Abstract

This paper addresses the problem of tomographic reconstruction of absorption and scattering parameters in the optical region from measurements of transilluminated light. Specifically, the question of the sensitivity of different measurement schemes on the boundary of an object to perturbations of the optical parameters within the object are addressed. The concept of a photon-sampling volume [Appl. Opt. 33, 448 (1994)] and a photon-hitting density [Appl. Opt. 32, 448 (1993)] is extended to a photon-measurement density function (PMDF). The PMDF is derived from the Green’s function of the diffusion equation and can be expressed for measurements such as the time-varying intensity, integrated intensity, temporal moments, and phase shift, as well as for both absorption and diffusion perturbations. Closed-form solutions are given for a number of these functions in infinite space, half-space, and slab geometries.

Example results are given in terms of three-dimensional images.

© 1995 Optical Society of America

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  2. S. R. Arridge, P. van der Zee, D. T. Delpy, M. Cope, “Aspects of clinical infra red absorption imaging,” in Medical Images: Formation, Handling, and Evaluation, Vol. 98 of NATO Advanced Study Institute Imaging SeriesF A. Todd Pokropek, M. A. Viergever, eds., (Springer-Verlag, Heidelberg, Germany, 1992), pp. 407–418.
  3. S. R. Arridge, P. van der Zee, M. Cope, D.T. Delpy “Reconstruction methods for infrared absorption imaging,” Time-Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204–215 (1991).
  4. S. R. Arridge, M. Schweiger, D. T. Delpy, “Iterative reconstruction of near infra-red absorption images,” in Inverse Problems in Scattering and Imaging, M.A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1767, 372–383 (1982).
  5. G. Jarry, S. Ghesquiere, J. M. Maarek, S. Debray, Bui-Mong-Hung, D. Laurent, “Imaging mammalian tissues and organs using laser collimated transillumination,” J. Biomed. Eng. 6, 70–74 (1984).
    [CrossRef] [PubMed]
  6. P. C. Jackson, P. H. Stevens, J. H. Smith, D. Kean, H. Key, P. N. T. Wells, “The development of a system for transillumination computed tomography,” Br. J. Rad. 60, 375–380 (1987).
    [CrossRef]
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  11. R. Berg, S. Andersson-Engels, O. Jarlman, S. Svanberg, “Time-resolved transillumination for medical diagnostics,” in Time-Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 110–119 (1991).
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    [CrossRef]
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    [CrossRef] [PubMed]
  21. 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. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
    [CrossRef] [PubMed]
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    [CrossRef]
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  26. 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]
  27. 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).
  28. 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]
  29. 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]
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  35. E. M. Sevick, C. L. Burch, J. K. Frisoli, J. R. Lakowicz, “Localization of absorbers in scattering media by use of frequency–domain measurements of time-dependent photon migration,” Appl. Opt. 33, 3562–3571 (1994).
    [CrossRef] [PubMed]
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  38. A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply embedded in tissues,” Med. Phys. 21, 185–191 (1994).
    [CrossRef] [PubMed]
  39. A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Theoretical study of resolution limits for time-resolved imaging of human breast,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases, R. R. Alfano, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2135, 176–185 (1994).
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  41. R. A. J. Groenhuis, H. A. Ferwerda, J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory,” Appl. Opt. 22, 2456–2462 (1983); R. A. J. Groenhuis, J. J. Ten Bosch, H. A. Ferwerda, “Scattering and absorption of turbid materials determined from reflection measurements. 2: Measuring method and calibration,” Appl. Opt. 22, 2463–2467 (1983).
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  45. S. R. Arridge, M. Schweiger, “Reconstruction in optical tomography using MRI based prior knowledge,” in Information Processing in Medical Imaging ’95, Y. Bizais, C. Barillot, R. di Paola, eds. (Kluwer Academic, Dordrecht, 1995), pp. 77–88.
  46. The program C++ used to generate the photon-measurement density functions, as described in Subsection 6.A is available by writing to S. Arridge, Department of Computer Science, University College of London, London WC1E 6BT, England, or by e-mail at arridge@CS.ucl.ac.uk.
  47. R. A. Robb, C. Barillot, “Interactive display and analysis of 3-D medical images,” IEEE Trans. Med. Imag. 8, 217–226 (1989).
    [CrossRef]

1994 (2)

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply embedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

E. M. Sevick, C. L. Burch, J. K. Frisoli, J. R. Lakowicz, “Localization of absorbers in scattering media by use of frequency–domain measurements of time-dependent photon migration,” Appl. Opt. 33, 3562–3571 (1994).
[CrossRef] [PubMed]

1993 (1)

1992 (1)

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

1991 (2)

T. R. Cheatle, L. A. Potter, M. Cope, D. T. Delpy, P. D. Coleridge-Smith, J. H. Scurr, “Near infra-red spectroscopy in peripheral vascular disease,” Br. J. Surg. 78, 405–408 (1991).
[CrossRef] [PubMed]

M. S. Patterson, J. D. Moulton, B. C. Wilson, K. E. Berndt, J. R. Lakowicz, “Frequency-domain reflectance for the determination of the scattering and optical properties of tissue,” Appl. Opt. 30, 4474–4476 (1991).
[CrossRef] [PubMed]

1990 (3)

J. R. Lakowicz, K. Berndt, “Frequency domain measurement of photon migration in tissues,” Chem. Phys. Lett. 166, 246–252 (1990).
[CrossRef]

J. R. Singer, F. A. Grünbaum, P. D. Kohn, J. P. Zubelli, “Imaging reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infra-red spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
[PubMed]

1989 (3)

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 noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

R. A. Robb, C. Barillot, “Interactive display and analysis of 3-D medical images,” IEEE Trans. Med. Imag. 8, 217–226 (1989).
[CrossRef]

1988 (4)

N. B. Hampson, C. A. Piantadosi, “Near infrared monitoring of human skeletal muscle oxygenation during forearm ischaemia,” J. Appl. Physiol. 64, 2449–2457 (1988).
[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 measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[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. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spectroscopy,” Lancet ii, 770–771 (1988).
[CrossRef]

1987 (4)

P. C. Jackson, P. H. Stevens, J. H. Smith, D. Kean, H. Key, P. N. T. Wells, “The development of a system for transillumination computed tomography,” Br. J. Rad. 60, 375–380 (1987).
[CrossRef]

M. Tamura, Y. Nomura, O. Hazeki, “Laser tissue spectroscopy—near-infrared CT,” Rev. Laser Eng. (Japan) 15, 74–82 (1987).

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).

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]

1986 (1)

J. S. Wyatt, M. Cope, D. T. Delpy, S. Wray, E. O. R. Reynolds, “Quantitation of cerebral oxygenation and haemodynamics in sick newborn infants by near infrared spectroscopy,” Lancet ii, 1063–1066 (1986).
[CrossRef]

1985 (1)

J. E. Brazy, D. V. Lewis, M. H. Mitnick, F. F. Jöbsis van der Vliet, “Noninvasive monitoring of cerebral oxygenation in preterm infants: preliminary observations,” Paediatrics 75, 217–225 (1985).

1984 (1)

G. Jarry, S. Ghesquiere, J. M. Maarek, S. Debray, Bui-Mong-Hung, D. Laurent, “Imaging mammalian tissues and organs using laser collimated transillumination,” J. Biomed. Eng. 6, 70–74 (1984).
[CrossRef] [PubMed]

1983 (2)

Abumi, R.

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abumi, K. Nagai, K. Nakagawa, M. Tamura, “Non-invasive haemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 284–293 (1991).

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]

Andersson-Engels, S.

R. Berg, S. Andersson-Engels, O. Jarlman, S. Svanberg, “Time-resolved transillumination for medical diagnostics,” in Time-Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 110–119 (1991).

Aronson, R.

H. L. Graber, J. Chang, J. Lubowsky, R. Aronson, R. L. Barbour, “Near infrared absorption imaging of dense scattering media by steady-state diffusion tomography,” in Photon Migration and Imaging in Random Media, R. R. Alfano, B. Chance, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1888, 372–386 (1993).

Arridge, S. R.

S. R. Arridge, M. Cope, D. T. Delpy, “Theoretical basis for the determination of optical pathlengths 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 measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

S. R. Arridge, P. van der Zee, D. T. Delpy, M. Cope, “Aspects of clinical infra red absorption imaging,” in Medical Images: Formation, Handling, and Evaluation, Vol. 98 of NATO Advanced Study Institute Imaging SeriesF A. Todd Pokropek, M. A. Viergever, eds., (Springer-Verlag, Heidelberg, Germany, 1992), pp. 407–418.

S. R. Arridge, M. Schweiger, “Reconstruction in optical tomography using MRI based prior knowledge,” in Information Processing in Medical Imaging ’95, Y. Bizais, C. Barillot, R. di Paola, eds. (Kluwer Academic, Dordrecht, 1995), pp. 77–88.

