The expression "Hierarchical fibre bundles" indicates structures with one principal direction. The structure is composed of multiple fibrillar elements, i.e. with one principal direction. Interfaces may or may not be present between the fibrillar elements. Finally, each fibrillar element is itself defined as a fibre bundle, made of fibrils with different mechanical and geometrical properties. The properties of each scale arise from the properties of the lower scale.
Hierarchical fibre bundles are abundant in nature across a variety of anisotropic biological materials, such as tendons, muscles and silk. These structures often exhibit exceptional fracture properties with outstanding toughness, thanks to the presence of structural proteins and/or to bio-mineralisation and their arrangement across different scales.
Analysing the fracture behaviour of complex fibre-based structures is still a challenging task, even when using advanced computational tools. Current numerical models often require a set of assumptions and (over-)simplifications. In particular, many models require the knowledge a priori of the fracture path. Biological materials, however, are characterised by complex internal structures with random features; moreover, for nano- and micro-bundles, it is often impossible to capture the fracture phenomenon empirically. The fracture path is thus often unknown, and it is one of the open questions which needs to be addressed using mechanical models. Here, we propose an alternative solution to model the fracture mechanics of fibre-based materials, which combines the mathematical Fibre Bundle Model (FBM) and Finite Element (FE) methods.
The Fibre Bundle model is a general theory of failure based on the statistical distribution of the critical elongations of the fibres forming the bundle. FBM assumes the bundle as a system of parallel one-dimensional elements, being clamped at their extremities and undergoing external forces along their main direction. Being a mathematical tool, the FBM is based on analytical formulations and thus it can be computed at practically instantaneous speed using computational solvers. In this work, we use the FBM to obtain general properties of the fracture behaviour of a fibre bundle undergoing elongation along its main axis, and to describe the stress-strain curves characterising the fibrils in the structure. The non-linear stress-strain curve obtained is utilised as an input for an FE approach, which aids us in comprehending the deformation and fracture behaviour of the fibrils and the interaction between different fibrils and the eventual surrounding interfaces.
We demonstrate the utility of the approach by applying it to spider-silk threads and bio-polymeric fibre bundles that are known for their non-linear properties. The enhanced numerical approach to the fracture mechanics of mixed fibre bundles will improve our understanding of the function of biological structures and bio-inspired materials across scales.

Constant demand for improvement of cardiovascular treatments and necessity for better understanding of soft tissue behaviour is frequently tackled with finite element (FE) modelling. The accuracy of FE simulations considerably depends on precise and reliable material parameters. They are obtained through the process of material characterization that consists of experimental testing and subsequent data analysis. When it comes to soft tissue, planar biaxial tensile tests are widely applied because they resemble in vivo conditions fairly well and material is assumed as anisotropic. In spite of planar biaxial tensile tests being the go-to type of experiment, they are still not standardized. The possible variations are numerous, from specimen size and different gripping mechanisms to the choice of preconditioning stretches, experimental protocols, and optimization procedures during data analysis. Limited availability of soft tissue requires simple and tiny square specimens. Before the start of experiment, specimens are mounted with hooks, rakes, or sutures and slightly preloaded to avoid the specimen hanging and to ensure that its orientation is perpendicular to loading axes. Application of the initial loads reflects in a different stress state compared to the reference configuration. This pre-stretching of the specimen cannot be measured due to the nature of experiment. Thus, measured initial force results in zero stretches during the material parameter estimation. However, these occurrences are mostly neglected. Either existing pre-stretches are disregarded or initial forces are annulled manually. Obviously, neither of the cases corresponds to real stress state.

In this work we developed the procedure to calculate mounting pre-stretches and consequently material parameters are determined more accurately. Standard optimization procedure is based on minimizing the least squares difference between experimental and model Cauchy stresses. Errors induced while calculating experimental Cauchy stresses lead to inaccurate material parameters. To avoid such errors, additional deformation gradient that represents mounted specimen in a state prior to experiment, has to be introduced. Since mounting pre-stretches are not measured, a new data fitting process is established with supplementary nested loop that will iteratively correct pre-stretches to satisfy equilibrium between initial experimental and model stresses. The procedures were first applied on the series of simulated virtual planar biaxial experiments where the exact material parameters can be set and compared to the obtained ones. Holzapfel-Gasser-Ogden strain energy function was used in virtual experiments while material parameters were based on the literature. Furthermore, data fitting procedures were applied to data gathered from biaxial experiments on aortic tissue samples. The analysis has shown significant difference between obtained material parameters, especially in cases where HGO structural parameters such as fibre family angle and in-plane dispersion are considered as fitting parameters. The rate of error increases with the amount of applied pre-stretches and also decreases with increase of maximum achieved experimental stretches. Finally, annulling initial force proved to be better approach if only standard optimization is to be applied.

