MOTIVATION: The intra-tumor heterogeneity of protein expression is well recognized and may provide important information for cancer prognosis and predicting treatment responses. Analytic methods that account for spatial heterogeneity remain methodologically complex and computationally demanding for single-cell protein expression. For many functional proteins, single-cell expressions vary independently of spatial localization in a substantial proportion of the tumor tissues, and incorporation of spatial information may not affect the prognostic value of such protein biomarkers. RESULTS: We developed a new framework for using the distributions of functional single-cell protein expression levels as cancer biomarkers. The quantile functions of single-cell expressions are used to fully capture the heterogeneity of protein expression across all cancer cells. The quantile index (QI) biomarker is defined as an integral of an unspecified function which may depend linearly or nonlinearly on a tissue-specific quantile function. Linear and nonlinear versions of QI biomarkers based on single-cell expressions of ER, Ki67, TS, and CyclinD3 were derived and evaluated as predictors of progression-free survival or high mitotic index in a large breast cancer dataset. We evaluated performance and demonstrated the advantages of nonlinear QI biomarkers through simulation studies. AVAILABILITY AND IMPLEMENTATION: The associated R package Qindex is available at https://CRAN.R-project.org/package=Qindex and R package hyper.gam is available at https://github.com/tingtingzhan/hyper.gam. Examples of R code and detailed instructions could be found in vignette quantile-index-predictor (https://CRAN.R-project.org/package=hyper.gam/vignettes/applications.html#quantile-index-predictor).

BACKGROUND: Protein biomarkers of cancer progression and response to therapy are increasingly important for improving personalized medicine. Advanced quantitative pathology platforms enable measurement of protein expression in tissues at the single-cell level. However, this rich quantitative cell-by-cell biomarker information is most often not exploited. Instead, it is reduced to a single mean across the cells of interest or converted into a simple proportion of binary biomarker-positive or -negative cells. RESULTS: We investigated the utility of retaining all quantitative information at the single-cell level by considering the values of the quantile function (inverse of the cumulative distribution function) estimated from a sample of cell signal intensity levels in a tumor tissue. An algorithm was developed for selecting optimal cutoffs for dichotomizing cell signal intensity distribution quantiles as predictors of continuous, categorical or survival outcomes. The proposed algorithm was used to select optimal quantile biomarkers of breast cancer progression based on cancer cells' cell signal intensity levels of nuclear protein Ki-67, Proliferating cell nuclear antigen, Programmed cell death 1 ligand 2, and Progesterone receptor. The performance of the resulting optimal quantile biomarkers was validated and compared to the standard cancer compartment mean signal intensity markers using an independent external validation cohort. For Ki-67, the optimal quantile biomarker was also compared to established biomarkers based on percentages of Ki67-positive cells. For proteins significantly associated with PFS in the external validation cohort, the optimal quantile biomarkers yielded either larger or similar effect size (hazard ratio for progression-free survival) as compared to cancer compartment mean signal intensity biomarkers. CONCLUSION: The optimal quantile protein biomarkers yield generally improved prognostic value as compared to the standard protein expression markers. The proposed methodology has a broad application to single-cell data from genomics, transcriptomics, proteomics, or metabolomics studies at the single cell level.

MOTIVATION: Quantitative immunofluorescence is often used for immunohistochemistry quantification of proteins that serve as cancer biomarkers. Advanced image analysis systems for pathology allow capturing expression levels in each individual cell or subcellular compartment. However, only the mean signal intensity within the cancer tissue region of interest is usually considered as biomarker completely ignoring the issue of tumor heterogeneity. RESULTS: We propose using immunohistochemistry image-derived information on the spatial distribution of cellular signal intensity (CSI) of protein expression within the cancer cell population to quantify both mean expression level and tumor heterogeneity of CSI levels. We view CSI levels as marks in a marked point process of cancer cells in the tissue and define spatial indices based on conditional mean and conditional variance of the marked point process. The proposed methodology provides objective metrics of cell-to-cell heterogeneity in protein expressions that allow discriminating between different patterns of heterogeneity. The prognostic utility of new spatial indices is investigated and compared to the standard mean signal intensity biomarkers using the protein expressions in tissue microarrays incorporating tumor tissues from 1000+ breast cancer patients. AVAILABILITY AND IMPLEMENTATION: THE R CODE FOR COMPUTING THE PROPOSED SPATIAL INDICES IS INCLUDED AS SUPPLEMENTARY MATERIAL: . SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.