Abstract
In this article, we aimed to reduce the effects of geometric errors and measurement noise on the inverse problem of Electrocardiography (ECG) solutions. We used the Kalman filter to solve the inverse problem in terms of epicardial potential distributions. The geometric errors were introduced into the problem via wrong determination of the size and location of the heart in simulations. An error model, which is called the enhanced error model (EEM), was modified to be used in inverse problem of ECG to compensate for the geometric errors. In this model, the geometric errors are modeled as additive Gaussian noise and their noise variance is added to the measurement noise variance. The Kalman filter method includes a process noise component, whose variance should also be estimated along with the measurement noise. To estimate these two noise variances, two different algorithms were used: (1) an algorithm based on residuals, (2) expectation maximization algorithm. The results showed that it is important to use the correct noise variances to obtain accurate results. The geometric errors, if ignored in the inverse solution procedure, yielded incorrect epicardial potential distributions. However, even with a noise model as simple as the EEM, the solutions could be significantly improved.
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References
Aydin U, Serinagaoglu Y (2008) Use of activation time based Kalman filtering in inverse problem of ECG. In: Sloten JV, Verdonck P, Nyssen M, Haueisen J (eds) Proceedings of 4th European Conference International Federation for Medical and Biological Engineering, vol 22. Antwerp, IFMBE, pp 1200–1203
Aydin U, Serinagaoglu Y (2009) Statistical modeling of the geometric error in cardiac electrical imaging. In: Proceedings of The Sixth IEEE International Symposium on Biomedical Imaging (ISBI 2009), Boston, pp 442–445
Berrier KL, Sorensen DC, Khoury DS (2004) Solving the inverse problem of electrocardiography using a Duncan and Horn formulation of the Kalman filter. IEEE Trans Biomed Eng 51:507–515
Brooks DH, Ahmad GF, Macleod RS, Maratos GM (1999) Inverse electrocardiography by simultaneous imposition of multiple constraints. IEEE Trans Biomed Eng 46(1):3–18
Cheng LK, Bradley CP, Pullan AJ (2003) Effects of experimental and modeling errors on electrocardiographic inverse formulations. IEEE Trans Biomed Eng 50:23–32
El-Jakl J, Champagnat F, Goussard Y (1995) Time-space regularization of the inverse problem of electrocardiography. In: Proceedings of 17th Annual International Conference IEEE EMBS, Montreal, pp 213–214
Ghanem RN, Ramanathan C, Jia P, Rudy Y (2003) Heart-surface reconstruction and ecg electrodes localization using fluoroscopy, epipolar geometry and stereovision: application to noninvasive imaging of cardiac electrical activity. IEEE Trans Med Imaging 22(10):1307–1318
Ghodrati A, Brooks DH, Tadmor G, MacLeod RS (2006) Wavefront-based models for inverse electrocardiography. IEEE Trans Biomed Eng 53(9):1821–1831
Greensite F (2003) The temporal prior in bioelectromagnetic source imaging problems. IEEE Trans Biomed Eng 50(10):1152–1159
Gulrajani RM (1998) The forward and inverse problems of electrocardiography. IEEE Eng Med Bio 17:84–101
Hansen PC, OLeary DP (1993) The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J Sci Comput 14:1487–1503
He B, Wu D (2001) Imaging and visualization of 3-D cardiac electrical activity. IEEE Trans Inf Technol Biomed 5(3):181–186
He B, Li G, Zhang X (2002) Noninvasive imaging of cardiac transmembrane potentials within three-dimensional myocardium by means of a realistic geometry anisotropic heart model. Phys Med Biol 47:4063–4078
Heino J, Somersalo E, Kaipio JP (2005) Compensation for geometric mismodelling by anisotropies in optical tomography. Opt Express 13(1):296–308
Hoekema R, Uijen GJH, van Erning L, van Oosterom A (1999) Interindividual variability of multilead electrocardiographic recordings: influence of heart position. J Electrocardiol 32(2):137–148
Huiskamp G, van Oosterom A (1989) Tailored versus realistic geometry in the inverse problem of electrocardiography. IEEE Trans Biomed Eng 36(8):827–835
Jazwinski AH, Bailie AE (1967) Adaptive filtering interim report, Report no. 67-6 Contract NAS 5-9085
Jazwinski AH (2007) Stochastic processes and filtering theory, 2nd edn. Dover Publications, Mineola
Jiang M, Xia L, Shou G, Tang M (2007) Combination of the LSQR method and a genetic algorithm for solving the electrocardiography inverse problem. Phys Med Biol 52:1277–1294
Jiang Y, Farina D, Doessel O (2008) Effect of heart motion on the solutions of forward and inverse electrocardiographic problem - a simulation study. In: Murray A (ed) Proceedings of 35th Annual International Conference on Computers in Cardiology, vol 35. Bologna, pp 365–368
