Abstract

We present analytic solutions for fluorescent diffuse photon density waves originating from fluorophores distributed in thick turbid media. Solutions are derived for a homogeneous turbid medium containing a uniform distribution of fluorophores and for a system that is homogeneous except for the presence of a single spherical inhomogeneity. Generally the inhomogeneity has fluorophore concentration, and lifetime and optical properties that differ from those of the background. The analytic solutions are verified by numerical calculations and are used to determine the fluorophore lifetime and concentration changes required for the accurate detection of inhomogeneities in biologically relevant systems. The relative sensitivities of absorption and fluorescence methods are compared.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today31–36 (Mar.1995) and references therein.
  2. E. M. Sevick-Muraca, C. L. Burch, “Origin of phosphorescence signals reemitted from tissues,” Opt. Lett. 19, 1928–1930 (1994).
    [Crossref] [PubMed]
  3. M. S. Patterson, B. W. Pogue, “Mathematical model for time-resolved and frequency-domain fluorescence spectros-copy in biological tissues,” Appl. Opt. 33, 1963–1974 (1994).
    [Crossref] [PubMed]
  4. A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Acousto-optic scanning and interfering photon density waves for precise localization of an absorbing (or fluorescent) body in a turbid medium,” Rev. Sci. Instrum. 64, 638–644 (1993).
    [Crossref]
  5. D. A. Boas, M. A. O'Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Res. E 47, R2999–R3002 (1993).
    [Crossref]
  6. M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Luminesc. 60-1, 281–286 (1994).
    [Crossref]
  7. X. D. Li, B. Beauvoit, R. White, S. Nioka, B. Chance, A. G. Yodh, “Tumor localization using fluorescence of indocyanine green (ICG) in rat models,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 789–797 (1995).
    [Crossref]
  8. J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, M.S. Feld, “Time-resolved multichannel imaging of fluorescent objects embedded in turbid media,” Opt. Lett. 20, 489–491 (1995).
    [Crossref] [PubMed]
  9. S. B. Bambot, J. R. Lakowicz, G. Rao, “Potential applications of lifetime-based, phase-modulation fluorimetry in bio-process and clinical monitoring,” Trends Biotechnol. 13, 106– 115 (1995).
    [Crossref] [PubMed]
  10. W. L. Rumsey, J. M. Vanderkooi, D. F. Wilson, “Imaging of phosphorescence: a novel method for measuring oxygen distribution in perfused tissue,” Science 241, 1649–1651 (1988).
    [Crossref] [PubMed]
  11. H. Szmacinski, J. R. Lakowicz, “Optical measurements of pH using fluorescence lifetimes and phase-modulation fluorometry,” Anal. Chem. 65, 1668–1674 (1993).
    [Crossref] [PubMed]
  12. J. R. Lakowicz, H. Szmacinski, K. Nowaczyk, M. L. Johnson, “Fluorescence lifetime imaging of calcium using Quin-2,” Cell Calcium 13, 131–147 (1992).
    [Crossref] [PubMed]
  13. J. Folkman, “Introduction: angiogenesis and cancer,” Semin. Cancer Biol. 3, 65–71 (1992).
    [PubMed]
  14. P. W. Vaupel, S. Frinak, H. I. Bicher, “Heterogeneous oxygen partial pressure and pH distribution in C3H mouse mammary adenocarcinoma,” Cancer Res. 41, 2008–2013 (1981).
    [PubMed]
  15. C. L. Hutchinson, T. L. Troy, E. M. Sevick-Muraca, “Fluorescence-lifetime spectroscopy and imaging in random media,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds. Proc. SPIE2389, 274–283 (1995).
    [Crossref]
  16. C. L. Hutchinson, J. R. Lakowicz, E. M. Sevick-Muraca, “Fluorescence lifetime-based sensing in tissues: a computational study,” Biophys. J. 68, 1574–1582 (1995).
    [Crossref] [PubMed]
  17. J. Wu, M. S. Feld, R. P. Rava, “Analytical model for extracting intrinsic fluorescence in turbid media,” Appl. Opt. 32, 3585–3595 (1993).
    [Crossref] [PubMed]
  18. We are aware that faster finite difference methods exist. However, none of these methods is expected to be faster than calculations of the analytic solutions.
  19. M. S. Patterson, B. Chance, B. C. Wilson, “Time-resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2335 (1989).
    [Crossref] [PubMed]
  20. J. B. Fishkin, E. Gratton, “Propagation of photon density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
    [Crossref] [PubMed]
  21. B. J. Tromberg, S. Madsen, C. Chapman, L. O. Svaasand, R. C. Haskell, “Frequency-domain photon migration in turbid media,” in Advances in Optical Imaging and Photon Migration, Vol. 21 of OSA Proceedings (Optical Society of America, Washington, D.C., 1994), pp. 93–95.
  22. 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 solutions and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
    [Crossref] [PubMed]
  23. R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [Crossref]
  24. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
    [Crossref]
  25. S. J. Madsen, M. S. Patterson, B. C. Wilson, S. M. Jaywant, A. Othonos, “Numerical modeling and experimental studies of light pulse propagation in inhomogeneous random media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 90–102 (1993).
    [Crossref]
  26. P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), pp. 218–222.
  27. J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Plenum, New York, 1983), pp. 258–266.
  28. G. Valduga, E. Reddi, G. Jori, R. Cubeddu, P. Taroni, G. Valentini, “Steady state and time-resolved spectroscopic studies on zinc(II) phthalocynine in liposomes,” J. Photochem. Photobiol. B: Biol. 16, 331–340 (1992);R. Cubeddu, G. Canti, P. Taroni, G. Valentini, “Time-gated fluorescence imaging for the diagnosis of tumors in a murine model,” Photochem. Photobiol. 57, 480–485 (1993).
    [Crossref] [PubMed]
  29. M.A. O'Leary, D.A. Boas, X. D. Li, B. Chance, A. G Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. 21, 158–160 (1996).
    [Crossref]

1996 (1)

1995 (5)

R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
[Crossref]

S. B. Bambot, J. R. Lakowicz, G. Rao, “Potential applications of lifetime-based, phase-modulation fluorimetry in bio-process and clinical monitoring,” Trends Biotechnol. 13, 106– 115 (1995).
[Crossref] [PubMed]

C. L. Hutchinson, J. R. Lakowicz, E. M. Sevick-Muraca, “Fluorescence lifetime-based sensing in tissues: a computational study,” Biophys. J. 68, 1574–1582 (1995).
[Crossref] [PubMed]

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today31–36 (Mar.1995) and references therein.

