Inflammation plays a significant role in the occurrence and development of acute kidney injury (AKI). Evidence regarding the prognostic effect of the systemic immune-inflammation index (SII) in critically ill patients with AKI is scarce. The aim of this study was to assess the association between SII and all-cause mortality in these patients. Detailed clinical data were extracted from the Medical Information Mart for Intensive Care Database (MIMIC)-IV. The primary outcome was set as the in-hospital mortality. A total of 10,764 AKI patients were enrolled in this study. The restricted cubic splines analyses showed a J-shaped curve between SII and the risk of in-hospital and ICU mortality. After adjusting for relevant confounders, multivariate Cox regression analysis showed that both lower and higher SII levels were associated with an elevated risk of in-hospital all-cause mortality. A similar trend was observed for ICU mortality. In summary, we found that the SII was associated in a J-shaped pattern with all-cause mortality among critically ill patients with AKI. SII appears to be have potential applications in the clinical setting as a novel and easily accessible biomarker for predicting the prognosis of AKI patients.

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Persistent homology is an important methodology from topological data analysis which adapts theory from algebraic topology to data settings and has been successfully implemented in many applications. It produces a statistical summary in the form of a persistence diagram, which captures the shape and size of the data. Despite its widespread use, persistent homology is simply impossible to implement when a dataset is very large. In this paper we address the problem of finding a representative persistence diagram for prohibitively large datasets. We adapt the classical statistical method of bootstrapping, namely, drawing and studying smaller multiple subsamples from the large dataset. We show that the mean of the persistence diagrams of subsamples-taken as a mean persistence measure computed from the subsamples-is a valid approximation of the true persistent homology of the larger dataset. We give the rate of convergence of the mean persistence diagram to the true persistence diagram in terms of the number of subsamples and size of each subsample. Given the complex algebraic and geometric nature of persistent homology, we adapt the convexity and stability properties in the space of persistence diagrams together with random set theory to achieve our theoretical results for the general setting of point cloud data. We demonstrate our approach on simulated and real data, including an application of shape clustering on complex large-scale point cloud data.

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