Abstract

Time-resolved diffuse optical methods have been applied to detect hemodynamic changes induced by cerebral activity. We describe a near infrared spectroscopic (NIRS) reconstruction free method which allows retrieving depth-related information on absorption variations. Variations in the absorption coefficient of tissues have been computed over the duration of the whole experiment, but also over each temporal step of the time-resolved optical signal, using the microscopic Beer-Lambert law. Finite element simulations show that time-resolved computation of the absorption difference as a function of the propagation time of detected photons is sensitive to the depth profile of optical absorption variations. Differences in deoxyhemoglobin and oxyhemoglobin concentrations can also be calculated from multi-wavelength measurements. Experimental validations of the simulated results have been obtained for resin phantoms. They confirm that time-resolved computation of the absorption differences exhibited completely different behaviours, depending on whether these variations occurred deeply or superficially. The hemodynamic response to a short finger tapping stimulus was measured over the motor cortex and compared to experiments involving Valsalva manoeuvres. Functional maps were also calculated for the hemodynamic response induced by finger tapping movements.

© 2006 Optical Society of America

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  1. B. Chance, Z. Zhuang, C. Unah, C. Alter and L. Lipton, "Cognition-activated low-frequency modulation of light absorption in human brain," Proc. Natl. Acad. Sci. USA 90, 3770-3774 (1993).
    [CrossRef] [PubMed]
  2. A. Villringer and B. Chance, "Non-invasive optical spectroscopy and imaging of human brain function," Trends in Neurosciences 20,435-442 (1997).
    [CrossRef] [PubMed]
  3. D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray and J. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1988).
    [CrossRef] [PubMed]
  4. J. P. Culver, A. M. Siegel, J. J. Stott and D. A. Boas, "Volumetric diffuse optical tomography of brain activity," Opt. Lett. 28, 2061-2063 (2003).
    [CrossRef] [PubMed]
  5. A.Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev and A. H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, part 1: hypercapnia," J. Biomed. Opt. 9, 1046-1062 (2004).
    [CrossRef] [PubMed]
  6. J. C. Hebden, A. Gibson, T. Austin, R. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek and J. S. Wyatt, "Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography," Phys. Med. Biol. 49,1117-1130 (2004).
    [CrossRef] [PubMed]
  7. B. W. Pogue, T. O. McBride, O. L. Osterberg and K. D. Paulsen, "Comparison of imaging geometries for diffuse optical tomography of tissue," Opt. Express 4,270-286 (1999).
    [CrossRef] [PubMed]
  8. C. H. Schmitz, H. L. Graber, Y. L. Pei, M. Farber, M. Stewart, R. D. Levina, M. B. Levin, Y. Xu and R. L. Barbour, "Dynamic studies of small animals with a four-color diffuse optical tomography imager," Rev. Sci. Instrum. 76,094302 (2005).
    [CrossRef]
  9. H. Koizumi, T. Yamamoto, A. Maki, Y. Yamashita, H. Sato, H. Kawaguchi and N. Ichikawa, "Optical topography: practical problems and new applications," Appl. Opt. 42,3054-3062 (2003).
    [CrossRef] [PubMed]
  10. Y. Hoshi, "Functional near-infrared optical imaging: Utility and limitations in human brain mapping," Psychophysiology 40,511-520 (2003).
    [CrossRef] [PubMed]
  11. S. Del Bianco, F. Martelli and G. Zaccanti, "Penetration depth of light re-emitted by a diffusive medium: theoretical and experimental investigation," Phys. Med. Biol. 47,4131-4144 (2002).
    [CrossRef] [PubMed]
  12. J. Selb, J. J. Stott, M. A. Franceschini, A. G. Sorensen and D. A. Boas, "Improved sensitivity to cerebral hemodynamics during brain activation with a time-gated optical system: analytical model and experimental validation," J. Biomed. Opt. 10,011013 (2005).
    [CrossRef]
  13. B. Montcel, R. Chabrier and P. Poulet, "Detection of cortical activation with time-resolved diffuse optical methods," Appl. Opt. 44, 1942-1947 (2005).
    [CrossRef] [PubMed]
  14. J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer and H. Rinneberg, "Determining changes in NIR absorption using a layered model of the human head," Phys. Med. Biol. 46, 879-896 (2001).
    [CrossRef] [PubMed]
  15. A. Liebert, H. Wabnitz, J. Steinbrink, H. Obrig, M. Möller, R. Macdonald, A. Villringer and H. Rinneberg, "Time-resolved multidistance near-infrared spectroscopy of the adult head: intracerebral and extracerebral absorption changes from moments of distribution of times of flight of photons," Appl. Opt. 43, 3037-3047 (2004).
    [CrossRef] [PubMed]
  16. Y. Nomura, O. Hazeki and M. Tamura, "Exponential attenuation of light along nonlinear path through the biological model," Adv. Exp. Med. Biol. 248, 77-80 (1989).
    [CrossRef] [PubMed]
  17. M. Oda, Y. Yamashita, G. Nishimura and M. Tamura, "A simple and novel algorithm for time resolved multiwavelength oximetry," Phys. Med. Biol. 41,551-562 (1996).
    [CrossRef] [PubMed]
  18. Y. Tsuchiya, "Photon path distribution and optical responses of turbid media: theoretical analysis based on the microscopic Beer-Lambert law," Phys. Med. Biol. 46, 2067-2084 (2001).
    [CrossRef] [PubMed]
  19. Y. Nomura, O. Hazeki and M. Tamura, "Relationship between time-resolved and non-time-resolved Beer-Lambert law in turbid media," Phys. Med. Biol. 42,1009-1022 (1997).
    [CrossRef] [PubMed]
  20. F. P. Bolin, L. E. Preuss, C. Taylor and R. J. Ference, "Refractive index of some mammalian tissues using a fiber optic cladding method," Appl. Opt. 28, 2297-2303 (1989).
    [CrossRef] [PubMed]
  21. M. S. Patterson, B. Chance and B. C. Wilson, "Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties," Appl. Opt. 28, 2331-2336 (1989).
    [CrossRef] [PubMed]
  22. S. R. Arridge, "Photon-measurement density functions Part I: Analytical forms," Appl. Opt. 34, 7395-7409 (1995).
    [CrossRef] [PubMed]
  23. Y. Fukui, Y. Ajichi and E. Okada "Monte Carlo prediction of near-infrared light propagation in realistic adult and neonatal head models," Appl. Opt. 42, 2881-2887 (2003).
    [CrossRef] [PubMed]
  24. P. Poulet, C. V. Zint, M. Torregrossa, W. Uhring and B. Cunin, "Comparison of two time-resolved detectors for diffuse optical tomography: photomultiplier tube-time-correlated single photon counting and multichannel streak camera," in Optical Tomography and Spectroscopy of Tissue V, B. Chance, R. R. Alfano, B. J. Tromberg, M. Tamura and E. M. Sevick-Muraca, eds., Proc. SPIE 4955, 154-163 (2003).
    [CrossRef]
  25. L. A. Henderson, P. M. Macey, K. E. Macey, R. C. Frysinger, M. A. Woo, R. K. Harper, J. R. Alger, F. L. Yan-Go and R. M. Harper, "Brain responses associated with the Valsalva maneuver revealed by functional magnetic resonance imaging," J Neurophysiol. 88, 3477-3486 (2002).
    [CrossRef] [PubMed]
  26. J. Mobley and T. Vo-Dinh, "Optical properties of tissue," in Biomedical Photonics Handbook, T. Vo-Dinh ed. (CRC Press, Boca Raton, Fla., 2003).
  27. A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521-528 (2006).
    [CrossRef]
  28. D. Contini, A. Torricelli, A. Pifferi, L. Spinelli, F. Paglia, R. Cubeddu, "Multi-channel time-resolved system for functional near infrared spectroscopy," Opt. Express 14, 5418-5432 (2006).
    [CrossRef] [PubMed]