S. R. Arridge, M. Schweiger, D. T. Delpy, “Iterative reconstruction of near infra-red absorption images,” in Inverse Problems in Scattering and Imaging, M.A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1767, 372–383 (1982).

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

S. R. Arridge, M. Cope, P. van der Zee, P. J. Hillson, D. T. Delpy, “Visualisation of the oxygenation state of brain and muscle in newborn infants by near infra-red transillumination,” in Information Processing in Medical Imaging, S. L. Bacharach, ed. (Martinus Nijhoff, Amsterdam, 1985), pp. 155–176.

S. R. Arridge, M. Schweiger, “The use of multiple data types in time-resolved optical absorption and scatter 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).

Barbour, R. L.

H. L. Graber, J. Chang, J. Lubowsky, R. Aronson, R. L. Barbour, “Near infrared absorption imaging of dense scattering media by steady-state diffusion tomography,” in Photon Migration and Imaging in Random Media, R. R. Alfano, B. Chance, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1888, 372–386 (1993).

Barillot, C.

R. A. Robb, C. Barillot, “Interactive display and analysis of 3-D medical images,” IEEE Trans. Med. Imag. 8, 217–226 (1989).
[CrossRef]

Berg, R.

R. Berg, S. Andersson-Engels, O. Jarlman, S. Svanberg, “Time-resolved transillumination for medical diagnostics,” in Time-Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 110–119 (1991).

Berndt, K.

J. R. Lakowicz, K. Berndt, “Frequency domain measurement of photon migration in tissues,” Chem. Phys. Lett. 166, 246–252 (1990).
[CrossRef]

Berndt, K. E.

Bonner, R. F.

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply embedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

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]

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Theoretical study of resolution limits for time-resolved imaging of human breast,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases, R. R. Alfano, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2135, 176–185 (1994).

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. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Brazy, J. E.

J. E. Brazy, D. V. Lewis, M. H. Mitnick, F. F. Jöbsis van der Vliet, “Noninvasive monitoring of cerebral oxygenation in preterm infants: preliminary observations,” Paediatrics 75, 217–225 (1985).

Bui-Mong-Hung,

G. Jarry, S. Ghesquiere, J. M. Maarek, S. Debray, Bui-Mong-Hung, D. Laurent, “Imaging mammalian tissues and organs using laser collimated transillumination,” J. Biomed. Eng. 6, 70–74 (1984).
[CrossRef] [PubMed]

Burch, C. L.

E. M. Sevick, C. L. Burch, J. K. Frisoli, J. R. Lakowicz, “Localization of absorbers in scattering media by use of frequency–domain measurements of time-dependent photon migration,” Appl. Opt. 33, 3562–3571 (1994).
[CrossRef] [PubMed]

E. M. Sevick, C. L. Burch, J. K. Frisoli, M. L. Johnson, K. Nowaczyk, H. Szmacinski, J. R. Lakowicz, “The physical basis of photon migration imaging using frequency–domain measurements,” in Medical Optical Tomography: Functional Imaging and Monitoring, Vol. 11 of SPIE Institute Series, G. Müller, ed. (SPIE Press, Bellingham, Wash., 1993), pp. 485–512.

Carpi, A.

M. Ferrari, E. Zanette, I. Giannini, G. Sideri, C. Fieschi, A. Carpi, “Effects of carotid artery compression test on regional cerebral blood volume, haemoglobin oxygen saturation and cytochrome-c-oxidase redox level in cerebrovascular patients ” in Oxygen Transport to Tissue VIII, I. A. Longmuir, ed. (Plenum, New York, 1986), pp. 213–222.
[CrossRef]

Chance, B.

M. S. Patterson, B. Chance, B. C. Wilson, “Time-resolved reflectance and transmittance for the noninvasive 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. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

B. Chance, M. Maris, J. Sorge, M. Z. Zhang, “A phase modulation system for dual wavelength difference spectroscopy of haemoglobin deoxygenation in tissue,” in Time-Resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1204, 481–491 (1990).

E. M. Sevik, B. Chance, “Photon migration in a model of the head measured using time and frequency domain techniques: potentials of spectroscopy and imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 84–96 (1991).

B. Chance, “Multielement phased arrays for phase modulation imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1888, 254–259 (1993).

J. Haselgrove, J. Leigh, C. Lee, N.-G. Wang, M. Maris, B. Chance, “Monte Carlo and diffusion calculations of photon migration in non-infinite highly scattering media,” in Time Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 30–41 (1991).

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time resolved light scattering measurements to photodynamic therapy dosimetry,” in Photodynamic Therapy: Mechanisms, II, T. J. Dougherty, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1203, 62–75 (1990).

Chang, J.

H. L. Graber, J. Chang, J. Lubowsky, R. Aronson, R. L. Barbour, “Near infrared absorption imaging of dense scattering media by steady-state diffusion tomography,” in Photon Migration and Imaging in Random Media, R. R. Alfano, B. Chance, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1888, 372–386 (1993).

Cheatle, T. R.

T. R. Cheatle, L. A. Potter, M. Cope, D. T. Delpy, P. D. Coleridge-Smith, J. H. Scurr, “Near infra-red spectroscopy in peripheral vascular disease,” Br. J. Surg. 78, 405–408 (1991).
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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. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
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Coleridge-Smith, P. D.

T. R. Cheatle, L. A. Potter, M. Cope, D. T. Delpy, P. D. Coleridge-Smith, J. H. Scurr, “Near infra-red spectroscopy in peripheral vascular disease,” Br. J. Surg. 78, 405–408 (1991).
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Cope, M.

S. R. Arridge, M. Cope, D. T. Delpy, “Theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
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T. R. Cheatle, L. A. Potter, M. Cope, D. T. Delpy, P. D. Coleridge-Smith, J. H. Scurr, “Near infra-red spectroscopy in peripheral vascular disease,” Br. J. Surg. 78, 405–408 (1991).
[CrossRef] [PubMed]

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infra-red spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
[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 measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
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A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spectroscopy,” Lancet ii, 770–771 (1988).
[CrossRef]

J. S. Wyatt, M. Cope, D. T. Delpy, S. Wray, E. O. R. Reynolds, “Quantitation of cerebral oxygenation and haemodynamics in sick newborn infants by near infrared spectroscopy,” Lancet ii, 1063–1066 (1986).
[CrossRef]

S. R. Arridge, P. van der Zee, D. T. Delpy, M. Cope, “Aspects of clinical infra red absorption imaging,” in Medical Images: Formation, Handling, and Evaluation, Vol. 98 of NATO Advanced Study Institute Imaging SeriesF A. Todd Pokropek, M. A. Viergever, eds., (Springer-Verlag, Heidelberg, Germany, 1992), pp. 407–418.

S. R. Arridge, M. Cope, P. van der Zee, P. J. Hillson, D. T. Delpy, “Visualisation of the oxygenation state of brain and muscle in newborn infants by near infra-red transillumination,” in Information Processing in Medical Imaging, S. L. Bacharach, ed. (Martinus Nijhoff, Amsterdam, 1985), pp. 155–176.

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

Debray, S.

G. Jarry, S. Ghesquiere, J. M. Maarek, S. Debray, Bui-Mong-Hung, D. Laurent, “Imaging mammalian tissues and organs using laser collimated transillumination,” J. Biomed. Eng. 6, 70–74 (1984).
[CrossRef] [PubMed]

Delpy, D. T.

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

T. R. Cheatle, L. A. Potter, M. Cope, D. T. Delpy, P. D. Coleridge-Smith, J. H. Scurr, “Near infra-red spectroscopy in peripheral vascular disease,” Br. J. Surg. 78, 405–408 (1991).
[CrossRef] [PubMed]

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infra-red spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
[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 measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spectroscopy,” Lancet ii, 770–771 (1988).
[CrossRef]

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).

J. S. Wyatt, M. Cope, D. T. Delpy, S. Wray, E. O. R. Reynolds, “Quantitation of cerebral oxygenation and haemodynamics in sick newborn infants by near infrared spectroscopy,” Lancet ii, 1063–1066 (1986).
[CrossRef]

S. R. Arridge, P. van der Zee, D. T. Delpy, M. Cope, “Aspects of clinical infra red absorption imaging,” in Medical Images: Formation, Handling, and Evaluation, Vol. 98 of NATO Advanced Study Institute Imaging SeriesF A. Todd Pokropek, M. A. Viergever, eds., (Springer-Verlag, Heidelberg, Germany, 1992), pp. 407–418.

S. R. Arridge, M. Cope, P. van der Zee, P. J. Hillson, D. T. Delpy, “Visualisation of the oxygenation state of brain and muscle in newborn infants by near infra-red transillumination,” in Information Processing in Medical Imaging, S. L. Bacharach, ed. (Martinus Nijhoff, Amsterdam, 1985), pp. 155–176.

