Abstract

Images produced in six different geometries with diffuse optical tomography simulations of tissue have been compared using a finite element-based algorithm with iterative refinement provided by the Newton-Raphson approach. The source-detector arrangements studied include (i) fan-beam tomography, (ii) full reflectance and transmittance tomography, as well as (iii) sub-surface imaging, where each of these three were examined in a circular and a flat slab geometry. The algorithm can provide quantitatively accurate results for all of the tomographic geometries investigated under certain circumstances. For example, quantitatively accurate results occur with sub-surface imaging only when the object to be imaged is fully contained within the diffuse projections. In general the diffuse projections must sample all regions around the target to be characterized in order for the algorithm to recover quantitatively accurate results. Not only is it important to sample the whole space, but maximal angular sampling is required for optimal image reconstruction. Geometries which do not maximize the possible sampling angles cause more noise artifact in the reconstructed images. Preliminary simulations using a mesh of the human brain confirm that optimal images are produced from circularly symmetric source-detector distributions, but that quantitatively accurate images can be reconstructed even with a sub-surface imaging, although spatial resolution is modest.

© Optical Society of America

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References

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  1. M. Cope, and D. T. Delpy, "System for long-term measurement of cerebral blood and tissue oxygenation on newborn infants by near-infrared transillumination," Med. Biol. Eng. Comp. 26, 289-294 (1988).
    [CrossRef]
  2. B. Chance, Q. Luo, S. Nioka, D. C. Alsop and J. A. Detre, "Optical investigations of physiology: a study of intrinsic and extrinsic biomedical contrast," Phil. Trans. R. Soc. Lond. B 352, 707-716 (1997).
    [CrossRef]
  3. M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, H., W. W. Mantulin, M. Seeber, P. Schlag and M. Kaschke, "Frequency-domain techniques enhance optical mammography: initial clinical results," Proc. Nat. Acad. Sci USA 94, 6468-73 (1997).
    [CrossRef] [PubMed]
  4. S. R. Arridge and M. Schweiger, "Sensitivity to prior knowledge in optical tomographic resconstruction," Proc. SPIE 2389, 378-388 (1995).
    [CrossRef]
  5. V. Ntziachristos, M. O,Leary, B. Chance and A. G. Yodh, "Coregistration of images from diffusive wave with other imaging modalities to enhance specificity," in OSA TOPS on Advances in Optical Imaging and Photon Migration II. (OSA publications, Orlando, FL, 1996).
  6. J. Chang, H. Graber, P. Koo, R. Aronson, S. S. Barbour, R. L. Barbour, "Optical imaging of anatomical maps derived from magnetic resonance images using time independent optical sources," IEEE Trans. Med. Imag. 16, 68-77 (1997).
    [CrossRef]
  7. B. W. Pogue and K.D. Paulsen, "High resolution near infrared tomographic imaging simulations of rat cranium using apriori MRI structural information," Opt. Lett. 23, 1716-8 (1998).
    [CrossRef]
  8. C. L. Hutchinson, J.R. Lakowicz, and E.M. Sevick-Muraca, "Fluorescence lifetime-based sensing in tissues: a computational study," Biophys. J. 68, 1574-1582 (1995).
    [CrossRef] [PubMed]
  9. J. Chang, H.L. Graber, and R.L. Barbour, "Luminescence optical tomography of dense scattering media," J. Opt. Soc. Am. A 14, 288-99 (1998).
