Abstract

As Part V in our series, this paper examines steady-state fluorescence photon diffusion in a homogenous medium that contains a homogenous distribution of fluorophores, and is enclosed by a “concave” circular cylindrical applicator or is enclosing a “convex” circular cylindrical applicator, both geometries being infinite in the longitudinal dimension. The aim is to predict by analytics and examine with the finite-element method the changing characteristics of the fluorescence-wavelength photon-fluence rate and the ratio (sometimes called the Born ratio) of it versus the excitation-wavelength photon-fluence rate, with respect to the source–detector distance. The analysis is performed for a source and a detector located on the medium–applicator interface and aligned either azimuthally or longitudinally in both concave and convex geometries. When compared to its steady-state counterparts on a semi-infinite medium–applicator interface with the same line-of-sight source–detector distance, the fluorescence-wavelength photon-fluence rate reduces faster along the longitudinal direction and slower along the azimuthal direction in the concave geometry, and conversely in the convex geometry. However, the Born ratio increases slower in both azimuthal and longitudinal directions in the concave geometry and faster in both directions in the convex geometry, respectively, when compared to that in the semi-infinite geometry.

© 2013 Optical Society of America

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2012 (3)

2011 (2)

2010 (1)

2009 (4)

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
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[CrossRef]

A. Da Silva, M. Leabad, C. Driol, T. Bordy, M. Debourdeau, J. Dinten, P. Peltié, and P. Rizo, “Optical calibration protocol for an x-ray and optical multimodality tomography system dedicated to small-animal examination,” Appl. Opt. 48, D151–D162 (2009).
[CrossRef]

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

2008 (1)

Y. Tan and H. Jiang, “DOT guided fluorescence molecular tomography of arbitrarily shaped objects,” Med. Phys. 35, 5703–5707 (2008).
[CrossRef]

2007 (3)

C. Li, R. Liengsawangwong, H. Choi, and R. Cheung, “Using a priori structural information from magnetic resonance imaging to investigate the feasibility of prostate diffuse optical tomography and spectroscopy: a simulation study,” Med. Phys. 34, 266–274 (2007).
[CrossRef]

A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15, 6696–6716 (2007).
[CrossRef]

Y. Lin, H. Gao, O. Nalcioglu, and G. Gulsen, “Fluorescence diffuse optical tomography with functional and anatomical a priori information: feasibility study,” Phys. Med. Biol. 52, 5569–5585 (2007).
[CrossRef]

2006 (2)

A. Liebert, H. Wabnitz, H. Obrig, R. Erdmann, M. Möller, R. Macdonald, H. Rinneberg, A. Villringer, and J. Steinbrink, “Noninvasive detection of fluorescence from exogenous chromophores in the adult human brain,” NeuroImage 31, 600–608 (2006).
[CrossRef]

A. Soubret and V. Ntziachristos, “Fluorescence molecular tomography in the presence of background fluorescence,” Phys. Med. Biol. 51, 3983–4001 (2006).
[CrossRef]

2005 (2)

M. Gao, G. Lewis, G. M. Turner, A. Soubret, and V. Ntziachristos, “Effects of background fluorescence in fluorescence molecular tomography,” Appl. Opt. 44, 5468–5474 (2005).
[CrossRef]

V. Ntziachristos, G. Turner, J. Dunham, S. Windsor, A. Soubret, J. Ripoll, and H. A. Shih, “Planar fluorescence imaging using normalized data,” J. Biomed. Opt. 10, 064007 (2005).
[CrossRef]

2004 (1)

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488–496 (2004).
[CrossRef]

2003 (3)

A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, and R. P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. 42, 3081–3094 (2003).
[CrossRef]

V. Ntziachristos, C. Bremer, and R. Weissleder, “Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging,” Eur. J. Radiol. 13, 195–208 (2003).
[CrossRef]

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with indocyanine green,” Med. Phys. 30, 1039–1047 (2003).
[CrossRef]

2001 (1)

M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time-resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
[CrossRef]

1999 (1)

1997 (1)

1996 (1)

1995 (2)

R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
[CrossRef]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

1994 (1)

1989 (1)

Abascal, J. F.

J. F. Abascal, J. Aguirre, J. Chamorro-Servent, M. Schweiger, S. Arridge, J. Ripoll, J. J. Vaquero, and M. Desco, “Influence of absorption and scattering on the quantification of fluorescence diffuse optical tomography using normalized data,” J. Biomed. Opt. 17, 036013 (2012).
[CrossRef]

Aguirre, J.

J. F. Abascal, J. Aguirre, J. Chamorro-Servent, M. Schweiger, S. Arridge, J. Ripoll, J. J. Vaquero, and M. Desco, “Influence of absorption and scattering on the quantification of fluorescence diffuse optical tomography using normalized data,” J. Biomed. Opt. 17, 036013 (2012).
[CrossRef]

Aronson, R.

Arridge, S.

J. F. Abascal, J. Aguirre, J. Chamorro-Servent, M. Schweiger, S. Arridge, J. Ripoll, J. J. Vaquero, and M. Desco, “Influence of absorption and scattering on the quantification of fluorescence diffuse optical tomography using normalized data,” J. Biomed. Opt. 17, 036013 (2012).
[CrossRef]

Arridge, S. R.

A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15, 6696–6716 (2007).
[CrossRef]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

Bérubé-Lauzière, Y.

Boas, D. A.

Bordy, T.

Bouman, C. A.

Boutet, J.

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

Bremer, C.

V. Ntziachristos, C. Bremer, and R. Weissleder, “Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging,” Eur. J. Radiol. 13, 195–208 (2003).
[CrossRef]

Bunting, C. F.

Carpenter, C. M.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

Chamorro-Servent, J.

J. F. Abascal, J. Aguirre, J. Chamorro-Servent, M. Schweiger, S. Arridge, J. Ripoll, J. J. Vaquero, and M. Desco, “Influence of absorption and scattering on the quantification of fluorescence diffuse optical tomography using normalized data,” J. Biomed. Opt. 17, 036013 (2012).
[CrossRef]

Chance, B.

