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Applying Constraints to the Electrocardiographic Inverse Problem

Applying Constraints to the Electrocardiographic Inverse Problem. Rob MacLeod Dana Brooks. Electrocardiography. Electrocardiographic Mapping. Bioelectric Potentials Goals Higher spatial density Imaging modality Measurements Body surface Heart surfaces Heart volume.

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Applying Constraints to the Electrocardiographic Inverse Problem

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  1. Applying Constraints to the Electrocardiographic Inverse Problem Rob MacLeodDana Brooks

  2. Electrocardiography

  3. Electrocardiographic Mapping • Bioelectric Potentials • Goals • Higher spatial density • Imaging modality • Measurements • Body surface • Heart surfaces • Heart volume

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

  5. Cardiac Mapping • Coverage • Sampling Density • Surface or volume

  6. A Forward B Inverse Inverse Problems in Electrocardiography + - + - + - C

  7. Definition Estimate sources from remote measurements Motivation Noninvasive detection of abnormalities Spatial smoothing and attenuation Forward Inverse Epicardial Inverse Problem

  8. Forward problem Epicardial/Endocardial Activation Time Geometric Model Body Surface Potentials Inverse problem Forward/Inverse Problem Thom Oostendorp, Univ. of Nijmegen

  9. Sample Problem: Activation Times Measured Thom Oostendorp, Univ. of Nijmegen Computed

  10. Sample Problem: PTCA

  11. Elements of the Inverse Problem • Components • Source description • Geometry/conductivity • Forward solution • “Inversion” method (regularization) • Challenges • Inverse is ill-posed • Solution ill-conditioned

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

  13. 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

  14. Problem formulation Constraint Solution Tikhonov Approach

  15. For k constraints with solution Multiple Constraints

  16. For two spatial constraints: Dual Spatial Constraints Note: two regularization factors required

  17. Redefine y, h, A: And write a new minimization equation: Joint Time-Space Constraints

  18. General solution: For a single space and time constraint: Note: two regularization factors and implicit temporal factor Joint Time-Space Constraints

  19. 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

  20. Rhl Thl Ahl- y L-Surface • Natural extension of single constraint approach • “Knee” point becomes a region

  21. Joint Regularization Results with Fixed Laplacian Parameter RMSError Energy Regularization Parameter with Fixed Energy Parameter RMSError Laplacian Regularization Parameter

  22. Admissible Solution Approach Admissible Solution Region Constraint 3 (differentiable) Constraint 1 (non-differentiable but convex) Constraint 4 (differentiable) Constraint 2 (non-differentiable but convex)

  23. Define f(x) s.t. with the constraint such that that satisfies the convex condition Single Constraint

  24. Define multiple constraints fi(x) so that the set of these represents the intersection of all constraints. When they satisfy the joint condition Then the resulting x is theadmissible solution Multiple Constraints

  25. Examples of Constraints • Residual contsraint • Regularization contstraints • Tikhonov constraints • Spatiotemporal contraints • Weighted constraints • Novel constraints

  26. Ellipsoid Algorithm

  27. Subgradients Ci Ci+1 + + Ei Ei+1 Constraint set Normal hyperplane Ellipsoid Algorithm

  28. Admissible Solution Results AdmissibleSolution Original Regularized

  29. New Opportunity • Catheter mapping • provides source information • limited sites

  30. Endocardial Epicardial Venous Catheter Mapping

  31. 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

  32. CVRTI Bruno Taccardi Rich Kuenzler Bob Lux Phil Ershler Yonild Lian CDSP Dana Brooks Ghandi Ahmad Agknowledgements

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