    [CrossRef]
  10. S. R. Arridge, "Forward and inverse problems in time-resolved infrared imaging," in Medical Optical Tomography: Functional Imaging and Monitoring, Ed. G. Muller, (SPIE Optical Eng. Press, Bellingham, WA, 1993) pp. 35-64.
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    [CrossRef] [PubMed]
  12. H. Jiang, K. D. Paulsen, U. L. Osterberg and M. S. Patterson, "Frequency-domain optical image reconstruction in turbid media: an experimental study of single-target detectability," Appl. Opt. 36, 52-63 (1997).
    [CrossRef] [PubMed]
  13. H. Jiang, K. D. Paulsen, U. L. Osterberg, and M. S. Patterson, "Frequency-domain optical image reconstruction for breast imaging: initial evaluation in multi-target tissue-like phantoms," Med. Phys. 25, 183-193 (1997).
    [CrossRef]
  14. B. W. Pogue, M. Testorf, U. L. Osterberg and K. D. Paulsen, "Instrumentation and design of a frequency-domain diffuse optical tomography imager for breast cancer detection," Opt. Express 1, 391-403 (1997). (http://epubs.osa.org/oearchive/source/2827.htm)
    [CrossRef] [PubMed]
  15. B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and K. D. Paulsen, "Spatially variant regularization improves diffuse optical tomography," Appl. Opt. 38, (in press) 1999.
    [CrossRef]
  16. T. O. McBride, B. W. Pogue, E. Gerety, S. Poplack, U. L. Osterberg, and K. D. Paulsen, "Spectroscopic diffuse optical tomography for quantitatively assessing hemoglobin concentration and oxygenation in tissue," (submitted, 1999).
  17. M. S. Patterson, B. Chance and B. C. Wilson, "Time resolved reflectance and transmittances for the non-invasive measurement of tissue optical properties," Appl. Opt. 28, 2331-2336 (1989).
    [CrossRef] [PubMed]
  18. S. R. Arridge, M. Cope and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 37, 1531-1560 (1992).
    [CrossRef] [PubMed]
  19. A. H. Gandjbakhche, V. Chernomordik, R.F. Bonner, J. C. Hebden, R. Nossal, "Use of time-dependent contrast functions to discriminate between the scattering and absorption properties of abnormal regions hidden within a tissue-like phantom," Proc. SPIE 2979, 211-225 (1997).
    [CrossRef]
  20. J. A. Moon, R. Mahon, M. D. Duncan and J. Reintjes, "Resolution limits for imaging through turbid media with diffuse light," Opt. Lett. 18, 1591-1593 (1993).
    [CrossRef] [PubMed]
  21. A. H. Gandjbakhche, R. Nossal and R. F. Bonner, "Resolution limits for optical transillumination of abnormalities deeply embedded in tissues," Med. Phys. 21, 185-91 (1994).
    [CrossRef] [PubMed]
  22. D. A. Boas, M. A. OLeary, B. Chance and A. G. Yodh, "Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis," Appl. Opt. 36, 75-92 (1997).
    [CrossRef] [PubMed]
  23. H. Jess, H. Erdl, T. Moesta, S. Fantini, M. A. Francecshini, E. Gratton and M. Kaschke, "Intensity modulated breast imaging: Technology and clinical pilot study results," Adv. in Optical Imaging and Photon Migration. (OSA publications, Orlando, FL, 1996).
  24. S. Fantini, O. Schutz, J. Edler, M. A. Franceschini, S. Heywang-Krbrunner, L. Gotz and H. Siebold, "Performance of N-Images and spectral features in frequency-domain optical mammography," SPIE Technical Abstract Digest. (SPIE Press, San Jose, CA, 1999).
  25. Y. Painchaud, A. Mailloux, E. Harvey, S. Verrault, J. Frechette, C. Gilbert, M. L. Vernon and P. Beaudry, "Multi-port time-domain laser mammography: results on solid phantoms and volunteers," SPIE BiOS Technical Abstract Digest. (SPIE Press, San Jose, CA, 1999).
