Skip to main content
SpringerLink
Log in

The inverse problem of bioelectricity: an evaluation

  • Review Article
  • Published:
Medical & Biological Engineering & Computing Aims and scope Submit manuscript

Abstract

This invited paper presents a personal view on the current status of the solution to the inverse problem of bioelectricity. Its focus lies on applications in the field of electrocardiography. The topic discussed is also relevant in other medical domains, such as electroencephalography, electroneurography and electromyography. In such domains the methodology involved rests on the same basic principles of physics and electrophysiology as well as on the applied techniques of signal analysis and numerical analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barnard ACL, Merrill AJ, Holt JH, Kramer JO (1970) Progress in the evaluation of multiple dipole electrocardiography in a clinical environment. In: Proceedings of the XIth international vectorcardiography symposium. North-Holland, Amsterdam

  2. Barr RC, Ramsey M, Spach MS (1977) Relating epicardial to body surface potentials by means of transfer coefficients based on geometry measurements. IEEE Trans Biomed Eng BME-24:1–11

    Article  Google Scholar 

  3. Beck JV, Arnold KJ (1977) Parameter estimation in engineering and science. Wiley, New York

  4. Berger T, Fischer G, Pfeifer B, Modre R, Hanser F, Trieb T, Roithinger FX, Stuehlinger M, Pachinger O, Tilg B, Hintringer F (2006) Single-beat noninvasive imaging of cardiac electrophysiology of ventricular pre-excitation. J Am Coll Cardiol 48:2045–2052

    Google Scholar 

  5. Brody DA (1956) A theoretical analysis of intracavitary blood mass influence on the heart–lead relationship. Circ Res IV:731–738

    Google Scholar 

  6. Burger HC, Milaan JBv (1946) Heart vector and leads. Br Heart J 8:157–161

    Article  Google Scholar 

  7. Cadzow JA (1990) Signal processing via least squares error modeling. IEEE ASSP Mag, pp 12–31

  8. Cheng LK, Bodley JM, Pullan AJ (2001) Comparison of potential- and activation-based formulations for the inverse problem of electrocardiology. IEEE Trans Biomed Eng 50:11–22

    Article  Google Scholar 

  9. Colli-Franzone PC, Guerri L, Taccardi B, Viganotti C (1979) The direct and inverse potential problems in electrocardiology. Numerical aspects of some regularization methods and application to data collected in dog heart experiments. I.A.N.-C.N.R., Pavia

  10. Cuppen JJM, van Oosterom A (1984) Model studies with the inversely calculated isochrones of ventricular depolarization. IEEE Trans Biomed Eng BME-31:652–659

    Article  Google Scholar 

  11. de Munck JC (1992) A linear discretization of the volume conductor boundary integral equation using analytically integrated elements. IEEE Trans Biomed Eng BME-39:986–990

    Article  Google Scholar 

  12. Duchêne C, Lemay M, Vesin J-M, van Oosterom A (2009) Estimation of atrial multiple reentrant circuits from surface ECG signals based on a vectorcardiographic approach. In: Ayache N, Delinggett H, Sermesant M (eds) Functional modelling of the heart. Springer, Nice, pp 277–284

  13. Durrer D, van Dam RT, Freud GE, Janse MJ, Meijler FL, Arzbaecher RC (1970) Total excitation of the isolated human heart. Circulation 41:899–912

    Article  PubMed  CAS  Google Scholar 

  14. Einthoven W, Fahr G, de Waart A (1913) Űber die Richtung und die manifeste Grősse der Potential Schwankungen im menschlichen Herzen und űber den Einfluss der Herzlage auf die Form des Elektrokardiogramms. Pflugers Arch 150:275–315 (translated: Am Heart J, 1950;40:163–211)

    Google Scholar 

  15. Forsythe GE, Malcolm MA, Moler CB (1977) Computer methods for mathematical computations. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  16. Frank E (1956) An accurate, clinically practical system for spatial vectorcardiography. Circulation 13:737–749

    Article  PubMed  CAS  Google Scholar 

  17. Gabor D, Nelson CV (1954) The determination of the resultant dipole of the heart from measurements on the body surface. J Appl Phys 25:413–416

