This abstract is focused on the use of the fractal dimension as a global descriptor of the white matter. The fractal nature of diffusion MRI images has been studied before^1,2. However, to the best of our knowledge, fractal descriptors have not been employed as markers of disease. Simple fractal measures, such as the fractal dimension as estimated with the box counting technique, can provide simple yet powerful procedure to obtain information about the complexity of the white matter architecture.

Four groups of subjects from an Alzheimer study were analyzed, containing a healthy control group (group A: N = 17, age= 74.5 ± 3.5y), patients with mild cog- nitive impairment (group B: N = 13, age= 76.3 ± 1.1y), patients with mild Alzheimer’s disease (group C: N = 19, age= 76.1 ± 2.7y) and patients with moderate Alzheimer’s disease (group D: N = 7, age= 76.6±1.4y). Differences in age were not significant between the cohorts. Patients were diagnosed according to NINCDS-ADRDA Alzheimer's Criteria.

Diffusion weighted images were acquired in a GE Signa 1.5 T MRI unit at QDiagnóstica, Valladolid, Spain. The parameters of the acquisition protocol were the following: 25 gradient directions, one baseline volume, b = 1000 s/mm2, 1.015 × 1.015 × 3 mm3 of voxel size,TR = 13,000 ms, TE = 85.5 ms, 256 × 256 matrix, NEX = 2 and 39 slices covering the entire brain.

After preprocessing, including the removal of non-brain structures such as the skull, diffusion tensors were estimated using a least squares method [27]. From the tensor volume, Fractional Anisotropy (FA), Mean Diffusivity (MD) and Radial Diffusivity (RD) maps were computed. The FA maps were afterwards slightly eroded in order to remove possible outliers. From them, binary maps can be obtained simply by thresholding the FA maps.

The Hausdoff fractal dimension can describe how much a certain pattern changes when the scale at which it is measured also changes. While the most simple objects can have an integer Hausdoff dimension (1 for a line, 2 for a square, 3 for a cube), more complex objects can have non-integer Hausdorff dimensions. Among the many techniques for the calculation of fractal properties, the box counting method, which approximates the Hausdorff fractal dimension, is the most commonly employed. Using this method, the (3D) space is partitioned in equal boxes of size r. Then, N(r) is the number of boxes of size r that contain at least a non-zero voxel. The estimation of the fractal dimension, FD, is performed by computing the slope of N(r), when plotted in a double logarithmic scale.

The notion of fractal dimension can also be extended to gray-level images. In this case, N(r) is the mean value inside each box, instead of the number of non-zero voxels.

FD values for the binarized FA maps (using a threshold of FA=0.3) and for the gray-level FA maps were computed for all subjects, together with mean values over the white matter of the FA, MD and RD maps. A one-way Anova test was performed to investigate whether the four groups belong to the same distribution. When they did not, bilateral t-tests were applied to check for pairwise differences.

Figure 2 collects the p-values corresponding to the Anova and t-tests carried out. Significant differences were found for the FD over the binarized (FD 0.3) and gray-level FA (FA gray) maps, while no significant differences were found using the mean FA values. There is extensive literature indicating that MD and RD are more powerful descriptors of the changes within the white matter in Alzheimer’s disesase and, accordingly, significant differences were also found for the mean values of these maps.

With regard to the pairwise comparisons, the FD 0.3 and, to a lesser extent, the FD gray showed a considerable capacity to differenciate subjects at different stages of Alzheimer’s disease. Notably, both FD measures found significant differences between groups C and D, while mean MD and mean RD discovered sifnificant differences between groups B and C (RD was close to statistical significance). Although further investigation is needed, this is a possible indication of different mechanisms of neurodegeneration taking place at different stages of Alzheimer’s disease and thus affecting different diffusion properties as measured by diffusion MRI.

Fractal dimension of FA maps is a simple yet powerful method for providing global descriptors of the white matter architecture. In the case of a group study on Alzheimer’s disease, FD was able to reveal significant differences between subjects at different stages of the disease.