Abstract

A number of investigators have recently claimed, based on both analysis from transport theory and transport-theory-based Monte Carlo calculations, that the diffusion coefficient for photon migration should be taken to be independent of absorption. We show that these analyses are flawed and that the correct way of extracting diffusion theory from transport theory gives an absorption-dependent diffusion coefficient. Experiments by two different sets of investigators give conflicting results concerning whether the diffusion coefficient depends on absorption. The discrepancy between theory and the earlier set of experiments poses an interesting challenge.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. L. Barbour, H. Graber, R. Aronson, J. Lubowsky, “Model for 3-D optical imaging of tissue,” in Proceedings of the International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, Piscataway, New Jersey, 1990), Vol. II, pp. 1295–1399.
  2. A. G. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
    [CrossRef]
  3. S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
    [CrossRef] [PubMed]
  4. M. Bassani, F. Martelli, G. Zaccanti, D. Contini, “Independence of the diffusion coefficient from absorption: experimental and numerical evidence,” Opt. Lett. 22, 853–855 (1997).
    [CrossRef] [PubMed]
  5. K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
    [CrossRef]
  6. K. Furutsu, “Pulse wave scattering by an absorber and integrated attenuation in the diffusion approximation,” J. Opt. Soc. Am. A 14, 267–274 (1997).
    [CrossRef]
  7. T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 60–63.
  8. T. Durduran, A. G. Yodh, B. Chance, D. A. Boas, “Does the photon-diffusion coefficient depend on absorption?” J. Opt. Soc. Am. A 14, 3358–3365 (1997).
    [CrossRef]
  9. K. Rinzema, L. H. P. Murrer, W. M. Star, “Direct experimental verification of light transport theory in an optical phantom,” J. Opt. Soc. Am. A 15, 2078–2088 (1998).
    [CrossRef]
  10. See, for instance, S. Chandrasekhar, “Stochastic Problems in physics and astronomy,” Rev. Mod. Phys. 15, 1–89 (1943);M. C. Wang, G. E. Uhlenbeck, “On the theory of Brownian motion II,” Rev. Mod. Phys. 17, 323–342.Both of these papers are reprinted in Selected Papers on Noise and Stochastic Processes, N. Wax, ed. (Dover, New York, 1964).
    [CrossRef]
  11. See, for instance, J. R. Lamarsh, Nuclear Reactor Theory (Addison-Wesley, Reading, Mass., 1966), pp. 125–128.
  12. E. W. Larsen, J. B. Keller, “Asymptotic solution of neutron transport problems for small mean free paths,” J. Math. Phys. 15, 75–81 (1974);E. W. Larsen, “Diffusion theory as an asymptotic limit of transport theory for nearly critical systems with small mean paths,” Ann. Nucl. Energy 7, 249–255 (1980).
    [CrossRef]
  13. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).
  14. J. R. Mika, “Neutron transport with anisotropic scattering,” Nucl. Sci. Eng. 11, 415–427 (1961).
  15. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Table 4.2, p. 91.
  16. K. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion, Vol. I (Los Alamos Scientific Laboratory, Los Alamos, N. Mex., 1953).
  17. G. Holte, “On a method of calculating the density of neutrons emitted from a point source in an infinite medium,” Ark. Mat. Astron. Fys. 35A(36), 1–9 (1948). Holte’s notation is somewhat different from ours.
  18. I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991). This paper gives Holte’s result17 in the form where we quote here.
    [CrossRef]
  19. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  20. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass.1967), Chap. 7.
  21. G. Caroll, R. Aronson, “One-speed neutron transport problems—part II: slab transmission and reflection and finite reflector critical problems,” Nucl. Sci. Eng. 51, 166–179 (1973).
  22. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 2, Chap. 10.
  23. G. Zaccanti, Dipartimento di Fisica dell’Università degli Studi di Firenze, Via Santa Marta 3, 50139 Firenze, Italy (personal communication), 1998.
  24. D. J. Durian, “The diffusion coefficient depends on absorption,” Opt. Lett. 23, 1502–1504 (1998).
    [CrossRef]
  25. K. Rinzema, Department of Physics/Institute for Physics Education, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands (personal communication, 1998).

