Pathologists routinely classify breast tumors according to recurring patterns of nuclear grades, cytoplasmic coloration, and large-scale morphological formations (i.e. streams of spindle cells, adenoid islands, etc.). The fact that there are large-scale morphological formations suggest that tumor cells still possess the genetic programming to arrange themselves in orderly patterns. However, small regions of order or subtle patterns of order are invisible to the human eye. The ability to detect subtle regions of order and correlate them with clinical outcome and resistance to treatment can enhance diagnostic efficacy. By measuring the acute angle that results when the line extending from the longest length within a nucleus intersects with the corresponding line of an adjacent nucleus, the degree of alignment between two adjacent nuclei can be measured. Through a series of systematic transformations, subtle regions of order and disorder within a tumor image can be quantified and visualized in the form of a heat map. This numerical transformation of spatial relationships between nuclei within tumors allows for the detection of subtly ordered regions.
The fact that pathologists can classify neoplastic lesions by recurring large-scale morphological features suggests that there is a certain degree of order within the architecture of a tumor [1]. However, beyond broad classifications such as adenocarcinoma, apocrine carcinoma, metaplastic carcinoma, medullary carcinoma, and other categories (Figure 1), the human eye cannot detect smaller regions of order within a tumor slice visualized by hematoxylin & eosin (H&E) stains [2]. Subtle changes in architecture are hard with measure. Without numerical analysis, statistical approaches cannot quantify subtle changes in local regions within a tumor.
Normal mammary epithelia have their nuclei aligned side-by-side, with the longest length nearly parallel with each other (Figure 2). However, since neighboring nuclei are not uniformly aligned, the lines that extend from their longest length intersect each other at acute angles. By measuring the angles formed by the lines that extend from the maximum nuclear axis of two adjacent nuclei, the visual near-uniformity can be represented numerically in the form of degrees. This quantitative approach allows for detection of subtle changes in nuclear packing that are invisible to the human eye. This method will be referred to as the Nearest-Neighbor Angular Profile (N-NAP).
Breast tumors are a mixture of multiple cell types, including ductal epithelia, fibroblasts, vasculature, nerves, and immune cells [3]. Though the nuclear morphology of epithelial cells in breast cancers are distinct from other cells types and can easily be distinguished by a pathologist, the arrangement of cells and their nuclei can appear random or highly complex, depending on other visual features present in normal mammary ducts.
Though arranged in ways that do not intuitively suggest a pattern, the alignment of adjacent nuclei in tumors can be quantified using the N-NAP approach. Pathologists routinely identify large morphological features in tumors (Figure 1), such papillary (finger-like) projections, adenoid (circular clusters) foci, and cells arranged as if flowing in a fluid path (referred to as the “streaming” effect) [2]. These large pathological features are the result of tumor cells attempting to do what normal epithelial cells in the mammary gland do. They are trying to form hollow, branching ducts [4]. Tumor cells that are not forming the large pathological features are assumed to be in a random or nonordered pattern.
However, if breast tumor cells still possess the programming to be like normal breast epithelial cells, though the programming is not fully functional, it can be hypothesized that the tumor cells are attempting to order themselves on a micro scale. This scale would be too small to be detected by the human eye as more than random arrangements. However, a precise quantitative approach applied to a large sample size would be able to detect subtle regions of order amid a sea of seeming randomness. Quantitative approaches have successfully detected morphological subtly in nuclei and tissues [5,6]. Nuclei that are adjacent to each other and aligned in the same direction have longest length axes that intersect to form acute angles. The closer to having parallel alignment, the smaller the angle formed by the longest length axes will be (green angles). The misalignment of cells resulting from ductal hyperplasia will cause the average angle size to be larger than that found in normal ducts. The nearestneighbor alignment profile is useful for detecting subtle changes in alignment.
In order to be systematic, a nucleus’ nearest neighbors are defined as those within a radius extending from the edge of the nucleus. The nucleus of interest is referred to as the “central nucleus,” while the nuclei around it are referred to as neighbors. To objectively define a ring for detecting the nearest neighbors around a central nucleus, the ring’s width is defined to be the longest length within the central nucleus (Figure 3A,C).
Since there may be fewer neighboring nuclei around some central nuclei due to the total size of the cells in that location, a second ring was defined in order to include more neighboring nuclei. The width of the second ring, which is concentric to the first ring, is also defined as the longest length within the central nucleus (Figure 3B,D).
Plotting the measure of the angles between a central nucleus and its primary and secondary nearest neighbors creates a representation of the spatial relationships between nuclei that cluster near each other. (Figure 4B,D). Since the nuclei of luminal epithelial cells are packed nearly parallel with each other (Figure 5C), the angle between adjacent nuclei will tend to be <45°.
Slope of Line = 32
Angles < 45°Angles > 45°F igure 5. The arrangement of two adjacent nuclei can be quantified by the acute angle that results when the lines from their longest axes intersect. (A) The measures of each angle between nearest-neighbor nuclei and the
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