Abstract

In this paper we describe a photon migration approach for modeling fluorescence in an optically thick, turbid medium such as human tissue. In such a medium the intrinsic fluorescence spectrum of the fluorophores ϕ can be distorted by the interplay of many factors, including scattering and absorption, excitation and collection geometries, and boundary conditions. The model provides an analytical relationship between the bulk fluorescence spectrum F and the diffuse reflectance spectrum R for arbitrary geometries and boundary conditions. We demonstrate that the distortion can be simply and accurately removed by measuring R from the optically thick medium over the same wavelength range and in the same manner as F. Over a wide range of tissue parameters this relationship may be written as ϕ∝ F/Reff, with Reff a corrected form of the measured diffuse reflectance. The validity of this approach is demonstrated in both laboratory experiments on human aortic media and by comparison with Monte Carlo simulations and diffusion theory. Connection with a previous algorithm for extracting intrinsic fluorescence is also discussed.

© 1993 Optical Society of America

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References

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  1. R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. B. Ratliff, J. R. Kramer, M. S. Feld, “A one-layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
    [CrossRef] [PubMed]
  2. L. I. Deckelbaum, J. K. Lam, H. S. Cabin, K. S. Clubb, M. B. Long, “Discrimination of normal and atherosclerotic aorta by laser induced fluorescence,” Lasers Surg. Med. 7, 330–335 (1987).
    [CrossRef] [PubMed]
  3. M. Sartori, R. Sauerbrey, S. Kubodera, F. Tittel, R. Robert, P. Henry, “Autofluorescence maps of atherosclerotic human arteries—a new technique in medical imaging,” IEEE J. Quantum Electron. 23, 1794–1797 (1987).
    [CrossRef]
  4. R. R. Alfano, G. C. Pradhan, G. C. Tang, B. B. Das, K. M. Yoo, “Optical spectroscopy may offer novel diagnostic approaches for the medical profession,” in Laser Non-Surgical Medicine: New Challenges for an Old Application, L. Goldman, ed. (Technomic, Lancaster, Pa., 1991), and references therein.
  5. R. Nossal, R. F. Bonner, G. H. Weiss, “Influence of path length on remote optical sensing of properties of biological tissue,” Appl. Opt. 28, 2238–2244 (1989).
    [CrossRef] [PubMed]
  6. J. Wu, F. Partovi, M. S. Feld, R. P. Rava, “Diffuse reflectance from turbid media: an analytical model of photon migration,” Appl. Opt. 32, 1115–1121 (1993).
    [CrossRef] [PubMed]
  7. S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. Miller, D. Sliney, R. Potter, eds. Proc. Soc. Photo-Opt. Instrum. Eng.IS5, 102–111 (1989).
  8. S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).
  9. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  10. S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas at Austin, Austin, Tex., 1988).
  11. W. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron 26, 2166–2185 (1990).
    [CrossRef]
  12. J. Wu, “Light–tissue interactions: applications to diffuse reflectance and fluorescence spectroscopy of human tissue,” M.S. thesis (Department of Physics, Massachusetts Institute of Technology, Cambridge, Mass., 1992).
  13. S. K. Ma, Statistical Mechanics (World Scientific, Philadelphia, Pa., 1985).
  14. W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979).
  15. M. Keijzer, R. R. Richards-Kortum, S. L. Jacques, M. S. Feld, “Fluorescence spectroscopy of turbid media: autofluorescence of human aorta,” Appl. Opt. 28, 4286–4292 (1989).
    [CrossRef] [PubMed]
  16. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978), Vol. 1.
  17. J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
    [CrossRef]
  18. R. P. Rava, J. J. Baraga, M. S. Feld, “Near infrared Fourier transform Raman spectroscopy of human artery,” Spectrochim. Acta A 47, 509–512 (1991).
    [CrossRef]

1993

1991

R. P. Rava, J. J. Baraga, M. S. Feld, “Near infrared Fourier transform Raman spectroscopy of human artery,” Spectrochim. Acta A 47, 509–512 (1991).
[CrossRef]

1990

W. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron 26, 2166–2185 (1990).
[CrossRef]

