Bimodal collagen fibril diameter distributions direct age-related variations in tendon resilience and resistance to rupture

Abstract

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 (D_D1) and larger (D_D2) mean Ds, respectively. Both D_D1 and D_D2 increase from 1.6 to 4.0 mo but decrease thereafter. From tensile tests to rupture, two strain energy densities were calculated: 1) u_E [from initial loading until the yield stress (σ_Y)], which contributes primarily to tendon resilience, and 2) u_F [from σ_Y through the maximum stress (σ_U) until rupture], which relates primarily to resistance of the tendons to rupture. As measured by the normalized strain energy densities u_E/σ_Y and u_F/σ_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 u_E/σ_Y (resilience energy) are associated with decreases in D_D1 and increases in D_D2, whereas u_F/σ_U (rupture energy) is associated with increases in D_D1 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.

tendons may be regarded as biological fiber composites comprising 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 biological 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 morphology (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 profiles 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 (regression) 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 modulus 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 conflicting findings 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 influence 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.

MATERIALS AND METHODS

An energy approach to modeling the structure-property relationship 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 elastically, 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 electrostatics 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 responsible 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 β, 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 (τ_β), which is constant throughout the interface. Mode β 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 plastically 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 corresponds 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, τ_RP (>τ_β), is constant throughout the fibril/PG-matrix interface during fibril rupture and fibril pull-out from the ECM (Fig. 2B).

Fig. 1.

Model of a collagen fibril embedded in a coaxial cylinder of hydrated PG-rich matrix (shaded region). Here, 2l_f denotes 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.

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 u_E and u_F, respectively; the strain energy density to fracture u₀ = u_E + u_F.

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 (u_E), which contributes primarily to tendon resilience (regulated by fibrils undergoing elastic deformation), and the essential energy (u_F), 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), u_E 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 u_F corresponds to the area under the curve from the PI to the point of fracture. From the scaling relationships, we find that u_E = μ_β(D) (see Eq. A6), and u_F = μ_R(D) + μ_P(D) (by combining Eqs. A8 and A10; see below). Here, μ represents the strain energy density (symbol in adjacent parentheses indicates the argument of μ), and subscripts β, R, and P denote mode β, fibril rupture, and fibril pull-out, respectively. In principle, the equations for u_E and u_F would 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 hereafter 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 (D_D1) and larger (D_D2) mean Ds. We find that

$$\begin{array}{c}{u}_E={\mu }_{\beta }\left(D_{\text{D}1}\right)+{\mu }_{\beta }\left(D_{\text{D}2}\right)\\ u_F=\left[{\mu }_R\left(D_{\text{D}1}\right)+{\mu }_{\beta }\left(D_{\text{D}1}\right)\right]+\left[{\mu }_R\left(D_{\text{D}2}\right)+{\mu }_P\left(D_{\text{D}2}\right)\right]\end{array}.$$ (1)

From Eq. 1, we arrive at

$$u_E/{\sigma }_Y=c_{E1}{D}_{\text{D}1}+c_{E2}{D}_{\text{D}2},$$ (2)

$$u_F/{\sigma }_U=c_{RP1}{D}_{\text{D}1}+c_{RP2}{D}_{\text{D}2},$$ (3)

where c_E1 and c_E2 are constants of proportionality for Eq. A6 (see below), and c_RP1 and c_RP2 describe the summation of the constants of proportionality c_R1 and c_R2 (see Eq. A8) and c_P1 and c_P2 (see Eq. A10); i.e.

$$\begin{array}{c}{c}_{RP1}=c_R+c_{P1}\\ c_{RP2}=c_R+c_{P2}\end{array}.$$ (4)

We note that c_E1 and c_E2 are expressed in terms of the fibril yield stress (σ_f^y), fibril stiffness (E_f), τ_β, and a scale factor for length (L; see Eq. A6); c_R1 and c_R2 are expressed in terms of the fibril fracture stress (σ_f^u), E_f, τ_RP, and L (see Eq. A8); c_P1 and c_P2 are expressed in terms of only σ_f^u, τ_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 inhibition 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 absorption 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 D_D1 and D_D2.

