Elasto-damage mechanics of osteons: A bottom-up multiscale approach

Abstract

In this paper, a multiscale rationale is applied to develop a bottom-up modelling strategy for analysing the elasto-damage response of osteons, resulting in a first step towards a refined mechanical description of cortical bone tissue at the macroscale. Main structural features over multiple length scales are encompassed. A single osteon is described by considering a multi-layered arrangement of cylindrical lamellae and accounting for both lacunar micro-voids and thin interlamellar regions, these latter modelled as soft interfaces. A multi-step homogenization procedure has been conceived and numerically applied to describe the equivalent mechanical response of osteon constituents, upscaling dominant subscale mechanisms. A progressive stress-based damage approach has been implemented via a finite-element technique, allowing to describe interlaminar and/or intralaminar brittle failure modes. Proposed approach has been successfully validated by numerically reproducing available experimental tests of isolated osteons under different loading conditions. Present histologically-oriented multiscale model revealed to be sound and consistent, opening towards further insights about the influence on bone biomechanics of through-the-scales biophysical/biochemical alterations, possibly related to ageing or diseases.

Introduction

Bone is a complex heterogeneous material consisting of different organized hierarchical structures over multiple length scales (Cowin et al., 2001). All the bone constituents are combined in a remarkably effective way so as to form a properly-organized and nature-optimized material with superior mechanical properties combining high stiffness, strength and toughness (Hamed and Jasiuk, 2013, Barthelat and Rabiei, 2011). At the organ level, two distinct types of bone tissues, characterized by different mechanical properties, can be distinguished, namely the cortical (or compact) bone, dense and stiff, and the trabecular (or spongy) bone, porous and tough (Rho et al., 1998, Reznikov et al., 2014). Stiffness and loading-bearing capacity of bone structures are mainly associated to cortical tissue. At the microscale, cortical bone is constituted by hollow quasi-cylindrical systems, called osteons or Haversian systems (Rho et al., 1998, Weiner and Traub, 1992, Ural and Vashishth, 2014), running approximately parallel to the main direction of the bone structure. Osteons have an average diameter in the range of $200÷250$ µm and are characterized by a central canal (called Haversian canal) having diameter in the order of 40 µm (see Fig. 1). At an intermediate subscale between the micro- and the nano-scale, osteons are made up of several coaxial pseudo-cylindrical layers, about 3 µm in thickness and wrapping around the Haversian canal, called lamellae (Weiner et al., 1999, Reznikov et al., 2014). Each lamella exhibits a defined pattern of sub-microstructural units (the so-called sublamellae), characterized by an ordered arrangement of mineralized collagen fibrils (MCF) (Giraud-Guille, 1988, Weiner et al., 1999). In turn, at the nanoscale, MCFs are constituted by Type-I collagen fibrils strengthened by crystals of a mineral phase mainly constituted by hydroxyapatite (HA). Minor quantities of water, non-collagenous organic proteins (NCPs) and impurities are also present (Weiner and Traub, 1992, Landis et al., 1993, Cowin et al., 2001, Olszta et al., 2007).

Bone material composition and structural organization at each length scale highly affect its mechanical behaviour and failure mechanisms at the macroscale level. In this framework, it is well-documented that bone is characterized by an anisotropic constitutive response, and that osteons have a key role for the propagation of cracks at organ level (Zioupos and Currey, 1998, Parsamian and Norman, 2001, Vashishth, 2007). As such, a proper understanding of onset and evolution of bone fracture mechanisms at the macroscale requires a suitable characterization of osteon mechanics and related damage features (Sabet et al., 2016, Ural and Vashishth, 2014).

To this aim, a consistent multiscale rationale, able to account for the histological arrangement of tissue constituents through multiple length scales, revealed to be highly effective for accurately investigating the mechanical response of biological tissues (Maceri et al., 2010, Marino and Vairo, 2014b, Marino and Vairo, 2014a, Bianchi et al., 2017, Pandolfi et al., 2019, Concha and Hurtado, 2020, Morin et al., 2021, Kwon and Clumpner, 2018). Since a structured multiscale strategy (differently from phenomenological approaches) is defined via model parameters with a clear physical meaning, it also allows to describe the influence of possible physiopathological alterations related to ageing or diseases by straightforwardly setting such parameters through consistent histological evidence (Maceri et al., 2013, Marino et al., 2017, Gierig et al., 2021, Bianchi et al., 2016, Gizzi et al., 2021).

