The anterior cruciate ligament’s (ACL) mechanics is an important factor governing the ligament’s integrity and, hence, the knee joint’s response. Despite many investigations in this area, the cause and effect of injuries remain unclear or unknown. This may be due to the complexity of the direct link between macro- and micro-scale damage mechanisms. In the first part of this investigation, a three-dimensional coarse-grained model of collagen fibril (type I) was developed using a bottom-up approach to investigate deformation mechanisms under tensile testing.
Collagens type I that form fibrils are present as the main contributor to the integrity of the joint ligaments via a hierarchical extension over many length scales. Collagen is made up of amino-acid sequences organized in a polypeptide helix and combined into a set of three supercoils that produce a molecule of tropocollagen (TC) . An X-ray diffraction experiment was employed to determine the topography of short peptide fragments representing the collagen molecules. This molecule was characterized by a weight of 300 KDa, length of 300 nm, and 1 to 2 nm diameter. The TC molecules aggregate into fibrils via intermolecular adhesion and covalent crosslinks at their ends with respect to well-specified axial and center-to-center locations offset . The mechanical behavior of the collagenous structures has received significant attention from molecular to aggregate level, either experimentally or computationally via molecular dynamic (MD) simulation . A clear agreement has been reported that these superstructures lead to helpful mechanical behavior characterized by high strength and extensibility of up to 100% strain before breakage . However, most previous investigations were limited to nano- and micro-scales. The reasons behind this limitation were the intimate coupling between chemistry, biology, and mechanical deformation and their structural texture that involves specific implementation via different scales.
On the larger scale (millimeter and upscale), the soft tissues’ mechanical behavior was considered either elastic, basic hyperelastic, hyperelastic, or viscoelastic fiber-reinforced composite materials . Most of these constitutive models were limited to the elastic behavior and have been able to reproduce the mechanical response of the ligaments measured during experimental tensile testing. A very limited number of studies considered softening elasticity behavior yet restricted to the macro-mechanical scale . On the side of the model application, the anterior cruciate ligament (ACL) was the focus of most of the developed computational frames since it is more prone to injury than any other knee soft tissue. Recent investigations have shown that around 50% of lower limb ligament injuries involve the ACL . ACL damage’s pathogenesis and pathophysiological underpinnings were well connected to the molecular level . This process of damage was described microscopely as collagen degradation . Thereafter, exploring how molecular-level interactions combine, including micro-degradation mechanism, to create a tissue-level mechanical response of collagenous materials is a potential path for the advancement of ACL injury or trauma treatments.
The principal aim of this work was to highlight the link between ACL molecular structure and its aggregate mechanical behavior. At one extreme, such a link uses molecular dynamic simulation to characterize the fibril structure properties as a function of crosslink degradation. At the other extreme, the link uses a multilevel hyper-elastoplastic fibril-reinforced model to simultaneously investigate how changes in the fibril structure are manifested at the ligament level.
Figure 1 shows the obtained stress–strain curve for a single molecule. The curve can be divided into three zones describing three different mechanisms. Zone I : in this phase, the molecule was stretched along its principal axis. The small resulting stress was due to the change of the angle energy. This phase is hardly visible on the stress–strain curve since it needed an extremely slow strain rate to allow for atoms to relax. Zone II : in this phase, the molecule was uniformly stretched. The elastic constant was ~7.9 GPa and resulted mainly from the bond stretching below the hyperelastic critical distance r1. Zone III : in this phase, a sharp change in strength occurred as the bond distance reached the hyperelastic critical distance r1 (elastic constant was ~47.1 GPa). The molecule continued to stretch uniformly until reaching the breaking point. The molecule broke when interatomic distances reached the breaking distance rb
Current computed stress–strain variation in the single TC molecule along with Buehler et al. [42
] model prediction.
Figure 2 shows the stress–strain curve for the collagen fibril for different crosslink densities. The overall trend of the curves is consistent with previous work: increasing crosslink densities increased both the ultimate tensile strength and the ultimate tensile strain of the fibril. Similar to the single molecule, the angles on the fibril were first straightened since the angle energy was much smaller than the bond and pairwise energies. Then, with increasing strain, the internal stress started building up in the structure up to a threshold where the pairwise forcefield could no longer resist the shear forces induced between molecules. As a result, the molecules started sliding and stress started to decrease. When crosslinks were present in the fibril, they provided additional resistance to the shearing between molecules and, therefore, retarded the sliding threshold, increasing the ultimate tensile strain and stress.
