Aspectos Socioculturales

This chapter presented a brief overview of the literature regarding equine metacarpal bone fractures, possible methods of data acquisition for related finite element models, techniques and approaches used in numerical modelling of bone adaptation and bone fractures. The observations made in this chapter are an essential prerequisite, which enables the choice of numerical methods used in the developed framework described in the following chapters.

Chapter 3

Bone imaging and material mapping

In this chapter, two methods for efficient approximation of bone CT scan data onto finite

element meshes are presented. The first approach is a simple but robust L2-projection. The

method is subsequently used for determining density gradients from CT-scans of 18 horses from three different cohorts. The second presented method is Moving Weighted Least Squares approximation. It provides the same capability of approximating the noisy CT-scan data onto

the smooth field as L2-projection and also allows for the computation of derivatives. This

feature will be later exploited in configurational force driven crack propagation framework discussed in Chapter 6. Additionally, a Partial Volume Artifacts correction technique is proposed for improved accuracy. The performance of both methods is tested on simple examples.

3.1

Introduction

The first step when building finite element models for analysing bones is to generate accurate geometry from three-dimensional imaging techniques like computed tomography (CT) or magnetic resonance imaging (MRI). Many studies in the past have established that experimentally validated bone FEM models require both high resolution in meshing and heterogeneous material mapping of bone density (Pakdel et al., 2016). The most widely adopted and validated method for patient-specific finite element modelling of bone is BONEMAT program (Viceconti et al., 2004). The algorithm implemented therein interpolates the CT-scan data (radiopacity) of the voxels mapped from the CT image to the volume of the FEM mesh, assigning the density and elastic modulus to each individual element. This results in a constant density within elements leading to a very noisy distribution of material properties with unrealistically sharp gradients. However, as it will be shown later in Chapter 6

for efficient and accurate simulation of crack propagation, the heterogeneities within the body should be smooth. Moreover, smoothly varying density allows for calculations of gradients which can help to identify potential points of crack initiation.

It is hypothesised that changes in subchondral bone mineral density, induced by the repetitive cyclical loading in racehorses, are increasing the propensity of fatal injuries such as lateral condyle fracture of 3rd metacarpal bones. Many researchers over the years reported significantly higher bone mineral density at the distal articulating surface in bones of trained horses or with an already fractured limb (Loughridge et al., 2017; Riggs et al., 1999; Whitton et al., 2010). Some of them observed the presence of associated high-density gradient at the parasagittal grooves. However, such gradients have never been quantified. They may be predisposing the site of the fracture, since rapid changes in mechanical properties within the bone may lead to concentration of shear forces, causing the localisation of micro-cracks (Riggs and Boyde, 1999). In this section, numerical tools based on Finite Element Method for mapping CT data and subsequently determining density gradients from CT scans are presented. Developed techniques are utilised to characterise gradients at the sagittal grooves of the third metacarpal bone in racehorses with and without lateral condylar fractures.

In FEM simulations, each finite element in the mesh is assigned with material properties. In the case of analysing bones, it is beneficial to utilise CT scan data that was used to generate the geometry. Each voxel (3D pixel) of a CT scan contains information about measured radiopacity in so-called Hounsfield units (HU), which are directly related to the average stiffness of the bone part enclosed by the voxels. Depending on this relation HU values of a CT image can be used for density mapping and for determining elastic properties based on density-elasticity relationships. It was shown that the apparent density can be related to the mechanical properties of the bones using power-laws.

Subsequently, such relationships can be used to correlate bone density to Young’s modulus. Poisson’s ratio is usually assumed to be constant ν = 0.3 in hard tissues. It is not clear which power-law relationship is the most accurate. Many proposed relationships in the literature often significantly differ from each other (Helgason et al., 2008a). Some of them have already been used in FEM, validated by experiments and resulted in satisfactory agreement (Eberle et al., 2013). However, there was only one relation found, that considered equine MC3 (Les et al., 1994). Over three hundred bone specimens from the MC3 were harvested in that

study. Subsequently, CT scans, along with a Cann-Genant K2HPO4calibration phantom were

obtained (Cann and Genant, 1980). The specimens were compressed until failure to estimate elastic modulus, which resulted in the following empiric relation.

E = 15100 · ρQCT2.25 [MPa] (3.1)

Where ρQCT is dipotassium phosphate K2HPO4equivalent density. Equation 3.1 will be used

throughout this thesis to translate density data into elastic modulus.

Another important issue regarding the assignment of density and mechanical properties is whether one single value should be assigned to every element or rather vary within the same

Figure 3.1: Example data set: Finite Element mesh inside voxels lattice from MetaIO file. element. A comparison between both approaches showed that the single-value approach results in slightly more accurate simulation results as compared to the modified method (Helgason et al., 2008a). Surprisingly, regardless of the discontinuous field of densities (and further elastic modulus) and a limited number of different material properties, numerical outcomes were in better agreement with experiments for the less accurate approach. It shows that methods employing non-constant distribution of material properties within element still need to be improved. As was previously shown, the choice of the mapping algorithm might be critical (Poelert et al., 2013) for estimating stresses. Therefore, in this study new approaches

for this application are proposed: L2-projection and MWLS approximation.