## A numerical assessment of architectural parameters for anisotropic behavior in idealised trabecular structures

Master Thesis

2018

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University of Cape Town

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Bones macroscopically consist of two major constituents; namely cortical and trabecular (also known as cancellous) bone. Cortical bone is the hard and dense outer layer of bone, which holds majority of the load bearing capacity. Trabecular bone is the porous internal bone, which distributes loads at joints by allowing for a larger bearing surface and acts as an energy absorber. Trabecular bone has a complex, heterogeneous, anisotropic open cell lattice structure with a large variation in mechanical properties across anatomic site, species, sex, age, normal loading direction and disease state. A common attempt to account for this variation is to correlate the structure of the trabecular bone sample to the mechanical response, which requires a means of quantifying the structure. Microstructural indices such as bone volume vs. total volume (BV/TV), trabecular thickness (Tb.Th), trabecular separation (Tb.Sp), structural modal index (SMI) and mean intercept length (MIL) have been widely used to find correlations between structure and properties. Early studies only considered densitometric indices, which accounted for much of the variation however cross study correlations did not agree, leading to an interest in capturing non-scalar valued indices to account for features such as the anisotropy of the bone. The structural anisotropy varies from fully equiaxed to highly directional based on where the trabecular bone is located and what the function would be. The mean intercept length has been proposed as a measure of the structural anisotropy, specifically the interfacial anisotropy of the sample, which is commonly used to account for the mechanical anisotropy. This research falls within a longer term goal of investigating and understanding the mechanical anisotropy of trabecular bone. To that end, the anisotropy of regular lattice structures was investigated, with the particular goal that the investigated lattices were simple analogues for the more complex structures seen in trabecular bone. A framework for assessing the structure-property relations of trabecular bone is created, with focus on anisotropy. The mechanical anisotropy of idealised trabecular structures is quantified using well known microstructural indices, which are compared to the numerically determined mechanical response. The modelling methodology initially investigated 2D lattices that have very well known responses, such that the modelled approach could be verified. Three 2D lattices were used to do this, with the aim that the 3D lattices would be their analogues. Specifically a 2D square, hexagonal and triangular lattice were investigated. The square lattice is highly anisotropic as is the cubic lattice. The hexagonal lattice is isotropic with a large constraint effect as is the Kelvin cell, and the triangular lattice is isotropic with a small constraint effect. The octet-truss was the closest analogue to the triangular lattice, having a small constraint effect and being less anisotropic than the cubic lattice. The three 3D lattices were chosen to represent highly directional trabecular bone (using a cubic lattice) and more equiaxed trabecular bone, with the fully isotropic Kelvin cell lattice (also known as a tetrakaidecahedron) and the octet-truss lattice which has a lower degree of anisotropy than the cubic. Two confinement arrangements were also investigated as analogues for the trabecular bone at the free surface and at the cortical surface. To assess the mean intercept length analysis as a measure of mechanical anisotropy, this research performed the analysis on three 3D periodic lattice structures and compared the results to mechanical properties which were numerically determined using finite element analysis. The mean intercept analysis was performed by generating 3D images for the lattices, similar to the output of (mu)CT images, using a combination of open-source software and custom code, and performing the analysis in BoneJ, an open-source software package. The mechanical response was determined using two methods, namely discrete and continuum modelling approaches. The discrete approach characterised the lattice with each strut modelled as a Timoshenko beam element solved in LS-DYNA. To capture the anisotropy, the lattice had to be loaded at arbitrary angles, which was achieved by a rotating the whole lattice and cropping it to a specified test region using custom code. The continuum modelling approach used a homogenisation approach by treating the lattice as a solid material with effective properties, this was solved in a custom implicit solver written in MATLAB using solid elements. The anisotropy was modelled by transforming the elasticity tensor to arbitrary coordinate systems to load the model in arbitrary directions. The discrete modelling approach suffered from high computational costs and difficulty in removing the boundary effects, all of which would be worsened for models of real trabecular bone. However the discrete approach did accurately captured the mechanical behaviour of the lattices tested. The continuum approach accurately captured some of the responses but failed to capture all behaviour caused by confinement. The continuum model could not capture the switch in predominant deformation mode of the 2D hexagonal lattice caused by lateral confinement, and failed to accurately capture the symmetry of the highly anisotropic 3D cubic lattice. The mean intercept length analysis failed to capture the anisotropic response of simple periodic lattices, showing no significant difference between the octet-truss and cubic lattices, despite them having a very large difference in mechanical anisotropy. It also showed that the Kelvin cell lattice had the highest degree of geometric anisotropy, which is compared to having the lowest mechanical anisotropy being the only fully isotropic 3D lattice investigated. The mechanical investigation showed that the lateral confinement has a large effect, significantly scaling the response of isotropic lattices whilst distinctly changing the anisotropic behaviour of the cubic and octet-truss lattice. The mean intercept length analysis cannot capture the mechanical confinement effect from geometry alone, and thus fails to capture the mechanical response due to confinement Overall, the continuum modelling approach showed difficulty in capturing the confinement effect in all lattices and thus a more robust method is required. The mean intercept analysis proved unsuccessful in capturing the mechanical response of three periodic idealised trabecular structures. A new microstructural index that can capture the mechanical anisotropy is required, with the ability to consider the effects of confinement on the structure.

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Moore, K. 2018. A numerical assessment of architectural parameters for anisotropic behavior in idealised trabecular structures. University of Cape Town.