  26. J. P. Van Houten, D. A. Benaron, S. Spilman and D. K. Stevenson, "Imaging brain injury using time-resolved near infrared light scanning," Pediat. Res. 39, 470-6 (1996).
    [CrossRef] [PubMed]
  27. M. R. Stankovic, A. Fujii, D. Maulik and D. Boas, "Optical monitoring of cerebral hemodynamics and oxygenation in the neonatal piglet," J. Matern-Fetal Inves. 8, 71-8 (1998).
  28. A. Siegel, J. Marota, J. Mandeville, B. Rosen, and D. Boas, "Diffuse optical tomography of rat brain function," in SPIE Technical Abstract Digest. (SPIE Press, San Jose, CA, 1999).
  29. S. Fantini, S. Walker, M. A. Franceschini, M. Kaschke, P. M. Schlag and K. T. Moesta, "Assessment of the size, position, and optical properties of breast tumors in vivo by noninvasive optical methods," Appl. Opt. 37, 1982- 89 (1998).
    [CrossRef]
  30. V. Quaresima, S. J. Matcher and M. Ferrari, "Identification and quantification of intrinsic optical contrast for near-infrared mammography," Photochem. Photobiol. 67, 4-14 (1998).
    [CrossRef] [PubMed]
  31. X. Li, J. Culver, J., T. Durduran, B. Chance, A. G. Yodh and D. N. Pattanayak, "Diffraction tomography with diffuse-photon density waves: clinical studies and background subtraction," in Advances in Optical Imaging and Photon Migration. (OSA publications, Orlando, FL, 1993).
  32. S. B. Colak, D. G. Papaioannou, G. W. t'Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen and N. A. A. J. van Asten, "Tomographic image reconstruction from optical projections in light-diffusing media," Appl. Opt. 36, 180-213 (1997).
    [CrossRef] [PubMed]
  33. S. R. Arridge and M. Schwieger, "Gradient-based optimisation scheme for optical tomography," Opt. Exp. 2, 212-226 (1998). (http://epubs.osa.org/oearchive/source/4014.htm)
    [CrossRef]
  34. J. C. Hebden, F. E. W. Schmidt, M. E. Fry, M. Schweiger, E. M. C. Hillman, and D. T. Delpy, "Simultaneous reconstruction of absorption and scattering images using multi-channel measurement of purely temporal data," Opt. Lett, 1999.
    [CrossRef]
  35. B. J. Tromberg, O. Coquoz, J. B. Fishkin, T. Pham, E. R. Anderson, J. Butler, M. Cahn, J. D. Gross, V. Venugopalan and D. Pham, "Non-invasive measurements of breast tissue optical properties using frequency-domain photon migration," Phil. Trans. R. Soc. Lond. B. 352, 661-668 (1997).
  36. S. R. Arridge and M. Schweiger, "Inverse methods for optical tomography," in Information Processing in Medical Imaging (Springer-Verlag, Flagstaff, AZ, 1993).
    [CrossRef] [PubMed]
  37. K. D. Paulsen and H. Jiang "Spatially varying optical property reconstruction using a finite element diffusion equation approximation," Med. Phys. 22, 691-701 (1995).
    [CrossRef]
  38. S. R. Arridge and M. Schweiger, "Image reconstruction in optical tomography," Phil. Trans. R. Soc. Lond. B. 352, 717-726 (1997).
    [CrossRef]
  39. A. Neumaier, "Solving ill-conditioned and singular linear systems: a tutorial on regularization," SIAM Rev. 40, 636-666 (1998).
    [CrossRef] [PubMed]
  40. T. J. Yorkey, J. G. Webster and W. J. Tompkins, "Comparing reconstruction algorithms for electrical impedance tomography," IEEE Trans. Biomed. Eng. 34, 843-852 (1987).
    [CrossRef]
  41. B. W. Pogue, T. O. McBride, C. Nwaigwe, U. L. Osterberg, J. F. Dunn, K. D. Paulsen, "Near-infrared diffuse tomography with apriori MRI structural information: testing a hybrid image reconstruction methodology with functional imaging of the rat cranium," Proc. SPIE 3597 (in press), (1999).

