Abstract

We describe a novel Monte Carlo code for photon migration through 3D media with spatially varying optical properties. The code is validated against analytic solutions of the photon diffusion equation for semi-infinite homogeneous media. The code is also cross-validated for photon migration through a slab with an absorbing heterogeneity. A demonstration of the utility of the code is provided by showing time-resolved photon migration through a human head. This code, known as ‘tMCimg’, is available on the web and can serve as a resource for solving the forward problem for complex 3D structural data obtained by MRI or CT.

© 2002 Optical Society of America

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    [Crossref]
  4. M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, Kidney D., J. Butler, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical mammography with ultrasound localization in a human subject,” J Biomed Opt 5, 237–47. (2000).
    [Crossref] [PubMed]
  5. D. A. Benaron, W. F. Cheong, and D. K. Stevenson, “Tissue Optics,” Science 276, 2002–2003 (1997).
    [Crossref] [PubMed]
  6. D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W. F. Cheong, and D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J Cereb Blood Flow Metab 20, 469–77 (2000).
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    [PubMed]
  8. V. Ntziachristos and B. Chance, “Probing physiology and molecular function using optical imaging: applications to breast cancer,” Breast Cancer Res 3, 41–6 (2001).
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  9. V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc Natl Acad Sci U S A 97, 2767–72 (2000).
    [Crossref] [PubMed]
  10. M. A. Franceschini, V. Toronov, M. Filiaci, E. Gratton, and S. Fanini, “On-line optical imaging of the human brain with 160-ms temporal resolution,” Opt. Express 6, 49–57 (2000). http://www.opticsexpress.org/oearchive/source/18957.htm
    [Crossref] [PubMed]
  11. S. R. Hintz, D. A. Benaron, A. M. Siegel, A. Zourabian, D. K. Stevenson, and D. A. Boas, “Bedside functional imaging of the premature infant brain during passive motor activation,” J Perinat Med 29, 335–43 (2001).
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    [Crossref] [PubMed]
  13. C. S. Robertson, S. P. Gopinath, and B. Chance, “A new application for near-infrared spectroscopy: Detection of delayed intracranial hematomas after head injury,” Journal of Neurotrauma 12, 591–600 (1995).
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  14. S. R. Hintz, W. F. Cheong, J. P. van Houten, D. K. Stevenson, and D. A. Benaron, “Bedside imaging of intracranial hemorrhage in the neonate using light: comparison with ultrasound, computed tomography, and magnetic resonance imaging,” Pediatr Res 45, 54–9. (1999).
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    [Crossref] [PubMed]
  21. M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys Med Biol 41, 767–83. (1996).
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  22. E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope, and D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. 36, 21–31 (1997).
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    [Crossref] [PubMed]
  25. S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, “The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions,” Med. Phys. 27, 252–64. (2000).
    [Crossref] [PubMed]
  26. J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J Opt Soc Am A Opt Image Sci Vis 17, 1671–81. (2000).
    [Crossref] [PubMed]
  27. O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Problems 14, 1107–1130 (1998).
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  28. J. J. Stott and D. A. Boas, tMCimg: Monte Carlo code for photon migration through general 3D Media. http://www.nmr.mgh.harvard.edu/DOT
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    [Crossref]
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    [Crossref] [PubMed]
  35. T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
    [Crossref] [PubMed]
  36. A. Kienle and M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from semi-infinite turbid medium,” Journal of the Optical Society of America 14, 246–254 (1997).
    [Crossref] [PubMed]
  37. K. Furutsu and Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys.Rev.E 50, 3634 (1994).
    [Crossref]
  38. T. Durduran, B. Chance, A. G. Yodh, and D. A. Boas, “Does the photon diffusion coefficient depend on absorption?,” J. Opt. Soc. Am A 14, 3358–3365 (1997).
    [Crossref]
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  40. 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, 1531–60 (1992).
    [Crossref] [PubMed]
  41. S. R. Arridge and M. Schweiger, “Photon-measurement density functions. Part2: Finite-element-method calculations,” Appl. Opt. 34, 8026–8037 (1995).
    [Crossref] [PubMed]
  42. S. R. Arridge and M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express 2, 213–226 (1998). http://www.opticsexpress.org/oearchive/source/4014.htm
    [Crossref] [PubMed]
  43. A. M. Dale, B. Fischl, and M. I. Sereno, “Cortical surface-based analysis. I. Segmentation and surface reconstruction,” Neuroimage 9, 179–94 (1999).
    [Crossref] [PubMed]

