Abstract

A diffusion model of noninvasive absorption spectroscopy was used to determine how the change in signal resulting from a point absorber depends on the position of that absorber relative to the source and detector. This is equivalent to calculating the relative probability that a photon will visit a certain location in tissue before its detection. Experimental mapping of the point-target response in tissue-simulating materials confirmed the accuracy of the model. For steady-state spectroscopy a simple relation was derived between the mean depth visited by detected photons, the source–detector separation, and the optical penetration depth. It was also demonstrated theoretically that combining a pulsed source with time-gated detection provides additional control over the spatial distribution of the photon-visit probability.

© 1995 Optical Society of America

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    [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

1993 (4)

1992 (3)

E. M. Sevick, J. R. Lakowicz, H. Smacinski, K. Nowaczyk, M. L. Johnson, “Frequency domain imaging of absorbers obscured by scattering,” J. Photochem. Photobiol. B: Biol. 16, 169–185 (1992).
[CrossRef]

S. J. Madsen, M. S. Patterson, B. C. Wilson, “The use of India ink as an optical absorber in tissue-simulating phantoms,” Phys. Med. Biol. 37, 985–993 (1992).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

1991 (4)

1990 (3)

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue. I: Models of radiation transport and their application,” Lasers Med. Sci. 6, 155–168 (1990).
[CrossRef]

J. M. Schmitt, G. X. Zhou, E. C. Walker, R. T. Wall, “Multilayer model of photon diffusion in skin,” J. Opt. Soc. Am. A 7, 2141–2153 (1990).
[CrossRef] [PubMed]

1989 (2)

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

G. H. Weiss, R. Nossal, R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
[CrossRef]

1988 (3)

M. Keijzer, W. M. Star, P. R. Storchi, “Optical diffusion in layered media,” Appl. Opt. 27, 1820–1824 (1988).
[CrossRef] [PubMed]

M. Cope, D. T. Delpy, “System for long-term measurement of cerebral blood and tissue oxygenation on newborn infants by near infrared transillumination,” Med. Biol. Eng. Comput. 26, 289–294 (1988).
[CrossRef] [PubMed]

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

1987 (2)

M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihematoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[CrossRef] [PubMed]

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

1986 (2)

B. C. Wilson, M. S. Patterson, D. M. Burns, “The effect of photosensitizer concentration on the penetration depth of photoactivating light,” Lasers Med. Sci. 1, 235–244 (1986).
[CrossRef]

J. M. Steinke, A. P. Shepherd, “Diffuse reflectance of whole blood: model for a diverging light beam,” IEEE Trans. Biomed. Eng. BME-34, 826–833 (1986).
[CrossRef]

1983 (1)

1977 (1)

F. F. Jobsis, J. H. Keizer, J. C. LaManna, M. Rosenthal, “Reflectance spectroscopy of cytochrome aa3 in vivo,” J. Appl. Physiol. 43, 858–872 (1977).
[PubMed]

1943 (1)

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1–88 (1943).
[CrossRef]

Alfano, R. R.

Berg, R.

Berndt, K. W.

Bonner, R. F.

G. H. Weiss, R. Nossal, R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
[CrossRef]

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

Burns, D. M.

M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihematoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[CrossRef] [PubMed]

B. C. Wilson, M. S. Patterson, D. M. Burns, “The effect of photosensitizer concentration on the penetration depth of photoactivating light,” Lasers Med. Sci. 1, 235–244 (1986).
[CrossRef]

Chance, B.

Chandrasekhar, S.

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1–88 (1943).
[CrossRef]

Cheong, W. F.

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Cope, M.

M. Cope, D. T. Delpy, “System for long-term measurement of cerebral blood and tissue oxygenation on newborn infants by near infrared transillumination,” Med. Biol. Eng. Comput. 26, 289–294 (1988).
[CrossRef] [PubMed]

Cui, W.

Das, B. B.

Delpy, D. T.

M. Cope, D. T. Delpy, “System for long-term measurement of cerebral blood and tissue oxygenation on newborn infants by near infrared transillumination,” Med. Biol. Eng. Comput. 26, 289–294 (1988).
[CrossRef] [PubMed]

Duderstadt, J. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Farrell, T. J.

T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Feather, J. W.

M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihematoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[CrossRef] [PubMed]

Ferwerda, H. A.