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with indocyanine green,” Med. Phys. 30, 1039–1047 (2003).
[CrossRef]

X. D. Li, M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescent diffuse photon density waves in homogeneous and heterogeneous turbid media: analytic solutions and applications,” Appl. Opt. 35, 3746–3758 (1996).
[CrossRef]

Chen, Y.

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with indocyanine green,” Med. Phys. 30, 1039–1047 (2003).
[CrossRef]

Cheung, R.

C. Li, R. Liengsawangwong, H. Choi, and R. Cheung, “Using a priori structural information from magnetic resonance imaging to investigate the feasibility of prostate diffuse optical tomography and spectroscopy: a simulation study,” Med. Phys. 34, 266–274 (2007).
[CrossRef]

Choe, R.

Choi, H.

C. Li, R. Liengsawangwong, H. Choi, and R. Cheung, “Using a priori structural information from magnetic resonance imaging to investigate the feasibility of prostate diffuse optical tomography and spectroscopy: a simulation study,” Med. Phys. 34, 266–274 (2007).
[CrossRef]

Contini, D.

Corlu, A.

Da Silva, A.

Daluwatte, C.

Davis, S. C.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

Debourdeau, M.

A. Da Silva, M. Leabad, C. Driol, T. Bordy, M. Debourdeau, J. Dinten, P. Peltié, and P. Rizo, “Optical calibration protocol for an x-ray and optical multimodality tomography system dedicated to small-animal examination,” Appl. Opt. 48, D151–D162 (2009).
[CrossRef]

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

Dehghani, H.

D. S. Kepshire, S. L. Gibbs-Strauss, J. A. O’Hara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt. 14, 030501 (2009).
[CrossRef]

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

Del Bianco, S.

Delpy, D. T.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

Desco, M.

J. F. Abascal, J. Aguirre, J. Chamorro-Servent, M. Schweiger, S. Arridge, J. Ripoll, J. J. Vaquero, and M. Desco, “Influence of absorption and scattering on the quantification of fluorescence diffuse optical tomography using normalized data,” J. Biomed. Opt. 17, 036013 (2012).
[CrossRef]

Di Ninni, P.

Dinten, J.

Dinten, J.-M.

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

Domínguez, J. B.

Driol, C.

Duboeuf, F.

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

Dunham, J.

V. Ntziachristos, G. Turner, J. Dunham, S. Windsor, A. Soubret, J. Ripoll, and H. A. Shih, “Planar fluorescence imaging using normalized data,” J. Biomed. Opt. 10, 064007 (2005).
[CrossRef]

Durduran, T.

Eames, M. E.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

Eppstein, M. J.

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488–496 (2004).
[CrossRef]

Erdmann, R.

A. Liebert, H. Wabnitz, H. Obrig, R. Erdmann, M. Möller, R. Macdonald, H. Rinneberg, A. Villringer, and J. Steinbrink, “Noninvasive detection of fluorescence from exogenous chromophores in the adult human brain,” NeuroImage 31, 600–608 (2006).
[CrossRef]

Feng, T.

Gao, H.

Y. Lin, H. Gao, O. Nalcioglu, and G. Gulsen, “Fluorescence diffuse optical tomography with functional and anatomical a priori information: feasibility study,” Phys. Med. Biol. 52, 5569–5585 (2007).
[CrossRef]

Gao, M.

Gibbs-Strauss, S. L.

D. S. Kepshire, S. L. Gibbs-Strauss, J. A. O’Hara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt. 14, 030501 (2009).
[CrossRef]

Godavarty, A.

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488–496 (2004).
[CrossRef]

Gulsen, G.

Y. Lin, H. Gao, O. Nalcioglu, and G. Gulsen, “Fluorescence diffuse optical tomography with functional and anatomical a priori information: feasibility study,” Phys. Med. Biol. 52, 5569–5585 (2007).
[CrossRef]

Gurfinkel, M.

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488–496 (2004).
[CrossRef]

Guyon, L.

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

Haskell, R. C.

Herve, L.

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

Hiraoka, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

Hutchins, M.

D. S. Kepshire, S. L. Gibbs-Strauss, J. A. O’Hara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt. 14, 030501 (2009).
[CrossRef]

Intes, X.

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with indocyanine green,” Med. Phys. 30, 1039–1047 (2003).
[CrossRef]

Ishimaru, A.

Jiang, H.

Y. Tan and H. Jiang, “DOT guided fluorescence molecular tomography of arbitrarily shaped objects,” Med. Phys. 35, 5703–5707 (2008).
[CrossRef]

Kepshire, D. S.

D. S. Kepshire, S. L. Gibbs-Strauss, J. A. O’Hara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt. 14, 030501 (2009).
[CrossRef]

Khayat, M.

D. S. Kepshire, S. L. Gibbs-Strauss, J. A. O’Hara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt. 14, 030501 (2009).
[CrossRef]

Kumar, S.

M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time-resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
[CrossRef]

Leabad, M.

Leblond, F.

D. S. Kepshire, S. L. Gibbs-Strauss, J. A. O’Hara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt. 14, 030501 (2009).
[CrossRef]

Lewis, G.

Li, C.

C. Li, R. Liengsawangwong, H. Choi, and R. Cheung, “Using a priori structural information from magnetic resonance imaging to investigate the feasibility of prostate diffuse optical tomography and spectroscopy: a simulation study,” Med. Phys. 34, 266–274 (2007).
[CrossRef]

Li, X. D.

Liebert, A.

A. Liebert, H. Wabnitz, H. Obrig, R. Erdmann, M. Möller, R. Macdonald, H. Rinneberg, A. Villringer, and J. Steinbrink, “Noninvasive detection of fluorescence from exogenous chromophores in the adult human brain,” NeuroImage 31, 600–608 (2006).
[CrossRef]

Liengsawangwong, R.

C. Li, R. Liengsawangwong, H. Choi, and R. Cheung, “Using a priori structural information from magnetic resonance imaging to investigate the feasibility of prostate diffuse optical tomography and spectroscopy: a simulation study,” Med. Phys. 34, 266–274 (2007).
[CrossRef]

Lin, Y.

