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Purpose: Using mathematical equations, to allow for the differentiation of different eye shapes of varying refractive error. Ultimately being able to identify a prolate, oblate and spherical eye from each other.
Method: Using previous data, which has measured the peripheral refraction out to 40 degrees in the horizontal meridian. Once this data has been collected, created theoretical model eyes of -8.00,-4.00, 0.00, 4.00, 8.00D and make a prolate, oblate and spherical eye model for them.
Results: The equation y=ax2, proved to be the most successful, within only one co-efficient and high r2 values, it was the only equation tried that was able to differentiate between the different eye shapes. The other two equations had 2 co-efficients, one (y=ax2+bx) with higher r2 values and one (y= a + b ln x) with lower r2 values, but showed similar values between the eye shapes, making differentiation hard extremely difficult.
Conclusion: y=ax2, was able to tick the boxes which were needed for the study to be a success. However the study is purely theoretical. The next step in determining if eye shape can be determined by a mathematical equation is to repeat the study on real human eyes of the same refractive error.
The refractive error of the eye can be determined by three main structures; the curvature of the cornea, the power of the crystalline lens and the axial length of the eye. How these three components perform is linked together in a process called emmetropisation. After birth the eye undergoes the perceived process, using a number of visual stimuli the eye can determine changes that must occur for it to become emmetropic (Sorsby et al 1961)
There are two different thoughts on the mechanism of emmetropisation. The first, 'active' emmetropisation deals with the ability of a visual feedback system to control the process, i.e. the relationship between the eye and brain to deal with the defocus which is being presented to the eye, this theory doesn't include the role of the crystalline lens. (Wildsoet CF et al 1997) showed data which supported the theory of active emmetropisation, the study found that if an eye was given myopic correction then eye growth would be accelerated, this was due to the presence of hyperopic blur. Accommodation was also looked at in this study and was found that poor accommodation appeared to accompany the development of myopia rather than be the cause of it. Active emmetropisation has been supported by its presence in animal studies, showing the ability of the eye to be able to detect and compensate for forced refractive error (Schaffel et al 1988). Whereas the passive model of emmetropisation, suggests that the relationship between the globe and the lens during eye growth form the overall spherical power of the eye (Mutti et al 1998). Within this study it was noticed that as the child developed the lens radii curvature flattened, there was a subsequent decrease in the refractive power of the lens and that the lens thinned. The changes in myopia which are seen around the age of 10 and the events which happen to the lens around this time suggest that the limitation of eye growth is therefore responsible for the 21.3% increase in myopia between the ages of 10 and 14.
The basis of thought behind 'active' emmetropisation is that blur present within the eye is decoded by the brain into either hyperopic blur or myopic blur. If hyperopic, the eye would grow. If myopic, growth would be inhibited. However it has been well documented that the axial length relationship between myopic and hyperopic eyes is considerably different (Strang et al 1998, Atchison et al 2004), myopic eyes having longer axial lengths (Lourdes Llorente et al 2004).
As the spherical refraction of the eye is determined by the fovea, then it might be thought that the blur presented to the fovea contributes to the emmetropisation process. This is however untrue, in a trial with monkeys where the fovea was ablated, emmetropisation still occurred (Smith et al 2007). The light that hits the peripheral retina must contribute to the process. The peripheral retina has been found to have the ability of detection acuity (Artal et al 1995). Within these early months detection acuity is thought to be highly developed, and it has been found that it can be considerably compromised by peripheral defocus (Wang et al 1997).
The overall transmission of nerve signals from the fovea stands at around 10% of the total amount for the eye therefore the theory that the peripheral retina plays a large part in emmetropisation would make sense. The total input of nerve activity to the brain from the eye would largely be coming from the peripheral retina, i.e. not the foveal region. Hence the larger weight for effects of defocus on the peripheral retina leading to the dramatic outcome of emmetropisation.
As ideal a process as emmetropisation is there are problems. These problems contribute to 15% of 15 year olds having myopia (Mutti et al 2000). If myopic blur is present within the eye the brain should inhibit growth. However in myopic patients it has been found that the axial length is longer than emmetropic patients (Lourdes Llorente et al 2004). For some reason the eye continues to grow. One of the reasons for the failure of this process and the development of myopia is due to the eye having a relatively hyperopic peripheral retina. This would therefore cause hyperopic blur being presented to the brain, this blur, would therefore stimulate the eye to grow.
