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Edited by: Charles David Cox, Victor Chang Cardiac Research Institute, Australia

Reviewed by: Elisabetta M. Zanetti, University of Perugia, Italy; Nicola Francesco Lopomo, University of Brescia, Italy

This article was submitted to Biomechanics, a section of the journal Frontiers in Bioengineering and Biotechnology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In the current work, we investigate the effect of aging on the viscosity of tendon subunits. To that scope, we make use of experimental relaxation curves of healthy and aged tendon fascicles and fibers, upon which we identify the viscosity parameters characterizing the time-dependent behavior of each tendon subunit. We subsequently combine the obtained results with analytical viscoelastic homogenization analysis methods to extract information on the effective viscous contribution of the embedding matrix substance at the fiber scale. The results suggest that the matrix substance plays a significant role in the relaxation process of the upper tendon subunits both for aged and healthy specimens. What is more, the viscosity coefficients computed for the fibrillar components indicate that aging leads to a viscosity reduction that is statistically significant for both fascicles and fibers. Its impact is more prominent for the lower hierarchical scale of fibers. As such, the reduced stress relaxation capability at the tendon macroscale is to be primarily attributed to the modified viscosity of its inner fibrillar subunits rather than to the matrix substance.

The multiscale structure of tendons plays a functional role in the transfer of forces from the muscles to the bones (Maceri et al.,

Tendon fibers immersed in a matrix substance

In order to characterize the tendon's mechanical attributes, tensile experiments have been carried out at different inner tendon hierarchies. In particular, quasi-static stress-strain curves have been used to provide estimates for the elastic modulus of fascicles and fibers (Kato et al.,

It is up to now well-established that the aging process, as well as deceases such as diabetes, result in functional changes, which have been directly related to increased tendon damage and injury (Dressler et al.,

While the qualitative effects of aging on tendons have been long identified and relevant stress measurements are available, quantitative estimates of the resulting modifications in the effective viscosities of the different inner tendon scales remain to be provided. Information of the kind is of primal importance, not only for the understanding of the mechanical behavior of the tendon's inner fibrillar components (Karathanasopoulos and Ganghoffer,

In the current work, we make use of experimental data which we combine with viscoelastic mechanical models to provide quantitative estimates for the effective viscosity of healthy and aged tendon subunits. In particular, in section Methodology, we summarize the theoretical framework to compute the relaxation behavior of tendon subunits. Thereupon, we compute the viscoelastic parameters that characterize the relaxation behavior of healthy and aged tendon fascicles and fibers, quantifying the relevant experimental uncertainty (see sections Relaxing Healthy and Aging Fascicles and Fiber Scale Aging Relaxation) and the statistical significance of the observed alterations (see Statistical Significance of the Effective Viscosity Alterations Upon Aging). Combing the mechanical data with analytical, homogenization analysis techniques, we furnish estimates for the effective viscosity of the embedding matrix substance at the fiber scale both for the healthy (control) and for the aged tendon specimens (see section Effective Viscosity Contributions of the Embedding Matrix). In section Discussion, we comment on the obtained results, providing considerable insights in the effect of aging at the different tendon inner scales and conclude in section Conclusions.

The fibrous, matrix-embeded structure of fascicles and fibers (_{r} can be defined (

Each material phase can be considered to be in the general case of viscoelastic nature, with elastic and viscous properties _{f}, _{m} and η_{f}, η_{m} for the fibrillar and matrix components accordingly. As such, the effective homogenized viscoelastic response of the matrix embedded tendon fibers is characterized by the following constitutive equation (Ganghoffer et al.,

Where in Equation (1), components with a bar stand for the homogenized strain and stress

Given the homogenized elastic and viscous material parameters, the time-dependent response of the tendon subunits is characterized by Maxwell-type relaxation kernels (

where in Equation (3), _{r} stands for the relaxing modulus part, which is equal to the substraction of the elastic modulus part _{el} (the modulus part remaining at the end of the relaxation experiment) from the initial modulus _{r} = _{el} (_{fasc} and η_{f} accordingly.

While the initial modulus can be directly retrieved out of experimental testing, information on the viscous modulus η of Equation (3) needs to be extracted making use of the relaxation curves (_{s} and final time _{f} of each relaxation experiment, upon the following loss function:

For the minimization of Equation (4), a standart derivative-free method has been employed using Matlab 2018a. In the Sections to follow, we make use of the definitions of Equations (1)–(4) along with experimental data provided in Li et al. (

In order to assess the significance of the identified viscosity changes among the control and aged tendon fascicles and fibers, we carry out a Welch's

where μ_{c} and μ_{a} in Equation (5) stand for the mean value (μ) of a certain control (c) and aged (a) quantity, while σ and

In the current section, we identify the viscosity coefficient η of relaxing healthy and aged fascicles, using Equation (3) along with the experimental data provided in Gautieri et al. (_{c} = 892_{a} = 942

Probability density function of the viscosity coefficients of healthy and aged tendon fascicles (left) and the percentage reduction of its mean value (right).

