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Wavefront-based models for inverse electrocardiography Alireza Ghodrati (Draeger Medical) Dana Brooks, Gilead Tadmor (Northeastern University) Rob MacLeod (University of Utah)

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Inverse ECG Basics Problem statement: Problem statement: Estimate sources from remote measurements Source model: Potential sources on epicardium Activation times on endo- and epicardium Volume conductor Inhomogeneous Three dimensional Challenge: Challenge: Spatial smoothing and attenuation An ill-posed problem Forward Inverse

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Source Models A) Epi (Peri)cardial potentials A) Epi (Peri)cardial potentials Higher order problem Higher order problem Few assumptions (hard to include assumptions) Few assumptions (hard to include assumptions) Numerically extremely ill-posed Numerically extremely ill-posed Linear problem Linear problem A - + B - + + - B) Surface activation times B) Surface activation times Lower order parameterization Lower order parameterization Assumptions of potential shape (and other features) Assumptions of potential shape (and other features) Numerically better posed Numerically better posed Nonlinear problem Nonlinear problem

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Combine Both Approaches? WBCR: wavefront based curve reconstruction WBCR: wavefront based curve reconstruction Surface activation is time evolving curve Surface activation is time evolving curve Predetermined cardiac potentials lead to torso potentials Predetermined cardiac potentials lead to torso potentials Extended Kalman filter to correct cardiac potentials Extended Kalman filter to correct cardiac potentials WBPR: wavefront based potential reconstruction WBPR: wavefront based potential reconstruction Estimate cardiac potentials Estimate cardiac potentials Refine them based on body surface potentials Refine them based on body surface potentials Equivalent to using estimated potentials as a constraint to inverse problem Equivalent to using estimated potentials as a constraint to inverse problem A - + B - + + - Use phenomenological data as constraints!

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The Forward model Laplace’s equation in the source free medium where forward matrix created by Boundary Element Method or Finite Element Method torso potentials heart surface potentials

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Spatial Assumptions Three regions: activated, inactive and transition Three regions: activated, inactive and transition Potential values of the activated and inactive regions are almost constant Potential values of the activated and inactive regions are almost constant Complex transition region Complex transition region Potential surface Projection to a plane

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Temporal Assumptions Propagation is mostly continuous Propagation is mostly continuous The activated region remains activated during the depolarization period The activated region remains activated during the depolarization period

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Wavefront-based Curve Reconstruction Approach (WBCR) Formulate the potential based model in terms of the activation wavefront (parameterized continuous curve) Formulate the potential based model in terms of the activation wavefront (parameterized continuous curve) Decrease the order of parameterization Decrease the order of parameterization Apply spatial-temporal constraints Apply spatial-temporal constraints State-space model State-space model Wavefront curve is the state variable and evolves on the surface of the heart Wavefront curve is the state variable and evolves on the surface of the heart State evolution model based on phenomenological study of physiological properties State evolution model based on phenomenological study of physiological properties Heart surface potentials predicted from wavefront position using a three-zone model (activated, inactive, and transition) and study of recorded data Heart surface potentials predicted from wavefront position using a three-zone model (activated, inactive, and transition) and study of recorded data

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: time instant : activation wavefront which is state variable (continuous curve) : measurements on the body : state evolution function : potential function : state model error (Gaussian white noise) : forward model error (Gaussian white noise) ncnc yfguwyfguw WBCR Formulation

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Curve Evolution Model, f( ) : speed in the normal direction at point s on the heart surface and time t. : spatial factor : coefficients of the fiber direction effect : angle between fiber direction and normal to the wavefront Speed of the wavefront:

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Surface fiber directions Auckland heart fiber directions Utah heart electrodes Geometry matching Utah heart with approximated fiber directions

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Wavefront Potential, g( ) Potential at a point on the heart surface Potential at a point on the heart surface Second order system step response Second order system step response Function of distance of each point to the wavefront curve Function of distance of each point to the wavefront curve Negative inside wavefront curve and zero outside plus reference potential Negative inside wavefront curve and zero outside plus reference potential

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Setting Model Parameters Goal : Find rules for propagation of the activation wavefront Goal : Find rules for propagation of the activation wavefront Study of the data Study of the data Dog heart in a tank simulating human torso Dog heart in a tank simulating human torso 771 nodes on the torso, 490 nodes on the heart 771 nodes on the torso, 490 nodes on the heart 6 beats paced on the left ventricle & 6 beats paced on the right ventricle 6 beats paced on the left ventricle & 6 beats paced on the right ventricle