Tissue preparation and mechanical testing.

The tissue preparations, 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 accorded 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, u_E, and u_F, were determined. In addition, we have determined σ_Y and σ_U from 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 instrument (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 4 μm × 5 μm. 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 computed using the Semper5 image analysis package (Synoptics, Cambridge, 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.

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 subpopulation), 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, respectively, 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—D_D1 and D_D2. 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, u_E/σ_Y, and u_F/σ_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 D_D1 and D_D2 with u_E/σ_Y and u_F/σ_U, respectively; the results were reported as means ± SE. Significance was defined as P < 0.05. Multicollinearity of D_D1 vs. D_D2 was assessed using the Pearson correlation coefficient test (1). All statistical analyses were performed using Minitab commercial software (version 14; Minitab, State College, PA).

RESULTS

Strain energy density.

Figure 3 shows the plot of u₀ vs. age. The ANOVA test reveals that not all of the means of u₀ among the different age groups are equal (P = 0.001; F = 5.84). The Fisher LSD test shows that statistical differences occur between 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. Correspondingly, the plot of u₀ vs. age describes an increasing u₀ from 1.7 to 4.0 mo, fluctuation of u₀ with no appreciable trend from 4.0 to 29.0 mo, and a decreasing u₀ from 29.0 to 35.3 mo.

Fig. 3.

Graph of strain energy density to fracture (u₀) vs. age. The vertical bars represent SE.

Figure 4A shows the normalized strain energy density (u_E/σ_Y) vs. age. Included in this figure are plots of the respective parameters—u_E (Fig. 4B) and σ_Y (Fig. 4C)—vs. age for informational purpose. The ANOVA test reveals that not all of the means of u_E/σ_Y are 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 u_E/σ_Y vs. age describes fluctuations in u_E/σ_Y from 1.7 to 4.0 mo with no appreciable trend; thereafter, u_E/σ_Y increases from 4.0 to 23.0 mo and fluctuates from 23.0 to 35.3 mo.

Fig. 4.

Graphs of (A) normalized strain energy density (u_E/σ_Y), (B) strain energy density contributing to tendon resilience (u_E), and (C) stress generated by the tendon at yield point (σ_Y) vs. age. The vertical bars represent SE.

Figure 5A shows the normalized strain energy density (u_F/σ_U) vs. age. Also included in this figure are plots of the respective parameters—u_F (Fig. 5B) and σ_U (Fig. 5C)—vs. age for information. [Interestingly, it is observed that the plots of σ_Y (Fig. 4C) and σ_U (Fig. 5C) vs. age reveal very similar profiles to u₀ vs. age (Fig. 3).] The ANOVA test reveals that not all of the means of u_F/σ_U are 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 u_F/σ_U vs. age reveals no appreciable change in u_F/σ_U from 1.7 to 2.6 mo, increase in u_F/σ_U from 2.6 at 4.0 mo, decrease in u_F/σ_U from 4.0 to 11.5 mo, and fluctuations with no appreciable change thereafter. Clearly, the trend exhibited by u_F/σ_U is a contrast to that exhibited by the u_E/σ_Y. We note that whereas results from the Fisher test show that the variation in u₀ with respect to age is accounted for by four out of the seven sets of statistical differences between the means, the variation in u_E/σ_Y (and also u_F/σ_U) is accounted for by three out of the seven sets of statistical differences between the means.

Fig. 5.

Graphs of (A) normalized strain energy density (u_F/σ_U), (B) strain energy density contributing to tendon resistance to rupture (u_F), and (C) strength of the tendon (σ_U) vs. age. The vertical bars represent SE.

Fibril diameter distribution.

Figure 6 shows an array of histograms of normalized frequency vs. D and the corresponding (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 irregularity) dominate.

Fig. 6.

