Abstract

The choice of the diffusion coefficient to be used in photon diffusion theory has been a subject of discussion in recent publications on tissue optics. We compared several diffusion coefficients with the apparent diffusion coefficient from the more fundamental transport theory, Dapp. Application to point sources in turbid media, for which exact solutions are available, showed that Dapp has to be preferred. We give a simple equation to approximate Dapp for several phase functions that apply to tissue optics. Reasons for the remaining discrepancies in diffusion coefficients applied to time-resolved and time-averaged descriptions of photon propagation in homogeneous turbid media are discussed.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  2. R. A. J. Groenhuis, H. A. Ferwerda, and J. J. ten Bosch, Appl. Opt. 22, 2456 (1983).
    [CrossRef] [PubMed]
  3. R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, and H. W. Jentink, Appl. Opt. 26, 2273 (1989).
    [CrossRef]
  4. R. A. Bolt and J. J. ten Bosch, Waves Random Media 4, 233 (1994).
    [CrossRef]
  5. K. Rinzema, B. J. Hoenders, H. A. Ferwerda, and J. J. ten Bosch, Pure Appl. Opt. 4, 629 (1995).
    [CrossRef]
  6. T. J. Farrell, M. S. Patterson, and B. C. Wilson, Med. Phys. 19, 879 (1992).
    [CrossRef] [PubMed]
  7. M. S. Patterson, B. Chance, and B. C. Wilson, Appl. Opt. 28, 2331 (1989).
    [CrossRef] [PubMed]
  8. M. Ferrari, Q. Wei, L. Carraresi, R. A. de Blasi, and G. Zaccanti, J. Photochem. Photobiol. 16, 141 (1992).
  9. S. R. Arridge, M. Cope, and D. T. Delpy, Phys. Med. Biol. 37, 1531 (1992).
    [CrossRef] [PubMed]
  10. A. H. Hielscher, H. Liu, B. Chance, F. K. Tittel, and S. L. Jacques, Appl. Opt. 35, 719 (1996).
    [CrossRef] [PubMed]
  11. K. Furutsu and Y. Yamada, Phys. Rev. E 50, 3634 (1994).
    [CrossRef]
  12. K. Furutsu, J. Opt. Soc. Am. A 14, 267 (1997).
    [CrossRef]
  13. V. G. Kolinko, F. F. M. de Mul, J. Greve, and A. V. Priezzhev, Appl. Opt. 35, 4541 (1996).
    [CrossRef] [PubMed]
  14. M. Bassani, F. Martelli, G. Zaccanti, and D. Contini, Opt. Lett. 22, 853 (1997).
    [CrossRef] [PubMed]
  15. T. Nakai, G. Nishimura, K. Yamamoto, and M. Tamura, Phys. Med. Biol. 42, 2541 (1997).
    [CrossRef]
  16. T. Durduran, A. G. Yodh, B. Chance, and D. A. Boas, J. Opt. Soc. Am. A 14, 3358 (1997).
    [CrossRef]
  17. K. Rinzema, L. H. P. Murrer, and W. M. Star, J. Opt. Soc. Am. A 15, 2078 (1998).
    [CrossRef]
  18. R. Aronson and N. Corngold, J. Opt. Soc. Am. A 16, 1066 (1999).
    [CrossRef]
  19. J. R. Mika, Nucl. Sci. Eng. 11, 415 (1961).
  20. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 87–93.
  21. H. C. van de Hulst, Multiple Light Scattering, Tables, Formulas and Applications (Academic, New York, 1980).
  22. H. C. van de Hulst and R. Graaff, Phys. Med. Biol. 41, 2519 (1996).
    [CrossRef] [PubMed]
  23. R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, and H. W. Jentink, Opt. Eng. 32, 244 (1993).
    [CrossRef]
  24. K. M. Case, F. de Hoffman, and G. Placzek, Introduction to the Theory of Neutron Diffusion (U.S. Government Printing Office, Washington, D.C., 1953), Vol. 1.
  25. D. J. Durian, Opt. Lett. 23, 1502 (1998).
    [CrossRef]

