Abstract

We approach the perturbative solution to the diffusion equation for the case of absorbing inclusions embedded in a heterogeneous scattering medium by using general properties of the radiative transfer equation and the solution of the Fredholm equation of the second kind given by the Neumann series. The terms of the Neumann series are used to obtain the expression of the moments of the generalized temporal point-spread function derived in transport theory. The moments are calculated independently by using Monte Carlo simulations for validation of the theory. While the mixed moments are correctly derived from the theory by using the solution of the diffusion equation in the geometry of interest, in order to obtain the self moments we should reframe the problem in transport theory and use a suitable solution of the radiative transfer equation for the calculation of the multiple integrals of the corresponding Neumann series. Since the rigorous theory leads to impractical formulas, in order to simplify and speed up the calculation of the self moments, we propose a heuristic method based on the calculation of only a single integral and some scaling parameters. We also propose simple quadrature rules for the calculation of the mixed moments for speeding up the computation of perturbations due to multiple defects. The theory can be developed in the continuous-wave domain, the time domain, and the frequency domain. In a companion paper [J. Opt. Soc. Am. A 23, 2119 (2006) ] we discuss the conditions of applicability of the theory in practical cases found in diffuse optical imaging of biological tissues.

© 2006 Optical Society of America

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2006 (1)

2005 (2)

2004 (1)

2003 (3)

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, and G. Zaccanti, "Solution of the time-dependent diffusion equation for layered random media by the eigenfunction method," Phys. Rev. E 67, 056623 (2003).
[CrossRef]

G. S. Abdoulaev and A. H. Hielscher, "Three-dimensional optical tomography with the equation of radiative transfer," J. Electron. Imaging 12, 594-600 (2003).
[CrossRef]

G. Strangman, M. A. Franceschini, and D. A. Boas, "Factors affecting the accuracy of near-infrared spectroscopy concentration calculations for focal changes in oxygenation parameters," Neuroimage 18, 865-879 (2003).
[CrossRef] [PubMed]

2002 (2)

2001 (4)

S. Carraresi, T. S. Mohamed Shatir, F. Martelli, and G. Zaccanti, "Accuracy of a perturbation model to predict the effect of scattering and absorbing inhomogeneities on photon migration," Appl. Opt. 40, 4622-4632 (2001).
[CrossRef]

R. L. Barbour, H. L. Graber, Y. Pei, S. Zhong, and C. H. Schmitz, "Optical tomographic imaging of dynamic features of dense-scattering media," J. Opt. Soc. Am. A 18, 3018-3036 (2001).
[CrossRef]

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]

R. Graaf and K. Rinzema, "Practical improvements on photon diffusion theory: application to isotropic scattering," Phys. Med. Biol. 46, 3043-3050 (2001).
[CrossRef]

1999 (1)

1998 (2)

1997 (7)

1996 (2)

1995 (8)

M. A. O'Leary, D. A. Boas, B. Chance, and A. G. Yodh, "Experimental images of inhomogeneous turbid media by frequency-domain diffusing- photon tomography," Opt. Lett. 20, 426-428 (1995).
[CrossRef] [PubMed]

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995), p. 934.

S. Feng, F. Zeng, and B. Chance, "Photon migration in the presence of a single defect: a perturbation analysis," Appl. Opt. 34, 3826-3837 (1995).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, "Photon-measurement density functions. Part 1. Analytical forms," Appl. Opt. 34, 7395-7409 (1995).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, "Photon-measurement density functions. Part 2. Finite-element-method calculations," Appl. Opt. 34, 8026-8037 (1995).
[CrossRef] [PubMed]

K. Rinzema, B. J. Hoenders, H. A. Ferwerda, and J. J. ten Bosch, "Analytic calculation of the radiance in an anisotropically scattering turbid medium close to a source," Pure Appl. Opt. 4, 629-642 (1995).
[CrossRef]

B. W. Pogue, M. S. Patterson, H. Jiang, and K. D. Paulsen, "Initial assessment of a simple system for frequency-domain diffuse optical tomography," Phys. Med. Biol. 40, 1709-1729 (1995).
[CrossRef] [PubMed]

L. Wang, S. L. Jacques, and L. Zheng, "MCML-Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

1994 (2)

K. Furutsu and Y. Yamada, "Diffusion approximation for a dissipative random medium and the applications," Phys. Rev. E 50, 3634-3640 (1994).
[CrossRef]

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, "Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: Analytic solution and applications," Proc. Natl. Acad. Sci. U.S.A. 91, 4887-4891 (1994).
[CrossRef] [PubMed]

1993 (4)

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, "A finite element approach to modelling 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, and 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]

R. Graaf, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, and J. G. Aarnoudse, "Condensed Monte Carlo simulations for the description of light transport," Appl. Opt. 32, 426-434 (1993).
[CrossRef]

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

1992 (1)

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

1989 (1)

R. Graaf, J. G. Aarnoudse, F. F. de Mul, and H. W. Jentink, "Light propagation parameters for anisotropically scattering media based on a rigorous solution of the transport equation," Appl. Opt. 26, 2273-2279 (1989).
[CrossRef]

1983 (1)

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

1979 (1)

J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).

