**1. Introduction**

Several biological/physical systems involve two-dimensional spatial arrangements with elements such as molecules, or cellular tissue, and the profiling in space has important repercussions to biological and physical investigations. The analysis of biological tissues often requires quantitative measurements to explore cell organization, in order to understand, identify, and monitor changes during pathology development and healing processes.

Measurements of cellular organization might not be always able to distinguish healthy and diseased tissues from each other [1]. It has also been revealed that the existing density measurements in cells present multiple restrictions [2–7]. In

particular, some studies have reported that the density of retinal cones is not an appropriate parameter to identify pathological states in early stages of retina diseases [8, 9].

Maps of cellular distribution have to be considered linked to morphogenesis, mechanical structural stabilities and functional state of a given healthy tissue. Finding the connection that links form to disorder and is based on distance networks partition constructed from the position of cells when defined as mathematical nodes generators of Voronoi tessellations, then they can be used to construct summary functions [10], see **Table 1**. For example, Sudbø and Marcelponil Reith developed 27 algorithms based on Voronoi diagrams to describe the architecture of tissues [11]. Chiu showed that the minimum angles and areas of Delaunay triangles are responsive parameters when it concerns cellular distributions [12]. Voronoi analysis has also been used to determine the packing arrangement of cones at different retinal storage bins displayed with adaptive optics (AO) and hexagonal packing was found [13, 14]. Other authors have proposed to measure the regularity of convex polygons by successive measurements of irregular polygonal reconfiguration until regularity [15].

However, side number, distance between cones, according to proportion, any measurements derivative from the least distance will cause statistics to fail to identify healthy tissues and pathological tissues, which yield identical statistical densities measured from the shortest distances [1]. Other serious constraints developed from the method of counting the number of Voronoi cell sides result

## **Table 1.**

*Voronoi polygons and Delaunay triangles in a retinal tissue.*

*Analysis and Modeling of Polygonality in Retinal Tissue Based on Voronoi Diagram… DOI: http://dx.doi.org/10.5772/intechopen.106178*

from the fact that the number of cell sides will not change except for gross perturbations of the particle system. In addition, in Delaunay triangulation, no significant differences are observed by using Delaunay segments and Delaunay areas in tissues [16].

The general aim of this chapter is to establish an analysis for a comparison of the distribution of PRs in tissues with retinopathy diabetic, which is an important problem to mechanically understanding of the processes that lead to the experimental observations. This research presents only the model of the photoreceptor's spatial location (cones and sticks) considering photoreceptors small enough, in the suitable scale, to contemplate them as mathematical points. Then, Voronoi polygons and Delaunay triangles are formed by using these points. Therefore, Voronoi polygons do not represent the photoreceptor's shape. So, in this chapter, Voronoi polygons are not used to model biological cells nor their surrounding tissues.

This chapter is organized as follows: in Section 2, the description of several metrics using Voronoi polygon and Delaunay triangles and its integrated platform based on computational geometry, is given; in Section 3, multiphoton microscopy and image analysis is provided; in Section 4 we continue with a detailed description of retinal tissues, using Voronoi and Delaunay metrics described in Section 2; the Discussion is presented in Section 5, and finally the Conclusions are in Section 6.