The reason why myopia suddenly appear when the child is around 10-15 years old maybe due to the axial length growth being compensated, within the first 10 years, mostly by the lens rather than the cornea. Even though the cornea is responsible for the majority of refraction, it undergoes a slight structural change early in life, of only around 3-5D (Mutti et al 2000). Therefore it can't be responsible for overcoming the progression of myopia. The lens can compensate for this progression of myopia due to the fact that as time goes on the lens can flatten its curvature and thin equatorially, hence focusing the image onto the retina. After the age of ten however, it is unable to continue to flatten and therefore the child will start to experience problems
The general relationship between the different refractive powers and relative axial length, is myopic eyes are longer than hyperopic or emmetropic eyes. Elongation of the eye is the usual finding, however it has been stated that all dimensions of the eye increase in size however the elongation aspect is usually the more considerate change (Atchinson et al 2004). The limitation of the other dimensions of growth has been explained in this paper as being the responsibility of the orbit itself. The dimensions of the eye are more restricted by the orbit on either side of the eye rather than at the back. Therefore restriction is less at the back than it is at the sides, hence the notable axial change located within myopic eyes. There were differences between this study and Cheng et al 1992, who found that the dimensions increased in the order of horizontal, axial and then vertical compared with axial, vertical then horizontal. One notable reason for this difference was Cheng at al 1992 had taken dimensions which included sclera and choroid; however Atchinson et al 2004 had only taken retinal points. Within the latter study, there was a notable uniform thinning in the sclera and chorodial structures for myopic eyes compared with hyperopic or emmetropic eyes (Cheng et al 1992).
The association of corneal asphericity has also been linked to the development of the axial length; it has been found that for myopic eyes there is a significant correlation between corneal asphericity vs. vitreous chamber depth and also vs. axial length (Carney et al 1997). (Carney et al 1998) however found no significant correlation for the same things when using hyperopic subjects.
The problem that is presented is to clearly define what the shape of the eye is for that subsequent refractive error. There has been much deliberation between whether the eye is of prolate or oblate shape. In trying to determine this, the refractive error of the peripheral retina has been used. In a spherical eye all dimensions would be the same as the axial length of the eye. In a prolate eye the retina would flatten towards the equator, whereas in an oblate eye the retina would be steepen towards the equator. Chen. X et al 2010, conducted a study on around 82 subjects, all of varying refractive error, and assessed the peripheral refractive errors of these subjects. The results suggested the presence of myopic shifts within patients of a 'low hyperopic' refractive error, and a relatively hyperopic shift in 'moderately myopic' refractive error patients. The documentation of a hyperopic shift within the periphery of myopic subjects has been presented within other studies, (David A. Atchison et al 2004)
There have been many different studies into determining the different shapes of the eye, each study has used different types of imaging system to try and obtain the end result. (Hitzenberger et al 1991) used a Doppler laser, (Schmid et al 2003) used an optical low coherence reflectometry, and (Cheng et al 1992) employed the use of an MRI machine to examine the physical and optical properties of the eye.
These recent MRI studies have varied in sample size; Cheng et al 1992 used small sample groups and found that there is no clear relationship between myopia and eye shape, it did support the theory that the axial length of the myopic eye was larger than that of the hyperopic eye. In another study,(Atchinson et al 2005) that has carried out experiments on larger sample studies have also found such a relationship for both hyperopic and myopic patients and axial length. Within a study on hyperopic eyes, (Strang et al 1998) it was found that the combination of corneal curvature with axial length contributes to 83.9% of refractive error. Measurements were not carried out on the lens, but it would conclude that it could only contribute 16.1% in to determining refractive error. Within a study carried out on anisomyopia and isomyopia patients, it was found that there was no overall pattern between the degree of myopia and corneal curvature, anterior depth or lens thickness. However there was a pattern between the vitreous depth and the refractive error of these patients (Logan et al 2004).