Y. Lin, H. Gao, O. Nalcioglu, and G. Gulsen, “Fluorescence diffuse optical tomography with functional and anatomical a priori information: feasibility study,” Phys. Med. Biol. 52, 5569–5585 (2007).
[CrossRef]

Macdonald, R.

A. Liebert, H. Wabnitz, H. Obrig, R. Erdmann, M. Möller, R. Macdonald, H. Rinneberg, A. Villringer, and J. Steinbrink, “Noninvasive detection of fluorescence from exogenous chromophores in the adult human brain,” NeuroImage 31, 600–608 (2006).
[CrossRef]

Martelli, F.

McAdams, M. S.

Millane, R. P.

Milstein, A. B.

Mincu, N.

D. S. Kepshire, S. L. Gibbs-Strauss, J. A. O’Hara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt. 14, 030501 (2009).
[CrossRef]

Möller, M.

A. Liebert, H. Wabnitz, H. Obrig, R. Erdmann, M. Möller, R. Macdonald, H. Rinneberg, A. Villringer, and J. Steinbrink, “Noninvasive detection of fluorescence from exogenous chromophores in the adult human brain,” NeuroImage 31, 600–608 (2006).
[CrossRef]

Nalcioglu, O.

Y. Lin, H. Gao, O. Nalcioglu, and G. Gulsen, “Fluorescence diffuse optical tomography with functional and anatomical a priori information: feasibility study,” Phys. Med. Biol. 52, 5569–5585 (2007).
[CrossRef]

Nioka, S.

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with indocyanine green,” Med. Phys. 30, 1039–1047 (2003).
[CrossRef]

Ntziachristos, V.

A. Soubret and V. Ntziachristos, “Fluorescence molecular tomography in the presence of background fluorescence,” Phys. Med. Biol. 51, 3983–4001 (2006).
[CrossRef]

M. Gao, G. Lewis, G. M. Turner, A. Soubret, and V. Ntziachristos, “Effects of background fluorescence in fluorescence molecular tomography,” Appl. Opt. 44, 5468–5474 (2005).
[CrossRef]

V. Ntziachristos, G. Turner, J. Dunham, S. Windsor, A. Soubret, J. Ripoll, and H. A. Shih, “Planar fluorescence imaging using normalized data,” J. Biomed. Opt. 10, 064007 (2005).
[CrossRef]

V. Ntziachristos, C. Bremer, and R. Weissleder, “Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging,” Eur. J. Radiol. 13, 195–208 (2003).
[CrossRef]

O’Hara, J. A.

D. S. Kepshire, S. L. Gibbs-Strauss, J. A. O’Hara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt. 14, 030501 (2009).
[CrossRef]

O’Leary, M. A.

Obrig, H.

A. Liebert, H. Wabnitz, H. Obrig, R. Erdmann, M. Möller, R. Macdonald, H. Rinneberg, A. Villringer, and J. Steinbrink, “Noninvasive detection of fluorescence from exogenous chromophores in the adult human brain,” NeuroImage 31, 600–608 (2006).
[CrossRef]

Oh, S.

Paulsen, K. D.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

Peltie, P.

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

Peltié, P.

Piao, D.

Pogue, B. W.

A. Zhang, G. Xu, C. Daluwatte, G. Yao, C. F. Bunting, B. W. Pogue, and D. Piao, “Photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator. II. Quantitative examinations of the steady-state theory,” J. Opt. Soc. Am. A 28, 66–75 (2011).
[CrossRef]

A. Zhang, D. Piao, C. F. Bunting, and B. W. Pogue, “Photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator. I. Steady-state theory,” J. Opt. Soc. Am. A 27, 648–662 (2010).
[CrossRef]

D. S. Kepshire, S. L. Gibbs-Strauss, J. A. O’Hara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt. 14, 030501 (2009).
[CrossRef]

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

Rinneberg, H.

A. Liebert, H. Wabnitz, H. Obrig, R. Erdmann, M. Möller, R. Macdonald, H. Rinneberg, A. Villringer, and J. Steinbrink, “Noninvasive detection of fluorescence from exogenous chromophores in the adult human brain,” NeuroImage 31, 600–608 (2006).
[CrossRef]

Ripoll, J.

J. F. Abascal, J. Aguirre, J. Chamorro-Servent, M. Schweiger, S. Arridge, J. Ripoll, J. J. Vaquero, and M. Desco, “Influence of absorption and scattering on the quantification of fluorescence diffuse optical tomography using normalized data,” J. Biomed. Opt. 17, 036013 (2012).
[CrossRef]

V. Ntziachristos, G. Turner, J. Dunham, S. Windsor, A. Soubret, J. Ripoll, and H. A. Shih, “Planar fluorescence imaging using normalized data,” J. Biomed. Opt. 10, 064007 (2005).
[CrossRef]

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with indocyanine green,” Med. Phys. 30, 1039–1047 (2003).
[CrossRef]

Riseborough, P.

M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time-resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
[CrossRef]

Rizo, P.

Rosen, M. A.

Roy, R.

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488–496 (2004).
[CrossRef]

R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: part I theory and formulation,” Opt. Express 4, 353–371 (1999).
[CrossRef]

Sadoqi, M.

M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time-resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
[CrossRef]

Saroul, L.

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

Schnall, M. D.

Schweiger, M.

J. F. Abascal, J. Aguirre, J. Chamorro-Servent, M. Schweiger, S. Arridge, J. Ripoll, J. J. Vaquero, and M. Desco, “Influence of absorption and scattering on the quantification of fluorescence diffuse optical tomography using normalized data,” J. Biomed. Opt. 17, 036013 (2012).
[CrossRef]

A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15, 6696–6716 (2007).
[CrossRef]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

Sevick-Muraca, E. M.

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488–496 (2004).
[CrossRef]

R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: part I theory and formulation,” Opt. Express 4, 353–371 (1999).
[CrossRef]

Shih, H. A.

V. Ntziachristos, G. Turner, J. Dunham, S. Windsor, A. Soubret, J. Ripoll, and H. A. Shih, “Planar fluorescence imaging using normalized data,” J. Biomed. Opt. 10, 064007 (2005).
[CrossRef]

Soubret, A.