Histograms of normalized frequency vs. D distribution and the corresponding (representative) transmission 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 subpopulations, respectively. Original scale bars, 150 nm.

Plots of D_D1 and D_D2 vs. age (Fig. 7) reveal that the mean Ds are small at 1.7 mo but increase rapidly with age until 4.0 mo. Thereafter, D_D1 decreases (more rapidly than D_D2) with age. Interestingly, the SD of the D_D2 is 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 D_D1 is small at 1.7 mo, increases steadily with age until 4.0 mo, but decreases steadily with age thereafter. From the multicollinearity analysis of D_D1 vs. D_D2, we find that the Pearson correlation coefficient test = 0.599. We conclude that the D_D1 and D_D2 are 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).

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.

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 observations on the age-related variations in the mean D, the results on the age-related variations in the width of the bimodal distributions could be the focus for further investigation. Of considerable 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 D_D1 during 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 u_E/σ_Y and both D_D1 and D_D2. Here, it is observed that the two coefficients, c_E1 and c_E2, of Eq. 2 are different from zero (ANOVA, P = 0.000; F = 15.98). Table 1 summarizes the results of c_E1 (P = 0.000) and c_E2 (P = 0.000). The R² value indicates that the predictors, i.e., the D_D1 and D_D2, can explain 54.5% of the variation in u_E/σ_Y; the remaining 45.5% of the variation in u_E/σ_Y arises from random arrangement of the points on the u_E/σ_Y vs. D_D1 and D_D2 space. The model's y intercept [=(4.4 ± 5.7) × 10⁻³] is not significantly different from zero (P = 0.452), suggesting that u_E/σ_Y will not be significantly different from zero when D_D1 and D_D2 are zero.

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 c_E1 and c_E2 refer to the coefficients of Eq. 2 [constants of proportionality (see Eq. A6)], and those of c_RP1 and c_RP2 refer to the coefficients of Eq. 3 [summation of the constants of proportionality c_R (see Eq. A8) and c_P (see Eq. A10)].Values are means ± SE.

^*Significantly different, P < 0.05.

