On topological analysis of cells organization in biological images

On Topological Analysis of Cells Organization in Biological Images Maria-Jose Jimenez(B) Departamento de Matematica Aplicada I, Universidad de Sevilla, Seville, Spain [email protected] Abstract. There are many problems inside the field of biomedical image analysis that can be dealt from a topological (and geometric) point of view. One of them refers to the way in which cells are self-organised inside a tissue, mainly motivated by the changes that may occur in such an organization in case of disease. This problem can be faced from different perspectives in terms, first, of how to model the cells and their ‘connections’ and second, how to computationally characterise their organization. We will discuss some topological approaches to this topic as well as future lines of research. Keywords: Topological data analysis ·Persistent homology · Epithelial organization The present manuscript summarizes the main aspects treated in the invited talk, with the only purpose of giving some intuition on the main concepts, as well as a general overview. 1 Biological Setting Epithelia are packed tissues formed by tightly assembled cells. Their apical surfaces describe regions that are similar to convex polygons forming a natural tessellation. This way, each cell can be identified with a polygon with as many sides as neighboring cells. Hence, epithelial organization has been analyzed in several works (see, for example [9,18,19]) with a common approach based on the study of the properties and the distribution of the polygons (cells). In [18], they show that geometrical and network characteristics of each cell are dependent on the number of sides. In [19], they focused on the polygons distribution to describe a range of values (obtained from mathematical tesselations) that can be taken as a reference in terms of cell arrangements. A different approach was developed in [21], were the authors provided an image analysis tool implemented in the openaccess platform FIJI, to quantify epithelial organization based in computational Project PID2019-107339GB-I00, funded by: Ministerio de Ciencia e Innovaci´on – Agencia Estatal de Investigaci´on/10.13039/501100011033. https://doi.org/10.1007/978-3-030-76657-3_3 59 geometry and graph theory concepts. More specifically, they looked locally for specific motifs represented by small graphs (graplets). At this point, it seems interesting to introduce a measure that provides global topological information about the graph of cell contacts (not only about local configurations or number of neighbors), preserving part of the geometric information of the cells. 2 Topological Data Analysis Topological Data Analysis (TDA) studies the shape of data from a topological viewpoint, having persistent homology [7,23] as its main tool. Persistent homology studies the evolution of homology classes and their life-times (persistence) in an increasing nested sequence of spaces (that is called a filtration). For a basic reference, see [8]. Ak-simplex (or simplex of dimension k), for k=0,1,2,... is given by the convex hull of k+ 1 affinely independent points in Rd,k≤d. A 0-simplex is a vertex, a 1-simplex is an edge, a 2-simplex is a triangle and a k–simplex is the generalization to dimension k.Asimplicial complex is a finite collection of simplices, K, such that for any simplex in K, all its faces are also simplices in Kand the intersection of any two simplices is either empty or a simplex in K. The dimension of the simplicial complex is the maximum dimension of any of its simplices. In the case of the data dealt here, the topological spaces will be simplicial complexes of dimension 2 and homology classes will represent either connected components or holes. For persistent homology computation on K, it is necessary to have a filter function that induces an increasing sequence of simplicial complexes (a filtration) K0⊂K1⊂ ··· ⊂ Kn=K, along which the lifespan of connected components and holes will be measured. The output of persistent homology computation can be codified under the form of a set of intervals, barcode, encoding the birth and death times of each homology class arising in the increasing sequence of spaces. However, the nature of the barcode makes it hard to combine it with usual algorithms of machine learning. There are several works in the literature to construct different types of topological summaries (out of the barcode) that are easier to compute and represent a global topological feature of the space. Some examples are persistence landscapes [4], persistence images [1] or tropical coordinates [13]. Another topological summary is persistent entropy, which is a parameter that can be computed from the lengths of the intervals in the barcode. The concept first arose in [6] and can be described as an adaptation of Shannon entropy to this context. In [2], the authors proved that persistent entropy is robust to small perturbations of the input data. There have been several successful applications of persistent entropy for classification or characterization purposes, as, for example, [15], for detecting the transition between the preictal and ictal states in EEG signals and [17], for classifying long-length noisy signals of DC electrical motors. Lately, there has been a great proliferation of applications of TDA to many different areas, including the analysis of biomedical images (for example, [14,16]). 