2006

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521-528 (2006).
[CrossRef]

D. Contini, A. Torricelli, A. Pifferi, L. Spinelli, F. Paglia, R. Cubeddu, "Multi-channel time-resolved system for functional near infrared spectroscopy," Opt. Express 14, 5418-5432 (2006).
[CrossRef] [PubMed]

2005

C. H. Schmitz, H. L. Graber, Y. L. Pei, M. Farber, M. Stewart, R. D. Levina, M. B. Levin, Y. Xu and R. L. Barbour, "Dynamic studies of small animals with a four-color diffuse optical tomography imager," Rev. Sci. Instrum. 76,094302 (2005).
[CrossRef]

J. Selb, J. J. Stott, M. A. Franceschini, A. G. Sorensen and D. A. Boas, "Improved sensitivity to cerebral hemodynamics during brain activation with a time-gated optical system: analytical model and experimental validation," J. Biomed. Opt. 10,011013 (2005).
[CrossRef]

B. Montcel, R. Chabrier and P. Poulet, "Detection of cortical activation with time-resolved diffuse optical methods," Appl. Opt. 44, 1942-1947 (2005).
[CrossRef] [PubMed]

2004

A.Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev and A. H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, part 1: hypercapnia," J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

J. C. Hebden, A. Gibson, T. Austin, R. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek and J. S. Wyatt, "Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography," Phys. Med. Biol. 49,1117-1130 (2004).
[CrossRef] [PubMed]

A. Liebert, H. Wabnitz, J. Steinbrink, H. Obrig, M. Möller, R. Macdonald, A. Villringer and H. Rinneberg, "Time-resolved multidistance near-infrared spectroscopy of the adult head: intracerebral and extracerebral absorption changes from moments of distribution of times of flight of photons," Appl. Opt. 43, 3037-3047 (2004).
[CrossRef] [PubMed]

2003

2002

S. Del Bianco, F. Martelli and G. Zaccanti, "Penetration depth of light re-emitted by a diffusive medium: theoretical and experimental investigation," Phys. Med. Biol. 47,4131-4144 (2002).
[CrossRef] [PubMed]

L. A. Henderson, P. M. Macey, K. E. Macey, R. C. Frysinger, M. A. Woo, R. K. Harper, J. R. Alger, F. L. Yan-Go and R. M. Harper, "Brain responses associated with the Valsalva maneuver revealed by functional magnetic resonance imaging," J Neurophysiol. 88, 3477-3486 (2002).
[CrossRef] [PubMed]

