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.

© Optical Society of America

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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-interferingsource 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 RandomMedia 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, James G. 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).
    [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]

Other (36)

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]

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]

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]

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

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]

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

M. O'Leary, "Imaging with diffuse photon density waves," PhD University of Pennsylvania (1996).

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]

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]

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

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]

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]

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

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, 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]

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]

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Physics Today 48, 34-40 (1995).
[CrossRef]

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).

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]

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 RandomMedia 9, 489-499 (1999).
[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]

S. Arridge, "Photon-measurement density function.I Analytical forms," Appl. Opt. 34, 7395-7409 (1995).
[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, James G. Fujimoto and Michael S. Patterson, eds. (Optical Society of America, Washington, DC 1998), 266-270.

S. Arridge, "Optical tomography in medical imaging," Inverse Problems 15, R41-R93 (1999).
[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).

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]

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

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).
[CrossRef] [PubMed]

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. 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]

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]

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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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