Figure 2. Strain–stress curve of the collagen fibril as a function of the amount of crosslink (β) predicted using the coarse-grained model under axial tensile testing.
Nonlinear curve fitting of our data collected from molecular dynamics simulation was used to evaluate the material characteristics of the fibril for native and degraded tropocollagen crosslinks. The goodness-of-fit (GOF) value (coefficient of determination, given as mean, standard error (SE)) for the fitted data was 0.91 ± 0.04 for the crosslink failures. For all cases, the fitted curves provided satisfactory fits to the MDS data. Table 1 shows the average optimal input fibril parameters for degradation processes. The mechanical properties of the fibrils influenced the aggregate behavior of the ACL at full extension. Tropocollagen crosslinks degradation slightly influenced the elastic stress of the ACL as observed during the axial tension that ranged between 0 and 6%. The effect of this degradation became remarkable when the axial strain varied between 6 and 8% (Figure 5). However, the ACL yield stress was substantially influenced by tropocollagen crosslink failure. The yield stress was approximately reduced by 90% when β varied gradually from 100% to 0 (Figure 3 and Figure 4). However, a minimal variation was observed on the elastoplastic behavior of the ACL when the amount of crosslink failure was more than 80% (β = 20% or less). Under pre-yielding loading conditions (7.5% axial strain), ligaments stresses distributions were nonuniform at all degradation stages (Figure 5). The maximum axial stresses ranged from ~10MPa to ~20 MPa. The largest stress values were located at the posterolateral side near the tibial junction, and the smallest values were at the anteromedial area near the femoral junction (Figure 5).
Figure 3. (a) Finite-element model of the ACL with the associated boundary conditions considered to simulate the uniaxial tensile test. (b) Computed axial maximum stress–strain behavior of the ACL.
Figure 4. Computed ACL yield strength: (a) and strain; (b) as a function of the crosslink content (β).
Figure 5. Stress distribution in the ACL at 7.5% axial strain (elastic zone).
Table 1. Material parameters of the fibril found via the data-fitting procedure.
Table 1. Mesoscopic structural properties.
|Molecule number of atoms
|Molecule total mass [g/mol]
|Number of beads per molecule
|Mass of each bead [g/mol]
||Length of fibril [Å]
|Length along principal axis [Å]
||Hex. lattice constant [Å]
Plastic elongation was used as a plastic-change indicator in this study (λp). This variable specified when and where the plastic change occurred. Damage propagation (plastic change) of ligament structures was restricted to the collagen fiber’s surface layers (Figure 6). The maximum stress distribution was typically related to damage initiations and distributions. In most cases, the damage began in the highest stressed element and spread in the same direction of the stress distribution (Figure 6). ACL damage started between 8 and 17% axial strain with an associated range of axial stress varying between 13 MPa and 58 MPa when tropocollagen crosslinks failure ranged from no crosslink to 100% crosslink. The maximum plastic elongation computed ranged from 29% (100%) to 34% (0%). The increase in the crosslink’s failure between the tropocollagen molecules significantly increased the area of damage.
Figure 6. ACL-damage distribution (plastic change) at 3% post-yielding strain.
Researchers presents a comprehensive work incorporating the degradation mechanism of the type I collagen on the nano-scale into a description of the mechanical behavior of the ACL. The developed framework connects a meso-scale molecular dynamics-simulation model to a continuum fibril-reinforced hyper-elastoplastic model. The degradation mechanism was presented by the elimination of the enzymatical crosslinks. This was later well documented as a key element in the collagen's integrity and thereafter expressing a significant role in the yielding tissue level, and serves as an internal control for the integrity of our modeling approach. The proposed engineering frame was validated at two levels, molecular and continuum, by comparing the predictions with reported results in the literature. At the micro-level, the degradation in the enzymatical crosslinks affected the elastic stiffness of the second and third deformation regimes and the collagen failure strain. A higher crosslink density generated a well-connected structure of the collagen fibril. On the macro-scale and by means of the hierarchical model, a limited zone of the elastic and the complete plastic responses were influenced by the degradation of enzymatical crosslinks.