The application of large amplitude oscillatory stress in a study of fully formed fibrin clots

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I. INTRODUCTION
Characterisation of the nonlinear rheological properties of biopolymer gels has important implications for understanding their structure-function relationships under large stress and strain regimes.A salient example is a fibrin gel that provides the structural backbone of a blood clot and serves to perform a haemostatic function by preventing the loss of blood at a site of injury to a vessel. 1 A fibrin gel is formed following the conversion of fibrinogen to fibrin by the action of the catalytic enzyme thrombin and subsequent polymerisation of fibrin to produce a three-dimensional gel network of branching fibres. 1 The fibres are formed from protofibrils that may aggregate both laterally and longitudinally.Conditions that favour lateral aggregation produce thicker fibres with relatively low degree of branching, while conditions that inhibit lateral aggregation yield thin fibres with more branching. 13][4] In disease, the undesirable formation of fibrin clots can block blood vessels, thus preventing blood supply to tissues.Blockages in the veins are termed a Venous Thromboembolism (VTE) and are responsible for a significant global disease burden. 5a) Author to whom correspondence should be addressed: k.m.hawkins@ swansea.ac.uk Clots formed in the deep veins (i.e., a deep vein thrombosis) can embolise (i.e., break off), travel in the bloodstream, and lodge in the pulmonary arteries leading to a lack of blood supply to the lungs (i.e., a pulmonary embolism). 6,7n vivo, shear forces due to the flow of blood may induce fracture or breakage of the blood clot and is a potential mechanism for embolism. 8,9The wall shear stress in veins can span several magnitudes depending on the size, branching, and number of vessels and can be elevated in disease states. 10icrofluidic channel approaches studying clot growth have shown that, in a "pressure-relief mode" whereby the flow of blood is allowed to divert to other channels, the shear stress increases as the flow squeezes past the growing thrombus and then rapidly decreases at approximately 80% of full occlusion. 9Thus, it is important to consider the effects of both the vast range and progressive increase in stress experienced by fully formed clots.Marked changes in the structure and rheological properties of the clot can arise as a consequence of shear stress through several different mechanisms including strain-stiffening of the fibrin network, 11 strain-stiffening of individual fibrin fibres, 12,13 protein unfolding of stretched fibres, 14 unfolding of fibrin monomers, 15 yielding of branching points, 16 slippage of protofibrils within fibres, 17 and breakage of individual fibrin fibres. 18,19The propensity of the formed blood clot to macroscopically fracture may be influenced by all of the aforementioned mechanisms.
There have been several methods used to characterise the rheological properties of fibrin clots under relatively large stresses and strains.These include conventional rheological approaches involving oscillatory techniques under conditions that provoke strain-stiffening behaviour of the network and the subsequent reporting of storage moduli. 11,20However, it is important to recognise that, under such conditions, the resulting waveform is non-sinusoidal and thus does not pertain to the theory of linear viscoelasticity. 21Other workers have exploited inertio-elastic oscillations in controlled stress rheometry to provide a measurement of tangent moduli. 22Several studies have reported the application of a pre-stress followed by the superposition of small amplitude oscillatory shear stress to probe the rheological properties in terms of differential moduli. 15,23,24Using this approach, Piechocka et al. 15 identified distinct regimes of elastic response.Large Amplitude Oscillatory Shear (LAOS) has been used in a controlled strain manner to perform repeated large strain loadings on fibrin clots and the data were presented using Lissajous-Bowditch curves. 17,25Using repeated cyclic strain loadings, Münster et al. 17 showed the ability of un-crosslinked fibrin clots to adapt to stress through the slippage of protofibrils.Moreover, the LAOS studies of van Kempen et al. 25 revealed several nonlinear features of fibrin clots, including time-dependent softening, intra-cycle strain stiffening, and increased viscous dissipation with increased strain, illustrating the usefulness of LAOS in capturing rich nonlinear behaviour.
Controlled stress LAOS (LAOStress) is arguably a more suitable technique than controlled strain LAOS for studying the nonlinear properties of fully formed fibrin clots.LAOStress is physiologically relevant insofar that it affords progressive increases in stress to be imposed on the sample which can be related to the hemodynamic forces experienced by clots in vivo.LAOStress has been previously used to characterise the nonlinear rheology of polymer solutions and gels. 26,27In the present study, we investigate the suitability of LAOStress for studying the nonlinear behaviour of fibrin clots up to the point of macroscopic fracture.Furthermore, we combine alternating LAOStress and linear viscoelastic measurements in order to study reversible and irreversible changes in fibrin clot microstructure as a consequence of different levels of stress.