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Filtering the residual (Extended Kalman Filter) Error in the potential model is large Error in the potential model is large This error is low spatial frequency This error is low spatial frequency Thus we filtered the low frequency components in the residual error Thus we filtered the low frequency components in the residual error contains column k+1 to N of U

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Implementation Spherical coordinate ( , ) to represent the curve. Spherical coordinate ( , ) to represent the curve. B-spline used to define a continuous wavefront curve. B-spline used to define a continuous wavefront curve. Distance from the wavefront approximated as the shortest arc from a point to the wavefront curve. Distance from the wavefront approximated as the shortest arc from a point to the wavefront curve. Torso potentials simulated using the true data in the forward model plus white Gaussian noise (SNR=30dB) Torso potentials simulated using the true data in the forward model plus white Gaussian noise (SNR=30dB) Filtering : k=3 Filtering : k=3 Extended Kalman Filtering Extended Kalman Filtering

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Red : wavefront from true potential White: wavefront from Tikhonov solution Blue: wavefront reconstructed by WBCR Sample Results

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Study Observations Higher speed along the fibers than across the fibers in the early time of activation. Higher speed along the fibers than across the fibers in the early time of activation. Speed increases over time. Speed increases over time. Wavefront speed invariably increases in specific regions of the surface. Wavefront speed invariably increases in specific regions of the surface. Overshoots and undershoots in the transition area Overshoots and undershoots in the transition area Weak correlation between the slope of the potential and fiber direction in transition region. Weak correlation between the slope of the potential and fiber direction in transition region.

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Wavefront-based Potential Reconstruction Approach (WBPR) Previous reports were mostly focused on designing R, leaving We focus on approximation of an initial solution while R is identity Tikhonov (Twomey) solution:

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: Potential estimate of node i at time instant k : Distance of the node i from the wavefront : Negative value of the activated region : reference potential : -1 inside the wavefront curve, 1 outside the wavefront curve Potentials from wavefront V ref wavefront curve: boundary of the activated region on the heart surface

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Wavefront from thresholding previous time step solution Initial solution from wavefront curve Step 1: Step 2: Step 3: WBPR Algorithm

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Simulation study 490 lead sock data (Real measurements of dog heart in a tank simulating a human torso) 490 lead sock data (Real measurements of dog heart in a tank simulating a human torso) Forward matrix : 771 by 490 Forward matrix : 771 by 490 Measurements are simulated and white Gaussian noise was added (SNR=30dB) Measurements are simulated and white Gaussian noise was added (SNR=30dB) The initial wavefront curve: a circle around the pacing site with radius of 2cm The initial wavefront curve: a circle around the pacing site with radius of 2cm

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Original Forward Backward Tikhonov t=10ms t=20ms t=30ms t=40ms Results: WBPR with epicardially paced beat

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Original Forward Backward Tikhonov t=50ms t=60ms t=70ms Results: WBPR with epicardially paced beat

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OriginalForwardBackwardTikhonov t=1ms t=4ms t=10ms t=15ms Results: WBPR with supraventricularly paced beat

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Conclusions High complexity is possible and sometimes even useful High complexity is possible and sometimes even useful WBCR approach reconstructed better activation wavefronts than Tikhonov, especially at early time instants after initial activation WBCR approach reconstructed better activation wavefronts than Tikhonov, especially at early time instants after initial activation WBPR approach reconstructed considerably better epicardial potentials than Tikhonov WBPR approach reconstructed considerably better epicardial potentials than Tikhonov Using everyone’s brain is always best Using everyone’s brain is always best

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Future Plans Employ more sophisticated temporal constraints Employ more sophisticated temporal constraints Investigate the sensitivity of the inverse solution with respect to the parameters of the initial solution Investigate the sensitivity of the inverse solution with respect to the parameters of the initial solution Use real torso measurements to take the forward model error into account Use real torso measurements to take the forward model error into account Investigate certain conditions such as ischemia (the height of the wavefront changes on the heart) Investigate certain conditions such as ischemia (the height of the wavefront changes on the heart) Compare with other spatial-temporal methods Compare with other spatial-temporal methods

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