ELSEVIER

Time-dependent stress and displacement of the eye wall tissue of the human eye

Tim David*, Steve Smye-f, Teifi James-f and Tim Dabbst

*Department of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, UK iDepartments of Medical Physics and Opthalmolgy, St James’ University Hospital, L,eeds, UK

ABSTRACT Alyopia or short sightedne.ss, is the most important predisposing ,fLctor to retinal drtochmen t’ .“. The relative ritk o/

detac‘hment rises with increasing mvopii’. The model character&y that because the severity oJ myopia increase\ with

the axial length (anter(~osterior diameter) of the eyeball, the relative risk oJretina1 detachment rises with increa.sinCq

p1’e .tize. We present a mathematical model of the time-dependent shear .stre,ss forcr that occurs in the thin eye wall

.A~11 .~u#n-ting the vitreous humour inside the tye globe during the a&.eration and deceleration phase.7 oJ tarrad~c

ty~ movement. Kpsults .show that the shearforce increases as thr thickne,s.s o/” the qje wall derreuses. It is common /in

myo$e,rs to have thinnm GYP wall tissue than emmetropes. In addition, iJ account is taken of the increased /;UW

required to provide normal saccadic movement of myopic (large) ryes, then the shear force b up to seven time., Sgrea/~*r

than that ex@+nred ,jin emmetro+. 0 1997 lil$sylGr Scienc.e I,td 1;~ I&M.

Keywords: Retinal detachment, stress, eye

Med. E:ng. Phys., 1997, Vol. 19, 131-139, Marc-h

1. INTRODUCTION

When retinal detachment occurs as a result of breaks, holes or tears in the retina, they are known as rhegmatogenous detachments, distinguishing them from the much rarer tractional and exud- ative types. Epidemiological studies have shown that an eye with a refractive error of - 1 to -3 dioptres has a fourfold increased relative risk of retinal detachment compared with a non-myopic eye. When the refractive error is greater than -3 dioptres, the risk increases to tenfold3. Rhegma- togenous detachments may arise as a result of vit- reous traction on the retina causing a retinal tear with separation of the neural retina from its underlying pigment epithelium“. Fluid then accumulates under the detached retina with loss of retinal function due both to porous flow through the retinal body and as the eye moves saccadically causing vitreous flow across and into the retinal tear. Previously, it was thought that areas of degenerative change in myopic periph- eral retina rendered it vulnerable to tearing and subsequent detachment4-“. Retinal detachment is sight-threatening, although surgery is frequently successful. More important, however, is the fact that over half of non-traumatic detachments in

eyes without previous surgery appear to be linked to myopia’.

The nature of normal vitreous gel is well known1 ‘; it is mainly water (about 97%)) stabilized by an ordered meshwork of very fine collagen fibrils that are joined together in loosely parallel bundles by bridges of sulphated glycosaminogly- cans, either hyaluronan or chondroitin sulphate12. Peripherally, larger bundles of vitreous fibrils are attached to the basal laminae of retinal glial cells, and cords of vitreous collagen are inserted into gaps between the neuroglial cells of the retina’“. The structure of the peripheral vitreo-retinal interface has been likened to Velcro@ and explains the very strong vitreo-retinal adhesion’“. The retina is an elastic membrane, and Young’s modulus (for bovine retina) has been calculated to be approximately 2x10” Pa, which is about three orders of magnitude weaker than typical rubber15

Retinal tearing and subsequent detachment often occur at the same time as the process of vitreous gel separation, an event known as pos- terior vitreous detachment. This collapse of the vitreous gel is more common and happens at an earlier age in myopic eyes”. Sometimes retinal detachment occurs as a result of a giant retinal tear, and in a series of 100 such cases, over 70% occurred in myopic eyes’“. While normal eyes have an axial length (antero-posterior diameter)

Stress and displacemat of human eye wall tissue: T. David et al.

of about 22 mm, highly myopic eyes may be more than 30 mm long.

