Abstract

The diffusion approximation of the radiative transfer equation is a model used widely to describe photon migration in highly diffusing media and is an important matter in biological tissue optics. An analysis of the time-dependent diffusion equation together with its solutions for the slab geometry and for a semi-infinite diffusing medium are reported. These solutions, presented for both the time-dependent and the continuous wave source, account for the refractive index mismatch between the turbid medium and the surrounding medium. The results have been compared with those obtained when different boundary conditions were assumed. The comparison has shown that the effect of the refractive index mismatch cannot be disregarded. This effect is particularly important for the transmittance. The discussion of results also provides an analysis of the role of the absorption coefficient in the expression of the diffusion coefficient.

© 1997 Optical Society of America

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    [CrossRef] [PubMed]
  3. J. C. Hebden, D. T. Delpy, “Enhanced time-resolved imaging with a diffusion model of photon transport,” Opt. Lett. 19, 311–313 (1994).
    [CrossRef] [PubMed]
  4. G. Zaccanti, D. Contini, M. Gurioli, A. Ismaelli, H. Liszka, A. Sassaroli, “Detectability of inhomogeneities within highly diffusing 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, 755–762 (1995).
  5. S. Fantini, M. A. Franceschini, E. Gratton, “Quantitative determination of the absorption-spectra of cromophores in strongly scattered media. A light-emitting-diode based technique,” Appl. Opt. 33, 5204–5213 (1994).
    [CrossRef] [PubMed]
  6. M. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  7. S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
    [CrossRef] [PubMed]
  8. M. S. Patterson, S. J. Madsen, J. D. Moulton, B. C. Wilson, “Diffusion equation representation of photon migration in tissue,” in IEEE Microwave Theory and Techniques Symposium Digest, (IEEE, New York, 1991), Vol. BB-1, pp. 905–908.
  9. 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]
  10. K. Suzuki, Y. Yamashita, K. Ohta, M. Kaneko, M. Yoshida, B. Chance, “Quantitative measurements of optical parameters in normal breast using time-resolved spectroscopy: in vivo results of 30 Japanese women,” J. Biom. Opt. 1, 330–334 (1996).
    [CrossRef]
  11. U. Sukowski, F. Schubert, D. Grosenick, H. Rinneberg, “Preparation of solid phantoms with defined scattering and absorption properties for optical tomography,” Phys. Med. Biol. 41, 1823–1844 (1996).
    [CrossRef] [PubMed]
  12. F. Martelli, “Photon migration through highly scattering media and methodologies for measuring optical parameters of biological tissues” (in Italian), M.S. thesis (University of Florence, Florence, Italy, 1996).
  13. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Chap. 7, p. 157; Chap. 9, p. 175.
  14. S. Chandrasekhar, Radiative Transfer (Oxford, New York, 1969), Chap. 1, p. 9.
  15. K. Furutsu, “Diffusion equation derived from space–time transport equation,” J. Opt. Soc. Am. 70, 360–366 (1980).
    [CrossRef]
  16. E. P. Zege, A. I. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, New York, 1991), Chap. 2, p. 20.
  17. K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
    [CrossRef]
  18. S. Kumar, K. Mitra, Y. Yamada, “Hyperbolic damped-wave models for transient light-pulse propagation in scattering media,” Appl. Opt. 35, 3372–3378 (1996).
    [CrossRef] [PubMed]
  19. V. G. Kolinko, F. F. M. de Mul, J. Greve, A. V. Priezzhev, “Probabilistic model of multiple light scattering based on computation of the first and second moments of photon coordinates,” Appl. Opt. 35, 4541–4550 (1996).
    [CrossRef] [PubMed]
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  21. S. Glasstone, Principles of Nuclear Reactor Engineering (Macmillan, London, 1956), Chap. 3, p. 132.
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    [CrossRef] [PubMed]
  23. T. J. Farrel, M. S. Patterson, B. Wilson, “A diffusion theory model for spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888, (1992).
    [CrossRef]
  24. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), Chap. 3, p. 340.
  25. A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittel, “The influence of the boundary conditions on the accuracy of diffusion theory in the time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
    [CrossRef] [PubMed]
  26. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
    [CrossRef]
  27. B. Wilson, M. S. Patterson, D. M. Burns, “Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light,” Lasers Med. Sci. 1, 235–244 (1986).
    [CrossRef]

