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The authors have declared that no competing interests exist.

Conceived and designed the experiments: HK AG AH MKM PJ PE PA HI. Performed the experiments: HK AG AH MKM. Analyzed the data: HK AG AH MKM PJ PE PA HI. Contributed reagents/materials/analysis tools: PJ PE PA. Wrote the paper: HK AG AH MKM PJ PE PA HI.

Computational models of Achilles tendons can help understanding how healthy tendons are affected by repetitive loading and how the different tissue constituents contribute to the tendon’s biomechanical response. However, available models of Achilles tendon are limited in their description of the hierarchical multi-structural composition of the tissue. This study hypothesised that a poroviscoelastic fibre-reinforced model, previously successful in capturing cartilage biomechanical behaviour, can depict the biomechanical behaviour of the rat Achilles tendon found experimentally.

We developed a new material model of the Achilles tendon, which considers the tendon’s main constituents namely: water, proteoglycan matrix and collagen fibres. A hyperelastic formulation of the proteoglycan matrix enabled computations of large deformations of the tendon, and collagen fibres were modelled as viscoelastic. Specimen-specific finite element models were created of 9 rat Achilles tendons from an animal experiment and simulations were carried out following a repetitive tensile loading protocol. The material model parameters were calibrated against data from the rats by minimising the root mean squared error (RMS) between experimental force data and model output.

All specimen models were successfully fitted to experimental data with high accuracy (RMS 0.42-1.02). Additional simulations predicted more compliant and soft tendon behaviour at reduced strain-rates compared to higher strain-rates that produce a stiff and brittle tendon response. Stress-relaxation simulations exhibited strain-dependent stress-relaxation behaviour where larger strains produced slower relaxation rates compared to smaller strain levels. Our simulations showed that the collagen fibres in the Achilles tendon are the main load-bearing component during tensile loading, where the orientation of the collagen fibres plays an important role for the tendon’s viscoelastic response. In conclusion, this model can capture the repetitive loading and unloading behaviour of intact and healthy Achilles tendons, which is a critical first step towards understanding tendon homeostasis and function as this biomechanical response changes in diseased tendons.

The Achilles tendon is the largest tendon in the body and the most commonly injured tendon [

Tendons are composed of principally 70% water and 30% collagens [

Existing biomechanical models of tendons are often generalised and treat tendon and ligament behaviour concurrently as soft musculoskeletal collagenous tissues. However, studies have shown that although the tissues are similar in their structure and composition, their extracellular matrix that provides the mechanical response is function-dependent [

The importance of multi-structural models of tendons and soft tissues was first demonstrated by Lanir [

This study adapts the poroviscoelastic model developed by Wilson, van Donkelaar [

The results demonstrate that our novel multi-structural material model for the rat Achilles tendon has the capacity to consider the role that collagen fibril morphology and collagen viscoelasticity play in the overall biomechanical behaviour of the tissue. The study shows how different tissue constituents play a part in resisting tensile loading and predicts the relaxation and recovery response of rat Achilles tendons.

A constitutive model for the Achilles tendon was developed based on an existing material model for articular cartilage [_{s} is the stress in the solid matrix, _{f} and _{m} are the stresses in the collagen fibres and the non-fibrillar matrix respectively.

Collagen fibres exhibit a viscoelastic response during loading [

_{1}, _{2}, _{1}, _{2} and _{f} stands for the total fibre strain, _{v} is the strain in the dash pot and _{e} is the strain in the spring in the Maxwell element.

Assuming that the fibres only carry load in tension, the stress in the single spring system (top spring in _{1} and _{2} are the first Piola-Kirchoff stresses, _{1}, _{2}, _{1} and _{2} are stiffness constants and _{f}) equals the sum of the strains in the dashpot (_{v}) and in the spring (_{e}) in the Maxwell element. Thus, the total fibre stress (_{f}) is given by

Expressed as a function of the total fibre strain (_{f}), the total fibres stress becomes (

After time integration with the backward Euler method, the expression above (

Where

And
_{f}^{t+Δt} is the fibre stress in the current time increment and _{f}^{t} is the fibre stress from the previous time increment.

