Applying Constraints to the Electrocardiographic Inverse Problem

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Applying Constraints to the Electrocardiographic
Inverse Problem

• Rob MacLeodDana Brooks

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Electrocardiography
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Electrocardiographic Mapping

• Bioelectric Potentials
• Goals
• Higher spatial density
• Imaging modality
• Measurements
• Body surface
• Heart surfaces
• Heart volume

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Body Surface Potential Mapping
Taccardi et al, Circ., 1963
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Cardiac Mapping

• Coverage
•  Sampling Density
•  Surface or volume

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Inverse Problems in Electrocardiography

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C
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Epicardial Inverse Problem

• Definition
• Estimate sources from remote measurements

• Motivation
• Noninvasive detection of abnormalities
• Spatial smoothing and attenuation

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Forward/Inverse Problem
Thom Oostendorp, Univ. of Nijmegen
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Sample Problem Activation Times
Measured
Thom Oostendorp, Univ. of Nijmegen
Computed
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Sample Problem PTCA
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Elements of the Inverse Problem

• Components
• Source description
• Geometry/conductivity
• Forward solution
• Inversion method (regularization)
• Challenges
• Inverse is ill-posed
• Solution ill-conditioned

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Inverse Problem Research

• Role of geometry/conductivity
• Numerical methods
• Improving accuracy to clinical levels
• Regularization
• A priori constraints versus fidelityto
  measurements

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Regularization

• Current questions
• Choice of constraints/weights
• Effects of errors
• Reliability
• Contemporary approaches
• Multiple Constraints
• Time Varying Constraints
• Novel constraints (e.g., Spatial Covariance)
• Tuned constraints

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Tikhonov Approach
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Multiple Constraints
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Dual Spatial Constraints
Note two regularization factors required
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Joint Time-Space Constraints
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Joint Time-Space Constraints
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Determining Weights

• Based on a posteriori information
• Ad hoc schemes
• CRESO composite residual and smooth operator
• BNC bounded norm constraint
• AIC Akaike information criterion
• L-curve residual norm vs. solution seminorm

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L-Surface

• Natural extension of single constraint approach
• Knee point becomes a region

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Joint Regularization Results
with Fixed Laplacian Parameter
RMSError
Energy Regularization Parameter
with Fixed Energy Parameter
RMSError
Laplacian Regularization Parameter
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Admissible Solution Approach
Admissible Solution Region
Constraint 3 (differentiable)
Constraint 1 (non-differentiable but convex)
Constraint 4 (differentiable)
Constraint 2 (non-differentiable but convex)
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Single Constraint
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Multiple Constraints
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Examples of Constraints

• Residual contsraint
• Regularization contstraints
• Tikhonov constraints
• Spatiotemporal contraints
• Weighted constraints
• Novel constraints

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Ellipsoid Algorithm
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Ellipsoid Algorithm
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Admissible Solution Results
AdmissibleSolution
Original
Regularized
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New Opportunity

• Catheter mapping
• provides source information
• limited sites

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Catheter Mapping

• Endocardial
• Epicardial
• Venous

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New Opportunity

• Catheter mapping
• provides source information
• limited sites
• Problem
• how to include this information in the inverse
  solution
• where to look for best information
• Solutions?
• Admissible solutions, Tikhonov?
• Statistical estimation techniques

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Agknowledgements

• CVRTI
• Bruno Taccardi
• Rich Kuenzler
• Bob Lux
• Phil Ershler
• Yonild Lian

• CDSP
• Dana Brooks
• Ghandi Ahmad