Measure of three cell population co-localization for spatial protein imaging data analysis

Background: Spatial protein imaging technologies enable detailed study of the tumor microenvironment (TME) to characterize cell abundance and spatial architecture. Pairwise colocalization metrics (e.g., Ripley's K) miss higher-order patterns, while enumerating all triangles for three cell populations is computationally infeasible on whole-slide images. Metrics using a triangle's longest edge poorly distinguish compact triples from elongated or dispersed arrangements. Thus, we propose a new trivariate colocalization measure using triangle area. Methods: We developed a Horvitz-Thompson Monte Carlo estimator and benchmarked the unweighted and isoperimetrically weighted area variants against a longest-edge analogue on simulated data. These metrics were applied to a study of primary high-grade serous carcinoma (n=101) to quantify colocalization among T cells (CD3+), cytotoxic T cells (CD3+CD8+), B cells (CD19+), and macrophages (CD68+). Cox PH models evaluated associations between normalized clustering metrics and overall survival at radii of 15 and 25 μm - adjusted for diagnosis age, stage, and debulking. Results: Simulations showed unbiased, consistent colocalization estimates with lower small-radius variance for area-based measures. Monte Carlo sampling achieved a >10 5 -fold runtime reduction versus full enumeration. In the ovarian study, lower three-cell colocalization levels of lymphocytes with macrophages corresponded to small improvements in overall survival, though not significant (Table 1). Narrower confidence intervals were also observed for area-based estimators, with weighted and unweighted area variants performing similarly. Discussion: Our framework enables scalable quantification of higher-order spatial organization in the TME. Despite comparable associations across estimators in the ovarian study, the greater stability of area-based measures supports their use as surrogates of complex cellular architecture in future spatial studies. Table 1: Cox model hazard ratios and 95% confidence intervals for study cell triples at chosen radii Cell Triple Radius Longest-Edge Unweighted Area Weighted Area T Cell B Cell Macrophage 15 0.83 (0.61, 1.14) 0.93 (0.66, 1.31) 0.89 (0.65, 1.21) 25 0.90 (0.65, 1.23) 0.91 (0.67, 1.23) 0.93 (0.68, 1.27) Cytotoxic T Cell B Cell Macrophage 15 0.95 (0.73, 1.23) 0.90 (0.68, 1.17) 0.93 (0.73, 1.20) 25 0.85 (0.65, 1.12) 0.86 (0.66, 1.11) 0.84 (0.64, 1.09)