Finally, the Cauchy stress tensor for the collagen fibres in the solid tendon matrix (_{f} in _{f} is a one-dimensional unit vector describing the current fibre orientation [

Since tendons undergo large deformations the non-fibrillar component of the solid matrix was modelled as a compressible neo-Hookean material. The energy function (_{m}) and the shear (_{m}) moduli of the matrix are defined as
_{m} is the Young’s modulus and _{m} is Poisson’s ratio of the non-fibrillar matrix. The Cauchy stress in the non-fibrillar matrix (_{m}) was derived as

The permeability (_{0} is the initial permeability, _{k} a positive constant and _{0} the current and initial void ratios [_{f}) and the solid volume fraction (_{s} = 1 − _{f}). The total fluid volume fraction can be estimated from the water mass fraction (_{f,m}) [_{s} is the solid tissue density, calculated from the tendon experiments (data from Eliasson, Fahlgren [

Specimen-specific finite element models were created of 9 rat Achilles tendons using geometries measured in the experiments by Eliasson, Fahlgren [

Boundary conditions were modelled following the experimental set up in the mechanical tensile tests. The finite element nodes at the base of the tendon models (calcaneus bone end) were modelled with encastre boundary conditions to represent the clamp in the mechanical testing machine (

A) Mesh and boundary conditions. B) The experimental loading protocol cycles 1–3 for all 9 tendons interpolated over 2π for each load cycle (grey). The average loading protocol for cycles 1–3 (red). C) The loading protocols for cycle 1–3 for all 9 tendons in the time-domain (grey) illustrate the variability among experimental specimens. The average loading protocol for cycles 1–3 in time-domain (red).

The experimental data exhibited large temporal variation in the tendons’ response to cyclic loading. Therefore, to create an average tensile loading protocol the tensile data from the 9 tendons were translated to the frequency domain by interpolating each load cycle over 2π (^{2}).

The proposed constitutive material model was calibrated to the data from the mechanical tensile test following an iterative computational scheme (^{–5}. The tendon model was fitted to experimental data by optimising the nine unknown constitutive model parameters (_{1}, _{2}, _{1}, _{2}, _{0}, _{k}, _{m}, _{m}).

Three load cycles from different time points of the experiment were used to calibrate the data: (1) cycle 1–3 where large variations are seen between the cycles and the viscoelastic properties are very pronounced and (2) cycle 10–12 which assumingly correspond to preconditioned tendons. Displacement controlled loads were applied with a strain rate of 0.1 mm/s, corresponding to experimental protocol.

In the FE simulations, Abaqus’ total Lagrangian formulation and the NLGEOM key was used to account for the large deformations and the geometrical nonlinearities when implementing the material model for the fibre-reinforced solid matrix in the UMAT subroutine. In the initial configuration, all fibre direction vectors (^{0}) were directed in parallel with the tendon axis. The directional fibre vectors were updated based on the deformation gradient (

The fibre vectors defined the fibres stretch, (

The reaction forces from the Abaqus simulations were fed into Matlab where an unconstrained nonlinear minimisation algorithm was used for optimising the unknown material parameters. This was done by minimising the objective function (_{mod} is the reaction force from the finite element analysis, _{exp} is the reaction force measured in the experiment and _{i} is the number of data points in each load step.

