Ординатура / Офтальмология / Английские материалы / Orthokeratology Principles and Practice_Mountford, Ruston, Dave_2004

beyond the parameters of this chapter to delve into this advanced science, but it has been used to study the effects of contact lenses on the cornea.

Pye (1996) applied the finite element method to study the stress effects of contact lenses on the cornea. The model simulated eyelid force applied to the cornea under three conditions: no lens, an alignment fit, and a Contex OK-3. In all cases, the tear layer was included as an incompressible solid with no shear strength. The effects of lid force in the no-lens situation are shown in Figure lOA. Greater compressive forces occur in the posterior cornea than the anterior at the corneal apex. At approximately 1.8 mm from center, the compressive forces are equal at all layers, and then reverse, so that the compressive forces are greater in the anterior cornea than the posterior cornea. This pattern continues out to the limbus. When an alignment lens is placed on the eye, there is a dramatic change in the stress distribution (Fig. 10.5). The compressive force is greater in the anterior layers at the apex. There is some fluctuation in the stress until a point 2.0 mm from center is reached. The compressive force for all layers then remains constant to a distance of approximately 3.00 mm from the apex, where the anterior layers develop more compressive stress

Figure 10.5. The stress induced by an alignment fitting lens. Note the difference from the no-lens situation. Courtesy of David Pye.

than the posterior layers, and this then continues out to the limbus.

The addition of the OK-3 to the surface does not appear to change the compression centrally to any significant degree, indicating that the flat back optic zone radius (BOZR) has no greater effect than the alignment BOZR. However, the meridional stresses become constant and equal for all layers between 1.80 and 3.40 mm from

Figure 10.4 Finite element analysis of the stresses on the closed eye with no lens. The stress reverses at approximately 1.80 mm from center. Courtesy of David Pye.

Figure 10.6 The effect of an OK-3 reverse geometry lens on the eye. There is an increase in the stress between the alignment lens and the OK-3 in the area that corresponds with the back optic zone diameter. Courtesy of David Pye.

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center, which is a 60% greater surface area effect than that of the alignment lens (Fig. 10.6). The gradient of change from the 3.40 mm zone to the limbus is greater than that of the alignment lens, although the values become approximately equal at the limbus.

Pye concludes that the manner by which the reverse geometry lens may alter corneal shape is due to either one or both of the following factors:

1.The tear layer trapped between the back surface of the lens and the corneal surface exerts an effect that keeps the cornea in a constant compressive stress in all layers. This may prevent the cornea from maintaining its normal shape in the 1.80-3.40 mm area. This

also corresponds to the deepest part of the postlens tear layer, and the area of corneal shape change seen with the lens.

2.The gradient of change in meridional stress outside the 3.40 mm area from the corneal center is greater than the no-lens or alignment lens conditions. The steep gradient may act as a "driving force" to the corneal shape changes seen with reverse geometry wear.

The model was extended by Howard (2000), who manipulated the viscosity of the tear layer under the lens and also added a measurement of the displacement of the cornea caused by the induced stress of the lens. The model predicted that an applied load to the corneal apex would result in "downwards" deflection of the surface associated with an "upwards and outwards" dis-

Figure 10.7 The displacement induced by the stress of an OK-3. Note the central backward displacement, and the forward displacement at the back optic zone diameter. Courtesy of Michael Howard.

Figure 10.8 The stress on the anterior and posterior cornea. Compression exists anteriorly at the apex and near the edge of the lens. The posterior stromal stress occurs in the area of the reverse curve, Courtesy of Michael Howard.

placement of the cornea in the mid-periphery. The mid-peripheral deflection was due to the hydraulic forces acting on the surface. The displacement of corneal tissue by both a modeled alignment and OK-3lens is shown in Figure 10.7. The stresses acting on the cornea are shown in Figure 10.8. Note the compression centrally and at the edge of the lens and the tension on the posterior surface of the cornea at the optic zone / reverse curve junction.

