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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics


  • Editor: Gregory W. Faris
  • Vol. 4, Iss. 6 — May. 26, 2009

Higher-order perturbation theory for the diffusion equation in heterogeneous media: application to layered and slab geometries

Angelo Sassaroli, Fabrizio Martelli, and Sergio Fantini  »View Author Affiliations

Applied Optics, Vol. 48, Issue 10, pp. D62-D73 (2009)

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We apply a previously proposed perturbation theory of the diffusion equation for studying light propagation through heterogeneous media in the presence of absorbing defects. The theory is based on the knowledge of (a) the geometric characteristics of a focal inclusion, (b) the mean optical path length inside the inclusion, and (c) the optical properties of the inclusion. The potential of this method is shown in the layered and slab geometries, where calculations are carried out up to the fourth order. The relative changes of intensity with respect to the unperturbed (heterogeneous) medium are predicted by the theory to within 10% for a wide range of contrasts d Δ μ a (up to d Δ μ a 0.4 0.8 ), where d is the effective diameter of the defect and Δ μ a the absorption contrast between defect and local background. We also show how the method of Padé approximants can be used to extend the validity of the theory for a larger range of absorption contrasts. Finally, we study the possibility of using the proposed method for calculating the effect of a colocalized scattering and absorbing perturbation.

© 2009 Optical Society of America

OCIS Codes
(170.3660) Medical optics and biotechnology : Light propagation in tissues
(170.5280) Medical optics and biotechnology : Photon migration
(170.7050) Medical optics and biotechnology : Turbid media

Original Manuscript: September 2, 2008
Revised Manuscript: November 21, 2008
Manuscript Accepted: November 29, 2008
Published: January 15, 2009

Virtual Issues
Vol. 4, Iss. 6 Virtual Journal for Biomedical Optics

Angelo Sassaroli, Fabrizio Martelli, and Sergio Fantini, "Higher-order perturbation theory for the diffusion equation in heterogeneous media: application to layered and slab geometries," Appl. Opt. 48, D62-D73 (2009)