Other (41)

M. Cope, and D. T. Delpy, "System for long-term measurement of cerebral blood and tissue oxygenation on newborn infants by near-infrared transillumination," Med. Biol. Eng. Comp. 26, 289-294 (1988).
[CrossRef]

B. Chance, Q. Luo, S. Nioka, D. C. Alsop and J. A. Detre, "Optical investigations of physiology: a study of intrinsic and extrinsic biomedical contrast," Phil. Trans. R. Soc. Lond. B 352, 707-716 (1997).
[CrossRef]

M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, H., W. W. Mantulin, M. Seeber, P. Schlag and M. Kaschke, "Frequency-domain techniques enhance optical mammography: initial clinical results," Proc. Nat. Acad. Sci USA 94, 6468-73 (1997).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, "Sensitivity to prior knowledge in optical tomographic resconstruction," Proc. SPIE 2389, 378-388 (1995).
[CrossRef]

V. Ntziachristos, M. O,Leary, B. Chance and A. G. Yodh, "Coregistration of images from diffusive wave with other imaging modalities to enhance specificity," in OSA TOPS on Advances in Optical Imaging and Photon Migration II. (OSA publications, Orlando, FL, 1996).

J. Chang, H. Graber, P. Koo, R. Aronson, S. S. Barbour, R. L. Barbour, "Optical imaging of anatomical maps derived from magnetic resonance images using time independent optical sources," IEEE Trans. Med. Imag. 16, 68-77 (1997).
[CrossRef]

B. W. Pogue and K.D. Paulsen, "High resolution near infrared tomographic imaging simulations of rat cranium using apriori MRI structural information," Opt. Lett. 23, 1716-8 (1998).
[CrossRef]

C. L. Hutchinson, J.R. Lakowicz, and E.M. Sevick-Muraca, "Fluorescence lifetime-based sensing in tissues: a computational study," Biophys. J. 68, 1574-1582 (1995).
[CrossRef] [PubMed]

J. Chang, H.L. Graber, and R.L. Barbour, "Luminescence optical tomography of dense scattering media," J. Opt. Soc. Am. A 14, 288-99 (1998).
[CrossRef]

S. R. Arridge, "Forward and inverse problems in time-resolved infrared imaging," in Medical Optical Tomography: Functional Imaging and Monitoring, Ed. G. Muller, (SPIE Optical Eng. Press, Bellingham, WA, 1993) pp. 35-64.

D. Boas, "A fundamental limitation of linearized algorithms for diffuse optical tomography," Opt. Express 1, 404-413 (1997). (http://epubs.osa.org/oearchive/source/2831.htm)
[CrossRef] [PubMed]

H. Jiang, K. D. Paulsen, U. L. Osterberg and M. S. Patterson, "Frequency-domain optical image reconstruction in turbid media: an experimental study of single-target detectability," Appl. Opt. 36, 52-63 (1997).
[CrossRef] [PubMed]

H. Jiang, K. D. Paulsen, U. L. Osterberg, and M. S. Patterson, "Frequency-domain optical image reconstruction for breast imaging: initial evaluation in multi-target tissue-like phantoms," Med. Phys. 25, 183-193 (1997).
[CrossRef]

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

B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and K. D. Paulsen, "Spatially variant regularization improves diffuse optical tomography," Appl. Opt. 38, (in press) 1999.
[CrossRef]

T. O. McBride, B. W. Pogue, E. Gerety, S. Poplack, U. L. Osterberg, and K. D. Paulsen, "Spectroscopic diffuse optical tomography for quantitatively assessing hemoglobin concentration and oxygenation in tissue," (submitted, 1999).

M. S. Patterson, B. Chance and B. C. Wilson, "Time resolved reflectance and transmittances for the non-invasive measurement of tissue optical properties," Appl. Opt. 28, 2331-2336 (1989).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef] [PubMed]

A. H. Gandjbakhche, V. Chernomordik, R.F. Bonner, J. C. Hebden, R. Nossal, "Use of time-dependent contrast functions to discriminate between the scattering and absorption properties of abnormal regions hidden within a tissue-like phantom," Proc. SPIE 2979, 211-225 (1997).
[CrossRef]

J. A. Moon, R. Mahon, M. D. Duncan and J. Reintjes, "Resolution limits for imaging through turbid media with diffuse light," Opt. Lett. 18, 1591-1593 (1993).
[CrossRef] [PubMed]

A. H. Gandjbakhche, R. Nossal and R. F. Bonner, "Resolution limits for optical transillumination of abnormalities deeply embedded in tissues," Med. Phys. 21, 185-91 (1994).
[CrossRef] [PubMed]

D. A. Boas, M. A. OLeary, B. Chance and A. G. Yodh, "Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis," Appl. Opt. 36, 75-92 (1997).
[CrossRef] [PubMed]

H. Jess, H. Erdl, T. Moesta, S. Fantini, M. A. Francecshini, E. Gratton and M. Kaschke, "Intensity modulated breast imaging: Technology and clinical pilot study results," Adv. in Optical Imaging and Photon Migration. (OSA publications, Orlando, FL, 1996).

