Bimodal collagen fibril diameter distributions direct age-related variations
in tendon resilience and resistance to rupture
K. L. Goh,1D. F. Holmes,2Y. Lu,2P. P. Purslow,3K. E. Kadler,2D. Bechet,4and T. J. Wess5
1School of Chemical and Biomedical Engineering, Nanyang Technological University, Singapore;2Wellcome Trust Centre for Cell-Matrix Research, Faculty of Life Sciences, University of Manchester, Manchester, United Kingdom;3Department of Food Science, University of Guelph, Ontario, Canada;4Institut National de la Recherche Agronomique, Unité Mixte de Recherche 1019, Unité de Nutrition Humaine, Centre de Recherche en Nutrition Humaine d’Auvergne, Clermont-Ferrand, France; and5School of Optometry and Vision Sciences, Cardiff University, Cardiff, United Kingdom
Submitted 27 February 2012; accepted in final form 19 July 2012 Goh KL, Holmes DF, Lu Y, Purslow PP, Kadler KE, Bechet D, Wess TJ. Bimodal collagen fibril diameter distributions direct
age-related variations in tendon resilience and resistance to rup-ture. J Appl Physiol 113: 878 – 888, 2012. First published July 26, 2012; doi:10.1152/japplphysiol.00258.2012.—Scaling relationships have been formulated to investigate the influence of collagen fibril diameter (D) on age-related variations in the strain energy density of tendon. Transmission electron microscopy was used to quantify D in tail tendon from 1.7- to 35.3-mo-old (C57BL/6) male mice. Frequency histograms of D for all age groups were modeled as two normally distributed subpopulations with smaller (DD1) and larger (DD2) mean Ds, respectively. Both DD1and DD2increase from 1.6 to 4.0 mo but decrease thereafter. From tensile tests to rupture, two strain energy densities were calculated: 1) uE [from initial loading until the yield
stress (Y)], which contributes primarily to tendon resilience, and
2) uF[fromYthrough the maximum stress (U) until rupture], which
relates primarily to resistance of the tendons to rupture. As measured by the normalized strain energy densities uE/Yand uF/U, both the
resilience and resistance to rupture increase with increasing age and peak at 23.0 and 4.0 mo, respectively, before decreasing thereafter. Multiple regression analysis reveals that increases in uE/Y(resilience
energy) are associated with decreases in DD1and increases in DD2, whereas uF/U(rupture energy) is associated with increases in DD1 alone. These findings support a model where age-related variations in tendon resilience and resistance to rupture can be directed by subtle changes in the bimodal distribution of Ds.
stress transfer; strain energy density; work of fracture; finite mixture model
TENDONS MAY BE REGARDED ASbiological fiber composites com-prising highly parallel collagen fibrils, strong and stiff in tension, reinforcing a weak, hydrated extracellular matrix (ECM) rich in proteoglycans (PGs) (28). Increasingly, studies on tendon biolog-ical organizations from the molecular level (2, 7, 8, 29, 41) to the fibrillar level (2, 21b, 22, 25, 27) are revealing new insights for how fibrils respond to external loads. In principle, this is important because the insights could facilitate the development of a basic understanding of the structure-property relationship of tendon addressing the influence of fibrillar structure, e.g., size and mor-phology (22, 26), and alteration to the structure by specific genes (5, 11, 40, 55) on the age variation in the mechanical properties of tendon, e.g., strength and stiffness (9, 21a, 49, 52). In practice, the qualitative arguments (5, 12, 34, 37, 45, 54, 55) that have been
developed for explaining the structure-property relationship of tendon are not validated adequately by quantitative models (11, 26, 40).
The influence of the spread of fibril lateral sizes on the mechanical property of tendon is one of the key aspects for understanding the structure-property relationship. Studies have revealed that the frequency histograms of the fibril diameter (D), a parameter for the fibril lateral size, of tendon from postnatal growth stages to old age feature non-Gaussian pro-files that may be described as bimodal or even trimodal (11, 30, 37, 54, 55). Except in tissues from very young animals (namely mice at birth until, e.g., 2 wk old) that feature near-Gaussian profiles (37), the non-Gaussian profiles from growth to old age preclude any straight-forward analyses of D arising from such a population (30). Accordingly, studies using quantitative (re-gression) models (11, 40), which relate the mean D of the frequency histogram to the specimen mechanical property, have yielded conflicting findings. For instance, we note that the mean D correlates to the structural strength (maximum force) and structural modulus of the force-displacement curve but not to the material strength (maximum stress) and material modu-lus of the stress-strain curve (11, 40). Thus the justification for existing regression models is limited. Additionally, so far, only data from the postnatal growth phase of mice have been evaluated (11, 40), and the applicability of the structure-property relationship seems to be much less established for a wider age range, spanning from maturation to old age.
The intent of this paper is to present an investigation into the structure-property relationship of tendon to clarify the conflict-ing findconflict-ings of D-mediated variations in mechanical property of tendons during aging. To this end, we provide an elegant demonstration of an energy approach for modeling the influ-ence of collagen fibril lateral size to the age variation strain energy components for resilience and resistance to rupture of tail tendons from C57BL/6 mice. Unlike previous studies on the structure-property relationship, which focus on the age-related variations in the mechanical property at specific points of the loading process, the energy approach establishes a strategic framework for a comprehensive understanding of the various mechanical parameters throughout the entire loading process. Hypotheses are proposed to evaluate the significance of the structure-property relationships.
Glossary
D Collagen fibril diameter
DD1, DD2 Mean Ds of collagen fibril subpopulations, D1
and D2, respectively
Address for correspondence and present address: K. L. Goh, School of Mechanical and Systems Engineering, Newcastle Univ. (Singapore campus), 180 Ang Mo Kio Ave. 8, Block P, Rm. 220, Singapore 569830 (e-mail: kheng-lim.goh@ncl.ac.uk).
