Abstract

Starting from light transport theory, we present an ab initio calculation of the fluence rate that is due to an isotropic point source in an infinite, anisotropically scattering medium. To verify the results experimentally, a latex suspension with uniform particle size, corresponding to g=0.734, was prepared. In the suspension an isotropic light source and an isotropic light detector were placed, and the fluence rate as a function of distance was measured. We observed good agreement in absolute values between the calculated and the observed fluence rate over distances ranging from 1/4 to 8× the total mean free path, which corresponds to a fluence rate varying over some five orders of magnitude. Furthermore, the calculated fluence was obtained from measured values of μs and μa and a calculated phase function without any kind of fitting, and thus the calculation was completely independent from the measurement. This is the first time that the fluence was measured quantitatively with an isotropic probe and found to agree within experimental error with transport theory. The experimental results indicate that, far from the source, the behavior is diffusionlike, even for low albedos, albeit with a corrected effective extinction coefficient.

© 1998 Optical Society of America

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References

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  1. K. Rinzema, B. J. Hoenders, H. A. Ferwerda, J. J. Ten Bosch, “Analytic calculation of the radiance in an anisotropically scattering turbid medium close to a source,” Pure Appl. Opt. 4, 629–642 (1995).
    [Crossref]
  2. A. Ishimaru, R. L. T. Cheung, K. Shimizu, “Scattering and diffusion of a beam in randomly distributed scatterers,” J. Opt. Soc. Am. 73, 131–136 (1983).
    [Crossref]
  3. B. Davison, J. B. Sykes, “The spherical harmonics method for spherical geometries,” in Neutron Transport Theory, N. F. Mott, E. C. Ballard, eds. (Clarendon, Oxford, 1957), Sec. 11.1.
  4. M. Abramowitz, I. A. Stegun, “Bend functions of fractional order,” in Handbook of Mathematical Functions (Dover, New York, 1972), p. 437.
  5. I. S. Gradshteyn, I. M. Ryzhik, “8. Special functions,” in Tables of Integrals, Series, and Products, Yu. V. Geronimus, M. Yu. Tseytlin, eds. (Academic, San Diego, Calif., 1980), p. 1019.
  6. J. Mathews, R. L. Walker, “7. Special functions,” in Mathematical Methods of Physics, 2nd ed. (Addison-Wesley, Reading, Mass., 1970), p. 174.
  7. A. Ishimaru, “9. Diffusion approximation,” in Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978), Vol. 1, p. 178.
  8. J. P. A. Marijnissen, W. M. Star, “Calibration of isotropic light dosimetry detectors based on scattering bulbs in clear media,” Phys. Med. Biol. 41, 1191–1208 (1996).
    [Crossref] [PubMed]
  9. M. S. Patterson, E. Schwartz, B. C. Wilson, “Quantitative reflectance spectroscopy for the noninvasive measurement of photosensitizer concentration in tissue during photodynamic therapy,” in Photodynamic Therapy: Mechanisms, T. J. Dougherty, ed., Proc. SPIE1065, 115–122 (1989).
    [Crossref]
  10. W. M. Star, J. P. A. Marijnissen, M. J. C. V. Gemert, “Light dosimetry in optical phantoms and in tissues: I. multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
    [Crossref] [PubMed]
  11. W. M. Star, “Diffusion theory of light transport,” in Optical-Thermal Response of Laser Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), pp. 131–206.
  12. J. W. Goodwin, J. Hearn, C. C. Ho, R. H. Ottewil, “Studies on the preparation of monodisperse polystyrene latices,” Colloid Polym. Sci. 252, 464–471 (1974).
    [Crossref]
  13. J. R. Zijp, J. J. Ten Bosch, “Pascal program to perform Mie calculations,” Opt. Eng. 32, 1691–1695 (1993).
    [Crossref]
  14. L. Wang, S. L. Jacques, “Error estimation of measuring total interaction coefficients of turbid media using colli-mated light transmission,” Phys. Med. Biol. 39, 2349–2354 (1994).
    [Crossref] [PubMed]
  15. H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  16. S. T. Flock, B. C. Wilson, M. J. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
    [Crossref] [PubMed]
  17. J. P. A. Marijnissen, W. M. Star, “Quantitative light dosimetry in vitro and in vivo,” Lasers Med. Sci. 2, 235–242 (1987).
  18. H. J. Van Staveren, J. P. A. Marijnissen, M. C. G. Aalders, W. M. Star, “Construction, quality control and calibration of spherical isotropic fibre-optic light diffusers,” Lasers Med. Sci. 10, 137–147 (1996).
  19. L. H. P. Murrer, J. P. A. Marijnissen, W. M. Star, “Ex vivo light dosimetry and Monte Carlo simulations for endobronchial photodynamic therapy,” Phys. Med. Biol. 40, 1807–1817 (1995).
    [Crossref] [PubMed]
  20. W. M. Star, J. P. A. Marijnissen, “Calculating the response of isotropic light dosimetry probes as a function of the tissue refractive index,” Appl. Opt. 12, 2288–292 (1989).
    [Crossref]
  21. H. G. Kaper, “Application to the slab albedo problem. Part 1. Theory,” , Mathematics Department, University of Gronigen, Gronigen, The Netherlands, 1967), pp. 7–10.

