Abstract

We have developed a theoretical model for photon migration through scattering media in the presence of an absorbing inhomogeneity. A closed-form solution for the average diffuse intensity has been obtained through an iterative approximation scheme of the steady-state diffusion equation. The model describes absorbing defects in a wide range of values. Comparisons with the results of Monte Carlo simulations show that the error of the model is lower than 3% for size inclusion lower than 4 mm and absorption contrast up to the threshold value of the “black defect.” The proposed model provides a tractable mathematical basis for diffuse optical and photoacoustic tomographic reconstruction techniques.

© 2014 Optical Society of America

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  1. T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, Rep. Prog. Phys. 73, 076701 (2010).
    [CrossRef]
  2. B. Cox, S. Arridge, K. Kostli, and P. Beard, Appl. Opt. 45, 1866 (2006).
  3. A. Q. Bauer, R. E. Nothdurft, T. N. Erpelding, L. V. Wang, and J. P. Culver, J. Biomed. Opt. 16, 096016 (2011).
    [CrossRef]
  4. T. Correia, N. Ducros, C. D’Andrea, M. Schweiger, and S. Arridge, Opt. Lett. 38, 1903 (2013).
    [CrossRef]
  5. V. Ntziachristos and R. Weissleder, Opt. Lett. 26, 893 (2001).
    [CrossRef]
  6. M. R. Ostermeyer and S. L. Jacques, J. Opt. Soc. Am. A 14, 255 (1997).
    [CrossRef]
  7. S. Carraresi, T. S. M. Shatir, F. Martelli, and G. Zaccanti, Appl. Opt. 40, 4622 (2001).
    [CrossRef]
  8. S. De Nicola, R. Esposito, and M. Lepore, Phys. Rev. E 68, 21901 (2003).
    [CrossRef]
  9. S. Feng, F. A. Zeng, and B. Chance, Appl. Opt. 34, 3826 (1995).
    [CrossRef]
  10. D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, Phys. Rev. E 76, 061908 (2007).
    [CrossRef]
  11. A. Sassaroli, F. Martelli, and S. Fantini, Appl. Opt. 48, D62 (2009).
    [CrossRef]
  12. A. Sassaroli, F. Martelli, and S. Fantini, J. Opt. Soc. Am. A 27, 1723 (2010).
    [CrossRef]
  13. A. G. Gandjbakhche, R. F. Bonner, R. Nossal, and G. H. Weiss, Appl. Opt. 35, 1767 (1996).
    [CrossRef]
  14. V. Chernomordik, D. Hattery, A. H. Gandjbakhche, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, and R. Cubeddu, Opt. Lett. 25, 951 (2000).
    [CrossRef]
  15. G. Started, The Language of Technical Computing (Prentice Hall, 1997), Vol. L, p. 186.
  16. G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Elsevier, 1971), Vol. 39.
  17. F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE, 2010).
  18. R. Esposito, S. De Nicola, M. Lepore, and P. L. Indovina, J. Opt. Soc. Am. A 23, 1937 (2006).
    [CrossRef]

2013

2011

A. Q. Bauer, R. E. Nothdurft, T. N. Erpelding, L. V. Wang, and J. P. Culver, J. Biomed. Opt. 16, 096016 (2011).
[CrossRef]

2010

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, Rep. Prog. Phys. 73, 076701 (2010).
[CrossRef]

A. Sassaroli, F. Martelli, and S. Fantini, J. Opt. Soc. Am. A 27, 1723 (2010).
[CrossRef]

2009

2007

D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, Phys. Rev. E 76, 061908 (2007).
[CrossRef]

2006

2003

S. De Nicola, R. Esposito, and M. Lepore, Phys. Rev. E 68, 21901 (2003).
[CrossRef]

2001

2000

1997

1996

1995

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Elsevier, 1971), Vol. 39.

Arridge, S.

Baker, W. B.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, Rep. Prog. Phys. 73, 076701 (2010).
[CrossRef]

Bauer, A. Q.

A. Q. Bauer, R. E. Nothdurft, T. N. Erpelding, L. V. Wang, and J. P. Culver, J. Biomed. Opt. 16, 096016 (2011).
[CrossRef]

Beard, P.

Bonner, R. F.

Carraresi, S.

Chance, B.

Chernomordik, V.

Choe, R.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, Rep. Prog. Phys. 73, 076701 (2010).
[CrossRef]

Correia, T.

Cox, B.

Cubeddu, R.

Culver, J. P.

A. Q. Bauer, R. E. Nothdurft, T. N. Erpelding, L. V. Wang, and J. P. Culver, J. Biomed. Opt. 16, 096016 (2011).
[CrossRef]

D’Andrea, C.

De Nicola, S.

Del Bianco, S.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE, 2010).

Ducros, N.

Durduran, T.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, Rep. Prog. Phys. 73, 076701 (2010).
[CrossRef]

Erpelding, T. N.

A. Q. Bauer, R. E. Nothdurft, T. N. Erpelding, L. V. Wang, and J. P. Culver, J. Biomed. Opt. 16, 096016 (2011).
[CrossRef]

Esposito, R.

