Introduction
All biological tissues sustain some amount of fatigue or injury damage over time, even under normal physiological conditions. Most tissues also undergo continuous repair, though the corresponding rate of remodelling may not always keep up with the rate of damage, depending on various factors, including ageing and extent of injury. The ability to model the progression of mechanical damage in biological tissues may be valuable for better understanding the balance between damage and repair, and for elucidating the influence of pathomechanics of mechanically mediated progressive degenerative diseases on tissue composition and structure. For example, in articular cartilage, mechanical damage induced by loading may be the initiating factor in progressive osteoarthritic degeneration.
Introduction
In tissue mechanics, theoretical models of damage have been applied most extensively for the analysis of bone [1] [2] [3] but also for soft tissues such as tendon [4] and cartilage [5] . These models are generally based on the continuum damage mechanics framework originally developed by Kachanov [6] and Rabotnov [7] , and subsequently refined by LeMaitre [8, 9] and Chaboche [10] , Simo & Ju [11] , and others. As summarized in [11] , this framework for continuum damage mechanics is based on the thermodynamics of irreversible processes and internal state variable theory [12] , where damage is modelled as an internal or hidden (i.e. non-observable) state variable.
Introduction
Therefore, modern measurement techniques that can directly detect chemical bond modifications and disruptions in biological tissues, such as biochemical measurements for assessing cross-link density in collagenous tissues [13] , immunohistochemistry for assessing mechanically or enzymatically degraded collagens & 2015 The Author(s) Published by the Royal Society. All rights reserved. [14] [15] [16] [17] [18] [19] , and Raman spectroscopy for detecting alterations in chemical bonds [20, 21] , may not inform the traditional damage framework directly.
Introduction
The objective of this study was to demonstrate that this classical continuum damage mechanics framework may be reformulated using only observable scalar state variables for modelling damage in isotropic and anisotropic materials. These state variables represent the density of intact or broken bonds in the solid constituents of a tissue. Different bond species may exist in a tissue, each of which may regulate different components of the strain energy density function governing the anisotropic material response. In this framework, the evolution of bond densities with time is governed by the axiom of mass balance, with a mass supply term representing the exchange of mass between intact and broken bonds. The reaction representing the breaking of bonds is regulated by a failure criterion measure, such as strain energy density or effective stress, whose magnitude determines the probability of bonds breaking, based on a given probability density function (p.d.f.). The evolution of tissue ultrastructure with increasing bond damage may be taken into account by distinguishing the bond densities of fibril bundles oriented along different directions and allowing them to evolve independently based on each fibril bundle's state of strain. This approach differs from the alternative method of allowing the texture tensors of anisotropic materials from evolving with damage.
Introduction
We show that this reactive damage framework reproduces the classical damage mechanics formulation for isotropic materials, with the classical damage variable D actually representing the fraction of bonds that are broken. Using a bond-breaking framework for damage mechanics demonstrates that damage may still be represented by a scalar variable D even in the case of anisotropic materials. Because of its relevance to soft tissue mechanics, the damage response of fibrous materials is examined. First, the damage response of a single fibre bundle is analysed for various choices of p.d.f.'s, under homogeneous and inhomogeneous strain states. Then, the damage mechanics of continuous fibrous distributions are examined, where bundles oriented along different directions are associated with direction-dependent damage variables.
Introduction
Finally, this damage framework is used to examine an intriguing observation from our cartilage tissue engineering studies, regarding the temporal evolution of the composition and material properties of tissue constructs. Experiments reported here have shown that, in long-term culture, collagen and proteoglycan content grow continuously over time, even though the construct's compressive Young's modulus reaches a plateau. We use our damage framework to examine the hypothesis that collagen fibrils synthesized early in culture become progressively more damaged, as the increasing proteoglycan content causes greater Donnan osmotic swelling, thus shifting the structural integrity of the collagenous matrix to younger, less damaged, generations of synthesized collagens.