E Elastic modulus (stiffness) of tendon ECM Extracellular matrix
Ef, Em Elastic moduli of collagen fibril and
interfibril-lar matrix, respectively
G Work of elastic deformation of a fibril during
transitional stage (mode)
GP Work of fibril pull-out from ECM
GR Work of fibril rupture
L Scale factor for length
lz Embedded length of a fibril prior to pull-out
N Number of fibrils/unit cross-sectional area of tendon
PG Proteoglycan
uE, uF Strain energy densities contributing to tendon
resilience and resistance to rupture, respec-tively
Uf, Um Strain energies for elastic deformation of a
fibril and intefibrillar matrix, respectively
u0 Strain energy density to fracture of tendon
Vf Volume fraction of collagen fibrils in ECM
z Axial ordinate of the cylindrical polar coordi-nate system
␦ Axial deformation in fibril εf Axial strain in fibril
 Strain energy density for elastic deformation
of fibrils
P Strain energy density for fibril pull-out from
ECM
R Strain energy density for rupture of fibrils
f Axial stress in the fibril as a function of
dis-tance, z
m Average stress in interfibrillar ECM
U Maximum stress (strength) of tendon
Y Stress in the tendon at the point of yielding
fu Breaking stress of collagen fibril
fy Yield stress at the fibril center when the tendon
yields atY
 Shear stress generated at the fibril/PG-matrix
inter-face during the transitional stage (mode) RP Shear stress generated at the fibril/PG-matrix
interface during fibril rupture or pull-out 2lc Critical length of collagen fibril
2lf Collagen fibril length
MATERIALS AND METHODS
An energy approach to modeling the structure-property relation-ship of tendon. Consider tension to be applied to a tendon in the axial
direction of an assembly of parallel collagen fibrils. Figure 1 shows a single fibril embedded in and reinforcing ECM of the tendon. Such regions are observed in many tissues, such as ligament and articular cartilage; application of tensile stress also tends to orient fibrils (28). Loading begins with the elastic stress-transfer stage (22). According to a shear-lag analysis (20, 22), as interfibrillar ECM deforms elasti-cally, adhesion at the fibril/PG-matrix interface causes the fibrils to be recruited into tension. The fibrils stretch elastically, and the fibril-fibril lateral spacing decreases as the fibrils are increasingly drawn closer together. In the process of fibril deformation, intermolecular sliding of tropocollagen (18, 47) is resisted predominantly by the forces of interaction associated with covalent cross-links; a variety of other forces arising from hydrogen bonding, van der Waals, and electro-statics could also play an important part in the molecular mechanics of collagen, although the precise mechanism is not clear. Additionally,
hydrogen bonding, van der Waals, and electrostatics forces are re-sponsible for regulating the stress transfer to the fibrils (39); these forces account for the mechanics of interaction of decoron with collagen (36) as well as that of glycosaminoglycan (GAG) side chains associated with decorin PGs bound on adjacent fibrils (16, 20a, 39) and the other PGs in interfibrillar ECM (20a). As loading progresses from the elastic stress-transfer stage to mode, a transitional stage (20), stress transfer occurs by sliding friction between the fibril and the PG-matrix; at a molecular level, this corresponds to a continuous disruption and formation of GAG interaction. The frictional force is parameterized by the corresponding shear stress (), which is con-stant throughout the interface. Mode marks the transition from the elastic to the plastic stress-transfer stage (20, 22), where the fibril continues to deform elastically as interfibrillar ECM deforms plasti-cally and slides over the fibril/PG-matrix interface. After the plastic stress-transfer stage, the loading regime enters the failure stage, where macroscopic, permanent deformation of interfibrillar matrix corre-sponds at the molecular level to an extensive disruption of GAG interactions arising from appreciable displacement of the side chains. However, substantial interpenetration of the GAG side chains also occurs as the distance of adjacent fibrils decreases, and so, the frictional force increases correspondingly. Accordingly, the frictional shear stress,RP
(⬎), is constant throughout the fibril/PG-matrix interface during fibril rupture and fibril pull-out from the ECM (Fig. 2B).
Application of the principles of essential work of fracture to a fiber composite (53) yields the following partitioned components of the energy stored in the tendon: the nonessential energy (uE), which
contributes primarily to tendon resilience (regulated by fibrils under-going elastic deformation), and the essential energy (uF), which
relates primarily to resistance of the tendon to rupture (regulated by fibril rupture, leading to defibrillation and rupture of the interfibrillar matrix). On the stress-strain curve (Fig. 2C), uE corresponds to the
area under the curve from the origin to the point of inflection (PI; which marks the limit of the linear region), whereas uFcorresponds to
the area under the curve from the PI to the point of fracture. From the scaling relationships, we find that uE⫽ (D) (see Eq. A6), and uF⫽
R(D)⫹ P(D) (by combining Eqs. A8 and A10; see below). Here,
represents the strain energy density (symbol in adjacent parentheses indicates the argument of), and subscripts , R, and P denote mode , fibril rupture, and fibril pull-out, respectively. In principle, the equations for uEand uFwould be valid if all of the fibrils in the tendon
featured the same D. In reality, we find that the population of fibrils possesses a spread of D with a non-Gaussian profile. According to the finite mixture law (35), this profile may be described by two or more normally distributed subpopulations. The argument that follows here-after has been developed using the bimodal distribution. [This is not unrealistic, because it has been reported that the minimum number of subpopulations for the mouse tail tendon from growth to old age is two (37).] Let D1 and D2 represent the normal distributions of the respective subpopulations with the smaller (DD1) and larger (DD2) mean Ds. We find that
Fig. 1. Model of a collagen fibril embedded in a coaxial cylinder of hydrated PG-rich matrix (shaded region). Here, 2lfdenotes the length of the fibril and D
its diameter. The fibril center (O) defines the origin of the cylindrical polar coordinate system; the fibril axis defines the z-axis.
uE⫽ (DD1)⫹ (DD2)
uF⫽ [R(DD1)⫹ P(DD1)]⫹ [R(DD2)⫹ P(DD2)] . (1) From Eq. 1, we arrive at
uE⁄Y⫽ cE1DD1⫹ cE2DD2, (2) uF⁄U⫽ cRP1DD1⫹ cRP2DD2, (3) where cE1and cE2are constants of proportionality for Eq. A6 (see
below), and cRP1and cRP2describe the summation of the constants of
proportionality cR1and cR2 (see Eq. A8) and cP1and cP2(see Eq.
A10); i.e.
cRP1⫽ cR1⫹ cP1
cRP2⫽ cR2⫹ cP2. (4)
We note that cE1and cE2 are expressed in terms of the fibril yield
stress (fy), fibril stiffness (Ef),, and a scale factor for length (L; see Eq. A6); cR1and cR2are expressed in terms of the fibril fracture stress
(fu), Ef,RP, and L (see Eq. A8); cP1and cP2are expressed in terms
of onlyfu,RP, and L (see Eq. A10).
According to the postulates of Parry et al. (37), in the small-strain regime, where the elasticity of the tissue predominates, creep inhibi-tion is accomplished by the presence of small D fibrils (creep could otherwise yield a nonrecoverable strain). In the high-stress regime, for the tissue to withstand high stress, the strength of the tissue is accomplished by the presence of large D fibrils. By adapting these postulates for our energy-based argument for investigating how the bimodal distribution of D directs tendon resilience and resistance to rupture in tendons from growth to old age, we hypothesize that when the tendon is acted on by an external load: 1) at small deformation, fibrils of all diameters are recruited into action, absorbing strain energy and contributing to elastic tensile deformation (the resilience hypothesis), and 2) at very large deformation up until tissue fracture, larger D fibrils are responsible for regulating the strain energy ab-sorption to resist rupture (the resistance-to-rupture hypothesis). In this study, using in vitro data from a mouse model, we will test these hypotheses by evaluating the mathematical models of Eqs. 2 and 3 to find out how much of the age-related variations in the respective strain energy density components can be explained by a linear relationship with increasing DD1and DD2.