S. R. Arridge, M. Schweiger, D. T. Delpy, “Iterative reconstruction of near infra-red absorption images,” in Inverse Problems in Scattering and Imaging, M.A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1767, 372–383 (1982).

Delpy, D.T.

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

Devaney, A. J.

A. B. Weglein, A. J. Devaney, “The inverse source problem in the presence of external sources,” in Inverse Problems in Scattering and Imaging, M. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1767, 170–176 (1992).

Eda, H.

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abumi, K. Nagai, K. Nakagawa, M. Tamura, “Non-invasive haemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 284–293 (1991).

Edwards, A. D.

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infra-red spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
[PubMed]

A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spectroscopy,” Lancet ii, 770–771 (1988).
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Ferrari, M.

M. Ferrari, E. Zanette, I. Giannini, G. Sideri, C. Fieschi, A. Carpi, “Effects of carotid artery compression test on regional cerebral blood volume, haemoglobin oxygen saturation and cytochrome-c-oxidase redox level in cerebrovascular patients ” in Oxygen Transport to Tissue VIII, I. A. Longmuir, ed. (Plenum, New York, 1986), pp. 213–222.
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Ferwerda, H. A.

Fieschi, C.

M. Ferrari, E. Zanette, I. Giannini, G. Sideri, C. Fieschi, A. Carpi, “Effects of carotid artery compression test on regional cerebral blood volume, haemoglobin oxygen saturation and cytochrome-c-oxidase redox level in cerebrovascular patients ” in Oxygen Transport to Tissue VIII, I. A. Longmuir, ed. (Plenum, New York, 1986), pp. 213–222.
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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. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
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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).
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Frisoli, J. K.

E. M. Sevick, C. L. Burch, J. K. Frisoli, J. R. Lakowicz, “Localization of absorbers in scattering media by use of frequency–domain measurements of time-dependent photon migration,” Appl. Opt. 33, 3562–3571 (1994).
[CrossRef] [PubMed]

E. M. Sevick, C. L. Burch, J. K. Frisoli, M. L. Johnson, K. Nowaczyk, H. Szmacinski, J. R. Lakowicz, “The physical basis of photon migration imaging using frequency–domain measurements,” in Medical Optical Tomography: Functional Imaging and Monitoring, Vol. 11 of SPIE Institute Series, G. Müller, ed. (SPIE Press, Bellingham, Wash., 1993), pp. 485–512.

Gandjbakhche, A. H.

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply embedded in tissues,” Med. Phys. 21, 185–191 (1994).
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A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Theoretical study of resolution limits for time-resolved imaging of human breast,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases, R. R. Alfano, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2135, 176–185 (1994).

Ghesquiere, S.

G. Jarry, S. Ghesquiere, J. M. Maarek, S. Debray, Bui-Mong-Hung, D. Laurent, “Imaging mammalian tissues and organs using laser collimated transillumination,” J. Biomed. Eng. 6, 70–74 (1984).
[CrossRef] [PubMed]

Giannini, I.

M. Ferrari, E. Zanette, I. Giannini, G. Sideri, C. Fieschi, A. Carpi, “Effects of carotid artery compression test on regional cerebral blood volume, haemoglobin oxygen saturation and cytochrome-c-oxidase redox level in cerebrovascular patients ” in Oxygen Transport to Tissue VIII, I. A. Longmuir, ed. (Plenum, New York, 1986), pp. 213–222.
[CrossRef]

Graber, H. L.

H. L. Graber, J. Chang, J. Lubowsky, R. Aronson, R. L. Barbour, “Near infrared absorption imaging of dense scattering media by steady-state diffusion tomography,” in Photon Migration and Imaging in Random Media, R. R. Alfano, B. Chance, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1888, 372–386 (1993).

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. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Groenhuis, R. A. J.

Grünbaum, F. A.

J. R. Singer, F. A. Grünbaum, P. D. Kohn, J. P. Zubelli, “Imaging reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

F. A. Grünbaum, P. D. Kohn, G. A. Latham, J. R. Singer, J. P. Zubelli, “Diffuse tomography,” in Time Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 232–238 (1991).

Hampson, N. B.

N. B. Hampson, C. A. Piantadosi, “Near infrared monitoring of human skeletal muscle oxygenation during forearm ischaemia,” J. Appl. Physiol. 64, 2449–2457 (1988).
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Haselgrove, J.

J. Haselgrove, J. Leigh, C. Lee, N.-G. Wang, M. Maris, B. Chance, “Monte Carlo and diffusion calculations of photon migration in non-infinite highly scattering media,” in Time Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 30–41 (1991).

Haselgrove, J. C.

Havlin, R.

Hazeki, O.

M. Tamura, Y. Nomura, O. Hazeki, “Laser tissue spectroscopy—near-infrared CT,” Rev. Laser Eng. (Japan) 15, 74–82 (1987).

Hillson, P. J.

S. R. Arridge, M. Cope, P. van der Zee, P. J. Hillson, D. T. Delpy, “Visualisation of the oxygenation state of brain and muscle in newborn infants by near infra-red transillumination,” in Information Processing in Medical Imaging, S. L. Bacharach, ed. (Martinus Nijhoff, Amsterdam, 1985), pp. 155–176.

Ito, Y.

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abumi, K. Nagai, K. Nakagawa, M. Tamura, “Non-invasive haemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 284–293 (1991).

Jackson, P. C.

P. C. Jackson, P. H. Stevens, J. H. Smith, D. Kean, H. Key, P. N. T. Wells, “The development of a system for transillumination computed tomography,” Br. J. Rad. 60, 375–380 (1987).
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Jarlman, O.

R. Berg, S. Andersson-Engels, O. Jarlman, S. Svanberg, “Time-resolved transillumination for medical diagnostics,” in Time-Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 110–119 (1991).

Jarry, G.

G. Jarry, S. Ghesquiere, J. M. Maarek, S. Debray, Bui-Mong-Hung, D. Laurent, “Imaging mammalian tissues and organs using laser collimated transillumination,” J. Biomed. Eng. 6, 70–74 (1984).
[CrossRef] [PubMed]

Jöbsis van der Vliet, F. F.

J. E. Brazy, D. V. Lewis, M. H. Mitnick, F. F. Jöbsis van der Vliet, “Noninvasive monitoring of cerebral oxygenation in preterm infants: preliminary observations,” Paediatrics 75, 217–225 (1985).

Johnson, M. L.

E. M. Sevick, C. L. Burch, J. K. Frisoli, M. L. Johnson, K. Nowaczyk, H. Szmacinski, J. R. Lakowicz, “The physical basis of photon migration imaging using frequency–domain measurements,” in Medical Optical Tomography: Functional Imaging and Monitoring, Vol. 11 of SPIE Institute Series, G. Müller, ed. (SPIE Press, Bellingham, Wash., 1993), pp. 485–512.

Kaltenbach, J. P.

J. P. Kaltenbach, M. Kaschke, “Frequency- and time-domain modelling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Muller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds. (SPIE Press, Bellingham, Wa., 1993), pp. 65–86.

Kaschke, M.

J. P. Kaltenbach, M. Kaschke, “Frequency- and time-domain modelling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Muller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds. (SPIE Press, Bellingham, Wa., 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. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Kean, D.

P. C. Jackson, P. H. Stevens, J. H. Smith, D. Kean, H. Key, P. N. T. Wells, “The development of a system for transillumination computed tomography,” Br. J. Rad. 60, 375–380 (1987).
[CrossRef]

Key, H.

P. C. Jackson, P. H. Stevens, J. H. Smith, D. Kean, H. Key, P. N. T. Wells, “The development of a system for transillumination computed tomography,” Br. J. Rad. 60, 375–380 (1987).
[CrossRef]

Kohn, P. D.

J. R. Singer, F. A. Grünbaum, P. D. Kohn, J. P. Zubelli, “Imaging reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

F. A. Grünbaum, P. D. Kohn, G. A. Latham, J. R. Singer, J. P. Zubelli, “Diffuse tomography,” in Time Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 232–238 (1991).

Lakowicz, J. R.

E. M. Sevick, C. L. Burch, J. K. Frisoli, J. R. Lakowicz, “Localization of absorbers in scattering media by use of frequency–domain measurements of time-dependent photon migration,” Appl. Opt. 33, 3562–3571 (1994).
[CrossRef] [PubMed]

M. S. Patterson, J. D. Moulton, B. C. Wilson, K. E. Berndt, J. R. Lakowicz, “Frequency-domain reflectance for the determination of the scattering and optical properties of tissue,” Appl. Opt. 30, 4474–4476 (1991).
[CrossRef] [PubMed]

J. R. Lakowicz, K. Berndt, “Frequency domain measurement of photon migration in tissues,” Chem. Phys. Lett. 166, 246–252 (1990).
[CrossRef]

E. M. Sevick, C. L. Burch, J. K. Frisoli, M. L. Johnson, K. Nowaczyk, H. Szmacinski, J. R. Lakowicz, “The physical basis of photon migration imaging using frequency–domain measurements,” in Medical Optical Tomography: Functional Imaging and Monitoring, Vol. 11 of SPIE Institute Series, G. Müller, ed. (SPIE Press, Bellingham, Wash., 1993), pp. 485–512.

