Abstract

From analytical and numerical solutions that predict the scattering of diffuse photon density waves and from experimental measurements of changes in phase shift θ and ac amplitude demodulation M caused by the presence of single and double cylindrical heterogeneities, we show that second- and higher-order perturbations can affect the prediction of the propagation characteristics of diffuse photon density waves. Our experimental results for perfect absorbers in a lossless medium suggest that the performance of fast inverse-imaging algorithms that use first-order Born or Rytov approximations might have inherent limitations compared with inverse solutions that use iterative solutions of a linear perturbation equation or numerical solutions of the diffusion equation.

© 1997 Optical Society of America

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

1997 (2)

M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag, M. Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results.” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef]

Y. Yao, Y. Wang, Y. Pei, W. Zhu, 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]

1996 (3)

1995 (2)

1994 (3)

1993 (2)

E. Gratton, W. Mantulin, M. J. van de Ven, J. Fishkin, M. Maris, B. Chance, “A novel approach to laser tomography,” Bioimaging 1, 40–46 (1993).
[CrossRef]

B. B. Das, K. M. Yoo, R. R. Alfano, “Ultrafast time-gated imaging in thick tissues: a step toward optical mammography,” Opt. Lett. 18, 1092–1094 (1993).
[CrossRef] [PubMed]

1992 (2)

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

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

1991 (1)

1990 (1)

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Alfano, R. R.

Aronson, R.

R. L. Barbour, H. L. Graber, Y. Wang, J. H. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time resolved data,” in Fiber Optic Sensors: Engineering and Applications, A. J. Bruinsma, B. Culshaw, eds., Proc. SPIE1511, 87–120 (1993).

Arridge, S. R.

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

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near-infrared absorption and scatter imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

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

Barbour, R. L.

Y. Yao, Y. Wang, Y. Pei, W. Zhu, 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]

R. L. Barbour, H. L. Graber, Y. Wang, J. H. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time resolved data,” in Fiber Optic Sensors: Engineering and Applications, A. J. Bruinsma, B. Culshaw, eds., Proc. SPIE1511, 87–120 (1993).

Boas, D. A.

D. A. Boas, B. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytical solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

Boss, D. A.

Burch, C. L.

Chance, B.

M. A. O’Leary, D. A. Boss, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency domain diffusing photon tomography,” Opt. Lett. 20, 426–428 (1995).
[CrossRef]

D. A. Boas, B. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytical solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

E. Gratton, W. Mantulin, M. J. van de Ven, J. Fishkin, M. Maris, B. Chance, “A novel approach to laser tomography,” Bioimaging 1, 40–46 (1993).
[CrossRef]

Chang, J. H.

R. L. Barbour, H. L. Graber, Y. Wang, J. H. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time resolved data,” in Fiber Optic Sensors: Engineering and Applications, A. J. Bruinsma, B. Culshaw, eds., Proc. SPIE1511, 87–120 (1993).

Cope, M.

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

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

Das, B. B.

Delpy, D. T.

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

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

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near-infrared absorption and scatter imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

Demos, S. G.

S. G. Demos, H. Savage, A. S. Heerdt, S. Schantz, R. R. Alfano, “Time resolved degree of polarization for human breast tissue,” Opt. Commun. 124, 439–442 (1996).
[CrossRef]

Fantini, S.

M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag, M. Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results.” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef]

Fishkin, J.

E. Gratton, W. Mantulin, M. J. van de Ven, J. Fishkin, M. Maris, B. Chance, “A novel approach to laser tomography,” Bioimaging 1, 40–46 (1993).
[CrossRef]

Franceschini, M. A.

M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag, M. Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results.” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef]

Frisoli, J. K.

Gaida, G.

M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag, M. Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results.” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef]

Graber, H. L.

R. L. Barbour, H. L. Graber, Y. Wang, J. H. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time resolved data,” in Fiber Optic Sensors: Engineering and Applications, A. J. Bruinsma, B. Culshaw, eds., Proc. SPIE1511, 87–120 (1993).

Gratton, E.

M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag, M. Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results.” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef]

E. Gratton, W. Mantulin, M. J. van de Ven, J. Fishkin, M. Maris, B. Chance, “A novel approach to laser tomography,” Bioimaging 1, 40–46 (1993).
[CrossRef]

Grunbaum, F. A.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Hebden, J. C.

Heerdt, A. S.

S. G. Demos, H. Savage, A. S. Heerdt, S. Schantz, R. R. Alfano, “Time resolved degree of polarization for human breast tissue,” Opt. Commun. 124, 439–442 (1996).
[CrossRef]

Hiraoka, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near-infrared absorption and scatter imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

Ho, P. P.

Hutchinson, C. L.

Jess, H.

M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag, M. Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results.” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef]

Jiang, H.

Johnson, M. L.

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

Kaschke, M.

M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag, M. Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results.” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef]

Kohn, P.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Kruger, R. A.

Lakowicz, J. R.