The final thought about the development of myopia has been its link with nature and nurture. The nurture concept has been contributed to a substantial amount of near work during visual development, the idea that if the developing eye uses a near environment then with the help of accommodation the eye will become myopic (Flitcroft et al 1998). When a large sample size was used it clearly showed that juvenile onset myopia is a result of both parents of the child suffering from myopia themselves rather than on exposure to an environmental factor (Zadnik et al 1995). Therefore when it comes to the determination between whether is myopia is caused by nature or nurture it comes down to nature more so than nurture.
The aim of this study is to collect data for peripheral refraction out to 40 degrees, and to use this to formulate theoretical eye models of powers +8.00, +4.00, 0.00, -4.00 and -8.00. Once these eyes have been developed they will then be made prolate, oblate and spherical. By using mathematical equations for curve fitting, these shapes will try to be best explained by an equation which will then allow for that shape to be differentiated against all the others. It has been noted that the difference between what is prolate and what is oblate has been an area of confusion, therefore this study will try and shine some clarity on this grey area.
The concept of the eye that was being created was only to take into account the axial length and the refractive power of the eye. The lens structure was not going to be included in the study; therefore the properties and characteristics of it could be ignored. The eye was to be presented as a single refractive surface; therefore all the refractive power of the eye was going to be the responsibility of the cornea.
Figure - Angles within the eye
Figure1.1: Shows how light focused through the centre of the cornea will end up on the different parts of the peripheral retina. As light is being diffracted through the central part of the cornea it can be assured that the refractive error the light is subjected to is the same.
The experiment called for the development of only one theoretical eye for each different power and shape, -8.00, -4.00, 0.00, +4.00, +8.00 all of which would have prolate, oblate and spherical representation. These powers could give a wide spectrum of different eye shapes and should be able to provide enough overlap between each other to analyze if they can be separated out by an equation.
The nature of this project meant that being able to create the theoretical models of oblate, prolate and spherical eyes from different data samples was impossible. The literature had limitations in providing peripheral refraction out to 40 degrees which followed an Oblate or Prolate shape, especially for hypermetropic eyes. Therefore, data was collected from one set of literature (Atchinson et al 2006) and applied to all eyes. Having all eyes change by the same Dioptric value would be ideal for this study. As all that was needed was for the theoretical eye shapes to be developed. Making sure each eye was able to be considered a prolate or an oblate was the most important point of the study. The degree of refractive change as you moved to out to the 40 degree mark was not as important.
The study used to establish the change in refractive error when moving into the periphery, had taken a total number of 121 subjects with a refractive range of +0.75 to -12.83D. There were a number of exclusion criteria. The study was to create the optical models of myopic eyes. The only data taken that applied to this study was the measurements of the peripheral refractive error from 0 to 40 degrees out, all other measurements were not considered or taken. The results were taken from the graph that represented a -8.00 D eye. Only the horizontal meridian was to be considered for this study.
The changes in refractive error were as follows;
Degrees from centre
Table - Dioptric Change
Oblate eyes theoretically get steeper as you move towards the equator, the effect that this has on the refractive power is that the eye gets relatively hypermetropic as you move out from the central point. The opposite is true for prolate eyes, as you move towards the equator the eye gets relatively more myopic. Therefore, with the data above, for oblates the refractive error was added to the total spherical power of the eye, and vice versa for the prolate eyes.
Peripheral axial length = 4/3 / (60 + (spherical power of the eye + peripheral Rx))
Using the equation above meant that the axial length at the different peripheral points could therefore be calculated. Once these were known then it was only a matter of using trigonometry to find out where these points lay in relation to the central point of the eye, i.e. 0 degrees . To calculate the eye shape, two points would have to be calculated (X, Y). X would represent the horizontal distance from the central point to that part of the peripheral retina, and Y would represent the vertical displacement.
Figure : Theoretical Model of the Eye
Figure 1.2: Represents the theoretical model of the eye, it shows that the eye considered did not consist of a cornea or lens. The central axial length of the eye (C.A.L) and the physical representations of X and Y are also featured.
The first value to be calculated was the hypotenuse of the above diagram; this was the peripheral axial length. Using trigonometry;
Sin(angle) = opposite / hypotenuse or ; opposite = sin(angle) x hypotenuse
This gave the value for the letter X
To find the value for Y;
Tan (angle) = Opposite / Adjacent or; Adjacent = opposite / tan(angle)
[NB. For all these equations Sin and Tan were in degrees]
This value was Z was then placed into the equation;
Y= Central axial length - Z
The values of (X, Y) where then plotted on to a scatter graph.
