Abstract

We introduce a system of coupled time-dependent parabolic simplified spherical harmonic equations to model the propagation of both excitation and fluorescence light in biological tissues. We resort to a finite element approach to obtain the time-dependent profile of the excitation and the fluorescence light fields in the medium. We present results for cases involving two geometries in three-dimensions: a homogeneous cylinder with an embedded fluorescent inclusion and a realistically-shaped rodent with an embedded inclusion alike an organ filled with a fluorescent probe. For the cylindrical geometry, we show the differences in the time-dependent fluorescence response for a point-like, a spherical, and a spherically Gaussian distributed fluorescent inclusion. From our results, we conclude that the model is able to describe the time-dependent excitation and fluorescent light transfer in small geometries with high absorption coefficients and in nondiffusive domains, as may be found in small animal diffuse optical tomography (DOT) and fluorescence DOT imaging.

©2011 Optical Society of America

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Light transport in turbid media with non-scattering, low-scattering and high absorption heterogeneities based on hybrid simplified spherical harmonics with radiosity model

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2010 (1)

2009 (5)

J. Pichette, E. Lapointe, and Y. Bérubé-Lauzière, “Three-dimensional localization of discrete fluorescent inclusions from multiple tomographic projections in the time-domain,” Proc. SPIE 7174, 71741A, 71741A-10 (2009).
[Crossref]

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol. 54(8), 2493–2509 (2009).
[Crossref] [PubMed]

Y. Bérubé-Lauzière, V. Issa, and J. B. Domínguez, “Simplified spherical harmonics approximation of the time-dependent equation of radiative transfer for the forward problem in time-domain diffuse optical tomography,” Proc. SPIE 7174, 717403, 717403-11 (2009).
[Crossref]

R. E. Nothdurft, S. V. Patwardhan, W. Akers, Y. Ye, S. Achilefu, and J. P. Culver, “In vivo fluorescence lifetime tomography,” J. Biomed. Opt. 14(2), 024004 (2009).
[Crossref] [PubMed]

J. Klohs, J. Steinbrink, R. Bourayou, S. Mueller, R. Cordell, K. Licha, M. Schirner, U. Dirnagl, U. Lindauer, and A. Wunder, “Near-infrared fluorescence imaging with fluorescently labeled albumin: a novel method for non-invasive optical imaging of blood-brain barrier impairment after focal cerebral ischemia in mice,” J. Neurosci. Methods 180(1), 126–132 (2009).
[Crossref] [PubMed]

2008 (3)

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Utilizing the radiative transfer equation in optical tomography,” PIERS Proceedings 4, 730–735 (2008).

D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantitative fluorescence imaging of point-like sources in small animals,” Phys. Med. Biol. 53(20), 5797–5814 (2008).
[Crossref] [PubMed]

J. Pichette, E. Lapointe, and Y. Bérubé-Lauzière, “Time-domain 3D localization of fluorescent inclusions in a thick scattering medium,” Proc. SPIE 7099, 709907, 709907-12 (2008).
[Crossref]

2007 (10)

M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, “Approximate models of radiative transfer,” Bull. Inst. Math. Acad. Sin. 2, 409–432 (2007).

P. Kotiluoto, J. Pyyry, and H. Helminen, “Multitrans SP3 code in coupled photon-electron transport problems,” Radiat. Phys. Chem. 76(1), 9–14 (2007).
[Crossref]

M. Frank, A. Klar, E. Larsen, and S. Yasuda, “Time-dependent simplified PN approximation to the equations of radiative transfer,” J. Comput. Phys. 226(2), 2289–2305 (2007).
[Crossref]

J. Rao, A. Dragulescu-Andrasi, and H. Yao, “Fluorescence imaging in vivo: recent advances,” Curr. Opin. Biotechnol. 18(1), 17–25 (2007).
[Crossref] [PubMed]

E. M. C. Hillman and A. Moore, “All-optical anatomical co-registration for molecular imaging of small animals using dynamic contrast,” Nat. Photonics 1(9), 526–530 (2007).
[Crossref] [PubMed]

L. Hervé, A. Koenig, A. Da Silva, M. Berger, J. Boutet, J. M. Dinten, P. Peltié, and P. Rizo, “Noncontact fluorescence diffuse optical tomography of heterogeneous media,” Appl. Opt. 46(22), 4896–4906 (2007).
[Crossref] [PubMed]

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52(3), 577–587 (2007).
[Crossref] [PubMed]

M. Comtois and Y. Bérubé-Lauzière, “Optical surface metrology in 3D for small animal non-contact diffuse optical tomography,” Proc. SPIE 6796, 679605, 679605-9 (2007).
[Crossref]

H. Zhao, F. Gao, Y. Tanikawa, and Y. Yamada, “Time-resolved diffuse optical tomography and its application to in vitro and in vivo imaging,” J. Biomed. Opt. 12(6), 062107 (2007).
[Crossref] [PubMed]

T. Svensson, S. Andersson-Engels, M. Einarsdóttír, and K. Svanberg, “In vivo optical characterization of human prostate tissue using near-infrared time-resolved spectroscopy,” J. Biomed. Opt. 12(1), 014022 (2007).
[Crossref] [PubMed]

2006 (2)

V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8(1), 1–33 (2006).
[Crossref] [PubMed]

A. Klose and E. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441–470 (2006).
[Crossref]

2005 (1)

A. H. Hielscher, “Optical tomography imaging of small animals,” Curr. Opin. Biotechnol. 16(1), 79–88 (2005).
[Crossref]

2004 (1)

L. Martí-López, J. Bouza-Domínguez, and J. C. Hebden, “Interpretation of the failure of the time-independent diffusion equation near a point source,” Opt. Commun. 242(1-3), 23–43 (2004).
[Crossref]

2003 (5)

X. Intes, J. Ripoll, Y. Chen, Sh. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30(6), 1039–1047 (2003).
[Crossref] [PubMed]

R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nat. Med. 9(1), 123–128 (2003).
[Crossref] [PubMed]

Ch. Haritoglou, A. Gandorfer, M. Schaumberger, R. Tadayoni, A. Gandorfer, and A. Kampik, “Light-absorbing properties and osmolarity of indocyanine-green depending on concentration and solvent medium,” Invest. Ophthalmol. Vis. Sci. 44(6), 2722–2729 (2003).
[Crossref] [PubMed]

G. Abdoulaev and A. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imaging 12(4), 594–601 (2003).
[Crossref]

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(16), 3081–3094 (2003).
[Crossref] [PubMed]

2002 (1)

2001 (2)

R. Weissleder and U. Mahmood, “Molecular imaging,” Radiology 219(2), 316–333 (2001).
[PubMed]

V. Ntziachristos and B. Chance, “Accuracy limits in the determination of absolute optical properties using time-resolved NIR spectroscopy,” Med. Phys. 28(6), 1115–1124 (2001).
[Crossref] [PubMed]

2000 (2)

P. S. Brantley and E. W. Larsen, “The simplified P3 approximation,” Nucl. Sci. Eng. 134, 121 (2000).

L. V. Wang and S. L. Jacques, “Sources of error in calculation of optical diffuse reflectance from turbid media using diffusion theory,” Comput. Meth. Prog. Bio. 61(3), 163–170 (2000).
[Crossref]

1999 (2)

1998 (2)

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[Crossref] [PubMed]

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[Crossref] [PubMed]

1996 (2)

E. W. Larsen, J. E. Morel, and J. M. McGhee, “Asymptotic derivation of the multigroup P1 and simplified PN equations with anisotropic scattering,” Nucl. Sci. Eng. 123, 328 (1996).