Clearly, the fluid motion in the eye provides a possible mechanism for increasing the detach- ment of the retina from the retinal pigment epi- thelium once a tear has occurred. There are now three possibilities for the initial detachment; firstly, a mechanism wholly due to the vitreous traction exerted on the eye wall by the fibre bundles; secondly, the stress in the eye wall caused by the forced of the horizontal rectus muscle dur- ing normal eye movement; finally, a combination of both. As mentioned above, myopes have larger than average eye globes. It seems less than likely that the retina in this larger eye would experience an increased force due solely to the fibre bundles. We therefore initially investigate the stress occur- ring in the eye wall due to the muscle force and leave the more complex model of a combination of both traction and stress for further investi- gation. Development of this model will help to determine whether the geometry of the myopic eye may predispose an increased risk of retinal detachment due to the larger stresses produced in the sclerotic, choroid and retinal layers during ‘normal’ eye movement.

In normal movement, the horizontal rectus muscles act on the emmetropic eyeball turning it to the left or right, exerting forces on the eye in the range of 0.07-0.8 N I’ These forces increase . with the angular deviation of the eye. At the onset of an horizontal saccade, the muscles apply a force F to the outer aspect of the sclera, which results in the rotation of the eyeball about the vertical axis. The pair of ocular muscles responsible for this rotation are reciprocally innervated (Herrings law). As one contracts to produce rotation, the opposing muscle relaxes. The net force is trans- mitted to the rest of the eyeball across the neuro- retinal cleavage plane, which is assumed to be hemispherical. For an emmetropic eye (a normal- sized eye with no refractive error), estimates of F and R are 0.07 N and 1.1~10~~ m, respectively. High peak velocities recorded from human eye movements have been measured at up to 1000” s-l and a characteristic of saccades is that the peak veloci

4y is proportional to the size of the

eye movement l ,lg We can find no published evi- . dence that the angular velocities of myopic sac- cades are any different to those of emmetropic eyes. The mathematical model given below can assume either that the muscle force is constant or that the rectus muscle force, increases with eyeball size in order to maintain the angular velocity achi- eved during the initial acceleration phase. Towards the end of a saccadic movement, during the deceleration phase, the shear force across the vitreo-retinal interface is similarly radiusdepen- dent, provided the angle through which the decel- eration occurs is independent of eye size.

Retinal thinning may also accentuate the pre- disposition to retinal tearing as the thinner mem- brane will be less able to accommodate the increased shearing forces without rupture. The overall thickness of the wall of the eye depends on the thicknesses of the contributing ocular

structures, sclera, choroid and retina. Myopic eyes often have thinner sclera, and this often presents practical problems during surgical procedures. We now develop a dynamic displacement/stress model that includes the variation of eye wall thickness, eye radius and increased muscular force that accompanies myopic conditions in order to elucidate the differences in probabilities of retinal tearing in myopes compared with that of emmetropic eyes.

2. MATHEMATICAL MODEL

Figure I shows the general coordinate axes used in the analysis, where we take the positive z-axis as being the direction in which the light rays travel. 8 (the circumferential angle) measures the angle subtended around the equator, whilst cp measures the angle subtended from the north to the south pole (from the lens to the optic nerve). R meas- ures the radius of the eye globe. In order to obtain an analytical solution to the problem of strain and displacement in the wall tissue of the eye, we will need to make some simplifying assumptions. Firstly, it is assumed that the eye is spherical. Sec- ondly, we assume that the traction exerted by the muscle on the eye is independent of the circum- ferential angle 8. This essentially simplifies the external muscles attached to the outside of the eye globe to an axisymmetric band centred on the equator. This simplifies the analysis without sacri- ficing the essential mechanics. Relaxing this restriction would require an expensive numerical solution, and it is not unreasonable at this stage to evaluate initial models prior to those complex numerical investigations.

In addition, we need to make several assump- tions about the force exerted on the eye; initially, we restrict the model to that of a notional ‘force’ simply switching on in an impulsive fashion. The proposed model makes use of ‘foundation para- meters’ (A and k), which represent a damping of the displacement (from the equilibrium) of the sclera, choroid and retinal tissue due to the sur- rounding tissue in the eye socket. Given that there are no experimentally known values for these parameters, we initially assume an underdamped

Figure 1 Co-ordinate system for analysis

132

situation where these parameters have zero value. This can be thought of as the ‘no-damping con- dition’.

