Abstract

The attenuation of light by scattering and absorbing media is nonlinearly dependent upon the absorption coefficient, since detected light has experienced many different flight times. The frequency response of such a sample to modulated light is also nonlinear. We derive an expression for the attenuation that includes both the absorption coefficient and the modulation frequency. Its form is a power series whose coefficients are the cumulants of the temporal point-spread function (TPSF). Recasting this expression in terms of intensity leads to a similar expression with the cumulants replaced by the moments of the TPSF. A means of exploiting this relationship to produce estimates of the absolute concentration of the absorbing species is suggested.

© 2007 Optical Society of America

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  1. D. T. Delpy and M. Cope, "Quantification in tissue near-infrared spectroscopy," Philos. Trans. R. Soc. London, Ser. B 352, 649-659 (1997).
    [CrossRef]
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  5. J. Mayhew, Y. Zheng, Y. Hou, B. Vuksanovic, J. Berwick, S. Askew, and P. Coffey, "Spectroscopic analysis of changes in remitted illumination: the response to increased neural activity in brain," Neuroimage 10, 304-326 (1999).
    [CrossRef] [PubMed]
  6. S. J. Matcher, M. Cope, and D. T. Delpy, "Use of the water absorption spectrum to quantify tissue chromophore concentration changes in near-infrared spectroscopy," Phys. Med. Biol. 38, 177-196 (1993).
  7. S. J. Matcher and C. E. Cooper, "Absolute quantification of deoxyhaemoglobin concentration in tissue near infrared spectroscopy," Phys. Med. Biol. 39, 1295-1312 (1994).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  10. B. J. Tromberg, L. O. Svaasand, T.-T. Tsay, and R. C. Haskell, "Properties of photon density waves in multiple-scattering media," Appl. Opt. 32, 607-616 (1993).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  12. 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).
    [PubMed]
  13. Y. Hasegawa, Y. Yamada, M. Tamura, and Y. Nomura, "Monte Carlo simulation of light transmission through living tissues," Appl. Opt. 30, 4515-4520 (1991).
    [CrossRef] [PubMed]
  14. A. Sassaroli and S. Fantini, "Comment on the modified Beer-Lambert law for scattering media," Phys. Med. Biol. 49, N255-N257 (1997).
    [CrossRef]
  15. A. Sassaroli, F. Martelli, and S. Fantini, "Perturbation theory for the diffusion equation by use of the moments of the generalized temporal point-spread function. II. Continuous-wave results," J. Opt. Soc. Am. A 23, 2119-2131 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
  18. M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, "A Monte Carlo investigation of optical pathlength in homogeneous tissue and its application to near-infrared spectroscopy," Phys. Med. Biol. 38, 1859-1876 (1993).
    [CrossRef] [PubMed]
  19. D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. S. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1997).
    [CrossRef]
  20. P. Wobst, R. Wenzel, M. Kohl, H. Obriq, and A. Villringer, "Linear aspects of changes in deoxygenated hemoglobin concentration and cytochrome oxidase oxidation during brain activation," Neuroimage 13, 520-530 (2001).
    [CrossRef] [PubMed]

2006 (1)

2001 (2)

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]

P. Wobst, R. Wenzel, M. Kohl, H. Obriq, and A. Villringer, "Linear aspects of changes in deoxygenated hemoglobin concentration and cytochrome oxidase oxidation during brain activation," Neuroimage 13, 520-530 (2001).
[CrossRef] [PubMed]

1999 (1)

J. Mayhew, Y. Zheng, Y. Hou, B. Vuksanovic, J. Berwick, S. Askew, and P. Coffey, "Spectroscopic analysis of changes in remitted illumination: the response to increased neural activity in brain," Neuroimage 10, 304-326 (1999).
[CrossRef] [PubMed]

1997 (3)

D. T. Delpy and M. Cope, "Quantification in tissue near-infrared spectroscopy," Philos. Trans. R. Soc. London, Ser. B 352, 649-659 (1997).
[CrossRef]

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. S. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1997).
[CrossRef]