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, M.S. Feld, “Time-resolved multichannel imaging of fluorescent objects embedded in turbid media,” Opt. Lett. 20, 489–491 (1995).
[Crossref] [PubMed]

1994 (5)

E. M. Sevick-Muraca, C. L. Burch, “Origin of phosphorescence signals reemitted from tissues,” Opt. Lett. 19, 1928–1930 (1994).
[Crossref] [PubMed]

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 solutions and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[Crossref] [PubMed]

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
[Crossref]

M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Luminesc. 60-1, 281–286 (1994).
[Crossref]

M. S. Patterson, B. W. Pogue, “Mathematical model for time-resolved and frequency-domain fluorescence spectros-copy in biological tissues,” Appl. Opt. 33, 1963–1974 (1994).
[Crossref] [PubMed]

1993 (5)

J. B. Fishkin, E. Gratton, “Propagation of photon density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
[Crossref] [PubMed]

J. Wu, M. S. Feld, R. P. Rava, “Analytical model for extracting intrinsic fluorescence in turbid media,” Appl. Opt. 32, 3585–3595 (1993).
[Crossref] [PubMed]

A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Acousto-optic scanning and interfering photon density waves for precise localization of an absorbing (or fluorescent) body in a turbid medium,” Rev. Sci. Instrum. 64, 638–644 (1993).
[Crossref]

D. A. Boas, M. A. O'Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Res. E 47, R2999–R3002 (1993).
[Crossref]

H. Szmacinski, J. R. Lakowicz, “Optical measurements of pH using fluorescence lifetimes and phase-modulation fluorometry,” Anal. Chem. 65, 1668–1674 (1993).
[Crossref] [PubMed]

1992 (3)

J. R. Lakowicz, H. Szmacinski, K. Nowaczyk, M. L. Johnson, “Fluorescence lifetime imaging of calcium using Quin-2,” Cell Calcium 13, 131–147 (1992).
[Crossref] [PubMed]

J. Folkman, “Introduction: angiogenesis and cancer,” Semin. Cancer Biol. 3, 65–71 (1992).
[PubMed]

G. Valduga, E. Reddi, G. Jori, R. Cubeddu, P. Taroni, G. Valentini, “Steady state and time-resolved spectroscopic studies on zinc(II) phthalocynine in liposomes,” J. Photochem. Photobiol. B: Biol. 16, 331–340 (1992);R. Cubeddu, G. Canti, P. Taroni, G. Valentini, “Time-gated fluorescence imaging for the diagnosis of tumors in a murine model,” Photochem. Photobiol. 57, 480–485 (1993).
[Crossref] [PubMed]

1989 (1)

1988 (1)

W. L. Rumsey, J. M. Vanderkooi, D. F. Wilson, “Imaging of phosphorescence: a novel method for measuring oxygen distribution in perfused tissue,” Science 241, 1649–1651 (1988).
[Crossref] [PubMed]

1981 (1)

P. W. Vaupel, S. Frinak, H. I. Bicher, “Heterogeneous oxygen partial pressure and pH distribution in C3H mouse mammary adenocarcinoma,” Cancer Res. 41, 2008–2013 (1981).
[PubMed]

Aronson, R.

Bambot, S. B.

S. B. Bambot, J. R. Lakowicz, G. Rao, “Potential applications of lifetime-based, phase-modulation fluorimetry in bio-process and clinical monitoring,” Trends Biotechnol. 13, 106– 115 (1995).
[Crossref] [PubMed]

Barnes, R.

A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Acousto-optic scanning and interfering photon density waves for precise localization of an absorbing (or fluorescent) body in a turbid medium,” Rev. Sci. Instrum. 64, 638–644 (1993).
[Crossref]

Beauvoit, B.

X. D. Li, B. Beauvoit, R. White, S. Nioka, B. Chance, A. G. Yodh, “Tumor localization using fluorescence of indocyanine green (ICG) in rat models,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 789–797 (1995).
[Crossref]

Bicher, H. I.

P. W. Vaupel, S. Frinak, H. I. Bicher, “Heterogeneous oxygen partial pressure and pH distribution in C3H mouse mammary adenocarcinoma,” Cancer Res. 41, 2008–2013 (1981).
[PubMed]

Boas, D. A.

M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Luminesc. 60-1, 281–286 (1994).
[Crossref]

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 solutions and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[Crossref] [PubMed]

D. A. Boas, M. A. O'Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Res. E 47, R2999–R3002 (1993).
[Crossref]

Boas, D.A.

Burch, C. L.

Chance, B.

M.A. O'Leary, D.A. Boas, X. D. Li, B. Chance, A. G Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. 21, 158–160 (1996).
[Crossref]

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today31–36 (Mar.1995) and references therein.

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 solutions and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[Crossref] [PubMed]

M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Luminesc. 60-1, 281–286 (1994).
[Crossref]

D. A. Boas, M. A. O'Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Res. E 47, R2999–R3002 (1993).
[Crossref]

M. S. Patterson, B. Chance, B. C. Wilson, “Time-resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2335 (1989).
[Crossref] [PubMed]

X. D. Li, B. Beauvoit, R. White, S. Nioka, B. Chance, A. G. Yodh, “Tumor localization using fluorescence of indocyanine green (ICG) in rat models,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 789–797 (1995).
[Crossref]

Chapman, C.

B. J. Tromberg, S. Madsen, C. Chapman, L. O. Svaasand, R. C. Haskell, “Frequency-domain photon migration in turbid media,” in Advances in Optical Imaging and Photon Migration, Vol. 21 of OSA Proceedings (Optical Society of America, Washington, D.C., 1994), pp. 93–95.

Cubeddu, R.

