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Spatial-dependent regularization to solve the inverse problem in electromyometrial imaging

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Abstract

Recently, electromyometrial imaging (EMMI) was developed to non-invasively image uterine contractions in three dimensions. EMMI collects body surface electromyography (EMG) measurements and uses patient-specific body-uterus geometry generated from magnetic resonance images to reconstruct uterine electrical activity. Currently, EMMI uses the zero-order Tikhonov method with mean composite residual and smoothing operator (CRESO) to stabilize the underlying ill-posed inverse computation. However, this method is empirical and implements a global regularization parameter over all uterine sites, which is sub-optimal for EMMI given the severe eccentricity of body-uterus geometry. To address this limitation, we developed a spatial-dependent (SP) regularization method that considers both body-uterus eccentricity and EMG noise. We used electrical signals simulated with spherical and realistic geometry models to compare the reconstruction accuracy of the SP method to those of the CRESO and the L-Curve methods. The SP method reconstructed electrograms and potential maps more accurately than the other methods, especially in cases of high eccentricity and noise contamination. Thus, the SP method should facilitate clinical use of EMMI and can be used to improve the accuracy of other electrical imaging modalities, such as Electrocardiographic Imaging.

The spatial-dependent regularization (SP) technique was designed to improve the accuracy of Electromyometrial Imaging (EMMI). The top panel shows the eccentricity of body-uterus geometry and four representative body surface electrograms. The bottom panel shows boxplots of correlation coefficients and relative errors for the electrograms reconstructed with SP and two conventional methods, the L-Curve and mean CRESO methods.

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References

  1. Aster RC, Borchers B, Thurber CH (2013) Parameter estimation and inverse problems. Academic Press

  2. Barr RC, Ramsey M, Spach MS (1977) Relating epicardial to body surface potential distributions by means of transfer coefficients based on geometry measurements. IEEE Trans Biomed Eng 1–11

  3. Behar J, Andreotti F, Zaunseder S, Li Q, Oster J, Clifford GD (2014) An ECG simulator for generating maternal-foetal activity mixtures on abdominal ECG recordings. Physiol Meas 35:1537–1550. https://doi.org/10.1088/0967-3334/35/8/1537

    Article  PubMed  Google Scholar 

  4. Burnes JE, Taccardi B, MacLeod RS, Rudy Y (2000) Noninvasive ECG imaging of electrophysiologically abnormal substrates in infarcted hearts: a model study. Circulation 101:533–540

    Article  CAS  Google Scholar 

  5. Cluitmans M, Brooks DH, MacLeod R, Dössel O, Guillem MS, van Dam PM, Svehlikova J, He B, Sapp J, Wang L, Bear L (2018) Validation and opportunities of electrocardiographic imaging: from technical achievements to clinical applications. Front Physiol 9

  6. Cluitmans MJM, Bonizzi P, Karel JMH, Das M, Kietselaer BLJH, de Jong MMJ, Prinzen FW, Peeters RLM, Westra RL, Volders PGA (2017) In vivo validation of electrocardiographic imaging. JACC Clin Electrophysiol 3:232–242

    Article  Google Scholar 

  7. Devedeux D, Marque C, Mansour S, Germain G, Duchêne J (1993) Uterine electromyography: a critical review. Am J Obstet Gynecol 169:1636–1653

    Article  CAS  Google Scholar 

  8. Engl HW, Hanke M, Neubauer A (1996) Regularization of inverse problems. Springer Science & Business Media

  9. Euliano TY, Nguyen MT, Darmanjian S et al (2013) Monitoring uterine activity during labor: a comparison of 3 methods. Am J Obstet Gynecol 208(66):e1-66–e1-e6

    Google Scholar 

  10. Franzone PC, Guerri L, Taccardi B, Viganotti C (1985) Finite element approximation of regularized solutions of the inverse potential problem of electrocardiography and applications to experimental data. Calcolo 22:91–186

