Abstract

In our companion paper we presented a model to describe photon migration through a diffusing slab. The model, developed for a homogeneous slab, is based on the diffusion approximation and is able to take into account reflection at the boundaries resulting from the refractive index mismatch. In this paper the predictions of the model are compared with solutions of the radiative transfer equation obtained by Monte Carlo simulations in order to determine the applicability limits of the approximated theory in different physical conditions. A fitting procedure, carried out with the optical properties as fitting parameters, is used to check the application of the model to the inverse problem. The results show that significant errors can be made if the effect of the refractive index mismatch is not properly taken into account. Errors are more important when measurements of transmittance are used. The effects of using a receiver with a limited angular field of view and the angular distribution of the radiation that emerges from the slab have also been investigated.

© 1997 Optical Society of America

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References

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  1. D. Contini, F. Martelli, G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36, 4587–4599 (1996).
    [CrossRef]
  2. M. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  3. P. Bruscaglioni, G. Zaccanti, “Multiple scattering in dense media,” in Scattering in Volumes and Surfaces, M. N. Vesperinas, J. C. Dainty, eds. (Elsevier-North-Holland, Amsterdam, 1990), pp. 53–71.
  4. G. Zaccanti, “Monte Carlo study of light propagation in optically thick media: point-source case,” Appl. Opt. 30, 2031–2041 (1991).
    [CrossRef] [PubMed]
  5. S. Brandt, Statistical and Computational Methods in the Data Analysis (North-Holland, Amsterdam, 1976), Chap. 5.
  6. G. Zaccanti, E. Battistelli, P. Bruscaglioni, Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994).
    [CrossRef]
  7. T. J. Farrel, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–883 (1992).
    [CrossRef]
  8. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, (Cambridge U. Press, Cambridge, UK1988).
  9. 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.
  10. 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]
  11. 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. Biomed. Opt. 1, 330–334 (1996).
    [CrossRef] [PubMed]
  12. G. Mitic, J. Kolzer, J. Otto, E. Plies, G. Solkner, W. Zinth, “Time-gated transillumination of biological tissues and tissue-like phantom,” Appl. Opt. 33, 6699–6709 (1994).
    [CrossRef] [PubMed]
  13. 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]
  14. S. P. Proskurin, Y. Yamada, Y. Takahashi, “Absorption coefficient measurements of highly scattering media in slabs and cylindrical phantoms by means of time-resolved optical spectroscopy,” 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, 157–166 (1995).
  15. S. L. Jacques, A. H. Hielscher, L. Wang, F. K. Tittel, “How source/collector placement and subsurface absorbing layer affect time-resolved and phase/modulation-resolved photon migration,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 311–319 (1993).
  16. Y. Tsunazawa, I. Oda, H. Eda, M. Takada, “A new algorithm to determine absorption and scattering coefficient from time-resolved measurement,” 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, 75–86 (1995).
  17. S. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distributions in multiple scattering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 78–89 (1993).
    [CrossRef]
  18. M. Firbank, S. R. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with nonscattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
    [CrossRef] [PubMed]

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. Biomed. Opt. 1, 330–334 (1996).
[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]

M. Firbank, S. R. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with nonscattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

D. Contini, F. Martelli, G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36, 4587–4599 (1996).
[CrossRef]

1995 (1)

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 (2)

G. Zaccanti, E. Battistelli, P. Bruscaglioni, Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994).
[CrossRef]

G. Mitic, J. Kolzer, J. Otto, E. Plies, G. Solkner, W. Zinth, “Time-gated transillumination of biological tissues and tissue-like phantom,” Appl. Opt. 33, 6699–6709 (1994).
[CrossRef] [PubMed]

1992 (1)

T. J. Farrel, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–883 (1992).
[CrossRef]

1991 (1)

1989 (1)

Arridge, S. R.

M. Firbank, S. R. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with nonscattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

Battistelli, E.

G. Zaccanti, E. Battistelli, P. Bruscaglioni, Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994).
[CrossRef]

Brandt, S.

S. Brandt, Statistical and Computational Methods in the Data Analysis (North-Holland, Amsterdam, 1976), Chap. 5.

Bruscaglioni, P.

G. Zaccanti, E. Battistelli, P. Bruscaglioni, Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994).
[CrossRef]

P. Bruscaglioni, G. Zaccanti, “Multiple scattering in dense media,” in Scattering in Volumes and Surfaces, M. N. Vesperinas, J. C. Dainty, eds. (Elsevier-North-Holland, Amsterdam, 1990), pp. 53–71.

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. Biomed. Opt. 1, 330–334 (1996).
[CrossRef] [PubMed]

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

S. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distributions in multiple scattering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 78–89 (1993).
[CrossRef]

Contini, D.

