Comparison of five cluster validity indices performance in brain [¹⁸ F]FET-PET image segmentation using k-means

Dynamic [¹⁸ F]fluoro-ethyl-L-tyrosine positron emission tomography ([¹⁸ F]FET-PET) is used to identify tumor lesions for radiotherapy treatment planning, to differentiate glioma recurrence from radiation necrosis and to classify gliomas grading. To segment different regions in the brain k-means cluster analysis can be used. The main disadvantage of k-means is that the number of clusters must be pre-defined. In this study, we therefore compared different cluster validity indices for automated and reproducible determination of the optimal number of clusters based on the dynamic PET data.

The k-means algorithm was applied to dynamic [¹⁸ F]FET-PET images of 8 patients. Akaike information criterion (AIC), WB, I, modified Dunn's and Silhouette indices were compared on their ability to determine the optimal number of clusters based on requirements for an adequate cluster validity index. To check the reproducibility of k-means, the coefficients of variation CVs of the objective function values OFVs (sum of squared Euclidean distances within each cluster) were calculated using 100 random centroid initialization replications RCI₁₀₀ for 2 to 50 clusters. k-means was performed independently on three neighboring slices containing tumor for each patient to investigate the stability of the optimal number of clusters within them. To check the independence of the validity indices on the number of voxels, cluster analysis was applied after duplication of a slice selected from each patient. CVs of index values were calculated at the optimal number of clusters using RCI₁₀₀ to investigate the reproducibility of the validity indices. To check if the indices have a single extremum, visual inspection was performed on the replication with minimum OFV from RCI₁₀₀ .

The maximum CV of OFVs was 2.7 × 10^-2 from all patients. The optimal number of clusters given by modified Dunn's and Silhouette indices was 2 or 3 leading to a very poor segmentation. WB and I indices suggested in median 5, [range 4-6] and 4, [range 3-6] clusters, respectively. For WB, I, modified Dunn's and Silhouette validity indices the suggested optimal number of clusters was not affected by the number of the voxels. The maximum coefficient of variation of WB, I, modified Dunn's, and Silhouette validity indices were 3 × 10^-2 , 1, 2 × 10^-1 and 3 × 10^-3 , respectively. WB-index showed a single global maximum, whereas the other indices showed also local extrema.

From the investigated cluster validity indices, the WB-index is best suited for automated determination of the optimal number of clusters for [¹⁸ F]FET-PET brain images for the investigated image reconstruction algorithm and the used scanner: it yields meaningful results allowing better differentiation of tissues with higher number of clusters, it is simple, reproducible and has an unique global minimum.