Flatow, Automated image analysis method for quantifying damage accumulation in tendon, J. Biomech. 43 (13) (2010) 2641–2644. [78] Jennifer H. Shepherd, Graham P. Riley, Hazel R.C. Screen, Early stage fatigue damage occurs in bovine tendon fascicles in the absence of changes in mechanics at either the gross or micro-structural level, J. Mech. Behav. Biomed. Mater. 38 (2014) 163–172. [79] Jennifer H. Shepherd, Hazel R.C. Screen, Fatigue loading of tendon, Int. J. Exp. Pathol. 94 (2013) 260–270. [80] Jess G. Snedeker, Jasper Foolen, Tendon injury and repair – a perspective on the basic mechanisms of tendon disease and future clinical therapy, Acta Biomater. 63 (Supplement C) (2017) 18–36. [81] René B. Svensson, Tue Hassenkam, Philip Hansen, S. Peter Magnusson, Viscoelastic behavior of discrete human collagen fibrils, J. Mech. Behav. Biomed. Mater. 3 (1) (2010) 112–115. [82] Rene B. Svensson, Andreas Herchenhan, Tobias Starborg, Michael Larsen, Karl E. Kadler, Klaus Qvortrup, S. Peter Magnusson, Evidence of structurally continuous collagen fibrils in tendons, Acta Biomater. 50 (Supplement C) (2017) 293–301. [83] Spencer E. Szczesny, Dawn M. Elliott, Interfibrillar shear stress is the loading mechanism of collagen fibrils in tendon, Acta Biomater. 10 (6) (2014) 2582– 2590. [84] Spencer E. Szczesny, Kristen L. Fetchko, George R. Dodge, Dawn M. Elliott, Evidence that interfibrillar load transfer in tendon is supported by small diameter fibrils and not extrafibrillar tissue components, J. Orthop. Res. 35 (10) (2017) 2127–2134. [85] Chavaunne T. Thorpe, Graham P. Riley, Helen L. Birch, Peter D. Clegg, Hazel R.C. Screen, Fascicles and the interfascicular matrix show adaptation for fatigue resistance in energy storing tendons, Acta Biomater. 42 (2016) 308–315. [86] Sebastien G.M. Uzel, Markus J. Buehler, Molecular structure, mechanical behavior and failure mechanism of the C-terminal cross-link domain in type I collagen, J. Mech. Behav. Biomed. Mater. 4 (2) (2011) 153–161. [87] Samuel P. Veres, Julia M. Harrison, J. Michael Lee, Cross-link stabilization does not affect the response of collagen molecules, fibrils, or tendons to tensile overload, J. Orthop. Res. 31 (12) (2013) 1907–1913. [88] Samuel P. Veres, J. Michael Lee, Designed to fail: a novel mode of collagen fibril disruption and its relevance to tissue toughness, Biophys. J. 102 (12) (2012) 2876–2884. [89] Simone Vesentini, Alberto Redaelli, Franco M. Montevecchi, Estimation of the binding force of the collagen molecule-decorin core protein complex in collagen fibril, J. Biomech. 38 (3) (2005) 433–443. [90] Pramod B. Voleti, Mark R. Buckley, Louis J. Soslowsky, Tendon healing: repair and regeneration, Annu. Rev. Biomed. Eng. 14 (2012) 47–71. [91] P.N. Watton, N.A. Hill, Evolving mechanical properties of a model of abdominal aortic aneurysm, Biomech. Model. Mechanobiol. 8 (1) (2009) 25–42. [92] Tishya A.L. Wren, Derek P. Lindsey, Gary S. Beaupré, Dennis R. Carter, Effects of creep and cyclic loading on the mechanical properties and failure of human achilles tendons, Ann. Biomed. Eng. 31 (November) (2003) 710–717. [93] Jared L. Zitnay, Yang Li, Zhao Qin, Boi Hoa San, Baptiste Depalle, Shawn P. Reese, Markus J. Buehler, S. Michael Yu, Jeffrey A. Weiss, Molecular level detection and localization of mechanical damage in collagen enabled by collagen hybridizing peptides, Nat. Commun. 8 (2017) 14913.

Please cite this article in press as: K. Linka et al., Fatigue of soft fibrous tissues: Multi-scale mechanics and constitutive modeling, Acta Biomater. (2018), https://doi.org/10.1016/j.actbio.2018.03.010