1996 (2)

J. P. A. Marijnissen, W. M. Star, “Calibration of isotropic light dosimetry detectors based on scattering bulbs in clear media,” Phys. Med. Biol. 41, 1191–1208 (1996).
[Crossref] [PubMed]

H. J. Van Staveren, J. P. A. Marijnissen, M. C. G. Aalders, W. M. Star, “Construction, quality control and calibration of spherical isotropic fibre-optic light diffusers,” Lasers Med. Sci. 10, 137–147 (1996).

1995 (2)

L. H. P. Murrer, J. P. A. Marijnissen, W. M. Star, “Ex vivo light dosimetry and Monte Carlo simulations for endobronchial photodynamic therapy,” Phys. Med. Biol. 40, 1807–1817 (1995).
[Crossref] [PubMed]

K. Rinzema, B. J. Hoenders, H. A. Ferwerda, J. J. Ten Bosch, “Analytic calculation of the radiance in an anisotropically scattering turbid medium close to a source,” Pure Appl. Opt. 4, 629–642 (1995).
[Crossref]

1994 (1)

L. Wang, S. L. Jacques, “Error estimation of measuring total interaction coefficients of turbid media using colli-mated light transmission,” Phys. Med. Biol. 39, 2349–2354 (1994).
[Crossref] [PubMed]

1993 (1)

J. R. Zijp, J. J. Ten Bosch, “Pascal program to perform Mie calculations,” Opt. Eng. 32, 1691–1695 (1993).
[Crossref]

1989 (1)

1988 (1)

W. M. Star, J. P. A. Marijnissen, M. J. C. V. Gemert, “Light dosimetry in optical phantoms and in tissues: I. multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
[Crossref] [PubMed]

1987 (2)

S. T. Flock, B. C. Wilson, M. J. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[Crossref] [PubMed]

J. P. A. Marijnissen, W. M. Star, “Quantitative light dosimetry in vitro and in vivo,” Lasers Med. Sci. 2, 235–242 (1987).

1983 (1)

1974 (1)

J. W. Goodwin, J. Hearn, C. C. Ho, R. H. Ottewil, “Studies on the preparation of monodisperse polystyrene latices,” Colloid Polym. Sci. 252, 464–471 (1974).
[Crossref]

Aalders, M. C. G.

H. J. Van Staveren, J. P. A. Marijnissen, M. C. G. Aalders, W. M. Star, “Construction, quality control and calibration of spherical isotropic fibre-optic light diffusers,” Lasers Med. Sci. 10, 137–147 (1996).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, “Bend functions of fractional order,” in Handbook of Mathematical Functions (Dover, New York, 1972), p. 437.

Cheung, R. L. T.

Davison, B.

B. Davison, J. B. Sykes, “The spherical harmonics method for spherical geometries,” in Neutron Transport Theory, N. F. Mott, E. C. Ballard, eds. (Clarendon, Oxford, 1957), Sec. 11.1.

