Generalized index for spatial data sets as a measure of complete spatial randomness

Pair correlation functions for identifying spatial correlation in discrete domains

Identifying and quantifying spatial correlation are important aspects of studying the collective behavior of multiagent systems. Pair correlation functions (PCFs) are powerful statistical tools that can provide qualitative and quantitative information about correlation between pairs of agents. Despite the numerous PCFs defined for off-lattice domains, only a few recent studies have considered a PCF for discrete domains. Our work extends the study of spatial correlation in discrete domains by defining a new set of PCFs using two natural and intuitive definitions of distance for a square lattice: the taxicab and uniform metric. We show how these PCFs improve upon previous attempts and compare between the quantitative data acquired. We also extend our definitions of the PCF to other types of regular tessellation that have not been studied before, including hexagonal, triangular, and cuboidal. Finally, we provide a comprehensive PCF for any tessellation and metric, allowing investigation of spatial correlation in irregular lattices for which recognizing correlation is less intuitive.

Spatial intratumoral heterogeneity of proliferation in immunohistochemical images of solid tumors

PURPOSE

The interactions of neoplastic cells with each other and the microenvironment are complex. To understand intratumoral heterogeneity, subtle differences should be quantified. Main factors contributing to heterogeneity include the gradient ischemic level within neoplasms, action of microenvironment, mechanisms of intercellular transfer of genetic information, and differential mechanisms of modifications of genetic material/proteins. This may reflect on the expression of biomarkers in the context of prognosis/stratification. Hence, a rigorous approach for assessing the spatial intratumoral heterogeneity of histological biomarker expression with accuracy and reproducibility is required, since patterns in immunohistochemical images can be challenging to identify and describe.

METHODS

A quantitative method that is useful for characterizing complex irregular structures is lacunarity; it is a multiscale technique that exhaustively samples the image, while the decay of its index as a function of window size follows characteristic patterns for different spatial arrangements. In histological images, lacunarity provides a useful measure for the spatial organization of a biomarker when a sampling scheme is employed and relevant features are computed. The proposed approach quantifies the segmented proliferative cells and not the textural content of the histological slide, thus providing a more realistic measure of heterogeneity within the sample space of the tumor region. The aim is to investigate in whole sections of primary pancreatic neuroendocrine neoplasms (pNENs), using whole-slide imaging and image analysis, the spatial intratumoral heterogeneity of Ki-67 immunostains. Unsupervised learning is employed to verify that the approach can partition the tissue sections according to distributional heterogeneity.

RESULTS

The architectural complexity of histological images has shown that single measurements are often insufficient. Inhomogeneity of distribution depends not only on percentage content of proliferation phase but also on how the phase fills the space. Lacunarity curves demonstrate variations in the sampled image sections. Since the spatial distribution of proliferation in each case is different, the width of the curves changes too. Image sections that have smaller numerical variations in the computed features correspond to neoplasms with spatially homogeneous proliferation, while larger variations correspond to cases where proliferation shows various degrees of clumping. Grade 1 (uniform/nonuniform: 74%/26%) and grade 3 (uniform: 100%) pNENs demonstrate a more homogeneous proliferation with grade 1 neoplasms being more variant, while grade 2 tumor regions render a more diverse landscape (50%/50%). Hence, some cases show an increased degree of spatial heterogeneity comparing to others with similar grade. Whether this is a sign of different tumor biology and an association with a more benign/malignant clinical course needs to be investigated further. The extent and range of spatial heterogeneity has the potential to be evaluated as a prognostic marker.

CONCLUSIONS

The association with tumor grade as well as the rationale that the methodology reflects true tumor architecture supports the technical soundness of the method. This reflects a general approach which is relevant to other solid tumors and biomarkers. Drawing upon the merits of computational biomedicine, the approach uncovers salient features for use in future studies of clinical relevance.

Positive feedback can lead to dynamic nanometer-scale clustering on cell membranes

Clustering of molecules on biological membranes is a widely observed phenomenon. A key example is the clustering of the oncoprotein Ras, which is known to be important for signal transduction in mammalian cells. Yet, the mechanism by which Ras clusters form and are maintained remains unclear. Recently, it has been discovered that activated Ras promotes further Ras activation. Here we show using particle-based simulation that this positive feedback is sufficient to produce persistent clusters of active Ras molecules at the nanometer scale via a dynamic nucleation mechanism. Furthermore, we find that our cluster statistics are consistent with experimental observations of the Ras system. Interestingly, we show that our model does not support a Turing regime of macroscopic reaction-diffusion patterning, and therefore that the clustering we observe is a purely stochastic effect, arising from the coupling of positive feedback with the discrete nature of individual molecules. These results underscore the importance of stochastic and dynamic properties of reaction diffusion systems for biological behavior.