A. Materials
Purified, plasminogen depleted human fibrinogen at least 95% clottable and human thrombin were obtained from Enzyme Research Laboratories, UK.The fibrinogen was made up to a stock solution of 44.54 mg/ml by the addition of Trisbuffer saline, TBS (20 mM Tris, pH 7.4% and 0.9% NaCl, Sigma Aldrich, UK) and left to fully dissolve in a water bath at 37 • C. Thrombin was dissolved in distilled water and made up to a stock solution of 500 NIH unit/ml.These stock solutions were aliquoted and frozen at 80 • C until required.A 1M stock solution of CaCl 2 (Sigma Aldrich, UK) was also stored at 4 • C. The appropriate volumes of TBS, fibrinogen, and CaCl 2 were mixed together to make a (final) concentration of 4 mg/ml fibrinogen and 0.005M CaCl 2 .Clotting was initiated by adding the required volume of thrombin to produce a (final) concentration of 0.1 NIH/ml.After appropriate mixing, the sample was transferred onto the lower plate of the rheometer.

B. Rheometry
Rheological measurements were conducted using an AR-G2 controlled stress rheometer (TA Instruments, UK) fitted with a cone and plate (40 mm cone, 0.0351 rad) measuring system.The lower plate was maintained at a temperature of 24 • C for all measurements.In order to reduce effects of evaporation, the sample was sealed with a thin layer of low viscosity (10 mPa s) silicone oil.The sample was allowed to clot between the rheometer plates for a period equivalent to 40 gel times, after which the network is fully formed. 28,29AOStress measurements were performed on the fully formed clot by applying continuous stress waveforms at a stress amplitude in the nonlinear viscoelastic range of the sample and a frequency of 1 Hz until the sample reached a steady state.Following the acquisition of LAOStress data, the sample was probed by small amplitude oscillatory stress measurements at a stress level of 5 Pa and a frequency of 1 Hz until it reached a steady state in order to provide a value of linear elastic compliance J .Adherence of the Small Amplitude Oscillatory Shear (SAOS) measurements to the linear viscoelastic range of the sample was confirmed by the absence of a third harmonic response.The process of alternating LAOStress measurements at incrementally increasing stress amplitudes and SAOS measurements was performed until the sample macroscopically fractured leading the upper rheometer plate to overspeed.A criterion was imposed throughout all experiments that steady state was reached once the difference in the strain response between consecutive waveforms was less than 2%.This criterion ensured that point symmetry was observed in the Lissajous-Bowditch curves obtained from the LAOStress tests and that irreversible damage following relaxation of the sample was probed by SAOS.The sample stress and strain response waveform datasets were obtained by the rheometer software (TRIOS) in a transient data collection mode at the software's default sampling rate of 978 Hz.