Ferwerda, H. A.

K. Rinzema, B. J. Hoenders, H. A. Ferwerda, J. J. Ten Bosch, “Analytic calculation of the radiance in an anisotropically scattering turbid medium close to a source,” Pure Appl. Opt. 4, 629–642 (1995).
[Crossref]

Flock, S. T.

S. T. Flock, B. C. Wilson, M. J. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[Crossref] [PubMed]

Gemert, M. J. C. V.

W. M. Star, J. P. A. Marijnissen, M. J. C. V. Gemert, “Light dosimetry in optical phantoms and in tissues: I. multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
[Crossref] [PubMed]

Goodwin, J. W.

J. W. Goodwin, J. Hearn, C. C. Ho, R. H. Ottewil, “Studies on the preparation of monodisperse polystyrene latices,” Colloid Polym. Sci. 252, 464–471 (1974).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, “8. Special functions,” in Tables of Integrals, Series, and Products, Yu. V. Geronimus, M. Yu. Tseytlin, eds. (Academic, San Diego, Calif., 1980), p. 1019.

Hearn, J.

J. W. Goodwin, J. Hearn, C. C. Ho, R. H. Ottewil, “Studies on the preparation of monodisperse polystyrene latices,” Colloid Polym. Sci. 252, 464–471 (1974).
[Crossref]

Ho, C. C.

J. W. Goodwin, J. Hearn, C. C. Ho, R. H. Ottewil, “Studies on the preparation of monodisperse polystyrene latices,” Colloid Polym. Sci. 252, 464–471 (1974).
[Crossref]

Hoenders, B. J.

K. Rinzema, B. J. Hoenders, H. A. Ferwerda, J. J. Ten Bosch, “Analytic calculation of the radiance in an anisotropically scattering turbid medium close to a source,” Pure Appl. Opt. 4, 629–642 (1995).
[Crossref]

Ishimaru, A.

A. Ishimaru, R. L. T. Cheung, K. Shimizu, “Scattering and diffusion of a beam in randomly distributed scatterers,” J. Opt. Soc. Am. 73, 131–136 (1983).
[Crossref]

A. Ishimaru, “9. Diffusion approximation,” in Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978), Vol. 1, p. 178.

Jacques, S. L.

L. Wang, S. L. Jacques, “Error estimation of measuring total interaction coefficients of turbid media using colli-mated light transmission,” Phys. Med. Biol. 39, 2349–2354 (1994).
[Crossref] [PubMed]

Kaper, H. G.

H. G. Kaper, “Application to the slab albedo problem. Part 1. Theory,” , Mathematics Department, University of Gronigen, Gronigen, The Netherlands, 1967), pp. 7–10.

Marijnissen, J. P. A.

H. J. Van Staveren, J. P. A. Marijnissen, M. C. G. Aalders, W. M. Star, “Construction, quality control and calibration of spherical isotropic fibre-optic light diffusers,” Lasers Med. Sci. 10, 137–147 (1996).

J. P. A. Marijnissen, W. M. Star, “Calibration of isotropic light dosimetry detectors based on scattering bulbs in clear media,” Phys. Med. Biol. 41, 1191–1208 (1996).
[Crossref] [PubMed]

L. H. P. Murrer, J. P. A. Marijnissen, W. M. Star, “Ex vivo light dosimetry and Monte Carlo simulations for endobronchial photodynamic therapy,” Phys. Med. Biol. 40, 1807–1817 (1995).
[Crossref] [PubMed]

W. M. Star, J. P. A. Marijnissen, “Calculating the response of isotropic light dosimetry probes as a function of the tissue refractive index,” Appl. Opt. 12, 2288–292 (1989).
[Crossref]

W. M. Star, J. P. A. Marijnissen, M. J. C. V. Gemert, “Light dosimetry in optical phantoms and in tissues: I. multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
[Crossref] [PubMed]

J. P. A. Marijnissen, W. M. Star, “Quantitative light dosimetry in vitro and in vivo,” Lasers Med. Sci. 2, 235–242 (1987).

Mathews, J.

