Techniques for analysing pattern formation in populations of stem cells and their progeny

Quantification of spatial and phenotypic heterogeneity in an agent-based model of tumour-macrophage interactions

We introduce a new spatial statistic, the weighted pair correlation function (wPCF). The wPCF extends the existing pair correlation function (PCF) and cross-PCF to describe spatial relationships between points marked with combinations of discrete and continuous labels. We validate its use through application to a new 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. We also define a distinct 'PCF signature' that characterises each of the three Es of immunoediting, by combining wPCF measurements with the cross-PCF describing interactions between vessels and tumour cells. 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.

Combining multiple spatial statistics enhances the description of immune cell localisation within tumours

Digital pathology enables computational analysis algorithms to be applied at scale to histological images. An example is the identification of immune cells within solid tumours. Image analysis algorithms can extract precise cell locations from immunohistochemistry slides, but the resulting spatial coordinates, or point patterns, can be difficult to interpret. Since localisation of immune cells within tumours may reflect their functional status and correlates with patient prognosis, novel descriptors of their spatial distributions are of biological and clinical interest. A range of spatial statistics have been used to analyse such point patterns but, individually, these approaches only partially describe complex immune cell distributions. In this study, we apply three spatial statistics to locations of CD68+ macrophages within human head and neck tumours, and show that images grouped semi-quantitatively by a pathologist share similar statistics. We generate a synthetic dataset which emulates human samples and use it to demonstrate that combining multiple spatial statistics with a maximum likelihood approach better predicts human classifications than any single statistic. We can also estimate the error associated with our classifications. Importantly, this methodology is adaptable and can be extended to other histological investigations or applied to point patterns outside of histology.

The devil is in the mesoscale: Mechanical and behavioural heterogeneity in collective cell movement

Heterogeneity within cell populations can be an important aspect affecting their collective movement and tissue-mechanical properties, determining for example their effective viscoelasticity. Differences in cell-level properties and behaviour within a group of moving cells can give rise to unexpected and non-intuitive behaviours at the tissue level. Such emergent phenomena often manifest themselves through spatiotemporal patterns at an intermediate 'mesoscale' between cell and tissue scales, typically involving tens of cells. Focussing on the development of embryonic animal tissues, we review recent evidence for the importance of heterogeneity at the mesoscale for collective cell migration and convergence and extension movements. We further discuss approaches to incorporate heterogeneity into computational models to complement experimental investigations.

Understanding interactions between populations: Individual based modelling and quantification using pair correlation functions

Understanding the underlying mechanisms that produce the huge variety of swarming and aggregation patterns in animals and cells is fundamental in ecology, developmental biology, and regenerative medicine, to name but a few examples. Depending upon the nature of the interactions between individuals (cells or animals), a variety of different large-scale spatial patterns can be observed in their distribution; examples include cell aggregates, stripes of different coloured skin cells, etc. For the case where all individuals are of the same type (i.e., all interactions are alike), a considerable literature already exists on how the collective organisation depends on the inter-individual interactions. Here, we focus on the less studied case where there are two different types of individuals present. Whilst a number of continuum models of this scenario exist, it can be difficult to compare these models to experimental data, since real cells and animals are discrete. In order to overcome this problem, we develop an agent-based model to simulate some archetypal mechanisms involving attraction and repulsion. However, with this approach (as with experiments), each realisation of the model is different, due to stochastic effects. In order to make useful comparisons between simulations and experimental data, we need to identify the robust features of the spatial distributions of the two species which persist over many realisations of the model (for example, the size of aggregates, degree of segregation or intermixing of the two species). In some cases, it is possible to do this by simple visual inspection. In others, the features of the pattern are not so clear to the unaided eye. In this paper, we introduce a pair correlation function (PCF), which allows us to analyse multi-species spatial distributions quantitatively. We show how the differing strengths of inter-individual attraction and repulsion between species give rise to different spatial patterns, and how the PCF can be used to quantify these differences, even when it might be impossible to recognise them visually.

Identifying the necrotic zone boundary in tumour spheroids with pair-correlation functions

Automatic identification of the necrotic zone boundary is important in the assessment of treatments on in vitro tumour spheroids. This has been difficult especially when the difference in cell density between the necrotic and viable zones of a tumour spheroid is small. To help overcome this problem, we developed novel one-dimensional pair-correlation functions (PCFs) to provide quantitative estimates of the radial distance of the necrotic zone boundary from the centre of a tumour spheroid. We validate our approach on synthetic tumour spheroids in which the position of the necrotic zone boundary is known a priori It is then applied to nine real tumour spheroids imaged with light sheet-based fluorescence microscopy. PCF estimates of the necrotic zone boundary are compared with those of a human expert and an existing standard computational method.

Reconciling diverse mammalian pigmentation patterns with a fundamental mathematical model

Bands of colour extending laterally from the dorsal to ventral trunk are a common feature of mouse chimeras. These stripes were originally taken as evidence of the directed dorsoventral migration of melanoblasts (the embryonic precursors of melanocytes) as they colonize the developing skin. Depigmented ‘belly spots' in mice with mutations in the receptor tyrosine kinase Kit are thought to represent a failure of this colonization, either due to impaired migration or proliferation. Tracing of single melanoblast clones, however, has revealed a diffuse distribution with high levels of axial mixing—hard to reconcile with directed migration. Here we construct an agent-based stochastic model calibrated by experimental measurements to investigate the formation of diffuse clones, chimeric stripes and belly spots. Our observations indicate that melanoblast colonization likely proceeds through a process of undirected migration, proliferation and tissue expansion, and that reduced proliferation is the cause of the belly spots in Kit mutants.

How embryonic melanoblast behaviour influences adult pigmentation patterns and causes patterning defects is unclear. Here, Mort et al. construct a stochastic model parameterised experimentally to show that melanoblast migration is undirected and that reduced proliferation causes patterning defects.