Latham, G. A.

F. A. Grünbaum, P. D. Kohn, G. A. Latham, J. R. Singer, J. P. Zubelli, “Diffuse tomography,” in Time Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 232–238 (1991).

Laurent, D.

G. Jarry, S. Ghesquiere, J. M. Maarek, S. Debray, Bui-Mong-Hung, D. Laurent, “Imaging mammalian tissues and organs using laser collimated transillumination,” J. Biomed. Eng. 6, 70–74 (1984).
[CrossRef] [PubMed]

Lee, C.

J. Haselgrove, J. Leigh, C. Lee, N.-G. Wang, M. Maris, B. Chance, “Monte Carlo and diffusion calculations of photon migration in non-infinite highly scattering media,” in Time Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 30–41 (1991).

Leigh, J.

J. Haselgrove, J. Leigh, C. Lee, N.-G. Wang, M. Maris, B. Chance, “Monte Carlo and diffusion calculations of photon migration in non-infinite highly scattering media,” in Time Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 30–41 (1991).

Leigh, J. S.

J. C. Schotland, J. C. Haselgrove, J. S. Leigh, “Photon hitting density,” Appl. Opt. 32, 448–453 (1993).
[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. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

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. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Lewis, D. V.

J. E. Brazy, D. V. Lewis, M. H. Mitnick, F. F. Jöbsis van der Vliet, “Noninvasive monitoring of cerebral oxygenation in preterm infants: preliminary observations,” Paediatrics 75, 217–225 (1985).

Lubowsky, J.

H. L. Graber, J. Chang, J. Lubowsky, R. Aronson, R. L. Barbour, “Near infrared absorption imaging of dense scattering media by steady-state diffusion tomography,” in Photon Migration and Imaging in Random Media, R. R. Alfano, B. Chance, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1888, 372–386 (1993).

Maarek, J. M.

G. Jarry, S. Ghesquiere, J. M. Maarek, S. Debray, Bui-Mong-Hung, D. Laurent, “Imaging mammalian tissues and organs using laser collimated transillumination,” J. Biomed. Eng. 6, 70–74 (1984).
[CrossRef] [PubMed]

Maris, M.

B. Chance, M. Maris, J. Sorge, M. Z. Zhang, “A phase modulation system for dual wavelength difference spectroscopy of haemoglobin deoxygenation in tissue,” in Time-Resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1204, 481–491 (1990).

J. Haselgrove, J. Leigh, C. Lee, N.-G. Wang, M. Maris, B. Chance, “Monte Carlo and diffusion calculations of photon migration in non-infinite highly scattering media,” in Time Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 30–41 (1991).

Mitnick, M. H.

J. E. Brazy, D. V. Lewis, M. H. Mitnick, F. F. Jöbsis van der Vliet, “Noninvasive monitoring of cerebral oxygenation in preterm infants: preliminary observations,” Paediatrics 75, 217–225 (1985).

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. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Moulton, J. D.

M. S. Patterson, J. D. Moulton, B. C. Wilson, K. E. Berndt, J. R. Lakowicz, “Frequency-domain reflectance for the determination of the scattering and optical properties of tissue,” Appl. Opt. 30, 4474–4476 (1991).
[CrossRef] [PubMed]

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time resolved light scattering measurements to photodynamic therapy dosimetry,” in Photodynamic Therapy: Mechanisms, II, T. J. Dougherty, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1203, 62–75 (1990).

Nagai, K.

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abumi, K. Nagai, K. Nakagawa, M. Tamura, “Non-invasive haemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 284–293 (1991).

Nakagawa, K.

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abumi, K. Nagai, K. Nakagawa, M. Tamura, “Non-invasive haemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 284–293 (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. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Nomura, Y.

M. Tamura, Y. Nomura, O. Hazeki, “Laser tissue spectroscopy—near-infrared CT,” Rev. Laser Eng. (Japan) 15, 74–82 (1987).

Nossal, R.

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply embedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

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]

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Theoretical study of resolution limits for time-resolved imaging of human breast,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases, R. R. Alfano, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2135, 176–185 (1994).

Nowaczyk, K.

E. M. Sevick, C. L. Burch, J. K. Frisoli, M. L. Johnson, K. Nowaczyk, H. Szmacinski, J. R. Lakowicz, “The physical basis of photon migration imaging using frequency–domain measurements,” in Medical Optical Tomography: Functional Imaging and Monitoring, Vol. 11 of SPIE Institute Series, G. Müller, ed. (SPIE Press, Bellingham, Wash., 1993), pp. 485–512.

Oda, I.

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abumi, K. Nagai, K. Nakagawa, M. Tamura, “Non-invasive haemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 284–293 (1991).

Patterson, M. S.

M. S. Patterson, J. D. Moulton, B. C. Wilson, K. E. Berndt, J. R. Lakowicz, “Frequency-domain reflectance for the determination of the scattering and optical properties of tissue,” Appl. Opt. 30, 4474–4476 (1991).
[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]

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

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time resolved light scattering measurements to photodynamic therapy dosimetry,” in Photodynamic Therapy: Mechanisms, II, T. J. Dougherty, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1203, 62–75 (1990).

Piantadosi, C. A.

N. B. Hampson, C. A. Piantadosi, “Near infrared monitoring of human skeletal muscle oxygenation during forearm ischaemia,” J. Appl. Physiol. 64, 2449–2457 (1988).
[PubMed]

Potter, L. A.

T. R. Cheatle, L. A. Potter, M. Cope, D. T. Delpy, P. D. Coleridge-Smith, J. H. Scurr, “Near infra-red spectroscopy in peripheral vascular disease,” Br. J. Surg. 78, 405–408 (1991).
[CrossRef] [PubMed]

Reynolds, E. O. R.

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infra-red spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
[PubMed]

A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spectroscopy,” Lancet ii, 770–771 (1988).
[CrossRef]

J. S. Wyatt, M. Cope, D. T. Delpy, S. Wray, E. O. R. Reynolds, “Quantitation of cerebral oxygenation and haemodynamics in sick newborn infants by near infrared spectroscopy,” Lancet ii, 1063–1066 (1986).
[CrossRef]

Richardson, C. E.

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infra-red spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
[PubMed]

A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spectroscopy,” Lancet ii, 770–771 (1988).
[CrossRef]

Robb, R. A.

R. A. Robb, C. Barillot, “Interactive display and analysis of 3-D medical images,” IEEE Trans. Med. Imag. 8, 217–226 (1989).
[CrossRef]

Schotland, J. C.

Schweiger, M.

S. R. Arridge, M. Schweiger, D. T. Delpy, “Iterative reconstruction of near infra-red absorption images,” in Inverse Problems in Scattering and Imaging, M.A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1767, 372–383 (1982).

S. R. Arridge, M. Schweiger, “The use of multiple data types in time-resolved optical absorption and scatter 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).

S. R. Arridge, M. Schweiger, “Reconstruction in optical tomography using MRI based prior knowledge,” in Information Processing in Medical Imaging ’95, Y. Bizais, C. Barillot, R. di Paola, eds. (Kluwer Academic, Dordrecht, 1995), pp. 77–88.

Scurr, J. H.

T. R. Cheatle, L. A. Potter, M. Cope, D. T. Delpy, P. D. Coleridge-Smith, J. H. Scurr, “Near infra-red spectroscopy in peripheral vascular disease,” Br. J. Surg. 78, 405–408 (1991).
[CrossRef] [PubMed]

Sevick, E. M.

E. M. Sevick, C. L. Burch, J. K. Frisoli, J. R. Lakowicz, “Localization of absorbers in scattering media by use of frequency–domain measurements of time-dependent photon migration,” Appl. Opt. 33, 3562–3571 (1994).
[CrossRef] [PubMed]

E. M. Sevick, C. L. Burch, J. K. Frisoli, M. L. Johnson, K. Nowaczyk, H. Szmacinski, J. R. Lakowicz, “The physical basis of photon migration imaging using frequency–domain measurements,” in Medical Optical Tomography: Functional Imaging and Monitoring, Vol. 11 of SPIE Institute Series, G. Müller, ed. (SPIE Press, Bellingham, Wash., 1993), pp. 485–512.

Sevik, E. M.

E. M. Sevik, B. Chance, “Photon migration in a model of the head measured using time and frequency domain techniques: potentials of spectroscopy and imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 84–96 (1991).

Sideri, G.

M. Ferrari, E. Zanette, I. Giannini, G. Sideri, C. Fieschi, A. Carpi, “Effects of carotid artery compression test on regional cerebral blood volume, haemoglobin oxygen saturation and cytochrome-c-oxidase redox level in cerebrovascular patients ” in Oxygen Transport to Tissue VIII, I. A. Longmuir, ed. (Plenum, New York, 1986), pp. 213–222.
[CrossRef]

Singer, J. R.