C. Analysis of LAOStress waveforms
A quantitative framework for the analysis of LAOStress data in terms of deducing the nonlinear compliances has been described by Dimitriou et al. 26 and Läuger and Stettin. 27This includes the minimum-stress elastic compliance J M and the large-stress elastic compliance J L (see Fig. 1) which can be calculated from Lissajous-Bowditch curves of strain versus stress. 26or an imposed sinusoidal stress, σ (t) = σ 0 cos ωt, the minimum-stress elastic compliance, J M , can be defined by and the large-stress elastic compliance, J L , can be defined by where σ 0 is the applied stress amplitude, γ is the resulting strain, c n is the nth order Chebyshev coefficient, in which the letter "c" represents the compliance nonlinearities, and J n is the nth compliance. 26igure 1 shows an illustration of a Lissajous-Bowditch curve of stress versus strain data obtained from a LAOStress measurement.J M is calculated from the gradient of the tangent at zero stress.J L is calculated from the gradient of a straight line connecting the origin to the point of maximum stress.was calculated for all LAOStress measurements and In 0.05 throughout, indicating that instrument inertia alone did not significantly affect the total torque signal.However, at increasing levels of applied stress, the sample stress and strain response waveforms contain artefacts due to a source of ringing.To our knowledge, this is the first observation of ringing phenomena in a LAOS measurement.This ringing is possibly attributable to a resonance caused by a coupling of inertial and elastic forces at relatively large levels of applied stress. 30n an attempt to eliminate the effects of ringing, the sample stress and strain response waveform datasets (obtained from TRIOS) were smoothed to a cubic spline function (specifically a piecewise cubic polynomial computed from a smoothing parameter p = 0.999 992 52) using the Curve Fitting Toolbox in MATLAB.The datasets were then reconstructed from the smoothing function by sampling at a rate identical to the original dataset (978 Hz). Figure 3 displays an example of the resulting smoothed waveforms.The process of smoothing eliminates any noticeable effects of ringing.However, in order to ensure that nonlinearities due to the sample response (which manifest as odd harmonics) are not affected by the process of smoothing, the datasets were subjected to Fourier analysis and the resulting frequency spectra were compared.To this end, the strain and stress datasets were trimmed to contain 15 000 data points which contain more than 15 complete cycles.The datasets were windowed using a Hanning window function and zero padded with 5000 data points in order to adjust the resolution of the frequency spectrum to 0.05 Hz.The datasets were analysed in MATLAB using a discrete Fourier transform.

A. LAOStress data processing
Figure 4 shows the frequency spectra of the sample stress and strain response waveforms for the original (i.e., unprocessed) data and smoothed data at three different levels of applied stress (a) 100 Pa, (b) 500 Pa, and (c) 1500 Pa.The frequency spectra of the strain response are dominated by the fundamental (imposed) frequency and contain odd harmonics, confirming that the measurement is within the nonlinear viscoelastic regime. 21The magnitude of the even harmonics was negligible confirming that the sample was non-transient and had reached a steady state. 31It is noticeable that some harmonic distortion in the sample stress waveform is apparent, including at the lowest stress shown where there was no ringing encountered.These nonlinearities will be transmitted to, and replicated in, the corresponding strain response.However, the magnitude of the odd harmonics in the strain response is much greater than the sample stress indicating that the rich nonlinearity of the sample is being captured.Importantly, the frequency spectra of the smoothed waveforms are identical to their original waveforms providing evidence that the process of smoothing does not eliminate the essential information originating from the nonlinear response of the sample.Henceforth, we use the smoothed stress and strain waveforms as the basis of the Lissajous-Bowditch curves and calculation of nonlinear compliances.
Figure 5 shows a comparison of Lissajous-Bowditch curves constructed from original and smoothed datasets.
At relatively low levels of stress [Fig.5(a)], the Lissajous-Bowditch curves appear identical.Figures 5(b) and 5(c) show that the ringing encountered in the original waveforms introduces secondary loops in the corresponding Lissajous-Bowditch curves.These secondary loops are eliminated by the process of smoothing.Furthermore, at these relatively large stress levels, inspection of the original and smoothed curves shows profound differences around the region pertaining to the minimum stress.Thus, any calculated values of J M are highly sensitive to ringing artefacts.compliances J L and J M gradually decrease with increasing levels of stress and strain indicating stress stiffening of the sample.This stress stiffening is also captured in individual LAOStress measurements at discrete levels of stress by a dominance of J M over J L over the entire range of stress levels studied.This behaviour is expected given that individual fibrin fibres' strain stiffens and is highly extensible. 12,13,15Our findings are similar to the LAOS controlled strain studies of van Kempen et al. 25 that reported intra-cycle stiffening and softening during continuous application of large strain deformations to fibrin clots.However, an advantage of LAOStress measurements is that the fibrin clot is allowed to reach the steady state under a fixed level of oscillatory stress.This is more physiologically relevant insofar that the stress is imposed on the clot. 9,10nalysis of the nonlinear and linear viscoelastic behaviour of the sample [in Figs.6(c) and 6(d), respectively] reveals four distinct regions.In region 1 (σ 0 < 150 Pa), the sample shows increasing tendency to stress stiffen as evidenced by an increase in the difference between J M and J L with increasing levels of stress.The sample returns to its original linear viscoelastic behaviour following a decrease in applied stress indicating full recovery of the structure.In region 2 (150 < σ 0 < 400 Pa), the linear elastic compliance gradually increased following the application of increasing levels of LAOStress, indicating irreversible structural rearrangements.This stress-softening, which is also evident by close inspection of the curvature of the oscillatory strain response in the LAOStress experiments, is likely to be due to yielding of branching points leading to a slackening of the network. 16he behaviour encountered in region 3 (450 < σ 0 < 2000 Pa) is similar to region 1 insofar that the sample returns to a level of linear elastic compliance that is independent of the stress applied in the LAOStress measurement.This finding is similar to the study of Piechocka et al., 15 which showed that the differential modulus is independent of values of pre-stress within a specific stress regime.In region 3, the structure of the sample has been permanently re-arranged by the stresses encountered in region 2, resulting in a sample that has the ability to resist structural changes when exposed to further increases in stress.Furthermore, in the latter stages of region 3 (i.e., at long times), there is generally a decrease in linear elastic compliance following the application of increasing levels of LAOStress, indicating that effects due to ageing of the sample (possibly due to Factor XIII mediated cross-linking) are greater than any stress induced structural damage.In region 4 (σ 0 > 2000 Pa), the ability of the sample to resist structural changes at increasing levels of stress diminishes as evidenced by an increase in linear elastic compliance.Further increases in the applied levels of stress lead to fracture of the sample network causing the rheometer upper plate to overspeed.

