Biomech Model Mechanobiol (2015) 14:995–1006DOI 10.1007/s10237-015-0649-1

ORIGINAL PAPER

A constitutive model for the time-dependent, nonlinear stressresponse of fibrin networks

Thomas H. S. van Kempen · Gerrit W. M. Peters ·Frans N. van de Vosse

Received: 12 September 2014 / Accepted: 9 January 2015 / Published online: 25 January 2015© The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract Blood clot formation is important to preventblood loss in case of a vascular injury but disastrous whenit occludes the vessel. As the mechanical properties of theclot are reported to be related to many diseases, it is impor-tant to have a good understanding of their characteristics. Inthis study, a constitutive model is presented that describesthe nonlinear viscoelastic properties of the fibrin network,the main structural component of blood clots. The modelis developed using results of experiments in which the fib-rin network is subjected to a large amplitude oscillatoryshear (LAOS) deformation. The results show three dominat-ing nonlinear features: softening over multiple deformationcycles, strain stiffening and increasing viscous dissipationduring a deformation cycle. These features are incorporatedin a constitutive model based on the Kelvin–Voigt model. Anetwork state parameter is introduced that takes into accountthe influence of the deformation history of the network. Fur-thermore, in the period following the LAOS deformation, thestiffness of the networks increases which is also incorporatedin the model. The influence of cross-links created by factorXIII is investigated by comparing fibrin networks that havepolymerized for 1 and 2h. A sensitivity analysis providesinsights into the influence of the eight fit parameters. Themodel developed is able to describe the rich, time-dependent,nonlinear behavior of the fibrin network. The model is rela-tively simple which makes it suitable for computational sim-

T. H. S. van Kempen (B) · F. N. van de VosseDepartment of Biomedical Engineering, Eindhoven Universityof Technology, PO Box 513, 5600MB Eindhoven, The Netherlandse-mail: t.h.s.v.kempen@tue.nl

G. W. M. PetersDepartment of Mechanical Engineering, Eindhoven Universityof Technology, Eindhoven, The Netherlands

ulations of blood clot formation and is general enough to beused for other materials showing similar behavior.

Keywords Blood clotting · Constitutive modeling · Largeamplitude oscillatory shear (LAOS) ·Biopolymer networks ·Sensitivity analysis

1 Introduction

Blood clots form to prevent blood loss after a vascular injurybut can also lead to undesired complications such as thrombo-sis when forming intravascularly. The mechanical propertiesof the blood clot are of major of importance for its function-ing and have been related to many diseases (Weisel 2008) butremain elusive to model. The main structural component ofthe blood clot is fibrin, a fibrous network that forms withinthe blood clot and thereby provides strength to the plateletplug that forms as a provisional closure of the injury. The fib-rin network shows rich nonlinear mechanical behavior thatenables the network to perform its physiological function(Brown et al. 2009; Münster et al. 2013), but this behavioris not fully described by current constitutive models. There-fore, in this study, a constitutive model is developed for thenonlinear mechanical behavior of the fibrin network.

The fibrin network forms in multiple steps after the con-version of fibrinogen to fibrin monomers, enzymatically cat-alyzed by thrombin. Thesemonomers aggregate to form two-stranded structures known as protofibrils which subsequentlypolymerize into fibers that eventually form the fibrin net-work (Cilia La Corte et al. 2011). This network is further-more strengthened due to the presence of factor XIII (fXI-IIa) that creates cross-links within and between protofibrils(Ryan et al. 1999; Lorand 2005). It is the hierarchical struc-ture of the fibers that gives the fibrin network its remark-

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able, yet complicated, mechanical properties (Brown et al.2009; Piechocka et al. 2010; Münster et al. 2013). One ofthe most pronounced nonlinear mechanical effects is thatfibrin stiffens with an increasing deformation (strain stiffen-ing) (Shah and Janmey 1997; Brown et al. 2009; Kang et al.2009; Weigandt et al. 2011; Münster et al. 2013). Duringsuch a deformation, individual fibers can stretch to multi-ple times their own length (Liu et al. 2006). Upon repeateddeformation cycles, the fibers persistently lengthen, leadingto a lower stiffness at the same strain and hence a softeningeffect (Münster et al. 2013). The combination of these non-linear viscoelastic features makes it complicated to describethe mechanical properties of the fibrin network, yet they areessential for a realistic description of the mechanical (in-situ) behavior where complex loading histories may occur.Therefore, the goal of this study is to develop a constitutivemodel for the nonlinear viscoelastic, thixotropic behavior ofthe fibrin network. To our knowledge, such a model, suitablefor advanced numerical simulations of blood clot formation(e.g., Storti et al. 2014) and based on a continuummechanicsapproach, has not been developed before.

The nonlinear mechanical properties of fibrin, and net-works of biopolymers in general, have been studied usingmodels and various experimental techniques. Models havebeen used to show that the strain stiffening behavior can havean entropic or a nonaffine origin (Storm et al. 2005; Oncket al. 2005), while experimentally, it has been shown thatboth mechanisms play a role, most likely at different strainregimes (Brown et al. 2009; Piechocka et al. 2010; Weigandtet al. 2011). The intrinsic nonlinear behavior of single fibrinfibers has been studied (Liu et al. 2006; Averett et al. 2012)as well as its influence on network mechanics (Hudson et al.2010). Recently, it has been shown that parts of fibrin fiberscan relocate within the network which makes it a dynamicstructure that can remodel (Chernysh et al. 2012).