G. Valduga, E. Reddi, G. Jori, R. Cubeddu, P. Taroni, G. Valentini, “Steady state and time-resolved spectroscopic studies on zinc(II) phthalocynine in liposomes,” J. Photochem. Photobiol. B: Biol. 16, 331–340 (1992);R. Cubeddu, G. Canti, P. Taroni, G. Valentini, “Time-gated fluorescence imaging for the diagnosis of tumors in a murine model,” Photochem. Photobiol. 57, 480–485 (1993).
[Crossref] [PubMed]

Dasari, R. R.

Eberly, J. H.

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), pp. 218–222.

Feld, M. S.

Feld, M.S.

Feng, T. C.

Fishkin, J. B.

Folkman, J.

J. Folkman, “Introduction: angiogenesis and cancer,” Semin. Cancer Biol. 3, 65–71 (1992).
[PubMed]

Frinak, S.

P. W. Vaupel, S. Frinak, H. I. Bicher, “Heterogeneous oxygen partial pressure and pH distribution in C3H mouse mammary adenocarcinoma,” Cancer Res. 41, 2008–2013 (1981).
[PubMed]

Gratton, E.

Haskell, R. C.

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
[Crossref]

B. J. Tromberg, S. Madsen, C. Chapman, L. O. Svaasand, R. C. Haskell, “Frequency-domain photon migration in turbid media,” in Advances in Optical Imaging and Photon Migration, Vol. 21 of OSA Proceedings (Optical Society of America, Washington, D.C., 1994), pp. 93–95.

Hutchinson, C. L.

C. L. Hutchinson, J. R. Lakowicz, E. M. Sevick-Muraca, “Fluorescence lifetime-based sensing in tissues: a computational study,” Biophys. J. 68, 1574–1582 (1995).
[Crossref] [PubMed]

C. L. Hutchinson, T. L. Troy, E. M. Sevick-Muraca, “Fluorescence-lifetime spectroscopy and imaging in random media,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds. Proc. SPIE2389, 274–283 (1995).
[Crossref]

Itzkan, I.

Jaywant, S. M.

S. J. Madsen, M. S. Patterson, B. C. Wilson, S. M. Jaywant, A. Othonos, “Numerical modeling and experimental studies of light pulse propagation in inhomogeneous random media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 90–102 (1993).
[Crossref]

Johnson, M. L.

J. R. Lakowicz, H. Szmacinski, K. Nowaczyk, M. L. Johnson, “Fluorescence lifetime imaging of calcium using Quin-2,” Cell Calcium 13, 131–147 (1992).
[Crossref] [PubMed]

Jori, G.

G. Valduga, E. Reddi, G. Jori, R. Cubeddu, P. Taroni, G. Valentini, “Steady state and time-resolved spectroscopic studies on zinc(II) phthalocynine in liposomes,” J. Photochem. Photobiol. B: Biol. 16, 331–340 (1992);R. Cubeddu, G. Canti, P. Taroni, G. Valentini, “Time-gated fluorescence imaging for the diagnosis of tumors in a murine model,” Photochem. Photobiol. 57, 480–485 (1993).
[Crossref] [PubMed]

Knutson, J. R.

A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Acousto-optic scanning and interfering photon density waves for precise localization of an absorbing (or fluorescent) body in a turbid medium,” Rev. Sci. Instrum. 64, 638–644 (1993).
[Crossref]

Knüttel, A.

A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Acousto-optic scanning and interfering photon density waves for precise localization of an absorbing (or fluorescent) body in a turbid medium,” Rev. Sci. Instrum. 64, 638–644 (1993).
[Crossref]

Lakowicz, J. R.

S. B. Bambot, J. R. Lakowicz, G. Rao, “Potential applications of lifetime-based, phase-modulation fluorimetry in bio-process and clinical monitoring,” Trends Biotechnol. 13, 106– 115 (1995).
[Crossref] [PubMed]

C. L. Hutchinson, J. R. Lakowicz, E. M. Sevick-Muraca, “Fluorescence lifetime-based sensing in tissues: a computational study,” Biophys. J. 68, 1574–1582 (1995).
[Crossref] [PubMed]

H. Szmacinski, J. R. Lakowicz, “Optical measurements of pH using fluorescence lifetimes and phase-modulation fluorometry,” Anal. Chem. 65, 1668–1674 (1993).
[Crossref] [PubMed]

J. R. Lakowicz, H. Szmacinski, K. Nowaczyk, M. L. Johnson, “Fluorescence lifetime imaging of calcium using Quin-2,” Cell Calcium 13, 131–147 (1992).
[Crossref] [PubMed]

J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Plenum, New York, 1983), pp. 258–266.

Li, X. D.

M.A. O'Leary, D.A. Boas, X. D. Li, B. Chance, A. G Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. 21, 158–160 (1996).
[Crossref]

X. D. Li, B. Beauvoit, R. White, S. Nioka, B. Chance, A. G. Yodh, “Tumor localization using fluorescence of indocyanine green (ICG) in rat models,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 789–797 (1995).
[Crossref]

Madsen, S.

B. J. Tromberg, S. Madsen, C. Chapman, L. O. Svaasand, R. C. Haskell, “Frequency-domain photon migration in turbid media,” in Advances in Optical Imaging and Photon Migration, Vol. 21 of OSA Proceedings (Optical Society of America, Washington, D.C., 1994), pp. 93–95.

Madsen, S. J.

S. J. Madsen, M. S. Patterson, B. C. Wilson, S. M. Jaywant, A. Othonos, “Numerical modeling and experimental studies of light pulse propagation in inhomogeneous random media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 90–102 (1993).
[Crossref]

McAdams, M. S.

Milonni, P. W.

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), pp. 218–222.

Nioka, S.

X. D. Li, B. Beauvoit, R. White, S. Nioka, B. Chance, A. G. Yodh, “Tumor localization using fluorescence of indocyanine green (ICG) in rat models,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 789–797 (1995).
[Crossref]

Nowaczyk, K.

J. R. Lakowicz, H. Szmacinski, K. Nowaczyk, M. L. Johnson, “Fluorescence lifetime imaging of calcium using Quin-2,” Cell Calcium 13, 131–147 (1992).
[Crossref] [PubMed]

O'Leary, M. A.