1996 (4)

K. Suzuki, Y. Yamashita, K. Ohta, M. Kaneko, M. Yoshida, B. Chance, “Quantitative measurements of optical parameters in normal breast using time-resolved spectroscopy: in vivo results of 30 Japanese women,” J. Biom. Opt. 1, 330–334 (1996).
[CrossRef]

U. Sukowski, F. Schubert, D. Grosenick, H. Rinneberg, “Preparation of solid phantoms with defined scattering and absorption properties for optical tomography,” Phys. Med. Biol. 41, 1823–1844 (1996).
[CrossRef] [PubMed]

S. Kumar, K. Mitra, Y. Yamada, “Hyperbolic damped-wave models for transient light-pulse propagation in scattering media,” Appl. Opt. 35, 3372–3378 (1996).
[CrossRef] [PubMed]

V. G. Kolinko, F. F. M. de Mul, J. Greve, A. V. Priezzhev, “Probabilistic model of multiple light scattering based on computation of the first and second moments of photon coordinates,” Appl. Opt. 35, 4541–4550 (1996).
[CrossRef] [PubMed]

1995 (2)

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

A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittel, “The influence of the boundary conditions on the accuracy of diffusion theory in the time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

1994 (5)

1992 (2)

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

T. J. Farrel, M. S. Patterson, B. Wilson, “A diffusion theory model for spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888, (1992).
[CrossRef]

1989 (1)

1988 (1)

1986 (1)

B. Wilson, M. S. Patterson, D. M. Burns, “Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light,” Lasers Med. Sci. 1, 235–244 (1986).
[CrossRef]

1983 (1)

1980 (1)

Aronson, R.

Arridge, S. R.

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Burns, D. M.

B. Wilson, M. S. Patterson, D. M. Burns, “Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light,” Lasers Med. Sci. 1, 235–244 (1986).
[CrossRef]

Chance, B.

K. Suzuki, Y. Yamashita, K. Ohta, M. Kaneko, M. Yoshida, B. Chance, “Quantitative measurements of optical parameters in normal breast using time-resolved spectroscopy: in vivo results of 30 Japanese women,” J. Biom. Opt. 1, 330–334 (1996).
[CrossRef]

M. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford, New York, 1969), Chap. 1, p. 9.

Contini, D.

G. Zaccanti, D. Contini, M. Gurioli, A. Ismaelli, H. Liszka, A. Sassaroli, “Detectability of inhomogeneities within highly diffusing 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, 755–762 (1995).

Cope, M.

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

de Mul, F. F. M.

Delpy, D. T.

J. C. Hebden, D. T. Delpy, “Enhanced time-resolved imaging with a diffusion model of photon transport,” Opt. Lett. 19, 311–313 (1994).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Fantini, S.

Farrel, T. J.

T. J. Farrel, M. S. Patterson, B. Wilson, “A diffusion theory model for spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888, (1992).
[CrossRef]

Feng, T. C.

Ferwerda, H. A.

Franceschini, M. A.

Furutsu, K.

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

K. Furutsu, “Diffusion equation derived from space–time transport equation,” J. Opt. Soc. Am. 70, 360–366 (1980).
[CrossRef]

Glasstone, S.

S. Glasstone, Principles of Nuclear Reactor Engineering (Macmillan, London, 1956), Chap. 3, p. 132.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), Chap. 3, p. 340.

Gratton, E.

Greve, J.

Groenhuis, R. A.

Grosenick, D.