These steps where then repeated for all the different eye powers/shapes that were being plotted in this study.
The graph fitting program 'Curve expert' was then used to analyse the different retinal shapes. A variety of different equations were tried, the one that was able to give the best fit (r2 value â‰ˆ 1) whilst limiting the co-efficient number would be considered as the best result. The lower the number of co-efficients the more reliable the equation, due to the fact that the more co-efficients present within an equation the more difficult it would be to differentiate when all these co-efficients are liable to change.
The different equations used for this software where;
Y= ax2 + bx
Y= a + b ln x
The retinal shape data was entered into the graph fitting software. Using the first two equations the (X, Y) values were entered in the order that they appeared, however when using the third equation it was necessary to eliminate the (0, 0) point from the data and also the (X, Y) coordinates had to be switched round hence, (Y, X). The reason for this was to maximise the best possible fit that could be achieved with the Logarithm fitting function.
Whilst using the curve fitting software there was no ability to force the graph through the origin. Therefore the 'c' co-efficient was simply ignored for the purpose of equation 2, as it should clearly be 0. So instead of writing y=ax2+bx+c we are able to write y=ax2+bx
The creation of the model eyes was presented through the Microsoft Excel Program, the dimensions and figures calculated within this program where then taken and placed into the Curve expert program. Once the values were entered a variety of different equations were used. The first equation that to be analysed was;
y = ax2
For the spherical eye shapes of all powers there is a clear separation between the co-efficient values. The +8.00D spherical eye has the highest co-efficient number with the -8.00D eye has the smallest co-efficient number. All the other refractive powers fall in-between these two values in the order of most plus to most minus.
The prolate eye shapes show a different pattern of separation compared to the spherical eyes. This time the+4.00D prolate eye showed the smallest co-efficient value, the largest value coming from the emmetropic prolate eye. A clear separation between the positive and negative refractive errors has also developed. The degree of this separation is small, 0.31x101 units.
The co-efficient data for the oblate eyes shows yet another pattern. There is a substantial separation between the positive and negative refractive powers, a value of 1.14x101 units. The +8.00D and +4.00D hold the largest and second largest co-efficient values, respectively. With this set of data there is no longer such a clear separation between the different co-efficient values. The data for the spherical oblate and the -8.00D oblate are only separated by 0.09x101 units.
The r2 values recorded for this data showed that very high trends, none of which decreased lower than the 0.98 value. This allows for the determination that this equation fits the different retinal shapes to a very high standard.
y = ax2 + bx
This equation has 2 different co-efficients present within it, as it can be seen within the graph the values for co-efficient 'b' could be ignored as they all are lie around the same point and therefore can be considered as a constant. The values for co-efficient 'a' where the values that were important and these were going to be analysed.
For the spherical values of all refractive powers it shows a similar pattern to that of the first equation. All refractive powers are separated and the co-efficient values follow the order of highest to lowest running from +8.00D to -8.00D. The overall sizes of the co-efficient values are also around the same as in the first equation 2.00-3.50x101.
The values recorded for the co-efficient data for the prolate shaped eyes was different to that of the first equation. Although there was still a separation between the positive and negative Rx values, 0.24x101units, there was more overlap within these two boundaries, compared to the previous equation. Also the positive powers had kept large co-efficient values and the negative powers had remained with the small co-efficient values. The co-efficient values recorded for the -8.00 and -4.00D prolate eyes are only separated by 0.02x101 units, showing a hard differentiation between these two retinal shapes.
For the oblate eye shapes there was again a clear separation between the positive and negative co-efficients, a value of 0.45x101, however this one was greater than that of the prolate eye shape. The difference between the two positive power values has stayed constant, 0.18x101, however the emmetropic oblate value has decreased in comparison to the prolate value. Therefore the separation between the +8.00 and +4.00 can now be clearly seen on the graph.
The co-efficient values for the -8.00 and -4.00D oblate eyes showed a larger separation of 0.06x101 units, again leaving it hard to clearly differentiate between them.
An overall look at the graph shows that moving from spherical, prolate to oblate eye shapes of the same refractive power there is a considerable overlap. Therefore this equation has not helped in clearly identifying the difference between the various eye shapes.
