The run-times of different methods (in seconds) with the genes left after pruning based on different CV cut-offs.

λ1 and λ2 were respectively kept at 0.01 and 0.02. The mark “X” means that we could not run those methods due to inordinate amount of time required.</p

Illustration for SPI calculation.

A point pattern consisting of 30 cells, with 10 cells per each type A, B, and C, was simulated. Circles of radius 0.25 were drawn around each point to identify co-occurrences of cell types with the first distance range w1 = (0, 0.25]. Specifically, there were 1 AA, 1 BB, 3 CC, 0 AB, 0 AC, and 0 BC. (TIF)</p

FPCA results from SPI curves in NSCLC dataset.

Spatial heterogeneity in the tumor microenvironment (TME) plays a critical role in gaining insights into tumor development and progression. Conventional metrics typically capture the spatial differential between TME cellular patterns by either exploring the cell distributions in a pairwise fashion or aggregating the heterogeneity across multiple cell distributions without considering the spatial contribution. As such, none of the existing approaches has fully accounted for the simultaneous heterogeneity caused by both cellular diversity and spatial configurations of multiple cell categories. In this article, we propose an approach to leverage spatial entropy measures at multiple distance ranges to account for the spatial heterogeneity across different cellular organizations. Functional principal component analysis (FPCA) is applied to estimate FPC scores which are then served as predictors in a Cox regression model to investigate the impact of spatial heterogeneity in the TME on survival outcome, potentially adjusting for other confounders. Using a non-small cell lung cancer dataset (n = 153) as a case study, we found that the spatial heterogeneity in the TME cellular composition of CD14+ cells, CD19+ B cells, CD4+ and CD8+ T cells, and CK+ tumor cells, had a significant non-zero effect on the overall survival (p = 0.027). Furthermore, using a publicly available multiplexed ion beam imaging (MIBI) triple-negative breast cancer dataset (n = 33), our proposed method identified a significant impact of cellular interactions between tumor and immune cells on the overall survival (p = 0.046). In simulation studies under different spatial configurations, the proposed method demonstrated a high predictive power by accounting for both clinical effect and the impact of spatial heterogeneity.</div

Simulation scenarios.

(A) low additive noise. (B) medium additive noise. (C) large additive noise. Three levels of noise were added to the reference SPI curves (clustered vs. random) to generate subject-specific SPI curves. (TIF)</p

The networks between the hub-genes whose degree changed from IL and PL to LHB.

The top row corresponds to the genes whose degree decreased from IL and PL to LHB, while the bottom row corresponds to the genes whose degree increased.</p

FPCA results from SPI curves in ovarian cancer dataset.

(A) Mean function. (B) First two eigenfunctions. (TIF)</p

FPCA results from SPI curves in IHC COVID-19 dataset.

(A) Mean function. (B) First three eigenfunctions. (TIF)</p

Kaplan–Meier curves for the overall survival probability from the NSCLC dataset, stratified using the Mantel correlation.

Subjects were classified as clustered vs. random based on the permutation test of the Mantel correlation. P-value of 0.24 indicates non-significant difference in survival probability in two groups. (TIF)</p

Simulated spatial configurations.

Two reference spatial configurations: clustered (A) and random (B) of five different cell types: CD14+, CD19+, CD4+, CD8+, and CK+. (C): Corresponding spatial entropy at multiple distance ranges for each configuration.</p

Normalized root mean squared errors (NRMSE) across three different models, with four sample sizes (N = 100, 200, 500, 1000), and at three levels of additive noise (small, medium, and large).

Corresponding standard deviations are recorded in parentheses.</p