A. Soubret and V. Ntziachristos, “Fluorescence molecular tomography in the presence of background fluorescence,” Phys. Med. Biol. 51, 3983–4001 (2006).
[CrossRef]

M. Gao, G. Lewis, G. M. Turner, A. Soubret, and V. Ntziachristos, “Effects of background fluorescence in fluorescence molecular tomography,” Appl. Opt. 44, 5468–5474 (2005).
[CrossRef]

V. Ntziachristos, G. Turner, J. Dunham, S. Windsor, A. Soubret, J. Ripoll, and H. A. Shih, “Planar fluorescence imaging using normalized data,” J. Biomed. Opt. 10, 064007 (2005).
[CrossRef]

Srinivasan, S.

D. S. Kepshire, S. L. Gibbs-Strauss, J. A. O’Hara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt. 14, 030501 (2009).
[CrossRef]

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

Steinbrink, J.

A. Liebert, H. Wabnitz, H. Obrig, R. Erdmann, M. Möller, R. Macdonald, H. Rinneberg, A. Villringer, and J. Steinbrink, “Noninvasive detection of fluorescence from exogenous chromophores in the adult human brain,” NeuroImage 31, 600–608 (2006).
[CrossRef]

Svaasand, L. O.

Tan, Y.

Y. Tan and H. Jiang, “DOT guided fluorescence molecular tomography of arbitrarily shaped objects,” Med. Phys. 35, 5703–5707 (2008).
[CrossRef]

Thompson, A. B.

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488–496 (2004).
[CrossRef]

Tromberg, B. J.

Tsay, T.

Turner, G.

V. Ntziachristos, G. Turner, J. Dunham, S. Windsor, A. Soubret, J. Ripoll, and H. A. Shih, “Planar fluorescence imaging using normalized data,” J. Biomed. Opt. 10, 064007 (2005).
[CrossRef]

Turner, G. M.

Vaquero, J. J.

J. F. Abascal, J. Aguirre, J. Chamorro-Servent, M. Schweiger, S. Arridge, J. Ripoll, J. J. Vaquero, and M. Desco, “Influence of absorption and scattering on the quantification of fluorescence diffuse optical tomography using normalized data,” J. Biomed. Opt. 17, 036013 (2012).
[CrossRef]

Villringer, A.

A. Liebert, H. Wabnitz, H. Obrig, R. Erdmann, M. Möller, R. Macdonald, H. Rinneberg, A. Villringer, and J. Steinbrink, “Noninvasive detection of fluorescence from exogenous chromophores in the adult human brain,” NeuroImage 31, 600–608 (2006).
[CrossRef]

Vray, D.

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

Wabnitz, H.

A. Liebert, H. Wabnitz, H. Obrig, R. Erdmann, M. Möller, R. Macdonald, H. Rinneberg, A. Villringer, and J. Steinbrink, “Noninvasive detection of fluorescence from exogenous chromophores in the adult human brain,” NeuroImage 31, 600–608 (2006).
[CrossRef]

Webb, K. J.

Weissleder, R.

V. Ntziachristos, C. Bremer, and R. Weissleder, “Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging,” Eur. J. Radiol. 13, 195–208 (2003).
[CrossRef]

Windsor, S.

V. Ntziachristos, G. Turner, J. Dunham, S. Windsor, A. Soubret, J. Ripoll, and H. A. Shih, “Planar fluorescence imaging using normalized data,” J. Biomed. Opt. 10, 064007 (2005).
[CrossRef]

Xu, G.

Yalavarthy, P. K.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

Yao, G.

Yodh, A. G.

Zaccanti, G.

Zhang, A.

Zhang, C.

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488–496 (2004).
[CrossRef]

Zhang, Q.

Appl. Opt. (6)

Biomed. Opt. Express (2)

Commun. Numer. Methods Eng. (1)

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

Eur. J. Radiol. (1)

V. Ntziachristos, C. Bremer, and R. Weissleder, “Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging,” Eur. J. Radiol. 13, 195–208 (2003).
[CrossRef]

J. Biomed. Opt. (5)

D. S. Kepshire, S. L. Gibbs-Strauss, J. A. O’Hara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt. 14, 030501 (2009).
[CrossRef]

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488–496 (2004).
[CrossRef]

J. F. Abascal, J. Aguirre, J. Chamorro-Servent, M. Schweiger, S. Arridge, J. Ripoll, J. J. Vaquero, and M. Desco, “Influence of absorption and scattering on the quantification of fluorescence diffuse optical tomography using normalized data,” J. Biomed. Opt. 17, 036013 (2012).
[CrossRef]

V. Ntziachristos, G. Turner, J. Dunham, S. Windsor, A. Soubret, J. Ripoll, and H. A. Shih, “Planar fluorescence imaging using normalized data,” J. Biomed. Opt. 10, 064007 (2005).
[CrossRef]

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

J. Opt. Soc. Am. A (5)

Med. Phys. (4)

Y. Tan and H. Jiang, “DOT guided fluorescence molecular tomography of arbitrarily shaped objects,” Med. Phys. 35, 5703–5707 (2008).
[CrossRef]

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with indocyanine green,” Med. Phys. 30, 1039–1047 (2003).
[CrossRef]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

C. Li, R. Liengsawangwong, H. Choi, and R. Cheung, “Using a priori structural information from magnetic resonance imaging to investigate the feasibility of prostate diffuse optical tomography and spectroscopy: a simulation study,” Med. Phys. 34, 266–274 (2007).
[CrossRef]

NeuroImage (1)

A. Liebert, H. Wabnitz, H. Obrig, R. Erdmann, M. Möller, R. Macdonald, H. Rinneberg, A. Villringer, and J. Steinbrink, “Noninvasive detection of fluorescence from exogenous chromophores in the adult human brain,” NeuroImage 31, 600–608 (2006).
[CrossRef]

Opt. Express (2)

Phys. Med. Biol. (3)

Y. Lin, H. Gao, O. Nalcioglu, and G. Gulsen, “Fluorescence diffuse optical tomography with functional and anatomical a priori information: feasibility study,” Phys. Med. Biol. 52, 5569–5585 (2007).
[CrossRef]

M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time-resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
[CrossRef]

A. Soubret and V. Ntziachristos, “Fluorescence molecular tomography in the presence of background fluorescence,” Phys. Med. Biol. 51, 3983–4001 (2006).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

Illustrations of a medium of infinite geometry (A) and a medium of semi-infinite geometry (B). In the semi-infinite geometry, the directional source and the detector are positioned on the physical boundary of the medium, and it becomes convenient to assign the same radial and azimuthal coordinates to the source and detector. The implementation of the extrapolated zero-boundary condition introduces the image of the equivalent isotropic source and the image of the fluorophore element with respect to the extrapolated boundary.

