# Retinal Vessel Local Tortuosity under a Macula-to-Optic Disc Central-Framing Change

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Local Tortuosity Indices

#### 2.2. Participants and Equipment

#### 2.3. Region of Interest (ROI)

#### 2.4. Vessel Segmentation and Parametrization

#### 2.5. Vessel Curvature

## 3. Results

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Witt, N.; Wong, T.Y.; Hughes, A.D.; Chaturvedi, N.; Klein, B.E.; Evans, R.; McNamara, M.; Thom, S.A.M.; Klein, R. Abnormalities of retinal microvascular structure and risk of mortality from ischemic heart disease and stroke. Hypertension
**2006**, 47, 975–981. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cheung, C.Y.; Zheng, Y.; Hsu, W.; Lee, M.L.; Bcomp, Q.P.L.; Mitchell, P.; Mmed, J.J.W.; Klein, R.; Wong, T.Y. Retinal vascular tortuosity, blood pressure, and cardiovascular risk factors. Ophthalmology
**2011**, 118, 812–818. [Google Scholar] [CrossRef] [PubMed] - Sun, Z.; Yang, D.; Tang, Z.; Ng, D.S.; Cheung, C.Y. Optical coherence tomography angiography in diabetic retinopathy: An updated review. Eye Lond
**2021**, 35, 149–161. [Google Scholar] [CrossRef] [PubMed] - Veluchamy, A.; Ballerini, L.; Vitart, V.; Schraut, K.E.; Kirin, M.; Campbell, H.; Joshi, P.K.; Relan, D.; Harris, S.; Brown, E.; et al. Novel Genetic Locus Influencing Retinal Venular Tortuosity Is Also Associated With Risk of Coronary Artery Disease. Arterioscler. Thromb. Vasc. Biol.
**2019**, 39, 2542–2552. [Google Scholar] [CrossRef] [PubMed] - van de Kreeke, J.A.; Nguyen, H.T.; Konijnenberg, E.; Tomassen, J.; den Braber, A.; Kate, M.; Sudre, C.H.; Barkhof, F.; Boomsma, D.I.; Tan, H.S.; et al. Retinal and Cerebral Microvasculopathy: Relationships and Their Genetic Contributions. Investig. Ophthalmol. Vis. Sci.
**2018**, 59, 5025–5031. [Google Scholar] [CrossRef] [PubMed] [Green Version] - McClintic, B.R.; McClintic, J.I.; Bisognano, J.D.; Block, R.C. The relationship between retinal microvascular abnormalities and coronary heart disease: A review. Am. J. Med.
**2010**, 123, 374.e1–374.e7. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Stefánsson, E.; Olafsdottir, O.B.; Einarsdottir, A.B.; Eliasdottir, T.S.; Eysteinsson, T.; Vehmeijer, W.; Vandewalle, E.; Bek, T.; Hardarson, S.H. Retinal Oximetry Discovers Novel Biomarkers in Retinal and Brain Diseases. Invest. Ophthalmol. Vis. Sci.
**2017**, 58, BIO227–BIO233. [Google Scholar] [CrossRef] [PubMed] - Lotmar, W.; Freiburghaus, A.; Bracher, D. Measurement of vessel tortuosity on fundus photographs. Albrecht Graefes Arch. Klin. Exp. Ophthalmol.
**1979**, 211, 49–57. [Google Scholar] [CrossRef] [PubMed] - Hart, W.E.; Goldbaum, M.; Côté, B.; Kube, P.; Nelson, M.R. Measurement and classification of retinal vascular tortuosity. Int. J. Med. Inform.
**1999**, 53, 239–252. [Google Scholar] [CrossRef] [PubMed] - Grisan, E.; Foracchia, M.; Ruggeri, A. A novel method for the automatic grading of retinal vessel tortuosity. IEEE Trans. Med. Imaging
**2008**, 27, 310–319. [Google Scholar] [CrossRef] [PubMed] - Trucco, E.; Azegrouz, H.; Dhillon, B. Modeling the tortuosity of retinal vessels: Does caliber play a role? IEEE Trans. Biomed. Eng.
**2010**, 57, 2239–2247. [Google Scholar] [CrossRef] [PubMed] - Onkaew, D.; Turior, R.; Uyyanonvara, B.; Akinori, N.; Sinthanayothin, C. Automatic Retinal Vessel Tortuosity Measurement Using Curvature of Improved Chain Code; InECCE: Kuantan, Malaysia, 2011; pp. 183–186. [Google Scholar]
- Kalitzeos, A.A.; Lip, G.Y.; Heitmar, R. Retinal vessel tortuosity measures and their applications. Exp. Eye Res.
**2013**, 106, 40–46. [Google Scholar] [CrossRef] [PubMed] - Mapayi, T.; Tapamo, J.R.; Viriri, S.; Adio, A.O. Automatic Retinal Vessel Detection and Tortuosity Measurement. Image Anal. Stereol.
**2016**, 35, 117–135. [Google Scholar] [CrossRef] - Ramos, L.; Novo, J.; Rouco, J.; Romeo, S.; Álvarez, M.D.; Ortega, M. Computational assessment of the retinal vascular tortuosity integrating domain-related information. Sci. Rep.
**2019**, 9, 19940. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mok, K.H.; Lee, V.W. Disc-to-macula distance to disc-diameter ratio for optic disc size estimation. J. Glaucoma
**2002**, 11, 392–395. [Google Scholar] [CrossRef] [PubMed] - Ralló, M.; Millán, M.S. Connectivity-based segmentation of retinal vessels in eye fundus images. Opt. Pura Apl.
**2017**, 50, 359–368. [Google Scholar] [CrossRef] - Lewiner, T.; Gomes, J.D.; Lopes, H.; Craizer, M. Curvature and torsion estimators based on parametric curve fitting. Comp. Grap.
**2005**, 29, 641–655. [Google Scholar] [CrossRef] - Ramos, L.; Novo, J.; Rouco, J.; Romeo, S.; Álvarez, M.D.; Ortega, M. Retinal vascular tortuosity assessment: Inter-intra expert analysis and correlation with computational measurements. BMC Med. Res. Methodol.
**2018**, 18, 144. [Google Scholar] [CrossRef] [PubMed] - Aslam, T.; Fleck, B.; Patton, N.; Trucco, M.; Azegrouz, H. Digital image analysis of plus disease in retinopathy of prematurity. Acta Ophthalmol.
**2009**, 87, 368–377. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Illustration of the $DF$ index as the ratio between the arc length ($L$ ) and the chord length ($D$ ), the latter being the Euclidean distance between the two endpoints of the vessel.