The lid forces that were applied in each model differed dramatically, with the Pye model applying 3.9 kPa, and the Howard model only 0.46 Pa. However, both of these models do give some good information on the likely stress effects of the lens on the cornea. The modeled displacements do reflect what is currently known with respect to central corneal thinning and mid-peripheral thickening, and are reflected in the postwear topography plots of treated eyes. Future advances in the application of the finite element method may add more information on the exact interaction of the lens / cornea / tear layer system. Howard, like Pye, found that the overall flattening effects of the OK-3 lens were not any more significant than those produced by the standard contact lens model. However, the negative forces produced in the tear layer under the reverse geometry lens, which reach a maximum at the optic zone I reverse curve border, caused significantly greater changes in the mid-periphery.

A MODEl OF THE SQUEEZE FILM FORCE AND LID FORCE INTERACTION

The following model takes the squeeze film force formulas developed by Conway and adds them to the modeled lid force as defined by Lydon & Tait (1988). It must be stressed that this is a purely hypothetical model. However, the intention is to demonstrate graphically the forces under the lens, and then vary the lens design and assess the change in the model. A further extension will be to try to use the model to explain some of the observed outcomes of orthokeratology lens wear.

The concept of the squeeze film force model is shown in Figure 10.9. The lens approaches the surface of the eye due to the force of the lid, and is separated from the cornea by a tear layer. The

squeeze action creates negative force under the lens, with the escape channel being the difference in shape between the horizontal and vertical meridians. This is a quasistatic state, in that some movement of the lens towards the surface of the eye is occurring. Both the Allaire & Flack and Conway models assume this to be in the order of O.llJ.m/ s. As the movement of the lens towards the eye decreases, the viscosity of the tears increases.

Figure 10.10 shows the tear layer profiles of a trio of alignment lenses fitted according to sag philosophy to a cornea of Ro 7.80 mm and eccentricity of 0.50. The back optic zone diameter

Figure 10.9 A model of the squeeze film force. The lens approaches the eye by the force of the lid through a liquid tear layer. The squeeze film escape channel is dueto the difference between the flat and steep meridian. RGL. reverse geometry lens. Courtesy of Christina Eglund. Polymer Technology Corporation.

Figure 10.10 The tear layerprofile of a group of spherical back optic zone radius lenses on a typical aspheric cornea. The flat lens (7.90 mm) shows less apical clearance than the ideal [7.85 mm) or steep [7.70 mm) lens. TLT. tear layerthickness.

Figure 10.11 The force distribution underthe three lenses. The steep lens shows negative (tension) centrally, whilst the ideal fit shows little force centrally, and the flat lens shows central compression.

(BOZO) is 8.00 mm and the periphery is a tangent. The profile extends to the edge of the BOZO. The "ideal" fit, 7.85 mm BOZR, shows approximately 25 IJ.m of apical clearance, whilst the "steep" lens has 35 IJ.m and the "flat" lens 15 IJ.m. This is an accurate reflection of what is observed in clinical practice. The next step is to calculate the combined lid and squeeze film forces and observe what happens. The result is shown in Figure 10.11. Note that the 7.90 mm BOZR lens applies greater positive force centrally than the other two lenses.

As the BOZR of the lens is steepened, the force under the center of the lens becomes more negative, reflecting the clinical adage that steep lenses steepen the cornea by negative or suction force. The force at the edge of the BOZO is positive, and equals the lid force. In the open-eye situation, this positive force is balanced by the surface tension force around the edge of the lens. "Flat"-fitting lenses show greater central compression than do steep lenses, but they also show relative negative tension forces at the BOZO. Optometrists have always understood that flat lenses flatten the cornea by applying compressive force to the surface, and that steep lenses steepen the cornea due to the presence of negative or suction forces under the lens. The modeled forces reflect what is seen in practice.

The values for the force are Nm2 x 10-7, and this is the unit of measurement used for all successive force graphs.

The tear layer profile of a four-zone reverse geometry lens is shown in Figure 10.12. The lens has approximately 5 IJ.m of apical clearance with a BOZR fitted 3.50 0 flatter than Kf . The alignment curve has 10 IJ.m of clearance at the reverse curve / alignment curve junction, and comes into contact with the cornea at its outer edge.