M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Luminesc. 60-1, 281–286 (1994).
[Crossref]

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 solutions and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[Crossref] [PubMed]

D. A. Boas, M. A. O'Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Res. E 47, R2999–R3002 (1993).
[Crossref]

O'Leary, M.A.

Othonos, A.

S. J. Madsen, M. S. Patterson, B. C. Wilson, S. M. Jaywant, A. Othonos, “Numerical modeling and experimental studies of light pulse propagation in inhomogeneous random media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 90–102 (1993).
[Crossref]

Patterson, M. S.

M. S. Patterson, B. W. Pogue, “Mathematical model for time-resolved and frequency-domain fluorescence spectros-copy in biological tissues,” Appl. Opt. 33, 1963–1974 (1994).
[Crossref] [PubMed]

M. S. Patterson, B. Chance, B. C. Wilson, “Time-resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2335 (1989).
[Crossref] [PubMed]

S. J. Madsen, M. S. Patterson, B. C. Wilson, S. M. Jaywant, A. Othonos, “Numerical modeling and experimental studies of light pulse propagation in inhomogeneous random media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 90–102 (1993).
[Crossref]

Perelman, L.

Pogue, B. W.

Rao, G.

S. B. Bambot, J. R. Lakowicz, G. Rao, “Potential applications of lifetime-based, phase-modulation fluorimetry in bio-process and clinical monitoring,” Trends Biotechnol. 13, 106– 115 (1995).
[Crossref] [PubMed]

Rava, R. P.

Reddi, E.

G. Valduga, E. Reddi, G. Jori, R. Cubeddu, P. Taroni, G. Valentini, “Steady state and time-resolved spectroscopic studies on zinc(II) phthalocynine in liposomes,” J. Photochem. Photobiol. B: Biol. 16, 331–340 (1992);R. Cubeddu, G. Canti, P. Taroni, G. Valentini, “Time-gated fluorescence imaging for the diagnosis of tumors in a murine model,” Photochem. Photobiol. 57, 480–485 (1993).
[Crossref] [PubMed]

Rumsey, W. L.

W. L. Rumsey, J. M. Vanderkooi, D. F. Wilson, “Imaging of phosphorescence: a novel method for measuring oxygen distribution in perfused tissue,” Science 241, 1649–1651 (1988).
[Crossref] [PubMed]

Schmitt, J. M.

A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Acousto-optic scanning and interfering photon density waves for precise localization of an absorbing (or fluorescent) body in a turbid medium,” Rev. Sci. Instrum. 64, 638–644 (1993).
[Crossref]

Sevick-Muraca, E. M.

C. L. Hutchinson, J. R. Lakowicz, E. M. Sevick-Muraca, “Fluorescence lifetime-based sensing in tissues: a computational study,” Biophys. J. 68, 1574–1582 (1995).
[Crossref] [PubMed]

E. M. Sevick-Muraca, C. L. Burch, “Origin of phosphorescence signals reemitted from tissues,” Opt. Lett. 19, 1928–1930 (1994).
[Crossref] [PubMed]

C. L. Hutchinson, T. L. Troy, E. M. Sevick-Muraca, “Fluorescence-lifetime spectroscopy and imaging in random media,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds. Proc. SPIE2389, 274–283 (1995).
[Crossref]

Svaasand, L. O.

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
[Crossref]

B. J. Tromberg, S. Madsen, C. Chapman, L. O. Svaasand, R. C. Haskell, “Frequency-domain photon migration in turbid media,” in Advances in Optical Imaging and Photon Migration, Vol. 21 of OSA Proceedings (Optical Society of America, Washington, D.C., 1994), pp. 93–95.

Szmacinski, H.

H. Szmacinski, J. R. Lakowicz, “Optical measurements of pH using fluorescence lifetimes and phase-modulation fluorometry,” Anal. Chem. 65, 1668–1674 (1993).
[Crossref] [PubMed]

J. R. Lakowicz, H. Szmacinski, K. Nowaczyk, M. L. Johnson, “Fluorescence lifetime imaging of calcium using Quin-2,” Cell Calcium 13, 131–147 (1992).
[Crossref] [PubMed]

Taroni, P.

G. Valduga, E. Reddi, G. Jori, R. Cubeddu, P. Taroni, G. Valentini, “Steady state and time-resolved spectroscopic studies on zinc(II) phthalocynine in liposomes,” J. Photochem. Photobiol. B: Biol. 16, 331–340 (1992);R. Cubeddu, G. Canti, P. Taroni, G. Valentini, “Time-gated fluorescence imaging for the diagnosis of tumors in a murine model,” Photochem. Photobiol. 57, 480–485 (1993).
[Crossref] [PubMed]

Tromberg, B. J.

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
[Crossref]

B. J. Tromberg, S. Madsen, C. Chapman, L. O. Svaasand, R. C. Haskell, “Frequency-domain photon migration in turbid media,” in Advances in Optical Imaging and Photon Migration, Vol. 21 of OSA Proceedings (Optical Society of America, Washington, D.C., 1994), pp. 93–95.

Troy, T. L.

C. L. Hutchinson, T. L. Troy, E. M. Sevick-Muraca, “Fluorescence-lifetime spectroscopy and imaging in random media,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds. Proc. SPIE2389, 274–283 (1995).
[Crossref]

Tsay, T. T.

Valduga, G.

G. Valduga, E. Reddi, G. Jori, R. Cubeddu, P. Taroni, G. Valentini, “Steady state and time-resolved spectroscopic studies on zinc(II) phthalocynine in liposomes,” J. Photochem. Photobiol. B: Biol. 16, 331–340 (1992);R. Cubeddu, G. Canti, P. Taroni, G. Valentini, “Time-gated fluorescence imaging for the diagnosis of tumors in a murine model,” Photochem. Photobiol. 57, 480–485 (1993).
[Crossref] [PubMed]

Valentini, G.

