Abstract

A photon-transport forward model for image reconstruction in turbid media is derived that treats weak inhomogeneities through a Born approximation of the Boltzmann radiative transfer equation. This model can conveniently replace the commonly used diffusion approximation in optical tomography. An analytical expression of the background Green’s function is obtained from the cumulant solution of the Boltzmann equation. Our model provides the correct behavior of photon migration at early times and reduces at long times to the center-moved diffusion approximation. Numerical comparisons between this model and the standard and center-moved diffusion models are presented.

© 2001 Optical Society of America

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References

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  1. R. R. Alfano, S. G. Demos, and S. K. Gayen, Ann. N.Y. Acad. Sci. 820, 248 (1997).
    [CrossRef]
  2. A. G. Yodh, B. Tromberg, E. Sevick-Muraca, and D. Pine, eds., feature on Diffusing Photons in Turbid MediaJ. Opt. Soc. Am. A14, 136–342 (1997).
  3. S. R. Arridge and J. C. Hebden, Phys. Med. Biol. 42, 841 (1997).
    [CrossRef] [PubMed]
  4. K. Furutsu and Y. Yamada, Phys. Rev. E 50, 3634 (1994).
    [CrossRef]
  5. M. S. Patterson, B. Chance, and B. C. Wilson, Appl. Opt. 28, 2331 (1989).
    [CrossRef] [PubMed]
  6. Y. Tsuchiya, K. Ohta, and T. Urakami, Jpn. J. Appl. Phys. Part 1 34, 2495 (1995).
    [CrossRef]
  7. W. Cai, M. Lax, and R. R. Alfano, Phys. Rev. E 61, 3871 (2000).
    [CrossRef]
  8. W. Cai, M. Lax, and R. R. Alfano, J. Phys. Chem. B 104, 3996 (2000).
  9. S. R. Arridge, Appl. Opt. 34, 7395 (1995).
    [CrossRef] [PubMed]
  10. L. G. Henyey and J. L. Greenstein, Astrophys. J. 93, 70 (1941).
    [CrossRef]
  11. K. M. Yoo, F. Liu, and R. R. Alfano, Phys. Rev. Lett. 64, 2647 (1990).
    [CrossRef] [PubMed]

2000 (2)

W. Cai, M. Lax, and R. R. Alfano, Phys. Rev. E 61, 3871 (2000).
[CrossRef]

W. Cai, M. Lax, and R. R. Alfano, J. Phys. Chem. B 104, 3996 (2000).

1997 (2)

R. R. Alfano, S. G. Demos, and S. K. Gayen, Ann. N.Y. Acad. Sci. 820, 248 (1997).
[CrossRef]

S. R. Arridge and J. C. Hebden, Phys. Med. Biol. 42, 841 (1997).
[CrossRef] [PubMed]

1995 (2)

Y. Tsuchiya, K. Ohta, and T. Urakami, Jpn. J. Appl. Phys. Part 1 34, 2495 (1995).
[CrossRef]

S. R. Arridge, Appl. Opt. 34, 7395 (1995).
[CrossRef] [PubMed]

1994 (1)

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

1990 (1)

K. M. Yoo, F. Liu, and R. R. Alfano, Phys. Rev. Lett. 64, 2647 (1990).
[CrossRef] [PubMed]

1989 (1)

1941 (1)

L. G. Henyey and J. L. Greenstein, Astrophys. J. 93, 70 (1941).
[CrossRef]

Alfano, R. R.

W. Cai, M. Lax, and R. R. Alfano, Phys. Rev. E 61, 3871 (2000).
[CrossRef]

W. Cai, M. Lax, and R. R. Alfano, J. Phys. Chem. B 104, 3996 (2000).

R. R. Alfano, S. G. Demos, and S. K. Gayen, Ann. N.Y. Acad. Sci. 820, 248 (1997).
[CrossRef]

K. M. Yoo, F. Liu, and R. R. Alfano, Phys. Rev. Lett. 64, 2647 (1990).
[CrossRef] [PubMed]

Arridge, S. R.

S. R. Arridge and J. C. Hebden, Phys. Med. Biol. 42, 841 (1997).
[CrossRef] [PubMed]

S. R. Arridge, Appl. Opt. 34, 7395 (1995).
[CrossRef] [PubMed]

Cai, W.

W. Cai, M. Lax, and R. R. Alfano, J. Phys. Chem. B 104, 3996 (2000).

W. Cai, M. Lax, and R. R. Alfano, Phys. Rev. E 61, 3871 (2000).
[CrossRef]

Chance, B.

Demos, S. G.

R. R. Alfano, S. G. Demos, and S. K. Gayen, Ann. N.Y. Acad. Sci. 820, 248 (1997).
[CrossRef]

Furutsu, K.

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

Gayen, S. K.

R. R. Alfano, S. G. Demos, and S. K. Gayen, Ann. N.Y. Acad. Sci. 820, 248 (1997).
[CrossRef]

Greenstein, J. L.