2001 (5)

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, K. S. Osterman, U. L. Osterberg, and K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–6. (2001).
[PubMed]

V. Ntziachristos and B. Chance, “Probing physiology and molecular function using optical imaging: applications to breast cancer,” Breast Cancer Res 3, 41–6 (2001).
[Crossref] [PubMed]

S. R. Hintz, D. A. Benaron, A. M. Siegel, A. Zourabian, D. K. Stevenson, and D. A. Boas, “Bedside functional imaging of the premature infant brain during passive motor activation,” J Perinat Med 29, 335–43 (2001).
[Crossref] [PubMed]

A. Bluestone, G. Abdoulaev, C. Schmitz, R. Barbour, and A. Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Express 9, 272–286 (2001). http://www.opticsexpress.org/oearchive/source/34858.htm
[Crossref] [PubMed]

C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26, 1335–1337 (2001).
[Crossref]

2000 (6)

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, “The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions,” Med. Phys. 27, 252–64. (2000).
[Crossref] [PubMed]

J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J Opt Soc Am A Opt Image Sci Vis 17, 1671–81. (2000).
[Crossref] [PubMed]

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W. F. Cheong, and D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J Cereb Blood Flow Metab 20, 469–77 (2000).
[Crossref] [PubMed]

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc Natl Acad Sci U S A 97, 2767–72 (2000).
[Crossref] [PubMed]

M. A. Franceschini, V. Toronov, M. Filiaci, E. Gratton, and S. Fanini, “On-line optical imaging of the human brain with 160-ms temporal resolution,” Opt. Express 6, 49–57 (2000). http://www.opticsexpress.org/oearchive/source/18957.htm
[Crossref] [PubMed]

M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, Kidney D., J. Butler, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical mammography with ultrasound localization in a human subject,” J Biomed Opt 5, 237–47. (2000).
[Crossref] [PubMed]

1999 (5)

Q. Zhu, T. Durduran, V. Ntziachristos, M. Holboke, and A. G. Yodh, “Imager that combines near-infrared diffusive light and ultrasound,” Opt. Lett. 24, 1050–1052 (1999).
[Crossref]

M. Schweiger and S. R. Arridge, “Optical tomographic reconstruction in a complex head model using a priori region boundary information,” Phys Med Biol 44, 2703–21 (1999).
[Crossref] [PubMed]

A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26, 1698–707. (1999).
[Crossref] [PubMed]

S. R. Hintz, W. F. Cheong, J. P. van Houten, D. K. Stevenson, and D. A. Benaron, “Bedside imaging of intracranial hemorrhage in the neonate using light: comparison with ultrasound, computed tomography, and magnetic resonance imaging,” Pediatr Res 45, 54–9. (1999).
[Crossref] [PubMed]

A. M. Dale, B. Fischl, and M. I. Sereno, “Cortical surface-based analysis. I. Segmentation and surface reconstruction,” Neuroimage 9, 179–94 (1999).
[Crossref] [PubMed]

1998 (5)

S. R. Arridge and M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express 2, 213–226 (1998). http://www.opticsexpress.org/oearchive/source/4014.htm
[Crossref] [PubMed]

M. Firbank, Okada E., and D. T. Delpy, “A theoretical study of the signal contribution of regions of the adult head to near-infrared spectroscopy studies of visual evoked responses,” Neuroimage 8, 69–78. (1998).
[Crossref] [PubMed]

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys Med Biol 43, 1285–302. (1998).
[Crossref] [PubMed]

O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Problems 14, 1107–1130 (1998).
[Crossref]

B. W. Pogue and K. D. Paulsen, “High-resolution near-infrared tomographic imaging simulations of the rat cranium by use of a priori magnetic resonance imaging structural information,” Opt. Lett. 23, 1716–1718 (1998).
[Crossref]

1997 (5)

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

D. A. Benaron, W. F. Cheong, and D. K. Stevenson, “Tissue Optics,” Science 276, 2002–2003 (1997).
[Crossref] [PubMed]

E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope, and D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. 36, 21–31 (1997).
[Crossref] [PubMed]

A. Kienle and M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from semi-infinite turbid medium,” Journal of the Optical Society of America 14, 246–254 (1997).
[Crossref] [PubMed]

