Abstract

We formulate a perturbative solution for the heterogeneous diffusion equation which demonstrates how to use differential changes in diffuse light transmission to construct images of tissue absorption changes following contrast agent administration. The analysis exposes approximations leading to an intuitive and simplified inverse algorithm, shows explicitly why transmission geometries are less susceptible to error than the remission geometries, and why differential measurements are less susceptible to surface artifacts. These ideas about differential diffuse optical tomography are not only applicable to tumor detection and characterization using contrast agents, but also to functional activation studies with or without contrast agents and multi-wavelength measurements.

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References

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  1. B. W. Pogue, "Focus issue: Biomedical diffuse optical tomography - Introduction," Opt. Express 4, 230-230 (1999). http://www.opticsexpress.org/tocv4n8.htm
    [CrossRef] [PubMed]
  2. F. Kelcz, G. Santyr, "Gadolinium-Enhanced Breast MRI," Critical Reviews in Diagnostic Imaging 36, 287-338 (1995).
    [PubMed]
  3. F. Barkhof, J. Valk, O. R.Hommes, P. Scheltens, "Meningeal Gd-DTPA enhancement in multiple-sclerosis," Am. J. Neuroradiology 13, 397-400 (1992).
  4. C. Tilcock, "Delivery of contrast agents for magnetic resonance imaging, computed tomography, nuclear medicine and ultrasound," Adv Drug Deliver Rev 37, 33-51 (1999).
    [CrossRef]
  5. R. P. Kedar, D. Cosgrove, V. R. McCready, J. C. Bamber, E. R. Carter, "Microbubble contrast agent for color Doppler US: Effect on breast masses," Radiology 198, 679-686 (1996).
    [PubMed]
  6. M. L. Melany, E. G. Grant, S. Farooki, et al. "Effect of US contrast agents on spectral velocities: In vitro evaluation," Radiology 211, 427-431 (1999).
    [PubMed]
  7. S. E. Thompson, V. Raptopoulos, R. L. Sheiman, M. M. J McNicholas, P. Prassopoulos, "Abdominal helical CT: Milk as a low-attenuation oral contrast agent," Radiology 211, 870-875 (1999).
    [PubMed]
  8. R. Jain, S. Sawhney, P. Sahni, K. Taneja, M. Berry, "CT portography by direct intrasplenic contrast injection: a new technique," Abdominal Imaging 24, 272-277 (1999).
    [CrossRef] [PubMed]
  9. L. W. Nunes, M. D. Schnall, S. G. Orel, M. G. Hochman, C. P. Langlotz, C. A. Reynolds, M. H Torosian, "Correlation of lesion appearance and histologic findings for the nodes of a breast MR imaging interpretation model," Radiographics 19, 79-92 (1999).
    [PubMed]
  10. S. G. Orel, M. D. Schnall, V. A. Livolsi, R. H. Troupin, "Suspicious Breast-Lesions - MR-Imaging With Radiologic- Pathological Correlation," Radiology 190, 485-493 (1994).
    [PubMed]
  11. B. W. Pogue, M. Testorf, T. McBride, U. Osterberg, K. Paulsen, "Instrumentation and design of a frequency-domain diffuse optical tomography imager for breast cancer detection," Opt. Express 1, 391-403 (1997). http://www.opticsexpress.org/oearchive/source/2827.htm
    [CrossRef] [PubMed]
  12. V. Ntziachristos, X. H. Ma, B. Chance, "Time-correlated single photon counting imager for simultaneous magnetic resonance and near-infrared mammography," Rev. Sci. Instr. 69, 4221-4233 (1998).
    [CrossRef]
  13. V. Ntziachristos, A. G. Yodh, M. D. Schnall, B. Chance, "Comparison between intrinsic and extrinsic contrast for malignancy detection using NIR mammography," Proc. SPIE 3597, 565-570 (1999).
    [CrossRef]
  14. A. Ishimaru, Wave propagation and scattering in random media., (New York, Academic Press 1978).
  15. J. B. Fishkin, E. Gratton, "Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge," Opt. Soc. Am. A 10, 127-140 (1993).
    [CrossRef]
  16. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (New York, IEEE Press, 1988).
  17. S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, "Reconstruction methods for infra-red absorption imaging," Proc. SPIE 1431, 204-215 (1991).
    [CrossRef]
  18. M. S. Patterson, B. Chance, B. C. Wilson, "Time Resolved Reflectance and Transmittance for the Noninvasive Measurement of Tissue Optical Properties," J. Appl. Opt. 28, 2331-2336 (1989).
    [CrossRef]
  19. T. J. Farrell, M. S. Patterson, B. Wilson, "A diffusion-theory model of spatially resolved, steady-state diffuse reflectance for the non-invasive determination of tissue optical properties in-vivo," Med. Phys. 19, 879-888 (1992).
    [CrossRef] [PubMed]
  20. R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, B. J. Tromberg, "Boundary conditions for the diffusion equation in radiative transfer," J. Opt. Soc. Am. A 11, 2727-2741 (1994).
    [CrossRef]
  21. R. Aronson, "Boundary conditions for diffusion of light," J. Opt. Soc. Am. A 12, 2532-2539 (1995).
    [CrossRef]
  22. M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, "Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography," Opt. Lett. 20, 426-428 (1995).
    [CrossRef]
  23. V. Ntziachristos, A. Hielscher, A. G. Yodh, B. Chance, "Performance of perturbation tomography with highly heterogeneous media under the P1 approximation" in Biomedical Optics: Advances in Optical Imaging, Photon Migration and Tissue Optics, OSA Technical Digest CLEO/Europe AMB3-1: 211-213 (1999).
  24. V. Ntziachristos, X. H. Ma, A. G. Yodh, B. Chance, "Multichannel photon counting instrument for spatially resolved near infrared spectroscopy," Rev. Sci. Instr. 70, 193-201 (1999).
    [CrossRef]