To test the model’s ability to predict tendon biomechanical behaviour reported in literature, new FE simulations were carried out on the average tendon model using the optimised model parameters for the constitutive model. The following tests were performed:

The poroviscoelastic fibre-reinforced model developed in this study is able to capture the loading and unloading behaviour of rat Achilles tendons with good accuracy (Root Mean Square (RMS) between 0.42 and 1.02), see

The new poroviscoelastic model fitted to experimental data from cycle 1–3 of the tensile tests on rat Achilles tendons. The best (A) material model fit (RMS = 0.42) and the worst (B) material model fit (RMS = 1.02).

specimen | _{1} (MPa) |
_{2} (MPa) |
_{1} |
_{2} |
_{0} (mm/s) |
_{k} |
_{m} (MPa) |
_{m} |
||
---|---|---|---|---|---|---|---|---|---|---|

0.699 | 0.017 | 0.233 | 38.49 | 29.61 | 582.01 | 3.96E-07 | 0.21 | 0.56 | 0.24 | |

0.834 | 0.074 | 1.683 | 44.91 | 25.97 | 1103.10 | 5.87E-10 | 0.59 | 0.57 | 0.21 | |

0.869 | 0.017 | 0.380 | 44.95 | 35.78 | 595.58 | 1.82E-07 | 0.67 | 0.70 | 0.40 | |

0.421 | 0.648 | 6.251 | 25.85 | 9.14 | 336.59 | 7.65E-10 | 0.45 | 1.00 | 0.49 | |

0.854 | 0.023 | 0.452 | 48.50 | 37.01 | 943.17 | 5.47E-10 | 0.55 | 0.56 | 0.14 | |

1.021 | 0.003 | 0.030 | 36.13 | 38.03 | 537.40 | 1.39E-09 | 1.40 | 0.59 | 0.34 | |

0.819 | 0.027 | 3.001 | 42.81 | 16.50 | 484.60 | 3.04E-08 | 0.001 | 0.70 | 0.44 | |

0.920 | 0.001 | 0.054 | 44.54 | 34.45 | 770.67 | 6.22E-10 | 0.78 | 0.65 | 0.20 | |

0.731 | 0.022 | 0.452 | 42.96 | 31.86 | 897.18 | 1.39E-07 | 0.47 | 0.95 | 0.19 | |

- | 0.092 | 1.393 | 41.02 | 28.71 | 694.48 | 8.34E-08 | 0.57 | 0.70 | 0.29 | |

- | 0.209 | 2.063 | 6.77 | 9.93 | 248.73 | 1.36E-07 | 0.39 | 0.17 | 0.13 | |

- | 2.26 | 1.48 | 0.16 | 0.35 | 0.36 | 1.63 | 0.68 | 0.24 | 0.43 |

The mean (of the 9 optimised tendon models), the standard deviations (SD) and the coefficients of variation (CV) of the specimens are calculated.

The experimental data agreed well with the average tendon model and the average tensile loading protocol created in this study, see

The optimised result when the material model is fitted to the average tendon model and the average loading protocol. A) Loading cycle 1–3, RMS = 0.84 and B) during later loading cycles (cycle 10–12), RMS = 0.41.

cycle | _{1} (MPa) |
_{2} (MPa) |
_{1} |
_{2} |
_{0} (mm/s) |
_{k} |
_{m} (MPa) |
_{m} |
||
---|---|---|---|---|---|---|---|---|---|---|

0.837 | 0.023 | 0.443 | 40.00 | 31.06 | 609.34 | 1.19E-07 | 0.96 | 0.77 | 0.35 | |

0.413 | 0.024 | 7.009 | 38.00 | 11.63 | 413.23 | 1.75E-09 | 0.84 | 0.37 | 0.17 |

Furthermore, our model predicted the characteristic strain-stiffening behaviour of the Achilles tendons. The simulations demonstrated how increased strain-rate leads to a stiffer and more brittle biomechanical response compared to more compliant tendon behaviour during lower strain-rates (

The strain-stiffening behaviour captured by the poroviscoelastic model where the Achilles tendons subjected to higher strain-rates exhibit a stiffer and more brittle behaviour than when subjected to slower strain-rates.

Stress-relaxation response of the Achilles tendon, as predicted by the material model. Higher strains result in reduced relaxation compared to lower strain magnitudes (A), and demonstrate a slower relaxation rate, as illustrated in the log-log plot (B).

