Novel methods to analyze the tumor microenvironment (TME) are urgently needed to stratify melanoma patients for adjuvant immunotherapy. Tumor-infiltrating lymphocyte (TIL) analysis, by conventional pathologic methods, is predictive but is insufficiently precise for clinical application. Quantitative multiplex immunofluorescence (qmIF) allows for evaluation of the TME using multiparameter phenotyping, tissue segmentation, and quantitative spatial analysis (qSA). Given that CD3CD8 cytotoxic lymphocytes (CTLs) promote antitumor immunity, whereas CD68 macrophages impair immunity, we hypothesized that quantification and spatial analysis of macrophages and CTLs would correlate with clinical outcome. We applied qmIF to 104 primary stage II to III melanoma tumors and found that CTLs were closer in proximity to activated (CD68HLA-DR) macrophages than nonactivated (CD68HLA-DR) macrophages ( < 0.0001). CTLs were further in proximity from proliferating SOX10 melanoma cells than nonproliferating ones ( < 0.0001). In 64 patients with known cause of death, we found that high CTL and low macrophage density in the stroma ( = 0.0038 and = 0.0006, respectively) correlated with disease-specific survival (DSS), but the correlation was less significant for CTL and macrophage density in the tumor ( = 0.0147 and = 0.0426, respectively). DSS correlation was strongest for stromal HLA-DR CTLs ( = 0.0005). CTL distance to HLA-DR macrophages associated with poor DSS ( = 0.0016), whereas distance to Ki67 tumor cells associated inversely with DSS ( = 0.0006). A low CTL/macrophage ratio in the stroma conferred a hazard ratio (HR) of 3.719 for death from melanoma and correlated with shortened overall survival (OS) in the complete 104 patient cohort by Cox analysis ( = 0.009) and merits further development as a biomarker for clinical application. .

Glioblastoma multiforme (GBM) is an aggressive form of human brain cancer that is under active study in the field of cancer biology. Its rapid progression and the relative time cost of obtaining molecular data make other readily-available forms of data, such as images, an important resource for actionable measures in patients. Our goal is to utilize information given by medical images taken from GBM patients in statistical settings. To do this, we design a novel statistic-the smooth Euler characteristic transform (SECT)-that quantifies magnetic resonance images (MRIs) of tumors. Due to its well-defined inner product structure, the SECT can be used in a wider range of functional and nonparametric modeling approaches than other previously proposed topological summary statistics. When applied to a cohort of GBM patients, we find that the SECT is a better predictor of clinical outcomes than both existing tumor shape quantifications and common molecular assays. Specifically, we demonstrate that SECT features alone explain more of the variance in GBM patient survival than gene expression, volumetric features, and morphometric features. The main takeaways from our findings are thus twofold. First, they suggest that images contain valuable information that can play an important role in clinical prognosis and other medical decisions. Second, they show that the SECT is a viable tool for the broader study of medical imaging informatics.

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◥Patients with resected stage II-III melanoma have approximately a 35% chance of death from their disease. A deeper understanding of the tumor immune microenvironment (TIME) is required to stratify patients and identify factors leading to therapy resistance. We previously identified that the melanoma immune profile (MIP), an IFN-based gene signature, and the ratio of CD8 þ cytotoxic T lymphocytes (CTL) to CD68 þ macrophages both predict disease-specific survival (DSS). Here, we compared primary with metastatic tumors and found that the nuclei of tumor cells were significantly larger in metastases. The CTL/macrophage ratio was significantly different between primary tumors without distant metastatic recurrence (DMR) and metastases. Patients without DMR had higher degrees of clustering between tumor cells and CTLs, and between tumor cells and HLA-DR þ macrophages, but not HLA-DR À macrophages. The HLA-DR À subset coexpressed CD163 þ CSF1R þ at higher levels than CD68 þ HLA-DR þ macrophages, consistent with an M2 phenotype. Finally, combined transcriptomic and multiplex data revealed that densities of CD8 and M1 macrophages correlated with their respective cell phenotype signatures. Combination of the MIP signature with the CTL/macrophage ratio stratified patients into three risk groups that were predictive of DSS, highlighting the potential use of combination biomarkers for adjuvant therapy.Significance: These findings provide a deeper understanding of the tumor immune microenvironment by combining multiple modalities to stratify patients into risk groups, a critical step to improving the management of patients with melanoma.

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We show that an embedding in Euclidean space based on tropical geometry generates stable sufficient statistics for barcodes. In topological data analysis, barcodes are multiscale summaries of algebraic topological characteristics that capture the "shape" of data; however, in practice, they have complex structures that make them difficult to use in statistical settings. The sufficiency result presented in this work allows for classical probability distributions to be assumed on the tropical geometric representation of barcodes. This makes a variety of parametric statistical inference methods amenable to barcodes, all while maintaining their initial interpretations. More specifically, we show that exponential family distributions may be assumed and that likelihood functions for persistent homology may be constructed. We conceptually demonstrate sufficiency and illustrate its utility in persistent homology dimensions 0 and 1 with concrete parametric applications to human immunodeficiency virus and avian influenza data.

We study the problem of optimal transport in tropical geometry and define the Wasserstein-p distances in the continuous metric measure space setting of the tropical projective torus. We specify the tropical metric—a combinatorial metric that has been used to study of the tropical geometric space of phylogenetic trees—as the ground metric and study the cases of $$p=1,2$$ p = 1 , 2 in detail. The case of $$p=1$$ p = 1 gives an efficient computation of the infinitely-many geodesics on the tropical projective torus, while the case of $$p=2$$ p = 2 gives a form for Fréchet means and a general inner product structure. Our results also provide theoretical foundations for geometric insight a statistical framework in a tropical geometric setting. We construct explicit algorithms for the computation of the tropical Wasserstein-1 and 2 distances and prove their convergence. Our results provide the first study of the Wasserstein distances and optimal transport in tropical geometry. Several numerical examples are provided.

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.