Tissue preparation and mechanical testing. The tissue
prepara-tions, load-deformation curves, and transmission electron microscopy (TEM) obtained by Goh et al. (21a) were the starting point for the
re-analysis of data necessary for the modeling presented below. All procedures (21a) were approved by the UK Home Office and ac-corded with the UK Animals (Scientific Procedures) Act 1986. In summary, tail tendons were obtained from three to four male C57BL/6 mice at 1.7, 2.6, 4.0, 11.5, 23.0, 29.0, 31.5, and 35.3 mo of age, and 12 mm-long sections of the tendon were subjected to tensile tests in PBS (pH 7.2) at a displacement rate of 0.067 mm/s. Starting with the original records of the stress-strain data of each sample (21a), the areas under the stress-strain curve (Fig. 2C), corresponding to the parameters
u0, uE, and uF, were determined. In addition, we have determinedY
andUfrom the original stress-strain data (21a). Five to 10 samples/
tail were tested; averaging the values of the respective parameter for all samples/animal and for all animals within that age group yielded the representative value (within SE) for each age group.
The occurrence of yield in the tendon is associated with the point of inflexion lying between the toe region and the point of maximum stress on the stress-strain curve (Fig. 2C). Within this region, we find the gradient increases, peaks at the inflexion point, and decreases with increasing strain. To identify the inflexion point, we fitted an appropriate polynomial equation to the stress-strain data points from the origin to the maximum stress and evaluated the polynomial equation to determine the stress vs. strain corresponding to the peak gradient (21a).
TEM. Images from TEM obtained by Goh et al. (21a) were
re-analyzed. Briefly, the micrographs were taken of ultrathin sections of fixed and resin-embedded tendons on a Tecnai BioTWIN instru-ment (FEI, Eindhoven, The Netherlands) at an accelerating voltage of 80 kV. Digitized micrographs of near-transverse sections were used to measure D. The micrographs for measurements of D were at an instrumental magnification of ⫻15,000. The precise magnification was determined using a diffraction grating replica (2,160 lines/mm). Each sample area is an entire micrograph corresponding to a specimen rectangle of 4m ⫻ 5 m. The area from each tendon sample was an average of 10 specimen rectangles with all fibrils in a near-transverse section; these rectangular areas were selected randomly through a survey over widely separate locations across the tendon sample. The cross-section of each fibril was manually traced and the area com-puted using the Semper5 image analysis package (Synoptics, Cam-bridge, UK). Following approaches reported elsewhere (11, 30, 37), D was derived from the fibril cross-sectional area by modeling the shape of the fibril as circular. Data from all animals were then combined and considered representative of the collagen fibril profile in fascicles from that age group.
Fig. 2. Collagen fibril reinforcement in ECM. A, top row: interaction of glycosaminoglycans (GAGs) between 2 collagen fibrils. B: schematic of fibrillar failures in ECM. Rupture of interfibrillar ECM is depicted to have occurred with the crack tip propagating from left to right. Fibrils are represented by long rods featuring bands (representing the 67-nm periodicity). FB, fibril bridging the ruptured site in interfibrillar ECM; FR, fibril rupture; and FP, fibril pull-out. C, bottom row: a typical stress-strain plot from a mouse tail tendon. On the stress-strain curve, the point of inflection (PI) is where yielding has started. The areas of the lighter and darker shaded regions under the stress-strain curve correspond to uEand uF, respectively; the strain energy density to fracture u0⫽ uE⫹ uF.
Finite mixture modeling. Consider the lateral size population of the
fibrils to be a heterogeneous mixture of a few normally distributed subpopulations for the purpose of modeling the non-Gaussian profiles of the frequency histograms of D (35). By inspection, the smallest number of subpopulations in the mixture is two, and we shall refer to these subpopulations by D1 and D2 for simplicity. Finite mixture modeling was implemented using MatLab (version 7; MathWorks, Natick, MA). To determine the optimal solution to the mean and SD of D for the respective subpopulations, a simulated annealing (SA) optimization algorithm was developed to evaluate for possible profiles that best fit the primary distribution [for a similar approach using SA, see Goh et al. (21)]. To execute the SA algorithm, the “temperature” parameter was assigned a value of 0.5 with a reduction factor of 0.9. During each run, the maximum number of configurations that the SA algorithm could explore was fixed at 100; the number of temperature steps to be executed was fixed at 100; and the number of successes allowable before looping to the next temperature steps was fixed at 10. The configuration space addressed the fibril subpopulations D1 and D2. The Gaussian profile of these subpopulations was defined by the amplitude (which described the proportion of fibrils in the subpopu-lation), mean, and SD; the magnitudes of these parameters were assigned from a predefined range of values using an algorithm for randomizing the selection of values. An initial run was executed to obtain a preliminary range of values for the mean and SD, respec-tively, followed by a refinement run by narrowing the range of values. Linear regression was implemented as part of the the SA approach to fit the profiles of D1 and D2 to the primary distribution.
Statistical analysis. The nature of the tensile test data was checked
to ensure that it satisfied the normality and homogeneity of variance of residuals (6) with respect to age—DD1 and DD2. One-way ANOVA, complemented by the Fisher least square difference (LSD) test (6), was carried out to evaluate for significant differences in the age variations of u0, uE/Y, and uF/U. The Fisher LSD test was used
to compare the mean of one age group with the mean of another for statistical difference (at an individual error rate of 0.05) by generating (and comparing) seven sets of confidence intervals from the eight age groups used in this study. Multiple regression analysis (1, 6) was used to assess for significant correlation of DD1and DD2with uE/Yand
uF/U, respectively; the results were reported as means⫾ SE.
Sig-nificance was defined as P⬍ 0.05. Multicollinearity of DD1vs. DD2 was assessed using the Pearson correlation coefficient test (1). All statistical analyses were performed using Minitab commercial soft-ware (version 14; Minitab, State College, PA).
RESULTS
Strain energy density. Figure 3 shows the plot of u0vs. age.
The ANOVA test reveals that not all of the means of u0among
the different age groups are equal (P⫽ 0.001; F ⫽ 5.84). The Fisher LSD test shows that statistical differences occur be-tween the means only for: 1) 1.7 mo vs. the respective age group from 2.6 to 35.3 mo, 2) 2.6 vs. 4.0 and 23.0 mo, 3) 4.0 vs. 31.5 and 35.3 mo, and 4) 23.0 vs. 35.3 mo. Correspond-ingly, the plot of u0vs. age describes an increasing u0from 1.7
to 4.0 mo, fluctuation of u0with no appreciable trend from 4.0
to 29.0 mo, and a decreasing u0from 29.0 to 35.3 mo.
