A mathematical determination of foveal attachment in primary rhegmatogenous retinal detachment when obscured by bullous retina

In primary rhegmatogenous retinal detachment (RRD), the foveal attachment is an important prognostic factors for post-operative vision. When the fovea is obscured by the RRD, its attachment status is considered uncertain. Using a model of the reduced emmetropic and − 10 dioptre myopic eye and the physical properties of the detached retina, we aimed to mathematically ascertain if it is clinically possible for the fovea to be attached while it is obscured by the primary RRD. With the patient upright, a primary RRD due to a 12 o’clock break directly above the fovea was considered. Mathematically, once the trough of the RRD touches the visual axis the edge of the RRD nearest to fovea is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.77\,{\text{mm}}$$\end{document}2.77mm away from fovea in emmetropic eye and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.89\;\,{\text{mm}}$$\end{document}2.89mm in myopic eye. When the RRD reaches the fovea, its trough is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.20{\text{ mm}}$$\end{document}2.20mm below the visual axis in emmetropic eye and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.29{\text{ mm}}$$\end{document}2.29mm in myopic eye. However, in vivo the RRD makes an acute angle with the retinal pigment epithelium and the corrugation of the retina in RRD shortens the retina. When these in vivo constraints are considered, in both of the above situations the fovea will be detached. If the fovea is obscured by an RRD, the fovea is very likely to be detached. In idiomatic terms, if the fovea cannot be seen, the fovea cannot see. This is an important clinical diagnosis for appropriate triage of the patient. Supplementary Information The online version contains supplementary material available at 10.1186/s40942-022-00359-3.


Figure S2
The inner globe can be treated as a circle with centre and radius as shown in Figure S2.

Scenario 1:
In this scenario the RRD represented by the catenary arc extends from the ora serrata at point to fovea at point . We would like to calculate how much the trough of the RRD hangs below the visual axis given by 1 − ( Figure S3). With aid of Python Version 3.6 and 'math' module, we used trial and error method of iteration and calculated constant as = 6.714439009374067.
Next we need to calculate the coordinates of points ( 1 , 1 ) and ( 2 , 2 ) for this catenary.
( 1 , 1 ) Therefore, the distance of the trough of the curve below the visual axis is given by:
[ Therefore, when the trough of the RRD just touches the visual axis and starts to obscure the fovea, the closest edged of the RRD to fovea is 2.77 or 14.4° away from the fovea.

Figure S5
We would like to ascertain the position of the point on the Figure S5 when the first order superior arcade vessel is visible on the RRD at the its trough (point ).
First order superior arcade vessel visible on RRD at or anterior to point C Let's assume that the first order superior arcade vessel is 15° or 12 above the fovea. This means the arclength is given by, where is the horizontal distance between points and . The value of can be calculated as follows, = sinh −1 ( 77 12 ) = 6.7 … sinh −1 ( 77 12 × 6.7 … ) = 12.24737500590527 Therefore, the horizontal distance between points and ( ̅̅̅̅ ) is very close to, ̅̅̅̅̅ ≈ + cos − = 11 + 11 cos 3 − 12.2 … = 4.285335803611268 The arclength can be calculated as follows, We can calculate the distance between points and on the circle in angles ( ℎ + ℎ ) = (4.5 … + 77 12 )

= 2.249163505329795
Therefore the distance between points and on the circle in angle is Therefore, if the first order superior arcade vessel is visible on the RRD, even if it is at the trough of the RRD, the edge of the RRD is 8.8° or 1.7 below the fovea -i.e. the fovea is detached.

Shortened detached retina due to corrugation
If the RRD is starting from the ora serrata and extends to the fovea, how much the retina needs to be shortened by the corrugation before the trough of the RRD is at the level of the fovea on the visual axis, because the retina is lifted, as illustrated in Figure S6. Therefore, we need to calculate the value of in this scenario.

Myopic eye as an ellipsoid:
The dimensions of the myopic eye was taken from the study by Pope et al (2). For a -10 dioptre myopic eye, the height was calculated to be 25.13 and length was calculated to be 27.13 . Therefore, the sagittal section of a myopic eye is an ellipse ( Figure S7).
The length of the detached retinal arc for a myopic eye can be calculated using the equation for ellipse: The scenarios 1 and 2 for the myopic eye was calculated as described for emmetropic eye. However, we calculated the and coordinates and arclength for an ellipse as described above.