5.1.5.1 Using average model to find optimised VOI
To obtain quantification results using the average model, it must first be correctly aligned with the tibiae being measured. An example of the registration of the model to experimental data is shown in Figure 5.5. The width at the top and bottom of the bone, cortical wall thickness and orientation are taken into account when registering, using a correlative registration method (Weber and Ivanovic, 1994).
The registered model acts as a template to measure bone parameters for quantitative comparison of implant efficacy. An example of this is shown in Figure 5.6a where the model is registered to a tibia that has a defect partially healed by a 70S30C porous scaffold. In each 2D slice, the area bounded by the model, termed the ‘model region’ ( QRKIS), is separated from the area bounded inside QRKIS, which is defined as the ‘marrow cavity region’ ( QTLLRH) (Figure 5.6b).
Within each region, the proportion of bone, termed QRKIS (Figure 5.6c) and QTLLRH (Figure 5.6d) and implant, QRKIS (Figure 5.6e) and QTLLRH (Figure 5.6f) is measured i.e. the subscript b and i are used to denote bone and implant material, respectively.
Figure 5.5 The reconstructed average contra4lateral tibia model can be registered to experimental data (a) which takes into account the relative position of the defect in the tibia. Its separate constituent parts are shown for (b) the experimental data and (c) the average model. (d) and (e) show the fit of the model to experimental data in 2D. Scalebars are 1000 Km and 500 Km
Figure 5.6 The average model is registered to experimental datasets such as in (a). Once registered, the bone ( QRKIS) and marrow regions ( QTLLRH) of interest are identified as shown in
(b). The proportion of bone in QRKIS and QTLLRH is termed QRKIS and QTLLRH respectively and highlighted in blue (c) and red (d) respectively. (e) and (f) are the implant material defined in
Next a distinct ‘defect volume of interest’ was created to quantify the missing bone due to the defect. The segmented bone is shown in Figure 5.7a, showing the defect site. By registering the average model (in grey) to the bone (Figure 5.7b), the defect area can be approximated by the subtraction of the two volumes from each other (Figure 5.7c). From this, the volume representing the remaining defect area was isolated (green in Figure 5.7d). A cylinder of the original size of defect (in this case a 3 mm diameter, 1.8 mm height) was registered to this isolated volume fragment via correlation and orientated by principle component analysis (Yue et al., 2010) as shown in Figure 5.7e. The cylinder and bone, defining the original defect position (Figure 5.7f) can then be used to identify the defect VOI by the intersection area of the cylinder and the average model (Figure 5.7g). The volume fraction of bone ingrowth, , was measured using a marching cubes algorithm shown in blue in Figure 5.7h (Hege et al., 1997).
The percentage of bone ingrowth was calculated as the volume of bone ( RUI) divided by the volume of the VOI ( ) less the volume of implant ( QNSTUJ)
= − RUI
QNSTUJ
(5.1)
In some cases, bone may grow around an implant rather than through it, resulting in the need for a method to quantify curvature. A new parameter for quantifying the curvature of new bone at the defect site was developed using a sphere fitting technique as seen in Figure 5.8a and b. Similar to the quantification of bone ingrowth, the average model is used to identify a reference sub0volume to calculate bone curvature.
Figure 5.7 Summary of the procedure to identify correctly oriented defect VOI in 3D using the contra4lateral model as mask represented in 3D and 2D images concurrently. For any segmented
experimental data4set of bone with an unknown original defect area (a; bone – yellow), the contra4lateral tibia model in grey is overlaid as a template (b). (c) shows the subtraction result
between the average model and bone from which the defect region is isolated (in green (d)). A cylinder (pink) is registered to this region (e) which can define the region in bone that determines the orientation of the original defect (f). The defect VOI is shown in (g). The bone after the in vivo experiment that has grown in is classified as ‘bone ingrowth’ shown in light blue
When there is a contiguous segment of bone across the defect area, the sub0volume, Vcap, can be
modelled to a spherical cap of height, h.
VTN = 13 7ℎ/"3 − ℎ& (5.2)
where r is the radius of the sphere representing the spherical cap (Figure 5.8c and d). When Eqn. (5.2) is rearranged, this provides us the radius of the fitting sphere that correctly defines the defect curvature, where is found by:
= 7ℎVTN/+ ℎ3 (5.3)
The overall definition of curvature not only depends on the radius of curvature but the distance between the centroid of the created sphere and the centre of the defect VOI. This distance, termed , is the Euclidean distance measured between the centroid of the sphere to the centre of the defect VOI (Figure 5.8e and f). The ratio Z , provides a measure of the severity of the curvature
described. The curvature described by the sphere is minimised as the value of Z approaches 1.
Figure 5.8 An equivalent spherical descriptor of the defect area is used to produce parameters of curvature, C (from radius, ) and depth of bone delineation, by modelling the defect area as
a spherical cap of height ‘h’. The arrow indicates the length of with respect to the sphere radius. (Scale = 1000 Km)
5.1.5.2 Using conventional µCT VOI
Figure 5.9 Conventional quantification method in 3D by using convex hull to find the VOI of the implant volume (a) the bone (in yellow) bounded by the convex hull of implant (b) shown from a top4down view of the volume bounded by the convex hull (blue outline) 4 adapted from (Midha et
al., 2013a).
The implant and bone phase are identified using the method as described in section 5.1.4. In order to establish the VOI, a convex hull algorithm is used to find the volume that fully encloses the implant as shown in blue in Figure 5.9.
Bone ingrowth, V, was measured in the same method as in section 5.1.5.1 (bone volume/VOI volume by marching cubes algorithm). The bone implant contact area [ \V was also measured. This was done by first applying an 180direction 3D dilation algorithm on the implant phase. The cross product of this and bone, identifies the region around the implant that is in direct contact with bone. The area of this contact region was measured using marching cubes. The superscript ‘c’ is given to indicate that these results are from using a ‘conventional’ method and is only for comparison to the results obtained by the custom VOI method.