## 3 Results
### Real Brain Image Data Studies
#### 3.2.3 Clinical Applications of Segmentations with sFCM  
In clinical studies, DTI indices, such as Fractional anisotropy (FA), mean diffusivity (MD), and Radial diffusivity (RD), are used in the comparison of healthy and non-healthy brain images (often manually segmented). The efficacy of the new methods presented in the paper is demonstrated by using one of our automatically created segmentations (via sFCM with root Euclidean) and demonstrating that these indices can distinguish between healthy and SCA2 subjects, and they are sensitive to ageing effects. This reaffirms the results in the literature [12; 13; 15], but with the extra knowledge that the use of sFCM will have reduced the impact of noise. In addition to this, it can be seen that the determinant (DET) of the tensors, which is easy to compute, can also be used to distinguish between healthy and SCA2 subjects and is sensitive to ageing effects. That is, DET is shown to be a viable DTI index. First, recall the definitions of the DTI indices, which are functions of the eigenvalues of the diffusion tensors. FA measures the deviations from isotropic diffusion of water inside a voxel in the brain, and it is a fraction with FA equal to 1 for diffusion that is highly anisotropic (i.e., water diffuses in one direction) and FA equal to 0 for isotropic diffusion (i.e., water diffuses in all directions). Let \(\lambda_{1},\lambda_{2}\) and \(\lambda_{3}\) be the eigenvalues of diffusion tensor \(D\) and assume that \(\lambda_{1}\) is the largest eigenvalue. Then, FA [5] can be calculated as follows:  
\[\text{FA}=\sqrt{\frac{(\lambda_{1}-\lambda_{2})^{2}+(\lambda_{2}-\lambda_{3}) ^{2}+(\lambda_{3}-\lambda_{1})^{2}}{2(\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda _{3}^{2})}}. \tag{17}\]  
MD measures the average water diffusivity in a voxel in the brain. It is calculated as follows [5]:  
\[\text{MD}=\frac{\lambda_{1}+\lambda_{2}+\lambda_{3}}{3}. \tag{18}\]  
Radial diffusivity measures the perpendicular diffusion to the main diffusion of water. It is calculated as follows [5]:  
\[\text{RD}=\frac{\lambda_{2}+\lambda_{3}}{2}. \tag{19}\]  
These DTI indices, together with the DET are computed. The Mann-Whitney test is used to test the significant difference in FA, MD, RD, and DET between healthy and SCA2 subjects at the significance level 0.05. The results of FA, MD, RD, and DET are all significant at both baseline and post baseline. In detail:  
* FA values in SCA2 subjects are significantly lower than in healthy subjects (\(p\)-value at baseline = 0.0005, \(p\)-value at post baseline = 0.0004). * MD values in SCA2 subjects are significantly increased as compared to healthy subjects (\(p\)-value at baseline = 0.018, \(p\)-value at post baseline = 0.035). * RD values in SCA2 subjects are significantly increased as compared to healthy subjects (\(p\)-value at baseline = 0.0005, \(p\)-value at post baseline = 0.0007). * DET values in SCA2 subjects are significantly larger than in healthy subjects (\(p\)-value at baseline = 0.001, \(p\)-value at post baseline = 0.002). These results show that FA, MD, RD, and DET distinguish well between healthy and SCA2 subjects. The rate of change can be calculated as follows:  
\[\text{Rate of change}=\frac{\text{data}_{\text{baseline}}-\text{data}_{\text{ post baseline}}}{\text{number of days in between}} \tag{20}\]  
The rates of change of FA, MD, RD, and DET in SCA2 subjects are not significantly different from the rates of change in healthy subjects. The Wilcoxon Signed-Rank test is used to test the significant difference in FA, MD, RD, and DET at baseline and post baseline. The results of FA, RD, and DET were all significant. However, MD values were not significantly different at baseline and post baseline. The details are as follows:* FA values at post baseline were significantly lower than at baseline (\(p\)-value = 0.0001). * RD values at post baseline were significantly increased as compared to RD values at baseline (\(p\)-value = 0.0004). * DET values at post baseline were significantly larger than at baseline (\(p\)-value = 0.003).