Mathematical investigation of posterior corneal surface

1
Mathematical investigation of posterior corneal
surface elevation topography in regression
following LASIK for myopia M.Batterbury,
A.C.Fisher1, V.L.Kennedy1, S.B.Kaye St Paul's
Eye Unit and 1Dept of Clinical Engineering
(a.c.fisher_at_liv.ac.uk)

• Summary
• Changing the surface shape of the front of the
  eye (cornea) by Laser (the LASIK procedure) to
  treat short- or long- sight is very successful
• In Liverpool, we use a sophisticated technique
  (Orbscan video-topography) to measure the shape
  of the cornea before and after surgery on both
  the front and back (posterior untreated)
  surfaces
• This measurement, accurate to 1000th of an inch
  (25µm), can reveal subtle changes in the corneal
  shape (regression) in the months after surgery
• Understanding this regression would improve the
  design of LASIK operations
• This study sets out to develop a mathematical
  description of posterior surface regression and
  relate it to the improvement in vision

.... The number of records in the datasets did
not support dimensionalisation of the input layer
above 3 nodes. Thus, for both the 5-term models
(rotated-general ellipsotoric and Zernike), only
the most significant 3 parameters as inferred
from principal component analysis (PCA) were
chosen.
Question Can regression after LASIK for myopia
be related to shape changes on the posterior
corneal surface? Introduction Several
explanations for regression after LASIK have been
advanced, however no study has investigated the
posterior corneal topography in detail. Here we
characterise the posterior surface shape as
described by Orbscan (Bausch Lomb)
video-topographic elevation data using 3
mathematical models i. 3-term simple
aconic ii. 5-term general ellipsotoric (with
orthogonal principal axes but with rotation)
iii. 5-term Zernike polynomial decomposition The
topographic (sagittal height) data are regressed
(fitted) each of the models using an artificial
adaptive neural network with supervised output.
Two relationships are sought i.
discrimination between non-regression and
regression eyes ( binary outcome) ii.
continuous relationship between the parameterised
data and the degree of regression viz opposite ...
Method 258 records of patients whom had
undergone LASIK for myopia at St Paul's Eye Unit
were examined. Regression was defined as change
in mean spherical equivalent (MSE) subjective
refraction from week 1 to 3 months
post-operatively of at least 1D. All patients
received a complete pre and post-operative
examination including refraction and Orbscan
elevation topography. Fifty four eyes were
identified. Pre-operative MSE was -8.5D (range -3
to -18). Mean regression was 1.56D (range 1 to
3.25). A matched group of 50 non-regressed eyes
(lt 0.5D change) was identified.

• Results
• binary outcome (discriminating regression)
• the general ellipsotoric and Zernike models
  (where 3 of 5 parameters were selected by PCA)
  performed equally well (75 accuracy) in
  discriminating regression and non-regression
  groups
• the 3-term aconic model had only an accuracy of
  50
• continuous outcome (prediction of degree of
  regression)
• the Zernike model with 4 parameters achieved a
  fair correlation (r20.6, plt0.05) with degree of
  regression across the complete database
  (population)
• general
• an attempt to implement the optimal combination
  (by PCA AANN) of any 3 parameters selected from
  the 3 models produced a stable solution but did
  not improve upon either the general ellipsotoric
  or Zernike model alone
• Conclusions
• It is shown that post-operative changes at 3
  months in posterior corneal shape, as inferred
  from Orbscan elevation videotopography data,
  relates to changes in MSE subjective refraction.
• The analysis relates only to the population
  dataset recruited for this study but indicates
  the potential of an extended study with larger
  numbers of patients.
• The inclusion of other corneal topographic
  measurements, such as anterior corneal shape and
  pachymetry, should be considered.
• The shape parameter models used here have been
  relatively little explored for posterior
  elevations some further work is required.
• The Artificial Neural Network as used here did
  not expose the parameters selected by PCA nor did
  it define the weightings ascribed (relative
  importance) to chosen inputs this is a
  shortcoming to be addressed.

After screening for completeness and quality of
records, 2 groups were established and
demonstrated significantly different at plt0.001
as 29 regressors MSE change gt 1.0D 24 non-
regressors MSE change lt 0.5D Raw posterior
surface elevation data were exported from the
Orbscan at the time of examination using the
recorder function t a programming environment
supported by MatLab (Mathworks Ltd) with Signal
Processing and Neural Network extensions. Elevati
on data sets were completed for missing fields
using polynomial interpolation and a region of
interest defined as the 3mm radial field centred
at the anterior corneal apex.
Models were fitted graphically to the completed
data over the region of interest using the
MatLab general least squares (Fminsearch)
algorithm. General multiple regression of the 2
outcome scenarios (i. discrimant - binary
non-regressors versus regressors, and, ii.
regression as a continuous output variable)
against each of the 3 corneal shape models
(aconic, general ellipsotoric and Zernike
decomposition) was performed using a 3-layer
Adaptive Artificial Neural Network implemented in
MatLab. The threshold value for the
discrimination outcome was set on the basis of
optimal accuracy inferred from the Receiver
Operator Curve. The data were treated as a
population rather than a sample. .....
continued
2
Mathematical investigation of posterior corneal
surface elevation topography in regression
following LASIK for myopia M.Batterbury,
A.C.Fisher1, V.L.Kennedy1, S.B.Kaye St Paul's
Eye Unit and 1Dept of Clinical Engineering
(a.c.fisher_at_liv.ac.uk)

