Abstract

We report what to our knowledge is a novel perturbation approach for time-resolved transmittance imaging in diffusive media, based on the diffusion approximation with extrapolated boundary conditions. The model relies on the method of Padé approximants and consists of a nonlinear approximation of time-resolved transmittance curves in the presence of an inclusion. The proposed model is intended to extend the range of applicability of perturbation models when applied to inclusions that are non-point-like. We test the model on different tissue phantoms with scattering only, absorbing only, and both scattering and absorbing inclusions. Maps of the optical properties are displayed, and the results are compared with those obtained by means of the usual linear approximation of time-resolved transmittance curves. We found that the nonlinear approach gives a better prediction for absolute values of the scattering and absorption coefficients of inclusions, when the inclusion optical properties are higher than the surrounding background. Furthermore, better-resolved spots and a reduced cross talk between the two parameters are found in the reconstructed maps. Because the range of the optical properties spanned by the considered phantoms covers the values expected for optical mammography, the application of the reported reconstruction method to in vivo images of a breast appears promising from a diagnostic viewpoint.

© 2003 Optical Society of America

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References

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  1. For a recent review, see Biomedical Topical Meetings, Vol. 71 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002).
  2. http://www.optimamm.de/www.optimamm.de (EC grant QLG1-CT-2000-00690 “Optimamm”).
  3. R. Cubeddu, G. M. Danesini, E. Giambattistelli, F. Messina, L. Pallaro, A. Pifferi, P. Taroni, A. Torricelli, “Time-resolved optical mammograph for clinical studies beyond 900 nm,” in Biomedical Topical Meetings, Vol. 71 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 674–676.
  4. B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 1–15 (2000).
    [CrossRef]
  5. D. Grosenick, H. Wabnitz, H. Rinneberg, K. T. Moesta, P. M. Schlag, “Development of a time-domain optical mammograph and first in vivo applications,” Appl. Opt. 38, 2927–2943 (1999).
    [CrossRef]
  6. R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Time-resolved imaging on a realistic tissue phantom: μ′s and μa images versus time-integrated images,” Appl. Opt. 35, 4533–4540 (1996).
    [CrossRef] [PubMed]
  7. R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Imaging of optical inhomogeneities in highly diffusive media: discrimination between scattering and absorption contributions,” Appl. Phys. Lett. 69, 4162–4164 (1996).
    [CrossRef]
  8. J. C. Hebden, S. R. Arridge, “Imaging through scattering media by the use of an analytical model of perturbation amplitudes in the time domain,” Appl. Opt. 35, 6788–6796 (1996).
    [CrossRef] [PubMed]
  9. M. Morin, S. Verreault, A. Mailloux, J. Frechette, S. Chatigny, Y. Painchaud, P. Beaudry, “Inclusion characterization in a scattering slab with time-resolved transmittance measurements: perturbation analysis,” Appl. Opt. 39, 2840–2852 (2000).
    [CrossRef]
  10. S. Carraresi, T. S. M. Shatir, F. Martelli, 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]
  11. A. H. Gandjbakhche, V. Chernomordik, J. C. Hebden, R. Nossal, “Time-dependent contrast functions for quantitative imaging in time-resolved transillumination experiments,” Appl. Opt. 37, ∼1973–1981 (1998).
    [CrossRef]
  12. V. Chernomordik, D. Hattery, A. H. Gandjbakhche, A. Pifferi, A. Torricelli, P. Taroni, G. Valentini, R. Cubeddu, “Quantification by random walk of the optical parameters of nonlocalized abnormalities embedded within tissuelike phantoms,” Opt. Lett. 25, 951–953 (2000).
    [CrossRef]
  13. D. A. Boas, M. A. O’Leary, B. Chance, 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]
  14. G. A. Baker, P. Graves-Morris, Padé Approximants, 2nd ed. (Cambridge University Press, Cambridge, UK, 1996).
    [CrossRef]

2001

2000

1999

1998

A. H. Gandjbakhche, V. Chernomordik, J. C. Hebden, R. Nossal, “Time-dependent contrast functions for quantitative imaging in time-resolved transillumination experiments,” Appl. Opt. 37, ∼1973–1981 (1998).
[CrossRef]

1996

1994

D. A. Boas, M. A. O’Leary, B. Chance, 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]

Arridge, S. R.

Baker, G. A.

G. A. Baker, P. Graves-Morris, Padé Approximants, 2nd ed. (Cambridge University Press, Cambridge, UK, 1996).
[CrossRef]

Beaudry, P.