    Article  Google Scholar 

  11. Garcia-Casado J, Ye-Lin Y, Prats-Boluda G et al (2018) Electrohysterography in the diagnosis of preterm birth: a review. Physiol Meas 39:02TR01

    Article  CAS  Google Scholar 

  12. Garfield RE, Maner WL (2007) Physiology and electrical activity of uterine contractions. Semin Cell Dev Biol 18:289–295. https://doi.org/10.1016/J.SEMCDB.2007.05.004

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  13. Garfield RE, Maner WL (2007) Physiology and electrical activity of uterine contractions 18:289–295

  14. Gulrajani RM (1998) The forward and inverse problems of electrocardiography. IEEE Eng Med Biol Mag 17:84–101

    Article  CAS  Google Scholar 

  15. Hansen PC (1990) The discrete Picard condition for discrete ill-posed problems. BIT Numer Math 30:658–672

    Article  Google Scholar 

  16. Hansen PC, O’Leary DP (1993) The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J Sci Comput 14:1487–1503. https://doi.org/10.1137/0914086

    Article  Google Scholar 

  17. Harper LM, Shanks AL, Tuuli MG et al (2013) The risks and benefits of internal monitors in laboring patients. Am J Obstet Gynecol 209(38):e1-38–e1-e6

    Google Scholar 

  18. Khoruv DS, Marks GF (1995) Adaptive regularization of the inverse problem in electrocardiography. In: Engineering in medicine and biology society, 1995., IEEE 17th Annual Conference. IEEE, pp 211–212

  19. Levin E, Meltzer AY (2017) Estimation of the regularization parameter in linear discrete ill-posed problems using the Picard parameter. SIAM J Sci Comput 39:A2741–A2762

    Article  Google Scholar 

  20. Lucovnik M, Maner WL, Chambliss LR, et al (2011) Noninvasive uterine electromyography for prediction of preterm delivery. YMOB 204:228.e1-228.e10. 10.1016/j.ajog.2010.09.024

  21. Luo C, Rudy Y (1991) A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction. Circ Res 68:1501–1526

    Article  CAS  Google Scholar 

  22. MacLeod R, Buist M (2010) The forward problem of electrocardiography. In: Comprehensive electrocardiology. Springer, pp 247–298

  23. MacLeod RS, Brooks DH (1998) Recent progress in inverse problems in electrocardiology. IEEE Eng Med Biol Mag 17:73–83

    Article  CAS  Google Scholar 

  24. Oster HS, Rudy Y (1997) Regional regularization of the electrocardiographic inverse problem: a model study using spherical geometry. IEEE Trans Biomed Eng 44:188–199

    Article  CAS  Google Scholar 

  25. Oster HS, Taccardi B, Lux RL, Ershler PR, Rudy Y (1997) Noninvasive electrocardiographic imaging. Circulation 96:1012–1024

    Article  CAS  Google Scholar 

  26. Rabotti C, Mischi M (2015) Propagation of electrical activity in uterine muscle during pregnancy: a review. Acta Physiol 213:406–416. https://doi.org/10.1111/apha.12424

    Article  CAS  Google Scholar 

  27. Rabotti C, Mischi M, van Laar JOEH, Oei GS, Bergmans JWM (2009) Inter-electrode delay estimators for electrohysterographic propagation analysis. Physiol Meas 30:745–761. https://doi.org/10.1088/0967-3334/30/8/002

    Article  PubMed  Google Scholar 

  28. Ramanathan C, Jia P, Ghanem R, Calvetti D, Rudy Y (2003) Noninvasive electrocardiographic imaging (ECGI): application of the generalized minimal residual (GMRes) method. Ann Biomed Eng 31:981–994

    Article  Google Scholar 

  29. Romero R, Dey SK, Fisher SJ (2014) Preterm labor: one syndrome, many causes. Science 345(80):760–765

    Article  CAS  Google Scholar 

  30. Sikora J, Matonia A, Czabanski R et al (2011) Recognition of premature threatening labour symptoms from bioelectrical uterine activity signals. Arch Perinat Med 17:97–103