• Summary
• Changing the surface shape of the front of the
  eye (cornea) by Laser (the LASIK procedure) to
  treat short- or long- sight is very successful
• In Liverpool, we use a sophisticated technique
  (Orbscan video-topography) to measure the shape
  of the cornea before and after surgery on both
  the front and back (posterior untreated)
  surfaces
• This measurement, accurate to 1000th of an inch
  (25µm), can reveal subtle changes in the corneal
  shape (regression) in the months after surgery
• Understanding this regression would improve the
  design of LASIK operations
• This study sets out to develop a mathematical
  description of posterior surface regression and
  relate it to the improvement in vision

.... The number of records in the datasets did
not support dimensionalisation of the input layer
above 3 nodes. Thus, for both the 5-term models
(rotated-general ellipsotoric and Zernike), only
the most significant 3 parameters as inferred
from principal component analysis (PCA) were
chosen.
Question Can regression after LASIK for myopia
be related to shape changes on the posterior
corneal surface? Introduction Several
explanations for regression after LASIK have been
advanced, however no study has investigated the
posterior corneal topography in detail. Here we
characterise the posterior surface shape as
described by Orbscan (Bausch Lomb)
video-topographic elevation data using 3
mathematical models i. 3-term simple
aconic ii. 5-term general ellipsotoric (with
orthogonal principal axes but with rotation)
iii. 5-term Zernike polynomial decomposition The
topographic (sagittal height) data are regressed
(fitted) each of the models using an artificial
adaptive neural network with supervised output.
Two relationships are sought i.
discrimination between non-regression and
regression eyes ( binary outcome) ii.
continuous relationship between the parameterised
data and the degree of regression viz opposite ...
Method 258 records of patients whom had
undergone LASIK for myopia at St Paul's Eye Unit
were examined. Regression was defined as change
in mean spherical equivalent (MSE) subjective
refraction from week 1 to 3 months
post-operatively of at least 1D. All patients
received a complete pre and post-operative
examination including refraction and Orbscan
elevation topography. Fifty four eyes were
identified. Pre-operative MSE was -8.5D (range -3
to -18). Mean regression was 1.56D (range 1 to
3.25). A matched group of 50 non-regressed eyes
(lt 0.5D change) was identified.

• Results
• binary outcome (discriminating regression)
• the general ellipsotoric and Zernike models
  (where 3 of 5 parameters were selected by PCA)
  performed equally well (75 accuracy) in
  discriminating regression and non-regression
  groups
• the 3-term aconic model had only an accuracy of
  50
• continuous outcome (prediction of degree of
  regression)
• the Zernike model with 4 parameters achieved a
  fair correlation (r20.6, plt0.05) with degree of
  regression across the complete database
  (population)
• general
• an attempt to implement the optimal combination
  (by PCA AANN) of any 3 parameters selected from
  the 3 models produced a stable solution but did
  not improve upon either the general ellipsotoric
  or Zernike model alone
• Conclusions
• It is shown that post-operative changes at 3
  months in posterior corneal shape, as inferred
  from Orbscan elevation videotopography data,
  relates to changes in MSE subjective refraction.
• The analysis relates only to the population
  dataset recruited for this study but indicates
  the potential of an extended study with larger
  numbers of patients.
• The inclusion of other corneal topographic
  measurements, such as anterior corneal shape and
  pachymetry, should be considered.
• The shape parameter models used here have been
  relatively little explored for posterior
  elevations some further work is required.
• The Artificial Neural Network as used here did
  not expose the parameters selected by PCA nor did
  it define the weightings ascribed (relative
  importance) to chosen inputs this is a
  shortcoming to be addressed.

After screening for completeness and quality of
records, 2 groups were established and
demonstrated significantly different at plt0.001
as 29 regressors MSE change gt 1.0D 24 non-
regressors MSE change lt 0.5D Raw posterior
surface elevation data were exported from the
Orbscan at the time of examination using the
recorder function t a programming environment
supported by MatLab (Mathworks Ltd) with Signal
Processing and Neural Network extensions. Elevati
on data sets were completed for missing fields
using polynomial interpolation and a region of
interest defined as the 3mm radial field centred
at the anterior corneal apex.
Models were fitted graphically to the completed
data over the region of interest using the
MatLab general least squares (Fminsearch)
algorithm. General multiple regression of the 2
outcome scenarios (i. discrimant - binary
non-regressors versus regressors, and, ii.
regression as a continuous output variable)
against each of the 3 corneal shape models
(aconic, general ellipsotoric and Zernike
decomposition) was performed using a 3-layer
Adaptive Artificial Neural Network implemented in
MatLab. The threshold value for the
discrimination outcome was set on the basis of
optimal accuracy inferred from the Receiver
Operator Curve. The data were treated as a
population rather than a sample. .....
continued