Abstract

We analyze a role of a localized inclusion as a probe for spatial distributions of migrating photons in turbid media. We present new experimental data and two-dimensional analysis of radiance detection of a localized absorptive inclusion formed by gold nanoparticles in Intralipid-1% when the target is translated along the line connecting the light source and detector. Data are analyzed using the novel analytical expression for the relative angular photon distribution function for radiance developed by extending the perturbation approach for fluence. Obtained photon maps allow predicting conditions for detectability of inclusions for which proximity to the detector is essential.

© 2013 Optical Society of America

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2013

S. Grabtchak, E. Tonkopi, and W. M. Whelan, “Optical detection of gold nanoparticles in a prostate-shaped porcine phantom,” J. Biomed. Opt.18(7), 077005 (2013).
[CrossRef] [PubMed]

F. Martelli, A. Pifferi, D. Contini, L. Spinelli, A. Torricelli, H. Wabnitz, R. Macdonald, A. Sassaroli, and G. Zaccanti, “Phantoms for diffuse optical imaging based on totally absorbing objects, part 1: Basic concepts,” J. Biomed. Opt.18(6), 066014 (2013).
[CrossRef] [PubMed]

2012

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt.17(6), 067007 (2012).
[CrossRef] [PubMed]

S. Grabtchak, T. J. Palmer, I. A. Vitkin, and W. M. Whelan, “Radiance detection of non-scattering inclusions in turbid media,” Biomed. Opt. Express3(11), 3001–3011 (2012).
[CrossRef] [PubMed]

S. Grabtchak and W. M. Whelan, “Separation of absorption and scattering properties of turbid media using relative spectrally resolved cw radiance measurements,” Biomed. Opt. Express3(10), 2371–2380 (2012).
[CrossRef] [PubMed]

E. C. Dreaden, A. M. Alkilany, X. Huang, C. J. Murphy, and M. A. El-Sayed, “The golden age: gold nanoparticles for biomedicine,” Chem. Soc. Rev.41(7), 2740–2779 (2012).
[CrossRef] [PubMed]

C. J. Wen, L. W. Zhang, S. A. Al-Suwayeh, T. C. Yen, and J. Y. Fang, “Theranostic liposomes loaded with quantum dots and apomorphine for brain targeting and bioimaging,” Int. J. Nanomedicine7, 1599–1611 (2012).
[PubMed]

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics6(5), 283–292 (2012).
[CrossRef]

2011

M. G. Giacomelli and A. Wax, “Imaging beyond the ballistic limit in coherence imaging using multiply scattered light,” Opt. Express19(5), 4268–4279 (2011).
[CrossRef] [PubMed]

A. Laidevant, L. Hervé, M. Debourdeau, J. Boutet, N. Grenier, and J. M. Dinten, “Fluorescence time-resolved imaging system embedded in an ultrasound prostate probe,” Biomed. Opt. Express2(1), 194–206 (2011).
[CrossRef] [PubMed]

E. Hutter and D. Maysinger, “Gold Nanoparticles and Quantum Dots for Bioimaging,” Microsc. Res. Tech.74(7), 592–604 (2011).
[CrossRef] [PubMed]

A. Pellicer and M. C. Bravo, “Near-infrared spectroscopy: A methodology-focused review,” Semin. Fetal Neonatal Med.16(1), 42–49 (2011).
[CrossRef] [PubMed]

B. E. Schaafsma, J. S. D. Mieog, M. Hutteman, J. R. van der Vorst, P. J. Kuppen, C. W. Löwik, J. V. Frangioni, C. J. van de Velde, and A. L. Vahrmeijer, “The clinical use of indocyanine green as a near-infrared fluorescent contrast agent for image-guided oncologic surgery,” J. Surg. Oncol.104(3), 323–332 (2011).
[CrossRef] [PubMed]

A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.83(3), 036605 (2011).
[CrossRef] [PubMed]

S. R. Arridge, “Methods in diffuse optical imaging,” Philos Trans A Math Phys Eng Sci369(1955), 4558–4576 (2011).
[CrossRef] [PubMed]

S. Grabtchak, T. J. Palmer, and W. M. Whelan, “Detection of localized inclusions of gold nanoparticles in Intralipid-1% by point-radiance spectroscopy,” J. Biomed. Opt.16(7), 077003 (2011).
[CrossRef] [PubMed]

2010

M. Kacprzak, A. Liebert, P. Sawosz, N. Zolek, D. Milej, and R. Maniewski, “Time-resolved imaging of fluorescent inclusions in optically turbid medium - phantom study,” Opto-Electron. Rev.18(1), 37–47 (2010).
[CrossRef]

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys.73(7), 076701 (2010).
[CrossRef]

2009

A. Sassaroli, F. Martelli, and S. Fantini, “Higher-order perturbation theory for the diffusion equation in heterogeneous media: application to layered and slab geometries,” Appl. Opt.48(10), D62–D73 (2009).
[CrossRef] [PubMed]

L. C. L. Chin, B. Lloyd, W. M. Whelan, and I. A. Vitkin, “Interstitial point radiance spectroscopy of turbid media,” J. Appl. Phys.105(10), 102025 (2009).
[CrossRef]

