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Applying Constraints to the Electrocardiographic Inverse Problem

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Title: 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
Inverse Problems in Electrocardiography

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

8
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
Tikhonov Approach
15
Multiple Constraints
16
Dual Spatial Constraints
Note two regularization factors required
17
Joint Time-Space Constraints
18
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
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
Single Constraint
24
Multiple Constraints
25
Examples of Constraints
  • Residual contsraint
  • Regularization contstraints
  • Tikhonov constraints
  • Spatiotemporal contraints
  • Weighted constraints
  • Novel constraints

26
Ellipsoid Algorithm
27
Ellipsoid Algorithm
28
Admissible Solution Results
AdmissibleSolution
Original
Regularized
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
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