## 3 Methods
### Staging
#### 3.3.1 Locational Feature Extraction  
Locational features are essential parameters for identifying the stages of lung cancer. They can provide information about the local extent or invasion of the primary tumors. As described in Table 1, the stages of tumors are highly associated with their locations. T1-staged tumors are located inside the lungs and in the lobar bronchus; T2-staged tumors invade the main bronchus, visceral pleura, and hilar regions; T3-staged tumors spread to the chest wall, phrenic nerve, and parietal pericardium; and finally, T4-staged tumors invade the diaphragm, mediastinum, heart, great vessels, and trachea. Considering these clinical descriptions, we extracted the locational features of the tumor nodules by dividing the lungs into different locational zones. The segmentation of lung anatomical structures in Section 3.1 gives a great deal of help for diving the locational zones. Firstly, we divided the isotropic structure of the trachea and bronchial airway tree (Section 3.1.5) into three different zones. The first zone (Zone 1) represents the regions of the trachea; the second zone (Zone 2) represents the regions of the main bronchus and hilar regions, and the third zone (Zone 3) represents the region of the lobar bronchus. To determine these three zones, the 3D skeletonization method [27] was applied. This method iteratively erodes the image voxels until only a single-voxel centerline is achieved. Figure 10 depicts an isotropic image of the trachea and bronchial airway tree. In the figure, the red thin-centerline represents the skeleton of the isotropic image. From this centerline, we tried to detect the branching points, because the trachea branches into two main bronchi at a point called the carina. The first branching point of the centerline can be designated as the place of the carina. Based on this point, we determined the upper portion of the carina as the region of the trachea (Zone 1). Similarly, the regions between the carina point and the next branching  
Figure 10: Different zones of the trachea and bronchial airway. points were defined as Zone 2--the areas of the main bronchus and hilar. Then, the other portion was assigned as the Zone 3--the lobar bronchus. Once these three zones had been defined, the next location was the region of the chest wall (Zone 4). The chest wall is an organ that surrounds and protects the lungs. It is comprised of bones especially the ribs, sternum, and spine. This zone was easily defined because the ribs and bones had already been segmented, as described in Section 3.1.8. Another location was the pulmonary vessels (Zone 5). As in zone 4, this zone was easily defined using the vessel segments described in Section 3.1.6. Next, zone 6 included the regions of the mediastum, heart, phenic nerve, and parietal pericardium. We classified these organs together into Zone 6 because they are located inside the mediastum, the region between the two lungs. The right and left lungs segments can help with the identification of this zone, and the area between the two lungs can be regarded as Zone 6. Finally, Zone 7 was defined as the location of the diaphragm, which is the portion below the lungs. To identify zones 6 and 7, the segments of the lung regions described in Section 3.1.4 were applied. For a clear understanding, the locational zones and associated tumor stages are summarized in Table 2. Moreover, a graphical representation of these zones is also presented in Figure 11.  
The idea behind dividing the lungs into different locational zones is to check the invasion of the tumor nodules. This task can be accomplished by calculating the 3D convex hull and comparing the coordinate values. The convex hull of a region can be calculated by finding the smallest convex polygon which encloses all of the points of that region. In this study, we applied the Quick hull algorithm [28], because it can handle high dimension problems.