J. R. Singer, F. A. Grünbaum, P. D. Kohn, J. P. Zubelli, “Imaging reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

F. A. Grünbaum, P. D. Kohn, G. A. Latham, J. R. Singer, J. P. Zubelli, “Diffuse tomography,” in Time Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 232–238 (1991).

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. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Smith, J. H.

P. C. Jackson, P. H. Stevens, J. H. Smith, D. Kean, H. Key, P. N. T. Wells, “The development of a system for transillumination computed tomography,” Br. J. Rad. 60, 375–380 (1987).
[CrossRef]

Sorge, J.

B. Chance, M. Maris, J. Sorge, M. Z. Zhang, “A phase modulation system for dual wavelength difference spectroscopy of haemoglobin deoxygenation in tissue,” in Time-Resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1204, 481–491 (1990).

Stevens, P. H.

P. C. Jackson, P. H. Stevens, J. H. Smith, D. Kean, H. Key, P. N. T. Wells, “The development of a system for transillumination computed tomography,” Br. J. Rad. 60, 375–380 (1987).
[CrossRef]

Svanberg, S.

R. Berg, S. Andersson-Engels, O. Jarlman, S. Svanberg, “Time-resolved transillumination for medical diagnostics,” in Time-Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 110–119 (1991).

Szmacinski, H.

E. M. Sevick, C. L. Burch, J. K. Frisoli, M. L. Johnson, K. Nowaczyk, H. Szmacinski, J. R. Lakowicz, “The physical basis of photon migration imaging using frequency–domain measurements,” in Medical Optical Tomography: Functional Imaging and Monitoring, Vol. 11 of SPIE Institute Series, G. Müller, ed. (SPIE Press, Bellingham, Wash., 1993), pp. 485–512.

Takada, M.

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abumi, K. Nagai, K. Nakagawa, M. Tamura, “Non-invasive haemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 284–293 (1991).

Tamura, M.

M. Tamura, Y. Nomura, O. Hazeki, “Laser tissue spectroscopy—near-infrared CT,” Rev. Laser Eng. (Japan) 15, 74–82 (1987).

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abumi, K. Nagai, K. Nakagawa, M. Tamura, “Non-invasive haemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 284–293 (1991).

Tamura, T.

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abumi, K. Nagai, K. Nakagawa, M. Tamura, “Non-invasive haemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 284–293 (1991).

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 measurement,” 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, D. T. Delpy, M. Cope, “Aspects of clinical infra red absorption imaging,” in Medical Images: Formation, Handling, and Evaluation, Vol. 98 of NATO Advanced Study Institute Imaging SeriesF A. Todd Pokropek, M. A. Viergever, eds., (Springer-Verlag, Heidelberg, Germany, 1992), pp. 407–418.

S. R. Arridge, M. Cope, P. van der Zee, P. J. Hillson, D. T. Delpy, “Visualisation of the oxygenation state of brain and muscle in newborn infants by near infra-red transillumination,” in Information Processing in Medical Imaging, S. L. Bacharach, ed. (Martinus Nijhoff, Amsterdam, 1985), pp. 155–176.

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

Wang, N.-G.

J. Haselgrove, J. Leigh, C. Lee, N.-G. Wang, M. Maris, B. Chance, “Monte Carlo and diffusion calculations of photon migration in non-infinite highly scattering media,” in Time Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 30–41 (1991).

Weglein, A. B.

A. B. Weglein, A. J. Devaney, “The inverse source problem in the presence of external sources,” in Inverse Problems in Scattering and Imaging, M. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1767, 170–176 (1992).

Weiss, G. H.

Wells, P. N. T.

P. C. Jackson, P. H. Stevens, J. H. Smith, D. Kean, H. Key, P. N. T. Wells, “The development of a system for transillumination computed tomography,” Br. J. Rad. 60, 375–380 (1987).
[CrossRef]

Wilson, B. C.

M. S. Patterson, J. D. Moulton, B. C. Wilson, K. E. Berndt, J. R. Lakowicz, “Frequency-domain reflectance for the determination of the scattering and optical properties of tissue,” Appl. Opt. 30, 4474–4476 (1991).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, B. C. Wilson, “Time-resolved reflectance and transmittance for the noninvasive 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]

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time resolved light scattering measurements to photodynamic therapy dosimetry,” in Photodynamic Therapy: Mechanisms, II, T. J. Dougherty, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1203, 62–75 (1990).

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 measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

J. S. Wyatt, M. Cope, D. T. Delpy, S. Wray, E. O. R. Reynolds, “Quantitation of cerebral oxygenation and haemodynamics in sick newborn infants by near infrared spectroscopy,” Lancet ii, 1063–1066 (1986).
[CrossRef]

Wray, S. C.

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infra-red spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
[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 measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Wyatt, J. S.

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infra-red spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
[PubMed]

A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spectroscopy,” Lancet ii, 770–771 (1988).
[CrossRef]

J. S. Wyatt, M. Cope, D. T. Delpy, S. Wray, E. O. R. Reynolds, “Quantitation of cerebral oxygenation and haemodynamics in sick newborn infants by near infrared spectroscopy,” Lancet ii, 1063–1066 (1986).
[CrossRef]

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. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

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. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Zanette, E.

M. Ferrari, E. Zanette, I. Giannini, G. Sideri, C. Fieschi, A. Carpi, “Effects of carotid artery compression test on regional cerebral blood volume, haemoglobin oxygen saturation and cytochrome-c-oxidase redox level in cerebrovascular patients ” in Oxygen Transport to Tissue VIII, I. A. Longmuir, ed. (Plenum, New York, 1986), pp. 213–222.
[CrossRef]

Zhang, M. Z.

B. Chance, M. Maris, J. Sorge, M. Z. Zhang, “A phase modulation system for dual wavelength difference spectroscopy of haemoglobin deoxygenation in tissue,” in Time-Resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1204, 481–491 (1990).

Zubelli, J. P.

J. R. Singer, F. A. Grünbaum, P. D. Kohn, J. P. Zubelli, “Imaging reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

F. A. Grünbaum, P. D. Kohn, G. A. Latham, J. R. Singer, J. P. Zubelli, “Diffuse tomography,” in Time Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 232–238 (1991).

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. (5)

Br. J. Rad. (1)

P. C. Jackson, P. H. Stevens, J. H. Smith, D. Kean, H. Key, P. N. T. Wells, “The development of a system for transillumination computed tomography,” Br. J. Rad. 60, 375–380 (1987).
[CrossRef]

Br. J. Surg. (1)

T. R. Cheatle, L. A. Potter, M. Cope, D. T. Delpy, P. D. Coleridge-Smith, J. H. Scurr, “Near infra-red spectroscopy in peripheral vascular disease,” Br. J. Surg. 78, 405–408 (1991).
[CrossRef] [PubMed]

Chem. Phys. Lett. (1)

J. R. Lakowicz, K. Berndt, “Frequency domain measurement of photon migration in tissues,” Chem. Phys. Lett. 166, 246–252 (1990).
[CrossRef]

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]

IEEE Trans. Med. Imag. (1)

R. A. Robb, C. Barillot, “Interactive display and analysis of 3-D medical images,” IEEE Trans. Med. Imag. 8, 217–226 (1989).
[CrossRef]

J. Appl. Physiol. (2)

N. B. Hampson, C. A. Piantadosi, “Near infrared monitoring of human skeletal muscle oxygenation during forearm ischaemia,” J. Appl. Physiol. 64, 2449–2457 (1988).
[PubMed]

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infra-red spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
[PubMed]

J. Biomed. Eng. (1)

G. Jarry, S. Ghesquiere, J. M. Maarek, S. Debray, Bui-Mong-Hung, D. Laurent, “Imaging mammalian tissues and organs using laser collimated transillumination,” J. Biomed. Eng. 6, 70–74 (1984).
[CrossRef] [PubMed]

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

Lancet (2)

J. S. Wyatt, M. Cope, D. T. Delpy, S. Wray, E. O. R. Reynolds, “Quantitation of cerebral oxygenation and haemodynamics in sick newborn infants by near infrared spectroscopy,” Lancet ii, 1063–1066 (1986).
[CrossRef]

A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spectroscopy,” Lancet ii, 770–771 (1988).
[CrossRef]

Med. Phys. (2)

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]

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply embedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

Paediatrics (1)

J. E. Brazy, D. V. Lewis, M. H. Mitnick, F. F. Jöbsis van der Vliet, “Noninvasive monitoring of cerebral oxygenation in preterm infants: preliminary observations,” Paediatrics 75, 217–225 (1985).