Boas, D. A.

D. A. Boas, M. A. O’Leary, B. Chance, 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]

Butler, J.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 1–15 (2000).
[CrossRef]

Carraresi, S.

Cerussi, A.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 1–15 (2000).
[CrossRef]

Chance, B.

D. A. Boas, M. A. O’Leary, B. Chance, 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]

Chatigny, S.

Chernomordik, V.

V. Chernomordik, D. Hattery, A. H. Gandjbakhche, A. Pifferi, A. Torricelli, P. Taroni, G. Valentini, R. Cubeddu, “Quantification by random walk of the optical parameters of nonlocalized abnormalities embedded within tissuelike phantoms,” Opt. Lett. 25, 951–953 (2000).
[CrossRef]

A. H. Gandjbakhche, V. Chernomordik, J. C. Hebden, R. Nossal, “Time-dependent contrast functions for quantitative imaging in time-resolved transillumination experiments,” Appl. Opt. 37, ∼1973–1981 (1998).
[CrossRef]

Cubeddu, R.

V. Chernomordik, D. Hattery, A. H. Gandjbakhche, A. Pifferi, A. Torricelli, P. Taroni, G. Valentini, R. Cubeddu, “Quantification by random walk of the optical parameters of nonlocalized abnormalities embedded within tissuelike phantoms,” Opt. Lett. 25, 951–953 (2000).
[CrossRef]

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Time-resolved imaging on a realistic tissue phantom: μ′s and μa images versus time-integrated images,” Appl. Opt. 35, 4533–4540 (1996).
[CrossRef] [PubMed]

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Imaging of optical inhomogeneities in highly diffusive media: discrimination between scattering and absorption contributions,” Appl. Phys. Lett. 69, 4162–4164 (1996).
[CrossRef]

R. Cubeddu, G. M. Danesini, E. Giambattistelli, F. Messina, L. Pallaro, A. Pifferi, P. Taroni, A. Torricelli, “Time-resolved optical mammograph for clinical studies beyond 900 nm,” in Biomedical Topical Meetings, Vol. 71 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 674–676.

Espinoza, J.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 1–15 (2000).
[CrossRef]

Frechette, J.

Gandjbakhche, A. H.

V. Chernomordik, D. Hattery, A. H. Gandjbakhche, A. Pifferi, A. Torricelli, P. Taroni, G. Valentini, R. Cubeddu, “Quantification by random walk of the optical parameters of nonlocalized abnormalities embedded within tissuelike phantoms,” Opt. Lett. 25, 951–953 (2000).
[CrossRef]

A. H. Gandjbakhche, V. Chernomordik, J. C. Hebden, R. Nossal, “Time-dependent contrast functions for quantitative imaging in time-resolved transillumination experiments,” Appl. Opt. 37, ∼1973–1981 (1998).
[CrossRef]

Giambattistelli, E.

R. Cubeddu, G. M. Danesini, E. Giambattistelli, F. Messina, L. Pallaro, A. Pifferi, P. Taroni, A. Torricelli, “Time-resolved optical mammograph for clinical studies beyond 900 nm,” in Biomedical Topical Meetings, Vol. 71 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 674–676.

Graves-Morris, P.

G. A. Baker, P. Graves-Morris, Padé Approximants, 2nd ed. (Cambridge University Press, Cambridge, UK, 1996).
[CrossRef]

Grosenick, D.

Hattery, D.

Hebden, J. C.

A. H. Gandjbakhche, V. Chernomordik, J. C. Hebden, R. Nossal, “Time-dependent contrast functions for quantitative imaging in time-resolved transillumination experiments,” Appl. Opt. 37, ∼1973–1981 (1998).
[CrossRef]

J. C. Hebden, S. R. Arridge, “Imaging through scattering media by the use of an analytical model of perturbation amplitudes in the time domain,” Appl. Opt. 35, 6788–6796 (1996).
[CrossRef] [PubMed]

Lanning, R.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 1–15 (2000).
[CrossRef]

M. Danesini, G.

R. Cubeddu, G. M. Danesini, E. Giambattistelli, F. Messina, L. Pallaro, A. Pifferi, P. Taroni, A. Torricelli, “Time-resolved optical mammograph for clinical studies beyond 900 nm,” in Biomedical Topical Meetings, Vol. 71 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 674–676.

Mailloux, A.

Martelli, F.

Messina, F.