A. Sassaroli and S. Fantini, "Comment on the modified Beer-Lambert law for scattering media," Phys. Med. Biol. 49, N255-N257 (1997).
[CrossRef]

1995 (1)

A. Duncan, J. H. Meek, M. Clemence, C. E. Elwell, L. Tyszczuk, M. Cope, and D. Delpy, "Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy," Phys. Med. Biol. 40, 295-304 (1995).
[CrossRef] [PubMed]

1994 (1)

S. J. Matcher and C. E. Cooper, "Absolute quantification of deoxyhaemoglobin concentration in tissue near infrared spectroscopy," Phys. Med. Biol. 39, 1295-1312 (1994).
[CrossRef] [PubMed]

1993 (3)

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, "A Monte Carlo investigation of optical pathlength in homogeneous tissue and its application to near-infrared spectroscopy," Phys. Med. Biol. 38, 1859-1876 (1993).
[CrossRef] [PubMed]

S. J. Matcher, M. Cope, and D. T. Delpy, "Use of the water absorption spectrum to quantify tissue chromophore concentration changes in near-infrared spectroscopy," Phys. Med. Biol. 38, 177-196 (1993).

B. J. Tromberg, L. O. Svaasand, T.-T. Tsay, and R. C. Haskell, "Properties of photon density waves in multiple-scattering media," Appl. Opt. 32, 607-616 (1993).
[CrossRef] [PubMed]

1992 (1)

S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 47, 1531-1560 (1992).
[CrossRef]

1991 (1)

1989 (2)

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).
[PubMed]

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

1975 (2)

R. N. Pittman and B. R. Duling, "A new method for the measurement of percent oxyhemoglobin," J. Appl. Physiol. 38, 315-320 (1975).

D. W. Lübbers and R. Wodick, "Absolute reflection photometry applied to the measurement of capillary oxyhemoglobin saturation of the skin in man," in Oxygen Measurements in Biology and Medicine, J.P.Payne and D.W.Hill, eds. (Butterworths, 1975), pp. 85-110.

1970 (1)

1964 (1)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th Dover printing, 10th U.S. GPO printing ed. (Dover, 1964), p. 928.

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th Dover printing, 10th U.S. GPO printing ed. (Dover, 1964), p. 928.

Arridge, S. R.

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. S. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1997).
[CrossRef]

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, "A Monte Carlo investigation of optical pathlength in homogeneous tissue and its application to near-infrared spectroscopy," Phys. Med. Biol. 38, 1859-1876 (1993).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 47, 1531-1560 (1992).
[CrossRef]

Askew, S.

J. Mayhew, Y. Zheng, Y. Hou, B. Vuksanovic, J. Berwick, S. Askew, and P. Coffey, "Spectroscopic analysis of changes in remitted illumination: the response to increased neural activity in brain," Neuroimage 10, 304-326 (1999).
[CrossRef] [PubMed]

Berwick, J.

J. Mayhew, Y. Zheng, Y. Hou, B. Vuksanovic, J. Berwick, S. Askew, and P. Coffey, "Spectroscopic analysis of changes in remitted illumination: the response to increased neural activity in brain," Neuroimage 10, 304-326 (1999).
[CrossRef] [PubMed]

Chance, B.

Clemence, M.

A. Duncan, J. H. Meek, M. Clemence, C. E. Elwell, L. Tyszczuk, M. Cope, and D. Delpy, "Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy," Phys. Med. Biol. 40, 295-304 (1995).
[CrossRef] [PubMed]

Coffey, P.

J. Mayhew, Y. Zheng, Y. Hou, B. Vuksanovic, J. Berwick, S. Askew, and P. Coffey, "Spectroscopic analysis of changes in remitted illumination: the response to increased neural activity in brain," Neuroimage 10, 304-326 (1999).
[CrossRef] [PubMed]

Cooper, C. E.

S. J. Matcher and C. E. Cooper, "Absolute quantification of deoxyhaemoglobin concentration in tissue near infrared spectroscopy," Phys. Med. Biol. 39, 1295-1312 (1994).
[CrossRef] [PubMed]

Cope, M.

