The rPACI package
can read corneal topography files and then analyze them. This analysis
includes the computation of some Placido irregularity indices. The
indices computed by rPACI allow to discriminate between
normal and irregular corneas. Most of them were introduced and validated
with real datasets in the scientific publications Ramos-López et al. (2011) and Ramos-López et al. (2013). An additional index,
based on a naive Bayes classifier, was proposed later, and tested with a
different database, in Castro-Luna et al.
(2020). In these papers, all indices demonstrated a good
sensitivity for the detection of keratoconus, a corneal disease.
The purpose of this vignette is to explain the mathematical
definition and the meaning of the indices given by
rPACI. They can be split into two categories: primary
and combined indices. The primary indices are: \(PI_1\), \(PI_2\), \(PI_3\), \(SL\), \(AR_1\), \(AR_2\), \(AR_3\), \(AR_4\), \(AR_5\). They all measure certain
geometrical properties of the data distribution. Based on them, other
combined indices are computed: \(GLPI\)
(a generalized linear model) and \(NBI\) (naive Bayes index).
Data structure
First, we assume a data.frame with corneal data is
already available, which may have been loaded from a file using the
function readFile (see Workflow
of rPACI pacakge), simulated using simulateData (see Simulating corneal datasets), or by other
ways (as long as it meets the dataset requirements below).
The dataset consists of three columns: x, y (with the X and Y
Cartesian coordinates of data points), and ring index (1, 2, …). The
number of rows (number of data points) in the dataset depends on the
data availability (readFile by default only reads complete
rings, and simulateData also simulates complete rings), and
the chosen number of innermost rings to use (an optional parameter for
readFile, 15 by default).
dataset = simulateData(rings = 13)
plot(dataset$x,dataset$y)
If \(N\) is the number of complete
rings to be used and \(N_A\) is the
number of points per ring (in the example, \(N=13\) and \(N_A=256\), its default value), the format
in dataset is the following:
| \(x_1\) |
\(y_1\) |
\(1\) |
| \(x_2\) |
\(y_2\) |
\(1\) |
| \(x_3\) |
\(y_3\) |
\(1\) |
| … |
… |
… |
| \(x_{N_A}\) |
\(y_{N_A}\) |
\(1\) |
| \(x_{(N_A+1)}\) |
\(y_{(N_A+1)}\) |
\(2\) |
| \(x_{(N_A+2)}\) |
\(y_{(N_A+2)}\) |
\(2\) |
| … |
… |
… |
| \(x_{(2 N_A)}\) |
\(y_{(2 N_A)}\) |
\(2\) |
| … |
… |
… |
| \(x_{((N-1) \times N_A + 1)}\) |
\(y_{((N-1) \times N_A + 1)}\) |
\(N\) |
| … |
… |
… |
| \(x_{(N \times N_A)}\) |
\(y_{(N \times N_A)}\) |
\(N\) |
There are \(N_A\) rows per ring
(\(N\) rings), and thus, a total of
\(N \times N_A\) data points (in the
example, \(N \times N_A = 13 \times 256 =
3328\)). The dataset shall not contain missing data (this is
guaranteed if using readFile with default parameters or
simulateData).
Thus, we have \(N \times N_A\) data
points corresponding to \(N\) complete
rings, given in Cartesian coordinates. Let us define \(R_k\), the set of points belonging to the
\(k\)-th ring/mire (from the inside
out) and \(k \in \{1, \dots, N\}\) is
the ring index. Then, a single data point \(P_j=( x_j, y_j )\) (corresponding to the
j-th row of the dataframe) belongs to \(R_k\), i.e., \(P_j \in R_k\) if \(j \in J_k\), \(J_k:=\{ n_k, n_k+1, ..., n_k+(N_A-1) \}\),
with \(n_k = 1+N_A(k-1)\) (\(J_k\) is the set of indices \(j\) which correspond to the \(k\)-th mire \(R_k\), and \(n_k\) is the first (smallest) of them).
Primary indices
The first step to calculate the indices is to compute the
best-fitting circle (with center \(C_k\) and (average) radius \(AR_k\)) for each ring, with \(k \in \{1, \dots, N\}\). This is done by
least-squares fitting of a general circle equation to the Cartesian
coordinates of data points in ring \(k\) and then, the geometrical parameters
(center and radius) are computed from the coefficients (see Ahn et al. (2001) and Ramos-López et al. (2011) for more details).
These centers and radii contain information about the displacement or
off-centering of the mires, as well as their size and regularity. A high
level of drift or variability in these parameters is a signal of corneal
irregularity (Ramos-López et al.
(2011)).
From the best-fitting circle centers \(C_k\), \(k \in
\{1, \dots, N\}\), a regression line can be adjusted to their
coordinates (\(y\) over \(x\)), yielding a slope \(m\) and an intercept (which is not used).
