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Strain-rate sensitivity of the lateral collateral ligament of the knee

Timothy J Bonner1,2, Nicolas Newell2, Angelo Karunaratne2, Andy D Pullen3,

Andrew A Amis4,5, Anthony MJ Bull2, Spyros D Masouros2

1The Academic Department of Military Surgery and Trauma, The Royal Centre for Defence

Medicine, Birmingham, B15 2SQ

2Department of Bioengineering, Imperial College London, London SW7 2AZ, UK

3Department of Civil and Environmental Engineering, Imperial College London, London SW7

2AZ, UK

4Department of Mechanical Engineering, Imperial College London, London SW7 2AZ, UK

5Department of Musculoskeletal Surgery, Imperial College London, London W6 8RF, UK

Corresponding author:

Dr Spyros Masouros

Lecturer in Trauma Biomechanics

Department of Bioengineering,

Imperial College London,

London SW7 2BP,

United Kingdom

Tel: +44 20 7594 2645

Fax: +44 20 7594 9817

E-mail: [email protected]

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Abstract

The material properties of ligaments are not well characterized at rates of deformation

that occur during high-speed injuries. The aim of this study was to measure the material proper-

ties of lateral collateral ligament of the porcine stifle joint in a uniaxial tension model through

strain rates in the range from 0.01-100/s. Failure strain, tensile modulus and failure stress were

calculated. Across the range of strain rates, tensile modulus increased from 288 to 905 MPa and

failure stress increased from 39.9 to 77.3 MPa. The strain-rate sensitivity of the material proper-

ties decreased as deformation rates increased, and reached a limit at approximately 1/s, beyond

which there was no further significant change. In addition, time resolved microfocus small angle

X-ray scattering was used to measure the effective fibril modulus (stress/fibril strain) and fibril to

tissue strain ratio. The nano scale data suggest that the contribution of the collagen fibrils to-

wards the observed tissue-level deformation of ligaments diminishes as the loading rate in-

creases. These findings help to predict the patterns of limb injuries that occur at different speeds

and improve computational models used to assess and develop mitigation technology.

Keywords. Ligament, stress, modulus, strain rate, injury

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1. Introduction

Human ligament injuries are common and can cause significant morbidity and long-term

disablement [1]. The EUROCOST reference group estimate the incidence of knee ligament injur-

ies at 0.04% population/year with associated treatment costs alone at an average of €1,727 per

injury [2]. The types of joint injuries that occur may vary according to health, age, sex, anatomy,

mechanism of trauma and rate of loading [3]. Prevention of significant joint injuries requires us

to understand why different patterns of injury may occur with different traumatic mechanisms

and rates of loading. Computational modelling of joint injuries is one method that may be useful

to help predict different patterns of injury and assess mitigation technologies. However, reliable

biomechanical measurements of the connective tissues of joints are required if the models are to

be accurate and useful.

Ligaments are viscoelastic materials made of collagen fibres, which change in strength

and stiffness relative to their rate of loading [4,5]. Tensile modulus and failure stress are useful

measurements to compare the material properties of different ligaments. These material proper-

ties are important input parameters to computational models of human joint injuries. Previous

laboratory studies have found that the tensile modulus and failure stress of ligaments both in-

crease as the rate of loading increases [6-9]. However, many of the previous studies have focused

on the failure characteristics at quasi-static loading rates or assessed only a few different loading

rates [6,10-12].

This strain-rate dependent material behaviour of ligament tissue cannot be understood

without considering the hierarchical nature of the structure. Small angle X-ray diffraction

(SAXD) has been performed previously on collagenous tissues such as tendons, bones and cartil-

age in an attempt to quantify the viscoelastic properties of the tissues at a fibrillar level and util-

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ize them to explain their typical macroscopic behaviour [27-29]. Fratzl et al. proposed a simple

model explaining why the ratio of fibril-to-tissue strain increases with strain rate in the quasi-

static range. They suggest that the proteoglycan-rich matrix becomes stiffer due to an increase of

the viscous component as strain rate increases. Unfortunately, the maximum strain rate at which

they tested was 0.001/s and so their observations are not adequate to demonstrate or explain po-

tential changes in strength and modulus at strain rates experienced at injury.

Whilst work at slow rates is useful to understand behaviour in normal joint function and

to choose replacement grafts, the application of these results to high-speed injury modelling may

not be valid; significant error may occur if low strain-rate material properties are applied to sim-

ulations of traumatic injury. The limitations of previous work are likely to be caused by the tech-

nical difficulties of measuring stress and strain at rates that simulate high-speed injuries, such as

motor vehicle collisions or battlefield injuries due to blast [15,16].