    Article  Google Scholar 

  18. Geselowitz DB (1960) Multipole representation for an equivalent cardiac generator. Proc IRE 48:75–79

    Google Scholar 

  19. Geselowitz DB (1989) On the theory of the electrocardiogram. Proc IEEE 77/6:857–876

    Article  Google Scholar 

  20. Geselowitz DB (1992) Description of cardiac sources in anisotropic cardiac muscle: application of the bidomain model. J Electrocardiogr S25:65–67

    Article  Google Scholar 

  21. Ghanem RN, Jia P, Ramanathan C, Ryu K, Markowitz A, Rudy Y (2005) Noninvasive electrocardiographic imaging (ECGI): comparison to intraoperative mapping in patients. Heart Rhythm 2:339–354

    Article  PubMed  Google Scholar 

  22. Gill PE, Murray W, Wright MH (1981) Practical optimization. Academic Press, New York

    Google Scholar 

  23. Golub GH, Pereyra V (1973) The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate. SIAM J Numer Anal 10:413–432

    Article  Google Scholar 

  24. Golub GH, van Loan CF (1996) Matrix computations, 3rd edn. The Johns Hopkins University Press, Baltimore

    Google Scholar 

  25. Golub GH, Heath M, Wahba G (1979) Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21:215–223

    Article  Google Scholar 

  26. Greensite F (1992) A new method for regularization of the inverse problem of electrocardiography. Math Biosci 111:131–154

    Article  PubMed  CAS  Google Scholar 

  27. Gulrajani RM, Savard P, Roberge FA (1988) The inverse problem in electrocardiography: solution in terms of equivalent sources. CRC Crit Rev Biomed Eng 16:171–214

    CAS  Google Scholar 

  28. Gulrajani RM, Roberge FA, Savard P (1989) The inverse problem of electrocardiography. Compr Electrocardiol I:237–288

    Google Scholar 

  29. 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

    Article  Google Scholar 

  30. He B, Li G, Zhang X (2003) Noninvasive imaging of cardiac transmembrane potentials within three-dimensional myocardium by means of a realistic geometry anisotropic heart model. IEEE Trans Biomed Eng BME-50:1190–2003

    Google Scholar 

  31. Hoekema R, Uijen GJH, van Oosterom A (1999) The number of independent signals in body surface maps. Methods Inf Med 38/2:119–124

    Google Scholar 

  32. Huiskamp GJM (1991) Difference formulas for the surface Laplacian on a triangulated surface. J Comput Phys 95:477–496

    Article  Google Scholar 

  33. Huiskamp GJM, Greensite F (1997) A new method for myocardial activation imaging. IEEE Trans Biomed Eng 44:433–446

    Article  PubMed  CAS  Google Scholar 

  34. Huiskamp GJM, van Oosterom A (1989) The depolarization sequence of the human heart surface computed from measured body surface potentials. IEEE Trans Biomed Eng 35:1047–1058

    Article  Google Scholar 

  35. Khoury DS, Taccardi B, Lux RL, Ershler PR, Rudy Y (1995) Reconstruction of endocardial potentials and activation sequences from intracavitary probe measurements: localization of pacing sites and effects of myocardial structure. Circulation 91:845–863

    Article  PubMed  CAS  Google Scholar 

  36. Lawson CL, Hanson RJ (1974) Solving least squares problems. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  37. Macchi E, de Cola G, Marmiroli D, Musso E, Nicoli F, Stilli D, Taccardi B (1983) A solution of the inverse problem in terms of single or multiple dipoles in an in-homogeneous torso model. In: Yamada K, Harumi K, Musha T (eds) Advances in body surface potential mapping. University of Nagoya Press, Nagoya, pp 11–19

  38. Macfarlane PW, van Oosterom A, Pahlm O, Kligfield P, Janse MC, Camm J (2012) Basic electrocardiology. Springer, London

    Book  Google Scholar 

  39. MacLeod R, Buist M (2012) The forward problem of electrocardiography. In: Macfarlane PW, van Oosterom A, Pahlm O, Kligfield P, Janse MC, Camm J (eds) Basic electrocardiology. Springer, London, p 480

    Google Scholar 

  40. Marquardt DW (1963) An algorithm for least-squares estimation of non-linear parameters. J Soc Ind Appl Math 11:431–441