D. I. Tomasevic and E. W. Larsen, “The simplified P2 approximation,” Nucl. Sci. Eng. 122, 309–325 (1996).

1995 (4)

X. Li, B. Beauvoit, R. White, Sh. Nioka, B. Chance, and A. G. Yodh, “Tumor localization using fluorescence of indocyanine green (ICG) in rat models,” Proc. SPIE 2389, 789–797 (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(11), 1779–1792 (1995).
[Crossref] [PubMed]

K. D. Paulsen and H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22(6), 691–701 (1995).
[Crossref] [PubMed]

S. R. Arridge, “Photon-measurement density functions. Part I: Analytical forms,” Appl. Opt. 34(31), 7395–7409 (1995).
[Crossref] [PubMed]

1994 (1)

S. Mordon, J. M. Devoisselle, and V. Maunoury, “In vivo pH measurement and imaging of tumor tissue using a pH-sensitive fluorescent probe (5,6-carboxyfluorescein): instrumental and experimental studies,” Photochem. Photobiol. 60(3), 274–279 (1994).
[Crossref] [PubMed]

1993 (2)

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imaging Vis. 3(3), 263–283 (1993).
[Crossref]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[Crossref] [PubMed]

1990 (1)

W. Cheong, S. Prahl, and A. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26(12), 2166–2185 (1990).
[Crossref]

1978 (1)

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of back-scattering of light by blood,” J. Phys. D Appl. Phys. 11(10), 1463–1479 (1978).
[Crossref]

1976 (1)

M. L. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40(4), 575–583 (1976).
[PubMed]

1970 (1)

G. Paumgartner, P. Probst, R. Kraines, and C. M. Leevy, “Kinetics of indocyanine green removal from the blood,” Ann. N. Y. Acad. Sci. 170(1 The Hepatic C), 134–147 (1970).
[Crossref]

Abdoulaev, G.

G. Abdoulaev and A. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imaging 12(4), 594–601 (2003).
[Crossref]

Achilefu, S.

R. E. Nothdurft, S. V. Patwardhan, W. Akers, Y. Ye, S. Achilefu, and J. P. Culver, “In vivo fluorescence lifetime tomography,” J. Biomed. Opt. 14(2), 024004 (2009).
[Crossref] [PubMed]

Akers, W.

R. E. Nothdurft, S. V. Patwardhan, W. Akers, Y. Ye, S. Achilefu, and J. P. Culver, “In vivo fluorescence lifetime tomography,” J. Biomed. Opt. 14(2), 024004 (2009).
[Crossref] [PubMed]

Alcouffe, R. E.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[Crossref] [PubMed]

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[Crossref] [PubMed]

Andersson-Engels, S.

T. Svensson, S. Andersson-Engels, M. Einarsdóttír, and K. Svanberg, “In vivo optical characterization of human prostate tissue using near-infrared time-resolved spectroscopy,” J. Biomed. Opt. 12(1), 014022 (2007).
[Crossref] [PubMed]

Arridge, S. R.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Utilizing the radiative transfer equation in optical tomography,” PIERS Proceedings 4, 730–735 (2008).

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999).
[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(11), 1779–1792 (1995).
[Crossref] [PubMed]

S. R. Arridge, “Photon-measurement density functions. Part I: Analytical forms,” Appl. Opt. 34(31), 7395–7409 (1995).
[Crossref] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[Crossref] [PubMed]

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imaging Vis. 3(3), 263–283 (1993).
[Crossref]

Barbour, R. L.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[Crossref] [PubMed]

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[Crossref] [PubMed]

Beauvoit, B.

X. Li, B. Beauvoit, R. White, Sh. Nioka, B. Chance, and A. G. Yodh, “Tumor localization using fluorescence of indocyanine green (ICG) in rat models,” Proc. SPIE 2389, 789–797 (1995).
[Crossref]

Berger, M.

Bérubé-Lauzière, Y.

J. B. Domínguez and Y. Bérubé-Lauzière, “Diffuse light propagation in biological media by a time-domain parabolic simplified spherical harmonics approximation with ray-divergence effects,” Appl. Opt. 49(8), 1414–1429 (2010).
[Crossref] [PubMed]

J. Pichette, E. Lapointe, and Y. Bérubé-Lauzière, “Three-dimensional localization of discrete fluorescent inclusions from multiple tomographic projections in the time-domain,” Proc. SPIE 7174, 71741A, 71741A-10 (2009).
[Crossref]

Y. Bérubé-Lauzière, V. Issa, and J. B. Domínguez, “Simplified spherical harmonics approximation of the time-dependent equation of radiative transfer for the forward problem in time-domain diffuse optical tomography,” Proc. SPIE 7174, 717403, 717403-11 (2009).
[Crossref]

J. Pichette, E. Lapointe, and Y. Bérubé-Lauzière, “Time-domain 3D localization of fluorescent inclusions in a thick scattering medium,” Proc. SPIE 7099, 709907, 709907-12 (2008).
[Crossref]

M. Comtois and Y. Bérubé-Lauzière, “Optical surface metrology in 3D for small animal non-contact diffuse optical tomography,” Proc. SPIE 6796, 679605, 679605-9 (2007).
[Crossref]

Boas, D. A.

Bouman, C. A.

Bourayou, R.

J. Klohs, J. Steinbrink, R. Bourayou, S. Mueller, R. Cordell, K. Licha, M. Schirner, U. Dirnagl, U. Lindauer, and A. Wunder, “Near-infrared fluorescence imaging with fluorescently labeled albumin: a novel method for non-invasive optical imaging of blood-brain barrier impairment after focal cerebral ischemia in mice,” J. Neurosci. Methods 180(1), 126–132 (2009).
[Crossref] [PubMed]

Boutet, J.

Bouza-Domínguez, J.

L. Martí-López, J. Bouza-Domínguez, and J. C. Hebden, “Interpretation of the failure of the time-independent diffusion equation near a point source,” Opt. Commun. 242(1-3), 23–43 (2004).
[Crossref]

Brantley, P. S.

P. S. Brantley and E. W. Larsen, “The simplified P3 approximation,” Nucl. Sci. Eng. 134, 121 (2000).

Chance, B.

X. Intes, J. Ripoll, Y. Chen, Sh. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30(6), 1039–1047 (2003).
[Crossref] [PubMed]

V. Ntziachristos and B. Chance, “Accuracy limits in the determination of absolute optical properties using time-resolved NIR spectroscopy,” Med. Phys. 28(6), 1115–1124 (2001).
[Crossref] [PubMed]

X. Li, B. Beauvoit, R. White, Sh. Nioka, B. Chance, and A. G. Yodh, “Tumor localization using fluorescence of indocyanine green (ICG) in rat models,” Proc. SPIE 2389, 789–797 (1995).
[Crossref]

Chatziioannou, A. F.

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52(3), 577–587 (2007).
[Crossref] [PubMed]

Chen, Y.

X. Intes, J. Ripoll, Y. Chen, Sh. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30(6), 1039–1047 (2003).
[Crossref] [PubMed]

Cheong, W.

W. Cheong, S. Prahl, and A. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26(12), 2166–2185 (1990).
[Crossref]

Chu, M.

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol. 54(8), 2493–2509 (2009).
[Crossref] [PubMed]

Comsa, D. C.

D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantitative fluorescence imaging of point-like sources in small animals,” Phys. Med. Biol. 53(20), 5797–5814 (2008).
[Crossref] [PubMed]

Comtois, M.

M. Comtois and Y. Bérubé-Lauzière, “Optical surface metrology in 3D for small animal non-contact diffuse optical tomography,” Proc. SPIE 6796, 679605, 679605-9 (2007).
[Crossref]

Cordell, R.

