Abstract

To validate models of light propagation in biological tissue, experiments to measure the mean time of flight have been carried out on several solid cylindrical layered phantoms. The optical properties of the inner cylinders of the phantoms were close to those of adult brain white matter, whereas a range of scattering or absorption coefficients was chosen for the outer layer. Experimental results for the mean optical path length have been compared with the predictions of both an exact Monte Carlo (MC) model and a diffusion equation, with two differing boundary conditions implemented in a finite-element method (FEM). The MC and experimental results are in good agreement despite poor statistics for large fiber spacings, whereas good agreement with the FEM prediction requires a careful choice of proper boundary conditions.

© 1996 Optical Society of America

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    [CrossRef]

1995 (8)

M. Schweiger, S. R. Arridge, “Near-infrared imaging: photon measurement density functions,” Proc. SPIE 2389, 366–377 (1995).
[CrossRef]

S. R. Arridge, M. Schweiger, “Sensitivity to prior knowledge in optical tomographic reconstruction,” Proc. SPIE 2389, 378–388 (1995).
[CrossRef]

B. W. Pogue, M. S. Patterson, T. J. Farrel, “Forward and inverse calculations for 3-D frequency-domain diffuse optical tomography,” Proc. SPIE 2389, 328–339 (1995).
[CrossRef]

D. A. Benaron, J. P. Van Houten, W. F. Cheong, E. L. Kermit, R. A. King, “Early clinical results of time-of-flight optical tomography in a neonatal intensive care unit,” Proc. SPIE 2389, 582–596 (1995).
[CrossRef]

S. R. Arridge, M. Schweiger, “Photon-measurement density functions. Part II: finite-element-method calculations,” Appl. Opt. 34, 8026–8027 (1995).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys.1779–1792 (1995).
[CrossRef] [PubMed]

M. Firbank, M. Oda, D. T. Delpy, “An improved design for a stable and reproducible phantom material for use in near infrared spectroscopy and imaging,” Phys. Med. Biol. 40, 955–961 (1995).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, “Direct calculation of the moments of the distribution of photon time of flight in tissue with a finite-element method,” Appl. Opt. 34, 2683–2687 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (3)

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

M. Firbank, D. T. Delpy, “A design for a stable and reproducible phantom for use in near-infrared imaging and spectroscopy,” Phys. Med. Biol. 38, 847–853 (1993).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

1992 (4)

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

W. Cui, L. E. Ostrander, “The relationship of surface reflectance measurements to optical properties of layered biological media,” IEEE Trans. Biomed. Eng. 39, 194–201 (1992).
[CrossRef] [PubMed]

J. C. Haselgrove, J. C. Schotland, J. S. Leigh, “Long-time behavior of photon diffusion in an absorbing medium: application to time-resolved spectroscopy,” Appl. Opt. 31, 2678–2683 (1992).
[CrossRef] [PubMed]

K. M. Yoo, B. B. Das, R. R. Alfano, “Imaging of translucent objects hidden in highly scattering medium from the early portion of the diffuse component of a transmitted ultrafast laser pulse,” Opt. Lett. 17, 958–960 (1992).
[CrossRef] [PubMed]

1991 (3)

P. Donnelli, P. Bruscaglioni, A. Ismelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
[CrossRef]

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

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

1990 (3)

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

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]

P. van der Zee, S. R. Arridge, M. Cope, D. T. Delpy, “The effect of optode positioning on optical path length in near infrared spectroscopy of brain,” Adv. Exp. Med. Biol. 277, 79–84 (1990).
[PubMed]

1989 (1)

1988 (3)

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. S. Watt, “Estimation of optical path length through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (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]

M. Cope, D. T. Delpy, E. O. R. Reynolds, S. Wray, J. Wyatt, P. van der Zee, “Methods of quantitating cerebral near infrared spectroscopy data,” Adv. Exp. Med. Biol. 222, 183–189 (1988).
[CrossRef] [PubMed]

1987 (2)

O. Hazeki, A. Seyama, M. Tamura, “Near infrared spectrophotometric monitoring of hemoglobin and cytochrome aa3 in vivo,” Adv. Exp. Med. Biol. 215, 283–289 (1987).
[PubMed]

P. van der Zee, D. T. Delpy, “Simulation of the point spread function for light in tissue by a Monte Carlo technique,” Adv. Exp. Med. Biol. 215, 179–191 (1987).