1978 (1)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

1967 (1)

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

1953 (1)

K. M. Case, F. de Hoffman, and G. Placzek, Introduction to the Theory of Neutron Diffusion (U.S. Government Printing Office, 1953).

Aarnoudse, J. G.

R. Graaf, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, and J. G. Aarnoudse, "Condensed Monte Carlo simulations for the description of light transport," Appl. Opt. 32, 426-434 (1993).
[CrossRef]

R. Graaf, J. G. Aarnoudse, F. F. de Mul, and H. W. Jentink, "Light propagation parameters for anisotropically scattering media based on a rigorous solution of the transport equation," Appl. Opt. 26, 2273-2279 (1989).
[CrossRef]

Abdoulaev, G. S.

K. Ren, G. S. Abdoulaev, G. Bal, and A. H. Hielscher, "Algorithm for solving the equation of radiative transfer in the frequency domain," Opt. Lett. 29, 578-580 (2004).
[CrossRef] [PubMed]

G. S. Abdoulaev and A. H. Hielscher, "Three-dimensional optical tomography with the equation of radiative transfer," J. Electron. Imaging 12, 594-600 (2003).
[CrossRef]

Adam, G.

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

Alianelli, L.

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995), p. 934.

Aronson, R.

Arridge, S. R.

J. C. Hebden and S. R. Arridge, "Imaging through scattering media by the use of an analytical model of perturbation amplitudes in the time domain," Appl. Opt. 35, 6788-6796 (1996).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, "Photon-measurement density functions. Part 1. Analytical forms," Appl. Opt. 34, 7395-7409 (1995).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, "Photon-measurement density functions. Part 2. Finite-element-method calculations," Appl. Opt. 34, 8026-8037 (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 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, and D. T. Delpy, "A finite element approach to modelling photon transport in tissue," Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

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

Bal, G.

Barbour, R. L.

Bassani, M.

Blumetti, C.

Boas, D. A.

Carraresi, C.

Carraresi, S.

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

K. M. Case, F. de Hoffman, and G. Placzek, Introduction to the Theory of Neutron Diffusion (U.S. Government Printing Office, 1953).

Chance, B.

Contini, D.

Cope, 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 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, and D. T. Delpy, "The theoretical basis for the determination of optical pathlength in tissue: temporal and frequency analysis," Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef] [PubMed]

Culver, J. P.

Dassel, A. C. M.

de Hoffman, F.

K. M. Case, F. de Hoffman, and G. Placzek, Introduction to the Theory of Neutron Diffusion (U.S. Government Printing Office, 1953).

de Mul, F. F.

R. Graaf, J. G. Aarnoudse, F. F. de Mul, and H. W. Jentink, "Light propagation parameters for anisotropically scattering media based on a rigorous solution of the transport equation," Appl. Opt. 26, 2273-2279 (1989).
[CrossRef]

de Mul, F. F. M.

Del Bianco, S.

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, and G. Zaccanti, "Solution of the time-dependent diffusion equation for layered random media by the eigenfunction method," Phys. Rev. E 67, 056623 (2003).
[CrossRef]

Delpy, D. T.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, "A finite element approach to modelling 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, and 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, and D. T. Delpy, "The theoretical basis for the determination of optical pathlength in tissue: temporal and frequency analysis," Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef] [PubMed]

den Outer, P. N.

Duderstadt, J. J.

J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).

Dunn, A. K.

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 path length in inhomogeneous tissue and its application to near-infrared spectroscopy," Phys. Med. Biol. 38, 1859-1876 (1993).
[CrossRef] [PubMed]

Fantini, S.

Feng, S.

Ferwerda, H. A.

K. Rinzema, B. J. Hoenders, H. A. Ferwerda, and J. J. ten Bosch, "Analytic calculation of the radiance in an anisotropically scattering turbid medium close to a source," Pure Appl. Opt. 4, 629-642 (1995).
[CrossRef]

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 path length in inhomogeneous tissue and its application to near-infrared spectroscopy," Phys. Med. Biol. 38, 1859-1876 (1993).
[CrossRef] [PubMed]

Franceschini, M. A.

G. Strangman, M. A. Franceschini, and D. A. Boas, "Factors affecting the accuracy of near-infrared spectroscopy concentration calculations for focal changes in oxygenation parameters," Neuroimage 18, 865-879 (2003).
[CrossRef] [PubMed]

Furutsu, K.

K. Furutsu and Y. Yamada, "Diffusion approximation for a dissipative random medium and the applications," Phys. Rev. E 50, 3634-3640 (1994).
[CrossRef]

Gao, F.

Graaf, R.