1998 (2)

1997 (4)

1995 (1)

A. G. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[CrossRef]

1994 (1)

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

1991 (1)

I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991). This paper gives Holte’s result17 in the form where we quote here.
[CrossRef]

1974 (1)

E. W. Larsen, J. B. Keller, “Asymptotic solution of neutron transport problems for small mean free paths,” J. Math. Phys. 15, 75–81 (1974);E. W. Larsen, “Diffusion theory as an asymptotic limit of transport theory for nearly critical systems with small mean paths,” Ann. Nucl. Energy 7, 249–255 (1980).
[CrossRef]

1973 (1)

G. Caroll, R. Aronson, “One-speed neutron transport problems—part II: slab transmission and reflection and finite reflector critical problems,” Nucl. Sci. Eng. 51, 166–179 (1973).

1961 (1)

J. R. Mika, “Neutron transport with anisotropic scattering,” Nucl. Sci. Eng. 11, 415–427 (1961).

1948 (1)

G. Holte, “On a method of calculating the density of neutrons emitted from a point source in an infinite medium,” Ark. Mat. Astron. Fys. 35A(36), 1–9 (1948). Holte’s notation is somewhat different from ours.

1943 (1)

See, for instance, S. Chandrasekhar, “Stochastic Problems in physics and astronomy,” Rev. Mod. Phys. 15, 1–89 (1943);M. C. Wang, G. E. Uhlenbeck, “On the theory of Brownian motion II,” Rev. Mod. Phys. 17, 323–342.Both of these papers are reprinted in Selected Papers on Noise and Stochastic Processes, N. Wax, ed. (Dover, New York, 1964).
[CrossRef]

Aronson, R.

G. Caroll, R. Aronson, “One-speed neutron transport problems—part II: slab transmission and reflection and finite reflector critical problems,” Nucl. Sci. Eng. 51, 166–179 (1973).

R. L. Barbour, H. Graber, R. Aronson, J. Lubowsky, “Model for 3-D optical imaging of tissue,” in Proceedings of the International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, Piscataway, New Jersey, 1990), Vol. II, pp. 1295–1399.

Arridge, S. R.

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

Barbour, R. L.

R. L. Barbour, H. Graber, R. Aronson, J. Lubowsky, “Model for 3-D optical imaging of tissue,” in Proceedings of the International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, Piscataway, New Jersey, 1990), Vol. II, pp. 1295–1399.

Bassani, M.

Boas, D. A.

T. Durduran, A. G. Yodh, B. Chance, D. A. Boas, “Does the photon-diffusion coefficient depend on absorption?” J. Opt. Soc. Am. A 14, 3358–3365 (1997).
[CrossRef]

T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 60–63.

Caroll, G.

G. Caroll, R. Aronson, “One-speed neutron transport problems—part II: slab transmission and reflection and finite reflector critical problems,” Nucl. Sci. Eng. 51, 166–179 (1973).

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass.1967), Chap. 7.

K. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion, Vol. I (Los Alamos Scientific Laboratory, Los Alamos, N. Mex., 1953).

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Table 4.2, p. 91.

Chance, B.

T. Durduran, A. G. Yodh, B. Chance, D. A. Boas, “Does the photon-diffusion coefficient depend on absorption?” J. Opt. Soc. Am. A 14, 3358–3365 (1997).
[CrossRef]

A. G. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[CrossRef]

T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 60–63.

Chandrasekhar, S.

See, for instance, S. Chandrasekhar, “Stochastic Problems in physics and astronomy,” Rev. Mod. Phys. 15, 1–89 (1943);M. C. Wang, G. E. Uhlenbeck, “On the theory of Brownian motion II,” Rev. Mod. Phys. 17, 323–342.Both of these papers are reprinted in Selected Papers on Noise and Stochastic Processes, N. Wax, ed. (Dover, New York, 1964).
[CrossRef]

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Contini, D.

de Hoffmann, F.

K. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion, Vol. I (Los Alamos Scientific Laboratory, Los Alamos, N. Mex., 1953).

Durduran, T.

T. Durduran, A. G. Yodh, B. Chance, D. A. Boas, “Does the photon-diffusion coefficient depend on absorption?” J. Opt. Soc. Am. A 14, 3358–3365 (1997).
[CrossRef]

T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 60–63.

Durian, D. J.

Furutsu, K.

