# Four Severity Levels for Grading the Tortuosity of a Retinal Fundus Image

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## Abstract

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## 1. Introduction

- Calculated the tortuosity values of each of the fourteen tortuosity metrics to the vessels of each image in the large-scale AV-classification dataset [15].
- The result is two tortuosity feature sets, one at the image level and the other at each vessel-segment level.
- Collaborated with two ophthalmologists from RAK university of science and technology and Saqr hospital to label AV-classification dataset images to 4-levels of severity.
- The two ophthalmologists label the images of the AV classification dataset to classify each image to its tortuosity severity level (normal, mild, medium, and severe) based on their expert judgment as ground truth labels for tortuosity severity.
- The feature sets and the expert labels are used as input to machine learning to classify each retinal image into a class from (0 to 4) that tags each image with the severity of retinal vessels tortuosity, whether it is normal, mild, medium, or severe.
- The AV-classification dataset is extended by adding the 4-severity grades of each image.
- Finally, the new extended AV-classification comprehensive dataset is renamed to the retinal vessel morphometry (RVM) dataset. As it is a sizable data set and contains ground truth labels for each of these issues, the data set is made available for use by other researchers to study the tortuosity measures and other categories of retinal fundus image research problems, such as vessel segmentation, artery vein classification, and tortuosity severity.

## 2. Retinal Tortuosity Review

## 3. Materials and Method

#### 3.1. Materials

#### 3.2. Method

#### 3.3. Tortuosity Metrics

- Straight Line Distance (Chord): It is the straight-line-distance between two endpoints of the centrelines segment skeleton (Euclidian distance).$$Chord=\sqrt{{({x}_{2}-{x}_{1})}^{2}+{({y}_{2}-{y}_{1})}^{2}}$$
- Geodesic distance (Arc): The arc-length distance between the segment end- points, using the maximum non-infinity quasi-Euclidean distance between two endpoints of the segment center-line skeleton.$$d(\gamma \left(t2\right)-\gamma \left(t1\right))=v\ast \left|t2-t1\right|$$
- Distance Metric (DM): Although, the most straightforward dimensionless tortuosity measure used in the literature is the arc to chord ratio between the start and end points of the center line. It does not distinguish between the curvature of “S” and “C” shaped segments with equivalent arc-length. It is the most commonly used metric in the literature [46].$$DF=\frac{Arc}{Chord}$$
- Arc length/chord length: Called the distance factor (DF) as well. After dividing the segment into subsegments into chosen sample points (n) , (DF) is the summation arc to chord ratio of each subsegment for all subsegments.
- Tortuosity density (TD): After dividing the segment to subsegments into chosen sample points (n). TD is the summation of each subsegment’s arc to chord ratio for all sub-segments.$$TD=\frac{n-1}{n}\frac{1}{Arc}\sum _{i=1}^{n}[\frac{Lc{s}_{i}}{Lx{s}_{i}}-1]$$
- The curvature at a single point (t): For a point $t\left(x\right(t),y(t\left)\right)$ at vessel segment (s), the curvature at a point t is defined as the equation$$\mathit{C}\left(\mathit{t}\right)=\frac{{x}^{\prime}\left(t\right)\ast {y}^{\u2033}\left(t\right)-{y}^{\prime}\left(t\right)\ast {x}^{\u2033}\left(t\right)}{{({x}^{\prime}{\left(t\right)}^{2}+{y}^{\prime}{\left(t\right)}^{2})}^{\frac{3}{2}}}.$$
- Tortuosity density ($\tau 1$): After dividing the segment into subsegments on chosen sample points (n), tortuosity density, for straight line segment.$$\tau 1=DF-1$$
- Total curvature ($\tau 2$): Is the integration of $C\left(t\right)$.$$\tau 2={\int}_{{t}_{o}}^{{t}_{n}}C\left(t\right)dt$$
- Total Squared curvature ($\tau 3$): Is the integration of $C{\left(t\right)}^{2}$$$\tau 3={\int}_{{t}_{o}}^{{t}_{n}}C{\left(t\right)}^{2}dt$$
- Total curvature/Arc-length ($\tau 4$): Is the integration of $C\left(t\right)/Arc$$$\tau 4={\int}_{{t}_{o}}^{{t}_{n}}\frac{C\left(t\right)}{Arc}dt$$
- Total squared curvature/Arc-length ($\tau 5$): Is the integration of $\frac{C\left(t\right)}{Arc}$$$\tau 5={\int}_{{t}_{o}}^{{t}_{n}}\frac{C{\left(t\right)}^{2}}{Chird}dt$$
- Total curvature/Chord-length ($\tau 6$): Is the integration of $C\left(t\right)$/Chordlength$$\tau 6={\int}_{{t}_{o}}^{{t}_{n}}\frac{C\left(t\right)}{Chord}dt$$
- Total squared curvature/Chord-length ($\tau 7$): Is the integration of $C\left(t\right)\ast C\left(t\right)$/ Chordlength$$\tau 7={\int}_{{t}_{o}}^{{t}_{n}}\frac{C{\left(t\right)}^{2}}{Chord}dt$$
- Sum of angles metric (SOAM): It is used for measuring the strongly coiled vessels, and the SOAM is a result of measuring the angle between two vectors formed by each consecutive three-segment point. The normalized summation of all of these angles along the segment is measured by segment length [46]. The units of the SOAM measure are (radians/mm). This metric requires that points used from a segment to calculate it are evenly spaced.$$SOAM=\sum _{i=1}^{n}\frac{(180-{\alpha}_{i})}{Arc}$$
- Inflection count metric (ICM): It was extending the DM and is known as the $DM$ times the inflection points count along the segment.

