A robust, scalable K-statistic for quantifying immune cell clustering in spatial proteomics data

Technical Summary: A Robust, Scalable K-Statistic for Quantifying Immune Cell Clustering in Spatial Proteomics Data

Problem Statement
Spatial proteomics technologies enable the measurement of protein abundance and cell locations within tissue samples, preserving spatial context. A common approach to analyzing these data is Ripley's $K$ function, a metric used to quantify whether cells are spatially clustered or randomly dispersed. Standard applications of Ripley's $K$ rely on the assumption of Complete Spatial Randomness (CSR), where the expected number of cells in any subset is consistent across the sample, yielding a theoretical null value of $K_0(r) = \pi r^2$.

However, spatial proteomics data frequently violate this homogeneity assumption due to biological factors (e.g., tissue architecture) and technical artifacts (e.g., tissue degradation, tearing, or incomplete slide adherence during imaging). When the standard theoretical null is applied to inhomogeneous data, it leads to the overestimation of spatial clustering and colocalization. This bias can distort downstream analyses, such as linking cell interactions to patient survival, by conflating technical artifacts with biological signals. Existing methods to address inhomogeneity, such as inhomogeneous adaptations of Ripley's $K$ or permutation-based empirical nulls, suffer from limitations: the former may introduce bias by ignoring background cell information, while the latter becomes computationally prohibitive for large datasets (often containing >10,000 cells per sample) and introduces sampling error when approximated via Monte Carlo methods.

Methodology: KAMP
The authors propose KAMP (K adjustment by Analytical Moments of the Permutation distribution), a novel method designed to quantify spatial organization robustly against inhomogeneity while maintaining computational scalability.

1. Null Hypothesis and Permutation Framework:
   KAMP adopts a "proportional intensity" null hypothesis ($H_0$), positing that the spatial intensity of immune cells is proportional to that of the background cells ($\lambda_m(u) \propto \lambda_{n-m}(u)$). Under this hypothesis, immune cells are distributed as a random subset of the total cell population. The method constructs an empirical null benchmark by considering all possible permutations of cell labels (immune vs. background) while keeping cell locations fixed.

2. Analytical Moments:
   Instead of performing computationally expensive permutations, KAMP derives closed-form analytical expressions for the first and second moments (mean and variance) of the permutation distribution of Ripley's $K$.

   • Expectation: The expected value under the permutation null, $E(K_s(r))$, is shown to be equivalent to the univariate Ripley's $K$ computed using all cells in the sample (pooled immune and background). This allows existing software to compute the empirical null directly.
   • Variance: The authors derive a complex combinatorial formula for the variance, $Var(K_s(r))$, which accounts for pairwise and higher-order interactions within the permutation distribution.
   • Asymptotic Normality: Theorem 2.4 establishes that the test statistic $Z_{K_s(r)} = (\hat{K}_s(r) - E(K_s(r))) / \sqrt{Var(K_s(r))}$ converges to a standard normal distribution under the permutation null as the number of cells increases, provided certain regularity conditions regarding neighborhood contributions are met.

3. Computational Efficiency (KAMP Lite):
   To handle extremely large datasets (e.g., >1 million cells), the authors introduce KAMP Lite, which employs random thinning of points before computing the moments. Theoretical properties of point processes ensure that thinning preserves distributional properties, allowing for accurate approximations of the moments with significantly reduced computation time.

4. Inference:
   The method provides:

   • A corrected clustering metric: $\tilde{K}_s(r) = \hat{K}_s(r) - E(K_s(r))$.
   • Approximate p-values for within-sample hypothesis testing of positive excess clustering or colocalization.
   • A framework for across-sample analysis, where $\tilde{K}_s(r)$ serves as a covariate in survival models (e.g., Cox proportional hazards).

Key Results
The paper evaluates KAMP through simulations and an application to a High-Grade Serous Ovarian Cancer (HGSOC) dataset (128 patients, >1.5 million cells total).

• Simulation Studies:

  • Bias Correction: In inhomogeneous null scenarios (simulating tissue tearing), standard Ripley's $K$ and the inhomogeneous variant ($K_{hom}$) significantly overestimated clustering. KAMP, KAMP Lite, and the full permutation method correctly centered estimates around zero.
  • Power and Type I Error: KAMP maintained appropriate Type I error rates and demonstrated high power comparable to the full permutation method across varying cell densities and abundances.
  • Speed: KAMP was orders of magnitude faster than the permutation method. For the HGSOC dataset, KAMP took 3.6 minutes compared to 562 minutes for 10,000 permutations. KAMP Lite was even faster (0.6 minutes).

• HGSOC Application:

  • Within-Sample: KAMP effectively distinguished between true biological clustering and technical inhomogeneity. In samples with tissue tearing, the empirical null $E(K_s(r))$ was significantly higher than the theoretical $\pi r^2$, and KAMP successfully adjusted the observed $K$ values to reflect true clustering.
  • Across-Sample (Survival Analysis): The study focused on the colocalization of B cells and macrophages. Models using KAMP-derived colocalization metrics showed stronger protective associations with overall survival and higher concordance indices (c-index) compared to models using standard Ripley's $K$. Standard $K$ failed to detect significant associations, suggesting that uncorrected inhomogeneity biases hazard ratio estimates.

Significance and Claims
The paper claims that KAMP represents the first closed-form characterization of the first two permutation-null moments for univariate and bivariate Ripley's $K$, coupled with an asymptotic normal approximation.

• Robustness: By leveraging background cells and analytical moments, KAMP corrects for spatial inhomogeneity without the computational burden of permutations or the bias of ignoring background structure.
• Scalability: The analytical approach eliminates the need for Monte Carlo permutations, making it feasible to analyze large-scale spatial proteomics datasets (millions of cells) that were previously computationally intractable for rigorous permutation testing.
• Scientific Impact: The application to ovarian cancer demonstrates that failing to correct for sample inhomogeneity can lead to biased survival analyses. KAMP provides a statistically valid framework to link specific cell-cell interactions (e.g., B cell–macrophage colocalization) to clinical outcomes, revealing associations that standard methods miss.

The authors note that KAMP measures excess clustering relative to the pooled tissue structure. Consequently, if the biological signal of interest is shared by both immune and background cells (aligned with tissue architecture), KAMP may remove this shared structure, potentially reducing power for detecting such specific signals. However, for detecting cell-type-specific interactions distinct from the background, the method offers a robust and scalable solution.