**Purpose.**:
We aim to understand how mechanical causation influences retinal detachment and reattachment processes. In particular, myopes suffer retinal detachment more frequently than emmetropes, and following a retinal detachment, scleral buckling promotes retinal reattachment. We test the hypothesis that stresses arising from saccadic eye rotations are involved in the processes, and that the alteration in the stress due to the change in the vitreous chamber geometry is sufficient to explain the phenomena.

**Methods.**:
The vitreous chamber of the eye has an approximately spherical shape and it is filled with vitreous humor. We developed a mathematical model, treating the vitreous chamber in emmetropic and myopic eyes as a spheroid and in eyes subjected to scleral buckling as a sphere with a circumferential indentation. We assume that the eye performs prescribed small-amplitude, periodic, torsional rotations and we solve semi-analytically for the fluid pressure, velocity, and stress distributions.

**Results.**:
The shape of the vitreous chamber has a large effect on the retinal stress. The vitreous and the retina of a highly myopic eye continuously experience shear stresses significantly higher than those of an emmetropic eye. An eye fitted with a scleral buckle experiences large stress levels localized around the buckle.

**Conclusions.**:
Our results provide a mechanical explanation for the more frequent occurrence of posterior vitreous detachment and retinal detachment in myopic eyes. To understand how the stress distribution in a buckled eye facilitates reattachment, an additional model of the details of the reattachment process should be coupled to this model.

^{ 1–6 }The gel state is maintained by a network of collagen fibrils, which typically degrades with age, resulting in vitreous liquefaction. The mechanisms underlying this process are not fully understood.

^{ 7 }It is also known that both posterior vitreous detachment (PVD) and lattice degeneration, which are risk factors for RD, are more common in myopes.

^{ 8–10 }To our knowledge, the reasons for these findings are not fully understood. Myopia is typically associated with a shape change in the vitreous chamber; in particular, myopic eyes are on average larger than emmetropic eyes and more elongated in the anteroposterior direction.

^{ 11,12 }We postulate that this might promote the disintegration of the vitreous structure due to the generation of larger stresses in the vitreous humor and on the retina during eye movements, in turn increasing the risk of PVD and RD. The potential mechanical influence of such a shape change has been put forward by various authors in the past,

^{ 13,14 }but quantitative evaluation of the variation of the stress in the vitreous and on the retina in myopic eyes with respect to emmetropic ones has not been carried out. Evidence for mechanical causation of RD in myopes is provided by the fact that mechanical insult can induce RD in some circumstances. Furthermore, Pickett-Seltner et al.

^{ 15 }measured the chemical composition of the vitreous humor of chicken myopes, finding it to be similar to those with normal eyes. Finally, Meyer-Schwickerath and Gerke

^{ 16 }found that eyes with RD, whether myopic or emmetropic, are of different dimensions from normal eyes, another indication that mechanics plays a role.

^{ 17 }reviewed the types of implants that are in use in this surgery, both in terms of the materials used and in terms of the shape of the buckle. A mechanical model of the effect of scleral buckling was developed by Foster et al.,

^{ 18 }but a full understanding of the reattachment process is so far elusive.

^{ 19 }developed a mathematical model of an eye undergoing rotations, representing the vitreous chamber as a torsionally oscillating sphere. With this model, they calculated the fluid motion and the resulting stress on the retina. The stress grows with the radius of the eye; thus, for a given eye movement the retinas of myopes are subject to a greater shear stress, and the authors suggest this finding as a possible explanation for the increased incidence of RD in myopia.

^{ 20 }investigated and compared different rheological models of the vitreous humor,

^{ 2–4 }paying particular attention to the possible occurrence of resonance. They computed the natural frequencies of oscillation of the vitreous using the parameter values reported by various authors and showed that, if the fluid-filled sphere were forced by torsional oscillations of the appropriate frequency and of constant amplitude, the fluid motion would be significantly stronger than in the case of nonresonant forcing.

^{ 21 }studied the effect of the shape of the vitreous chamber on the flow of a purely viscous fluid during torsional oscillations. They found that the flow in the chamber indented by the lens, and consequently also the wall shear stress, is significantly different from that in a sphere. These changes are caused by additional vortices in the flow that appear near to the indentation.