The force graph for the same lens is shown in Figure 10.13. Note that the central area is positive or compression force, but this rapidly changes to tension or negative force at the edge of the BOZO. The force then becomes positive again at the beginning of the alignment curve. Reim (1998) has termed this type of lens construction "dual compression" design, and the force distribution confirms the statement.

Figure 10.12 The tear layer profile of a four-zone lens showing 5 IJ.m of apical clearance. TLT, tear layer thickness.

Figure 10.13 The forcedistribution underthe same lens as Figure 10.12. The central force (compression) is positive, while the negative force (tension) reachesa maximum at the back optic zone diameter.

Figure 10.14 The force under a steep lens and a reverse geometry lens (RGL). Both lenses are approximately

3.00 Dsteeper or flatter than K. Note that the flat lens shows central compression and the steep lens central tension.

Figure 10.15 The squeeze film forces under a group of varying tear layer profile lenses. The squeeze film force is always negative. The lid force is positive. Note that the squeeze force varies with changes to the tear layer thickness (TLn centrally and at the edge of the back optic zone diameter. The lens shown is a BE.

The difference between a very steep (3.00 D steeper than Kf ) spherical lens and a reverse geometry lens (3.00 D flatter than Kf ) over the BOZD is shown in Figure 10.14. Note that they are total opposites. The "steep" lens generates substantial negative force centrally whilst the RGL generates only positive force. However, the important thing to note is that there is a change in the forces acting across the corneal surface. The "steep" lens has negative or tension force centrally and positive force peripherally, whilst the "flat" lens has compressive force centrally and negative force peripherally. It is the differential between the areas of positive and negative forces that determines the tangential stress across the corneal surface. This is dependent on the TLT under the lens which, in tum, is responsible for the squeeze film force. The squeeze film force is dependent on the difference between the maximum and minimum TLT (IX factor), so a logical next step is to calculate the difference in force on the lens for differing TLTs. This is shown in Figure 10.15.

The squeeze film force varies with the change in thickness of the tear layer, either with increasing or decreasing thickness at the edge of the BOZD, or with alterations to the apical clearance. Note that the lid force (blue line) is constant over

the lens surface. The lid force is positive, and the squeeze film force negative.

This indicates that the variation of the effect seen with the lenses could be due to the change in relative negative force at the BOZD rather then the central compressive force from the lids, which remains relatively constant, as the degree of force centrally is very low.

There are two methods of altering the TLT: keeping the apical clearance constant and altering the depth of the tear layer at the BOZD, or, conversely, keeping the TLT at the BOZD constant, and changing the apical clearance.

In practical optometric terms, the first option is the Jessen factor. Here we change the BOZR as a means of controlling the refractive change. If the BOZR is flatter than the flat-K, the degree of flattening controls the depth of the tear layer at the edge of the BOZD.

This is shown in Figure 10.16, where a fourzone lens is fitted 2.00, 3.00, and 4.00 D flatter than K on the model eye with an apical radius (Ro) of 7.80 mm and an eccentricity of 0.50. The apical clearance is assumed to be 10,....,m in each case. Note that, as the BOZR is flattened, the negative force at the BOZD increases. However, the difference between the lens fitted 4.00 D flatter than Kf is very small when compared to the 3.00 D lens. The effect of flattening the BOZR to increase the TLT at the BOZD increases the IX

Force forJessen factor (10J.lm apical clearanca)

Figure 10.16 The forces generated by a four-zone lens fitted according to theJessen factor. The apical clearance isassumed to be 10 J.Lm. Note that there is little difference between the 3.00 Dand 4.00 Dchange lens.

factor. Also, the value of hoincreases, leading to a situation in the Conway formula where increased flattening of the BOZRdoes not produce a change in the overall force. In the example above, the apical clearance is assumed to be 10 J.Lm. However, what if it were less?