**Figure 2.**Original images. Retinographies of the pair (MR2/DR2) taken from subject 2′s right eye, with the frame centered on (

**a**) macula (MR2); (

**b**) optic disc (DR2).

**Figure 3.**Determination of the ROI for DR2 and MR2. Brightest (red) points in all three RGB components of (

**a**) DR2; (

**b**) MR2. Darkest points are marked in green in (

**a**). Optic disc midpoint (red) in (

**c**) DR2; (

**d**) MR2. Macula midpoint is marked in green in (

**c**). (

**e**) Intersection and common area (grey shaded) of DR2 and MR2. (

**f**) ROI of DR2. (

**g**) ROI of MR2. (See the text for more details).

**Figure 4.**Flow chart for vessel-ROI segmentation, binarization, and skeletonization: from the eye fundus ROI (

**a**), a vessel is manually selected (

**b**). The vessel ROI is segmented (

**c**), binarized (

**d**), and skeletonized (

**e**).

**Figure 5.**Vessel segment refinement, coordinate acquisition, and parametrization. (

**a**) Skeletonized element; Endpoints: principal (

**b**,

**d**), secondary (

**c**); (

**e**) 1-pixel wide curve of connected pixels running along the central line of the vessel segment for coordinate acquisition.

**Figure 7.**Individual tortuosity values (DF, T3, T5, and T7 indices) for the 10 selected vessels in DR2-ROI (D-retinography of subject 2′s right eye).