Fountain, M.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Greenfeld, R.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Groenhuis, R. A. J.

Hamilton, L. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Haselgrove, J. C.

Havlin, S.

Haw, T.

L. Lilge, T. Haw, B. C. Wilson, “Miniature isotropic optical fibre probes for quantitative light dosimetry in tissues,” Phys. Med. Biol. 38, 215–230 (1993).
[CrossRef] [PubMed]

Hebden, J. C.

Holtom, G.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

Jarlman, O.

Jobsis, F. F.

F. F. Jobsis, J. H. Keizer, J. C. LaManna, M. Rosenthal, “Reflectance spectroscopy of cytochrome aa3 in vivo,” J. Appl. Physiol. 43, 858–872 (1977).
[PubMed]

Johnson, M. L.

E. M. Sevick, J. R. Lakowicz, H. Smacinski, K. Nowaczyk, M. L. Johnson, “Frequency domain imaging of absorbers obscured by scattering,” J. Photochem. Photobiol. B: Biol. 16, 169–185 (1992).
[CrossRef]

Keijzer, M.

Keizer, J. H.

F. F. Jobsis, J. H. Keizer, J. C. LaManna, M. Rosenthal, “Reflectance spectroscopy of cytochrome aa3 in vivo,” J. Appl. Physiol. 43, 858–872 (1977).
[PubMed]

Kent, J.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Kruger, R. A.

Lakowicz, J. R.

E. M. Sevick, J. R. Lakowicz, H. Smacinski, K. Nowaczyk, M. L. Johnson, “Frequency domain imaging of absorbers obscured by scattering,” J. Photochem. Photobiol. B: Biol. 16, 169–185 (1992).
[CrossRef]

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

LaManna, J. C.

F. F. Jobsis, J. H. Keizer, J. C. LaManna, M. Rosenthal, “Reflectance spectroscopy of cytochrome aa3 in vivo,” J. Appl. Physiol. 43, 858–872 (1977).
[PubMed]

Leigh, J. S.

Lilge, L.

L. Lilge, T. Haw, B. C. Wilson, “Miniature isotropic optical fibre probes for quantitative light dosimetry in tissues,” Phys. Med. Biol. 38, 215–230 (1993).
[CrossRef] [PubMed]

Madsen, S. J.

S. J. Madsen, M. S. Patterson, B. C. Wilson, “The use of India ink as an optical absorber in tissue-simulating phantoms,” Phys. Med. Biol. 37, 985–993 (1992).
[CrossRef] [PubMed]

McCully, K.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Moulton, J. D.

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

J. D. Moulton, “Diffusion modelling of picosecond laser pulse propagation in turbid media,” M. Eng. thesis (McMaster University, Hamilton, Ontario, Canada, 1990).

Nioka, S.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Nossal, R.

G. H. Weiss, R. Nossal, R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
[CrossRef]

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

Nowaczyk, K.

E. M. Sevick, J. R. Lakowicz, H. Smacinski, K. Nowaczyk, M. L. Johnson, “Frequency domain imaging of absorbers obscured by scattering,” J. Photochem. Photobiol. B: Biol. 16, 169–185 (1992).
[CrossRef]

Patterson, M. S.

T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

S. J. Madsen, M. S. Patterson, B. C. Wilson, “The use of India ink as an optical absorber in tissue-simulating phantoms,” Phys. Med. Biol. 37, 985–993 (1992).
[CrossRef] [PubMed]

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

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue. I: Models of radiation transport and their application,” Lasers Med. Sci. 6, 155–168 (1990).
[CrossRef]

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

M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihematoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[CrossRef] [PubMed]

B. C. Wilson, M. S. Patterson, D. M. Burns, “The effect of photosensitizer concentration on the penetration depth of photoactivating light,” Lasers Med. Sci. 1, 235–244 (1986).
[CrossRef]

Pine, D. J.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Prahl, S. A.

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Pushka, W.

M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihematoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[CrossRef] [PubMed]

Rosenthal, M.

F. F. Jobsis, J. H. Keizer, J. C. LaManna, M. Rosenthal, “Reflectance spectroscopy of cytochrome aa3 in vivo,” J. Appl. Physiol. 43, 858–872 (1977).
[PubMed]

Schmitt, J. M.