Figure 4A shows the normalized strain energy density (uE/
Y) vs. age. Included in this figure are plots of the respective
parameters—uE(Fig. 4B) andY(Fig. 4C)—vs. age for
infor-mational purpose. The ANOVA test reveals that not all of the means of uE/Yare equal (P⫽ 0.009; F ⫽ 3.90). The Fisher
LSD test shows that statistical differences occur between the means for: 1) 1.7 mo, 2) 2.6 mo, and 3) 4.0 mo vs. the respective age group from 23.0 to 35.3 mo. Correspondingly, the plot of uE/Yvs. age describes fluctuations in uE/Yfrom 1.7
to 4.0 mo with no appreciable trend; thereafter, uE/Y increases
from 4.0 to 23.0 mo and fluctuates from 23.0 to 35.3 mo. Figure 5A shows the normalized strain energy density (uF/
U) vs. age. Also included in this figure are plots of the
respective parameters—uF(Fig. 5B) andU(Fig. 5C)—vs. age
for information. [Interestingly, it is observed that the plots of Y (Fig. 4C) and U (Fig. 5C) vs. age reveal very similar
profiles to u0vs. age (Fig. 3).] The ANOVA test reveals that
not all of the means of uF/Uare equal (P⫽ 0.049; F ⫽ 2.59).
The Fisher LSD test shows that statistical differences occur between the means for: 1) 1.7 vs. 4.0 mo, 2) 2.6 vs. 11.5 mo, and 3) 4.0 mo vs. the respective age groups from 11.5 to 35.3 mo. Correspondingly, the plot of uF/U vs. age reveals no
appreciable change in uF/U from 1.7 to 2.6 mo, increase in
Fig. 3. Graph of strain energy density to fracture (u0) vs. age. The vertical bars
represent SE.
Fig. 4. Graphs of (A) normalized strain energy density (uE/Y), (B) strain energy density
con-tributing to tendon resilience (uE), and (C) stress
generated by the tendon at yield point (Y) vs.
uF/Ufrom 2.6 at 4.0 mo, decrease in uF/Ufrom 4.0 to 11.5
mo, and fluctuations with no appreciable change thereafter. Clearly, the trend exhibited by uF/U is a contrast to that
exhibited by the uE/Y. We note that whereas results from the
Fisher test show that the variation in u0with respect to age is
accounted for by four out of the seven sets of statistical differences between the means, the variation in uE/Y(and also
uF/U) is accounted for by three out of the seven sets of
statistical differences between the means.
Fibril diameter distribution. Figure 6 shows an array of
histograms of normalized frequency vs. D and the correspond-ing (representative) TEMs of the cross-section of tendon for age group 1.7-35.3 mo. To the best of our knowledge, there has
been no systematic study to evaluate similar histograms of frequency vs. D of the C57BL/6 tail tendon from growth to old age. However, we note that our results for the growth phase reveal similar profiles to those reported elsewhere for 3- and 8-wk-old mice (11). By inspection, it is observed that the maximum D in each histogram fluctuates with age. At 1.7 mo, the spread of D values has a maximum D of⬃300 nm. At 4.0 mo thereafter, the maximum D fluctuates at close to 400 nm. Qualitatively, these observations implicate that the tendon of young mice is dominated by fibrils with small D (with a fairly regular morphology), but as the mice grow older, larger D fibrils (with increased morphological irreg-ularity) dominate.
Fig. 5. Graphs of (A) normalized strain energy density (uF/U), (B) strain energy density
con-tributing to tendon resistance to rupture (uF),
and (C) strength of the tendon (U) vs. age. The
vertical bars represent SE.
Fig. 6. Histograms of normalized frequency vs. D dis-tribution and the corresponding (representative) trans-mission electron micrographs of the cross-sections of tendons from (A) 1.7-, (B) 2.6-, (C) 4.0-, (D) 11.5-, (E) 23.0-, (F) 29.0-, (G) 31.5-, and (H) 35.3-mo-old mice. The number of C57BL/6 mice tested for the respective age group was: (A) 3, (B) 3, (C) 3, (D) 4, (E) 3, (F) 4, (G) 4, and (H) 4 mice. The data were combined from all animals for each age group. Dark and light solid curves in each histogram represent the D1 and D2 fibril sub-populations, respectively. Original scale bars, 150 nm.
Plots of DD1and DD2vs. age (Fig. 7) reveal that the mean Ds
are small at 1.7 mo but increase rapidly with age until 4.0 mo. Thereafter, DD1decreases (more rapidly than DD2) with age.
Interestingly, the SD of the DD2is small at 1.7 mo, increases
steadily with age until 29.0 mo, and shows no appreciable change from 29.0 to 35.3 mo. On the other hand, the SD of the
DD1is small at 1.7 mo, increases steadily with age until 4.0 mo,
but decreases steadily with age thereafter. From the multicol-linearity analysis of DD1 vs. DD2, we find that the Pearson
correlation coefficient test⫽ 0.599. We conclude that the DD1
and DD2are not correlated with each other, with a cautionary
note that the Pearson correlation coefficient test is marginally less than the tolerable threshold, i.e., 0.600 (1).
To the best of our knowledge, no quantitative models have been reported of fibril growth in vertebrate tissues that could account for all of the factors that control the fibril diameter distribution. Although this study was prompted by observa-tions on the age-related variaobserva-tions in the mean D, the results on the age-related variations in the width of the bimodal distribu-tions could be the focus for further investigation. Of consider-able value are the following points relating to these results:
1) the spread of these sizes is complicated by the presence of
both unipolar and bipolar fibrils as well as the contribution of interfibrillar fusion (27), and 2) the individual variations could play a part in contributing to the spread of D values. For instance, the smaller SD value of the DD1during old age could
implicate reduction in individual variability associated with the subpopulation.
Multiple regression analysis. Regression analysis reveals
strong evidence of a linear relationship between uE/Yand both
DD1and DD2. Here, it is observed that the two coefficients, cE1
and cE2, of Eq. 2 are different from zero (ANOVA, P⫽ 0.000;
F⫽ 15.98). Table 1 summarizes the results of cE1(P⫽ 0.000)
and cE2(P⫽ 0.000). The R2value indicates that the predictors,
i.e., the DD1 and DD2, can explain 54.5% of the variation in
uE/Y; the remaining 45.5% of the variation in uE/Y arises
from random arrangement of the points on the uE/Yvs. DD1
and DD2space. The model’s y intercept [⫽(4.4 ⫾ 5.7) ⫻ 10⫺3]
is not significantly different from zero (P⫽ 0.452), suggesting that uE/Y will not be significantly different from zero when
DD1and DD2are zero.
Similarly, regression analysis reveals strong evidence of a linear relationship between uF/U and DD1 and DD2. The
analysis suggests that at least one coefficient, i.e., cRP1, cRP2, or
both, is different from zero (ANOVA, P ⫽ 0.000; F ⫽ 4.74).
However, further examination shows that only cRP1 is
signifi-cantly different from zero (P⫽ 0.040), indicating that only DD1is
significantly related to uF/U. The R2 value indicates that the
predictors can only explain 29.2% of the variation in uF/U; in
other words, the remaining 70.8% of the variation in uF/U
arises from random arrangement of the points on the uF/Uvs.
