Quantification of spatial and phenotypic heterogeneity in an agent-based model of tumour-macrophage interactions
Authors: Bull Joshua A. , Byrne Helen M. 
Published: May 2022

We introduce a new spatial statistic, the weighted pair correlation function (wPCF). The wPCF extends the existing pair correlation function (PCF) to describe spatial relationships between points marked with combinations of discrete and continuous labels. We validate its use through application to an agent-based model (ABM) which simulates interactions between macrophages and tumour cells. These interactions are influenced by the spatial positions of the cells and by macrophage phenotype, a continuous variable that ranges from anti-tumour to pro-tumour. By varying model parameters that regulate macrophage phenotype, we show that the ABM exhibits behaviours which resemble the ‘three Es of cancer immunoediting’: Equilibrium, Escape, and Elimination.

We use the wPCF to analyse synthetic images generated by the ABM. We show that the wPCF generates a ‘human readable’ statistical summary of where macrophages with different phenotypes are located relative to both blood vessels and tumour cells. In combination with the cross-PCF (describing interactions between vessels and tumour cells), we show further that each of the three Es of immunoediting is characterised by a distinct ‘PCF signature’. By applying dimension reduction techniques to this signature, we identify its key features and train a support vector machine classifier to distinguish between simulation outputs based on their PCF signature. This proof-of-concept study shows how multiple spatial statistics can be combined to analyse the complex spatial features that the ABM generates, and to partition them into interpretable groups.

The intricate spatial features produced by the ABM are similar to those generated by state-of-the-art multiplex imaging techniques which distinguish the spatial distribution and intensity of multiple biomarkers in biological tissue regions. Applying methods such as the wPCF to multiplex imaging data would exploit the continuous variation in biomarker intensities and generate more detailed characterisation of the spatial and phenotypic heterogeneity in tissue samples.

Author summary

Multiplex images provide exquisitely detailed information about the spatial distribution and intensity of up to 40 biomarkers within two-dimensional tissue regions, creating challenges and opportunities for quantitative analysis. Although stain intensities are measured on a continuous scale, they are typically converted into discrete labels to simplify subsequent spatial analysis. In this paper we propose a new spatial statistic, the weighted pair correlation function (wPCF), which exploits, rather than neglects, the continuous variation in stain intensity contained in multiplex images, and can characterise both spatial and phenotypic heterogeneity.

As proof-of-principle, we apply the wPCF to synthetic data that resemble multiplex images of solid tumours. We generate data from an agent-based model (ABM) that simulates macrophage-tumour interactions. The wPCF shows how the continuous label describing macrophage phenotype is spatially related to categorical labels associated with tumour cells and blood vessels. We demonstrate that correlation functions can categorise spatial relationships in a manner which is interpretable and quantitative.

The methods we present can be used to analyse both ABM simulations and multiplex imaging data, with applications that go beyond macrophage phenotype to include other biological processes that exhibit continuous variation (e.g., cancer cell stemness, biomarkers for T-cell exhaustion, and levels of oxygenation).