Applying Constraints to the Electrocardiographic

Inverse Problem

Applying Constraints to the Electrocardiographic

Inverse Problem

Rob MacLeodDana Brooks

ElectrocardiographyElectrocardiography

Electrocardiographic Mapping

Electrocardiographic Mapping

• Bioelectric Potentials

• Goals– Higher spatial density– Imaging modality

• Measurements– Body surface– Heart surfaces– Heart volume

Body Surface Potential Mapping

Body Surface Potential Mapping

Taccardi et al,Circ., 1963

Cardiac MappingCardiac Mapping

• Coverage • Sampling Density• Surface or volume

Inverse Problems in ElectrocardiographyInverse Problems in Electrocardiography

Forward

Inverse

A

B

-

+

C

-+

+

-

Forward

Inverse

Epicardial Inverse ProblemEpicardial Inverse Problem• Definition

– Estimate sources from remote measurements

• Motivation– Noninvasive

detection of abnormalities

– Spatial smoothing and attenuation

Forward/Inverse ProblemForward/Inverse ProblemForward problem

Body Surface Potentials

GeometricModel

Epicardial/EndocardialActivation Time

Inverse problem

3

4

5

6

ra

7

8

9

10

11

12

13

14

15

16

17

18

v1

20

21

22

23

24

25

v2

27

28

29

30

31

32

33

34

35

36

37

38

39

40

v4

42

43

44

45

46

47

v5

49

50

51

la

52

53

v6

55

56

57

58

59

60

61

62

63

64

1

4

5

σ

2σ

3σ

σ

σ

70

6065

7580

6065 50

55

4550

55

60

5550

4035

30

85

65

7075

60

50

55

80

5055

50

40

35

25

55

45

60

50

55

55

front

back

Thom Oostendorp,

Univ. of Nijmegen

Sample Problem: Activation Times

Sample Problem: Activation Times

Measured

Computed Thom Oostendorp,

Univ. of Nijmegen

Sample Problem: PTCASample Problem: PTCA

• 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

Inverse Problem ResearchInverse Problem Research

• Role of geometry/conductivity

• Numerical methods

• Improving accuracy to clinical levels

• Regularization– A priori constraints versus fidelity

to measurements

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

Tikhonov ApproachTikhonov ApproachProblem formulation

Constraint

Solution

Multiple ConstraintsMultiple ConstraintsFor k constraints

with solution

Dual Spatial ConstraintsDual Spatial Constraints

For two spatial constraints:

Note: two regularization factors required

Joint Time-Space ConstraintsJoint Time-Space Constraints

Redefine y, h, A:

And write a new minimization equation:

Joint Time-Space ConstraintsJoint Time-Space ConstraintsGeneral solution:

For a single space and time constraint:

Note: two regularization factors and implicit temporal factor

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

L-SurfaceL-SurfaceRh

Th

Ahy

• Natural extension of single constraint approach

• “Knee” point becomes a region

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

Admissible Solution ApproachAdmissible Solution Approach

Constraint 1 (non-differentiablebut convex)

Constraint 2(non-differentiablebut convex)

Constraint 3(differentiable)

Admissible Solution Region

Constraint 4 (differentiable)

Single ConstraintSingle ConstraintDefine (x) s.t.

with the constraint such that

that satisfies the convex condition

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

Examples of ConstraintsExamples of Constraints

• Residual contsraint

• Regularization contstraints

• Tikhonov constraints

• Spatiotemporal contraints

• Weighted constraints

• Novel constraints

Ellipsoid AlgorithmEllipsoid Algorithm

Ellipsoid AlgorithmEllipsoid Algorithm

Ci

Ei

+ Ci+1

Ei+1

+

Constraint set

Normal hyperplane

Subgradients

Admissible Solution ResultsAdmissible Solution Results

Original Regularized AdmissibleSolution

New OpportunityNew Opportunity

• Catheter mapping– provides source information– limited sites

Catheter MappingCatheter Mapping• Endocardial• Epicardial• Venous

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

AgknowledgementsAgknowledgements

• CVRTI– Bruno Taccardi– Rich Kuenzler– Bob Lux– Phil Ershler– Yonild Lian

• CDSP– Dana Brooks– Ghandi Ahmad