A Two-Stage Approach for Segmenting Spatial Point Patterns Applied to Multiplex Imaging

Recent advances in multiplex imaging have enabled researchers to locate different types of cells within a tissue sample. This is especially relevant for tumor immunology, as clinical regimes corresponding to different stages of disease or responses to treatment may manifest as different spatial arrangements of tumor and immune cells. Spatial point pattern modeling can be used to partition multiplex tissue images according to these regimes. To this end, we propose a two-stage approach: first, local intensities and pair correlation functions are estimated from the spatial point pattern of cells within each image, and the pair correlation functions are reduced in dimension via spectral decomposition of the covariance function. Second, the estimates are clustered in a Bayesian hierarchical model with spatially-dependent cluster labels. The clusters correspond to regimes of interest that are present across subjects; the cluster labels segment the spatial point patterns according to those regimes. Through Markov Chain Monte Carlo sampling, we jointly estimate and quantify uncertainty in the cluster assignment and spatial characteristics of each cluster. Simulations demonstrate the performance of the method, and it is applied to a set of multiplex immunofluorescence images of diseased pancreatic tissue.

💡 Research Summary

The paper introduces a novel two‑stage statistical framework for segmenting spatial point patterns (SPPs) derived from multiplex immunofluorescence (mIF) images, with a focus on tumor immunology applications. In the first stage, each image is overlaid with a pre‑specified rectangular grid. Within every grid cell, the authors estimate local first‑order characteristics (intensity for each cell type) and second‑order characteristics, namely the pair correlation function (PCF), which captures distance‑dependent interaction between points. Because the PCF is a function of distance, it yields high‑dimensional functional data. To reduce dimensionality while preserving the main variation, the authors perform a spectral decomposition of the covariance function of the transformed PCF curves (square‑rooted and truncated). The leading eigenfunctions serve as functional principal components, and the inner products of each curve with these eigenfunctions produce a set of scores. These scores are concatenated with the intensity estimates to form a Q‑dimensional feature vector ξₙₗ for subject n and grid cell l, which is then centered and scaled across all subjects and cells.

The second stage clusters the collection of ξₙₗ vectors using a Bayesian hierarchical mixture model called the Potts Clustering Model (PCM). The PCM assumes a fixed number M of spatial regimes (clusters) and assigns a latent label Cₙₗ to each grid cell. Conditional on the label, ξₙₗ follows a multivariate normal distribution with regime‑specific mean µ_η and common covariance (often taken as identity after scaling). Crucially, the prior for the label field is a Potts model, which encourages neighboring grid cells to share the same label. The Potts prior is parameterized by an offset α_η (controlling overall prevalence of each regime) and a smoothness parameter ψ ≥ 0 that governs the strength of spatial dependence. The joint posterior over labels, regime means, and Potts parameters is explored via Markov chain Monte Carlo (MCMC) using Gibbs updates for the labels (with Metropolis‑Hastings steps to handle the intractable normalizing constant) and standard conjugate updates for the means and hyper‑parameters.

The authors compare their approach to existing model‑based methods that typically assume only two regimes (clutter vs. feature) and to LISA‑based two‑stage methods that cluster pointwise local statistics without explicit spatial regularization. Their simulations vary grid resolution, number of regimes, and noise levels, demonstrating that the PCM achieves higher clustering accuracy, more coherent spatial regions, and reliable uncertainty quantification. In a real‑world application, the method is applied to a cohort of pancreatic tissue samples imaged with mIF, comprising six clinically defined disease groups (including normal, acute pancreatitis, chronic pancreatitis, and various stages of pancreatic cancer). The pipeline identifies 3–4 spatial regimes per image, each characterized by distinct intensity profiles of tumor, immune, and stromal cells and by characteristic PCF shapes indicating attraction or repulsion at specific distances. The spatial distribution of these regimes correlates with disease status, offering biologically interpretable segmentation that could aid in prognosis or treatment stratification.

Key contributions include: (1) simultaneous use of first‑ and second‑order spatial summaries, (2) functional principal component analysis to compress PCF curves, (3) a Bayesian Potts‑based clustering that enforces spatial smoothness while allowing pooling across subjects, and (4) a full MCMC inference scheme that yields posterior distributions for cluster assignments and regime characteristics. Limitations noted are the dependence on a user‑chosen grid size (which must balance local homogeneity against statistical power), sensitivity of PCF estimation near image borders, and the current inability to model direct interactions among multiple cell types beyond the aggregated PCF. Future work may explore adaptive or irregular tessellations, hierarchical modeling of multi‑type interactions, and integration with downstream predictive tasks. Overall, the paper provides a robust, flexible, and interpretable framework for dissecting complex spatial organization in high‑dimensional multiplex imaging data.