K. K. H. Wang and T. C. Zhu, “Reconstruction of in-vivo optical properties for human prostate using interstitial diffuse optical tomography,” Opt. Express17(14), 11665–11672 (2009).
[CrossRef] [PubMed]

E. Boisselier and D. Astruc, “Gold nanoparticles in nanomedicine: preparations, imaging, diagnostics, therapies and toxicity,” Chem. Soc. Rev.38(6), 1759–1782 (2009).
[CrossRef] [PubMed]

2008

R. Wilson, “The use of gold nanoparticles in diagnostics and detection,” Chem. Soc. Rev.37(9), 2028–2045 (2008).
[CrossRef] [PubMed]

K. Bensalah, A. Tuncel, D. Peshwani, I. Zeltser, H. Liu, and J. Cadeddu, “Optical reflectance spectroscopy to differentiate renal tumor from normal parenchyma,” J. Urol.179(5), 2010–2013 (2008).
[CrossRef] [PubMed]

2007

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt.12(6), 064027 (2007).
[CrossRef] [PubMed]

S. K. Ghosh and T. Pal, “Interparticle coupling effect on the surface plasmon resonance of gold nanoparticles: From theory to applications,” Chem. Rev.107(11), 4797–4862 (2007).
[CrossRef] [PubMed]

2006

2005

X. Montet, V. Ntziachristos, J. Grimm, and R. Weissleder, “Tomographic fluorescence mapping of tumor targets,” Cancer Res.65(14), 6330–6336 (2005).
[CrossRef] [PubMed]

T. Kitai, T. Inomoto, M. Miwa, and T. Shikayama, “Fluorescence navigation with indocyanine green for detecting sentinel lymph nodes in breast cancer,” Breast Cancer12(3), 211–215 (2005).
[CrossRef] [PubMed]

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt.10(3), 034018 (2005).
[CrossRef] [PubMed]

M. Xu, M. Alrubaiee, S. K. Gayen, and R. R. Alfano, “Three-dimensional localization and optical imaging of objects in turbid media with independent component analysis,” Appl. Opt.44(10), 1889–1897 (2005).
[CrossRef] [PubMed]

M. E. Zevallos, S. K. Gayen, M. Alrubaiee, and R. R. Alfano, “Time-gated backscattered ballistic light imaging of objects in turbid water,” Appl. Phys. Lett.86(1), 011115 (2005).
[CrossRef]

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol.50(10), 2291–2311 (2005).
[CrossRef] [PubMed]

T. C. Zhu, J. C. Finlay, and S. M. Hahn, “Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in-vivo in human prostate during motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B79(3), 231–241 (2005).
[CrossRef] [PubMed]

2004

X. H. Gao, Y. Y. Cui, R. M. Levenson, L. W. K. Chung, and S. M. Nie, “In vivo cancer targeting and imaging with semiconductor quantum dots,” Nat. Biotechnol.22(8), 969–976 (2004).
[CrossRef] [PubMed]

2003

2001

2000

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A.97(6), 2767–2772 (2000).
[CrossRef] [PubMed]

M. Ono, Y. Kashio, M. Schweiger, H. Dehghani, S. R. Arridge, M. Firbank, and E. Okada, “Topographic distribution of photon measurement density functions on the brain surface by hybrid radiosity-diffusion method,” Opt. Rev.7(5), 426–431 (2000).
[CrossRef]

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt.Soc. Am. A.17, 1659–1670 (2000).

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol.45(5), 1359–1373 (2000).
[CrossRef] [PubMed]

F. Bevilacqua, A. J. Berger, A. E. Cerussi, D. Jakubowski, and B. J. Tromberg, “Broadband absorption spectroscopy in turbid media by combined frequency-domain and steady-state methods,” Appl. Opt.39(34), 6498–6507 (2000).
[CrossRef] [PubMed]

Y. Yamada, “Fundamental studies of photon migration in biological tissues and their application to optical tomography,” Opt. Rev.7(5), 366–374 (2000).
[CrossRef]

1999

1998

S. A. Walker, D. A. Boas, and E. Gratton, “Photon density waves scattered from cylindrical inhomogeneities: theory and experiments,” Appl. Opt.37(10), 1935–1944 (1998).
[CrossRef] [PubMed]

D. Dickey, O. Barajas, K. Brown, J. Tulip, and R. B. Moore, “Radiance modelling using the P3 approximation,” Phys. Med. Biol.43(12), 3559–3570 (1998).
[CrossRef] [PubMed]

1997

1996

A. M. Ballangrud, P. J. Wilson, K. Brown, G. G. Miller, R. B. Moore, M. S. McPhee, and J. Tulip, “Anisotropy of radiance in tissue phantoms and Dunning R3327 rat tumors: Radiance measurements with flat cleaved fiber probes,” Lasers Surg. Med.19(4), 471–479 (1996).
[CrossRef] [PubMed]

1995

S. C. Feng, F. A. Zeng, and B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt.34(19), 3826–3837 (1995).
[CrossRef] [PubMed]

S. Fantini, M. A. Franceschinifantini, J. S. Maier, S. A. Walker, B. Barbieri, and E. Gratton, “Frequency-domain mutlichannel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng.34(1), 32–42 (1995).
[CrossRef]

1993

B. Chance, K. Kang, L. He, J. Weng, and E. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. U.S.A.90(8), 3423–3427 (1993).
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B. J. Tromberg, L. O. Svaasand, T. T. Tsay, and R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt.32(4), 607–616 (1993).
[CrossRef] [PubMed]

1991

H. J. van Staveren, C. J. M. Moes, J. van Marie, S. A. Prahl, and M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm,” Appl. Opt.30(31), 4507–4514 (1991).
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M. S. Patterson, B. C. Wilson, and D. R. Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their application,” Lasers Med. Sci.6(2), 155–168 (1991).
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M. S. Patterson, B. C. Wilson, and D. R. Wyman, “The propagation of optical radiation in tissue II. Optical properties of tissues and resulting fluence distributions,” Lasers Med. Sci.6(4), 379–390 (1991).
[CrossRef]

’t Hooft, G. W.