Fig. 2.
Fig. 2.

Illustrations of a medium of concave geometry (A) and a medium of convex geometry (B). In both geometries, the directional source and the detector are positioned on the physical boundary of the medium. The implementation of extrapolated zero-boundary condition introduces the image of the equivalent isotropic source along the radial direction of the source, and the image of the fluorophore element along the radial direction of the element.

Fig. 3.
Fig. 3.

FEM results of the changes of CW photon-fluence rates at excitation wavelength (A), fluorescence wavelength (B), and Born ratio (C), in concave and convex geometries in comparison to semi-infinite geometry. In concave geometry, the CW photon-fluence rates at both the excitation wavelength and the fluorescence wavelength reduce more slowly in case-azi and more quickly in case-longi than those along a straight line on the semi-infinite interface. In convex geometry, the CW photon-fluence rates at both the excitation wavelength and the fluorescence wavelength reduce more quickly in case-azi and more slowly in case-longi than those along a straight line on the semi-infinite interface. Note that the dominant slopes of the excitation-wavelength and fluorescence-wavelength lines associated with the semi-infinite medium are identical. We also note that the lines associated with the fluorescence wavelength have a slower change than the lines associated with the excitation wavelength in the concave geometry, and conversely in the convex geometry. In concave geometry, the changes in the Born ratio in both case-azi and case-longi are smaller than those along a straight line on the semi-infinite interface. In convex geometry, the changes in the Born ratio in both case-azi and case-longi are greater than those along a straight line on the semi-infinite interface.

Equations (71)

