In this work we started from a set of 39 parameters including the subject's age, the total ocular refraction (
Ref M,
Ref J0,
Ref J45) written in the form of Thibos' Fourier power vectors,
31 the anterior keratometry (
K A,M,
K A,J0,
K A,J45), the anterior corneal eccentricity
Ecc A, the posterior keratometry (
K P,M,
K P,J0,
K P,J45) and eccentricity
Ecc P, the central corneal thickness
Pachy, the anterior chamber depth
ACD, the anterior and posterior curvature of the crystalline lens (
R LA,
R LP), the crystalline lens thickness
T, the crystalline lens power
P L, the ocular axial length
L, and the scotopic pupil size
S P.
As the Project Gullstrand data did not contain the lens thickness parameter, it was not possible to calculate the crystalline lens power directly using ray tracing or Bennett's formula.
32 Instead the lens power was estimated using the T2 formula,
33 an updated version of the SRK/T formula.
34 However, rather than the phakic lens power, this procedure provides an estimate of the pseudophakic lens power required to obtain a certain preset refraction after cataract surgery.
To remedy this problem we used the emmetropic group to find the correlation between the crystalline lens power calculated with the Bennett formula (P Bennett), as well as the pseudophakic lens power using the T2 formula (P T2). Through reduced major axis regression we found the following relationship: P Bennett = 1.133 P T2 − 1.386 (r2 = 0.922). Inserting P T2 into this formula allows an estimate of the real crystalline lens power to be made. Even though in individual cases the calculated crystalline lens power may deviate from the actual crystalline lens power, the calculated average and covariance values of the population would be correct.
Since the biometric parameters of left and right eyes are strongly correlated (see the Results section), combining both eyes of the same subject into the calculations may have a considerable influence on the covariance values in matrix C. Although the use of a linear mixed-effects model would account for such correlations, a different approach was chosen here. By including the biometry of right and left eyes into M and C separately, one has the opportunity to create a binocular eye model that leaves the correlation between both eyes intact for these parameters. Including this binocular aspect would introduce several interesting options, such as the study of aniseikonia and anisometropia.
As mentioned above, the mean and covariance values for the anterior and posterior curvature of the crystalline lens (R LA, R LP) and the crystalline lens thickness T were taken from the emmetropic data set. With the exception of the scotopic pupil size S P, the emmetropic data set contained all the model parameters included in the covariance matrix C. Hence most of the covariances between the lens parameters (R LA, R LP, T) and the other parameters could be inserted. However, the emmetropic data set did not include binocular information, so the covariance values between the lens parameters and the other parameters had to be used for both left and right eyes. Covariance values that could not be determined were given the default value of 10−5.
In practice, it proved to be necessary to slightly increase the covariance values between the K A,M, P L, and L parameters for both eyes, to obtain a more realistic correlation between the eyes. Example of the M and C matrices used in this work are given in the Appendix for the monocular version of the model (right eyes only).
One drawback of randomly generating a set of biometric parameters in this fashion is that, even though the values of the individual parameters are realistic and the correlations between them are correct, the parameters defining ocular refraction (i.e.,
K A,M,
P L, and
L) would not add up to the value of
Ref M that was randomly generated by the model. In a healthy real eye on the other hand the refraction calculated from the biometry and the measured refraction would match very closely. This problem can be solved by using ray tracing,
35 to calculate the refraction along the meridians of maximum and minimum corneal curvature, from which the resultant spherical and cylindrical refraction may be derived.
However, this process leads to a refraction with a Gaussian distribution (see the Results section), which does not correspond with reality. It is therefore necessary to filter the generated data in such a way that both distributions will match each other more closely. The filtering was achieved by dividing both the original and the random refraction data of the right eyes into bins of 1 D according to their refraction. Next, eyes were removed randomly from the bins of the generated data until the overall distribution matched that of the original data. This reduced the amount of usable generated data by a factor of 4 to 5.
After exclusion of the refractive parameters (i.e., Ref M, Ref J0, Ref J45 for each eye), 33 parameters were included in the following calculations.