Abstract

An analytical model to perform tomographic reconstructions for absorptive inclusions in highly scattering media using dual interfering sources was derived. A perturbation approach within the first order Rytov expansion was used to solve the heterogeneous diffusion equation. Analytical weight functions necessary to solve the inverse problem were obtained. The reconstructions performance was assessed using simulated data of breast-like media after contrast agent enhancement. We further investigated the reconstruction quality as a function of object depth location, modulation frequency and source separation. The ability of the algorithm to resolve multi-objects was also demonstrated.

© 2002 Optical Society of America

Full Article  |  PDF Article
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References

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  1. D. Hawrys and E. Sevick-Muraca, “Developments toward diagnostic breast cancer imaging using Near-Infrared optical measurements and fluorescent contrast agents,” Neoplasia 2, 388–417 (2000).
    [Crossref]
  2. T. McBride, B. Pogue, S. Jiang, U. Osterberg, and K. Paulsen, “Initial studies of in-vivo absorbing and scattering heterogeneity in near-infrared tomographic breast imaging,” Opt. Let. 26, 822–824 (2001).
    [Crossref]
  3. V. Ntziachristos and B. Chance, “Probing physiology and molecular function using optical imaging: applications to breast cancer,” Breast Cancer Research 3, 41–47 (2001).
    [Crossref] [PubMed]
  4. A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human function,” Trends Neurosci. 20, 435–442 (1997).
    [Crossref] [PubMed]
  5. M. O’Leary, D. Boas, B. Chance, and A. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing photon-tomography,” Opt. Lett. 20, 426–428 (1995).
    [Crossref]
  6. V. Ntziachristos, A. Yodh, M. Schnall, and B. Chance, “Concurent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc.Nat.Acad.Sci. USA 97, 2767–2772 (2000).
    [Crossref] [PubMed]
  7. A. Kak and M. Slaney, “Computerized tomographic Imaging,” IEEE Press, N-Y (1987).
  8. M. O’Leary, “Imaging with diffuse photon density waves,” PhD University of Pennsylvania (1996).
  9. V. Ntziachristos, B. Chance, and A. Yodh, “Differential diffuse optical tomography,” Opt. Express 5, 230–242 (1999). http://www.opticsexpress.org/opticsexpress/tocv5n10.htm
    [Crossref] [PubMed]
  10. A. Knuttel, J.M. Schmitt, and J.R. Knutson, “Spatial localization of absorbing bodies by interfering diffuse photon-density waves,” Appl. Opt. 32, 381–389 (1993).
    [Crossref] [PubMed]
  11. M. Erickson, J. Reynolds, and K. Webb, “Comparison of sensitivity for single-source and dual-interfering-source configurations in optical diffusion imaging,” J.Opt.Soc.Am.A 14, 3083–3092 (1997).
    [Crossref]
  12. Y. Chen, C. Mu, X. Intes, and B. Chance, “Signal-to-noise analysis for detection sensitivity of small absorbing heterogeneity in turbid media with single-source and dual-interfering-source”, Opt. Express 9, 212–224 (2001). http://www.opticsexpress.org/opticsexpress/tocv9n4.htm
    [Crossref] [PubMed]
  13. B. Chance, K. Kang, L. He, J. Weng, and E. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Nat. Acad. Sci. USA 90, 3423–3427 (1993).
    [Crossref] [PubMed]
  14. B. Chance and E. Conant, “A novel tumor imager using NIR light,” in preparation.
  15. Y. Chen, S. Zhou, C. Xie, S. Nioka, M. Delivoria-Papadopoulos, E. Anday, and B. Chance, “Preliminary evaluation of dual-wavelength phased array imaging on neonatal brain function,” Journal of Biomedical Optics 5, 206–213 (2000).
    [Crossref]
  16. V. Ntziachristos, XuHui Ma, and B. Chance, “Time-correlated single photon counting imager for simultaneaous magnetic resonance and near-infrared mammography,” Rev. Sci. Instrum. 69, 4221–4233 (1998).
    [Crossref]
  17. S. Morgan, M. Somekh, and K. Hopcraqft, “Probabilistic method for phased array detection in scattering media,” Opt. Eng. 37, 1618–1626 (1998).
    [Crossref]
  18. S. Morgan and K. Yong, “Controlling the phase response of a diffusive wave phased array system,” Opt. Express 7, 540–546 (2001). http://www.opticsexpress.org/opticsexpress/tocv7n13.htm
    [Crossref]
  19. A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Physics Today 48, 34–40 (1995).
    [Crossref]
  20. P. Morse and H. Feshbach, “Methods of theoretical physics,” Mc Graw Hill, N-Y (1953).
  21. A. Ishimaru, “Wave propagation and scattering in random media,” Vol. 1, Academic Press, N-Y (1980).
  22. K. Yoo, F. Liu, and R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?,” Phys. Rev. Lett. 24, 2647–2650 (1990).
    [Crossref]
  23. X. Intes, B. Le Jeune, F. Pellen, Y. Guern, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source”, Waves Random Media 9, 489–499 (1999).
    [Crossref]
  24. R. Haskell, L. Svaasand, TT. Tsay, Tc. Feng, M. McAdams, and B. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J.Opt.Soc.Am.A 11, 2727–2741 (1994).
    [Crossref]
  25. S. Arridge, “Photon-measurement density function.I Analytical forms,” Appl. Opt. 34, 7395–7409 (1995).
    [Crossref] [PubMed]
  26. S. Nioka, S. Colak, X. Li, Y. Yang, and B. Chance, “Breast tumor images of hemodynamics information using a contrast agent with backprojection and FFT enhancement”, OSA Trends in Optics and Photonics vol. 21, Advances in Optical imaging and Photon Migration, JamesG. Fujimoto and Michael S. Patterson, eds. (Optical Society of America, Washington, DC 1998), 266–270.
  27. S. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, R41–R93 (1999).
    [Crossref]
  28. T. Durduran, M. Holboke, J. Culver, L. Zubkov, R. Choe, D. Pattanayak, B. Chance, and A. Yodh, “Tissue bulk optical properties of breast and phantoms obtained with clinical optical imager,” in Biomedical Topical Meetings, OSA Technical Digest (Optical Society of America, Washington DC, 2000), 386–388 (2000).
  29. M. Patterson, B. Chance, and B. Wilson, “Time-resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [Crossref] [PubMed]
  30. L. Wang, “Rapid modeling of diffuse reflectance of light in turbid slabs,” J.Opt.Soc.Am.A 15, 936–944 (1998).
    [Crossref]
  31. D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36, 4587–4599 (1997).
    [Crossref] [PubMed]
  32. X. Intes, V. Ntziachristos, J. Culver, A. Yodh, and B. Chance, “Projection access order in Algebraic Reconstruction Technique for Diffuse Optical Tomography,” Phys. Med. Biol. 47, N1–N10 (2002).
    [Crossref]
  33. X. Intes, B. Chance, M. Holboke, and A. Yodh, “Interfering diffusive photon-density waves with an absorbing-fluorescent inhomogeneity,” Opt. Express 8, 223–231 (2001). http://www.opticsexpress.org/opticsexpress/tocv8n3.htm
    [Crossref] [PubMed]
  34. D. Papaioannou, G.’ tHoof, S. Colak, and J. Oostveen, “Detection limit in localizing objects hidden in turbid medium using an optically scanned phased array,” Journal of Biomedical Optics 1, 305–310 (1996).
    [Crossref] [PubMed]
  35. B. Pogue, T. Mc. Bride, J. Prewitt, U. Osterberg, and K. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. 38, 2950–2961 (1999).
    [Crossref]
  36. B. Chance, K. Kang, L. He, H. Liu, and S. Zhou, “Precision localization of hidden absorbers in body tissues with phased-array optical systems,” Rev. Sci. Instrum. 67, 4324–4331 (1996).
    [Crossref]