Sevick, E. M.

E. M. Sevick, J. R. Lakowicz, H. Smacinski, K. Nowaczyk, M. L. Johnson, “Frequency domain imaging of absorbers obscured by scattering,” J. Photochem. Photobiol. B: Biol. 16, 169–185 (1992).
[CrossRef]

Shepherd, A. P.

J. M. Steinke, A. P. Shepherd, “Diffuse reflectance of whole blood: model for a diverging light beam,” IEEE Trans. Biomed. Eng. BME-34, 826–833 (1986).
[CrossRef]

Shotland, J. C.

Smacinski, H.

E. M. Sevick, J. R. Lakowicz, H. Smacinski, K. Nowaczyk, M. L. Johnson, “Frequency domain imaging of absorbers obscured by scattering,” J. Photochem. Photobiol. B: Biol. 16, 169–185 (1992).
[CrossRef]

Star, W. M.

Steinke, J. M.

J. M. Steinke, A. P. Shepherd, “Diffuse reflectance of whole blood: model for a diverging light beam,” IEEE Trans. Biomed. Eng. BME-34, 826–833 (1986).
[CrossRef]

Storchi, P. R.

Svanberg, S.

Ten Bosch, J. J.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering Tables, Formulas and Applications (Academic, New York, 1980).

Walker, E. C.

Wall, R. T.

Wang, N.

Weiss, G. H.

G. H. Weiss, R. Nossal, R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
[CrossRef]

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

Weitz, D. A.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Welch, A. J.

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Wilson, B. C.

L. Lilge, T. Haw, B. C. Wilson, “Miniature isotropic optical fibre probes for quantitative light dosimetry in tissues,” Phys. Med. Biol. 38, 215–230 (1993).
[CrossRef] [PubMed]

S. J. Madsen, M. S. Patterson, B. C. Wilson, “The use of India ink as an optical absorber in tissue-simulating phantoms,” Phys. Med. Biol. 37, 985–993 (1992).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

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

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue. I: Models of radiation transport and their application,” Lasers Med. Sci. 6, 155–168 (1990).
[CrossRef]

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

M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihematoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[CrossRef] [PubMed]

B. C. Wilson, M. S. Patterson, D. M. Burns, “The effect of photosensitizer concentration on the penetration depth of photoactivating light,” Lasers Med. Sci. 1, 235–244 (1986).
[CrossRef]

Wong, K. S.

Wyman, D. R.

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue. I: Models of radiation transport and their application,” Lasers Med. Sci. 6, 155–168 (1990).
[CrossRef]

Yoo, K. M.

Zhou, G. X.

Zhu, J. X.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Anal. Biochem. (1)

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Appl. Opt. (7)

IEEE J. Quantum Electron. (1)

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

J. M. Steinke, A. P. Shepherd, “Diffuse reflectance of whole blood: model for a diverging light beam,” IEEE Trans. Biomed. Eng. BME-34, 826–833 (1986).
[CrossRef]

J. Appl. Physiol. (1)

F. F. Jobsis, J. H. Keizer, J. C. LaManna, M. Rosenthal, “Reflectance spectroscopy of cytochrome aa3 in vivo,” J. Appl. Physiol. 43, 858–872 (1977).
[PubMed]

J. Mod. Opt. (1)

G. H. Weiss, R. Nossal, R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Photochem. Photobiol. B: Biol. (1)

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[CrossRef]

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[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Geometry for calculation of the relative visit probability in absorption spectroscopy. The light source (S) is a collimated pencil beam incident on the tissue at the origin, and a detector (D) is located at r on the tissue surface (the xy plane). For the simple diffusion model it is assumed that all photons are isotropically scattered at a depth zo on the z axis. As explained in the text, the fluence rate is assumed to be zero on a plane parallel to and above the tissue surface at z = zp. The goal of the calculation is ton determine the change in signal at the detector resulting from a perturbation in the absorption coefficient in d V near P(ρ, θ, z).

Fig. 2
Fig. 2

Schematic diagram of the apparatus used to measure the signal resulting from the fluorescent target as a function of its position. PMT, photomultiplier tube.