J. Klohs, J. Steinbrink, R. Bourayou, S. Mueller, R. Cordell, K. Licha, M. Schirner, U. Dirnagl, U. Lindauer, and A. Wunder, “Near-infrared fluorescence imaging with fluorescently labeled albumin: a novel method for non-invasive optical imaging of blood-brain barrier impairment after focal cerebral ischemia in mice,” J. Neurosci. Methods 180(1), 126–132 (2009).
[Crossref] [PubMed]

Culver, J. P.

R. E. Nothdurft, S. V. Patwardhan, W. Akers, Y. Ye, S. Achilefu, and J. P. Culver, “In vivo fluorescence lifetime tomography,” J. Biomed. Opt. 14(2), 024004 (2009).
[Crossref] [PubMed]

Da Silva, A.

Dehghani, H.

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol. 54(8), 2493–2509 (2009).
[Crossref] [PubMed]

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(11), 1779–1792 (1995).
[Crossref] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[Crossref] [PubMed]

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imaging Vis. 3(3), 263–283 (1993).
[Crossref]

Devoisselle, J. M.

S. Mordon, J. M. Devoisselle, and V. Maunoury, “In vivo pH measurement and imaging of tumor tissue using a pH-sensitive fluorescent probe (5,6-carboxyfluorescein): instrumental and experimental studies,” Photochem. Photobiol. 60(3), 274–279 (1994).
[Crossref] [PubMed]

Dinten, J. M.

Dirnagl, U.

J. Klohs, J. Steinbrink, R. Bourayou, S. Mueller, R. Cordell, K. Licha, M. Schirner, U. Dirnagl, U. Lindauer, and A. Wunder, “Near-infrared fluorescence imaging with fluorescently labeled albumin: a novel method for non-invasive optical imaging of blood-brain barrier impairment after focal cerebral ischemia in mice,” J. Neurosci. Methods 180(1), 126–132 (2009).
[Crossref] [PubMed]

Dogdas, B.

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52(3), 577–587 (2007).
[Crossref] [PubMed]

Domínguez, J. B.

J. B. Domínguez and Y. Bérubé-Lauzière, “Diffuse light propagation in biological media by a time-domain parabolic simplified spherical harmonics approximation with ray-divergence effects,” Appl. Opt. 49(8), 1414–1429 (2010).
[Crossref] [PubMed]

Y. Bérubé-Lauzière, V. Issa, and J. B. Domínguez, “Simplified spherical harmonics approximation of the time-dependent equation of radiative transfer for the forward problem in time-domain diffuse optical tomography,” Proc. SPIE 7174, 717403, 717403-11 (2009).
[Crossref]

Dragulescu-Andrasi, A.

J. Rao, A. Dragulescu-Andrasi, and H. Yao, “Fluorescence imaging in vivo: recent advances,” Curr. Opin. Biotechnol. 18(1), 17–25 (2007).
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Durian, D. J.

Eason, G.

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of back-scattering of light by blood,” J. Phys. D Appl. Phys. 11(10), 1463–1479 (1978).
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Einarsdóttír, M.

T. Svensson, S. Andersson-Engels, M. Einarsdóttír, and K. Svanberg, “In vivo optical characterization of human prostate tissue using near-infrared time-resolved spectroscopy,” J. Biomed. Opt. 12(1), 014022 (2007).
[Crossref] [PubMed]

Farrell, T. J.

D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantitative fluorescence imaging of point-like sources in small animals,” Phys. Med. Biol. 53(20), 5797–5814 (2008).
[Crossref] [PubMed]

Frank, M.

M. Frank, A. Klar, E. Larsen, and S. Yasuda, “Time-dependent simplified PN approximation to the equations of radiative transfer,” J. Comput. Phys. 226(2), 2289–2305 (2007).
[Crossref]

M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, “Approximate models of radiative transfer,” Bull. Inst. Math. Acad. Sin. 2, 409–432 (2007).

Gandorfer, A.

Ch. Haritoglou, A. Gandorfer, M. Schaumberger, R. Tadayoni, A. Gandorfer, and A. Kampik, “Light-absorbing properties and osmolarity of indocyanine-green depending on concentration and solvent medium,” Invest. Ophthalmol. Vis. Sci. 44(6), 2722–2729 (2003).
[Crossref] [PubMed]

Ch. Haritoglou, A. Gandorfer, M. Schaumberger, R. Tadayoni, A. Gandorfer, and A. Kampik, “Light-absorbing properties and osmolarity of indocyanine-green depending on concentration and solvent medium,” Invest. Ophthalmol. Vis. Sci. 44(6), 2722–2729 (2003).
[Crossref] [PubMed]

Gao, F.

H. Zhao, F. Gao, Y. Tanikawa, and Y. Yamada, “Time-resolved diffuse optical tomography and its application to in vitro and in vivo imaging,” J. Biomed. Opt. 12(6), 062107 (2007).
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F. Gao, H. Zhao, and Y. Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt. 41(4), 778–791 (2002).
[Crossref] [PubMed]

Haritoglou, Ch.

Ch. Haritoglou, A. Gandorfer, M. Schaumberger, R. Tadayoni, A. Gandorfer, and A. Kampik, “Light-absorbing properties and osmolarity of indocyanine-green depending on concentration and solvent medium,” Invest. Ophthalmol. Vis. Sci. 44(6), 2722–2729 (2003).
[Crossref] [PubMed]

Hebden, J. C.

L. Martí-López, J. Bouza-Domínguez, and J. C. Hebden, “Interpretation of the failure of the time-independent diffusion equation near a point source,” Opt. Commun. 242(1-3), 23–43 (2004).
[Crossref]

Helminen, H.

P. Kotiluoto, J. Pyyry, and H. Helminen, “Multitrans SP3 code in coupled photon-electron transport problems,” Radiat. Phys. Chem. 76(1), 9–14 (2007).
[Crossref]

Hervé, L.

Hielscher, A.

G. Abdoulaev and A. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imaging 12(4), 594–601 (2003).
[Crossref]

Hielscher, A. H.

A. H. Hielscher, “Optical tomography imaging of small animals,” Curr. Opin. Biotechnol. 16(1), 79–88 (2005).
[Crossref]

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[Crossref] [PubMed]

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[Crossref] [PubMed]

Hillman, E. M. C.

E. M. C. Hillman and A. Moore, “All-optical anatomical co-registration for molecular imaging of small animals using dynamic contrast,” Nat. Photonics 1(9), 526–530 (2007).
[Crossref] [PubMed]

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(11), 1779–1792 (1995).
[Crossref] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[Crossref] [PubMed]

Intes, X.

X. Intes, J. Ripoll, Y. Chen, Sh. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30(6), 1039–1047 (2003).
[Crossref] [PubMed]

Issa, V.

Y. Bérubé-Lauzière, V. Issa, and J. B. Domínguez, “Simplified spherical harmonics approximation of the time-dependent equation of radiative transfer for the forward problem in time-domain diffuse optical tomography,” Proc. SPIE 7174, 717403, 717403-11 (2009).
[Crossref]

Jacques, S. L.

L. V. Wang and S. L. Jacques, “Sources of error in calculation of optical diffuse reflectance from turbid media using diffusion theory,” Comput. Meth. Prog. Bio. 61(3), 163–170 (2000).
[Crossref]

Jiang, H.

K. D. Paulsen and H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22(6), 691–701 (1995).
[Crossref] [PubMed]

Kaipio, J. P.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Utilizing the radiative transfer equation in optical tomography,” PIERS Proceedings 4, 730–735 (2008).

Kampik, A.

Ch. Haritoglou, A. Gandorfer, M. Schaumberger, R. Tadayoni, A. Gandorfer, and A. Kampik, “Light-absorbing properties and osmolarity of indocyanine-green depending on concentration and solvent medium,” Invest. Ophthalmol. Vis. Sci. 44(6), 2722–2729 (2003).
[Crossref] [PubMed]

Klar, A.

