É. Debreuve, G.T. Gullberg Medical Imaging Research Laboratory

1. Dense electrical map reconstruction from ECG/MCG measurements with known fiber structure and standard activation sequence É. Debreuve, G.T. Gullberg Medical Imaging Research Laboratory The University of Utah, Salt Lake City

2. Electrical activity of the myocardium • Myocardium contraction: • Electrical activity of the myocardium • Signal propagation (integral equations) • Electrical potential on the thorax • Magnetic field close to the thorax

3. Signal acquisition • Non-invasive measurements: • ECG (electrical potential) • Standard clinical exam: 12 electrodes • MCG (magnetic field) • E.g., 30 measurement sites Electrodes ECG SQUIDs MCG

4. Analysis of the measurements • Diagnosis of heart diseases: • Analysis of the ECG curves • Coarse defect localizations • Analysis of the MCG curves ? • Reconstruction of the electrical activity • Electrical model of the myocardium • Geometrical model of the thorax • Discretization of the propagation equations • Resolution of a system of equations • Analysis of the reconstructed electrical activity or

5. Forward model: General • Electrical model of the myocardium: • Equivalent current dipoles • Location, direction, magnitude (variable over time) • Tissue conductivities • Geometrical model of the thorax: • Piecewise constant isotropic conductivity volume • Triangulation of the boundaries

6. Forward model: Based on NCAT • NCAT phantom: • With myocardium+cavities, liver, lungs • Isotropic conductivities: • Blood, myocardium, liver, lungs, soft tissues • Discretization: • 2900 triangles • 1500 nodes BEM of the NCAT phantom, The University of North Carolina, Chapel Hill

7. Electrical activity reconstruction • Ill-posed inverse problem: • Too many unknowns • Location, direction, magnitude of each dipole • Too few measurements • Too much noise • Reconstruction of many dipoles • Need for regularization

8. Regularization of the problem Bioengineering Research Group, Auckland • Use of known (or a priori) information: • Voxelization of the myocardium  Known locations • Cardiac fibers  Known directions • Activation sequence + Action potential shape  Standard magnitudes at anytime during the cycle • Unknowns with a priori: Dipole magnitudes Fiber structure Dipole directions Activation sequence Action potential

9. Regularized reconstruction • Implementation: • System of equations: RX = M • R: Transformation matrix from dipole magnitudes to measurements • X: Unknown dipole magnitudes • M: Electrical potential and magnetic field measurements • Solution close to standard magnitudes: ( X - X ) = 0 • ( ): Function allowing half-quadratic regularization (commonly used for support or edge-preserving smoothing constraints) • X: Standard magnitudes • Criterion to be minimized: |RX - M|2 +  ( X - X ) • Polak-Ribiere conjugate gradient algorithm

10. Preliminary results: Data simulated w/o noise • Dipoles: • 700+ • Locations: Spaced every 3 mm in x, y, and z inside the myocardium • Directions & magnitudes: variations around a given dipole configuration • Magnitude interval: [0.75, 1.25] • Measurement sites: • Electrical potential: 250 (each node of the outer surface) • Magnetic field: 250 (close to each potential measurement sites) Front Back

11. Preliminary results: Reconstruction Relative error Relative error histogram • Without regularization: • With regularization: • No regularization Relative error histogram Relative error • With regularization

12. Future works • Using this forward model: • Measurements with noise • Improved forward model: • Bi-domain representation • Coupled boundary-element/finite-element model • Real data