Phys. Med. Biol. (2)

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 measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

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

Proc. Natl. 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. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Rev. Laser Eng. (Japan) (1)

M. Tamura, Y. Nomura, O. Hazeki, “Laser tissue spectroscopy—near-infrared CT,” Rev. Laser Eng. (Japan) 15, 74–82 (1987).

Science (1)

J. R. Singer, F. A. Grünbaum, P. D. Kohn, J. P. Zubelli, “Imaging reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Other (22)

F. A. Grünbaum, P. D. Kohn, G. A. Latham, J. R. Singer, J. P. Zubelli, “Diffuse tomography,” in Time Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 232–238 (1991).

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abumi, K. Nagai, K. Nakagawa, M. Tamura, “Non-invasive haemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 284–293 (1991).

R. Berg, S. Andersson-Engels, O. Jarlman, S. Svanberg, “Time-resolved transillumination for medical diagnostics,” in Time-Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 110–119 (1991).

J. Haselgrove, J. Leigh, C. Lee, N.-G. Wang, M. Maris, B. Chance, “Monte Carlo and diffusion calculations of photon migration in non-infinite highly scattering media,” in Time Resolved Spectroscopy and Imaging in Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 30–41 (1991).

S. R. Arridge, M. Cope, P. van der Zee, P. J. Hillson, D. T. Delpy, “Visualisation of the oxygenation state of brain and muscle in newborn infants by near infra-red transillumination,” in Information Processing in Medical Imaging, S. L. Bacharach, ed. (Martinus Nijhoff, Amsterdam, 1985), pp. 155–176.

S. R. Arridge, P. van der Zee, D. T. Delpy, M. Cope, “Aspects of clinical infra red absorption imaging,” in Medical Images: Formation, Handling, and Evaluation, Vol. 98 of NATO Advanced Study Institute Imaging SeriesF A. Todd Pokropek, M. A. Viergever, eds., (Springer-Verlag, Heidelberg, Germany, 1992), pp. 407–418.

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

S. R. Arridge, M. Schweiger, D. T. Delpy, “Iterative reconstruction of near infra-red absorption images,” in Inverse Problems in Scattering and Imaging, M.A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1767, 372–383 (1982).

B. Chance, M. Maris, J. Sorge, M. Z. Zhang, “A phase modulation system for dual wavelength difference spectroscopy of haemoglobin deoxygenation in tissue,” in Time-Resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1204, 481–491 (1990).

M. Ferrari, E. Zanette, I. Giannini, G. Sideri, C. Fieschi, A. Carpi, “Effects of carotid artery compression test on regional cerebral blood volume, haemoglobin oxygen saturation and cytochrome-c-oxidase redox level in cerebrovascular patients ” in Oxygen Transport to Tissue VIII, I. A. Longmuir, ed. (Plenum, New York, 1986), pp. 213–222.
[CrossRef]

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The program C++ used to generate the photon-measurement density functions, as described in Subsection 6.A is available by writing to S. Arridge, Department of Computer Science, University College of London, London WC1E 6BT, England, or by e-mail at arridge@CS.ucl.ac.uk.

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Figures (12)

Fig. 1
Fig. 1

Green’s function propagator in infinite space.

Fig. 2
Fig. 2

Green’s function propagator in infinite half-space. A negative mirror source is used to satisfy the boundary conditions on the tissue boundary.

Fig. 3
Fig. 3

Green’s function propagator in an infinite slab. Multiple positive and negative mirror sources are used to satisfy the boundary conditions on the surfaces z = 0 and z = d.

Fig. 4
Fig. 4

Phase relations for the determination of J (ψ).

Fig. 5
Fig. 5

J α ( Γ ) at 50 ps in a 20-mm slab, where μ a = 0.025 mm−1 and μ s ′ = 2 mm−1. The source is at (0, 0, 0), and the detector at (0, 0, 20). The function is sampled on a 64 × 64 × 16 grid inside a 20 mm × 20 mm × 20 mm volume centered on the source–detector pair and displayed by means of analyze tm in the maximum-intensity projection mode.

Fig. 6
Fig. 6

J α ( Γ ) at 200 ps. All other parameters are the same as for Fig. 5.

Fig. 7
Fig. 7

J ν ( Γ ) at 200 ps in half-space, where μ a = 0.025 mm−1 and μ s ′ = 2 mm−1, displayed as a set of 3D plots. The source is at (−8, 0, 0), and the detector at (8, 0, 0). The function is sampled on a 64 × 64 grid at successively greater depths into the half-space from 0.9 to 4 mm.

Fig. 8
Fig. 8

J α ( t ) in half-space. All other parameters are the same as for Fig. 7.

Fig. 9
Fig. 9

J ν ( t ) in half-space. All other parameters are the same as for Fig. 7.

Fig. 10
Fig. 10

J α ( ψ ) in a slab at 50 MHz and displayed as gray-scale images. The parameters and sampling density are the same as for Fig. 5.

Fig. 11
Fig. 11

J α ( ψ ) in a slab at 50 MHz for five sources and five detectors and displayed as gray-scale images. The parameters and sampling density are the same as for Fig. 5.

Fig. 12
Fig. 12

Diagram of the arrangement of the sources and the detectors to produce a phased-array simulation of Fig. 11. Five sources of equal strength are arranged in a cross with their relative phases indicated by arrows. A similar arrangement is made for the detectors with all signals combined to form the measurement.

Tables (2)

Tables Icon

Table 1 Definitions of the Notations Used in this Papera

Tables Icon

Table 2 Definitions of Auxiliary Functions

Equations (65)

