Integrating qPLM and biomechanical test data with an anisotropic fiber distribution model and predictions of TGF-β1 and IGF-1 regulation of articular cartilage fiber modulus

Multi-fidelity surrogate modeling through hybrid machine learning for biomechanical and finite element analysis of soft tissues

Biomechanical simulation enables medical researchers to study complex mechano-biological conditions, although for soft tissue modeling, it may apply highly nonlinear multi-physics theories commonly implemented by expensive finite element (FE) solvers. This is a significantly time-consuming process on a regular computer and completely inefficient in urgent situations. One remedy is to first generate a dataset of the possible inputs and outputs of the solver in order to then train an efficient machine learning (ML) model, i.e., the supervised ML-based surrogate, replacing the expensive solver to speed up the simulation. But it still requires a large number of expensive numerical samples. In this regard, we propose a hybrid ML (HML) method that uses a reduced-order model defined by the simplification of the complex multi-physics equations to produce a dataset of the low-fidelity (LF) results. The surrogate then has this efficient numerical model and an ML model that should increase the fidelity of its outputs to the level of high-fidelity (HF) results. Based on our empirical tests via a group of diverse training and numerical modeling conditions, the proposed method can improve training convergence for very limited training samples. In particular, while considerable time gains comparing to the HF numerical models are observed, training of the HML models is also significantly more efficient than the purely ML-based surrogates. From this, we conclude that this non-destructive HML implementation may increase the accuracy and efficiency of surrogate modeling of soft tissues with complex multi-physics properties in small data regimes.

A backward pre-stressing algorithm for efficient finite element implementation of in vivo material and geometrical parameters into fibril-reinforced mixture models of articular cartilage

Classical continuum mechanics has been widely used for implementation of the material models of articular cartilage (AC) mainly with the aid of the finite element (FE) method, which, in many cases, considers the stress-free configuration as the initial configuration. On the contrary, the AC experimental tests typically begin with the pre-stressed state of both material and geometrical properties. Indeed, imposing the initial pre-stress onto AC models with the in vivo values as the initial state would result in nonphysiologically expansion of the FE mesh due to the soft nature of AC. This change in the model configuration can also affect the material behavior kinematically in the mixture models of cartilage due to the intrinsic compressibility of the tissue. Although several different fixed-point backward algorithms, as the most straightforward pre-stressing methods, have already been developed to incorporate these initial conditions into FE models iteratively, such methods focused merely on the geometrical parameters, and they omitted the material variations of the anisotropic mixture models of AC. To address this issue, we propose an efficient algorithm generalizing the backward schemes to restore stress-free conditions by optimizing both the involving variables, and we hypothesize that it can affect the results considerably. To this end, a comparative simulation was implemented on an advanced and validated multiphasic model by the new and conventional algorithms. The results are in support of the hypothesis, as in our illustrative general AC model, the material parameters experienced a maximum error of 16% comparing to the initial in vivo data when the older algorithm was employed, and it led to a maximum variation of 44% in the recorded stresses comparing to the results of the new method. We conclude that our methodology enhanced the model fidelity, and it is applicable in most of the existing FE solvers for future mixture studies with accurate stress distributions.

Computational simulation of the multiphasic degeneration of the bone-cartilage unit during osteoarthritis via indentation and unconfined compression tests

It has been experimentally proposed that the discrete regions of articular cartilage, along with different subchondral bone tissues, known as the bone-cartilage unit, are biomechanically altered during osteoarthritis degeneration. However, a computational framework capturing all of the dominant changes in the multiphasic parameters has not yet been developed. This article proposes a new finite element model of the bone-cartilage unit by combining several validated, nonlinear, depth-dependent, fibril-reinforced, and swelling models, which can computationally simulate the variations in the dominant parameters during osteoarthritis degeneration by indentation and unconfined compression tests. The mentioned dominant parameters include the proteoglycan depletion, collagen fibrillar softening, permeability, and fluid fraction increase for approximately non-advanced osteoarthritis. The results depict the importance of subchondral bone tissues in fluid distribution within the bone-cartilage units by decreasing the fluid permeation and pressure (up to a maximum of 100 kPa) during osteoarthritis, supporting the notion that subchondral bones might play a role in the pathogenesis of osteoarthritis. Furthermore, the osteoarthritis composition-based studies shed light on the significant biomechanical role of the calcified cartilage, which experienced a maximum change of 70 kPa in stress, together with relative load contributions of articular cartilage constituents during osteoarthritis, in which the osmotic pressure bore around 70% of the loads after degeneration. To conclude, the new insights provided by the results reveal the significance of the multiphasic osteoarthritis simulation and demonstrate the functionality of the proposed bone-cartilage unit model.