^{ 20 }and Repetto et al.

^{ 21 }As the eye rotates in the course of everyday activities, its motion induces flow in the vitreous humor, leading to a time- and space-dependent stress distribution within it. We model the vitreous humor as a viscoelastic fluid and the vitreous chamber as a deformed sphere, and use this to predict the stress on the retina during saccades. We use this to find how the mechanical forces on the retina due to eye motion differ when the shape of the sclera is changed. We investigate whether mechanics can be responsible for the increased incidence of RD among myopes and whether it is involved in the reattachment process during scleral buckling.

*ρ*. We used the model to calculate the flow and pressure fields during idealized eye rotations, and we found, in particular, the stress on the retina. The details of the method are provided in the Appendix, and here we give a summary.

*G*(

*t*), which describes the strength of the memory of the state at a time

*t*ago. It can be shown that, for small time-sinusoidal displacements and velocities, this fluid behavior can be modeled by introducing a “complex viscosity,” in place of the usual “real” viscosity used for viscous fluids.

^{ 22 }The real part of this complex viscosity provides a measure of the viscosity of the fluid, whereas its imaginary part quantifies the elastic component.

*R*(

*ϑ*,

*φ*), where (

*r**,

*ϑ*,

*φ*) are spherical coordinates (radial, zenithal, and azimuthal, respectively), and we assume that

*R*is equal to a constant value,

*R*

_{0}, plus a relatively small perturbation, whose maximum size is

*δR*

_{0}; thus

*δ*(assumed small) characterizes the depth of the indentation or the height of the maximum projecting region.

*ε*radians (assumed small), and frequency

*ω*. In the discussion section, we remove this assumption and consider the effect of real saccadic rotations on the results.

*R*

_{0}can be calculated analytically as a closed form solution. This was done by David et al.

^{ 19 }for the case of a viscoelastic fluid described by a model with four mechanical elements and by Meskauskas et al.

^{ 20 }for the general case. One advantage of analytical, as opposed to numerical, solutions is that the solution is found for all parameter values, leading to a better understanding of the behavior. This can enable the trends as parameters change to be found, and also any qualitative changes in behavior to be determined.

*δ*, using the method developed by Repetto et al.,

^{ 21 }which involves decomposing the solution into vector spherical harmonics.

^{ 23 }The flow field can be used to find the stress at the wall, which is our estimate of the force per unit area exerted by the fluid on the retina. The normal component of the stress on the boundary is due to the fluid pressure, whereas the tangential component arises due to the fluid velocity.

^{ 11,12 }studied the shape of myopic eyes using magnetic resonance imaging. They provided regression lines for the vitreous chamber width, height, and length as a function of the refractive error (best sphere correction) (Fig. 4A of their article

^{ 12 }), and showed that the vitreous chamber shape can be approximated by an ellipsoid. They showed that in emmetropic eyes, the anteroposterior axis is typically shorter than the other two. As the degree of myopia increases, all axes grow in length, but the anteroposterior axis grows the most. Therefore, myopic eyes are both larger and closer to spherical in shape than emmetropic ones, whereas in severely myopic eyes the vitreous chamber may be prolate (i.e., the anteroposterior axis is the longest). Horizontal and vertical cross-sections of various eyes are illustrated in Figure 1, based on the regressions proposed by Atchison et al.

^{ 12 }As can be seen, the predicted shapes are close to spherical.

**Figure 1.**

**Figure 1.**

^{ 12 }

^{ 24 }studied the effect of scleral buckles that encircle the eyeball, or cerclages, on the scleral shape. They modeled the eye as a uniform spherical shell filled with a fluid and the buckle as an elastic band stretched around the equator that exerts an inward radial force on the shell. The authors calculated both the intraoperative shape and the postoperative shape, and compared their results to the shape change in an ennucleated eyeball. Wang et al.

^{ 25 }also investigated the change in shape of the eyeball due to a scleral buckle, using a finite element numerical model to calculate the predicted shape change and the change in refractive index.

