Mathematical Modeling and Classification of Eye Disease

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Mathematical Modeling and Classification of Eye
Disease

• Srinivasan Parthasarathy
• Joint work with M. Bullimore, K. Marsolo and
• M. Twa

Aspects of this work are funded by the NIH, NSF
and the DOE.
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Desiderata for Clinical Diagnosis

• Should be accurate and ideally interoperable
• Can we use mathematical modeling?
• Can we improve accuracy by meta-learning?
• Should be interpretable
• Can we visualize the decision making process
  effectively?
• Should be responsive
• Can we leverage distributed computing tools to
  speed up the process?

Synopsis of Approach
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Ocular Anatomy 101
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Case Study Keratoconus

• Progressive, degenerative, non-inflammatory
  disease.
• A leading cause of blindness and corneal
  transplant.
• Early detection is difficult important
• Has implications for eye surgery and
  control-of-disease
• Initial Symptoms Minor fluctuations in corneal
  shape
• Diagnosis procedure
• Video-keratography exam
• Manual analysis of results by clinician
• Challenges to detection
• Voluminous data
• one image is 1000s of data points representing
  corneal surface
• spatial and temporal (longitudinal)
• Features of interest small in scale to mean shape
• Leads to variance in prognosis

Late stage Keratoconus Normal (clinically
ideal)
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Raw Data Description

• Corneal surface represented as a 7000 point
  matrix output of video-keratographic device
• Fixed angula sampling in concentric circles
  around center
• Data stored in cylindrical coordinates
• Radius (?)
• Angle (?)
• Height / Elevation (z)
• 3 classes Keratoconus, surgical repaired
  (lasik), normal
• 508 eyes from 254 people (L-R normalization)
• Can we use this data to construct a 3-D surface
  of the cornea.
• How to model?

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Modeling

• Desired Properties
• Reflect General Shape and Structure of Entity of
  Interest
• Should capture important features? local
  harmonics
• Need for Compact Representation
• Should be capable of capturing important features
  such as local harmonics
• Capable of tuning to desired resolution
• Capable of dealing with multiple dimensions
• Options Evaluated
• Zernike Polynomials, Pseudo-Zernike Polynomials,
  Wavelets

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Modeling Zernike and Pseudo-Zernike Polynomials

• Hyper-geometric radial basis functions
• Each term (mode) in the series represents a 3D
  geometric surface.
• Value of each term represents the contribution of
  that mode to the overall surface (independent)
• Benefits
• Lower order modes show correlation to general
  surface features of the cornea.
• Higher order modes capture local harmonics
• Orthogonal
• Anatomic correspondence to clinical concepts
• Drawbacks
• Can be computationally expensive.
• Can model noise as well especially higher order

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Details

• Zernike (Z)

• Pseudo-Zernike (PZ)

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Z PZ Transformation Algorithm

• Compute least-squares fit between model and
  original data
• Then use coefficients as feature vector as input
  for classification

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Wavelet Modeling

• Convert to 1-Dimensional Signal
• A. Sample along concentric circles (Klyce
  Smolek)
• Use standard 1D Wavelets to model Signal and use
  coefficients to classify
• B. Sample along a space-filling curve (us)
• Key idea is to maintain spatial coherence
• Same as above to classify (works better than A)
• Apply 2-dimensional Wavelet Models (us)
• Use coefficients to classify (does not work as
  well as 1.B.)
• Pros
• Fast and efficient
• Cons
• Overall performance worse (5 -10) than
  Zernike-based approaches
• No anatomic correspondence difficult to
  interpret

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Experiments Model Fidelity

• Model error of Z vs. PZ?
• Model error on different patient classes?
• What set of parameters provides the best model
  fit?
• Transformation Parameters
• Polynomial Order 4th 10th
• Larger the order ? may model signal noise !
• Radius 2.0, 2.5, 3.0, 3.5mm (max)
• Larger Radius ? more number of points to model

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Results Model Fidelity

• General Trend
• Increasing polynomial order decreases error.
• Increasing transformation radius increases error.
  
• Same order Z gt PZ
• Same coefficients Z PZ
• Between patient classesKeratoconus gt LASIK gt
  Normal

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Classification and Clinical Decision Support

• Prefer transparent algorithms over black-box
  classifiers.
• Use simple classifiers and provide a way to
  visually explore the decision making process
• Desire high accuracy
• Use an ensemble of simple and interpretable
  classifiers
• Desire efficiency
• Use netsolve distributed computing tool

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Basic Classification Performance

• Accuracy of Classifier based on PZ vs. Z?
• Zernike works better
• Which classifiers work well
• C4.5 (84-85), Naïve Bayes (84), VFI (84),
• Neural Networks (81), one-vs-all SVM (82)
• SVMs and NN are also difficult to explain
  (interpretability)
• Performance of Ensemble Techniques
• Boosting, bagging and random forests
• All upgrade performance (3-4)
• Bagging prefered easy to interpret, performance
  marginally better
• More accurate model higher classification
  accuracy?
• C4.5 (4th order works best but others are not
  bad)
• NB/VFI/NN/SVM (higher orders do not work well
  noise or irrelevant features hampers performance)

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Ensemble Learning

• Combine results of multiple classification models
  built from different samples of dataset to
  improve accuracy.
• Training data represents a sample of the
  population.
• A classifier built on one sample can overfit
  and model noise.
• Constructing multiple models can filter noise and
  reduce generalization error Breiman, 1996.
• Traditional Methods
• Bootstrap Aggregation (Bagging)
• Boosting

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Spatial Averaging (SA)

• Use classifiers built on different resolutions
  and models of the dataset to improve accuracy.
• Build classifier for each spatial transformation
  and resolution.
• Take modal label of classifiers to reach final
  decision.
• Can view as a structured column bagging
  algorithm
• Intuition
• Lower order transformations result in more
  general, global model.
• Higher order transformations better at capturing
  local harmonics, but can model noise.
• If errors are uncorrelated, SA should smooth
  noise effects.

