ENHANCING INVERSE ELECTROCARDIOGRAPHY WITH
SPARSE  NOISY MEASUREMENTS OF EPICARDIAL POTENTIALS

Yesim Serinagaoglu¹, Dana H. Brooks¹, Robert S. MacLeod^2
1CDSP Center, ECE Dept., Northeastern University, Boston, MA, USA
²CVRTI, University of Utah, Salt Lake City, UT, USA

Abstract: The inverse problem of electrocardiography is ill-posed and requires the imposition of a priori constraints. Despite recent progress, solutions to this problem have not yet achieved clinical utility. Extra measurements from catheters inserted into cardiac veins, even though quite sparse, may help increase accuracy and robustness. In this paper, we study various methods to incorporate sparse epicardial measurements in solutions to the inverse problem.

INTRODUCTION

In inverse electrocardiography, the goal is to estimate cardiac electrical activity based on measured torso surface potentials and a geometric model of the torso. Due to attenuation and spatial smoothing that occurs in the thorax, the inverse problem is ill-posed [1,2] and the forward matrix is badly conditioned, so even small disturbances in the measurements lead to amplification of error in the inverse solution. The most common approach to overcome this problem is “regularization” where the solution is a trade-off between the estimate that best represents the data, and fidelity to an a priori regularization constraint imposed on the solution.

There has been some recent interest in statistical approaches to the solution of inverse electrocardiography. In [3], some statistical a priori information about the epicardial potentials is used to estimate the inverse solution in a Bayesian Maximum-a-posteriori (MAP) setting. In [4], spatial and temporal correlations of the epicardial potentials are used to estimate the epicardial potentials jointly at all time instants. Given a reasonable statistical prior model for the epicardial potentials, these methods often produce better results than heuristic methods. However, despite some notable recent results, solutions to this problem are still not easily applied in clinical settings.

Recent progress in the fabrication of multielectrode venous catheters permits simultaneous measurements from several catheters providing direct access to epicardial cardiac veins [5]. The coverage of these catheters on the epicardium is very sparse, but the extra measurements near the heart surface may be useful to improve the accuracy and the robustness of the inverse solution, especially if they can be used in a statistical setting. The goal in this study is to develop approaches to solve the inverse problem including the extra epicardial measurements coming from the sparsely located intra-venous catheters.

METHODS

The electrocardiographic inverse problem at a particular time instant can be defined as:

y = A x + n     (1)

where y is an Mx1 vector of torso potentials, x is the Nx1 vector of epicardial potentials, A is the Mx N matrix representing the forward solution, and n is the noise in the torso measurements of the same dimensions as y. We also assume that we have noisy epicardial measurements  at some subset of N_k (N_k < N) epicardial sites. The goal is to estimate x = [x_m^T  x_u^T]^T i.e. the union of the potentials at the measured and unmeasured sites using both the torso measurements, y, and the partial epicardial measurements, . The partial epicardial potentials are modeled as the sum of x_m with a noise term:  =  x_m +  e_m.

We consider the following methods:

1.       Let x_m  = , subtract this part from the overall problem, and solve the reduced inverse problem
(y - A_m ) = A_u x_u + n for x_u using Tikhonov regularization. (SUBT-M)

2.       A hybrid approach that first uses statistical estimation based on a priori training data to estimate all epicardial potentials from the sparse measurements followed by a second step that uses the results of the first step and solves the inverse problem also including the torso measurements. Step-1: We update the method suggested in [5] to estimate missing epicardial measurements ( ) using and training datasets that include values for both measured and unmeasured sites:  =  + C_um C_mm^-1 ( - ) where  and   are the mean values of  and  respectively, C_mm is the covariance of , and C_um is the cross-covariance between  and  (LLS-epi).     Step-2: We use the results of Step-1 in a Bayesian MAP estimation setting with n ~ N(0, σ_n² I),
e_m ~ N(0, σ_em² I), and x ~ N( , C_xx) where = [^{T   T}]^T and

where C_ww = C_uu – C_um C_mm^-1 C_um^T is the covariance of estimation error in (i.e. – ), C_uu is the covariance of . Then, the MAP estimate of x is: = ( A^TA + σ_n² C_xx^-1) ^–1 (A^Ty + σ_n² C_xx^-1 ) (MAP-hybrid). 