1989

R. Nossal, R. F. Bonner, G. H. Weiss, “Influence of path length on remote optical sensing of properties of biological tissue,” Appl. Opt. 28, 2238–2244 (1989).
[CrossRef] [PubMed]

M. Keijzer, R. R. Richards-Kortum, S. L. Jacques, M. S. Feld, “Fluorescence spectroscopy of turbid media: autofluorescence of human aorta,” Appl. Opt. 28, 4286–4292 (1989).
[CrossRef] [PubMed]

R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. B. Ratliff, J. R. Kramer, M. S. Feld, “A one-layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
[CrossRef] [PubMed]

1987

L. I. Deckelbaum, J. K. Lam, H. S. Cabin, K. S. Clubb, M. B. Long, “Discrimination of normal and atherosclerotic aorta by laser induced fluorescence,” Lasers Surg. Med. 7, 330–335 (1987).
[CrossRef] [PubMed]

M. Sartori, R. Sauerbrey, S. Kubodera, F. Tittel, R. Robert, P. Henry, “Autofluorescence maps of atherosclerotic human arteries—a new technique in medical imaging,” IEEE J. Quantum Electron. 23, 1794–1797 (1987).
[CrossRef]

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

1976

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Alfano, R. R.

R. R. Alfano, G. C. Pradhan, G. C. Tang, B. B. Das, K. M. Yoo, “Optical spectroscopy may offer novel diagnostic approaches for the medical profession,” in Laser Non-Surgical Medicine: New Challenges for an Old Application, L. Goldman, ed. (Technomic, Lancaster, Pa., 1991), and references therein.

Alter, C. A.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Baraga, J. J.

R. P. Rava, J. J. Baraga, M. S. Feld, “Near infrared Fourier transform Raman spectroscopy of human artery,” Spectrochim. Acta A 47, 509–512 (1991).
[CrossRef]

Bonner, R. F.

Cabin, H. S.

L. I. Deckelbaum, J. K. Lam, H. S. Cabin, K. S. Clubb, M. B. Long, “Discrimination of normal and atherosclerotic aorta by laser induced fluorescence,” Lasers Surg. Med. 7, 330–335 (1987).
[CrossRef] [PubMed]

Cheong, W.

W. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron 26, 2166–2185 (1990).
[CrossRef]

Clubb, K. S.

L. I. Deckelbaum, J. K. Lam, H. S. Cabin, K. S. Clubb, M. B. Long, “Discrimination of normal and atherosclerotic aorta by laser induced fluorescence,” Lasers Surg. Med. 7, 330–335 (1987).
[CrossRef] [PubMed]

Das, B. B.

R. R. Alfano, G. C. Pradhan, G. C. Tang, B. B. Das, K. M. Yoo, “Optical spectroscopy may offer novel diagnostic approaches for the medical profession,” in Laser Non-Surgical Medicine: New Challenges for an Old Application, L. Goldman, ed. (Technomic, Lancaster, Pa., 1991), and references therein.

Deckelbaum, L. I.

L. I. Deckelbaum, J. K. Lam, H. S. Cabin, K. S. Clubb, M. B. Long, “Discrimination of normal and atherosclerotic aorta by laser induced fluorescence,” Lasers Surg. Med. 7, 330–335 (1987).
[CrossRef] [PubMed]

Egan, W. G.

W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979).

Feld, M. S.

J. Wu, F. Partovi, M. S. Feld, R. P. Rava, “Diffuse reflectance from turbid media: an analytical model of photon migration,” Appl. Opt. 32, 1115–1121 (1993).
[CrossRef] [PubMed]

R. P. Rava, J. J. Baraga, M. S. Feld, “Near infrared Fourier transform Raman spectroscopy of human artery,” Spectrochim. Acta A 47, 509–512 (1991).
[CrossRef]

R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. B. Ratliff, J. R. Kramer, M. S. Feld, “A one-layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
[CrossRef] [PubMed]

M. Keijzer, R. R. Richards-Kortum, S. L. Jacques, M. S. Feld, “Fluorescence spectroscopy of turbid media: autofluorescence of human aorta,” Appl. Opt. 28, 4286–4292 (1989).
[CrossRef] [PubMed]

Fitzmaurice, M.

