Abstract

We present an analytical perturbation analysis for studying the sensitivity of diffusive photon flux to the addition of a small spherical defect object in multiple-scattering media such as human tissues. As a first simple application of our perturbation method, we derive analytically the photon migration path distributions and the shapes of the so-called banana regions in which the photon migration paths are concentrated. We then derive analytically the sensitivity of detected photon flux densities to the inclusion of small spherical defects in the multiple-scattering medium for both single-source and two-source configurations, at both steady-state (dc) and frequency-modulation conditions, and compare the results with Monte Carlo simulations.

© 1995 Optical Society of America

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  1. J. E. Brazy, D. V. Lewis, M. H. Mitnick, J. Vander Vliet F.F., “Noninvasive monitorin of cerebral oxygenation in preterm infants: preliminary observations,” Paediatrics 75, 217–225 (1985);A. D. Edwards, J. S. Wyatt, C. Richardson, D. T. Delpy, M. Cope, E. D. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spectroscopy,” Lancet 2, 770–771 (1988);N. B. Hampson, C. A. Piantodosi, “Near infrared monitoring of human skeletal muscle oxygenation during forearm ischaemia,” J. Appl. Physiol. 64, 2449–2457 (1988).
    [CrossRef] [PubMed]
  2. M. S. Patterson, B. Chance, C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  3. S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo—diffusion theory modeling of light distributions in tissue,” in Laser Interaction with Tissue,M. W. Berns, ed., Proc. Soc. Photo-Opt. Instrum. Eng.908, 20–28 (1988).
  4. B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, 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]
  5. M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
    [CrossRef]
  6. A. Knüttel, J. M. Schmitt, J. R. Knutson, “Spatial localization of absorbing bodies by interfering diffusive photon-density waves,” Appl. Opt. 32, 381–389 (1993).
    [CrossRef] [PubMed]
  7. B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object location in tissue models with linear inphase and antiphase multielement optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
    [CrossRef] [PubMed]
  8. P. N. den Outer, T. M. Nieuwenhuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am A 10, 1209–1218 (1993).
    [CrossRef]
  9. S. R. Arridge, M. Schweiger, D. T. Delpy, “Iterative reconstruction of new infra-red absorption images,” in Inverse Problems in Scattering and Imaging,M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1767, 372–383 (1992).
  10. J. C. Schotland, J. C. Haselgrove, J. S. Leigh, “Photon hitting density,” Appl. Opt. 32, 448–453 (1993).
    [CrossRef] [PubMed]
  11. W. Cui, C. Kumar, B. Chance, “Experimental study of migration depth for the photons measured at sample surface,” in Time-Resolved Spectroscopy and Imaging of Tissue,B. Chance, A. Katzir, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 180–191(1991).
  12. R. Aronson, “Extrapolation distance for diffusion of light,” in Photon Migration and Imaging in Random Media and Tissues,B. Chance, R. R. Alfano, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1888, 297–305 (1993).
  13. S. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distributions in multiple scatttering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1888, 78–89 (1993).

1993

B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object location in tissue models with linear inphase and antiphase multielement optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
[CrossRef] [PubMed]

P. N. den Outer, T. M. Nieuwenhuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am A 10, 1209–1218 (1993).
[CrossRef]

A. Knüttel, J. M. Schmitt, J. R. Knutson, “Spatial localization of absorbing bodies by interfering diffusive photon-density waves,” Appl. Opt. 32, 381–389 (1993).
[CrossRef] [PubMed]

J. C. Schotland, J. C. Haselgrove, J. S. Leigh, “Photon hitting density,” Appl. Opt. 32, 448–453 (1993).
[CrossRef] [PubMed]

1992

M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef]

1989

1988

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, 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]

1985

J. E. Brazy, D. V. Lewis, M. H. Mitnick, J. Vander Vliet F.F., “Noninvasive monitorin of cerebral oxygenation in preterm infants: preliminary observations,” Paediatrics 75, 217–225 (1985);A. D. Edwards, J. S. Wyatt, C. Richardson, D. T. Delpy, M. Cope, E. D. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spectroscopy,” Lancet 2, 770–771 (1988);N. B. Hampson, C. A. Piantodosi, “Near infrared monitoring of human skeletal muscle oxygenation during forearm ischaemia,” J. Appl. Physiol. 64, 2449–2457 (1988).
[CrossRef] [PubMed]

Aronson, R.