Finite Element (FE) modelling represents an effective tool for investigating bone biomechanics in health and disease at both the macro- and micro-scale (Falcinelli et al., 2019, Pisano and Fuschi, 2021, Gaziano et al., 2022, Hogan, 1992, Crolet et al., 1993, Prendergast and Huiskes, 1996, Giner et al., 2014). However, at present only few FE-based models have been developed with the aim of describing the micromechanical behaviour of osteonal elements. Hogan (1992) and Crolet et al. (1993) were among the first researchers proposing estimates of compact bone equivalent elastic properties through FE-based micromechanical approaches. In particular, Hogan (1992) investigated the dependence of the elastic moduli on the material properties of osteon substructures, finding a reasonable agreement with available experimental data. Crolet et al. (1993) conducted a similar study by accounting for collagen/HA distribution within osteonic lamellae, in the framework of a multiscale approach. Their results, furnished however only in terms of bone homogenized elastic properties, showed good agreement with the experimental data, corroborating the hypothesis of periodicity of collagen/HA distribution within each single osteonic lamella.

A more detailed numerical model of a single osteon was developed by Prendergast and Huiskes (1996), who investigated whether damage generates strains which may trigger bone remodelling processes. The authors provided important and useful evidence that local changes in the strain field are strictly related to the presence of microdamage and lacunar voids in Haversian bone microstructure. Nonetheless, the existence of damage was therein assumed a priori, not querying where and when the damage develops and evolves. In the studies of Guo et al. and Najafi et al. (2007), linearly-elastic fracture mechanics theory was employed to assess the role of microcracks at a single-osteon level, as well as their influence on the mechanical behaviour of cortical bone. However, their models revealed several limitations, since microstructural features of osteons were disregarded, and the assumption of homogeneous linearly-elastic materials was made. Hamed and Jasiuk (2013) numerically investigated the bone strength at multiple length scales by modelling damage onset and propagation through the cohesive element technique. The authors provided some evidence contributing towards the understanding of bone sub-macroscale failure mechanisms. Nonetheless, although constitutive microscale arrangement was considered to a certain extent, fracture mechanisms were driven by a-priori-defined initiation sites. Moreover, numerical models at different scales were decoupled each other, in the sense that mechanical features of bone structures at a given length scale were obtained not accounting for tissue features and structural organization at lower scales. More recently, Giner et al. (2014) developed a multiscale numerical model of a single osteon, investigating microdamage onset and evolution. Therein-proposed results, associated to a radial compressive loading scenario, were successfully compared with experimental test carried out by Ascenzi et al. (1973). However, the structural model of Giner et al. (2014), and all the other previously-described ones, considered a planar section of the actual three dimensional osteonal structure, thus allowing only for an in-plane strain and stress analysis as a simplifying assumption.

To the best of the authors’ knowledge, a detailed three-dimensional multiscale model accounting for coupled through-the-scales mechanical responses up to the failure is still lacking. Accordingly, present paper aims to propose a novel histological-based description of single osteons by accounting for three-dimensional multiscale structural arrangement, based on a bottom-up (from nano- to micro-scale) formulation. In depth, a refined constitutive description, also encompassing a multistep homogenization procedure, has been conceived and employed to identify dominant mechanical response of osteonal constituents at different scales. A progressive damage formulation allowing to describe interlaminar and/or intralaminar brittle failure modes, has been conceived and applied to model osteonal elasto-damage response and failure mechanisms. Proposed model has been implemented through a FE technique and it has been successfully validated by replicating available experimental tests carried out under different loading conditions (Ascenzi et al., 1990, Ascenzi and Bonucci, 1967, Ascenzi et al., 1994). Obtained results highlight soundness and accuracy of the proposed approach, that can therefore be considered as useful for understanding how microdamage processes affect the overall mechanical behaviour of cortical bone, also accounting for disease- or ageing-induced structural and morphological alterations through different length scales. In such a way, enhanced novel multiscale strategies could be developed up to the macroscale, furnishing suitable indications for the proper assessment of bone fracture risk, and thereby allowing for possible clinical applications related to effective etiology identification, diagnosis and treatment of bone-related pathologies.