**Figure 8.**Bland–Altman plots for each tortuosity index (DF, T1 … T7). Differences between the values obtained in D-retinographies and the values obtained in M-retinographies are represented on the vertical axis. In each plot, the orange line represents the mean value of the individual differences. Points of the same color correspond to the same eye. Dot colors represent eye and subject, being MR1/DR1 (orange), ML1/DL1 (red), MR2/DR2 (green), ML2/DL2 (blue). Point 14 represents vessel 4 in ML1/DL1, point 18 vessel 8 in ML1/DL1, point 23 vessel 3 in MR2/DR2, and point 36 vessel 6 in ML2/DL2.

**Figure 9.**Pairs of vessel segments. Point 14 is vessel 4 in ML1/DL1, point 18 is vessel 8 in ML1/DL1, point 23 is vessel 3 in MR2/DR2 (see Figure 6), and point 36 is vessel 6 in ML2/DL2). They fall off the limits of concordance in some Bland–Altman plots (Figure 8). Magenta (cyan) lines correspond to M-retinography (D-retinography).

**Figure 10.**Dendrogram showing the dissimilarity ($1-r)$ among tortuosity indices according to Pearson’s correlation coefficient and the resulting clusters.

Chord length | $D=\sqrt{{\left(x\left({t}_{1}\right)-x\left({t}_{0}\right)\right)}^{2}+{\left(y\left({t}_{1}\right)-y\left({t}_{0}\right)\right)}^{2}}$ |

Arc length | $L={{\displaystyle \int}}_{C}^{}1ds={{\displaystyle \int}}_{{t}_{0}}^{{t}_{1}}1\xb7\sqrt{{\left(x\u2019\left(t\right)\right)}^{2}+{\left(y\u2019\left(t\right)\right)}^{2}}dt$ |

Total curvature | $TK={{\displaystyle \int}}_{C}^{}\left|\kappa \right|ds={{\displaystyle \int}}_{{t}_{0}}^{{t}_{1}}\left|\kappa \left(t\right)\right|\xb7\sqrt{{\left(x\u2019\left(t\right)\right)}^{2}+{\left(y\u2019\left(t\right)\right)}^{2}}dt$ |

Total squared curvature | $TSK={{\displaystyle \int}}_{C}^{}{\kappa}^{2}ds={{\displaystyle \int}}_{{t}_{0}}^{{t}_{1}}{\kappa}^{2}\left(t\right)\xb7\sqrt{{\left(x\u2019\left(t\right)\right)}^{2}+{\left(y\u2019\left(t\right)\right)}^{2}}dt$ |

**Table 2.**Definition of eight common local tortuosity indices [13].

$DF=\frac{L}{D}$ | $T1=\frac{L}{D}-1$ | $T2=TK$ | $T3=TSK$ |

$T4=\frac{TK}{L}$ | $T5=\frac{TSK}{L}$ | $T6=\frac{TK}{D}$ | $T7=\frac{TSK}{D}$ |

Right Eye | Left Eye | |||
---|---|---|---|---|

Frame Center | Macula | Optic Disc | Macula | Optic Disc |

Subject 1 | MR1 | DR1 | ML1 | DL1 |

Subject 2 | MR2 | DR2 | ML2 | DL2 |

**Table 4.**Mean and standard deviation for the 40 pairs of selected vessel segments of each index and frame center (M, macula and D, optic disc).

Mean | Standard Deviation | |||
---|---|---|---|---|

Index | M | D | M | D |

DF | 1.1543 | 1.1621 | 0.1101 | 0.1169 |

T1 | 0.1543 | 0.1621 | 0.1101 | 0.1169 |

T2 | 4.935 | 5.063 | 3.307 | 3.427 |

T3 | 0.2085 | 0.2344 | 0.2412 | 0.2756 |

T4 | 0.01709 | 0.01742 | 0.00915 | 0.00927 |

T5 | 0.000694 | 0.000774 | 0.000718 | 0.000811 |

T6 | 0.02046 | 0.02105 | 0.01251 | 0.01292 |

T7 | 0.000860 | 0.000970 | 0.000952 | 0.001087 |

**Table 5.**Mean and standard deviation of individual differences. Limits of agreement and p-value of the paired t-test for the tortuosity indices. p-value > 0.05 appeared boldfaced.

