Nanomechanical properties of mineralized collagen microfibrils based on finite elements method: biomechanical role of cross-links. Abdelwahed Barkaoui, Ridha Hambli PRISME Laboratory, EA4229, University of Orleans Polytech’ Orléans, 8, Rue Léonard de Vinci 45072 Orléans, France Mail :
[email protected]
ABSTRACT: Hierarchical structures in bio-composites such as bone tissue have many scales or levels, and synergic interactions between the different levels. They also have a highly complex architecture in order to fulfil their biological and mechanical functions. In this study, a three-dimensional model based on the finite elements method (FEM) was developed to model the relationship between the hierarchical structure and the properties of the constituents at the sub-structure scale (mineralized collagen microfibrils) and to investigate their equivalent nanomechanical properties. The results of the proposed finite element simulations coupled to an Abaqus finite element (FE) code via the User MATerial (UMAT) subroutine and an optimization algorithm show that the elastic properties of microfibrils depend on different factors such as the number of cross-links, the mechanical properties and the volume fraction of phases. The results obtained under compression loading at a small deformation < 2% show that the microfibrils have a Young’s modulus (Ef) ranging from 0.4 to 1.16 GPa and a Poisson’s ratio ranging from 0.26 to 0.3. These results are in excellent agreement with experimental data (X-ray, AFM, MEMS) and molecular simulations. Keywords: Bone tissue; Mineralized collagen microfibrils; cross-links; Finite elements simulation; Nanomechanical properties. 1. Introduction At the ultra-structure level, mineral and tropocollagen (TC) are arranged in higher hierarchical levels to form microfibrils, fibrils and fibers [1-3]. The existence of substructures in collagen fibrils has been the subject of debate for years. Recent studies suggest the presence of microfibrils in fibrils. Experimental work by Hulmes, Orgel, Fratzl, Currey, Aladin and others shows that virtually all collagen-based tissues are organized in hierarchical structures, with the lowest hierarchical level consisting of triple helical collagen molecules [49]and that the multi-scale structure is defined as TC molecules-fibrils-fibers. A longitudinal microfibrillar structure with a width of 4 - 8 nm was visualized in both hydrated [10-11] and dehydrated [12] forms. Three-dimensional image reconstructions of 36 nm-diameter corneal collagen fibrils also showed a 4 nm repeat in a transverse section, which was related to the microfibrillar structure [13]. Using X-ray diffraction culminating in an electron density map, Orgel et al. [5] suggested the presence of right-handed super twisted microfibrillar structures in collagen fibrils. The microfibril shown in (Fig.1) is a helical assembly of five tropocollagen molecules (rotational symmetry of order 5) which are offset one another with an apparent periodicity of 67 nm. This periodic length is denoted D and is used as a primary reference scale in describing the structural levels. The helical length of a TC molecule is 4.34 D ≈ 291 nm and the discrete gap denoted G (hole zone) is 0.66 D ≈ 44 nm between two consecutive type-I TC molecules in a strand. These gaps in bone are the sites for the nucleation and deposition of hydroxyapatite (HA) crystals (the mineral component of bone tissue) [14].The five TC molecules create a cylindrical formation with a diameter of 3.5–4 nm and an unknown length
[15]. The orientation and axial arrangement of TC molecules in the microfibril have been deduced from electron-microscopy observations showing transverse striations with a period D. The origin of this streaking was a gradation in the arrangement of elements that are staggered TC with themselves at an interval D. These TC molecules are interconnected by cross-links. Previous research has established the existence of collagen microfibrils and their three-dimensional geometric shape. This structure is obtained by employing conventional crystallographic techniques in X-ray fiber.
