Abstract

We derive two mathematical relations between quantitative phase images of thin slices of inhomogeneous media and the scattering parameters of the bulk, i.e., scattering mean free path, ls, and anisotropy factor, g. The ls turns out to be inversely proportional to the spatial variance of the phase shift, and g is related to the variance of the phase gradient. These formulas, referred collectively to as the scattering-phase theorem, allow for extracting ls and g in a spatially resolved manner and across an entire tissue section, that is, mapping large cross sections of tissues in terms of ls and g.

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  1. J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis (Wiley, 1976).
  2. T. J. Farrell, M. S. Patterson, and B. Wilson, Med. Phys. 19, 879 (1992).
    [CrossRef] [PubMed]
  3. M. S. Patterson, B. Chance, and B. C. Wilson, Appl. Opt. 28, 2331 (1989).
    [CrossRef] [PubMed]
  4. J. M. Schmitt and G. Kumar, Appl. Opt. 37, 2788 (1998).
    [CrossRef]
  5. M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, G. Venkatesh, K. Badizadegan, G. D. Stoner, and M. S. Feld, Phys. Rev. Lett. 97, 138102 (2006).
    [CrossRef] [PubMed]
  6. S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, IEEE Trans. Biomed. Eng. 36, 1162 (1989).
    [CrossRef] [PubMed]
  7. A. Dunn and R. Richards-Kortum, IEEE J. Sel. Top. Quantum Electron. 2, 898 (1996).
    [CrossRef]
  8. H. Ding, Z. Wang, F. Nguyen, S. A. Boppart, and G. Popescu, Phys. Rev. Lett. 101, 238102 (2008).
    [CrossRef] [PubMed]
  9. H. Ding, F. Nguyen, S. A. Boppart, and G. Popescu, Opt. Lett. 34, 1372 (2009).
    [CrossRef] [PubMed]
  10. R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 2000).
  11. H. Ding, X. Liang, Z. Wang, S. A. Boppart, and G. Popescu, “Tissue scattering parameters from organelle to organ scale,” submitted to Opt. Lett.

2009 (1)

2008 (1)

H. Ding, Z. Wang, F. Nguyen, S. A. Boppart, and G. Popescu, Phys. Rev. Lett. 101, 238102 (2008).
[CrossRef] [PubMed]

2006 (1)

M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, G. Venkatesh, K. Badizadegan, G. D. Stoner, and M. S. Feld, Phys. Rev. Lett. 97, 138102 (2006).
[CrossRef] [PubMed]

1998 (1)

1996 (1)

A. Dunn and R. Richards-Kortum, IEEE J. Sel. Top. Quantum Electron. 2, 898 (1996).
[CrossRef]

1992 (1)

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

1989 (2)

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

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, IEEE Trans. Biomed. Eng. 36, 1162 (1989).
[CrossRef] [PubMed]

Backman, V.

M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, G. Venkatesh, K. Badizadegan, G. D. Stoner, and M. S. Feld, Phys. Rev. Lett. 97, 138102 (2006).
[CrossRef] [PubMed]

Badizadegan, K.

M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, G. Venkatesh, K. Badizadegan, G. D. Stoner, and M. S. Feld, Phys. Rev. Lett. 97, 138102 (2006).
[CrossRef] [PubMed]

Boone, C. W.

M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, G. Venkatesh, K. Badizadegan, G. D. Stoner, and M. S. Feld, Phys. Rev. Lett. 97, 138102 (2006).
[CrossRef] [PubMed]

Boppart, S. A.

H. Ding, F. Nguyen, S. A. Boppart, and G. Popescu, Opt. Lett. 34, 1372 (2009).
[CrossRef] [PubMed]

H. Ding, Z. Wang, F. Nguyen, S. A. Boppart, and G. Popescu, Phys. Rev. Lett. 101, 238102 (2008).
[CrossRef] [PubMed]

H. Ding, X. Liang, Z. Wang, S. A. Boppart, and G. Popescu, “Tissue scattering parameters from organelle to organ scale,” submitted to Opt. Lett.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 2000).