Delpy, D. T.

M. Firbank, S. R. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with nonscattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

Eda, H.

Y. Tsunazawa, I. Oda, H. Eda, M. Takada, “A new algorithm to determine absorption and scattering coefficient from time-resolved measurement,” 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, 75–86 (1995).

Farrel, T. J.

T. J. Farrel, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–883 (1992).
[CrossRef]

Feng, S.

S. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distributions in multiple scattering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 78–89 (1993).
[CrossRef]

Firbank, M.

M. Firbank, S. R. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with nonscattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, (Cambridge U. Press, Cambridge, UK1988).

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]

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]

S. L. Jacques, A. H. Hielscher, L. Wang, F. K. Tittel, “How source/collector placement and subsurface absorbing layer affect time-resolved and phase/modulation-resolved photon migration,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 311–319 (1993).

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]

S. L. Jacques, A. H. Hielscher, L. Wang, F. K. Tittel, “How source/collector placement and subsurface absorbing layer affect time-resolved and phase/modulation-resolved photon migration,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 311–319 (1993).

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. Biomed. Opt. 1, 330–334 (1996).
[CrossRef] [PubMed]

Kolzer, J.

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.

Mitic, G.

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.

Oda, I.

Y. Tsunazawa, I. Oda, H. Eda, M. Takada, “A new algorithm to determine absorption and scattering coefficient from time-resolved measurement,” 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, 75–86 (1995).

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. Biomed. Opt. 1, 330–334 (1996).
[CrossRef] [PubMed]

Otto, J.

Patterson, M.

Patterson, M. S.

T. J. Farrel, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–883 (1992).
[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.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, (Cambridge U. Press, Cambridge, UK1988).

Proskurin, S. P.

S. P. Proskurin, Y. Yamada, Y. Takahashi, “Absorption coefficient measurements of highly scattering media in slabs and cylindrical phantoms by means of time-resolved optical spectroscopy,” 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, 157–166 (1995).

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]

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]

Schweiger, M.

M. Firbank, S. R. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with nonscattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

Solkner, G.

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. Biomed. Opt. 1, 330–334 (1996).
[CrossRef] [PubMed]

Takada, M.

Y. Tsunazawa, I. Oda, H. Eda, M. Takada, “A new algorithm to determine absorption and scattering coefficient from time-resolved measurement,” 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, 75–86 (1995).

Takahashi, Y.

S. P. Proskurin, Y. Yamada, Y. Takahashi, “Absorption coefficient measurements of highly scattering media in slabs and cylindrical phantoms by means of time-resolved optical spectroscopy,” 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, 157–166 (1995).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, (Cambridge U. Press, Cambridge, UK1988).

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]

S. L. Jacques, A. H. Hielscher, L. Wang, F. K. Tittel, “How source/collector placement and subsurface absorbing layer affect time-resolved and phase/modulation-resolved photon migration,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 311–319 (1993).

Tsunazawa, Y.

Y. Tsunazawa, I. Oda, H. Eda, M. Takada, “A new algorithm to determine absorption and scattering coefficient from time-resolved measurement,” 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, 75–86 (1995).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, (Cambridge U. Press, Cambridge, UK1988).

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]

S. L. Jacques, A. H. Hielscher, L. Wang, F. K. Tittel, “How source/collector placement and subsurface absorbing layer affect time-resolved and phase/modulation-resolved photon migration,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 311–319 (1993).

Wei, Q. N.

G. Zaccanti, E. Battistelli, P. Bruscaglioni, Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994).
[CrossRef]

Wilson, B.

T. J. Farrel, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–883 (1992).
[CrossRef]

Wilson, B. C.

M. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive 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. P. Proskurin, Y. Yamada, Y. Takahashi, “Absorption coefficient measurements of highly scattering media in slabs and cylindrical phantoms by means of time-resolved optical spectroscopy,” 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, 157–166 (1995).

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. Biomed. Opt. 1, 330–334 (1996).
[CrossRef] [PubMed]

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. Biomed. Opt. 1, 330–334 (1996).
[CrossRef] [PubMed]

Zaccanti, G.

D. Contini, F. Martelli, G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36, 4587–4599 (1996).
[CrossRef]

G. Zaccanti, E. Battistelli, P. Bruscaglioni, Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994).
[CrossRef]

G. Zaccanti, “Monte Carlo study of light propagation in optically thick media: point-source case,” Appl. Opt. 30, 2031–2041 (1991).
[CrossRef] [PubMed]

P. Bruscaglioni, G. Zaccanti, “Multiple scattering in dense media,” in Scattering in Volumes and Surfaces, M. N. Vesperinas, J. C. Dainty, eds. (Elsevier-North-Holland, Amsterdam, 1990), pp. 53–71.