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. S. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1997).
[CrossRef]

D. T. Delpy and M. Cope, "Quantification in tissue near-infrared spectroscopy," Philos. Trans. R. Soc. London, Ser. B 352, 649-659 (1997).
[CrossRef]

A. Duncan, J. H. Meek, M. Clemence, C. E. Elwell, L. Tyszczuk, M. Cope, and D. Delpy, "Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy," Phys. Med. Biol. 40, 295-304 (1995).
[CrossRef] [PubMed]

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, "A Monte Carlo investigation of optical pathlength in homogeneous tissue and its application to near-infrared spectroscopy," Phys. Med. Biol. 38, 1859-1876 (1993).
[CrossRef] [PubMed]

S. J. Matcher, M. Cope, and D. T. Delpy, "Use of the water absorption spectrum to quantify tissue chromophore concentration changes in near-infrared spectroscopy," Phys. Med. Biol. 38, 177-196 (1993).

S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 47, 1531-1560 (1992).
[CrossRef]

Delpy, D.

A. Duncan, J. H. Meek, M. Clemence, C. E. Elwell, L. Tyszczuk, M. Cope, and D. Delpy, "Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy," Phys. Med. Biol. 40, 295-304 (1995).
[CrossRef] [PubMed]

Delpy, D. T.

D. T. Delpy and M. Cope, "Quantification in tissue near-infrared spectroscopy," Philos. Trans. R. Soc. London, Ser. B 352, 649-659 (1997).
[CrossRef]

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. S. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1997).
[CrossRef]

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, "A Monte Carlo investigation of optical pathlength in homogeneous tissue and its application to near-infrared spectroscopy," Phys. Med. Biol. 38, 1859-1876 (1993).
[CrossRef] [PubMed]

S. J. Matcher, M. Cope, and D. T. Delpy, "Use of the water absorption spectrum to quantify tissue chromophore concentration changes in near-infrared spectroscopy," Phys. Med. Biol. 38, 177-196 (1993).

S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 47, 1531-1560 (1992).
[CrossRef]

Duling, B. R.

R. N. Pittman and B. R. Duling, "A new method for the measurement of percent oxyhemoglobin," J. Appl. Physiol. 38, 315-320 (1975).

Duncan, A.

A. Duncan, J. H. Meek, M. Clemence, C. E. Elwell, L. Tyszczuk, M. Cope, and D. Delpy, "Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy," Phys. Med. Biol. 40, 295-304 (1995).
[CrossRef] [PubMed]

Elwell, C. E.

A. Duncan, J. H. Meek, M. Clemence, C. E. Elwell, L. Tyszczuk, M. Cope, and D. Delpy, "Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy," Phys. Med. Biol. 40, 295-304 (1995).
[CrossRef] [PubMed]

Essenpreis, M.

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, "A Monte Carlo investigation of optical pathlength in homogeneous tissue and its application to near-infrared spectroscopy," Phys. Med. Biol. 38, 1859-1876 (1993).
[CrossRef] [PubMed]

Fantini, S.

Firbank, M.

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, "A Monte Carlo investigation of optical pathlength in homogeneous tissue and its application to near-infrared spectroscopy," Phys. Med. Biol. 38, 1859-1876 (1993).
[CrossRef] [PubMed]

Hasegawa, Y.

Haskell, R. C.

Hazeki, O.

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).
[PubMed]

Hiraoka, M.

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, "A Monte Carlo investigation of optical pathlength in homogeneous tissue and its application to near-infrared spectroscopy," Phys. Med. Biol. 38, 1859-1876 (1993).
[CrossRef] [PubMed]

Hou, Y.

J. Mayhew, Y. Zheng, Y. Hou, B. Vuksanovic, J. Berwick, S. Askew, and P. Coffey, "Spectroscopic analysis of changes in remitted illumination: the response to increased neural activity in brain," Neuroimage 10, 304-326 (1999).
[CrossRef] [PubMed]

Kohl, M.

P. Wobst, R. Wenzel, M. Kohl, H. Obriq, and A. Villringer, "Linear aspects of changes in deoxygenated hemoglobin concentration and cytochrome oxidase oxidation during brain activation," Neuroimage 13, 520-530 (2001).
[CrossRef] [PubMed]

Lübbers, D. W.

