## ALGORITHMS \\

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## https://github.com/dh2nguyen/

Translating the Architectural Complexity of the Colon or Polyp into a Sinusoid Wave for Classification via the Fast Fourier Transform

David H. Nguyen, Ph.D.

Affiliate Scientist

Dept. of Cellular & Tissue Imaging

Division of Molecular Biophysics and Integrated Bioimaging

Lawrence Berkeley National Laboratory

Archived at: ArXiv:1801.06752 [q-bio.TO]

Abstract

There is no method to quantify the spatial complexity within colon polyps. This paper describes a spatial transformation that translates the tissue architecture within a polyp, or a normal colon lining, into a complex sinusoid wave composed of discrete points. This sinusoid wave can then undergo the Fast Fourier Transform to obtain a spectrum of frequencies that represents the sinusoid wave. This spectrum can then serve as a signature of the spatial complexity [an index] within the polyp. By overlaying vertical lines that radiate from the bottom middle [like a fold-out fan] of an image of a polyp stained by hematoxylin and eosin, the image is segmented into sectors. Each vertical line also forms an angle with the horizontal axis of the image, ranging from 0 degrees to 180 degrees rising counter clockwise. Each vertical line will intersect with various features of the polyp [border of lumens, border of epithelial lining]. Each of these intersections is a point that can be characterized by its distance from the origin [this distance is also a magnitude of that point]. Thus, each intersection between radial line and polyp feature can be mapped by polar coordinates [radius length, angle measure]. By summing the distance of all points along the same radial line, each radial line that divides the image becomes one value. Plotting these values [y variable] against the angle of each radial line from the horizontal axis [x variable] results in a sinusoid wave consisting of discrete points. This method is referred to as the Linearized Compressed Polar Coordinates [LCPC] Transform. The LCPC transform, in conjunction with the Fast Fourier Transform, can reduce the complexity of visually hidden histological grades in colon polyps into categories of similar wave frequencies [each histological grade has a signature consisting of a handful of frequencies].

Detecting Partial Rosettes in Tumor Histopathology Using the Cross Product

David H. Nguyen, Ph.D.

Affiliate Scientist

Dept. of Cellular & Tissue Imaging

Division of Molecular Biophysics and Integrated Bioimaging

Lawrence Berkeley National Laboratory

Archived at: arXiv:1710.06593 [q-bio.TO]

Abstract

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.

Quantifying and Visualizing Hidden Preferential Aggregations Amid Heterogeneity

David H. Nguyen, Ph.D.

Affiliate Scientist

Dept. of Cellular & Tissue Imaging

Division of Molecular Biophysics and Integrated Bioimaging

Lawrence Berkeley National Laboratory

Archived at: arXiv:1704.07567 [q-bio.TO]

Abstract

Biological systems often exhibit a heterogeneous arrangement of objects, such as assorted nuclear chromatin patterns in a tumor, assorted species of bacteria in biofilms, or assorted aggregates of subcellular particles. Principle Component Analysis (PCA) and Multiple Component Analysis (MCA) provide information about which features in multidimensional data aggregate, but do not provide in situ spatial information about these aggregations. This paper outlines the Numericized Histogram Score (NHS) algorithm, which converts the histogram distribution of shortest distances between objects into a continuous variable that can be represented as a spatial heatmap. A histogram can be transformed into an intensity value by assigning a weighting factor to each sequential bin. Each object in an image can be replaced by its NHS value, which when calibrated to a color scale results in a heatmap. These spatial heatmaps reveal regions of aggregation amid heterogeneity that would otherwise mask loco-regional spatial associations, which will be especially useful in the field of digital pathology. In addition to visualizing aggregations as heatmaps, the ability to calculate degrees of recurring patterns of aggregation allows investigators to stratify samples for further insights into clinical outcome, response to treatment, or omic subtypes (genomic, transcriptomic, proteomic, metabolomic, etc.).

Quantifying Hidden Architectural Patterns in Metaplastic Tumors by Calculating the Quadrant-Slope Index (QSI)

David H. Nguyen, Ph.D.

Affiliate Scientist

Dept. of Cellular & Tissue Imaging

Division of Molecular Biophysics and Integrated Bioimaging

Lawrence Berkeley National Laboratory

Archived at: arXiv:1704.07571 [q-bio.TO]

Abstract

The Quadrant-Slope Index (QSI) method was created in order to detect subtle patterns of organization in tumor images that have metaplastic elements, such as streams of spindle cells. However, metaplastic tumors also have nuclei that may be aligned like a stream but are not obvious to the pathologist because the shape of the cytoplasm is unclear. The previous method that I developed, the Nearest-Neighbor Angular Profile (N-NAP) method, is good for detecting subtle patterns of order based on the assumption that breast tumor cells are attempting to arrange themselves side-by-side (like bricks), as in the luminal compartment of a normal mammary gland. However, this assumption is not optimal for detecting cellular arrangements that are head-to-tail, such as in streams of spindle cells. Metaplastic carcinomas of the breast (i.e. basal-like breast cancers, triple-negative breast cancers) are believed to be derived from the stem or progenitor cells that reside in the basal/myoepithelial compartment of the normal mammary gland. Epithelial cells in the basal/myoepithelial compartment arrange themselves in an head-to-tail fashion, forming a net that surrounds the luminal compartment. If cancer cells in a metaplastic tumor are trying to be normal, the optimal way to detect subtle regions of them attempting to be ordered normally should highlight the head-to-tail alignment of cells.

Quantifying Subtle Regions of Order and Disorder in Tumor Architecture by Calculating the Nearest-Neighbor Angular Profile

David H. Nguyen, Ph.D.

Affiliate Scientist

Dept. of Cellular & Tissue Imaging

Division of Molecular Biophysics and Integrated Bioimaging

Lawrence Berkeley National Laboratory

Archived at: arXiv:1704.07567 [q-bio.TO]

Abstract

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.