Other

B. W. Pogue, "Focus issue: Biomedical diffuse optical tomography - Introduction," Opt. Express 4, 230-230 (1999). http://www.opticsexpress.org/tocv4n8.htm
[CrossRef] [PubMed]

F. Kelcz, G. Santyr, "Gadolinium-Enhanced Breast MRI," Critical Reviews in Diagnostic Imaging 36, 287-338 (1995).
[PubMed]

F. Barkhof, J. Valk, O. R.Hommes, P. Scheltens, "Meningeal Gd-DTPA enhancement in multiple-sclerosis," Am. J. Neuroradiology 13, 397-400 (1992).

C. Tilcock, "Delivery of contrast agents for magnetic resonance imaging, computed tomography, nuclear medicine and ultrasound," Adv Drug Deliver Rev 37, 33-51 (1999).
[CrossRef]

R. P. Kedar, D. Cosgrove, V. R. McCready, J. C. Bamber, E. R. Carter, "Microbubble contrast agent for color Doppler US: Effect on breast masses," Radiology 198, 679-686 (1996).
[PubMed]

M. L. Melany, E. G. Grant, S. Farooki, et al. "Effect of US contrast agents on spectral velocities: In vitro evaluation," Radiology 211, 427-431 (1999).
[PubMed]

S. E. Thompson, V. Raptopoulos, R. L. Sheiman, M. M. J McNicholas, P. Prassopoulos, "Abdominal helical CT: Milk as a low-attenuation oral contrast agent," Radiology 211, 870-875 (1999).
[PubMed]

R. Jain, S. Sawhney, P. Sahni, K. Taneja, M. Berry, "CT portography by direct intrasplenic contrast injection: a new technique," Abdominal Imaging 24, 272-277 (1999).
[CrossRef] [PubMed]

L. W. Nunes, M. D. Schnall, S. G. Orel, M. G. Hochman, C. P. Langlotz, C. A. Reynolds, M. H Torosian, "Correlation of lesion appearance and histologic findings for the nodes of a breast MR imaging interpretation model," Radiographics 19, 79-92 (1999).
[PubMed]

S. G. Orel, M. D. Schnall, V. A. Livolsi, R. H. Troupin, "Suspicious Breast-Lesions - MR-Imaging With Radiologic- Pathological Correlation," Radiology 190, 485-493 (1994).
[PubMed]

B. W. Pogue, M. Testorf, T. McBride, U. Osterberg, K. Paulsen, "Instrumentation and design of a frequency-domain diffuse optical tomography imager for breast cancer detection," Opt. Express 1, 391-403 (1997). http://www.opticsexpress.org/oearchive/source/2827.htm
[CrossRef] [PubMed]