The aim of this study was to investigate the material properties of ligaments in a uniaxial

tension model at strain rates in the range from 0.01-100/s. A porcine stifle joint ligament experi-

mental model was designed to simulate the strain rates that may occur during a full range of dif-

ferent human knee ligament loading. The hypothesis was formulated that the strain-rate sensitive

material properties of a ligament would diminish as strain rate increased. Studying ligament

properties over a large order of magnitude of strain rates also provides an insight into the differ-

ent structural explanations for their viscoelasticity. Furthermore, time resolved synchrotron small

angle X-ray scattering on human ligaments was used to investigate the deformation mechanisms

at the nanoscale in order an attempt to explain the strain-rate dependent behaviour of ligaments.

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2. Methods

2.1. Specimen preparation

Ligaments of the porcine stifle joint were selected because of their similarity in morpho-

logy, size, structure, material properties and physiological loading to the human knee joint [18].

Sixty porcine hind limbs were delivered to the laboratory on the day of slaughter from a local

abattoir. Excess muscle bulk was removed from the limbs, which were then stored at -20°C. All

limbs were utilized within one month of slaughter to minimize any potential deterioration in their

mechanical properties [17]. All limbs were from healthy female large-white pigs, aged between 9

and 12 months. The demographics of the pigs were controlled to limit the physiological variation

in material properties, which is known to occur between sexes, age groups, pig breeds and in un-

healthy subjects [19,20].

Each hind limb was thawed at room temperature on the day of testing. The lateral collat-

eral ligament (LCL) of the porcine stifle joint was isolated by removing skin, muscle, other joint

ligaments and tibia, thus leaving the femur, LCL and fibula intact. A hand saw was used to cut a

15 × 15 × 25 mm bone block around the femoral attachment of the LCL. A similar bone block

was created with the fibula by removing its rounded proximal margin and dividing it transversely

at 40 mm in length. A thin longitudinal round segment of ligament was isolated along the pos-

terior margin of each LCL, such that each ligament’s fascicles were easily aligned, similar in

length and would reliably fail in its mid-substance. The unwanted anterior segment was removed

by separating the ligament via blunt dissection in line with the fascicles, and divided transversely

both proximally and distally when the fascicles could no longer be easily separated; thus ensur-

ing no structural damage. This created a test specimen with a long thin middle section of a relat-

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ively constant cross-sectional area, with a broad anchor at either end made of the bone blocks

and fibrocartilage transition zone (Figure 1).

Cross-sectional area was measured using a previously validated technique for use in soft

tissues [21]. Each specimen was held under 1 N of tension and the mid-substance of the ligament

was cast in a quick-setting, stiff alginate paste (Blueprint® cremix, Dentsply DeTrey, Germany).

The solid alginate paste was cut perpendicular to the long axis of the construct after removal of

the ligament. Digital photographs were taken of the cut sections of alginate paste at three differ-

ent sites. Each photograph was converted into binary code, based on whether or not each pixel

contained an image of the ligament cast. The number of pixels in each photograph was counted

using a custom computer code (MatLAB, MathWorks Inc., Natick, MA, USA). The cross-sec-

tional area was calculated by comparing the mean of the three pixel counts against a calibration

photograph of a shape with a known cross section.

The bone blocks were placed into cylindrical aluminium pots and secured by alignment

screws. Positioning of the ligament sample was carefully adjusted such that it was coaxial with

the uniaxial tension test. The bone blocks were then set in polymethyl-methacrylate (PMMA)

bone cement. Petroleum jelly and saline soaked gauze were used to keep the ligament hydrated

and particularly to protect its attachments from the heat generated during cement polymerization.

Multiple small black dots were made across both ends of the ligament with permanent black ink

(Staedtler Ltd., UK) during set up on the tensile testing machines. The samples were then

sprayed with water to ensure they did not dry out before testing. The length of each ligament

sample was measured once with digital callipers before testing to provide an estimate of the de-

formation rates needed to achieve the required strain rates. All tests were performed at room tem-

perature.

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2.2. Tensile testing

Tests were carried out to achieve target strain rates of 0.01, 0.1, 1, 10, and 100/s.

Quasi-static tensile tests

A screw-driven electro-mechanical (5866; Instron, Canton, MA, USA) and a servo-hy-

draulic (8872; Instron, Canton, MA, USA) materials testing machines were used for tensile tests

at 0.01-0.1/s and 1/s, respectively. Pre-conditioning of the ligaments was performed for the

quasi-static tests with cyclic loading between 1 and 10 N at 10 mm/min, and repeated five times,

then held at 0N for 10 seconds [17]. The specimens were loaded to failure at extension rates of

0.47, 4.7 and 47 mm/s to achieve strain rates in the region of 0.01, 0.1, and 1/s, respectively.

