Mathematical Methods of Optical Coherence Tomography


In this chapter a general mathematical model of Optical Coherence Tomography (OCT) is presented on the basis of the electromagnetic theory. OCT produces high-resolution images of the inner structure of biological tissues. Images are obtained by measuring the time delay and the intensity of the backscattered light from the sample considering also the coherence properties of light. The scattering problem is considered for a weakly scattering medium located far enough from the detector. The inverse problem is to reconstruct the susceptibility of the medium given the measurements for different positions of the mirror. Different approaches are addressed depending on the different assumptions made about the optical properties of the sample. This procedure is applied to a full field OCT system and an extension to standard (time and frequency domain) OCT is briefly presented.



The authors would like to thank Wolfgang Drexler and Boris Hermann from the Medical University Vienna for their valuable comments and stimulating discussions. This work has been supported by the Austrian Science Fund (FWF) within the national research network Photoacoustic Imaging in Biology and Medicine, projects S10501-N20 and S10505-N20.


  1. 1.
    Ammari, H., Bao, G.: Analysis of the scattering map of a linearized inverse medium problem for electromagnetic waves. Inverse Prob. 17, 219–234 (2001)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Andersen, P.E., Thrane, L., Yura, H.T., Tycho, A., Jørgensen, T.M., Frosz, M.H.: Advanced modelling of optical coherence tomography systems. Phys. Med. Biol. 49, 1307–1327 (2004)CrossRefGoogle Scholar
  3. 3.
    Born, M., Wolf, E.: Principles of Optics. 7th Edn. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  4. 4.
    Bouma, B.E., Tearney, G.J.: Handbook of Optical Coherence Tomography. Marcel Dekker Inc., New York (2002)Google Scholar
  5. 5.
    Brezinski, M.E.: Optical Coherence Tomography Principles and Applications. Academic Press, New York (2006)Google Scholar
  6. 6.
    Brodsky, A., Thurber, S.R., Burgess, L.W.: Low-coherence interferometry in random media. i. theory. J. Opt. Soc. Am. A 17(11), 2024–2033 (2000)Google Scholar
  7. 7.
    Bruno, O., Chaubell, J.: One-dimensional inverse scattering problem for optical coherence tomography. Inverse Prob. 21, 499–524 (2005)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. 2nd edn, In: Applied Mathematical Sciences, vol. 93. Springer, Berlin (1998)Google Scholar
  9. 9.
    Dolin, L.S.: A theory of optical coherence tomography. Radiophys. Quantum Electron. 41(10), 850–873 (1998)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Drexler, W., Fujimoto, J.G.: Optical Coherence Tomography. Springer, Berlin (2008)CrossRefGoogle Scholar
  11. 11.
    Duan, L., Makita, S., Yamanari, M., Lim, Y., Yasuno, Y.: Monte-carlo-based phase retardation estimator for polarization sensitive optical coherence tomography. Opt. Express 19, 16330–16345 (2011)CrossRefGoogle Scholar
  12. 12.
    Feng, Y., Wang, R.K., Elder, J.B.: Theoretical model of optical coherence tomography for system optimization and characterization. J. Opt. Soc. Am. A 20(9), 1792–1803 (2003)CrossRefGoogle Scholar
  13. 13.
    Fercher, A.F.: Optical coherence tomography. J. Biomed. Opt. 1(2), 157–173 (1996)CrossRefGoogle Scholar
  14. 14.
    Fercher, A.F.: Optical coherence tomography - development, principles, applications. Z. Med. Phys. 20, 251–276 (2010)CrossRefGoogle Scholar
  15. 15.
    Fercher, A.F., Hitzenberger, C.K.: Optical Coherence Tomography. In: Progress in Optics. Elsevier Science B. V., Amsterdam (2002)Google Scholar
  16. 16.
    Fercher, A.F., Drexler, W., Hitzenberger, C.K., Lasser, T.: Optical coherence tomography - principles and applications. Rep. Prog. Phys. 66(2), 239–303 (2003)CrossRefGoogle Scholar
  17. 17.