1986 (2)

M. Ferrari, E. Zanette, I. Giannini, G. Sideri, C. Fieschi, A. Carpi, “Effect of carotid artery compression test on regional cerebral blood volume, hemoglobin oxygen saturation and cytochrome-c-oxidase redox level in cerebrovascular patients,” Adv. Exp. Med. Biol. 200, 213–222 (1986).
[CrossRef] [PubMed]

C. C. Piantadosi, T. M. Hemstreet, F. F. Jöbsis, “Near infrared spectrophotometric monitoring of oxygen distribution to intact brain and skeletal muscle tissue,” Crit. Care Med. 14, 698–706 (1986).
[CrossRef] [PubMed]

1983 (1)

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

1977 (1)

F. F. Jöbsis, “Noninvasive, infrared monitoring of cerebral and myocardial oxygen sufficiency and circulatory parameters,” Science 198, 1264–1267 (1977).
[CrossRef] [PubMed]

Adam, G.

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Alfano, R. R.

Arridge, S. R.

M. Schweiger, S. R. Arridge, “Near-infrared imaging: photon measurement density functions,” Proc. SPIE 2389, 366–377 (1995).
[CrossRef]

S. R. Arridge, M. Schweiger, “Sensitivity to prior knowledge in optical tomographic reconstruction,” Proc. SPIE 2389, 378–388 (1995).
[CrossRef]

S. R. Arridge, M. Schweiger, “Photon-measurement density functions. Part II: finite-element-method calculations,” Appl. Opt. 34, 8026–8027 (1995).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys.1779–1792 (1995).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, “Direct calculation of the moments of the distribution of photon time of flight in tissue with a finite-element method,” Appl. Opt. 34, 2683–2687 (1995).
[CrossRef] [PubMed]

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

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

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

P. van der Zee, S. R. Arridge, M. Cope, D. T. Delpy, “The effect of optode positioning on optical path length in near infrared spectroscopy of brain,” Adv. Exp. Med. Biol. 277, 79–84 (1990).
[PubMed]

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. S. Watt, “Estimation of optical path length through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

Benaron, D. A.

D. A. Benaron, J. P. Van Houten, W. F. Cheong, E. L. Kermit, R. A. King, “Early clinical results of time-of-flight optical tomography in a neonatal intensive care unit,” Proc. SPIE 2389, 582–596 (1995).
[CrossRef]

Bruscaglioni, P.

P. Donnelli, P. Bruscaglioni, A. Ismelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
[CrossRef]

Carpi, A.

M. Ferrari, E. Zanette, I. Giannini, G. Sideri, C. Fieschi, A. Carpi, “Effect of carotid artery compression test on regional cerebral blood volume, hemoglobin oxygen saturation and cytochrome-c-oxidase redox level in cerebrovascular patients,” Adv. Exp. Med. Biol. 200, 213–222 (1986).
[CrossRef] [PubMed]

Chance, B.

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

B. Chance, M. Maris, J. Sorge, M. Z. Zhang, “A phase modulation system for dual wavelength difference spectroscopy of hemoglobin deoxygenation in tissue,” in Time-Resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. SPIE1204, 481–491 (1990).
[CrossRef]

J. Haselgrove, J. Leigh, C. Yee, N. G. Wang, M. Maris, B. Chance, “Monte Carlo and diffusion calculations of photon migration in noninfinite highly scattering media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 30–41 (1991).
[CrossRef]

Cheong, W. F.

D. A. Benaron, J. P. Van Houten, W. F. Cheong, E. L. Kermit, R. A. King, “Early clinical results of time-of-flight optical tomography in a neonatal intensive care unit,” Proc. SPIE 2389, 582–596 (1995).
[CrossRef]

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

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

P. van der Zee, S. R. Arridge, M. Cope, D. T. Delpy, “The effect of optode positioning on optical path length in near infrared spectroscopy of brain,” Adv. Exp. Med. Biol. 277, 79–84 (1990).
[PubMed]

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. S. Watt, “Estimation of optical path length through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (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]

M. Cope, D. T. Delpy, E. O. R. Reynolds, S. Wray, J. Wyatt, P. van der Zee, “Methods of quantitating cerebral near infrared spectroscopy data,” Adv. Exp. Med. Biol. 222, 183–189 (1988).
[CrossRef] [PubMed]

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

Cui, W.