IV. CONCLUSIONS
We investigated the applicability of LAOStress in studying the nonlinear viscoelastic properties of fully formed fibrin clots.The stress and strain waveforms obtained from the LAOStress measurements contained artefacts due to a source of ringing.This is the first observation of ringing in a LAOS measurement.We showed that the artefacts due to ringing can be effectively eliminated whilst retaining the essential harmonic information related to the nonlinear viscoelastic response of the sample by fitting the original waveforms to a smoothing spline.This has allowed a characterisation of the nonlinear compliances of fibrin clots up to the point of fracture.Combined with supplementary SAOS measurements, various regions of nonlinear viscoelastic behaviour were identified which can be related to previous findings involving large scale deformations of fibrin clots.The quantitative framework of LAOStress provides additional information of fibrin clot nonlinear behaviour, and the approach is more physiologically relevant than LAOS in a controlled strain mode.Further work will involve the application of LAOStress in studies of abnormal clots such as those formed in blood taken from patients with venous thromboembolism.

FIG. 1 .
FIG.1.A Lissajous-Bowditch curve obtained from a LAOStress experiment and an illustration of the calculation of nonlinear compliances, the minimumstress elastic compliance J M (red) and the large-stress elastic compliance J L (blue).

Figure 2
Figure 2 displays an example of sample stress waveforms (upper) and strain response waveforms (lower) at an applied stress level of (a) 100 Pa (slightly into the nonlinear viscoelastic range of the sample) and two higher levels of applied stress (b) 500 Pa and (c) 1500 Pa.The dimensionless inertia

FIG. 2 .
FIG. 2. Sample stress (upper) and strain response (lower) waveforms obtained from the rheometer (TRIOS) software at input stress levels of (a) 100 Pa, (b) 500 Pa, and (c) 1500 Pa.These waveforms consist of discrete data points at a sampling rate of 978 Hz.

Figure 6
Figure6shows the results of the alternating LAOStress and SAOS measurements.Figure6(b) displays the magnitude of the strain response to the applied stress versus time profile shown in Fig.6(a).The time period for which each stress is applied depends on the duration of the measurement to reach a steady state.Figure6(c)shows that values of the nonlinear

Figure 6 (
Figure6shows the results of the alternating LAOStress and SAOS measurements.Figure6(b) displays the magnitude of the strain response to the applied stress versus time profile shown in Fig.6(a).The time period for which each stress is applied depends on the duration of the measurement to reach a steady state.Figure6(c)shows that values of the nonlinear