ABSTRACT Advancements in image analysis have recently made it possible to segment the cells and nuclei, of a wide variety of tissues, from 3D images collected using fluorescence confocal microscopy [Ortiz de Solorzano et al. J. Microscopy 193:212-226, 1999]. This has made it possible to analyze the spatial organization of individual cells and nuclei within the natural tissue context. We present here a spatial statistical method which examines an arbitrary 3D distribution of cells of two different types and determines the probability that the cells are randomly mixed, cells of one type are clustered, or cells of different types are preferentially associated. Beginning with a segmented 3D image of cells (or nuclei), the Voronoi diagram is calculated to indicate the nearest neighbour relationships of the cells. Then, in a test image of the same topology, cells are randomly assigned a type in the same proportions as in the actual specimen and the ratio of cells with nearest neighbours of the same type versus the other types is calculated. Repetition of this random assignment is used to generate a distribution function which is specific for the tissue image. Comparison of the ratios for the actual sample to this distribution assigns probabilities for the conditions defined above. The technique is being used to analyze the organization of genetically normal versus abnormal cells in cancer tissue.  Keywords:  3D spatial analysis, tissue organization, tissue segmentation, fluorescence confocal microscopy ------------------------------ INTRODUCTION Recent technologies in three dimensional (3D) fluorescence imaging, specific fluorescent bio-markers combined with an ever increasing computational power has ensured that acquisition of large, detailed, multi-probe, 3D images is becoming routine in biomedical research. This technology is providing images of the spatial relationships between cells in their natural tissue context together with details of their subcellular and extracelluler properties. However, what is lagging are the tools for analyzing such large and complex images. We present here a spatial statistical method which examines an arbitrary 3D distribution of cells of two different types, or cells exhibiting two different types of features, and determines the probability that the cells are randomly mixed, cells of one type are clustered, or cells of different types are preferentially associated. The analysis involves a process by which structures in the tissue are segmented and then individually scored based on measurements of a given property. The measured property of interest may involve the segmented structure itself, for example its size or directional polarity, or might involve the abundance or distribution of other cellular components. Currently, we are studying the spatial organization of genetically aberrant cells relative to genetically normal cells in cancer. The image analysis technique has allowed us to quantitate the high cell-to-cell heterogeneity observed in the copy numbers of specific DNA sequences in human breast tumors [1]. For this report, we use images from our breast cancer work to demonstrate the spatial statistical analysis technique and to show that, even in 2D, this technique easily detects spatial associations that are not obvious by eye. METHODS & RESULTS The input for this analysis is a 3D image containing individually segmented structures which have been classified into one of two types in terms of a measured property. In the case of our breast cancer work, the structures segmented were cell nuclei and the measured property was the normal or abnormal copy number of specific DNA sequences. For illustration, the implementation of our spatial statistical analysis technique is done on a 2D slice from a 3D image stack of ToPro (Molecular Probes, Eugene, OR) labeled DNA in human breast tissue (Figure 1A). Staining the total DNA allowed the nuclei within the tissue to be segmented by a semi-automated algorithm [2]. Segmentation produces an image mask, referred to as a labeled image (Figure 1B), where all pixels within each nuclei are assigned a constant but unique value in a black background. This value is used to identify individual nuclei by number. FIGURE 1 To determine the nearest-neighbour relationships specific for these nuclei, segmented nuclei in the labeled image are dilated, one voxel layer at a time, for a fixed number of iterations. Dilation is only allowed into the background, not into other structures and thus their shape was not necessarily conserved. The Voronoi diagram-like surfaces that form between two dilated nuclei (Figure 2A) are used to infer which pairs of nuclei are nearest neighbours. For the breast tumor tissue, nuclei were large compared to the size of the cells and thus a relatively small number of dilation iterations were required to detect the nearest neighbour interactions.

Generally, the number of dilation steps equals the radius, in voxels, of an average nuclei. The set of nearest-neighbour pairs are then tabulated in a triangular look-up-table (LUT) (Figure 2B). Rows and columns of the LUT are labeled with the numbers representing the segmented nuclei and elements of the LUT are assigned the value 1 when the nuclei corresponding to the row and column number are neighbours. This provides quick reference to which nuclei are adjacent to which others. FIGURE 2 Nuclei in the labeled image are then scored as either dark (d), for normal nuclei with 2 copies of each chromosome, or light (l) for an aberrant number of chromosomes (Figure 3). To quantitate the spatial organization of the dark and light nuclei in the image, we compute the fractions (Fdd, Fll, and Fdl) of dark-dark, light-light and dark-light nearest-neighbour interactions by dividing the number of each such interaction, respectively, by the total number of interactions. As illustrated in figure 3, there are 18 nuclei, of which 9 are dark (normal nuclei) and 9 are light (aberrant nuclei). There are 22 nearest-neighbour interactions (the number of non-zero elements in the LUT), 1 dark-dark interaction, 4 light-light interactions and 17 dark-light interactions. These result in: Fdd=0.05, Fll=0.18 and Fdl=0.77, respectively. Note that Fdd + Fll + Fdl =1. FIGURE 3 We then ask how these values would compare if the labeled image had the same numbers of dark and light nuclei but the assignment to specific nuclei was random. To answer this, the labeled image is randomly assigned to generate an ensemble of random states. For the breast cancer work, this was done 1000 times. For each random assignment the fraction of each type of nearest-neighbour interaction, Fdd, Fll and Fdl, is calculated and used to generate probability histograms (Figures 4A, 4B, 4C) respectively. Each distribution shows the range and most probable fraction of nearest-neighbour interactions. A typical random scoring of the topology is shown in figure 4D. FIGURE 4 In order to determine if nuclei of the same type are clustered, randomly mixed or nuclei of different types are preferentially associated, the values, Fdd, Fll and Fdl, from the actual image are compared to their respective distributions. If the fraction of any interaction type Fxy, for example, is significantly above its random distribution it would imply that nuclei of type x and y are preferentially clustered. If Fxy falls within the distribution then nuclei types x and y are randomly distributed. However, if Fxy is significantly below the distribution, it implies that there are fewer x-y type neighbouring interactions than in the random case and that x and y nuclei are distributed so that they are not likely to be neighbours. Integrating the random probability distributions, from zero to their appropriate fractions Fxy, result in confidence probabilities. Low confidence probabilities indicate that the distribution of nuclei is not random because the nuclei are organized such that there are few x-y type interactions. High values (close to 1) of the confidence probability indicate that the nuclei are organized such that there are many x-y type interactions. Intermediate values indicate randomness. For the image in figure 3 the fractions of nearest-neighbour interactions (Fdd=0.05, Fll=0.18 and Fdl=0.77) result in confidence probabilities of 0.002, 0.3712 and 0.9964 respectively. These clearly indicate that the distribution of the two types of nuclei is far from random.