Two specimens were extended at 0.39 mm/s and 0.55 mm/s because the ligaments were shorter

and longer, respectively, than the rest. Force-extension data was measured simultaneously using

the built in load cell and extensometer at a sampling rate of 50 Hz, then recorded with the ma-

chine’s accompanying materials testing software (Bluehill® v2.11, Instron, High Wycombe,

UK)

Impact tensile tests

Tensile tests at strain rates in the region of 10 and 100 /s were performed using an Instron

Dynatup 9250-HV spring-assisted drop-weight rig (Instron, High Wycombe, UK). A custom-

made impact tensile adaptor (ITA) was manufactured for the experiment (Figure 2). The upper

aluminium pot was attached by a spherical rod-end connector to a fixed steel crossbeam. The

lower pot was attached to an aluminium rectangular fixture, free to move vertically in the axis of

the tensile test, which was rapidly accelerated by a 7.45 kg impactor with a 50 mm diameter head

for each test. The velocity of the impactor could be altered by changing its drop height, or with

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the addition of accelerator springs. A pair of diametrically-opposed bi-axial strain gauge rosettes

(FCA-6; Techni Measure, Warwickshire, UK) were bonded to the upper aluminium pot in a

Wheatstone full-bridge configuration in order to measure load. The strain gauge output was cal-

ibrated in a series of preliminary tests against known loads. A PXIe data acquisition system in

conjunction with a custom-written LabVIEW® software program (NI instruments, Austin, TX,

USA) was used to record strain gauge output with a sampling rate of 16 kHz – 2 MHz. Strain

rates in the region of 10 and 100/s were achieved with an impactor velocity at impact of 1 m/s

and 8-12 m/s, respectively. The impactor velocity for the 100/s tests was increased from 8 to

12 m/s after six tests because the average strain rate was less than the target rate.

2.3. Data Analysis

Strain was measured directly from the mid-substance of each ligament using high-speed

photography (Phantom V12.1, frame rate 27 - 47000 fps). Two points were selected at either end

of the ligament and were tracked across frames using digital tracking software (Phantom Camera

Control Application 1.3, Vision Research, NJ, USA) by noting their pixel position at every

frame. For each frame, strain was calculated as the change in pixel separation of the pair of

points, divided by the pixel separation of the two points in a reference frame. This procedure was

repeated for an additional 2 pairs of points, and ligament strain was taken as the mean value of

the 3 pairs.

Strain rate was calculated as the gradient of the linear part of the strain-time curve for

each test. Stress was calculated as the recorded force over the calculated original area. Stress-

strain curves were created and used to calculate the tensile modulus, and stress and strain at fail-

ure for each sample. Pearson’s correlation coefficient was calculated to establish the linear rela-

tionship between time and strain before failure. Changes across the range of strain rates of strain

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at failure, stress at failure and tensile modulus were examined using one-way analysis of vari-

ance (ANOVA) with post-hoc Bonferonni testing; p<0.05 was taken to indicate a significant

change.

2.4. Nano-scale experiments

In an attempt to understand the strain-rate sensitivity at a fibrillar level we conducted

mechanical testing utilizing synchrotron SAXD on human LCLs. Ethical approval had been se-

cured prior to testing from the local research ethics committee. We used the same experimental

protocol to previous studies [27,28,30,31] to measure fibrillar deformation at 0.001, 0.005, 0.01,

and 0.05/s. Twenty human LCL samples were prepared (5 per strain rate) and gripped in a micro-

tensile tester [30] that was mounted on a 2 axis motorised stage beam-line I22 at Diamond Light

Source, UK. A synchrotron X-ray beam (wavelength 0.886 Å, beam cross section 10 12 µm)

was used to measure the SAXD patterns, which were collected by a Pilatus detector system. The

sample-to-detector distance was 1 m. For each strain rate a SAXD pattern with 1 s exposure time

was collected at every 1% applied external (grip-to-grip) strain up to failure. The fibril strain was

measured as described elsewhere [27,28,29] by tracking the change in the D-periodicity

(~ 67 nm) of the meridional banding patterns in the collagen fibrils arising from the intrafibrillar

tropocollagen packing (Figure 3).

3. Results

The data-sets resulting from 41 ligament tests were available for analysis; the remaining

data-sets were not used due to either trigger failure during testing, inadequate image capture,

early sample failure around the bone block, or use in preliminary testing. One further sample was

excluded during the analysis because the sample was a gross outlier and is likely to have failed in

an atypical way.