M. Frank, A. Klar, E. Larsen, and S. Yasuda, “Time-dependent simplified PN approximation to the equations of radiative transfer,” J. Comput. Phys. 226(2), 2289–2305 (2007).
[Crossref]

M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, “Approximate models of radiative transfer,” Bull. Inst. Math. Acad. Sin. 2, 409–432 (2007).

Klohs, J.

J. Klohs, J. Steinbrink, R. Bourayou, S. Mueller, R. Cordell, K. Licha, M. Schirner, U. Dirnagl, U. Lindauer, and A. Wunder, “Near-infrared fluorescence imaging with fluorescently labeled albumin: a novel method for non-invasive optical imaging of blood-brain barrier impairment after focal cerebral ischemia in mice,” J. Neurosci. Methods 180(1), 126–132 (2009).
[Crossref] [PubMed]

Klose, A.

A. Klose and E. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441–470 (2006).
[Crossref]

Klose, A. D.

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol. 54(8), 2493–2509 (2009).
[Crossref] [PubMed]

Koenig, A.

Kolehmainen, V.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Utilizing the radiative transfer equation in optical tomography,” PIERS Proceedings 4, 730–735 (2008).

Kotiluoto, P.

P. Kotiluoto, J. Pyyry, and H. Helminen, “Multitrans SP3 code in coupled photon-electron transport problems,” Radiat. Phys. Chem. 76(1), 9–14 (2007).
[Crossref]

Kraines, R.

G. Paumgartner, P. Probst, R. Kraines, and C. M. Leevy, “Kinetics of indocyanine green removal from the blood,” Ann. N. Y. Acad. Sci. 170(1 The Hepatic C), 134–147 (1970).
[Crossref]

Kwant, G.

M. L. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40(4), 575–583 (1976).
[PubMed]

Landsman, M. L.

M. L. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40(4), 575–583 (1976).
[PubMed]

Lapointe, E.

J. Pichette, E. Lapointe, and Y. Bérubé-Lauzière, “Three-dimensional localization of discrete fluorescent inclusions from multiple tomographic projections in the time-domain,” Proc. SPIE 7174, 71741A, 71741A-10 (2009).
[Crossref]

J. Pichette, E. Lapointe, and Y. Bérubé-Lauzière, “Time-domain 3D localization of fluorescent inclusions in a thick scattering medium,” Proc. SPIE 7099, 709907, 709907-12 (2008).
[Crossref]

Larsen, E.

M. Frank, A. Klar, E. Larsen, and S. Yasuda, “Time-dependent simplified PN approximation to the equations of radiative transfer,” J. Comput. Phys. 226(2), 2289–2305 (2007).
[Crossref]

A. Klose and E. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441–470 (2006).
[Crossref]

Larsen, E. W.

M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, “Approximate models of radiative transfer,” Bull. Inst. Math. Acad. Sin. 2, 409–432 (2007).

P. S. Brantley and E. W. Larsen, “The simplified P3 approximation,” Nucl. Sci. Eng. 134, 121 (2000).

E. W. Larsen, J. E. Morel, and J. M. McGhee, “Asymptotic derivation of the multigroup P1 and simplified PN equations with anisotropic scattering,” Nucl. Sci. Eng. 123, 328 (1996).

D. I. Tomasevic and E. W. Larsen, “The simplified P2 approximation,” Nucl. Sci. Eng. 122, 309–325 (1996).

Leahy, R. M.

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52(3), 577–587 (2007).
[Crossref] [PubMed]

Leevy, C. M.

G. Paumgartner, P. Probst, R. Kraines, and C. M. Leevy, “Kinetics of indocyanine green removal from the blood,” Ann. N. Y. Acad. Sci. 170(1 The Hepatic C), 134–147 (1970).
[Crossref]

Li, X.

X. Li, B. Beauvoit, R. White, Sh. Nioka, B. Chance, and A. G. Yodh, “Tumor localization using fluorescence of indocyanine green (ICG) in rat models,” Proc. SPIE 2389, 789–797 (1995).
[Crossref]

Licha, K.

J. Klohs, J. Steinbrink, R. Bourayou, S. Mueller, R. Cordell, K. Licha, M. Schirner, U. Dirnagl, U. Lindauer, and A. Wunder, “Near-infrared fluorescence imaging with fluorescently labeled albumin: a novel method for non-invasive optical imaging of blood-brain barrier impairment after focal cerebral ischemia in mice,” J. Neurosci. Methods 180(1), 126–132 (2009).
[Crossref] [PubMed]

Lindauer, U.

J. Klohs, J. Steinbrink, R. Bourayou, S. Mueller, R. Cordell, K. Licha, M. Schirner, U. Dirnagl, U. Lindauer, and A. Wunder, “Near-infrared fluorescence imaging with fluorescently labeled albumin: a novel method for non-invasive optical imaging of blood-brain barrier impairment after focal cerebral ischemia in mice,” J. Neurosci. Methods 180(1), 126–132 (2009).
[Crossref] [PubMed]

Mahmood, U.

R. Weissleder and U. Mahmood, “Molecular imaging,” Radiology 219(2), 316–333 (2001).
[PubMed]

Martí-López, L.

L. Martí-López, J. Bouza-Domínguez, and J. C. Hebden, “Interpretation of the failure of the time-independent diffusion equation near a point source,” Opt. Commun. 242(1-3), 23–43 (2004).
[Crossref]

Maunoury, V.

S. Mordon, J. M. Devoisselle, and V. Maunoury, “In vivo pH measurement and imaging of tumor tissue using a pH-sensitive fluorescent probe (5,6-carboxyfluorescein): instrumental and experimental studies,” Photochem. Photobiol. 60(3), 274–279 (1994).
[Crossref] [PubMed]

McGhee, J. M.

E. W. Larsen, J. E. Morel, and J. M. McGhee, “Asymptotic derivation of the multigroup P1 and simplified PN equations with anisotropic scattering,” Nucl. Sci. Eng. 123, 328 (1996).

Millane, R. P.

Milstein, A. B.

Mook, G. A.

M. L. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40(4), 575–583 (1976).
[PubMed]

Moore, A.

E. M. C. Hillman and A. Moore, “All-optical anatomical co-registration for molecular imaging of small animals using dynamic contrast,” Nat. Photonics 1(9), 526–530 (2007).
[Crossref] [PubMed]

Mordon, S.

S. Mordon, J. M. Devoisselle, and V. Maunoury, “In vivo pH measurement and imaging of tumor tissue using a pH-sensitive fluorescent probe (5,6-carboxyfluorescein): instrumental and experimental studies,” Photochem. Photobiol. 60(3), 274–279 (1994).
[Crossref] [PubMed]

Morel, J. E.

E. W. Larsen, J. E. Morel, and J. M. McGhee, “Asymptotic derivation of the multigroup P1 and simplified PN equations with anisotropic scattering,” Nucl. Sci. Eng. 123, 328 (1996).

Mueller, S.

J. Klohs, J. Steinbrink, R. Bourayou, S. Mueller, R. Cordell, K. Licha, M. Schirner, U. Dirnagl, U. Lindauer, and A. Wunder, “Near-infrared fluorescence imaging with fluorescently labeled albumin: a novel method for non-invasive optical imaging of blood-brain barrier impairment after focal cerebral ischemia in mice,” J. Neurosci. Methods 180(1), 126–132 (2009).
[Crossref] [PubMed]

Nioka, Sh.