R. Aronson, “Extrapolation distance for diffusion of light,” in Photon Migration and Imaging in Random Media and Tissues,B. Chance, R. R. Alfano, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1888, 297–305 (1993).

Arridge, S. R.

S. R. Arridge, M. Schweiger, D. T. Delpy, “Iterative reconstruction of new infra-red absorption images,” in Inverse Problems in Scattering and Imaging,M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1767, 372–383 (1992).

Boas, D. A.

M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef]

Boretsky, R.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, 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, J. Vander Vliet F.F., “Noninvasive monitorin of cerebral oxygenation in preterm infants: preliminary observations,” Paediatrics 75, 217–225 (1985);A. D. Edwards, J. S. Wyatt, C. Richardson, D. T. Delpy, M. Cope, E. D. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spectroscopy,” Lancet 2, 770–771 (1988);N. B. Hampson, C. A. Piantodosi, “Near infrared monitoring of human skeletal muscle oxygenation during forearm ischaemia,” J. Appl. Physiol. 64, 2449–2457 (1988).
[CrossRef] [PubMed]

Chance, B.

B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object location in tissue models with linear inphase and antiphase multielement optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
[CrossRef] [PubMed]

M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef]

M. S. Patterson, B. Chance, 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. Kaufmann, W. Levy, M. Young, P. Cohen, 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]

S. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distributions in multiple scatttering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1888, 78–89 (1993).

W. Cui, C. Kumar, B. Chance, “Experimental study of migration depth for the photons measured at sample surface,” in Time-Resolved Spectroscopy and Imaging of Tissue,B. Chance, A. Katzir, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 180–191(1991).

Cohen, P.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, 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]

Cui, W.

W. Cui, C. Kumar, B. Chance, “Experimental study of migration depth for the photons measured at sample surface,” in Time-Resolved Spectroscopy and Imaging of Tissue,B. Chance, A. Katzir, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 180–191(1991).

Delpy, D. T.

S. R. Arridge, M. Schweiger, D. T. Delpy, “Iterative reconstruction of new infra-red absorption images,” in Inverse Problems in Scattering and Imaging,M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1767, 372–383 (1992).

den Outer, P. N.

P. N. den Outer, T. M. Nieuwenhuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am A 10, 1209–1218 (1993).
[CrossRef]

Feng, S.

S. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distributions in multiple scatttering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1888, 78–89 (1993).

Finander, M.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, 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]

Flock, S. T.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo—diffusion theory modeling of light distributions in tissue,” in Laser Interaction with Tissue,M. W. Berns, ed., Proc. Soc. Photo-Opt. Instrum. Eng.908, 20–28 (1988).

Greenfeld, R.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, 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]

Haselgrove, J. C.

He, L.

B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object location in tissue models with linear inphase and antiphase multielement optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
[CrossRef] [PubMed]

Kang, K.

B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object location in tissue models with linear inphase and antiphase multielement optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
[CrossRef] [PubMed]

Kaufmann, K.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, 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]

Knutson, J. R.

Knüttel, A.

Kumar, C.

W. Cui, C. Kumar, B. Chance, “Experimental study of migration depth for the photons measured at sample surface,” in Time-Resolved Spectroscopy and Imaging of Tissue,B. Chance, A. Katzir, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 180–191(1991).

Lagendijk, A.

P. N. den Outer, T. M. Nieuwenhuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am A 10, 1209–1218 (1993).
[CrossRef]

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. Kaufmann, W. Levy, M. Young, P. Cohen, 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. Kaufmann, W. Levy, M. Young, P. Cohen, 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, J. Vander Vliet F.F., “Noninvasive monitorin of cerebral oxygenation in preterm infants: preliminary observations,” Paediatrics 75, 217–225 (1985);A. D. Edwards, J. S. Wyatt, C. Richardson, D. T. Delpy, M. Cope, E. D. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spectroscopy,” Lancet 2, 770–771 (1988);N. B. Hampson, C. A. Piantodosi, “Near infrared monitoring of human skeletal muscle oxygenation during forearm ischaemia,” J. Appl. Physiol. 64, 2449–2457 (1988).
[CrossRef] [PubMed]

Mitnick, M. H.