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

Qfl=ημaflΨex(r⃗r⃗fl),
k0=μatotalDtotal,
2Ψinfex(r⃗)k02Ψinfex(r⃗)=Sex(r⃗)D,
2Ψinffl(r⃗)k02Ψinffl(r⃗)=Qfl(r⃗)D.
Ψsemiex(r⃗)=Ψinfex(r⃗realr⃗)Ψinfex(r⃗imagr⃗)=0forr⃗Ω.
Ψsemifl(r⃗)=Ψinffl(r⃗flr⃗)Ψinffl(r⃗imagflr⃗)=0forr⃗Ω.
Ψinfex(r⃗r⃗)=S4πDexp(k0|r⃗r⃗|)|r⃗r⃗|.
Ψinffl(r⃗flr⃗)=Q4πDexp(k0|r⃗flr⃗|)|r⃗flr⃗|.
Qfl=ημaflΨinfex(r⃗r⃗fl)=ημaflS4πDexp(k0|r⃗r⃗fl|)|r⃗r⃗fl|.
Ψinffl(r⃗r⃗flr⃗)=Qfl4πDexp(k0|r⃗flr⃗|)|r⃗flr⃗|=ημaflS(4πD)2exp(k0|r⃗r⃗fl|)|r⃗r⃗fl|exp(k0|r⃗flr⃗|)|r⃗flr⃗|.
Ψinffl(r⃗r⃗flr⃗)=Ψinffl(r⃗r⃗flr⃗)d3r⃗fl=ημaflS8πk0D2exp(k0|r⃗r⃗|),
exp(k0|r⃗r⃗fl|)|r⃗r⃗fl|exp(k0|r⃗flr⃗|)|r⃗flr⃗|d3r⃗fl=2πk0exp(k0|r⃗r⃗|).
Ψsemiex(r⃗r⃗)=Ψinfex(r⃗realr⃗)Ψinfex(r⃗imagr⃗)=S4πDexp(k0|r⃗realr⃗|)|r⃗realr⃗|S4πDexp(k0|r⃗imagr⃗|)|r⃗imagr⃗|.
Ψsemiex(r⃗r⃗fl)=Ψinfex(r⃗realr⃗fl)Ψinfex(r⃗imagr⃗fl)=S4πDexp(k0|r⃗realr⃗fl|)|r⃗realr⃗fl|S4πDexp(k0|r⃗imagr⃗fl|)|r⃗imagr⃗fl|.
Qfl=ημaflΨsemiex(r⃗r⃗fl)=ημafl[Ψinfex(r⃗realr⃗fl)Ψinfex(r⃗imagr⃗fl)].
Ψsemifl(r⃗flr⃗)=Ψinffl(r⃗flr⃗)Ψinffl(r⃗imagflr⃗)=Qfl4πDexp(k0|r⃗flr⃗|)|r⃗flr⃗|Qimagfl4πDexp(k0|r⃗imagflr⃗|)|r⃗imagflr⃗|.
Qimagfl=[Qfl]imag=[ημaflΨsemiex(r⃗r⃗fl)]imag=ημafl[Ψinfex(r⃗realr⃗fl)Ψinfex(r⃗imagr⃗fl)]imag=ημafl[Ψinfex(r⃗imagr⃗imagfl)Ψinfex(r⃗realr⃗imagfl)],
Ψsemifl(r⃗r⃗flr⃗)=ημafl4πD[Ψinfex(r⃗realr⃗fl)]exp(k0|r⃗flr⃗|)|r⃗flr⃗|ημafl4πD[Ψinfex(r⃗imagr⃗fl)]exp(k0|r⃗flr⃗|)|r⃗flr⃗|ημafl4πD[Ψinfex(r⃗imagr⃗imagfl)]exp(k0|r⃗imagflr⃗|)|r⃗imagflr⃗|+ημafl4πD[Ψinfex(r⃗realr⃗imagfl)]exp(k0|r⃗imagflr⃗|)|r⃗imagflr⃗|=Ψinffl(r⃗realr⃗flr⃗)Ψinffl(r⃗imagr⃗flr⃗)+Ψinffl(r⃗realr⃗imagflr⃗)Ψinffl(r⃗imagr⃗imagflr⃗),
Ψsemifl(r⃗r⃗flr⃗)=ρfl=ρΨsemifl(r⃗r⃗flr⃗)d3r⃗fl=ρfl=ρΨinffl(r⃗realr⃗flr⃗)d3r⃗flρfl=ρΨinffl(r⃗imagr⃗flr⃗)d3r⃗fl+ρfl=ρΨinffl(r⃗realr⃗imagflr⃗)d3r⃗flρfl=ρΨinffl(r⃗imagr⃗imagflr⃗)d3r⃗fl.
ρfl=ρΨinffl(r⃗realr⃗flr⃗)d3r⃗fl=Ψinffl(r⃗realr⃗flr⃗)ρfl=ρ2RbρΨinffl(r⃗realr⃗flr⃗)d3r⃗flρfl=ρ2RbΨinffl(r⃗realr⃗flr⃗)d3r⃗fl.
ρfl=ρΨinffl(r⃗imagr⃗flr⃗)d3r⃗fl=Ψinffl(r⃗imagr⃗flr⃗)ρfl=ρ2RbρΨinffl(r⃗imagr⃗flr⃗)d3r⃗flρfl=ρ2RbΨinffl(r⃗imagr⃗flr⃗)d3r⃗fl.
Ψsemifl(r⃗r⃗flr⃗)=Ψinffl(r⃗realr⃗flr⃗)Ψinffl(r⃗imagr⃗flr⃗)ρfl=ρ2RbρΨinffl(r⃗realr⃗flr⃗)d3r⃗flρfl=ρ2RbΨinffl(r⃗realr⃗flr⃗)d3r⃗fl+ρfl=ρ2RbρΨinffl(r⃗imagr⃗flr⃗)d3r⃗fl+ρfl=ρ2RbΨinffl(r⃗imagr⃗flr⃗)d3r⃗fl+ρfl=ρΨinffl(r⃗realr⃗imagflr⃗)d3r⃗flρfl=ρΨinffl(r⃗imagr⃗imagflr⃗)d3r⃗fl.
Ψsemifl(r⃗r⃗flr⃗)=Ψinffl(r⃗realr⃗flr⃗)Ψinffl(r⃗imagr⃗flr⃗)ρfl=ρ2Rbρ[Ψinffl(r⃗realr⃗flr⃗)Ψinffl(r⃗imagr⃗flr⃗)]d3r⃗fl.
ΨconCex(r⃗r⃗)=Ψinfex(r⃗realr⃗)Ψinfex(r⃗imagr⃗)=S4π2Ddk{eik(zz)m=Im[keff(R0Ra)]Km(keffR0)eim(ϕϕ)}S4π2Ddk{eik(zz)m=Im[keff(R0Ra)]Im(keffR0)Km[keff(R0+Rb)]Im[keff(R0+Rb)]eim(ϕϕ)},
keff=k2+k02.
ΨconCex(r⃗r⃗fl)=Ψinfex(r⃗realr⃗fl)Ψinfex(r⃗imagr⃗fl),
Qfl=ημaflΨconCex(r⃗r⃗fl)=ημafl[Ψinfex(r⃗realr⃗fl)Ψinfex(r⃗imagr⃗fl)].
ΨconCfl(r⃗flr⃗)=Ψinffl(r⃗flr⃗)Ψinffl(r⃗imagflr⃗)=Qfl4π2Ddk{eik(zflz)m=Im[keff(R0Rfl)]Km(keffR0)eim(ϕflϕ)}Qfl4π2Ddk{eik(zflz)m=Im[keff(R0Rfl)]Im(keffR0)Km[keff(R0+Rb)]Im[keff(R0+Rb)]eim(ϕflϕ)}.