E. M. Sevick, J. K. Frisoli, C. L. Burch, J. R. Lakowicz, “Localization of absorbers in scattering media by use of frequency-domain measurements of time-dependent photon migration,” Appl. Opt. 33, 3562–3570 (1994).
[CrossRef] [PubMed]

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

Mantulin, W.

E. Gratton, W. Mantulin, M. J. van de Ven, J. Fishkin, M. Maris, B. Chance, “A novel approach to laser tomography,” Bioimaging 1, 40–46 (1993).
[CrossRef]

Mantulin, W. W.

M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag, M. Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results.” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef]

Maris, M.

E. Gratton, W. Mantulin, M. J. van de Ven, J. Fishkin, M. Maris, B. Chance, “A novel approach to laser tomography,” Bioimaging 1, 40–46 (1993).
[CrossRef]

Moesta, K. T.

M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag, M. Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results.” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef]

Nowaczyk, K.

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

O’Leary, B.

D. A. Boas, B. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytical solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

O’Leary, M. A.

Osterberg, U. L.

Patterson, M. S.

Paulsen, K. D.

Pei, Y.

Pogue, B. W.

Savage, H.

S. G. Demos, H. Savage, A. S. Heerdt, S. Schantz, R. R. Alfano, “Time resolved degree of polarization for human breast tissue,” Opt. Commun. 124, 439–442 (1996).
[CrossRef]

Schantz, S.

S. G. Demos, H. Savage, A. S. Heerdt, S. Schantz, R. R. Alfano, “Time resolved degree of polarization for human breast tissue,” Opt. Commun. 124, 439–442 (1996).
[CrossRef]

Schlag, P. M.

M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag, M. Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results.” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef]

Schweiger, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near-infrared absorption and scatter imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

Seeber, M.

M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag, M. Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results.” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef]

Sevick, E. M.

E. M. Sevick, J. K. Frisoli, C. L. Burch, J. R. Lakowicz, “Localization of absorbers in scattering media by use of frequency-domain measurements of time-dependent photon migration,” Appl. Opt. 33, 3562–3570 (1994).
[CrossRef] [PubMed]

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

Sevick-Muraca, E. M.

Singer, J. R.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Szmacinski, H.

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

Troy, T. L.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

van de Ven, M. J.

E. Gratton, W. Mantulin, M. J. van de Ven, J. Fishkin, M. Maris, B. Chance, “A novel approach to laser tomography,” Bioimaging 1, 40–46 (1993).
[CrossRef]

van der Zee, P.

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

Wang, L. M.

Wang, Y.

Y. Yao, Y. Wang, Y. Pei, W. Zhu, 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]

R. L. Barbour, H. L. Graber, Y. Wang, J. H. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time resolved data,” in Fiber Optic Sensors: Engineering and Applications, A. J. Bruinsma, B. Culshaw, eds., Proc. SPIE1511, 87–120 (1993).

Wong, K. S.

Yao, Y.

Yodh, A. G.

M. A. O’Leary, D. A. Boss, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency domain diffusing photon tomography,” Opt. Lett. 20, 426–428 (1995).
[CrossRef]

D. A. Boas, B. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytical solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

Yoo, K. M.

Zhu, W.

Zubelli, J. P.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Appl. Opt. (4)

Bioimaging (1)

E. Gratton, W. Mantulin, M. J. van de Ven, J. Fishkin, M. Maris, B. Chance, “A novel approach to laser tomography,” Bioimaging 1, 40–46 (1993).
[CrossRef]

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

J. Photochem. Photobiol. B (1)

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

Opt. Commun. (1)

S. G. Demos, H. Savage, A. S. Heerdt, S. Schantz, R. R. Alfano, “Time resolved degree of polarization for human breast tissue,” Opt. Commun. 124, 439–442 (1996).
[CrossRef]

Opt. Lett. (3)

Phys. Med. Biol. (1)

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

Proc. Natl. Acad. Sci. USA (2)

D. A. Boas, B. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytical solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag, M. Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results.” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef]

Science (1)

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Other (5)

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

R. L. Barbour, H. L. Graber, Y. Wang, J. H. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time resolved data,” in Fiber Optic Sensors: Engineering and Applications, A. J. Bruinsma, B. Culshaw, eds., Proc. SPIE1511, 87–120 (1993).

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

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near-infrared absorption and scatter imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

D. A. Boas, http://dpdw.eotc.tufts.edu/boas/PMI/pmi.html .

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Fig. 1
Fig. 1

Schematic illustrating the incident wave Φinc originating from the source at position ρs (dashed lines); the first-order scattered wave Φscat,k1n=1 arising from the first heterogeneity (solid lines); and the second-order wave Φscat,k2n = 2 arising from rescatter of the first-order wave off the second heterogeneity (dotted lines).

Fig. 2
Fig. 2

Schematic detailing the geometry used in the calculation of scattered waves from analytical expression. The centroid of the cylinder is the origin, with z denoting the length, angle ϑ denoting the angle in the plane containing the source and the detector, and r denoting the radial direction.

Fig. 3
Fig. 3

Schematic illustrating the geometry of the experimental measurements.

