Corrigendum to "A preliminary study into the emergence of tendon microstructure during postnatal development" [Matrix Biol. Plus 21 (2024) 100142].

[This corrects the article DOI: 10.1016/j.mbplus.2024.100142.].

A preliminary study into the emergence of tendon microstructure during postnatal development.

Tendons maintain mechanical function throughout postnatal development whilst undergoing significant microstructural changes. We present a study of postnatal tendon growth and characterise the major changes in collagen fibril architecture in mouse tail tendon from birth to eight weeks by analysing the geometries of cross-sectional transmission electron microscopy images. This study finds that a bimodal distribution of fibril diameters emerges from a unimodal distribution of narrow fibrils as early as the eighth day postnatal, and three distinct fibril populations are visible at around 14 days. The tendons in this study do not show evidence of precise hexagonal packing, even at birth, and the spaces between the fibrils remain constant throughout development. The fibril number in the tissue stabilises around day 28, and the fibril area fraction stabilises around day 26. This study gives coarse-grained insight into the transition periods in early tendon development.

A three-dimensional active cloaking strategy for the Helmholtz equation that exploits the symmetry of the platonic solids.

An active cloaking strategy for the scalar Helmholtz equation in three dimensions is developed by placing active sources at the vertices of Platonic solids. In each case, a "silent zone" is created interior to the Platonic solid and only the incident field remains in a defined region exterior to this zone. This distribution of sources ensures that implementation of the cloaking strategy is efficient: once the multipole source amplitudes at a single source location are determined, the other amplitudes are calculated by multiplying the multipole source vector by a rotation matrix. The technique is relevant to any scalar wave field.

Bayesian inference on a microstructural, hyperelastic model of tendon deformation.

Microstructural models of soft-tissue deformation are important in applications including artificial tissue design and surgical planning. The basis of these models, and their advantage over their phenomenological counterparts, is that they incorporate parameters that are directly linked to the tissue's microscale structure and constitutive behaviour and can therefore be used to predict the effects of structural changes to the tissue. Although studies have attempted to determine such parameters using diverse, state-of-the-art, experimental techniques, values ranging over several orders of magnitude have been reported, leading to uncertainty in the true parameter values and creating a need for models that can handle such uncertainty. We derive a new microstructural, hyperelastic model for transversely isotropic soft tissues and use it to model the mechanical behaviour of tendons. To account for parameter uncertainty, we employ a Bayesian approach and apply an adaptive Markov chain Monte Carlo algorithm to determine posterior probability distributions for the model parameters. The obtained posterior distributions are consistent with parameter measurements previously reported and enable us to quantify the uncertainty in their values for each tendon sample that was modelled. This approach could serve as a prototype for quantifying parameter uncertainty in other soft tissues.

Correction to 'A modified formulation of quasi-linear viscoelasticity for transversely isotropic materials under finite deformation'.

[This corrects the article DOI: 10.1098/rspa.2018.0231.][This corrects the article DOI: 10.1098/rspa.2018.0231.].

Collagen fibril assembly: New approaches to unanswered questions.

Collagen fibrils are essential for metazoan life. They are the largest, most abundant, and most versatile protein polymers in animals, where they occur in the extracellular matrix to form the structural basis of tissues and organs. Collagen fibrils were first observed at the turn of the 20th century. During the last 40 years, the genes that encode the family of collagens have been identified, the structure of the collagen triple helix has been solved, the many enzymes involved in the post-translational modifications of collagens have been identified, mutations in the genes encoding collagen and collagen-associated proteins have been linked to heritable disorders, and changes in collagen levels have been associated with a wide range of diseases, including cancer. Yet despite extensive research, a full understanding of how cells assemble collagen fibrils remains elusive. Here, we review current models of collagen fibril self-assembly, and how cells might exert control over the self-assembly process to define the number, length and organisation of fibrils in tissues.

A microstructural model of tendon failure.