W. Cui, L. E. Ostrander, “The relationship of surface reflectance measurements to optical properties of layered biological media,” IEEE Trans. Biomed. Eng. 39, 194–201 (1992).
[CrossRef] [PubMed]

Das, B. B.

Delpy, D. T.

M. Firbank, M. Oda, D. T. Delpy, “An improved design for a stable and reproducible phantom material for use in near infrared spectroscopy and imaging,” Phys. Med. Biol. 40, 955–961 (1995).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys.1779–1792 (1995).
[CrossRef] [PubMed]

J. C. Hebden, D. T. Delpy, “Enhanced time-resolved imaging with a diffusion model of photon transport,” Opt. Lett. 19, 311–313 (1994).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

M. Firbank, D. T. Delpy, “A design for a stable and reproducible phantom for use in near-infrared imaging and spectroscopy,” Phys. Med. Biol. 38, 847–853 (1993).
[CrossRef]

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

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

P. van der Zee, S. R. Arridge, M. Cope, D. T. Delpy, “The effect of optode positioning on optical path length in near infrared spectroscopy of brain,” Adv. Exp. Med. Biol. 277, 79–84 (1990).
[PubMed]

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. S. Watt, “Estimation of optical path length through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

M. Cope, D. T. Delpy, E. O. R. Reynolds, S. Wray, J. Wyatt, P. van der Zee, “Methods of quantitating cerebral near infrared spectroscopy data,” Adv. Exp. Med. Biol. 222, 183–189 (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]

P. van der Zee, D. T. Delpy, “Simulation of the point spread function for light in tissue by a Monte Carlo technique,” Adv. Exp. Med. Biol. 215, 179–191 (1987).

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

P. van der Zee, M. Essenpreis, D. T. Delpy, “Optical properties of brain tissue,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 454–465 (1993).
[CrossRef]

Donnelli, P.

P. Donnelli, P. Bruscaglioni, A. Ismelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
[CrossRef]

Egan, W. G.

W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979).

Essenpreis, M.

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

P. van der Zee, M. Essenpreis, D. T. Delpy, “Optical properties of brain tissue,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 454–465 (1993).
[CrossRef]

Farrel, T. J.

B. W. Pogue, M. S. Patterson, T. J. Farrel, “Forward and inverse calculations for 3-D frequency-domain diffuse optical tomography,” Proc. SPIE 2389, 328–339 (1995).
[CrossRef]

Ferrari, M.

M. Ferrari, E. Zanette, I. Giannini, G. Sideri, C. Fieschi, A. Carpi, “Effect of carotid artery compression test on regional cerebral blood volume, hemoglobin oxygen saturation and cytochrome-c-oxidase redox level in cerebrovascular patients,” Adv. Exp. Med. Biol. 200, 213–222 (1986).
[CrossRef] [PubMed]

Fieschi, C.

M. Ferrari, E. Zanette, I. Giannini, G. Sideri, C. Fieschi, A. Carpi, “Effect of carotid artery compression test on regional cerebral blood volume, hemoglobin oxygen saturation and cytochrome-c-oxidase redox level in cerebrovascular patients,” Adv. Exp. Med. Biol. 200, 213–222 (1986).
[CrossRef] [PubMed]

Firbank, M.

M. Firbank, M. Oda, D. T. Delpy, “An improved design for a stable and reproducible phantom material for use in near infrared spectroscopy and imaging,” Phys. Med. Biol. 40, 955–961 (1995).
[CrossRef] [PubMed]

M. Firbank, D. T. Delpy, “A design for a stable and reproducible phantom for use in near-infrared imaging and spectroscopy,” Phys. Med. Biol. 38, 847–853 (1993).
[CrossRef]

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

M. Firbank, “The design, calibration and usage of a solid scattering and absorbing phantom for near infra red spectroscopy,” Ph.D. dissertation (Department of Medical Physics and Bioengineering, University College of London, London, 1994).

Giannini, I.

M. Ferrari, E. Zanette, I. Giannini, G. Sideri, C. Fieschi, A. Carpi, “Effect of carotid artery compression test on regional cerebral blood volume, hemoglobin oxygen saturation and cytochrome-c-oxidase redox level in cerebrovascular patients,” Adv. Exp. Med. Biol. 200, 213–222 (1986).
[CrossRef] [PubMed]

Hasegawa, Y.

Haselgrove, J.