Theoretical formulation
Biological tissues often consist of heterogeneous mixtures of constituents. For example, in soft connective tissues, the solid matrix may consist of a variety of collagen types, elastin and other macromolecular species such as proteoglycans and hyaluronan. Each specific substance may exhibit a variety of bond species, which may get damaged under different conditions. Therefore, the theoretical formulation first addresses the damage response of a single-bond species, followed by a generalization to multiple bond species. Furthermore, this presentation only considers putative strong bonds that break permanently, thus producing damage, in contrast with putative weak bonds that break and reform spontaneously, thus producing viscoelasticity [22, 23] . Lastly, damaged bonds in biological tissues may ultimately be reincorporated (rebound) due to a repair response. Such phenomena are not featured here, although such behaviours are particularly well accommodated within the theoretical framework. The framework for this theoretical formulation is the theory of mixtures [24] , as applied specifically to constrained mixtures of solid constituents [25, 26] . In this notation, the subscript 0 implies that all bonds are intact for that variable, whereas the subscript 'r' implies that only some of the bonds remain intact; for both subscripts, the corresponding variable is normalized by the volume of the material in the reference configuration. Now consider that bonds in that species have a probability of failure determined by a failure criterion measure J b ðFÞ, where the definition of J b ðFÞ follows the classical framework for damage [27] . In this study, it is proposed that the probability that intact bonds of species b will fail at a value of 
Theoretical formulation
Let the failure measure be given by J b s ¼ J b ðFðsÞÞ at a particular state of deformation F(s), and let the maximum value of J b s over the history up until the current time t be given by [27] represents the fraction of all bonds that have failed up until the current time t, leaving 
Thermodynamics
To examine the thermodynamics of this process, consider that damage represents a reaction that transforms intact bonds into broken bonds consisting of two or more fragment species c,
Thermodynamics
ð2:6Þ
Thermodynamics
The evolving referential density of broken fragments is r In this reactive damage framework, it is assumed that the specific strain energy of broken bond fragments is zero, c c ¼ 0; therefore, the net free energy C r of the mixture of b and c species is simply b is not a function of any of the bond densities r a r . (In physical chemistry of solutions, this same assumption produces ideal solutions; we may thus describe our bond mixture model as an ideal mixture.) The rate at which the intact bond density changes over time is governed by the axiom of mass balance for that constituent [26] , is a positive semi-definite function of the strain. Thus, the thermodynamic constraint simply states that intact bond species may break spontaneously with loading (as their mass supply must be negative) but may not reform spontaneously in this damage framework.
Damage criterion
As described in the prior damage mechanics literature [27] , a damage criterion needs to be defined such as
Damage criterion
where w b represents a damage surface whose tensorial normal is
Damage criterion
The damage criterion of equation (2.11) presents the following alternatives: w b , 0, if the current state of deformation is below the current damage threshold, or w b ¼ 0 and the deformation is receding from, or tangent to, the damage (N : _ F 0), or increasing the damage (N : _ F . 0). Substituting equation (2.4) into the mass balance of equation (2.9) produces the damage evolution constitutive relation, 
Stress
For a single bond species b, the state of stress may be evaluated from C r ¼ C b r using the usual relations of hyperelasticity, recalling that r b r and F are independent state variables [26] . Thus, for the Cauchy stress,
Stress
where D b is evaluated from equation (2.3) and With decreasing s c and increasing a c , these two distributions exhibit opposite biases about the value of m c : in the lognormal distribution, damage progresses more rapidly with increasing J when J , m c , and more slowly when J . m c ; the opposite trend is found in the Weibull distribution. Alternatively, when modelling fracture (onset of complete damage at a single threshold value of J ), the p.d.f. may be given by the Dirac delta function dðÁÞ, with its c.d.f. given by the Heaviside unit step function HðÁÞ, f ðJÞ ¼ dðJ À m c Þ and
Stress
where m c is the critical value of J at which fracture occurs.
Multiple intact bond species
This reactive damage framework may be generalized to any number of intact bond species b, each associated with its own strain energy density C b 0 , such that the net free energy density of the mixture is
Multiple intact bond species
ð2:23Þ
Multiple intact bond species
and the mixture Cauchy stress is 
rsfs.royalsocietypublishing.org Interface Focus 6: 20150063
This approach can accommodate anisotropic materials and anisotropic damage, by allowing damage to evolve differently for various terms of the strain energy density function C r . For example, in a transversely isotropic material whose strain energy density depends on five strain invariants, damage may evolve according to This form of splitting damage among various terms is only an illustration (a constitutive assumption) and does not represent a general form (which does not exist). As the molecular structure of a material permits any number of bond species, constitutive relations for damage may include any number of bond-breaking reactions in response to loading. where T 0 represents the stress at the current state of strain, in the absence of damage, and T is the actual stress [29, 30] . For isotropic materials, D ¼ DI , where I ¼ II is the fourthorder identity tensor and I is the second-order identity tensor, and MðDÞ ¼ ð1 À DÞI , so that T ¼ ð1 À DÞT 0 , consistent with our formulation in equation (2.15), though derived using a different framework. A constitutive relation D is also needed to model the evolution of damage, such that the damage time rate of change is given by
Comparison with classical damage mechanics
This theoretical framework requires the formulation of fourth-order tensor relations for MðDÞ andD in the general case of anisotropic materials [29, 30] . Furthermore, as an internal variable, D is strictly non-observable [12] ; therefore, it cannot be measured experimentally.