Ψinfex(r⃗imagr⃗fl)=Ψinfex(r⃗realr⃗imagfl)andΨinfex(r⃗realr⃗fl)=Ψinfex(r⃗imagr⃗imagfl),
ΨconCfl(r⃗r⃗flr⃗)=Ψinffl(r⃗realr⃗flr⃗)Ψinffl(r⃗imagr⃗flr⃗)+Ψinffl(r⃗realr⃗imagflr⃗)Ψinffl(r⃗imagr⃗imagflr⃗),
ΨconCfl(r⃗r⃗flr⃗)=ρfl=0R0ΨconCfl(r⃗r⃗flr⃗)d3r⃗fl=ρfl=0R0Ψinffl(r⃗realr⃗flr⃗)d3r⃗flρfl=0R0Ψinffl(r⃗imagr⃗flr⃗)d3r⃗fl+ρfl=0R0Ψinffl(r⃗realr⃗imagflr⃗)d3r⃗flρfl=0R0Ψinffl(r⃗imagr⃗imagflr⃗)d3r⃗fl.
ρfl=0R0Ψinffl(r⃗realr⃗flr⃗)d3r⃗fl=Ψinffl(r⃗realr⃗flr⃗)ρfl=R0R0+2RbΨinffl(r⃗realr⃗flr⃗)d3r⃗flρfl=R0+2RbΨinffl(r⃗realr⃗flr⃗)d3r⃗fl.
ρfl=0R0Ψinffl(r⃗imagr⃗flr⃗)d3r⃗fl=Ψinffl(r⃗imagr⃗flr⃗)ρfl=R0R0+2RbΨinffl(r⃗imagr⃗flr⃗)d3r⃗flρfl=R0+2RbΨinffl(r⃗imagr⃗flr⃗)d3r⃗fl.
ΨconCfl(r⃗r⃗flr⃗)=Ψinffl(r⃗realr⃗flr⃗)Ψinffl(r⃗imagr⃗flr⃗)ρfl=R0R0+2RbΨinffl(r⃗realr⃗flr⃗)d3r⃗flρfl=R0+2RbΨinffl(r⃗realr⃗flr⃗)d3r⃗fl+ρfl=R0R0+2RbΨinffl(r⃗imagr⃗flr⃗)d3r⃗fl+ρfl=R0+2RbΨinffl(r⃗imagr⃗flr⃗)d3r⃗fl+ρfl=0R0Ψinffl(r⃗realr⃗imagflr⃗)d3r⃗flρfl=0R0Ψinffl(r⃗imagr⃗imagflr⃗)d3r⃗fl.
ΨconCfl(r⃗r⃗flr⃗)=Ψinffl(r⃗realr⃗flr⃗)Ψinffl(r⃗imagr⃗flr⃗)ρfl=R0R0+2Rb[Ψinffl(r⃗realr⃗flr⃗)Ψinffl(r⃗imagr⃗flr⃗)]d3r⃗fl.
ΨconVex(r⃗r⃗)=Ψinfex(r⃗realr⃗)Ψinfex(r⃗imagr⃗)=S4π2Ddk{eik(zz)m=Im(keffR0)Km[keff(R0+Ra)]eim(ϕϕ)}S4π2Ddk{eik(zz)m=Im[keff(R0Rb)]Km[keff(R0Rb)]Km(keffR0)Km[keff(R0+Ra)]eim(ϕϕ)}.
ΨconVfl(r⃗r⃗flr⃗)=Ψinffl(r⃗realr⃗flr⃗)Ψinffl(r⃗imagr⃗flr⃗)ρfl=R02RbR0[Ψinffl(r⃗realr⃗flr⃗)Ψinffl(r⃗imagr⃗flr⃗)]d3r⃗fl.
Ψinfex(r⃗r⃗)=S4πDexp(k0d)d,
Ψinffl(r⃗r⃗flr⃗)=ημaflS8πk0D2exp(k0d),
|r⃗realr⃗|=lreal=d1+Ra2d2;|r⃗imagr⃗|=limag=d1+(Ra+2Rb)2d2.
Ψsemiex(r⃗r⃗)=S2πDexp(k0d)d2k0Rb(Ra+Rb).
ρfl=ρ2Rbρ[Ψinffl(r⃗realr⃗flr⃗)Ψinffl(r⃗imagr⃗flr⃗)]d3r⃗fl=ημaflS8πk0D2exp(k0lreal)lrealk0Rb2.
Ψsemifl(r⃗rflr⃗)=ημaflS8πk0D2[exp(k0lreal)exp(k0limag)exp(k0lreal)lrealk0Rb2]=ημaflS4πD2Rb(Ra+Rb)exp(k0d)d[1Rb2(Ra+Rb)(112k0dRa2d2)].
lreal=d[1+Ra2d2RaR0(cosα)2];limag=d[1+(Ra+2Rb)2d2+Ra+2RbR0(cosα)2],
ΨconCex(r⃗r⃗)=S4πDexp(k0lreal)lrealS4πDexp(k0limag)limagR0+Ra+2RbR0Ra=S2πD[k0Rb(Ra+Rb)]1d2exp{[k0d+d2k0Rb(R0Ra)2R0Ra+2Rb4R0Rb(R0Ra)(cosα)2d2]}.
ρfl=R0R0+2Rb[Ψinffl(r⃗realr⃗flr⃗)Ψinffl(r⃗imagr⃗flr⃗)]d3r⃗fl=ημaflS8πk0D2exp(k0lreal)lrealk0Rb2,
Ψinffl(r⃗imagr⃗flr⃗)=ημaflS8π2k0Dexp(k0|r⃗imagr⃗|semiR0RaR0+Ra+2Rb),
ΨconCfl(r⃗r⃗flr⃗)=ημaflS8πk0D2[exp(k0lr)exp(k0liR0RaR0+Ra+2Rb)exp(k0lr)lrk0Rb2]=ημaflS4πD2Rb(Ra+Rb)1dexp{[k0d+d22Rb(R0+Ra+2Rb)2(R0+Ra+2Rb)+(Ra+2Rb)k0d4R0Rb(R0+Ra+2Rb)(cosα)2d2]}ημaflS4πD2Rb(Ra+Rb)exp(k0d)dRb2(Ra+Rb)(112k0d[Ra2d2RaR0(cosα)2]).
lreal=d[1+Ra2d2+RaR0(cosα)2];limag=d[1+(Ra+2Rb)2d2Ra+2RbR0(cosα)2],
ΨconVex(r⃗r⃗)=S4πDexp(k0lreal)lrealS4πDexp(k0limag)limagR0Ra2RbR0+Ra=S2πD[k0Rb(Ra+Rb)]1d2exp{[k0dd2k0Rb(R0+Ra)+2R0+Ra2Rb4R0Rb(R0+Ra)(cosα)2d2]}.
ρfl=R02RbR0[Ψinffl(r⃗realr⃗flr⃗)Ψinffl(r⃗imagr⃗flr⃗)]d3r⃗fl=ημaflS8πk0D2exp(k0lreal)lrealk0Rb2,
Ψinffl(r⃗imagr⃗flr⃗)=ημaflS8π2k0Dexp(k0|r⃗imagr⃗|semiR0+RaR0Ra2Rb),
