Abstract

We report a cross-referencing method to quickly and accurately characterize the optical properties of nanoparticles including the extinction, scattering, absorption and backscattering cross sections by using an OCT system alone. Among other applications, such a method is particularly useful for developing nanoparticle-based OCT imaging contrast agents. The method involves comparing two depth-dependent OCT intensity signals collected from two samples (with one having and the other not having the nanoparticles), to extract the extinction and backscattering coefficient, from which the absorption coefficient can be further deduced (with the help of the established scattering theories for predicting the ratio of the backscattering to total scattering cross section). The method has been experimentally validated using test nanoparticles and was then applied to characterizing gold nanocages. With the aid of this method, we were able to successfully synthesize scattering dominant gold nanocages for the first time and demonstrated the highest contrast enhancement ever achieved by the gold nanocages (and by any nanoparticles of a similar size and concentration) in an in vivo mouse tumor model. This method also enables quantitative analysis of contrast enhancement and provides a general guideline on choosing the optimal concentration and optical properties for the nanoparticle-based OCT contrast agents.

©2013 Optical Society of America

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  1. J. K. Barton, J. B. Hoying, and C. J. Sullivan, “Use of microbubbles as an optical coherence tomography contrast agent,” Acad. Radiol. 9(Suppl 1), S52–S55 (2002).
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  17. J. M. Schmitt, A. R. Knuettel, A. H. Gandjbakhche, and R. F. Bonner, “Optical characterization of dense tissues using low-coherence interferometry,” Proc. SPIE 1889, 197–211 (1993).
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  18. T. S. Ralston, D. L. Marks, P. Scott Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
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  21. P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine,” J. Phys. Chem. B 110(14), 7238–7248 (2006).
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2011 (1)

N. Bosschaart, D. J. Faber, T. G. van Leeuwen, and M. C. G. Aalders, “Measurements of wavelength dependent scattering and backscattering coefficients by low-coherence spectroscopy,” J. Biomed. Opt. 16(3), 030503 (2011).
[Crossref] [PubMed]

2010 (1)

2007 (4)

T. S. Ralston, D. L. Marks, P. Scott Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref]

A. M. Gobin, M. H. Lee, N. J. Halas, W. D. James, R. A. Drezek, and J. L. West, “Near-infrared resonant nanoshells for combined optical imaging and photothermal cancer therapy,” Nano Lett. 7(7), 1929–1934 (2007).
[Crossref] [PubMed]

T. S. Troutman, J. K. Barton, and M. Romanowski, “Optical coherence tomography with plasmon resonant nanorods of gold,” Opt. Lett. 32(11), 1438–1440 (2007).
[Crossref] [PubMed]

F. Hao, C. L. Nehl, J. H. Hafner, and P. Nordlander, “Plasmon resonances of a gold nanostar,” Nano Lett. 7(3), 729–732 (2007).
[Crossref] [PubMed]

2006 (1)

P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine,” J. Phys. Chem. B 110(14), 7238–7248 (2006).
[Crossref] [PubMed]

2005 (2)

H. Cang, T. Sun, Z.-Y. Li, J. Chen, B. J. Wiley, Y. Xia, and X. D. Li, “Gold nanocages as contrast agents for spectroscopic optical coherence tomography,” Opt. Lett. 30(22), 3048–3050 (2005).
[Crossref] [PubMed]

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold nanocages: bioconjugation and their potential use as optical imaging contrast agents,” Nano Lett. 5(3), 473–477 (2005).
[Crossref] [PubMed]

2004 (2)

2003 (2)

T. G. van Leeuwen, D. J. Faber, and M. C. Aalders, “Measurement of the axial point spread function in scattering media using single-mode fiber-based optical coherence tomography,” IEEE J. Sel. Top. Quantum Electron. 9(2), 227–233 (2003).
[Crossref]

T. M. Lee, A. L. Oldenburg, S. Sitafalwalla, D. L. Marks, W. Luo, F. J.-J. Toublan, K. S. Suslick, and S. A. Boppart, “Engineered microsphere contrast agents for optical coherence tomography,” Opt. Lett. 28(17), 1546–1548 (2003).
[Crossref] [PubMed]

2002 (1)

J. K. Barton, J. B. Hoying, and C. J. Sullivan, “Use of microbubbles as an optical coherence tomography contrast agent,” Acad. Radiol. 9(Suppl 1), S52–S55 (2002).
[Crossref] [PubMed]

1998 (1)

1994 (1)

1993 (1)

J. M. Schmitt, A. R. Knuettel, A. H. Gandjbakhche, and R. F. Bonner, “Optical characterization of dense tissues using low-coherence interferometry,” Proc. SPIE 1889, 197–211 (1993).
[Crossref]

Aalders, M. C.