V. Ntziachristos, X. H. Ma, B. Chance, "Time-correlated single photon counting imager for simultaneous magnetic resonance and near-infrared mammography," Rev. Sci. Instr. 69, 4221-4233 (1998).
[CrossRef]

V. Ntziachristos, A. G. Yodh, M. D. Schnall, B. Chance, "Comparison between intrinsic and extrinsic contrast for malignancy detection using NIR mammography," Proc. SPIE 3597, 565-570 (1999).
[CrossRef]

A. Ishimaru, Wave propagation and scattering in random media., (New York, Academic Press 1978).

J. B. Fishkin, E. Gratton, "Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge," Opt. Soc. Am. A 10, 127-140 (1993).
[CrossRef]

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (New York, IEEE Press, 1988).

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, "Reconstruction methods for infra-red absorption imaging," Proc. SPIE 1431, 204-215 (1991).
[CrossRef]

M. S. Patterson, B. Chance, B. C. Wilson, "Time Resolved Reflectance and Transmittance for the Noninvasive Measurement of Tissue Optical Properties," J. Appl. Opt. 28, 2331-2336 (1989).
[CrossRef]

T. J. Farrell, M. S. Patterson, B. Wilson, "A diffusion-theory model of spatially resolved, steady-state diffuse reflectance for the non-invasive determination of tissue optical properties in-vivo," Med. Phys. 19, 879-888 (1992).
[CrossRef] [PubMed]

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, B. J. Tromberg, "Boundary conditions for the diffusion equation in radiative transfer," J. Opt. Soc. Am. A 11, 2727-2741 (1994).
[CrossRef]

R. Aronson, "Boundary conditions for diffusion of light," J. Opt. Soc. Am. A 12, 2532-2539 (1995).
[CrossRef]

M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, "Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography," Opt. Lett. 20, 426-428 (1995).
[CrossRef]

V. Ntziachristos, A. Hielscher, A. G. Yodh, B. Chance, "Performance of perturbation tomography with highly heterogeneous media under the P1 approximation" in Biomedical Optics: Advances in Optical Imaging, Photon Migration and Tissue Optics, OSA Technical Digest CLEO/Europe AMB3-1: 211-213 (1999).

V. Ntziachristos, X. H. Ma, A. G. Yodh, B. Chance, "Multichannel photon counting instrument for spatially resolved near infrared spectroscopy," Rev. Sci. Instr. 70, 193-201 (1999).
[CrossRef]

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Figures (7)

Figure 1.
Figure 1.

(a) Amplitude of the term e i(k′-k″)R(r⃗) as a function of the average absorption coefficient of the post-ICG breast, (b) Phase of the term e i(k′-k″)R(r⃗) as a function of the average absorption coefficient of the post-ICG breast, (c) Test geometry for calculations in (a) and (b).

Figure 2.
Figure 2.

(a) Amplitude deviation and (b) Phase shift introduced in the field measured in Eq. 16 when S is assumed zero. The calculation is done as a function of the distance a., for the geometry shown in Fig. 1c, assuming 200MHz, background µa0=0.05cm-1, µa0=0.05cm-1, µs=10 cm-1, δµarel =0.05 cm-1 and δµ a (r⃗)=0.01 cm-1

Figure 3.
Figure 3.

(a) T1-weighted MR coronal slice of a human breast, (b) Gd-DTPA distribution (in color) of the same coronal slice. A ductal carcinoma appears in yellow.

Figure 4.
Figure 4.

Absorption maps used for the simulation. (a) pre-ICG breast (b) post-ICG breast.

Figure 5.
Figure 5.

Sources and detector arrangement used for the simulation. The region reconstructed is outlined with a green double line.

Figure 6.
Figure 6.

Reconstruction results imaging the region indicated with the green double line on Fig.5 assuming an infinite slab geometry. a)The reconstructed image using Eq.20. b)The reconstructed image using Eq.22. c)The reconstructed image using Eq.23. d) The result of subtracting an image of the post-ICG breast (reconstructed relative to a homogeneous baseline medium) from an image of the pre-ICG breast (reconstructed relatively to the same baseline medium). The optical properties of the homogeneous medium were µ a0=0.03cm-1 and µs =8cm-1.

Figure A1.
Figure A1.