2001

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer and H. Rinneberg, "Determining changes in NIR absorption using a layered model of the human head," Phys. Med. Biol. 46, 879-896 (2001).
[CrossRef] [PubMed]

Y. Tsuchiya, "Photon path distribution and optical responses of turbid media: theoretical analysis based on the microscopic Beer-Lambert law," Phys. Med. Biol. 46, 2067-2084 (2001).
[CrossRef] [PubMed]

1999

1997

Y. Nomura, O. Hazeki and M. Tamura, "Relationship between time-resolved and non-time-resolved Beer-Lambert law in turbid media," Phys. Med. Biol. 42,1009-1022 (1997).
[CrossRef] [PubMed]

A. Villringer and B. Chance, "Non-invasive optical spectroscopy and imaging of human brain function," Trends in Neurosciences 20,435-442 (1997).
[CrossRef] [PubMed]

1996

M. Oda, Y. Yamashita, G. Nishimura and M. Tamura, "A simple and novel algorithm for time resolved multiwavelength oximetry," Phys. Med. Biol. 41,551-562 (1996).
[CrossRef] [PubMed]

1995

1993

B. Chance, Z. Zhuang, C. Unah, C. Alter and L. Lipton, "Cognition-activated low-frequency modulation of light absorption in human brain," Proc. Natl. Acad. Sci. USA 90, 3770-3774 (1993).
[CrossRef] [PubMed]

1989

1988

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray and J. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1988).
[CrossRef] [PubMed]

Abdoulaev, G. S.

A.Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev and A. H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, part 1: hypercapnia," J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

Ajichi, Y.

Alger, J. R.

L. A. Henderson, P. M. Macey, K. E. Macey, R. C. Frysinger, M. A. Woo, R. K. Harper, J. R. Alger, F. L. Yan-Go and R. M. Harper, "Brain responses associated with the Valsalva maneuver revealed by functional magnetic resonance imaging," J Neurophysiol. 88, 3477-3486 (2002).
[CrossRef] [PubMed]

Alter, C.

B. Chance, Z. Zhuang, C. Unah, C. Alter and L. Lipton, "Cognition-activated low-frequency modulation of light absorption in human brain," Proc. Natl. Acad. Sci. USA 90, 3770-3774 (1993).
[CrossRef] [PubMed]

Arridge, S.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray and J. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1988).
[CrossRef] [PubMed]

Arridge, S. R.

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521-528 (2006).
[CrossRef]

J. C. Hebden, A. Gibson, T. Austin, R. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek and J. S. Wyatt, "Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography," Phys. Med. Biol. 49,1117-1130 (2004).
[CrossRef] [PubMed]

S. R. Arridge, "Photon-measurement density functions Part I: Analytical forms," Appl. Opt. 34, 7395-7409 (1995).
[CrossRef] [PubMed]

Austin, T.

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521-528 (2006).
[CrossRef]

J. C. Hebden, A. Gibson, T. Austin, R. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek and J. S. Wyatt, "Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography," Phys. Med. Biol. 49,1117-1130 (2004).
[CrossRef] [PubMed]

Barbour, R. L.

C. H. Schmitz, H. L. Graber, Y. L. Pei, M. Farber, M. Stewart, R. D. Levina, M. B. Levin, Y. Xu and R. L. Barbour, "Dynamic studies of small animals with a four-color diffuse optical tomography imager," Rev. Sci. Instrum. 76,094302 (2005).
[CrossRef]

Bluestone, A.Y.

A.Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev and A. H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, part 1: hypercapnia," J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

Boas, D. A.

J. Selb, J. J. Stott, M. A. Franceschini, A. G. Sorensen and D. A. Boas, "Improved sensitivity to cerebral hemodynamics during brain activation with a time-gated optical system: analytical model and experimental validation," J. Biomed. Opt. 10,011013 (2005).
[CrossRef]

J. P. Culver, A. M. Siegel, J. J. Stott and D. A. Boas, "Volumetric diffuse optical tomography of brain activity," Opt. Lett. 28, 2061-2063 (2003).
[CrossRef] [PubMed]

Bolin, F. P.

Chabrier, R.

Chance, B.

A. Villringer and B. Chance, "Non-invasive optical spectroscopy and imaging of human brain function," Trends in Neurosciences 20,435-442 (1997).
[CrossRef] [PubMed]

B. Chance, Z. Zhuang, C. Unah, C. Alter and L. Lipton, "Cognition-activated low-frequency modulation of light absorption in human brain," Proc. Natl. Acad. Sci. USA 90, 3770-3774 (1993).
[CrossRef] [PubMed]

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

Contini, D.

Cope, M.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray and J. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1988).
[CrossRef] [PubMed]

Cubeddu, R.

Culver, J. P.

Del Bianco, S.

S. Del Bianco, F. Martelli and G. Zaccanti, "Penetration depth of light re-emitted by a diffusive medium: theoretical and experimental investigation," Phys. Med. Biol. 47,4131-4144 (2002).
[CrossRef] [PubMed]

Delpy, D. T.