Alfano, R. R.

M. Xu, M. Alrubaiee, S. K. Gayen, and R. R. Alfano, “Three-dimensional localization and optical imaging of objects in turbid media with independent component analysis,” Appl. Opt.44(10), 1889–1897 (2005).
[CrossRef] [PubMed]

M. E. Zevallos, S. K. Gayen, M. Alrubaiee, and R. R. Alfano, “Time-gated backscattered ballistic light imaging of objects in turbid water,” Appl. Phys. Lett.86(1), 011115 (2005).
[CrossRef]

Alianelli, L.

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol.45(5), 1359–1373 (2000).
[CrossRef] [PubMed]

Alkilany, A. M.

E. C. Dreaden, A. M. Alkilany, X. Huang, C. J. Murphy, and M. A. El-Sayed, “The golden age: gold nanoparticles for biomedicine,” Chem. Soc. Rev.41(7), 2740–2779 (2012).
[CrossRef] [PubMed]

Alrubaiee, M.

M. Xu, M. Alrubaiee, S. K. Gayen, and R. R. Alfano, “Three-dimensional localization and optical imaging of objects in turbid media with independent component analysis,” Appl. Opt.44(10), 1889–1897 (2005).
[CrossRef] [PubMed]

M. E. Zevallos, S. K. Gayen, M. Alrubaiee, and R. R. Alfano, “Time-gated backscattered ballistic light imaging of objects in turbid water,” Appl. Phys. Lett.86(1), 011115 (2005).
[CrossRef]

Al-Suwayeh, S. A.

C. J. Wen, L. W. Zhang, S. A. Al-Suwayeh, T. C. Yen, and J. Y. Fang, “Theranostic liposomes loaded with quantum dots and apomorphine for brain targeting and bioimaging,” Int. J. Nanomedicine7, 1599–1611 (2012).
[PubMed]

Arridge, S. R.

S. R. Arridge, “Methods in diffuse optical imaging,” Philos Trans A Math Phys Eng Sci369(1955), 4558–4576 (2011).
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M. Ono, Y. Kashio, M. Schweiger, H. Dehghani, S. R. Arridge, M. Firbank, and E. Okada, “Topographic distribution of photon measurement density functions on the brain surface by hybrid radiosity-diffusion method,” Opt. Rev.7(5), 426–431 (2000).
[CrossRef]

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt.Soc. Am. A.17, 1659–1670 (2000).

Astruc, D.

E. Boisselier and D. Astruc, “Gold nanoparticles in nanomedicine: preparations, imaging, diagnostics, therapies and toxicity,” Chem. Soc. Rev.38(6), 1759–1782 (2009).
[CrossRef] [PubMed]

Baker, W. B.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys.73(7), 076701 (2010).
[CrossRef]

Ballangrud, A. M.

O. Barajas, A. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol.42(9), 1675–1687 (1997).
[CrossRef] [PubMed]

A. M. Ballangrud, P. J. Wilson, K. Brown, G. G. Miller, R. B. Moore, M. S. McPhee, and J. Tulip, “Anisotropy of radiance in tissue phantoms and Dunning R3327 rat tumors: Radiance measurements with flat cleaved fiber probes,” Lasers Surg. Med.19(4), 471–479 (1996).
[CrossRef] [PubMed]

Barajas, O.

D. Dickey, O. Barajas, K. Brown, J. Tulip, and R. B. Moore, “Radiance modelling using the P3 approximation,” Phys. Med. Biol.43(12), 3559–3570 (1998).
[CrossRef] [PubMed]

O. Barajas, A. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol.42(9), 1675–1687 (1997).
[CrossRef] [PubMed]

Barbieri, B.

S. Fantini, M. A. Franceschinifantini, J. S. Maier, S. A. Walker, B. Barbieri, and E. Gratton, “Frequency-domain mutlichannel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng.34(1), 32–42 (1995).
[CrossRef]

Bargo, P. R.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt.10(3), 034018 (2005).
[CrossRef] [PubMed]

Bassani, M.

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol.45(5), 1359–1373 (2000).
[CrossRef] [PubMed]

Bensalah, K.

K. Bensalah, A. Tuncel, D. Peshwani, I. Zeltser, H. Liu, and J. Cadeddu, “Optical reflectance spectroscopy to differentiate renal tumor from normal parenchyma,” J. Urol.179(5), 2010–2013 (2008).
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Berger, A. J.

Bevilacqua, F.

Blair, G.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt.10(3), 034018 (2005).
[CrossRef] [PubMed]

Boas, D. A.