Figure 1. (a) Real collagen fiber with cross-links [16], (b) Schematic illustration of the tropocollagen triple helix molecules and the formation of the microfibril [17-18]. In summary, microfibrils are a bio-composite material containing an inorganic mineral (mature mineral + immature mineral) HA reinforced by organic phase TC molecules connected by cross-links. The mechanical behavior of a composite material is necessarily linked to the biomechanical properties of each phase. Experimental and numerical studies on bone have shown the effect of each component (tropocollagen, mineral and cross-links), and have reported the following general results: Collagen is biomechanically important in bone, providing the plastic, ductile properties, whereas the mineral confers stiffness [19].The torsional stability and tensile strength of collagen lead to the stability and integrity of bone tissue [20-21]. TC molecules provide mechanical stability, elasticity, and strength to connective tissues [2].Bone strength is governed by the characteristics of collagen, including collagen cross-links which are important in the reinforcement of bone strength. The biomechanical effects of collagen depend largely on cross-linking [22-23]. The strength and stability during maturation of the microfibrils are achieved by the development of intermolecular cross-links[24-25].Experiments in vitro [26-27]and in vivo [26, 23, 28] have documented that increases in cross-linking are associated with the enhancement of certain mechanical properties (strength and stiffness) and a reduction in others (energy absorption). Experimental evidence demonstrates that collagen cross-linking in bone tissue significantly influences its deformation and failure behavior, yet difficulties exist in determining the independent biomechanical effects of collagen cross-linking using in vitro and in vivo experiments [29].Considering a macroscopic response of bone, enzymatic cross-linking has been linked to improved mechanical properties [30] whereas nonenzymatic cross-linking prevents energy absorption by microdamage formation and may accelerate brittle fracture [31-34]. Natural cross-linking gives collagen a high tensile strength and proteolytic resistance
[35].The properties related to bone strength include rate of bone turnover, bone mineral density, geometry, microarchitecture, and mean degree of mineralization. These properties (with or without bone density) are sometimes collectively referred to as bone quality, that's why a study at the nanoscale of the mineralized collagen microfibril was conducted, a study that combines the mechanical and structural parameters of all phases. Except for some recent atomistic modeling [2] and experimental work [36-39] on the behavior of microfibrils, studies at this level of scale are scarce, and information about mineralized collagen microfibrils sparse. The question arises as to whether the microfibrillar structures play a role in the mechanical properties of single mineralized collagen fibrils and in bone. In this study, we expand upon previous models dealing with bone ultra-structure modeling in three respects: (i) creation of a realistic, lower level (microfibril) 3D finite element model to represent the structure of the mineralized collagen microfibril with three constituents (mineral, TC molecules, and cross-links); (ii) investigation of the microfibril mechanical behavior in relation to the mechanical properties of its constituents (tropocollagen, mineral) and the number of cross-links; (iii) validation of the model based on experimental data and a numerical study from the literature. The proposed 3D finite element model of mineralized collagen microfibrils enables the bottom-up investigation of structureproperty relationships in human bone. This model can be applied to study the effects of biochemical properties related to tropocollagen or the mineral components on the strength of human bone. 2. Method and tools 2.1 model geometry In this work a 3D finite element model for the collagen mineralized microfibril (Fig.1) was developed choosing a repetitive, periodic and symmetric portion [18, 40]. (Fig.2) shows the real 3D form of the collagen microfibril with unfolding of the cylindrical form in the plane. This shows the organization and unfolding of the particular arrangement of TC molecules and how the cross-links are arranged. It also shows the chosen dimension of the model, the hole zones (gaps), and the overlap. It illustrates the distribution of the five fractions of TC molecules and their dimensions.
Figure 2. Geometric model of collagen microfibril: (a) reel 3D form of mineralized collagen microfibril, (b) unfolding in plane of the chosen portion (c) unfolding of microfibril in plan.