Various experimental protocols have been used to probenonlinear mechanical properties of fibrin and other biopoly-mers, including strain ramps (Schmoller et al. 2010), com-pression (Kim et al. 2014), differential prestress (Piechockaet al. 2010) and large amplitude oscillatory shear (LAOS)(Münster et al. 2013). Each protocol probes different aspectsof the nonlinear viscoelastic behavior of the material anda combination is useful to obtain a complete description(Semmrich et al. 2008; Broedersz et al. 2010). In this study,a LAOS deformation is used to develop a constitutive modelfor the nonlinear viscoelastic behavior of the fibrin network.An advantage of LAOS is that it is suitable to probe the richnonlinear viscoelastic behavior of a material, while it is stillpossible to distinguish the various features observed. Therepeated oscillatory deformation provides insights into thethixotropic behavior of the fibrin network. Furthermore, theLAOSdeformation is a straightforward extensionof the smallamplitude oscillatory shear (SAOS) deformation that is usu-

ally used to study viscoelastic behavior. Also, the responseof the fibrin network to a large oscillatory deformation mim-ics the physiological deformation due to an oscillatory bloodflow that these networks have towithstand.ThismakesLAOSideally suited for the development of a constitutive modelof the fibrin network. LAOS deformations have been usedbefore to study the nonlinear viscoelastic properties of fibrin(Münster et al. 2013). In this study, the experimental dataare used to unravel the viscoelastic response of the fibrinnetworks and develop a constitutive model.

In the next section, experiments are introduced, the resultsof which are used subsequently to develop the constitutivemodel. The model is then used to describe and predict theresults of the experiments, followed by a discussion of theoutcome.

2 Experimental methods

2.1 Fibrin network formation

Fibrin networks are formed by adding 0.5U/ml humanthrombin to 1 mg/ml human fibrinogen (Kordia, Leiden, TheNetherlands) after which the sample is quickly transferred tothe titanium cone-plate geometry (25mm diameter, 0.02 radcone angle) of an ARES rheometer (Rheometric Scientific,USA). To follow the network formation, a SAOS deforma-tion with a frequency of 1Hz and strain amplitude of 0.01is imposed for 2h. This deformation is within the linear vis-coelastic regime and does not interfere with the network for-mation (van Kempen et al. 2014). The networks are formedat 37 ◦C and a layer of mineral oil is applied to the sampleedge to minimize evaporation. The response of the networksduring the SAOS deformation is predominantly elastic (vanKempen et al. 2014), indicated by an elastic modulus G ′ thatexceeds the viscous modulus G ′′ many times. Therefore, theSAOSdata are presented in terms of the elasticmodulus only.

2.2 Large amplitude oscillatory shear (LAOS) experiment

After the network formation, the nonlinear viscoelastic prop-erties are studied by imposing a LAOS deformation. Whilethe frequency of the oscillatory deformation is always main-tained at 1Hz, the strain amplitude is increased in discretesteps of 60 s to 0.05, 0.1, 0.25, 0.5, 0.75 and 1.0. After theLAOS deformation, the response of the network is followedfor 2h by imposing the same SAOS deformation as before.Subsequently, the LAOS sequence is repeated. The firstLAOS sequence is used to determine parameter values of thedifferent parts of the constitutive model, as explained later.The second sequence is used to test the predicting capabilitiesof the model. An overview of this protocol is shown in Fig. 1.For convenience, in the remainder of this paper, the start ofa LAOS sequence is defined as starting time i.e., t = 0.

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Fig. 1 The elastic modulus, G ′, in time is measured during a SAOSexperiment of 2h (a). Then, a LAOS experiment is performed in whichthe strain amplitude is increased (b), and the resulting stress ismeasured(c). This procedure is repeated afterward

The raw data from the rotation and torque signals are col-lected using an analog-to-digital converter (ADC) (Wilhelm2002) and converted to strain and stress. The data obtained inthis way during the SAOS experiment contain a large amountof noise, due to the low torque. Therefore, the strain andstress during the SAOS experiment are not obtained fromthe raw data but using the data provided by the rheometersoftware. These data, provided in terms of strain amplitudeand the elastic and viscous modulus, are used to reconstructthe strain and stress signals in time.

The data are analyzed in terms of Lissajous–Bowditchplots in which the strain is plotted versus the stress, showingclosed loops that illustrate the nonlinear viscoelastic behaviorof the fibrin network (Ewoldt et al. 2008; Hyun et al. 2011).Different aspects of these plots are used as a guideline fordeveloping the constitutive model.

To examine the influence of cross-links formed by fXIIIa,present in the fibrinogen stock solution as shown by SDS-PAGE, someof the samples are allowed to polymerize for oneinstead of 2h. Since the cross-linking occurs over a relativelylong time scale comparedwith the network formation itself, anetwork polymerized for 1h has less cross-links (Ryan et al.1999; Lorand 2005). It is expected that this will influence thenonlinear viscoelastic behavior (Münster et al. 2013).

3 Model development

The constitutive model is an extension of a Kelvin–Voigtmodel, a relatively simple model for a viscoelastic solid(Barnes et al. 1989). In a previous study, the model has been

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Fig. 2 The Lissajous–Bowditch plots shown in panel a illustrate thenonlinear behavior (a). Note that every fifth cycle is plotted for clar-ity. Zooming in on the maximal stress values illustrates that the stressdecreases over multiple deformation cycles (b). Zooming in on the ori-gin illustrates the same effect (c). The slopes of the dashed lines in panelc correspond to the estimated minimal strain modulus,G0. The increas-ing viscous dissipation during the deformation cycle is illustrated by theobservation of a single loop (d). The colors correspond to the strainsshown in Fig. 1b

used to describe the maturation of the fibrin network (vanKempen et al. 2014). The model relates the shear stress τ tothe strain γ and its temporal derivative, the strain rate γ , as

τ = G γ + η γ . (1)

The shear modulus G and viscosity η have been related tostructural quantities of the network during its maturation(van Kempen et al. 2014). Although this connection to struc-tural quantities is still useful, this relation is not used in thisstudy explicitly. Instead, the shear modulus and viscositybecome dependent on the strain history, G = G (γ, t) andη = η (γ, t). In this way, the Kelvin–Voigt model is extendedby including the nonlinear features that are observed in theresults of the LAOS experiments.