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F ( M ) [ r m , q ; p ] ,
J p ( M ) ( r m , q ; r ) lim Δ p 0 F ( M ) [ r m , q ; p + Δ p ( r ) ] - F ( M ) [ r m , q ; p ] Δ p ( r ) .
{ · κ ( r ) - γ ( r ) - t } Φ ( r , t ) = - q ( 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 ) ,
M ( X ) = M Y X [ y ( Y ) ] ,
E ( ξ ) = E ( Ω × R ) Ω [ Γ ( ξ , t ) ] = 0 Γ ( ξ , t ) d t
T n ( ξ ) = T ( Ω × R ) Ω n [ Γ ( ξ , t ) ] = 0 t n Γ ( ξ , t ) d t
t n ( ξ ) = t n ( Ω × R ) Ω [ Γ ( ξ , t ) ] = T n ( ξ ) E ( ξ ) = 0 t n Γ ( ξ , t ) d t 0 Γ ( ξ , t ) d t
ψ ( ξ , ω ) = ψ ( Ω × R ) ( Ω × R ) [ Γ ^ ( ξ , ω ) ] = Arg [ - Γ ( ξ , t ) exp ( - i ω t ) d t ]
A ( ξ , ω ) = A ( Ω × R ) ( Ω × R ) [ Γ ^ ( ξ , ω ) ] = Abs [ - Γ ( ξ , t ) exp ( - i ω t ) d t ]
G ( t n ) ( ξ , r ) = t n [ G ( Γ ) ( ξ , r , t ) ] .
Δ Γ ^ ( ξ , ω ) - Ω d 3 r G ^ ( Γ ) ( ξ , r , ω ) α ( r ) G ^ ( Φ ) ( r , r q , ω ) - Ω d 3 r ν ( r ) r G ^ ( Γ ) ( ξ , r , ω ) · r G ^ ( Φ ) ( r , r q , ω ) .
Δ Γ ( ξ , t ) - Ω d 3 r - d t [ g ( Γ ) ( ξ , r , t ) α ( r ) × g ( Φ ) ( r , r q , t - t ) - ν ( r ) r g ( Γ ) ( ξ , r , t ) · r g ( Φ ) ( r , r q , t - t ) ] .
J p ( M ) [ Γ ] = lim Δ p 0 M [ Γ + Δ Γ ] - M [ Γ ] Δ p ,
f ( x , t ) = ( 2 κ t ) - 3 / 2 exp ( - γ t - x 2 4 κ t ) ,
f ( x , t ) F - 1 F f ^ ( x , ω ) = 1 2 κ exp ( - σ x ) x .
g inf ( Φ ) ( r 3 , r 1 , t ) = ( 2 π ) - 3 / 2 f ( ρ 13 , t )
G ^ inf ( Φ ) ( r 3 , r 1 , ω ) = ( 2 π ) - 3 / 2 f ^ ( ρ 13 , ω )
J ^ α , inf ( Φ ) ( r 3 , r 1 , ω ; r 2 ) = G ^ inf ( Φ ) ( r 3 , r 2 , ω ) G ^ inf ( Φ ) ( r 2 , r 1 , ω ) = 1 4 ( 2 π ) 3 κ 2 exp [ - σ ( ρ 12 + ρ 23 ) ] ρ 12 ρ 23 = 1 2 ( 2 π ) 3 κ ( 1 ρ 12 + 1 ρ 23 ) f ^ ( ρ 12 + ρ 23 , ω ) .
J ^ α , inf ( Γ ) ( r 3 , r 1 , ω ; r 2 ) = [ s ˜ ( r 3 ) · ρ ˜ 23 ] S ^ ( Γ ) ( ρ 12 , ρ 23 , ω ) .
J ^ ν , inf ( Φ ) ( r 3 , r 3 , ω ; r 2 ) = r 2 G ^ ( Φ ) ( r 1 , r 2 , ω ) · r 2 G ^ ( Φ ) ( r 2 , r 1 , ω ) = - ( ρ 12 · ρ 23 ) 4 ( 2 π ) 3 κ 2 ( 1 + σ ρ 12 ) ( 1 + σ ρ 23 ) exp [ - σ ( ρ 12 + ρ 23 ) ] ρ 12 3 ρ 23 3 = - ( ρ 12 · ρ 23 ) κ T ^ ( Γ ) ( ρ 12 , ρ 13 , ω ) ,
a ρ b a = a ρ a b = - ρ ˜ b a = ρ ˜ a b , b ρ b a = b ρ a b = ρ ˜ b a = - ρ ˜ a b ,
J ^ ν , inf ( Γ ) ( r 3 , r 1 , ω ; r 2 ) = s ˜ ( r 3 ) · [ T ^ ( Γ ) ( ρ 12 , ρ 23 , ω ) ρ 12 - ( ρ ˜ 12 · ρ ˜ 23 ) W ^ ( Γ ) ( ρ 12 , ρ 23 , ω ) ρ 23 ] .
z 1 = b [ ( 1 - f ¯ ) μ s ] - 1 ,
G ^ half ( Φ ) ( r 2 , r 1 , ω ) = 1 2 ( 2 π ) 3 / 2 κ [ exp [ - σ ρ 12 - ] ρ 12 - - exp [ - σ ρ 12 + ] ρ 12 + ]
G ^ half ( Γ ) ( ξ 3 , r 2 , ω ) = z 23 ( 1 + σ ρ 23 ) exp [ - σ ρ 23 ] ( 2 π ) 3 / 2 ρ 23 3 .
J ^ α , half ( Γ ) ( ξ 3 , r 1 , ω ; r 2 ) = 2 z 23 ρ 23 [ S ^ ( Γ ) ( ρ 12 - , ρ 23 , ω ) - S ^ ( Γ ) ( ρ 12 + , ρ 23 , ω ) ] ,
J ^ ν , half ( Γ ) ( ξ 3 , r 1 , ω ; r 2 ) = 2 { [ - z 12 - T ^ ( Γ ) ( ρ 12 - , ρ 23 , ω ) + z 23 ( ρ ˜ 12 - · ρ ˜ 23 ) × W ^ ( Γ ) ( ρ 12 - , ρ 23 , ω ) ] - [ - z 12 + T ^ ( Γ ) ( ρ 12 + , ρ 23 , ω ) + z 23 ( ρ ˜ 12 + · ρ ˜ 23 ) W ^ ( Γ ) ( ρ 12 + , ρ 23 , ω ) ] } .
G ^ slab ( Φ ) ( r 2 , r 1 , ω ) = 1 2 κ ( 2 π ) 3 / 2 n = - n = [ exp ( - σ ρ 12 + n ) ρ 12 + n - exp ( - σ ρ 12 - n ) ρ 12 - n ] ,
G ^ slab ( Γ ) ( ξ 3 , r 2 , ω ) = 1 2 ( π ) 3 / 2 n = 0 n = [ z 23 + n ρ 23 + n 3 ( 1 + σ ρ 23 + n ) exp [ - σ ρ 23 + n ] - z 23 - n ρ 23 - n 3 ( 1 + σ ρ 23 - n ) exp [ - σ ρ 23 - n ] ] ,
J ^ α , slab ( Γ ) ( ξ 3 , r 1 , ω ; r 2 ) = 2 m = - , n = 0 m = , n = { z 23 + n ρ 23 + n [ S ^ ( Γ ) ( ρ 12 + m , ρ 23 + n , ω ) - S ^ ( Γ ) ( ρ 12 - m , ρ 23 + n , ω ) ] + z 23 - n ρ 23 - n [ S ^ ( Γ ) ( ρ 12 - m , ρ 23 - n , ω ) - S ^ ( Γ ) ( ρ 12 + m , ρ 23 - n , ω ) ] } ,
J ^ ν , slab ( Γ ) ( ξ 3 , r 1 , ω ; r 2 ) = 2 m = - , n = 0 m = , n = { z 12 + m [ T ^ ( Γ ) ( ρ 12 + m , ρ 23 + n , ω ) - T ^ ( Γ ) ( ρ 12 + m , ρ 23 - n , ω ) ] - [ z 23 + n ( ρ ˜ 12 + m · ρ ˜ 23 + n ) W ^ ( Γ ) ( ρ 12 + m , ρ 23 + n , ω ) - z 23 - n ( ρ ˜ 12 + m · ρ ˜ 23 - n ) W ^ ( Γ ) ( ρ 12 + m , ρ 23 - n , ω ) ] + z 12 - m [ T ^ ( Γ ) ( ρ 12 - m , ρ 23 - n , ω ) - T ^ ( Γ ) ( ρ 12 - m , ρ 23 + n , ω ) ] + [ z 23 + n ( ρ ˜ 12 - m · ρ ˜ 23 + n ) W ^ ( Γ ) ( ρ 12 - m , ρ 23 + n , ω ) - z 23 - n ( ρ ˜ 12 - m · ρ ˜ 23 - n ) W ^ ( Γ ) ( ρ 12 - m , ρ 23 - n , ω ) ] } .
f ^ ( x , y , ω ) f ( x , y , t ) , S ^ ( Γ ) ( x , y , ω ) S ( Γ ) ( x , y , t ) , T ^ ( Γ ) ( x , y , ω ) T ( Γ ) ( x , y , t ) , W ^ ( Γ ) ( x , y , ω ) W ( Γ ) ( x , y , t ) .
J α , inf ( Φ ) ( r 3 , r 1 , t ; r 2 ) = η ( r 3 , t ) α ( r 2 ) = 1 2 κ ( 2 π ) 3 ( 1 ρ 12 + 1 ρ 23 ) f ( ρ 12 + ρ 23 , t )
J α , inf ( Γ ) ( r 3 , r 1 , t ; r 2 ) = - κ s ˜ ( r 3 ) · r 3 η ( r 3 , t ) α ( r 2 ) = Δ Γ ( r 3 , t ) α ( r 2 ) = ( s ˜ ( r 3 ) · ρ ˜ 23 ) S ( Γ ) ( ρ 12 , ρ 23 , t ) .
- h ( t ) d t = 2 π H ^ ( ω ) ω = 0 .
J ^ p ( R ) ( r 3 , r 1 , ω ; r 2 ) = J ^ p ( Γ ) ( r 3 , r 1 , ω ; r 2 ) G ^ ( Γ ) ( r 3 , r 1 , ω ) ,
J α , half ( R ) ( ξ 3 , r 1 , ω ; r 2 ) = 1 2 ( 2 π ) 3 / 2 κ z 23 ρ 13 3 z 1 ρ 23 3 ( 1 + σ ρ 23 1 + σ ρ 13 ) × { 1 ρ 12 - exp [ - σ ( ρ 12 - + ρ 23 - ρ 13 ) ] - 1 ρ 12 + exp [ - σ ( ρ 12 + + ρ 23 - ρ 13 ) ] } .