R. Cubeddu, G. M. Danesini, E. Giambattistelli, F. Messina, L. Pallaro, A. Pifferi, P. Taroni, A. Torricelli, “Time-resolved optical mammograph for clinical studies beyond 900 nm,” in Biomedical Topical Meetings, Vol. 71 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 674–676.

Moesta, K. T.

Morin, M.

Nossal, R.

A. H. Gandjbakhche, V. Chernomordik, J. C. Hebden, R. Nossal, “Time-dependent contrast functions for quantitative imaging in time-resolved transillumination experiments,” Appl. Opt. 37, ∼1973–1981 (1998).
[CrossRef]

O’Leary, M. A.

D. A. Boas, M. A. O’Leary, B. Chance, 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]

Painchaud, Y.

Pallaro, L.

R. Cubeddu, G. M. Danesini, E. Giambattistelli, F. Messina, L. Pallaro, A. Pifferi, P. Taroni, A. Torricelli, “Time-resolved optical mammograph for clinical studies beyond 900 nm,” in Biomedical Topical Meetings, Vol. 71 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 674–676.

Pham, T.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 1–15 (2000).
[CrossRef]

Pifferi, A.

V. Chernomordik, D. Hattery, A. H. Gandjbakhche, A. Pifferi, A. Torricelli, P. Taroni, G. Valentini, R. Cubeddu, “Quantification by random walk of the optical parameters of nonlocalized abnormalities embedded within tissuelike phantoms,” Opt. Lett. 25, 951–953 (2000).
[CrossRef]

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Time-resolved imaging on a realistic tissue phantom: μ′s and μa images versus time-integrated images,” Appl. Opt. 35, 4533–4540 (1996).
[CrossRef] [PubMed]

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Imaging of optical inhomogeneities in highly diffusive media: discrimination between scattering and absorption contributions,” Appl. Phys. Lett. 69, 4162–4164 (1996).
[CrossRef]

R. Cubeddu, G. M. Danesini, E. Giambattistelli, F. Messina, L. Pallaro, A. Pifferi, P. Taroni, A. Torricelli, “Time-resolved optical mammograph for clinical studies beyond 900 nm,” in Biomedical Topical Meetings, Vol. 71 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 674–676.

Rinneberg, H.

Schlag, P. M.

Shah, N.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 1–15 (2000).
[CrossRef]

Shatir, T. S. M.

Svaasand, L.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 1–15 (2000).
[CrossRef]

Taroni, P.

V. Chernomordik, D. Hattery, A. H. Gandjbakhche, A. Pifferi, A. Torricelli, P. Taroni, G. Valentini, R. Cubeddu, “Quantification by random walk of the optical parameters of nonlocalized abnormalities embedded within tissuelike phantoms,” Opt. Lett. 25, 951–953 (2000).
[CrossRef]

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Time-resolved imaging on a realistic tissue phantom: μ′s and μa images versus time-integrated images,” Appl. Opt. 35, 4533–4540 (1996).
[CrossRef] [PubMed]

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Imaging of optical inhomogeneities in highly diffusive media: discrimination between scattering and absorption contributions,” Appl. Phys. Lett. 69, 4162–4164 (1996).
[CrossRef]

R. Cubeddu, G. M. Danesini, E. Giambattistelli, F. Messina, L. Pallaro, A. Pifferi, P. Taroni, A. Torricelli, “Time-resolved optical mammograph for clinical studies beyond 900 nm,” in Biomedical Topical Meetings, Vol. 71 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 674–676.

Torricelli, A.

V. Chernomordik, D. Hattery, A. H. Gandjbakhche, A. Pifferi, A. Torricelli, P. Taroni, G. Valentini, R. Cubeddu, “Quantification by random walk of the optical parameters of nonlocalized abnormalities embedded within tissuelike phantoms,” Opt. Lett. 25, 951–953 (2000).
[CrossRef]

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Time-resolved imaging on a realistic tissue phantom: μ′s and μa images versus time-integrated images,” Appl. Opt. 35, 4533–4540 (1996).
[CrossRef] [PubMed]

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Imaging of optical inhomogeneities in highly diffusive media: discrimination between scattering and absorption contributions,” Appl. Phys. Lett. 69, 4162–4164 (1996).
[CrossRef]

R. Cubeddu, G. M. Danesini, E. Giambattistelli, F. Messina, L. Pallaro, A. Pifferi, P. Taroni, A. Torricelli, “Time-resolved optical mammograph for clinical studies beyond 900 nm,” in Biomedical Topical Meetings, Vol. 71 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 674–676.