Both Fdd and Fdl lie outside their corresponding random probability distribution functions and indicate that in the actual image the fraction of normal-normal interactions, Fdd, is too low and the fraction of normal-aberrant interactions, Fdl, is too high for this scoring to be random. Rather, these data clearly indicate that normal and aberrant nuclei are preferentially anti-clustered. In contrast, for a typical random image in figure 4D Fdd=0.14, Fll=0.32 and Fdl=0.54 and the corresponding confidence probabilities are 0.20, 0.88 and 0.66, respectively. Since none of these data are close to either probabilities of zero or one, they demonstrate that the image in figure 4D is random.  DISCUSSION In this study we implemented an algorithm that calculates a statistical probability that cells of one type are clustered, randomly mixed with cells of another type or cells of different types are spatially associated with each other. What distinguishes this technique from other spatial statistical analysis methods [3 & 4] is that it makes no assumptions about how the objects are distributed in space. This makes it particularly suitable for spatial analysis of cell in tissue. Further, errors resulting from the analysis (for example undetected nuclei, assignment of nuclei to the wrong type, or selection of a region of tissue dominated by one type), bias the results towards the random case. Consequently, this gives additional confidence that a non-random result is correct.  We foresee several useful extensions to the current method. These include analysis of three or more cell types, use of the area of the surface between dilated segmented structures as a measure of the strength of the nearest neighbour interaction and the spatial analysis of non nearest-neighbours. Non-random, non-nearest-neighbours could conceivably exist in complex tissue, yet would be difficult to observe visually. The current application of this technique is the analysis of the spatial patterns of genetically aberrant versus genetically normal cells in breast cancer tissue [1]. Spatial clustering of the genetically aberrantly cells may be indicative of their clonal expansion. On the other hand the deliberate association of genetically aberrant and normal cells should not be unexpected, given the high level of cell-to-cell communication that takes place in normal and cancerous tissue. Other potential applications could be to study the pattern of progesterone receptor and estrogen receptor positive cells, which are known to be heterogeneously distributed in mouse mammary epithelium [5 & 6]. ACKNOWLEDGMENTS This work has been supported by the Director, Office of Energy Research, Office of Health and Environmental Research of the U.S. Department of Energy under contract number DE-AC03-76SF00098, NIH grant CA-67412 and a contract with Carl Zeiss Inc. REFERENCES [1] S.J. Lockett, E. G.Rodriguez, C. Ortiz de Solorzano, A Jones, K. Chin, J Gray 1999 Technology for histogenetic analysis of breast cancer Proc. Amer. Ass. for Cancer Res. 40:#480 Chin K, Kuo w-l, Garcia Rodriquez E, Ortiz de Solorzano C, Lockett S, Gray JW 1999 Rates of genome instability in human breast cancers: FISH analysis in vitro and in vivo. Breast Cancer Reserch and Treatment 57, 1:133 [2] C. Ortiz de Solorzano, E.G. Rodriguez, A. Jones, D. Pinkel, J.W. Gray, D. Sudar, S.J. Lockett 1999 Segmentation of confocal microscope images of cell nuclei in thick tissue sections. J. Microscopy 193:212-226 [3] R. Albert, T. Schindewolf, I Baumann, H. Harms 1992 Three-dimensional image processing for morphometric analysis of epithelium sections Cytometry 13:759-765 [4] B. Weyn, G. van de Wouwer, S. Kumar-Singh, A. van Daele, P. Scheunders, E. van Mark, W. Jacob 1999 Computer-assisted differential diagnosis of malignant mesothelioma based on syntactic structure analysis Cytometry 35:23-29 [5] Shyamala, G. 1999 Progesterone signaling and mammary gland morphogenesis. J Mammary Gland Biol Neoplasia, 4(1):89-104. [6] Silberstein, GB; Van Horn, K; Shyamala, G; Daniel, CW. 1996 Progesterone receptors in the mouse mammary duct: distribution and developmental regulation. Cell Growth and Differentiation, 7(7):945-52.