Abstract

Modeling of the full temporal behavior of photons propagating in diffusive materials is computationally costly. Rather than deriving intensity as a function of time to fine sampling, we may consider methods that derive a transform of this function. To derive the Fourier transform involves calculation in the (complex) frequency domain and relates to intensity-modulated experiments. We consider instead the Mellin transform and show that this relates to the moments of the original temporal distribution. A derivation of the Mellin transform given the Fourier transform that permits closed-form derivations of the temporal moments for various simple geometries is presented. For general geometries a finite-element method is presented, and it is demonstrated that the computational cost to produce the nth moment is the same as producing the first n temporal samples of the original function.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurment,” Phys. Med. Biol. 33, 1433–1442 (1988).
    [CrossRef] [PubMed]
  2. B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohne, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Nat. Acad. Sci. USA 85, 4971–4975 (1985).
    [CrossRef]
  3. S. J. Madsen, M. S. Patterson, B. C. Wilson, Y. D. Park, J. D. Moulton, S. L. Jaques, Y. Hefetz, “Time-resolved diffuse relectance and transmittance studies in tissue simulating phantoms: a comparison between theory and experiment,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 42–51 (1991).
  4. S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204–215 (1991).
  5. B. C. Wilson, G. Adam, “A Monte-Carlo model for the absorption and flux distribution of light in tissue,” Med. Phys. 10, 824–830 (1983).
    [CrossRef] [PubMed]
  6. P. van der Zee, D. T. Delpy, “Simulation of the point-spread function for light in tissue,” Adv. Exp. Med. Biol. 215, 179–192 (1987).
  7. S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues—I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
    [CrossRef] [PubMed]
  8. R. F. Bonner, R. Nossal, R. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
    [CrossRef] [PubMed]
  9. S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlength in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
    [CrossRef] [PubMed]
  10. S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
    [CrossRef] [PubMed]
  11. S. R. Arridge, M. Schweiger, “The use of multiple data types in Time-resolved Optical Absorption and Scattering Tomography (TOAST),” in Mathematical Methods in Medical Imaging II, D. C. Wilson, J. N. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2035, 218–229 (1993).
  12. J. P. Kaltenbach, M. Kaschke, “Frequency- and time-domain modeling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Karchke, B. Masters, S. Svonberg, P. Van der Zee, eds. (Society of Photo-Optical and Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.
  13. R. A. J. Groenhuis, H. A. Ferwada, J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements (parts 1 and 2),” Appl. Opt. 22, 2456–2467 (1983).
    [CrossRef] [PubMed]
  14. M. S. Patterson, B. Chance, 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]
  15. G. Strang, G. J. Fix, An Analysis of the Finite Element Method (Prentice-Hall, Englewood Cliffs, N.J., 1973).

1993 (1)

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

1992 (1)

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

1989 (2)

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues—I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, 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]

1988 (1)

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurment,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

1987 (2)

R. F. Bonner, R. Nossal, R. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
[CrossRef] [PubMed]

P. van der Zee, D. T. Delpy, “Simulation of the point-spread function for light in tissue,” Adv. Exp. Med. Biol. 215, 179–192 (1987).

1985 (1)

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohne, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Nat. Acad. Sci. USA 85, 4971–4975 (1985).
[CrossRef]

1983 (2)

Adam, G.

B. C. Wilson, G. Adam, “A Monte-Carlo model for the absorption and flux distribution of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Arridge, S. R.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

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

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurment,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204–215 (1991).

S. R. Arridge, M. Schweiger, “The use of multiple data types in Time-resolved Optical Absorption and Scattering Tomography (TOAST),” in Mathematical Methods in Medical Imaging II, D. C. Wilson, J. N. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2035, 218–229 (1993).

Bonner, R. F.

Boretsky, R.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohne, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Nat. Acad. Sci. USA 85, 4971–4975 (1985).
[CrossRef]

Chance, B.

M. S. Patterson, B. Chance, 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]

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohne, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Nat. Acad. Sci. USA 85, 4971–4975 (1985).
[CrossRef]

Cohne, P.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohne, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Nat. Acad. Sci. USA 85, 4971–4975 (1985).
[CrossRef]

Cope, M.

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

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurment,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204–215 (1991).

Delpy, D. T.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

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

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurment,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

P. van der Zee, D. T. Delpy, “Simulation of the point-spread function for light in tissue,” Adv. Exp. Med. Biol. 215, 179–192 (1987).

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204–215 (1991).

Ferwada, H. A.

Finander, M.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohne, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Nat. Acad. Sci. USA 85, 4971–4975 (1985).
[CrossRef]

Fix, G. J.

G. Strang, G. J. Fix, An Analysis of the Finite Element Method (Prentice-Hall, Englewood Cliffs, N.J., 1973).

Flock, S. T.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues—I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

Greenfeld, R.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohne, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Nat. Acad. Sci. USA 85, 4971–4975 (1985).
[CrossRef]

Groenhuis, R. A. J.

Havlin, R.

Hefetz, Y.