R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. B. Ratliff, J. R. Kramer, M. S. Feld, “A one-layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
[CrossRef] [PubMed]

Henry, P.

M. Sartori, R. Sauerbrey, S. Kubodera, F. Tittel, R. Robert, P. Henry, “Autofluorescence maps of atherosclerotic human arteries—a new technique in medical imaging,” IEEE J. Quantum Electron. 23, 1794–1797 (1987).
[CrossRef]

Hilgeman, T. W.

W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978), Vol. 1.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Jacques, S. L.

M. Keijzer, R. R. Richards-Kortum, S. L. Jacques, M. S. Feld, “Fluorescence spectroscopy of turbid media: autofluorescence of human aorta,” Appl. Opt. 28, 4286–4292 (1989).
[CrossRef] [PubMed]

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. Miller, D. Sliney, R. Potter, eds. Proc. Soc. Photo-Opt. Instrum. Eng.IS5, 102–111 (1989).

Joseph, J. H.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Keijzer, M.

M. Keijzer, R. R. Richards-Kortum, S. L. Jacques, M. S. Feld, “Fluorescence spectroscopy of turbid media: autofluorescence of human aorta,” Appl. Opt. 28, 4286–4292 (1989).
[CrossRef] [PubMed]

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. Miller, D. Sliney, R. Potter, eds. Proc. Soc. Photo-Opt. Instrum. Eng.IS5, 102–111 (1989).

Kramer, J. R.

R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. B. Ratliff, J. R. Kramer, M. S. Feld, “A one-layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
[CrossRef] [PubMed]

Kubodera, S.

M. Sartori, R. Sauerbrey, S. Kubodera, F. Tittel, R. Robert, P. Henry, “Autofluorescence maps of atherosclerotic human arteries—a new technique in medical imaging,” IEEE J. Quantum Electron. 23, 1794–1797 (1987).
[CrossRef]

Lam, J. K.

L. I. Deckelbaum, J. K. Lam, H. S. Cabin, K. S. Clubb, M. B. Long, “Discrimination of normal and atherosclerotic aorta by laser induced fluorescence,” Lasers Surg. Med. 7, 330–335 (1987).
[CrossRef] [PubMed]

Long, M. B.

L. I. Deckelbaum, J. K. Lam, H. S. Cabin, K. S. Clubb, M. B. Long, “Discrimination of normal and atherosclerotic aorta by laser induced fluorescence,” Lasers Surg. Med. 7, 330–335 (1987).
[CrossRef] [PubMed]

Ma, S. K.

S. K. Ma, Statistical Mechanics (World Scientific, Philadelphia, Pa., 1985).

Nossal, R.

Partovi, F.

Pradhan, G. C.

R. R. Alfano, G. C. Pradhan, G. C. Tang, B. B. Das, K. M. Yoo, “Optical spectroscopy may offer novel diagnostic approaches for the medical profession,” in Laser Non-Surgical Medicine: New Challenges for an Old Application, L. Goldman, ed. (Technomic, Lancaster, Pa., 1991), and references therein.

Prahl, S. A.

W. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron 26, 2166–2185 (1990).
[CrossRef]

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas at Austin, Austin, Tex., 1988).

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. Miller, D. Sliney, R. Potter, eds. Proc. Soc. Photo-Opt. Instrum. Eng.IS5, 102–111 (1989).

Ratliff, N. B.

R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. B. Ratliff, J. R. Kramer, M. S. Feld, “A one-layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
[CrossRef] [PubMed]

Rava, R. P.

J. Wu, F. Partovi, M. S. Feld, R. P. Rava, “Diffuse reflectance from turbid media: an analytical model of photon migration,” Appl. Opt. 32, 1115–1121 (1993).
[CrossRef] [PubMed]

R. P. Rava, J. J. Baraga, M. S. Feld, “Near infrared Fourier transform Raman spectroscopy of human artery,” Spectrochim. Acta A 47, 509–512 (1991).
[CrossRef]

R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. B. Ratliff, J. R. Kramer, M. S. Feld, “A one-layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
[CrossRef] [PubMed]

Richards-Kortum, R.