T. Durduran, B. Chance, A. G. Yodh, and D. A. Boas, “Does the photon diffusion coefficient depend on absorption?,” J. Opt. Soc. Am A 14, 3358–3365 (1997).
[Crossref]

1996 (2)

E. Okada, M. Schweiger, S. R. Arridge, M. Firbank, and D. T. Delpy, “Experimental validation of Monte Carlo and finite-element methods of estimation of the optical path length in inhomogeneous tissue,” Appl. Opt. 35, 3362–3371 (1996).
[Crossref] [PubMed]

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys Med Biol 41, 767–83. (1996).
[Crossref] [PubMed]

1995 (5)

1994 (2)

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

K. Furutsu and Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys.Rev.E 50, 3634 (1994).
[Crossref]

1992 (2)

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, 1531–60 (1992).
[Crossref] [PubMed]

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[Crossref] [PubMed]

1989 (1)

Abdoulaev, G.

Alcouffe, R. E.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys Med Biol 43, 1285–302. (1998).
[Crossref] [PubMed]

Arridge, S. R.

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, “The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions,” Med. Phys. 27, 252–64. (2000).
[Crossref] [PubMed]

J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J Opt Soc Am A Opt Image Sci Vis 17, 1671–81. (2000).
[Crossref] [PubMed]

M. Schweiger and S. R. Arridge, “Optical tomographic reconstruction in a complex head model using a priori region boundary information,” Phys Med Biol 44, 2703–21 (1999).
[Crossref] [PubMed]

S. R. Arridge and M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express 2, 213–226 (1998). http://www.opticsexpress.org/oearchive/source/4014.htm
[Crossref] [PubMed]

E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope, and D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. 36, 21–31 (1997).
[Crossref] [PubMed]

E. Okada, M. Schweiger, S. R. Arridge, M. Firbank, and D. T. Delpy, “Experimental validation of Monte Carlo and finite-element methods of estimation of the optical path length in inhomogeneous tissue,” Appl. Opt. 35, 3362–3371 (1996).
[Crossref] [PubMed]

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys Med Biol 41, 767–83. (1996).
[Crossref] [PubMed]

S. R. Arridge and M. Schweiger, “Photon-measurement density functions. Part2: Finite-element-method calculations,” Appl. Opt. 34, 8026–8037 (1995).
[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, 1531–60 (1992).
[Crossref] [PubMed]

Barbour, R.

Barbour, R. L.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys Med Biol 43, 1285–302. (1998).
[Crossref] [PubMed]

Benaron, D. A.

S. R. Hintz, D. A. Benaron, A. M. Siegel, A. Zourabian, D. K. Stevenson, and D. A. Boas, “Bedside functional imaging of the premature infant brain during passive motor activation,” J Perinat Med 29, 335–43 (2001).
[Crossref] [PubMed]

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W. F. Cheong, and D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J Cereb Blood Flow Metab 20, 469–77 (2000).
[Crossref] [PubMed]

S. R. Hintz, W. F. Cheong, J. P. van Houten, D. K. Stevenson, and D. A. Benaron, “Bedside imaging of intracranial hemorrhage in the neonate using light: comparison with ultrasound, computed tomography, and magnetic resonance imaging,” Pediatr Res 45, 54–9. (1999).
[Crossref] [PubMed]

D. A. Benaron, W. F. Cheong, and D. K. Stevenson, “Tissue Optics,” Science 276, 2002–2003 (1997).
[Crossref] [PubMed]

Bevilacqua, F.

Bluestone, A.

Boas, D.

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W. F. Cheong, and D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J Cereb Blood Flow Metab 20, 469–77 (2000).
[Crossref] [PubMed]

Boas, D. A.

S. R. Hintz, D. A. Benaron, A. M. Siegel, A. Zourabian, D. K. Stevenson, and D. A. Boas, “Bedside functional imaging of the premature infant brain during passive motor activation,” J Perinat Med 29, 335–43 (2001).
[Crossref] [PubMed]

T. Durduran, B. Chance, A. G. Yodh, and D. A. Boas, “Does the photon diffusion coefficient depend on absorption?,” J. Opt. Soc. Am A 14, 3358–3365 (1997).
[Crossref]

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

J. J. Stott and D. A. Boas, tMCimg: Monte Carlo code for photon migration through general 3D Media. http://www.nmr.mgh.harvard.edu/DOT

Butler, J.