*R̂*with two circular indentations of radius

*R*

_{ i }on opposite sides of the circle (the lines around the intersections between the circles are smoothed). The depth of the indentation with respect to the circle of radius

*R̂*is denoted by

*d*. The three-dimensional shape is obtained by rotating this cross section about the

*z*-axis.

*R*

_{0}is chosen as the radius of the sphere with the same volume as the actual domain.

**Figure 2.**

**Figure 2.**

*R*

_{ i }/

*R̂*and

*d*/

*R̂*, and investigate the role of these parameters on the stress distribution and intensity within the vitreous chamber.

*δ*, we obtain the solution for the motion of a fluid in a sphere, originally found by David et al.

^{0}^{ 19 }and recently reconsidered by Meskauskas et al.

^{ 20 }We focus in this article on the effect of the shape of the vitreous chamber.

*x*,

*z*) orthogonal to the axis of rotation,

*y*. In the figure,

*φ*= 0 corresponds to the anterior part of the vitreous chamber and

*φ*=

*π*to the back pole. The normal and tangential stresses are scaled by

*ερω*

^{2}$ R 0 , e 2 $, where

*R*

_{0,e }denotes the radius of the sphere with the same volume as the emmetropic eye in all plots. Figures 3 and 4 are obtained using the rheological properties of the vitreous humor proposed by Swindle et al.

^{ 4 }Note that the maximum stress on the retina also represents the maximum value of the stress attained in the vitreous body. The tangential stress is significantly larger than the pressure, and it appears at leading order, whereas the pressure first appears at the next order (

*δ*) (see equations [16] and [21] in the Appendix). Therefore, the normal stress is close to zero in a domain with an (almost) spherical shape.

**Figure 3.**

**Figure 3.**

**Figure 4.**

**Figure 4.**

^{ 3–5 }In spite of the large differences of the rheological properties of the vitreous humor provided by the authors, the curves are relatively close to one another. Figure 5a shows that the maximum shear stress on the retina increases monotonically with increasing degree of myopia. On the other hand, the normal stress on the retina has a minimum for eyes with a refractive error of approximately −11 D, which corresponds to the point at which the shape of the vitreous chamber is closest to spherical.

**Figure 5.**

**Figure 5.**

^{ 12 }The shape has a significant effect on the shear stress intensity on the retina, and departure from sphericity invariably induces an increase of stress magnitude.

**Figure 6.**

**Figure 6.**

*ω*and the dependence of the maximum stress on the retina upon the frequency is shown in Figure 7. The figure shows that the wall shear stress in highly myopic eyes is significantly higher than that in emmetropic eyes for the whole range of frequencies spanned.

**Figure 7.**

**Figure 7.**

*ε ρω*

^{2}$ R 0 2 $, in a geometry approximating the vitreous chamber of an eye subjected to scleral buckling by a cerclage. The wall shear stress has intense peaks at the locations of maximum indentation of the wall. The value of the maximum shear stress is approximately twice as large as the maximum shear stress generated in a sphere, indicating that the buckle has a significant effect.

**Figure 8.**

**Figure 8.**

*d*, and buckle width on the stress magnitude, we consider a range of various indentations with different dimensionless depths

*d*/

*R̂*and different shapes (in each case the cross section of the indentation is an arc of a circle, but we change the value of

*R*/

_{i}*R̂*). In the plots we report the stress scaled with

*ε ρω*

^{2}$ R 0 2 $, and we keep the volume of the vitreous chamber constant (i.e.,

*R*

_{0}has been fixed). Following the discussion reported in Keeling et al.,

^{ 24 }this case is representative of short times after surgery; however, we also ran the model for the case in which the surface area of the vitreous chamber was kept constant. This case corresponds approximately to the state of the eye in the long term after surgery. We found very little quantitative (and no qualitative) difference on replotting Figure 9 for a fixed scleral surface area.

**Figure 9.**

**Figure 9.**

*d*/

*R̂*(thick curves), and there is a slight decrease as

*R*/

_{i}*R̂*increases. We note that our model is not valid for very small values of

*R*/

_{i}*R̂*, due to the asymptotic method used; see also Repetto et al.

