Stable Topological Summaries for Analyzing the Organization of Cells in a Packed Tissue

Nieves Atienza,Manuel Soriano-Trigueros,Maria-Jose Jimenez

doi:10.3390/math9151723

Abstract

We use topological data analysis tools for studying the inner organization of cells in segmented images of epithelial tissues. More specifically, for each segmented image, we compute different persistence barcodes, which codify the lifetime of homology classes (persistent homology) along different filtrations (increasing nested sequences of simplicial complexes) that are built from the regions representing the cells in the tissue. We use a complete and well-grounded set of numerical variables over those persistence barcodes, also known as topological summaries. A novel combination of normalization methods for both the set of input segmented images and the produced barcodes allows for the proven stability results for those variables with respect to small changes in the input, as well as invariance to image scale. Our study provides new insights to this problem, such as a possible novel indicator for the development of the drosophila wing disc tissue or the importance of centroids’ distribution to differentiate some tissues from their CVT-path counterpart (a mathematical model of epithelia based on Voronoi diagrams). We also show how the use of topological summaries may improve the classification accuracy of epithelial images using a Random Forest algorithm.

Highlights

Epithelia morphogenesis is key to understanding the development of tissues and organs
The concept of centroidal Voronoi tessellation (CVT) was used, which is a Voronoi diagram where the point generating each region coincides with its centroid
A different approach was developed in [4], where the authors provided an image analysis tool implemented in the open-access platform FIJI, to quantify epithelial organization based in computational geometry and graph theory

Summary

Introduction

Epithelia morphogenesis is key to understanding the development of tissues and organs. Looking for methods that can quantify the arrangement of cells is still an open and interesting problem [4]. These tissues are formed by tightly assembled cells, with almost no intercellular spaces. The study of epithelial organization has been mainly focused on the polygon distributions [1], that is, the distribution of the number of neighbors (sides) of the cells (polygons). The concept of centroidal Voronoi tessellation (CVT) was used, which is a Voronoi diagram where the point generating each region coincides with its centroid. The authors compared the polygon distributions of images of natural packed tissues with those of the CVT-path and showed that the former fit to the polygon distribution of specific. Considering the contact graph, that is, the graph generated by the cells (vertices) and the cell-to-cell contacts (edges), they searched locally for specific motifs represented by small subgraphs (graphlets) to characterize the tissue

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