#### 3.4. Preparing the New Feature-Set

#### 3.5. Tortuosity Labeling Approach

#### 3.6. Tortuosity Labeling Methodology

#### 3.7. Tortuosity Severity Levels Identification Using Machine Learning

#### 3.7.1. J48 Decision Tree

#### 3.7.2. Rotation Forest

#### 3.7.3. Distributed Random Forest

## 4. Results

#### 4.1. Tortuosity Classification Results

#### 4.1.1. Results of the Tortuosity Grading of Severity Levels Using J48 Decision Tree Model

#### 4.1.2. Results of the Tortuosity Grading Using Ensemble Rotation Forest Model

#### 4.1.3. Results of the Tortuosity Grading Using Distributed Random Forest Model

#### 4.2. From ‘AV Classification’ to RVM Data Set

#### Visualization of the Tortuosity Classification Results

## 5. Discussion

#### 5.1. The Increase in the Number of Trees Impact

#### 5.2. Comparison with the Other Methods

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CRVO | Central retinal vein occlusion |

ROP | Retinopathy of prematurity |

AVR | Arteriovenous ratio |

RVM | Retinal vessel morphometry |

VTI | Vessel tortuosity index |

TI | Tortuosity index |

TD | Tortuosity density |

SOAM | Sum of angles metric |

ICM | Inflection count metric |

DM | Distance metric |

DF | Distance factor |

RM | Resampling method |

CTC | Consolidated tree construction |

DRF | Distributed random forest |

RMSE | Root-mean-square error |

MSE | Mean square error |

MAE | Mean absolute error |

RMSLE | Root mean squared logarithmic error |

Se | Sensitivity |

PR | Precision |

TPR | True positive rate |

FPR | False positive rate |

RAKMHSU | RAK Medical and Health Sciences University |

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**Figure 3.**Illustration of the tortuosity calculation steps: (

**A**) the fundus image, (

**B**) vessels extraction, (

**C**) intersection points identification, (

**D**) skeletonization, (

**E**) vessel segments of fragments segmentation, (

**F**) tortuosity calculation for the fourteen metrics.

**Figure 4.**Illustration for arc and chord length in: (

**a**) tortuosity index (TI), (

**b**) tortuosity density, (

**c**) sum of angles metric (SOAM), and (

**d**) Inflection count metric (ICM).

**Figure 7.**Illustration of the distributed random forest method used with feature set created from the AV-classification dataset.

**Figure 9.**A retinal images sample at every tortuosity grade: (

**A**) normal, (

**B**) mild, (

**C**) moderate, and (

**D**) severe.

**Figure 10.**A comparison graph of the loss reported from using the DRF model of 50 Trees versus a 105 trees DRF model loss, in terms of (MSE, RMSE, MAE, RMSLE, and mean residual deviance) loss measures.

**Figure 11.**The model training and validation scoring results of RMSE (

**a**) vs. number of trees (

**b**) vs. number of EPOCS (

**c**) and MAE vs. number of EPOCS.

Image | Seg. | 1-A | Arc | Chord | DM | SOAM | ICMn | ICMb | SDavc | Navc | $\mathit{\tau}$1 | $\mathit{\tau}$2 | $\mathit{\tau}$3 | $\mathit{\tau}$4 | $\mathit{\tau}$5 | $\mathit{\tau}$6 | $\mathit{\tau}$7 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

No. | No. | 2-V | |||||||||||||||

2 | 42 | 2 | 0.12 | 0.09 | 0.0001 | 0.13 | 0.0007 | 0.0008 | 0.01 | 0.0014 | 0.0006 | 0.01 | 0.0003 | 0.12 | 0.1 | 0.005 | 0 |

2 | 44 | 2 | 0.18 | 0.14 | 0.0002 | 0.36 | 0.0011 | 0.0009 | 0.01 | 0.0022 | 0.0011 | 0.01 | 0.0001 | 0.12 | 0.1 | 0.005 | 0 |

2 | 46 | 2 | 0.06 | 0.04 | 0.0002 | 0.13 | 0.0003 | 0.0002 | 0.01 | 0.0009 | 0.0014 | 0.08 | 0.05 | 0.25 | 0.3 | 0.009 | 0.004 |