Fig. 4
Fig. 4

Experimental values of θk1,k2–θinc the phase difference (in degrees) relative to the absence condition as a function of distance from the source–detector pair. The symbols denote individual measurements in the presence of two cylinders separated by distances of (a) 6, (b) 10, and (c) 20 mm, whereas the line connects predictions from Eq. (7) and measurements of θk1, θk2, and θinc. The error bars denote the propagation of measurement errors (standard deviation) associated with θk1, θk2, and θinc. The x axis is reported as the distance between the wall and the first cylinder (k1). The asterisks denote significant difference (p < 0.005, paired Student’s t-test) between the values experimentally measured and those obtained from Eq. (7).

Fig. 5
Fig. 5

Experimental values of Mk1,k2/Minc (in arbitrary units) relative to the absence condition as a function of distance from the source–detector pair. The symbols denote individual measurements in the presence of two cylinders separated by distances of (a) 6 (b) 10, and (c) 20 mm, whereas the line connects predictions from Eq. (8) and measurements of Mk1, Mk2, and Minc. The error bars denote the propagation of measurement errors (standard deviation) associated with Mk1, Mk2, and Minc. The x axis is reported as the distance between the wall and the first cylinder (k1).

Fig. 6
Fig. 6

Values of θk1,k2–θinc (in degrees) calculated from the analytical prediction of first-order (solid line) and inclusive of second-order (dashed line)-scattering effects as a function of distance (cm) from the source–detector pair for two absorbing cylinders separated by (a) 6, (b) 10, and (c) 20 mm. The x axis is reported as the distance between the wall and the first cylinder (k1) and the phase shift reported relative to the absence case.

Fig. 7
Fig. 7

Values of θk1,k2–θinc (in degrees) calculated from analytical prediction of first order (solid curve) and inclusive of second order (dashed curve) for two absorbing cylinders separated by 6 mm and interrogated at 80, 160, and 240 MHz.

Fig. 8
Fig. 8

Finite element computations of θk1,k2–θinc (in degrees) relative to an absence condition for considering only first-order (solid line) and including second-order (dashed line) perturbations as a function of distance (cm) from the source–detector pair for absorbing cylinders separated by (a) 6, (b) 10, and (c) 20 mm. The x axis is reported as the distance between the wall and the first cylinder (k1) and the phase shift reported relative to the absence case.

Fig. 9
Fig. 9

Finite element computations of Mk1,k2/Minc (in arbitrary units) referenced to an absence condition for both the first-order (solid line) and the second-order (dashed line) perturbations as a function of distance (cm) from the source–detector pair for absorbing cylinders separated by (a) 6, (b) 10, and (c) 20 mm. The x axis is reported as the distance between the wall and the first cylinder (k1) and the modulation ratio is reported relative to the absence case.

Equations (17)

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Φρs, ρd=Φincρs, ρd+Gρ, ρdΦ0ρs, ρΔρ3ρ,
Gρ, ρD=14πρ-ρD expi-μa+iω/cn/D1/2ρ-ρD,
Φρ=Φincρ+n=1k=1mΦscat,knρ.
Φk1ρd=Φincρd+Φscat,k1n=1ρd,
Φk2ρd=Φincρd+Φscat,k2n=1ρd,
Φk1,k2ρd=Φk1ρd+Φk2ρd-Φincρd+Φscat,k1,k2n=2ρd+Φscat,k1,k2n=3ρd+.
Φk1,k2ρd=Φk=1ρd+Φk=2ρd-Φincρd.
θk1,k2ρd=tan-1Mk1 sin θk1+Mk2 sin θk2-Minc sin θincMk1 cos θk1+Mk2 cos θk2-Minc cos θinc,
Mk1,k2ρd=Mk1 cos θk1+Mk2 cos θk2-Minc cos θinc2+Mk1 sin θk1+Mk2 sin θk2-Minc sin θinc21/2.
Φincρ=Ssourceρsexpi-μa+iω/cnD1/2ρ-ρd4πDρ-ρd,
D=1/3μa+μs.
Φincρd=Ssourceρs14πD×exp-μa+iω/cnD1/2ρs-ρd2+z-z021/2ρs-ρd2+z-z021/2-exp-μa+iω/cnD1/2ρs-ρd2+z+z021/2ρs-ρd2+z+z021/2,
Φscat,k1n=1ρd=-Φincm=10cosmϑcospz×Kmp2+-μa+iω/cnD1/2ρd×Kmp2+-μa+iω/cnD1/2ρs×DxImxImy-Dk1yImxImyDxKmxImy-Dk1yKmxImydp,
x=p2+-μa+iω/cnD1/2ak,  y=p2+-μa+iω/cnDk1/2ak;
Φscat,k1n=2ρd=-Φscat,k2n=1m=10cosmϑcospz×Kmp2+-μa+iw/cnD1/2ρd×Kmp2+-μa+iω/cnD1/2ρs×DxImxImy-Dk1yImxImyDxKmxImy-Dk1yKmxImydp,
θρd=tan-1ImΦρdReΦρd
Mρd=ImΦρd2+ReΦρd21/2.

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