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521-528 (2006).
[CrossRef]

J. C. Hebden, A. Gibson, T. Austin, R. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek and J. S. Wyatt, "Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography," Phys. Med. Biol. 49,1117-1130 (2004).
[CrossRef] [PubMed]

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray and J. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1988).
[CrossRef] [PubMed]

Everdell, N.

J. C. Hebden, A. Gibson, T. Austin, R. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek and J. S. Wyatt, "Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography," Phys. Med. Biol. 49,1117-1130 (2004).
[CrossRef] [PubMed]

Everdell, N. L.

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521-528 (2006).
[CrossRef]

Farber, M.

C. H. Schmitz, H. L. Graber, Y. L. Pei, M. Farber, M. Stewart, R. D. Levina, M. B. Levin, Y. Xu and R. L. Barbour, "Dynamic studies of small animals with a four-color diffuse optical tomography imager," Rev. Sci. Instrum. 76,094302 (2005).
[CrossRef]

Ference, R. J.

Franceschini, M. A.

J. Selb, J. J. Stott, M. A. Franceschini, A. G. Sorensen and D. A. Boas, "Improved sensitivity to cerebral hemodynamics during brain activation with a time-gated optical system: analytical model and experimental validation," J. Biomed. Opt. 10,011013 (2005).
[CrossRef]

Frysinger, R. C.

L. A. Henderson, P. M. Macey, K. E. Macey, R. C. Frysinger, M. A. Woo, R. K. Harper, J. R. Alger, F. L. Yan-Go and R. M. Harper, "Brain responses associated with the Valsalva maneuver revealed by functional magnetic resonance imaging," J Neurophysiol. 88, 3477-3486 (2002).
[CrossRef] [PubMed]

Fukui, Y.

Gibson, A.

J. C. Hebden, A. Gibson, T. Austin, R. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek and J. S. Wyatt, "Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography," Phys. Med. Biol. 49,1117-1130 (2004).
[CrossRef] [PubMed]

Gibson, A. P.

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521-528 (2006).
[CrossRef]

Graber, H. L.

C. H. Schmitz, H. L. Graber, Y. L. Pei, M. Farber, M. Stewart, R. D. Levina, M. B. Levin, Y. Xu and R. L. Barbour, "Dynamic studies of small animals with a four-color diffuse optical tomography imager," Rev. Sci. Instrum. 76,094302 (2005).
[CrossRef]

Harper, R. K.

L. A. Henderson, P. M. Macey, K. E. Macey, R. C. Frysinger, M. A. Woo, R. K. Harper, J. R. Alger, F. L. Yan-Go and R. M. Harper, "Brain responses associated with the Valsalva maneuver revealed by functional magnetic resonance imaging," J Neurophysiol. 88, 3477-3486 (2002).
[CrossRef] [PubMed]

Harper, R. M.

L. A. Henderson, P. M. Macey, K. E. Macey, R. C. Frysinger, M. A. Woo, R. K. Harper, J. R. Alger, F. L. Yan-Go and R. M. Harper, "Brain responses associated with the Valsalva maneuver revealed by functional magnetic resonance imaging," J Neurophysiol. 88, 3477-3486 (2002).
[CrossRef] [PubMed]

Hazeki, O.

Y. Nomura, O. Hazeki and M. Tamura, "Relationship between time-resolved and non-time-resolved Beer-Lambert law in turbid media," Phys. Med. Biol. 42,1009-1022 (1997).
[CrossRef] [PubMed]

Y. Nomura, O. Hazeki and M. Tamura, "Exponential attenuation of light along nonlinear path through the biological model," Adv. Exp. Med. Biol. 248, 77-80 (1989).
[CrossRef] [PubMed]

Hebden, J. C.

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521-528 (2006).
[CrossRef]

J. C. Hebden, A. Gibson, T. Austin, R. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek and J. S. Wyatt, "Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography," Phys. Med. Biol. 49,1117-1130 (2004).
[CrossRef] [PubMed]

Henderson, L. A.

L. A. Henderson, P. M. Macey, K. E. Macey, R. C. Frysinger, M. A. Woo, R. K. Harper, J. R. Alger, F. L. Yan-Go and R. M. Harper, "Brain responses associated with the Valsalva maneuver revealed by functional magnetic resonance imaging," J Neurophysiol. 88, 3477-3486 (2002).
[CrossRef] [PubMed]

Hielscher, A. H.

A.Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev and A. H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, part 1: hypercapnia," J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

Hoshi, Y.

Y. Hoshi, "Functional near-infrared optical imaging: Utility and limitations in human brain mapping," Psychophysiology 40,511-520 (2003).
[CrossRef] [PubMed]

Ichikawa, N.

Kawaguchi, H.

Koizumi, H.

Lasker, J.

A.Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev and A. H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, part 1: hypercapnia," J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

Levin, M. B.

C. H. Schmitz, H. L. Graber, Y. L. Pei, M. Farber, M. Stewart, R. D. Levina, M. B. Levin, Y. Xu and R. L. Barbour, "Dynamic studies of small animals with a four-color diffuse optical tomography imager," Rev. Sci. Instrum. 76,094302 (2005).
[CrossRef]

Levina, R. D.