Boisselier, E.

E. Boisselier and D. Astruc, “Gold nanoparticles in nanomedicine: preparations, imaging, diagnostics, therapies and toxicity,” Chem. Soc. Rev.38(6), 1759–1782 (2009).
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Boutet, J.

Bravo, M. C.

A. Pellicer and M. C. Bravo, “Near-infrared spectroscopy: A methodology-focused review,” Semin. Fetal Neonatal Med.16(1), 42–49 (2011).
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Brooksby, B. A.

B. W. Pogue, S. C. Davis, X. M. Song, B. A. Brooksby, H. Dehghani, and K. D. Paulsen, “Image analysis methods for diffuse optical tomography,” J. Biomed. Opt.11(3), 033001 (2006).
[CrossRef] [PubMed]

Brown, K.

D. Dickey, O. Barajas, K. Brown, J. Tulip, and R. B. Moore, “Radiance modelling using the P3 approximation,” Phys. Med. Biol.43(12), 3559–3570 (1998).
[CrossRef] [PubMed]

A. M. Ballangrud, P. J. Wilson, K. Brown, G. G. Miller, R. B. Moore, M. S. McPhee, and J. Tulip, “Anisotropy of radiance in tissue phantoms and Dunning R3327 rat tumors: Radiance measurements with flat cleaved fiber probes,” Lasers Surg. Med.19(4), 471–479 (1996).
[CrossRef] [PubMed]

Cadeddu, J.

K. Bensalah, A. Tuncel, D. Peshwani, I. Zeltser, H. Liu, and J. Cadeddu, “Optical reflectance spectroscopy to differentiate renal tumor from normal parenchyma,” J. Urol.179(5), 2010–2013 (2008).
[CrossRef] [PubMed]

Carraresi, S.

Cerussi, A. E.

Chance, B.

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A.97(6), 2767–2772 (2000).
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D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: A signal-to-noise analysis,” Appl. Opt.36(1), 75–92 (1997).
[CrossRef] [PubMed]

S. C. Feng, F. A. Zeng, and B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt.34(19), 3826–3837 (1995).
[CrossRef] [PubMed]

B. Chance, K. Kang, L. He, J. Weng, and E. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. U.S.A.90(8), 3423–3427 (1993).
[CrossRef] [PubMed]

Chin, L. C. L.

L. C. L. Chin, B. Lloyd, W. M. Whelan, and I. A. Vitkin, “Interstitial point radiance spectroscopy of turbid media,” J. Appl. Phys.105(10), 102025 (2009).
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L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt.12(6), 064027 (2007).
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L. C. L. Chin, W. M. Whelan, and I. A. Vitkin, “Information content of point radiance measurements in turbid media: implications for interstitial optical property quantification,” Appl. Opt.45(9), 2101–2114 (2006).
[CrossRef] [PubMed]

Choe, R.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys.73(7), 076701 (2010).
[CrossRef]

Chung, L. W. K.

X. H. Gao, Y. Y. Cui, R. M. Levenson, L. W. K. Chung, and S. M. Nie, “In vivo cancer targeting and imaging with semiconductor quantum dots,” Nat. Biotechnol.22(8), 969–976 (2004).
[CrossRef] [PubMed]

Colak, S. B.

Contini, D.

F. Martelli, A. Pifferi, D. Contini, L. Spinelli, A. Torricelli, H. Wabnitz, R. Macdonald, A. Sassaroli, and G. Zaccanti, “Phantoms for diffuse optical imaging based on totally absorbing objects, part 1: Basic concepts,” J. Biomed. Opt.18(6), 066014 (2013).
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Cui, Y. Y.

X. H. Gao, Y. Y. Cui, R. M. Levenson, L. W. K. Chung, and S. M. Nie, “In vivo cancer targeting and imaging with semiconductor quantum dots,” Nat. Biotechnol.22(8), 969–976 (2004).
[CrossRef] [PubMed]

Davis, S. C.

B. W. Pogue, S. C. Davis, X. M. Song, B. A. Brooksby, H. Dehghani, and K. D. Paulsen, “Image analysis methods for diffuse optical tomography,” J. Biomed. Opt.11(3), 033001 (2006).
[CrossRef] [PubMed]

Debourdeau, M.

Dehghani, H.

D. Piao, H. Xie, W. L. Zhang, J. S. Krasinski, G. L. Zhang, H. Dehghani, and B. W. Pogue, “Endoscopic, rapid near-infrared optical tomography,” Opt. Lett.31(19), 2876–2878 (2006).
[CrossRef] [PubMed]

B. W. Pogue, S. C. Davis, X. M. Song, B. A. Brooksby, H. Dehghani, and K. D. Paulsen, “Image analysis methods for diffuse optical tomography,” J. Biomed. Opt.11(3), 033001 (2006).
[CrossRef] [PubMed]

M. Ono, Y. Kashio, M. Schweiger, H. Dehghani, S. R. Arridge, M. Firbank, and E. Okada, “Topographic distribution of photon measurement density functions on the brain surface by hybrid radiosity-diffusion method,” Opt. Rev.7(5), 426–431 (2000).
[CrossRef]

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt.Soc. Am. A.17, 1659–1670 (2000).

Delpy, D. T.