Representative results of a LAOS experiment, illustratingthe nonlinear response of the fibrin network, are shown inFig. 2 and subsequently used to explain the development ofthe model.

3.1 Nonlinear features and underlying assumptions

The Lissajous–Bowditch plots shown in Fig. 2a clearly devi-ate from an ellipse, indicating the nonlinear viscoelasticbehavior of the fibrin network. Three nonlinear features can

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be distinguished by observing the plots in detail. The first fea-ture is a cycle-dependent softening effect that takes place overmultiple deformation cycles. This is visible as a decrease inthe stress at maximal strain (Fig. 2b) and also as a decrease inthe slope of the curve at minimum strain (Fig. 2c). The sec-ond feature is strain stiffening, illustrated by the increasedslope of the curves with increasing strain (Fig. 2d). The thirdfeature is that the viscous dissipation increases with increas-ing strain, shown by the broadening of the cycles throughoutthe cycle (Fig. 2d).

Besides these nonlinear phenomena observed during theLAOS experiment, a fourth phenomenon observed is thatthe elastic modulus increases during the 2h after the LAOSexperiment (Fig. 1a). This increase is attributed partially torecovery of the network after the large deformation and to thecontinuous creation of new cross-links due to fXIIIa (Ryanet al. 1999; Lorand 2005). This effect, and the three non-linear features are incorporated in the Kelvin-Voigt model.The focus is on the description of the mechanical propertiesof the fibrin network during the first LAOS experiment andto a lesser extent on the response during the second LAOSexperiment.

3.1.1 Softening and subsequent recovery

As shown in Fig. 2b, c, the stiffness of the fibrin networkdecreases during the multiple deformation cycles of theLAOS experiment. This effect is attributed to the lengtheningof fibers due to the increasing cyclic deformation that leadsto a lower stiffness (Münster et al. 2013).When subsequentlythe amplitude of the strain is decreased, the stiffness of thenetwork rises, as shown by the increasingG ′ value in Fig. 1a.Both effects are included in the model using a network stateparameter (NSP), x , that describes the change in mechani-cal properties of the fibrin network over time based on itsdeformation history. This is simply modeled by making thelow-strain shear modulus proportional to the NSP,

G0 = G00 x, (2)

with G0 the modulus at low strain and G00 the low-strainmodulus of the virgin network, which is the stiffness at areference state, chosen as the start of the first LAOS exper-iment. The form of Eq. (2) implies that the NSP is alwaysnonnegative and at the reference state equal to one. Morecomplicated expressions can be used if desired but a linearrelation is used here as a convenient start to limit the com-plexity of the model. In the following, an evolution equationfor x is developed based on the behavior of G0, but it is firstoutlined how this behavior is extracted from the experimentaldata.

Although the softening effect that takes place during mul-tiple deformation cycles is best illustrated by observing the

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Fig. 3 The NSP, x , in time during LAOS (a). The colors correspondto the strains shown in Fig. 1b. After a change in strain, the NSP levelsoff to a new value denoted by x∞ that decreases with the strain (b). Thedashed lines are obtained with the model [see Eqs. (5) and (6)]

stress values at maximal strain (τ0) as shown in Fig. 2b, thiseffect influences the stress throughout the entire deforma-tion cycle. To quantify the softening, values of τ0 could beused, but a drawback is that the stress at maximal strain isinfluenced by both the softening effect and strain stiffening.Therefore, the modulus at minimal strain, G0, is used (seeFig. 2c) to be able to quantify the kinetics of the softeningfrom the data, independent of the other nonlinear effects.

The values of G0 can be interpreted as a local derivativeof the stress with respect to strain, at the strain γ = 0,

G0 (t, γ ) = ∂τ

∂γ

∣∣∣∣γ=0

, (3)

and are estimated by a linearization of the positive stressesat the strains of γ = ±0.05. For the cycles with a strainamplitude of 0.01, the stresses at maximal strain are usedfor the linearization. The accuracy of this method increaseswith increasing strain amplitude, because the correspondingstresses are higher and the signals contain less noise, as vis-ible in Fig. 2c. Values for the NSP are subsequently foundusing the relation between the shear modulus and the NSP(Eq. (2)),

x (t, γ ) = G0

G00. (4)

Results for the NSP during a LAOS experiment are shown inFig. 3 with the colors corresponding to the LAOS protocolin Fig. 1b.

Note that this implies that the NSP obtained in this wayis intrinsically noisy since it is determined using the stress atthe lowest possible strain. This especially plays a role duringthe lowest strains applied (Fig. 3a).

As shown in Fig. 3a, the NSP decreases due to the appliedstrain, which should be described by the evolution equationfor x . After increasing the strain, the NSP decreases toward

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A constitutive model for the time-dependent, nonlinear stress response of fibrin networks 999

a value that depends on the corresponding strain amplitude.The value towhich theNSP levels off is referred to as x∞, andthe decrease toward this value goes with a time constant cd .As observed during the SAOS deformation imposed after theLAOS sequence, the stiffness of the network rises againwhenthe deformation amplitude is decreased (see Fig. 1a). Thismeans that x increases, in principle to the x∞ value relatedto that lower strain. This increase is modeled with a timeconstant ci . The above thus leads to an evolution equationfor x ,

x ={

−cd (x − x∞) x > x∞,

−ci (x − x∞) x ≤ x∞.(5)

The values of x∞ at which the NSP levels off are related tothe imposed strain, x∞ = x∞ (γ (t)). An expression for thisfunction is found by observing the values of the NSP at theend of each strain step during the LAOS protocol, illustratedby the circles in Fig. 3b. These values are not necessarilythe steady-state value that x would reach if the strain ampli-tude is imposed for a longer period than 60 s. Nevertheless,they provide useful information about the expression thatthe equation for x∞ should have. Requirements are that theexpression is non-negative, independent of the direction ofstrain and equal to one at minimal strain. A suitable expres-sion for x∞ is,

x∞ = e−a|γ |b , (6)

with a and b fit parameters. The expression for x∞ is shownin Fig. 3b together with experimental results.