^{ 21 }In Figure 9a we also show with thin lines the maximum (in time) of the shear stress at the back pole of the eye. This is only slightly affected by the presence of the buckle, and it is invariably significantly smaller than the stress at the indentation. Finally, we note that the pressure at the back pole of the vitreous chamber is always equal to zero.

**Figure 10.**

**Figure 10.**

^{ 19 }showed that the maximum amplitude of the wall shear stress grows with the eyeball diameter; however, in that article the shape of the eye was not considered, and, most crucially from the clinical point of view, the relationship between the degree of myopia and the stress on the retina was not quantified.

^{ 19 }in which only spheres were considered.

^{ 26 }and thus more prone to damage. Also, the pumping efficiency of the retina in myopic eyes is likely to be different, and this possibly influences the strength of adhesion between the neural retina and retinal pigment epithelium.

^{ 18,27 }To this end, it is essential to understand the stress distribution on the retina in the presence of a scleral buckle. We believe that the pressure distribution is particularly relevant, as the dynamics of the detached retina are likely to be strongly influenced by the pressure difference between the two sides of the flap. Our model shows that the buckle has a significant influence on the stress exerted on the retina. In particular, we showed that the maximum values of the pressure are located at the sides of the indentation produced by the buckle; moreover, we found that, for a given indentation depth, the stress on the retina does not depend significantly on the radius of the indentation produced by the buckle, which, for simplicity and generality, has been assumed to have a cross section consisting of an arc of a circle.

^{ 28 }we model the angular velocity of the eyeball during a saccadic rotation with a fourth-order polynomial, the coefficients of which are based on the empirical relationships proposed by Becker

^{ 29 }(see equation [1] in Repetto et al.

^{ 28 }). In Figure 11a we show the eyeball angular velocity versus time during saccades of different amplitudes.

**Figure 11.**

**Figure 11.**

*T*, separated by a resting time of duration 19

*T*. We expand the angular velocity corresponding to this motion in Fourier harmonics and superimpose the solution (in terms of velocity and wall shear stress) induced by each of these harmonics. Owing to the dissipative nature of the system, the velocity in the vitreous humor decreases to very small values after each saccadic rotation; therefore, the solution can be thought of as the response to a single eye rotation (the more so the higher the separation time between two successive rotations). We note that, to study vitreous dynamics induced by this motion of the eye, we had to make an assumption concerning the dependence of the complex modulus

*G*on the frequency

*ω*. Following Meskauskas et al.,

^{ 20 }we modeled the vitreous according to a two-parameter Kelvin model.

^{ 21 }have shown (in the case of a purely viscous fluid) that the presence of the lens predominantly influences the fluid motion in the anterior segment of the vitreous chamber, while the flow in the posterior pole is only slightly affected, meaning the predictions of our model in the posterior part of the vitreous chamber are expected to hold.

^{ 19 }added strength to this argument by showing that the stress at the retina increases with the size of the eye. In this article, we have shown that the nonspherical shape of the vitreous chamber is also significant, and were able to relate the degree of myopia to the excess of stress on the retina. We found that the vitreous humor and the retina in highly myopic eyes are continuously subjected to significantly larger shear stresses than in emmetropic eyes and postulate that this might provide a mechanical explanation for the more frequent occurrence of disruption of the structure of the vitreous humor, PVD, and RD in myopia.

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*ρ*, occupying a near-spherical region of approximate radius

*R*

_{0}. The fluid motion is governed by Here superscript asterisks are used to denote dimensional variables that will later be made dimensionless, and

*** is the velocity,**

*u**t** is time and

*** is the Cauchy stress tensor. In a linear viscoelastic fluid the stress tensor is given by**

*σ*^{22}where

*p** is the fluid pressure,

*G*is the relaxation modulus, is the rate-of-strain tensor, and

**is the identity matrix. Equations (1**

*I**a,b*) are to be solved subject to no-slip boundary conditions.