G. Valduga, E. Reddi, G. Jori, R. Cubeddu, P. Taroni, G. Valentini, “Steady state and time-resolved spectroscopic studies on zinc(II) phthalocynine in liposomes,” J. Photochem. Photobiol. B: Biol. 16, 331–340 (1992);R. Cubeddu, G. Canti, P. Taroni, G. Valentini, “Time-gated fluorescence imaging for the diagnosis of tumors in a murine model,” Photochem. Photobiol. 57, 480–485 (1993).
[Crossref] [PubMed]

Vanderkooi, J. M.

W. L. Rumsey, J. M. Vanderkooi, D. F. Wilson, “Imaging of phosphorescence: a novel method for measuring oxygen distribution in perfused tissue,” Science 241, 1649–1651 (1988).
[Crossref] [PubMed]

Vaupel, P. W.

P. W. Vaupel, S. Frinak, H. I. Bicher, “Heterogeneous oxygen partial pressure and pH distribution in C3H mouse mammary adenocarcinoma,” Cancer Res. 41, 2008–2013 (1981).
[PubMed]

Wang, Y.

White, R.

X. D. Li, B. Beauvoit, R. White, S. Nioka, B. Chance, A. G. Yodh, “Tumor localization using fluorescence of indocyanine green (ICG) in rat models,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 789–797 (1995).
[Crossref]

Wilson, B. C.

M. S. Patterson, B. Chance, B. C. Wilson, “Time-resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2335 (1989).
[Crossref] [PubMed]

S. J. Madsen, M. S. Patterson, B. C. Wilson, S. M. Jaywant, A. Othonos, “Numerical modeling and experimental studies of light pulse propagation in inhomogeneous random media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 90–102 (1993).
[Crossref]

Wilson, D. F.

W. L. Rumsey, J. M. Vanderkooi, D. F. Wilson, “Imaging of phosphorescence: a novel method for measuring oxygen distribution in perfused tissue,” Science 241, 1649–1651 (1988).
[Crossref] [PubMed]

Wu, J.

Yodh, A.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today31–36 (Mar.1995) and references therein.

Yodh, A. G

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 solutions and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[Crossref] [PubMed]

M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Luminesc. 60-1, 281–286 (1994).
[Crossref]

D. A. Boas, M. A. O'Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Res. E 47, R2999–R3002 (1993).
[Crossref]

X. D. Li, B. Beauvoit, R. White, S. Nioka, B. Chance, A. G. Yodh, “Tumor localization using fluorescence of indocyanine green (ICG) in rat models,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 789–797 (1995).
[Crossref]

Anal. Chem. (1)

H. Szmacinski, J. R. Lakowicz, “Optical measurements of pH using fluorescence lifetimes and phase-modulation fluorometry,” Anal. Chem. 65, 1668–1674 (1993).
[Crossref] [PubMed]

Appl. Opt. (3)

Biophys. J. (1)

C. L. Hutchinson, J. R. Lakowicz, E. M. Sevick-Muraca, “Fluorescence lifetime-based sensing in tissues: a computational study,” Biophys. J. 68, 1574–1582 (1995).
[Crossref] [PubMed]

Cancer Res. (1)

P. W. Vaupel, S. Frinak, H. I. Bicher, “Heterogeneous oxygen partial pressure and pH distribution in C3H mouse mammary adenocarcinoma,” Cancer Res. 41, 2008–2013 (1981).
[PubMed]

Cell Calcium (1)

J. R. Lakowicz, H. Szmacinski, K. Nowaczyk, M. L. Johnson, “Fluorescence lifetime imaging of calcium using Quin-2,” Cell Calcium 13, 131–147 (1992).
[Crossref] [PubMed]

J. Luminesc. (1)

M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Luminesc. 60-1, 281–286 (1994).
[Crossref]

J. Opt. Soc. Am. A (3)

J. Photochem. Photobiol. B: Biol. (1)

G. Valduga, E. Reddi, G. Jori, R. Cubeddu, P. Taroni, G. Valentini, “Steady state and time-resolved spectroscopic studies on zinc(II) phthalocynine in liposomes,” J. Photochem. Photobiol. B: Biol. 16, 331–340 (1992);R. Cubeddu, G. Canti, P. Taroni, G. Valentini, “Time-gated fluorescence imaging for the diagnosis of tumors in a murine model,” Photochem. Photobiol. 57, 480–485 (1993).
[Crossref] [PubMed]

Opt. Lett. (3)

Phys. Res. E (1)

D. A. Boas, M. A. O'Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Res. E 47, R2999–R3002 (1993).
[Crossref]

Phys. Today (1)

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today31–36 (Mar.1995) and references therein.

Proc. Natl. Acad. Sci. USA (1)

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 solutions and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[Crossref] [PubMed]

Rev. Sci. Instrum. (1)

A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Acousto-optic scanning and interfering photon density waves for precise localization of an absorbing (or fluorescent) body in a turbid medium,” Rev. Sci. Instrum. 64, 638–644 (1993).
[Crossref]

Science (1)

W. L. Rumsey, J. M. Vanderkooi, D. F. Wilson, “Imaging of phosphorescence: a novel method for measuring oxygen distribution in perfused tissue,” Science 241, 1649–1651 (1988).
[Crossref] [PubMed]

Semin. Cancer Biol. (1)

J. Folkman, “Introduction: angiogenesis and cancer,” Semin. Cancer Biol. 3, 65–71 (1992).
[PubMed]

Trends Biotechnol. (1)

S. B. Bambot, J. R. Lakowicz, G. Rao, “Potential applications of lifetime-based, phase-modulation fluorimetry in bio-process and clinical monitoring,” Trends Biotechnol. 13, 106– 115 (1995).
[Crossref] [PubMed]

Other (7)

X. D. Li, B. Beauvoit, R. White, S. Nioka, B. Chance, A. G. Yodh, “Tumor localization using fluorescence of indocyanine green (ICG) in rat models,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 789–797 (1995).
[Crossref]

C. L. Hutchinson, T. L. Troy, E. M. Sevick-Muraca, “Fluorescence-lifetime spectroscopy and imaging in random media,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds. Proc. SPIE2389, 274–283 (1995).
[Crossref]

We are aware that faster finite difference methods exist. However, none of these methods is expected to be faster than calculations of the analytic solutions.