n ω n Γ ^ ( ω ) = 1 2 π n ω n - Γ ( t ) exp ( - i ω t ) d t = ( - i ) n 2 π 0 t n Γ ( t ) exp ( - i ω t ) d t ,
J ^ α , inf ( T n ) ( r 3 , r 1 , ω ; r 2 ) = ( - i ) n n ω n [ G ^ inf ( Γ ) ( r 3 , r 2 , ω ) G ^ inf ( Φ ) ( r 2 , r 1 , ω ) ] ω = 0 = ( - i ) n j = 0 n ( n j ) [ j ω j G ^ inf ( Γ ) ( r 3 , r 2 , ω ) ] ω = 0 × ( n - j ) ω ( n - j ) G ^ inf ( Φ ) ( r 2 , r 1 , ω ) ] ω = 0 ] .
J α ( t n ) ( r 3 , r 1 , ω ; r 2 ) = lim α 0 0 d t t n [ Γ ( r 3 , t ) + Δ Γ ( r 3 , t ) ] 0 d t [ Γ ( r 3 , t ) + Δ Γ ( r 3 , t ) - 0 d t t n [ Γ ( r 3 , t ) ] 0 d t Γ ( r 3 , t ) J α ( T n ) [ Γ ] E [ Γ ] - t n [ Γ ] J α ( E ) [ Γ ] E [ Γ ]
J α , inf ( T ) ( r 3 , r 1 ; r 2 ) = ( s ˜ ( r 3 ) · ρ ˜ 23 ) i ω S ^ ( Γ ) ( ρ 12 , ρ 23 , ω ) ω = 0 = [ s ˜ ( r 3 ) · ρ ˜ 23 ] S ( T ) ( ρ 12 , ρ 23 ) .
S ( T ) ( x , y ) = 1 8 ( 2 π ) 3 κ 2 σ 0 x + σ 0 ( x + y ) y x y 2 exp [ - σ 0 ( x + y ) ] .
t inf ( Γ ) ( x ) = 1 2 κ x 2 ( 1 + σ 0 x ) ,
J α , inf ( t ) ( r 3 , r 1 ; r 2 ) = s ˜ ( r 3 ) · ρ ˜ 23 G ( E ) ( r 3 , r 1 ) [ S ( T ) ( ρ 12 , ρ 23 ) - t inf ( Γ ) ( ρ 13 ) S ( Γ ) ( ρ 12 , ρ 23 ) ] ,
J α , inf ( t ) ( r 3 , r 1 ; r 2 ) = 1 8 κ 2 ( 2 π ) 3 / 2 ( s ˜ ( r 3 ) · ρ ˜ 23 s ˜ ( r 3 ) · ρ ˜ 23 ) ρ 13 2 1 + σ 0 ρ 13 × [ ρ 12 + σ 0 ρ 23 ( ρ 12 + ρ 23 ) σ 0 ρ 12 ρ 23 2 - ρ 12 2 ( 1 + σ 0 ρ 23 ) ( 1 + σ 0 ρ 12 ) ρ 12 ρ 23 2 ] × exp [ - σ 0 ( ρ 12 + ρ 23 - ρ 13 ) ] .
J α , half ( T ) ( ξ 3 , r 1 ; r 2 ) = 2 z 23 ρ 23 [ S ( T ) ( ρ 12 - , ρ 23 ) - S ( T ) ( ρ 12 + , ρ 23 ) ] ,
J α , half ( t ) ( ξ 3 , r 1 ; r 2 ) = z 23 ρ 23 1 G half ( E ) ( ξ 3 , r 1 ) { [ S ( T ) ( ρ 12 - , ρ 23 ) - S ( T ) ( ρ 12 + , ρ 23 ) ] - t half ( Γ ) ( ρ 13 ) [ S ( Γ ) ( ρ 12 - , ρ 23 ) - S ( Γ ) ( ρ 12 + , ρ 23 ) ] } .
T ( T ) ( x , y ) = 1 8 ( 2 π ) 3 κ 2 [ ( x 2 + y 2 ) + σ 0 ( x 2 y + x y 2 ) x 3 y 3 ] exp [ - σ ( x + y ) ] ,
W ( T ) ( x , y ) = 1 8 ( 2 π ) 3 κ 2 × [ 3 ( x 2 + y 2 ) + σ 0 ( x y 2 + 3 x 2 y + y 3 ) + σ 0 2 ( x 2 y 2 + x y 3 ) x 2 y 4 ] × exp [ - σ ( x + y ) ] .
J ν , half ( t ) ( ξ 3 , r 1 ; r 2 ) = 1 G half ( E ) ( ξ 3 , r 1 ) × [ J ν , half ( T ) ( ξ 3 , r 1 ; r 2 ) - t half ( Γ ) ( ρ 13 ) J ν , half ( Γ ) ( ξ 3 , r 1 ; r 2 ) ] ,
J ^ ν , half ( T ) ( ξ 3 , r 1 , ω ; r 2 ) = { [ - z 12 - T ^ ( T ) ( ρ 12 - , ρ 23 , ω ) + z 23 ( ρ ^ 12 · ρ ˜ 23 ) W ^ ( T ) ( ρ 12 - , ρ 23 , ω ) ] - [ - z 12 + T ^ ( T ) ( ρ 12 + , ρ 23 , ω ) + z 23 ( ρ ˜ 12 + · ρ ˜ 23 ) W ^ ( T ) ( ρ 12 + , ρ 23 , ω ) ] } .
sin ( Δ ψ ) Δ Γ = sin ( β ) Γ ,
sin ( Δ ψ ) [ 1 Δ Γ + cos ( τ - ψ ) Γ ] = sin ( τ - ψ ) Γ cos ( Δ ψ ) ,
sin ( Δ ψ ) Δ Γ Γ sin ( τ - ψ ) × [ 1 + Δ Γ Γ cos ( τ - ψ ) ] - 1 .
Δ ψ sin ( Δ ψ ) Δ Γ Γ sin ( τ - ψ ) = Im [ Δ Γ Γ ] .
J α ( ψ ) ( r 3 , r 1 , ω ; r 2 ) Im [ J ^ α ( Γ ) ( r 3 , r 1 , ω ; r 2 ) G ^ ( Γ ) ( r 3 , r 1 , ω ) ] = Im [ J α ( R ) ( r 3 , r 1 , ω ; r 2 ) ] ,
J α , half ( ψ ) ( ξ 3 , r 1 , ω ; r 2 ) = 1 2 ( 2 π ) 3 / 2 κ z 23 ρ 13 3 z 1 ρ 23 3 × Im ( ( 1 + σ ρ 23 1 + σ ρ 13 ) { 1 ρ 12 - exp [ - σ ( ρ 12 - + ρ 23 - ρ 13 ) ] - 1 ρ 12 - exp [ - σ ( ρ 12 - + ρ 23 - ρ 13 ) ] } )
σ = σ re + i σ im , σ re = ( γ 2 + ω 2 ) 1 / 4 κ 1 / 2 cos ( 1 2 tan - 1 ω γ ) , σ im = ( γ 2 + ω 2 ) 1 / 4 κ 1 / 2 sin ( 1 2 tan - 1 ω γ ) ,
J α , half ( ψ ) ( ξ 3 , r 1 , ω ; r 2 ) = 1 2 ( 2 π ) 3 / 2 κ z 23 ρ 13 3 z 1 ρ 23 3 × { B ( ρ 13 , ρ 23 ) ρ 12 - exp [ - σ re ( ρ 12 - + ρ 23 - ρ 13 ) ] × cos [ σ im ( ρ 12 - + ρ 23 - ρ 13 ) ] - A ( ρ 13 , ρ 23 ) ρ 12 - exp [ σ re ( ρ 12 - + ρ 23 - ρ 13 ) ] × sin [ σ im ( ρ 12 - + ρ 23 - ρ 13 ) ] - B ( ρ 12 , ρ 23 ) ρ 12 + exp [ - σ re ( ρ 12 + + ρ 23 - ρ 13 ) ] × cos [ - σ im ( ρ 12 + + ρ 23 - ρ 13 ) ] + A ( ρ 13 , ρ 23 ) ρ 12 + exp [ - σ im ( ρ 12 + + ρ 23 - ρ 13 ) ] × sin [ σ im ( ρ 12 + + ρ 23 - ρ 13 ) ] } .
A ( x , y ) = ( 1 + σ re x ) ( 1 + σ re y ) + σ im 2 x y , B ( x , y ) = σ im ( y - x ) .
J p ( R ) ( { r 3 } , { r 1 } , ω ; r 2 ) = i , k C i D k J ^ p ( Γ ) ( r 3 , k , r 1 , i , ω ; r 2 ) i , k C i D k G ^ ( Γ ) ( r 3 , k , r 1 , i , ω ) .
[ Δ M ζ 1 ( ξ 1 ) Δ M ζ 1 ( ξ 2 ) Δ M ζ i ( ξ k ) Δ M ζ S ( ξ N ) ] = [ J p ( M ) ( ξ 1 , ζ 1 ; r 1 ) J p ( M ) ( ξ 1 , ζ 1 ; r j ) J p ( M ) ( ξ 1 , ζ 1 ; r L ) J p ( M ) ( ξ 2 , ζ 1 ; r 1 ) J p ( M ) ( ξ 2 , ζ 1 ; r j ) J p ( M ) ( ξ 2 , ζ 1 ; r L ) J p ( M ) ( ξ k , ζ i ; r 1 ) J p ( M ) ( ξ k , ζ i ; r j ) J p ( M ) ( ξ k , ζ i ; r L ) J p ( M ) ( ξ N , ζ i ; r 1 ) J p ( M ) ( ξ N , ζ S ; r j ) J p ( M ) ( ξ N , ζ S ; r L ) ] [ Δ p ( r 1 ) Δ p ( r j ) Δ p ( r L ) ] .
exp ( - σ x ) 2 κ x f ( x , t ) , σ exp ( - σ x ) [ ( x 2 t - 2 κ ) ] f ( x , t ) , σ 2 exp ( - σ x ) ( x 2 2 κ t 2 - 3 x t ) f ( x , t ) , σ 3 exp ( - σ x ) [ x 4 ( 2 κ ) 2 t 3 - 3 x 2 κ t 2 + 3 t ] f ( x , t ) σ n exp ( - σ x ) [ κ ( 1 4 κ t ) n - 1 H n + 1 ( x 4 κ t ) ] f ( x , t ) ,

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