The dura mater is the outermost layer of the meninges, a layered membrane that covers and protects the brain and spinal cord. It is a dense and fibrous connective tissue that is composed of collagen and elastin fibers, proteoglycans, and other extracellular matrix components [1]. The dura mater plays an important role in maintaining the stability and integrity of the brain and spinal cord, as well as in regulating the cerebrospinal fluid dynamics and protecting the neural tissue from mechanical stress and injury [2]. Despite its important role in, e.g., traumatic brain injury pathology it is frequently neglected in computational and physical human head models. Until now, the biomechanical failure tests have not yielded information about the anisotropic mechanical behavior of various locations of the human dura mater under physiological conditions. Additionally, limited data exists on the orientation of collagen fibers, which has not been thoroughly quantified.

Therefore, this study aimed to investigate the mechanical and structural anisotropy of different locations of the human dura mater and to correlate mechanical data with the quantification of the orientation of collagen fibers. To achieve this objective, sixty samples from six donors were subjected to quasi-static, uniaxial extension tests until failure in a heated tissue bath, utilizing a Z020 torsion multi-axis testing system (ZwickRoell AG, Ulm, Germany) together with an Aramis image correlation system (GOM, Braunschweig, Germany). Additionally, the collagen microstructure of samples from four donors was analyzed using second-harmonic generation imaging.

The data obtained were used to determine the failure stress and strain, E-moduli, and a microstructurally motivated material model was employed to examine local differences in both structure and mechanics. Among others, significant differences in both collagen fiber dispersion and main fiber orientation were found. The structural parameters obtained from only four out of six donors yielded good fitting results to the structurally motivated mechanical model for the remaining two donors in all directions. This study establishes a foundation for further research on the microstructure and mechanical properties of the human dura mater, enabling realistic modeling and prediction of tissue failure.

References

[1] J. T. Maikos, R. A. Elias, and D. I. Shreiber, "Mechanical properties of dura mater from the rat brain and spinal cord," J Neurotrauma, vol. 25, no. 1, pp. 38-51, Jan 2008, doi: 10.1089/neu.2007.0348.

[2] D. J. Patin, E. C. Eckstein, K. Harum, and V. S. Pallares, "Anatomic and biomechanical properties of human lumbar dura mater," Anesth Analg, vol. 76, no. 3, pp. 535-40, Mar 1993, doi: 10.1213/00000539-199303000-00014

Cardiovascular diseases are still one of the major cause of death worldwide. They are responsible for significant morbidity and therefore both noninvasive and especially invasive treatments are still being developed. Our research group is working in the field of new materials for cardiovascular surgery.

There are several procedures in the field of cardiovascular surgery which are based on the mechanical interaction of the arterial wall and the external support. A typical example is Pulmonary Artery Banding (PAB) in infant patients. PAB is a palliative procedure that reduces pulmonary over-circulation in neonates suffering from certain congenital heart diseases and constitutes the first stage of intervention prior to the complete repair of cardiac defects. The principle of PAB consists in the implantation of a band around the pulmonary artery, which results in the reduction of blood flow.

Our goal is to develop resorbable fabric based on composite material consisting of a polylactide and polycaprolactone (PLA/PCL) copolymer nanofibrous reinforcement combined with a collagen matrix for use in cardiovascular surgery as the pulmonary artery banding material. The resulting composite composed of these polymers prepared by electrospinning followed by the impregnation of the layers with collagen (type I) dispersion is then collagen-poly[L-lactide/caprolactone] (COLL-PLCL). This compound is bioresorbable and gives the advantage of PAB as a one-step procedure.

In our work, we present the results of the comparison of the mechanical properties of our COLL-PLCL material with two kinds of commercially available Gore material that can be used in pulmonary artery banding. They are the Gore Pericardial Membrane and the Gore Cardiovascular Patch, and both materials are chemically based on the form of polytetrafluoroethylene (PTFE).

The strips of all three materials underwent uniaxial tensile tests. In the case of the COLL-PLCL material, it was a monotonic tensile test, whereas the Gore materials were tested cyclically. The COLL-PLCL materials were tested in the hydrated state.