D. W. Lübbers and R. Wodick, "Absolute reflection photometry applied to the measurement of capillary oxyhemoglobin saturation of the skin in man," in Oxygen Measurements in Biology and Medicine, J.P.Payne and D.W.Hill, eds. (Butterworths, 1975), pp. 85-110.

Martelli, F.

Matcher, S. J.

S. J. Matcher and C. E. Cooper, "Absolute quantification of deoxyhaemoglobin concentration in tissue near infrared spectroscopy," Phys. Med. Biol. 39, 1295-1312 (1994).
[CrossRef] [PubMed]

S. J. Matcher, M. Cope, and D. T. Delpy, "Use of the water absorption spectrum to quantify tissue chromophore concentration changes in near-infrared spectroscopy," Phys. Med. Biol. 38, 177-196 (1993).

Mayhew, J.

J. Mayhew, Y. Zheng, Y. Hou, B. Vuksanovic, J. Berwick, S. Askew, and P. Coffey, "Spectroscopic analysis of changes in remitted illumination: the response to increased neural activity in brain," Neuroimage 10, 304-326 (1999).
[CrossRef] [PubMed]

Meek, J. H.

A. Duncan, J. H. Meek, M. Clemence, C. E. Elwell, L. Tyszczuk, M. Cope, and D. Delpy, "Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy," Phys. Med. Biol. 40, 295-304 (1995).
[CrossRef] [PubMed]

Nomura, Y.

Y. Hasegawa, Y. Yamada, M. Tamura, and Y. Nomura, "Monte Carlo simulation of light transmission through living tissues," Appl. Opt. 30, 4515-4520 (1991).
[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).
[PubMed]

Obriq, H.

P. Wobst, R. Wenzel, M. Kohl, H. Obriq, and A. Villringer, "Linear aspects of changes in deoxygenated hemoglobin concentration and cytochrome oxidase oxidation during brain activation," Neuroimage 13, 520-530 (2001).
[CrossRef] [PubMed]

Patterson, M. S.

Pittman, R. N.

R. N. Pittman and B. R. Duling, "A new method for the measurement of percent oxyhemoglobin," J. Appl. Physiol. 38, 315-320 (1975).

Sassaroli, A.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th Dover printing, 10th U.S. GPO printing ed. (Dover, 1964), p. 928.

Svaasand, L. O.

Tamura, M.

Y. Hasegawa, Y. Yamada, M. Tamura, and Y. Nomura, "Monte Carlo simulation of light transmission through living tissues," Appl. Opt. 30, 4515-4520 (1991).
[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).
[PubMed]

Tromberg, B. J.

Tsay, T.-T.

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]

Twersky, V.

Tyszczuk, L.

A. Duncan, J. H. Meek, M. Clemence, C. E. Elwell, L. Tyszczuk, M. Cope, and D. Delpy, "Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy," Phys. Med. Biol. 40, 295-304 (1995).
[CrossRef] [PubMed]

van der Zee, P.

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. S. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1997).
[CrossRef]

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, "A Monte Carlo investigation of optical pathlength in homogeneous tissue and its application to near-infrared spectroscopy," Phys. Med. Biol. 38, 1859-1876 (1993).
[CrossRef] [PubMed]

Villringer, A.

P. Wobst, R. Wenzel, M. Kohl, H. Obriq, and A. Villringer, "Linear aspects of changes in deoxygenated hemoglobin concentration and cytochrome oxidase oxidation during brain activation," Neuroimage 13, 520-530 (2001).
[CrossRef] [PubMed]

Vuksanovic, B.

J. Mayhew, Y. Zheng, Y. Hou, B. Vuksanovic, J. Berwick, S. Askew, and P. Coffey, "Spectroscopic analysis of changes in remitted illumination: the response to increased neural activity in brain," Neuroimage 10, 304-326 (1999).
[CrossRef] [PubMed]

Wenzel, R.