Comparison with classical damage mechanics
In our mixture framework where damage is tracked by the mass fraction of bonds that have broken, anisotropic damage is subsumed in the assumption that multiple bond species may coexist in a material, each having its own damage behaviour, as covered in the previous section and summarized in equations (2.23) and (2.24). The mixture approach employs bond kinetics to formulate the constitutive relation forr b r in the mass balance relation of equation (2.9), whereas the damage evolution relation in equation (2.27) is not based on any of the classical balance axioms because it employs non-observable variables.
Fibrous solid matrix

Unidirectional fibre bundle
Consider a fibrous biological soft tissue whose ground substance is modelled as a compressible neo-Hookean solid whose free energy density is [31] 
Unidirectional fibre bundle
where l g and m g are material constants equivalent to the Lamé constants in the limit of infinitesimal strains,
Unidirectional fibre bundle
, where C is the right Cauchy-Green tensor. Consider that the fibrous matrix consists of a unidirectional fibre bundle with a single bond species, whose free energy density is
Unidirectional fibre bundle
where j f is a material constant related to the fibre stiffness, b f is a unitless exponent (b f ! 2), and I n ¼ n Á C Á n is the square of the stretch ratio along the referential fibre direction n. The Heaviside unit step function H( . ) ensures that the fibres contribute to the tissue response only when they are in tension (I n . 1).
Unidirectional fibre bundle
For illustrative purposes, consider that only the fibres' bonds may get damaged in this tissue, such that the tissue free energy density is
Unidirectional fibre bundle
where D is the damage variable for the single-bond species of the fibres. Let the failure criterion measure be the fibre free energy density, J ¼ C In the first analysis, consider that the fibrous tissue is subjected to a homogeneous state of uniaxial tensile stress with traction-free lateral boundaries, where the axial stretch ratio is increased from 1 to 2. The material properties for the ground substance are m g ¼ 0.05 megapascals (MPa), l g ¼ 0.075 MPa; and for the matrix, j f ¼ 1 MPa and b f ¼ 2. These properties are representative of a fibrous tissue such as articular cartilage [33] [34] [35] [36] . The fibres are oriented along the loading direction. In the absence of damage, the value of C r hharrell@example.com mJ mm 23 when the stretch ratio is 2, with a small contribution from the ground substance, C g 0 ¼ 0:05 mJ mm À3 ; thus, almost all the strain energy density is stored in the fibres.
Unidirectional fibre bundle
When damage is modelled using a lognormal c.d.f., with m c ¼ 1 mJ mm 23 and s c ¼ 1/2, 1/4 or 1/8, the free energy density C r begins to deviate from the intact response at values below m c ( figure 3) . Smaller values of s c exhibit a later onset and quicker propagation of damage than larger values of this parameter, as evidenced in the profiles of C r and the axial stress. As the ground substance was not assumed to get damaged in this example, a small residual strain energy density and axial stress are found to persist even after complete failure of all fibres.
Unidirectional fibre bundle
A similar response is observed when damage is modelled with a Weibull c.d.f., with m c ¼ 1 mJ mm 23 and a c ¼ 2.5, 5 or 10, where larger values of a c exhibit a later onset and quicker propagation of damage than smaller values of this parameter (figure 4).
Multiple fibre bundles
Many fibrous biological soft tissues exhibit a continuous fibre distribution density about one or more predominant fibre directions, including articular cartilage, arterial wall and rsfs.royalsocietypublishing.org Interface Focus 6: 20150063 skin [37] [38] [39] . As the tissue is loaded, fibres get recruited along the loading direction, producing a load response that is significantly dependent on the initial fibre distribution and recruitment [40] . To examine the damage response of such fibre distributions, first consider the example of a tissue with three coplanar fibre bundles, oriented at u b relative to the loading direction (b ¼ 1, 2, 3). The free energy density of this tissue is given by
Multiple fibre bundles
where C The axial stress-stretch responses for these three cases exhibit distinct behaviours (figure 5a). In all cases, as one fibre bundle gets damaged, fibres from the remaining bundles sustain greater stress, showing progressive rises and falls in the stress-stretch profile. For case (a), three consecutive failure events may be noted in the response, indicating that the three aligned bundles contribute distinctly at different levels of the prescribed axial deformation; the peak stress reached with increasing damage in the second bundle is higher than that in the first bundle. For cases (b) and (c), only two consecutive failure events are observed, as two of the fibre bundles (u b ¼ +p/6) are oriented symmetrically about the third bundle which coincides with the loading direction; thus, damage progresses identically in these symmetric bundles, explaining the presence of only two peaks. In (b), the second peak is higher than the first peak, whereas in (c) the opposite occurs. An average measure of the damage in all three fibres is also shown, evaluated as D ; 1=N P N b¼1 D b with N¼ 3 (figure 5b). The plateaus D represents the increasing damage in the remaining fibre bundles being recruited, until they fail as well.