R. Graaf and K. Rinzema, "Practical improvements on photon diffusion theory: application to isotropic scattering," Phys. Med. Biol. 46, 3043-3050 (2001).
[CrossRef]

R. Graaf, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, and J. G. Aarnoudse, "Condensed Monte Carlo simulations for the description of light transport," Appl. Opt. 32, 426-434 (1993).
[CrossRef]

R. Graaf, J. G. Aarnoudse, F. F. de Mul, and H. W. Jentink, "Light propagation parameters for anisotropically scattering media based on a rigorous solution of the transport equation," Appl. Opt. 26, 2273-2279 (1989).
[CrossRef]

Graber, H. L.

Hebden, J. C.

Hielscher, A. H.

K. Ren, G. S. Abdoulaev, G. Bal, and A. H. Hielscher, "Algorithm for solving the equation of radiative transfer in the frequency domain," Opt. Lett. 29, 578-580 (2004).
[CrossRef] [PubMed]

G. S. Abdoulaev and A. H. Hielscher, "Three-dimensional optical tomography with the equation of radiative transfer," J. Electron. Imaging 12, 594-600 (2003).
[CrossRef]

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 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, and D. T. Delpy, "A finite element approach to modelling photon transport in tissue," Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

Hoenders, B. J.

K. Rinzema, B. J. Hoenders, H. A. Ferwerda, and J. J. ten Bosch, "Analytic calculation of the radiance in an anisotropically scattering turbid medium close to a source," Pure Appl. Opt. 4, 629-642 (1995).
[CrossRef]

Homma, K.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Ismaelli, A.

Jacques, S. L.

M. R. Ostermeyer and S. L. Jacques, "Perturbation theory for diffuse light transport in complex biological tissues," J. Opt. Soc. Am. A 14, 255-261 (1997).
[CrossRef]

L. Wang, S. L. Jacques, and L. Zheng, "MCML-Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Jentink, H. W.

R. Graaf, J. G. Aarnoudse, F. F. de Mul, and H. W. Jentink, "Light propagation parameters for anisotropically scattering media based on a rigorous solution of the transport equation," Appl. Opt. 26, 2273-2279 (1989).
[CrossRef]

Jiang, H.

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, "Optical image reconstruction using frequency-domain data: simulations and experiments," J. Opt. Soc. Am. A 13, 253-266 (1996).
[CrossRef]

B. W. Pogue, M. S. Patterson, H. Jiang, and K. D. Paulsen, "Initial assessment of a simple system for frequency-domain diffuse optical tomography," Phys. Med. Biol. 40, 1709-1729 (1995).
[CrossRef] [PubMed]

Kienle, A.

Koelink, M. H.

Lagendijk, A.

Martelli, F.

Martin, W. R.

J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).

Murrer, L. H. P.

Nieuwenhuizen, Th. M.

Obrig, H.

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]

O'Leary, M. A.

Osterberg, U. L.

Ostermeyer, M. R.

Paasschens, J. C. J.

J. C. J. Paasschens, "Solution of the time-dependent Boltzmann equation," Phys. Rev. E 56, 1135-1141 (1997).
[CrossRef]

Patterson, M. S.

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, "Optical image reconstruction using frequency-domain data: simulations and experiments," J. Opt. Soc. Am. A 13, 253-266 (1996).
[CrossRef]

B. W. Pogue, M. S. Patterson, H. Jiang, and K. D. Paulsen, "Initial assessment of a simple system for frequency-domain diffuse optical tomography," Phys. Med. Biol. 40, 1709-1729 (1995).
[CrossRef] [PubMed]

Paulsen, K. D.

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, "Optical image reconstruction using frequency-domain data: simulations and experiments," J. Opt. Soc. Am. A 13, 253-266 (1996).
[CrossRef]

B. W. Pogue, M. S. Patterson, H. Jiang, and K. D. Paulsen, "Initial assessment of a simple system for frequency-domain diffuse optical tomography," Phys. Med. Biol. 40, 1709-1729 (1995).
[CrossRef] [PubMed]

Pei, Y.

Placzek, G.

K. M. Case, F. de Hoffman, and G. Placzek, Introduction to the Theory of Neutron Diffusion (U.S. Government Printing Office, 1953).

Pogue, B. W.

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, "Optical image reconstruction using frequency-domain data: simulations and experiments," J. Opt. Soc. Am. A 13, 253-266 (1996).
[CrossRef]

B. W. Pogue, M. S. Patterson, H. Jiang, and K. D. Paulsen, "Initial assessment of a simple system for frequency-domain diffuse optical tomography," Phys. Med. Biol. 40, 1709-1729 (1995).
[CrossRef] [PubMed]

Ren, K.

Rinneberg, H.

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]

Rinzema, K.