U. Sukowski, F. Schubert, D. Grosenick, H. Rinneberg, “Preparation of solid phantoms with defined scattering and absorption properties for optical tomography,” Phys. Med. Biol. 41, 1823–1844 (1996).
[CrossRef] [PubMed]

Gurioli, M.

G. Zaccanti, D. Contini, M. Gurioli, A. Ismaelli, H. Liszka, A. Sassaroli, “Detectability of inhomogeneities within highly diffusing 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, 755–762 (1995).

Haskell, R. C.

Hebden, J. C.

Hielscher, A. H.

A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittel, “The influence of the boundary conditions on the accuracy of diffusion theory in the time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Chap. 7, p. 157; Chap. 9, p. 175.

Ismaelli, A.

G. Zaccanti, D. Contini, M. Gurioli, A. Ismaelli, H. Liszka, A. Sassaroli, “Detectability of inhomogeneities within highly diffusing 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, 755–762 (1995).

Ivanov, A. I.

E. P. Zege, A. I. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, New York, 1991), Chap. 2, p. 20.

Jacques, S. L.

A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittel, “The influence of the boundary conditions on the accuracy of diffusion theory in the time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

Kaneko, M.

K. Suzuki, Y. Yamashita, K. Ohta, M. Kaneko, M. Yoshida, B. Chance, “Quantitative measurements of optical parameters in normal breast using time-resolved spectroscopy: in vivo results of 30 Japanese women,” J. Biom. Opt. 1, 330–334 (1996).
[CrossRef]

Katsev, I. L.

E. P. Zege, A. I. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, New York, 1991), Chap. 2, p. 20.

Keijzer, M.

Kolinko, V. G.

Kolzer, J.

Kumar, S.

Liszka, H.

G. Zaccanti, D. Contini, M. Gurioli, A. Ismaelli, H. Liszka, A. Sassaroli, “Detectability of inhomogeneities within highly diffusing 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, 755–762 (1995).

Madsen, S. J.

M. S. Patterson, S. J. Madsen, J. D. Moulton, B. C. Wilson, “Diffusion equation representation of photon migration in tissue,” in IEEE Microwave Theory and Techniques Symposium Digest, (IEEE, New York, 1991), Vol. BB-1, pp. 905–908.

Martelli, F.

F. Martelli, “Photon migration through highly scattering media and methodologies for measuring optical parameters of biological tissues” (in Italian), M.S. thesis (University of Florence, Florence, Italy, 1996).

McAdams, M. S.

Mitic, G.

Mitra, K.

Moulton, J. D.

M. S. Patterson, S. J. Madsen, J. D. Moulton, B. C. Wilson, “Diffusion equation representation of photon migration in tissue,” in IEEE Microwave Theory and Techniques Symposium Digest, (IEEE, New York, 1991), Vol. BB-1, pp. 905–908.

Ohta, K.

K. Suzuki, Y. Yamashita, K. Ohta, M. Kaneko, M. Yoshida, B. Chance, “Quantitative measurements of optical parameters in normal breast using time-resolved spectroscopy: in vivo results of 30 Japanese women,” J. Biom. Opt. 1, 330–334 (1996).
[CrossRef]

Otto, J.

Patterson, M.

Patterson, M. S.

T. J. Farrel, M. S. Patterson, B. Wilson, “A diffusion theory model for spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888, (1992).
[CrossRef]

B. Wilson, M. S. Patterson, D. M. Burns, “Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light,” Lasers Med. Sci. 1, 235–244 (1986).
[CrossRef]

M. S. Patterson, S. J. Madsen, J. D. Moulton, B. C. Wilson, “Diffusion equation representation of photon migration in tissue,” in IEEE Microwave Theory and Techniques Symposium Digest, (IEEE, New York, 1991), Vol. BB-1, pp. 905–908.

Plies, E.

Priezzhev, A. V.

Rinneberg, H.