C. H. Schmitz, H. L. Graber, Y. L. Pei, M. Farber, M. Stewart, R. D. Levina, M. B. Levin, Y. Xu and R. L. Barbour, "Dynamic studies of small animals with a four-color diffuse optical tomography imager," Rev. Sci. Instrum. 76,094302 (2005).
[CrossRef]

Liebert, A.

Lipton, L.

B. Chance, Z. Zhuang, C. Unah, C. Alter and L. Lipton, "Cognition-activated low-frequency modulation of light absorption in human brain," Proc. Natl. Acad. Sci. USA 90, 3770-3774 (1993).
[CrossRef] [PubMed]

Macdonald, R.

Macey, K. E.

L. A. Henderson, P. M. Macey, K. E. Macey, R. C. Frysinger, M. A. Woo, R. K. Harper, J. R. Alger, F. L. Yan-Go and R. M. Harper, "Brain responses associated with the Valsalva maneuver revealed by functional magnetic resonance imaging," J Neurophysiol. 88, 3477-3486 (2002).
[CrossRef] [PubMed]

Macey, P. M.

L. A. Henderson, P. M. Macey, K. E. Macey, R. C. Frysinger, M. A. Woo, R. K. Harper, J. R. Alger, F. L. Yan-Go and R. M. Harper, "Brain responses associated with the Valsalva maneuver revealed by functional magnetic resonance imaging," J Neurophysiol. 88, 3477-3486 (2002).
[CrossRef] [PubMed]

Maki, A.

Martelli, F.

S. Del Bianco, F. Martelli and G. Zaccanti, "Penetration depth of light re-emitted by a diffusive medium: theoretical and experimental investigation," Phys. Med. Biol. 47,4131-4144 (2002).
[CrossRef] [PubMed]

McBride, T. O.

Meek, J. H.

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521-528 (2006).
[CrossRef]

J. C. Hebden, A. Gibson, T. Austin, R. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek and J. S. Wyatt, "Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography," Phys. Med. Biol. 49,1117-1130 (2004).
[CrossRef] [PubMed]

Möller, M.

Montcel, B.

Nishimura, G.

M. Oda, Y. Yamashita, G. Nishimura and M. Tamura, "A simple and novel algorithm for time resolved multiwavelength oximetry," Phys. Med. Biol. 41,551-562 (1996).
[CrossRef] [PubMed]

Nomura, Y.

Y. Nomura, O. Hazeki and M. Tamura, "Relationship between time-resolved and non-time-resolved Beer-Lambert law in turbid media," Phys. Med. Biol. 42,1009-1022 (1997).
[CrossRef] [PubMed]

Y. Nomura, O. Hazeki and M. Tamura, "Exponential attenuation of light along nonlinear path through the biological model," Adv. Exp. Med. Biol. 248, 77-80 (1989).
[CrossRef] [PubMed]

Obrig, H.

Oda, M.

M. Oda, Y. Yamashita, G. Nishimura and M. Tamura, "A simple and novel algorithm for time resolved multiwavelength oximetry," Phys. Med. Biol. 41,551-562 (1996).
[CrossRef] [PubMed]

Okada, E.

Osterberg, O. L.

Paglia, F.

Patterson, M. S.

Paulsen, K. D.

Pei, Y. L.

C. H. Schmitz, H. L. Graber, Y. L. Pei, M. Farber, M. Stewart, R. D. Levina, M. B. Levin, Y. Xu and R. L. Barbour, "Dynamic studies of small animals with a four-color diffuse optical tomography imager," Rev. Sci. Instrum. 76,094302 (2005).
[CrossRef]

Pifferi, A.

Pogue, B. W.

Poulet, P.

Preuss, L. E.

Rinneberg, H.

Sato, H.

Schmitz, C. H.

C. H. Schmitz, H. L. Graber, Y. L. Pei, M. Farber, M. Stewart, R. D. Levina, M. B. Levin, Y. Xu and R. L. Barbour, "Dynamic studies of small animals with a four-color diffuse optical tomography imager," Rev. Sci. Instrum. 76,094302 (2005).
[CrossRef]

Schweiger, M.

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521-528 (2006).
[CrossRef]

Selb, J.

J. Selb, J. J. Stott, M. A. Franceschini, A. G. Sorensen and D. A. Boas, "Improved sensitivity to cerebral hemodynamics during brain activation with a time-gated optical system: analytical model and experimental validation," J. Biomed. Opt. 10,011013 (2005).
[CrossRef]

Siegel, A. M.

Sorensen, A. G.

J. Selb, J. J. Stott, M. A. Franceschini, A. G. Sorensen and D. A. Boas, "Improved sensitivity to cerebral hemodynamics during brain activation with a time-gated optical system: analytical model and experimental validation," J. Biomed. Opt. 10,011013 (2005).
[CrossRef]

Spinelli, L.

Steinbrink, J.

Stewart, M.