Zeng, F.

S. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distributions in multiple scattering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 78–89 (1993).
[CrossRef]

Zinth, W.

Appl. Opt. (4)

J. Biomed. 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. Biomed. Opt. 1, 330–334 (1996).
[CrossRef] [PubMed]

Med. Phys. (1)

T. J. Farrel, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–883 (1992).
[CrossRef]

Phys. Med. Biol. (3)

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]

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]

M. Firbank, S. R. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with nonscattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

Pure Appl. Opt. (1)

G. Zaccanti, E. Battistelli, P. Bruscaglioni, Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994).
[CrossRef]

Other (8)

S. Brandt, Statistical and Computational Methods in the Data Analysis (North-Holland, Amsterdam, 1976), Chap. 5.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, (Cambridge U. Press, Cambridge, UK1988).

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.

S. P. Proskurin, Y. Yamada, Y. Takahashi, “Absorption coefficient measurements of highly scattering media in slabs and cylindrical phantoms by means of time-resolved optical spectroscopy,” 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, 157–166 (1995).

S. L. Jacques, A. H. Hielscher, L. Wang, F. K. Tittel, “How source/collector placement and subsurface absorbing layer affect time-resolved and phase/modulation-resolved photon migration,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 311–319 (1993).

Y. Tsunazawa, I. Oda, H. Eda, M. Takada, “A new algorithm to determine absorption and scattering coefficient from time-resolved measurement,” 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, 75–86 (1995).

S. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distributions in multiple scattering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 78–89 (1993).
[CrossRef]

P. Bruscaglioni, G. Zaccanti, “Multiple scattering in dense media,” in Scattering in Volumes and Surfaces, M. N. Vesperinas, J. C. Dainty, eds. (Elsevier-North-Holland, Amsterdam, 1990), pp. 53–71.

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Figures (16)

Fig. 1
Fig. 1

Comparison of the results from the DE (noiseless curves) and the MC simulations. The time-resolved transmittance at ρ = 0 for a nonabsorbing slab 40 mm thick with μs′ = 0.5 mm-1 and n2 = 1.4 is reported for two values of the relative refractive index, n = 1.4 and n = 0.7143.

Fig. 2
Fig. 2

Comparison of the results from the DE (noiseless curves) and the MC simulations for the time-resolved reflectance R and transmittance T for a nonabsorbing slab 40 mm thick with μs′ = 0.75 mm-1, n2 = 1.4, and n = 1.33. Both the transmittance and the reflectance were evaluated at a distance ρ = 30 mm from the light beam.

Fig. 3
Fig. 3

Comparison of the results from the DE (noiseless curves) and the MC simulations for the total time-resolved transmittance. Data refer to a nonabsorbing slab 40 mm thick with μs′ = 0.5 mm-1, n2 = 1.4, and are reported for two values of the relative refractive index, n = 1.4 and n = 1.

Fig. 4
Fig. 4

Comparison of the results from the DE and the MC simulations (marks) for a cw source. The transmittance is reported for a nonabsorbing slab 40 mm thick with μs′ = 0.5 mm-1 for two values of the relative refractive index, n = 1.4 and n = 1.

Fig. 5
Fig. 5

Comparison of the results from the DE and the MC simulations for the reflectance of a slab 40 mm thick with μs′ = 0.5 mm-1, μa = 0.01 mm-1, and n = 1.4: (a) cw response versus the distance from the light beam (MC results are reported for both a pencil beam at z = 0 and an isotropic point source at z = z0); (b) percentage of difference between the DE and the MC results; (c) comparison for the mean path length followed by received photons for the pencil beam.

Fig. 6
Fig. 6

Comparison of the results from the DE and the MC simulations for a slab with moderate optical thickness: (a) time-resolved transmittance at ρ = 0 for a nonabsorbing slab 40 mm thick with μs′ = 0.2 mm-1, n2 = 1.4, and n = 1. To point out the dependence of earlier received photons on the single-scattering properties of the medium and on the characteristics of the source, the MC results are reported also for a scattering function with g = 0.5 and for an isotropic source at z = z0; (b) percentages of difference between the DE and the MC results.

Fig. 7
Fig. 7

Comparison between the results from the DE and the MC simulations for a nonabsorbing slab 40 mm thick with μs′ = 0.5 mm-1, n2 = 1.4, and n = 1.4: (a), (b) examples of time-resolved reflectance for various values of ρ. The MC results are reported for both the pencil beam at z = 0 and the isotropic point source at z = z0; (c), (d) percentages of difference between the DE and the MC results for ρ = 4.2 and 20.4 mm, respectively.