X. Intes, J. Ripoll, Y. Chen, Sh. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30(6), 1039–1047 (2003).
[Crossref] [PubMed]

X. Li, B. Beauvoit, R. White, Sh. Nioka, B. Chance, and A. G. Yodh, “Tumor localization using fluorescence of indocyanine green (ICG) in rat models,” Proc. SPIE 2389, 789–797 (1995).
[Crossref]

Nisbet, R. M.

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of back-scattering of light by blood,” J. Phys. D Appl. Phys. 11(10), 1463–1479 (1978).
[Crossref]

Nothdurft, R. E.

R. E. Nothdurft, S. V. Patwardhan, W. Akers, Y. Ye, S. Achilefu, and J. P. Culver, “In vivo fluorescence lifetime tomography,” J. Biomed. Opt. 14(2), 024004 (2009).
[Crossref] [PubMed]

Ntziachristos, V.

V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8(1), 1–33 (2006).
[Crossref] [PubMed]

R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nat. Med. 9(1), 123–128 (2003).
[Crossref] [PubMed]

V. Ntziachristos and B. Chance, “Accuracy limits in the determination of absolute optical properties using time-resolved NIR spectroscopy,” Med. Phys. 28(6), 1115–1124 (2001).
[Crossref] [PubMed]

Oh, S.

Patterson, M. S.

D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantitative fluorescence imaging of point-like sources in small animals,” Phys. Med. Biol. 53(20), 5797–5814 (2008).
[Crossref] [PubMed]

Patwardhan, S. V.

R. E. Nothdurft, S. V. Patwardhan, W. Akers, Y. Ye, S. Achilefu, and J. P. Culver, “In vivo fluorescence lifetime tomography,” J. Biomed. Opt. 14(2), 024004 (2009).
[Crossref] [PubMed]

Paulsen, K. D.

K. D. Paulsen and H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22(6), 691–701 (1995).
[Crossref] [PubMed]

Paumgartner, G.

G. Paumgartner, P. Probst, R. Kraines, and C. M. Leevy, “Kinetics of indocyanine green removal from the blood,” Ann. N. Y. Acad. Sci. 170(1 The Hepatic C), 134–147 (1970).
[Crossref]

Peltié, P.

Pichette, J.

J. Pichette, E. Lapointe, and Y. Bérubé-Lauzière, “Three-dimensional localization of discrete fluorescent inclusions from multiple tomographic projections in the time-domain,” Proc. SPIE 7174, 71741A, 71741A-10 (2009).
[Crossref]

J. Pichette, E. Lapointe, and Y. Bérubé-Lauzière, “Time-domain 3D localization of fluorescent inclusions in a thick scattering medium,” Proc. SPIE 7099, 709907, 709907-12 (2008).
[Crossref]

Prahl, S.

W. Cheong, S. Prahl, and A. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26(12), 2166–2185 (1990).
[Crossref]

Probst, P.

G. Paumgartner, P. Probst, R. Kraines, and C. M. Leevy, “Kinetics of indocyanine green removal from the blood,” Ann. N. Y. Acad. Sci. 170(1 The Hepatic C), 134–147 (1970).
[Crossref]

Pyyry, J.

P. Kotiluoto, J. Pyyry, and H. Helminen, “Multitrans SP3 code in coupled photon-electron transport problems,” Radiat. Phys. Chem. 76(1), 9–14 (2007).
[Crossref]

Rao, J.

J. Rao, A. Dragulescu-Andrasi, and H. Yao, “Fluorescence imaging in vivo: recent advances,” Curr. Opin. Biotechnol. 18(1), 17–25 (2007).
[Crossref] [PubMed]

Ripoll, J.

X. Intes, J. Ripoll, Y. Chen, Sh. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30(6), 1039–1047 (2003).
[Crossref] [PubMed]

Rizo, P.

Rudnick, J.

Schaumberger, M.

Ch. Haritoglou, A. Gandorfer, M. Schaumberger, R. Tadayoni, A. Gandorfer, and A. Kampik, “Light-absorbing properties and osmolarity of indocyanine-green depending on concentration and solvent medium,” Invest. Ophthalmol. Vis. Sci. 44(6), 2722–2729 (2003).
[Crossref] [PubMed]

Schirner, M.

J. Klohs, J. Steinbrink, R. Bourayou, S. Mueller, R. Cordell, K. Licha, M. Schirner, U. Dirnagl, U. Lindauer, and A. Wunder, “Near-infrared fluorescence imaging with fluorescently labeled albumin: a novel method for non-invasive optical imaging of blood-brain barrier impairment after focal cerebral ischemia in mice,” J. Neurosci. Methods 180(1), 126–132 (2009).
[Crossref] [PubMed]

Schweiger, 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(11), 1779–1792 (1995).
[Crossref] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[Crossref] [PubMed]

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imaging Vis. 3(3), 263–283 (1993).
[Crossref]

Steinbrink, J.

J. Klohs, J. Steinbrink, R. Bourayou, S. Mueller, R. Cordell, K. Licha, M. Schirner, U. Dirnagl, U. Lindauer, and A. Wunder, “Near-infrared fluorescence imaging with fluorescently labeled albumin: a novel method for non-invasive optical imaging of blood-brain barrier impairment after focal cerebral ischemia in mice,” J. Neurosci. Methods 180(1), 126–132 (2009).
[Crossref] [PubMed]

Stout, D.

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52(3), 577–587 (2007).
[Crossref] [PubMed]

Svanberg, K.

T. Svensson, S. Andersson-Engels, M. Einarsdóttír, and K. Svanberg, “In vivo optical characterization of human prostate tissue using near-infrared time-resolved spectroscopy,” J. Biomed. Opt. 12(1), 014022 (2007).
[Crossref] [PubMed]

Svensson, T.

T. Svensson, S. Andersson-Engels, M. Einarsdóttír, and K. Svanberg, “In vivo optical characterization of human prostate tissue using near-infrared time-resolved spectroscopy,” J. Biomed. Opt. 12(1), 014022 (2007).
[Crossref] [PubMed]

Tadayoni, R.

Ch. Haritoglou, A. Gandorfer, M. Schaumberger, R. Tadayoni, A. Gandorfer, and A. Kampik, “Light-absorbing properties and osmolarity of indocyanine-green depending on concentration and solvent medium,” Invest. Ophthalmol. Vis. Sci. 44(6), 2722–2729 (2003).
[Crossref] [PubMed]

Tanikawa, Y.

H. Zhao, F. Gao, Y. Tanikawa, and Y. Yamada, “Time-resolved diffuse optical tomography and its application to in vitro and in vivo imaging,” J. Biomed. Opt. 12(6), 062107 (2007).
[Crossref] [PubMed]

Tarvainen, T.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Utilizing the radiative transfer equation in optical tomography,” PIERS Proceedings 4, 730–735 (2008).

Tomasevic, D. I.

D. I. Tomasevic and E. W. Larsen, “The simplified P2 approximation,” Nucl. Sci. Eng. 122, 309–325 (1996).

Turnbull, F. W.

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of back-scattering of light by blood,” J. Phys. D Appl. Phys. 11(10), 1463–1479 (1978).
[Crossref]

Vauhkonen, M.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Utilizing the radiative transfer equation in optical tomography,” PIERS Proceedings 4, 730–735 (2008).

Veitch, A. R.

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of back-scattering of light by blood,” J. Phys. D Appl. Phys. 11(10), 1463–1479 (1978).
[Crossref]

Vishwanath, K.

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol. 54(8), 2493–2509 (2009).
[Crossref] [PubMed]

Wang, L. V.

L. V. Wang and S. L. Jacques, “Sources of error in calculation of optical diffuse reflectance from turbid media using diffusion theory,” Comput. Meth. Prog. Bio. 61(3), 163–170 (2000).
[Crossref]

Webb, K. J.

Weissleder, R.