Similarly, regression analysis reveals strong evidence of a linear relationship between u_F/σ_U and D_D1 and D_D2. The analysis suggests that at least one coefficient, i.e., c_RP1, c_RP2, or both, is different from zero (ANOVA, P = 0.000; F = 4.74). However, further examination shows that only c_RP1 is significantly different from zero (P = 0.040), indicating that only D_D1 is significantly related to u_F/σ_U. The R² value indicates that the predictors can only explain 29.2% of the variation in u_F/σ_U; in other words, the remaining 70.8% of the variation in u_F/σ_U arises from random arrangement of the points on the u_F/σ_U vs. D_D1 and D_D2 space. Interestingly, the model's y intercept [=(1.2 ± 0.2) × 10⁻¹] is significantly different from zero (P = 0.000), suggesting that u_F/σ_U will be significantly greater than zero when D_D1 is zero. However, this must be meaningless, since we cannot have a nonzero value for u_F/σ_U when D_D1 is zero. It may well be that if our sample included smaller values of D_D1, then we would find that the relationship was curved. Note also that the magnitude of c_RP1 is one order of magnitude greater than that of c_E1; this is consistent with the (order of magnitude) difference between u_F/σ_U and u_E/σ_Y. The results of the derived coefficients c_RP1 and c_RP2 are summarized 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 u₀ as 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 u₀ but have been established previously for σ_U and E (21a). For the model tendon used in this study, growth is characterized by a rapid increase in u₀ until maturation; the transition from postnatal growth to maturation occurs at ∼4 mo. Throughout the maturation phase, u₀ fluctuates 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 u₀ of 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 significance of the data in Viidik's work. It is necessary, however, to consider the molecular, cellular-ECM, physiological, and population 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 u₀ is the result of changes in u_E and u_F. In turn, u_E and u_F are 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 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) suggest 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 u_E/σ_Y vs. D_D1 and D_D2 (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 loading. From Table 1, the results suggest that an increasing D_D2 contributes to increasing the tissue resilience, but an increasing D_D1 contributes 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 u_F/σ_U vs. D_D1 and D_D2 (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 applicability 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 discussion 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 τ_β and τ_RP from 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 τ_RP to τ_β, which is a parameterization of the relative stress transfer. Let E_m and σ_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 (V_f) on E and σ_U leads to E = [E_f − E_m]V_f + E_m, and σ_U = [σ_f^u − σ_m]V_f + σ_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 predominance of the role of collagen for regulating E_f and σ_U can be satisfied to order of magnitude (see appendix) by setting these y intercepts to zero. One then finds that E = E_fV_f, and σ_U = σ_f^uV_f; additionally, σ_Y = σ_f^yV_f. Linear regression analysis of E = E_fV_f (P < 0.001; F = 768.9), σ_Y = σ_f^yV_f (P < 0.001; F = 440.6), and σ_U = σ_f^uV_f (P < 0.001; F = 571.8) yields the gradients (i.e., E_f, σ_f^y, and σ_f^u) of the respective equations (Table 2). We note that the order of magnitude estimates for E_f falls with the range of 0.3–1.6 GPa derived from experiments (14, 44); the order of magnitude estimate for σ_f^y is smaller than the lower limit (70 MPa) obtained by experiment (44); and the order of magnitude estimate for σ_f^u is smaller than the cyclic loading fracture stress (200 MPa) reported by Shen et al. (44). To develop our analysis further, we note that c_R1, c_P1, and c_E1 are related to σ_f^y, σ_f^u, τ_β, τ_RP, and E_f, according to Eqs. A6, A8, and A10 (see below). From Eqs. 2 to 4, we find that c_RP1/c_E1 = [c_R1 + c_P1]/c_E1 leads to a mathematical expression for the ratio of τ_RP to τ_β; i.e., τ_RP/τ_β = Kσ_f^u{σ_f^u + [3/4]E_f}/[σ_f^y]², where K = c_E1/c_RP1. By considering the upper (mean + SE) and lower (mean − SE) limits of c_E1, c_RP1, E_f, σ_f^y, and σ_f^u (Tables 1 and 2), we obtain an estimate of τ_RP/τ_β in the range of 15–87. This estimate is important in terms of the mechanism of stress transfer by frictional sliding (as quantified by τ_β and τ_RP, respectively) for small and large deformation effects; the argument demonstrates that τ_RP is 10–10² times (to order of magnitude) higher than τ_β.

Table 2.

List of derived parameters for the order of magnitude analysis of τ_RP/τ_β

Parameter	Predicted Value (MPa)
Ef	728.8 ± 29.7
σfy	29.0 ± 1.5
σfu	65.8 ± 3.0

Values are means ± SE.

The τ_RP/τ_β analysis lends to further predictions of τ_β and τ_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 σ_f^u[πD²/4] = 10⁻⁷ N for typical fibrils having D = 10⁻⁷ m and length not exceeding the critical value of 2l_c = 10⁻⁶ m (34). Second, from shear-sliding analysis (see appendix), we have 2l_c/D = σ_f^u/2τ_RP; the shear action τ_RP is of order of magnitude σ_f^u[πD²/4]/{πl_cD} = 10⁵ Pa. Given that τ_RP/τ_β ranges from 10 to 10² (from the argument established in the previous paragraph), it follows that τ_β ranges from 10³ to 10⁴ Pa. Of note, this range of estimates for τ_β is 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 10⁻¹¹ N (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 × 10⁻¹¹ N)/[π(10⁻⁷ m)(68 × 10⁻⁹ m)] ≈ 7,489 Pa, which lies within the estimated limits of τ_β. Both decorin and biglycan PGs compete for collagen binding and the possibility 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 regulate the stress transfer (maintaining the values of τ_β and τ_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 enabled 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 featuring a Gly-X1-X2 amino acid sequence pattern and a standard 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 σ_f^y is the measure of the force of attraction arising from covalent bonds within (and between) collagen molecules/unit volume; E_f is the measure of the rate of change of the force of attraction/unit volume with respect to strain; and σ_f^u is 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., σ_f^y, σ_f^u, and E_f, 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 σ_f^y, σ_f^u, and E_f 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-hydroxylysinonorleucine (HHL) (4). In addition, deH-HLNL may react with another cross-link or lysyl aldehyde from an adjacent fibril to form interfibrillar cross-links. The presence of these interfibrillar cross-links may contribute to increasing u_F as 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 of σ_f^y, σ_f^u, and E_f; this could challenge our underlying 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 glycation end-products (AGEs) by the glycation 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 influence σ_f^y, σ_f^u, and E_f during 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 arguments 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 fibrillar 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 u_E (Eq. 2) and u_F (Eq. 3).