The geometric parameters of the microfibril model are given in Table 1: Table1. geometric parameters of proposed model Length of FE model : L Length of TC molecule: l radius of model: R TC molecule radius: r Gap: G Periodicity:D transverse spacing: j
340 nm 300 nm 2 to 2.5 nm 0.65 à 075 nm
40 nm 67 nm 1 to 3 nm 27 nm
Overlap : h
In order to permit the repetitiveness of the portion, the TC molecule fraction length was calculated as in equation (1) assuming that the length of TC molecule l = 300 nm and D = 67 nm. (1) The lengths of the portions of TC molecules xi illustrated in (Fig.2) are calculated by the formula: (2) In the mineralization phase the pure mineral occupies the gap zones. After 10 days of this phase the mineral apatite in its amorphous form, together with other substances, is deposited between the TC molecules, and fills the rest of the empty cylinder which has a radius R and length L. The microfibril can be considered as a composite material composed of three phases (i) collagen; (ii) mature mineral; (iii) immature mineral. The distribution volumes of the three phases are calculated by the following formulas: Total volume: (3) Volume of tropocollagen phase: ∑
(4)
Volume of mature mineral: (5) Volume of immature mineral: (6) 2.2
Collagen mineralized microfibril composition
The elementary components of the microfibril can be distinguished as follows: (a) Tropocollagen: Long cylindrically shaped TC molecules formed of three polypeptide strands with a diameter of about 1.5 nm and a length of about 300 nm [41, 1, 2], selfassembled in the form of microfibrils. (b) Mineral: Bone mineral is composed of poorly crystalline hydroxyapatite (HA). HA is calcium phosphate [Ca10(PO4)6(OH)2] and recent studies using transmission electron microscopy (TEM) and scanning small-angle X-ray scattering (SAXS) have shown
that HA is plate-like in shape. Fratzl et al. [42] determined the most probable size of the particle to be 15-200 nm long, 10-80 nm wide and 2-5 nm thick. (c) Cross-links: They join two TC molecules. The microfibril structure is stabilized through intermolecular cross-links, formed between telopeptides and adjacent triple helical chains through lysine–lysine covalent bonding [22]. Cross-linking is either enzymatically or non-enzymatically mediated [43-44]. (a) Different non-collagenous organic molecules: These are predominantly lipids and proteins which regulate HA mineralization, probably by proteins supporting or inhibiting mineralization, possibly also by lipids [45-47]. (b) Water: Provides the liquid environment for the biochemical activity of the noncollagenous organic matter. To the best of our knowledge, in the most earlier multiscale modeling studies, the mineralized collagen fibrils is considered as the smallest component in the cortical bone. Previous numerical studies of mineralized collagen fibrils were based on the model proposed by Jager and Fratzl [48]. This model assumes that the fibril is a collagen matrix reinforced by mineral crystals with the following assumptions [49]: -
-
In the matrix, in addition to type-I collagen, there are hundreds of non-collagenous proteins [50]. However, because of their small volume fraction (altogether 25). This plateau can also be explained by the fact that whenever the number of cross links increases, their stiffness increases up to a threshold value at which the behaviour of all collagen cross-links becomes insensitive to the number [18]. In the macroscopic response of bone, cross-linking has been linked to improved mechanical properties [30] and prevents energy absorption by micro-damage formation and may accelerate brittle fracture [32, 33, 34]. In this study the same results are found for the nano-scale collagen microfibril. 3.2 Influence of mineral fraction volume The current section is devoted to studying the influence of the volume fraction of phases on the biomechanical behaviour of the microfibril. Three finite element models with different phase distributions were developed for this study. The idea was to vary the TC molecule diameter from 1.2 nm to 1.5 nm. For this study we chose a constant Young's modulus of the phases: Ec = 2.7GPa and Em = 114GPa.
Figure 8. Elastics properties of microfibril under varying fraction volume and number of cross-links: (a) Young’s modulus of microfibril as function of number of cross-links under varying Vm; (b) Poisson’s ratio of microfibril as function of number of cross-links under varying Vm.
(Fig.8) shows the influence of phase distribution. If the mineral volume fraction increases, the microfibril becomes more rigid and solid and its breaking strength and increased ability to dissipate this energy decreases. A microfibril which contains 67% mineral is one and a half times more rigid than a microfibril with 48% mineral. 4. Conclusion The fundamental understanding of the mechanical behaviour of bone is based on a thorough understanding of the behaviour of nanoscale structures, mineralized collagen microfibrils, which are the basic elements of the construction of bone [88, 89]. This study has focused on the relationship between the structure and the elastic properties of mineralized collagen microfibrils. A three-dimensional FE model was created to represent the main structural features of microfibrils, composed at the nanoscale of TC molecules connected by cross-links embedded in a matrix of crystalline and amorphous mineral. The finite element method was used to simulate the mechanical response of this model. Wee valuated how the geometrical structure of the characteristic parameters (volume fraction, number of cross-links) and material properties influence the elastic behaviour of mineralized collagen microfibrils. The modeling results are compared with published experimental data obtained by various techniques (X-ray diffraction, AFM, MEMS), and the results obtained by atomistic modeling. The numerical finite element model and experimental results are in good agreement, thus validating this approach and justifying the digital 3D model adopted. Acknowledgements
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