J. E. Brazy, D. V. Lewis, M. H. Mitnick, J. Vander Vliet F.F., “Noninvasive monitorin of cerebral oxygenation in preterm infants: preliminary observations,” Paediatrics 75, 217–225 (1985);A. D. Edwards, J. S. Wyatt, C. Richardson, D. T. Delpy, M. Cope, E. D. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spectroscopy,” Lancet 2, 770–771 (1988);N. B. Hampson, C. A. Piantodosi, “Near infrared monitoring of human skeletal muscle oxygenation during forearm ischaemia,” J. Appl. Physiol. 64, 2449–2457 (1988).
[CrossRef] [PubMed]

Miyake, H.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, 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]

Nieuwenhuizen, T. M.

P. N. den Outer, T. M. Nieuwenhuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am A 10, 1209–1218 (1993).
[CrossRef]

Nioka, S.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, 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]

O'Leary, M. A.

M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef]

Patterson, M. S.

M. S. Patterson, B. Chance, 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, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo—diffusion theory modeling of light distributions in tissue,” in Laser Interaction with Tissue,M. W. Berns, ed., Proc. Soc. Photo-Opt. Instrum. Eng.908, 20–28 (1988).

Schmitt, J. M.

Schotland, J. C.

Schweiger, M.

S. R. Arridge, M. Schweiger, D. T. Delpy, “Iterative reconstruction of new infra-red absorption images,” in Inverse Problems in Scattering and Imaging,M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1767, 372–383 (1992).

Sevick, E.

B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object location in tissue models with linear inphase and antiphase multielement optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
[CrossRef] [PubMed]

Smith, D. S.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, 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]

Vander Vliet F.F., J.

J. E. Brazy, D. V. Lewis, M. H. Mitnick, J. Vander Vliet F.F., “Noninvasive monitorin of cerebral oxygenation in preterm infants: preliminary observations,” Paediatrics 75, 217–225 (1985);A. D. Edwards, J. S. Wyatt, C. Richardson, D. T. Delpy, M. Cope, E. D. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spectroscopy,” Lancet 2, 770–771 (1988);N. B. Hampson, C. A. Piantodosi, “Near infrared monitoring of human skeletal muscle oxygenation during forearm ischaemia,” J. Appl. Physiol. 64, 2449–2457 (1988).
[CrossRef] [PubMed]

Weng, J.

B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object location in tissue models with linear inphase and antiphase multielement optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
[CrossRef] [PubMed]

Wilson, B. C.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo—diffusion theory modeling of light distributions in tissue,” in Laser Interaction with Tissue,M. W. Berns, ed., Proc. Soc. Photo-Opt. Instrum. Eng.908, 20–28 (1988).

Wilson, C.

Yodh, A. G.

M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef]

Yoshioka, H.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, 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. Kaufmann, W. Levy, M. Young, P. Cohen, 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]

Zeng, F.

S. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distributions in multiple scatttering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1888, 78–89 (1993).

Appl. Opt.

J. Opt. Soc. Am A

P. N. den Outer, T. M. Nieuwenhuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am A 10, 1209–1218 (1993).
[CrossRef]

Paediatrics

J. E. Brazy, D. V. Lewis, M. H. Mitnick, J. Vander Vliet F.F., “Noninvasive monitorin of cerebral oxygenation in preterm infants: preliminary observations,” Paediatrics 75, 217–225 (1985);A. D. Edwards, J. S. Wyatt, C. Richardson, D. T. Delpy, M. Cope, E. D. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spectroscopy,” Lancet 2, 770–771 (1988);N. B. Hampson, C. A. Piantodosi, “Near infrared monitoring of human skeletal muscle oxygenation during forearm ischaemia,” J. Appl. Physiol. 64, 2449–2457 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett.