The difference in force with a change in apical clearance while keeping all other factors constant is shown in Figure 10.17. The simple act of changing the apical clearance from 10 to 5 J.Lm causes a major change in the squeeze film force for the same eye and lens design. The force for a 4.00 0 refractive change is now a maximum of -20 x 10-7 Nm2 compared to +40 x 10-7 Nm 2 for the greater apical clearance. Therefore it appears that the Jessen factor works by altering the squeeze force by changing the difference between the tear layer at the apex and the BOZO. It is more effective if the apical clearance is reduced. The limit is approximately 4.50 0 flatter than Kit as any greater flattening increases the ho to the stage where it has a negative impact on the force. Interestingly enough, Reim (1998) states that the limit of effectiveness for the Dreimlens occurs when the BOZR is approximately 4.750 flatter than Kf"

The effect that keeping the TLT constant at the BOZO and changing the apical clearance exerts on the force is shown in Figure 10.18. This is the method behind the BElens design and fitting phi-

Figure 10.17 The forces under a four-zone lens with the apical clearance decreased to 5 J.Lm. Note the marked increase in tension at the back optic zone diameter (BOZD). The central compression remains unaltered. Once again, there is little difference between the 3.00 D and 4.00 Dlens. The apical clearance is fixed, and the tear layer thickness at the BOZD variable with the back optic zone radius.

Figure 10.18 The variation in force produced under BE lenses by altering the apical clearance while keeping the tear layer thickness (TLT) at the back optic zone diameter (BOZD) constant. Note that the central compression is unaltered, and that only the force at the edge of the BOZD changes.

losophy. The tension force changes depending on the level of apical clearance. An apical clearance of 1 J.Lm, for example, generates over -2000 x 10-7 Nm2 of force at the edge of the BOZO.

However, the compressive force does not vary with changes to the negative or tension force with the exception that the diameter over which it acts is reduced as the negative force increases. This also occurs with the Jessen factor method of fitting, in that the greater the generation of negative force (and thereby, theoretically, the refractive change), the smaller the area of positive compression.

The clinical implication of this is immediately obvious: the greater the refractive change, the smaller the treatment zone (TxZ) diameter and the greater the tension force generated by the tear film, the smaller the area of action of the compression zone. The relationship between the compression zone and the TxZ diameter requires further investigation.

The finite element analysis shows that the compressive effect of the BOZR is of little concern with respect to the stress induced on the cornea. The above model supports the finding, and shows that the real initiator of the shape change is the change in squeeze film force generated by alterations to tear film thickness. In effect, all lenses produce the same degree of compression, but the tension varies with the tear layer profile. This change can be initiated by either flattening the BOZR whilst keeping the apical clearance constant (Jessen factor) or by keeping the tear layer at the BOZD constant and changing the apical clearance (BE). Orthokeratology does not compress the eye into a flatter shape, but rather tends to "suck" the midperipheral areas outwards.

However, changes to TLT come at a price, and that is accuracy. Chapter 4 gives the simple rule for fitting reverse geometry lenses:

Lens sag = corneal sag + TLT

The entire premise of fitting lenses according to sag philosophy is that a more accurate fitting relationship between the lens and the cornea than that possible with keratometry will occur. As stated previously in this book, orthokeratology lenses are fitted for an outcome, and not judged solely on the appearance of the fluorescein pattern. The result aimed for is dependent on the accuracy of the fit, particularly with respect to the apical clearance that is the prime factor in controlling the squeeze film forces under the lens. The

difference between the central compressive force and the negative tension force at the BOZD sets up the tangential stress that causes the epithelium to change. If, according to the model, the apical clearance varies, so will the tangential stress and so, theoretically, will the corneal shape change.

This then leads to a few other interesting scenarios with the model. What happens if the original topography data are incorrect or the lens manufacture is incorrect?

An example will be used to show the effect of instrument error (Fig. 10.19). Assuming that the corneal eccentricity on which the lens design is based is 0.50, but the "real" eccentricity value varies, then topography overor underestimation of the eccentricity will affect the apical clearance values and hence the force.

If the "true" eccentricity is greater than that used to design the lens, then the original data overestimate the sag and the lens will have "excessive" apical clearance. This decreases the force difference under the lens, as shown by the pink and grey lines. Even greater underestimations of the eccentricity lead to less squeeze film force differentials with a correspondingly low corneal change. In extreme cases, central islands occur (see later). If the "true" eccentricity is less

Figure 10.19 The effect of instrument errorin calculating corneal eccentricity. Assuming the true eccentricity is 0.50, then if the errorproduces a lens with less apical clearance, the tension is increased. The opposite occurs if the eccentricity is overestimated against the actualvalue, resulting in increased apical clearance and less tension.