J. Mathews, R. L. Walker, “7. Special functions,” in Mathematical Methods of Physics, 2nd ed. (Addison-Wesley, Reading, Mass., 1970), p. 174.

Murrer, L. H. P.

L. H. P. Murrer, J. P. A. Marijnissen, W. M. Star, “Ex vivo light dosimetry and Monte Carlo simulations for endobronchial photodynamic therapy,” Phys. Med. Biol. 40, 1807–1817 (1995).
[Crossref] [PubMed]

Ottewil, R. H.

J. W. Goodwin, J. Hearn, C. C. Ho, R. H. Ottewil, “Studies on the preparation of monodisperse polystyrene latices,” Colloid Polym. Sci. 252, 464–471 (1974).
[Crossref]

Patterson, M. J.

S. T. Flock, B. C. Wilson, M. J. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[Crossref] [PubMed]

Patterson, M. S.

M. S. Patterson, E. Schwartz, B. C. Wilson, “Quantitative reflectance spectroscopy for the noninvasive measurement of photosensitizer concentration in tissue during photodynamic therapy,” in Photodynamic Therapy: Mechanisms, T. J. Dougherty, ed., Proc. SPIE1065, 115–122 (1989).
[Crossref]

Rinzema, K.

K. Rinzema, B. J. Hoenders, H. A. Ferwerda, J. J. Ten Bosch, “Analytic calculation of the radiance in an anisotropically scattering turbid medium close to a source,” Pure Appl. Opt. 4, 629–642 (1995).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, “8. Special functions,” in Tables of Integrals, Series, and Products, Yu. V. Geronimus, M. Yu. Tseytlin, eds. (Academic, San Diego, Calif., 1980), p. 1019.

Schwartz, E.

M. S. Patterson, E. Schwartz, B. C. Wilson, “Quantitative reflectance spectroscopy for the noninvasive measurement of photosensitizer concentration in tissue during photodynamic therapy,” in Photodynamic Therapy: Mechanisms, T. J. Dougherty, ed., Proc. SPIE1065, 115–122 (1989).
[Crossref]

Shimizu, K.

Star, W. M.

J. P. A. Marijnissen, W. M. Star, “Calibration of isotropic light dosimetry detectors based on scattering bulbs in clear media,” Phys. Med. Biol. 41, 1191–1208 (1996).
[Crossref] [PubMed]

H. J. Van Staveren, J. P. A. Marijnissen, M. C. G. Aalders, W. M. Star, “Construction, quality control and calibration of spherical isotropic fibre-optic light diffusers,” Lasers Med. Sci. 10, 137–147 (1996).

L. H. P. Murrer, J. P. A. Marijnissen, W. M. Star, “Ex vivo light dosimetry and Monte Carlo simulations for endobronchial photodynamic therapy,” Phys. Med. Biol. 40, 1807–1817 (1995).
[Crossref] [PubMed]

W. M. Star, J. P. A. Marijnissen, “Calculating the response of isotropic light dosimetry probes as a function of the tissue refractive index,” Appl. Opt. 12, 2288–292 (1989).
[Crossref]

W. M. Star, J. P. A. Marijnissen, M. J. C. V. Gemert, “Light dosimetry in optical phantoms and in tissues: I. multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
[Crossref] [PubMed]

J. P. A. Marijnissen, W. M. Star, “Quantitative light dosimetry in vitro and in vivo,” Lasers Med. Sci. 2, 235–242 (1987).

W. M. Star, “Diffusion theory of light transport,” in Optical-Thermal Response of Laser Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), pp. 131–206.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, “Bend functions of fractional order,” in Handbook of Mathematical Functions (Dover, New York, 1972), p. 437.

Sykes, J. B.

B. Davison, J. B. Sykes, “The spherical harmonics method for spherical geometries,” in Neutron Transport Theory, N. F. Mott, E. C. Ballard, eds. (Clarendon, Oxford, 1957), Sec. 11.1.

Ten Bosch, J. J.