DD1 and DD2 space. Interestingly, the model’s y intercept
[⫽(1.2 ⫾ 0.2) ⫻ 10⫺1] is significantly different from zero (P⫽ 0.000), suggesting that uF/Uwill be significantly greater
than zero when DD1is zero. However, this must be
meaning-less, since we cannot have a nonzero value for uF/Uwhen DD1
is zero. It may well be that if our sample included smaller values of DD1, then we would find that the relationship was
curved. Note also that the magnitude of cRP1 is one order of
magnitude greater than that of cE1; this is consistent with the
(order of magnitude) difference between uF/Uand uE/Y. The
results of the derived coefficients cRP1 and cRP2 are
summa-rized in Table 1. DISCUSSION
Aging changes in the fracture toughness of tendon reflect phases of development and influence from different biological organizational levels. The variations in u0as a function of age
may be divided into three phases corresponding to postnatal growth, maturation, and old age (32, 51). These phases of aging are not unique to u0but have been established previously
for Uand E (21a). For the model tendon used in this study,
growth is characterized by a rapid increase in u0until
matura-tion; the transition from postnatal growth to maturation occurs at⬃4 mo. Throughout the maturation phase, u0fluctuates with
no appreciable trend. The transition from maturation to old age occurs for ⬃23 mo; thereafter, u0 decreases with increasing
age during old age. Overall, this suggests that tendon appears to be optimally toughened at maturation. Viidik (51) reported that the u0of rat tail tendon (spanning 1.0 –35.0 mo) features
trends similar to those reported here, albeit that there is an absence of justifications for the underlying statistical signifi-cance of the data in Viidik’s work. It is necessary, however, to consider the molecular, cellular-ECM, physiological, and pop-ulation levels of biological organization to understand the effects of one level on the next-higher level in the presence of aging (17). This study argues that at the physiological level, age variation in u0is the result of changes in uEand uF. In turn,
uEand uFare influenced by the structural changes at the ECM
level. The implications of this will be discussed further in the next paragraph, where we address the use of the scaling relationships for evaluating, at the ECM level, the influence of
Fig. 7. Graph of D vs. age. Here, circles and squares correspond to the D1 and D2 fibril subpopulations, respectively. Results are reported as means⫾ SD; vertical bars represent SD.
Table 1. Multiple regression analysis
Coefficient Predicted Value⫻ 10⫺4(nm⫺1)
cE1 ⫺1.2 ⫾ 0.2*
cE2 1.3⫾ 0.3*
cRP1 1.8⫾ 0.8*
cRP2 ⫺0.7 ⫾ 1.2
Here, the predicted values of cE1and cE2refer to the coefficients of Eq. 2
[constants of proportionality (see Eq. A6)], and those of cRP1and cRP2refer to
the coefficients of Eq. 3 [summation of the constants of proportionality cR(see Eq. A8) and cP(see Eq. A10)].Values are means⫾ SE. *Significantly different, P⬍ 0.05.
collagen fibril lateral size on the strain energy density absorbed by the tendon in the presence of aging.
Predictions from the scaling relationships (Eqs. 2 and 3) sug-gest a new meaning for the influence of collagen fibril lateral size on resilience and resistance to rupture in connective tissue from growth to old age. Applications of the energy argument, in contrast to the previous stress argument (9, 10, 26, 37, 54), have allowed a deeper understanding of the influence of the absorbed energy on the depth of disruption of the fibrillar fine structure as quantified by the scaling relationships. Multiple regression analysis of uE/Yvs. DD1and DD2(Eq. 2) indicates
support for the resilience hypothesis, which holds that fibrils of all sizes are responsible for directing the absorption of the strain energy imparted to the deforming tissue at initial load-ing. From Table 1, the results suggest that an increasing DD2
contributes to increasing the tissue resilience, but an increasing
DD1contributes to decreasing the tissue resilience. On the other
hand, the resistance-to-rupture hypothesis is rejected based on evidence from multiple regression analysis of the uF/U vs.
DD1 and DD2 (Eq. 3). Alternatively, we speculate that the
smaller D fibrils direct further absorption of strain energy by the deforming tissue until fracture. Compared with fibrils with small D, the fine structure of fibrils with large D will have to be disrupted to a greater depth for fracture to occur. Thus the failure of ECM favors fibrils with smaller Ds; these fibrils fracture easily because of the lower strain energy absorbed. Similarly, fibrils with small D will also debond easily, owing to the smaller area available between the fibril-PG interface for molecular interactions.
Although these findings lend to possible broader applicabil-ity of the scaling relationships to the other connective tissues of similar concern, e.g., aorta (3) and cornea (13), we highlight the following cautionary notes. First, considering that tail tendons are useful for aging studies (11, 21a, 40, 49, 52)—they are not weight bearing and may be influenced by systemic effects of aging (21a, 51)—we acknowledge that the findings may not apply to all tendons (or to connective tissues of all species). For instance, the different limb tendons that are directly involved in locomotion would experience a different loading environment and could respond differently to aging (9, 10, 26). Second, although the findings suggest that fibril lateral size influences the age variation in the strain energy densities, we do not rule out the possibility of the fibril lateral size and strain energy densities responding independently to collagen cross-linking (4, 5) or an unknown third factor. Further dis-cussion is out of the scope of this paper, but this issue has been targeted for investigation in future studies.
Fibril/PG-matrix interactions. The transfer of stress by 
andRPfrom the interfibrillar matrix to the fibril is central to
the theoretical framework of the scaling relationships. This section presents simple order-of-magnitude estimates of the ratio of RPto, which is a parameterization of the relative
stress transfer. Let Em and m be the stiffness and average
stress in interfibrillar ECM, respectively. Applying the rule of mixtures to modeling the influence of the fibril volume fraction (Vf) on E andUleads to E⫽ [Ef ⫺ Em]Vf⫹ Em, and U⫽
[fu⫺ m]Vf⫹ m(21a); linear regression analysis reveals that
the y intercepts from the respective relationships are both negative and not significant (P ⬎ 0.05). Thus the predomi-nance of the role of collagen for regulating EfandUcan be
satisfied to order of magnitude (seeAPPENDIX) by setting these
y intercepts to zero. One then finds that E⫽ EfVf, andU⫽
fuVf; additionally, Y⫽ fyVf. Linear regression analysis of
E⫽ EfVf(P⬍ 0.001; F ⫽ 768.9), Y⫽ fyVf(P⬍ 0.001; F ⫽
440.6), and U ⫽ fuVf (P ⬍ 0.001; F ⫽ 571.8) yields the
gradients (i.e., Ef, fy, and fu) of the respective equations
(Table 2). We note that the order of magnitude estimates for Ef
falls with the range of 0.3–1.6 GPa derived from experiments (14, 44); the order of magnitude estimate forfyis smaller than
the lower limit (70 MPa) obtained by experiment (44); and the order of magnitude estimate for fuis smaller than the cyclic
loading fracture stress (200 MPa) reported by Shen et al. (44). To develop our analysis further, we note that cR1, cP1, and cE1
are related tofy,fu,,RP, and Ef, according to Eqs. A6, A8,
and A10 (see below). From Eqs. 2 to 4, we find that cRP1/cE1⫽
[cR1⫹ cP1]/cE1leads to a mathematical expression for the ratio
of RPto; i.e.,RP/⫽ Kfu{fu⫹ [3/4]Ef}/[fy]2, where
K ⫽ cE1/cRP1. By considering the upper (mean ⫹ SE) and
lower (mean⫺ SE) limits of cE1, cRP1,Ef,fy, andfu(Tables
1 and 2), we obtain an estimate ofRP/in the range of 15– 87.