This line contains information about the direction of the displacement
of the centers, which can be a signal of keratoconus (more specifically,
the upper-bottom or north-south asymmetry is a marker for keratoconus,
see Ramos-López et al. (2013)).
One can find also the best-fitting ellipses (with centers \(\tilde{C}_k\) and axis ratio \(c_k\), i.e., ratio or quotient between the
major and the minor axis) for each ring (\(k
\in \{1, \dots, N\}\)). This process is also based on a
least-squares fit of the general ellipse equation to the Cartesian
coordinates of data points of mire \(k\) (see Ahn et al.
(2001) and Ramos-López et al.
(2011) for more details). A high level of variability in the axis
ratio is also a signal of corneal irregularity (Ramos-López et al. (2011)).
Using the quantities, we can define four corneal irregularity
indices, labeled as \(PI_1, PI_2,
PI_3\) (from Placido Irregularity indices) and \(SL\):
\(PI_1\) \[ PI_1 = \frac{1}{N} \max_{1 \leq n,m \leq N}{
\|C_n - C_m \| } \] This index measures the diameter of the set
of centers \(C_k\) (normalized by the
total number of rings \(N\)), where
\(\| \cdot \|\) is the standard
Euclidean norm in \(\mathbb R^2\). A
high value of this index shows a large displacement or off-centering
between the mires.
\(PI_2\) \[ PI_2 = \frac{1}{N-1} \sum_{1 \leq n \leq N-1}{
\|C_{n+1} - C_{n} \| } \] corresponds to the total length of the
path connecting consecutive centers. \(PI_2\) measures the accumulated drift
between each pair of mires. In some cases, \(PI_1\) (diameter) can be relatively small
but the centers drift a lot from a mire to the next. \(PI_2\) is able to capture this kind of
irregularity, which is an abnormal sign.
\(PI_3\) \[ PI_3 = \sqrt{ \frac{1}{N} \sum_{1 \leq k
\leq N} { (c_k - \overline{c})^2 } }, \quad \text{where}
\quad \overline{c}=\frac{1}{N} \sum_{1 \leq k \leq N} c_k \] is
the standard deviation of the distribution of axis ratios, so that it
measures the variability of the axis ratios or eccentricities of the
individual rings. A large value of \(PI_3\) indicates a high variability in the
shapes of the rings, and thus it is a signal of irregularity.
\(SL\) \[ SL = |m| \] is the absolute value of the
slope of the best-fitting line to the coordinates of the centers \(C_k\). High values of this index correspond
to vertical mire displacements whereas low values correspond to
horizontal displacements.
Apart from these 4 indices, we also take as irregularity indices the
values of \(AR_k\) (average radius of
the \(k\)-th ring) corresponding to the
first (innermost) 5 rings. These values also showed a certain
sensitivity for detecting keratoconus (Ramos-López et al. (2011)).
Normalization of the primary indices
In order to obtain indices with values in the same range (more easily
comparable and to prevent scale problems when combining them), a
normalization of each primary index was performed (Ramos-López et al. (2011) and Ramos-López et al. (2013)). To do it, these
indices were tested on a real dataset of corneas of different
conditions. Then, their values were linearly translated to \([0,100]\), so that the value corresponding
to the best possible condition (less irregularity) was transformed to
\(0\), and the worst (more
irregularity) to \(100\). Therefore, in
the test dataset, the values of all primary indices were between \(0\) (best-case) and \(100\) (worst-case).
In practice, after the transformation, most index values for any
cornea will be in \([0,100]\), but in
some particular cases, they may be outside that interval. To facilitate
their interpretation, we can assume that values below zero correspond to
zero, and the maximum irregularity is attained at a certain value
(e.g. 150, this value is discussed in Ramos-López
et al. (2011)).
The normalization coefficients used in the previous studies (Ramos-López et al. (2011) and Ramos-López et al. (2013)) were the following
(for notation simplicity, values on the left-hand side of the
expressions represent normalized values whereas values on the right-hand
side are values before the normalization (as computed by their
definitions above)):
\[ PI_1 = 12368.3980719706 PI_1 -
12.5200561911951 \] \[ PI_2 =
9699.23915314471 PI_2 - 18.2916611541283\] \[ PI_3 = 5233.70399537826 PI_3 -
13.7375152045819\] \[ SL = 50
SL\]
\[ AR_1 = -1961.92346976023 AR_1 +
444.770836424576 \] \[ AR_2 =
-562.8465909984122 AR_2 + 279.7430721547980 \] \[ AR_3 = -486.9218093835968 AR_3 +
331.3137688964757 \] \[ AR_4 =
-318.192614292727 AR_4 + 298.064078007947 \] \[ AR_5 = -288.4037960143595 AR_5 +
321.4180137915255 \]
Thus, after the normalization (and possible truncation of extremely
high or low values), all indices values lay in \([0,150]\), with \(0\) corresponding to the most normal case
and \(150\) to the most irregular case.