R. Graaf and K. Rinzema, "Practical improvements on photon diffusion theory: application to isotropic scattering," Phys. Med. Biol. 46, 3043-3050 (2001).
[CrossRef]

K. Rinzema, L. H. P. Murrer, and W. M. Star, "Direct experimental verification of light transport theory in an optical phantom," J. Opt. Soc. Am. A 15, 2078-2088 (1998).
[CrossRef]

K. Rinzema, B. J. Hoenders, H. A. Ferwerda, and J. J. ten Bosch, "Analytic calculation of the radiance in an anisotropically scattering turbid medium close to a source," Pure Appl. Opt. 4, 629-642 (1995).
[CrossRef]

Sassaroli, A.

Schmitz, C. H.

Schweiger, M.

Shatir, T. S. Mohamed

Star, W. M.

Steinbrink, J.

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]

Stott, J. J.

Strangman, G.

G. Strangman, M. A. Franceschini, and D. A. Boas, "Factors affecting the accuracy of near-infrared spectroscopy concentration calculations for focal changes in oxygenation parameters," Neuroimage 18, 865-879 (2003).
[CrossRef] [PubMed]

Tanikawa, Y.

ten Bosch, J. J.

K. Rinzema, B. J. Hoenders, H. A. Ferwerda, and J. J. ten Bosch, "Analytic calculation of the radiance in an anisotropically scattering turbid medium close to a source," Pure Appl. Opt. 4, 629-642 (1995).
[CrossRef]

Tsuchiya, Y.

van der Zee, P.

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 path length in inhomogeneous tissue and its application to near-infrared spectroscopy," Phys. Med. Biol. 38, 1859-1876 (1993).
[CrossRef] [PubMed]

Villringer, A.

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]

Wabnitz, H.

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]

Wang, L.

L. Wang, S. L. Jacques, and L. Zheng, "MCML-Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Wang, Y.

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995), p. 934.

Wilson, B. C.

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

Yamada, Y.

H. Zhao, F. Gao, Y. Tanikawa, K. Homma, and Y. Yamada, "Time-resolved diffuse optical tomography imaging for the provision of both anatomical and functional information about biological tissue," Appl. Opt. 44, 1905-1916 (2005).
[CrossRef] [PubMed]

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, and G. Zaccanti, "Solution of the time-dependent diffusion equation for layered random media by the eigenfunction method," Phys. Rev. E 67, 056623 (2003).
[CrossRef]

K. Furutsu and Y. Yamada, "Diffusion approximation for a dissipative random medium and the applications," Phys. Rev. E 50, 3634-3640 (1994).
[CrossRef]

Yao, Y.

Yodh, A. G.

Zaccanti, G.

Zeng, F.

Zhao, H.

Zheng, L.

L. Wang, S. L. Jacques, and L. Zheng, "MCML-Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Zhong, S.

Zhu, W.

Zijlstra, W. G.

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

Appl. Opt. (10)

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, "Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal to noise analysis," Appl. Opt. 36, 75-92 (1997).
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R. Graaf, J. G. Aarnoudse, F. F. de Mul, and H. W. Jentink, "Light propagation parameters for anisotropically scattering media based on a rigorous solution of the transport equation," Appl. Opt. 26, 2273-2279 (1989).
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S. R. Arridge and M. Schweiger, "Photon-measurement density functions. Part 2. Finite-element-method calculations," Appl. Opt. 34, 8026-8037 (1995).
[CrossRef] [PubMed]

H. Zhao, F. Gao, Y. Tanikawa, K. Homma, and Y. Yamada, "Time-resolved diffuse optical tomography imaging for the provision of both anatomical and functional information about biological tissue," Appl. Opt. 44, 1905-1916 (2005).
[CrossRef] [PubMed]

R. Graaf, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, and J. G. Aarnoudse, "Condensed Monte Carlo simulations for the description of light transport," Appl. Opt. 32, 426-434 (1993).
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A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaelli, and G. Zaccanti, "Monte Carlo procedure for investigating light propagation and imaging of highly scattering media," Appl. Opt. 37, 7392-7400 (1998).
[CrossRef]

S. R. Arridge and M. Schweiger, "Photon-measurement density functions. Part 1. Analytical forms," Appl. Opt. 34, 7395-7409 (1995).
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J. C. Hebden and S. R. Arridge, "Imaging through scattering media by the use of an analytical model of perturbation amplitudes in the time domain," Appl. Opt. 35, 6788-6796 (1996).
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S. Carraresi, T. S. Mohamed Shatir, F. Martelli, and G. Zaccanti, "Accuracy of a perturbation model to predict the effect of scattering and absorbing inhomogeneities on photon migration," Appl. Opt. 40, 4622-4632 (2001).
[CrossRef]

S. Feng, F. Zeng, and B. Chance, "Photon migration in the presence of a single defect: a perturbation analysis," Appl. Opt. 34, 3826-3837 (1995).
[CrossRef] [PubMed]

Comput. Methods Programs Biomed. (1)

L. Wang, S. L. Jacques, and L. Zheng, "MCML-Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

J. Electron. Imaging (1)

G. S. Abdoulaev and A. H. Hielscher, "Three-dimensional optical tomography with the equation of radiative transfer," J. Electron. Imaging 12, 594-600 (2003).
[CrossRef]