Structure and collagen crimp patterns of functionally distinct equine tendons, revealed by quantitative polarised light microscopy (qPLM)

Structure-function relationships in tendons are directly influenced by the arrangement of collagen fibres. However, the details of such arrangements in functionally distinct tendons remain obscure. This study demonstrates the use of quantitative polarised light microscopy (qPLM) to identify structural differences in two major tendon compartments at the mesoscale: fascicles and interfascicular matrix (IFM). It contrasts functionally distinct positional and energy storing tendons, and considers changes with age. Of particular note, the technique facilitates the analysis of crimp parameters, in which cutting direction artefact can be accounted for and eliminated, enabling the first detailed analysis of crimp parameters across functionally distinct tendons.

IFM shows lower birefringence (0.0013 ± 0.0001 [−]), as compared to fascicles (0.0044 ± 0.0005 [−]), indicating that the volume fraction of fibres must be substantially lower in the IFM. Interestingly, no evidence of distinct fibre directional dispersions between equine energy storing superficial digital flexor tendons (SDFTs) and positional common digital extensor tendons (CDETs) were noted, suggesting either more subtle structural differences between tendon types or changes focused in the non-collagenous components.

By contrast, collagen crimp characteristics are strongly tendon type specific, indicating crimp specialisation is crucial in the respective mechanical function. SDFTs showed much finer crimp (21.1 ± 5.5 µm) than positional CDETs (135.4 ± 20.1 µm). Further, tendon crimp was finer in injured tendon, as compared to its healthy equivalents. Crimp angle differed strongly between tendon types as well, with average of 6.5 ± 1.4° in SDFTs and 13.1 ± 2.0° in CDETs, highlighting a substantially tighter crimp in the SDFT, likely contributing to its effective recoil capacity.

Statement of Significance

This is the first study to quantify birefringence in fascicles and interfascicular matrix of functionally distinct energy storing and positional tendons. It adopts a novel method – quantitative polarised light microscopy (qPLM) to measure collagen crimp angle, avoiding artefacts related to the direction of histological sectioning, and provides the first direct comparison of crimp characteristics of functionally distinct tendons of various ages.

A comparison of matched picrosirius red stained and unstained tendons sections identified non-homogenous staining effects, and leads us to recommend that only unstained sections are analysed in the quantitative manner.

qPLM is successfully used to assess birefringence in soft tissue sections, offering a promising tool for investigating the structural arrangements of fibres in (soft) tissues and other composite materials.

An evolutionary model of osteoarthritis including articular cartilage damage, and bone remodeling in a computational study

With osteoarthritis, a complex set of progressive chemical, biological, and mechanical changes occur in both cartilage and bone. The aim of this study is to develop a high-fidelity computational model of the complete bone-cartilage unit to study the evolution of osterarthritis-induced articular cartilage (AC) damage and remodeling of subchondral cortical bone (SCB) and subchondral trabecular bone (STB). A finite element model of spherical indentation was developed with a depth-dependent anisotropic model of degenerating articular cartilage, a calcified cartilage (CC) zone, and SCB and STB remodeling regions. Calcified tissue (CC, SCB, and STB) and AC material regions were integrated to form an evolutionary bone-cartilage unit model. Results indicate that with indentation loading, articular cartilage damage occurs at the articular surface. Furthermore, bone remodeling was predicted to occur with a net stiffening of the subchondral bone plate. Changes in indentation force were minimal (<2%) between initial and final peak indentation loading. However, additional degradation and wear of AC and/or alterations in loading may have more pronounced effects on the mechanical response of the bone-cartilage unit. Bone remodeling and articular cartilage damage predictions are consistent with experimental observations that cartilage damage begins at the articular surface and subchondral bone experiences a thickening (i.e., stiffening) response with osteoarthritis. Our results provide insight into the early-term initiation behavior of osteoarthritis; the potential consequences of evolutions in AC, SCB, and STB with disease progression; and may guide future experimental and computational studies to elucidate mechanisms of osteoarthritis progression.

A quantitative interpretation of the response of articular cartilage to atomic force microscopy-based dynamic nanoindentation tests

In this paper, a quantitative interpretation for atomic force microscopy-based dynamic nanoindentation (AFM-DN) tests on the superficial layers of bovine articular cartilage (AC) is provided. The relevant constitutive parameters of the tissue are estimated by fitting experimental results with a finite element model in the frequency domain. Such model comprises a poroelastic stress-strain relationship for a fibril reinforced tissue constitution, assuming a continuous distribution of the collagen network orientations. The identification procedure was first validated using a simplified transversely isotropic constitutive relationship; then, the experimental data were manually fitted by using the continuous distribution fibril model. Tissue permeability is derived from the maximum value of the phase shift between the input harmonic loading and the harmonic tissue response. Tissue parameters related to the stiffness are obtained from the frequency response of the experimental storage modulus and phase shift. With this procedure, an axial to transverse stiffness ratio (anisotropy ratio) of about 0.15 is estimated.