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The time to failure of the ligaments ranged from less than 1 ms at the higher strain rates,

to greater than 20 s at the lower strain rates. The cross-sectional areas of the samples was

3.6 (SD = 1.1) mm2. The relationship between strain and time was linear for all tests (mean

R2 = 0.99 (SD = 0.02), p < 0.001); hence constant strain rate was achieved across all experi-

mental setups. Figure 4a presents typical stress-strain curves for each strain rate. A non-linear –

often called ‘toe’ – region was pronounced at lower strain rates, requiring up to 3-4% strain be-

fore a linear gradient was observed. A ‘yield’ point could be observed with some samples at the

two lowest strain rates, but complete structural failure occurred shortly afterwards with minimal

further deformation. Toe region and ‘yield’ point were not observed at the two fastest strain

rates. There was a general trend towards a negative relationship between the failure strain and

strain rate (Table 1). Figure 4b presents the mean curves at the five target strain rates of the

study. Failure stress was found to increase at the three slowest strain rates, but further change

was insignificant after approximately 1/s. This was supported by ANOVA tests, which showed

that there were statistically significant increases in tensile modulus and stress at failure between

each consecutive strain-rate group up to 0.94/s (p<0.05), but not beyond. This suggests that a

strain-rate sensitivity limit occurs at approximately 1/s.

Tensile modulus and failure stress as a function of strain rate are shown in Figure 5.

Tensile modulus and failure stress of the ligament increased by an average of approximately 3-

and 2-fold respectively over the strain rates tested, but the change occurred almost entirely over

the three slowest strain rates. The relationship between tensile modulus and failure stress with

strain rate was fitted with logarithmic and bilinear curves. The bilinear relationship was found to

fit the data with a smaller error than the logarithmic. Specifically:

Tensile modulus, E={384 ε̇+292 , ε̇<ε̇ 0=1.3/ s0.68 ( ε̇−ε̇ 0 )+790 , ε̇>ε̇ 0

in MPa (R2 = 0.76).

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Failure stress, σ f ={32 ε̇+44 , ε̇< ε̇0=0.97/s0.05 ( ε̇− ε̇0 )+75 , ε̇> ε̇0

in MPa (R2 = 0.68).

Tensile modulus, E=158.55 log10 ε̇+584.76 in MPa (R2 = 0.73).

Failure stress, σ f =9.97 log10 ε̇+63.88 in MPa (R2 = 0.62).

Figure 6 presents stress – tissue strain and fibril strain – tissue strain relationships for hu-

man ligament tissue from the SAXD experiments. Fibril strain for measured tissue strain signi-

ficantly reduced at higher strain rates compared to quasi static strain rates. Finally, Figure 7

shows the increase in collagen-fibril modulus (slope of the stress – fibril-strain curve) with strain

rate.

4. Discussion

The material properties of the lateral collateral ligament of the porcine stifle joint were

found to be sensitive to strain rates up to a limit of approximately 1/s, beyond which this effect

became insignificant. This is the first study to report the strain-rate dependency of the material

properties of ligament tissue across 5 orders of magnitude. The results at low strain rates (<1/s)

are in broad agreement with previous laboratory studies on the strain-rate sensitivity of animal

and human ligaments [6-14,22]. For example, Woo et al. found that the tensile failure stress of

the lapine medial collateral ligament (MCL) increased by up to 40% from 0.0001-2.2/s [13]. Ng

et al. reported that the failure stress of murine flexor digitorum superficialis tendon increased

from 50 to 70 MPa from 0.0005 to 1/s, respectively [9]. However, the strain-rate sensitivity limit

at high strain rates (>1/s) observed in this study has not been reported previously.

Earlier studies have recognized that the strain-rate sensitivity of ligaments diminishes as

they approach realistic traumatic injury strain rates [7,14]. Crisco et al. recognized that the stiff-

ness, k and failure load Ff of lapine stifle joint ligaments were insensitive to strain rate at two dif-

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ferent high loading rates (at 36.6/s: k = 145 N/mm, Ff = 434 N; at 140/s: k = 136 N/mm, Ff

= 443 N) [14]. Elsewhere, a logarithmic relationship between strain rate and failure stress, σf up

to strain rates of an order of magnitude less than in the current study has been reported for the

MCL of the lapine stifle joint (σ f =14.2 log10 ε̇+85.9) [7]. Interestingly, the logarithmic fit of our

data results in coefficients that are not very dissimilar to the study on the lapine MCL [7].