J. Opt. Soc. Am. A (10)

M. R. Ostermeyer and S. L. Jacques, "Perturbation theory for diffuse light transport in complex biological tissues," J. Opt. Soc. Am. A 14, 255-261 (1997).
[CrossRef]

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]

R. L. Barbour, H. L. Graber, Y. Pei, S. Zhong, and C. H. Schmitz, "Optical tomographic imaging of dynamic features of dense-scattering media," J. Opt. Soc. Am. A 18, 3018-3036 (2001).
[CrossRef]

K. Rinzema, L. H. P. Murrer, and W. M. Star, "Direct experimental verification of light transport theory in an optical phantom," J. Opt. Soc. Am. A 15, 2078-2088 (1998).
[CrossRef]

Y. Yao, Y. Wang, Y. Pei, W. Zhu, and R. L. Barbour, "Frequency domain optical imaging of absorption and scattering distributions by a Born iterative method," J. Opt. Soc. Am. A 14, 325-342 (1997).
[CrossRef]

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, "Optical image reconstruction using frequency-domain data: simulations and experiments," J. Opt. Soc. Am. A 13, 253-266 (1996).
[CrossRef]

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

A. Kienle, "Light diffusion through a turbid parallelepiped," J. Opt. Soc. Am. A 22, 1883-1888 (2005).
[CrossRef]

R. Aronson, "Radiative transfer implies a modified reciprocity relation," J. Opt. Soc. Am. A 14, 486-490 (1997).
[CrossRef]

Y. Tsuchiya, "Photon path distribution in inhomogeneous turbid media: theoretical analysis and a method of calculation," J. Opt. Soc. Am. A 19, 1383-1389 (2002).
[CrossRef]

Med. Phys. (2)

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, "A finite element approach to modelling photon transport in tissue," Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

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

Neuroimage (1)

G. Strangman, M. A. Franceschini, and D. A. Boas, "Factors affecting the accuracy of near-infrared spectroscopy concentration calculations for focal changes in oxygenation parameters," Neuroimage 18, 865-879 (2003).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (4)

Phys. Med. Biol. (5)

B. W. Pogue, M. S. Patterson, H. Jiang, and K. D. Paulsen, "Initial assessment of a simple system for frequency-domain diffuse optical tomography," Phys. Med. Biol. 40, 1709-1729 (1995).
[CrossRef] [PubMed]

R. Graaf and K. Rinzema, "Practical improvements on photon diffusion theory: application to isotropic scattering," Phys. Med. Biol. 46, 3043-3050 (2001).
[CrossRef]

S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlength 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, and 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]

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]

Phys. Rev. E (3)

K. Furutsu and Y. Yamada, "Diffusion approximation for a dissipative random medium and the applications," Phys. Rev. E 50, 3634-3640 (1994).
[CrossRef]

J. C. J. Paasschens, "Solution of the time-dependent Boltzmann equation," Phys. Rev. E 56, 1135-1141 (1997).
[CrossRef]

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, and G. Zaccanti, "Solution of the time-dependent diffusion equation for layered random media by the eigenfunction method," Phys. Rev. E 67, 056623 (2003).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, "Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: Analytic solution and applications," Proc. Natl. Acad. Sci. U.S.A. 91, 4887-4891 (1994).
[CrossRef] [PubMed]

Pure Appl. Opt. (1)

K. Rinzema, B. J. Hoenders, H. A. Ferwerda, and J. J. ten Bosch, "Analytic calculation of the radiance in an anisotropically scattering turbid medium close to a source," Pure Appl. Opt. 4, 629-642 (1995).
[CrossRef]

Other (5)

K. M. Case, F. de Hoffman, and G. Placzek, Introduction to the Theory of Neutron Diffusion (U.S. Government Printing Office, 1953).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995), p. 934.

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Figures (1)

Fig. 1
Fig. 1

Absorbing and scattering medium having arbitrary boundary. A pencil beam is impinging in r = r s and a pointlike detector is placed at r = r b . The position of the pointlike source equivalent to the pencil beam is at r 0 = 1 μ s 0 (where μ s 0 is the reduced scattering coefficient of the voxel struck by the pencil beam). The medium is divided into N elementary regions or voxels characterized by different values of the optical properties and volumes V i , i = 1 , 2 , , N .

Equations (69)

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

I ( r , s ) = I 0 ( s ) exp ( μ a l ) ,
R N + f 0 ( t 1 , , t N ) d t 1 d t N = A 0 ( r b , μ a = 0 ) ,
f i n ( t 1 , , t N ) = f 0 ( t 1 , , t N ) exp ( i = 1 N μ a i v t i ) .
R N + f i n ( t 1 , , t N ) d t 1 d t N = A i n ( r b , μ a ) .
R N + f i n ( t 1 , , t N ) exp ( i = 1 N v t i Δ μ a i ) d t 1 d t N = A f i ( r b , μ a + Δ μ a ) .
A f i ( r b , μ a + Δ μ a ) = A i n ( r b , μ a ) k 1 , k 2 , , k N = 0 ( 1 ) k 1 + + k N ( Δ μ a 1 ) k 1 ( Δ μ a N ) k N k 1 ! k N ! l 1 k 1 l N k N ,
l 1 k 1 l N k N = R N + ( v t 1 ) k 1 ( v t N ) k N f i n ( t 1 , , t N ) d t 1 d t N R N + f i n ( t 1 , , t N ) d t 1 d t N ,
l 1 k 1 l N k N = ( 1 ) k 1 + k 2 + + k N 1 A i n ( r b , μ a ) k 1 + k 2 + + k N A i n ( r b , μ a ) μ a 1 k 1 μ a 2 k 2 μ a N k N ,
k 1 + k 2 + + k N > 0 .
[ D ( r ) ϕ i n ( r ) ] + μ a ( r ) ϕ i n ( r ) = δ ( r r 0 ) ,
{ D ( r ) [ Δ ϕ ( r ) ] } + μ a ( r ) Δ ϕ ( r ) = Δ μ a ( r ) ϕ f i ( r ) ,
ϕ f i ( r ) = ϕ i n ( r ) V Δ μ a ( r 1 ) ϕ i n ( r , r 1 ) ϕ f i ( r 1 ) d r 1 ,
ϕ f i ( r ) = lim n ϕ n ( r ) ,
ϕ 0 ( r ) = ϕ i n ( r ) , ϕ n ( r ) = k = 0 n ( 1 ) k u k ( r ) ,
u n ( r ) = V V V K ( r , r 1 ) K ( r 1 , r 2 ) K ( r n 1 , r n ) ϕ 0 ( r n ) d r n d r 1 ,
K ( r , r 1 ) = Δ μ a ( r 1 ) ϕ 0 ( r , r 1 ) .
A ( r b ) = D ( r b ) ϕ ( r ) r = r b n ,
A 0 ( r b ) = A i n ( r b ) , A n ( r b ) = k = 0 n ( 1 ) k u ̃ k ( r b ) ,
u ̃ n ( r b ) = V V V K ̃ ( r b , r 1 ) K ( r 1 , r 2 ) K ( r n 1 , r n ) ϕ 0 ( r n ) d r n d r 1 .
K ̃ ( r b , r 1 ) = Δ μ a ( r 1 ) A 0 ( r b , r 1 ) = Δ μ a ( r 1 ) D ( r b ) ϕ 0 ( r , r 1 ) r = r b n ,
A 0 ( r b ) = D ( r b ) ϕ 0 ( r ) r = r b n .
A n ( r b ) = A n 1 ( r b ) + ( 1 ) n V Δ μ a ( r 1 ) A 0 ( r b , r 1 ) V Δ μ a ( r 2 ) ϕ 0 ( r 1 , r 2 ) V Δ μ a ( r n 1 ) ϕ 0 ( r n 2 , r n 1 ) V Δ μ a ( r n ) ϕ 0 ( r n 1 , r n ) ϕ 0 ( r n ) d r n d r 1 .
A 1 ( r b ) = A 0 ( r b ) i = 1 N V i Δ μ a ( r 1 ) A 0 ( r b , r 1 ) ϕ 0 ( r 1 ) d r 1 .
A 1 ( r b ) A 0 ( r b ) A 0 ( r b ) = i = 1 N Δ μ a ( r i ) V i A 0 ( r b , r 1 ) ϕ 0 ( r 1 ) d r 1 A 0 ( r b ) = k 1 + + k N = 1 l 1 k 1 l N k N k 1 ! k N ! [ Δ μ a ( r 1 ) ] k 1 [ Δ μ a ( r N ) ] k N ,
l i = V i A 0 ( r b , r 1 ) ϕ 0 ( r 1 ) d r 1 A 0 ( r b ) .
l i A 0 ( r b , r i ) ϕ 0 ( r i ) V i A 0 ( r b ) .
A 2 ( r b ) = A 1 ( r b ) + i = 1 N Δ μ a ( r i ) V i A 0 ( r b , r 1 ) d r 1 j = 1 N Δ μ a ( r j ) V j ϕ 0 ( r 1 , r 2 ) ϕ 0 ( r 2 ) d r 2 .
A 2 ( r b ) A 0 ( r b ) A 0 ( r b ) = i = 1 N l i Δ μ a ( r i ) + k 1 + + k N = 2 [ Δ μ a ( r 1 ) ] k 1 , , [ Δ μ a ( r N ) ] k N k 1 ! , , k N ! l 1 k 1 l N k N ,
l i k i l j k j = k i ! k j ! P ( i , j ) V i A 0 ( r b , r 1 ) d r 1 V j ϕ 0 ( r 1 , r 2 ) ϕ 0 ( r 2 ) d r 2 A 0 ( r b ) .
l i l j P ( i , j ) A 0 ( r b , r i ) ϕ 0 ( r i , r j ) ϕ 0 ( r j ) V i V j A 0 ( r b ) ,
l i 2 = 2 V i A 0 ( r b , r 1 ) d r 1 V i ϕ 0 ( r 1 , r 2 ) ϕ 0 ( r 2 ) d r 2 A 0 ( r b ) .
l i 2 2 l i V i V i d r 1 V i ϕ 0 ( r 1 , r 2 ) d r 2 ,
l i k i l j k j l m k m = k i ! k j ! k m ! P ( i , j , m ) V i A 0 ( r b , r 1 ) d r 1 V j ϕ 0 ( r 1 , r 2 ) d r 2 V m ϕ 0 ( r 2 , r 3 ) ϕ 0 ( r 3 ) d r 3 A 0 ( r b ) .
l i l j l m P ( i , j , m ) A 0 ( r b , r i ) ϕ 0 ( r i , r j ) ϕ 0 ( r j , r m ) ϕ 0 ( r m ) V i V j V m A 0 ( r b ) ,
l i 3 3 ! l i V i V i d r 1 V i ϕ 0 ( r 1 , r 2 ) d r 2 V i ϕ 0 ( r 2 , r 3 ) d r 3 ,
l i n n ! l i V i V i d r 1 V i ϕ 0 ( r 1 , r 2 ) d r 2 V i ϕ 0 ( r 2 , r 3 ) d r 3 V i ϕ 0 ( r n 1 , r n ) d r n .
l i n c n 1 l i ( V i ϕ 0 ( r , r i ) d r ) n 1 ,
V i d r 1 V i ϕ 0 ( r 1 , r 2 ) d r 2 V i ϕ 0 ( r n 1 , r n ) d r n c n 1 V i n ! ( V i ϕ 0 ( r , r i ) d r ) n 1 .
l i n c n 1 l i ( V i ϕ 0 ( r r i ) d r ) n 1 ,
ϕ 0 ( r r i ) = exp ( μ eff r r i ) 4 π D r r i ,
ϕ 0 ( r r i ) = 1 4 π [ a exp ( k 0 r r i ) D app r r i + exp ( μ t r r i ) r r i 2 ] .
a arctanh ( k 0 μ t ) k 0 μ t = 1 ,
l 1 k 1 l n k n = k 1 ! k n ! P ( i 1 , , i n ) V i 1 A 0 ( r b , r i 1 ) d r i 1 V i 2 ϕ 0 ( r i 1 , r i 2 ) d r i 2 V i n ϕ 0 ( r i n 1 , r i n ) ϕ 0 ( r i n ) d r i n A 0 ( r b ) .
σ N f 0 ( t 1 , , t N ) d σ = R 0 ( r b , t , μ a = 0 ) ,
R f i ( r b , t , μ a + Δ μ a ) = R i n ( r b , t , μ a ) k 1 , k 2 , , k N = 0 ( 1 ) k 1 + k 2 + + k N ( Δ μ a 1 ) k 1 ( Δ μ a N ) k N k 1 ! k N ! l 1 k 1 , , l N k N ( t ) ,
l 1 k 1 l N k N ( t ) = σ N ( v t 1 ) k 1 , , ( v t N ) k N f i n ( t 1 , , t N ) d σ σ N f i n ( t 1 , , t N ) d σ .
1 v ϕ i n ( r , t ) t [ D ( r ) ϕ i n ( r , t ) ] + μ a ( r ) ϕ i n ( r , t ) = δ ( r r 0 ) δ ( t ) ,
l i ( t ) = V i d r 1 0 R 0 ( r b , r 1 , t t 1 ) ϕ 0 ( r 1 , t 1 ) d t 1 R 0 ( r b , t ) ,
l i k i l j k j ( t ) = 1 R 0 ( r b , t ) k i ! k j ! P ( i , j ) V i d r 1 0 R 0 ( r b , r 1 , t t 1 ) d t 1 × V j d r 2 0 ϕ 0 ( r 1 , r 2 , t 1 t 2 ) ϕ 0 ( r 2 , t 2 ) d t 2 .
{ D ( r ) [ ϕ ̃ ( r , ω ) ] } + [ μ a ( r ) + i ω ν ] ϕ ̃ ( r , ω ) = δ ( r r 0 ) ,
R ̃ f i ( r b , ω , μ a + Δ μ a ) = R ̃ i n ( r b , ω , μ a ) × k 1 , k 2 , , k N = 0 ( 1 ) k 1 + k 2 + + k N ( Δ μ a 1 ) k 1 ( Δ μ a N ) k N k 1 ! k N ! l 1 k 1 l 2 k 2 l N k N ( ω ) ,