J. Haselgrove, J. Leigh, C. Yee, N. G. Wang, M. Maris, B. Chance, “Monte Carlo and diffusion calculations of photon migration in noninfinite highly scattering media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 30–41 (1991).
[CrossRef]

Haselgrove, J. C.

Hazeki, O.

O. Hazeki, A. Seyama, M. Tamura, “Near infrared spectrophotometric monitoring of hemoglobin and cytochrome aa3 in vivo,” Adv. Exp. Med. Biol. 215, 283–289 (1987).
[PubMed]

Hebden, J. C.

Hemstreet, T. M.

C. C. Piantadosi, T. M. Hemstreet, F. F. Jöbsis, “Near infrared spectrophotometric monitoring of oxygen distribution to intact brain and skeletal muscle tissue,” Crit. Care Med. 14, 698–706 (1986).
[CrossRef] [PubMed]

Hilgeman, T. W.

W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979).

Hiraoka, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys.1779–1792 (1995).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

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

Ho, P. P.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Ishimaru, A.

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

Ismelli, A.

P. Donnelli, P. Bruscaglioni, A. Ismelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
[CrossRef]

Jöbsis, F. F.

C. C. Piantadosi, T. M. Hemstreet, F. F. Jöbsis, “Near infrared spectrophotometric monitoring of oxygen distribution to intact brain and skeletal muscle tissue,” Crit. Care Med. 14, 698–706 (1986).
[CrossRef] [PubMed]

F. F. Jöbsis, “Noninvasive, infrared monitoring of cerebral and myocardial oxygen sufficiency and circulatory parameters,” Science 198, 1264–1267 (1977).
[CrossRef] [PubMed]

Kermit, E. L.

D. A. Benaron, J. P. Van Houten, W. F. Cheong, E. L. Kermit, R. A. King, “Early clinical results of time-of-flight optical tomography in a neonatal intensive care unit,” Proc. SPIE 2389, 582–596 (1995).
[CrossRef]

King, R. A.

D. A. Benaron, J. P. Van Houten, W. F. Cheong, E. L. Kermit, R. A. King, “Early clinical results of time-of-flight optical tomography in a neonatal intensive care unit,” Proc. SPIE 2389, 582–596 (1995).
[CrossRef]

Leigh, J.

J. Haselgrove, J. Leigh, C. Yee, N. G. Wang, M. Maris, B. Chance, “Monte Carlo and diffusion calculations of photon migration in noninfinite highly scattering media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 30–41 (1991).
[CrossRef]

Leigh, J. S.

Liu, C.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Maris, M.

B. Chance, M. Maris, J. Sorge, M. Z. Zhang, “A phase modulation system for dual wavelength difference spectroscopy of hemoglobin deoxygenation in tissue,” in Time-Resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. SPIE1204, 481–491 (1990).
[CrossRef]

J. Haselgrove, J. Leigh, C. Yee, N. G. Wang, M. Maris, B. Chance, “Monte Carlo and diffusion calculations of photon migration in noninfinite highly scattering media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 30–41 (1991).
[CrossRef]

Nomura, Y.

Oda, M.

M. Firbank, M. Oda, D. T. Delpy, “An improved design for a stable and reproducible phantom material for use in near infrared spectroscopy and imaging,” Phys. Med. Biol. 40, 955–961 (1995).
[CrossRef] [PubMed]

Ostrander, L. E.

W. Cui, L. E. Ostrander, “The relationship of surface reflectance measurements to optical properties of layered biological media,” IEEE Trans. Biomed. Eng. 39, 194–201 (1992).
[CrossRef] [PubMed]

Patterson, M. S.

B. W. Pogue, M. S. Patterson, T. J. Farrel, “Forward and inverse calculations for 3-D frequency-domain diffuse optical tomography,” Proc. SPIE 2389, 328–339 (1995).
[CrossRef]

M. S. Patterson, B. C. Wilson, D. 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]

Piantadosi, C. C.

C. C. Piantadosi, T. M. Hemstreet, F. F. Jöbsis, “Near infrared spectrophotometric monitoring of oxygen distribution to intact brain and skeletal muscle tissue,” Crit. Care Med. 14, 698–706 (1986).
[CrossRef] [PubMed]

Pogue, B. W.

B. W. Pogue, M. S. Patterson, T. J. Farrel, “Forward and inverse calculations for 3-D frequency-domain diffuse optical tomography,” Proc. SPIE 2389, 328–339 (1995).
[CrossRef]

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]

Reynolds, E. O. R.