Fantini, S.

Feng, S.

Gandjbakhche, A. G.

Gandjbakhche, A. H.

Grosenick, D.

D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, Phys. Rev. E 76, 061908 (2007).
[CrossRef]

Hattery, D.

Indovina, P. L.

Ismaelli, A.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE, 2010).

Jacques, S. L.

Kostli, K.

Kummrow, A.

D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, Phys. Rev. E 76, 061908 (2007).
[CrossRef]

Lepore, M.

Macdonald, R.

D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, Phys. Rev. E 76, 061908 (2007).
[CrossRef]

Martelli, F.

A. Sassaroli, F. Martelli, and S. Fantini, J. Opt. Soc. Am. A 27, 1723 (2010).
[CrossRef]

A. Sassaroli, F. Martelli, and S. Fantini, Appl. Opt. 48, D62 (2009).
[CrossRef]

S. Carraresi, T. S. M. Shatir, F. Martelli, and G. Zaccanti, Appl. Opt. 40, 4622 (2001).
[CrossRef]

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE, 2010).

Nossal, R.

Nothdurft, R. E.

A. Q. Bauer, R. E. Nothdurft, T. N. Erpelding, L. V. Wang, and J. P. Culver, J. Biomed. Opt. 16, 096016 (2011).
[CrossRef]

Ntziachristos, V.

Ostermeyer, M. R.

Pifferi, A.

Rinneberg, H.

D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, Phys. Rev. E 76, 061908 (2007).
[CrossRef]

Sassaroli, A.

Schlag, P. M.

D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, Phys. Rev. E 76, 061908 (2007).
[CrossRef]

Schweiger, M.

Shatir, T. S. M.

Started, G.

G. Started, The Language of Technical Computing (Prentice Hall, 1997), Vol. L, p. 186.

Taroni, P.

Torricelli, A.

Valentini, G.

Wang, L. V.

A. Q. Bauer, R. E. Nothdurft, T. N. Erpelding, L. V. Wang, and J. P. Culver, J. Biomed. Opt. 16, 096016 (2011).
[CrossRef]

Weiss, G. H.

Weissleder, R.

Yodh, A. G.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, Rep. Prog. Phys. 73, 076701 (2010).
[CrossRef]

Zaccanti, G.

S. Carraresi, T. S. M. Shatir, F. Martelli, and G. Zaccanti, Appl. Opt. 40, 4622 (2001).
[CrossRef]

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE, 2010).

Zeng, F. A.

Appl. Opt.

J. Biomed. Opt.

A. Q. Bauer, R. E. Nothdurft, T. N. Erpelding, L. V. Wang, and J. P. Culver, J. Biomed. Opt. 16, 096016 (2011).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Phys. Rev. E

S. De Nicola, R. Esposito, and M. Lepore, Phys. Rev. E 68, 21901 (2003).
[CrossRef]

D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, Phys. Rev. E 76, 061908 (2007).
[CrossRef]

Rep. Prog. Phys.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, Rep. Prog. Phys. 73, 076701 (2010).
[CrossRef]

Other

G. Started, The Language of Technical Computing (Prentice Hall, 1997), Vol. L, p. 186.

G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Elsevier, 1971), Vol. 39.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE, 2010).

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Figures (3)

Fig. 1.
Fig. 1.

Schematic of the heterogeneous slab with the absorptive inclusion.

Fig. 2.
Fig. 2.

Relative contrast |ΔI/I0| for the transmittance through a slab plotted against the absorption contrast Δμa between a spherical absorptive inclusion and the background medium.

Fig. 3.
Fig. 3.

Relative contrast as a function of the inclusion volume V of the black defect. The red dotted curve is the model prediction and the solid black curve is the MC simulation. The solid blue curve shows the relative discrepancies. The slab geometry and the optical parameters are the same as in Fig. 2(a).

Equations (12)

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

D2φ(r,r0)(μa+Δμa(r))φ(r,r0)=δ(rr0)4π,
φ(r,r0)=φ0(r,r0)+n=1φn(r,r0).
φn(r,r0)=ΔμaVdrG(rr)φn1(r,r0),
φ1(r,r0)=Δμa4πVdrG(rr)G(rr0),
φ1(r,r0)=ΔμaV4πG(rcr0)G¯(r),
G¯(r)=1VVdrG(rr).
φ2(r,r0)=Δμa2V24πG¯(rc)G(rcr0)G¯(r),
φn(r,r0)=(ΔμaVG¯(rc))n4πG¯(rc)G(rcr0)G¯(r).
Δφ(r,r0;Δμa)=(ΔμaV+Δμa2V2G¯(rc)1+ΔμaVG¯(rc))G(rcr0)G¯(r)4π,
Δφ(r,r0;Δμaμa)=G(rcr0)G¯(r)4πG¯(rc).
G(rr0)=14πDn=exp(μaDρ+)ρ+exp(μaDρ)ρ.
G¯(rc)=1Vμa[1eRα(1+Rα)],

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