K. Furutsu, “Pulse wave scattering by an absorber and integrated attenuation in the diffusion approximation,” J. Opt. Soc. Am. A 14, 267–274 (1997).
[CrossRef]

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

Graber, H.

R. L. Barbour, H. Graber, R. Aronson, J. Lubowsky, “Model for 3-D optical imaging of tissue,” in Proceedings of the International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, Piscataway, New Jersey, 1990), Vol. II, pp. 1295–1399.

Hebden, J. C.

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

Holte, G.

G. Holte, “On a method of calculating the density of neutrons emitted from a point source in an infinite medium,” Ark. Mat. Astron. Fys. 35A(36), 1–9 (1948). Holte’s notation is somewhat different from ours.

Keller, J. B.

E. W. Larsen, J. B. Keller, “Asymptotic solution of neutron transport problems for small mean free paths,” J. Math. Phys. 15, 75–81 (1974);E. W. Larsen, “Diffusion theory as an asymptotic limit of transport theory for nearly critical systems with small mean paths,” Ann. Nucl. Energy 7, 249–255 (1980).
[CrossRef]

Kušcer, I.

I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991). This paper gives Holte’s result17 in the form where we quote here.
[CrossRef]

Lamarsh, J. R.

See, for instance, J. R. Lamarsh, Nuclear Reactor Theory (Addison-Wesley, Reading, Mass., 1966), pp. 125–128.

Larsen, E. W.

E. W. Larsen, J. B. Keller, “Asymptotic solution of neutron transport problems for small mean free paths,” J. Math. Phys. 15, 75–81 (1974);E. W. Larsen, “Diffusion theory as an asymptotic limit of transport theory for nearly critical systems with small mean paths,” Ann. Nucl. Energy 7, 249–255 (1980).
[CrossRef]

Lubowsky, J.

R. L. Barbour, H. Graber, R. Aronson, J. Lubowsky, “Model for 3-D optical imaging of tissue,” in Proceedings of the International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, Piscataway, New Jersey, 1990), Vol. II, pp. 1295–1399.

Martelli, F.

McCormick, N. J.

I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991). This paper gives Holte’s result17 in the form where we quote here.
[CrossRef]

Mika, J. R.

J. R. Mika, “Neutron transport with anisotropic scattering,” Nucl. Sci. Eng. 11, 415–427 (1961).

Murrer, L. H. P.

Placzek, G.

K. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion, Vol. I (Los Alamos Scientific Laboratory, Los Alamos, N. Mex., 1953).

Rinzema, K.

K. Rinzema, L. H. P. Murrer, W. M. Star, “Direct experimental verification of light transport theory in an optical phantom,” J. Opt. Soc. Am. A 15, 2078–2088 (1998).
[CrossRef]

K. Rinzema, Department of Physics/Institute for Physics Education, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands (personal communication, 1998).

Star, W. M.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 2, Chap. 10.

Yamada, Y.

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

Yodh, A. G.

T. Durduran, A. G. Yodh, B. Chance, D. A. Boas, “Does the photon-diffusion coefficient depend on absorption?” J. Opt. Soc. Am. A 14, 3358–3365 (1997).
[CrossRef]

A. G. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[CrossRef]

T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 60–63.

Zaccanti, G.

M. Bassani, F. Martelli, G. Zaccanti, D. Contini, “Independence of the diffusion coefficient from absorption: experimental and numerical evidence,” Opt. Lett. 22, 853–855 (1997).
[CrossRef] [PubMed]

G. Zaccanti, Dipartimento di Fisica dell’Università degli Studi di Firenze, Via Santa Marta 3, 50139 Firenze, Italy (personal communication), 1998.

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass.1967), Chap. 7.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Table 4.2, p. 91.

Ark. Mat. Astron. Fys. (1)

G. Holte, “On a method of calculating the density of neutrons emitted from a point source in an infinite medium,” Ark. Mat. Astron. Fys. 35A(36), 1–9 (1948). Holte’s notation is somewhat different from ours.

