Solving the Inverse Problem of Electrocardiography Using a Duncan and Horn Formulation of the Kalman Filter

Numeric regularization methods most often used to solve the ill-posed inverse problem of electrocardiography are spatial and ignore the temporal nature of the problem. In this study, a reformulation of the Kalman filter was used to incorporate temporal information in regularizing the inverse problem. The inverse problem was solved in the left ventricle by reconstructing endocardial surface electrograms based on cavitary electrograms measured by a noncontact, multielectrode probe. The results were validated based on electrograms measured in situ at the same endocardial locations using an integrated, multielectrode basket-catheter. A three-dimensional, probe-endocardium model was determined from multiplane fluoroscopic images. The boundary element method was applied to solve the boundary value problem and determine a linear relationship between endocardial and probe potentials. The Duncan and Horn formulation of the Kalman filter was employed and was compared to the commonly used zero- and first-order Tikhonov regularization as well as Twomey regularization. Endocardial electrograms were reconstructed during both sinus and paced rhythms. The Paige and Saunders solution of the Duncan and Horn formulation reconstructed endocardial electrograms at an amplitude relative error of 13% (potential amplitude) which was superior to solutions obtained with zero-order Tikhonov (relative error, 31%), first-order Tikhonov (relative error, 19%), and Twomey regularization (relative error, 44%). Likewise, activation error in the inverse solution using the Duncan and Horn formulation (2.9 ms) was smaller than that of zero-order Tikhonov (4.8 ms), first-order Tikhonov (5.4 ms), and Twomey regularization (5.8 ms). Therefore, temporal egularization based on the Duncan and Horn formulation of the Kalman filter improves the solution of the inverse problem of electrocardiography