M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef]

Proc. Natl. Acad. Sci. USA

B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object location in tissue models with linear inphase and antiphase multielement optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
[CrossRef] [PubMed]

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, 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]

Other

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo—diffusion theory modeling of light distributions in tissue,” in Laser Interaction with Tissue,M. W. Berns, ed., Proc. Soc. Photo-Opt. Instrum. Eng.908, 20–28 (1988).

W. Cui, C. Kumar, B. Chance, “Experimental study of migration depth for the photons measured at sample surface,” in Time-Resolved Spectroscopy and Imaging of Tissue,B. Chance, A. Katzir, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 180–191(1991).

R. Aronson, “Extrapolation distance for diffusion of light,” in Photon Migration and Imaging in Random Media and Tissues,B. Chance, R. R. Alfano, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1888, 297–305 (1993).

S. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distributions in multiple scatttering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1888, 78–89 (1993).

S. R. Arridge, M. Schweiger, D. T. Delpy, “Iterative reconstruction of new infra-red absorption images,” in Inverse Problems in Scattering and Imaging,M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1767, 372–383 (1992).

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Figures (10)

Fig. 1
Fig. 1

Geometry of a semi-infinite sample. Z > 0 defines the multiple-scattering sample being studied. The source fiber is located at the origin, and the detector fiber is located at rd = (d, 0, 0).

Fig. 2
Fig. 2

Boundary of the banana-shape region. We define such a boundary as the cross section for each X plane where the normalized photon path density r(X, Y, Z) is a factor of B = 0.87 of its peak value unity, (a) We take the realistic values μs = 0.002/mm, μs = 4/mm, g = 0.75, with the source–detector distance d = 10 mm. This gives κd = 0.78; i.e., we are in the relatively low absorption limit, (b) We change the parameters to μa = 0.02/mm, d = 30 mm, which gives κd = 7.3; i.e., we are in the strong absorption regime. We use the same value of B = 0.87 to define the boundary of the banana region.

Fig. 3
Fig. 3

Sensitivity of the detected photon flux density J1 as a function of the position of the defect r in the μ a μ a case so that q ≠ 0. We take the parameters μa = 0.002/mm, μ s = μ s = 4 / mm, g = g = 0.75, μ a = 0.008 / mm, and the size a = 2 mm. The distance between the source and detector is d = 10 mm. The plot is the three-dimensional isosurface contour for the detected photon-intensity change at the value of J1 = −2.6 × 10−4 cm2 photon.

Fig. 4
Fig. 4

Sensitivity of the detected photon flux density J1 versus r for the μ a = μ a case so that q = 0; but μ s μ s so that p ≠ 0. The various parameters are μ a = μ a = 0.002 / mm, μs = 4/mm, g = g = 0.75, μ s = 8 / mm, a = 2 mm, and d = 10 mm. The isosurface plotted corresponds to the value J1 = − 1.1 × 10−3 cm2 photon.

Fig. 5
Fig. 5

Comparison of analytical results for the normalized detected intensity change caused by a very strongly absorbing spherical defect placed at r = (x, y, z), in the single-source geometry, with Monte Carlo simulations. We take a large value of μ a = 30 / mm. We fix the scattering properties for the medium as μs = 4.0/mm and g = 0.75. In each plot we vary the depth of the defect position Z while keeping the other coordinates, x = d/2 and y = 0, fixed. We normalize all the results by dividing out the maximum value for J1. (a) We take μa = 0.002/mm, a = 2 mm, and d = 10 mm. The parameter κd = 0.78, which signifies that we are in an intermediate absorption regime, (b) We chance d = 20 mm with all other parameters fixed. This gives κd = 1.6. (c) We increase d to 30 mm, making κd = 2.3; i.e., we are in the strong absorption regime, (d) We take a more absorbing medium with μa = 0.02/mm with a = 2 mm and d = 10 mm. This gives κd = 2.4. (e) We take μa = 0.02/mm but with a larger source–detector distance d = 20 mm with κd = 4.8.

Fig. 6
Fig. 6

Geometry for the symmetric double-source, single-detector configration; the two sources are placed at a distance d from the detector, which is placed at the origin. This geometry is chosen so that in finite-frequency-modulation conditions the detected signal from the homogeneous medium is identically zero. The sample region is that of Z > 0.