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt.Soc. Am. A.17, 1659–1670 (2000).

Dickey, D.

D. Dickey, O. Barajas, K. Brown, J. Tulip, and R. B. Moore, “Radiance modelling using the P3 approximation,” Phys. Med. Biol.43(12), 3559–3570 (1998).
[CrossRef] [PubMed]

Dickey, D. J.

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol.46(9), 2359–2370 (2001).
[CrossRef] [PubMed]

Dimofte, A.

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol.50(10), 2291–2311 (2005).
[CrossRef] [PubMed]

Dinten, J. M.

Dreaden, E. C.

E. C. Dreaden, A. M. Alkilany, X. Huang, C. J. Murphy, and M. A. El-Sayed, “The golden age: gold nanoparticles for biomedicine,” Chem. Soc. Rev.41(7), 2740–2779 (2012).
[CrossRef] [PubMed]

Durduran, T.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys.73(7), 076701 (2010).
[CrossRef]

El-Sayed, M. A.

E. C. Dreaden, A. M. Alkilany, X. Huang, C. J. Murphy, and M. A. El-Sayed, “The golden age: gold nanoparticles for biomedicine,” Chem. Soc. Rev.41(7), 2740–2779 (2012).
[CrossRef] [PubMed]

Fang, J. Y.

C. J. Wen, L. W. Zhang, S. A. Al-Suwayeh, T. C. Yen, and J. Y. Fang, “Theranostic liposomes loaded with quantum dots and apomorphine for brain targeting and bioimaging,” Int. J. Nanomedicine7, 1599–1611 (2012).
[PubMed]

Fantini, S.

A. Sassaroli, F. Martelli, and S. Fantini, “Higher-order perturbation theory for the diffusion equation in heterogeneous media: application to layered and slab geometries,” Appl. Opt.48(10), D62–D73 (2009).
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S. Fantini, M. A. Franceschinifantini, J. S. Maier, S. A. Walker, B. Barbieri, and E. Gratton, “Frequency-domain mutlichannel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng.34(1), 32–42 (1995).
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Feng, S. C.

Fink, M.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics6(5), 283–292 (2012).
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Finlay, J. C.

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol.50(10), 2291–2311 (2005).
[CrossRef] [PubMed]

T. C. Zhu, J. C. Finlay, and S. M. Hahn, “Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in-vivo in human prostate during motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B79(3), 231–241 (2005).
[CrossRef] [PubMed]

Firbank, M.

M. Ono, Y. Kashio, M. Schweiger, H. Dehghani, S. R. Arridge, M. Firbank, and E. Okada, “Topographic distribution of photon measurement density functions on the brain surface by hybrid radiosity-diffusion method,” Opt. Rev.7(5), 426–431 (2000).
[CrossRef]

Foschum, F.

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt.17(6), 067007 (2012).
[CrossRef] [PubMed]

Franceschinifantini, M. A.

S. Fantini, M. A. Franceschinifantini, J. S. Maier, S. A. Walker, B. Barbieri, and E. Gratton, “Frequency-domain mutlichannel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng.34(1), 32–42 (1995).
[CrossRef]

Frangioni, J. V.

B. E. Schaafsma, J. S. D. Mieog, M. Hutteman, J. R. van der Vorst, P. J. Kuppen, C. W. Löwik, J. V. Frangioni, C. J. van de Velde, and A. L. Vahrmeijer, “The clinical use of indocyanine green as a near-infrared fluorescent contrast agent for image-guided oncologic surgery,” J. Surg. Oncol.104(3), 323–332 (2011).
[CrossRef] [PubMed]

Gao, X. H.

X. H. Gao, Y. Y. Cui, R. M. Levenson, L. W. K. Chung, and S. M. Nie, “In vivo cancer targeting and imaging with semiconductor quantum dots,” Nat. Biotechnol.22(8), 969–976 (2004).
[CrossRef] [PubMed]

Gayen, S. K.

M. Xu, M. Alrubaiee, S. K. Gayen, and R. R. Alfano, “Three-dimensional localization and optical imaging of objects in turbid media with independent component analysis,” Appl. Opt.44(10), 1889–1897 (2005).
[CrossRef] [PubMed]

M. E. Zevallos, S. K. Gayen, M. Alrubaiee, and R. R. Alfano, “Time-gated backscattered ballistic light imaging of objects in turbid water,” Appl. Phys. Lett.86(1), 011115 (2005).
[CrossRef]

Ghosh, S. K.

S. K. Ghosh and T. Pal, “Interparticle coupling effect on the surface plasmon resonance of gold nanoparticles: From theory to applications,” Chem. Rev.107(11), 4797–4862 (2007).
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Giacomelli, M. G.

Goodell, T. T.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt.10(3), 034018 (2005).
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Grabtchak, S.