We assume that the eyeball is represented as a spherically thin shell whose material properties are isotropic. Hence, we have neglected any influence resulting from the vitreous fluid motion. In addition, we will treat the vibrations and subsequent stress calculations as being elastic in nature (there are no permanent deformations). Under these assumptions, we may use thin shell theory, and the reader is referred to Klaus”’ for full details of the theory. We use a similar set of notation to that of Klaus so as to avoid confusion.

The displacements w, u (away from equilibrium) in both the radial (normal) and the polar (cp) co-ordinates, respectively, are taken as being represented by a spectral expansion’using the fundamental free vibration modes (of fre- quency o,J of a sphere. In addition, the presented model provides for the inclusion of resistive forces proportional to displacement and velocity of the reference surface of the eye wall, the respective coefficients being k and A. In order to develop the basic model, we nondimensionalize the displace- ments uV, ue, w in the cp, 8 and normal directions, respectively, the time t, the foundation parameters

k and h and finally the modal frequencies w,. Hence:

where R is the eyeball radius, p the tissue density, M a characteristic moment exerted on the eye, E the Young’s Modulus of the sclerotic tissue and &, a characteristic pressure. Then, we can write the polar and normal displacements as a sum of fun- damental displacements u,,(cp), w,, (cp): thus:

(2a)

133

Stress and displacmmt of human eye wall tissue: T. David et al.

Tangential Stress R=l lmm, h=l mm

phi (radians)

time (ascs)

Figure 3 Timedependent surface plot for the tangential stress v~, kll mm and kl mm

where m, are the mode participation factors (defined below), &(r) is the time-dependent force component and r, represents the system con- dition of being either under-, over- or critically damped. Here, the fundamental displacements u,,, (cp), w,(q) are written as functions of the Legen- dre polynomials P,, and using a ratio of displace- ments to eliminate a constant of integration, we can show thatzO

(2b) cal about the circumferential coordinate 8 and noting that the force is only time dependent, then

At this point, we need to specify the forces exerted on the eye. The analysis dictates that we separate the temporal and spatial components such that, for an external force, H(cp,O,7) then

WcpJA4 =f(4 G(O). (4) The mode participation factors then have the

integral representation given by

Using the fact that the surface force is symmetri-

(6)

where E is the angle subtended (centred about the equator) by the farthest positions of attachment of the external rectus muscle at the surface of the eye; hence, we can simplify the mode partici- pation factor in the following manner.

m, = $ JJG((p,f3) u,,(cp)R?sin~d~dO 0 12 3

( 1-n:

)

;+e

1+ (1+ $2; 23M[sin(pP,(coscp)

77 2 s

(7)

;+,

+ I P,(cosp)coscpdq].

Although anatomically, the rectus muscle sits anteriorly to the equator, this initial model places the muscle at the equator of the eye. This simpli-

134

phi (radians)

Figure 4

Stress (Pa)

3-

2.5 t

2

time (sets)

fication is unlikely to make any significant differ- ence to the outcome, and in fact, the model can tolerate any position on the globe. For the present case, the general integral is given by

where

= PTR? I

[ uqnZ + w,‘]sincpdq

(8) where

[K+ 12(1-u”)R. > h”]

II

since both u and w are independent of 13. We may also simplify the integral for the forcing

function F,,(T); under the assumption that the external rectus muscle ‘immediately’ switches on

’ 0 for 60 such that ft n(~) =

1 for 9-20,’ then we can write

for the underdamped case (where, for example, A and K are both zero)

= ~,~e-S2T[Rsin(lYn7) + lY,,cos(r,,~)]

AZ + my

(9)

and for the overdamped case, which is deter- mined by

[K+ 12(1-zr’)&,“<h”,]

we have

i

F,!(T) = I

r~*(‘-~)sin(r,,(7-rl))drl

0

= ; { $* [/p‘,,--rZP-1)

[ +i;i’l-. p (‘r, + *)T- 1 I}.