S. Fantini, O. Schutz, J. Edler, M. A. Franceschini, S. Heywang-Krbrunner, L. Gotz and H. Siebold, "Performance of N-Images and spectral features in frequency-domain optical mammography," SPIE Technical Abstract Digest. (SPIE Press, San Jose, CA, 1999).

Y. Painchaud, A. Mailloux, E. Harvey, S. Verrault, J. Frechette, C. Gilbert, M. L. Vernon and P. Beaudry, "Multi-port time-domain laser mammography: results on solid phantoms and volunteers," SPIE BiOS Technical Abstract Digest. (SPIE Press, San Jose, CA, 1999).

J. P. Van Houten, D. A. Benaron, S. Spilman and D. K. Stevenson, "Imaging brain injury using time-resolved near infrared light scanning," Pediat. Res. 39, 470-6 (1996).
[CrossRef] [PubMed]

M. R. Stankovic, A. Fujii, D. Maulik and D. Boas, "Optical monitoring of cerebral hemodynamics and oxygenation in the neonatal piglet," J. Matern-Fetal Inves. 8, 71-8 (1998).

A. Siegel, J. Marota, J. Mandeville, B. Rosen, and D. Boas, "Diffuse optical tomography of rat brain function," in SPIE Technical Abstract Digest. (SPIE Press, San Jose, CA, 1999).

S. Fantini, S. Walker, M. A. Franceschini, M. Kaschke, P. M. Schlag and K. T. Moesta, "Assessment of the size, position, and optical properties of breast tumors in vivo by noninvasive optical methods," Appl. Opt. 37, 1982- 89 (1998).
[CrossRef]

V. Quaresima, S. J. Matcher and M. Ferrari, "Identification and quantification of intrinsic optical contrast for near-infrared mammography," Photochem. Photobiol. 67, 4-14 (1998).
[CrossRef] [PubMed]

X. Li, J. Culver, J., T. Durduran, B. Chance, A. G. Yodh and D. N. Pattanayak, "Diffraction tomography with diffuse-photon density waves: clinical studies and background subtraction," in Advances in Optical Imaging and Photon Migration. (OSA publications, Orlando, FL, 1993).

S. B. Colak, D. G. Papaioannou, G. W. t'Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen and N. A. A. J. van Asten, "Tomographic image reconstruction from optical projections in light-diffusing media," Appl. Opt. 36, 180-213 (1997).
[CrossRef] [PubMed]

S. R. Arridge and M. Schwieger, "Gradient-based optimisation scheme for optical tomography," Opt. Exp. 2, 212-226 (1998). (http://epubs.osa.org/oearchive/source/4014.htm)
[CrossRef]

J. C. Hebden, F. E. W. Schmidt, M. E. Fry, M. Schweiger, E. M. C. Hillman, and D. T. Delpy, "Simultaneous reconstruction of absorption and scattering images using multi-channel measurement of purely temporal data," Opt. Lett, 1999.
[CrossRef]

B. J. Tromberg, O. Coquoz, J. B. Fishkin, T. Pham, E. R. Anderson, J. Butler, M. Cahn, J. D. Gross, V. Venugopalan and D. Pham, "Non-invasive measurements of breast tissue optical properties using frequency-domain photon migration," Phil. Trans. R. Soc. Lond. B. 352, 661-668 (1997).