M. Cope, D. T. Delpy, E. O. R. Reynolds, S. Wray, J. Wyatt, P. van der Zee, “Methods of quantitating cerebral near infrared spectroscopy data,” Adv. Exp. Med. Biol. 222, 183–189 (1988).
[CrossRef] [PubMed]

Schotland, J. C.

Schweiger, M.

S. R. Arridge, M. Schweiger, “Photon-measurement density functions. Part II: finite-element-method calculations,” Appl. Opt. 34, 8026–8027 (1995).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys.1779–1792 (1995).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, “Direct calculation of the moments of the distribution of photon time of flight in tissue with a finite-element method,” Appl. Opt. 34, 2683–2687 (1995).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, “Near-infrared imaging: photon measurement density functions,” Proc. SPIE 2389, 366–377 (1995).
[CrossRef]

S. R. Arridge, M. Schweiger, “Sensitivity to prior knowledge in optical tomographic reconstruction,” Proc. SPIE 2389, 378–388 (1995).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

Seyama, A.

O. Hazeki, A. Seyama, M. Tamura, “Near infrared spectrophotometric monitoring of hemoglobin and cytochrome aa3 in vivo,” Adv. Exp. Med. Biol. 215, 283–289 (1987).
[PubMed]

Sideri, G.

M. Ferrari, E. Zanette, I. Giannini, G. Sideri, C. Fieschi, A. Carpi, “Effect of carotid artery compression test on regional cerebral blood volume, hemoglobin oxygen saturation and cytochrome-c-oxidase redox level in cerebrovascular patients,” Adv. Exp. Med. Biol. 200, 213–222 (1986).
[CrossRef] [PubMed]

Sorge, J.

B. Chance, M. Maris, J. Sorge, M. Z. Zhang, “A phase modulation system for dual wavelength difference spectroscopy of hemoglobin deoxygenation in tissue,” in Time-Resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. SPIE1204, 481–491 (1990).
[CrossRef]

Tamura, M.

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

O. Hazeki, A. Seyama, M. Tamura, “Near infrared spectrophotometric monitoring of hemoglobin and cytochrome aa3 in vivo,” Adv. Exp. Med. Biol. 215, 283–289 (1987).
[PubMed]

van der Zee, P.

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

P. van der Zee, S. R. Arridge, M. Cope, D. T. Delpy, “The effect of optode positioning on optical path length in near infrared spectroscopy of brain,” Adv. Exp. Med. Biol. 277, 79–84 (1990).
[PubMed]

M. Cope, D. T. Delpy, E. O. R. Reynolds, S. Wray, J. Wyatt, P. van der Zee, “Methods of quantitating cerebral near infrared spectroscopy data,” Adv. Exp. Med. Biol. 222, 183–189 (1988).
[CrossRef] [PubMed]

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. S. Watt, “Estimation of optical path length through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

P. van der Zee, D. T. Delpy, “Simulation of the point spread function for light in tissue by a Monte Carlo technique,” Adv. Exp. Med. Biol. 215, 179–191 (1987).

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

P. van der Zee, M. Essenpreis, D. T. Delpy, “Optical properties of brain tissue,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 454–465 (1993).
[CrossRef]

Van Houten, J. P.

D. A. Benaron, J. P. Van Houten, W. F. Cheong, E. L. Kermit, R. A. King, “Early clinical results of time-of-flight optical tomography in a neonatal intensive care unit,” Proc. SPIE 2389, 582–596 (1995).
[CrossRef]

Wang, L.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Wang, N. G.

J. Haselgrove, J. Leigh, C. Yee, N. G. Wang, M. Maris, B. Chance, “Monte Carlo and diffusion calculations of photon migration in noninfinite highly scattering media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 30–41 (1991).
[CrossRef]

Watt, J. S.

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. S. Watt, “Estimation of optical path length through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[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.

M. S. Patterson, B. C. Wilson, D. 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]

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Wray, S.

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. S. Watt, “Estimation of optical path length through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

M. Cope, D. T. Delpy, E. O. R. Reynolds, S. Wray, J. Wyatt, P. van der Zee, “Methods of quantitating cerebral near infrared spectroscopy data,” Adv. Exp. Med. Biol. 222, 183–189 (1988).
[CrossRef] [PubMed]

Wyatt, J.