P. Wobst, R. Wenzel, M. Kohl, H. Obriq, and A. Villringer, "Linear aspects of changes in deoxygenated hemoglobin concentration and cytochrome oxidase oxidation during brain activation," Neuroimage 13, 520-530 (2001).
[CrossRef] [PubMed]

Wilson, B. C.

Wobst, P.

P. Wobst, R. Wenzel, M. Kohl, H. Obriq, and A. Villringer, "Linear aspects of changes in deoxygenated hemoglobin concentration and cytochrome oxidase oxidation during brain activation," Neuroimage 13, 520-530 (2001).
[CrossRef] [PubMed]

Wodick, R.

D. W. Lübbers and R. Wodick, "Absolute reflection photometry applied to the measurement of capillary oxyhemoglobin saturation of the skin in man," in Oxygen Measurements in Biology and Medicine, J.P.Payne and D.W.Hill, eds. (Butterworths, 1975), pp. 85-110.

Wray, S.

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. S. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1997).
[CrossRef]

Wyatt, J. S.

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. S. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1997).
[CrossRef]

Yamada, Y.

Zheng, Y.

J. Mayhew, Y. Zheng, Y. Hou, B. Vuksanovic, J. Berwick, S. Askew, and P. Coffey, "Spectroscopic analysis of changes in remitted illumination: the response to increased neural activity in brain," Neuroimage 10, 304-326 (1999).
[CrossRef] [PubMed]

Adv. Exp. Med. Biol. (1)

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).
[PubMed]

Appl. Opt. (3)

J. Appl. Physiol. (1)

R. N. Pittman and B. R. Duling, "A new method for the measurement of percent oxyhemoglobin," J. Appl. Physiol. 38, 315-320 (1975).

J. Opt. Soc. Am. (1)

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

Neuroimage (2)

P. Wobst, R. Wenzel, M. Kohl, H. Obriq, and A. Villringer, "Linear aspects of changes in deoxygenated hemoglobin concentration and cytochrome oxidase oxidation during brain activation," Neuroimage 13, 520-530 (2001).
[CrossRef] [PubMed]

J. Mayhew, Y. Zheng, Y. Hou, B. Vuksanovic, J. Berwick, S. Askew, and P. Coffey, "Spectroscopic analysis of changes in remitted illumination: the response to increased neural activity in brain," Neuroimage 10, 304-326 (1999).
[CrossRef] [PubMed]

Philos. Trans. R. Soc. London, Ser. B (1)

D. T. Delpy and M. Cope, "Quantification in tissue near-infrared spectroscopy," Philos. Trans. R. Soc. London, Ser. B 352, 649-659 (1997).
[CrossRef]

Phys. Med. Biol. (8)

S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 47, 1531-1560 (1992).
[CrossRef]

A. Duncan, J. H. Meek, M. Clemence, C. E. Elwell, L. Tyszczuk, M. Cope, and D. Delpy, "Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy," Phys. Med. Biol. 40, 295-304 (1995).
[CrossRef] [PubMed]

A. Sassaroli and S. Fantini, "Comment on the modified Beer-Lambert law for scattering media," Phys. Med. Biol. 49, N255-N257 (1997).
[CrossRef]

S. J. Matcher, M. Cope, and D. T. Delpy, "Use of the water absorption spectrum to quantify tissue chromophore concentration changes in near-infrared spectroscopy," Phys. Med. Biol. 38, 177-196 (1993).

S. J. Matcher and C. E. Cooper, "Absolute quantification of deoxyhaemoglobin concentration in tissue near infrared spectroscopy," Phys. Med. Biol. 39, 1295-1312 (1994).
[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]

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, "A Monte Carlo investigation of optical pathlength in homogeneous tissue and its application to near-infrared spectroscopy," Phys. Med. Biol. 38, 1859-1876 (1993).
[CrossRef] [PubMed]

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. S. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1997).
[CrossRef]

Other (2)

D. W. Lübbers and R. Wodick, "Absolute reflection photometry applied to the measurement of capillary oxyhemoglobin saturation of the skin in man," in Oxygen Measurements in Biology and Medicine, J.P.Payne and D.W.Hill, eds. (Butterworths, 1975), pp. 85-110.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th Dover printing, 10th U.S. GPO printing ed. (Dover, 1964), p. 928.