1999 (1)

1998 (2)

1997 (4)

1996 (3)

1995 (1)

K. Rinzema, B. J. Hoenders, H. A. Ferwerda, and J. J. ten Bosch, Pure Appl. Opt. 4, 629 (1995).
[CrossRef]

1994 (2)

R. A. Bolt and J. J. ten Bosch, Waves Random Media 4, 233 (1994).
[CrossRef]

K. Furutsu and Y. Yamada, Phys. Rev. E 50, 3634 (1994).
[CrossRef]

1993 (1)

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, and H. W. Jentink, Opt. Eng. 32, 244 (1993).
[CrossRef]

1992 (3)

T. J. Farrell, M. S. Patterson, and B. C. Wilson, Med. Phys. 19, 879 (1992).
[CrossRef] [PubMed]

M. Ferrari, Q. Wei, L. Carraresi, R. A. de Blasi, and G. Zaccanti, J. Photochem. Photobiol. 16, 141 (1992).

S. R. Arridge, M. Cope, and D. T. Delpy, Phys. Med. Biol. 37, 1531 (1992).
[CrossRef] [PubMed]

1989 (2)

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, and H. W. Jentink, Appl. Opt. 26, 2273 (1989).
[CrossRef]

M. S. Patterson, B. Chance, and B. C. Wilson, Appl. Opt. 28, 2331 (1989).
[CrossRef] [PubMed]

1983 (1)

1961 (1)

J. R. Mika, Nucl. Sci. Eng. 11, 415 (1961).

Aarnoudse, J. G.

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, and H. W. Jentink, Opt. Eng. 32, 244 (1993).
[CrossRef]

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, and H. W. Jentink, Appl. Opt. 26, 2273 (1989).
[CrossRef]

Aronson, R.

Arridge, S. R.

S. R. Arridge, M. Cope, and D. T. Delpy, Phys. Med. Biol. 37, 1531 (1992).
[CrossRef] [PubMed]

Bassani, M.

Boas, D. A.

Bolt, R. A.

R. A. Bolt and J. J. ten Bosch, Waves Random Media 4, 233 (1994).
[CrossRef]

Carraresi, L.

M. Ferrari, Q. Wei, L. Carraresi, R. A. de Blasi, and G. Zaccanti, J. Photochem. Photobiol. 16, 141 (1992).

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 87–93.

K. M. Case, F. de Hoffman, and G. Placzek, Introduction to the Theory of Neutron Diffusion (U.S. Government Printing Office, Washington, D.C., 1953), Vol. 1.

Chance, B.

Contini, D.

Cope, M.

S. R. Arridge, M. Cope, and D. T. Delpy, Phys. Med. Biol. 37, 1531 (1992).
[CrossRef] [PubMed]

Corngold, N.

de Blasi, R. A.

M. Ferrari, Q. Wei, L. Carraresi, R. A. de Blasi, and G. Zaccanti, J. Photochem. Photobiol. 16, 141 (1992).

de Hoffman, F.

K. M. Case, F. de Hoffman, and G. Placzek, Introduction to the Theory of Neutron Diffusion (U.S. Government Printing Office, Washington, D.C., 1953), Vol. 1.

de Mul, F. F. M.

V. G. Kolinko, F. F. M. de Mul, J. Greve, and A. V. Priezzhev, Appl. Opt. 35, 4541 (1996).
[CrossRef] [PubMed]

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, and H. W. Jentink, Opt. Eng. 32, 244 (1993).
[CrossRef]

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, and H. W. Jentink, Appl. Opt. 26, 2273 (1989).
[CrossRef]

Delpy, D. T.

S. R. Arridge, M. Cope, and D. T. Delpy, Phys. Med. Biol. 37, 1531 (1992).
[CrossRef] [PubMed]

Durduran, T.

Durian, D. J.

Farrell, T. J.

T. J. Farrell, M. S. Patterson, and B. C. Wilson, Med. Phys. 19, 879 (1992).
[CrossRef] [PubMed]

Ferrari, M.

