netPCF: geometry-aware pair correlation functions for spatial biology

Preprint

Spatial organisation is a defining feature of biological systems, underpinning cellular interactions, tissue function, disease progression and therapeutic response. Identifying and quantifying spatial organisation may require methods that resolve relationships across spatial scales. The pair correlation function (PCF) quantifies spatial dependence between points across multiple length scales, but its standard Euclidean formulation is poorly suited to data defined on irregular, curved or otherwise structured domains, where tissue geometry may constrain biological organisation and distort Euclidean distances. Here, we introduce netPCF, a geometry-aware extension of the PCF for quantifying spatial organisation on complex biological domains. By representing tissue structures, anatomical surfaces and other constrained geometries as spatial networks, netPCF generalises the PCF beyond extrinsic Euclidean settings. The framework derives the expected behaviour of the statistic under complete spatial randomness using interpretable finite-support kernels, provides bootstrap-based uncertainty quantification, and includes practical criteria for assessing domain discretisation adequacy. We further extend netPCF to marked (labelled) biological data using feature kernels for categorical and continuous attributes, enabling unified analysis of cell identities, marker intensities, phenotypic states, gene expression and other quantitative features on structured domains in any spatial dimension. All methods are implemented in the open-source Python package spacenet.

Synthetic studies show that netPCF recovers classical Euclidean behaviour on sufficiently resolved networks and is robust to common imaging noise. We demonstrate its utility in two biological applications. In three-dimensional imaging mass cytometry data from HER2+ breast carcinoma, netPCF separates tissue architecture-driven proximity from biologically meaningful endothelial and immune cell organisation. In reconstructed surfaces of developing murine embryos, netPCF identifies a transition in the Wnt1 –Wnt6 relationship from short-range co-localisation at E9.5 to spatial exclusion at E11.5, a pattern of ectodermal boundary refinement not captured by prior voxel-wise co-expression analysis. Overall, netPCF provides a statistically grounded and practical framework for quantifying spatial organisation on complex biological domains.

Authors: Moore, JW; Bull, JA; Byrne, HM

• Publication status: Published
• Peer review status: Not peer reviewed
• Preprint server: bioRxiv
• Publication date: 2026-07-07
• DOI: 10.64898/2026.07.02.736020
• EISSN: 2692-8205
• Language: English
• Keywords: developmental biology; point pattern analysis; spatial statistics; spatial analysis; spatial biology; spatial omics; pair correlation function; spatial networks; tumour microenvironment