Discrete fibre distributions
The results from the above analysis of discrete fibre bundles suggest that a continuous fibre distribution may similarly require making constitutive assumptions about the elastic and damage properties along various directions. Normally, for continuous fibre distributions, variations in elastic properties with direction are taken into account using a fibre density distribution function R(n), where n represents the fibre direction in the reference configuration [39, 40] . The function R(n) typically scales the free energy density and stress responses such that
Discrete fibre distributions
where the integration is performed over the unit sphere, for three-dimensional distributions, or the unit circle, for twodimensional distributions. The fibre density distribution function must satisfy Ð A RðnÞ dA ¼ 1. In the integrand, C n 0 represents the strain energy density of the fibre bundle oriented along n. For example, C n 0 may have the form given in equation (3.2).
Discrete fibre distributions
To account for distinct damage responses along each fibre bundle direction n, let the free energy density of the tissue be given by 
Discrete fibre distributions
In practice, the integral of equation (3.6) may be approximated by a summation, ð
Discrete fibre distributions
where the unit sphere (for three-dimensional distributions) or unit circle (for two-dimensional distributions) is discretized into N finite elements, each with area DA i , whose centroid is located along n i , with
Discrete fibre distributions
A variety of methods are available for discretizing the unit sphere into finite elements with nearly uniform DA i values across the surface [41, 42] , including a geodesic dome representation [43] . We may refer to the discretized form in equation ( A common choice for R(n) in biological soft tissues is the p-periodic von Mises distribution [44] . For three-dimensional fibre distributions with a principal orientation along the unit vector m, Using the discretization of equation (3.7), the number N of fibre bundles needed to reproduce the conditions of a continuous fibre distribution is first investigated for the case b ¼ 0, using N ¼ 196, 596, 1196 and 1796. Results show that the axial stress versus stretch response is nearly identical for all N values up to a stretch ratio of approximately 2, when no significant damage has yet occurred, as also suggested by the response of an intact (i.e. non-damageable) continuous fibre distribution (figure 6a). This outcome indicates that the stress calculation in the absence of damage has converged with increasing N, producing the solution for a continuous fibre distribution. Beyond this stretch ratio, however, the onset of damage varies with N, such that the peak value of the axial stress (a measure of damage resilience) increases with increasing N. A plot of the average damage D in the fibres shows that increasing N decreases the average damage per fibre at a given stretch ratio (figure 6b).
Discrete fibre distributions
A similar behaviour is observed in the case of a concentrated transversely isotropic fibre distribution (b ¼ 5), with the main distinctions being that the stress-stretch response exhibits a significantly higher effective modulus relative to the isotropic fibre distribution, and the progression of damage occurs more rapidly with increasing stretch, after the peak axial stress has been achieved ( figure 7 ).
Discrete fibre distributions
These results demonstrate that the damage response is sensitive to the number of fibre bundles; increasing the number of fibre bundles increases the resilience of the fibrous tissue to damage. A plot of the peak axial stress achieved for each N indicates that this relation is nearly linear, for both isotropic and transversely isotropic fibre distributions (figure 8).
Discrete fibre distributions
As the damage resilience increases as OðNÞ, a truly continuous fibre distribution (where N ! 1) would exhibit damage only as the stress goes to infinity. We conclude that seeking a converged numerical solution for the damage response as N ! 1 is nonsensical: damage is in fact critically dependent on the number of fibre bundles. In these two illustrations, fibres oriented along different directions increasingly get recruited toward the loading direction with increasing stretch and damage, providing further resilience.
Cartilage tissue engineering
In the experimental component of this study, we examine the topic of collagen damage in a growing engineered cartilage tissue construct. The concept that growth may cause damage rsfs.royalsocietypublishing.org Interface Focus 6: 20150063 is perhaps counterintuitive, but this hypothesis was motivated by experimental observations from our tissue engineering studies, as described here.
Cartilage tissue engineering
Cartilage tissue engineering combines cells and a biocompatible scaffold to synthesize tissue constructs for the purpose of repairing defects in osteoarthritic articular layers. When cultured in a chemically defined medium, constructs synthesize native extracellular matrix proteins, principally glycosaminoglycans (GAG) and collagen. Sulfated GAG imparts a negative fixedcharge density to the tissue, causing it to swell due to the resulting Donnan osmotic pressure; conversely, a cross-linked collagen network functions to resist this osmotic swelling [45] . Together, the collagen network and GAG give rise to the robust mechanical properties observed in native cartilage.