2002 (1)

X. Intes, V. Ntziachristos, J. Culver, A. Yodh, and B. Chance, “Projection access order in Algebraic Reconstruction Technique for Diffuse Optical Tomography,” Phys. Med. Biol. 47, N1–N10 (2002).
[Crossref]

2001 (5)

2000 (3)

V. Ntziachristos, A. Yodh, M. Schnall, and B. Chance, “Concurent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc.Nat.Acad.Sci. USA 97, 2767–2772 (2000).
[Crossref] [PubMed]

Y. Chen, S. Zhou, C. Xie, S. Nioka, M. Delivoria-Papadopoulos, E. Anday, and B. Chance, “Preliminary evaluation of dual-wavelength phased array imaging on neonatal brain function,” Journal of Biomedical Optics 5, 206–213 (2000).
[Crossref]

D. Hawrys and E. Sevick-Muraca, “Developments toward diagnostic breast cancer imaging using Near-Infrared optical measurements and fluorescent contrast agents,” Neoplasia 2, 388–417 (2000).
[Crossref]

1999 (4)

X. Intes, B. Le Jeune, F. Pellen, Y. Guern, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source”, Waves Random Media 9, 489–499 (1999).
[Crossref]

S. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, R41–R93 (1999).
[Crossref]

V. Ntziachristos, B. Chance, and A. Yodh, “Differential diffuse optical tomography,” Opt. Express 5, 230–242 (1999). http://www.opticsexpress.org/opticsexpress/tocv5n10.htm
[Crossref] [PubMed]

B. Pogue, T. Mc. Bride, J. Prewitt, U. Osterberg, and K. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. 38, 2950–2961 (1999).
[Crossref]

1998 (3)

L. Wang, “Rapid modeling of diffuse reflectance of light in turbid slabs,” J.Opt.Soc.Am.A 15, 936–944 (1998).
[Crossref]

V. Ntziachristos, XuHui Ma, and B. Chance, “Time-correlated single photon counting imager for simultaneaous magnetic resonance and near-infrared mammography,” Rev. Sci. Instrum. 69, 4221–4233 (1998).
[Crossref]

S. Morgan, M. Somekh, and K. Hopcraqft, “Probabilistic method for phased array detection in scattering media,” Opt. Eng. 37, 1618–1626 (1998).
[Crossref]

1997 (3)

A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human function,” Trends Neurosci. 20, 435–442 (1997).
[Crossref] [PubMed]

M. Erickson, J. Reynolds, and K. Webb, “Comparison of sensitivity for single-source and dual-interfering-source configurations in optical diffusion imaging,” J.Opt.Soc.Am.A 14, 3083–3092 (1997).
[Crossref]

D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36, 4587–4599 (1997).
[Crossref] [PubMed]

1996 (2)

D. Papaioannou, G.’ tHoof, S. Colak, and J. Oostveen, “Detection limit in localizing objects hidden in turbid medium using an optically scanned phased array,” Journal of Biomedical Optics 1, 305–310 (1996).
[Crossref] [PubMed]

B. Chance, K. Kang, L. He, H. Liu, and S. Zhou, “Precision localization of hidden absorbers in body tissues with phased-array optical systems,” Rev. Sci. Instrum. 67, 4324–4331 (1996).
[Crossref]

1995 (3)

1994 (1)

R. Haskell, L. Svaasand, TT. Tsay, Tc. Feng, M. McAdams, and B. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J.Opt.Soc.Am.A 11, 2727–2741 (1994).
[Crossref]

1993 (2)

B. Chance, K. Kang, L. He, J. Weng, and E. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Nat. Acad. Sci. USA 90, 3423–3427 (1993).
[Crossref] [PubMed]

A. Knuttel, J.M. Schmitt, and J.R. Knutson, “Spatial localization of absorbing bodies by interfering diffuse photon-density waves,” Appl. Opt. 32, 381–389 (1993).
[Crossref] [PubMed]

1990 (1)

K. Yoo, F. Liu, and R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?,” Phys. Rev. Lett. 24, 2647–2650 (1990).
[Crossref]

1989 (1)

1987 (1)

A. Kak and M. Slaney, “Computerized tomographic Imaging,” IEEE Press, N-Y (1987).

1980 (1)

A. Ishimaru, “Wave propagation and scattering in random media,” Vol. 1, Academic Press, N-Y (1980).

Alfano, R.

K. Yoo, F. Liu, and R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?,” Phys. Rev. Lett. 24, 2647–2650 (1990).
[Crossref]

Anday, E.