R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nat. Med. 9(1), 123–128 (2003).
[Crossref] [PubMed]

R. Weissleder and U. Mahmood, “Molecular imaging,” Radiology 219(2), 316–333 (2001).
[PubMed]

Welch, A.

W. Cheong, S. Prahl, and A. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26(12), 2166–2185 (1990).
[Crossref]

White, R.

X. Li, B. Beauvoit, R. White, Sh. Nioka, B. Chance, and A. G. Yodh, “Tumor localization using fluorescence of indocyanine green (ICG) in rat models,” Proc. SPIE 2389, 789–797 (1995).
[Crossref]

Wunder, A.

J. Klohs, J. Steinbrink, R. Bourayou, S. Mueller, R. Cordell, K. Licha, M. Schirner, U. Dirnagl, U. Lindauer, and A. Wunder, “Near-infrared fluorescence imaging with fluorescently labeled albumin: a novel method for non-invasive optical imaging of blood-brain barrier impairment after focal cerebral ischemia in mice,” J. Neurosci. Methods 180(1), 126–132 (2009).
[Crossref] [PubMed]

Yamada, Y.

H. Zhao, F. Gao, Y. Tanikawa, and Y. Yamada, “Time-resolved diffuse optical tomography and its application to in vitro and in vivo imaging,” J. Biomed. Opt. 12(6), 062107 (2007).
[Crossref] [PubMed]

F. Gao, H. Zhao, and Y. Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt. 41(4), 778–791 (2002).
[Crossref] [PubMed]

Yao, H.

J. Rao, A. Dragulescu-Andrasi, and H. Yao, “Fluorescence imaging in vivo: recent advances,” Curr. Opin. Biotechnol. 18(1), 17–25 (2007).
[Crossref] [PubMed]

Yasuda, S.

M. Frank, A. Klar, E. Larsen, and S. Yasuda, “Time-dependent simplified PN approximation to the equations of radiative transfer,” J. Comput. Phys. 226(2), 2289–2305 (2007).
[Crossref]

M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, “Approximate models of radiative transfer,” Bull. Inst. Math. Acad. Sin. 2, 409–432 (2007).

Ye, Y.

R. E. Nothdurft, S. V. Patwardhan, W. Akers, Y. Ye, S. Achilefu, and J. P. Culver, “In vivo fluorescence lifetime tomography,” J. Biomed. Opt. 14(2), 024004 (2009).
[Crossref] [PubMed]

Yodh, A. G.

X. Intes, J. Ripoll, Y. Chen, Sh. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30(6), 1039–1047 (2003).
[Crossref] [PubMed]

X. Li, B. Beauvoit, R. White, Sh. Nioka, B. Chance, and A. G. Yodh, “Tumor localization using fluorescence of indocyanine green (ICG) in rat models,” Proc. SPIE 2389, 789–797 (1995).
[Crossref]

Zhang, Q.

Zhao, H.

H. Zhao, F. Gao, Y. Tanikawa, and Y. Yamada, “Time-resolved diffuse optical tomography and its application to in vitro and in vivo imaging,” J. Biomed. Opt. 12(6), 062107 (2007).
[Crossref] [PubMed]

F. Gao, H. Zhao, and Y. Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt. 41(4), 778–791 (2002).
[Crossref] [PubMed]

Zijlstra, W. G.

M. L. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40(4), 575–583 (1976).
[PubMed]

Ann. N. Y. Acad. Sci. (1)

G. Paumgartner, P. Probst, R. Kraines, and C. M. Leevy, “Kinetics of indocyanine green removal from the blood,” Ann. N. Y. Acad. Sci. 170(1 The Hepatic C), 134–147 (1970).
[Crossref]

Annu. Rev. Biomed. Eng. (1)

V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8(1), 1–33 (2006).
[Crossref] [PubMed]

Appl. Opt. (5)

Bull. Inst. Math. Acad. Sin. (1)

M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, “Approximate models of radiative transfer,” Bull. Inst. Math. Acad. Sin. 2, 409–432 (2007).

Comput. Meth. Prog. Bio. (1)

L. V. Wang and S. L. Jacques, “Sources of error in calculation of optical diffuse reflectance from turbid media using diffusion theory,” Comput. Meth. Prog. Bio. 61(3), 163–170 (2000).
[Crossref]

Curr. Opin. Biotechnol. (2)

J. Rao, A. Dragulescu-Andrasi, and H. Yao, “Fluorescence imaging in vivo: recent advances,” Curr. Opin. Biotechnol. 18(1), 17–25 (2007).
[Crossref] [PubMed]

A. H. Hielscher, “Optical tomography imaging of small animals,” Curr. Opin. Biotechnol. 16(1), 79–88 (2005).
[Crossref]

IEEE J. Quantum Electron. (1)

W. Cheong, S. Prahl, and A. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26(12), 2166–2185 (1990).
[Crossref]

Inverse Probl. (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999).
[Crossref]

Invest. Ophthalmol. Vis. Sci. (1)

Ch. Haritoglou, A. Gandorfer, M. Schaumberger, R. Tadayoni, A. Gandorfer, and A. Kampik, “Light-absorbing properties and osmolarity of indocyanine-green depending on concentration and solvent medium,” Invest. Ophthalmol. Vis. Sci. 44(6), 2722–2729 (2003).
[Crossref] [PubMed]

J. Appl. Physiol. (1)

M. L. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40(4), 575–583 (1976).
[PubMed]

J. Biomed. Opt. (3)

R. E. Nothdurft, S. V. Patwardhan, W. Akers, Y. Ye, S. Achilefu, and J. P. Culver, “In vivo fluorescence lifetime tomography,” J. Biomed. Opt. 14(2), 024004 (2009).
[Crossref] [PubMed]

H. Zhao, F. Gao, Y. Tanikawa, and Y. Yamada, “Time-resolved diffuse optical tomography and its application to in vitro and in vivo imaging,” J. Biomed. Opt. 12(6), 062107 (2007).
[Crossref] [PubMed]

T. Svensson, S. Andersson-Engels, M. Einarsdóttír, and K. Svanberg, “In vivo optical characterization of human prostate tissue using near-infrared time-resolved spectroscopy,” J. Biomed. Opt. 12(1), 014022 (2007).
[Crossref] [PubMed]

J. Comput. Phys. (2)

A. Klose and E. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441–470 (2006).
[Crossref]

M. Frank, A. Klar, E. Larsen, and S. Yasuda, “Time-dependent simplified PN approximation to the equations of radiative transfer,” J. Comput. Phys. 226(2), 2289–2305 (2007).
[Crossref]

J. Electron. Imaging (1)

G. Abdoulaev and A. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imaging 12(4), 594–601 (2003).
[Crossref]

J. Math. Imaging Vis. (1)

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imaging Vis. 3(3), 263–283 (1993).
[Crossref]

J. Neurosci. Methods (1)

J. Klohs, J. Steinbrink, R. Bourayou, S. Mueller, R. Cordell, K. Licha, M. Schirner, U. Dirnagl, U. Lindauer, and A. Wunder, “Near-infrared fluorescence imaging with fluorescently labeled albumin: a novel method for non-invasive optical imaging of blood-brain barrier impairment after focal cerebral ischemia in mice,” J. Neurosci. Methods 180(1), 126–132 (2009).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (1)

J. Phys. D Appl. Phys. (1)

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of back-scattering of light by blood,” J. Phys. D Appl. Phys. 11(10), 1463–1479 (1978).
[Crossref]

Med. Phys. (5)

X. Intes, J. Ripoll, Y. Chen, Sh. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30(6), 1039–1047 (2003).
[Crossref] [PubMed]

V. Ntziachristos and B. Chance, “Accuracy limits in the determination of absolute optical properties using time-resolved NIR spectroscopy,” Med. Phys. 28(6), 1115–1124 (2001).
[Crossref] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[Crossref] [PubMed]

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(11), 1779–1792 (1995).
[Crossref] [PubMed]

K. D. Paulsen and H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22(6), 691–701 (1995).
[Crossref] [PubMed]

Nat. Med. (1)

R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nat. Med. 9(1), 123–128 (2003).
[Crossref] [PubMed]

Nat. Photonics (1)

E. M. C. Hillman and A. Moore, “All-optical anatomical co-registration for molecular imaging of small animals using dynamic contrast,” Nat. Photonics 1(9), 526–530 (2007).
[Crossref] [PubMed]

Nucl. Sci. Eng. (3)

E. W. Larsen, J. E. Morel, and J. M. McGhee, “Asymptotic derivation of the multigroup P1 and simplified PN equations with anisotropic scattering,” Nucl. Sci. Eng. 123, 328 (1996).