Fig. 8
Fig. 8

Comparison between the results from the DE (noiseless curves) and the MC simulations for the time-resolved reflectance from a semi-infinite nonabsorbing medium with μs′ = 0.5 mm-1, n2 = 1.4, and n = 1. Results are reported for two distances from the light beam.

Fig. 9
Fig. 9

Comparison between the results from the DE and the MC simulations for the cw reflectance from a semi-infinite nonabsorbing medium with μs′ = 0.5 mm-1 and n = 1.4. The results refer to a pencil beam.

Fig. 10
Fig. 10

Examples of fits performed on the time-resolved transmittance obtained with MC simulations. Data are reported for two values of the relative refractive index and refer to ρ = 0 for a nonabsorbing slab 40 mm thick with μs′ = 0.5 mm-1 and n2 = 1.4. The noiseless curves are obtained from the DE with the optical parameters μs′ = 0.497 mm-1 and μa = -4 × 10-5 mm-1 for n = 0.7143, μs′ = 0.501 mm-1 and μa = -2.9 × 10-5 mm-1 for n = 1.4, coming from the fitting procedure.

Fig. 11
Fig. 11

Results of the two-parameters fitting procedure carried out on the time-resolved transmittance at ρ = 0 to determine the (a) reduced scattering coefficients, (b) absorption. The TPSF’s were obtained with MC simulations for a nonabsorbing slab 40 mm thick with μs′ = 0.5 mm-1 and n2 = 1.4 for different values of the relative refractive index. Diamond shapes and square shapes represent the results obtained by means of fitting with Eq. (39) from Ref.1 and Eq. (13) from Ref. 2, respectively. The error bars (reported only when they are larger than the mark) indicate 1 standard deviation.

Fig. 12
Fig. 12

Results of the two-parameters fitting procedure carried out on the time-resolved reflectance at ρ = 40 mm for determining the (a) reduced scattering coefficients, (b) absorption. The TPSF’s were obtained by MC simulations for a nonabsorbing slab 40 mm thick with μs′ = 0.5 mm-1 and n2 = 1.4 for various values of the relative refractive index. Diamond shapes and square shapes represent the results obtained by means of fitting with Eq. (36) from Ref.1 and Eq. (12) from Ref. 2, respectively. The error bars (reported only when they are larger than the mark) indicate 1 standard deviation.

Fig. 13
Fig. 13

Effect of the receiver angular field of view η on the TPSF. The three curves refer to η = 90°, 60°, and 30° (from upper to lower curve, respectively). The transmittance for a coaxial receiver was obtained by a MC simulation for a nonabsorbing slab 40 mm thick with μs′ = 0.5 mm-1 and n2 = n = 1. The curves obtained from a three-parameters fitting procedure (noiseless curves) are also reported.

Fig. 14
Fig. 14

Results obtained for the (a) reduced scattering coefficient, (b) absorption coefficient from the three-parameters fitting procedure carried out on the time-resolved transmittance. The results are reported for various values of η and refer to the same slab that was described for Fig. 13. The horizontal dashed lines represent the actual values of the optical parameters used for MC simulations. The error bars indicate 1 standard deviation.

Fig. 15
Fig. 15

Effect of the receiver angular field of view on the total time-resolved transmittance. The curves refer to η = 90° and 4° (upper and lower curve, respectively) and were obtained by a MC simulation for a nonabsorbing slab 40 mm thick with μs′ = 0.5 mm-1, n2 = 1.4, and n = 1. The curves obtained from a three-parameters fitting procedure (continuous noiseless curves) are also reported.

Fig. 16
Fig. 16

Effect of the receiver angular field of view η on the received energy: The relative transmittance f(η), evaluated by MC simulations (marks), is reported for a nonabsorbing slab 40 mm thick with μs′ = 0.5 mm-1. The results are reported for two values of the relative refractive index. The dashed curve represents the behavior expected for a Lambertian surface. The solid curve was obtained with the model described in Ref. 18.

Tables (2)

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Table 1 Fraction Incident Energy Reflected and Transmitted: Comparison of DE and MC Simulationsa

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Table 2 Optical Properties Obtained from Fitting Proceduresa

Equations (4)

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σ2nuiN=1N2nui-nui2NnuiN2.
R =12 exp-3μaμs1/21+exp-43A 3μaμs1/2.
Tfitρ, t=α exp-μavt-ρ24Dvt24πDv3/2t5/2×m=-+ z1,m exp-z21,m4Dvt-z2,m exp-z22,m4Dvt,
z1,m=s1-2m-4mze-z0  z2,m=s1-2m-4m-2ze+z0.

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