Y. Chen, S. Zhou, C. Xie, S. Nioka, M. Delivoria-Papadopoulos, E. Anday, and B. Chance, “Preliminary evaluation of dual-wavelength phased array imaging on neonatal brain function,” Journal of Biomedical Optics 5, 206–213 (2000).
[Crossref]

Arridge, S.

S. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, R41–R93 (1999).
[Crossref]

S. Arridge, “Photon-measurement density function.I Analytical forms,” Appl. Opt. 34, 7395–7409 (1995).
[Crossref] [PubMed]

Boas, D.

Bride, T. Mc.

Chance, B.

X. Intes, V. Ntziachristos, J. Culver, A. Yodh, and B. Chance, “Projection access order in Algebraic Reconstruction Technique for Diffuse Optical Tomography,” Phys. Med. Biol. 47, N1–N10 (2002).
[Crossref]

X. Intes, B. Chance, M. Holboke, and A. Yodh, “Interfering diffusive photon-density waves with an absorbing-fluorescent inhomogeneity,” Opt. Express 8, 223–231 (2001). http://www.opticsexpress.org/opticsexpress/tocv8n3.htm
[Crossref] [PubMed]

Y. Chen, C. Mu, X. Intes, and B. Chance, “Signal-to-noise analysis for detection sensitivity of small absorbing heterogeneity in turbid media with single-source and dual-interfering-source”, Opt. Express 9, 212–224 (2001). http://www.opticsexpress.org/opticsexpress/tocv9n4.htm
[Crossref] [PubMed]

V. Ntziachristos and B. Chance, “Probing physiology and molecular function using optical imaging: applications to breast cancer,” Breast Cancer Research 3, 41–47 (2001).
[Crossref] [PubMed]

V. Ntziachristos, A. Yodh, M. Schnall, and B. Chance, “Concurent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc.Nat.Acad.Sci. USA 97, 2767–2772 (2000).
[Crossref] [PubMed]

Y. Chen, S. Zhou, C. Xie, S. Nioka, M. Delivoria-Papadopoulos, E. Anday, and B. Chance, “Preliminary evaluation of dual-wavelength phased array imaging on neonatal brain function,” Journal of Biomedical Optics 5, 206–213 (2000).
[Crossref]

V. Ntziachristos, B. Chance, and A. Yodh, “Differential diffuse optical tomography,” Opt. Express 5, 230–242 (1999). http://www.opticsexpress.org/opticsexpress/tocv5n10.htm
[Crossref] [PubMed]

V. Ntziachristos, XuHui Ma, and B. Chance, “Time-correlated single photon counting imager for simultaneaous magnetic resonance and near-infrared mammography,” Rev. Sci. Instrum. 69, 4221–4233 (1998).
[Crossref]

A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human function,” Trends Neurosci. 20, 435–442 (1997).
[Crossref] [PubMed]

B. Chance, K. Kang, L. He, H. Liu, and S. Zhou, “Precision localization of hidden absorbers in body tissues with phased-array optical systems,” Rev. Sci. Instrum. 67, 4324–4331 (1996).
[Crossref]

M. O’Leary, D. Boas, B. Chance, and A. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing photon-tomography,” Opt. Lett. 20, 426–428 (1995).
[Crossref]

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

B. Chance, K. Kang, L. He, J. Weng, and E. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Nat. Acad. Sci. USA 90, 3423–3427 (1993).
[Crossref] [PubMed]

M. Patterson, B. Chance, and B. Wilson, “Time-resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[Crossref] [PubMed]

B. Chance and E. Conant, “A novel tumor imager using NIR light,” in preparation.

T. Durduran, M. Holboke, J. Culver, L. Zubkov, R. Choe, D. Pattanayak, B. Chance, and A. Yodh, “Tissue bulk optical properties of breast and phantoms obtained with clinical optical imager,” in Biomedical Topical Meetings, OSA Technical Digest (Optical Society of America, Washington DC, 2000), 386–388 (2000).

S. Nioka, S. Colak, X. Li, Y. Yang, and B. Chance, “Breast tumor images of hemodynamics information using a contrast agent with backprojection and FFT enhancement”, OSA Trends in Optics and Photonics vol. 21, Advances in Optical imaging and Photon Migration, JamesG. Fujimoto and Michael S. Patterson, eds. (Optical Society of America, Washington, DC 1998), 266–270.

Chen, Y.

Y. Chen, C. Mu, X. Intes, and B. Chance, “Signal-to-noise analysis for detection sensitivity of small absorbing heterogeneity in turbid media with single-source and dual-interfering-source”, Opt. Express 9, 212–224 (2001). http://www.opticsexpress.org/opticsexpress/tocv9n4.htm
[Crossref] [PubMed]

Y. Chen, S. Zhou, C. Xie, S. Nioka, M. Delivoria-Papadopoulos, E. Anday, and B. Chance, “Preliminary evaluation of dual-wavelength phased array imaging on neonatal brain function,” Journal of Biomedical Optics 5, 206–213 (2000).
[Crossref]

Choe, R.

T. Durduran, M. Holboke, J. Culver, L. Zubkov, R. Choe, D. Pattanayak, B. Chance, and A. Yodh, “Tissue bulk optical properties of breast and phantoms obtained with clinical optical imager,” in Biomedical Topical Meetings, OSA Technical Digest (Optical Society of America, Washington DC, 2000), 386–388 (2000).

Colak, S.

D. Papaioannou, G.’ tHoof, S. Colak, and J. Oostveen, “Detection limit in localizing objects hidden in turbid medium using an optically scanned phased array,” Journal of Biomedical Optics 1, 305–310 (1996).
[Crossref] [PubMed]

S. Nioka, S. Colak, X. Li, Y. Yang, and B. Chance, “Breast tumor images of hemodynamics information using a contrast agent with backprojection and FFT enhancement”, OSA Trends in Optics and Photonics vol. 21, Advances in Optical imaging and Photon Migration, JamesG. Fujimoto and Michael S. Patterson, eds. (Optical Society of America, Washington, DC 1998), 266–270.

Conant, E.

B. Chance and E. Conant, “A novel tumor imager using NIR light,” in preparation.

Contini, D.

Culver, J.