M. Ferrari, Q. Wei, L. Carraresi, R. A. de Blasi, and G. Zaccanti, J. Photochem. Photobiol. 16, 141 (1992).

Ferwerda, H. A.

K. Rinzema, B. J. Hoenders, H. A. Ferwerda, and J. J. ten Bosch, Pure Appl. Opt. 4, 629 (1995).
[CrossRef]

R. A. J. Groenhuis, H. A. Ferwerda, and J. J. ten Bosch, Appl. Opt. 22, 2456 (1983).
[CrossRef] [PubMed]

Furutsu, K.

K. Furutsu, J. Opt. Soc. Am. A 14, 267 (1997).
[CrossRef]

K. Furutsu and Y. Yamada, Phys. Rev. E 50, 3634 (1994).
[CrossRef]

Graaff, R.

H. C. van de Hulst and R. Graaff, Phys. Med. Biol. 41, 2519 (1996).
[CrossRef] [PubMed]

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, and H. W. Jentink, Opt. Eng. 32, 244 (1993).
[CrossRef]

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, and H. W. Jentink, Appl. Opt. 26, 2273 (1989).
[CrossRef]

Greve, J.

Groenhuis, R. A. J.

Hielscher, A. H.

Hoenders, B. J.

K. Rinzema, B. J. Hoenders, H. A. Ferwerda, and J. J. ten Bosch, Pure Appl. Opt. 4, 629 (1995).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Jacques, S. L.

Jentink, H. W.

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, and H. W. Jentink, Opt. Eng. 32, 244 (1993).
[CrossRef]

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, and H. W. Jentink, Appl. Opt. 26, 2273 (1989).
[CrossRef]

Kolinko, V. G.

Liu, H.

Martelli, F.

Mika, J. R.

J. R. Mika, Nucl. Sci. Eng. 11, 415 (1961).

Murrer, L. H. P.

Nakai, T.

T. Nakai, G. Nishimura, K. Yamamoto, and M. Tamura, Phys. Med. Biol. 42, 2541 (1997).
[CrossRef]

Nishimura, G.

T. Nakai, G. Nishimura, K. Yamamoto, and M. Tamura, Phys. Med. Biol. 42, 2541 (1997).
[CrossRef]

Patterson, M. S.

T. J. Farrell, M. S. Patterson, and B. C. Wilson, Med. Phys. 19, 879 (1992).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and B. C. Wilson, Appl. Opt. 28, 2331 (1989).
[CrossRef] [PubMed]

Placzek, G.

K. M. Case, F. de Hoffman, and G. Placzek, Introduction to the Theory of Neutron Diffusion (U.S. Government Printing Office, Washington, D.C., 1953), Vol. 1.

Priezzhev, A. V.

Rinzema, K.

K. Rinzema, L. H. P. Murrer, and W. M. Star, J. Opt. Soc. Am. A 15, 2078 (1998).
[CrossRef]

K. Rinzema, B. J. Hoenders, H. A. Ferwerda, and J. J. ten Bosch, Pure Appl. Opt. 4, 629 (1995).
[CrossRef]

Star, W. M.

Tamura, M.

T. Nakai, G. Nishimura, K. Yamamoto, and M. Tamura, Phys. Med. Biol. 42, 2541 (1997).
[CrossRef]

ten Bosch, J. J.

K. Rinzema, B. J. Hoenders, H. A. Ferwerda, and J. J. ten Bosch, Pure Appl. Opt. 4, 629 (1995).
[CrossRef]

R. A. Bolt and J. J. ten Bosch, Waves Random Media 4, 233 (1994).
[CrossRef]

R. A. J. Groenhuis, H. A. Ferwerda, and J. J. ten Bosch, Appl. Opt. 22, 2456 (1983).
[CrossRef] [PubMed]

Tittel, F. K.

van de Hulst, H. C.

H. C. van de Hulst and R. Graaff, Phys. Med. Biol. 41, 2519 (1996).
[CrossRef] [PubMed]

H. C. van de Hulst, Multiple Light Scattering, Tables, Formulas and Applications (Academic, New York, 1980).

Wei, Q.