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Spatial Averaging
Org.Data
1. Transform Data
2. Classify
3. Tally Votes
Spatial Transformation(s)
4Z
6Z
8Z
10Z
10PZ
8PZ
6PZ
4PZ
5Z
7Z
9Z
9PZ
7PZ
5PZ
5
0
2
4
2
1
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Spatial Averaging with Sub-Selection (Combined)
Org.Data
1. Transform Data
2. Classify
3. Tally Votes
Spatial Transformation(s)
4Z
10Z
4PZ
7Z
7PZ
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Experiments

• How does SA compare to a single decision tree?
• How does SA compare to traditional
  ensemble-learning methods?
• Can SA be combined with ensemble-learning methods
  to further improve results?
• Ensemble Methods Evaluated Include
• Bagging, Boosting, Random Forests

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Spatial Averaging vs. C4.5

• 10-fold c.v.
• Zernike-based SA
• 7 trees (4th to 10th order)
• 3-5 over individual tree.
• PZ-based based SA
• Up to 7 improvement
• Combined SA classifier (5 trees)
• accuracy of 91.1, 6-10 improvement.
• Rationale Clinically it appears that PZ and Z do
  better on different varieties of Keratoconus

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SA and Traditional Ensemble-Learning

• Combined SA (5 trees) outperforms Boosted (10) or
  Bagged C4.5 (10)
• Bagging does marginally better than Boosting RF
  (not shown)
• Combined Bagging (5)
• 94.1 accuracy
• However it trades off interpretability (5X5
  trees) for accuracy

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Visualization of Results

• Task Visualize results to provide decision
  support for clinicians.
• Give intuition as to why a group of patients are
  classified the way they are.
• Contrast an individual patient with others in the
  same group
• How?
• Modes of Zernike/Pseudo-Zernike polynomial
  correspond to specific features of the cornea.

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Patient-Specific Decision Surface

• Treat each path through the decision tree as a
  rule.
• Cluster training data by rule.
• Compute average coefficient values for each
  cluster.
• Given a patient, classify and keep the rule
  coefficients, set others to zero.
• Construct overall surface and rule surface

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Patient-Specific Decision Surface

• Create surfaces using
• All patient coefficients.
• All rule mean coefficients.
• Patient coefficients used in the classifying rule
  (rest zero).
• Rule mean coefficients used in the classifying
  rule (rest zero).
• Also
• Bar chart with relative error between patient and
  rule mean coefficients.

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Visualization Strongest Rules
Rule 1 - Keratoconus
Rule 8 - Normal
Rule 4 - LASIK
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High Performance Results

• Optimize and parallelize (5 nodes) key steps of
  the code over a grid environment
• M unoptimized algorithm
• NS netsolve version
• Times shown for computing one decision tree using
  particular model (Z/PZ) and includes model
  building time.

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Case Study Glaucoma

• Progressive neuropathy of the optic nerve
• Disease characteristics
• Symptom free
• Elevated intraocular pressure
• Structural loss of retinal ganglion cells
• Gradual restriction of the visual field from
  periphery to center

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Clinical Management of Glaucoma

• Monitoring Intraocular pressure
• Static threshold visual field sensitivity
• Observations of structural change at the optic
  nerve head

Glaucoma
Normal
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Topographic Modeling

• Objectives
• Feature reduction
• Preservation of spatial correlation
• Polynomial Modeling
• Zernike
• Pseudo-Zernike
• Spline Modeling
• Knot locations, coefficients
• Wavelet Modeling
• 1D vs. 2D

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Concluding Remarks

• Modeling and Classifying Corneal Shape
• Low-order Zernike polynomials provide adequate
  model of corneal surface.
• Higher order polynomials begin to model noise but
  may contain a few useful features for
  classification
• Decision trees provide classification accuracy
  greater than or equal to other classification
  methods.
• Accuracy can be further improved by using
  SA-strategy.
• Visualization
• Using classification attributes as basis for
  visualization provides method of decision support
  for clinicians.
• High Performance Implementations can help
• Modeling and Classifying Glaucoma ongoing

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General thoughts on Interdisciplinary
Collaboration

• Steep learning curve
• Need to learn language and requirements
• Need to express results in domain language
• Patience, patience, patience
• Communities are inertia bound
• Often difficult to make headway
• Potential for incredible rewards
• Scientific/medical implications
• Good working relationship essential
• Equal partners

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900µm diameter fit
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How is RMS related to Class?

• Greater variance in glaucoma
• Higher mean in glaucoma

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Conclusions
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Single Decision Tree
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Decision Surface
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Data

• 254 Patient Records
• 3 Patient Categories
• Normal (119)
• Diseased (99)
• Post-LASIK (36)

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Imaging Scripting Crop Routines
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Imaging Scripting Centering Routines

• 2D Mean SD
• What is the best center point (disc vs cup)?
• Failure associated with class assignment
• Normals fail more often

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Re-centered