T. G. van Leeuwen, D. J. Faber, and M. C. Aalders, “Measurement of the axial point spread function in scattering media using single-mode fiber-based optical coherence tomography,” IEEE J. Sel. Top. Quantum Electron. 9(2), 227–233 (2003).
[Crossref]

Aalders, M. C. G.

N. Bosschaart, D. J. Faber, T. G. van Leeuwen, and M. C. G. Aalders, “Measurements of wavelength dependent scattering and backscattering coefficients by low-coherence spectroscopy,” J. Biomed. Opt. 16(3), 030503 (2011).
[Crossref] [PubMed]

Au, L.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold nanocages: bioconjugation and their potential use as optical imaging contrast agents,” Nano Lett. 5(3), 473–477 (2005).
[Crossref] [PubMed]

Barton, J. K.

T. S. Troutman, J. K. Barton, and M. Romanowski, “Optical coherence tomography with plasmon resonant nanorods of gold,” Opt. Lett. 32(11), 1438–1440 (2007).
[Crossref] [PubMed]

J. K. Barton, J. B. Hoying, and C. J. Sullivan, “Use of microbubbles as an optical coherence tomography contrast agent,” Acad. Radiol. 9(Suppl 1), S52–S55 (2002).
[Crossref] [PubMed]

Bonner, R. F.

J. M. Schmitt, A. R. Knuettel, A. H. Gandjbakhche, and R. F. Bonner, “Optical characterization of dense tissues using low-coherence interferometry,” Proc. SPIE 1889, 197–211 (1993).
[Crossref]

Boppart, S.

Boppart, S. A.

Bosschaart, N.

N. Bosschaart, D. J. Faber, T. G. van Leeuwen, and M. C. G. Aalders, “Measurements of wavelength dependent scattering and backscattering coefficients by low-coherence spectroscopy,” J. Biomed. Opt. 16(3), 030503 (2011).
[Crossref] [PubMed]

Cang, H.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold nanocages: bioconjugation and their potential use as optical imaging contrast agents,” Nano Lett. 5(3), 473–477 (2005).
[Crossref] [PubMed]

H. Cang, T. Sun, Z.-Y. Li, J. Chen, B. J. Wiley, Y. Xia, and X. D. Li, “Gold nanocages as contrast agents for spectroscopic optical coherence tomography,” Opt. Lett. 30(22), 3048–3050 (2005).
[Crossref] [PubMed]

Chen, J.

H. Cang, T. Sun, Z.-Y. Li, J. Chen, B. J. Wiley, Y. Xia, and X. D. Li, “Gold nanocages as contrast agents for spectroscopic optical coherence tomography,” Opt. Lett. 30(22), 3048–3050 (2005).
[Crossref] [PubMed]

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold nanocages: bioconjugation and their potential use as optical imaging contrast agents,” Nano Lett. 5(3), 473–477 (2005).
[Crossref] [PubMed]

Cobb, M. J.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold nanocages: bioconjugation and their potential use as optical imaging contrast agents,” Nano Lett. 5(3), 473–477 (2005).
[Crossref] [PubMed]

Do, M.

Draine, B. T.

Drezek, R. A.

A. M. Gobin, M. H. Lee, N. J. Halas, W. D. James, R. A. Drezek, and J. L. West, “Near-infrared resonant nanoshells for combined optical imaging and photothermal cancer therapy,” Nano Lett. 7(7), 1929–1934 (2007).
[Crossref] [PubMed]

El-Sayed, I. H.

P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine,” J. Phys. Chem. B 110(14), 7238–7248 (2006).
[Crossref] [PubMed]

El-Sayed, M. A.

P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine,” J. Phys. Chem. B 110(14), 7238–7248 (2006).
[Crossref] [PubMed]

Faber, D. J.

N. Bosschaart, D. J. Faber, T. G. van Leeuwen, and M. C. G. Aalders, “Measurements of wavelength dependent scattering and backscattering coefficients by low-coherence spectroscopy,” J. Biomed. Opt. 16(3), 030503 (2011).
[Crossref] [PubMed]

T. G. van Leeuwen, D. J. Faber, and M. C. Aalders, “Measurement of the axial point spread function in scattering media using single-mode fiber-based optical coherence tomography,” IEEE J. Sel. Top. Quantum Electron. 9(2), 227–233 (2003).
[Crossref]

Flatau, P. J.