    Article  Google Scholar 

  41. Martin RO (1970) Inverse electrocardiography. PhD thesis. Duke University, Duke

  42. Mosher J, Lewis P, Leahy R (1992) Multiple dipole modeling and localization from spatio-temporal MEG data. IEEE Eng Med Biol BME 39:541–557

    Google Scholar 

  43. Oostendorp TF, van Oosterom A (1989) Source parameter estimation in inhomogeneous volume conductors of arbitrary shape. IEEE Trans Biomed Eng BME-36:382–391

    Article  Google Scholar 

  44. Oostendorp TF, van Oosterom A (1993) Decoupling linear and non-linear parameters in biolectric source estimation. Presented at proceedings of the annual international conference of the IEEE Engineering in Medicine and Biology Society, New York

  45. Oster HS, Rudy Y (1992) The use of temporal information in the regularization of the inverse problem of electrocardiography. IEEE Trans Biomed Eng BME 39:65–75

    Article  Google Scholar 

  46. Oster HS, Taccardi B, Lux RL, Ershler PR, Rudy Y (1997) Noninvasive electrocardiographic imaging: reconstruction of epicardial potentials, electrograms, and isochrones and localization of single and multiple electrocardiac events. Circulation 96:1012–1024

    Article  PubMed  CAS  Google Scholar 

  47. Panofski WKH, Phillips M (1962) Classical electricity and magnetism. Addison-Wesley, London

    Google Scholar 

  48. Plonsey R, Barr RC (2011) Bioelectricity: a quantitative approach. Springer, New York

    Google Scholar 

  49. Plonsey R, Heppner D (1967) Considerations of quasi-stationarity in electrophysiological systems. Bull Math Biophys 29:657–664

    Article  PubMed  CAS  Google Scholar 

  50. 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

    Article  PubMed  CAS  Google Scholar 

  51. Rudy Y, Burns JE (1999) Noninvasive electrocardiographic imaging. Ann Noninvasive Electrocardiol 4:340–359

    Article  Google Scholar 

  52. Rudy Y, Messinger-Rapport BJ (1988) The inverse problem in electrocardiology: solutions in terms of epicardial potentials. CRC Crit Rev Biomed Eng 16:215–268

    CAS  Google Scholar 

  53. Salu Y (1978) Relating the multipole moments of the heart to activated parts of the epicardium and endocardium. Ann Biomed Eng 6:492–505

    Article  PubMed  CAS  Google Scholar 

  54. Scher AM, Young AC (1957) Ventricular depolarization and the genesis of the QRS. Ann NY Acad Sci 65:768–778

    Article  PubMed  CAS  Google Scholar 

  55. Serinağaoğlu Y, MacLeod RS, Yilmaz B, Brooks DH (2002) Multielectrode venous catheter mapping as a high quality constraint for electrocardiographic solution. J Electrocardiol 35S:65–73

    Google Scholar 

  56. Snellen HA (1977) Selected papers on electrocardiography of Willem Einthoven. Leiden University Press, Leiden

    Book  Google Scholar 

  57. Taccardi B (1963) Distribution of heart potentials on the thoracic surface of normal human subjects. Circ Res 12:341–352

    Google Scholar 

  58. van Bladel J (1964) Electromagnetic fields in a spherical cavity embedded in a dissipative medium. IEEE Trans AP 12:110–118

    Google Scholar 

  59. van Bladel J (1964) Electromagnetic fields. McGraw-Hill, New York

    Google Scholar 

  60. van Dam PM, van Oosterom A (2003) Atrial excitation assuming uniform propagation. J Cardiovasc Electrophysiol 14:S166–S171

    Article  PubMed  Google Scholar 

  61. van Dam PM, Oostendorp TF, Linnenbank AC, van Oosterom A (2009) Non-invasive imaging of cardiac activation and recovery. Ann Biomed Eng 37:1739–1756

    Article  PubMed  Google Scholar 

  62. van Oosterom A (1999) The use of the spatial covariance in computing pericardial potentials. IEEE Trans Biomed Eng BME-46/7:778–787

