Multiscale modeling of elastic properties of cortical bone

We model cortical bone as a composite material with hierarchical structure. At a nanostructural level, bone is composed of cross-linked collagen molecules, containing water and non-collagenous proteins in their gaps, reinforced with hydroxyapatite-like nanocrystals. Such a nanocomposite structure represents a mineralized collagen fibril, which serves as a primary building block of bone. At a sub-microstructural level (few microns), the mineralized collagen fibrils are embedded in an extrafibrillar hydroxyapatite matrix to form a single lamella, which also contains the lacunar cavities. At a microstructural level (hundreds of microns) one can distinguish two lamellar structures in the mature cortical bone: osteons, made of concentric layers of lamellae surrounding long hollow Haversian canals, and interstitial lamellae made of remnants of old osteons. At a mesostructural level (several millimeters), the cortical bone is represented by a random collection of osteons and resorption cavities in the interstitial lamellae. A macrostructural level is the whole bone level containing both the cortical (compact) and trabecular (spongy) bone types. In this paper, we predict analytically the effective elastic constants of cortical bone by modeling its elastic response at these different scales, spanning from the nanostructural to mesostructural levels, using micromechanics methods and composite materials laminate theories. The results obtained at a lower scale serve as inputs for the modeling at a higher scale. The predictions are in good agreement with the experimental data reported in literature.