S. J. Madsen, M. S. Patterson, B. C. Wilson, Y. D. Park, J. D. Moulton, S. L. Jaques, Y. Hefetz, “Time-resolved diffuse relectance and transmittance studies in tissue simulating phantoms: a comparison between theory and experiment,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 42–51 (1991).

Hiraoka, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

Jaques, S. L.

S. J. Madsen, M. S. Patterson, B. C. Wilson, Y. D. Park, J. D. Moulton, S. L. Jaques, Y. Hefetz, “Time-resolved diffuse relectance and transmittance studies in tissue simulating phantoms: a comparison between theory and experiment,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 42–51 (1991).

Kaltenbach, J. P.

J. P. Kaltenbach, M. Kaschke, “Frequency- and time-domain modeling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Karchke, B. Masters, S. Svonberg, P. Van der Zee, eds. (Society of Photo-Optical and Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.

Kaschke, M.

J. P. Kaltenbach, M. Kaschke, “Frequency- and time-domain modeling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Karchke, B. Masters, S. Svonberg, P. Van der Zee, eds. (Society of Photo-Optical and Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.

Kaufman, K.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohne, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Nat. Acad. Sci. USA 85, 4971–4975 (1985).
[CrossRef]

Leigh, J. S.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohne, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Nat. Acad. Sci. USA 85, 4971–4975 (1985).
[CrossRef]

Levy, W.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohne, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Nat. Acad. Sci. USA 85, 4971–4975 (1985).
[CrossRef]

Madsen, S. J.

S. J. Madsen, M. S. Patterson, B. C. Wilson, Y. D. Park, J. D. Moulton, S. L. Jaques, Y. Hefetz, “Time-resolved diffuse relectance and transmittance studies in tissue simulating phantoms: a comparison between theory and experiment,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 42–51 (1991).

Miyake, H.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohne, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Nat. Acad. Sci. USA 85, 4971–4975 (1985).
[CrossRef]

Moulton, J. D.

S. J. Madsen, M. S. Patterson, B. C. Wilson, Y. D. Park, J. D. Moulton, S. L. Jaques, Y. Hefetz, “Time-resolved diffuse relectance and transmittance studies in tissue simulating phantoms: a comparison between theory and experiment,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 42–51 (1991).

Nioka, S.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohne, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Nat. Acad. Sci. USA 85, 4971–4975 (1985).
[CrossRef]

Nossal, R.

Park, Y. D.

S. J. Madsen, M. S. Patterson, B. C. Wilson, Y. D. Park, J. D. Moulton, S. L. Jaques, Y. Hefetz, “Time-resolved diffuse relectance and transmittance studies in tissue simulating phantoms: a comparison between theory and experiment,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 42–51 (1991).

Patterson, M. S.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues—I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, 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]

S. J. Madsen, M. S. Patterson, B. C. Wilson, Y. D. Park, J. D. Moulton, S. L. Jaques, Y. Hefetz, “Time-resolved diffuse relectance and transmittance studies in tissue simulating phantoms: a comparison between theory and experiment,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 42–51 (1991).

Schweiger, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, “The use of multiple data types in Time-resolved Optical Absorption and Scattering Tomography (TOAST),” in Mathematical Methods in Medical Imaging II, D. C. Wilson, J. N. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2035, 218–229 (1993).

Smith, D. S.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohne, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Nat. Acad. Sci. USA 85, 4971–4975 (1985).
[CrossRef]

Strang, G.

G. Strang, G. J. Fix, An Analysis of the Finite Element Method (Prentice-Hall, Englewood Cliffs, N.J., 1973).

Ten Bosch, J. J.

van der Zee, P.

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurment,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

P. van der Zee, D. T. Delpy, “Simulation of the point-spread function for light in tissue,” Adv. Exp. Med. Biol. 215, 179–192 (1987).

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204–215 (1991).

Weiss, G. H.

Wilson, B. C.

M. S. Patterson, B. Chance, 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]

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues—I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

B. C. Wilson, G. Adam, “A Monte-Carlo model for the absorption and flux distribution of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

S. J. Madsen, M. S. Patterson, B. C. Wilson, Y. D. Park, J. D. Moulton, S. L. Jaques, Y. Hefetz, “Time-resolved diffuse relectance and transmittance studies in tissue simulating phantoms: a comparison between theory and experiment,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 42–51 (1991).

Wray, S.

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurment,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Wyatt, J.

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurment,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Wyman, D. R.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues—I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

Yoshioka, H.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohne, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Nat. Acad. Sci. USA 85, 4971–4975 (1985).
[CrossRef]

Young, M.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohne, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Nat. Acad. Sci. USA 85, 4971–4975 (1985).
[CrossRef]

Adv. Exp. Med. Biol. (1)

P. van der Zee, D. T. Delpy, “Simulation of the point-spread function for light in tissue,” Adv. Exp. Med. Biol. 215, 179–192 (1987).