X. Intes, V. Ntziachristos, J. Culver, A. Yodh, and B. Chance, “Projection access order in Algebraic Reconstruction Technique for Diffuse Optical Tomography,” Phys. Med. Biol. 47, N1–N10 (2002).
[Crossref]

T. Durduran, M. Holboke, J. Culver, L. Zubkov, R. Choe, D. Pattanayak, B. Chance, and A. Yodh, “Tissue bulk optical properties of breast and phantoms obtained with clinical optical imager,” in Biomedical Topical Meetings, OSA Technical Digest (Optical Society of America, Washington DC, 2000), 386–388 (2000).

Delivoria-Papadopoulos, M.

Y. Chen, S. Zhou, C. Xie, S. Nioka, M. Delivoria-Papadopoulos, E. Anday, and B. Chance, “Preliminary evaluation of dual-wavelength phased array imaging on neonatal brain function,” Journal of Biomedical Optics 5, 206–213 (2000).
[Crossref]

Durduran, T.

T. Durduran, M. Holboke, J. Culver, L. Zubkov, R. Choe, D. Pattanayak, B. Chance, and A. Yodh, “Tissue bulk optical properties of breast and phantoms obtained with clinical optical imager,” in Biomedical Topical Meetings, OSA Technical Digest (Optical Society of America, Washington DC, 2000), 386–388 (2000).

Erickson, M.

M. Erickson, J. Reynolds, and K. Webb, “Comparison of sensitivity for single-source and dual-interfering-source configurations in optical diffusion imaging,” J.Opt.Soc.Am.A 14, 3083–3092 (1997).
[Crossref]

Feng, Tc.

R. Haskell, L. Svaasand, TT. Tsay, Tc. Feng, M. McAdams, and B. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J.Opt.Soc.Am.A 11, 2727–2741 (1994).
[Crossref]

Feshbach, H.

P. Morse and H. Feshbach, “Methods of theoretical physics,” Mc Graw Hill, N-Y (1953).

Guern, Y.

X. Intes, B. Le Jeune, F. Pellen, Y. Guern, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source”, Waves Random Media 9, 489–499 (1999).
[Crossref]

Haskell, R.

R. Haskell, L. Svaasand, TT. Tsay, Tc. Feng, M. McAdams, and B. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J.Opt.Soc.Am.A 11, 2727–2741 (1994).
[Crossref]

Hawrys, D.

D. Hawrys and E. Sevick-Muraca, “Developments toward diagnostic breast cancer imaging using Near-Infrared optical measurements and fluorescent contrast agents,” Neoplasia 2, 388–417 (2000).
[Crossref]

He, L.

B. Chance, K. Kang, L. He, H. Liu, and S. Zhou, “Precision localization of hidden absorbers in body tissues with phased-array optical systems,” Rev. Sci. Instrum. 67, 4324–4331 (1996).
[Crossref]

B. Chance, K. Kang, L. He, J. Weng, and E. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Nat. Acad. Sci. USA 90, 3423–3427 (1993).
[Crossref] [PubMed]

Holboke, M.

X. Intes, B. Chance, M. Holboke, and A. Yodh, “Interfering diffusive photon-density waves with an absorbing-fluorescent inhomogeneity,” Opt. Express 8, 223–231 (2001). http://www.opticsexpress.org/opticsexpress/tocv8n3.htm
[Crossref] [PubMed]

T. Durduran, M. Holboke, J. Culver, L. Zubkov, R. Choe, D. Pattanayak, B. Chance, and A. Yodh, “Tissue bulk optical properties of breast and phantoms obtained with clinical optical imager,” in Biomedical Topical Meetings, OSA Technical Digest (Optical Society of America, Washington DC, 2000), 386–388 (2000).

Hopcraqft, K.

S. Morgan, M. Somekh, and K. Hopcraqft, “Probabilistic method for phased array detection in scattering media,” Opt. Eng. 37, 1618–1626 (1998).
[Crossref]

Intes, X.

X. Intes, V. Ntziachristos, J. Culver, A. Yodh, and B. Chance, “Projection access order in Algebraic Reconstruction Technique for Diffuse Optical Tomography,” Phys. Med. Biol. 47, N1–N10 (2002).
[Crossref]

X. Intes, B. Chance, M. Holboke, and A. Yodh, “Interfering diffusive photon-density waves with an absorbing-fluorescent inhomogeneity,” Opt. Express 8, 223–231 (2001). http://www.opticsexpress.org/opticsexpress/tocv8n3.htm
[Crossref] [PubMed]

Y. Chen, C. Mu, X. Intes, and B. Chance, “Signal-to-noise analysis for detection sensitivity of small absorbing heterogeneity in turbid media with single-source and dual-interfering-source”, Opt. Express 9, 212–224 (2001). http://www.opticsexpress.org/opticsexpress/tocv9n4.htm
[Crossref] [PubMed]

X. Intes, B. Le Jeune, F. Pellen, Y. Guern, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source”, Waves Random Media 9, 489–499 (1999).
[Crossref]

Ishimaru, A.

A. Ishimaru, “Wave propagation and scattering in random media,” Vol. 1, Academic Press, N-Y (1980).

Jeune, B. Le

X. Intes, B. Le Jeune, F. Pellen, Y. Guern, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source”, Waves Random Media 9, 489–499 (1999).
[Crossref]

Jiang, S.

T. McBride, B. Pogue, S. Jiang, U. Osterberg, and K. Paulsen, “Initial studies of in-vivo absorbing and scattering heterogeneity in near-infrared tomographic breast imaging,” Opt. Let. 26, 822–824 (2001).
[Crossref]

Kak, A.

A. Kak and M. Slaney, “Computerized tomographic Imaging,” IEEE Press, N-Y (1987).

Kang, K.

B. Chance, K. Kang, L. He, H. Liu, and S. Zhou, “Precision localization of hidden absorbers in body tissues with phased-array optical systems,” Rev. Sci. Instrum. 67, 4324–4331 (1996).
[Crossref]

B. Chance, K. Kang, L. He, J. Weng, and E. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Nat. Acad. Sci. USA 90, 3423–3427 (1993).
[Crossref] [PubMed]

Knutson, J.R.

Knuttel, A.