1

(10)

Having found the displacements, u and w (both tangential and normal to the eye globe, respectively) as functions of both time and polar

135

Stress and displacement of human eye wall tissue: T. David et al.

Tangential Stress R=l5, h=0.7, F=l.83N

0.004

Figure 5 Timedependent surface plot for the tangential stress aq, R=lfi

co-ordinate, we are able to compute the stresses along the reference surface of the eye. Thus,

(11)

Here, the strain in both polar and circumfer- ential coordinate at the reference surface need to be known, and we can write these as

E*=-. R

By virtue of the model being axisymmetric (independent in the 8 direction) the torsional stresses (and strains) are identically zero. The value of the derivative required in the definition of the reference surface strain is obtained by the simple second order difference formula

(12)

3. RESULTS

We show below the extent of the differences in the tangential stress that occur when we alter both

mm and kO.7 mm, fil.83 N

the radius and the thickness whilst holding the muscular force constant.

Figure 2 is a three-dimensional surface plot of stress, aV, as a function of both time and polar angle q. For this case, we have used the follow- ing values

muscle force: F = 0.5 (Newtons) eye globe radius: R = 0.011 (metres) tissue thickness: h = 0.001 (metres)

K=h=o.

It is clear that, without any damping effects, there is a vibrational wave moving up and down the lines of longitude, rebounding at the poles (which correspond roughly to the positions of the cornea and optic nerve, respectively). Large stresses occur at this point along with a travelling stress wave propagating through the ocular tissue. In Figure 3, we repeat the experiment (same values of F, R and h), but choose a shorter time scale to clarify the wave reflection phenomenon.

Our results indicate that both in-plane stresses have the same form but with slightly differing magnitudes. In relation to the stress measure- ments in the polar direction, Figure 3 shows that the stress increases at the equator over a period of approximately 1 ms whilst the stress wave propagates along the ocular tissue until it reaches the posterior portion (relating to the position of the optic nerve) where it rebounds (causing an

136

Tangential Stress R=15, h=0.7, 4000

stress (Pa) 7-

0.004

Figure 6 ‘Time&pendent surface plot for the tangential stress CT*, with damping parameters as in and K=A=IOOO .Um 5 ’

Table I Maximum stress as a function of R, h and F

K (mm) 11 (mm) F (Newtons) A (Nb ‘) k (Nrn ’ s) Max stress (Pa)

II 1 0.5 II 0 I 3 11 0.7 0.5 0 0 2 I.3 1 0.5 0 (I ‘,

1 5 0.7 0.5 0 0 2.6 15 0.7 1.8 0 I) IO

I .5 0.7 1 .x 1000 1000 ti

approximate doubling of the stress) and then pro- pagating back to the front of the eye. This oscil- lation will proceed indefinitely in the absence of any damping. In the light of this, we should con- centrate, therefore, on the first half of the stress wave cycle. The maximum value of the stress (neglecting the rebound) is approximately 1.5 Pa.

We now change the radius and thickness to model the myopic condition. Thus, we choose

muscle force: F= 0.5 (Newtons) eye globe radius: R = 0.015 (metres) tissue thickness: h = 0.0007 (metres)

Notice that we have maintained a constant mus- cle force I;. Epre 4 shows the stress, aq, for the above myopic condition.

For this case, the travelhng stress wave has an amplitude approximately twice that of the non- myopic (emmetropic) case. The profile is similar to the emmetropic solution. The wave takes longer to reach the rear of the eye because of the increased circumferential distance. The timescale is similar to Figure 3 and the rebound, although still existing, is not shown. As stated in the intro- duction, there is no reason to suggest that myopic eyes have a lower saccadic velocity and/or acceler- ation. Thus, assuming an increase in mass that is proportional to p, then the rotational moment of inertia increases as a function of W, and the corresponding muscle force (assumed to be con- stant throughout saccadic movement) can increase to values of the order of 1.8 N. Thus, if we simply increase the muscle for-cc, which we

137

Stress and displacement of hum.an eye wall tissue T. David et al.