The results of our experiments show that all three materials exhibit an approximately linear mechanical response under primary loading. Cyclic loading led to so called preconditioned state, in which the Gore materials exhibited some non-linearities in their response. The preconditioned Gore-Tex materials were also considerably stiffer than those in the pristine state. Gore materials showed higher stress at failure than COLL-PLCL material whereas strain at failure was approximately the same. What is significant is that the COLL-PLCL material was much more compliant than Gore. We hypothesize that it will therefore cause a less mechanobiological response to compliance mismatch during banding.

Although the biomechanics of the circulatory system is a very dynamically developing field, the study of venous aneurysms is somewhat outside the mainstream. Although aneurysm, a bulge of the blood vessel wall, is mainly known as a disease of the arteries, it can also be encountered in veins. It occurs mainly at the site of an arterio-venous shunt, which is artificially created as a so-called vascular access for hemodialysis. Other venous aneurysms are quite rare. However, the number of patients for dialysis is increasing, so the need to study tissue biomechanics in vascular access is a current scientific problem.

The exact mechanism of arterio-venous shunt aneurysm formation, apart from the rather vague claim that it is a consequence of altered hemodynamic conditions, is still unknown. The same applies to the constitutive models and their parameters found for such affected vein walls. However, it should be emphasized that not only the change in hemodynamics due to arterialization of the vein, but also the repeated needling of the tissue during the hemodialysis procedure are involved in the formation of the arteriovenous shunt aneurysm. These lead to significant scarring of the vessel wall, which results in a change in mechanical properties compared to healthy vein. As with any scar, we should expect a significant increase in connective tissue content.

The long-term goal of our study is to experimentally identify material models for pathologically affected veins and compare them with the behavior of healthy tissue. We then intend to use the constitutive models for the arterio-venous shunt aneurysm tissue in computational fluid dynamics simulations with a deformable wall to determine the changes that occur in the hemodynamics of the aneurysm at the vascular access.

As a first step in our study, we perform tensile experiments with vein wall samples, both healthy ones and veins with a created arteriovenous shunt. We chose to model the tissue behavior as elastic after preconditioning. The constitutive model used for the vein wall is adopted from the work of Gasser Holzapfel and Ogden (2006), or we model the vein wall as an incompressible, anisotropic, hyperelastic material that exhibits significant material nonlinearity. Because aneurysmal vein wall samples are collected during surgery from living patients, it is not possible to compare them with other donor veins, as they are not naturally available. Comparisons will be made against the characteristics of veins taken to create an aorto-coronary bypass graft. The number of experimentally measured samples in both groups is not yet large enough to achieve significant population differences. Therefore, our present results should be considered as preliminary.

Tissue fibrosis, classically associated with excessive accumulation of extracellular matrix (ECM) by activated fibroblasts, can occur in several pathological conditions. Fibroblasts are mechanosensitive cells and increased stiffness of the ECM activates fibrotic pathways [1]. To mimic the three-dimensional cell microenvironment in vitro, fibroblasts are cultured in 3D spheroids. Mechanical assessment of cell spheroids is accomplished via parallel plate compression and mechanical properties in a continuum sense are determined from force vs displacement (F-δ) curves. Current analysis methods are often limited to Hertzian theory and its modifications, that only account for small deformations [2]. However, spheroids exhibit low stiffness, thus undergoing large deformations at small external forces, and therefore current contact mechanics models fail to describe such behavior [3,4]. Here, we use the Tatara model, which considers both nonlinear elasticity and large deformations to describe the mechanics of cell spheroids under parallel-plate compression.

Cell spheroids consisting of primary human fibroblasts cultured for four days were subjected to parallel-plate compression testing (MicroSquisher, CellScale) fitted with a round tungsten cantilever and accompanying SquisherJoy V5.23 software (CellScale, Ontario, Canada). The fluid bath test chamber was filled with sterile phosphate buffered saline (pH=7.4). Stage and optics were calibrated according to manufacturer’s instructions. Samples were compressed up to 50% apparent linear strain at different displacement rates. F–δ data was fitted using linear least squares regression on the Hertz and Tatara model and its extended version (custom MATLAB code) with fully constrained contact points (F =0, δ =0). Images were captured via two digital cameras during compression and static states to determine contact radius and lateral expansion of cell spheroids over a range of deformations.

