Applying Constraints to the Electrocardiographic Inverse Problem

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