Conclusion.

There have been several studies to establish the structure-property relationship of connective tissues in the presence 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 improvement 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 application 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 relationship 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 increase in the D_D2 and a decrease in the D_D1 contribute to an increase in u_E/σ_Y and vice versa. On the other hand, only D1 contributes to age varations in tendon resistance to rupture. Accordingly, the larger the D_D1, the higher is the u_F/σ_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.

GRANTS

Support for this work was provided by grants from the European Union Framework Programme QLK6-CT-2001-00175 and Merlion-France Programme 5.03.07 under the French Ministere des Affaires Etrangeres et Europeennes 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.

Glossary

D

Collagen fibril diameter

D_D1, D_D2

Mean Ds of collagen fibril subpopulations, D1 and D2, respectively

E

Elastic modulus (stiffness) of tendon

ECM

Extracellular matrix

E_f, E_m

Elastic moduli of collagen fibril and interfibrillar matrix, respectively

G_β

Work of elastic deformation of a fibril during transitional stage (mode β)

G_P

Work of fibril pull-out from ECM

G_R

Work of fibril rupture

L

Scale factor for length

l_z

Embedded length of a fibril prior to pull-out

N

Number of fibrils/unit cross-sectional area of tendon

PG

Proteoglycan

u_E, u_F

Strain energy densities contributing to tendon resilience and resistance to rupture, respectively

U_f, U_m

Strain energies for elastic deformation of a fibril and intefibrillar matrix, respectively

u₀

Strain energy density to fracture of tendon

V_f

Volume fraction of collagen fibrils in ECM

z

Axial ordinate of the cylindrical polar coordinate system

δ

Axial deformation in fibril

ε_f

Axial strain in fibril

μ_β

Strain energy density for elastic deformation of fibrils

μ_P

Strain energy density for fibril pull-out from ECM

μ_R

Strain energy density for rupture of fibrils

σ_f

Axial stress in the fibril as a function of distance, z

σ_m

Average stress in interfibrillar ECM

σ_U

Maximum stress (strength) of tendon

σ_Y

Stress in the tendon at the point of yielding

σ_f^u

Breaking stress of collagen fibril

σ_f^y

Yield stress at the fibril center when the tendon yields at σ_Y

τ_β

Shear stress generated at the fibril/PG-matrix interface during the transitional stage (mode β)

τ_RP

Shear stress generated at the fibril/PG-matrix interface during fibril rupture or pull-out

2l_c

Critical length of collagen fibril

2l_f

Collagen fibril length

APPENDIX

Modeling the fibril elastic deformation scaling relationship.