U. Sukowski, F. Schubert, D. Grosenick, H. Rinneberg, “Preparation of solid phantoms with defined scattering and absorption properties for optical tomography,” Phys. Med. Biol. 41, 1823–1844 (1996).
[CrossRef] [PubMed]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), Chap. 3, p. 340.

Sassaroli, A.

G. Zaccanti, D. Contini, M. Gurioli, A. Ismaelli, H. Liszka, A. Sassaroli, “Detectability of inhomogeneities within highly diffusing 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, 755–762 (1995).

Schubert, F.

U. Sukowski, F. Schubert, D. Grosenick, H. Rinneberg, “Preparation of solid phantoms with defined scattering and absorption properties for optical tomography,” Phys. Med. Biol. 41, 1823–1844 (1996).
[CrossRef] [PubMed]

Solkner, G.

Star, W. M.

Storchi, P. R. M.

Sukowski, U.

U. Sukowski, F. Schubert, D. Grosenick, H. Rinneberg, “Preparation of solid phantoms with defined scattering and absorption properties for optical tomography,” Phys. Med. Biol. 41, 1823–1844 (1996).
[CrossRef] [PubMed]

Suzuki, K.

K. Suzuki, Y. Yamashita, K. Ohta, M. Kaneko, M. Yoshida, B. Chance, “Quantitative measurements of optical parameters in normal breast using time-resolved spectroscopy: in vivo results of 30 Japanese women,” J. Biom. Opt. 1, 330–334 (1996).
[CrossRef]

Svaasand, L. O.

ten Bosch, J. J.

Tittel, F. K.

A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittel, “The influence of the boundary conditions on the accuracy of diffusion theory in the time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

Tromberg, B. J.

Tsay, T. T.

Wang, L.

A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittel, “The influence of the boundary conditions on the accuracy of diffusion theory in the time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

Wilson, B.

T. J. Farrel, M. S. Patterson, B. Wilson, “A diffusion theory model for spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888, (1992).
[CrossRef]

B. Wilson, M. S. Patterson, D. M. Burns, “Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light,” Lasers Med. Sci. 1, 235–244 (1986).
[CrossRef]

Wilson, B. C.

M. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

M. S. Patterson, S. J. Madsen, J. D. Moulton, B. C. Wilson, “Diffusion equation representation of photon migration in tissue,” in IEEE Microwave Theory and Techniques Symposium Digest, (IEEE, New York, 1991), Vol. BB-1, pp. 905–908.

Yamada, Y.

S. Kumar, K. Mitra, Y. Yamada, “Hyperbolic damped-wave models for transient light-pulse propagation in scattering media,” Appl. Opt. 35, 3372–3378 (1996).
[CrossRef] [PubMed]

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Yamashita, Y.

K. Suzuki, Y. Yamashita, K. Ohta, M. Kaneko, M. Yoshida, B. Chance, “Quantitative measurements of optical parameters in normal breast using time-resolved spectroscopy: in vivo results of 30 Japanese women,” J. Biom. Opt. 1, 330–334 (1996).
[CrossRef]

Yoshida, M.

K. Suzuki, Y. Yamashita, K. Ohta, M. Kaneko, M. Yoshida, B. Chance, “Quantitative measurements of optical parameters in normal breast using time-resolved spectroscopy: in vivo results of 30 Japanese women,” J. Biom. Opt. 1, 330–334 (1996).
[CrossRef]

Zaccanti, G.

G. Zaccanti, D. Contini, M. Gurioli, A. Ismaelli, H. Liszka, A. Sassaroli, “Detectability of inhomogeneities within highly diffusing 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, 755–762 (1995).

Zege, E. P.

E. P. Zege, A. I. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, New York, 1991), Chap. 2, p. 20.

Zinth, W.

