Multiscale modelling of trabecular bone: from micro to macroscale

Abstract

Trabecular bone has a complex and porous microstructure. This study develops approaches to determine the mechanical behaviour of this material at the macroscopic level through the use of homogenisation-based multiscale methods using micro-finite element simulations. In homogenisation-based finite element methods, a simulation involving a representative volume element of the microstructure of the considered material is performed with a specific set of boundary conditions. The macroscopic stresses and strains are retrieved as averaged quantities defined over this domain. Most of the homogenisation-based work on trabecular bone has been performed to study its macroscopic elastic regime, and therefore define its constant macroscopic stiffness tensor. The rod and plate-shaped microstructure of trabecular bone can be precisely identified with advanced scanning tools, such as micro-computed tomography devices. Taking into account the size requirements to achieve a certain independence of boundary conditions for trabecular bone in a homogenisation-based multiscale setting, the resulting stack of images can have around ten million solid voxels after binarisation. Although a completely linear finite element simulation with such a large system may be feasible with commercial packages (with the proper time and memory requirements), it is not possible to perform a nonlinear simulation for such a mesh in a reasonable time frame, and the amount of required memory may not be available. A highly scalable parallel driver program which solves finite strain elastoplastic systems was developed within the framework of the existing parallel code ParaFEM. This code was used throughout this study to evaluate the yield and post-yield properties of trabecular bone. It was run on cutting edge high performance computing platforms (BlueGene/Q at the Hartree Centre, Science and Technology Facilities Council; and ARCHER, UK National Supercomputing Service, at Edinburgh Parallel Computing Centre). Micro-finite element simulations require definition of properties at the microscopic scale and it is unclear how these properties affect the macroscopic response. This study examines the effect of compressive hydrostatic yield at the microscopic scale on the macroscopic behaviour. Two different microscopic yield criteria, one permitting yielding at compressive hydrostatic stresses and the other not, were considered. A large number of load cases were examined. It was found that these two microscopic yield criteria only influence macroscopic yield behaviour in load scenarios which are compression-dominated; for other load cases, macroscopic response is insensitive to the choice of the microscopic yield criterion, provided it has an appropriate strength asymmetry. Also, in compression-dominated load cases, high density bone is much more sensitive as it is more like a continuum, resulting in the microscopic properties being more directly upscaled. Only a few previous studies have employed homogenisation to evaluate the macroscopic yield criterion of trabecular bone. However, they either used a simplified microscopic yield surface or examined only a small number of load cases. A thorough multiaxial evaluation of the macroscopic yield surface was performed by applying a wide range of loading scenarios (160 load cases) on trabecular bone samples. Closed-form yield surfaces with different symmetries (isotropy, orthotropy and full anisotropy) were fitted to the numerically obtained macroscopic yield points in strain space, and the fitting errors were evaluated in detail for different subsets of load cases. Although orthotropy and full anisotropy showed the smallest fitting errors, they were not significantly superior to the isotropic fit. Thus, isotropy in strain space presents itself as the most suitable option due to the simplicity of its implementation. The study showed that fitting errors do depend on the chosen set of load cases and that shear load cases are extremely important as it was found that even for these highly aligned samples, trabecular bone presents some degree of shear asymmetry, i.e. different strength in clockwise and counter-clockwise shear directions. There have been no previous attempts to evaluate the post-yield behaviour of trabecular bone through homogenisation-based studies on detailed micro-finite element trabecular bone meshes. A damage and plasticity constitutive law for the microscale based on existing data in the literature was considered. A homogenisation-based multiscale approach was used to evaluate the hardening and stiffness reduction at the macroscale when uniaxial load scenarios are applied to trabecular bone samples, for a small range of plastic strain Euclidean norms. Results show that damage progression at the macroscale for trabecular bone is not isotropic, which is contrary to what has been assumed previously, and that both the evolution of the yield surface and damage are different for tension, compression and shear. Nonetheless, they can be correlated with plastic strain Euclidean norms by using linear relationships. It was also observed that macroscopic damage in a specific load case affects differently the on-axis orthotropic stiffness and the off-axis orthotropic stiffness components. The findings of this study will permit the use of a more rigorous definition of the post-elastic macroscopic behaviour of trabecular bone in finite element settings.