    Fercher, A.F., Hitzenberger, C.K., Drexler, W., Kamp, G., Sattmann, H.: In vivo optical coherence tomography. Am. J. Ophthalmol. 116, 113–114 (1993)CrossRefGoogle Scholar
  18. 18.
    Fercher, A.F., Hitzenberger, C.K., Kamp, G., El Zaiat, S.Y.: Measurement of intraocular distances by backscattering spectral interferometry. Opt. Commun. 117, 43–48 (1995)CrossRefGoogle Scholar
  19. 19.
    Fercher, A.F., Sander, B., Jørgensen, T.M., Andersen, P.E.: Optical Coherence Tomography. In: Encyclopedia of Analytical Chemistry. John Wiley & Sons Ltd., Chichester (2009)Google Scholar
  20. 20.
    Friberg, A.T., Wolf, E.: Angular spectrum representation of scattered electromagnetic fields. J. Opt. Soc. Am. 73(1), 26–32 (1983)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Hecht, E.: Optics. 4th edn. Addison Wesley, San Francisco (2002)Google Scholar
  22. 22.
    Hellmuth, T.: Contrast and resolution in optical coherence tomography. In: Bigio, I.J., Grundfest, W.S., Schneckenburger, H., Svanberg K., Viallet P.M., (eds.) Optical Biopsies and Microscopic Techniques. Proceedings of SPIE, vol 2926, pp 228–237 (1997)CrossRefGoogle Scholar
  23. 23.
    Hohage, T.: Fast numerical solution of the electromagnetic medium scattering problem and applications to the inverse problem. J. Comput. Phys. 214, 224–238 (2006)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Huang, D., Swanson, E.A., Lin, C.P., Schuman, J.S., Stinson, G., Chang, W., Hee, M.R., Flotte, T., Gregory, K., Puliafito, C.A., Fujimoto, J.G.: Optical coherence tomography. Science 254(5035), 1178–1181 (1991)CrossRefGoogle Scholar
  25. 25.
    Izatt, J.A., Choma, M.A.: Theory of optical coherence tomography. In: Drexler, W., Fujimoto, J.G. (eds.) In: Optical Coherence Tomography, pp. 47–72. Springer, Berlin (2008)CrossRefGoogle Scholar
  26. 26.
    Kirillin, M., Meglinski, I., Kuzmin, V., Sergeeva, E., Myllylä, R.: Simulation of optical coherence tomography images by monte carlo modeling based on polarization vector approach. Opt. Express 18(21), 21714–21724 (2010)CrossRefGoogle Scholar
  27. 27.
    Knüttel, A., Schork, R., Böcker, D.: Analytical modeling of spatial resolution curves in turbid media acquired with optical coherence tomography (oct). In: Cogwell, C.J., Kino, G.S., Wilson, T. (eds.) Three- Dimensional Microscopy: Image Acquisition and Processing III, Proceedings of SPIE, vol 2655, pp. 258–270 (1996)CrossRefGoogle Scholar
  28. 28.
    Marks, D.L., Davis, B.J., Boppart, S.A., Carney, P.S.: Partially coherent illumination in full-field interferometric synthetic aperture microscopy. J. Opt. Soc. Am. A 26(2), 376–386 (2009)CrossRefGoogle Scholar
  29. 29.
    Marks, D.L., Ralston, T.S., Boppart, S.A., Carney, P.S.: Inverse scattering for frequency-scanned full-field optical coherence tomography. J. Opt. Soc. Am. A 24(4), 1034–1041 (2007)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Orfanidis, S.J.: Electromagnetic Waves and Antennas. Rutgers University Press, NJ (2002)Google Scholar
  31. 31.
    Pan, Y., Birngruber, R., Rosperich, J., Engelhardt, R.: Low-coherence optical tomography in turbid tissue: theoretical analysis. App. Opt. 34(28), 6564–6574 (1995)CrossRefGoogle Scholar
  32. 32.
    Podoleanu, A.G.: Optical coherence tomography. Br. J. Radiol. 78, 976–988 (2005)CrossRefGoogle Scholar
  33. 33.