K. Rinzema, B. J. Hoenders, H. A. Ferwerda, J. J. Ten Bosch, “Analytic calculation of the radiance in an anisotropically scattering turbid medium close to a source,” Pure Appl. Opt. 4, 629–642 (1995).
[Crossref]

J. R. Zijp, J. J. Ten Bosch, “Pascal program to perform Mie calculations,” Opt. Eng. 32, 1691–1695 (1993).
[Crossref]

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Van Staveren, H. J.

H. J. Van Staveren, J. P. A. Marijnissen, M. C. G. Aalders, W. M. Star, “Construction, quality control and calibration of spherical isotropic fibre-optic light diffusers,” Lasers Med. Sci. 10, 137–147 (1996).

Walker, R. L.

J. Mathews, R. L. Walker, “7. Special functions,” in Mathematical Methods of Physics, 2nd ed. (Addison-Wesley, Reading, Mass., 1970), p. 174.

Wang, L.

L. Wang, S. L. Jacques, “Error estimation of measuring total interaction coefficients of turbid media using colli-mated light transmission,” Phys. Med. Biol. 39, 2349–2354 (1994).
[Crossref] [PubMed]

Wilson, B. C.

S. T. Flock, B. C. Wilson, M. J. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[Crossref] [PubMed]

M. S. Patterson, E. Schwartz, B. C. Wilson, “Quantitative reflectance spectroscopy for the noninvasive measurement of photosensitizer concentration in tissue during photodynamic therapy,” in Photodynamic Therapy: Mechanisms, T. J. Dougherty, ed., Proc. SPIE1065, 115–122 (1989).
[Crossref]

Zijp, J. R.

J. R. Zijp, J. J. Ten Bosch, “Pascal program to perform Mie calculations,” Opt. Eng. 32, 1691–1695 (1993).
[Crossref]

Appl. Opt. (1)

Colloid Polym. Sci. (1)

J. W. Goodwin, J. Hearn, C. C. Ho, R. H. Ottewil, “Studies on the preparation of monodisperse polystyrene latices,” Colloid Polym. Sci. 252, 464–471 (1974).
[Crossref]

J. Opt. Soc. Am. (1)

Lasers Med. Sci. (2)

J. P. A. Marijnissen, W. M. Star, “Quantitative light dosimetry in vitro and in vivo,” Lasers Med. Sci. 2, 235–242 (1987).

H. J. Van Staveren, J. P. A. Marijnissen, M. C. G. Aalders, W. M. Star, “Construction, quality control and calibration of spherical isotropic fibre-optic light diffusers,” Lasers Med. Sci. 10, 137–147 (1996).

Med. Phys. (1)

S. T. Flock, B. C. Wilson, M. J. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[Crossref] [PubMed]

Opt. Eng. (1)

J. R. Zijp, J. J. Ten Bosch, “Pascal program to perform Mie calculations,” Opt. Eng. 32, 1691–1695 (1993).
[Crossref]

Phys. Med. Biol. (4)

L. Wang, S. L. Jacques, “Error estimation of measuring total interaction coefficients of turbid media using colli-mated light transmission,” Phys. Med. Biol. 39, 2349–2354 (1994).
[Crossref] [PubMed]

L. H. P. Murrer, J. P. A. Marijnissen, W. M. Star, “Ex vivo light dosimetry and Monte Carlo simulations for endobronchial photodynamic therapy,” Phys. Med. Biol. 40, 1807–1817 (1995).
[Crossref] [PubMed]

J. P. A. Marijnissen, W. M. Star, “Calibration of isotropic light dosimetry detectors based on scattering bulbs in clear media,” Phys. Med. Biol. 41, 1191–1208 (1996).
[Crossref] [PubMed]

W. M. Star, J. P. A. Marijnissen, M. J. C. V. Gemert, “Light dosimetry in optical phantoms and in tissues: I. multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
[Crossref] [PubMed]

Pure Appl. Opt. (1)

K. Rinzema, B. J. Hoenders, H. A. Ferwerda, J. J. Ten Bosch, “Analytic calculation of the radiance in an anisotropically scattering turbid medium close to a source,” Pure Appl. Opt. 4, 629–642 (1995).
[Crossref]

Other (9)

H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

W. M. Star, “Diffusion theory of light transport,” in Optical-Thermal Response of Laser Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), pp. 131–206.