3.       We solve the inverse problem using the Bayesian MAP estimation method suggested in [3] which assumes x ~ N( , R_xx)  where R_xx is the correlation matrix of x (MAP-CZM). Note that this method does not take advantage of the extra measurements and is included here for comparison.

RESULTS

We simulated sparse epicardial measurements from canine epicardial potentials recorded from sites on the heart surface by adding normally distributed zero mean {\em i.i.d.} noise and simulated torso potentials using a boundary element solution to Laplace's equation for a human shaped torso tank in which the heart was suspended and adding noise. We have N=490, M=771 and we set N_k=42.

We obtained the a priori information for the statistical methods from a database using a “leave-one-beat-out” protocol where we excluded the test beat from the training dataset [5,6]. In the training dataset, we included beats from the same experiment with the same pacing site as the test beat. The same training database was used for all statistical methods.

Fig. 1 shows a sample case of isosurface maps of the epicardial potentials.  We compare the estimated epicardial potentials to the original map in the Top-Left panel. The statistical methods (i.e., MAP-CZM, LLS-epi and MAP-hybrid) perform better than SUBT-M: the wavefront is better preserved in these methods. LLS-epi and MAP-hybrid create similar results, and they are better than those from MAP-CZM: the tight wavefront around 10:00-11:00 o'clock position in the original map is better preserved in LLS-epi and MAP-hybrid.  Table 1 shows relative error (RE) averaged over QRS region. On the average, MAP-hybrid has the lowest and SUBT-M has the highest RE.

Table I
Average of RE over QRS region (mean±std)

Figure 1. Isosurface maps of epicardial potentials. Top-left: Original epicardial maps, Mid-left: SUBT-M, Bottom-Left: MAP-CZM, Top-Right: LLS-epi, Mid-Right: MAP-hybrid. Display is an apical projection of the epicardial electrode array with the apex in the center and the left anterior descending artery at the 12:00 o'clock position.

DISCUSSION

In this paper, we studied various methods to include sparse epicardial measurements in the solution of inverse problem of electrocardiography. Using prior information alone (MAP-CZM) in this scenario performed  better than using sparse measurements with no prior information (SUBT-M). Including both types of information further improves the performance (MAP-hybrid). The ``leave-one-beat-out'' protocol is simple and useful to test the performance of the statistical algorithms when we have a training dataset that represents the test data well. But it is also unrealistic, as one rarely has access to beats from the same experiment in realistic clinical settings. Future work will include testing of the methods with different datasets and improving the methods to better use the measured subsets of epicardial potentials and a priori information.

Acknowledgments: This work was supported by National Center for Research Resources (NCRR) and the Whitaker Foundation. Y.S. thanks Turkish Higher Education Council for their support of her graduate studies.

REFERENCES

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[2] R. S. MacLeod and D. H. Brooks, “Recent progress in inverse problems in Electrocardiography,” IEEE Eng. in Med. and Bio. Soc. Mag., vol. 17, pp. 73-83, 1998.

[3] A. van Oosterom, “The use of spatial covariance in computing pericardial potentials,” IEEE Trans. on Biomed. Eng., vol. 46, no. 7, pp. 778-787, 1999.

[4] F. Greensite, “Myocardial activation imaging,” in Computational Inverse Problems in Electrocardiography (P. R. Johnston, ed.), pp. 143-190, WITpress, 2001.

[5] R. O. Kuenzler, et. al., “Estimation of epicardial activation maps from intravascular recordings,” Journal of Electrocardiology, vol. 32, pp. 77--92, 1999.

[6] R. S. MacLeod, et. al., “Direct and inverse methods for cardiac mapping using multielectrode catheter measurements,” in Biomedizinische Technik, NFSI, 2001.