S. R. Arridge and M. Schweiger, "Inverse methods for optical tomography," in Information Processing in Medical Imaging (Springer-Verlag, Flagstaff, AZ, 1993).
[CrossRef] [PubMed]

K. D. Paulsen and H. Jiang "Spatially varying optical property reconstruction using a finite element diffusion equation approximation," Med. Phys. 22, 691-701 (1995).
[CrossRef]

S. R. Arridge and M. Schweiger, "Image reconstruction in optical tomography," Phil. Trans. R. Soc. Lond. B. 352, 717-726 (1997).
[CrossRef]

A. Neumaier, "Solving ill-conditioned and singular linear systems: a tutorial on regularization," SIAM Rev. 40, 636-666 (1998).
[CrossRef] [PubMed]

T. J. Yorkey, J. G. Webster and W. J. Tompkins, "Comparing reconstruction algorithms for electrical impedance tomography," IEEE Trans. Biomed. Eng. 34, 843-852 (1987).
[CrossRef]

B. W. Pogue, T. O. McBride, C. Nwaigwe, U. L. Osterberg, J. F. Dunn, K. D. Paulsen, "Near-infrared diffuse tomography with apriori MRI structural information: testing a hybrid image reconstruction methodology with functional imaging of the rat cranium," Proc. SPIE 3597 (in press), (1999).

Supplementary Material (11)

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Figures (15)

Fig. 1.
Fig. 1.

Schematic of three typical source-detector geometries used in diffuse optical tomography applications. The projection-shadow geometry (a) is used in several optical mammography scanners [3, 23–25], while the circular-tomography geometry (b) is used for both breast and brain imaging [16, 26], as is the subsurface imaging geometry (c) [2, 27, 28].

Fig. 2.
Fig. 2.

Geometries used in this study are shown, where the outward arrows indicate detector locations and the inward arrows denote source locations. The geometries include (a) circular reflectance and transmittance imaging which uses circular symmetry for sampling angles (b) circular fan-beam imaging (where the source-detector array is rotated around the tissue), (c) sub-surface tomography which only uses a sub-set of projections from the surface of the tissue, (d) flat slab reflectance and transmittance imaging using maximal sampling from both surfaces, (e) flat slab fan-beam which does not use any reflectance data, and (e) sub-surface imaging with a flat upper surface. Together these six geometries, in various forms, define the possible configurations for imaging tissue with DOT.

Fig. 3.
Fig. 3.

Reconstructed set of images (video sequence) of the test target at different vertical positions within the circular mesh, using source-detector geometry in Fig. 2 (a) with a circularly symmetric distribution of 16 source and 16 detectors for reflectance and transmittance imaging. The color bar on the right is in absorption coefficient units of mm-1. [Media 1]

Fig. 4.
Fig. 4.

Reconstructed set of images (video sequence) showing the test target at different vertical positions within the circular mesh, using the fan beam geometry with 16 sources transmitting to 8 detectors. The color bar on the right is in absorption coefficient units of mm-1. [Media 2]

Fig. 5.
Fig. 5.

Reconstructed set of images of the test target within a circular mesh using the sub-surface orientation with (a) 8 sources and 8 detectors in an arc, and (b) 4 sources and 4 detectors in a smaller arc on the surface (as shown in Fig. 2(c)). Color bar at right is in absorption coefficient units of mm-1. [Media 3] [Media 4]

Fig. 6.
Fig. 6.

Reconstructed set of images of the test target at different (a) vertical positions (b) lateral positions along the central line, and (c) lateral positions along the surface, using the slab reflectance and transmittance geometry of sources and detectors as shown in Fig. 2 (d). [Media 5] [Media 6] [Media 7]

Fig. 7.
Fig. 7.

Reconstructed set of images of the test target at different (a) vertical positions (b) lateral positions along the central line, and (c) lateral positions along the surface similar to those in Fig. 6, using fan-beam slab imaging. Color bar at right is in absorption coefficient units of mm-1. [Media 8] [Media 9] [Media 10]

Fig. 8.
Fig. 8.