M. Cope, D. T. Delpy, E. O. R. Reynolds, S. Wray, J. Wyatt, P. van der Zee, “Methods of quantitating cerebral near infrared spectroscopy data,” Adv. Exp. Med. Biol. 222, 183–189 (1988).
[CrossRef] [PubMed]

Wyman, D.

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

Yamada, Y.

Yee, C.

J. Haselgrove, J. Leigh, C. Yee, N. G. Wang, M. Maris, B. Chance, “Monte Carlo and diffusion calculations of photon migration in noninfinite highly scattering media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 30–41 (1991).
[CrossRef]

Yoo, K. M.

Zaccanti, G.

P. Donnelli, P. Bruscaglioni, A. Ismelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
[CrossRef]

Zanette, E.

M. Ferrari, E. Zanette, I. Giannini, G. Sideri, C. Fieschi, A. Carpi, “Effect of carotid artery compression test on regional cerebral blood volume, hemoglobin oxygen saturation and cytochrome-c-oxidase redox level in cerebrovascular patients,” Adv. Exp. Med. Biol. 200, 213–222 (1986).
[CrossRef] [PubMed]

Zhang, G.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Zhang, M. Z.

B. Chance, M. Maris, J. Sorge, M. Z. Zhang, “A phase modulation system for dual wavelength difference spectroscopy of hemoglobin deoxygenation in tissue,” in Time-Resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. SPIE1204, 481–491 (1990).
[CrossRef]

Adv. Exp. Med. Biol. (5)

O. Hazeki, A. Seyama, M. Tamura, “Near infrared spectrophotometric monitoring of hemoglobin and cytochrome aa3 in vivo,” Adv. Exp. Med. Biol. 215, 283–289 (1987).
[PubMed]

M. Ferrari, E. Zanette, I. Giannini, G. Sideri, C. Fieschi, A. Carpi, “Effect of carotid artery compression test on regional cerebral blood volume, hemoglobin oxygen saturation and cytochrome-c-oxidase redox level in cerebrovascular patients,” Adv. Exp. Med. Biol. 200, 213–222 (1986).
[CrossRef] [PubMed]

M. Cope, D. T. Delpy, E. O. R. Reynolds, S. Wray, J. Wyatt, P. van der Zee, “Methods of quantitating cerebral near infrared spectroscopy data,” Adv. Exp. Med. Biol. 222, 183–189 (1988).
[CrossRef] [PubMed]

P. van der Zee, D. T. Delpy, “Simulation of the point spread function for light in tissue by a Monte Carlo technique,” Adv. Exp. Med. Biol. 215, 179–191 (1987).

P. van der Zee, S. R. Arridge, M. Cope, D. T. Delpy, “The effect of optode positioning on optical path length in near infrared spectroscopy of brain,” Adv. Exp. Med. Biol. 277, 79–84 (1990).
[PubMed]

Appl. Opt. (5)

Crit. Care Med. (1)

C. C. Piantadosi, T. M. Hemstreet, F. F. Jöbsis, “Near infrared spectrophotometric monitoring of oxygen distribution to intact brain and skeletal muscle tissue,” Crit. Care Med. 14, 698–706 (1986).
[CrossRef] [PubMed]

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)

W. Cui, L. E. Ostrander, “The relationship of surface reflectance measurements to optical properties of layered biological media,” IEEE Trans. Biomed. Eng. 39, 194–201 (1992).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

P. Donnelli, P. Bruscaglioni, A. Ismelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
[CrossRef]

Lasers Med. Sci. (1)

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

Med. Biol. Eng. Comput. (1)

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]

Med. Phys. (3)

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys.1779–1792 (1995).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Med. Biol. (5)

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. S. Watt, “Estimation of optical path length through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

M. Firbank, D. T. Delpy, “A design for a stable and reproducible phantom for use in near-infrared imaging and spectroscopy,” Phys. Med. Biol. 38, 847–853 (1993).
[CrossRef]

M. Firbank, M. Oda, D. T. Delpy, “An improved design for a stable and reproducible phantom material for use in near infrared spectroscopy and imaging,” Phys. Med. Biol. 40, 955–961 (1995).
[CrossRef] [PubMed]

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

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

Proc. SPIE (4)

M. Schweiger, S. R. Arridge, “Near-infrared imaging: photon measurement density functions,” Proc. SPIE 2389, 366–377 (1995).
[CrossRef]