This value of \(150\) is arbitrary and
could be changed (increased or lowered), but we found it performed well
in practice (Ramos-López et al. (2011) and
Ramos-López et al. (2013)).
Combined indices
The primary Placido indices described above, \(PI_1\), \(PI_2\), \(PI_3\), \(SL\), and \(AR_k\), show sensitivity to detecting
various irregularities in the cornea, such as keratoconus (Ramos-López et al. (2011), Ramos-López et al. (2013)). However, a single
index does not reach sufficient accuracy in the task, and compound
indices have been proposed and tested (Ramos-López et al. (2013), Castro-Luna et al. (2020)) in order to improve
the performance and precision of the individual indices. These compound
indices showed a significant improvement in accuracy when predicting
keratoconus.
A brief description of them is given as follows (see the
aforementioned references for more details):
GLPI index
The \(GLPI\) index (from Generalized
Linear Placido Index) (Ramos-López et al.
(2013)) is a generalized linear model computed using some of the
primary indices, in the following way: \[
GLPI = 100 \Phi \left( \left(-224.90 + 1.69 PI_1 + 1.28 PI_3 + 1.89
AR(4) + 0.19 SL \right)/20 \right) \]
where \(\Phi\) stands for the
cumulative distribution function of the standard normal distribution.
This corresponds to a generalized linear regression model with the
probit (\(\Phi^{-1}\)) link
function. Therefore, \(GLPI\) has
values between \(0\) and \(100\). The coefficients in the formula
above were obtained by fitting a real-world dataset of normal and
keratoconic corneas (Ramos-López et al.
(2013)). The index \(GLPI\) was
validated giving an excellent discrimination capability of irregular
corneas.
NBI index
The compound index \(NBI\) (from
Naive Bayes Index), introduced in Castro-Luna et
al. (2020), is a Bayesian classifier. More precisely, it is a
Bayesian network with the naïve Bayes structure depicted below.
This Bayesian network is a conditional linear Gaussian (CLG) network,
with its root node (\(KC\)) being a
discrete binary variable and the rest of the nodes (which are some of
the primary indices introduced above) being continuous. The parameters
of the model (their conditional probability distributions) were reported
in Castro-Luna et al. (2020). This model
can be used for predicting whether a specific cornea (whose values of
the primary indices are known) is a normal cornea or a keratoconic
cornea.
Moreover, the probability of being one type or another can be
computed as well, either using exact inference or approximate inference
with algorithms such as evidence weighting, likelihood weighting, or
more generally, importance sampling (Cheng and
Druzdel (2000)). Even though exact inference is feasible in this
case, approximate inference is easier to implement and to generalize to
more complex network structures. In a nutshell, these algorithms consist
of simulating a large number of samples from the network, according to
the evidence (i.e., variable values that are known a priori), and
averaging their likelihoods to estimate the probability of each state of
the target variable (in this case, \(KC\)). For a deeper explanation, we refer
the readers to Castro-Luna et al. (2020)
and its references.
Thus, the posterior probability estimation can be computed
efficiently with several approaches. In rPACI, we make
use of the R package bnlearn
(Scutari (2010)), which includes an
implementation of the likelihood weighting algorithm, for computing the
index NBI from the primary indices values of the bottom nodes.
References
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“Least-Squares Orthogonal Distances Fitting of Circle, Sphere,
Ellipse, Hyperbola, and Parabola.” Pattern Recognition
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https://doi.org/10.1016/S0031-3203(00)00152-7.
Castro-Luna, Gracia M., Andrei Martínez-Finkelshtein, and Darío
Ramos-López. 2020.
“Robust Keratoconus Detection with Bayesian
Network Classifier for Placido Based Corneal Indices.”
Contact Lens and Anterior Eye 43 (4): 366–72.
https://doi.org/10.1016/j.clae.2019.12.006.
Cheng, Jian, and Marek J. Druzdel. 2000.
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Ramos-López, Darío, Andrei Martínez-Finkelshtein, Gracia M. Castro-Luna,
et al. 2013.
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335–43.
https://doi.org/10.1097/opx.0b013e3182843f2a.
Ramos-López, Darío, Andrei Martínez-Finkelshtein, Gracia M. Castro-Luna,
David Piñero, and Jorge L. Alió. 2011.
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Corneal Irregularity.” Optometry and Vision Science 88
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https://doi.org/10.1097/opx.0b013e3182279ff8.
Scutari, Marco. 2010.
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