M. Ferrari, Q. Wei, L. Carraresi, R. A. de Blasi, and G. Zaccanti, J. Photochem. Photobiol. 16, 141 (1992).

Wilson, B. C.

T. J. Farrell, M. S. Patterson, and B. C. Wilson, Med. Phys. 19, 879 (1992).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and B. C. Wilson, Appl. Opt. 28, 2331 (1989).
[CrossRef] [PubMed]

Yamada, Y.

K. Furutsu and Y. Yamada, Phys. Rev. E 50, 3634 (1994).
[CrossRef]

Yamamoto, K.

T. Nakai, G. Nishimura, K. Yamamoto, and M. Tamura, Phys. Med. Biol. 42, 2541 (1997).
[CrossRef]

Yodh, A. G.

Zaccanti, G.

M. Bassani, F. Martelli, G. Zaccanti, and D. Contini, Opt. Lett. 22, 853 (1997).
[CrossRef] [PubMed]

M. Ferrari, Q. Wei, L. Carraresi, R. A. de Blasi, and G. Zaccanti, J. Photochem. Photobiol. 16, 141 (1992).

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 87–93.

Appl. Opt. (5)

J. Opt. Soc. Am. A (4)

J. Photochem. Photobiol. (1)

M. Ferrari, Q. Wei, L. Carraresi, R. A. de Blasi, and G. Zaccanti, J. Photochem. Photobiol. 16, 141 (1992).

Med. Phys. (1)

T. J. Farrell, M. S. Patterson, and B. C. Wilson, Med. Phys. 19, 879 (1992).
[CrossRef] [PubMed]

Nucl. Sci. Eng. (1)

J. R. Mika, Nucl. Sci. Eng. 11, 415 (1961).

Opt. Eng. (1)

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, and H. W. Jentink, Opt. Eng. 32, 244 (1993).
[CrossRef]

Opt. Lett. (2)

Phys. Med. Biol. (3)

H. C. van de Hulst and R. Graaff, Phys. Med. Biol. 41, 2519 (1996).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, and D. T. Delpy, Phys. Med. Biol. 37, 1531 (1992).
[CrossRef] [PubMed]

T. Nakai, G. Nishimura, K. Yamamoto, and M. Tamura, Phys. Med. Biol. 42, 2541 (1997).
[CrossRef]

Phys. Rev. E (1)

K. Furutsu and Y. Yamada, Phys. Rev. E 50, 3634 (1994).
[CrossRef]

Pure Appl. Opt. (1)

K. Rinzema, B. J. Hoenders, H. A. Ferwerda, and J. J. ten Bosch, Pure Appl. Opt. 4, 629 (1995).
[CrossRef]

Waves Random Media (1)

R. A. Bolt and J. J. ten Bosch, Waves Random Media 4, 233 (1994).
[CrossRef]

Other (4)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

K. M. Case, F. de Hoffman, and G. Placzek, Introduction to the Theory of Neutron Diffusion (U.S. Government Printing Office, Washington, D.C., 1953), Vol. 1.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 87–93.

H. C. van de Hulst, Multiple Light Scattering, Tables, Formulas and Applications (Academic, New York, 1980).

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.


Figures (2)

Fig. 1
Fig. 1

μtDapp for isotropic scattering (squares, g=0), HG scattering (crosses, g=0.875), and RG scattering (diamonds, g=0.875). Results with Eqs. (3) and (4) are given by the remaining solid curves.

Fig. 2
Fig. 2

Exact solution (symbols) for lnrϕr/μt as a function of μtr for a source with unit strength in a turbid medium with isotropic scatterers for a=0.6 and a=0.9. Solid curves, the results for diffusion theory [Eq. (2)], with D according to Eqs. (3), (4), and (6).

Equations (7)

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

ϕr,t=cexp-r2/4Dctexp-μact4πDct3/2,
ϕr=14πDrexp-rμa/D,
D=1/3μt,
D=1/3μs,
ϕz=ϕ0exp-k0z.
Dapp=μa/k02.
Dapp1/μa+3am-1μs,

Metrics