Gandjbakhche, A. H.

J. M. Schmitt, A. R. Knuettel, A. H. Gandjbakhche, and R. F. Bonner, “Optical characterization of dense tissues using low-coherence interferometry,” Proc. SPIE 1889, 197–211 (1993).
[Crossref]

Gobin, A. M.

A. M. Gobin, M. H. Lee, N. J. Halas, W. D. James, R. A. Drezek, and J. L. West, “Near-infrared resonant nanoshells for combined optical imaging and photothermal cancer therapy,” Nano Lett. 7(7), 1929–1934 (2007).
[Crossref] [PubMed]

Hafner, J. H.

F. Hao, C. L. Nehl, J. H. Hafner, and P. Nordlander, “Plasmon resonances of a gold nanostar,” Nano Lett. 7(3), 729–732 (2007).
[Crossref] [PubMed]

Halas, N. J.

A. M. Gobin, M. H. Lee, N. J. Halas, W. D. James, R. A. Drezek, and J. L. West, “Near-infrared resonant nanoshells for combined optical imaging and photothermal cancer therapy,” Nano Lett. 7(7), 1929–1934 (2007).
[Crossref] [PubMed]

Hao, F.

F. Hao, C. L. Nehl, J. H. Hafner, and P. Nordlander, “Plasmon resonances of a gold nanostar,” Nano Lett. 7(3), 729–732 (2007).
[Crossref] [PubMed]

Hoying, J. B.

J. K. Barton, J. B. Hoying, and C. J. Sullivan, “Use of microbubbles as an optical coherence tomography contrast agent,” Acad. Radiol. 9(Suppl 1), S52–S55 (2002).
[Crossref] [PubMed]

Jain, P. K.

P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine,” J. Phys. Chem. B 110(14), 7238–7248 (2006).
[Crossref] [PubMed]

James, W. D.

A. M. Gobin, M. H. Lee, N. J. Halas, W. D. James, R. A. Drezek, and J. L. West, “Near-infrared resonant nanoshells for combined optical imaging and photothermal cancer therapy,” Nano Lett. 7(7), 1929–1934 (2007).
[Crossref] [PubMed]

Kimmey, M. B.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold nanocages: bioconjugation and their potential use as optical imaging contrast agents,” Nano Lett. 5(3), 473–477 (2005).
[Crossref] [PubMed]

Knuettel, A. R.

J. M. Schmitt, A. R. Knuettel, A. H. Gandjbakhche, and R. F. Bonner, “Optical characterization of dense tissues using low-coherence interferometry,” Proc. SPIE 1889, 197–211 (1993).
[Crossref]

Kumar, G.

Lee, K. S.

P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine,” J. Phys. Chem. B 110(14), 7238–7248 (2006).
[Crossref] [PubMed]

Lee, M. H.

A. M. Gobin, M. H. Lee, N. J. Halas, W. D. James, R. A. Drezek, and J. L. West, “Near-infrared resonant nanoshells for combined optical imaging and photothermal cancer therapy,” Nano Lett. 7(7), 1929–1934 (2007).
[Crossref] [PubMed]

Lee, T. M.

Li, X. D.

H. Cang, T. Sun, Z.-Y. Li, J. Chen, B. J. Wiley, Y. Xia, and X. D. Li, “Gold nanocages as contrast agents for spectroscopic optical coherence tomography,” Opt. Lett. 30(22), 3048–3050 (2005).
[Crossref] [PubMed]

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold nanocages: bioconjugation and their potential use as optical imaging contrast agents,” Nano Lett. 5(3), 473–477 (2005).
[Crossref] [PubMed]

Li, Z.-Y.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold nanocages: bioconjugation and their potential use as optical imaging contrast agents,” Nano Lett. 5(3), 473–477 (2005).
[Crossref] [PubMed]

H. Cang, T. Sun, Z.-Y. Li, J. Chen, B. J. Wiley, Y. Xia, and X. D. Li, “Gold nanocages as contrast agents for spectroscopic optical coherence tomography,” Opt. Lett. 30(22), 3048–3050 (2005).
[Crossref] [PubMed]

Luo, W.