The evolution equation for x fitted to the experimentaldata is shown in Fig. 3a.

3.1.2 Strain stiffening

The stiffness of the fibrin network increases with strain dur-ing a deformation cycle. Several physical explanations canbe given for this strain stiffening behavior, such as entropicstretching (Stormet al. 2005), non-affine deformations (Oncket al. 2005) and protein unfolding (Brown et al. 2009). Inthis study, the strain stiffening is incorporated in the modelin a phenomenological way by making the shear modulusG a function of the strain γ . To obtain an expression basedon the experimental data, the values of the stress at max-imal strain during each deformation cycle are used. Whenthe strain reaches its maximal value during the deformationcycle, the strain rate is instantaneously zero and the viscouscontribution in theKelvin-Voigtmodel vanishes [seeEq. (1)].Using Eqs. (1, 2), the stress is given by,

τ0 = G00 x f (γ0) γ0, (7)

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Fig. 4 The function that describes the increase of the stiffness as func-tion of the strain (a). The experimental values are shown in coloredcircles, and the model description with a dashed line. Panel b shows asingle loop illustrating the nonlinear viscous dissipation as measured(colored line) and the modeled (dashed line)

with τ0 the stress at maximal strain γ0 and f (γ ) a func-tion of the strain that describes the strain stiffening behavior.Since all quantities, except f (γ0), are known, Eq. (7) can berewritten to,

f (γ0) = τ0

G00 x γ0, (8)

to obtain an expression for f (γ ) (Fig. 4a).The expression for f (γ ) should equal one at low strain,

increasewith increasing strain and be symmetricwith respectto strain. Furthermore, it should be a function of the invariantsof the deformation tensor (Macosko 1994). A function thatsatisfies all these criteria, and gives large flexibility with onlytwo parameters, is the function,

f (γ ) =(

1 + k1 γ 2)n1

, (9)

with k1 and n1 as fit parameters. As shown in Fig. 4a, thisfunction describes the data reasonably well.

3.1.3 Nonlinear viscous dissipation

The viscous dissipation changes throughout a deforma-tion cycle, illustrated by the broadening of the Lissajous–Bowditch plots with increasing strain (Fig. 4b). This featureis incorporated in the model by making the viscous contri-bution an increasing function of the strain,

η = η0 g (γ ) , (10)

with η0 the viscosity at minimal strain, and g (γ ) the functiondescribing the strain dependency. The same expression asused to describe the strain stiffening is chosen,

g (γ ) =(

1 + k2 γ 2)n2

, (11)

with k2 and n2 fit parameters.

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3.2 Overview of the constitutive model

Combining the above, the constitutive model reads,

τ = G00 x(

1 + k1 γ 2)n1

γ + η0

(

1 + k2 γ 2)n2

γ (12)

x ={

−cd (x − x∞) x > x∞,

−ci (x − x∞) x ≤ x∞,(13)

x∞ = e−a|γ |b , (14)

with cd , ci , a, b, k1, n1, k2 and n2 fitting parameters that areobtained from the data using a stepwise fitting procedure asexplained next.

3.3 Numerical procedures and parameter optimization

The constitutive model contains eight fit parameters thatare used to describe the nonlinear phenomena observed inthe Lissajous–Bowditch plots. The evolution equation of theNSP, x , contains four parameters, cd , ci ,a andb. The functionused to describe strain stiffening contains two parameters, k1and n1, similarly to the function that describes the viscousdissipation, k2 and n2. These parameter sets are determinedin three consecutive steps, based on the first LAOS sequenceand the subsequent SAOS measurement of 2h. This para-meter set is then used to study the predicting capabilities ofthe model using the second LAOS sequence. A flow chart ofthe fitting procedure is shown in Fig. 5, where experimentalvalues are denoted by a subscript e and model values by m.

The strain and stress signals obtained during an experi-ment are analyzed using an algorithm implemented in MAT-LAB (The MathWorks, Natick, MA). First, values for the(linear) shear modulus G00 and viscosity η0 are obtainedfrom the elastic and viscous modulus (van Kempen et al.2014) using the data from the strain of 0.01 at the beginningof the LAOS sequence. Subsequently, for every deformationcycle the maximum strain amplitude γ0, the correspondingstress τ0, the zero strainmodulusG0 and the stresses at 0.5 γ0and 1

2

√2 γ0 are determined. The values of G0 are used to

determine the NSP from the experimental data, denoted byxe, to which the evolution equation for the NSP is fit. TheNSP describes the decrease of the modulus during the LAOSdeformation and the recovery afterward. To avoid that theSAOS measurement of 2h dominates the fitting procedureit is given a relative weight of 0.01 compared with the sixmeasurements of 60 s each of the LAOS sequence. In thisway, values for the parameters cd , ci , a and b are obtained.

Using the obtainedNSPand the values for τ0,γ0 , the strainstiffening function fe (γ ) is determined for every cycle andused to find the parameters k1 and n1 in a second fitting step.Finally, the parameters that describe the nonlinear viscousdissipation k2 and n2 are determined by calculating the stressat strains of 1/2 γ0 and 1

2

√2 γ0 and comparing them with

Raw data:

Determine:

Minimize:

Determine:

Minimize:

Determine:

Determine for every cycle:

Determine:

Minimize:

Fig. 5 Flowchart of the parameter optimization process

the respective experimental stresses. The strains of 1/2 γ0and 1

2

√2 γ0 are chosen because at those values the viscous

contribution to the stress, and the stress itself, are relativelylarge.