Li, X.

S. Nioka, S. Colak, X. Li, Y. Yang, and B. Chance, “Breast tumor images of hemodynamics information using a contrast agent with backprojection and FFT enhancement”, OSA Trends in Optics and Photonics vol. 21, Advances in Optical imaging and Photon Migration, JamesG. Fujimoto and Michael S. Patterson, eds. (Optical Society of America, Washington, DC 1998), 266–270.

Liu, F.

K. Yoo, F. Liu, and R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?,” Phys. Rev. Lett. 24, 2647–2650 (1990).
[Crossref]

Liu, H.

B. Chance, K. Kang, L. He, H. Liu, and S. Zhou, “Precision localization of hidden absorbers in body tissues with phased-array optical systems,” Rev. Sci. Instrum. 67, 4324–4331 (1996).
[Crossref]

Lotrian, J.

X. Intes, B. Le Jeune, F. Pellen, Y. Guern, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source”, Waves Random Media 9, 489–499 (1999).
[Crossref]

Ma, XuHui

V. Ntziachristos, XuHui Ma, and B. Chance, “Time-correlated single photon counting imager for simultaneaous magnetic resonance and near-infrared mammography,” Rev. Sci. Instrum. 69, 4221–4233 (1998).
[Crossref]

Martelli, F.

McAdams, M.

R. Haskell, L. Svaasand, TT. Tsay, Tc. Feng, M. McAdams, and B. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J.Opt.Soc.Am.A 11, 2727–2741 (1994).
[Crossref]

McBride, T.

T. McBride, B. Pogue, S. Jiang, U. Osterberg, and K. Paulsen, “Initial studies of in-vivo absorbing and scattering heterogeneity in near-infrared tomographic breast imaging,” Opt. Let. 26, 822–824 (2001).
[Crossref]

Morgan, S.

S. Morgan and K. Yong, “Controlling the phase response of a diffusive wave phased array system,” Opt. Express 7, 540–546 (2001). http://www.opticsexpress.org/opticsexpress/tocv7n13.htm
[Crossref]

S. Morgan, M. Somekh, and K. Hopcraqft, “Probabilistic method for phased array detection in scattering media,” Opt. Eng. 37, 1618–1626 (1998).
[Crossref]

Morse, P.

P. Morse and H. Feshbach, “Methods of theoretical physics,” Mc Graw Hill, N-Y (1953).

Mu, C.

Nioka, S.

Y. Chen, S. Zhou, C. Xie, S. Nioka, M. Delivoria-Papadopoulos, E. Anday, and B. Chance, “Preliminary evaluation of dual-wavelength phased array imaging on neonatal brain function,” Journal of Biomedical Optics 5, 206–213 (2000).
[Crossref]

S. Nioka, S. Colak, X. Li, Y. Yang, and B. Chance, “Breast tumor images of hemodynamics information using a contrast agent with backprojection and FFT enhancement”, OSA Trends in Optics and Photonics vol. 21, Advances in Optical imaging and Photon Migration, JamesG. Fujimoto and Michael S. Patterson, eds. (Optical Society of America, Washington, DC 1998), 266–270.

Ntziachristos, V.

X. Intes, V. Ntziachristos, J. Culver, A. Yodh, and B. Chance, “Projection access order in Algebraic Reconstruction Technique for Diffuse Optical Tomography,” Phys. Med. Biol. 47, N1–N10 (2002).
[Crossref]

V. Ntziachristos and B. Chance, “Probing physiology and molecular function using optical imaging: applications to breast cancer,” Breast Cancer Research 3, 41–47 (2001).
[Crossref] [PubMed]

V. Ntziachristos, A. Yodh, M. Schnall, and B. Chance, “Concurent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc.Nat.Acad.Sci. USA 97, 2767–2772 (2000).
[Crossref] [PubMed]

V. Ntziachristos, B. Chance, and A. Yodh, “Differential diffuse optical tomography,” Opt. Express 5, 230–242 (1999). http://www.opticsexpress.org/opticsexpress/tocv5n10.htm
[Crossref] [PubMed]

V. Ntziachristos, XuHui Ma, and B. Chance, “Time-correlated single photon counting imager for simultaneaous magnetic resonance and near-infrared mammography,” Rev. Sci. Instrum. 69, 4221–4233 (1998).
[Crossref]

O’Leary, M.

Oostveen, J.

D. Papaioannou, G.’ tHoof, S. Colak, and J. Oostveen, “Detection limit in localizing objects hidden in turbid medium using an optically scanned phased array,” Journal of Biomedical Optics 1, 305–310 (1996).
[Crossref] [PubMed]

Osterberg, U.

T. McBride, B. Pogue, S. Jiang, U. Osterberg, and K. Paulsen, “Initial studies of in-vivo absorbing and scattering heterogeneity in near-infrared tomographic breast imaging,” Opt. Let. 26, 822–824 (2001).
[Crossref]

B. Pogue, T. Mc. Bride, J. Prewitt, U. Osterberg, and K. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. 38, 2950–2961 (1999).
[Crossref]

Papaioannou, D.

D. Papaioannou, G.’ tHoof, S. Colak, and J. Oostveen, “Detection limit in localizing objects hidden in turbid medium using an optically scanned phased array,” Journal of Biomedical Optics 1, 305–310 (1996).
[Crossref] [PubMed]

Pattanayak, D.

T. Durduran, M. Holboke, J. Culver, L. Zubkov, R. Choe, D. Pattanayak, B. Chance, and A. Yodh, “Tissue bulk optical properties of breast and phantoms obtained with clinical optical imager,” in Biomedical Topical Meetings, OSA Technical Digest (Optical Society of America, Washington DC, 2000), 386–388 (2000).

Patterson, M.

Paulsen, K.

T. McBride, B. Pogue, S. Jiang, U. Osterberg, and K. Paulsen, “Initial studies of in-vivo absorbing and scattering heterogeneity in near-infrared tomographic breast imaging,” Opt. Let. 26, 822–824 (2001).
[Crossref]

B. Pogue, T. Mc. Bride, J. Prewitt, U. Osterberg, and K. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. 38, 2950–2961 (1999).
[Crossref]

Pellen, F.