B. J. Tromberg, S. Madsen, C. Chapman, L. O. Svaasand, R. C. Haskell, “Frequency-domain photon migration in turbid media,” in Advances in Optical Imaging and Photon Migration, Vol. 21 of OSA Proceedings (Optical Society of America, Washington, D.C., 1994), pp. 93–95.

S. J. Madsen, M. S. Patterson, B. C. Wilson, S. M. Jaywant, A. Othonos, “Numerical modeling and experimental studies of light pulse propagation in inhomogeneous random media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 90–102 (1993).
[Crossref]

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), pp. 218–222.

J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Plenum, New York, 1983), pp. 258–266.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Diagrammatic representation of fluorescent DPDW generated by fluorophores (a) in a homogeneous system and (b)–(f) in a heterogeneous system. (a) The fluorophore is excited by the excitation DPDW, and the reemission reaches the detector. (b) The fluorophore inside is excited, and the reemission reaches the detector. (c) The fluorophore outside is excited by the incident DPDW, and the reemission reaches the detector without being scattered by the sphere. (d) The fluorophore outside is excited by the incident DPDW, but the reemission is scattered by the sphere before reaching the detector. (e) The fluorophore outside is excited by the scattered DPDW, and the reemission reaches the detector without being scattered by the sphere. (f) The fluorophore outside is excited by the scattered DPDW, but the reemission is scattered by the sphere before reaching the detector.

Fig. 2
Fig. 2

Two source configurations in a heterogeneous medium containing a spherical object. (a) The source is outside the sphere, and the solution of DPDW outside the sphere [Eq. (7)] is the sum of the homogeneous incident part, ϕ10, and the scattered part, ϕ1 sc . The solution inside the sphere is ϕ2 as given by Eq. (8). (b) The source is inside the sphere, and the solution of DPDW outside the sphere is Φ1 [Eq. (12)]. The solution inside the sphere is the sum of the homogeneous incident part, Φ20, and the scattered part, Φ2 sc , as given by Eq. (13).

Fig. 3
Fig. 3

Comparison of the analytic solution [Eq. (27)] and the finite difference method for a semi-infinite homogeneous system. The source and detector are placed on the surface of the medium 2 cm apart. (a) Amplitude normalized by the dc amplitude (f = 0 MHz). The solid curve and the open circles represent normalized amplitudes for τ = 1 ns as calculated by the analytic solutions and the finite difference method, respectively. The dashed curve and filled circles represent normalized amplitudes for τ = 2 ns as calculated by the analytic solutions and the finite difference method, respectively. (b) Ratio of the normalized amplitude calculated by the finite difference methods and the normalized amplitude calculated by analytic solutions (solid curve, τ = 1 ns; dashed curve, τ = 2 ns). (c) Phase with respect to the dc phase (which is zero). The solid curve and the open circles represent phases for τ = 1 ns calculated by analytic solutions and the finite difference method, respectively. The dashed curve and filled circles represent phases for τ = 2 ns calculated by analytic solutions and the finite difference method, respectively. (d) Phase residues that are the difference between the phase calculated by the finite difference method and the phase calculated by analytic solutions (solid curve, τ = 1 ns; dashed curve, τ = 2 ns).

Fig. 4
Fig. 4

Fluorescent DPDW [Eq. (6)] versus the modulation frequency (f) in an infinite homogeneous turbid medium for three different lifetimes (τ = 1, 2, and 5 ns): (a) normalized amplitude, (b) phase shift. The amplitude is normalized by the amplitude at zero modulation frequency (f = 0) for each curve at the corresponding fluorophore lifetime, and the phase is phase shifted with respect to the phase at the zero modulation frequency. Solid curves, τ = 1 ns; dashed curves, τ = 2 ns; dash–dotted curve, τ = 5 ns. Source–detector separation rsd , 6 cm. The optical properties (μ a 1 c , μ s 1′ for incident DPDW at λ ex and μ a 1 f c , μ s 1 f ′ for fluorescent DPDW at λ fl ) are given in Table 1.

Fig. 5
Fig. 5

Contour plots in a plane that contains the source and the object illustrating the required phase and amplitude precision to detect and localize an object of 0.6-cm radius. (a) Contours of the normalized amplitude. (b) Contours of the phase shift. The optical properties are given in Table 1. Fluorophore lifetimes inside and outside the sphere are assumed to be equal (τ1 = τ2 = 1 ns), and the fluorophore concentration contrast is assumed to be 5.

Fig. 6
Fig. 6

Fluorescent DPDW versus fluorophore concentration contrast in a heterogeneous system. The source and the detector are separated by 6 cm, and the object is halfway between them. In the simulation the lifetimes inside and outside the sphere are assumed to be equal (τ1 = τ2 = 1 ns). The system parameters are given in Table 1. The amplitude is normalized by the amplitude in a homogeneous system, and the phase shift is with respect to the phase in a homogeneous system.

Fig. 7
Fig. 7

Fluorescent DPDW versus a fluorophore lifetime change in a heterogeneous system. An object is centered between the source and the detector for a source–detector separation, rsd = 6 cm. We assume that the lifetime outside the sphere is τ1 = 1 ns and that the fluorophore concentration contrast, N 2/N 1, is 5. The parameters are given in Table 1. The amplitude is normalized by the amplitude in a homogeneous system, and the phase shift is with respect to the phase in a homogeneous system. The nosie level is indicated by the dashed–dotted curves.

Fig. 8
Fig. 8

Comparison of absorption and fluorescence of fluorophores in a heterogeneous medium. An object is centered between the source and the detector for rsd = 6 cm. We assume that the lifetimes outside and inside the sphere are equal (τ1 = τ2 = 1 ns) and that the fluorophore concentration contrast, N 2/N 1, is 5. The parameters are given in Table 1. The amplitude is normalized by the corresponding amplitude in a homogeneous system, and the phase shift is with respect to the corresponding phase in a homogeneous system.

