Primal and dual conic representable sets: a fresh view on multiparametric analysis

This paper introduces the concepts of the primal and dual conic (linear inequality) representable sets and applies them to explore a novel kind of duality in multiparametric conic linear optimization. Such a kind of duality may be described by the set-valued mappings between the primal and dual conic representable sets, which allows us to generalize as well as treat previous results for mulitparametric analysis in a unified framework. In particular, it leads to the invariant region decomposition of a conic representable set that is more general than the known results in the literatures. We develop the classical duality theory in conic linear optimization and obtain the multiparametric KKT conditions. As their applications, we then discuss the behaviour of the optimal partition of a conic representable set and investigate the multiparametric analysis of conic linear optimization problems. All results are corroborated by examples having correlation.Comment: 38 pages, 3 figur

Tumor-infiltrating lymphocytes provides recent survival information for early-stage HER2-low-positive breast cancer: a large cohort retrospective study

PurposeIt has been reported that breast cancer (BC) with low expression of human epidermal growth factor receptor 2 (HER2) might be a distinct subtype of BC. However, the prognostic effect of low HER2 expression on BC patients remains controversial. We aim to conduct this single-institution retrospective analysis to assess HER2-low-positive BC outcomes in Chinese women and the prognostic role of TILs in HER2-low-positive early-stage BC.MethodWe retrospectively enrolled 1,763 BC patients treated in a single institution from 2017 to 2018. TILs are regarded as continuous variables and are divided into low TILs (≤10%) and high TILs (>10%) for statistical analysis. Univariate and multivariable Cox proportional hazards regression models were used to test the associations between TILs and disease-free survival (DFS) with adjustment for clinicopathologic characteristics.ResultHigh TIL levels (>10%) were associated with tumor size (>2 cm, p = 0.042), age at diagnosis (p = 0.005), Ki-67 index (>25%; p <0.001), HR (hormone receptor) status (positive, p <0.001), advanced pathological stage (p = 0.043), subtype (p <0.001), and HER2 status (p <0.001). The Kaplan−Meier analysis indicated that no significant difference in DFS (p = 0.83) could be found between HER2-positive, HER2-low-positive, and HER2-0 BC. The DFS of HER2-low-positive BC and HER2-nonamplified BC with high levels of TILs was statistically better than that of patients with low levels of TILs (p = 0.015; p = 0.047). In HER2-low-positive BC patients with high TIL levels (>10%), DFS was significantly improved in both the univariate (HR = 0.44, 95% CI 0.22–0.87, P = 0.018) and multivariate (HR = 0.47, 95% CI 0.23–0.95, P = 0.035) Cox models. For further subgroup analysis, HR (+)/HER2-low-positive BC with high TIL (>10%) levels was associated with improved DFS in both the univariate (HR = 0.41, 95% CI 0.19–0.90, P = 0.025) and multivariate (HR = 0.42, 95% CI 0.19–0.93, P = 0.032) Cox models. The HR (−)/HER2-0 BC with high TIL (>10%) level was not statistically significant in the univariate Cox model, but it was statistically significant in the multivariate (HR = 0.16, 95% CI 0.28–0.96, P = 0.045) Cox model.ConclusionAmong early-stage BC, no significant survival difference could be found between the HER2-positive, HER2-low-positive, and HER2-0 cohorts. High levels of TILs were significantly associated with improved DFS in HER2-low-positive patients, especially in the HR (+)/HER2-low-positive subtype