Tortuosity Indices | Mean Difference | Standard Deviation | Limits of Agreement | Limits of Agreement | |
---|---|---|---|---|---|

Lower | Higher | ||||

DF | 0.00768 | 0.016570 | −0.02480 | 0.04016 | 0.006 |

T1 | 0.00768 | 0.016570 | −0.02480 | 0.04016 | 0.006 |

T2 | 0.12910 | 0.384400 | −0.62430 | 0.88252 | 0.040 |

T3 | 0.02591 | 0.058390 | −0.08850 | 0.14035 | 0.008 |

T4 | 0.00033 | 0.001285 | −0.00220 | 0.00285 | 0.112 |

T5 | 0.00008 | 0.000169 | −0.00025 | 0.00041 | 0.005 |

T6 | 0.00060 | 0.001764 | −0.00285 | 0.00405 | 0.038 |

T7 | 0.00011 | 0.000234 | −0.00035 | 0.00057 | 0.005 |

**Table 6.**Geometrical features of the vessels in Figure 9.

Vessel | Frame Center | D | L | TK | TSK |
---|---|---|---|---|---|

14 | DR2 | 339.86 | 457.30 | 12.57 | 1.007 |

MR2 | 340.51 | 446.45 | 11.81 | 0.752 | |

18 | DR2 | 187.58 | 271.50 | 8.31 | 0.520 |

MR2 | 190.69 | 261.84 | 7.93 | 0.466 | |

23 | DR2 | 308.30 | 377.31 | 7.79 | 0.627 |

MR2 | 318.91 | 383.10 | 6.92 | 0.439 | |

36 | DR2 | 102.46 | 139.02 | 5.09 | 0.356 |

MR2 | 103.79 | 138.37 | 4.75 | 0.280 |

**Table 7.**Spearman rank correlation ($\rho $ ) and Pearson correlation ($r$ ) coefficients computed for pairs of the tortuosity indices.

DF | T1 | T2 | T3 | T4 | T5 | T6 | ||
---|---|---|---|---|---|---|---|---|

T1 | $\rho $ $r$ | 1.000 1.000 | ||||||

T2 | $\rho $ $r$ | 0.830 0.800 | 0.830 0.800 | |||||

T3 | $\rho $ $r$ | 0.827 0.758 | 0.827 0.758 | 0.975 0.924 | ||||

T4 | $\rho $ $r$ | 0.775 0.758 | 0.775 0.758 | 0.869 0.826 | 0.921 0.817 | |||

T5 | $\rho $ $r$ | 0.790 0.760 | 0.790 0.760 | 0.890 0.815 | 0.954 0.919 | 0.983 0.932 | ||

T6 | $\rho $ $r$ | 0.814 0.834 | 0.814 0.834 | 0.890 0.846 | 0.938 0.843 | 0.995 0.990 | 0.985 0.943 | |

T7 | $\rho $ $r$ | 0.808 0.791 | 0.808 0.791 | 0.896 0.814 | 0.957 0.922 | 0.982 0.918 | 0.998 0.946 | 0.987 0.941 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ramírez, N.; Ralló, M.; Millan, M.S.
Retinal Vessel Local Tortuosity under a Macula-to-Optic Disc Central-Framing Change. *Diagnostics* **2023**, *13*, 1030.
https://doi.org/10.3390/diagnostics13061030

**AMA Style**

Ramírez N, Ralló M, Millan MS.
Retinal Vessel Local Tortuosity under a Macula-to-Optic Disc Central-Framing Change. *Diagnostics*. 2023; 13(6):1030.
https://doi.org/10.3390/diagnostics13061030

**Chicago/Turabian Style**

Ramírez, Natalia, Miquel Ralló, and Maria S. Millan.
2023. "Retinal Vessel Local Tortuosity under a Macula-to-Optic Disc Central-Framing Change" *Diagnostics* 13, no. 6: 1030.
https://doi.org/10.3390/diagnostics13061030