X. Intes, B. Le Jeune, F. Pellen, Y. Guern, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source”, Waves Random Media 9, 489–499 (1999).
[Crossref]

Pogue, B.

T. McBride, B. Pogue, S. Jiang, U. Osterberg, and K. Paulsen, “Initial studies of in-vivo absorbing and scattering heterogeneity in near-infrared tomographic breast imaging,” Opt. Let. 26, 822–824 (2001).
[Crossref]

B. Pogue, T. Mc. Bride, J. Prewitt, U. Osterberg, and K. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. 38, 2950–2961 (1999).
[Crossref]

Prewitt, J.

Reynolds, J.

M. Erickson, J. Reynolds, and K. Webb, “Comparison of sensitivity for single-source and dual-interfering-source configurations in optical diffusion imaging,” J.Opt.Soc.Am.A 14, 3083–3092 (1997).
[Crossref]

Schmitt, J.M.

Schnall, M.

V. Ntziachristos, A. Yodh, M. Schnall, and B. Chance, “Concurent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc.Nat.Acad.Sci. USA 97, 2767–2772 (2000).
[Crossref] [PubMed]

Sevick, E.

B. Chance, K. Kang, L. He, J. Weng, and E. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Nat. Acad. Sci. USA 90, 3423–3427 (1993).
[Crossref] [PubMed]

Sevick-Muraca, E.

D. Hawrys and E. Sevick-Muraca, “Developments toward diagnostic breast cancer imaging using Near-Infrared optical measurements and fluorescent contrast agents,” Neoplasia 2, 388–417 (2000).
[Crossref]

Slaney, M.

A. Kak and M. Slaney, “Computerized tomographic Imaging,” IEEE Press, N-Y (1987).

Somekh, M.

S. Morgan, M. Somekh, and K. Hopcraqft, “Probabilistic method for phased array detection in scattering media,” Opt. Eng. 37, 1618–1626 (1998).
[Crossref]

Svaasand, L.

R. Haskell, L. Svaasand, TT. Tsay, Tc. Feng, M. McAdams, and B. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J.Opt.Soc.Am.A 11, 2727–2741 (1994).
[Crossref]

tHoof, G.’

D. Papaioannou, G.’ tHoof, S. Colak, and J. Oostveen, “Detection limit in localizing objects hidden in turbid medium using an optically scanned phased array,” Journal of Biomedical Optics 1, 305–310 (1996).
[Crossref] [PubMed]

Tromberg, B.

R. Haskell, L. Svaasand, TT. Tsay, Tc. Feng, M. McAdams, and B. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J.Opt.Soc.Am.A 11, 2727–2741 (1994).
[Crossref]

Tsay, TT.

R. Haskell, L. Svaasand, TT. Tsay, Tc. Feng, M. McAdams, and B. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J.Opt.Soc.Am.A 11, 2727–2741 (1994).
[Crossref]

Villringer, A.

A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human function,” Trends Neurosci. 20, 435–442 (1997).
[Crossref] [PubMed]

Wang, L.

L. Wang, “Rapid modeling of diffuse reflectance of light in turbid slabs,” J.Opt.Soc.Am.A 15, 936–944 (1998).
[Crossref]

Webb, K.

M. Erickson, J. Reynolds, and K. Webb, “Comparison of sensitivity for single-source and dual-interfering-source configurations in optical diffusion imaging,” J.Opt.Soc.Am.A 14, 3083–3092 (1997).
[Crossref]

Weng, J.

B. Chance, K. Kang, L. He, J. Weng, and E. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Nat. Acad. Sci. USA 90, 3423–3427 (1993).
[Crossref] [PubMed]

Wilson, B.

Xie, C.

Y. Chen, S. Zhou, C. Xie, S. Nioka, M. Delivoria-Papadopoulos, E. Anday, and B. Chance, “Preliminary evaluation of dual-wavelength phased array imaging on neonatal brain function,” Journal of Biomedical Optics 5, 206–213 (2000).
[Crossref]

Yang, Y.

S. Nioka, S. Colak, X. Li, Y. Yang, and B. Chance, “Breast tumor images of hemodynamics information using a contrast agent with backprojection and FFT enhancement”, OSA Trends in Optics and Photonics vol. 21, Advances in Optical imaging and Photon Migration, JamesG. Fujimoto and Michael S. Patterson, eds. (Optical Society of America, Washington, DC 1998), 266–270.

Yodh, A.

X. Intes, V. Ntziachristos, J. Culver, A. Yodh, and B. Chance, “Projection access order in Algebraic Reconstruction Technique for Diffuse Optical Tomography,” Phys. Med. Biol. 47, N1–N10 (2002).
[Crossref]

X. Intes, B. Chance, M. Holboke, and A. Yodh, “Interfering diffusive photon-density waves with an absorbing-fluorescent inhomogeneity,” Opt. Express 8, 223–231 (2001). http://www.opticsexpress.org/opticsexpress/tocv8n3.htm
[Crossref] [PubMed]

V. Ntziachristos, A. Yodh, M. Schnall, and B. Chance, “Concurent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc.Nat.Acad.Sci. USA 97, 2767–2772 (2000).
[Crossref] [PubMed]

V. Ntziachristos, B. Chance, and A. Yodh, “Differential diffuse optical tomography,” Opt. Express 5, 230–242 (1999). http://www.opticsexpress.org/opticsexpress/tocv5n10.htm
[Crossref] [PubMed]

M. O’Leary, D. Boas, B. Chance, and A. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing photon-tomography,” Opt. Lett. 20, 426–428 (1995).
[Crossref]

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

T. Durduran, M. Holboke, J. Culver, L. Zubkov, R. Choe, D. Pattanayak, B. Chance, and A. Yodh, “Tissue bulk optical properties of breast and phantoms obtained with clinical optical imager,” in Biomedical Topical Meetings, OSA Technical Digest (Optical Society of America, Washington DC, 2000), 386–388 (2000).

Yong, K.

Yoo, K.

K. Yoo, F. Liu, and R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?,” Phys. Rev. Lett. 24, 2647–2650 (1990).
[Crossref]

Zaccanti, G.

Zhou, S.