Tromberg, B. J.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 1–15 (2000).
[CrossRef]

Valentini, G.

Verreault, S.

Wabnitz, H.

Yodh, A. G.

D. A. Boas, M. A. O’Leary, B. Chance, 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]

Zaccanti, G.

Appl. Opt.

Appl. Phys. Lett.

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Imaging of optical inhomogeneities in highly diffusive media: discrimination between scattering and absorption contributions,” Appl. Phys. Lett. 69, 4162–4164 (1996).
[CrossRef]

Neoplasia

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 1–15 (2000).
[CrossRef]

Opt. Lett.

Proc. Natl. Acad. Sci. USA

D. A. Boas, M. A. O’Leary, B. Chance, 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]

Other

G. A. Baker, P. Graves-Morris, Padé Approximants, 2nd ed. (Cambridge University Press, Cambridge, UK, 1996).
[CrossRef]

For a recent review, see Biomedical Topical Meetings, Vol. 71 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002).

http://www.optimamm.de/www.optimamm.de (EC grant QLG1-CT-2000-00690 “Optimamm”).

R. Cubeddu, G. M. Danesini, E. Giambattistelli, F. Messina, L. Pallaro, A. Pifferi, P. Taroni, A. Torricelli, “Time-resolved optical mammograph for clinical studies beyond 900 nm,” in Biomedical Topical Meetings, Vol. 71 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 674–676.

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Figures (6)

Fig. 1
Fig. 1

Geometry and dimensions of the tissue phantoms with three inclusions. The scan area is displayed as a white contour.

Fig. 2
Fig. 2

Fitting of the 2-D plot of the scattering coefficient reported in Fig. 4(a) with a 2-D Gaussian, in order to have an estimation of the inclusion scattering coefficient and spatial resolution of the image (FWHM). Two orthogonal sections of experimental data (symbols) and fitting Gaussian (solid curve) are shown in (a) and (b), respectively.

Fig. 3
Fig. 3

Plots of scattering (left column) and absorption (right column) coefficients for the phantoms with a single inclusion, obtained by fitting of the time-resolved transmittance data with the linear model [see Eq. (3)]: (a) phantom 1, (b) phantom 2, (c) phantom 3, (d) phantom 4.

Fig. 4
Fig. 4

As in Fig. 3, except that here the time-resolved transmittance data were fitted with the method of Padé approximants [see Eq. (5)]: (a) phantom 1, (b) phantom 2, (c) phantom 3, (d) phantom 4.

Fig. 5
Fig. 5

Plots of scattering (left column) and absorption (right column) coefficients for the phantoms with multiple inclusions, obtained by fitting of the time-resolved transmittance data with the linear model [see Eq. (3)]: (a) phantom 5, (b) phantom 6, (c) phantom 7.

Fig. 6
Fig. 6

As in Fig. 5, except that here the time-resolved transmittance data were fitted with the method of Padé approximants [see Eq. (5)]: (a) phantom 5, (b) phantom 6, (c) phantom 7.

Tables (2)

Tables Icon

Table 1 Description of the Phantoms Used for Measuresa

Tables Icon

Table 2 Optical Properties of Different Inclusions Obtained with the Two Perturbative Methods

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

Tt; μa, D=T0t; μa0, D0+δμaJat; μa0, D0+δDJDt; μa0, D0,
T0t; μa0, D0=0.54πνD0-3/2t-5/2 exp-μa0νt×n=-+zn+ exp-zn+24νD0t-zn- exp-zn-24νD0t,
CLt; μa, D=T0t; μa0, D0-Tt; μa, DT0t; μa0, D0=-δμaJat; μa0, D0T0t; μa0, D0+δD JDt; μa0, D0T0t; μa0, D0.
Tt; μa, D=T0t; μa0, D0+δμaJat; μa0, D0+δDJDt; μa0, D0+=T0t; μa0, D0n=0+δμaJat; μa0, D0T0t; μa0, D0+δD JDt; μa0, D0T0t; μa0, D0n=T0t; μa0, D01-δμaJat; μa0, D0T0t; μa0, D0+δD JDt; μa0, D0T0t; μa0, D0-1.
CNLt; μa, D=T0t; μa0, D0-Tt; μa, DTt; μa, D=-δμaJat; μa0, D0T0t; μa0, D0+δD JDt; μa0, D0T0t; μa0, D0.

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