3 | 2 | 2 | 0.08 | 0.05 | 0.0002 | 0.19 | 0.0002 | 0.0003 | 0.02 | 0.0015 | 0.0011 | 0.01 | 0.0001 | 0.12 | 0.1 | 0.005 | 0 |

3 | 4 | 2 | 0.32 | 0.27 | 0.0009 | 0.13 | 0.0024 | 0.0024 | 0.1 | 0.0041 | 0.002 | 0.14 | 0.14 | 0.17 | 0.2 | 0.006 | 0.002 |

3 | 5 | 1 | 0.02 | 0.01 | 0.0002 | 0.08 | 0.0001 | 0.0002 | 0.01 | 0.0004 | 0.0009 | 0.01 | 0.0001 | 0.12 | 0.1 | 0.005 | 0 |

44 | 117 | 1 | 0.2 | 0.16 | 0.0012 | 0.08 | 0.0008 | 0.0012 | 0.01 | 0.0021 | 0.0017 | 0.26 | 0.26 | 0.28 | 0.3 | 0.01 | 0.005 |

44 | 119 | 1 | 0.003 | 0.0073 | 0.03 | 0.01 | 0.0011 | 0.0005 | 0.0059 | 0.0001 | 0.07 | 0.07 | 0.06 | 0.53 | 0.5 | 0.06 | 0.06 |

45 | 1 | 1 | 0.1 | 0.07 | 0.0001 | 0.13 | 0.0006 | 0.0006 | 0.01 | 0.0014 | 0.0009 | 0.01 | 0.0003 | 0.12 | 0.1 | 0.005 | 0.004 |

45 | 2 | 2 | 0.0078 | 0.0076 | 0.02 | 0.02 | 0.002 | 0.0008 | 0.14 | 0.0017 | 0.02 | 0.04 | 0.02 | 0.32 | 0.3 | 0.01 | 0.01 |

45 | 3 | 1 | 0.02 | 0.01 | 0.0011 | 0.25 | 0.0005 | 0.0004 | 0.03 | 0.0011 | 0.0015 | 0.01 | 0.0004 | 0.13 | 0.1 | 0.005 | 0 |

45 | 4 | 2 | 0.02 | 0.0079 | 0.0002 | 0.05 | 0 | 0 | 0.02 | 0.0004 | 0.0012 | 0.01 | 0 | 0.11 | 0.1 | 0.005 | 0 |

Loss Function | 50 Trees | 105 Trees |
---|---|---|

MSE | 0.00000863 | $3.75\times {10}^{-12}$ |

RMSE | 0.00293811 | $1.94\times {10}^{-6}$ |

MAE | 0.00276098 | $1.82\times {10}^{-6}$ |

RMSLE | 0.00097426 | $5.15\times {10}^{-7}$ |

Mean-Residual-Deviance | 0.00000863 | $3.75\times {10}^{-12}$ |

Model | TPR | FPR | Pr | Se | Fscore | ROC |
---|---|---|---|---|---|---|

Decision Tree (J48) | 0.927 | 0.084 | 0.879 | 0.927 | 0.988 | 0.994 |

Rotation Forest | 0.869 | 0.132 | 0.667 | 0.869 | 0.484 | 0.953 |

Predicted as → | Normal | Mild | Moderate | Severe |
---|---|---|---|---|

Actual ↓ | ||||

Normal | 2 | 20 | 0 | 0 |

Mild | 0 | 269 | 0 | 0 |

Moderate | 0 | 14 | 196 | 0 |

Severe | 0 | 3 | 0 | 0 |

**Table 5.**Comparing the loss in distributed random forest with the loss in the J48 decision tree and rotation forest’ results.

Loss Function | J48 Decision Tree | Rotation Forest | DRF 105 Trees Model |
---|---|---|---|

RMSE | 0.05 | 0.18 | 0.00000194 |

MAE | 0.01 | 0.01 | 0.00000182 |

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**MDPI and ACS Style**

Badawi, S.A.Q.; Takruri, M.; Albadawi, Y.; Khattak, M.A.K.; Nileshwar, A.K.; Mosalam, E. Four Severity Levels for Grading the Tortuosity of a Retinal Fundus Image. *J. Imaging* **2022**, *8*, 258.
https://doi.org/10.3390/jimaging8100258

**AMA Style**

Badawi SAQ, Takruri M, Albadawi Y, Khattak MAK, Nileshwar AK, Mosalam E. Four Severity Levels for Grading the Tortuosity of a Retinal Fundus Image. *Journal of Imaging*. 2022; 8(10):258.
https://doi.org/10.3390/jimaging8100258

**Chicago/Turabian Style**

Badawi, Sufian Abdul Qader, Maen Takruri, Yaman Albadawi, Muazzam A. Khan Khattak, Ajay Kamath Nileshwar, and Emad Mosalam. 2022. "Four Severity Levels for Grading the Tortuosity of a Retinal Fundus Image" *Journal of Imaging* 8, no. 10: 258.
https://doi.org/10.3390/jimaging8100258