Cartilage tissue engineering
In our model system, we have used 2% agarose hydrogel as the scaffold and primary juvenile bovine chondrocytes as the cell source [46] . In recent years, we have adopted a chemically defined (serum-free) chondrogenic medium, supplemented with 10 ng ml 21 transforming growth factor (TGF)-b3 for the first 14 days in culture [47] ; this temporary supplementation of growth factor has been shown to double the rates of GAG and collagen synthesis after day 14 [48, 49] . This methodology has consistently produced GAG levels that meet or exceed those of native juvenile bovine cartilage, producing equilibrium compressive Young's moduli E Y that similarly meet or exceed levels in native tissue [47] . For example, in our recent study using a chondrocyte seeding density of 30 Â 10 6 cells ml 21 , GAG content was 5.7 + 1.6% of wet weight (%ww) and Young's modulus was 483 + 65 kPa at day 45 [49] . In comparison, native GAG content in juvenile bovine cartilage is 2.8 + 0.2%ww and E Y ¼ 400 + 200 kPa [50] . By contrast, collagen levels in our tissue constructs have fallen consistently below native levels [47, 49] ; in the same tissue engineering study quoted above, collagen content was 2.4 + 0.8%ww [49] , whereas the native level is 12.3 + 0.4%ww [50] . The relatively low level of collagen in our constructs was reflected in their low tensile equilibrium modulus (E T ¼ 703 + 315 kPa) when compared with native tissue levels (E T ¼ 17 + 5 MPa [51] ).
Cartilage tissue engineering
To address this deficiency in collagen content, we previously investigated the effect of digesting GAG using chondroitinase-ABC (CABC) at intermediate time points in culture, to let collagen synthesis catch up with GAG synthesis [52, 53] . These results produced higher collagen content than undigested controls, but did not reach native levels.
Cartilage tissue engineering
In this study, we opted to increase the cell seeding density to 120 Â 10 6 cells ml
21
, to match native levels in immature tissue (100-300 Â 10 6 cells ml 21 [54] ). We also increased media volume per construct to account for the nutrient needs of such elevated cell numbers, consistent with our recent findings [55] . Finally, we extended the culture duration to 105 days to provide more time for matrix synthesis, and included a CABC treatment group with the digestion occurring from day 14 to 16 using 0.15 U ml 21 as per our previous protocol [53] . Cylindrical constructs had an initial diameter of 3 mm and height of 2.3 mm. GAG, collagen and cell content, and equilibrium compressive Young's modulus, were assessed using standard protocols described in our previous studies [55] . Two-way analysis of variance (ANOVA) (a ¼ 0.05) was used to detect statistical differences ( p 0.05) for the factors of treatment (control and CABC) and time (14/16, 35, 56, 77 and 105 days for control/CABC), with n ¼ 4-6 samples per group; Tukey-Kramer correction was used for post hoc testing of the means.
21
One of the most striking outcomes of this experimental study was the considerable tissue construct volume growth or expansion from the size of the day-0 construct (figure 9). The control group specimens saw a nearly ninefold increase in their wet weight (and volume) relative to day 0, whereas CABC-treated specimens increased more than fivefold (figure 10). Because of this very large swelling ratio J, specifically defined as construct wet weight at a given time point normalized to the day-0 construct wet weight, GAG and collagen compositional measures are reported using normalization by current day wet weight (final day in culture) to produce conventional fractional measures (%ww, figure 11) , and normalization by day 0 wet weight to assess total matrix synthesis (%D0ww, figure 12) .
21
Using conventional measures, GAG content rose significantly above day 14/16 levels to nearly 7%ww by day 56 ( p , 10 ) and collagen ( p , 10
24
) content in the CABC group were smaller than control.
24
Total cell content per construct did not exhibit a monotonic trend, though significant differences were found with time in culture ( p , 10
24
); the CABC group exhibited consistently lower cell content than control ( p , 10 24 ) (figure 13).
24
Fractional fluid content w w in constructs decreased over time until day 35 ( p , 10 24 ), then remained nearly constant until day 105 ( figure 14) ; at most time points, CABC constructs had lower fluid content than controls ( p , 10 24 ).
24
Finally, the compressive equilibrium Young's modulus rose significantly above day 14/16 levels until day 35, then remained statistically constant until day 105 ( figure 15 ). At two time points, E Y was higher in the CABC group compared with control ( p , 0.01).