Variation of Max. Stress with thickness

8

3

0.4 0.6 0.8 1 1.2 1.4 1.6

eye wall thickness (mm)

Figure 7 Variation of maximum stress as a function of eye wall thickness, h

assume to be a condition of myopes, then the stress wave amplitude according to our model will increase by a factor of approximately 7, compared to that predicted for emmetropes. Table 1 shows the maximum stress obtained over all space and time for a variety of values of radius R, eye globe tissue thickness h and muscle force F.

Figure 5 shows the stress for values of F = 1.83 N and similar radius and wall thickness as in Figure 4. As predicted, the maximum stress value now exceeds 10 Pa.

Although we have no definite values for the damping parameters of A and k, we show below in Fi re 6 the results for A = lo3 Nm-’ s,

T k= 10 Nm-’ which are somewhat arbitrary but do provide a view of the damping effect. For this case, the external rectus muscle force, F= 1.83 N.

As predicted, the stress wave is damped signifi- cantly as 4 increases. In fact, as time increases, then the timedependent solution will tend to the static solution. Notice that, for this case, the maximum stress is marginally reduced.

Figure 7 shows the relationship between maximum stress occurring in the eye wall tissue and eye wall thickness for constant eye radius and muscle force. As the wall tissue thickness decreases, there is a significant increase in maximum stress. This maximum stress occurs at, or around, cp = 1.2 (slightly anterior to the equator). It should be noted that the thin shell

theory used in this analysis is based upon Love’s postulates, and hence no conclusions can be obtained from the investigation of stress variations in a direction normal to the eye wall surface. The stress evaluated is that occurring at the reference surface of the tissue (situated at the mid-plane of the eye wall). However, it is clear that, according to the model defined above, myopes will suffer considerably higher wall stress than emmetropes.

4. DISCUSSION

The results shown above suggest that the geometry does have a considerable affect on the stress patterns occurring in the eye wall tissue dur- ing normal saccadic motion. The analysis also sug- gests that the thickness is the most sensitive para- meter determining the stress (apart from the obvious muscle force), and reducing the thickness of the eye wall increases the stress. Although the stress wave is of a small amplitude, it should noted that the retina has an extremely weak mechanical attachment to the RPE*‘, and even small displace- ments, such as those that occur in this case, may prove to be a determining factor in the causation of a retinal tear or even the lifting off of the retina from the RPE.

As has been stated in the introduction, the col- lagen fibres located in the vitreous are oriented in a well-ordered fashion normal to the eye wall

138

at positions immediately anterior of the equator and at the optic nerve. The present model takes no account of this spatially varying restraint on the eye wall, since the collagen fibres insert normally to the eye wall, and they will act as a set of damped cantilevers and therefore possibly prevent any sig- nificant displacement in the polar direction. How- ever, the model can be altered to accomplish this by introducing damping factors k(q) and A(q) that are f&ctions of the polar angle rather than being simple constants, as in the case presented here. In addition, the forces exerted 011 the eye by the external rectus muscles may be modelled easily with more complexity (in terms of their tem- poral variation) since the integral used in defining the forcing term may be easily evaluated numerl- tally rather than analytically as at present. Additionally, in later life, the vitreous can become completely detached from the eye wall at specific sites. This will enhance the stress wave amplitude and may well increase the probability of retinal damage at or arolmd the site.

5. CONCLUSIONS

The above rnodel has shown that larger stresses may occur in the eye wall tissue purely as a result of variations in geometry normally found between myopes and emmetropes. This may be a determin- ing factor in the higher incidence of retinal detachments fotmd in myopic patients. Account- ing only for the changes in radius and wall thick- ness, the resultant stress along the eyeball refer- ence surface for myopes is approximately twice that of emmetropes. When the size of the eyeball >md the assumption of constant velocity and accel- eration of the eye are taken into account for both myopes and emmetropic eyes, then the resultant stress is approximately seven times larger for myopes than for emmetropes due to larger exter- nal muscle forces. Initial investigations seem to suggest that the model predicts a higher inci- dence of stress at a position slightly posterior of the equator. However, the muscle geometry has been idealized in this situation, and care sho~~lcl he taken with this interpretation.

ACKNOWLEDGEMENTS

Teifi James is supported by a Wellcome Trust Clinical Research Training Fellowship.

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