Finite element simulations

The mechanical problem, described by four equations (9) – (12) has to be solved to calculate the local strains within the callus. The geometrical relation between strain and spatial displacement of material is known by assuming infinitesimal strains (9).

Additionally, there were no sources of stresses within the investigated callus (10). At the geometric boundaries of the callus, either the displacement or the force were known (11). Linear elastic behaviour (12) of the tissues was assumed for this model.

Furthermore, isotropic behaviour was assumed which reduces the non-zero entries within the stiffness tensor C to two. The two entries are the elastic modulus E and Poisson ratio , which are known parameters. The displacement u, strain ε and stress

 remain as unknown parameters.

)

A commercial FE solver (ABAQUS v6.6, Hibbit Karlsson & Sorensen, Inc., RI, USA) was used in order to solve the mechanical equations (9) – (12). The details on the FE modelling can be found in the section Material and Methods of the according manuscript (Vetter et al. 2010b), which is found in the appendix. For each of the six healing stages, three models were implemented to study the influence of the mechanical heterogeneity of the hard callus and the softening of the cortex. The first model was the heterogeneous (“normal”) model with the heterogonous mechanical data. Additionally, two hypothetical cases for each stage were modelled as references: i) assuming a homogeneous hard callus and ii) assuming a homogeneous cortex. In the case of a homogeneous hard callus (as an example see Figure II.2), the elastic modulus of each hard callus element was set to the corresponding mean value of the hard callus (Figure II.3). In the homogeneous cortex, the elastic modulus of each element in the cortex was set to the corresponding cortical mean value. Other tissue types than hard callus develop during the healing too. Due to the lack of specific material data, the other tissues were described by standard, time-independent literature values for linear elastic behaviour (Table 3).

Table 3: Elastic modulus of the tissue material Emat, Poisson ratio ν and empirical exponent κ, see (1), of the different tissues within the callus.

[a) Hori and Lewis (1982), b) Isaksson et al. (2006), c) Schaffler and Burr (1988), d) Claes and Heigele (1999), e) Morgan et al. (2003), f) Manjubala et al. (2009)]

Material Emat  

Fibrous tissue, marrow 2 MPa ^(a,b) 0.167 ^(b) --

Cartilage 10 MPa ^(c) 0.167 ^(b) --

Hard callus Emat (t) ^(f) – (Figure I.4) 0.3 ^(d) 1.83 ^(e)

Cortex 20 GPa ^(d) 0.3 ^(d) 1.83 ^(e)

Results

The calculated stress and strain tensors were reduced to a scalar quantity for further investigations. One investigated invariant was the strain energy density (13) which is the product of strain and stress vector and describes the energy uptake of a material during deformation. This value is often used as failure criteria. The second investigated invariant was the maximum of absolute values of the principle strains (14). This value takes only the volume change of an element into account. The third invariant was the maximal shear strain (15). This value describes the shape deformation of an element. The results for the two investigated strain invariants were qualitatively the same.

As an example, the maximal shear strains of the three FE models at stage III are shown in Figure II.4 - a figure with all stages can be found in the attached second manuscript in the appendix (Vetter et al. 2010b, Figure 3). The left image shows the calculated maximal shear strains and indicates a strain in the fracture gap of around 22%. The middle image shows the difference of the strains calculated by the heterogeneous model and the strains calculated by the model assuming a homogeneous hard callus. This comparison shows that the mechanical heterogeneity of the hard callus influences strongly the strains within the cartilage region at this stage of healing. The cartilage close to the gap is higher deformed in the heterogeneous case (strains there are lower about 2.5% - 5%) while the outer part of the cartilage region is less deformed (strains there are higher about 2% - 4%).

From a mechanobiological viewpoint, this means that an ossification of the cartilage is more likely to occur at the outer fringe of the cartilage and this tendency is

cortex (Figure II.4, right). At stage III, the influence of the mechanical heterogeneity of the cortex is less pronounced than the influence of the hard callus heterogeneity (note the different orders of magnitude in Figure II.4).

Figure II.4: Spatial distributions of the maximal shear strain (left) and comparison between the heterogenous case and the two homogeneous cases as reference.

Left: maximal shear strains calculated for stage III during bone healing considering the heterogeneity of the stiffness of the hard callus and cortex. Middle: strain differences of the heterogeneous case and a hypothetical homogeneous hard callus. Right: strain differences of the heterogeneous case and a hypothetical homogeneous cortex. In the comparison between heterogeneous and homogeneous cases, areas with reddish colours indicate higher strains in the heterogeneous case while greenish colours indicate higher strains in the homogeneous case. Grey lines indicate the boundaries of the cortex and the hard callus. The region of cartilage is marked with a white dotted line. The orders of magnitude of the displayed values are provided on the right of each image. All values are given in dimensionless units of strain.

To transform to µstrain values, the given values have to be multiplied with 10⁶.

In the following chapters 6.3 to 6.6 additional data and interpretation of the data concerning the heterogeneity of the hard callus is presented, which is not part of the manuscript (Vetter et al. 2010b). First, the strain distributions calculated with the heterogeneous FE-models are investigated and in Chapter 6.6 the comparison with the homogeneous FE-models is summarized.