Abstract. We show that any graph, in the sequence given by Haagerup in 1991 as that of candidates of principal graphs of subfactors, is not realized as a principal graph except for the smallest two. This settles the remaining case of a previous work of the first author.

PurposeThe aim of our retrospective study was to determine the time to progression to castration-resistant prostate cancer (CRPC) in prostate cancer patients who undergo combined androgen blockade (CAB), as well as their prognoses.Materials and MethodsWe examined the overall survival (OS) and disease-specific survival rates, as well as the time to CRPC development, in 387 patients who were treated with CAB for prostate cancer. The disease-specific survival rate and time to CRPC were stratified by prostate-specific antigen (PSA) levels, Gleason score (GS), and presence of metastasis at diagnosis. We designated high-risk patients as those satisfying at least two of the following three criteria: extent of disease of bone metastasis grade ≥2, presence of metastasis at diagnosis, and a GS ≥8.ResultsThe 10- and 15-year OS rates were 74.0% and 50.4%, respectively, while the corresponding disease-specific survival rates were both 86.8%. Metastasis at diagnosis was an independent prognostic factor for disease-specific survival. The median time to CRPC development was 140.7 months. A PSA level ≥20 ng/mL, a GS ≥8, and the presence of metastasis at diagnosis were independent predictors of a shorter time to CRPC development. The 10-year disease-specific survival rate in the high-risk group was significantly lower than that in the low-risk group (approximately 74% vs. 98%), and the time to CRPC development was significantly shorter (median: 20.5 months vs. not reached).ConclusionsThe time to CRPC development was shorter in high-risk prostate cancer patients with metastases. Such patients require alternative novel treatment modalities.

ObjectivesTo investigate the necessity of stratifying patients in the intermediate-risk group of the Memorial Sloan Kettering Cancer Center (MSKCC) criteria in a real-world population of patients with metastatic renal cell carcinoma.Patients and MethodsWe retrospectively analyzed 234 consecutively treated patients who had received molecular targeted drugs. We examined the difference between progression-free survival and overall survival among patients in the intermediate-risk group of MSKCC criteria. We divided the intermediate group into two subgroups as follows: patients positive for only one risk factor (Int-1) and those positive for two risk factors (Int-2) including performance status, serum hemoglobin level, time from diagnosis to treatment, and corrected calcium and lactate dehydrogenase levels. Next, we evaluated the association between the number of metastatic organs, the presence of pancreatic metastasis, Int-1 or Int-2 grouping, and overall survival.ResultsThe median overall survival was 41.2 months. The median overall survival of the favorable-, intermediate-, and poor-risk groups of the MSKCC criteria were 91.0, 33.6, and 15.2 months, respectively. Patient characteristics were similar between the Int-1 and Int-2 groups. Increased positivity for risk factors of MSKCC classification between the two groups was for performance status and serum hemoglobin level. Progression-free survival and overall survival of the Int-1 group were significantly higher than those of the Int-2 group. In Cox proportional stepwise multivariate analysis, the Int-1 and Int-2 classification was an independent risk factor for overall survival.ConclusionPatients in the intermediate-risk group had different prognoses depending on the number of positive risk factors.

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