Damage in growing tissue constructs
On first pass, results of this experimental study were substantially consistent with our past results [49, 52] , producing comparable GAG and collagen %ww levels and compressive moduli, despite the fact that the cell seeding density was considerably higher. No differences were found in the conventional GAG and collagen measures between control and CABC treatments, though E Y was significantly higher in the CABC group. Figure 11 . GAG and collagen content in engineered constructs, normalized to wet weight on final day of culture (%ww). Different letters indicate significant differences across time points ( p , 10 24 for GAG and collagen) and asterisks indicate significant differences between control and CABC at that time point ( p , 10 23 for GAG only). Figure 12 . GAG and collagen content in engineered constructs, normalized to day 0 wet weight (%D0ww). Different letters indicate significant differences across time points ( p , 10 24 for GAG and collagen) and asterisks indicate significant differences between control and CABC at that time point ( p , 10 24 for GAG and p , 10 23 for collagen). However, once we accounted for the swelling ratio, our interpretation of these results was considerably refined: using a cell seeding density comparable with native juvenile tissue, it was in fact possible to synthesize native levels of collagen in our tissue constructs (15%D0ww in the control group at day 105), indicating that the primary chondrocytes used in this culture system had not lost their ability to produce sufficient collagen. However, the supra-physiological levels of GAG synthesized over the same time period (54%D0ww, compared with the native GAG content of 2.8%ww) produced so much swelling, resulting from the excessive GAG deposition-induced swelling pressure, that the collagen content was diluted (2%ww). It was experimentally evident that the swelling was caused primarily by the GAG (consistent with Donnan swelling theory [56] ), as lower GAG %D0ww levels in the CABC group (figure 12a) produced a lower swelling ratio (figure 10).
Damage in growing tissue constructs
The next intriguing outcome was that E Y was higher in the CABC treatment than control, even though %ww levels of GAG and collagen were not different, and %D0ww levels were in fact smaller in the CABC group. As GAG %ww usually correlates strongly with E Y in tissue constructs [57] , this surprising outcome led us to the hypothesis that the large swelling ratios observed in this study could in fact damage the collagen network, whose integrity is critical to prevent uncontrolled swelling (as the agarose hydrogel breaks apart early in the growth process based on visual inspection).
Damage in growing tissue constructs
We hypothesized the following specific mechanism: GAG and collagen are synthesized continuously during the growth process, each at nearly constant rates. The charged GAG produces a swelling pressure that expands the tissue construct; this expansion is primarily resisted by the collagen network that was synthesized at the start of culture until the current time. Collagen synthesized at earlier times points is then subjected to more swelling than collagen synthesized most recently, under the assumption that collagen is incorporated into the network in a stress-free (unswollen) state. Therefore, as time progresses, earlier generations of collagen are subjected to considerable stretching (up to ninefold swelling for the earliest collagen generation in this study, figure 10 ), causing them to become damaged. Only the most recent collagen generations remain intact or nearly intact, providing some resistance to swelling while maintaining a regular construct shape (figure 9). At some time point during the culture (e.g. day 35 based on the data), the rate of matrix synthesis and damage equalizes, such that GAG and collagen %ww, and E Y , all reach a plateau.
Damage in growing tissue constructs
Based on our earlier study [49] , nearly 90% of synthesized GAG and 85% of synthesized collagen is retained in the constructs throughout the culture period; therefore, in our hypothesized mechanism, we expect that damaged collagen mostly remains in the construct, possibly contributing to the structural integrity when newly synthesized generations cross-link with undamaged segments of those collagen fibrils.
Damage in growing tissue constructs
Under this hypothesized mechanism, the CABC-treated constructs are subjected to less damage in their earlier generations because of the reduced swelling resulting from the GAG digestion. Therefore, the collagen network in the CABC group has a higher tensile stiffness, which effectively translates to greater resistance to compression from a swollen state as shown in our earlier studies on native cartilage [36] .
