Detecting Partial Rosettes in Tumor Histopathology Using The Cross Product

Tumors of the eye and nervous system often exhibit an arch-like arrangement of nuclei, called rosettes. Pathologists are able to identify rosettes [full circles] and the presence of partial rosettes [semi-circles] and interpret this as a sign of differentiation in a tumor. However, there is no objective method to quantitate the many partial rosettes that are obvious or not obvious to the naked eye. This paper proposes a mathematical algorithm to computationally detect the presence of obvious or non-obvious partial rosettes, henceforth referred to as Nguyen-Wu Partial Rosettes. Quantifying the degree of partial rosettes present in a tumor may allow pathologists to stratify tumors into more refined groups that may respond better to therapy or have different clinical outcomes. The Midline Cross Product [MCP] algorithm calculates the magnitude of two cross products and adds them together to obtain one value. Each of the two cross products results from [1] the line that connects the midpoints of longest lengths of two neighboring ovals, and [2] the line that extends from the midpoint from one longest length and is perpendicular to that longest length. The MCP algorithm makes nuclei that are arranged in consecutive rows and arches quantitatively distinct from nuclei that are arranged next to each other in a disorderly manner.