Chance, B.

Ding, H.

H. Ding, F. Nguyen, S. A. Boppart, and G. Popescu, Opt. Lett. 34, 1372 (2009).
[CrossRef] [PubMed]

H. Ding, Z. Wang, F. Nguyen, S. A. Boppart, and G. Popescu, Phys. Rev. Lett. 101, 238102 (2008).
[CrossRef] [PubMed]

H. Ding, X. Liang, Z. Wang, S. A. Boppart, and G. Popescu, “Tissue scattering parameters from organelle to organ scale,” submitted to Opt. Lett.

Duderstadt, J. J.

J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis (Wiley, 1976).

Dunn, A.

A. Dunn and R. Richards-Kortum, IEEE J. Sel. Top. Quantum Electron. 2, 898 (1996).
[CrossRef]

Farrell, T. J.

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

Feld, M. S.

M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, G. Venkatesh, K. Badizadegan, G. D. Stoner, and M. S. Feld, Phys. Rev. Lett. 97, 138102 (2006).
[CrossRef] [PubMed]

Flock, S. T.

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, IEEE Trans. Biomed. Eng. 36, 1162 (1989).
[CrossRef] [PubMed]

Hamilton, L. J.

J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis (Wiley, 1976).

Hunter, M.

M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, G. Venkatesh, K. Badizadegan, G. D. Stoner, and M. S. Feld, Phys. Rev. Lett. 97, 138102 (2006).
[CrossRef] [PubMed]

Kalashnikov, M.

M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, G. Venkatesh, K. Badizadegan, G. D. Stoner, and M. S. Feld, Phys. Rev. Lett. 97, 138102 (2006).
[CrossRef] [PubMed]

Kumar, G.

Liang, X.

H. Ding, X. Liang, Z. Wang, S. A. Boppart, and G. Popescu, “Tissue scattering parameters from organelle to organ scale,” submitted to Opt. Lett.

Nguyen, F.

H. Ding, F. Nguyen, S. A. Boppart, and G. Popescu, Opt. Lett. 34, 1372 (2009).
[CrossRef] [PubMed]

H. Ding, Z. Wang, F. Nguyen, S. A. Boppart, and G. Popescu, Phys. Rev. Lett. 101, 238102 (2008).
[CrossRef] [PubMed]

Patterson, M. S.

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

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, IEEE Trans. Biomed. Eng. 36, 1162 (1989).
[CrossRef] [PubMed]

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

Popescu, G.

H. Ding, F. Nguyen, S. A. Boppart, and G. Popescu, Opt. Lett. 34, 1372 (2009).
[CrossRef] [PubMed]

H. Ding, Z. Wang, F. Nguyen, S. A. Boppart, and G. Popescu, Phys. Rev. Lett. 101, 238102 (2008).
[CrossRef] [PubMed]

M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, G. Venkatesh, K. Badizadegan, G. D. Stoner, and M. S. Feld, Phys. Rev. Lett. 97, 138102 (2006).
[CrossRef] [PubMed]

H. Ding, X. Liang, Z. Wang, S. A. Boppart, and G. Popescu, “Tissue scattering parameters from organelle to organ scale,” submitted to Opt. Lett.

Richards-Kortum, R.

A. Dunn and R. Richards-Kortum, IEEE J. Sel. Top. Quantum Electron. 2, 898 (1996).
[CrossRef]

Schmitt, J. M.

Stoner, G. D.

M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, G. Venkatesh, K. Badizadegan, G. D. Stoner, and M. S. Feld, Phys. Rev. Lett. 97, 138102 (2006).
[CrossRef] [PubMed]

Venkatesh, G.

M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, G. Venkatesh, K. Badizadegan, G. D. Stoner, and M. S. Feld, Phys. Rev. Lett. 97, 138102 (2006).
[CrossRef] [PubMed]

Wang, Z.