Tables (1)

Tables Icon

Table 1 Chromophore Optical Properties at λ ex and λ fl and Other Parameters a

Equations (48)

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

2 ϕ ( r , t ) υ μ a D ϕ ( r , t ) 1 D ϕ ( r , t ) t = υ D S ( r , t ) .
ϕ 0 ( r , r s ) = υ M 0 D exp ( i k | r r s | ) 4 π | r r s | .
N e ( r , r s ) = σ N t η Γ i ω ϕ 0 ( r , r s ) ,
S f ( r , r s ) = Γ N e ( r , r s ) = σ N t 1 i ω τ ϕ 0 ( r , r s ) .
ϕ 0 f ( r , r s ) = S f ( r 1 , r s ) υ D f exp ( i k f | r r 1 | ) 4 π | r r 1 | d r 1 .
ϕ 0 f ( r , r s ) = υ 2 M 0 D D f σ N t 1 i ω τ 1 k 2 k f 2 × [ exp ( i k | r r s | ) 4 π | r r s | exp ( i k f | r r s | ) 4 π | r r s | ] .
ϕ 1 ( r , r s ) = ϕ 10 ( r , r s ) + ϕ 1 sc ( r , r s ) = ϕ 10 ( r , r s ) + lm A lm ( r s ) h l ( 1 ) ( k 1 r ) Y lm ( Ω ) ,
ϕ 2 ( r , r s ) = lm B lm ( r s ) j l ( k 2 r ) Y lm ( Ω ) ,
ϕ 10 ( r , r s ) = υ M 0 D 1 exp ( i k 1 | r r s | ) 4 π | r r s | ,
ϕ 1 ( a , r s ) = ϕ 2 ( a , r s ) , D 1 ϕ 1 ( r , r s ) r | r = a = D 2 ϕ 2 ( r , r s ) r | r = a .
A lm ( r s ) = υ i k 1 D 1 Q l h l ( 1 ) ( k 1 r s ) Y lm * ( Ω s ) M 0 ,
B lm ( r s ) = υ R l h l ( 1 ) ( k 1 r s ) Y lm * ( Ω s ) M 0 ,
Φ 1 ( r , r s ) = lm C lm ( r s ) h l ( 1 ) ( k 1 r ) Y lm ( Ω ) ,
Φ 2 ( r , r s ) = Φ 20 ( r , r s ) + Φ 2 sc ( r , r s ) = Φ 20 ( r , r s ) + lm D lm ( r s ) j l ( k 2 r ) Y lm ( Ω ) .
C lm ( r s ) = υ R l j l ( k 2 r s ) Y lm * ( Ω s ) M 0 ,
D lm ( r s ) = υ i k 2 D 2 S l j l ( k 2 r s ) Y lm * ( Ω s ) M 0 ,
S 1 f ( r 1 , r s ) = σ N 1 1 i ω τ 1 ϕ 1 ( r 1 , r s ) ,
S 2 f ( r 2 , r s ) = σ N 2 1 i ω τ 2 ϕ 2 ( r 2 , r s ) ,
δ Φ 1 f ( r , r 2 , r s ) = lm δ C lm f ( r 2 ) h l ( 1 ) ( k 1 f r ) Y lm ( Ω ) ,
δ C lm f ( r 2 ) = υ R l f j l ( k 2 f r 2 ) Y lm * ( Ω 2 ) S 2 f ( r 2 , r s ) d r 2 ,
Φ 1 f ( r , r s ) = Inside δ Φ 1 f ( r , r 2 , r s ) = M 0 υ 2 σ N 2 1 i ω τ 2 a 2 k 2 2 k 2 f 2 × lm { [ k 2 f j l ( k 2 a ) j l ( k 2 f a ) k 2 j l ( k 2 a ) × j l ( k 2 f a ) ] R l R l f h l ( 1 ) ( k 1 f r ) h l ( 1 ) ( k 1 r s ) × Y lm ( Ω ) Y lm * ( Ω s ) } ,
δ ϕ 1 f ( r , r 1 , r s ) = δ ϕ 10 f ( r , r 1 , r s ) + δ ϕ 1 scf ( r , r 1 , r s ) = δ ϕ 10 f ( r , r 1 , r s ) + lm δ A lm f ( r 1 ) × h l ( 1 ) ( k 1 f r ) Y lm ( Ω ) ,
δ ϕ 10 f ( r , r 1 , r s ) = δ ϕ 10 f ( r , r 1 ) S 1 f ( r 1 , r s ) d r 1 = υ D 1 f exp ( i k 1 f | r r 1 | ) 4 π | r r 1 | × S 1 f ( r 1 , r s ) d r 1 ,
δ A lm f ( r 1 ) = ϕ 1 scf ( r , r 1 ) S 1 f ( r 1 , r s ) d r 1 = [ υ i k 1 f D 1 f Q l f h l ( 1 ) ( k 1 f r 1 ) Y lm * ( Ω 1 ) ] × S 1 f ( r 1 , r s ) d r 1 ,
ϕ 1 f ( r , r s ) = outside [ ϕ 10 f ( r , r 1 ) + ϕ 1 scf ( r , r 1 ) ] σ N 1 1 i ω τ 1 × [ ϕ 10 ( r 1 , r s ) + ϕ 1 sc ( r 1 , r s ) ] d r 1 = υ 2 M 0 D 1 D 1 f σ N 1 1 i ω τ 1 1 k 1 2 k 1 f 2 × { [ exp ( i k 1 | r r s | ) 4 π | r r s | exp ( i k 1 f | r r s | ) 4 π | r r s | ] + [ ( i k 1 f ) lm Q l f h l ( 1 ) ( k 1 f r ) h l ( 1 ) ( k 1 f r s ) + ( i k 1 ) lm Q l h l ( 1 ) ( k 1 r ) h l ( 1 ) ( k 1 r s ) + ( i k 1 ) lm Q l E l h l ( 1 ) ( k 1 f r ) h l ( 1 ) ( k 1 r s ) + ( i k 1 ) lm F l h l ( 1 ) ( k 1 f r ) h l ( 1 ) ( k 1 r s ) ] × Y lm * ( Ω s ) Y lm ( Ω ) } ,
ϕ hetero ( r , r s ) f l = ϕ 1 f ( r , r s ) + Φ 1 f ( r , r s ) .