M. S. Patterson, E. Schwartz, B. C. Wilson, “Quantitative reflectance spectroscopy for the noninvasive measurement of photosensitizer concentration in tissue during photodynamic therapy,” in Photodynamic Therapy: Mechanisms, T. J. Dougherty, ed., Proc. SPIE1065, 115–122 (1989).
[Crossref]

B. Davison, J. B. Sykes, “The spherical harmonics method for spherical geometries,” in Neutron Transport Theory, N. F. Mott, E. C. Ballard, eds. (Clarendon, Oxford, 1957), Sec. 11.1.

M. Abramowitz, I. A. Stegun, “Bend functions of fractional order,” in Handbook of Mathematical Functions (Dover, New York, 1972), p. 437.

I. S. Gradshteyn, I. M. Ryzhik, “8. Special functions,” in Tables of Integrals, Series, and Products, Yu. V. Geronimus, M. Yu. Tseytlin, eds. (Academic, San Diego, Calif., 1980), p. 1019.

J. Mathews, R. L. Walker, “7. Special functions,” in Mathematical Methods of Physics, 2nd ed. (Addison-Wesley, Reading, Mass., 1970), p. 174.

A. Ishimaru, “9. Diffusion approximation,” in Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978), Vol. 1, p. 178.

H. G. Kaper, “Application to the slab albedo problem. Part 1. Theory,” , Mathematics Department, University of Gronigen, Gronigen, The Netherlands, 1967), pp. 7–10.

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Figures (7)

Fig. 1
Fig. 1

Contour in the complex k plane for the calculation of the radiance in real space.

Fig. 2
Fig. 2

Directional characteristics of the probe used in our experiment. Also indicated are 15% boundaries and 100% reference level.

Fig. 3
Fig. 3

Schematic drawing of the experimental setup.

Fig. 4
Fig. 4

Comparison of experimental results with exact theory (solid curves) and diffusion (dashed curves); case 1, weakly scattering phantom (μt=0.573 cm-1) and total mean free path (μt-1=1.75 cm). Vessel dimensions are length and width 60 cm, height 44 cm. The enlarged portion shows the measurements taken at distances of <3 cm. Quantities and units are as in the main graph.

Fig. 5
Fig. 5

Comparison of experimental results with exact theory (solid curve) and diffusion (dashed curve); case 2, strongly scattering phantom (μt=5.01 cm-1) and total mean free path (μt-1=0.200 cm). Vessel dimensions are length 35 cm, width 24 cm, height 16 cm.

Fig. 6
Fig. 6

Discrepancies between measurements and exact theory (circles) and between measurements and diffusion approximation (triangles); case 1, weakly scattering phantom.

Fig. 7
Fig. 7

Discrepancies between measurements and exact theory (circles) and between measurements and diffusion approximation (triangles); case 2, strongly scattering phantom.

Tables (2)

Tables Icon

Table 1 Optical Parameters of the Two Suspensions

Tables Icon

Table 2 Attenuation Coefficients in cm-1 As Predicted by Rigorous Theory (κ0) and Diffusion Theory [μeff=(3μaμt)1/2] and Obtained by a Fit to the Measured Data (κ0,exp)

Equations (93)