(a) Amplitude and (b) Phase of R as a function of the average absorption coefficient of the post-ICG breast, plotted for the geometry of Fig 1c.

Tables (1)

Tables Icon

TABLE I Absolute optical properties of the different structures used for the simulations.

Equations (24)

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· [ D ( r ) U ( r ) ] + ( j ω c μ a ( r ) ) · U ( r ) = c S ( r ) .
U ' ( r s , r d , ω ) = U ' 0 ( r s , r d , ω ) e ϕ ' sc ( r s , r d , ω ) ,
ϕ sc ( r s , r d , ω ) = V [ W ' a ( r s , r d , r , μ ' a 0 , D ' 0 , ω ) δ μ ' a ( r ) + W ' s ( r s , r d , r , μ ' a 0 , D ' 0 , ω ) δ D ' ( r ) ] d r ,
W ' a ( r s , r d , r , μ a 0 ' , D ' 0 , ω ) = c 4 π ( D ' 0 ) 2 · r s r d r d r · r s r · e i k ' r d r · e i k ' r s r e i k ' r s r d ,
W ' s ( r s , r d , r , μ a 0 ' , D ' 0 , ω ) = r s r d 4 π ( D ' 0 ) 2 · e i k ' r d r r d r · e i k ' r s r r s r e i k ' r s r d ,
k ' = υ μ a 0 ' + i ω D ' 0 .
U " ( r s , r d , ω ) = U " 0 ( r s , r d , ω ) e ϕ sc " ( r s , r d , ω ) .
ϕ " sc ( r s , r d , ω ) = V [ W " a ( r s , r d , r , μ " a 0 , D " 0 , ω ) δ μ " a ( r ) + W " s ( r s , r d , r , μ " a 0 , D " 0 , ω ) δ D " ( r ) ] d r ,
W " a ( r s , r d , r , μ " a 0 , D " 0 , ω ) = c 4 π ( D " 0 ) 2 · r s r d r d r · r s r · e i k " r d r · e i k " r s r e i k " r s r d ,
W " s ( r s , r d , r , μ a 0 " , D " 0 , ω ) = r s r d 4 π ( D " 0 ) 2 · e i k " r d r r d r · e i k " r s r r s r e i k " r s r d ,
k " = υ μ " a 0 + j ω D " 0 .
ϕ sc = ϕ " sc ϕ ' sc = ln ( U " U ' · U ' 0 U " 0 ) .
ϕ sc ( r s , r d , ω ) = V [ W " a ( r s , r d , r , μ " a 0 , D " 0 , ω ) δ μ " a ( r ) W ' a ( r s , r d , r , μ ' a 0 , D ' 0 , ω ) δ μ ' a ( r ) ] d r .
μ " a ( r ) = μ " a 0 + δ μ " a ( r ) = μ ' a 0 + δ μ ' a ( r ) + δ μ a ICG ( r ) ,
δ μ a ( r ) = μ ' a 0 μ " a 0 + δ μ a ICG ( r ) + δ μ ' a ( r ) .
ϕ sc ( r s , r d , ω ) = V W " a δ μ a rel ( r ) d r + V ( W " a W ' a ) δ μ ' a ( r ) d r .
S = V ( W " a W ' a ) δ μ ' a ( r ) d r ,
S = c 4 π ( D " 0 ) 2 · V W " a · r s r d r d r · r s r · ( 1 e i ( k ' k " ) R ( r ) ) · δ μ ' a ( r ) d r ,
R ( r ) = r d r + r s r r s r d .
ϕ sc ( r s , r d , ω ) = ln ( U " U ' U ' 0 U " 0 ) V W " 0 δ μ a rel ( r ) d r .
[ ϕ sc 1 ( r s 1 , r d 1 , ω 1 ) ϕ sc m ( r so , r dp , ω q ) ] = [ W " a 11 W " a 1 n W " a m 1 W " a mn ] [ δ μ a rel ( r 1 ) δ μ a rel ( r n ) ] .
ϕ " sc ( r s , r d , ω ) = ln ( U " U ' ) = V W ' a δ μ a ICG ( r ) d r .
ϕ sc ( r s , r d , ω ) S = V W " a δ μ a rel ( r ) d r .
R = W ' s ( r s , r d , r , μ ' a 0 , D ' 0 , ω ) W " s ( r s , r d , r , μ " a 0 , D 0 , ω ) ,

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