This estimate is important in terms of the mechanism of stress transfer by frictional sliding (as quantified by  and RP,
respectively) for small and large deformation effects; the ar-gument demonstrates that RP is 10 –102 times (to order of
magnitude) higher than .
The RP/ analysis lends to further predictions of  and
RP. These parameters can be estimated to order of magnitude
according to two contentions as follow. First, from fibril pull-out tests carried out using an atomic force microscope (25), the force to pull out a fibril is of order of magnitude fu[D2/4]⫽ 10⫺7N for typical fibrils having D⫽ 10⫺7 m
and length not exceeding the critical value of 2lc ⫽ 10⫺6 m
(34). Second, from shear-sliding analysis (see APPENDIX), we have 2lc/D ⫽ fu/2RP; the shear action RP is of order of
magnitude fu[D2/4]/{lcD} ⫽ 105 Pa. Given that RP/
ranges from 10 to 102(from the argument established in the
previous paragraph), it follows thatranges from 103to 104
Pa. Of note, this range of estimates foris important since it
encompasses estimates derived from alternative arguments. For instance, the force of interaction between GAG side chains from decorin PGs bound on adjacent fibrils is of order of magnitude 10⫺11N (33). It can be assumed that the fibrils are packed tetragonally so that the distribution of PGs over the fibril surface is such that there are four PGs for every 68 nm along the fibril (39). By this argument, it follows that the shear stress is 4(4⫻ 10⫺11N)/[(10⫺7m)(68⫻ 10⫺9m)]⬇ 7,489 Pa, which lies within the estimated limits of. Both decorin
and biglycan PGs compete for collagen binding and the pos-sibility of targeting identical or adjacent binding sites on the fibril (42, 43). If the population of decorin PGs were to decrease, this could lead to an increased binding of biglycan to fibrils to compensate for the lack of decorin (55) and to
Table 2. List of derived parameters for the order of
magnitude analysis ofRP/
Parameter Predicted Value (MPa)
Ef 728.8⫾ 29.7
fy 29.0⫾ 1.5
fu 65.8⫾ 3.0
regulate the stress transfer (maintaining the values of  and
RP) throughout the different phases of aging. To summarize,
the argument presented in this paragraph reveals that in tendon from postnatal growth to old age, the fibril/PG-matrix interface experiences weaker GAG interactions during initial loading than at subsequent stages of loading following yielding.
Possible model limitations. Our energy argument has
en-abled the formulation of the scaling relationships for studying the age variation in structure-property relationship in tendon. As we weigh the implications of our approach, it is important to acknowledge the following possible model limitations.
To begin, we note that tendon is predominantly composed of type I collagen (4). This is a trimeric biomacromolecule fea-turing a Gly-X1-X2 amino acid sequence pattern and a stan-dard triple-helix throughout their ⬃1,000 residue triple-helix domain (Gly ⫽ glycyl residue; X1 ⫽ proline residue; X2 ⫽ 4-hydroxyproline residue). So far, it is not known if type I collagen triple-helix features crystal imperfections. Tendon also contains small amounts of one or more of the other minor collagens, such as type XXII fibril-associated collagen, which is located on the surface of fibrils at the myotendinous junction (31); it features interruptions (i.e., crystal imperfections) in the amino acid sequence pattern. An interrupted domain may involve one (or more) missing residue, in which case, the Gly residues in the interruptions would be separated by one (or more) non-Gly residue (48). Furthermore, the residues within the interruptions are predominantly hydrophobic; some of these residues also possess a net charge (48). We speculate that these interruptions may contribute to the microinhomogeneity of strains within the fibrils. Consequently, plastic microstrains could develop within these domains even when the tendon is nominally within the elastic limit (15). At a molecular level, we note that fy is the measure of the force of attraction arising
from covalent bonds within (and between) collagen molecules/ unit volume; Efis the measure of the rate of change of the force
of attraction/unit volume with respect to strain; andfuis the
measure of the force of attraction/unit volume for resisting bond disruption within (and between) collagen molecules. In the energy argument leading to the scaling (structure-property) relationships, we have assumed that these fibrillar mechanical parameters, i.e.,fy,fu, and Ef, are constants of age and that
fibril rupture occurs at the fibril center, where the stress is highest (see APPENDIX). However, variation in the proportions of type I and type XXII collagens has been difficult to assess with age. Indeed, if the proportion of type XXII collagen increases with age relative to type I collagen, these fibrillar mechanical properties would be affected by changes to the density and orientation of the crystal imperfections (44, 46). These imperfections could influence the mechanical stability of the fibril by regulating the proportion of the energy absorbed by the deforming fibril; the strain energy increases as the tendon deforms with increasing strain. From the theory of local average bonding (46), it is predicted that eventually, a crack will be initiated when a bond within the domain of the imperfection is disrupted, and this will propagate within the fibril as the nearby bonds become energetically unstable. We do not rule out the possibility of the influence of collagen imperfections on fy, fu, and Ef with increasing age. These
discussions are important because they implicate the effects of structure on the mechanical properties of fibrils as reported by Shen et al. (44) and provide the basis for directing future
studies on the structural effects on the mechanics of fibrils in the presence of aging.
Another concern addresses the changes in the intermolecular cross-links of collagen with age. According to Bailey (4), there are two major cross-link processes: 1) the initial stabilization of the fibrils through lysyl oxidase during growth and maturation and 2) the subsequent process arising from the reaction with glucose or its metabolites that occurs with age (during old age), as the turnover of the collagen is reduced to a minimum. Bailey (4) argued that the divalent dehydro-hydroxylysinonorleucine (deH-HLNL) cross-link, which binds two collagen molecules end to end along the length of the fibril, predominates in young tendon (i.e., the first cross-link process). At maturation, deH-HLNL could react with histidine to form the trivalent (mature) cross-link, which is known as histidino-hydroxylysinonorleu-cine (HHL) (4). In addition, deH-HLNL may react with an-other cross-link or lysyl aldehyde from an adjacent fibril to form interfibrillar cross-links. The presence of these interfibril-lar cross-links may contribute to increasing uFas more energy
would be required for fibril pull-out. During maturation, as the rate of turnover of collagen decreases, the proportion of new collagen stabilized by the deH-HLNL decreases, thus leading to a build-up of the HHL (4). An appreciable increase in HHL density with age may lead to a corresponding increase in the magnitudes offy,fu, and Ef; this could challenge our
under-lying assumption that these parameters are age invariant (21a). Following maturation, whereas there is little change in the concentration of HHL and deH-HLNL, new intermolecular cross-links could result from the production of advanced gly-cation end-products (AGEs) by the glygly-cation process [i.e., the second cross-link process (4, 10)]. The higher concentration of cross-links in the tendon at old age (vs. maturation) suggests a possible mechanism for maintaining— by counteracting the decrease in collagen concentration—the mechanical properties of tendon (10). Nevertheless, as acknowledged in an earlier study (21a), how the cross-links, i.e., HHL, deH-HLNL, and AGEs, cooperate to influencefy,fu, and Efduring old age is
not well understood. The scaling relationships of Eqs. 2 and 3 do not take into account any possible influence of changing cross-link profiles or densities with age.