D. I. Tomasevic and E. W. Larsen, “The simplified P2 approximation,” Nucl. Sci. Eng. 122, 309–325 (1996).

P. S. Brantley and E. W. Larsen, “The simplified P3 approximation,” Nucl. Sci. Eng. 134, 121 (2000).

Opt. Commun. (1)

L. Martí-López, J. Bouza-Domínguez, and J. C. Hebden, “Interpretation of the failure of the time-independent diffusion equation near a point source,” Opt. Commun. 242(1-3), 23–43 (2004).
[Crossref]

Photochem. Photobiol. (1)

S. Mordon, J. M. Devoisselle, and V. Maunoury, “In vivo pH measurement and imaging of tumor tissue using a pH-sensitive fluorescent probe (5,6-carboxyfluorescein): instrumental and experimental studies,” Photochem. Photobiol. 60(3), 274–279 (1994).
[Crossref] [PubMed]

Phys. Med. Biol. (5)

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[Crossref] [PubMed]

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[Crossref] [PubMed]

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol. 54(8), 2493–2509 (2009).
[Crossref] [PubMed]

D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantitative fluorescence imaging of point-like sources in small animals,” Phys. Med. Biol. 53(20), 5797–5814 (2008).
[Crossref] [PubMed]

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52(3), 577–587 (2007).
[Crossref] [PubMed]

PIERS Proceedings (1)

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Utilizing the radiative transfer equation in optical tomography,” PIERS Proceedings 4, 730–735 (2008).

Proc. SPIE (5)

M. Comtois and Y. Bérubé-Lauzière, “Optical surface metrology in 3D for small animal non-contact diffuse optical tomography,” Proc. SPIE 6796, 679605, 679605-9 (2007).
[Crossref]

J. Pichette, E. Lapointe, and Y. Bérubé-Lauzière, “Time-domain 3D localization of fluorescent inclusions in a thick scattering medium,” Proc. SPIE 7099, 709907, 709907-12 (2008).
[Crossref]

J. Pichette, E. Lapointe, and Y. Bérubé-Lauzière, “Three-dimensional localization of discrete fluorescent inclusions from multiple tomographic projections in the time-domain,” Proc. SPIE 7174, 71741A, 71741A-10 (2009).
[Crossref]

Y. Bérubé-Lauzière, V. Issa, and J. B. Domínguez, “Simplified spherical harmonics approximation of the time-dependent equation of radiative transfer for the forward problem in time-domain diffuse optical tomography,” Proc. SPIE 7174, 717403, 717403-11 (2009).
[Crossref]

X. Li, B. Beauvoit, R. White, Sh. Nioka, B. Chance, and A. G. Yodh, “Tumor localization using fluorescence of indocyanine green (ICG) in rat models,” Proc. SPIE 2389, 789–797 (1995).
[Crossref]

Radiat. Phys. Chem. (1)

P. Kotiluoto, J. Pyyry, and H. Helminen, “Multitrans SP3 code in coupled photon-electron transport problems,” Radiat. Phys. Chem. 76(1), 9–14 (2007).
[Crossref]

Radiology (1)

R. Weissleder and U. Mahmood, “Molecular imaging,” Radiology 219(2), 316–333 (2001).
[PubMed]

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A. D. Klose, “Radiative transfer of luminescence light in biological tissue,” in Light Scattering Reviews 4 Single Light Scattering and Radiative Transfer, A. A. Kokhanovsky, ed., (Springer, 2009), pp 293–345.

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

L. Wang and H. Wu, in Biomedical Optics: Principles and Imaging (Wiley-Interscience, 2007).

T. Tarvainen, Computational Methods for Light Transport in Optical Tomography, PhD Thesis, University of Kuopio, (2006).

http://www.art.ca/en/preclinical/optix-mx3/acquisition-system.php , Web page consulted on 15 Dec. 2010.

S. Prahl, “Optical absorption of Indocyanine Green,” http://omlc.ogi.edu/spectra/icg/index.html .

R. Klette, K. Schlüns, and A. Koschan, Computer Vision – Three-Dimensional Data from Images, (Springer, 1998).

P. Gallant, A. Belenkov, G. Ma, F. Lesage, Y. Wang, D. Hall, and L. McIntosh, “A quantitative time-domain optical imager for small animals in vivo fluorescence studies” in Biomedical Topical Meeting, OSA Technical Digest (Optical Society of America, 2004), paper WD2.

E. Hillman, Experimental and theoretical investigations of near infrared tomographic imaging methods and clinical applications, Ph.D. Thesis, University College London, (2002).

D. Hyde, Improving Forward Matrix Generation and Utilization for Time Domain Diffuse Optical Tomography, MSc Thesis, Dept. Elect. Comp. Eng., Northeastern University, (2004)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2003).

J. Mobley and T. Vo-Dinh, “Optical properties of tissue,” in Biomedical Photonics Handbook, T. Vo-Dinh ed. (CRC Press, 2003), pp. 20–95.

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Figures (9)

Fig. 1
Fig. 1

Volume of biological tissue. Black dots represent a distribution of discrete light sources illuminating the volume.

Fig. 2
Fig. 2

Mesh of the cylinder.

Fig. 3
Fig. 3

Excitation and fluorescence fluence spatial distributions for a fluorescent point inclusion (upper and lower rows) in the source plane for 0.15, 0.3 and 1.5 ns (left to right).

Fig. 4
Fig. 4

Mesh of the cylinder cut at half of its height. The spherical absorbing inclusion is shown in grey.

Fig. 5
Fig. 5

The excitation and the fluorescent fluence profiles for a fluorescent spherical inclusion (upper and lower rows) at the source plane for 0.15, 0.3 and 1.5 ns (left to right).

Fig. 6
Fig. 6

The excitation and the fluorescent fluence profiles for a fluorescent distributed inclusion (upper and lower rows) at the source plane for 0.15, 0.3 and 1.5 ns (left to right).

Fig. 7
Fig. 7

Fluorescent measurement temporal profiles at the nearest detector to the center of the fluorescent inclusions; (a) point inclusion, (b) spherical inclusion, (c) Gaussian distributed inclusion.

Fig. 8
Fig. 8

Representation of the torso of the small animal.

Fig. 9
Fig. 9

Excitation and fluorescence fluence profiles (upper and lower rows respectively) in a plane at a height of 1 cm (see Fig. 6) for 0.3, 0.6 and 1.0 ns (left to right).