    Google Scholar 

  31. Team Rs (2016) RStudio: integrated development environment for R. RStudio, Inc

  32. Wang H, Wu W, Talcott M, McKinstry RC, Woodard PK, Macones GA, Schwartz AL, Cuculich P, Cahill AG, Wang Y (2020) Accuracy of electromyometrial imaging of uterine contractions in clinical environment. Comput Biol Med 116:103543. https://doi.org/10.1016/j.compbiomed.2019.103543

    Article  PubMed  Google Scholar 

  33. Wang Y, Rudy Y (2006) Application of the method of fundamental solutions to potential-based inverse electrocardiography. Ann Biomed Eng 34:1272–1288

    Article  Google Scholar 

  34. Wray S (1993) Uterine contraction and physiological mechanisms of modulation

  35. Wu W, Wang H, Zhao P, et al (2019) Noninvasive high-resolution electromyometrial imaging of uterine contractions in a translational sheep model. Sci Transl Med 11:eaau1428. https://doi.org/10.1126/scitranslmed.aau1428

  36. Xu XG, Taranenko V, Zhang J, Shi C (2007) A boundary-representation method for designing whole-body radiation dosimetry models: pregnant females at the ends of three gestational periods—RPI-P3, -P6 and -P9. Phys Med Biol 52:7023–7044. https://doi.org/10.1088/0031-9155/52/23/017

    Article  PubMed  Google Scholar 

  37. Yochum M, Laforêt J, Marque C (2016) An electro-mechanical multiscale model of uterine pregnancy contraction. Comput Biol Med 77:182–194

    Article  Google Scholar 

  38. Zhang M, Tidwell V, La Rosa PS et al (2016) Modeling magnetomyograms of uterine contractions during pregnancy using a multiscale forward electromagnetic approach. PLoS One 11:e0152421

    Article  Google Scholar 

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Acknowledgments

We thank Dr. Deborah Frank for editing the manuscript and Zichao Wen for discussion of the mathematical representations.

Funding

This work was supported by the March of Dimes (March of Dimes Prematurity Research Center, PI Macones) and, in part, by grants from NIH/National Institute of Child Health and Human Development (RO1HD094381; PIs Wang/Cahill); the NIH/National Institute of Aging (RO1AG053548; PIs Benzinger/Wang); and the BrightFocus Foundation (A2017330S; PI Wang).

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Authors and Affiliations

  1. Department of Physics, Washington University, St. Louis, MO, 63130, USA

    Hui Wang

  2. Center for Reproductive Health Sciences, Washington University School of Medicine, St. Louis, MO, 63110, USA

    Hui Wang & Yong Wang

  3. Department of Obstetrics and Gynecology, Washington University School of Medicine, St. Louis, MO, 63110, USA

    Hui Wang & Yong Wang

  4. Department of Biomedical Engineering, Washington University, St. Louis, MO, 63130, USA

    Yong Wang

  5. Mallinckrodt Institute of Radiology, Washington University School of Medicine, St. Louis, MO, 63110, USA

    Yong Wang

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  1. Hui Wang
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Contributions

H.W. and Y.W. designed this study. H.W. developed the method and did the simulation study and data analysis. Both authors wrote and revised the paper.

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Correspondence to Yong Wang.

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ESM 1

Data file S1. CC value of optimal p and l. (XLSX 29 kb) (XLSX 29 kb)

ESM 2

Data file S2. Information on test data set of Fig. 5. (XLSX 10 kb) (XLSX 10 kb)

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(DOC 5462 kb)

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Wang, H., Wang, Y. Spatial-dependent regularization to solve the inverse problem in electromyometrial imaging. Med Biol Eng Comput 58, 1651–1665 (2020). https://doi.org/10.1007/s11517-020-02183-z

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