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  1. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999). [CrossRef]
  2. S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlength in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531-1560(1992). [CrossRef] [PubMed]
  3. D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887-4891(1994). [CrossRef] [PubMed]
  4. D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal to noise analysis,” Appl. Opt. 36, 75-92 (1997). [CrossRef] [PubMed]
  5. P. N. den Outer, Th. M. Nieuwenhuizen, and A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209-1218 (1993). [CrossRef]
  6. S. A. Walker, D. A. Boas, and E. Gratton, “Photon density waves scattered from cylindrical inhomogeneities: theory and experiments,” Appl. Opt. 37, 1935-1944 (1998). [CrossRef]
  7. A. Kienle, T. Glanzmann, G. Wagnieres, and H. V. Bergh, “Investigation of two-layered turbid medium with time-resolved reflectance,” Appl. Opt. 37, 6852-6862 (1998). [CrossRef]
  8. F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, and G. Zaccanti, “Solution of the time-dependent diffusion equation for layered random media by the eigenfunction method,” Phys. Rev. E 67, 056623 (2003). [CrossRef]
  9. J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, A. R. Arridge, and J. Ripoll, “Diffuse photon propagation in multilayered geometries,” Phys. Med. Biol. 51, 497-516 (2006). [CrossRef] [PubMed]
  10. J. Ripoll, V. Ntziachristos, J. P. Culver, D. N. Pattanayak, A. G. Yodh, and M. Nieto-Vesperinas, “Recovery of optical parameters in multiple-layered diffusive media: theory and experiments,” J. Opt. Soc. Am. A 18, 821-830 (2001). [CrossRef]
  11. K. Ren, G. S. Abdoulaev, G. Bal, and A. H. Hielscher, “Algorithm for solving the equation of radiative transfer in the frequency domain,” Opt. Lett. 29, 578-580 (2004). [CrossRef] [PubMed]
  12. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach to modelling photon transport in tissue,” Med. Phys. 20, 299-309 (1993). [CrossRef] [PubMed]
  13. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulations and experiments,” J. Opt. Soc. Am. 13, 253-266 (1996). [CrossRef]
  14. R. L. Barbour, H. L. Graber, Y. Pei, S. Zhong, and C. H. Schmitz, “Optical tomographic imaging of dynamic features of dense-scattering media,” J. Opt. Soc. Am. 18, 3018-3036 (2001). [CrossRef]
  15. Y. Yao, Y. wang, Y. Pei, W. Zhu, and R. L. Barbour, “Frequency domain optical imaging of absorption and scattering distributions by a Born iterative method,” J. Opt. Soc. Am. A 14, 325-342 (1997). [CrossRef]
  16. D. A. Boas, J. P. Culver, J. J. Stott, and A. K. Dunn, “Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head,” Opt. Express 10, 159-170 (2002). [PubMed]
  17. L. Wang, S. L. Jacques, and L. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995). [CrossRef] [PubMed]
  18. R. Graaf, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, and J. G. Aarnoudse, “Condensed Monte Carlo simulations for the description of light transport,” Appl. Opt. 32, 426-434 (1993). [CrossRef]
  19. G. Zaccanti, “Monte Carlo study of light propagation in optically thick media: point source case,” Appl. Opt. 30, 2031-2041 (1991). [CrossRef] [PubMed]
  20. M. R. Ostermeyer and S. L. Jacques, “Perturbation theory for diffuse light transport in complex biological tissues,” J. Opt. Soc. Am. A 14, 255-261 (1997). [CrossRef]
  21. D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Opt. Express 1, 404-413(1997). [CrossRef] [PubMed]
  22. A. Sassaroli, F. Martelli, and S. Fantini, “Perturbation theory for the diffusion equation by use of the moments of the generalized temporal point-spread function: I. Theory,” J. Opt. Soc. Am. A 23, 2105-2118 (2006). [CrossRef]
  23. A. Sassaroli, F. Martelli, and S. Fantini, “Perturbation theory for the diffusion equation by use of the moments of the generalized temporal point-spread function: II. Continuous wave results,” J. Opt. Soc. Am. A 23, 2119-2131 (2006). [CrossRef]
  24. B. Wassermann, “Limits of high-order perturbation in time-domain optical mammography,” Phys. Rev. E 74, 031908(2006). [CrossRef]
  25. D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, “Evaluation of higher-order time domain perturbation theory of photon diffusion on breast-equivalent phantoms and optical mammograms,” Phys. Rev. E 76, 061908 (2007). [CrossRef]
  26. V. Chernomordik, D. Hattery, A. Gandjbakhche, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, and R. Cubeddu, “Quantification by random walk of the optical parameters of nonlocalized abnormalities embedded within tissuelike phantoms,” Opt. Lett. 25, 951-953 (2000). [CrossRef]
  27. M. Xu, W. Cai, and R. R. Alfano, “Multiple passages of light through an absorption inhomogeneity in optical imaging of turbid media,” Opt. Lett. 29, 1757-1759(2004). [CrossRef] [PubMed]
  28. L. Spinelli, A. Torricelli, A. Pifferi, P. Taroni, and R. Cubeddu, “Experimental tests of a perturbation model for time-resolved imaging of diffusive media,” Appl. Opt. 42, 3145-3153 (2003). [CrossRef] [PubMed]
  29. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, 1992).
  30. F. Martelli, S. Del Bianco, and G. Zaccanti, “Perturbation model of light propagation through diffusive layered media,” Phys. Med. Biol. 50, 2159-2166 (2005). [CrossRef] [PubMed]
  31. S. Carraresi, T. S. M. Shatir, F. Martelli, and G. Zaccanti, “Accuracy of a perturbation model to predict the effect of scattering and absorbing inhomogeneities on photon migration,” Appl. Opt. 40, 4622-4632 (2001). [CrossRef]
  32. S. Fantini, E. L. Heffer, V. E. Pera, A. Sassaroli, and N. Liu, “Spatial and spectral information in optical mammography,” Technol. Cancer Res. Treat. 4, 471-482 (2005). [PubMed]
  33. P. Taroni, D. Comelli, A. Pifferi, A. Torricelli, and R. Cubeddu, “Absorption of collagen: effects on the estimate of breast composition and related diagnostic implications,” J Biomed. Opt. 12, 014021 (2007). [CrossRef] [PubMed]
  34. A. D. Kim and J. C. Schotland, “Self-consistent scattering theory for the radiative transport equation,” J. Opt. Soc. Am. A 23, 596-602 (2006). [CrossRef]

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