S. Grabtchak, E. Tonkopi, and W. M. Whelan, “Optical detection of gold nanoparticles in a prostate-shaped porcine phantom,” J. Biomed. Opt.18(7), 077005 (2013).
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S. Grabtchak and W. M. Whelan, “Separation of absorption and scattering properties of turbid media using relative spectrally resolved cw radiance measurements,” Biomed. Opt. Express3(10), 2371–2380 (2012).
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S. Grabtchak, T. J. Palmer, I. A. Vitkin, and W. M. Whelan, “Radiance detection of non-scattering inclusions in turbid media,” Biomed. Opt. Express3(11), 3001–3011 (2012).
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S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt.17(6), 067007 (2012).
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S. Grabtchak, T. J. Palmer, and W. M. Whelan, “Detection of localized inclusions of gold nanoparticles in Intralipid-1% by point-radiance spectroscopy,” J. Biomed. Opt.16(7), 077003 (2011).
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S. Grabtchak, T. J. Palmer, and W. Whelan, “Radiance spectroscopy tool box for characterizing Au nanoparticles in tissue mimicking phantoms as applied to prostate,” J. Cancer Sci. Ther.S1–008 (2011).

Gratton, E.

S. A. Walker, D. A. Boas, and E. Gratton, “Photon density waves scattered from cylindrical inhomogeneities: theory and experiments,” Appl. Opt.37(10), 1935–1944 (1998).
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S. Fantini, M. A. Franceschinifantini, J. S. Maier, S. A. Walker, B. Barbieri, and E. Gratton, “Frequency-domain mutlichannel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng.34(1), 32–42 (1995).
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Grenier, N.

Grimm, J.

X. Montet, V. Ntziachristos, J. Grimm, and R. Weissleder, “Tomographic fluorescence mapping of tumor targets,” Cancer Res.65(14), 6330–6336 (2005).
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Hahn, S. M.

T. C. Zhu, J. C. Finlay, and S. M. Hahn, “Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in-vivo in human prostate during motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B79(3), 231–241 (2005).
[CrossRef] [PubMed]

Haskell, R. C.

He, L.

B. Chance, K. Kang, L. He, J. Weng, and E. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. U.S.A.90(8), 3423–3427 (1993).
[CrossRef] [PubMed]

Hervé, L.

Huang, X.

E. C. Dreaden, A. M. Alkilany, X. Huang, C. J. Murphy, and M. A. El-Sayed, “The golden age: gold nanoparticles for biomedicine,” Chem. Soc. Rev.41(7), 2740–2779 (2012).
[CrossRef] [PubMed]

Hutteman, M.

B. E. Schaafsma, J. S. D. Mieog, M. Hutteman, J. R. van der Vorst, P. J. Kuppen, C. W. Löwik, J. V. Frangioni, C. J. van de Velde, and A. L. Vahrmeijer, “The clinical use of indocyanine green as a near-infrared fluorescent contrast agent for image-guided oncologic surgery,” J. Surg. Oncol.104(3), 323–332 (2011).
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E. Hutter and D. Maysinger, “Gold Nanoparticles and Quantum Dots for Bioimaging,” Microsc. Res. Tech.74(7), 592–604 (2011).
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Inomoto, T.

T. Kitai, T. Inomoto, M. Miwa, and T. Shikayama, “Fluorescence navigation with indocyanine green for detecting sentinel lymph nodes in breast cancer,” Breast Cancer12(3), 211–215 (2005).
[CrossRef] [PubMed]

Jacques, S. L.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt.10(3), 034018 (2005).
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Jakubowski, D.

Kacprzak, M.

M. Kacprzak, A. Liebert, P. Sawosz, N. Zolek, D. Milej, and R. Maniewski, “Time-resolved imaging of fluorescent inclusions in optically turbid medium - phantom study,” Opto-Electron. Rev.18(1), 37–47 (2010).
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Kang, K.

B. Chance, K. Kang, L. He, J. Weng, and E. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. U.S.A.90(8), 3423–3427 (1993).
[CrossRef] [PubMed]

Kashio, Y.

M. Ono, Y. Kashio, M. Schweiger, H. Dehghani, S. R. Arridge, M. Firbank, and E. Okada, “Topographic distribution of photon measurement density functions on the brain surface by hybrid radiosity-diffusion method,” Opt. Rev.7(5), 426–431 (2000).
[CrossRef]

Kienle, A.

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt.17(6), 067007 (2012).
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A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.83(3), 036605 (2011).
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Kitai, T.

T. Kitai, T. Inomoto, M. Miwa, and T. Shikayama, “Fluorescence navigation with indocyanine green for detecting sentinel lymph nodes in breast cancer,” Breast Cancer12(3), 211–215 (2005).
[CrossRef] [PubMed]

Koval, G.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt.10(3), 034018 (2005).
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[CrossRef] [PubMed]

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt.17(6), 067007 (2012).
[CrossRef] [PubMed]

S. Grabtchak, T. J. Palmer, and W. M. Whelan, “Detection of localized inclusions of gold nanoparticles in Intralipid-1% by point-radiance spectroscopy,” J. Biomed. Opt.16(7), 077003 (2011).
[CrossRef] [PubMed]

L. C. L. Chin, B. Lloyd, W. M. Whelan, and I. A. Vitkin, “Interstitial point radiance spectroscopy of turbid media,” J. Appl. Phys.105(10), 102025 (2009).
[CrossRef]

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt.12(6), 064027 (2007).
[CrossRef] [PubMed]

L. C. L. Chin, W. M. Whelan, and I. A. Vitkin, “Information content of point radiance measurements in turbid media: implications for interstitial optical property quantification,” Appl. Opt.45(9), 2101–2114 (2006).
[CrossRef] [PubMed]

Wilson, B. C.