Reconstructed set of images where the test target was translated vertically from the middle of the tissue volume (25 mm down) up to the surface, using 16 sources and 16 detectors alternated and equally spaced along the upper surface of the region, as illustrated in Fig. 2 (f). [Media 11]

Fig. 9.
Fig. 9.

Calculated (left-top graph) peak absorption coefficient values for the object at different positions within the simulated phantoms, and (right-top graph) reconstructed noise in the background region of the image measured by sampling of random regions of interest outside the target zone. The true value in the first graph is shown with a solid horizontal line, and the 20% region of acceptability is defined by the dotted lines. Calculated (left-bottom graph) target location as a depth from the upper surface, and (right-bottom graph) the calculated target full width at half maximum (FWHM) height as a function of depth in the medium, here taken as an average of the lateral and vertical directions.

Fig. 10.
Fig. 10.

Calculated (left-top graph) peak absorption coefficient values for the target zone at different positions within the simulated phantoms, and (right-top graph) reconstructed noise in the homogeneous background region of the image outside the target zones. Calculated (left-bottom graph) target location as a depth from the upper surface, and (right-bottom graph) the calculated target full width at half maximum (FWHM) height as a function of depth in the medium, measured here both in the lateral and vertical directions.

Fig.11.
Fig.11.

Reconstructed images of three localized targets within a homogeneous background field, (schematics shown in the left column) using geometries of (upper middle) circular reflectance & transmittance (upper right) circular fan-beam, (lower middle) slab reflectance & transmittance, and (lower right) slab fan-beam. The test targets (left) were all 2:1 absroption contrast from the background, and the two on the right were 20 mm diameter, while the one on the left is 10 mm in diameter.

Fig. 12.
Fig. 12.

Reconstructed images of the test fields (shown in the left colum) are displayed at right. The test fields contained two 10 mm diameter objects with one on the surface and the other 5 mm below the surface. In the top row, the middle image shows the result of the 4 source-4 detector array while the right image shows the result from the 8 source-8 detector array. In the bottom row, the result from the slab geometry is shown using the 16 source-16 detector array.

Fig. 13
Fig. 13

Tomographic reconstructions of a layered tested field with a single target included for both circular (top row) and slab (bottom row) geometries, with two 5 mm thick layers of μa = 0.01 and μa = 0.02, with the interior at μa = 0.015 mm-1, and a 20 mm diameter object at μa = 0.02 mm-1. The images in the left column represent the test field, while the middle row of images show the full reflectance and transmittance images, and the right column of images show the fan-beam images.

Fig. 14.
Fig. 14.

Reconstructions of the test fields shown on in the left column using the sub-surface imaging geometires, including the curved boundary with (upper middle image) 4 sources and 4 detectors, (upper right image) 8 sources and 8 detectors in an arc, and the slab boundary (lower right image) 16 sources and 16 detectors in a flat plane along the surface.

Fig. 15.
Fig. 15.

Reconstructed simulations of a human cranium with the test field (upper left), showing absorption coefficients of skin, μa = 0.01, skull, μa = 0.025, grey matter, μa = 0.02, and white matter, μa = 0.01 mm-1, with a fixed scattering coefficient of μ′s = 1.0 mm-1. An artificial inclusion having μa = 0.03 was included in the upper left of the test field. The reconstructed images are shown using full reflectance and transmittance imaging (upper right), fan beam imaging (lower left), and 8×8 sub-surface imaging with the detectors in an arc around the upper half of the head (lower right). Note that the sub-surface geometry is not expected to recover properties in the lower half of the image.

Equations (5)

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· D ( r ) Φ ( r , ω ) [ μ a ( r ) + i ( ω c ) ] Φ ( r , ω ) = S 0 ( r , ω )
Φ ( ξ ) + k Φ ( ξ , ω ) · n ̂ = 0
χ ( μ ) = ( ϕ C ϕ O ) T ( ϕ C ϕ O ) + λ F ( μ )
0 = ( ϕ C ϕ O ) T J + Δ μ ( J T J + λ I ) +
Δ μ = ( ϕ C ϕ O ) T J [ J T J + λ I ] 1

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