    Article  Google Scholar 

  63. van Oosterom A (2001) Genesis of the T wave as based on an equivalent surface source model. J Electrocardiogr 34:217–227

    Article  Google Scholar 

  64. van Oosterom A (2002) Solidifying the solid angle. J Electrocardiol 35S:181–192

    Article  Google Scholar 

  65. van Oosterom A (2010) Vectorcardiography based analysis of atrial fibrillation. In: Sobieszczanska M, Jagielski J, Macfarlane PW (eds) Cardiology 2009. JAKS Publishing Company, Wroclaw, pp 39–55

    Google Scholar 

  66. van Oosterom A (2012) The equivalent double layer; source models for repolarization. In: Macfarlane PW, van Oosterom A, Pahlm O, Kligfield P, Janse MC, Camm J (eds) Basic electrocardiology. Springer, London, pp 227–246

    Chapter  Google Scholar 

  67. van Oosterom A (2012) Closed-form analytical expressions for the potential fields generated by triangular monolayers with linearly distributed source strength. Med Biol Eng Comput 55(1):1–9

    Article  Google Scholar 

  68. van Oosterom A, Huiskamp GJM (1989) The effect of torso inhomogeneities on body surface potentials. J Electrocardiol 22/1:1–20

    Google Scholar 

  69. van Oosterom A, Jacquemet V (2005) A parameterized description of transmembrane potentials used in forward and inverse procedures. In: Electrocardiology’05, vol 12. Folia Cardiologica, Gdansk, pp 111–113

  70. van Oosterom A, Jacquemet V (2005) Genesis of the P wave: atrial signals as generated by the equivalent double layer source model. Europace 7:S21–S29

    Article  Google Scholar 

  71. van Oosterom A, Oostendorp TF (1992) On computing pericardial potentials and current densities. J Electrocardiol 25(Suppl):102–106

    Article  PubMed  Google Scholar 

  72. van Oosterom A, Oostendorp TF (1993) On computing pericardial potentials and current densities in inverse electrocardiography. J Electrocardiol 25S:102–106

    Google Scholar 

  73. van Oosterom A, Oostendorp TF (2004) ECGSIM: an interactive tool for studying the genesis of QRST waveforms. Heart 90:165–168

    Article  PubMed  Google Scholar 

  74. van Oosterom A, Oostendorp TF (2010) Cardiac simulation for education: the electrocardiogram according to ECGSIM. In: Pahlm O, Wagner GS (eds) Cardiovascular multimodal image guided diagnosis and therapy. McGraw Hill, New York, pp 263–280

    Google Scholar 

  75. Walsh GR (1975) Methods of optimization. Wiley, London

    Google Scholar 

  76. Wang L, Zhang H, Wong KCL, Liu H, Shi P (2010) Physiological-model-constrained noninvasive reconstruction of volumetric myocardial transmembrane potentials. IEEE Trans Biomed Eng BME-57:296–315

    Article  Google Scholar 

  77. Wilson FN, Macleod AG, Barker PS (1933) The distribution of action currents produced by the heart muscle and other excitable tissues immersed in conducting media. J Gen Physiol 16:423–456

    Article  PubMed  CAS  Google Scholar 

  78. Zhang X, Ramachandra I, Liu Z, Muneer B, Pogwizd SM, He B (2005) Noninvasive three-dimensional electrocardiographic imaging of ventricular activation sequence. Am J Physiol Heart Circ Physiol 289:H2724–H2732

    Article  PubMed  CAS  Google Scholar 

Download references

Acknowledgments

The author gratefully acknowledges the contribution to this contribution of the two anonymous reviewers for their close proof reading and their most valuable suggestions for rephrasing and extending some of the topics addressed.

Author information

Authors and Affiliations

  1. Professor Emeritus, Radboud University Nijmegen, Nijmegen, The Netherlands

    Adriaan van Oosterom

Authors
  1. Adriaan van Oosterom
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adriaan van Oosterom.

Rights and permissions

Reprints and permissions

About this article

Cite this article

van Oosterom, A. The inverse problem of bioelectricity: an evaluation. Med Biol Eng Comput 50, 891–902 (2012). https://doi.org/10.1007/s11517-012-0941-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11517-012-0941-5

Keywords

Search

Navigation