Improved bounds for the crossing numbers of K_m,n and K_n

It has been long--conjectured that the crossing number cr(K_m,n) of the complete bipartite graph K_m,n equals the Zarankiewicz Number Z(m,n):= floor((m-1)/2) floor(m/2) floor((n-1)/2) floor(n/2). Another long--standing conjecture states that the crossing number cr(K_n) of the complete graph K_n equals Z(n):= floor(n/2) floor((n-1)/2) floor((n-2)/2) floor((n-3)/2)/4. In this paper we show the following improved bounds on the asymptotic ratios of these crossing numbers and their conjectured values: (i) for each fixed m >= 9, lim_{n->infty} cr(K_m,n)/Z(m,n) >= 0.83m/(m-1); (ii) lim_{n->infty} cr(K_n,n)/Z(n,n) >= 0.83; and (iii) lim_{n->infty} cr(K_n)/Z(n) >= 0.83. The previous best known lower bounds were 0.8m/(m-1), 0.8, and 0.8, respectively. These improved bounds are obtained as a consequence of the new bound cr(K_{7,n}) >= 2.1796n^2 - 4.5n. To obtain this improved lower bound for cr(K_{7,n}), we use some elementary topological facts on drawings of K_{2,7} to set up a quadratic program on 6! variables whose minimum p satisfies cr(K_{7,n}) >= (p/2)n^2 - 4.5n, and then use state--of--the--art quadratic optimization techniques combined with a bit of invariant theory of permutation groups to show that p >= 4.3593.Comment: LaTeX, 18 pages, 2 figure

Proportion Differences Using the Beta-binomial Distribution

When estimating the difference between two proportions with overdispersion due to correlation within the trials, the usual asymptotic confidence interval based on the maximum likelihood estimators generally has lower than desired coverage rates for small sample sizes. Consequently, the purpose of this study is to construct confidence intervals in this setting that exhibit near nominal coverage even for small sample sizes. The beta-binomial model is one possible way to model correlated 0-1 data. This model is used to develop two new intervals, referred to as the Haldane and Jeffreys-Perks intervals. The paper then compares these two new intervals with four existing competitors and evaluates their performance via simulations.The usual asymptotic interval based on the maximum likelihood estimators is discouraged for cases when the sample sizes are small or the proportions are close to zero or one. The Haldane interval is an improvement over the usual interval but still has many cases with less than desireable coverage probability. The Jeffreys-Perks interval provides significant improvement over the usual interval as do the existing intervals referred to as the extended Beal, extended Newcombe, and extended Peskun intervals. In particular, the Jeffreys-Perks interval is generally the best choice in terms of coverage probability for cases where the difference between the proportions is large. In specific cases when the two proportions are equal, or close to equal, the extended Newcombe and extended Beal generally have the best results. In many other cases, the extended Newcombe, extended Beal, and the Jeffreys-Perks intervals provide very similar results.Department of Statistic

Mutation patterns identify adult patients with de novo acute myeloid leukemia aged 60 years or older who respond favorably to standard chemotherapy: An analysis of Alliance studies

© 2018 Macmillan Publishers Limited, part of Springer Nature. Thus far, only 5-15% of AML patients aged ≥60 years are cured with chemotherapy. Identification of patients who are less (more) likely to respond to standard chemotherapy might enable early risk stratification toward alternative treatment regimens. We used a next-generation sequencing panel of 80 cancer- and/or leukemia-associated genes to profile molecularly 423 older patients with de novo AML. Using variables identified in multivariable models and co-occurring mutations in NPM1-mutated AML, we classified the patients into good-, intermediate-, and poor-risk groups for complete remission (CR) attainment, disease-free (DFS), and overall survival (OS). Whereas 81% of good-risk patients (comprising NPM1-mutated patients harboring mutations in chromatin remodeling, cohesin complex, methylation-related, spliceosome, and/or RAS pathway genes, FLT3-TKD, and/or patients without FLT3-ITD) achieved a CR, only 32% of poor-risk patients (with U2AF1, WT1 mutations and/or complex karyotype) did. Intermediate-risk patients had a 50% CR rate. Similarly, using NPM1 co-mutation patterns and SF1 mutation status, we identified patients with favorable DFS and OS 3-year rates of 46% and 45%, respectively. Patients with adverse genetic features had DFS and OS rates of only 2% and 4%. We show that application of our proposed criteria may refine the 2017 European LeukemiaNet classification for older patients treated with chemotheapy

GAS6 expression identifies high-risk adult AML patients: potential implications for therapy

Emerging data demonstrate important roles for the TYRO3/AXL/MERTK receptor tyrosine kinase (TAM RTK) family in diverse cancers. We investigated the prognostic relevance of GAS6 expression, encoding the common TAM RTK ligand, in 270 adults (n=71 aged \u3c60 \u3eyears; n=199 aged \u3e= 60 years) with de novo cytogenetically normal acute myeloid leukemia (CN-AML). Patients expressing GAS6 (GAS6+), especially those aged \u3e= 60 years, more often failed to achieve a complete remission (CR). In all patients, GAS6+ patients had shorter disease-free (DFS) and overall (OS) survival than patients without GAS6 expression (GAS6-). After adjusting for other prognostic markers, GAS6+ predicted CR failure (P=0.02), shorter DFS (P=0.004) and OS (P=0.04). To gain further biological insights, we derived a GAS6-associated gene-expression signature (

