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 paper, a Kalman filter reformulation incorporated temporal information to regularize the inverse problem, and was applied to reconstruct left ventricular endocardial electrograms based on cavitary electrograms measured by a noncontact, multielectrode probe. These results were validated against in situ electrograms measured with 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 spatial regularization as well as the Twomey temporal regularization method. 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 time 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 regularization based on the Duncan and Horn formulation of the Kalman filter improves the solution of the inverse problem of electrocardiography.