Y. Chen, S. Zhou, C. Xie, S. Nioka, M. Delivoria-Papadopoulos, E. Anday, and B. Chance, “Preliminary evaluation of dual-wavelength phased array imaging on neonatal brain function,” Journal of Biomedical Optics 5, 206–213 (2000).
[Crossref]

B. Chance, K. Kang, L. He, H. Liu, and S. Zhou, “Precision localization of hidden absorbers in body tissues with phased-array optical systems,” Rev. Sci. Instrum. 67, 4324–4331 (1996).
[Crossref]

Zubkov, L.

T. Durduran, M. Holboke, J. Culver, L. Zubkov, R. Choe, D. Pattanayak, B. Chance, and A. Yodh, “Tissue bulk optical properties of breast and phantoms obtained with clinical optical imager,” in Biomedical Topical Meetings, OSA Technical Digest (Optical Society of America, Washington DC, 2000), 386–388 (2000).

Appl. Opt. (5)

Breast Cancer Research (1)

V. Ntziachristos and B. Chance, “Probing physiology and molecular function using optical imaging: applications to breast cancer,” Breast Cancer Research 3, 41–47 (2001).
[Crossref] [PubMed]

IEEE Press (1)

A. Kak and M. Slaney, “Computerized tomographic Imaging,” IEEE Press, N-Y (1987).

Inverse Problems (1)

S. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, R41–R93 (1999).
[Crossref]

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

L. Wang, “Rapid modeling of diffuse reflectance of light in turbid slabs,” J.Opt.Soc.Am.A 15, 936–944 (1998).
[Crossref]

R. Haskell, L. Svaasand, TT. Tsay, Tc. Feng, M. McAdams, and B. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J.Opt.Soc.Am.A 11, 2727–2741 (1994).
[Crossref]

M. Erickson, J. Reynolds, and K. Webb, “Comparison of sensitivity for single-source and dual-interfering-source configurations in optical diffusion imaging,” J.Opt.Soc.Am.A 14, 3083–3092 (1997).
[Crossref]

Journal of Biomedical Optics (2)

Y. Chen, S. Zhou, C. Xie, S. Nioka, M. Delivoria-Papadopoulos, E. Anday, and B. Chance, “Preliminary evaluation of dual-wavelength phased array imaging on neonatal brain function,” Journal of Biomedical Optics 5, 206–213 (2000).
[Crossref]

D. Papaioannou, G.’ tHoof, S. Colak, and J. Oostveen, “Detection limit in localizing objects hidden in turbid medium using an optically scanned phased array,” Journal of Biomedical Optics 1, 305–310 (1996).
[Crossref] [PubMed]

Neoplasia (1)

D. Hawrys and E. Sevick-Muraca, “Developments toward diagnostic breast cancer imaging using Near-Infrared optical measurements and fluorescent contrast agents,” Neoplasia 2, 388–417 (2000).
[Crossref]

Opt. Eng. (1)

S. Morgan, M. Somekh, and K. Hopcraqft, “Probabilistic method for phased array detection in scattering media,” Opt. Eng. 37, 1618–1626 (1998).
[Crossref]

Opt. Express (4)

Opt. Let. (1)

T. McBride, B. Pogue, S. Jiang, U. Osterberg, and K. Paulsen, “Initial studies of in-vivo absorbing and scattering heterogeneity in near-infrared tomographic breast imaging,” Opt. Let. 26, 822–824 (2001).
[Crossref]

Opt. Lett. (1)

Phys. Med. Biol. (1)

X. Intes, V. Ntziachristos, J. Culver, A. Yodh, and B. Chance, “Projection access order in Algebraic Reconstruction Technique for Diffuse Optical Tomography,” Phys. Med. Biol. 47, N1–N10 (2002).
[Crossref]

Phys. Rev. Lett. (1)

K. Yoo, F. Liu, and R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?,” Phys. Rev. Lett. 24, 2647–2650 (1990).
[Crossref]

Physics Today (1)

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

Proc. Nat. Acad. Sci. USA (1)

B. Chance, K. Kang, L. He, J. Weng, and E. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Nat. Acad. Sci. USA 90, 3423–3427 (1993).
[Crossref] [PubMed]

Proc.Nat.Acad.Sci. USA (1)

V. Ntziachristos, A. Yodh, M. Schnall, and B. Chance, “Concurent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc.Nat.Acad.Sci. USA 97, 2767–2772 (2000).
[Crossref] [PubMed]

Rev. Sci. Instrum. (2)

V. Ntziachristos, XuHui Ma, and B. Chance, “Time-correlated single photon counting imager for simultaneaous magnetic resonance and near-infrared mammography,” Rev. Sci. Instrum. 69, 4221–4233 (1998).
[Crossref]

B. Chance, K. Kang, L. He, H. Liu, and S. Zhou, “Precision localization of hidden absorbers in body tissues with phased-array optical systems,” Rev. Sci. Instrum. 67, 4324–4331 (1996).
[Crossref]

Trends Neurosci. (1)

A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human function,” Trends Neurosci. 20, 435–442 (1997).
[Crossref] [PubMed]

Waves Random Media (1)

X. Intes, B. Le Jeune, F. Pellen, Y. Guern, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source”, Waves Random Media 9, 489–499 (1999).
[Crossref]

Other (6)

S. Nioka, S. Colak, X. Li, Y. Yang, and B. Chance, “Breast tumor images of hemodynamics information using a contrast agent with backprojection and FFT enhancement”, OSA Trends in Optics and Photonics vol. 21, Advances in Optical imaging and Photon Migration, JamesG. Fujimoto and Michael S. Patterson, eds. (Optical Society of America, Washington, DC 1998), 266–270.

T. Durduran, M. Holboke, J. Culver, L. Zubkov, R. Choe, D. Pattanayak, B. Chance, and A. Yodh, “Tissue bulk optical properties of breast and phantoms obtained with clinical optical imager,” in Biomedical Topical Meetings, OSA Technical Digest (Optical Society of America, Washington DC, 2000), 386–388 (2000).