The three fitting procedures are performed using the non-linear least-squares solver lsqnonlin with a trust-region-reflective algorithm as implemented in the Global Optimiza-tion Toolbox of MATLAB. The fitting procedures are per-formed multiple times with initial parameter values cho-sen randomly from a broad interval using the Multistartalgorithm. In this way, it is avoided that local minima arefound.

3.4 Sensitivity analysis

The constitutive model developed contains eight parametersthat are estimated using experimental data. To get a deeperinsight into the sensitivity of the model to these parame-ters, a sensitivity analysis is performed. A global variance-based method (Sobol 2001) is applied here, that consid-ers the total output variance and determines the contribu-tion of each model parameter to this variance by itself and

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Fig. 6 Illustration of the stresses that are used for the outputs of thesensitivity analysis. Softening is quantified using the decrease of thestress during the strain interval (a). Stiffening is quantified using themaximal stress τmax. The output for the nonlinear viscous behavior isthe difference between the stresses at the strain γ = 1

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√2 γ0 (b)

through interactions with other parameters. These contribu-tions are expressed as sensitivity indices that are estimatedusing Saltelli’s method (Saltelli 2002). A sensitivity index isthe contribution of a parameter to the variance of that output,relative to the total variance of the output. For every output,two sensitivity indices for every parameter are defined.Themain sensitivity index, Si , quantifies the direct influence ofparameter i on the output, whereas the total sensitivity index,STi , describes this main effect but in addition also the influ-ence on the output due to all higher-order contributions inwhich parameter i is involved (Huberts et al. 2014). Themain index can be used to show which parameters are mostrewarding to determinemore accurately, while the total indexshows which parameters could be fixed. More details aboutthis method can be found elsewhere (Saltelli 2002; Hubertset al. 2014).

The output quantities, on which the influence of the vari-ance for each parameter is obtained, are inspired by the threemain features observed in the nonlinear viscoelastic responseof the fibrin network. The first output quantifies the soften-ing effect, and is defined as the relative decrease of the stressat maximal strain during the strain interval with amplitude,γ0 = 1,

Oso = τmax − τmin

τmax, (15)

with τmax and τmin the maximal stress during the first andlast full cycle, respectively (see Fig. 6a).

Strain stiffening is quantifiedusing themaximal stress dur-ing this interval, which occurs at the maximum strain ampli-tude of the first full cycle, τmax. This value is normalizedwiththe low-strain modulus of the virgin network G00,

Oss = τmax

G00. (16)

The third output quantifies the nonlinear viscous dissipationand is defined as the difference between the stresses at γ =12

√2 γ0 for the increasing and decreasing part of the first full

cycle of the interval with γ0 = 1,

Ovi = τup − τdown, (17)

with τup and τdown the corresponding stresses as shown inFig. 6b.

The outputs defined above are obtained using a parame-ter set drawn from a specified uncertainty range using Latinhypercube sampling (Saltelli 2002). The parameter range isbased on the values found from the results of three networksthat have polymerized for 2h and defined as the mean values± two standard deviations of this parameter set. The analysisis based on 5 · 104 model runs, which is five times the min-imum advised for a model with eight parameters (Saltelli2002).

4 Results

In this section, results are presented to illustrate the perfor-mance of the model. Representative results are shown forone sample and discussed in detail. Parameter values of mul-tiple samples are shown to illustrate the variation betweensamples. The model is first used to describe the first LAOSsequence and the recovery that takes place during the follow-ing 2h. Subsequently, the parameter set obtained is used topredict the outcome of the second LAOS sequence. Finally,the model is used to describe the response for networks thathave polymerized for 1h instead of 2h, to study the influenceof the presence of cross-links created by fXIIIa.

4.1 Describing LAOS results

The experimental and numerical results for the LAOSsequence are shown in Fig. 7 as Lissajous-Bowditch plots(panel A,B) and for the stress in time (panel C,D). The threenonlinear features observed in the experimental results, beingsoftening, strain stiffening and nonlinear viscous dissipation,are all described accurately by themodel. Themaximal stressvalues during a deformation cycle agree well, including thesoftening effect that occurs overmultiple deformation cycles,also visible from the NSP during the LAOS sequence alreadyshown in Fig. 3a. The nonlinear viscous dissipation is presentin the model and agrees qualitatively with the experimentalresult, although there is room for improvement.

After the LAOS sequence, a SAOS deformation isimposed for 2h to be able to observe the recovery of thenetwork during this period. The results of this measurementare shown in terms of the NSP, x , in Fig. 8a. Both the experi-mentally and numerically determined x increase in time. The

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Fig. 7 Experimental results of the first LAOS sequence (a, c) and thecorresponding model description (b, d). Both the Lissajous-Bowditchplots (a, b) and the stress in time (c, d) show that the model captures thenonlinear viscoelastic behavior of the fibrin networks that have poly-merized for 2h

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model describes an exponential increase, while experimen-tally, a fast initial rise is followed by a slower increase. Amore advanced kinetic equation could improve this but thishas no priority for this study.

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Fig. 9 Experimental (a, c) results of the second LAOS sequence. Themodel results (b, d) are obtained using the parameters obtained froma model fit to the first LAOS sequence. This shows that the model candescribe the second LAOS sequence using the same parameters thatdescribe the first LAOS sequence

4.2 Predicting LAOS results

The parameter set found byfitting themodel to the first LAOSsequence and the subsequent recovery, as described in theprevious section, is used to predict the outcome of the secondLAOS sequence. The results shown in Fig. 9 show that thestress during the second LAOS sequence is qualitatively thesame as during the first sequence, but the maximal stress val-ues during the cycles are slightly lower. The model describethis behavior well, but overestimates the maximal stress val-ues.

4.3 Influence of fXIIIa

To study the influence of cross-links created by fXIIIa, thesame LAOS sequence as used before is applied to networksthat have polymerized for 1h instead of 2h.