R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. B. Ratliff, J. R. Kramer, M. S. Feld, “A one-layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
[CrossRef] [PubMed]

Richards-Kortum, R. R.

Robert, R.

M. Sartori, R. Sauerbrey, S. Kubodera, F. Tittel, R. Robert, P. Henry, “Autofluorescence maps of atherosclerotic human arteries—a new technique in medical imaging,” IEEE J. Quantum Electron. 23, 1794–1797 (1987).
[CrossRef]

Sartori, M.

M. Sartori, R. Sauerbrey, S. Kubodera, F. Tittel, R. Robert, P. Henry, “Autofluorescence maps of atherosclerotic human arteries—a new technique in medical imaging,” IEEE J. Quantum Electron. 23, 1794–1797 (1987).
[CrossRef]

Sauerbrey, R.

M. Sartori, R. Sauerbrey, S. Kubodera, F. Tittel, R. Robert, P. Henry, “Autofluorescence maps of atherosclerotic human arteries—a new technique in medical imaging,” IEEE J. Quantum Electron. 23, 1794–1797 (1987).
[CrossRef]

Tang, G. C.

R. R. Alfano, G. C. Pradhan, G. C. Tang, B. B. Das, K. M. Yoo, “Optical spectroscopy may offer novel diagnostic approaches for the medical profession,” in Laser Non-Surgical Medicine: New Challenges for an Old Application, L. Goldman, ed. (Technomic, Lancaster, Pa., 1991), and references therein.

Tittel, F.

M. Sartori, R. Sauerbrey, S. Kubodera, F. Tittel, R. Robert, P. Henry, “Autofluorescence maps of atherosclerotic human arteries—a new technique in medical imaging,” IEEE J. Quantum Electron. 23, 1794–1797 (1987).
[CrossRef]

Tong, L. L.

R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. B. Ratliff, J. R. Kramer, M. S. Feld, “A one-layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
[CrossRef] [PubMed]

Weinman, J. A.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Weiss, G. H.

Welch, A. J.

W. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron 26, 2166–2185 (1990).
[CrossRef]

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. Miller, D. Sliney, R. Potter, eds. Proc. Soc. Photo-Opt. Instrum. Eng.IS5, 102–111 (1989).

Wiscombe, W. J.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Wu, J.

J. Wu, F. Partovi, M. S. Feld, R. P. Rava, “Diffuse reflectance from turbid media: an analytical model of photon migration,” Appl. Opt. 32, 1115–1121 (1993).
[CrossRef] [PubMed]

J. Wu, “Light–tissue interactions: applications to diffuse reflectance and fluorescence spectroscopy of human tissue,” M.S. thesis (Department of Physics, Massachusetts Institute of Technology, Cambridge, Mass., 1992).

Yoo, K. M.

R. R. Alfano, G. C. Pradhan, G. C. Tang, B. B. Das, K. M. Yoo, “Optical spectroscopy may offer novel diagnostic approaches for the medical profession,” in Laser Non-Surgical Medicine: New Challenges for an Old Application, L. Goldman, ed. (Technomic, Lancaster, Pa., 1991), and references therein.

Appl. Opt.

IEEE J. Quantum Electron

W. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron 26, 2166–2185 (1990).
[CrossRef]

IEEE J. Quantum Electron.

M. Sartori, R. Sauerbrey, S. Kubodera, F. Tittel, R. Robert, P. Henry, “Autofluorescence maps of atherosclerotic human arteries—a new technique in medical imaging,” IEEE J. Quantum Electron. 23, 1794–1797 (1987).
[CrossRef]

IEEE Trans. Biomed. Eng.

R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. B. Ratliff, J. R. Kramer, M. S. Feld, “A one-layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
[CrossRef] [PubMed]

J. Atmos. Sci.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Lasers Life Sci.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Lasers Surg. Med.