Appl. Opt. (2)

IEEE Trans. Biomed. Eng. (1)

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues—I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

Med. Phys. (2)

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

B. C. Wilson, G. Adam, “A Monte-Carlo model for the absorption and flux distribution of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Phys. Med. Biol. (2)

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

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurment,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

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

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohne, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Nat. Acad. Sci. USA 85, 4971–4975 (1985).
[CrossRef]

Other (5)

S. J. Madsen, M. S. Patterson, B. C. Wilson, Y. D. Park, J. D. Moulton, S. L. Jaques, Y. Hefetz, “Time-resolved diffuse relectance and transmittance studies in tissue simulating phantoms: a comparison between theory and experiment,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 42–51 (1991).

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204–215 (1991).

S. R. Arridge, M. Schweiger, “The use of multiple data types in Time-resolved Optical Absorption and Scattering Tomography (TOAST),” in Mathematical Methods in Medical Imaging II, D. C. Wilson, J. N. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2035, 218–229 (1993).

J. P. Kaltenbach, M. Kaschke, “Frequency- and time-domain modeling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Karchke, B. Masters, S. Svonberg, P. Van der Zee, eds. (Society of Photo-Optical and Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.

G. Strang, G. J. Fix, An Analysis of the Finite Element Method (Prentice-Hall, Englewood Cliffs, N.J., 1973).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Γ(t) at an angle of 86° from δ function source on a two-dimensional circle, radius 25 mm, μ a = 0.025 mm−1, μ s ′ = 2.0 mm−1.

Fig. 2
Fig. 2

First four weighted integrals of the TPSF, plotted as a function of angle, for a two-dimensional circle, radius 25 mm, μ a = 0.025 mm−1, μ s ′ = 2 mm−1. Solid curve, FEM; dashed curve, analytic method.

Fig. 3
Fig. 3

Relative difference of (direct minus numerically integrated) results for first four weighted integrals of the TPSF for the same parameters as in Fig. 2.

Tables (1)

Tables Icon

Table 1 Time in Seconds for FEM Calculationsa

Equations (20)

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

Γ * ( s ) = M [ Γ ( t ) ; s ] = 0 t s - 1 Γ ( t ) d t .
t n = 0 t n Γ ( t ) d t 0 Γ ( t ) d t = Γ * ( n + 1 ) Γ * ( 1 ) , σ 0 = Γ * ( 1 ) ,             σ 1 = t , σ n = 0 ( t - t ) n Γ ( t ) d t 0 Γ ( t ) d t = j = 0 n ( - 1 ) ( n - j ) ( n j ) t j σ 1 ( n - j )             ( n > 1 ) .
{ · κ ( r ) - μ a ( r ) c - t } Φ ( r , t ) = - q 0 ( r , t ) ,
κ ( r ) = c 3 [ μ a ( r ) + μ s ( r ) ] ,
Γ ( ξ ; t ) = - κ ( ξ ) n Φ ( r , t ) Ω ( ξ ) = - κ ( ξ ) n ^ ( ξ ) · Φ ( r , t ) Ω ( ξ ) ,
[ K ( κ ) + C ( μ a c ) ] Φ ( t ) + B Φ ( t ) t = Q ( t ) ,
Φ ( t ) = [ Φ 1 ( t ) , Φ 2 ( t ) , Φ D ( t ) ] T .
[ θ K ( κ ) + θ C ( μ a c ) + 1 Δ t B ] Φ n + 1 + [ ( 1 - θ ) K ( κ ) + ( 1 - θ ) C ( μ a c ) - 1 Δ t B ] Φ n = θ Q n + 1 + ( 1 - θ ) Q n ,
m n = g * ( n + 1 ) = M [ g ( t ) ; n + 1 ] = 0 t n g ( t ) d t .
- g ( t ) d t = 2 π G ^ ( ω ) ω = 0 ,
n ω n G ^ ( ω ) = 1 2 π n ω n - g ( t ) exp ( - i ω t ) d t = ( - i ) n 2 π - t n g ( t ) exp ( - i ω t ) d t .
m n = i n 2 π n ω n G ^ ( ω ) ω = 0 ,
t n = m n m 0 .
{ · κ - μ a c } Φ ( r ) = - q 0 ( r ) .
[ K ( κ ) + C ( μ a c ) ] Φ = Q ,
[ K ( κ ) + C ( μ a c ) + i ω B ] Φ ^ ( ω ) = Q ^ ( ω ) .
[ K ( κ ) + C ( μ a c ) + i ω B ] n Φ ^ ( ω ) ω n + i n B n - 1 Φ ^ ( ω ) ω n - 1 = n Q ^ ( ω ) ω n .
[ K ( κ ) + C ( μ a c ) ] n Φ ^ ( ω ) ω n | ω = 0 = - i n B n - 1 Φ ^ ( ω ) ω n - 1 | ω = 0 ,
m n = n [ K ( κ ) + C ( μ a c ) ] - 1 B m n - 1 .
G ^ ( r , ξ , ω ) = 1 ( 2 π ) 3 / 2 a n = - Cos ( n θ ) I n ( α r ) I n ( α a ) ,

Metrics