    Potthast, R.: Integral equation methods in electromagnetic scattering from anisotropic media. Math. Methods Appl. Sci. 23, 1145–1159 (2000)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Ralston, T.S.: Deconvolution methods for mitigation of transverse blurring in optical coherence tomography. IEEE Trans. Image Process. 14(9), 1254–1264 (2005)CrossRefGoogle Scholar
  35. 35.
    Ralston, T.S., Marks, D.L., Carney, P.S., Boppart, S.A.: Inverse scattering for optical coherence tomography. J. Opt. Soc. Am. A 23(5), 1027–1037 (2006)CrossRefGoogle Scholar
  36. 36.
    Schmitt, J.M.: Optical coherence tomography (OCT): A review. IEEE J. Quantum Electron. 5, 1205–1215 (1999)CrossRefGoogle Scholar
  37. 37.
    Schmitt, J.M., Knüttel, A.: Model of optical coherence tomography of heterogeneous tissue. J. Opt. Soc. Am. A 14(6), 1231–1242 (1997)CrossRefGoogle Scholar
  38. 38.
    Schmitt, J.M., Knüttel, A., Bonner, R.F.: Measurement of optical properties of biological tissues by low-coherence reflectometry. Appl. Opt. 32, 6032–6042 (1993)CrossRefGoogle Scholar
  39. 39.
    Schmitt, J.M., Xiang, S.H., Yung, K.M.: Differential absorption imaging with optical coherence tomography. J. Opt. Soc. Amer. A 15, 2288–2296 (1998)CrossRefGoogle Scholar
  40. 40.
    Smithies, D.J., Lindmo, T., Chen, Z., Nelson, J.S., Milner, T.E.: Signal attenuation and localization in optical coherence tomography studied by monte carlo simulation. Phys. Med. Biol. 43, 3025–3044 (1998)CrossRefGoogle Scholar
  41. 41.
    Swanson, E.A., Izatt, J.A., Hee, M.R., Huang, D., Lin, C.P., Schuman, J.S., Puliafito, C.A., Fujimoto, J.G.: In vivo retinal imaging by optical coherence tomography. Opt. Lett. 18, 1864–1866 (1993)CrossRefGoogle Scholar
  42. 42.
    Thomsen, J.B., Sander, B., Mogensen, M., Thrane, L., Jørgensen, T.M., Martini, T., Jemec, G.B.E., Andersen, P.E.: Optical coherence tomography: Technique and applications. In: Advanced Imaging in Biology and Medicine, pp. 103–129. Springer, Berlin (2009)Google Scholar
  43. 43.
    Thrane, L., Yura, H.T., Andersen, P.E.: Analysis of optical coherence tomography systems based on the extended huygens - fresnel principle. J. Opt. Soc. Am. A 17(3), 484–490 (2000)CrossRefMathSciNetGoogle Scholar
  44. 44.
    Tomlins, P.H., Wang, R.K.: Theory, developments and applications of optical coherence tomography. J. Phys. D: Appl. Phys. 38, 2519–2535 (2005)CrossRefGoogle Scholar
  45. 45.
    Turchin, I.V., Sergeeva, E.A., Dolin, L.S., Kamensky, V.A., Shakhova, N.M., Richards Kortum, R.: Novel algorithm of processing optical coherence tomography images for differentiation of biological tissue pathologies. J. Biomed. Opt. 10(6) 064024, (2005)CrossRefGoogle Scholar
  46. 46.
    Xu, C., Marks, D.L., Do, M.N., Boppart, S.A.: Separation of absorption and scattering profiles in spectroscopic optical coherence tomography using a least-squares algorithm. Opt. Express 12(20), 4790–4803 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Peter Elbau
    • 1
  • Leonidas Mindrinos
    • 1
  • Otmar Scherzer
    • 2
    • 3
  1. 1.Computational Science CenterUniversity of ViennaViennaAustria
  2. 2.Computational Science CenterUniversity of ViennaViennaAustria
  3. 3.RICAMAustrian Academy of SciencesLinzAustria

Personalised recommendations