Conjugacy classes and growth conditions

AbstractIn this paper we show using a purely combinatorial argument that a finitely generated infinite group such that fE(n)⩽ans, where a is a constant, admits for every ϵ a sequence {gi,ϵ} of non-unit elements whose centralizer contains more than i1/2−ϵ elements of length less than i. Of course, the interest of this result is in the fact that it excludes the possibility that the group is a pure torsion group, since otherwise the existence of the sequence {gi,ϵ} is obvious. As an application of this result, we show that, in the case where r<3/2, there exists an element whose centralizer has finite index in G

Prediction of the outcome of preoperative chemotherapy in breast cancer using DNA probes that provide information on both complete and incomplete responses

<p>Abstract</p> <p>Background</p> <p>DNA microarray technology has emerged as a major tool for exploring cancer biology and solving clinical issues. Predicting a patient's response to chemotherapy is one such issue; successful prediction would make it possible to give patients the most appropriate chemotherapy regimen. Patient response can be classified as either a pathologic complete response (PCR) or residual disease (NoPCR), and these strongly correlate with patient outcome. Microarrays can be used as multigenic predictors of patient response, but probe selection remains problematic. In this study, each probe set was considered as an elementary predictor of the response and was ranked on its ability to predict a high number of PCR and NoPCR cases in a ratio similar to that seen in the learning set. We defined a valuation function that assigned high values to probe sets according to how different the expression of the genes was and to how closely the relative proportions of PCR and NoPCR predictions to the proportions observed in the learning set was. Multigenic predictors were designed by selecting probe sets highly ranked in their predictions and tested using several validation sets.</p> <p>Results</p> <p>Our method defined three types of probe sets: 71% were mono-informative probe sets (59% predicted only NoPCR, and 12% predicted only PCR), 25% were bi-informative, and 4% were non-informative. Using a valuation function to rank the probe sets allowed us to select those that correctly predicted the response of a high number of patient cases in the training set and that predicted a PCR/NoPCR ratio for validation sets that was similar to that of the whole learning set. Based on DLDA and the nearest centroid method, bi-informative probes proved more successful predictors than probes selected using a t test.</p> <p>Conclusion</p> <p>Prediction of the response to breast cancer preoperative chemotherapy was significantly improved by selecting DNA probe sets that were successful in predicting outcomes for the entire learning set, both in terms of accurately predicting a high number of cases and in correctly predicting the ratio of PCR to NoPCR cases.</p

Promoter hypermethylation of HS3ST2, SEPTIN9 and SLIT2 combined with FGFR3 mutations as a sensitive/specific urinary assay for diagnosis and surveillance in patients with low or high-risk non-muscle-invasive bladder cancer

International audienceBackgroundNon-muscle-invasive bladder cancer (NMIBC) is a high incidence form of bladder cancer (BCa), where genetic and epigenetic alterations occur frequently. We assessed the performance of associating a FGFR3 mutation assay and a DNA methylation analysis to improve bladder cancer detection and to predict disease recurrence of NMIBC patients.MethodsWe used allele specific PCR to determine the FGFR3 mutation status for R248C, S249C, G372C, and Y375C. We preselected 18 candidate genes reported in the literature as being hypermethylated in cancer and measured their methylation levels by quantitative multiplex-methylation specific PCR. We selected HS3ST2, SLIT2 and SEPTIN9 as the most discriminative between control and NMIBC patients and we assayed these markers on urine DNA from a diagnostic study consisting of 167 NMIBC and 105 controls and a follow-up study consisting of 158 NMIBC at diagnosis time’s and 425 at follow-up time. ROC analysis was performed to evaluate the diagnostic accuracy of each assay alone and in combination.ResultsFor Diagnosis: Using a logistic regression analysis with a model consisting of the 3 markers’ methylation values, FGFR3 status, age and known smoker status at the diagnosis time we obtained sensitivity/specificity of 97.6 %/84.8 % and an optimism-corrected AUC of 0.96. With an estimated BCa prevalence of 12.1 % in a hematuria cohort, this corresponds to a negative predictive value (NPV) of 99.6 %. For Follow-up: Using a logistic regression with FGFR3 mutation and the CMI at two time points (beginning of the follow-up and current time point), we got sensitivity/specificity/NPV of 90.3 %/65.1 %/97.0 % and a corrected AUC of 0.84. We also tested a thresholding algorithm with FGFR3 mutation and the two time points as described above, obtaining sensitivity/specificity/NPV values of, respectively, 94.5 %/75.9 %/98.5 % and an AUC of 0.82.ConclusionsWe showed that combined analysis of FGFR3 mutation and DNA methylation markers on urine can be a useful strategy in diagnosis, surveillance and for risk stratification of patients with NMIBC. These results provide the basis for a highly accurate noninvasive test for population screening and allowing to decrease the frequency of cystoscopy, an important feature for both patient quality of life improvement and care cost reduction