M. O’Leary, “Imaging with diffuse photon density waves,” PhD University of Pennsylvania (1996).

B. Chance and E. Conant, “A novel tumor imager using NIR light,” in preparation.

P. Morse and H. Feshbach, “Methods of theoretical physics,” Mc Graw Hill, N-Y (1953).

A. Ishimaru, “Wave propagation and scattering in random media,” Vol. 1, Academic Press, N-Y (1980).

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Figures (6)

Fig. 1:
Fig. 1:

Perturbation reaching the detection plane for 3 positions of sources-systems and for the model of Fig. 5 (a) with sources separated by 2cm and for a 50MHz frequency. The 64 detectors were located at the geometrical markers.

Fig. 2:
Fig. 2:

Sensitivity profile for a sources-detector pair: (a) ν=50MHZ – d=2cm; (b) ν=50MHZ – d=2cm; (c) ν=200MHZ – d=1cm; (d) ν=200MHZ – d=1cm

Fig. 3:
Fig. 3:

One object reconstruction for a 50MHz modulation and a 2cm-separation between the sources. 17 couples of sources and 64 detectors were considered with 80×50 voxels: (a) model – (b) reconstruction.

Fig. 4:
Fig. 4:

Correlation coefficient ε1 and Euclidian distance ε2 for (a)-(b) d=1cm versus frequency and (c)-(d) υ=50MHz versus sources separation.

Fig. 5:
Fig. 5:

Differential absorption maps: (a) Simulated – (b) d=1cm; υ=50MHz, (c ) d=1cm; υ=200MHz – (d) d=2cm; υ=200MHz.

Fig. 6:
Fig. 6:

Differential absorption maps: (a) ±0.5% – ± 0.25 ° – (b) ±2.5% – ± 1 °.

Equations (27)

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

[ 2 + k 2 ) ] U ( r , r s ) = AS ( r s ) / D ,
[ 2 + k 2 + O ( r ) ] U ( r , r s ) = AS ( r s ) / D ,
O ( r ) = μ a ( r ) D ,
U ( r , r s ) = U 0 ( r , r s ) e Φ sc ( r , r s )
S ( r s ) = δ ( r s 1 ) + δ ( r s 2 ) · e i π
2 Φ o ( r , r s ) + 2 Φ sc ( r , r s ) + ( Φ o ( r , r s ) ) 2 +
( Φ sc ( r , r s ) ) 2 + k 0 2 + O ( r ) + 2 Φ o ( r , r s ) · Φ sc ( r , r s ) = 1 U 0 ( r , r s ) AS ( r s ) D
2 Φ o ( r , r s ) + ( Φ o ( r , r s ) ) 2 + k 0 2 = 1 U 0 ( r , r s ) AS ( r s ) D
2 Φ sc ( r , r s ) = 2 Φ o ( r , r s ) · Φ sc ( r , r s ) = ( Φ sc ( r , r s ) ) 2 O ( r ) ,
2 ( U 0 ( r , r s ) Φ sc ( r , r s ) )
= Φ sc ( r , r s ) 2 U 0 ( r , r s ) + 2 U 0 ( r , r s ) · Φ sc ( r , r s ) + U 0 ( r , r s ) 2 Φ sc ( r , r s )
2 ( U 0 ( r , r s ) Φ sc ( r , r s ) ) Φ sc ( r , r s ) 2 U 0 ( r , r s ) = U 0 ( r , r s ) ( ( Φ sc ( r , r s ) ) 2 + O ( r ) )
{ [ 2 + k 2 ] U 0 1 ( r , r s 1 ) = ( r s 1 ) / D [ 2 + k 2 ] U 0 1 ( r , r s 2 ) = ( r s 2 ) / D
1 U 0 ( r , r s ) = 2 ( U 0 1 ( r , r s 1 ) + U 0 2 ( r , r s 2 ) ) = k 2 U 0 1 ( r , r s 1 ) k 2 U 0 2 ( r , r s 2 ) = k 2 U 0 ( r , r s )
[ 2 + k 2 ] U 0 ( r , r s ) Φ sc ( r , r s ) = U 0 ( r , r s ) ( ( Φ sc ( r , r s ) ) 2 + O ( r ) )
[ 2 + k 2 ] U 0 ( r , r s ) Φ sc ( r , r s ) = U 0 ( r , r s ) O ( r ) )
Φ sc ( r , r s ) = 1 U 0 ( r , r s ) G ( r , r ) μ a ( r ) D U 0 ( r , r s ) dr
G ( r , r ) = 1 4 π e ik r r r r
U 0 ( r , r s ) = A 4 πvD · [ e ik r s 1 r r s 1 r + e ik r s 2 r + i π r s 2 r ]
Φ sc ( r s , r d ) = ln [ U heterogene ous ( r s , r d ) U hom ogeneous ( r s , r d ) ]
[ Φ sc ( r s 1 , r d 1 ) Φ sc ( r sm , r dm ) ] = [ W 11 W 1 n W m 1 W mn ] [ δ μ a ( r 1 ) δ μ a ( r n ) ]
W ij = v h 3 D · G ( r di , r j ) × [ U 0 1 ( r j , r sli ) + U 0 2 ( r j , r s 2 i ) ] [ U 0 1 ( r di , r sli ) + U 0 2 ( r di , r s 2 i ) ]
W ij = v h 3 D × { n = 1 ( G ( r di , r n j + ) G ( r di , r n j ) ) } .
[ { n = 1 ( U 0 1 ( r j , r n s 1 i + ) U 0 1 ( r j , r n s 1 i ) ) } + { n = 1 ( U 0 2 ( r j , r n s 2 i + ) U 0 2 ( r j , r n s 2 i ) ) } ] [ { n = 1 ( U 0 1 ( r di , r n sli + ) U 0 1 ( r di , r n sli ) ) } + { n = 1 ( U 0 2 ( r di , r n s 2 i + ) U 0 2 ( r di , r n s 2 i ) ) } ]
x j ( k + 1 ) = x j ( k ) + λ i ( b i j W ij x j ( k ) j W ij ) a ij i W ij
ε 1 ( k ) = i ( t i t ¯ ) ( x i ( k ) x ¯ ( k ) ) [ i ( t i t ¯ ) 2 ( x i ( k ) x ¯ ( k ) ) 2 ] 1 / 2
ε 2 ( k ) = [ i ( x i ( k ) t i ) 2 i ( t i t ¯ ) 2 ] 1 / 2

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