M. S. Patterson, B. C. Wilson, and D. R. Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their application,” Lasers Med. Sci.6(2), 155–168 (1991).
[CrossRef]

M. S. Patterson, B. C. Wilson, and D. R. Wyman, “The propagation of optical radiation in tissue II. Optical properties of tissues and resulting fluence distributions,” Lasers Med. Sci.6(4), 379–390 (1991).
[CrossRef]

Wilson, P. J.

A. M. Ballangrud, P. J. Wilson, K. Brown, G. G. Miller, R. B. Moore, M. S. McPhee, and J. Tulip, “Anisotropy of radiance in tissue phantoms and Dunning R3327 rat tumors: Radiance measurements with flat cleaved fiber probes,” Lasers Surg. Med.19(4), 471–479 (1996).
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Wilson, R.

R. Wilson, “The use of gold nanoparticles in diagnostics and detection,” Chem. Soc. Rev.37(9), 2028–2045 (2008).
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L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt.12(6), 064027 (2007).
[CrossRef] [PubMed]

Wyman, D. R.

M. S. Patterson, B. C. Wilson, and D. R. Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their application,” Lasers Med. Sci.6(2), 155–168 (1991).
[CrossRef]

M. S. Patterson, B. C. Wilson, and D. R. Wyman, “The propagation of optical radiation in tissue II. Optical properties of tissues and resulting fluence distributions,” Lasers Med. Sci.6(4), 379–390 (1991).
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Y. Yamada, “Fundamental studies of photon migration in biological tissues and their application to optical tomography,” Opt. Rev.7(5), 366–374 (2000).
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T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys.73(7), 076701 (2010).
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Zangheri, L.

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol.45(5), 1359–1373 (2000).
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Zeltser, I.

K. Bensalah, A. Tuncel, D. Peshwani, I. Zeltser, H. Liu, and J. Cadeddu, “Optical reflectance spectroscopy to differentiate renal tumor from normal parenchyma,” J. Urol.179(5), 2010–2013 (2008).
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C. J. Wen, L. W. Zhang, S. A. Al-Suwayeh, T. C. Yen, and J. Y. Fang, “Theranostic liposomes loaded with quantum dots and apomorphine for brain targeting and bioimaging,” Int. J. Nanomedicine7, 1599–1611 (2012).
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Biomed. Opt. Express

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Int. J. Nanomedicine

C. J. Wen, L. W. Zhang, S. A. Al-Suwayeh, T. C. Yen, and J. Y. Fang, “Theranostic liposomes loaded with quantum dots and apomorphine for brain targeting and bioimaging,” Int. J. Nanomedicine7, 1599–1611 (2012).
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L. C. L. Chin, B. Lloyd, W. M. Whelan, and I. A. Vitkin, “Interstitial point radiance spectroscopy of turbid media,” J. Appl. Phys.105(10), 102025 (2009).
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S. Grabtchak, E. Tonkopi, and W. M. Whelan, “Optical detection of gold nanoparticles in a prostate-shaped porcine phantom,” J. Biomed. Opt.18(7), 077005 (2013).
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S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt.17(6), 067007 (2012).
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S. Grabtchak, T. J. Palmer, and W. M. Whelan, “Detection of localized inclusions of gold nanoparticles in Intralipid-1% by point-radiance spectroscopy,” J. Biomed. Opt.16(7), 077003 (2011).
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J. Photochem. Photobiol. B

T. C. Zhu, J. C. Finlay, and S. M. Hahn, “Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in-vivo in human prostate during motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B79(3), 231–241 (2005).
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B. E. Schaafsma, J. S. D. Mieog, M. Hutteman, J. R. van der Vorst, P. J. Kuppen, C. W. Löwik, J. V. Frangioni, C. J. van de Velde, and A. L. Vahrmeijer, “The clinical use of indocyanine green as a near-infrared fluorescent contrast agent for image-guided oncologic surgery,” J. Surg. Oncol.104(3), 323–332 (2011).
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J. Urol.

K. Bensalah, A. Tuncel, D. Peshwani, I. Zeltser, H. Liu, and J. Cadeddu, “Optical reflectance spectroscopy to differentiate renal tumor from normal parenchyma,” J. Urol.179(5), 2010–2013 (2008).
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Lasers Med. Sci.

M. S. Patterson, B. C. Wilson, and D. R. Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their application,” Lasers Med. Sci.6(2), 155–168 (1991).
[CrossRef]

M. S. Patterson, B. C. Wilson, and D. R. Wyman, “The propagation of optical radiation in tissue II. Optical properties of tissues and resulting fluence distributions,” Lasers Med. Sci.6(4), 379–390 (1991).
[CrossRef]

Lasers Surg. Med.

A. M. Ballangrud, P. J. Wilson, K. Brown, G. G. Miller, R. B. Moore, M. S. McPhee, and J. Tulip, “Anisotropy of radiance in tissue phantoms and Dunning R3327 rat tumors: Radiance measurements with flat cleaved fiber probes,” Lasers Surg. Med.19(4), 471–479 (1996).
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Opto-Electron. Rev.