Fig. 3
Fig. 3

Relative signal resulting from the fluorescent target as a function of its position. (a), (b), (c) Data acquired at different depths by scanning the target parallel to the x axis in the xz plane (see Fig. 1). (d), (e), (f) Scans made parallel to the y axis at different depths in the plane midway between the source and detector. The optical properties of the scattering medium and source–detector separation are in Table 2. (a), (d) Run 1; (b), (e) run; 2; (c), (f) run 3. Note that each division on the ordinate scale corresponds to a factor of 10.

Fig. 4
Fig. 4

Results of numerical calculation of the mean photon-visit depth during steady-state noninvasive absorption spectroscopy according to Eq. (14). Isodepth curves (in millimeters) are shown in the plane defined by the reduced scattering coefficient and absorption coefficient of the tissue. The contour interval is 0.2 mm, and the source–detector separation is 10 mm.

Fig. 5
Fig. 5

Plots of the mean photon-visit depth during steady-state absorption spectroscopy versus the square root of the optical penetration depth in the tissue for three different source–detector separations. The points were obtained by the same sort of calculation used to generate Fig. 4 for three different source–detector separations. The lines are the empirical relaxation 〈zr = (rδ)1/2/2.

Fig. 6
Fig. 6

Dependence of the mean photon-visit depth on the square root of the detection time when the source is an impulse and the detector is time gated. For this example, μs′ = 1.0 mm−1, μa = 0.01 mm−1, c = 0.214 mm ps−1, and r = 10 mm.

Tables (2)

Tables Icon

Table 1 Reduced (or Transport) Scattering Coefficient for 1-wt. % TiO2 Particles in Ethylene Glycol Determined by Various Methods

Tables Icon

Table 2 Experimental Conditions for Measurement of the Point Response Functions in Fig. 3a

Equations (15)

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Δ R ( r ) = Δ μ a ϕ ( r ) d V E ( r , r ) .
Δ F ( r , λ x , λ m ) = Δ μ a Φ ϕ ( r , λ x ) d V E ( r , r , λ m ) ,
1 c t ϕ ( r , t ) - D 2 ϕ ( r , t ) + μ a ϕ ( r , t ) = S ( r , t ) ,
D = 1 3 ( μ a + μ s ) .
ϕ ( ρ , z ) = 1 4 π D ( exp { - μ eff [ ( z - z o ) 2 + ρ 2 ] 1 / 2 } [ ( z - z o ) 2 + ρ 2 ] 1 / 2 - exp { - μ eff [ ( z - z p ) 2 + ρ 2 ] 1 / 2 } [ ( z - z p ) 2 + ρ 2 ] 1 / 2 ) ,
( - D d d z ϕ ) z = 0 .
E ( ρ , θ , z , r ) = 1 4 π [ z exp ( - μ eff k ) k 2 ( μ eff + 1 k ) - z p exp ( - μ eff l ) l 2 ( μ eff + 1 l ) ] ,
k 2 = ( r - ρ cos θ ) 2 + ρ 2 sin 2 θ + z 2 ,
l 2 = ( r - ρ cos θ ) 2 + ρ 2 sin 2 θ + z p 2 .
Δ R ( r , t , t ) d t = Δ μ a ϕ ( r , t ) d V E ( r , r , t - t ) d t .
Δ R ( r , t ) = Δ μ a d V r / c t - r - r / c ϕ ( r , t ) E ( r , r , t - t ) d t .
ϕ ( ρ , z , t ) = c ( 4 π D c ) - 3 / 2 t - 3 / 2 exp ( - μ a c t ) × { exp [ - ( z - z o ) 2 + ρ 2 4 D c t ] - exp [ - ( z - z p ) 2 + ρ 2 4 D c t ] } ,
E ( ρ , θ , z , r , t - t ) = 1 2 ( 4 π D c ) - 3 / 2 ( t - t ) - 5 / 2 × exp [ - μ a c ( t - t ) ] × { z exp [ - k 2 4 D c ( t - t ) ] - z p exp [ - l 2 4 D c ( t - t ) ] } ,
z r = V ϕ ( ρ , θ , z ) E ( ρ , θ , z , r ) z d V V ϕ ( ρ , θ , z ) E ( ρ , θ , z , r ) d V ,
z r , t = V z d V r / c t - r - r / c ϕ ( r , t ) E ( r , r , t - t ) d t V d V r / c t - r - r / c ϕ ( r , t ) E ( r , r , t - t ) d t ,

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