C. H. Schmitz, H. L. Graber, Y. L. Pei, M. Farber, M. Stewart, R. D. Levina, M. B. Levin, Y. Xu and R. L. Barbour, "Dynamic studies of small animals with a four-color diffuse optical tomography imager," Rev. Sci. Instrum. 76,094302 (2005).
[CrossRef]

A.Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev and A. H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, part 1: hypercapnia," J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

Stott, J. J.

J. Selb, J. J. Stott, M. A. Franceschini, A. G. Sorensen and D. A. Boas, "Improved sensitivity to cerebral hemodynamics during brain activation with a time-gated optical system: analytical model and experimental validation," J. Biomed. Opt. 10,011013 (2005).
[CrossRef]

J. P. Culver, A. M. Siegel, J. J. Stott and D. A. Boas, "Volumetric diffuse optical tomography of brain activity," Opt. Lett. 28, 2061-2063 (2003).
[CrossRef] [PubMed]

Tamura, M.

Y. Nomura, O. Hazeki and M. Tamura, "Relationship between time-resolved and non-time-resolved Beer-Lambert law in turbid media," Phys. Med. Biol. 42,1009-1022 (1997).
[CrossRef] [PubMed]

M. Oda, Y. Yamashita, G. Nishimura and M. Tamura, "A simple and novel algorithm for time resolved multiwavelength oximetry," Phys. Med. Biol. 41,551-562 (1996).
[CrossRef] [PubMed]

Y. Nomura, O. Hazeki and M. Tamura, "Exponential attenuation of light along nonlinear path through the biological model," Adv. Exp. Med. Biol. 248, 77-80 (1989).
[CrossRef] [PubMed]

Taylor, C.

Torricelli, A.

Tsuchiya, Y.

Y. Tsuchiya, "Photon path distribution and optical responses of turbid media: theoretical analysis based on the microscopic Beer-Lambert law," Phys. Med. Biol. 46, 2067-2084 (2001).
[CrossRef] [PubMed]

Unah, C.

B. Chance, Z. Zhuang, C. Unah, C. Alter and L. Lipton, "Cognition-activated low-frequency modulation of light absorption in human brain," Proc. Natl. Acad. Sci. USA 90, 3770-3774 (1993).
[CrossRef] [PubMed]

Van der Zee, P.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray and J. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1988).
[CrossRef] [PubMed]

Villringer, A.

A. Liebert, H. Wabnitz, J. Steinbrink, H. Obrig, M. Möller, R. Macdonald, A. Villringer and H. Rinneberg, "Time-resolved multidistance near-infrared spectroscopy of the adult head: intracerebral and extracerebral absorption changes from moments of distribution of times of flight of photons," Appl. Opt. 43, 3037-3047 (2004).
[CrossRef] [PubMed]

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer and H. Rinneberg, "Determining changes in NIR absorption using a layered model of the human head," Phys. Med. Biol. 46, 879-896 (2001).
[CrossRef] [PubMed]

A. Villringer and B. Chance, "Non-invasive optical spectroscopy and imaging of human brain function," Trends in Neurosciences 20,435-442 (1997).
[CrossRef] [PubMed]

Wabnitz, H.

Wilson, B. C.

Woo, M. A.

L. A. Henderson, P. M. Macey, K. E. Macey, R. C. Frysinger, M. A. Woo, R. K. Harper, J. R. Alger, F. L. Yan-Go and R. M. Harper, "Brain responses associated with the Valsalva maneuver revealed by functional magnetic resonance imaging," J Neurophysiol. 88, 3477-3486 (2002).
[CrossRef] [PubMed]

Wray, S.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray and 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.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray and 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.

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521-528 (2006).
[CrossRef]

J. C. Hebden, A. Gibson, T. Austin, R. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek and J. S. Wyatt, "Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography," Phys. Med. Biol. 49,1117-1130 (2004).
[CrossRef] [PubMed]

Xu, Y.

C. H. Schmitz, H. L. Graber, Y. L. Pei, M. Farber, M. Stewart, R. D. Levina, M. B. Levin, Y. Xu and R. L. Barbour, "Dynamic studies of small animals with a four-color diffuse optical tomography imager," Rev. Sci. Instrum. 76,094302 (2005).
[CrossRef]

Yamamoto, T.

Yamashita, Y.

H. Koizumi, T. Yamamoto, A. Maki, Y. Yamashita, H. Sato, H. Kawaguchi and N. Ichikawa, "Optical topography: practical problems and new applications," Appl. Opt. 42,3054-3062 (2003).
[CrossRef] [PubMed]

M. Oda, Y. Yamashita, G. Nishimura and M. Tamura, "A simple and novel algorithm for time resolved multiwavelength oximetry," Phys. Med. Biol. 41,551-562 (1996).
[CrossRef] [PubMed]

Yan-Go, F. L.

L. A. Henderson, P. M. Macey, K. E. Macey, R. C. Frysinger, M. A. Woo, R. K. Harper, J. R. Alger, F. L. Yan-Go and R. M. Harper, "Brain responses associated with the Valsalva maneuver revealed by functional magnetic resonance imaging," J Neurophysiol. 88, 3477-3486 (2002).
[CrossRef] [PubMed]

Yusof, R.

J. C. Hebden, A. Gibson, T. Austin, R. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek and J. S. Wyatt, "Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography," Phys. Med. Biol. 49,1117-1130 (2004).
[CrossRef] [PubMed]

Zaccanti, G.