The r2 values recorded using this equation were slightly higher, than that of the previous equation, not dropping below 0.99, this indicates that the fits are slightly better, however the co-efficient values are not as successful in separating the co-efficients as that of the first equation.
y = a + b ln x
Logarithm function was used; this meant that the X and Y co-ordinates had to be reversed to ensure a greater fit correlation .
Again this equation had 2 co-efficient data within it, only 'a' value was going to be analysed, as the 'b' values were fairly similar in value which meant that they could be thought of as a constant and could be ignored. Overall the co-efficient values were considerably smaller than that of the previous other equations. The values lay around the 2.00-3.50x10-2 units.
The spherical values presented within this graph showed clear separation between all the refractive powers. The co-efficient values however were reversed compared to the previous graphs, +8.00 and +4.00D co-efficient values were the smallest.
Watching the pattern of the -8.00D values for all the different eye shapes it takes an unexpected decline when it comes to the oblate co-efficient it has the exact same value of a +8.00D oblate eye.
The r2 values of this equation are especially lower than that of the other equations, with the lowest value representing 0.83. Not only does this equation have a low r2 value it also does not allow for the differentiation between the co-efficient values.
The identification of whether and eye is prolate, spherical or oblate is becoming of more and more interest after the work of (Smith et al 2005) when peripheral refraction was labelled as a possible cause of axial elongation, (Smith et al 2007) worked on the foveal involvement of emmetropisation in monkeys, showing that after the fovea was disabled within monkeys emmetropisation still took place. Previous investigations have involved the peripheral retina of different test subjects, chickens (Schaeffel et al 1988), monkeys (Logan et al 2004) mice (Xiangtian et al 2008) and humans (Chen x et al 2010, Atchinson et al 2004, 2006, Mutti et al 2000). Within the studies, which were focused on humans, evidence that the peripheral retina is relatively hyperopic compared to the central spherical refraction of myopic eyes. They also found that with hyperopic (Strang et al 1998, Mutti et al 2000) and emmetropic patients' (Atchinson et al 2004) peripheral retinas were shown to be relatively myopic. (Mutti et al 2000) commented on the shapes of these different eye shapes, labelling the myopic eyes 'prolate' and the hyperopic eyes 'oblate'.
The degree of hyperopic shift within the periphery has been under some question. (Calver et al 2007) conducted a study on a population of emmetropic and myopic patients. It was found that within the emmetropic patients there was a significant myopic shift when moving out to 30 degrees of eccentricity. When it came to measuring the peripheral refraction of the myopic patients out to 30 degrees of eccentricity there was no significant change in hyperopic peripheral refractive error. It has also been stated that (Mutti et al 2000) only recorded a shift of 0.8D, (Millodot et al 1981) recorded values of 0.5D. (Atchinson et al 2006) showed that for his -4D myopes there was a hyperopic change of around +1D, the difference between them and his -3D subject was significant, a value of only +0.5D was found. This is evidence to support the theory that peripheral refractive changes are dependent on the central spherical power of the eye.
The limited research which has been carried out on hyperopic eyes (Strang et al 1998) means that evidence of any sort of eye shape on these hyperopic eyes is hard to find. Even with some studies being completed on hyperopic eyes, (Mutti et al 2000) these do not cover all the refractive powers needed for this study. This leads to the justification for the creation of a number of these eye shapes out of one set of data. The advantages of being able to create eye models for large spread of refractive errors outweighed limiting the study to a handful of eye shapes with specific data.
When the theoretical eyes were created from the published data of (Atchison et al 2006) myopic eye models, it was easy to make the different eye shapes. However the practicality of these models is debatable. It has been shown that as the myopia increases the rate at which the peripheral retina becomes relatively hyperopic also increases (Atchison et al 2006), within our study the same changes in peripheral refraction were applied to each eye model. Therefore the -4 and -8 D models would show the same hyperopic changes. This is probably not the case if the data was collected from a variety of human subjects with the desired refractive correction.
