## 4 Methods
### Feature Computation  
For each patient, several quantitative biomarkers were extracted from all the VOIs. The features were computed using an in-house software tool coded in MATLAB R2019a (The MathWorks, Inc., Natick, MA, USA) that calculates the first order statistical features and two families of textural features: 3D Grey Level Co-occurrence Matrix (GLCM) and Three Orthogonal Planes-Local Binary Patterns (TOP-LBP). The TOP-LBP implementation of 3D LPBs computationally simplifies the extraction process, as discussed below. First-order features: all of these features are computed from the grey levels' histogram of a ROI and therefore try to describe the statistical distribution of tissue's density inside the volume. From such histograms, we extracted 12 descriptors, which are the moments from first to fourth-order, namely the mean, the standard deviation, the skewness and the kurtosis, the histogram width, the energy, the entropy, the value of the histogram absolute maximum and the corresponding grey-level value, the energy around such maximum, and the number of relative maxima in the histogram and their energy. 3D Grey Level Co-occurrence Matrix: the healthy and the cancerous tissues usually present different structural patterns: for this reason, we go beyond the grey levels distribution of the VOIs voxels performing an additional textural analysis. In this respect, we computed the 3D Grey Level Co-occurrence Matrix, which generalises the 2D GLCM to the third dimension, being able to catch the differences of the tissues at the micro scale. Given a 3D grey-scale image \(I\) and a 3D Cartesian reference system \(O(x,y,z)\), whose origin is located in the top-front-left corner of \(I\), the position of each voxel can be identified by a vector \(\mathbf{p}=p_{x}\hat{i}+p_{y}\hat{j}+p_{z}\hat{k}\), with \(p_{x}\), \(p_{y}\) and \(p_{z}\in\mathbb{N}\). We also denote a displacement vector \(\mathbf{d}=d_{x}\hat{i}+d_{y}\hat{j}+d_{z}\hat{k}\), with \(d_{x},d_{y}\) and \(d_{z}\in\mathbb{N}\). If we define \(m\) equal to the number of bit used to represent \(I\), a \(GLCM3\) is a square matrix of size \(N=2^{m}\), where each entry \((g_{i},g_{j})\), with both \(g_{i}\) and \(g_{j}\in[0,2^{m}-1]\), represents the number of times a voxel in \(\mathbf{p}\) with intensity \(g_{i}\) is separated by a displacement \(\mathbf{d}\) from another voxel with intensity \(g_{j}\), therefore located in \(\mathbf{p}+\mathbf{d}\). Hence, a GLCM3 matrix synthesises the count of how many values co-occur between neighbouring pixels according to specific displacements. Those quantities vary, depending on the VOI structure. Denoting as \(d_{h}\) the \(h\)-\(th\) component of \(\mathbf{d}\), in order to assess all possible directions, we considered the combination of \(d_{h}\in\{-1,0,1\}\) as displacements, without considering the \((0,0,0)\) vector. This leads to 26 different displacement directions. Furthermore, when considering that each direction \(+\mathbf{d}\) produces a \(GLCM3\) matrix that is the transpose of the one produced by its opposite direction \(-\mathbf{d}\), thus giving a redundant informative contribution, we only considered the 13 different directions given by \(+\mathbf{d}\). More details on GLCM3 can be deepened in [48]. Finally, from each GLCM3, we extract seven second order statistical features, referred to as Haralick descriptors [49]. Namely, they are autocorrelation, homogeneity, entropy, energy, covariance, inertia, and absolute contrast. Their formal definition is reported in the Appendix A. Concatenating such measures for each GLCM3 we get \(13\times 7=91\) textural descriptors per patient. Three Orthogonal Planes-Local Binary Patterns: this is a relatively new descriptor in radiomics scenario, and it is one of the possible generalisation of the well-known 2D LBP to the 3D. Now, we will briefly recall the computation of a basic bi-dimensional LBP.