M. Kacprzak, A. Liebert, P. Sawosz, N. Zolek, D. Milej, and R. Maniewski, “Time-resolved imaging of fluorescent inclusions in optically turbid medium - phantom study,” Opto-Electron. Rev.18(1), 37–47 (2010).
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A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol.50(10), 2291–2311 (2005).
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T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys.73(7), 076701 (2010).
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Figures (11)

Fig. 1
Fig. 1

a) Experimental set-up for radiance measurements of gold target in a Intralipid-1%, b) conceptual top-view diagram illustrating the principle of distance dependent experiments.

Fig. 2
Fig. 2

a) Effective attenuation coefficient of Intralipid-1% and absorption coefficient of 5-nm Au NPs colloidal solutions with a) 3.95x1013 particles/mL and b) 7x1014 particles/mL (marked as 20x) concentrations.

Fig. 3
Fig. 3

a) Spectro-angular contour plot of raw radiance data for Intralipid-1% ; b) spectro-angular contour plot of raw radiance for Intralipid-1% with 3.5 mm inclusion with Au NPs located at 0° and 5mm from the detector. Color bars have units of counts per second. Source-detector separation 30 mm.

Fig. 4
Fig. 4

a) Spectro-angular contour plot of the radiance extinction ratio (RER) the Au NPs inclusion. b) Spectro-angular surface plot of the radiance extinction ratio (RER) of the Au NPs inclusion. Inclusion: 3.5 mm capillary with 5nm Au NPs at 0° at 5 mm from the detector. Source-detector separation 30 mm.

Fig. 5
Fig. 5

Spectro-angular contour plots of the radiance extinction ratio (RER) the Au NPs inclusion located at a) 10 mm, b) 15 mm, c) 20 mm, d) 25 mm from the detector at 0°. Inclusion: 3.5 mm capillary with 5nm Au NPs. Source-detector separation 30 mm.

Fig. 6
Fig. 6

Variation of maximal RER values observed along 0° and full-width-half-maximum (FWHM) measured at 520 nm with distance extracted from Figs. 4(a), 5(a)-5(d).

Fig. 7
Fig. 7

Experiment for 3.5 mm target with 3.95x1013 Au NPs/mL: a) spatio-angular contour plot of RER measured at 520 nm, b) spectro-spatial contour plot of RER measured along 0°. Plots created from data in Figs. 4(a), 5(a)-5(d).

Fig. 8
Fig. 8

Simulated spatio-angular contour plot of RER at 520 nm using Eq. (9) for 3.5 mm diameter perfect absorber.

Fig. 9
Fig. 9

Simulation of a realistic absorber (3.5 mm diameter with 3.95x1013 Au NPs/mL): a) spatio-angular contour plot of RER at 520 nm using Eq. (9), b) spectro-spatial contour plot of RER at 0° using Eq. (9)

Fig. 10
Fig. 10

Experiment for 2.0 mm target with 7x1014 Au NPs/mL: a) spatio-angular contour plot of RER measured at 530 nm, b) spectro-spatial contour plot of RER measured along 0°.

Fig. 11
Fig. 11

Simulation of the realistic absorber (2.0 mm diameter with 7x1014 Au NPs/mL (i.e.,20x)): a) spatio-angular contour plot of RER at 530 nm using Eq. (9), b) spectro-spatial contour plot of RER at 0° using Eq. (9)

Equations (9)

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

I 0 (| r S r D |,θ,λ)= S 0 4 2 π 2 D(λ) [ 1+3( D(λ) | r S r D | + μ eff (λ)D(λ) )cosθ ] exp( μ eff (λ)| r S r D | ) | r S r D |
I pert (| r S r D |,θ,λ)= 1 4π Φ pert (| r S r D |,λ)+ 3 4π D(λ) Φ pert (| r S r D |,λ) s ^
Φ pert (| r S r D |,λ)= Φ 0 (| r S r D |,λ)+ Φ 1 (| r S r D |,r,λ)
Φ 0 (| r S r D |,λ)= S 0 4πD(λ) exp( μ eff (λ)| r S r D | ) | r S r D |
Φ 1 (| r S r D |,r,λ)=k a S 0 4πD(λ) exp( μ eff (λ)| r r S | ) | r r S | exp( μ eff (λ)( | r D r |a ) ) | r D r |
I pert (| r S r D |,θ,λ)= 1 4π Φ 0 (| r S r D |,λ)+ 1 4π Φ 1 (| r S r D |,r,λ) + 3 4π D(λ) Φ 0 (| r S r D |,λ) s ^ + 3 4π D(λ) Φ 1 (| r S r D |,r,λ) s ^
I pert (d,θ,λ)= S 0 4 2 π 2 D(λ) [ exp( μ eff (λ)d ) d ak exp( μ eff (λ)( da ) ) x(dx) 3akD(λ) (d2x)exp( μ eff (λ)( da ) ) (dx) 2 x 2 cosθ ]
I 0 (d,θ,λ)= S 0 4 2 π 2 D(λ) [ 1+3( D(λ) d + μ eff (λ)D(λ) )cosθ ] exp( μ eff (λ)d ) d
RER= I 0 (d,θ,λ) I pert (d,θ,λ) 1+3( D(λ) d + μ eff (λ)D(λ) )cosθ 1 adk x(dx) exp( μ eff (λ)( da ) ) 3aD(λ)(d2x)dk (dx) 2 x 2 exp( μ eff (λ)( da ) )cosθ

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