Qualitatively, the results are the same as for the networksthat polymerized for 2h, and the model describes this well(Fig. 10). A difference is that the network that has poly-merized for 1h reaches a higher stress during the LAOSsequence (170 vs. 110Pa), while the difference between thelow-strain modulus before the LAOS sequence is smaller(11.6 vs. 8.4Pa). Themodel overestimates themaximal stressfor the largest strain amplitude by 11%, but the agreementis better for the lower strains, e.g., 7% for a strain amplitudeof 0.75.

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The recovery after the LAOS sequence, also visible in thenetwork that has polymerized for 1h, shows similar behavioras discussed before but increases faster (Fig. 8b).

The observations that the networks that have polymerizedfor 1h show more strain stiffening and recover faster alsofollows from the parameter values found by the model. Fig-ure 11 shows the values for the eight parameters of the modelfor the networks that polymerized for 2h (purple) and 1h(yellow), for three samples of each condition and the meanvalue with standard deviation. The parameters that describethe strain stiffening are k1 and n1. Although there is consid-erable variation between samples it is clear that the valuesof k1 are higher for the networks that polymerized for 1h.An exception to this is sample 1, which has a relatively lowvalue of k1, but this is balanced by a high value for n1, whichalso implies more strain stiffening.

The faster increase after the LAOS sequence is describedby the parameter ci . This parameter indeed has a higher valuefor the networks that polymerized for 1h, indicating that thethey recover faster from the LAOS sequence.

A third difference that is visible from the parameter valuesis the nonlinear viscous dissipation, described by k2 and n2.The networks that polymerized for 2h, have lower values fork2 (4.5 vs. 26) and higher values forn2 (1.9 vs. 1.1). The resultis that the viscous contribution increases more with increas-ing strain for the networks that have polymerized for 1h,which is not directly visible from the Lissajous–Bowditchplots.

4.4 Sensitivity analysis

A sensitivity analysis is performed to assess how the para-meters influence the output uncertainty of the three definedoutputs. For every output, the contribution of the parametersto the variance of the outputs is determined. These contri-butions are shown as a main effect, Si and a total effect, STi(Fig. 12).

The variance of the output related to softening, Oso, is toa large extent determined by the parameters a, cd and ci , asshown by their large contributions of both the main and totalindices (Fig. 12). This is reasonable because these parametersare used to describe the NSP (x) that governs the softeningbehavior. However, the parameter that is involved in strainstiffening, k1, also has a relatively large contribution to thesoftening.

As expected, the variance of the output for the strain stiff-ening behavior, Oss, is dominated by the parameters thatdescribe stiffening, k1 and n1. The sum of the main indices isless than one, which is an indication that higher-order effectscontribute significantly to the total variance (Huberts et al.2014). This is also shown by the relatively large values of thetotal indices in comparison with the main indices.

The variance of the output related to the nonlinear viscousdissipation, Ovi, is for a large part determined by the parame-ter k2 that describes this behavior, but also for a large portionby k1, and n1 that describe the strain stiffening behavior.

5 Discussion

The constitutive model proposed in this study describes thebehavior of the fibrin networks during a LAOS deforma-tion. Using the Lissajous–Bowditch plots, three nonlinearviscoelastic features have been distinguished and are sub-sequently incorporated in a Kelvin–Voigt model. Althoughthe features are modeled in a phenomenological way, it ispossible to relate them to structural changes in the network.The softening observed during the LAOS sequence origi-nates from the semi-permanent elongation of fibers due to theimposed deformation (Münster et al. 2013). During a repeat-ing deformation, fibers become longer in the rest state, lead-ing to a lower stiffness when the same deformation is reachedagain. This process explains why the NSP, that relates thedeformation history to the stiffness of the network, decreasesduring an increasing strain. Fibers become longer during thedeformation but this effect disappears when all fibers haveadapted to the current strain, which explains why the NSPlevels off to a values corresponding with that strain. Besides,it might be possible that breakage of connections betweenfibers occurs, which can also lead to an exponential decreaseof the stiffness (Abhilash et al. 2012). Both lengthening offibers and breakage of connections is inhibited by the cross-

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Fig. 11 Parameter valuesobtained by fitting the model toexperimental results fornetworks that have polymerizedfor 2 (purple) or 1 (yellow) h.Results of three independentmeasurements are shown percondition (1,2,3), together withthe mean value (m). Theerrorbar indicates the standarddeviation. The parameter valuesillustrate that the networks thatpolymerized 1h show morestrain stiffening (k1, n1), fasterrecovery after the LAOSsequence (c1) and more viscousdissipation (k2, n2) than thenetworks that polymerized 2h

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links created by fXIIIa (Münster et al. 2013). This can explainwhy the parameters b and cd that describe the decrease ofthe NSP and the corresponding time scale, respectively, haveslightly higher values for the samples that polymerized for1h instead of two.

During the SAOS experiment that follows the LAOSsequence,the stiffness of the network increases. Experimen-tally, two time scales are observed. A fast initial rise in stiff-

ness is followed by a slower increase for a longer time. Thelatter is attributed to the creation of cross-links by fXIIIa(Ryan et al. 1999; Lorand 2005) while the former mightbe caused by fibers that have the opportunity to becomeshorter again due to the smaller deformation. Recently, ithas been shown that the fibrin network is a dynamic struc-ture and that parts of fibers rearrange within the network(Chernysh et al. 2012). This remodelingmight take place andcan explain the increase in stiffness, and also why this hap-pens on a slower time scale when fXIIIa is present since thecreation of cross-links decreases the amount of rearrange-ments (Chernysh et al. 2012). Both phenomena could beincorporated in more detail using more structural informa-tion but this is considered outside the scope of the currentstudy.