ΨconVfl(r⃗r⃗flr⃗)=ημaflS8πk0D2[exp(k0lr)exp(k0liR0+RaR0Ra2Rb)exp(k0lr)lrk0Rb2]=ημaflS4πD2Rb(Ra+Rb)1dexp{[k0dd22Rb(R0Ra2Rb)+2(R0Ra2Rb)(Ra+2Rb)k0d4R0Rb(R0Ra2Rb)(cosα)2d2]}ημaflS4πD2Rb(Ra+Rb)exp(k0d)dRb2(Ra+Rb)(112k0d[Ra2d2+RaR0(cosα)2]).
ΨconCex(r⃗r⃗)=S2πD[k0Rb(Ra+Rb)]1d2exp{[k0d+d2k0Rb(R0Ra)]},
ΨconCfl(r⃗r⃗flr⃗)=ημaflS4πD2Rb(Ra+Rb)1dexp{[k0d+d22Rb(R0+Ra+2Rb)]}ημaflS4πD2Rb(Ra+Rb)exp(k0d)dRb2(Ra+Rb)(112k0d[Ra2d2]).
ΨconCex(r⃗r⃗)=S2πD[k0Rb(Ra+Rb)]1d2exp{[k0d+2R0d(2R0Ra+2Rb)d·k0d4k0RbR0(R0Ra)]},
ΨconCfl(r⃗r⃗flr⃗)=ημaflS4πD2Rb(Ra+Rb)1dexp{[k0d(Ra+2Rb)(2+k0d)d24RbR0(R0+Ra+2Rb)]}ημaflS4πD2Rb(Ra+Rb)exp(k0d)dRb2(Ra+Rb)(112k0d[Ra2d2RaR0]).
ΨconVex(r⃗r⃗)=S2πD[k0Rb(Ra+Rb)]1d2exp{[k0dd2k0Rb(R0+Ra)]},
ΨconVfl(r⃗r⃗flr⃗)=ημaflS4πD2Rb(Ra+Rb)1dexp{[k0dd22Rb(R0Ra2Rb)]}ημaflS4πD2Rb(Ra+Rb)exp(k0d)dRb2(Ra+Rb)(112k0d[Ra2d2]).
ΨconVex=S2πD[k0Rb(Ra+Rb)]1d2exp{[k0d2R0d(2R0+Ra2Rb)d·k0d4k0RbR0(R0+Ra)]},
ΨconVfl(r⃗r⃗flr⃗)=ημaflS4πD2Rb(Ra+Rb)1dexp{[k0d+(Ra+2Rb)(2+k0d)d24RbR0(R0Ra2Rb)]}ημaflS4πD2Rb(Ra+Rb)exp(k0d)dRb2(Ra+Rb)(112k0d[Ra2d2+RaR0]).
Rinf(r⃗r⃗)=Ψinffl(r⃗r⃗flr⃗)Ψinfex(r⃗r⃗)=ημafl2k0D|r⃗r⃗|=ημafl2k0Dd,
Rsemi(r⃗r⃗)=Ψsemifl(r⃗r⃗flr⃗)Ψsemiex(r⃗r⃗)=ημafl2k0Dd[1Rb2(Ra+Rb)(112k0Ra2d)].
RconC(r⃗r⃗)=ΨconCfl(r⃗r⃗flr⃗)ΨconCex(r⃗r⃗)=ημafl2k0Dd{AconCRb2(Ra+Rb)(112k0d[Ra2d2RaR0(cosα)2])BconC},
RconV(r⃗r⃗)=ΨconVfl(r⃗r⃗flr⃗)ΨconVex(r⃗r⃗)=ημafl2k0Dd{AconVRb2(Ra+Rb)(112k0d[Ra2d2RaR0(cosα)2])BconV},
Ψinfex(r⃗imagr⃗fl)=S4π2Ddk·eik(zzfl)eim(ϕϕfl)ηm*Im(keffρfl)Km(keffρ?),
Ψinfex(r⃗realr⃗imagfl)=S4π2Ddk·eik(zzfl)eim(ϕϕfl)ηmflIm[keff(R0Ra)]Km(keffρfl?).
ηm*Km(keffρ?)=Im[keff(R0Ra)]Km[keff(R0+Rb)]Im[keff(R0+Rb)]andηmflKm(keffρfl?)=Im(keffρfl)Km[keff(R0+Rb)]Im[keff(R0+Rb)],
S4π2Ddk·eik(zzfl)eim(ϕϕfl)Im[keff(R0Ra)]Im(keffρfl)Km[keff(R0+Rb)]Im[keff(R0+Rb)].
ρfl=R0R0+2Rb[Ψinffl(r⃗realr⃗flr⃗)Ψinffl(r⃗imagr⃗flr⃗)]d3r⃗fl=ημaflS(4πD)21π2dzfldϕflρfl=R0R0+2Rbρfldρfldkexeikex(zzfl)eim(ϕϕfl)Im[keffex(R0Ra)]Km(keffexρfl)dkfleikfl(zflz)ein(ϕflϕ)Im(keffflR0)Km(keffflρfl)ημaflS(4πD)21π2dzfldϕflρfl=R0R0+2Rbρfldρfldkexeikex(zzfl)eim(ϕϕfl)Im[keffex(R0Ra)]Im(keffexR0)Km[keffex(R0+Rb)]Im[keffex(R0+Rb)]dkfleikfl(zflz)ein(ϕflϕ)Im(keffflR0)Km(keffflρfl)=ημaflS(4πD)2dk·eik(zz)eim(ϕϕ)Im[keff(R0Ra)]Im(keffR0)ρfl=R0R0+2Rbρfl[Km(keffρfl)]2dρflημaflS(4πD)2dk·eik(zz)eim(ϕϕ)Im[keff(R0Ra)]Im(keffR0)Km[keff(R0+Rb)]Im[keff(R0+Rb)]ρfl=R0R0+2RbρflIm(keffρfl)Km(keffρfl)dρfl=ημaflS(4πD)2dk·eik(zz)eim(ϕϕ)Im[keff(R0Ra)]Km(keffR0)Im(keffR0)Km(keffR0)ρfl=R0R0+2Rbρfl{[Km(keffρfl)]2Km[keff(R0+Rb)]Im[keff(R0+Rb)]Im(keffρfl)Km(keffρfl)}dρfl.
ρfl=R0R0+2Rb[Ψinffl(r⃗realr⃗flr⃗)Ψinffl(r⃗imagr⃗flr⃗)]d3r⃗fl=ημaflS(4πD)2dk·eik(zz)eim(ϕϕ)Im[keff(R0Ra)]Km(keffR0)e2keffR02keffρfl=R0R0+2Rb[e2keffρfle2keff(R0+Rb)]dρfl=ημaflS(4πD)2dk·eik(zz)eim(ϕϕ)Im[keff(R0Ra)]Km(keffR0)[1(2keff)2e4keffRb(2keff)2Rbkeffe2keffRb]ημaflS(4πD)2dk·eik(zz)eim(ϕϕ)Im[keff(R0Ra)]Km(keffR0)[2(Rb)2]=ημaflS8πk0D2exp(k0|r⃗realr⃗|)|r⃗realr⃗|[k0(Rb)2].

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