Fig. 7
Fig. 7

Modulation |J1| for the two symmetrically placed sources. These two sources are modulated at the same frequency, but they have exactly the opposite phase. This guarantees that the measured intensity in the absence of the defect is identically zero.

Fig. 8
Fig. 8

Phase shift of the measured intensity for the double-source, opposite-phase, frequency-modulated configuration, calculated from our perturbation theory. The phase shift is defined as ϕ ≡ tan−1[Im(J1)/Re(J1)]. The phase angle is plotted as a function of the x position of the defect, while z = 3 mm is kept and a few different values of y are sampled. We choose the q ≠ 0 case for a defect sphere that is more absorbing than the surrounding medium. The parameters are the modulation frequency f = 200 MHz, μa = 0.002/mm, μ a = 0.008 / mm, μs = 4/mm, and μ s = 8 / mm, g = g = 0.75, a = 2 mm, and d = 10 mm. This gives κ(ω) = (0.26 − 0.25i)/ mm, κ ( ω ) = ( 0.39 0.32 i ) / mm, q = (−0.006 + 0.032i) mm, and p = (0.26 − 0.009i) mm3. (a) y = 8 mm, (b) y = 5 mm, (c) y = 3 mm, (d) y = 0.

Fig. 9
Fig. 9

Frequency dependence of the modulation |J1| for a fixed position of the defect at x = d/2 = 5 mm, y = 0, and z = 3 mnm. The other parameters are μa = 0.002/mm, μ a = 0.008 / mm, μs = 4/mm, μ s = 8 / mm, g = g = 0.75, a = 2 mm, and d = 10 mm.

Fig. 10
Fig. 10

Modulation |J1| as a function of the defect position r for the q = 0 case but p ≠ 0. The parameters are the modulation frequency f = 100 MHz, μ a = μ a = 0.002 / mm, μs = 4/mm, μ s = 8 / mm, g = g = 0.75, a = 2 mm, d = 10 mm. This gives κ(ω) = (0.19 − 0.17i)/mm, κ ( ω ) = ( 0.26 0.24 i ) / mm, q = 0, and p = (−1.6 − 0.14i) mm3. The isosurface is for |J1| = 3.0 × 10−4 cm2 photon.

Equations (33)