S. Del Bianco, F. Martelli and G. Zaccanti, "Penetration depth of light re-emitted by a diffusive medium: theoretical and experimental investigation," Phys. Med. Biol. 47,4131-4144 (2002).
[CrossRef] [PubMed]

Zhuang, Z.

B. Chance, Z. Zhuang, C. Unah, C. Alter and L. Lipton, "Cognition-activated low-frequency modulation of light absorption in human brain," Proc. Natl. Acad. Sci. USA 90, 3770-3774 (1993).
[CrossRef] [PubMed]

Adv. Exp. Med. Biol.

Y. Nomura, O. Hazeki and M. Tamura, "Exponential attenuation of light along nonlinear path through the biological model," Adv. Exp. Med. Biol. 248, 77-80 (1989).
[CrossRef] [PubMed]

Appl. Opt.

J Neurophysiol.

L. A. Henderson, P. M. Macey, K. E. Macey, R. C. Frysinger, M. A. Woo, R. K. Harper, J. R. Alger, F. L. Yan-Go and R. M. Harper, "Brain responses associated with the Valsalva maneuver revealed by functional magnetic resonance imaging," J Neurophysiol. 88, 3477-3486 (2002).
[CrossRef] [PubMed]

J. Biomed. Opt.

J. Selb, J. J. Stott, M. A. Franceschini, A. G. Sorensen and D. A. Boas, "Improved sensitivity to cerebral hemodynamics during brain activation with a time-gated optical system: analytical model and experimental validation," J. Biomed. Opt. 10,011013 (2005).
[CrossRef]

A.Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev and A. H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, part 1: hypercapnia," J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

Neuroimage

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521-528 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray and J. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1988).
[CrossRef] [PubMed]

J. C. Hebden, A. Gibson, T. Austin, R. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek and J. S. Wyatt, "Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography," Phys. Med. Biol. 49,1117-1130 (2004).
[CrossRef] [PubMed]

S. Del Bianco, F. Martelli and G. Zaccanti, "Penetration depth of light re-emitted by a diffusive medium: theoretical and experimental investigation," Phys. Med. Biol. 47,4131-4144 (2002).
[CrossRef] [PubMed]

M. Oda, Y. Yamashita, G. Nishimura and M. Tamura, "A simple and novel algorithm for time resolved multiwavelength oximetry," Phys. Med. Biol. 41,551-562 (1996).
[CrossRef] [PubMed]

Y. Tsuchiya, "Photon path distribution and optical responses of turbid media: theoretical analysis based on the microscopic Beer-Lambert law," Phys. Med. Biol. 46, 2067-2084 (2001).
[CrossRef] [PubMed]

Y. Nomura, O. Hazeki and M. Tamura, "Relationship between time-resolved and non-time-resolved Beer-Lambert law in turbid media," Phys. Med. Biol. 42,1009-1022 (1997).
[CrossRef] [PubMed]

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer and H. Rinneberg, "Determining changes in NIR absorption using a layered model of the human head," Phys. Med. Biol. 46, 879-896 (2001).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA

B. Chance, Z. Zhuang, C. Unah, C. Alter and L. Lipton, "Cognition-activated low-frequency modulation of light absorption in human brain," Proc. Natl. Acad. Sci. USA 90, 3770-3774 (1993).
[CrossRef] [PubMed]

Psychophysiology

Y. Hoshi, "Functional near-infrared optical imaging: Utility and limitations in human brain mapping," Psychophysiology 40,511-520 (2003).
[CrossRef] [PubMed]

Rev. Sci. Instrum.

C. H. Schmitz, H. L. Graber, Y. L. Pei, M. Farber, M. Stewart, R. D. Levina, M. B. Levin, Y. Xu and R. L. Barbour, "Dynamic studies of small animals with a four-color diffuse optical tomography imager," Rev. Sci. Instrum. 76,094302 (2005).
[CrossRef]

Trends in Neurosciences

A. Villringer and B. Chance, "Non-invasive optical spectroscopy and imaging of human brain function," Trends in Neurosciences 20,435-442 (1997).
[CrossRef] [PubMed]

Other

J. Mobley and T. Vo-Dinh, "Optical properties of tissue," in Biomedical Photonics Handbook, T. Vo-Dinh ed. (CRC Press, Boca Raton, Fla., 2003).

P. Poulet, C. V. Zint, M. Torregrossa, W. Uhring and B. Cunin, "Comparison of two time-resolved detectors for diffuse optical tomography: photomultiplier tube-time-correlated single photon counting and multichannel streak camera," in Optical Tomography and Spectroscopy of Tissue V, B. Chance, R. R. Alfano, B. J. Tromberg, M. Tamura and E. M. Sevick-Muraca, eds., Proc. SPIE 4955, 154-163 (2003).
[CrossRef]

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Figures (10)

Fig.1.
Fig.1.