J. Math. Phys. (1)

E. W. Larsen, J. B. Keller, “Asymptotic solution of neutron transport problems for small mean free paths,” J. Math. Phys. 15, 75–81 (1974);E. W. Larsen, “Diffusion theory as an asymptotic limit of transport theory for nearly critical systems with small mean paths,” Ann. Nucl. Energy 7, 249–255 (1980).
[CrossRef]

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

Nucl. Sci. Eng. (2)

J. R. Mika, “Neutron transport with anisotropic scattering,” Nucl. Sci. Eng. 11, 415–427 (1961).

G. Caroll, R. Aronson, “One-speed neutron transport problems—part II: slab transmission and reflection and finite reflector critical problems,” Nucl. Sci. Eng. 51, 166–179 (1973).

Opt. Lett. (2)

Phys. Med. Biol. (1)

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[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]

Phys. Today (1)

A. G. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[CrossRef]

Rev. Mod. Phys. (1)

See, for instance, S. Chandrasekhar, “Stochastic Problems in physics and astronomy,” Rev. Mod. Phys. 15, 1–89 (1943);M. C. Wang, G. E. Uhlenbeck, “On the theory of Brownian motion II,” Rev. Mod. Phys. 17, 323–342.Both of these papers are reprinted in Selected Papers on Noise and Stochastic Processes, N. Wax, ed. (Dover, New York, 1964).
[CrossRef]

Transp. Theory Stat. Phys. (1)

I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991). This paper gives Holte’s result17 in the form where we quote here.
[CrossRef]

Other (11)

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass.1967), Chap. 7.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 2, Chap. 10.

G. Zaccanti, Dipartimento di Fisica dell’Università degli Studi di Firenze, Via Santa Marta 3, 50139 Firenze, Italy (personal communication), 1998.

K. Rinzema, Department of Physics/Institute for Physics Education, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands (personal communication, 1998).

See, for instance, J. R. Lamarsh, Nuclear Reactor Theory (Addison-Wesley, Reading, Mass., 1966), pp. 125–128.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

R. L. Barbour, H. Graber, R. Aronson, J. Lubowsky, “Model for 3-D optical imaging of tissue,” in Proceedings of the International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, Piscataway, New Jersey, 1990), Vol. II, pp. 1295–1399.

T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 60–63.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Table 4.2, p. 91.

K. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion, Vol. I (Los Alamos Scientific Laboratory, Los Alamos, N. Mex., 1953).

Cited By

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

Alert me when this article is cited.


Tables (1)

Tables Icon

Table 1 Diffusion Coefficient D; Parameter α Defined by Eq. (18); Diffusion Attenuation Length ν; and Attenuation Length ν2 of Transient Modes for Henyey–Greenstein Scattering, All As Functions of Single-Scattering Probability ϖ and Anisotropy Parameter g

Equations (20)

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

Ωˆ·I(r, Ωˆ)+μtI(r, Ωˆ)
=μsp(Ωˆ·Ωˆ)I(r, Ωˆ)dΩˆ+S(r, Ωˆ)
1cI(r, Ωˆ, t)t+Ωˆ·I(r, Ωˆ, t)+μtI(r, Ωˆ, t)
=μsp(Ωˆ·Ωˆ)I(r, Ωˆ, t)dΩˆ+S(r, Ωˆ, t).
p(x)=l=0L 2l+14πflPl(x),
1cϕ(r, t)t+·J(r, t)+μaϕ(r, t)=S(r, t),
1c J(r, t) t+13ϕ(r, t)+μtr J(r, t)=0.
μ tr=(1-g)μs+μaμs+μa,
1ctϕ(r, t)-D 2ϕ(r, t)+μaϕ(r, t)=S(r, t),
D=1/3μ tr.
f (μa)=f (μa=0) exp(-μact).
D=1/3μs.
ϖν2ln ν+1ν-1=1,
1/ν2=3(1-ϖ)1-45(1-ϖ)+4175(1-ϖ)2+4175(1-ϖ)3+7556336,875(1-ϖ)4+471,84421,896,875(1-ϖ)5+,
1/D=3μt1-45μaμt=3(μs+0.2μa).
1ν2=h0h11-4h0h2+4h0h221-94h1h3-4h0h23×1-274h1h3+8116h1216h32+9 h12h32h2h4+,
hl=(2l+1)(1-ϖ f l),
ϖκ2(1+s/c)ln κ+1κ-1=1,
κ=(1+s/c)ν.
D=13(μs+αμ a).

Metrics