Many of the previous studies investigated the failure characteristics of a whole bone-liga-

ment-bone structural complex, rather than the material properties of the ligament itself. This ap-

proach renders some comparisons with this study difficult because the material properties of lig-

ament tissue could have been underestimated if the structures failed at the bony attachment of the

ligament or within the bone. Yamamoto et al. showed that the tibial attachment of the lapine

MCL demonstrated strain-rate sensitivity, but the failure load was less than that of the mid-sub-

stance of the ligament at all strain rates [8]. The strain-rate sensitivity limit described in our

study helps us to understand the changes in the site of failure in a ligament structural complex,

which has been described at different strain rates [10]. For example, a cross-over from bone fail-

ure to ligament failure may occur as strain rate increases if the tensile failure stress of a ligament

reaches its strain-rate sensitivity limit, but the strength of the surrounding bone continues to

change relative to strain rate. This finding is supported by the clinical literature, which shows

that the failure characteristics of ligaments vary amongst different ligaments and with different

mechanisms. For example, mid-substance tears of the anterior cruciate ligament of the knee are

very common in low speed sporting injuries, but both mid-substance rupture and avulsion frac-

tures are common in high speed injuries associated with knee dislocations after motor vehicle

collisions [23,24]. In contrast, Noyes et al. found in a laboratory study that avulsion fractures oc-

curred at very slow strain rates and mid-substance tears of the anterior cruciate ligament (ACL)

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were more common at higher speeds [10,25]. This apparent anomaly may be explained by either

differences in experimental design or, perhaps, due to variable strain-rate sensitivity between dif-

ferent tissues, ligaments, species, joints and health of the specimens. This might be supported by

studies that have observed that the strain-rate sensitivity varies between different ligaments from

around the same joint [7,12,26].

The stress-strain curves from the low strain-rate macroscale experiments presented here

are typical of ligament behaviour; they exhibit an initial non-linear, toe region up to 3-4% strain

followed by a linear region (Figure 4a). It has been suggested that the toe region is due to the

macroscopic uncrimping of collagen fibrils followed by sequential straightening of molecular

kinks (Figure 6a) within the gap regions of collagen fibrils [32-34]. The linear part of the stress-

strain curves corresponds to the elongation of the collagen fibrils (Figure 6b) and sliding between

collagen fibrils and fibres, probably controlled by the surrounding non-collagenous matrix [35].

The increase in axial periodicity is due to the stretching of the collagen triple helices and their

cross-links, with side-by-side gliding of adjacent molecules [36,37]. Furthermore, the pro-

teoglycan matrix between collagen fibrils also deforms during this linear phase and becomes

stiffer when the applied strain rate increases [29]. Therefore, the matrix is likely to be playing an

important role on the strain-rate sensitivity of ligaments [38]. We observed a reduction in the size

of the toe region with increasing strain rate (Figure 6b&c); this is likely due to the respective de-

crease in macroscale crimping; in contrast to the slower strain-rate synchrotron experiments we

conducted, at the higher strain rates elongation of the collagen fibrils was observed immediately

as the fibrils started to stretch (Figure 6c). Crucially, the synchrotron experiments we conducted

show that the collagen-fibril modulus increases with increasing strain rate in a logarithmic-type

fashion (Figure 7); this enables us to speculate the relationship between strain rate and tissue-

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level tensile modulus up to 1/s. At the higher loading rates the collagen fibrils debond from the

highly viscous matrix and fibrillar sliding occurs leading to macroscopic failure [29]. Collagen-

fibril load sharing is dependent on matrix crosslinking [39], and so the rapid disruption of inter-

connections would lead to a poor transmission of tensile stresses via crosslinking. This in turn

would promote fibrillar sliding, hence reduced fibrillar strain (Figure 6c).

The structure-function relationship for the strain-rate sensitivity of ligaments is also sup-

ported by evidence that increasing hydration increases the strain-rate sensitivity of the human pa-

tellar tendon [40]. The authors explain these findings by the interaction of the fluid and solid

phases of the tissue, leading to ‘hydraulic stiffening’ as strain rate increases.

An explanation for the observed strain-rate sensitivity limit requires further investigation.

However, the importance of the structural arrangement of fibrils, cross-links, water and pro-

teoglycan matrix at lower strain rates indicates that this may also determine why a strain-rate

sensitivity limit occurs.