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

Ω·L(r, Ω)=-μtL(r, Ω)+μs4πL(r, Ω)f(Ω, Ω)dΩ+S0δ(r),
rr/μt,μt,
δ(r)μt3δ(r),L(r, Ω)μt2L(r, Ω),
Ω·L(r, Ω)+L(r, Ω)
=a4πL(r, Ω)f(Ω, Ω)dΩ+δ(r).
L(r, Ω)=L(r, rˆ·Ω),
L(r, rˆ·Ω)=l=0m=-llLl(r)Ylm(Ω)Ylm*(rˆ).
(1+ik·Ω)L(k, Ω)
=a4πL(k, Ω)f(Ω, Ω)dΩ+1,
L(k, Ω)=drL(r, Ω)exp(-ik·r).
L(k, Ω)=L(k, kˆ·Ω)=l=0(2l+1)Ll(k)Pl(μΩ),
Ll(k)=(-i)l0drr2jl(kr)Ll(r),
f(Ω·Ω)=l=0N(2l+1)flPl(Ω·Ω)=l=0Nm=-llflYlm(Ω)Ylm*(Ω).
(1+ikμΩ)L(k, μΩ)=a/2l=0N(2l+1)×flLl(k)Pl(μΩ)+1.
(2q+1)μΩPq(μΩ)=(q+1)Pq+1(μΩ)+qPq-1(μΩ),
(q+1)/(2q+1)Lq+1(k)
=(1-afq)(i/k)Lq(k)-q/(2q+1)(i/k)
×Lq-1(k)-(i/k)δq0.
Lq(k)=l=0NTql(k)Ll(k)+Sq(k),
Tql(k)=(a/2)(2l+1)fl-11dμΩPl(μΩ)Pq(μΩ)1+ikμΩ,
Sq(k)=(1/2)-11dμΩ Pq(μΩ)1+ikμΩ.
[I-T(k)]L(k)=S(k),
L0(k)=|C0(k)|/|I-T(k)|,
|I-T(k)|=1-a(i/k)l=0N(2l+1)flϕl(k)Ql(i/k),
|C0(k)|=(i/k)Q0(i/k)-a(i/k)l=1N(2l+1)×flQl(i/k)χl(k),
ϕ0(k)=1,
ϕ1(k)=(i/k)(1-a),
(q+1)/(2q+1)ϕq+1(k)=(1-afq)(i/k)ϕq(k)-q/(2q+1)ϕq-1(k),
χ1(k)=(i/k),
χ2(k)=3/2(i/k)2(1-af1),
(q+1)/(2q+1)χq+1(k)=(1-afq)(i/k)χq(k)-q/(2q+1)χq-1(k).
Ql(z)=12Pl(z)log z+1z-1-Wn-1(z),
Wn-1(z)=k=1nPk-1(z)Pn-k(z).
12-11 Pn(μ)1+ikμdμ=(i/k)Qn(i/k).
Ll(r)=(2il/π)0Ll(k)jl(kr)k2dk.
Ll(r)=(il/π)-+Ll(k)jl(kr)k2dk
=Re(il/π)-Ll(k)hl(1)(kr)k2dk,
L0(r)=2/rj=1M |C0(iκj)|ddk|I-T(k)|k=iκj×exp(-κjr)κj+L0nonas,
L0±(k)=p=0L0,p±×(i/k)p,k=is,s.
L0(r)nonas=-i/(πr)p=1[L0,p+-L0,p-]×1s-p+1 exp(-sr)ds
=-i/(πr)p=1[L0,p+-L0,p-]Ep-1(r),
L0(r)=A1 exp[-κ1r]/r+A2 exp(-κ2r)/r++exp(-r)/r2+i=1C2iE2i(r)/r,
Aj=2 |C0(kj)|ddk|I-T(k)|k=iκj,
C2i=-(i/π)(L0,2i+1+-L0,2i+1-).
rμtr,L0(r)4πψ(r)/μt2,
κiκi/μt,AiAi/μt,C2iC2i/μt,
ψ(r)=(S0/4π)A1 exp(-κ1r)/r+A2 exp(-κ2r)/r++exp(-μtr)/r2+i=1C2iE2i(μtr)/r.
2ψd(r)-μeff2ψd(r)
=-34π(μt-Ω·)(r,Ω)dΩ
-3μs4π(μt-gΩ·)Lrr(r, Ω)dΩ,
ψ(r)=(3μt/4πr)exp[-μeff r].
ψ(r)=(3μtS0)/(4πr)exp(-μeffr)+(3μsS0[(1+g)μa+μs)])(8πμeffr)×exp(μtr1){exp(μeff r)E1[(μt+μeffr)]-exp(-μeff r)E1[(μt-μeff)r]+(S0/4πr2)exp[-μt(r-r1)].