A) Model prediction of the creep behaviour in Achilles tendons when subjected to different stress magnitudes. B) The log-log plot shows almost no stress-dependent creep rate behaviour.

The influence of the different tissue constituents (collagen fibres, non-fibrillar matrix and water) on tensile load bearing was investigated by looking at how much each component contributes to fluid velocities and total stress in the tendon (see

A) The stress in the collagen fibres and B) the stress in the non-fibrillar matrix, during one load cycle. C) Fluid velocities in the tendon during one load cycle (output from highlighted element at the centre edge of the tendon mesh, see

In this study, we have proposed a new material model for tendon that can capture the biomechanical behaviour of the rat Achilles tendon and the role of its structural components during tensile loading. To the best of our knowledge, this is the first poroviscoelastic structural model developed for tendons. The model has been calibrated to experimental data with high accuracy. Moreover, simulations of other loading conditions than those of the experiment (such as strain-rate, stress-relaxation and creep) show that the model can accurately capture the general biomechanical phenomena often reported in tendon experiments.

When comparing with experimental observations, there are some interesting aspects of the predicted biomechanical behaviour of the rat Achilles tendons in this study that need further attention. For example, the stress-relaxation test showed increased rate of relaxation for low strain levels compared to higher strains. Although similar magnitudes of strain (6% as reported in this study) have been reported for alternate stress-relaxation rates [

When creep was tested, the model showed almost no strain-dependent recovery rate. Reduced recovery rates compared to relaxation rates have previously also been reported in experiments of for example digital flexor tendons [

The simulations were able to capture the strain-stiffening response of tendons when subjected to increased loading rates. This phenomena has been reported in animal experiments [

The simulations in this study showed clearly that the viscoelastic behaviour of the collagen fibres in the rat Achilles tendons were responsible for the main biomechanical response during tensile loading. This agrees with experimental findings that have reported a much higher tissue elastic modulus in tendons along the fibre-aligned direction compared to the transverse [_{1}, _{2}, _{1}, _{2} and

In general, tensile loading of the tendon demonstrated a volume increase and a decrease in pore pressure, which resulted in positive fluid flux into the tendon. This feature has also been observed in other poroelastic models of soft tissues [

The non-fibrillar matrix showed viscoelastic behaviour due to the biphasic description and demonstrated fluid flow in the matrix (simulations with horizontally aligned collagen fibres). In tendons, the non-fibrillar matrix is believed to play an essential role in tendon micro-mechanics (although not in the overall tendon response) as the proteoglycan links are responsible for the sliding that occurs between the collagen fibrils when the tissue is stretched [

In this study, specimen-specific FE models were developed for each of the experimental samples with unique loading protocol recorded from the experiments. The large variability in the experiments was captured with good accuracy by the results of the tendon simulations. By using specimen-specific models we can capture biological variability instead of only using an average tendon model that may not be representative to an animal population. Variability in our models also enables quantitative analysis of our results and the application of statistical tools on computational results when comparing different simulation groups. In this manner, not only do we get quantitative experimental data but also quantitative computational output, such as constitutive constants and fibre configurations that contain information about the mechanical behaviour of the system. Cook and Purdam [

In summary, we have presented a novel multi-structural biomechanical model for tendon that can consider the role which the different tissue components play in tensile load bearing. The model presented is able to capture the biomechanical behaviour of rat Achilles tendons under tensile loading with good accuracy, when fitted to specific experimental data. It is able to predict general tendon biomechanical behaviour observed in other experiments, such as stress-relaxation, strain-stiffening and to some extent creep. Our simulations indicate that collagen fibres are the main load-bearing component in rat Achilles tendons under tensile loading, and that stress-relaxation rate in these tendons is inversely correlated to level of strain. This multi-structural model has allowed us to deconstruct and test the different components of the tissue in order to better understand how they interact to create the overall biomechanical response of the tendon.