## 3 Results  
Results are presented using the Protege user interface [28] with automatic reasoning for the different possible scenarios described in Section 2.4 and in reference to the flow diagram of Figure 1 representing the MPT validation-check process. Text with highlighted yellow background in the following figures is information inferred by the reasoner. Figures 3-6 show the inferences made by the reasoner on the basis of the axioms described in Section 2.4 for the three tumour-couplet permutations p1_tc1, p1_tc2, and p1_tc3. Figure 3 reveals that one of the associated morphologies is a haematological one whereas the other is not and since the non-haematological morphology is not in the group that is common for all, the morphology groups are considered independent. The two topographies (C150 and C269) are however determined to be equivalent due to topography code C269 being in a group (group C) that is common to the group containing the topography code C150 (group 8). Since the morphology groups are independent, tumours t1 and t2 are accepted as multiple primaries. Figure 3: Results of the reasoning operation for the tumour couplet p1_tc1. Only the topography codes are equivalent (classified under DuplicateTopographyGroup) and the couplet is a valid MPT. Figure 4 shows that the two morphologies are not in the same group. Even though the topographies are equivalent (since C768 is in the group S that is common to all other topography groups), the two tumours t1 and t3 are accepted as multiple primaries. In Figure 5, the two morphologies are the same and therefore automatically in the same group, and one of the topographies (C768) is in group S that is common to all other topography groups). Thus, both morphologies and both topographies are in duplicate groups and the tumours are considered as a single entity. Figure 6 shows the result of Axiom (A2), which flags the fact that patient p1 contains at least one invalid MPT. Figures 7-10 illustrate the results for four other hypothetical patients with multiple tumours showing some of the other possible scenarios. Figure 7 is a valid MPT case (p2_tcl) in which: (a) the two topography codes are both subclasses of the topography code C26 that is contained only within one or more common topography groups and therefore considered equivalent, c.f. Axiom (5); (b) the two morphology codes are both subclasses of the carcinoma group of morphologies but are also in different subgroups of carcinomas (and therefore not equivalent). Figure 8 is a valid MPT case (p3_tcl) in which: (a) the two topography codes are subclassed from C39, which from Table A1 is considered a duplicate in both a main group (group 22), and a common group (group H); (b) the two morphology codes are independent since one is of haematological type but the other not. Figure 9 is an invalid MPT case (p4_tcl) in which: (a) the two topography codes are C398 and C761 which are considered single entities from the perspective of a main group (group 22) and its common group (group I), or through a duplication in a common group (group H); (b) the two morphology codes are contained within the same non-haematological main group (adenocarcinoma) and therefore considered a single entity. Figure 4: Results of the reasoning operation for the tumour couplet p1_tc2. Although the topography codes are equivalent via Axiom (4), the morphology codes are independent and the couplet is a valid MPT. Finally, Figure 10 is an invalid MPT case (p5_tc1) in which: (a) the two topography codes are in different main groups and therefore independent; (b) the two morphology groups are contained within the same main haematological group (T-cell and NK-cell neoplasms) which will result in a duplicate primary tumour regardless of the independence between the topography groups. Figure 5: Results of the reasoning operation for the tumour couplet p1_tc3. Since the topography codes are equivalent and the morphology codes (of non-haematological type) are identical, the couplet is an invalid MPT and is classified as a duplicate primary tumour.