*β*(

*t**) = −

*ε*cos(

*ωt**), where

*ε*≪ 1. The assumption

*ε*≪ 1 allows us to neglect the nonlinear inertial term in equation (1

*a*), meaning the equations become linear and allowing us to seek solutions of the form

***(**

*u****,**

*x**t**) =

*** (**

*U****)**

*x**e*

^{iωt}^{*}+

*c.c.*,

*p**(

***,**

*x**t**) =

*P**(

***)**

*x**e*

^{iωt}^{*}+

*c.c.*and

***(**

*σ****,**

*x**t**) =

***(**

*σ****)**

*x**e*

^{iωt}^{*}+

*c.c*., where

*c.c*. denotes the complex conjugate. The variables are related by where the complex modulus of the fluid is given by and the governing equations (1

*a,b*) become Following the approach developed by Repetto et al.,

^{21}we model the vitreous chamber as a weakly deformed sphere and assume that the eyeball has radius where the constant

*R*

_{0}is the radius of the sphere whose volume equals that of the vitreous chamber and

*δ*is chosen so that the maximum absolute value of

*R*

_{1}is 1, and (

*r**,

*ϑ*,

*ϕ*) are spherical coordinates fixed in space, with the line

*ϑ*= 0,

*π*being the inferior-superior axis of the eye (the axis of rotation),

*ϕ*= 0 being the anterior direction, and we define

*φ*=

*ϕ*−

*β*(

*t**), so that

*φ*is the azimuthal coordinate rotating with the domain. The origin of the shape is coincident with its center of mass.

*a,b*) become The no-slip boundary condition gives where

**denotes the unit vector in the direction of its subscript. From a clinical point of view, it is of particular interest to evaluate the stress exerted by the vitreous humor on the retina, which is given by**

*e******

*σ***=**

*n**ρω*

^{2}$R02$(

**e**

*T**+*

^{it}*c.c.*), where

**is the unit normal vector of the surface and**

*n**δ*

*O*(

*δ*

^{2}) and higher. Where necessary in the following, we denote by

*U*,

*V*, and

*W*the components of

**in the directions of increasing**

*U**r*,

*ϑ*, and

*φ*, respectively, and we use the subscripts 0 and 1 to denote their contributions to

*U*_{0}and

*U*_{1}, respectively.

*O*(

*δ*

^{0}), the problem is in the spherical geometry

*r*≤ 1, and it was solved by David et al.

^{19}We briefly recap the solution: The continuity equation (11

*b*) and the

*r*- and

*ϑ*-components of equation (11

*a*) decouple from the

*φ*-component of (11

*a*), and thus since the right-hand side of the boundary condition (12) is purely in the

*φ*-direction, only the

*φ*-component of the velocity is nonzero. The solution is where

*a*=

*α*

_{c}e^{−iπ/4}. Substituting into equations (9) and (13) gives the leading-order contribution to the stress as At

*O*(

*δ*

^{1}) we use the fact that any smooth scalar function can be written as a sum of scalar spherical harmonics,

*Y*, and a vector-valued function as a sum of vector spherical harmonics,

_{mn}

*P**,*

_{mn}

*B**, and*

_{mn}

*C**. The reader is referred to other literature*

_{mn}^{23}for a detailed definition of the vectors

*P**,*

_{mn}

*B**, and*

_{mn}

*C**, but here we remark that for a particular*

_{mn}*m*and

*n*they are pairwise orthogonal, with

*P**always in the radial direction and*

_{mn}

*B**and*

_{mn}

*C**spanning the zenithal and azimuthal directions. We expand the velocity and pressure as where*

_{mn}*U*

_{1}

*(*

^{mn}*r*),

*V*

_{1}

*(*

^{mn}*r*),

*W*

_{1}

*(*

^{mn}*r*), and $P1mn$(

*r*) are functions to be found.

*a,b*) we obtain a system of ordinary differential equations that can be solved to obtain for n > 0, and $P100$ = $U100$ = 0, where

*J*denotes the Bessel function of order

_{k}*k*,

*s*= $n(n+1)$, and the boundary condition (18) implies The stress at order

_{n}*δ*can be calculated from the formula