**4. Retinal tissues analysis**

We use some metric/algorithms of Section 2.1 to analyze and model the polygonality in retinal tissues, especially with DR.

## **4.1 Mean-square deviation of angles**

Using the metric of mean-square deviation of angles *ε* (Eq. (3)), to know the angular distributions of photoreceptors in the rodent retina in order to identify healthy tissues (C) from tissues with pathology (R). The results are shown in **Figure 14**.

Results obtained during this analysis reported that at an eccentricity of 270 μm, there is a greater angular variation than 6 degrees in 4-sided polygons in tissues with pathology (R) with respect to its counterpart of healthy tissues (C). At an eccentricity of 810 μm a greater angular variation is observed in polygons of 10 and 9 sides, in addition to polygons of 6 and 7 degrees, respectively, in tissues with pathology (R). At an eccentricity of 1350 μm, an angular variation is presented up to 12 degrees in polygons of nine sides in tissues with pathology (R). At an eccentricity of 1890 μm, again the 9-sided polygons present an angular variation of 9 degrees in tissues with pathology (R). Likewise, we identify that as the eccentricity increases; the training of a greater number of sides of polygons is increasing in tissues with pathology which does not present healthy tissues (not presented here).

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

### **Figure 14.**

*Distribution of angular variation as eccentricity in retinal tissues, both healthy tissues (C) and diabetic retinopathy tissues (R): (A) 270 μm, (B) 810 μm, (C) 1350 μm, (D) 1890 μm.*


## **Table 2.**

P*-value\* to 5, 6, and 7 Voronoi ties.*

Is observed that if the angular variation is minor to 6.15 degrees, the value *P* is minor to 0.05. It is identified, then that to 810 and 1890 μm the angles of the polygons are minor to 30 degrees. For photography retinal, it is observed that to major eccentricity, the spacing of the photoreceptors is major, **Table 2**.

## **4.2 Detection of modal spacing**

Using the metrics/algorithm of radius in a circumscribed circle for estimating photoreceptor spacing based on the circumradius of Delaunay triangle, as a metric that allows knowing how is the density/separation of photoreceptors in retinal healthy tissues and with diabetic retinopathy. This metric is an alternative form at the Yellott's ring [18]. The circumradius provides an estimation of the modal spacing in the retinal image of the photoreceptors. The distributions of density grouped by circumradius interval are: (a) 1.12–2.24 μm, (b) 2.244–3.36 μm, (c) 3.364–4.48 μm, (d)

### **Figure 15.**

*Distribution of spacing based on the circumradius grouped by eccentricity in retinal tissues: (A) 270 μm, (B) 810 μm, (C) 1350 μm, (D) 1890 μm.*

4.485–5.601 μm, (e) 5.605–6.721 μm, (f) 6.725–7.842 μm, (g) 7.846–9.62 μm. The results are shown in **Figure 15**.

To distribute the density by spacing of photoreceptors, we have proposed a standard criterion for arbitrary intervals of circumradius: 1.12–2.24 μm, 2.244–3.36 μm, 3.364–4.48 μm, 4.485–5.601 μm, 5.605–6.721 μm, 6.725–7.842 μm, and 7.846–9.62 μm. By using these intervals with the metrics/algorithms of circumscribed circle to obtain each circumradius, our results allow us to observe that the two dominant distributions of grouped density of photoreceptors are at intervals between 3.364–4.48 μm and 2.244–3.36 μm.

With these results, it has been observed that the percentage of circumradius in the range of 2.244–3.36 μm increases in tissues with DR as eccentricity increases. These results show a density of separating healthy photoreceptors to 810 μm of eccentricity with circumradius in the range of 3.364–4.48 μm (42.51%) and in the range of 2.244–3.36 μm (37.68%), **Table 3**.

It has been also identified that in cellular damaged tissues with DR the training of circumradius to greater eccentricity and also with a larger radius increases, with


## **Table 3.** *Percent density by eccentricity intervals.*

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


**Table 4.**

*Percentage of circumradius in the interval [3.364–5.601] μm.*

respect to the control tissue, this is because they increase the spaces between photoreceptors that stop emitting fluorescence. It is also appreciated that the clustered density decreases in the photoreceptors of tissues with diabetic retinopathy, between the intervals of 2.244–3.36 μm and in 3.364–4.48 μm, as eccentricity.

However, in the interval [3.364–5.601] μm, that represents 55.8% of circumradius, they have a higher percentage behavior in control tissues than in tissues with DR, **Table 4**. Therefore, this shows that frequency of circumradius decreases in tissues with DR.

## **4.3 Mean distances average**

To measure distances between neighbors that form each Voronoi cell is employed the metric/algorithm of mean distances average (Eq. (1)), as a measure for expansion or contraction of the polygon of Voronoi. The results are shown in **Figure 16**. This is a modification to the metric Ulam tree in each polygon. A measurement of spacing (or contraction) between neighbors is explained in terms of a bi-dimensional space with reference to the populations on flat surfaces. This metric allows us to measure the space between neighboring cells, as a result of leakage in photoreceptors.

