Figure 2
From: Imaging-based representation and stratification of intra-tumor heterogeneity via tree-edit distance
(a) Curves displaying the filtered heights of the trees’ vertices for the three groups. Operationally, curves were built as follows: for any fixed height (x-axis), for any tree in the selected group, we count the number of nodes whose height value is greater than the fixed one (y-axis). For the step-by-step procedure see Section 3 of Supplementary Materials online. The curves in the plot represent the pointwise within-group means of such counts, and the shaded regions cover an area of 1 standard deviation around the means. The values of such counting process result in a monotonically non-increasing function detecting information about trees’ heterogeneity. In fact, higher values of such function, especially as the height threshold becomes bigger and bigger, correspond to a greater number of heterogeneous lesions in the patients. Patients of group 0 (blue line) are characterized by a very homogeneous disease where trees branches are on average less and very short compared to the other groups; patients of group 1 (orange line) tend to exhibit more lesions than patients belonging to group 0, lesions which are intermediately heterogeneous, as their representation trees display both short branches and longer branches than group 0; patients in group 2 (green line) are associated to very heterogeneous diseases, displaying a similar number of lesions to group 0, but with the associated branches being much longer. A synthetic example of tree per each group is displayed in Fig. 7, elucidating the differences with a graphical support. (b) Functional comparison between curves: in order to test the hypothesis that curves belonging to different groups are different, we use the ANOVA procedure proposed in22. It outputs an interval-wise adjusted p-value function. Depending on the sort and level \(\alpha\) of Type-I error control, significant intervals can be selected. Here, we highlighted in grey the region of significance. Of note, the curves appear different for what homogeneity-heterogeneity balance is concerned; they lose significance as they approach very big height values.