A recent study which was undertaken by Atchinson et al 2005, it looked at 'shape of the retinal surface in emmetropia and myopia'. The results of this study have questioned the theories that the myopic eye reflects a more prolate shape compared to that of an emmetropic or hyperopic eye, which follows the oblate eye shape. The study agrees with the aspect that emmetropic eyes are oblate in shape, but makes the conclusion that myopic eyes follow a less oblate shape. Only 12% of the myopic eyes featured within this study followed the principles of being a prolate eye, i.e. the length of the eye is of a greater value than that of the equator. This new thought on the shape of the eye shows the difficulties in identification of which eye is which.
Figure : Co-efficient Graph
Figure 2.1: This is a graph of the co-efficient values of the 'y=ax2' equation. The two bold lines are to represent the separation between the co-efficient values of the different eye shapes with the same spherical power. Emphasising the differentiation between these shapes
Using this equation's co-efficient values meant that a spherical, prolate and oblate eye with positively spherical correction could be differentiated. The shapes which overlap with a positive spherical eye, i.e. similar co-efficient values, are both prolate eyes of a negative Rx and to some degree a spherical prolate eye. This will not present a problem; as long as the spherical power of the eye is known then the shape of the eye will be able to be determined. The differences between all the other spherical powers is similar to that of the positive pattern, the conclusion which can therefore be taken from this equation is that as long as the spherical power of the eye is known, then by using this equation it can be determined along with peripheral refraction, whether the shape of that eye lies within the prolate, oblate or spherical boundaries.
Comparing this equation to the others it can be shown that neither of them can offer the same sort of differentiating ability as that of 'y=ax2'. Both have two co-efficient values which we have determined can be ignored due to the similarity of them, therefore allowing us to focus on the one , a, co-efficient. When viewing the 'a' co-efficient it showed to overlap the other eye shape values of the same refractive powers. Even though 'y=ax2+bx' had a minimally greater r2 value compared with the 'y=ax2' it was not as successful an equation in being able to identify which eye shape was which.
The logarithm equation was the least desirable equation that was tried, even after flipping the X and Y co-ordinates this still ensured some poor r2 values. Values lower than the rest and co-efficient values which could not lead to any clarity when it came to differentiating the different eye shapes. It can easily be labelled as an unsuccessful equation when it comes to dealing with peripheral refraction and the identifying of eye shape.
The graph associated with the logarithm showed a reverse in size of co-efficient values, this would be expected due to the fact that the co-ordinates where reversed. Moving along the graph viewing different shapes of the same refractive power there is a considerable degree of overlap between these values. The retinal shape of these three eyes displays a similar curve fit when linked with this equation. The plot of the oblate -8.00D eye is somewhat different to what would be expected. When viewing the two other eye shapes, the -8.00D eye has a higher co-efficient value than that of the -4.00D, however when it comes to the oblate shape the -8.00D eye has the same value as that of a +8.00D eye. This unexplained decline in co-efficient value is also linked with the lowest r2 value calculated within this study, a value of 0.837.
The study has flagged up many different thoughts when it comes to the theoretical development of model eyes. There are clear advantages and disadvantages to this approach taken within this study. Creation of these models allowed for the analyses of eyes that have had no previous investigation. Even though some of the eyes created within this study are clearly 'theoretical' compared to others it is better to take into account all aspects of what could happen compared to limiting the scope of possibility due to the limitations in the literature. 'Prolate hyperopic eyes' and 'prolate spherical eyes' are examples of eyes which fell out with the literature. The similarities between these shapes and others included in the study, allow for the possibility of confusion when these sorts of eye shapes are seem. The disadvantages of this model has been highlighted previously, this data is purely theoretical. The next stage in trying to identify the differences between different sorts of eye shapes is by using the same method, but instead of collecting data from previous studies and applying it to all eyes, the way in which it should be tackled is to find human subjects of the desired refractive error and recording their peripheral refraction out to the 40 degree mark.
When trying to identify eye shape it has been proved that using the equation 'y=ax2' is the most successful way of doing this. This study has highlighted that this equation is the only one, out of three, that was able to keep the co-efficient number to a minimum of 1 and still be able to five high r2 values. The parabola shape matched the eye shape well enough that these results can be taken. The success of this study is however minimal, it only allows for the differentiation between these theoretical eye shapes and powers. The next step would be to collect the peripheral data for a number of emmetropic and hyperopic eyes, plot these eye shapes and then see if using the same equation showed the same degree of success.
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