60 M.-J. Jimenez 3 Topological Approaches to Describe the Organization of Cells In a 2D image of a packed tissue, the cells are represented by different regions whose boundaries are always in contact with other cells. Once the image has been properly segmented, it becomes a map of regions tiling the whole image, as in Fig. 1. A first approach that was used to model the structure of the tissue is by a simple abstract graph where each vertex represents a cell and each edge represents a cell-to-cell (physical) contact. Notice that the structure reflects only neighboring relations, but nothing about the geometry of the cells [18,19,21]. Apart from the already mentioned papers, there have been also several works in which the contact graph (a simplicial complex of dimension 1) has been provided with a filter function for the computation of persistent homology. In the case of [22], the filter was given by the number of contacts of each cell; in the case of [11] persistent entropy was first used as a topological summary to characterise topologically the cell arrangements. Voronoi tessellations are commonly used for estimating the morphology of cells in epithelial tissues (see Fig. 2) by building the diagrams from the set of centroids of the cells. Indeed, in [12], the authors evaluate the suitability of approximating the cells by such a mathematical construction. A natural construction, from the Voronoi diagram on the centroids of the cells, is the Delaunay triangulation that is dually associated to the Voronoi diagram. Hence, the Delaunay complex arise as a 2-dimensional simplicial complex that may represent the cells (vertices) and the neighboring relations between Fig. 1. Segmented images of tissues of Neuroepithelium from chicken embryos, on the left, and Wing imaginal disc from Drosophila, on the right. Fig. 2. From left to right, a portion of segmented image of epithelial cells with centroids marked in red; Voronoi diagram of their centroids superimposed in blue; Voronoi diagrams and centroids of cells; Delaunay triangulation superimposed in purple; Delaunay triangulation representing the cells arrangement. For the color version we address the reader to the online publication. (Color figure online) 61 them (edges and triangles). Notice that this representation, unlike the previous one, keeps spatial information of the cells on the plane. Filter values to the edges and triangles will be assigned in terms of the distances between the centroids of the cells. This is, hence, a topological and geometrical model that is also a suitable input for persistent homology computation. A first work in this line was [3], also using persistent entropy as topological feature. However, other topological summaries can be explored in the search for new insights in the characterisation of different tissues as well as mathematical tessellations. In the previous representation, there is, in fact an approximation of the cells by Voronoi regions, but sometimes there might be an important difference between the real and the Voronoi region. So it is natural to think about a way of keeping more information about the morphology of each cell. For this aim, the distance transform could be used to provide a filter function to all the pixels of each region in the segmented image, in which case, persistent homology could also be computed. This is an ongoing work that is closer to digital topology, since every pixel on the boundary (between cells) should be taken into account for the computation. 4 Future Work A forward extension of the work introduced here is to deal with segmented 3D images of epithelia. In [10], the authors recently described the geometry of this tissues. The main drawback in this case is the huge amount of data that needs to be processed, what makes more difficult to get a database of segmented images to work with. A different problem in the field is the analysis of images derived from clonogenic studies, aiming to investigate the formation of colonies in cell culture assays. A recent collaboration with CITRE, a Biotechnology research center focused on drug discovery, has provided a new context to learn features from those cell colonies in the segmented images. Persistent homology has been successfully applied to related problems such as clustering [5] and particles aggregation [20], pointing to a potential application of TDA methods to the study of colony forming assays. Acknowledgments. The work introduced here has been (and/or is being) developed in collaboration with several colleagues: Luis M. Escudero and his team, from Instituto de Biomedicina de Sevilla (IBiS), Spain, who proposed the topological study of epithelia organization; Matteo Rucco, from United Technology Research Center, Italy; Nieves Atienza, Belen Medrano and Manuel Soriano-Trigueros (with a special mention to the latter, for his implication), from Universidad de Sevilla, Spain; and CITRE – A Bristol Myers Squibb Company. References 1. Adams, H., et al.: Persistence images: a stable vector representation of persistent homology. J. Mach. Learn. Res. 18, 1–35 (2017) 62 M.-J. Jimenez 2. 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