S. R. Arridge, M. Schweiger, “Sensitivity to prior knowledge in optical tomographic reconstruction,” Proc. SPIE 2389, 378–388 (1995).
[CrossRef]

B. W. Pogue, M. S. Patterson, T. J. Farrel, “Forward and inverse calculations for 3-D frequency-domain diffuse optical tomography,” Proc. SPIE 2389, 328–339 (1995).
[CrossRef]

D. A. Benaron, J. P. Van Houten, W. F. Cheong, E. L. Kermit, R. A. King, “Early clinical results of time-of-flight optical tomography in a neonatal intensive care unit,” Proc. SPIE 2389, 582–596 (1995).
[CrossRef]

Science (2)

F. F. Jöbsis, “Noninvasive, infrared monitoring of cerebral and myocardial oxygen sufficiency and circulatory parameters,” Science 198, 1264–1267 (1977).
[CrossRef] [PubMed]

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Other (7)

B. Chance, M. Maris, J. Sorge, M. Z. Zhang, “A phase modulation system for dual wavelength difference spectroscopy of hemoglobin deoxygenation in tissue,” in Time-Resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. SPIE1204, 481–491 (1990).
[CrossRef]

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

M. Firbank, “The design, calibration and usage of a solid scattering and absorbing phantom for near infra red spectroscopy,” Ph.D. dissertation (Department of Medical Physics and Bioengineering, University College of London, London, 1994).

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

W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979).

J. Haselgrove, J. Leigh, C. Yee, N. G. Wang, M. Maris, B. Chance, “Monte Carlo and diffusion calculations of photon migration in noninfinite highly scattering media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 30–41 (1991).
[CrossRef]

P. van der Zee, M. Essenpreis, D. T. Delpy, “Optical properties of brain tissue,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 454–465 (1993).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the design of the inhomogeneous phantoms.

Fig. 2
Fig. 2

Diagram of the optical system for the time-of-flight measurements.

Fig. 3
Fig. 3

Plots of the mean times of flight and the corresponding mean optical path lengths as functions of the detection angles estimated with two-dimensional (2D) and three-dimensional (3D) MC modeling and the FEM with either Dirichlet or Robin boundary conditions, together with the experimental results (the means of three measurements ±1 SD): (a) homogeneous phantom A, (b) inhomogeneous phantom B with a low-µ s ′ outer layer, (c) inhomogeneous phantom C with a high-µ s ′ outer layer, (d) inhomogeneous phantom D with a low-µ a outer layer, (e) inhomogeneous phantom E with a high-µ a outer layer.

Fig. 4
Fig. 4

Plots of the mean optical path length 〈L〉 divided by the chord length, which approximates the DPF, as a function of the detection angles corresponding to Fig. 3: (a) homogeneous phantom A, (b) inhomogeneous phantom B with a low-µ s ′ outer layer, (c) inhomogeneous phantom C with a high-µ s ′ outer layer, (d) inhomogeneous phantom D with a low-µ a outer layer, and (e) inhomogeneous phantom E with a high-µ a outer layer.

Fig. 5
Fig. 5

Plots of the relative mean optical path lengths in the inhomogeneous phantoms B–E as functions of the detection angle, normalized by the mean optical path length in the homogeneous phantom A.

Fig. 6
Fig. 6

Fraction of the total optical path length spent in the inner cylinder plotted as a function of the detection angle.

Tables (1)

Tables Icon

Table 1 Optical Properties of the Phantoms

Equations (10)

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

DP = Δ OD / Δ μ a L = c t ,
DPF = DP / D L / D = c t / D ,
PDP i = ( Δ OD ) i / Δ μ a i L i = c t i .
l i = ln ( P ) / μ s i ,
W = W 0 exp [ ( i = 1 2 μ a i L i ) ] ,
· κ ( r ) Φ ( r ) + μ a ( r ) Φ ( r ) = q 0 ( r ) ,
κ = 1 / { 3 [ μ a ( r ) + μ s ( r ) ] } ,
Φ ( ξ ) = 0 , ξ Ω ,
Φ ( ξ ) + 2 κ e n · Φ ( ξ ) = R [ Φ ( ξ ) 2 κ e n · Φ ( ξ ) ] , ξ Ω ,
R 1.4399 n 2 + 0.7099 n 1 + 0.6681 + 0.0636 n ,

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