We highlight three recent findings that support the argu-ments addressing the influence of covalent cross-link on the molecular mechanics of collagen: 1) the presence of the cross-link (C-terminal) between two collagen molecules serves to resist intermolecular slippage during deformation (50); 2) the overlap regions act as a buffer by preventing the stress from concentrating along the portion of the strands containing the cross-link (50); and 3) rupture and sliding of tropocollagen molecules in deforming fibrils are strongly influenced by fibril-lar length, width, and cross-linking density (18, 47). Thus increasing fibrillar width leads to increase in the stiffness, maximum stress generated, and strain energy absorbed within the fibril (47); accordingly, the scaling relationships have predicted how these parameters interplay to regulate uE(Eq. 2)
and uF(Eq. 3).
Conclusion. There have been several studies to establish the
structure-property relationship of connective tissues in the pres-ence of aging since the work of Parry et al. (37). These studies have included qualitative and quantitative models. The different approaches are inevitable because of the continuing improve-ment in the techniques used in these studies, such as the finite
mixture modeling (35) of the fibril size distribution (30) and decorin-deficient mouse model for studying the altered fibril structure (11, 40, 55), as a result of experience in the applica-tion of these techniques to an increasingly wide range of problems, such as the regulation of fibrillogenesis and tendon development (55). To this end, our study has made a number of contributions to understanding the structure-property relation-ship of tendon in the presence of aging. It has centrally addressed an energy-based argument to formulate the scaling relationships for modeling the influence of D on the strain energy absorbed to fracture by the tendon. These findings indicate the important contributions of the D1 and D2 fibril subpopulations to age variations in tendon resilience. An in-crease in the DD2and a decrease in the DD1contribute to an
increase in uE/Yand vice versa. On the other hand, only D1
contributes to age varations in tendon resistance to rupture. Accordingly, the larger the DD1, the higher is the uF/U and
vice versa. This study provides important first evidence of a possible role for the bimodal D distribution in directing age-related variations in tendon resilience and resistance to rupture. APPENDIX
Modeling the fibril elastic deformation scaling relationship.
Con-sider a long fibril (of length 2lfand diameter D) embedded within the
ECM (Fig. 1) in the direction of an applied load acting on the tendon. As loading progresses from the elastic stress-transfer stage to a transitional stage, mode  (20), the frictional sliding action at the fibril/PG-matrix interface generates, which is constant of distance (z) along the fibril axis (20). Here z⫽ 0 corresponds to the fibril center, and z⫽ lfcorresponds to the fibril end. In response to the shear action,
the fibril deforms, and an axial tensile stress (f), a function of z, is
generated within the fibril (20)
f(z)⫽ 4
兵
lf⁄ D其
[1⫺ z ⁄ lf]. (A1) Thus f is maximum at z ⫽ 0. Assuming symmetric loading, theelastic energy (Uf) stored in an infinitesimal length (⌬z) on one-half of
the fibril is
⌬Uf⫽ D2f2⌬z ⁄ 8Ef. (A2) Substituting the expression offfrom Eq. A1 into Eq. A2 gives⌬Uf⫽
2lf2{2/Ef}[1⫺ z/lf]2⌬z. On the other hand, work done by the fibril
element against interfibrillar ECM is
⌬Um⫽ D␦⌬z (A3)
where␦, the axial deformation in the fibril, is a function of z given by ␦ ⫽
兰
zlffdz (A4)
andεfis the corresponding elastic strain in the fibril (also a function
of z), which relates linearly tofsuch thatf⫽ Efεf. Substituting the
expression forf(Eq. A1) intof⫽ Efεf, we find thatεf⫽ 4[lf/D]{/ Ef}[1⫺ z/lf]. As the fibril end-face (z⫽ lf) is free of stress (20), thus
εfand␦ are reduced to zero at this point. Evaluating the integral in Eq.
A4 leads to␦ ⫽ ⌬Uf⫽ 2[lf2/D]{/Ef}[1⫺ z/lf]2; upon substituting
the expression of␦ into Eq. A3, we find ⌬Um⫽ 2lf2{2/Ef}[1⫺
z/lf]2⌬z ⫽ ⌬Uf. It follows that⌬Uf⫹ ⌬Um⫽ 2⌬Uf. Alternatively,
summing the integrals of the respective infinitesimal elements, ⌬Uf
and⌬Um, from z⫽ lfto 0 leads to the total work done, i.e., Uf⫹ Um⫽
[4/3]lf32/Ef. We assume that the distribution of the fibrils is a truly
random one. This implies that all cross-sections of the tissue normal to the tensile axis are identical, i.e., that in any small length of the tissue, the number of fibril centers/unit area of cross-section must be constant. Let N be the number of fibrils/unit cross-sectional area of the composite; the volume fraction of the fibrils is approximated by Vf⬇
ND2/4. The work of elastic deformation up to the point of yielding of the tendon, i.e., G⫽ N[Uf⫹ Um]⫽ {4/3}Nlf3[2/Ef]. At z⫽
0, assuming the fibrils have yielded, we have f ⫽ fy, and lf ⫽
D[fy/]/4 (Eq. A1). Thus we find G⫽[f
y ]3VfD
12Ef . (A5)
Comparing the dimensions of G(Eq. A5) and the associated strain energy density (), we find that G scales to if we arbitrarily divide Eq. A5 throughout by a scale factor (L), which to order of magnitude, may be identified with the “grip-to-grip” length of the test sample (which can be fixed for all samples). By applying the rule of mixtures (21a) to modelingY, assuming that to order of magnitude,
fyVf ⬎⬎ m[1 ⫺ Vf] at PI (Fig. 2C), it follows thatY⬇ fyVf.
ReplacingfyVfbyYin Eq. A5 and recalling from previous studies
that(3),fy, and Ef(21a) are constants and independent of age, we
find
⁄Y⫽ cED, (A6)
where cE⫽ [fy]2/{6Ef}.
We highlight two important findings concerning fibrillar yielding in light of recent reports. First, the occurrence of yield has been ex-plained by observing the point of deviation from linearity of the stress-strain curve in a single fibril test using a microelectromechani-cal system (44). Second, at the fibrillar level, yielding addresses a change in the mechanism of stress transfer. We attribute the change to two possible processes: 1) disruption of the bonds between adjacent fibrils and between the fibrils and PG matrix, where a sufficiently large, relative displacement of the fibrils occurs (36, 39), and 2) yielding in fibrils as slippage between tropocollagen molecules occurs (18, 47).