Damage in growing tissue constructs
To validate this hypothesized mechanistic interpretation of our experimental observations, we performed a theoretical analysis that captured its salient points. Cartilage constructs were modelled as a mixture of collagen, charged GAG, and an interstitial fluid containing water and monovalent salt counter-ions (Na þ and Cl
2
). In this case, the mixture stress is
2
where p is the interstitial fluid pressure, I is the identity tensor, and T e is the collagen matrix stress. Under steadystate conditions, when interstitial fluid flow and ion diffusion have subsided, the interstitial fluid pressure is given by the Donnan osmotic pressure relative to ambient bath conditions,
2
where R is the universal gas constant, u is the absolute temperature, F is the osmotic coefficient describing the deviation from ideal Donnan law (F ¼ 1 for ideal conditions), c* is the external NaCl bath concentration and c F is the fixedcharge density arising from the GAG. The parameter c F was calculated directly from the measured GAG %D0ww using
2
where z GAG is the charge number and M GAG is the molar mass of GAG (z GAG ¼ 22 and M GAG ¼ 513 g mol 21 , recognizing that GAG content was measured using a dimethylmethylene blue assay with chondroitin sulfate as the standard); r T is the construct mass density (taken to be r T ¼ 1 g ml
21
) and w w is the measured fractional fluid content. The only adjustable parameter in this relation is thus F. The collagen matrix stress T e was evaluated using the multigeneration growth model formulated in our earlier study [26] : each generation of collagen was assumed to be deposited in a stress-free reference configuration X g , coinciding with the current configuration at the start of the generation, X g ¼ xðX, t g Þ, where x is the motion, X is the reference configuration of the first generation (start of day 1 in culture), and t g is the time at the start of generation g. The deformation gradient of each generation is thus F g ¼ @x=@X g and the collagen in generation g is subjected to strains calculated from ) and asterisks indicate significant differences between control and CABC at that time point ( p , 10 22 ). E Y ¼ 12 + 5 kPa on day 0. Solid curves represent the model fitted to control data (fit) and the prediction of CABC data ( predict).
21
continuous, a computational scheme employs finite time increments and we chose to have 105 generations, corresponding to the start of each day in culture.
21
Each collagen generation g was modelled using a fibre distribution with N ¼ 210 fibre bundles corresponding to an isotropic distribution (R(n i ) ¼ 1/4p in equation (3.7)), reflecting the random arrangement of collagen fibres evident in free-swelling tissue constructs. Each bundle was modelled using the constitutive relation of equation (3. 2), with b f ¼ 2 and j f left as a single adjustable parameter (same j f value for all bundles and all generations). The lognormal c.d.f. of equation (2.20) was used to describe the probability of damage in each bundle of each generation, with adjustable parameters m c and s c (same values for all bundles). The failure criterion measure was the Simo criterion of the fibre bundle, equation (2.19), calculated for each collagen generation using F g for that generation.
21
This model was implemented in the FEBio finite-element environment to predict construct swelling and damage, and construct equilibrium compressive modulus under conditions equivalent to the experimental testing protocol (unconfined compression at 10% strain). Using FEBio predictions, the four adjustable material parameters, F, j f , m c and s c , were fitted simultaneously to the mean experimental results for swelling ratio J versus time in culture (figure 10) and compressive modulus E Y versus time in culture (figure 15), using control group data only
21
0.5 and s c ¼ 1.7).
21
These fits produced coefficients of determination R The successful curve-fits of control group data, and faithful theoretical predictions of CABC group data, provide a mechanistic validation of our hypothesis that growth is causing collagen damage in our engineered tissue constructs. The fact that the model could predict an initial rise in E Y followed by a plateau (figure 15), whereas the swelling ratio J rose continuously (figure 10), indicates that the onset of damage with growth can explain the observed responses.
21
The agreement between theory and experimental datasets (control and CABC groups) is also evident in a scatter plot of J versus GAG %D0ww data, showing a strong correlation with the theoretical model (figure 16a). A scatter plot of E Y versus GAG %ww from the control and CABC groups here and our prior study of [49] produces a moderate correlation (figure 16b).
Observable damage
Based on the general agreement between experimental results and our hypothesized damage-mechanism model of tissue growth, we investigated methods of measuring the damage directly and quantitatively. In 1997, Bank et al. [58] introduced a simplified measurement of collagen fibre denaturation using a-chymotrypsin, which reports the fraction of all collagen in that tissue that has been degraded (i.e. whose triple helix is unwound). They demonstrated that this assay could predict higher collagen degradation in cartilage from human osteoarthritic joints (10.4 + 5.6%) versus normal tissue (2.1 + 1.0%) suggesting that the technique would serve as an indicator of for collagen fibre damage.
Observable damage
We used this assay to assess damage in the control and CABC groups of the current study, as well as damage in native juvenile bovine cartilage from which we harvested chondrocytes for our tissue engineering studies. Native juvenile cartilage (n ¼ 5 samples) exhibited 10.1 + 0.9% denatured collagen, consistent with the approximately 14% value reported previously for young adult bovine cartilage [17] . Engineered cartilage constructs exhibited damage that increased over time, ranging from 38.9 + 4.8% at day 14 to 53.9 + 5.7% at day 105 for the control group, and from 22.4 + 4.6% at day 16 to 63.8 + 4.4% at day 105 for the CABC group ( figure 17) . A two-way ANOVA for the factors Figure 17. Fractional collagen damage measured using a-chymotrypsin digestion, as a function of time and treatment (bars). Different letters indicate significant differences across time points ( p , 10 24 ) and asterisks indicate significant differences between control and CABC at that time point ( p , 10 22 ). Solid curves represent the fractional damage from the model fitted to control data (fit) and the model prediction for CABC data ( predict).