Consider a long fibril (of length 2l_f and 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 β (20), the frictional sliding action at the fibril/PG-matrix interface generates τ_β, which is constant of distance (z) along the fibril axis (20). Here z = 0 corresponds to the fibril center, and z = l_f corresponds to the fibril end. In response to the shear action, the fibril deforms, and an axial tensile stress (σ_f), a function of z, is generated within the fibril (20)

$${\sigma }_f\left(z\right)=4\left\{l_f/D\right\}{\tau }_{\beta }\left[1-z/l_f\right].$$ (A1)

Thus σ_f is maximum at z = 0. Assuming symmetric loading, the elastic energy (U_f) stored in an infinitesimal length (Δz) on one-half of the fibril is

$$\Delta U_f=\pi D^2{\sigma }_f^2\Delta z/8E_f.$$ (A2)

Substituting the expression of σ_f from Eq. A1 into Eq. A2 gives ΔU_f = 2πl_f²{τ_β²/E_f}[1 − z/l_f]²Δz. On the other hand, work done by the fibril element against interfibrillar ECM is

$$\Delta U_m=\pi D{\tau }_{\beta }\delta \Delta z$$ (A3)

where δ, the axial deformation in the fibril, is a function of z given by

$$\delta ={\displaystyle {\int }_z^{l_f}{\epsilon }_f\text{d}z}$$ (A4)

and ε_f is the corresponding elastic strain in the fibril (also a function of z), which relates linearly to σ_f such that σ_f = E_fε_f. Substituting the expression for σ_f (Eq. A1) into σ_f = E_fε_f, we find that ε_f = 4[l_f/D]{τ_β/E_f}[1 − z/l_f]. As the fibril end-face (z = l_f) is free of stress (20), thus ε_f and δ are reduced to zero at this point. Evaluating the integral in Eq. A4 leads to δ = ΔU_f = 2[l_f²/D]{τ_β/E_f}[1 − z/l_f]²; upon substituting the expression of δ into Eq. A3, we find ΔU_m = 2πl_f²{τ_β²/E_f}[1 − z/l_f]²Δz = ΔU_f. It follows that ΔU_f + ΔU_m = 2ΔU_f. Alternatively, summing the integrals of the respective infinitesimal elements, ΔU_f and ΔU_m, from z = l_f to 0 leads to the total work done, i.e., U_f + U_m = [4/3]πl_f³τ_β²/E_f. 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 V_f ≈ NπD²/4. The work of elastic deformation up to the point of yielding of the tendon, i.e., G_β = N[U_f + U_m] = {4/3}Nπl_f³[τ_β²/E_f]. At z = 0, assuming the fibrils have yielded, we have σ_f = σ_f^y, and l_f = D[σ_f^y/τ_β]/4 (Eq. A1). Thus we find

$$G_{\beta }=\frac{{\left[{\sigma }_f^y\right]}^3{V}_{f}D}{12E_f{\tau }_{\beta }}.$$ (A5)

Comparing the dimensions of G_β (Eq. A5) and the associated strain energy density (μ_β), we find that G_β scales to μ_β 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 modeling σ_Y, assuming that to order of magnitude, σ_f^yV_f >> σ_m[1 − V_f] at PI (Fig. 2C), it follows that σ_Y ≈ σ_f^yV_f. Replacing σ_f^yV_f by σ_Y in Eq. A5 and recalling from previous studies that τ_β (3), σ_f^y, and E_f (21a) are constants and independent of age, we find

$${\mu }_{\beta }/{\sigma }_Y=c_{E}D,$$ (A6)

where c_E = [σ_f^y]²/{6E_fτ_β}.