Modeling the fibril rupture scaling relationship. As the applied
load increases, eventually, microcracks initiate in interfibrillar ECM (Fig. 2B). Consider fibrils bridging a crack region of interfibrillar ECM (Fig. 2B). Around the crack site, ECM can no longer take up load effectively, and the bulk of the load is transferred to the fibrils. As the crack opens,RPis generated as ECM shear slides over the
fibrils; strain energy accumulates in each fibril as it deforms in response to RP. For fibrils whose lengths are ⱖ2lc, eventually the
stored energy reaches a level that is sufficient to fracture the fibril at the point, i.e., z⫽ 0 of maximum f⫽ fu(⫽4RPlc/D; from Eq. A1).
Suppose we apply a conservative assumption that one out of every two fibrils randomly selected from the ECM has a lengthⱖ2lc; thus
50% of the population of fibrils across the crack could rupture, whereas the rest would be pulled out. Consider the elastic energy stored within a region from the crack region of interfibrillar ECM of the yielded fibril. On each side of the crack, the plane retracts and slips relative to the fibril (Fig. 2B). For an element (of infinitesimal length ⌬z) along this region, the energy takes the same form as Eq. A2. Recalling an argument from the elastic deformation analysis, we can determine the correspondingεffromf⫽ Efεfwhenf⫽ fu. The
elastic energy ⌬Uf absorbed by the small element is obtained by
substituting the expression of f into Eq. A2 to give ⌬Uf ⫽
2lf2{RP2/Ef}[1⫺ z/lf]2⌬z (38). Following from Eq. A3, the work
done by the element in sliding against ECM is given by ⌬Um ⫽
DRP␦⌬z2(38). Following from this expression of⌬Umand Eq. A4,
we proceed to evaluate␦. Noting that the values of εfand␦ arising
from the cracked-induced stress are reduced to zero at z ⫽ lc, on
evaluating the integral of Eq. A4, we find that ␦ ⫽ 2{RP/Ef}[lc⫺
z]2/D (38). The arguments leading up to here also implicate that ⌬Um⫽ ⌬Ufwhen fibrils (bridging interfibrillar ECM crack) rupture
(38). The sum of⌬Umand⌬Uf, by integrating from z⫽ lcto 0, gives
the total work, i.e., Uf ⫹ Um ⫽ {8RP2/Ef}兰lc
0l
c ⫺ z]2dz ⫽ [4/
3]lc3{RP2/Ef} (38). Suppose the matrix crack extends across the
cross-section of the tendon. We note that the number of ruptured fibril/unit cross-sectional area ⫽ N/2; the volume fraction of these fibrils is given by ND2/8 (38). With the use of the mathematical
model described by Eq. A1 but replacing lfby lc[⫽D(fu/RP)/4] and
byRP, noting thatf⫽ fuat z⫽ 0, the corresponding work of
rupture of the fibril (38) GR[⫽2(N/2)(Uf⫹ Um)⫽ N(8/3)lc3RP2/Ef]
becomes
GR⫽ [fu]2V
fD
6EfRP (A7)
Similarly, GR (Eq. A7) scales to the corresponding strain energy
density (R) by arbitrarily dividing Eq. A7 throughout by L. Applying
the rule of mixtures to modelU, in the limitfuVf⬎⬎ m[1⫺ Vf]
(21a), we find that U⬇ fuVf. Here, we have modeled fu as an
invariant of age (21a). Let cR⫽ [fu]2/{6EfRP}. ReplacingfuVfby
Uin Eq. A7 leads to
R⁄U⫽ cRD (A8)
Modeling the fibril pull-out scaling relationship. Across the crack
within interfibrillar ECM, fibrils shorter than 2lcare not expected to
rupture (38). Instead, as the applied load increases, eventually, these fibrils will be pulled out from the ECM when the surfaces of the crack are completely displaced apart (Fig. 2B). To calculate the work of pull-out, we assume on order of magnitude that the interfacial shear stress generated isRPand is constant during pull-out. Let lzbe the
embedded length of a fibril. The force needed to pull the fibril out is equal to the force resisted by the fibril, i.e., D2
f/4⫽ Dlz RP,
wherefis a function of fibril axial distance z whose origin has now
been designated to begin at the point adjacent to the crack plane. At any instance during the pull-out process, the profile of falong the
embedded length is such that it decreases linearly with distance; consequently, consistent with the earlier assumption of a stress-free end-face, we findf⫽ 0 at the fibril end. Consider an infinitesimally
small element along the embedded fibril at distance z from the cracked plane. The work of pull-out is Uf⫽
兰
lz0D
RPzdz⫽ {1/2}Dlz2RP
(38). We note that the number of fibrils pulled out across the unit area of crack is N/2 (38). Let the number of fibrils, with embedded lengths ranging from lz and lz ⫹ ⌬lz on one side of the crack, which are
involved in the pull-out process, be {N/2}⌬lz/2lz(38). Since we are
considering one side of the crack, the work of fibril pull-out becomes
GP/2. The work can be determined by summing the values of Uffor
the fibrils being pulled out; i.e., GP/2⫽ 兰0
lz[N/2]U
fdlz/2lz ⫽ VfDlf2
RP2/24 (38). Thus GP attains a maximum value when lf ⫽ lc
[⫽D(fu/RP)/4] (38), which is GP⫽ [f u ]2VfD 8RP (A9)
GP scales to the corresponding strain energy density (P) by
arbitrarily dividing Eq. A9 throughout by L. Let cP ⫽ fu/{8RP}.
Recalling an earlier argument, which invokes the rule of mixtures to establish the lower limit ofU⫽ fuVf, it follows that replacingfuVf
byUin Eq. A9 leads to
P⁄U⫽ cPD. (A10)
GRANTS
Support for this work was provided by grants from the European Union Framework Programme QLK6-CT-2001-00175 and Merlion-France Pro-gramme 5.03.07 under the French Ministere des Affaires Etrangeres et Euro-peennes and Collier Charitable Trust COLCHAGA09. D. F. Holmes and Y. Lu were funded by the Wellcome Trust 091840/Z/10/Z (to K. E. Kadler).
DISCLOSURES
No conflicts of interest, financial or otherwise, are declared by the authors.
AUTHOR CONTRIBUTIONS
Author contributions: K.L.G., P.P.P., D.F.H., K.E.K., T.J.W., and D.B. conception and design of research; K.L.G., P.P.P., D.F.H., and Y.L. performed experiments; K.L.G. and D.F.H. analyzed data; K.L.G., P.P.P., D.F.H., D.B.,
and K.E.K. interpreted results of experiments; K.L.G. and D.F.H. prepared figures; K.L.G. and D.F.H. drafted manuscript; K.L.G., P.P.P., D.F.H., K.E.K., T.J.W., and D.B. edited and revised manuscript; K.L.G., P.P.P., D.F.H., K.E.K., T.J.W., D.B., and Y.L. approved final version of manuscript.
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