In **Figure 16**, it can be seen that for the 5, 6 and 7-sided polygons, they have a similar spacing. In the same way it is observed that for polygons of four, eight and nine sides have a particular behavior of distance separating, the distances between diabetic cells increases, in relation to the healthy cells, because the distances between the cells increase because of the diabetic retinopathy.

**Figure 16** depicts the mean averaged distance (Eq. (1)), as a function of the number of sides of Voronoi polygons for both control and pathological retinas. Since it

### **Figure 16.**

*Distribution of cellular spacing in healthy tissue and with diabetic retinopathy according to the eccentricity measured from the optic nerve.*

has been just showed that the polygon frequency distribution does not change with retinal eccentricity (**Figure 20**), the data have been also averaged across all locations for better comparisons. Differences between both groups of retinas were statistically significant (paired *t*-test, *P* < 0.05).

Using the Ulam tree modified for distances, it is possible to characterize the environment of each cell and in a more general way, to study cellular interactions. This metric/algorithm appears to be useful for analyzing the effects of the cell surrounding on a given cellular function and vice-versa.

A topic as suitable to determinate of the average type of the spatial occupation, following isoperimetric inequality [19], see Eq. (5),

$$\left(L(X)^2 - 4\mu A(X)^2 \ge 0\right) \tag{5}$$

Where *A X*ð Þ is the area and *L X*ð Þ<sup>2</sup> is the perimeter for a convex set.

## **4.4 Angular polygonality**

With this metric/algorithm P*Nk i*¼1 *χ*j j *<sup>i</sup>*�*β <sup>n</sup>* is quantified that the Voronoi polygons of five, six, seven, and eight sides in healthy tissues and with diabetic retinopathy maintain a maximum range of angular values between 110 and 130 degrees measured from an arbitrary horizontal axis, defining an angular cluster of radio 10 degrees. It is shown that the Voronoi polygons of five, six, seven, and eight sides are sensitives.

However, the polygons of Voronoi in healthy tissues and with diabetic retinopathy of 3, 4, 9, 10, 11 and 12 sides are very irregular angularly, and the range of angular variation is from 60 to 180 degrees, defining an angular cluster radius of 60 degrees, **Figure 17**.

Other topics as suitable to capture regularity in convex Voronoi polygons is to measure how well they fit in a regular polygon or a regular polygon fits in them. This proposal has the characteristics to providing a way of fitting with convex polygons in

**Figure 17.** *Distribution of the angular polygonality relative to the side number in the Voronoi polygon.*

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

**Figure 18.**

*In (A), a pentagon that almost looks like an equilateral triangle, (B) hexagon that almost looks like an octagon, (C) square that almost looks like a rhombus shape.*

which they are very similar to regular polygons with a different number of edges [20], see **Figure 18**.

The procedure consists in transforming the given irregular polygon into a regular one, while measuring the amount of deformation required in such a process. The measurement of the angle variation gives a first parameter to consider in the final measurement of the regularity of Voronoi polygon. Using the Ulam tree modified to distances and the Ulam tree modified to angles gives a measure of the amount of deformation produced. Combining both procedures will be obtaining a measure of the regularity of the Voronoi tessellations on retinal tissues.

## **4.5 Frequency of Voronoi polygon**

In order to quantify the distribution of polygons according to the eccentricity, frequency graphs were generated. In these it is observed that five and six sides polygons predominate, in healthy retinal tissues (C) and with diabetic retinopathy (R). In healthy tissues, with eccentricities of 270 and 810 μm, the frequency of polygons of five and six sides is greater, with respect to pathological tissues. However, with eccentricities of 1350 and 1890 μm, predominate 5-sided polygons, in tissues with diabetic retinopathy, **Figure 19**.

### **Figure 19.**

*Distribution of frequencies in Voronoi polygons by eccentricity in healthy retinal tissues (C) and with diabetic retinopathy (R).*

## *Eye Diseases - Recent Advances, New Perspectives and Therapeutic Options*

**Figure 20.** *Distribution of maximum angles χ<sup>i</sup> relative to the side number (4,5,6, … ) in the Voronoi polygon.*

We found that the spatial sampling of the images with S window, even using resizing of the recorded images of 90 90 μm, has a significant impact on the performance of the metric/algorithms, but also that an excessive upsizing does not substantially improve the measurements. We observed that the percentage of Voronoi polygon is the parameter in which is most affected by errors in photoreceptors detection, and for this reason the combined measurements of more parameters could be a better choice in order to characterize different retinal regions and the different subjects' retinas.

## **4.6 Minimum and maximum angles**

Control (C) and diabetic retinopathy (R) tissues were characterized based on measuring the maximum angles *χ<sup>i</sup>* and minimum angles *χ<sup>i</sup>* of the Delaunay triangles in Voronoi polygons (**Figure 5**). Greater sensitivity is identified when quantifying maximum angles than minimum angles; see **Figure 20** and **Table 5**.