We highlight two important findings concerning fibrillar yielding in light of recent reports. First, the occurrence of yield has been explained by observing the point of deviation from linearity of the stress-strain curve in a single fibril test using a microelectromechanical 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, τ_RP is generated as ECM shear slides over the fibrils; strain energy accumulates in each fibril as it deforms in response to τ_RP. For fibrils whose lengths are ≥2l_c, eventually the stored energy reaches a level that is sufficient to fracture the fibril at the point, i.e., z = 0 of maximum σ_f = σ_f^u (=4τ_RPl_c/D; from Eq. A1). Suppose we apply a conservative assumption that one out of every two fibrils randomly selected from the ECM has a length ≥2l_c; 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 Δz) along this region, the energy takes the same form as Eq. A2. Recalling an argument from the elastic deformation analysis, we can determine the corresponding ε_f from σ_f = E_fε_f when σ_f = σ_f^u. The elastic energy ΔU_f absorbed by the small element is obtained by substituting the expression of σ_f into Eq. A2 to give ΔU_f = 2πl_f²{τ_RP²/E_f}[1 − z/l_f]²Δz (38). Following from Eq. A3, the work done by the element in sliding against ECM is given by ΔU_m = πDτ_RPδΔz² (38). Following from this expression of ΔU_m and Eq. A4, we proceed to evaluate δ. Noting that the values of ε_f and δ arising from the cracked-induced stress are reduced to zero at z = l_c, on evaluating the integral of Eq. A4, we find that δ = 2{τ_RP/E_f}[l_c − z]²/D (38). The arguments leading up to here also implicate that ΔU_m = ΔU_f when fibrils (bridging interfibrillar ECM crack) rupture (38). The sum of ΔU_m and ΔU_f, by integrating from z = l_c to 0, gives the total work, i.e., U_f + U_m = {8πτ_RP²/E_f}∫_lc⁰l_c − z]²dz = [4/3]πl_c³{τ_RP²/E_f} (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 NπD²/8 (38). With the use of the mathematical model described by Eq. A1 but replacing l_f by l_c [=D(σ_f^u/τ_RP)/4] and τ_β by τ_RP, noting that σ_f = σ_f^u at z = 0, the corresponding work of rupture of the fibril (38) G_R [=2(N/2)(U_f + U_m) = N(8/3)πl_c³τ_RP²/E_f] becomes

$$G_{\beta }=\frac{{\left[{\sigma }_f^u\right]}^2{V}_{f}D}{6E_f{\tau }_{RP}}$$ (A7)

Similarly, G_R (Eq. A7) scales to the corresponding strain energy density (μ_R) by arbitrarily dividing Eq. A7 throughout by L. Applying the rule of mixtures to model σ_U, in the limit σ_f^uV_f >> σ_m[1 − V_f] (21a), we find that σ_U ≈ σ_f^uV_f. Here, we have modeled σ_f^u as an invariant of age (21a). Let c_R = [σ_f^u]²/{6E_f τ_RP}. Replacing σ_f^uV_f by σ_U in Eq. A7 leads to

$${\mu }_R/{\sigma }_U=c_{R}D$$ (A8)

Modeling the fibril pull-out scaling relationship.

Across the crack within interfibrillar ECM, fibrils shorter than 2l_c are 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 is τ_RP and is constant during pull-out. Let l_z be 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., πD²σ_f/4 = πDl_z τ_RP, where σ_f is 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 σ_f along the embedded length is such that it decreases linearly with distance; consequently, consistent with the earlier assumption of a stress-free end-face, we find σ_f = 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 U_f = ∫_lz⁰ πD τ_RP zdz = {1/2}πDl_z² τ_RP (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 l_z and l_z + Δl_z on one side of the crack, which are involved in the pull-out process, be {N/2}Δl_z/2l_z (38). Since we are considering one side of the crack, the work of fibril pull-out becomes G_P/2. The work can be determined by summing the values of U_f for the fibrils being pulled out; i.e., G_P/2 = ∫₀^l_z[N/2]U_fdl_z/2l_z = V_fDl_f² τ_RP²/24 (38). Thus G_P attains a maximum value when l_f = l_c [=D(σ_f^u/τ_RP)/4] (38), which is

$$G_P=\frac{{\left[{\sigma }_f^u\right]}^2{V}_{f}D}{8{\tau }_{RP}}$$ (A9)

G_P scales to the corresponding strain energy density (μ_P) by arbitrarily dividing Eq. A9 throughout by L. Let c_P = σ_f^u/{8τ_RP}. Recalling an earlier argument, which invokes the rule of mixtures to establish the lower limit of σ_U = σ_f^uV_f, it follows that replacing σ_f^uV_f by σ_U in Eq. A9 leads to

$${\mu }_P/{\sigma }_U=c_{P}D.$$ (A10)