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than that given by the instrument, the sag of the lens will be less than that required, leading to less apical clearance. If the true eccentricity were 0.49, or as little as 0.01 difference from that given by topography, there is a marked increase in the force differences between the center and the BOZD (blue line). Greater underestimations of the eccentricity values lead to zero apical clearance and lens decentration.

The variation in force caused by instrument error raises an interesting question. If the required lens is represented by the green line, but error causes the force represented by the pink line and a less than optimal refractive change, could it explain the concept of "poor or slow responder?" Alternatively, if a lens that produced the blue line were fitted instead of the optimal lens, would this be classed as a "fast or good responder"? This interesting concept will be discussed in greater detail later.

Let it now be assumed that the topography data are perfect, and the lens designed from it is made to the highest standards. The commonly accepted standard of reproducibility for computer numeric-controlled (CNC) lathes is in the order of ± 0.01 mm or ± 2 urn in sag. What effect does this have on the modeled force?

The result of an error of ± 2 um steeper or flatter than ideal is shown in Figure 10.20. A lens that is 2 p.m flatter than required produces a greater force than either the ideal lens or the 2 urn steeper lens.

The difference in force between the two is considerable, and could help to explain some of the variation in responses when a patient is supplied with a "duplicate" lens that either performs better than the original or worse. Also, the difference could explain why some patients are classified as "good responders" whilst others are rated as being "poor responders." In those cases of higher refractive change where the apical clearance becomes critical, this level of error can lead to either overor undercorrection. Once again, the question of accuracy in fitting comes into the equation. This reaches its zenith when the difference between "what is mathematically perfect" is compared to "what is actually produced." In the examples shown above, the computer program used to produce the lens

Figure 10.20 The effect of manufacturing erroron the squeeze film force. If the final lens is 2.00 p.m too steep (grey line), the force is decreased, and increased if the lens is 2.00 urn too flat (blue line) in sag.

prescriptions is set to three decimal places, in that if a specific tear layer profile is desired, the program then calculates out the exact curves required to fulfill that requirement. This is not reproducible in reality, so there is an inherent error present, even in the model.

The theoretical differences between the "calculated lens" (lens 1) where the accuracy is three decimal places, and the manufactured lens (lens 2), where "rounding off" takes place, is shown in Figure 10.21. There is a marked difference between the two, indicating that the simple expedient of rounding off can have an impact on

Figure 10.21 The effect of "rounding off'can also have an effect on the force generated under the lens. This can reach significantlevels when small apical clearances of 5 um or less are required.

the results achieved with the lens. In the example, the "calculated lens" has the parameters

8.039:6.00/6.665:7.20/7.673:8.40, whilst the manufactured lens is rounded off to 8.04:6.00/ 6.66:7.20/7.67:8.40. However, in other cases of modeled changes due to rounding oft there is very little effect. In those cases where the rounding off has a minimal impact on the values, the manufactured lens will cause the same forces as the modeled lens. However, the fact that rounding off does have an effect in certain cases is yet another example of the sometimes unpredictable nature of the process.

A simple rule may be to ensure that the rounding off compensates for any change in sag at the common chord of contact by altering either the reverse curve or alignment curve in order to maintain sagittal equivalency. The differences between the two only become important when very small apical clearances of less than 5.00 urn are required. As stated previously, an apical clearance of 1 urn will theoretically generate a tension force of -2000 Pa at the edge of the BOZD. This is an impossible task, as the limit of manufacturing tolerance at the 95% confidence level is ± 4 urn. Also, the TLT is normally approximately 3 urn and the lens is assumed to rest on the tear layer.

The limit of the squeeze film force is therefore set by some boundary conditions in that the theoretically possible minimal apical clearance is in the order of 3 !-Lm whilst the maximum that will cause a difference in relative force between the center and the BOZD is approximately 20 urn. The model would therefore indicate that the "window" in which orthokeratology can be effective is between apical clearance values of 4 and 20 um, This is shown by the difference in force generated by a four-zone lens fitted 4.50 D flatter than Kf (0.50 D compression factor) with 5 um of apical clearance. If the apical clearance is increased to 20 urn, the force differential between compression and tension is negligible (Fig. 10.22).