Collagen fibrils are the most important structural component of tendons. Their crimped structure and parallel arrangement within the tendon lead to a distinctive non-linear stress-strain curve when a tendon is stretched. Microstructural models can be used to relate microscale collagen fibril mechanics to macroscale tendon mechanics, allowing us to identify the mechanisms behind each feature present in the stress-strain curve. Most models in the literature focus on the elastic behaviour of the tendon, and there are few which model beyond the elastic limit without introducing phenomenological parameters. We develop a model, built upon a collagen recruitment approach, that only contains microstructural parameters. We split the stress in the fibrils into elastic and plastic parts, and assume that the fibril yield stretch and rupture stretch are each described by a distribution function, rather than being single-valued. By changing the shapes of the distributions and their regions of overlap, we can produce macroscale tendon stress-strain curves that generate the full range of features observed experimentally, including those that could not be explained using existing models. These features include second linear regions occurring after the tendon has yielded, and step-like failure behaviour present after the stress has peaked. When we compare with an existing model, we find that our model reduces the average root mean squared error from 4.53MPa to 2.29MPa, and the resulting parameter values are closer to those found experimentally. Since our model contains only parameters that have a direct physical interpretation, it can be used to predict how processes such as ageing, disease, and injury affect the mechanical behaviour of tendons, provided we can quantify the effects of these processes on the microstructure.

Circadian control of the secretory pathway maintains collagen homeostasis.

Collagen is the most abundant secreted protein in vertebrates and persists throughout life without renewal. The permanency of collagen networks contrasts with both the continued synthesis of collagen throughout adulthood and the conventional transcriptional/translational homeostatic mechanisms that replace damaged proteins with new copies. Here, we show circadian clock regulation of endoplasmic reticulum-to-plasma membrane procollagen transport by the sequential rhythmic expression of SEC61, TANGO1, PDE4D and VPS33B. The result is nocturnal procollagen synthesis and daytime collagen fibril assembly in mice. Rhythmic collagen degradation by CTSK maintains collagen homeostasis. This circadian cycle of collagen synthesis and degradation affects a pool of newly synthesized collagen, while maintaining the persistent collagen network. Disabling the circadian clock causes abnormal collagen fibrils and collagen accumulation, which are reduced in vitro by the NR1D1 and CRY1/2 agonists SR9009 and KL001, respectively. In conclusion, our study has identified a circadian clock mechanism of protein homeostasis wherein a sacrificial pool of collagen maintains tissue function.

A Recruitment Model of Tendon Viscoelasticity That Incorporates Fibril Creep and Explains Strain-Dependent Relaxation.

Soft tissues exhibit complex viscoelastic behavior, including strain-rate dependence, hysteresis, and strain-dependent relaxation. In this paper, a model for soft tissue viscoelasticity is developed that captures all of these features and is based upon collagen recruitment, whereby fibrils contribute to tissue stiffness only when taut. We build upon existing recruitment models by additionally accounting for fibril creep and by explicitly modeling the contribution of the matrix to the overall tissue viscoelasticity. The fibrils and matrix are modeled as linear viscoelastic and each fibril has an associated critical strain (corresponding to its length) at which it becomes taut. The model is used to fit relaxation tests on three rat tail tendon fascicles and predict their response to cyclic loading. It is shown that all of these mechanical tests can be reproduced accurately with a single set of constitutive parameters, the only difference between each fascicle being the distribution of their fibril crimp lengths. By accounting for fibril creep, we are able to predict how the fibril length distribution of a fascicle changes over time under a given deformation. Furthermore, the phenomenon of strain-dependent relaxation is explained as arising from the competition between the fibril and matrix relaxation functions.

A modified formulation of quasi-linear viscoelasticity for transversely isotropic materials under finite deformation.

The theory of quasi-linear viscoelasticity (QLV) is modified and developed for transversely isotropic (TI) materials under finite deformation. For the first time, distinct relaxation responses are incorporated into an integral formulation of nonlinear viscoelasticity, according to the physical mode of deformation. The theory is consistent with linear viscoelasticity in the small strain limit and makes use of relaxation functions that can be determined from small-strain experiments, given the time/deformation separability assumption. After considering the general constitutive form applicable to compressible materials, attention is restricted to incompressible media. This enables a compact form for the constitutive relation to be derived, which is used to illustrate the behaviour of the model under three key deformations: uniaxial extension, transverse shear and longitudinal shear. Finally, it is demonstrated that the Poynting effect is present in TI, neo-Hookean, modified QLV materials under transverse shear, in contrast to neo-Hookean elastic materials subjected to the same deformation. Its presence is explained by the anisotropic relaxation response of the medium.

Four-Dimensional Imaging of Soft Tissue and Implanted Biomaterial Mechanics: A Barbed Suture Case Study for Tendon Repair.