The results from this study are important to understand the mechanics of human ligament

failure, to aid traumatic injury modelling and to develop mitigation strategies. If a strain-rate

sensitivity limit is common among all ligaments, then high strain-rate material properties of any

ligament can be measured at a one or two deformation rates, so long as the strain rate is known to

be above this limit. This would decrease significantly the number of future experiments that

would need to be performed, decrease the number of cadaveric samples required, and increase

the delivery of reliable biomechanical measurements for traumatic injury modelling. However,

the application of the observed relationships and strain-rate sensitivity limit to human injury

modelling requires some caution. The geometry of the ligament due to joint position affects sub-

stantially the number of collagen fibres recruited and should be considered in injury modelling

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[41]. Furthermore, the values for tensile modulus and failure stress of the porcine ligament we

obtained in this study are broadly similar to previous human studies. There is no concrete evid-

ence to support a strain-rate sensitivity limit of human ligaments at this time; nonetheless, the

nanoscale experiments presented here suggest that the human ligament might exhibit a similar

strain-rate sensitivity limit. Unfortunately, synchrotron experiments at strain rates high enough to

investigate the material behaviour at the nanoscale beyond the strain-rate dependency limit we

found in our macroscale experiments are not yet possible as current imaging technology is lim-

ited in terms of sampling rates. Synchronous testing at high strain rates and appropriate imaging

of the microstructure would elucidate the fibre-matrix relations that are likely responsible for the

macroscale strain-rate sensitivity limit we observed.

5. Conclusion

This study found that the material properties of the lateral collateral ligament of the por-

cine stifle joint were strain-rate dependent up to a limit of approximately 1/s, beyond which there

was no further significant change. This limit might be due to hydraulic stiffening and rapid dis-

ruption of matrix-fibril bonding at strain rates beyond 1/s that would impede the transmission of

tensile stresses through the collagen fibres. As human ligament injury occurs at high strain rates,

our findings and experimental techniques can form the basis for capturing human ligament ma-

terial properties at injurious strain rates.

6. Acknowledgements

Part of this work was conducted under the auspices of the The Royal British Legion

Centre for Blast Injury Studies at Imperial College London. Therefore the financial support of

the Royal British Legion for NN, AK, SM and AMJB is gratefully acknowledged. The financial

support of the Defence Medical Services (DMS) for TJB, of the Royal Centre for Defence Medi-

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cine (RCDM) for the acquisition of equipment and consumables, of BBSRC for NN, and of ABF

– The Soldiers’ Charity for SM are kindly acknowledged.

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18 Xerogeanes, J. W., Fox, R. J., Takeda, Y., Kim, H. S., Ishibashi, Y., Carlin, G. J. & Woo, S. L. 1998 A functional comparison of animal anterior cruciate ligament models to the human anterior cruciate ligament. Ann. Biomed. Eng. 26, 345–352. (doi:10.1114/1.91)

19 Germscheid, N. M., Thornton, G. M., Hart, D. A. & Hildebrand, K. A. 2011 A biomechani-cal assessment to evaluate breed differences in normal porcine medial collateral ligaments. J. Biomech. 44, 725–731. (doi:10.1016/j.jbiomech.2010.10.036)

20 Noyes, F. R. & Grood, E. S. 1976 The strength of the anterior cruciate ligament in humans and Rhesus monkeys. J. Bone Joint Surg. Am. 58, 1074–1082.

21 Goodship, A. E. & Birch, H. L. 2005 Cross sectional area measurement of tendon and liga-ment in vitro: a simple, rapid, non-destructive technique. J. Biomech. 38, 605–608. (doi:10.1016/j.jbiomech.2004.05.003)

22 Schenck, R. C., Kovach, I. S., Agarwal, A., Brummett, R., Ward, R. A., Lanctot, D. & Athanasiou, K. A. 1999 Cruciate injury patterns in knee hyperextension: a cadaveric model. Arthroscopy 15, 489–495. (doi:10.1053/ar.1999.v15.0150481)

23 Kennedy, J. C., Weinberg, H. W. & Wilson, A. S. 1974 The anatomy and function of the anterior cruciate ligament. As determined by clinical and morphological studies. J. Bone Joint Surg. Am. 56, 223–235.

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24 Twaddle, B. C., Bidwell, T. A. & Chapman, J. R. 2003 Knee dislocations: where are the le-sions? A prospective evaluation of surgical findings in 63 cases. J. Orthop. Trauma 17, 198–202.

25 Noyes, F. R. 1977 Functional properties of knee ligaments and alterations induced by im-mobilization: a correlative biomechanical and histological study in primates. Clin. Orthop. Relat. Res., 210–242.