Quantifying two-dimensional filamentous and invasive growth spatial patterns in yeast colonies

The top-view, two-dimensional spatial patterning of non-uniform growth in a Saccharomyces cerevisiae yeast colony is considered. Experimental images are processed to obtain data sets that provide spatial information on the cell-area that is occupied by the colony. A method is developed that allows for the analysis of the spatial distribution with three metrics. The growth of the colony is quantified in both the radial direction from the centre of the colony and in the angular direction in a prescribed outer region of the colony. It is shown that during the period of 100–200 hours from the start of the growth of the colony there is an increasing amount of non-uniform growth. The statistical framework outlined in this work provides a platform for comparative quantitative assays of strain-specific mechanisms, with potential implementation in inferencing algorithms used for parameter-rate estimation.

In nutrient-depleted environments, it is commonly observed that strains of the yeast Saccharomyces cerevisiae forage by the mechanisms of filamentous and invasive growth. How do we quantify this spatial patterning of outward growth from a yeast colony? Previous studies have primarily relied on measuring the amount of growth, but do not take into account the spatial distribution of this highly non-uniform process. We fill this void by providing a statistical approach that enables the quantification of this important spatial information. This approach enables a more detailed mathematical analysis of the growth process and should allow the precise definition of mutant phenotypes, thus enabling a detailed analysis of the genetic control of morphogenesis.

Spectral analysis of pair-correlation bandwidth: application to cell biology images

Images from cell biology experiments often indicate the presence of cell clustering, which can provide insight into the mechanisms driving the collective cell behaviour. Pair-correlation functions provide quantitative information about the presence, or absence, of clustering in a spatial distribution of cells. This is because the pair-correlation function describes the ratio of the abundance of pairs of cells, separated by a particular distance, relative to a randomly distributed reference population. Pair-correlation functions are often presented as a kernel density estimate where the frequency of pairs of objects are grouped using a particular bandwidth (or bin width), Δ>0. The choice of bandwidth has a dramatic impact: choosing Δ too large produces a pair-correlation function that contains insufficient information, whereas choosing Δ too small produces a pair-correlation signal dominated by fluctuations. Presently, there is little guidance available regarding how to make an objective choice of Δ. We present a new technique to choose Δ by analysing the power spectrum of the discrete Fourier transform of the pair-correlation function. Using synthetic simulation data, we confirm that our approach allows us to objectively choose Δ such that the appropriately binned pair-correlation function captures known features in uniform and clustered synthetic images. We also apply our technique to images from two different cell biology assays. The first assay corresponds to an approximately uniform distribution of cells, while the second assay involves a time series of images of a cell population which forms aggregates over time. The appropriately binned pair-correlation function allows us to make quantitative inferences about the average aggregate size, as well as quantifying how the average aggregate size changes with time.

Distinguishing between mechanisms of cell aggregation using pair-correlation functions

Many cell types form clumps or aggregates when cultured in vitro through a variety of mechanisms including rapid cell proliferation, chemotaxis, or direct cell-to-cell contact. In this paper we develop an agent-based model to explore the formation of aggregates in cultures where cells are initially distributed uniformly, at random, on a two-dimensional substrate. Our model includes unbiased random cell motion, together with two mechanisms which can produce cell aggregates: (i) rapid cell proliferation and (ii) a biased cell motility mechanism where cells can sense other cells within a finite range, and will tend to move towards areas with higher numbers of cells. We then introduce a pair-correlation function which allows us to quantify aspects of the spatial patterns produced by our agent-based model. In particular, these pair-correlation functions are able to detect differences between domains populated uniformly at random (i.e. at the exclusion complete spatial randomness (ECSR) state) and those where the proliferation and biased motion rules have been employed - even when such differences are not obvious to the naked eye. The pair-correlation function can also detect the emergence of a characteristic inter-aggregate distance which occurs when the biased motion mechanism is dominant, and is not observed when cell proliferation is the main mechanism of aggregate formation. This suggests that applying the pair-correlation function to experimental images of cell aggregates may provide information about the mechanism associated with observed aggregates. As a proof of concept, we perform such analysis for images of cancer cell aggregates, which are known to be associated with rapid proliferation. The results of our analysis are consistent with the predictions of the proliferation-based simulations, which supports the potential usefulness of pair correlation functions for providing insight into the mechanisms of aggregate formation.

Quantifying spatial structure in experimental observations and agent-based simulations using pair-correlation functions

We define a pair-correlation function that can be used to characterize spatiotemporal patterning in experimental images and snapshots from discrete simulations. Unlike previous pair-correlation functions, the pair-correlation functions developed here depend on the location and size of objects. The pair-correlation function can be used to indicate complete spatial randomness, aggregation, or segregation over a range of length scales, and quantifies spatial structures such as the shape, size, and distribution of clusters. Comparing pair-correlation data for various experimental and simulation images illustrates their potential use as a summary statistic for calibrating discrete models of various physical processes.