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1 c ϕ ( r , t ) t D 2 ϕ ( r , t ) + μ a ( r ) ϕ ( r , t ) = S ( r , t ) .
S ( r , t ) = S 0 δ ( r r 0 ) ,
S ( r , t ) = S 0 [ δ ( r z 0 ) δ ( r + z 0 ) ]
ϕ 0 ( r ) = S 0 4 π D exp ( κ | r r 0 | ) | r r 0 | ,
ϕ ( r d ) = ϕ 0 ( r d ) + ϕ 1 ( r d , r ) .
ϕ 1 ( r d , r ) = q exp ( κ | r d r | ) | r d r | ,
ϕ 1 ( r d , r ) = a S 0 4 π D exp [ κ ( | r d r | a ) ] | r d r | × exp ( κ | r d r 0 | ) | r r 0 | .
P ( r ) = exp ( κ | r d r | ) | r d r | exp ( κ | r d r 0 | ) | r r 0 | .
P ( x , ρ ) = exp ( [ κ { [ ( d x ) 2 + ρ 2 ] 1 / 2 + ( x 2 + ρ 2 ) } 1 / 2 ) { [ ( d x ) 2 + ρ 2 ] ( x 2 + ρ 2 ) } 1 / 2 .
P max ( x ) = exp ( κ d ) ( d x ) x .
r ( x , ρ ) = { ( d x ) 2 x 2 [ ( d x ) 2 + ρ 2 ] ( x 2 + ρ 2 ) } 1 / 2 × exp ( κ { [ ( d x ) 2 + ρ 2 ] 1 / 2 + ( x 2 + ρ 2 ) 1 / 2 d } ) .
ρ 1 / 2 ( { [ x 2 + ( d x ) 2 ] 2 + 12 ( d x ) 2 x 2 } 1 / 2 [ x 2 + ( d x ) 2 ] 2 ) 1 / 2 .
ρ 1 / 2 ( 2 ln 2 ) x ( d x ) κ d .
ϕ 0 ( r ) = S 0 4 π D [ exp ( κ | r z 0 | ) | r z 0 | exp ( κ | r + z 0 | ) | r + z 0 | ] 2 z 0 S 0 4 π D ( z ) exp ( κ | r | ) | r | .
ϕ 1 ( r d ) = a ϕ 0 ( r ) × [ exp ( κ | r d r | ) | r d r | exp ( κ | r d r * | ) | r d r * | ] ,
P ( x , y , z ) = z 2 exp ( κ { ( x 2 + y 2 + z 2 ) 1 / 2 + [ ( d x ) 2 + y 2 + z 2 ] 1 / 2 } ) ( x 2 + y 2 + z 2 ) 3 / 2 [ ( d x ) 2 + y 2 + z 2 ] 3 / 2 × [ κ ( x 2 + y 2 + z 2 ) 1 / 2 + 1 ] { κ [ ( d x ) 2 + y 2 + z 2 ] 1 / 2 + 1 } .
z 0 ( x ) [ ( { [ x 2 + ( d x ) 2 ] 2 + 32 x 2 ( d x ) 2 } 1 / 2 x 2 ( d x ) 2 ) ] 1 / 2 .
z 0 max 2 d 4 .
z 0 ( x ) [ 2 x ( d x ) κ d ] 1 / 2 ,
r ( x , y B , z B ) = P ( x , y B , z B ) / P ( x , 0 , z 0 ) = B ,
ϕ 0 ( r ) = 2 z 0 z S 0 4 π D ( κ r + 1 ) r 3 exp ( κ r ) ,
E 0 ( r ) ϕ 0 ( r ) = z 0 S 0 2 π D ( 3 κ z r r 4 + 3 z r r 5 κ r 2 r 3 + κ 2 z r r 3 ) exp ( κ r ) .
q = a exp ( κ a ) D B ( 1 μ a / μ a ) D ( 1 + κ a ) sinh ( κ a ) / ( κ a ) + D B .
p = a 3 exp ( κ a ) { D A D B D ( 1 + κ a ) A + D [ 2 + 2 κ a + ( κ a ) 2 ] B } ,
A = 2 sinh ( κ a ) / ( κ a ) + κ a sinh ( κ a ) 2 cosh ( κ a ) ,
B = cosh ( κ a ) sinh ( κ a ) / ( κ a ) .
J 1 D | E 1 ( r d ) | = 2 D q ( κ | r r d | + 1 ) z | r r d | 3 × exp ( κ | r r d | ) ϕ 0 ( r ) 2 D p × { [ ( r r d ) · E 0 ( r ) ] ( κ | r r d | + 3 ) z | r r d | 5 E 0 z ( r ) | r r d | 3 } × exp ( κ | r r d | ) .
J 1 J 0 a d ,
J 1 J 0 ( a d ) 3 .
κ ( ω ) = ( μ a D i ω D c ) 1 / 2 = κ ( ω ) i k ( ω ) ,
J 1 = 2 D q [ κ ( ω ) | r | + l ] z | r | 3 exp [ κ ( ω ) | r | ] ϕ 0 ( r ) 2 D p { [ r · E 0 ( r ) ] [ κ ( ω ) | r | + 3 ] z | r | 5 E 0 z ( r ) | r | 3 } × exp ( κ | r | ) .
ϕ 0 ( r ) = 2 z 0 z S 0 4 π D { κ ( ω ) | r r 0 | + 1 | r r 0 | 3 exp [ κ ( ω ) | r r 0 | ] κ ( ω ) | r + r 0 | + 1 | r + r 0 | 3 exp [ κ ( ω ) | r + r 0 | ] } ,
E 0 ( r ) = { z 0 S 0 2 π D [ 3 κ ( ω ) z ( r r 0 ) | r r 0 | 4 + 3 z ( r r 0 ) | r r 0 | 5 κ ( ω ) | r r 0 | 2 | r r d | 3 + κ ( ω ) 2 z ( r r d ) | r r d | 3 ] × exp [ κ ( ω ) | r r d | ] } ( r 0 r 0 ) .

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