Appl. Opt. (7)

J. Biom. Opt. (1)

K. Suzuki, Y. Yamashita, K. Ohta, M. Kaneko, M. Yoshida, B. Chance, “Quantitative measurements of optical parameters in normal breast using time-resolved spectroscopy: in vivo results of 30 Japanese women,” J. Biom. Opt. 1, 330–334 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Lasers Med. Sci. (1)

B. Wilson, M. S. Patterson, D. M. Burns, “Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light,” Lasers Med. Sci. 1, 235–244 (1986).
[CrossRef]

Med. Phys. (1)

T. J. Farrel, M. S. Patterson, B. Wilson, “A diffusion theory model for spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888, (1992).
[CrossRef]

Opt. Lett. (1)

Phys. Med. Biol. (3)

A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittel, “The influence of the boundary conditions on the accuracy of diffusion theory in the time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

U. Sukowski, F. Schubert, D. Grosenick, H. Rinneberg, “Preparation of solid phantoms with defined scattering and absorption properties for optical tomography,” Phys. Med. Biol. 41, 1823–1844 (1996).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Phys. Rev. E (1)

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Other (9)

S. Glasstone, Principles of Nuclear Reactor Engineering (Macmillan, London, 1956), Chap. 3, p. 132.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), Chap. 3, p. 340.

M. S. Patterson, S. J. Madsen, J. D. Moulton, B. C. Wilson, “Diffusion equation representation of photon migration in tissue,” in IEEE Microwave Theory and Techniques Symposium Digest, (IEEE, New York, 1991), Vol. BB-1, pp. 905–908.

B. Chance, R. R. Alfano, eds., Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE2389, (1995).

G. Zaccanti, D. Contini, M. Gurioli, A. Ismaelli, H. Liszka, A. Sassaroli, “Detectability of inhomogeneities within highly diffusing 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, 755–762 (1995).

F. Martelli, “Photon migration through highly scattering media and methodologies for measuring optical parameters of biological tissues” (in Italian), M.S. thesis (University of Florence, Florence, Italy, 1996).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Chap. 7, p. 157; Chap. 9, p. 175.

S. Chandrasekhar, Radiative Transfer (Oxford, New York, 1969), Chap. 1, p. 9.

E. P. Zege, A. I. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, New York, 1991), Chap. 2, p. 20.

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Figures (5)

Fig. 1
Fig. 1

Slab geometry and some notations.

Fig. 2
Fig. 2

Positions of source dipoles.

Fig. 3
Fig. 3

Comparison between the solutions of the DE on the basis of the extrapolated boundary condition (continuous lines) for two values of the relative refractive index n = 1.4 (higher curve) and n = 1 (lower curve), and the zero boundary condition (dashed curve). Data refer to (a) the time-resolved transmittance at ρ = 0 and (b) the time-resolved reflectance at ρ = 40 mm for a slab 40-mm thick with μs′ = 0.5 mm-1 and μa = 0.

Fig. 4
Fig. 4

Comparison among the values of A obtained with different models. Values obtained with Eq. (53) (dotted curve), Eq. (57) (thin continuous curve) and Eqs. (27) and (29) (thick continuous curve) are reported.

Fig. 5
Fig. 5

Comparison between the results obtained from MC simulations and from the DE when D or is used. Data refer to the reflectance from a semi-infinite medium with μs′ = 1 mm-1, and n = 1.4: (a) TPSF at ρ = 30 mm for μa = 0.1 mm-1, the dashed noisy curve refers to MC results, the thin and thick curves refer to D and , respectively; (b) percentage of difference between the reflectance evaluated with the DE and the MC results for the cw source. Dashed and continuous curves refer to D and , respectively. The results are reported for three values of μa: 0.01, 0.05, and 0.1 mm-1 (lower to upper curve for D, upper to lower curve for ).