26 Schenck, R., Kovach, I. S., Agarwal, A. & Brummett, R. 1998 Strain rate sensitivity of combined knee ligament injuries. Proceedings of The 17th Southern Biomedical Engineering Conference, 55. (doi:10.1109/SBEC.1998.666661)

27 Fratzl, P., Misof K., Zizak I., Rapp G., Amenitsch H., Bernstorff S. 1997 Fibrillar Structure and Mechanical Properties of Collagen. J. Struct. Biol. 122, 119-122. (doi:10.1006/jsbi.1998.3966)

28 Gupta, H.S., Seto, J., Krauss, S., Boesecke, P., Screen, H.R. 2010 In situ multi-level analysis of viscoelastic deformation mechanisms in tendon collagen. J. Struct. Biol. 169, 183-191. (doi: 10.1016/j.jsb.2009.10.002)

29 Puxkandl, R., Zizak, I., Paris, O., Keckes, J., Tesch, W., Bernstorff, S., Purslow, P. & Fratzl, P. 2002 Viscoelastic properties of collagen: synchrotron radiation investigations and structural model. Philos. Trans. R. Soc. Lond, B. 357, 191–197. (doi:10.1098/rstb.2001.1033)

30 Karunaratne, A., Esapa C.R., Hiller, J., Boyde, A., Head, R., Bassett J.H., et al., Significant deterioration in nanomechanical quality occurs through incomplete extrafibrillar mineraliza-tion in rachitic bone: Evidence from in-situ synchrotron X-ray scattering and backscattered electron imaging 2012 J. Bone Miner. Res. 27, 876-890. (doi: 10.1002/jbmr.1495)

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33 Misof, K., Rapp, G. & Fratzl, P. 1997 A new molecular model for collagen elasticity based on synchrotron X-ray scattering evidence. Biophys. J. 72, 1376–1381. (doi:10.1016/S0006-3495(97)78783-6)

34 Liao, H. & Belkoff, S. M. 1999 A failure model for ligaments. J. Biomech. 32, 183–188. (doi:10.1016/S0021-9290(98)00169-9)

35 Screen, H.R.C., Lee, D.A., Bader, D.L., Shelton J.C. 2004 An investigation into the effects of the hierarchical structure of tendon fascicles on micromechanical properties. Proc. Inst. Mech. Eng. H. 218 109-119. (doi: 10.1243/095441104322984004)

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38 Scott, J. E. & Thomlinson, A. M. 1998 The structure of interfibrillar proteoglycan bridges (shape modules') in extracellular matrix of fibrous connective tissues and their stability in various chemical environments. J. Anat. 192 ( Pt 3), 391–405. (doi:10.1046/j.1469-7580.1998.19230391.x)

39 Fessel, G. & Snedeker, J.G. 2011 Equivalent stiffness after glycosaminoglycan depletion in tendon — an ultra-structural finite element model and corresponding experiments. J. Theor. Biol. 268 77-83. (doi: 10.1016/j.jtbi.2010.10.007)

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41 Sidles, J. A., Clark, J. M. & Garbini, J. L. 1991 A geometric theory of the equilibrium me-chanics of fibers in ligaments and tendons. J. Biomech. 24, 943–949. (doi:10.1016/0021-9290(91)90172-J)

42 Nelder, J. A. & Mead R. 1965 A simplex method for function minimisation. Comput. J. 7, 308-313. (doi: 10.1093/comjnl/7.4.308)

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8. Annex

The data (tensile modulus and stress at failure) were fitted to bilinear and logarithmic

mathematical models using the Nelder-Mead (or polytope) algorithm for function minimization

[42]. The objective function (the one to be minimized) was defined as the squared difference

between the mathematical model prediction and the experimental point for every sample tested,

whereby the independent variable is strain rate and the dependent variable is tensile modulus and

stress at failure for each fit, respectively.

OF=∑i=1

N

( y i−f ( x i ))2

for N data points (= number of samples tested),

where yi is the experimental dependent variable (i.e. tensile modulus or stress at failure),

xi is the experimental independent variable (i.e. strain rate)

f (x) is the mathematical model to be fitted. This is

f ( x )=a1 log x+a2 for the logarithmic fit, and

f ( x )={ a1 x+a2 , x<x0

a3 ( x−x0 )+a4 , x≥ x0 for the bilinear fit, with the constraint x0=

a4−a2

a1 in order for the

two lines to meet at x = x0.

The parameters to be fitted are the ai’s for both mathematical models.

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9. Figure captions

Figure 1: Photograph of one ligament sample.

Figure 2: Diagram of the drop rig apparatus and impact tensile adapter used for measuring the

material properties at the highest strain rates.

Figure 3: (a) Schematic view of the synchrotron experimental setup. (b) Detail of the tensile

testing apparatus. (c) The ligament sample ends are potted in acrylonitrile butadiene styrene

molds using dental cement (Filtek Supreme XT, 3M ESPE, USA). (d) By irradiating the collagen

fibril with X-rays a series of Bragg reflections appears on the 2D scattering pattern due to the

staggered axial arrangement of tropocollagen molecules.