Prognostic and biologic significance of DNMT3B expression in older patients with cytogenetically normal primary acute myeloid leukemia

DNMT3B encodes a DNA methyltransferase implicated in aberrant epigenetic changes contributing to leukemogenesis. We tested whether DNMT3B expression, measured by NanoString nCounter assay, associates with outcome, gene and microRNA expression and DNA methylation profiles in 210 older (60 years) adults with primary, cytogenetically normal acute myeloid leukemia (CN-AML). Patients were dichotomized into high versus low expressers using median cut. Outcomes were assessed in the context of known CN-AML prognosticators. Gene and microRNA expression, and DNA methylation profiles were analyzed using microarrays and MethylCap-sequencing, respectively. High DNMT3B expressers had fewer complete remissions (CR; P=0.002) and shorter disease-free (DFS; P=0.02) and overall (OS;

Encorafenib Plus Cetuximab as a New Standard of Care for Previously Treated BRAF V600E–Mutant Metastatic Colorectal Cancer: Updated Survival Results and Subgroup Analyses from the BEACON Study

PURPOSE: BEACON CRC evaluated encorafenib plus cetuximab with or without binimetinib versus investigators' choice of irinotecan or FOLFIRI plus cetuximab in patients with BRAFV600E-mutant metastatic colorectal cancer (mCRC), after progression on 1-2 prior regimens. In the previously reported primary analysis, encorafenib, binimetinib plus cetuximab (ENCO/BINI/CETUX; triplet) and encorafenib plus cetuximab (ENCO/CETUX; doublet) regimens improved overall survival (OS) and objective response rate (ORR; by blinded central review) versus standard of care. The purpose of this analysis was to report updated efficacy and safety data. METHODS: In this open-label, phase III trial, 665 patients with BRAF V600E-mutant mCRC were randomly assigned 1:1:1 to receive triplet, doublet, or control. Primary end points were OS and independently reviewed ORR comparing triplet to control. OS for doublet versus control was a key secondary end point. Updated analyses include 6 months of additional follow-up and ORR for all randomized patients. RESULTS: Patients received triplet (n = 224), doublet (n = 220), or control (n = 221). Median OS was 9.3 months (95% CI, 8.2 to 10.8) for triplet and 5.9 months (95% CI, 5.1 to 7.1) for control (hazard ratio [HR], 0.60 [95% CI, 0.47 to 0.75]). Median OS for doublet was 9.3 months (95% CI, 8.0 to 11.3) (HR v control, 0.61 [95% CI, 0.48 to 0.77]). Confirmed ORR was 26.8% (95% CI, 21.1% to 33.1%) for triplet, 19.5% (95% CI, 14.5% to 25.4%) for doublet, and 1.8% (95% CI, 0.5% to 4.6%) for control. Adverse events were consistent with the prior primary analysis, with grade ≥ 3 adverse events in 65.8%, 57.4%, and 64.2% for triplet, doublet, and control, respectively. CONCLUSION: In the BEACON CRC study, encorafenib plus cetuximab improved OS, ORR, and progression-free survival in previously treated patients in the metastatic setting compared with standard chemotherapy. Based on the primary and updated analyses, encorafenib plus cetuximab is a new standard care regimen for previously treated patients with BRAF V600E mCRC

Encorafenib, Binimetinib, and Cetuximab in BRAF V600E-Mutated Colorectal Cancer

BACKGROUND: Patients with metastatic colorectal cancer with the BRAF V600E mutation have a poor prognosis, with a median overall survival of 4 to 6 months after failure of initial therapy. Inhibition of BRAF alone has limited activity because of pathway reactivation through epidermal growth factor receptor signaling. METHODS: In this open-label, phase 3 trial, we enrolled 665 patients with BRAF V600E–mutated metastatic colorectal cancer who had had disease progression after one or two previous regimens. Patients were randomly assigned in a 1:1:1 ratio to receive encorafenib, binimetinib, and cetuximab (triplet-therapy group); encorafenib and cetuximab (doublet-therapy group); or the investigators’ choice of either cetuximab and irinotecan or cetuximab and FOLFIRI (folinic acid, fluorouracil, and irinotecan) (control group). The primary end points were overall survival and objective response rate in the triplet-therapy group as compared with the control group. A secondary end point was overall survival in the doublet-therapy group as compared with the control group. We report here the results of a prespecified interim analysis. RESULTS: The median overall survival was 9.0 months in the triplet-therapy group and 5.4 months in the control group (hazard ratio for death, 0.52; 95% confidence interval [CI], 0.39 to 0.70; P<0.001). The confirmed response rate was 26% (95% CI, 18 to 35) in the triplet-therapy group and 2% (95% CI, 0 to 7) in the control group (P<0.001). The median overall survival in the doublet-therapy group was 8.4 months (hazard ratio for death vs. control, 0.60; 95% CI, 0.45 to 0.79; P<0.001). Adverse events of grade 3 or higher occurred in 58% of patients in the triplet-therapy group, in 50% in the doublet-therapy group, and in 61% in the control group. CONCLUSIONS: A combination of encorafenib, cetuximab, and binimetinib resulted in significantly longer overall survival and a higher response rate than standard therapy in patients with metastatic colorectal cancer with the BRAF V600E mutation. (Funded by Array BioPharma and others; BEACON CRC ClinicalTrials.gov number, NCT02928224. opens in new tab; EudraCT number, 2015-005805-35. opens in new tab.