Equations (32)

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

[ η x c t + s ^ + μ t x ( r ) + μ a x m ( r ) ] L x ( r , s ^ , t ) = μ s x ( r ) 4 π p x ( r , s ^ , s ^ ' ) L x ( r , s ^ ' , t ) d Ω ' ,
L x ( r , s ^ , t ) = B T x ( r , s ^ ) δ ( t ) + R F x ( s ^ ' n ) L x ( r , s ^ ' , t ) ,   r V ,   s ^ ' = s ^ ' 2 ( s ^ n ^ ) ,   s ^ n ^ < 0 ,
J n x = s ^ n ^ > 0 [ 1 R F x ( s ^ n ^ ) ]    s ^ n ^ L x ( r , s ^ , t )  d Ω ,
η x c L r x ( r , s ^ , t ) t + s ^ L r x ( r , s ^ , t ) = [ μ t x ( r ) + μ a x m ( r ) ] L r x ( r , s ^ , t ) ,
L r x ( r , s ^ , t ) = B T x ( r , s ^ ) δ ( t ) ,   r V ,   s ^ n ^ < 0.
[ η x c t + s ^ + μ t x ( r ) + μ a x m ( r ) ] L d x ( r , s ^ , t ) = μ s x ( r ) 4 π p x ( r , s ^ , s ^ ' ) L d x ( r , s ^ ' , t ) d Ω ' + Q x ( r , t ) ,
L d x ( r , s ^ , t ) = R F x ( s ^ ' n ) L d x ( r , s ^ ' , t ) ,   r V ,   s ^ ' = s ^ ' 2 ( s ^ n ^ ) ,   s ^ n ^ < 0 ,
Q x ( r , t ) = μ s x ( r ) 4 π p x ( r , s ^ , s ^ ' ) L r x ( r , s ^ ' , t ) d Ω ' .
p x ( r , s ^ , s ^ ' ) = 1 ( g x ( r ) ) 2 4 π [ 1 + ( g x ( r ) ) 2 2 ( g x ( r ) ) 2 ( s ^ s ^ ' ) ] ,
[ C x + η x c t T ] Φ x ( r , t ) + D x Φ x ( r , t ) = Q x ( r , t ) ,
A x Φ x ( r , t ) + B x n ^ Φ x ( r , t ) = 0 ,   r V ,
Ψ x ( r , t ) = T Φ x ( r , t ) .
d i a g 0 ( C x ) = [ μ 0 x ,   ( 4 / 9 ) μ 0 x + ( 5 / 9 ) μ 2 x ,   ( 64 / 225 ) μ 0 x + ( 16 / 45 ) μ 2 x + ( 9 / 25 ) μ 4 x , ( 256 / 1225 ) μ 0 x + ( 64 / 245 ) μ 2 x + ( 324 / 1225 ) μ 4 x + ( 13 / 49 ) μ 6 x ] d i a g 1 ( C x ) = [ ( 2 / 3 ) μ 0 x ,   ( 16 / 45 ) μ 0 x ( 4 / 9 ) μ 2 x ,   ( 128 / 525 ) μ 0 x ( 32 / 105 ) μ 2 x ( 54 / 175 ) μ 4 x ] , d i a g 2 ( C x ) = [ ( 8 / 15 ) μ 0 x ,   ( 32 / 105 ) μ 0 x + ( 8 / 21 ) μ 2 x ] ,   d i a g 3 ( C x ) = [ ( 16 / 35 ) μ 0 x ] ,
T 1 = [ 1 2 0 0 0 3 4 0 0 0 5 6 0 0 0 7 ] .
( D x ) i , i = [ 1 ( 4 i 1 ) μ 2 i 1 ] ,   i = 1 l N .
Q x ( r , t ) = [ Q x ( r , t ) ,  - ( 2 / 3 ) Q x ( r , t ) ,   ( 8 / 15 ) Q x ( r , t ) ,  - ( 16 / 35 ) Q x ( r , t ) ] T ,
J n x = [ j 1 x j 2 x ( B x ) 1 A x ] ( Φ x ( r , t ) | r V ) ,
[ K ˜ x + M ˜ x + Π ˜ x + ( η c ) T ˜ t ] Φ ˜ x ( t ) = F ˜ x ( t ) .
K ˜ k x ( i , j ) = V 1 ( 4 k 1 ) μ 2 k 1 x u i ( r ) u j ( r ) dV ,   k = 1 l N ,   i , j = 1 d .
M ˜ k 1 , k 2 x ( i , j ) = V C x ( k 1 , k 2 ) u i ( r ) u j ( r ) dV ,   k 1 , k 2 = 1 l N ,   i , j = 1 d ,
Π ˜ k 1 , k 2 x ( i , j ) = V Θ x ( k 1 , k 2 ) ( 4 k 1 1 ) μ 2 k 1 1 x u i ( r ) u j ( r ) d σ ,   k 1 , k 2 = 1 l N ,   i , j = 1 d ,
T ˜ k 1 , k 2 ( i , j ) = V T ( k 1 , k 2 ) u i ( r ) u j ( r ) dV ,   k 1 , k 2 = 1 l N ,   i , j = 1 d ,
F ˜ k x ( t ) = V Q x ( k ) u i ( r ) dV ,   k = 1 l N ,   i = 1 d ,
[ ρ K ˜ x + ρ M ˜ x + ρ Π ˜ x + 1 Δ t ( η x c ) T ˜ ] Φ ˜ x , ( n + 1 ) + [ ( 1 ρ ) K ˜ x + ( 1 ρ ) M ˜ x + ( 1 ρ ) Π ˜ x 1 Δ t η x c T ˜ ] Φ ˜ x , ( n + 1 ) = ρ F ˜ x , ( n + 1 ) + ( 1 ρ ) F ˜ x , ( n ) ,
[ η m c t + s ^ + μ t m ( r ) + ( s ^ ) ] L m ( r , s ^ , t ) = μ s m ( r ) 4 π p ( r , s ^ , s ^ ' ) L d m ( r , s ^ ' , t ) d Ω ' + Q m ( r , t ) ,
L m ( r , s ^ , t ) = R F m ( s ^ ' n ^ ) L m ( r , s ^ ' , t ) ,   r V ,   s ^ ' = s ^ ' 2 ( s ^ n ^ ) ,   s ^ n ^ < 0 ,
Q m ( r , t ) = ς μ a x m ( r ) τ t ' = 0 t ' = t ϕ x ( r , t ' ) exp ( t ' t τ ) dt' ,
[ C m + η m c t T ] Φ m ( r , t ) + D x Φ m ( r , t ) = Q m ( r , t ) ,
A m Φ m ( r , t ) + B m n Φ m ( r , t ) = 0 ,   r V .
[ ρ K ˜ m + ρ M ˜ m + ρ Π ˜ m + 1 Δ t ( η m c ) T ˜ ] Φ ˜ m , ( n + 1 ) + [ ( 1 ρ ) K ˜ m + ( 1 ρ ) M ˜ m + ( 1 ρ ) Π ˜ m 1 Δ t η m c T ˜ ] Φ ˜ m , ( n ) = ρ F ˜ m , ( n + 1 ) + ( 1 ρ ) F ˜ m , ( n ) .
j 1 i = [ 1 / 4 + J 0 ,   ( 1 / 4 + J 0 ) ( 2 / 3 ) + ( 5 / 16 + J 2 ) ( 1 / 3 ) ,   ( 1 / 4 + J 0 ) ( 8 / 15 ) + ( 5 / 16 + J 2 ) ( 4 / 15 ) + ( 3 / 32 + J 4 ) ( 1 / 5 ) , ( 1 / 4 + J 0 ) ( 15 / 35 ) + ( 5 / 16 + J 2 ) ( 8 / 35 ) + ( 3 / 32 + J 4 ) ( 6 / 35 ) + ( 13 / 256 + J 6 ) ( 1 / 7 ) ] ,
j 2 i = [ ( 0.5 + J 1 3 μ a 1 i ) ,    ( J 3 7 μ a 3 i ) ,    ( J 5 11 μ a 5 i ) ,    ( J 7 15 μ a 7 i ) ]    ,

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