H. Ding, Z. Wang, F. Nguyen, S. A. Boppart, and G. Popescu, Phys. Rev. Lett. 101, 238102 (2008).
[CrossRef] [PubMed]

H. Ding, X. Liang, Z. Wang, S. A. Boppart, and G. Popescu, “Tissue scattering parameters from organelle to organ scale,” submitted to Opt. Lett.

Wax, A.

M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, G. Venkatesh, K. Badizadegan, G. D. Stoner, and M. S. Feld, Phys. Rev. Lett. 97, 138102 (2006).
[CrossRef] [PubMed]

Wilson, B.

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

Wilson, B. C.

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, IEEE Trans. Biomed. Eng. 36, 1162 (1989).
[CrossRef] [PubMed]

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

Wyman, D. R.

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, IEEE Trans. Biomed. Eng. 36, 1162 (1989).
[CrossRef] [PubMed]

Appl. Opt. (2)

IEEE J. Sel. Top. Quantum Electron. (1)

A. Dunn and R. Richards-Kortum, IEEE J. Sel. Top. Quantum Electron. 2, 898 (1996).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, IEEE Trans. Biomed. Eng. 36, 1162 (1989).
[CrossRef] [PubMed]

Med. Phys. (1)

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

Opt. Lett. (1)

Phys. Rev. Lett. (2)

M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, G. Venkatesh, K. Badizadegan, G. D. Stoner, and M. S. Feld, Phys. Rev. Lett. 97, 138102 (2006).
[CrossRef] [PubMed]

H. Ding, Z. Wang, F. Nguyen, S. A. Boppart, and G. Popescu, Phys. Rev. Lett. 101, 238102 (2008).
[CrossRef] [PubMed]

Other (3)

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 2000).

H. Ding, X. Liang, Z. Wang, S. A. Boppart, and G. Popescu, “Tissue scattering parameters from organelle to organ scale,” submitted to Opt. Lett.

J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis (Wiley, 1976).

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Figures (1)

Fig. 1
Fig. 1

Light scattering by a thin tissue slice. (a)  U 0 , incident field; U 0 , unscattered component of the transmitted field; L, thickness of the tissue slice; l s , scattering mean free path; k 0 , incident wave vector. (b)  k s , scattering wave vector; q, momentum transfer; θ, scattering angle. (c) Position vector in the phase image.

Equations (16)

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

l s = L Δ ϕ 2 ( r ) r ,
g = 1 ( l s L ) 2 | [ ϕ ( r ) ] | 2 r 2 k 0 2 .
I 0 = I 0 e L / l s .
U ( r ) = U 0 + U 1 ( r ) .
U 0 = U 0 · e i ϕ ( r ) r .
U 0 = U 0 2 π ( Δ ϕ ) 2 e i ϕ e ϕ 2 2 Δ ϕ 2 r d ϕ = U 0 e Δ ϕ 2 r 2 .
l s = L Δ ϕ 2 ( r ) r .
g = cos θ θ .
cos θ = 1 q 2 2 k 0 2 , q = 2 k 0 sin θ 2 .
g = 1 1 2 k 0 2 ( q x 2 + q y 2 ) P ( q x , q y ) d q x d q y .
P ( q x , q y ) = | U ˜ ( q x , q y ) | 2 | U ˜ ( q x , q y ) | 2 d q x d q y .
g = 1 1 2 k 0 2 [ | i q x U ( q x , q y ) | 2 + | i q y U ( q x , q y ) | 2 ] d q x d q y | U ˜ ( q x , q y ) | 2 d q x d q y .
| i q α U ( q x , q y ) | 2 d q x d q y = | U ( x , y ) / α | 2 d x d y ,
g = 1 | [ ϕ l s ( r ) ] | 2 r 2 k 0 2 ,
g = 1 ( l s L ) 2 | [ ϕ ( r ) ] | 2 r 2 k 0 2 .
g = 1 1 2 k 0 2 | [ ϕ ( r ) ] | 2 r Δ ϕ 2 ( r ) r 2 .

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