The left subplot of _{fasc} between the healthy and aged tendon tissue is in the order of 60% (_{fasc} are provided for completeness in

At the fiber scale, we identify the viscosity coefficient η_{f} using Equation (3), so that Equation (3) simplifies to

Probability distribution of the viscosity coefficients η of healthy and aged tendon fibers (left) along with the percentage change upon aging of the mean viscous moduli value for fibers and fascicles (right).

The range of the effective viscosity of the embedding matrix –depicted in red- for control and aged tendon fascicle specimens and an _{r} value in between 0.3 and 0.7 (left), along with the ratio of the fiber to the embedding matrix viscosities for a fiber fraction of _{r} = 0.3 (right).

For the viscous contribution of the embedding matrix substance to be quantified, we make use of the previously obtained viscosity results at the fiber and fascicle scale (_{fasc} = 〈η〉 at the fascicle scale (_{m}, using the results of _{r} Equation (2) to vary in between 0.3 and 0.7 (Maceri et al., _{m} for control and aged specimens, while in _{m}/η_{f}) for a fibrillar fraction of _{r} = 0.3.

_{m} at the tendon fiber scale, both for the control and aged tendons. For the former, a matrix viscosity value between 18 and 22 _{r} (e.g., for _{r} = 0.3, corresponding to the minimum value of the bars in _{m}/η_{f} in

We subsequently assess the significance of the computed viscosity changes among the control and aged tendon fascicles and fibers, using the _{fasc} and standard deviation values provided in section Relaxing Healthy and Aging Fascicles. The values relate to a ^{−4} within a 5% significance interval, indicating a highly significant difference between the control and aged tendon fiber groups. The results are summarized in

Statistical significance of the alterations in the viscosity parameters computed for control and aged tendon fascicles (see section Relaxing Healthy and Aging Fascicles) and fibers (see section Fiber Scale Aging Relaxation) using t-test statistics.

Fascicle | 7.98 | 7.95 | <0.01 |

Fiber | 31.5 | 5.7 | <0.0001 |

The results of sections Relaxing Healthy and Aging Fascicles, Fiber Scale Aging Relaxation, and Effective Viscosity Contributions of the Embedding Matrix provide experimentally-based, quantitative estimates of the effect of aging on the time-dependent, viscous properties of tendon subunits. In particular, the viscosities at the scale of fascicles (see section Relaxing Healthy and Aging Fascicles) and fibers (see section Fiber Scale Aging Relaxation), as well as the effective viscoelastic contribution of the fiber embedding matrix substance (see section Effective Viscosity Contributions of the Embedding Matrix) are assessed.

The experimental data at the fascicle scale (

While the mean viscosity of control tendon fibers (

The range of matrix viscosity values η_{m} reported in section Effective Viscosity Contributions of the Embedding Matrix constitute the first estimates –to the author's best knowledge- of the effective viscous contribution of the embedding matrix substance at the fiber scale that is based on experimental data. It needs to be noted that data-based estimates of the kind can be primarily obtained through the coupling of multiscale mechanical information (Karathanasopoulos et al., _{m} (

We note that the current analysis has been restricted to available experimental studies on the effect of aging on the relaxation behavior of both lower and upper tendon subunits subunits (Li et al.,

Overall, the viscosity parameters computed in sections Relaxing Healthy and Aging Fascicles and Fiber Scale Aging Relaxation have provided primal, data-based quantitative estimates of the effect of aging on the time-dependent behavior of fascicles and fibers. It has been shown that the viscosity coefficients η_{fasc} and η_{f} are subject to statistically significant reductions as a result of the aging process (see section Statistical Significance of the Effective Viscosity Alterations Upon Aging). What is more, the effective viscous contribution of the embedding matrix substance η_{m} has been quantified, using a multi-scale mechanical analysis framework (see section Effective Viscosity Contributions of the Embedding Matrix). It has been shown that the matrix effective viscosity η_{m} is comparable to the one computed for the tendon's fibrillar components, while its viscoelastic contribution is higher for aged rather than for control, healthy tendon subunits. The results can be used as reference viscosity mechanical parameters, factors of primal importance for the understanding of the tendon's tissue mechanics, as well as for its regeneration (Sandri et al.,

NK: conception, design, computations, and main editing. JG: analysis and interpretation of results.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

NK would like to acknowledge the support of the Freenovation Grant 2017 along with the support of the ETH CSE-lab in the writing of the corresponding application.

The Supplementary Material for this article can be found online at: