Abstract

We apply a previously proposed perturbation theory of the diffusion equation for studying light propagation through heterogeneous media in the presence of absorbing defects. The theory is based on the knowledge of (a) the geometric characteristics of a focal inclusion, (b) the mean optical path length inside the inclusion, and (c) the optical properties of the inclusion. The potential of this method is shown in the layered and slab geometries, where calculations are carried out up to the fourth order. The relative changes of intensity with respect to the unperturbed (heterogeneous) medium are predicted by the theory to within 10% for a wide range of contrasts dΔμa (up to dΔμa0.40.8), where d is the effective diameter of the defect and Δμa the absorption contrast between defect and local background. We also show how the method of Padé approximants can be used to extend the validity of the theory for a larger range of absorption contrasts. Finally, we study the possibility of using the proposed method for calculating the effect of a colocalized scattering and absorbing perturbation.

© 2009 Optical Society of America

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2007 (2)

D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, “Evaluation of higher-order time domain perturbation theory of photon diffusion on breast-equivalent phantoms and optical mammograms,” Phys. Rev. E 76, 061908 (2007).
[CrossRef]

P. Taroni, D. Comelli, A. Pifferi, A. Torricelli, and R. Cubeddu, “Absorption of collagen: effects on the estimate of breast composition and related diagnostic implications,” J Biomed. Opt. 12, 014021 (2007).
[CrossRef] [PubMed]

2006 (5)

2005 (2)

F. Martelli, S. Del Bianco, and G. Zaccanti, “Perturbation model of light propagation through diffusive layered media,” Phys. Med. Biol. 50, 2159-2166 (2005).
[CrossRef] [PubMed]

S. Fantini, E. L. Heffer, V. E. Pera, A. Sassaroli, and N. Liu, “Spatial and spectral information in optical mammography,” Technol. Cancer Res. Treat. 4, 471-482 (2005).
[PubMed]

2004 (2)

2003 (2)

L. Spinelli, A. Torricelli, A. Pifferi, P. Taroni, and R. Cubeddu, “Experimental tests of a perturbation model for time-resolved imaging of diffusive media,” Appl. Opt. 42, 3145-3153 (2003).
[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]

2002 (1)

2001 (3)

2000 (1)

1999 (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

1998 (2)

1997 (4)

1996 (1)

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. 13, 253-266 (1996).
[CrossRef]

1995 (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]

1994 (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. USA 91, 4887-4891(1994).
[CrossRef] [PubMed]

1993 (3)

1992 (2)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, 1992).

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]

1991 (1)

Aarnoudse, J. G.

Abdoulaev, G. S.

Alfano, R. R.

Arridge, A. R.

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, A. R. Arridge, and J. Ripoll, “Diffuse photon propagation in multilayered geometries,” Phys. Med. Biol. 51, 497-516 (2006).
[CrossRef] [PubMed]

Arridge, S. R.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

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.

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. 18, 3018-3036 (2001).
[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]

Bergh, H. V.

Boas, D. A.

Cai, W.

Carraresi, S.

Chance, B.

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).
[CrossRef] [PubMed]

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. USA 91, 4887-4891(1994).
[CrossRef] [PubMed]

Chernomordik, V.

Comelli, D.

P. Taroni, D. Comelli, A. Pifferi, A. Torricelli, and R. Cubeddu, “Absorption of collagen: effects on the estimate of breast composition and related diagnostic implications,” J Biomed. Opt. 12, 014021 (2007).
[CrossRef] [PubMed]

Cope, M.

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]

Cubeddu, R.

Culver, J. P.

Dassel, A. C. M.

de Mul, F. F. M.

Del Bianco, S.

F. Martelli, S. Del Bianco, and G. Zaccanti, “Perturbation model of light propagation through diffusive layered media,” Phys. Med. Biol. 50, 2159-2166 (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]

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]

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.

Douiri, A.

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, A. R. Arridge, and J. Ripoll, “Diffuse photon propagation in multilayered geometries,” Phys. Med. Biol. 51, 497-516 (2006).
[CrossRef] [PubMed]

Dunn, A. K.

Fantini, S.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, 1992).

Gandjbakhche, A.

Glanzmann, T.

Graaf, R.

Graber, H. L.

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. 18, 3018-3036 (2001).
[CrossRef]

Gratton, E.

Grosenick, D.

D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, “Evaluation of higher-order time domain perturbation theory of photon diffusion on breast-equivalent phantoms and optical mammograms,” Phys. Rev. E 76, 061908 (2007).
[CrossRef]

Hattery, D.

Heffer, E. L.

S. Fantini, E. L. Heffer, V. E. Pera, A. Sassaroli, and N. Liu, “Spatial and spectral information in optical mammography,” Technol. Cancer Res. Treat. 4, 471-482 (2005).
[PubMed]

Hielscher, A. H.

Hiraoka, M.

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]

Horesh, L.

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, A. R. Arridge, and J. Ripoll, “Diffuse photon propagation in multilayered geometries,” Phys. Med. Biol. 51, 497-516 (2006).
[CrossRef] [PubMed]

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]

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. 13, 253-266 (1996).
[CrossRef]

Kienle, A.

Kim, A. D.

Koelink, M. H.

Kummrow, A.

D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, “Evaluation of higher-order time domain perturbation theory of photon diffusion on breast-equivalent phantoms and optical mammograms,” Phys. Rev. E 76, 061908 (2007).
[CrossRef]

Lagendijk, A.

Liu, N.

S. Fantini, E. L. Heffer, V. E. Pera, A. Sassaroli, and N. Liu, “Spatial and spectral information in optical mammography,” Technol. Cancer Res. Treat. 4, 471-482 (2005).
[PubMed]

Macdonald, R.

D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, “Evaluation of higher-order time domain perturbation theory of photon diffusion on breast-equivalent phantoms and optical mammograms,” Phys. Rev. E 76, 061908 (2007).
[CrossRef]

Martelli, F.

Nieto-Vesperinas, M.

Nieuwenhuizen, Th. M.

Ntziachristos, V.

O'Leary, M. A.

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).
[CrossRef] [PubMed]

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. USA 91, 4887-4891(1994).
[CrossRef] [PubMed]

Osterberg, U. L.

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. 13, 253-266 (1996).
[CrossRef]

Ostermeyer, M. R.

Pattanayak, D. N.

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. 13, 253-266 (1996).
[CrossRef]

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. 13, 253-266 (1996).
[CrossRef]

Pei, Y.

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. 18, 3018-3036 (2001).
[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]

Pera, V. E.

S. Fantini, E. L. Heffer, V. E. Pera, A. Sassaroli, and N. Liu, “Spatial and spectral information in optical mammography,” Technol. Cancer Res. Treat. 4, 471-482 (2005).
[PubMed]

Pifferi, A.

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. 13, 253-266 (1996).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, 1992).

Ren, K.

Rinneberg, H.

D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, “Evaluation of higher-order time domain perturbation theory of photon diffusion on breast-equivalent phantoms and optical mammograms,” Phys. Rev. E 76, 061908 (2007).
[CrossRef]

Ripoll, J.

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, A. R. Arridge, and J. Ripoll, “Diffuse photon propagation in multilayered geometries,” Phys. Med. Biol. 51, 497-516 (2006).
[CrossRef] [PubMed]

J. Ripoll, V. Ntziachristos, J. P. Culver, D. N. Pattanayak, A. G. Yodh, and M. Nieto-Vesperinas, “Recovery of optical parameters in multiple-layered diffusive media: theory and experiments,” J. Opt. Soc. Am. A 18, 821-830 (2001).
[CrossRef]

Sassaroli, A.

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]

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: I. Theory,” J. Opt. Soc. Am. A 23, 2105-2118 (2006).
[CrossRef]

S. Fantini, E. L. Heffer, V. E. Pera, A. Sassaroli, and N. Liu, “Spatial and spectral information in optical mammography,” Technol. Cancer Res. Treat. 4, 471-482 (2005).
[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]

Schlag, P. M.

D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, “Evaluation of higher-order time domain perturbation theory of photon diffusion on breast-equivalent phantoms and optical mammograms,” Phys. Rev. E 76, 061908 (2007).
[CrossRef]

Schmitz, C. H.

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. 18, 3018-3036 (2001).
[CrossRef]

Schotland, J. C.

Schweiger, M.

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, A. R. Arridge, and J. Ripoll, “Diffuse photon propagation in multilayered geometries,” Phys. Med. Biol. 51, 497-516 (2006).
[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]

Shatir, T. S. M.

Sikora, J.

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, A. R. Arridge, and J. Ripoll, “Diffuse photon propagation in multilayered geometries,” Phys. Med. Biol. 51, 497-516 (2006).
[CrossRef] [PubMed]

Spinelli, L.

Stott, J. J.

Taroni, P.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, 1992).

Torricelli, A.

Valentini, G.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, 1992).

Wagnieres, G.

Walker, S. A.

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.

Wassermann, B.

B. Wassermann, “Limits of high-order perturbation in time-domain optical mammography,” Phys. Rev. E 74, 031908(2006).
[CrossRef]

Xu, M.

Yamada, Y.

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]

Yao, Y.

Yodh, A. G.

Zaccanti, G.

F. Martelli, S. Del Bianco, and G. Zaccanti, “Perturbation model of light propagation through diffusive layered media,” Phys. Med. Biol. 50, 2159-2166 (2005).
[CrossRef] [PubMed]

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Appl. Opt. (7)

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]

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[CrossRef]

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Phys. Med. Biol. (3)

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Phys. Rev. E (3)

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Figures (10)

Fig. 1
Fig. 1

Schematic of two-layer medium and cylindrical defects used for the simulations. The thickness of the first layer is 10 mm , while the second layer is infinite. The diameter and height of cylinder A are 6 mm , while the diameter and height of cylinder B are 11.2 and 3 mm , respectively. The input point of the light source is the origin of the reference system, and the two detectors have coordinates ( x , y , z ) = ( 0 , 15 , 0 ) mm and ( x , y , z ) = ( 0 , 30 , 0 ) mm , respectively. The absorption and reduced scattering coefficient of the first and second layers are μ a 1 , μ s 1 ' and μ a 2 , μ s 2 ' , respectively. Only one of the two defects is present in any one test case.

Fig. 2
Fig. 2

The changes of relative intensity ( Δ I / I 0 ), obtained with different orders of perturbation theory, with Padé approximants and with MC, are plotted against the changes of absorption contrast Δ μ a between cylinder A (Fig. 1) and the local background medium. The thickness of the first layer is 10 mm , and the optical properties are μ a 1 = 0.01 mm 1 , μ s 1 ' = 0.5 mm 1 , and μ a 2 = 0.015 mm 1 , μ s 2 ' = 1 mm 1 , for the first and second layer, respectively. The defect is located in the top layer, with the center at ( x , y , z ) = ( 6.5 , 8 , 0 ) mm . Shown are the changes of intensity at the (a) shortest ( 15 mm ) and (b) farthest ( 30 mm ) source–detector distance. In (c) and (d) the same curves are plotted on an expanded scale.

Fig. 3
Fig. 3

As in Fig. 2, but the defect is located in the bottom layer, with the center at ( x , y , z ) = ( 13.5 , 8 , 0 ) mm .

Fig. 4
Fig. 4

As in Fig. 2, but the defect is cylinder B of Fig. 1, located in the top layer, with the center at ( x , y , z ) = ( 7.5 , 8 , 0 ) mm .

Fig. 5
Fig. 5

As in Fig. 2, but the defect is cylinder B of Fig. 1, located in the bottom layer, with the center at ( x , y , z ) = ( 12.5 , 8 , 0 ) mm .

Fig. 6
Fig. 6

Results obtained for a three-layer medium where the thicknesses of the first and second layers are 8 and 2 mm , respectively. The optical properties are μ a 1 = 0.01 mm 1 , μ s 1 ' = 0.5 mm 1 ; μ a 2 = 0.002 mm 1 , μ s 2 ' = 0.1 mm 1 ; and μ a 3 = 0.015 mm 1 , μ s 3 ' = 0.8 mm 1 for the first, second, and third layers, respectively. The defect (cylinder B) is located in the third layer, with the center at ( x , y , z ) = ( 12 , 8 , 0 ) mm . Changes of intensity are shown (a) at the shortest ( 15 mm ) and (b) farthest ( 30 mm ) source–detector distance. In panels (c) and (d) the same curves are plotted on an expanded scale.

Fig. 7
Fig. 7

In (a) we considered a background medium composed of the two-layer medium of Fig. 2 and also cylinder A (Fig. 1), which was located in the first layer with the center at ( x , y , z ) = ( 6.5 , 6.8 , 4.25 ) mm , and had reduced scattering coefficient of μ sc ' = 1 mm 1 . In (b) we considered a two-layer medium having the thickness of the first layer of 10 mm and optical properties μ a 1 = 0.01 mm 1 , μ s 1 ' = 1 mm 1 and μ a 2 = 0.015 mm 1 , μ s 2 ' = 1.5 mm 1 for the first and second layer, respectively. The background media also included also cylinder B located in the second layer with the center at ( x , y , z ) = ( 12 , 6.8 , 4.25 ) mm and having a reduced scattering coefficient of μ sc ' = 1 mm 1 . The results refer to a source–detector distance d = 30 mm .

Fig. 8
Fig. 8

Distribution of internal path lengths calculated with 10,000 detected photons for a background medium and a defect as in Fig. 2, but both nonabsorbing. The distributions refer to two values of the reduced scattering coefficient of the cylindrical defect, as indicated.

Fig. 9
Fig. 9

Schematic of the heterogeneous slab geometry composed by a slab with μ a = 0.005 mm 1 , μ s = 0.5 mm 1 , and a spherical region (1) with μ a 1 = 0.1 mm 1 and the same reduced scattering coefficient. The spherical region is located with the center at ( x , y , z ) = ( 12 , 7 , 0 ) mm . Another spherical region (2), located with the center at ( x , y , z ) = ( 20 , 0 , 0 ) mm was considered an absorbing perturbation. Both spherical regions have the same diameter of 10 mm . The thickness of the slab is 40 mm . Three detectors are placed in transmittance as indicated in the figure.

Fig. 10
Fig. 10

For the slab geometry the changes of intensities Δ I / I 0 are plotted at the detector (a)  d 1 and (b)  d 3 (Fig. 9).

Tables (1)

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Table 1 Calculated Values of the Moments a

Equations (7)

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l i n c n 1 l i ( V i ϕ 0 ( r , r i ) d r ) n 1 , n > 1 ,
Δ I ( Δ μ a ) I 0 l i Δ μ a + 1 2 ! l i 2 Δ μ a 2 1 3 ! l i 3 Δ μ a 3 + 1 4 ! l i 4 Δ μ a 4 .
Δ I ( Δ μ a ) I 0 P M / N ( Δ μ a ) = k = 0 M a k Δ μ a k 1 + k = 1 N b k Δ μ a k .
( Δ I / I 0 ) ( Δ μ a ) 0.
· { D ( r ) [ Δ ϕ ( r ) ] } + μ a ( r ) Δ ϕ ( r ) = Δ μ a ( r ) ϕ f i ( r ) ,
ϕ fi ( r ) = ϕ in ( r ) V Δ μ a ( r 1 ) ϕ i n ( r , r 1 ) ϕ fi ( r 1 ) d r 1 ,
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 ,

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