The dependence of compression modulus on the displacement rate was examined. The Hertzian (and extended) theory was successfully applied to the small strain regime of F-δ curves to extract the compression modulus. In contrast to the Hertzian theory, for compressive displacements over 10-25% (depending on displacement rate), where the force follows the third and fifth power of the displacement, the Tatara numerical analysis theory better described the non-linear behavior of cell spheroids. The predicted contact radius of cell spheroids was found to be in good agreement with the data obtained from image analysis. However, the lateral expansion was underestimated by the Tatara model, a difference which may arise from the adopted constant-volume assumption.

The Hertzian theory and its modifications can be applied for strain in the range of 10%-25% while, for larger deformations (above 25% strain), the extended Tatara model better describes the deformation behavior of cell spheroids also providing their lateral expansion. While Hertzian contact theory and its modifications is applicable for small deformations, hyperelastic and nonlinear elasticity theory should be employed to better describe the data at larger deformations.

1. M. Jones et al, eLife, 7 :1-24, 2018

2. Y. Efremov et al, Scientific Reports, 7:1-14, 2017.

3. Y Tatara et al, JSME INT A-SOLID M, 36:190-196, 1993.

4. K. Liu et al, J. Phys. D, 31 :294-303, 1998

Background. Despite biomechanical factors playing a pivotal role in the onset and development of disease [1], little is known concerning the fracture behaviour of most biological tissues. Most experimental testing techniques fail to acquire data for the mechanical fracture characterisation of soft tissues. Therefore, we propose the novel symmetry-constraint Compact Tension (symconCT) test for aortic tissue fracture testing. It allows for stable crack propagation and the in-depth exploration of fracture processes. Finite Element Method (FEM) modelling was combined with Digital Image Correlation (DIC) to identify the specimen-specific constitutive properties.

Material and Methods. The classical CT test was augmented with an elastic metal beam connecting the two clamps [2]. It pre-strained the 30x35 mm² specimens and forced the fracture to develop in the middle of the specimen. Specimens with a 10 mm pre-notch were obtained from the intima-media compound of the porcine abdominal aorta. Tensile loading, either in the axial or circumferential direction, was applied orthogonal to the pre-notch at 3 mm min^-1 (ADMET eXpert 4000 Universal Testing System). 2D FEM models were developed (Abaqus 2019) for six specimens (representing mean ± standard deviation of recorded symconCT data sets), incorporating a viscoelastic Yeoh model and an isotropic bilinear cohesive zone model. Whilst the viscous properties were fixed, the two elastic Yeoh coefficients, c₁ and c₂, were calibrated against the clamp force and DIC-based Green-Lagrange strains. The cohesive strength and the fracture energy were also calibrated against the force.

Results. Given the load aligned with the main collagen fibre direction in the media, circumferentially loaded specimens reached higher forces, with a mean of 5.76 ± 0.69 N against 3.89 ± 0.57 N of the axially loaded specimens (t-test, p<0.05). Whilst circumferential loading resulted in a zig-zag crack, axial loading provoked a straight fracture. The DIC showed the maximum principal strain being orthogonal to the crack tip. The identified elastic properties were c₁ = 34.70 ± 1.52 kPa and c₂ = 98.70 ± 5.63 kPa under circumferential loading and c₁ = 14.54 ± 5.24 kPa and c₂ = 67.34 ± 6.65 kPa under axial loading. The cohesive strength resulted in 235.62 ± 24.04 kPa and 242.40±72.48 kPa, the fracture energy in 1.57 ± 0.82 kJ m² and 0.96 ± 0.34 kJ m² under circumferential and axial loading, respectively. Given the calibrated FEM models, the FEM crack extensions closely matched the experimental ones.

Conclusions. The symconCT test allowed for stable crack propagation, permitting the exploration of the fracture behaviour ahead of the crack tip. The DIC-based principal strains revealed mode-I failure dominating the fracture in both loading conditions. The FEM models accurately simulated the tissue fracture. Although inertial effects were overaccelerated (mass-scaling), good agreement with the experimental recordings was observed. Additionally, the vessel wall shows anisotropic mechanical properties [3], characteristics which could have further improved our results.

References.

[1] N. Baeyens, M. A. Schwartz, Mol. Biol. Cell, vol. 27, pp. 7-11, 2016.

[2] M. Alloisio, M. Chatziefraimidou, J. Roy, T. Gasser, Acta Biomaterialia (under review).

[3] T.C. Gasser, Vascular Biomechanics, ISBN: 978-3-030-70965-5. Springer 2021.