Fibrin networks show a remarkable amount of strain stiff-ening (Storm et al. 2005; Onck et al. 2005; Brown et al. 2009;Piechocka et al. 2010;Weigandt et al. 2011)which ismodeledhere as an increase of the shear modulus as function of thestrain. Several mechanisms have been suggested to explainthe origin of this stiffening, including entropic stretching(Storm et al. 2005), non-affine deformation of fibers (Oncket al. 2005) and protein unfolding (Brown et al. 2009). It isexpected that a combination of thesemechanisms takes place(Piechocka et al. 2010; Weigandt et al. 2011).

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The relation betweenmechanical and structural propertiesof the fibrin network has been studied under various condi-tions (Brown andBarker 2014). Thefiberswithin the networkalign and subsequently stretch due to an increasing deforma-tion (Brown et al. 2009). Using a computational analysis, ithas been shown that the individual fiber properties and theiralignment can be related to the stiffness of the network (Kimet al. 2011). Further structural clues can be obtained by thebehavior of the fibrin network under compression, where thenetwork first softens due to buckling and bending of fibers,followed by stiffening due to a densification of the networkat larger compressions (Kim et al. 2014). Such structuralinformation can be used to extent the model and describethe mechanical properties of the fibrin network based on itsstructure. However, the current phenomenological descrip-tion already shows satisfying results and more complexity isnot wanted in view of the desired application of advancednumerical simulations of blood clot formation (Storti et al.2014).

The constitutive model is used to describe the behaviorof the fibrin network during a LAOS deformation and theincreasing stiffness observed afterward. The results showthat during these 2h, a relatively fast increase of the stiffnessis followed by a more gradual increase. These two effects,attributed to fiber remodeling and creation of cross-links,respectively, are not described by the model explicitly, sincea single time constant is used. This explains why the descrip-tion of the stiffness during this phase is relatively poor com-pared with the description during the LAOS deformation.However, the model is such that it can easily be extended tobetter describe this phase, but this has not been done hereto focus on the nonlinear mechanical behavior of the fibrinnetwork during the LAOS sequence.

The model developed contains eight fitting parameters.The values for these parameters are found in three separatefitting procedures, using the data of the entire LAOS defor-mation. Therefore, an obtained parameter set describes theLAOS experiment as a whole and is not necessarily the bestparameter set to describe a single deformation cycle. How-ever, as shown in Fig. 4b, the agreement for a single cycle isreasonably well.

A sensitivity analysis has been performed to study theinfluence of variations in parameter values. For three out-puts, related to the three observed nonlinear features, thecontribution of every parameter to the variance of this out-put is determined. The expectation that the parameters usedto model one of the effects have the largest influence on theoutput of this effect is confirmed. An example is that theparameter k1 and n1, used to describe strain stiffening, havethe largest contribution to the variance of the output Oss.However, it is also found that interactions exist between thethree effects. An example is that the parameters for the strainstiffening behavior have a large contribution to the output

Ovi that quantifies the nonlinear viscous dissipation. Twoparameters, b and n2, have small contributions to all the out-puts as shown by small total indices (Fig. 12). This is anindication that they could be fixed within their uncertaintyinterval.

In this study, a LAOS deformation is used to study thenonlinear viscoelastic properties of fibrin. Other protocolscould have been used, such as strain ramps (Schmolleret al. 2010) or prestress (Piechocka et al. 2010). An advan-tage of the LAOS deformation used here is that it mim-ics the large oscillatory deformation occurring in bloodvessels. Furthermore, the analysis in terms of Lissajous–Bowditch plots applied here illustrates that this method canbe used to distinguish various nonlinear effects that occur.The model is developed such that it can describe any arbi-trary deformation history because it uses the time-dependentstrain as an input. Therefore, the model could be used todescribe other deformation protocols as mentioned above aswell.

The model developed works well in describing the non-linear behavior of the fibrin networks. The model is flexibleand can easily be adapted to describe the mechanical behav-ior of other biopolymers that show similar behavior such ascollagen (Kurniawan et al. 2012; Münster et al. 2013), hag-fish slime (Ewoldt et al. 2011), keratin filaments (Ma et al.1999), gluten gel (Ng et al. 2011) and also soft tissues such asskin (Lamers et al. 2013). Although the mechanical behav-ior of these materials is complex, the model is deliberatelykept simple to make it suitable for numerical simulationsof blood clot formation (Storti et al. 2014) and to combineit with a model for fibrin network maturation (van Kempenet al. 2014).

6 Conclusion

A constitutive model has been developed to describe thetime-dependent, nonlinear viscoelastic behavior of fibrin net-works. The model is developed using experimental results ofa LAOS deformation and describes the observed softening,strain stiffening and increasing viscous dissipation that occurduringmultiple deformation cycles. Furthermore, an increas-ing stiffness that takes place after the LAOS deformation iscaptured. The model is able to describe all these features,and its generality makes it suitable to be applied to othermaterials showing similar behavior.

Acknowledgments We thank Wouter Donders and Wouter Hubertsfrom Maastricht University for help with the sensitivity analysis.

OpenAccess This article is distributed under the terms of theCreativeCommons Attribution License which permits any use, distribution, andreproduction in any medium, provided the original author(s) and thesource are credited.