Figure 4: (a) Typical stress – strain curves for 5 different specimens, each being an order of

magnitude apart in strain rate. (b) Mean stress-strain curves per strain rate. The vertical error bars

without end caps represent 1 SD. The horizontal and vertical crossed error bars with the circular

end-caps represent the mean ±1 SD stress and strain at failure per strain rate.

Figure 5: (a) Failure stress and (b) tensile modulus of the porcine LCL across strain rates. The

solid line represents a bilinear fit and the dashed straight line a logarithmic (with base 10) fit to

the experimental data. Note that a plateau would be observed at high strain rates when the curves

are plotted with a linear horizontal axis (inset).

Figure 6: (a) Two different deformation mechanisms of intra- and inter-fibrillar structures

proposed for low (top) and high (bottom) strain rates. At low strain rates fibrils start from an

unloaded state (i), go through the toe region (ii), and exhibit intra-fibrillar gliding at high strains

(iii). At higher strain rates intra-fibrillar sliding occurs due to the early debonding of the matrix

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from the collagen fibrils. (b) Stress-tissue strain curve (filled symbols) and fibril strain-tissue

strain curve (open symbols) for low and (c) high strain rate.

Figure 7: Fibril modulus of the human LCL across strain rates (5 samples at each strain rate).

Error bars represent ±1 SD.

10. Tables

Strain rate (/s) # of samples

Tensile modulus

(MPa)

Failure stress

(MPa) Failure strain (%)

0.01 (0) 7 288 (83.9)* 39.9 (11.0) § 17 (3)

0.1 (0) 7 364 (86.6) 56.5 (8.2) 18 (3)

0.94 (0.11) 7 656 (82.4)* 72.8 (11.1) § 14 (2)

10.6 (2.2) 7 763 (141.3) 75.9 (9.6) 11 (3)

129.9 (52.8) 12 906 (195.2) 77.4 (15.4) 9 (2)

Table 1: Material properties of the groups of ligament samples; mean (standard deviation).

There was a statistically significant difference in tensile modulus (*) and failure stress (§) as

strain rate increased from 0.01 to 0.94/s (p<0.01).

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Figure 1: Photograph of one

ligament sample.

Figure 2: Diagram of the drop rig apparatus and impact

tensile adapter used for measuring the material properties at

the highest strain rates.

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Linear motors

X-ray beam outlet

Fluid chamber

X-ray detector

Tensile testing machine

Load cell

Sample

Sample gripsRapid prototyped molds

Dental cement

Trimmed sample

D

Staggered collagen molecules

Tensile loading

Incident X-ray beam

X-ray image

Fibrillar reflection

Diffracted X-ray beam

(a)

(b)

(c)(d)

Figure 3: (a) Schematic view of the synchrotron experimental setup. (b) Detail of the tensile

testing apparatus. (c) The ligament sample ends are potted in acrylonitrile butadiene styrene

molds using dental cement (Filtek Supreme XT, 3M ESPE, USA). (d) By irradiating the

collagen fibril with X-rays a series of Bragg reflections appears on the 2D scattering pattern

due to the staggered axial arrangement of tropocollagen molecules.

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Figure 4: (a) Typical stress – strain curves for 5 different specimens, each being an order of

magnitude apart in strain rate. (b) Mean stress-strain curves per strain rate. The vertical error

bars without end caps represent 1 SD. The horizontal and vertical crossed error bars with the

circular end-caps represent the mean ±1 SD stress and strain at failure per strain rate.

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Figure 5. (a) Failure stress and (b) tensile modulus of the porcine LCL across strain rates. The

solid line represents a bilinear fit and the dashed straight line a logarithmic fit to the

experimental data. Note that a clear plateau would be observed at high strain rates if the

curves were plotted with a linear horizontal axis.

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Figure 6: (a) Two different deformation mechanisms of intra- and inter-fibrillar structures

proposed for low (top) and high (bottom) strain rates. At low strain rates fibrils start from an

unloaded state (i), go through the toe region (ii), and exhibit intra-fibrillar gliding at high

strains (iii). At higher strain rates intra-fibrillar sliding occurs due to the early debonding of

the matrix from the collagen fibrils. (b) Stress-tissue strain curve (filled symbols) and fibril

strain-tissue strain curve (open symbols) for low and (c) high strain rate.

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Figure 7: Fibril modulus of the human LCL across strain rates (5 samples at each strain rate).

Error bars represent ±1 SD..

7

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