L. I. Deckelbaum, J. K. Lam, H. S. Cabin, K. S. Clubb, M. B. Long, “Discrimination of normal and atherosclerotic aorta by laser induced fluorescence,” Lasers Surg. Med. 7, 330–335 (1987).
[CrossRef] [PubMed]

Spectrochim. Acta A

R. P. Rava, J. J. Baraga, M. S. Feld, “Near infrared Fourier transform Raman spectroscopy of human artery,” Spectrochim. Acta A 47, 509–512 (1991).
[CrossRef]

Other

R. R. Alfano, G. C. Pradhan, G. C. Tang, B. B. Das, K. M. Yoo, “Optical spectroscopy may offer novel diagnostic approaches for the medical profession,” in Laser Non-Surgical Medicine: New Challenges for an Old Application, L. Goldman, ed. (Technomic, Lancaster, Pa., 1991), and references therein.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978), Vol. 1.

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. Miller, D. Sliney, R. Potter, eds. Proc. Soc. Photo-Opt. Instrum. Eng.IS5, 102–111 (1989).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas at Austin, Austin, Tex., 1988).

J. Wu, “Light–tissue interactions: applications to diffuse reflectance and fluorescence spectroscopy of human tissue,” M.S. thesis (Department of Physics, Massachusetts Institute of Technology, Cambridge, Mass., 1992).

S. K. Ma, Statistical Mechanics (World Scientific, Philadelphia, Pa., 1985).

W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979).

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Figures (7)

Fig. 1
Fig. 1

Monte Carlo simulations of fn(g) for different collection geometries and boundary conditions. The collection geometries and boundary conditions are summarized in Table 1.

Fig. 2
Fig. 2

Comparison of Eq. (2′) with Monte Carlo calculations for two fiber geometries. The discrete points are Monte Carlo data, and the solid curves are the curve fits according to Eq. (2′). In 1(f), k0 = 0.0072, 1/S′ = 5.2 and in 1(h), k0 = 0.0056, 1/S′ = 1.8.

Fig. 3
Fig. 3

Photon migration picture of fluorescence.

Fig. 4
Fig. 4

Comparison of photon migration model with Monte Carlo for (a) intima and (b) media. The open circles are data points from Eq. (6), and the solid circles and Monte Carlo simulations. The curves are cubic spline curve fits to the data points.

Fig. 5
Fig. 5

Comparison of diffusion theory models with Monte Carlo for (a) intima and (b) media. The solid circles are from Monte Carlo simulations, the open circles and solid squares are from diffusion theory, Eddington approximation, and δ-Eddington approximation, respectively. The curves are cubic spline curve fits.

Fig. 6
Fig. 6

Experimental data from human aortic media soaked in saline, (a) Comparison of optically thick fluorescence (crosses), reflectance spectrum (circles), and thin fluorescence (squares). (b) Comparison of the intrinsic fluorescence extracted by using the model (crosses) and measured from thin sample (squares); residuals (circles) are included.

Fig. 7
Fig. 7

Experimental data from human aortic media soaked in lysed blood, (a) Comparison of optically thick fluorescence (crosses), reflectance spectrum (circles), and thin fluorescence (squares), (b) Comparison of the intrinsic fluorescence extracted by using the model (crosses) and measured from thin sample (squares); residuals (circles) are included.

Tables (2)

Tables Icon

Table 1 Summary of the Geometries and Boundary Conditions for Calculating the fn(g = 0.8) Curves Shown in Fig. 1

Tables Icon

Table 2 Summary of the Aortic Tissue Optical Parameters from the Literature15 Utilized in the Monte Carlo Simulations

Equations (29)