Timely, recent developments in X-ray microcomputed tomography (XµCT) imaging such as increased resolution and improved sample preparation enable nondestructive time-lapse imaging of polymeric biomaterials when implanted in soft tissue, which we demonstrate herein. Imaging the full three-dimensional (3D) structure of an implanted biomaterial provides new opportunities to assess the micromechanics of the interface between the implant and tissues and how this changes over time as force is applied in load-bearing musculoskeletal applications. In this paper, we present a case study demonstrating in situ XµCT and finite element analysis, using a dynamically loaded barbed suture repair for its novel use in tendon tissue. The aim of this study was to identify the distribution of stress in the suture and tendon as load is applied. The data gained demonstrate a clear 3D visualization of microscale features in both the tissue and implant in wet conditions. XµCT imaging has revealed, for the first time, pores around the suture, preventing full engagement of all the barbs with the tendon tissue. Subsequent finite element analysis reveals the localized stress and strain, which are not evenly distributed along the suture, or throughout the tissue. This case study demonstrates for the first time a powerful in situ mechanical imaging tool, which could be readily adapted by other laboratories to interrogate and optimize the interface between the implanted biomaterials and the soft tissue.

Correction to 'Antiplane wave scattering from a cylindrical cavity in pre-stressed nonlinear elastic media'.

[This corrects the article DOI: 10.1098/rspa.2015.0450.].

The relative compliance of energy-storing tendons may be due to the helical fibril arrangement of their fascicles.

A nonlinear elastic microstructural model is used to investigate the relationship between structure and function in energy-storing and positional tendons. The model is used to fit mechanical tension test data from the equine common digital extensor tendon (CDET) and superficial digital flexor tendon (SDFT), which are used as archetypes of positional and energy-storing tendons, respectively. The fibril crimp and fascicle helix angles of the two tendon types are used as fitting parameters in the mathematical model to predict their values. The outer fibril crimp angles were predicted to be 15.1° ± 2.3° in the CDET and 15.8° ± 4.1° in the SDFT, and the average crimp angles were predicted to be 10.0° ± 1.5° in the CDET and 10.5° ± 2.7° in the SDFT. The crimp angles were not found to be statistically significantly different between the two tendon types (p = 0.572). By contrast, the fascicle helix angles were predicted to be 7.9° ± 9.3° in the CDET and 29.1° ± 10.3° in the SDFT and were found to be statistically highly significantly different between the two tendon types (p < 0.001). This supports previous qualitative observations that helical substructures are more likely to be found in energy-storing tendons than in positional tendons and suggests that the relative compliance of energy-storing tendons may be directly caused by these helical substructures.

Three-dimensional visualisation of soft biological structures by X-ray computed micro-tomography.

Whereas the two-dimensional (2D) visualisation of biological samples is routine, three-dimensional (3D) imaging remains a time-consuming and relatively specialised pursuit. Current commonly adopted techniques for characterising the 3D structure of non-calcified tissues and biomaterials include optical and electron microscopy of serial sections and sectioned block faces, and the visualisation of intact samples by confocal microscopy or electron tomography. As an alternative to these approaches, X-ray computed micro-tomography (microCT) can both rapidly image the internal 3D structure of macroscopic volumes at sub-micron resolutions and visualise dynamic changes in living tissues at a microsecond scale. In this Commentary, we discuss the history and current capabilities of microCT. To that end, we present four case studies to illustrate the ability of microCT to visualise and quantify: (1) pressure-induced changes in the internal structure of unstained rat arteries, (2) the differential morphology of stained collagen fascicles in tendon and ligament, (3) the development of Vanessa cardui chrysalises, and (4) the distribution of cells within a tissue-engineering construct. Future developments in detector design and the use of synchrotron X-ray sources might enable real-time 3D imaging of dynamically remodelling biological samples.

Optimal Contrast Agent Staining of Ligaments and Tendons for X-Ray Computed Tomography.

X-ray computed tomography has become an important tool for studying the microstructures of biological soft tissues, such as ligaments and tendons. Due to the low X-ray attenuation of such tissues, chemical contrast agents are often necessary to enhance contrast during scanning. In this article, the effects of using three different contrast agents--iodine potassium iodide solution, phosphotungstic acid and phosphomolybdic acid--are evaluated and compared. Porcine anterior cruciate ligaments, patellar tendons, medial collateral ligaments and lateral collateral ligaments were used as the basis of the study. Three samples of each of the four ligament/tendon types were each assigned a different contrast agent (giving a total of twelve samples), and the progression of that agent through the tissue was monitored by performing a scan every day for a total period of five days (giving a total of sixty scans). Since the samples were unstained on day one, they had been stained for a total of four days by the time of the final scans. The relative contrast enhancement and tissue deformation were measured. It was observed that the iodine potassium iodide solution penetrated the samples fastest and caused the least sample shrinkage on average (although significant deformation was observed by the time of the final scans), whereas the phosphomolybdic acid caused the greatest sample shrinkage. Equations describing the observed behaviour of the contrast agents, which can be used to predict optimal staining times for ligament and tendon X-ray computed tomography, are presented.