The force profiles of two different lens designs are shown in Figure 10.23. Both lenses were calculated using the best standard of lens manufacture possible, and the closest-fitting sag match allowing for apical clearance. The graph shows the force developed under a Dreimlens and a BE lens

Figure 10.22 A four-zone lens is shown with the relative limits of the squeeze film force and apical clearance. The minimum apical clearance is between 4 and 5 urn,whilst the maximum appears to be approximately 20 um before there is little or notension developed at the edge of the opticzone (back optic zone diameter). AC, alignment curve.

Figure 10.23 The difference in squeeze film force between two differentdesigns, the four-curve design (Dreimlens) and the BE, for the same expected refractive change. The Dreimlens is a "dual compression" design, whereas the BE is based on maximizing the tangential stress across the corneal surface.

for a cornea requiring a 3.00 D refractive change. Note that the force difference between the center and the BOZD is different, and that the compression gradient in the periphery is totally different. The main difference between the designs is the intersection of the compression zone from

284 ORTHOKERATOLOGY

positive to the negative tension area. This could theoretically result in a difference in TxZ diameters between the designs. Also, the Oreimlens is a "dual compression" lens, whilst the BE is designed under the assumption that the tangential stress is maximized and then extended over a larger area.

The modeled forces under the lens show the sensitivity of the system to changes in the tear layer profile, with particular reference to the apical clearance. The important thing to note is that the values generated by the model are purely relative, and are not absolute values. However, it is the trend that accuracy is essential in fitting that is of major importance. The a value gives an independent method of assessing the difference between the "calculated" and actual apical clearance of a lens, and can be used as a method of refining the lens fit.

In developing the model, the same assumptions about the velocity of the lens towards the corneal surface and tear viscosity made by both Allaire & Flack (1980) and Conway (1982) were used. In theory, as the velocity of the lens decreases, the viscosity of the tears increases. Conway points out that the inclusion of the a factor is vital, and that all other parameters are basically independent of it. The force will also change depending on the lens diameter and the modeled lid force.

In this model, the force is proportional to the thickness of the lens, so some variation in thickness will lead to a variation in the force. If the lens center thickness is decreased to 0.15 mm, the lid force changes from 68 x to 52 X 10-7 Nm2, whilst an increase in center thickness to 0.30 mm would increase the force to 81 x 10-7 Nm2• Alterations to the lid force would simply alter the central compressive force, with no real difference to the tension force at the edge of the optic zone or BOZO (Fig. 10.24). However, the tangential stress exerted on the surface is a derivative of the difference between the maximums of the compression and tension values. The slope of the tangential stress would therefore be affected by alteration to lens center thickness, with a hypothesized decreased effect with thinner lenses and an increased effect with thicker lenses. The limitation to this is the Dk/t of the lens. There is little point

Figure 10.24 The effect of altering the central thickness of the lens. If the central thickness is increased, the central compression also increases. Decreasing the central thickness decreases the central compression, but neither affects the tension at the edge of the back optic zone diameter. The limit is really set bythe reverse curve, which determines the overall central thickness of the lens.

in making the lens thicker if the result is an increased risk of edema.

MODELED FORCE AND CORNEAL SHAPE CHANGES

The test of a model is to compare the theoretical results with those that occur in practice. The following section compares the calculated force distribution under the lens to the results seen clinically to ascertain the value of the model. The scenarios are: corneal sphericalization, bull's-eye plots, smiley faces, central islands, and the mystery of astigmatism.

CORNEAL SPHERICALIZATION

One of the simple basic laws of fluid hydraulics and physics is that nature is always seeking equilibrium. The forces beneath a reverse geometry lens in a quasistatic state are not in equilibrium. Compressive forces exist centrally with relative tension forces at the edge of the BOZO. The tear layer itself is an incompressible fluid, but the cornea is not. As a result, the tangential stress