ϕ 0 f s i z . b . c ( r , r s ) = ϕ 0 f ( r , r s ) ϕ 0 f ( r , r s zbc _ )
ϕ 0 f s i e . b . c ( r , r s ) = ϕ 0 f ( r , r s ) ϕ 0 f ( r , r s ebc _ ) ,
θ 21 = tan 1 [ ω ( τ 2 τ 1 ) 1 + ω 2 τ 1 τ 2 ] = tan 1 ( ω τ 2 ) tan 1 ( ω τ 1 ) ,
G ( r 1 , r 2 , k ) = exp ( i k | r 1 r 2 | ) 4 π | r 1 r 2 | = i k lm j l ( k r < ) × h l ( 1 ) ( k r > ) Y lm * ( Ω 1 ) Y lm ( Ω 2 ) ,
4 π Y lm ( Ω ) Y l m * ( Ω ) d Ω = δ l , l δ m , m .
f l ( k 1 r ) g l ( k 2 r ) r 2 d r = r 2 k 1 2 k 2 2 [ k 2 f l ( k 1 r ) g l ( k 2 r ) k 1 f l ( k 1 r ) g l ( k 2 r ) ] ,
Q l = D 2 k 2 j l ( k 1 a ) j l ( k 2 a ) D 1 k 1 j l ( k 1 a ) j l ( k 2 a ) D 2 k 2 h l ( 1 ) ( k 1 a ) j l ( k 2 a ) D 1 k 1 h l ( 1 ) ( k 1 a ) j l ( k 2 a ) ,
Q l f = D 2 f k 2 f j l ( k 1 f a ) j l ( k 2 f a ) D 1 f k 1 f j l ( k 1 f a ) j l ( k 2 f a ) D 2 f k 2 f h l ( 1 ) ( k 1 f a ) j l ( k 2 f a ) D 1 f k 1 f h l ( 1 ) ( k 1 f a ) j l ( k 2 f a ) ,
R l = 1 a 2 [ D 2 k 2 h l ( 1 ) ( k 1 a ) j l ( k 2 a ) D 1 k 1 h l ( 1 ) ( k 1 a ) j l ( k 2 a ) ] ,
R l f = 1 a 2 [ D 2 f k 2 f h l ( 1 ) ( k 1 f a ) j l ( k 2 f a ) D 1 f k 1 f h l ( 1 ) ( k 1 f a ) j l ( k 2 f a ) ] ,
S l = D 2 k 2 h l ( 1 ) ( k 1 a ) h l ( 1 ) ( k 2 a ) D 1 k 1 h l ( 1 ) ( k 1 a ) h l ( 1 ) ( k 2 a ) D 2 k 2 h l ( 1 ) ( k 1 a ) j l ( k 2 a ) D 1 k 1 h l ( 1 ) ( k 1 a ) j l ( k 2 a ) ,
E l = D 2 f k 2 f h l ( 1 ) ( k 1 a ) j l ( k 2 f a ) D 1 f k 1 h l ( 1 ) ( k 1 a ) j l ( k 2 f a ) D 2 f k 2 f h l ( 1 ) ( k 1 f a ) j l ( k 2 f a ) D 1 f k 1 f h l ( 1 ) ( k 1 f a ) j l ( k 2 f a ) ,
F l = D 2 k 2 f j l ( k 1 a ) j l ( k 2 f a ) D 1 f k 1 j l ( k 1 a ) j l ( k 2 f a ) D 2 f k 2 f h l ( 1 ) ( k 1 f a ) j l ( k 2 f a ) D 1 f k 1 f h l ( 1 ) ( k 1 f a ) j l ( k 2 f a ) ,
ϕ 0 s i ( r , r s ) = υ M 0 D [ exp ( i k | r r s | ) 4 π | r r s | exp ( i k | r r s _ | ) 4 π | r r s _ | ] .
S f real ( r , r s ) = σ N t 1 i ω τ ϕ 0 s i ( r , r s ) .
S f img ( r _ , r s ) = S f real ( r , r s ) = σ N t 1 i ω τ ϕ 0 s i ( r , r s ) .
δ ϕ 0 f s i z . b . c . ( r , r , r s ) = [ υ S f real ( r , r s ) D f exp ( i k f | r r | ) 4 π | r r | + υ S f img ( r _ , r s ) D f exp ( i k f | r r _ | ) 4 π | r r _ | ] d r .
ϕ 0 f s i z . b . c . ( r , r s ) = z > 0 [ υ S f real ( r , r s ) D f exp ( i k f | r r | ) 4 π | r r | + υ S f img ( r _ , r s ) D f exp ( i k f | r r _ | ) 4 π | r r _ | ] d r = υ 2 M 0 D D f σ N t 1 i ω τ d x d y d z × [ exp ( i k | r r s | ) 4 π | r r s | exp ( i k f | r r | ) 4 π | r r | exp ( i k | r r s _ | ) 4 π | r r s _ | exp ( i k f | r r | ) 4 π | r r | exp ( i k | r r s | ) 4 π | r r s | exp ( i k f | r r _ | ) 4 π | r r _ | + exp ( i k | r r s _ | ) 4 π | r r s _ | exp ( i k f | r r _ | ) 4 π | r r _ | .
υ 2 M 0 D D f σ N t 1 i ω τ d x d y 0 d z × exp ( i k | r r s | ) 4 π | r r s | exp ( i k f | r r | ) 4 π | r r | .
( 1 ) + ( 4 ) = υ 2 M 0 D D f σ N t 1 i ω τ d r exp ( i k | r r s | ) 4 π | r r s | × exp ( i k f | r r | ) 4 π | r r | = ϕ 0 f ( r , r s ) .
( 2 ) + ( 3 ) = υ 2 M 0 D D f σ N t 1 i ω τ d r exp ( i k | r r s _ | ) 4 π | r r s _ | × exp ( i k f | r r | ) 4 π | r r | = ϕ 0 f ( r , r s _ ) .
ϕ o f s i z . b . c . ( r , r s ) = ϕ o f ( r , r s ) ϕ o f ( r , r s _ ) ,

Metrics