Antiplane wave scattering from a cylindrical cavity in pre-stressed nonlinear elastic media.

The effect of a longitudinal stretch and a pressure-induced inhomogeneous radial deformation on the scattering of antiplane elastic waves from a cylindrical cavity is determined. Three popular nonlinear strain energy functions are considered: the neo-Hookean, the Mooney-Rivlin and a two-term Arruda-Boyce model. A new method is developed to analyse and solve the governing wave equations. It exploits their properties to determine an asymptotic solution in the far-field, which is then used to derive a boundary condition to numerically evaluate the equations local to the cavity. This method could be applied to any linear ordinary differential equation whose inhomogeneous coefficients tend to a constant as its independent variable tends to infinity. The effect of the pre-stress is evaluated by considering the scattering cross section. A longitudinal stretch is found to decrease the scattered power emanating from the cavity, whereas a compression increases it. The effect of the pressure difference depends on the strain energy function employed. For a Mooney-Rivlin material, a cavity inflation increases the scattered power and a deflation decreases it; for a neo-Hookean material, the scattering cross section is unaffected by the radial deformation; and for a two-term Arruda-Boyce material, both inflation and deflation are found to decrease the scattered power.

A new strain energy function for modelling ligaments and tendons whose fascicles have a helical arrangement of fibrils.

A new strain energy function for the hyperelastic modelling of ligaments and tendons whose fascicles have a helical arrangement of fibrils is derived. The stress-strain response of a single fascicle whose fibrils exhibit varying levels of crimp throughout its radius is calculated and used to determine the form of the strain energy function. The new constitutive law is used to model uniaxial extension test data for human patellar tendon and is shown to provide an excellent fit, with the average relative error being 9.8%. It is then used to model shear and predicts that the stresses required to shear a tendon are much smaller than those required to uniaxially stretch it to the same strain level. Finally, the strain energy function is used to model ligaments and tendons whose fascicles are helical, and the relative effects of the fibril helix angle, the fascicle helix angle and the fibril crimp variable are compared. It is shown that they all have a significant effect; the fibril crimp variable governs the non-linearity of the stress-strain curve, whereas the helix angles primarily affect its stiffness. Smaller values of the helix angles lead to stiffer tendons; therefore, the model predicts that one would expect to see fewer helical sub-structures in stiff positional tendons, and more in those that are required to be more flexible.

A new strain energy function for the hyperelastic modelling of ligaments and tendons based on fascicle microstructure.

A new strain energy function for the hyperelastic modelling of ligaments and tendons based on the geometrical arrangement of their fibrils is derived. The distribution of the crimp angles of the fibrils is used to determine the stress-strain response of a single fascicle, and this stress-strain response is used to determine the form of the strain energy function, the parameters of which can all potentially be directly measured via experiments - unlike those of commonly used strain energy functions such as the Holzapfel-Gasser-Ogden (HGO) model, whose parameters are phenomenological. We compare the new model with the HGO model and show that the new model gives a better match to existing stress-strain data for human patellar tendon than the HGO model, with the average relative error in matching this data when using the new model being 0.053 (compared with 0.57 when using the HGO model), and the average absolute error when using the new model being 0.12 MPa (compared with 0.31 MPa when using the HGO model).

X-ray computed tomography of the anterior cruciate ligament and patellar tendon.

The effect of phosphotungstic acid (PTA) and iodine solution (IKI) staining was investigated as a method of enhancing contrast in the X-ray computed tomography of porcine anterior cruciate ligaments (ACL) and patellar tendons (PT). We show that PTA enhanced surface contrast, but was ineffective at penetrating samples, whereas IKI penetrated more effectively and enhanced contrast after 70 hours of staining. Contrast enhancement was compared when using laboratory and synchrotron based X-ray sources. Using the laboratory source, PT fascicles were tracked and their alignment was measured. Individual ACL fascicles could not be identified, but identifiable features were evident that were tracked. Higher resolution scans of fascicle bundles from the PT and ACL were obtained using synchrotron imaging techniques. These scans exhibited greater contrast between the fascicles and matrix in the PT sample, facilitating the identification of the fascicle edges; however, it was still not possible to detect individual fascicles in the ACL.