Jiang M, Xia L, Shou G, Wei Q, Liu F, Crozier S (2009) Effect of cardiac motion on solution of the electrocardiography inverse problem. IEEE Trans Biomed Eng 56(4):923–931
Johnson CR, MacLeod RS (1994) Nonuniform spatial mesh adaptation using a posteriori error estimates: applications to forward and inverse problems. Appl Numer Math 14:311–326
Johnston PR, Gulrajani RM (1997) A new method for regularization parameter determination in the inverse problem of electrocardiography. IEEE Trans Biomed Eng 44(1):19–39
Joly D, Goussard Y, Savard P (1993) Time-recursive solution to the inverse problem of electrocardiography a model-based approach. In: Proceedings of the 15th Annual International Conference on IEEE EMBC and CMBEC, vol 15. New Jersey, pp 767–768
Cheng LK, Sands GB, French RL, Withy SJ, Wong SP, Legget ME, Smith WM, Pullan AJ (2005) Rapid construction of a patient-specific torso model from 3D ultrasound for non-invasive imaging of cardiac electrophysiology. Med Biol Eng Comput 46(3):325–330
Kaipio JP, Somersalo E (2004) Computational and statistical methods for inverse problems applied mathematical sciences. Springer-Verlag, New York
Macleod RS, Lux RL, Tacardi B (1997) A possible mechanism for electrocardiographically silent changes in cardiac repolarization. J Electrocardiol 30:114–121
Macleod RS, Ni O, Punske B, Ershler PR, Yilmaz B, Tacardi B (2000) Effects of heart position on the body-surface electrocardiogram. J Electrocardiol 33:229–237
Macleod RS, Johnson CR (1997) Map3d: interactive scientific visualization for bioengineering data. In: Proceedings of EMBS 15th Annual International Conference, vol 44, pp 196–208
Messinger-Rapport BJ, Rudy Y (1986) The inverse problem in electrocardiography: a model study of the effects of geometry and conductivity parameters on the reconstruction of epicardial potentials. IEEE Trans Biomed Eng 33(7):667–676
Okawa S, Honda S (2005) Reduction of noise from magnetoencephalography. Med Biol Eng Comput 43:630–637
Pullan AJ, Bradley CP (1996) A coupled cubic hermite finite element/boundary element procedure for electrocardiographic problems. Comput Mech 18(5):356–368
Pullan AJ, Paterson D, Greensite F (2001) Noninvasive imaging of cardiac electrophysiology. Phil Trans R Soc Lond A 359:1277–1286
Ramanathan C, Rudy Y (2001) Electrocardiographic imaging: I. effect of torso inhomogeneities on body surface electrocardiographic potentials. J Cardiovasc Electrophysiol 12:229–240
Ramanathan C, Rudy Y (2001) Electrocardiographic imaging: II. effect of torso inhomogeneities on noninvasive reconstruction of epicardial potentials, electrograms, and isochrones. J Cardiovasc Electrophysiol 12:241–252
Serinagaoglu Y, Brooks DH, MacLeod RS (2006) Improved performance of Bayesian solutions for inverse electrocardiography using multiple information sources. IEEE Trans Biomed Eng 53(10):2024–2034
Shou G, Xia L, Jiang M, Wei Q, Liu F, Crozier S (2008) Truncated total least squares: a new regularization method for the solution of ECG inverse problems. IEEE Trans Biomed Eng 55(4):1327–1335
Shumway RH, Stoffer DS (2006) Time series analysis and its applications: with R examples, 2nd edn. Springer, New York, pp 330–345
Stanley PC, Pilkington TC, Morrow MN (1986) The effects of thoracic inhomogeneities on the relationship between epicardial and torso potentials. IEEE Trans Biomed Eng 33(3):273–284
Throne RD, Olson LG (1995) The effects of errors in assumed conductivities and geometry on numerical solutions to the inverse problem of electrocardiology. IEEE Trans Biomed Eng 42(12):1192–1200
Tikhonov AN, Arsenin VY (1977) Solution of ill-posed problems. Winston and Sons, Washington
Tsui CSL, Gan JQ, Roberts SJ (2009) A self-paced brain-computer interface for controlling a robot simulator: an online event labeling paradigm and an extended Kalman filter based algorithm for online training. Med Biol Eng Comput 47:257–265
van Oosterom A (1999) The use of spatial covariance in computing pericardial potentials. IEEE Trans Biomed Eng 46(7):778–787
Acknowledgments
This study was supported by The Scientific and Technological Research Council of Turkey, grant number 105E070. The authors would like to thank Dr. Robert S. Macleod from University of Utah University of Utah, Nora Eccles Harrison Cardiovascular Research and Training Institute for the data used in this study. This study was made possible in part by software (Map3d) from the NIH/NCRR Center for Integrative Biomedical Computing, P41-RR12553-10.
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Aydin, U., Dogrusoz, Y.S. A Kalman filter-based approach to reduce the effects of geometric errors and the measurement noise in the inverse ECG problem. Med Biol Eng Comput 49, 1003–1013 (2011). https://doi.org/10.1007/s11517-011-0757-8
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DOI: https://doi.org/10.1007/s11517-011-0757-8