1. Applying Constraints to the Electrocardiographic Inverse Problem Rob MacLeod Dana 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. Inverse Problems in Electrocardiography Forward Inverse A B - + C - + + -

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

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

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

10. Sample Problem: PTCA

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

12. Inverse Problem Research Role of geometry/conductivity Numerical methods Improving accuracy to clinical levels Regularization –A priori constraints versus fidelity to 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. Tikhonov Approach Problem formulation Constraint Solution

15. Multiple Constraints For k constraints with solution

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

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

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

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. L-Surface Rh Th Ah  y Natural extension of single constraint approach “Knee” point becomes a region

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

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

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

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

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

26. Ellipsoid Algorithm

27. CiCi EiEi + C i+1 E i+1 + Constraint set Normal hyperplane Subgradients

28. Admissible Solution Results OriginalRegularized Admissible Solution

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

30. Catheter Mapping Endocardial Epicardial Venous

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