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

F ( λ x , λ m ) = μ a ( λ x ) ϕ ( λ x , λ m ) × [ 4 π A ( λ x ) + F 0 μ t ( λ x ) + κ d ( λ m ) + 4 π C 2 ( λ x ) κ d ( λ x ) + κ d ( λ m ) ] C ( λ m ) ,
R ( a , g ) = n = 1 a n f n ( g ) .
R ( μ a , μ s , g ) = 1 1 1 S ln ( a ) 1 g ,
R k 0 = 1 1 1 S ln ( a ) 1 g ,
{ a n ( λ x ) | m = 0 n 1 a m ( λ x ) [ 1 a ( λ x ) ] ϕ ( λ x , λ m ) a n m 1 ( λ m ) } .
F ( λ x , λ m ) = n = 1 + f n ( g ) { m = 0 n 1 a m ( λ x ) [ 1 a ( λ x ) ] ϕ ( λ x , λ m ) a n m 1 ( λ m ) } = [ 1 a ( λ x ) ] ϕ ( λ x , λ m ) R [ a ( λ x ) , g ] R [ a ( λ m ) , g ] a ( λ x ) a ( λ m ) ,
R ( a , g ) = n = 1 a n f n ( g ) .
g eff = ( N 1 ) g N .
a eff = a [ μ t 2 ( λ m ) ] / [ μ t 2 ( λ x ) ]
a eff = r d a + ( 1 r d ) a [ μ t 2 ( λ m ) ] / [ μ t 2 ( λ x ) ] ,
F ( λ x , λ m ) = [ 1 a ( λ x ) ] ϕ ( λ x , λ m ) × R [ a ( λ x ) , g eff ] R [ a eff ( λ m ) , g eff ] a ( λ x ) a eff ( λ m ) .
F ( λ x , λ m ) = ϕ ( λ x , λ m ) { 1 R [ a ( λ x ) , g eff ] } R [ a eff ( λ m ) , g eff ] ,
ϕ ( λ m ) = q F ( λ m ) R [ a eff ( λ m ) , g eff ] ,
1 R eff 1 1 R 1 = 1 g 1 g eff [ r d + ( 1 r d ) μ t 2 ( λ m ) μ t 2 ( λ x ) ] 1 g 1 g eff [ r d + ( 1 r d ) μ s 2 ( λ m ) μ s 2 ( λ x ) ] .
ϕ ( λ m ) = i c i ( μ a ) i μ a φ i ( λ m ) = i c i φ i ( λ m ) ,
F ( λ m ) = k μ a ϕ ( λ m ) / A ( λ x , λ m ) ,
I r i ( z , ω ) = F 0 exp ( μ t z ) δ ( ω ω z ) ,
d 2 U d ( z ) d z 2 κ d 2 U d ( z ) = Q 0 exp ( μ t z ) .
U d ( z ) ( mW mm 2 Hz 1 ) = 1 4 π 4 π I d d ω ( mW mm 2 Hz 1 ) , Q 0 = 3 F 0 ( μ s μ t r + g μ s μ t ) 4 π , κ d = ( 3 μ t r μ a ) 1 / 2 .
U d ( z ) h d U d ( z ) d z + Q 1 2 π = 0 at z = 0 , U d ( z ) = 0 at z = + ,
U d ( z ) = A exp ( μ t z ) + C 2 exp ( κ d z ) ,
A = Q 0 ( μ t 2 κ d 2 ) , C 2 = A ( 1 + μ t h ) ( 1 + κ d h ) Q 1 2 π ( 1 + κ d h ) .
I total ( z ) = 4 π U d ( z ) + I r i ( z ) .
d 2 U d ( z ) d z 2 κ d 2 U d ( z ) = 3 4 π μ t r P 0 δ ( z = 0 ) ,
U d ( z ) + h d U d ( z ) d z = 0 at z = z 0 , U d ( z ) = 0 at z = .
U d ( z ) = B exp ( κ d z ) + 3 P 0 8 π μ t r κ d exp ( κ d z ) , 0 < z < z 0 ,
B = P 0 4 π exp ( 2 κ d z 0 ) 1 3 μ tr / 2 κ d 1 + 2 κ d / 3 μ tr .
F d = 4 π 3 μ t r d U d ( z ) d z = C P 0 exp ( κ d z 0 ) ,
F ( λ x , λ m ) = 0 + d z ( { [ 4 π A ( λ x ) + F 0 ] × exp [ μ t ( λ x ) z ] + 4 π C 2 ( λ x ) exp [ κ d ( λ x ) z ] } × μ a ( λ x ) ϕ ( λ x , λ m ) C ( λ m ) exp [ κ d ( λ m ) z ] ) = μ a ( λ x ) ϕ ( λ x , λ m ) × [ 4 π A ( λ x ) + F 0 μ t ( λ x ) + κ d ( λ m ) + 4 π C 2 ( λ x ) κ d ( λ x ) + κ d ( λ m ) ] C ( λ m ) ,

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