Equations (61)

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

1v t Ir, t, sˆ+sˆ·Ir, t, sˆ=-μtIr, t, sˆ+μt4π 4π psˆ, sˆIr, t, sˆdω+εr, t, sˆ,
4π psˆ, sˆds=4πμsμt.
εr, t, sˆ=δrδsˆδt.
Īr¯, t¯, sˆ=μ¯tμt3 Ir, t, sˆ,
r¯=rμtμ¯t,  t¯=tμtμ¯t,
Ir, t, sˆ=exp-μavtIr, t, sˆ|μa=0
Ir, t, sˆ=Udr, t+34π Fdr, t·sˆ,
Udr, t=14π 4π Ir, t, sˆdω
Fdr, t=4π Ir, t, sˆsˆdω=Fdr, tsˆf
1vμs Fdr, ttFdr, t.
Fdr, t=-4πDUdr, t,
1v t-D2Udr, t=Qr, t,
D=13μs1-g=13μs,
g=cosθ=4π sˆ·sˆpsˆ·sˆds4π psˆ·sˆds
Udr, t=v exp-r-r24Dvt4π4πDvt3/2.
Udr, t=v exp-r-r24Dvt-μavt4π4πDvt3/2.
Ūdr¯, t¯=μ¯sμs3 Udr, t
r¯=rμsμ¯s and t¯=tμsμ¯s
D˜=13μs+μa.
Ir, t, sˆ=0,
sˆ·qˆ>0 Ir, t, sˆsˆ·qˆdω=0,
sˆ·qˆ>0 Ir, t, sˆsˆ·qˆdω=sˆ·qˆ<0 RsˆIr, t, sˆsˆ·qˆdω,
Rsˆ=12n cosθi-cosθtn cosθi+cosθt2+cosθi-n cosθtcosθi+n cosθt2.
Udr, t+ A2π Fdr, t·qˆ=0,
A=1+3 0π/2 Rθicos2θisinθidθi1-2 0π/2 Rθicosθisinθidθi.
Udr, t-2ADqˆ·Udr, t=0,
A=1+ 3281-n23/2105n3-n-128+32n+52n2+13n3105n31+n2+r1n+r2n+r3n1--3+7n+13n2+9n3-7n4+3n5+n6+n73n-1n+12n2+12-r4n,
r1n=-4+n-4n2+25n3-40n4-6n5+8n6+30n7-12n8+n9+n113nn2-12n2+13,  r2n=2n33+2n4n2-12n2+17/2 logn2n-1+n21/22+n2+21+n21/2n+1+n21/2-2+n4-21-n41/2,  r3n=41-n21/21+12n4+2n83nn2-12n2+13,  r4n=1+6n4+n8 log1-n1+n+4n2+n6log1+nn21-nn2-12n2+13.
A=1+32 21-1n23/23+t1n+t2n+1+6n4+n81-n2-13/2n33n4-12+t3n1-2+2n-3n2+7n3-15n4-19n5-7n6+3n7+3n8+3n93n2n-1n+12n2+12-t4n,
t1n=4-1-n2+6n3-10n4-3n5+2n6+6n7-3n8-6n2+9n6n2-11/23nn2-12n2+13,  t2n=-8+28n2+35n3-140n4+98n5-13n7+13nn2-131-1/n21/2105n3n2-12,  t3n=2n33+2n4logn-1+n21/22+n2+21+n21/2n2+n4-11/2n2n+1+n21/2-n2+n4-11/2n2-12n2+17/2,  t4n=1+6n4+n8log-1+n1+n+4n2+n6logn21+nn-1n2-12n2+13.
ze=2AD.
z+,m=2ms+2ze+z0for positive sourcesz-,m=2ms+2ze-2ze-z0for negativesources,  m=0,±1,±2,.
Udz, ρ, t=Ud+z, ρ, t+Ud-z, ρ, t,Ud+z, ρ, t=vexp-μavt-ρ24Dvt4π4πDvt3/2 m=-m=+ exp-z-z+,m24Dvt,Ud-z, ρ, t=-vexp-μavt-ρ24Dvt4π4πDvt3/2 m=-m=+ exp-z-z-,m24Dvt.