(a). Time-resolved absorption difference for 9 different depths of the inclusion (1, 4, 7, 10, 13, 16, 19, 22 and 25 mm below the surface). Source-detector separation 30mm. The simulated TPSF is also plotted, with a different vertical scale, for easier reading. The arrow indicates the different time-resolved absorption difference curves from 1 to 25 mm. (b). Time position of the maximum of the time-resolved absorption difference as a function of the depth of the inclusion. The values for depth greater than 20 mm are saturated to 5 ns only because of the limited time window of the simulations.

Fig. 2.
Fig. 2.

FEM simulation (solid line) and experimentally measured (dots) time-resolved absorption difference between rest and activation for an absorbing inclusion in the “gray matter” layer of the head-like phantom. Measurements at 690 nm for a source-detector distance of 30 mm.

Fig. 3.
Fig. 3.

FEM simulation (solid line) and experimentally measured (dots) time-resolved absorption difference between rest and activation for an absorbing inclusion in the “scalp layer” of the head-like phantom. Measurements at 690 nm for a source-detector distance of 30 mm.

Fig. 4.
Fig. 4.

(a). Finger tapping stimulus, (b). Valsalva manoeuvre. Statistical significance of the differences between the two time-resolved absorption differences at 830 nm, computed after binning the data by 2. The first sample is for the activation period and the second one for the rest period. The retained time window corresponding to a threshold of 5 % (p-value < 0.05) is indicated by the vertical dotted lines.

Fig. 5.
Fig. 5.

Finger tapping stimulus: Experimental absorption differences at 690 nm (x) and 830 nm (dots). Filtered absorption differences at 690 nm (dashed line) and 830 nm (solid line). The short finger tapping period started at the time 0 (vertical dashed line).

Fig. 6.
Fig. 6.

Finger tapping stimulus: Time-resolved absorption difference maps at 690 nm (a) and 830 nm (b). The short finger tapping period started at the time 0 (vertical dashed line). Valsalva manoeuvre: Time-resolved absorption difference maps at 690 nm (c) and 830 nm (d). The Valsalva manoeuvre was performed between 0 and 5 s (vertical dashed lines).

Fig. 7.
Fig. 7.

Valsalva manoeuvre: Absorption differences at 690 nm (x) and 830 nm (dots). Filtered absorption differences at 690 nm (dashed line) and 830 nm (solid line). The Valsalva manoeuvre was performed between 0 and 5 s (vertical dashed lines).

Fig. 8.
Fig. 8.

Finger tapping experiment: Bottom: Time-resolved HbO2-concentration differences for the stimulation period and for the rest period (error bars=2 times the standard error). Up: Statistical significance of the difference between these two time-resolved HbO2-concentration differences. The retained time window (0.3 to 1.4 ns) corresponding to a threshold of 1 % (p-value < 0.01) is indicated by the vertical dotted lines.

Fig. 9.
Fig. 9.

Finger tapping period: HbO2- (dot) and Hb- (+) concentration differences. Filtered HbO2- (dashed line) and Hb- (solid line) concentration differences. The finger tapping period started at the time 0 and ended at 31 s (vertical dashed lines).

Fig. 10.
Fig. 10.

Finger tapping period: Time-resolved HbO2- (a) and Hb- (b) concentration difference maps. The finger tapping period started at the time 0 and ended at 31 s (vertical dashed lines).

Tables (1)

Tables Icon

Table 1. Absorption coefficient μ a, transport scattering coefficient μs and thickness of each tissue layer of the headlike phantom, for a wavelength equal to 690 nm.

Equations (10)

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

I λ ( t , T ) = I 0 λ ( t ) exp ( μ ( t , T ) vt )
Δ A λ ( t , T T 0 ) = log ( I λ ( t , T 0 ) I λ ( t , T ) ) = Δ μ ( t , T T 0 ) vt ln ( 10 )
I ( r s , r d , t ) = ( 4 π D v ) 3 2 z 0 t 5 2 exp ( ρ s d 2 + z 0 2 4 D v t μ a vt )
Δ I ( r s , r d , r pert , t ) = z pert ( Δ μ abs V ) ( 2 π ) 3 2 ρ pert d [ S ( ρ s pert , ρ pert d , t ) S ( ρ s + pert , ρ pert d , t ) ]
S ( x , y , t ) = ( 1 y 2 + x + y 2 Dvt ( 1 x + 1 y ) ) G ( x + y , t )
G ( r , t ) = v ( 4 π D v t ) 3 2 exp ( r 2 4 D v t μ a v t )
Δ I I = vtz pert ( Δ μ abs V ) ( 2 π ) 3 2 ρ pert d z 0 [ f ( ρ s pert , ρ pert d , t ) f ( ρ s + pert , ρ pert d , t ) ]
f ( x , y , t ) = ( 1 y 2 + x + y 2 D v t ( 1 x + 1 y ) ) exp [ r 2 + z 0 2 ( x + y ) 2 4 D v t ]
Δ μ a ( t ) = ln ( 1 Δ I I ) vt
[ ε HbO 2 λ 1 ε Hb λ 1 ε HbO 2 λ 2 ε Hb λ 2 ] [ Δ C HbO 2 ( t , T T 0 ) Δ C Hb ( t , T T 0 ) ] = [ Δ μ 1 ( t , T T 0 ) Δ μ 2 ( t , T T 0 ) ]

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