l 1 k 1 l 2 k 2 l N k N ( ω ) = G ̃ ( k 1 , k 2 , , k N , ω ) R ̃ i n ( r b , ω , μ a ) .
l i ( ω ) = V i R ̃ 0 ( r b , r 1 , ω ) ϕ ̃ 0 ( r 1 , ω ) d r 1 R ̃ 0 ( r b , ω ) ,
l i k i l j k j ( ω ) = k i ! k j ! P ( i , j ) V i R ̃ 0 ( r b , r 1 , ω ) d r 1 V j ϕ ̃ 0 ( r 1 , r 2 , ω ) ϕ ̃ 0 ( r 2 , ω ) d r 2 R ̃ 0 ( r b , ω ) .
l i 2 l k = 2 A 0 ( r b ) [ V i A 0 ( r b , r 1 ) d r 1 V i ϕ 0 ( r 1 , r 2 ) d r 2 V k ϕ 0 ( r 2 , r 3 ) ϕ 0 ( r 3 ) d r 3 + V i A 0 ( r b , r 1 ) d r 1 V k ϕ 0 ( r 1 , r 2 ) d r 2 V i ϕ 0 ( r 2 , r 3 ) ϕ 0 ( r 3 ) d r 3 + V k A 0 ( r b , r 1 ) d r 1 V i ϕ 0 ( r 1 , r 2 ) d r 2 V i ϕ 0 ( r 2 , r 3 ) ϕ 0 ( r 3 ) d r 3 ] .
l i 2 l k 2 A 0 ( r b ) A 0 ( r b , r i ) ϕ 0 ( r i , r k ) ϕ 0 ( r k ) V k V i d r 1 V i ϕ 0 ( r 1 , r 2 ) d r 2 + 2 A 0 ( r b ) A 0 ( r b , r i ) ϕ 0 ( r i , r k ) ϕ 0 ( r k , r i ) ϕ 0 ( r i ) V i 2 V k + 2 A 0 ( r b ) A 0 ( r b , r k ) ϕ 0 ( r k , r i ) ϕ 0 ( r i ) V k V i d r 2 V i ϕ 0 ( r 2 , r 3 ) d r 3 ,
l i 2 l k c 1 l i l k V i ϕ 0 ( r r i ) d r + 2 l i [ ϕ 0 ( r i , r k ) ] 2 V i V k ,
s I ( r , s ) + [ μ a ( r ) + μ s ( r ) ] I ( r , s ) = μ s ( r ) 4 π p ( s , s ) I ( r , s ) d ω + S ( r , s ) .
I f i ( r , s ) = I 0 ( r , s ) V d r 1 4 π I 0 ( r , s , r 1 , s 1 ) Δ μ a ( r 1 ) I f i ( r 1 , s 1 ) d ω 1 .
A n ( r b ) = 2 π I n ( r b , s ) s n d ω .
l i k i l i k j = k i ! k j ! A 0 ( r b ) P ( i , j ) V i d r 1 V j d r 2 2 π s n d ω 4 π I 0 ( r b , s , r 1 , s 1 ) d ω 1 × 4 π I 0 ( r 1 , s 1 , r 2 , s 2 ) I 0 ( r 2 , s 2 ) d ω 2 .
A 0 ( r b , r 1 ) = 2 π I 0 ( r b , s , r 1 ) s n d ω ,
ϕ 0 ( r 1 , r 2 ) = 4 π d ω 1 4 π I 0 ( r 1 , s 1 , r 2 , s 2 ) 4 π d ω 2 .
l i k i l j k j l m k m = k i ! k j ! k m ! A 0 ( r b ) P ( i , j , m ) V i d r 1 V j d r 2 V m d r 3 2 π s n d ω 4 π I 0 ( r b , s , r 1 , s 1 ) d ω 1 × 4 π I 0 ( r 1 , s 1 , r 2 , s 2 ) d ω 2 4 π I 0 ( r 2 , s 2 , r 3 , s 3 ) I 0 ( r 3 , s 3 ) d ω 3 .
l i k i l j k j l m k m = k i ! k j ! k m ! A 0 ( r b ) P ( i , j , m ) V i A 0 ( r b , r 1 ) d r 1 V j d r 2 V m ϕ 0 ( r 3 ) d r 3 4 π d ω 1 4 π I 0 ( r 1 , s 1 , r 2 , s 2 ) d ω 2 4 π 1 4 π I 0 ( r 2 , s 2 , r 3 , s 3 ) d ω 3 .
ϕ ( r ) = ϕ 0 ( r ) k = 0 ( 1 ) k l k ϕ k ! ( Δ μ a ) k ,
ϕ ( r ) = ϕ 0 ( r ) exp [ α ( r ) ] ,
α ( r ) = Δ μ a V ϕ 0 ( r , r 1 ) ϕ 0 ( r 1 ) d r 1 ϕ 0 ( r ) = l ϕ Δ μ a .
A ( r b ) = A 0 ( r b ) exp ( l ϕ Δ μ a ) { ( 1 l ϕ ) ( Δ μ a ) 2 [ l A l ϕ ] } .

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