Rρ, t=sˆ·qˆ<0 1-Rsˆ)]Iρ, z=0, t, sˆ-qˆ·sˆdω,
Rρ, t=-qˆ·Fdρ, z=0, t=4πD z Udρ, z=0, t.
Rρ, t=-exp-μavt-ρ24Dvt24πDv3/2t5/2×m=-+ z3,m exp-z3,m24Dvt-z4,m exp-z4,m24Dvt,
z1,m=s1-2m-4mze-z0,z2,m=s1-2m-4m-2ze+z0,z3,m=-2ms-4mze-z0,z4,m=-2ms-4m-2ze+z0.
Tρ, t=-qˆ·Fdρ, z=s, t=-4πD z Udρ, z=s, t,
Tρ, t=exp-μavt-ρ24Dvt24πDv3/2t5/2 m=-+ z1,m exp-z1,m24Dvt-z2,m exp-z2,m24Dvt.
Rt=0+ Rρ, t2πρdρ=-exp-μavt24πDv1/2t3/2×m=-+ z3,m exp-z3,m24Dvt-z4,m exp-z4,m24Dvt,
Tt=0+ Tρ, t2πρdρ=exp-μavt24πDv1/2t3/2×m=-+ z1,m exp-z1,m24Dvt-z2,m exp-z2,m24Dvt.
0+ tv-1 exp-Bt-γtdt=2Bγv/2 Kv2γB1/2,
Kv+1/2x=K-v-1/2x  v=0, 1, 2, ,
π2x1/2 K1/2x=π2x exp-xπ2x1/2 K3/2x=π2x 1+x-1exp-x.
Rρ=-14π m=-+ z3,mρ2+z3,m2-3/2×1+μaρ2+z3,m2D1/2×exp-μaρ2+z3,m2D1/2-z4,mρ2+z4,m2-3/2×1+μaρ2+z4,m2D1/2×exp-μaρ2+z4,m2D1/2,
Tρ=14π m=-+ z1,mρ2+z1,m2-3/2×1+μaρ2+z1,m2D1/2×exp-μaρ2+z1,m2D1/2-z2,m×ρ2+z2,m2-3/2×1+μaρ2+z2,m2D1/2×exp-μaρ2+z2,m2D1/2.
lρR=v 0+ tRρ, tdt 0+ Rρ, tdt=-18πDRρ m=-+×z3mρ2+z3m2-1/2×exp-μaρ2+z3m2D1/2-z4mρ2+z4m2-1/2×exp-μaρ2+z4m2D1/2,
lρT=v 0+ tTρ, tdt0+ Tρ, tdt=18πDTρ m=-+×z1mρ2+z1m2-1/2×exp-μaρ2+z1m2D1/2-z2mρ2+z2m2-1/2×exp-μaρ2+z2m2D1/2.
R=-12 m=-+ sgnz3,mexp-μaD1/2z3,m-sgnz4,mexp-μaD1/2z4,m,
T=12 m=-+ sgnz1,mexp-μaD1/2z1,m-sgnz2,mexp-μaD1/2z2,m,
Rsiρze+z02πρ3 1+ρμaD1/2exp-ρμaD1/2,
lρsiρ22D 11+ρμa/D1/2.
AGFB=1+rd1-rd,  rd=-1.4399n2+0.7099n+0.6681+0.0636n.
Rstepsˆ=1for 0cosϑcosϑcR0for cosϑc<cosϑ1,
Rexpsˆ=exp-b cosϑ,
b=-2 loga1+a22,
a1=n1-34n21/2-12n1-34n21/2+122,  a2=n2-1-34n21/2n2+1-34n21/22.
AKSS=n221-R0-1+cos3ϑcforn1expbb3-3b2+2b+1-expbexpbb3+2bb+1-expbforn<1.
limn1- An=limn1+ An=1.
Afit=3.084635-6.531194n+8.357854n2-5.082751n3+1.171382n4  for n1,
Afit=504.332889-2641.00214n+5923.699064n2-7376.355814n3+5507.53041n4-2463.357945n5+610.956547n6-64.8047n7  for n>1.

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