L. G. Henyey and J. L. Greenstein, Astrophys. J. 93, 70 (1941).
[CrossRef]

Hebden, J. C.

S. R. Arridge and J. C. Hebden, Phys. Med. Biol. 42, 841 (1997).
[CrossRef] [PubMed]

Henyey, L. G.

L. G. Henyey and J. L. Greenstein, Astrophys. J. 93, 70 (1941).
[CrossRef]

Lax, M.

W. Cai, M. Lax, and R. R. Alfano, Phys. Rev. E 61, 3871 (2000).
[CrossRef]

W. Cai, M. Lax, and R. R. Alfano, J. Phys. Chem. B 104, 3996 (2000).

Liu, F.

K. M. Yoo, F. Liu, and R. R. Alfano, Phys. Rev. Lett. 64, 2647 (1990).
[CrossRef] [PubMed]

Ohta, K.

Y. Tsuchiya, K. Ohta, and T. Urakami, Jpn. J. Appl. Phys. Part 1 34, 2495 (1995).
[CrossRef]

Patterson, M. S.

Tsuchiya, Y.

Y. Tsuchiya, K. Ohta, and T. Urakami, Jpn. J. Appl. Phys. Part 1 34, 2495 (1995).
[CrossRef]

Urakami, T.

Y. Tsuchiya, K. Ohta, and T. Urakami, Jpn. J. Appl. Phys. Part 1 34, 2495 (1995).
[CrossRef]

Wilson, B. C.

Yamada, Y.

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

Yoo, K. M.

K. M. Yoo, F. Liu, and R. R. Alfano, Phys. Rev. Lett. 64, 2647 (1990).
[CrossRef] [PubMed]

Ann. N.Y. Acad. Sci. (1)

R. R. Alfano, S. G. Demos, and S. K. Gayen, Ann. N.Y. Acad. Sci. 820, 248 (1997).
[CrossRef]

Appl. Opt. (2)

Astrophys. J. (1)

L. G. Henyey and J. L. Greenstein, Astrophys. J. 93, 70 (1941).
[CrossRef]

J. Phys. Chem. B (1)

W. Cai, M. Lax, and R. R. Alfano, J. Phys. Chem. B 104, 3996 (2000).

Jpn. J. Appl. Phys. Part 1 (1)

Y. Tsuchiya, K. Ohta, and T. Urakami, Jpn. J. Appl. Phys. Part 1 34, 2495 (1995).
[CrossRef]

Phys. Med. Biol. (1)

S. R. Arridge and J. C. Hebden, Phys. Med. Biol. 42, 841 (1997).
[CrossRef] [PubMed]

Phys. Rev. E (2)

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

W. Cai, M. Lax, and R. R. Alfano, Phys. Rev. E 61, 3871 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

K. M. Yoo, F. Liu, and R. R. Alfano, Phys. Rev. Lett. 64, 2647 (1990).
[CrossRef] [PubMed]

Other (1)

A. G. Yodh, B. Tromberg, E. Sevick-Muraca, and D. Pine, eds., feature on Diffusing Photons in Turbid MediaJ. Opt. Soc. Am. A14, 136–342 (1997).

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Figures (2)

Fig. 1
Fig. 1

Photon-distribution profiles (a) I0r,z^,t at position r=0,0,5 mm and (b) I0r,-z^,t at position r=0,2,0 mm.

Fig. 2
Fig. 2

Changes of photon-distribution function (a) -δIr,z^,t at position r=0,0,5 mm and (b) δIr,-z^,t at position r=0,2,0 mm.

Equations (6)

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

tIr,s,t+cs·rIr,s,t+cμsr+μarIr,s,t=cμsrdsPs,sIr,s,tds+qr,s,t,
I0(r,s,t|r0,s0,t0)=N0(r,t|r0,s0,t0)F0(s,t|s0,t0)-34πDt-t0s·rN0(r,t|r0,s0,t0),
Dt=Dxx+Dyy+Dzz3=c3ttg1-1-exp-g1tg12-1-exp-g1t22g12.
N0(r,t|r0,s0,t0)=14πDt-t0ct-t03/2×exp-r-r0-s0Δt-t024Dt-t0ct-t0exp-μact-t0,
δI(r,s,t|r0,s0,t0)=dtdrdsI0(r,-s,t-t|r,-s)×{dscδμsPs,s,rI0(r,s,t-t0|r0,s0)-cδμsr+δμarI0(r,s,t-t0|r0,s0)},
δI(r,s,t|r0,s0,t0)=-14πdtdrcδμar×N0(r,t-t|r,-s)N0(r,t-t0|r0,s0)+3c4πdtdrDt-tDt-t0δμar+δμsr×rN0(r,t-t|r,-s)·rN0(r,t-t0|r0,s0)+3c4πdtdrDt-tδμar+δμsrexp-g1t×{N0(r,t-t0|r,-s)s·rN0(r,t-t|r0,s0)-s0·rN0(r,t-t|r,-s)N0(r,t-t0|r0,s0)}

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