Applying Constraints to the Electrocardiographic Inverse Problem Rob MacLeod Dana Brooks.
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Transcript of Applying Constraints to the Electrocardiographic Inverse Problem Rob MacLeod Dana Brooks.
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Applying Constraints to the Electrocardiographic
Inverse Problem
Applying Constraints to the Electrocardiographic
Inverse Problem
Rob MacLeodDana Brooks
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ElectrocardiographyElectrocardiography
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Electrocardiographic Mapping
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
Body Surface Potential Mapping
Taccardi et al,Circ., 1963
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Cardiac MappingCardiac Mapping
• Coverage • Sampling Density• Surface or volume
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Inverse Problems in ElectrocardiographyInverse Problems in Electrocardiography
Forward
Inverse
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Forward
Inverse
Epicardial Inverse ProblemEpicardial Inverse Problem• Definition
– Estimate sources from remote measurements
• Motivation– Noninvasive
detection of abnormalities
– Spatial smoothing and attenuation
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Forward/Inverse ProblemForward/Inverse ProblemForward problem
Body Surface Potentials
GeometricModel
Epicardial/EndocardialActivation Time
Inverse problem
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front
back
Thom Oostendorp,
Univ. of Nijmegen
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Sample Problem: Activation Times
Sample Problem: Activation Times
Measured
Computed Thom Oostendorp,
Univ. of Nijmegen
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Sample Problem: PTCASample Problem: PTCA
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• Components– Source description– Geometry/conductivity– Forward solution– “Inversion” method (regularization)
• Challenges– Inverse is ill-posed– Solution ill-conditioned
Elements of the Inverse Problem
Elements of the Inverse Problem
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Inverse Problem ResearchInverse Problem Research
• Role of geometry/conductivity
• Numerical methods
• Improving accuracy to clinical levels
• Regularization– A priori constraints versus fidelity
to measurements
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RegularizationRegularization• 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 ApproachTikhonov ApproachProblem formulation
Constraint
Solution
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Multiple ConstraintsMultiple ConstraintsFor k constraints
with solution
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Dual Spatial ConstraintsDual Spatial Constraints
For two spatial constraints:
Note: two regularization factors required
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Joint Time-Space ConstraintsJoint Time-Space Constraints
Redefine y, h, A:
And write a new minimization equation:
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Joint Time-Space ConstraintsJoint Time-Space ConstraintsGeneral solution:
For a single space and time constraint:
Note: two regularization factors and implicit temporal factor
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Determining WeightsDetermining 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-SurfaceL-SurfaceRh
Th
Ahy
• Natural extension of single constraint approach
• “Knee” point becomes a region
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Joint Regularization ResultsJoint Regularization Results
Energy Regularization Parameter
Laplacian Regularization Parameter
RM
SE
rror
RM
SE
rror
with Fixed Laplacian Parameter
with Fixed Energy Parameter
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Admissible Solution ApproachAdmissible Solution Approach
Constraint 1 (non-differentiablebut convex)
Constraint 2(non-differentiablebut convex)
Constraint 3(differentiable)
Admissible Solution Region
Constraint 4 (differentiable)
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Single ConstraintSingle ConstraintDefine (x) s.t.
with the constraint such that
that satisfies the convex condition
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Multiple ConstraintsMultiple ConstraintsDefine multiple constraints i(x)
so that the set of these
represents the intersection of all constraints. When they satisfythe joint condition
Then the resulting x is theadmissible solution
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Examples of ConstraintsExamples of Constraints
• Residual contsraint
• Regularization contstraints
• Tikhonov constraints
• Spatiotemporal contraints
• Weighted constraints
• Novel constraints
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Ellipsoid AlgorithmEllipsoid Algorithm
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Ellipsoid AlgorithmEllipsoid Algorithm
Ci
Ei
+ Ci+1
Ei+1
+
Constraint set
Normal hyperplane
Subgradients
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Admissible Solution ResultsAdmissible Solution Results
Original Regularized AdmissibleSolution
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New OpportunityNew Opportunity
• Catheter mapping– provides source information– limited sites
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Catheter MappingCatheter Mapping• Endocardial• Epicardial• Venous
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New OpportunityNew 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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AgknowledgementsAgknowledgements
• CVRTI– Bruno Taccardi– Rich Kuenzler– Bob Lux– Phil Ershler– Yonild Lian
• CDSP– Dana Brooks– Ghandi Ahmad