The effects of projection geometry and trabecular texture on estimated fractal dimensions in two alveolar bone models

To study the effects of trabecular texture and projection geometry on estimates of fractal dimensions from simulated alveolar bone.

Two models, one scale-variant and the other relatively scale-invariant, were used. Projections of 2-D slices through both the models were calculated, a uniform series of projections computed for the angular orientations between (0 degree and 180 degrees). Results of the 128 stacked projections are the projection of the 3-D object from different angles. Power spectra were produced by calculating the squared magnitude from the data and plotted as the logarithm of the power against the logarithm of the frequency, to give the fractal dimension.

In the scale-variant model, the alveolar trabecular texture produced with a relatively thicker cylindrical diameter yielded a smaller fractal dimension (t = 8.44, P = 0.00). Linear regression analysis showed that the correlation between projection angle and the resulting fractal dimension was low for both the thicker cylinder (r = 0.02, P = 0.82), and for thinner cylinder (r = 0.02, P = 0.84). The scale-invariant model, also yielded a relatively smaller estimated fractal dimension (t = 7.23, P = 0.00). However, the correlation between projection angle and the resulting estimate of the fractal dimension was found to be low for both trabecular configurations (r = 0.10, P = 0.29 and r = 0.07, P = 0.45 respectively).

Fractal dimensions increased when the diameter of the simulated cylindrical trabeculae decreased irrespective of the differences in self-similarity. The degree to which they varied with projection angle was relatively less for the scale-invariant (self-similar) model. However, fractal dimensions changed significantly with projection geometry.