M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, Kidney D., J. Butler, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical mammography with ultrasound localization in a human subject,” J Biomed Opt 5, 237–47. (2000).
[Crossref] [PubMed]

Chance, B.

V. Ntziachristos and B. Chance, “Probing physiology and molecular function using optical imaging: applications to breast cancer,” Breast Cancer Res 3, 41–6 (2001).
[Crossref] [PubMed]

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc Natl Acad Sci U S A 97, 2767–72 (2000).
[Crossref] [PubMed]

M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, Kidney D., J. Butler, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical mammography with ultrasound localization in a human subject,” J Biomed Opt 5, 237–47. (2000).
[Crossref] [PubMed]

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

T. Durduran, B. Chance, A. G. Yodh, and D. A. Boas, “Does the photon diffusion coefficient depend on absorption?,” J. Opt. Soc. Am A 14, 3358–3365 (1997).
[Crossref]

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

C. S. Robertson, S. P. Gopinath, and B. Chance, “A new application for near-infrared spectroscopy: Detection of delayed intracranial hematomas after head injury,” Journal of Neurotrauma 12, 591–600 (1995).
[Crossref] [PubMed]

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

Cheong, W. F.

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S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, “The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions,” Med. Phys. 27, 252–64. (2000).
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Supplementary Material (1)

» Media 1: AVI (1000 KB)     

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Figures (7)

Figure 1.
Figure 1.

(A) A comparison of the flux of photons remitted from a semi-infinite medium as calculated by diffusion theory and Monte Carlo. (B) A comparison of the photon fluence within a semi-infinite medium. The index-matched surface is at a depth of 0 mm.

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Figure 2.
Figure 2.

(A) A comparison of diffusion theory and Monte Carlo for the temporal response to a pulse of light as measured on the surface and within the medium. The black points is for the remitted flux of light 15 mm from the source. The red points indicate the fluence within the medium at a depth of 10 mm and displaced laterally 15 mm. (B) The comparison of the iso-contours for diffusion theory and Monte Carlo within the medium at 0.1, 0.5, 1.0, 1.5, and 2.0 ns after the pulse of light.

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Figure 3.
Figure 3.

(A) The geometry for the Monte Carlo simulation with an absorbing inclusion. (B) The relative decrease in the detected photon flux is shown as the absorption coefficient of the inclusion is increased. A comparison of 4 methods is made. See text for details.

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Figure 4.
Figure 4.

The photon fluence within a 40 mm thick homogeneous slab is shown in (A). The change in fluence due to an absorbing inclusion with μa = 0.025 mm-1 relative to

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Figure 5.
Figure 5.

The photon sensitivity profile for a source-detector pair separated by 3 cm. Profiles are given for (A) a continuous-wave measurement, (B) an amplitude measurement at 200 MHz, and time-gated measurements at (C) 500 ps and (D) 2000 ps.

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Figure 6.
Figure 6.

A movie of the propagation of a pulse of light through a 3D human head. The color scale is logarithmic and spans 10 orders of magnitude from a peak in the dark red to a minimum in the dark blue. The AVI movie file size is 1.0 Mega-Bytes.

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Figure 7.
Figure 7.

An estimate of the SNR by running 108 photons on the 3D human head model.

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Equations (7)

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Σ surface J out ( r i ) A i + Σ volume Φ ( r i ) μ a ( r i ) V voxel = 1
Φ ( t ) = 1 N photons ( t ) Δ t ∑ i = 1 N photons ( t ) ∏ j = 1 N regions exp ( − μ a , j L i , j )
Φ ( r s , r d ) = v S 4 π D [ exp ⁡ ( − 3 μ s ' μ a ∣ r s − r d ∣ ) ∣ r s − r d ∣ − exp ⁡ ( − 3 μ s ' μ a ∣ r s , i − r d ∣ ) ∣ r s , i − r d ∣ ]
Φ ( r s , r d , t ) = v S ( 4 π D t ) 3 / 2 [ exp ⁡ ( − ∣ r s − r d ∣ 2 ( 4 D t ) ) − exp ⁡ ( − ∣ r s , i − r d ∣ 2 ( 4 D t ) ) ] exp ⁡ ( − ν μ a t ) .
Φ = Φ o + Φ pert
Φ = Φ o exp ⁡ ( Φ pert ) .
Φ pert ( r s , r d ) = − 1 Φ o ( r s , r d ) ∫ Φ o ( r s , r d ) ν δ μ a ( r ) D o G ( r , r d ) d r .

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