fn=-11f(cos θ)Pn(cos θ)d cos θ.
ψ(r)=10.44 exp(-0.2566r)/r+2.640 exp(-0.5710r)/r+24.75 exp(-0.573r)/r2+0.9152E2(0.573r)/r+,mWcm-2,
ψ(r)=97.80 exp(-1.096r)/r+42.57 exp(-4.832r)/r+24.75 exp[-5.01r]/r2+4.485E2(5.01r)/r+,mWcm-2.
ψ(r)=17.75 exp(-0.2909r)/r+41.74[exp(0.2909r)E1(0.8639)/r-exp(-0.2909r)E1(0.2821r)/r]+26.97 exp(-0.573r)/r2,mWcm-2,
ψ(r)=114.3 exp(-1.139r)/r+1705[exp(1.139r)E1(6.150r)/r-exp(-1.139r)E1(3.872r)/r]+52.48 exp(-5.011r)/r2,mWcm-2.
4πμa0r2ψ(r)dr=S0,
12a(2l+1)fl-11 Pl(μ)μPq-1(μ)1+ikμdμ
=(i/k)-aflδl,q-1+(i/k)Tq-1,l(k),
q/(2q-1)Tql-(i/k)Tq-1,l+(q-1)/(2q-1)Tq-2,l
=-(i/k)afq-1δl,q-1,q2.
T1l-(i/k)T0l=-(i/k)af0δl0.
DN,0l=δ0l,
DN,rl=rδrl-(i/k)(2r-1)δr-1,l+(r-1)δr-2,l,r1,
DN=1000(i/k)-100-13(i/k)-20000(2N-1)(i/k)-N.
[DN(I-T)]rs=l=0NDN,rlTls=δ0s-T0s(k),r=0,
[DN(I-T)]rs=rTrs-(2r-1)(i/k)Tr-1,s+(r-1)Tr-2,s-(r-1)δr-2,s+(2r-1)(i/k)δr-1,s-rδrs=-(r-1)δr-2,s+(2r-1)×(1-afr-1)(i/k)δr-1,s-rδrs.
DN(I-T)=1-T00-T01-T02-T0N(1-af0)(i/k)-100-13(1-af1)(i/k)-200-25(1-af2)(i/k)0000-N.
ϕl+1(k)=1/(l+1)!×(1-af0)(i/k)-100-13(1-af1)(i/k)-200-25(1-af2)(i/k)0000-l(2l+1)(1-afl)(i/k),
lϕl=-(2r-1)(afr-1-1)×(i/k)ϕl-1(k)-(l-1)ϕl-2(k).
ϕ0(k)=1,ϕ1(k)=(1-af0)(i/k),
|DN[I-T(k)]|=(-1)NN!1-l=0NT0lϕl(k).
|I-T(k)|=1-l=0N(2l+1)aflϕl(k)(i/k)Ql(i/k),
[DNC0(k)]00=(1/a)T00,
[DNC0(k)]10=i/k,
[DNC0(k)]20=0,
[DNC0(k)]rs={DN[I-T(k)]}rs,s>0,
DNC0=1/aT00-T01-T02-T0N(i/k)-10003(1-af1)(i/k)-200-25(1-af2)(i/k)000-30000-N.
L0±(k)=a(i/k)l=0N(2l+1)flϕl(k)Ql±(i/k)L0±(k)+(i/k)Q0±(i/k)-a(i/k)l=1N(2l+1)×flχl(k)Ql±(i/k).
λ±(k)=a(i/k)l=0N(2l+1)flϕl(k)Ql±(i/k),
σ±(k)=(i/k)Q0±(i/k)-a(i/k)l=1N(2l+1)×flχl(k)Ql±(i/k),
L0±(k)=λ±(k)L0±(k)+σ±(k),
σ±(s)=n=0σn±×(1/s)n,
λ±(s)=n=0λn±×(1/s)n.
Qn(i/k)=12Pn(i/k)log 1-i/k1+i/kiπ+Wn-1(i/k),
log 1-1/s1+1/s=-2[1/s+1/s3+1/s5+],
λ0±=σ0±=0,λ1±,σ1±0,
L0,0±=0
L0,1±=σ1±
L0,2±=λ1±σ1±+σ2±
L0, n±=m=1n-1(λm±σn-m±)+σn±.

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