In this paper, we have characterized the spatial distributions of the nuclei of both benign brain white matter cells and infiltrating glioma cells via a variety of nontrivial statistical microstructural descriptors, including the pair correlation function, structure factor and various nearest neighbor functions that have been profitably utilized in statistical mechanics and material science. To the best of our knowledge, this is the first time that such spatial statistics has been applied in the analysis of histological images. Our primary data was derived from images of clinical microscopic slides and we focused our analysis on cell nuclei because in the CNS, glial and glioma cell borders are not well delineated in routine hematoxylin and eosin stained stained sections. While our GBM images were chosen in areas of high viable tumor cellularity, and histologically apparent non-tumoral structural heterogeneities were subtracted, the minority of non-tumoral nuclei present within our processed GBM images are treated as equal in our analysis. This is a limitation of our study that precludes definitive assignment of the relative contribution of non-malignant cells to the microstructural descriptors we observe as unique to GBM. Addressing this potential limitation requires future identification of molecular markers or other methods that identify malignant cells within GBM with a high degree of specificity (and ideally high sensitivity) across a series of randomly selected GBMs. An alternative approach involves comparative studies following methodical immunohistochemical detection and subtraction of each of the non-malignant cell types within GBM.
We note that GBM masses generally have a small fraction of cells that are multinucleated (i.e., with multiple nuclei in a single cell) 
. Since such nuclei are confined within single cells, their contributions to the spatial statistics are mainly associated with small-distance values and do not significantly affect the correlations on large length-scales. Although this multinucleation would cause certain discrepancies between the statistics associated with the distributions of cell nuclei and the cells themselves, we expect the discrepancies to be negligibly small on large length-scales. In addition, since the nuclei of the two types of cells appear to be similar in size, we believe that any artificial effects due to sectioning should be small. We also note that small perturbations of the individual nucleus positions do not affect the overall statistics associated with the distributions. Since distributions of both normal and abnormal cell nuclei are statistically homogeneous and isotropic, the conclusions based on the evaluations of the particular correlation functions of the 2D histological images examined in this paper should also apply in 3D nuclei distributions 
. Although the 3D Voronoi statistics will be quantitatively different than those in 2D, in terms of the deficiency of not being able to capture long-range correlations, our conclusion also holds.
For comparison purposes, we have also investigated the Voronoi statistics associated with the nuclei distributions. We have demonstrated that while Voronoi statistics cannot clearly distinguish structural differences between normal and abnormal cell nuclei beyond length scale associated with single cells, their salient structural distinctions are very well captured by the aforementioned correlation functions. In particular, by comparing the statistics of the nuclei distributions to the corresponding Poisson reference systems and by directly comparing properly scaled distributions of the nuclei, we have shown that there exist effective repulsions between both normal and abnormal cell nuclei; and that the repulsions between the abnormal cell nuclei are much stronger than that between the normal cell nuclei. This repulsion could simply result from exclusion-volume effects of the cytoplasm (i.e., one cell cannot occupy the same space as another cell) or it could be caused by the competition between local cells for nutritional needs. In addition, abnormal cell nuclei pack considerably more densely and are more spatially correlated than the normal cell nuclei, which is not completely surprising given the corresponding differences in their proliferation rates, nutritional needs and motilities. This in turn leads to deviations between their correlation functions at small length scales (i.e., the characteristic neighbor distances).
Importantly, we found that abnormal cell nuclei possess nontrivial spatial correlations on intermediate and large length scales, as manifested by the strong suppression of cell-density fluctuations on these length scales. This observation is revealing and appears to be new and biologically significant. Such long-range correlations can hardly arise from local packing effects determined by cell shapes and sizes. Possible mechanisms for these long-range correlations include altered structural or cellular components of the tumoral microenvironments. For example, subpopulations of glioblastoma cells can organize around a vascular niche 
. Alternatively, as glial cells are known to generate complex networks of cellular processes 
, the spatial correlations may be maintained by the ultrastructure of glial-derived processes. These possibilities enable a “mutualism” mechanism in which abnormal cells can survive in the stressful tumor environment based on “common goods” principles. There is increasing evidence that cooperative and collective cell behavior plays an important role in the invasion and metastasis of malignant tumors. The observed long-range spatial correlations between abnormal cell nuclei clearly supports the view that tumors are complex dynamic and self-organizing systems rather than a random (unorganized) collection of cells.
This work also provides the structural characteristics of brain glioma cells and sensitive statistical descriptors, which can have potential applications in cancer diagnosis. Recently, analysis of the alterations in nuclear structure 
and wavelet methods 
have been employed to analyze histological samples of prostate cancer and the obtained statistics can be used to devise a classification scheme of the malignancy of the tumor. Our analysis suggests that characterizing distributions of cell nuclei via correlation functions provides a complementary way to analyze histological samples and may have utility in advancing the development of computer assisted diagnostic pathology technologies. The unique patterns of cell nuclei distributions may be a measurable bio-marker of tumor behavior and tumor phenotypes over larger length scales and therefore, may have applications in assessing the extent of infiltration and margin status from a limited sample.
Finally, we note that the specific correlation functions employed here are just a small subset of the zoology of known sophisticated statistically descriptors, including those that have recently been fruitfully applied to characterize the microstructure of heterogeneous media 
. Our studies lay the groundwork for future biological investigations that seek to quantify the relative roles of tumors cells and non-neoplastic cells in shaping the organization of tumoral microenvironments via the descriptors reported here or even more sophisticated correlation functions. The ability to assay the collective behavior of cancer cells provides new opportunities to impede malignant progression through the targeting of tumor self-organization. Moreover, these microstructural descriptors may also have fruitful applications in the study of morphogenesis, for which understanding the spatial correlations among cells is crucial.