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Figures (3)

Fig. 1
Fig. 1

(a) Graph showing A versus μ a generated directly from Monte Carlo data using Eq. (7) (solid curve) and from the cumulants of the TPSF generated from these data using Eq. (15) for two terms (dotted curve) and eight terms (dashed curve). (b) Graph showing the absolute difference A versus μ a at ω = 0 between the directly calculated value from Eq. (7) and the approximation of the relationship between A and μ a using Eq. (15) with different numbers of terms. Key: solid curves, one term; dotted curves, two terms; dotted–dashed curves, five terms; dashed curves, eight terms.

Fig. 2
Fig. 2

(a) A versus μ a , (b) d A d μ a versus μ a , (c) d 2 A d μ a 2 versus μ a , (d) d 3 A d μ a 3 versus μ a , using Eq. (7) and its derivatives (solid curves) and the first nine terms of Eq. (16) (circles).

Fig. 3
Fig. 3

Amplitude and phase plots in the frequency domain for increasing levels of absorber. The solid curves are obtained by Fourier transformation of h ( l , μ a ) calculated using Eq. (3) with the dotted curves calculated using the first nine terms in Eq. (13). Each phase plot is offset successively by 0.25 rad to improve graph clarity. Key: circles, μ a = 0 mm 1 ; crosses, μ a = 0.08 mm 1 ; diamonds, μ a = 0.16 mm 1 ; triangles, μ a = 0.24 mm 1 .

Equations (23)

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E s = 0 S ( t ) d t < E .
A s = ln ( E E s ) .
h ( t , μ a ) = S ( t ) exp ( μ a c t )
= E s s ( t ) exp ( μ a c t ) ,
E a = 0 h ( t , μ a ) d t E s .
A a = ln ( E s E a )
= ln [ 0 s ( t ) exp ( μ a c t ) d t ] .
H ( ω , μ a ) = E s 0 s ( t ) exp ( μ a c t ) exp ( j ω t ) d t = E s 0 s ( t ) exp [ t ( j ω μ a c ) ] d t .
H ( ω , μ a ) E s = 0 [ 1 + t ( j ω μ a c ) + 1 2 ! t 2 ( j ω μ a c ) 2 + ] s ( t ) d t = 1 + ( j ω μ a c ) m 1 + 1 2 ! ( j ω μ a c ) 2 m 2 + 1 3 ! ( j ω μ a c ) 3 m 3 + = n = 0 1 n ! ( j ω μ a c ) n m n ,
A a ( ω , μ a ) = ln 0 s ( t ) exp [ t ( j ω μ a ) ] d t
= ln [ H ( ω , μ a ) E s ]
= ln [ n = 0 1 n ! ( j ω μ a c ) n m n ] .
A a ( ω , μ a ) = n = 1 1 n ! ( j ω μ a c ) n κ n .
Δ A ( ω , Δ μ a ) = n = 1 1 n ! ( j ω Δ μ a c ) n κ n ,
A a ( 0 , μ a ) = n = 1 1 n ! ( μ a c ) n κ n = μ a c κ 1 1 2 ( μ a c ) 2 κ 2 + 1 6 ( μ a c ) 3 κ 3 .
d m A a ( 0 , μ a ) d μ a m = n = m ( 1 ) n c n μ a ( n m ) κ n ( n m ) ! .
d m A a ( 0 , 0 ) d μ a m = ( 1 ) m + 1 c m κ m ,
A ( 0 , μ a ) = μ a c t ,
A ( 0 , μ a ) = μ a c t ( μ a c ) 2 σ 2 2 ,
d A d μ a = c t c 2 σ 2 μ a .
A a ( ω , 0 ) = j ω t + ω 2 σ 2 2 .
Δ μ a 1 n = i = 1 n α ( i , λ 0 ) i = 1 n α ( i , λ 1 n ) C i
= i = 1 n ( α ( i , λ 0 ) α ( i , λ 1 n ) ) C i ,

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