Observable damage
of time in culture and treatment showed significant differences across all factors ( p , 10 24 for time, indicating a significant increase in damage over time; p ¼ 0.002 for treatment, indicating that damage was lower in the CABC group versus control on average; and p , 10 24 for day Â treatment, indicating that damage was lower in the CABC group only on day 14/16, figure 17 ). These experimental results confirmed that damage did occur in tissue constructs, well above the baseline level of denatured collagen found in native juvenile cartilage. For comparison purposes, the fractional damage predicted from the model is also shown in figure 17 ). This damage was calculated by taking into account the multigenerational growth process: the average of Dðn i Þ over all N fibre bundles of generation g was denoted by D g , which represents the damage in collagen generation g; as each day produced a new generation, the fractional damage at the end of day d was evaluated from the cumulative damage until that day,
Observable damage
1Þ
Observable damage
The trend of this theoretical D versus time in culture was consistent with the biochemical assay results for both control and CABC groups, though the magnitudes did not match exactly ( figure 17 ). This qualitative agreement was similarly supportive of our hypothesis that growth and swelling cause damage in the collagen matrix of engineered cartilage constructs. The fact that D did not agree exactly with the measured collagen degradation was not surprising, as the model did not propose an explicit constitutive relation between the fraction of collagen that is denatured and the fraction of molecular bonds that are broken.
Discussion
The objectives of this study were to formulate a damage mechanics framework that employs observable state variables, instead of internal hidden variables as in the prior literature, while recovering the classical formulation for damage in isotropic materials; to show that this new framework was applicable to anisotropic materials without appealing to a tensorial damage measure; to demonstrate its application to fibrous tissues with unidirectional or multi-directional fibre orientations and to use this framework to test the hypothesis that growth of cartilage constructs can lead to damage of the synthesized collagen matrix.
Discussion
Most biological tissues exhibit viscoelasticity, which significantly hinders the ability to assess their damage exclusively from mechanical responses (except when the tissue exhibits gross failure). Similarly, damage assays (e.g. biochemistry or Raman spectroscopy) are not always easy to interpret on their own, as they may depend on complex interactions that are not fully understood. Using observable variables to track damage implies that predictions from damage models may be validated experimentally, and experimental damage measures may be interpreted mechanistically by validated models. This combination of theory and experiments can enhance our confidence when testing hypotheses, as demonstrated in the tissue engineering study presented here.
Discussion
Although we have conducted cartilage tissue engineering studies for nearly two decades, we had not initially suspected that newly synthesized extracellular matrix could rapidly become damaged during the growth process. In retrospect, the first indication that excessive GAG swelling could cause damage to engineered cartilage was reported in our earlier study, where constructs were found to crack at their center [59] , a one-time finding that we attributed to special circumstances. In this study, no cracks were observed in the constructs, but the temporal evolution of construct mechanical properties and swelling led us to hypothesize a more pervasive collagen damage process that we proceeded to investigate more quantitatively.
Discussion
Had we only performed the biochemical assay (using a-chymotrypsin digestion [58] ), our results would have remained somewhat ambiguous because this assay had not been previously used on engineered tissue constructs. Thus, the elevated levels of denatured collagen found here (figure 17) could have been attributed to the fact that newly synthesized cartilage has relatively high levels of immature collagen that may be responsible for these findings. (Indeed, it was somewhat surprising to us that healthy native juvenile and young adult bovine cartilage exhibit between 10% and 14% damage [17] , whereas healthy adult human cartilage only exhibits 2% damage [58] ). It was the combination of modelling and experiments that increased our confidence in the testing of our hypothesis.
Discussion
The results of this study have significant implications for our ongoing cartilage tissue engineering efforts: increasing cell seeding density to levels comparable with neonatal cartilage can produce collagen synthesis that matches native levels; however, we must develop tissue engineering strategies that prevent damage of newly synthesized collagens. This may be achieved by either preventing excessive swelling during the growth process (e.g. suitably constraining the constructs without compromising access to nutrients), or reducing GAG synthesis in the early stages of tissue growth using alternative strategies to CABC (e.g. small interfering RNA to temporarily suppress aggrecan synthesis). These alternative strategies are currently under consideration.
Discussion
Data accessibility. Experimental data for engineered cartilage constructs reported in this manuscript are available for download from https:// academiccommons.columbia.edu/catalog/ac%3A187860 and the model is available on the FEBio forum at http://mrlforums.sci.utah. edu/forums/.

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