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References

Abhilash A, Purohit P, Joshi S (2012) Stochastic rate-dependent elastic-ity and failure of soft fibrous networks. Soft Matter 8:7004–7016

Averett R, Menn B, Lee E, Helms C, Barker T, Guthold M (2012) Amodular fibrinogen model that captures the stress-strain behaviorof fibrin fibers. Biophys J 103:1537–1544

Barnes H, Hutton J, Walters K (1989) An introduction to rheology.Elsevier, Amsterdam

Broedersz C, Kasza K, Jawerth L, Münster S, Weitz D, MacKin-tosh F (2010) Measurement of nonlinear rheology of cross-linkedbiopolymer gels. Soft Matter 6:4120–4127

Brown A, Barker T (2014) Fibrin-based biomaterials: modulation ofmacroscopic properties through rational design at the molecularlevel. Acta Biomaterialia 10:1502–1514

Brown A, Litvinov R, Discher D, Purohit P, Weisel J (2009) Multiscalemechanics of fibrin polymer: gel stretching with protein unfoldingand loss of water. Science 325:741–744

Chernysh I, Nagaswami C, Purohit P, Weisel J (2012) Fibrin clotsare equilibrium polymers that can be remodeled with proteolyticdigestion. Scientific Reports

Cilia La Corte A, Philippou H, Ariëns R (2011) Role of fibrin structurein thrombosis and vascular disease. Adv Protein Chem Struct Biol83:75–127

Ewoldt R, Hosoi A, McKinley G (2008) Newmeasures for characteriz-ing nonlinear viscoelasticity in large amplitude oscillatory shear.J Rheol 52(6):1427–1458

Ewoldt R, Winegard T, Fudge D (2011) Non-linear viscoelasticity ofhagfish slime. Int J Non Linear Mech 46:627–636

Huberts W, Donders W, Delhaas T, van de Vosse F (2014) Applicabil-ity of the polynomial chaos expansion method for personalizationof a cardiovascular pulse wave propagation model. Int J NumerMethods Biomed Eng 30(12):1679–1704

Hudson N, Houser J, O’Brien III ET, Taylor II RM, Superfine R, LordS, Falvo M (2010) Stiffening of individual fibrin fibers equitablydistributes strain and strengthens networks. Biophys J 98:1632–1640

Hyun K, Wilhelm M, Klein CO, Cho KS, Nam JG, Ahn KH, Lee SJ,Ewoldt RH, McKinley GH (2011) A review of nonlinear oscilla-tory shear tests: analysis and application of large amplitude oscil-latory shear (laos). Prog Polym Sci 36(12):1697–1753

Kang H, Wen Q, Janmey P, Tang J, Conti E, MacKintosh F (2009)Nonlinear elasticity of stiff filament networks: strain stiffening,negative normal stress, and filament alignment in fibrin gels. JPhys Chem B 113(12):3799–3805

van Kempen T, Bogaerds A, Peters G, van de Vosse F (2014) A consti-tutive model for a maturing fibrin network.Biophys J 107:504–513

Kim E, Kim O, Machlus K, Liu X, Kupaev T, Lioi J, Wolberg A, ChenD, Rosen E, Xu Z, Alber M (2011) Correlation between fibrinnetwork structure and mechanical properties: an experimental andcomputational analysis. Soft Matter 7:4983–4992

Kim O, Litvinov R, Weisel J, Alber M (2014) Structural basis for thenonlinear mechanics of fibrin networks under compression. Bio-materials 35(25):6739–6749

Kurniawan N, Wong L, Rajagopalan R (2012) Early stiffening andsoftening of collagen: interplay of deformation mechanisms inbiopolymer networks. Biomacromolecules 13:691–698

Lamers E, van Kempen T, Baaijens F, Peters G, Oomens C (2013)Large amplitude oscillatory shear properties of human skin. JMechBehav Biomed Mater 26:462–470

Liu W, Jawerth L, Sparks E, Falvo M, Hantgan R, Superfine R, LordS, Guthold M (2006) Fibrin fibers have extraordinary extensibilityand elastictiy. Science 313:634

Lorand L (2005) Factor XIII and the clotting of fibrinogen: from basicresearch to medicine. J Thromb Haemost 3:1337–1348

Ma L, Xu J, Coulombe P, Wirtz D (1999) Keratin filament suspen-sions show unique micromechanical properties. J Biol Chem274(27):19,145–19,151

Macosko C (1994) Rheology: principles, measurements and applica-tions. Wiley, London

Münster S, Jawerth L, Leslie B, Weitz J, Fabry B, Weitz D (2013)Strain history dependence of the nonlinear stress response of fibrinand collagen networks. Proc Natl Acad Sci USA 110(30):12,197–12,202

Ng T, McKinley G, Ewoldt R (2011) Large amplitude oscillatory shearflow of gluten dough: a model power-law gel. J Rheol 55(3):627–654

Onck P, van Dillen TKT, van der Giessen E (2005) Alternative explana-tion of stiffening in cross-linked semiflexible networks. Phys RevLett 95(178):102

Piechocka I, Bacabac R, Potters M, MacKintosh F, Koenderink G(2010) Structural hierarchy governs fibrin gel mechanics. BiophysJ 98:2281–2289

Ryan E, Mockros L, Weisel J, Lorand L (1999) Structural origins offibrin clot rheology. Biophys J 77:2813–2826

Saltelli A (2002) Making best use of model evaluations to computesensitivity indices. Comput Phys Commun 145:280–297

Schmoller K, Fernández P, Arevalo R, Blair D, Bausch A (2010) Cyclichardening in bundled actin networks. Nat Commun 1(134)

SemmrichC, LarsenR,BauschA (2008)Nonlinearmechanics of entan-gled F-actin solutions. Soft Matter 4:1675–1680

Shah J, Janmey P (1997) Strain hardening of fibrin gels and plasmaclots. Rheol Acta 36(3):262–268

Sobol I (2001) Global sensitivity indices for nonlinear mathemati-cal models and their monte carlo estimates. Math Comput Simul55:271–280

Storm C, Pastore J, MacKintosh F, Lubensky T, Janmey P (2005) Non-linear elasticity in biological gels. Nature 435:191–194

Storti F, van Kempen T, van de Vosse F (2014) A continuum modelfor platelet plug formation and growth. Int J Numer Meth BiomedEng 30:634–658

WeigandtK, Porcar L, PozzoD (2011) In situ neutron scattering study ofstructural transitions in fibrin networks under shear deformation.Soft Matter 7:9992–10000

Weisel J (2008) Enigmas of blood clot elasticity. Science 320:456–457Wilhelm M (2002) Fourier-transform rheology. Macromol Mater Eng

287:83–105

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