Marks, D.

Marks, D. L.

Nehl, C. L.

F. Hao, C. L. Nehl, J. H. Hafner, and P. Nordlander, “Plasmon resonances of a gold nanostar,” Nano Lett. 7(3), 729–732 (2007).
[Crossref] [PubMed]

Nordlander, P.

F. Hao, C. L. Nehl, J. H. Hafner, and P. Nordlander, “Plasmon resonances of a gold nanostar,” Nano Lett. 7(3), 729–732 (2007).
[Crossref] [PubMed]

Oldenburg, A. L.

Ralston, T. S.

T. S. Ralston, D. L. Marks, P. Scott Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref]

Robles, F. E.

Romanowski, M.

Saeki, F.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold nanocages: bioconjugation and their potential use as optical imaging contrast agents,” Nano Lett. 5(3), 473–477 (2005).
[Crossref] [PubMed]

Schmitt, J. M.

J. M. Schmitt and G. Kumar, “Optical scattering properties of soft tissue: a discrete particle model,” Appl. Opt. 37(13), 2788–2797 (1998).
[Crossref] [PubMed]

J. M. Schmitt, A. R. Knuettel, A. H. Gandjbakhche, and R. F. Bonner, “Optical characterization of dense tissues using low-coherence interferometry,” Proc. SPIE 1889, 197–211 (1993).
[Crossref]

Scott Carney, P.

T. S. Ralston, D. L. Marks, P. Scott Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref]

Sitafalwalla, S.

Sullivan, C. J.

J. K. Barton, J. B. Hoying, and C. J. Sullivan, “Use of microbubbles as an optical coherence tomography contrast agent,” Acad. Radiol. 9(Suppl 1), S52–S55 (2002).
[Crossref] [PubMed]

Sun, T.

Suslick, K. S.

Toublan, F. J.-J.

Troutman, T. S.

van Leeuwen, T. G.

N. Bosschaart, D. J. Faber, T. G. van Leeuwen, and M. C. G. Aalders, “Measurements of wavelength dependent scattering and backscattering coefficients by low-coherence spectroscopy,” J. Biomed. Opt. 16(3), 030503 (2011).
[Crossref] [PubMed]

T. G. van Leeuwen, D. J. Faber, and M. C. Aalders, “Measurement of the axial point spread function in scattering media using single-mode fiber-based optical coherence tomography,” IEEE J. Sel. Top. Quantum Electron. 9(2), 227–233 (2003).
[Crossref]

Wax, A.

West, J. L.

A. M. Gobin, M. H. Lee, N. J. Halas, W. D. James, R. A. Drezek, and J. L. West, “Near-infrared resonant nanoshells for combined optical imaging and photothermal cancer therapy,” Nano Lett. 7(7), 1929–1934 (2007).
[Crossref] [PubMed]

Wiley, B. J.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold nanocages: bioconjugation and their potential use as optical imaging contrast agents,” Nano Lett. 5(3), 473–477 (2005).
[Crossref] [PubMed]

H. Cang, T. Sun, Z.-Y. Li, J. Chen, B. J. Wiley, Y. Xia, and X. D. Li, “Gold nanocages as contrast agents for spectroscopic optical coherence tomography,” Opt. Lett. 30(22), 3048–3050 (2005).
[Crossref] [PubMed]

Xia, Y.

H. Cang, T. Sun, Z.-Y. Li, J. Chen, B. J. Wiley, Y. Xia, and X. D. Li, “Gold nanocages as contrast agents for spectroscopic optical coherence tomography,” Opt. Lett. 30(22), 3048–3050 (2005).
[Crossref] [PubMed]

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold nanocages: bioconjugation and their potential use as optical imaging contrast agents,” Nano Lett. 5(3), 473–477 (2005).
[Crossref] [PubMed]

Xu, C.

Ye, J.

Zhang, H.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold nanocages: bioconjugation and their potential use as optical imaging contrast agents,” Nano Lett. 5(3), 473–477 (2005).
[Crossref] [PubMed]

Acad. Radiol. (1)

J. K. Barton, J. B. Hoying, and C. J. Sullivan, “Use of microbubbles as an optical coherence tomography contrast agent,” Acad. Radiol. 9(Suppl 1), S52–S55 (2002).
[Crossref] [PubMed]

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

T. G. van Leeuwen, D. J. Faber, and M. C. Aalders, “Measurement of the axial point spread function in scattering media using single-mode fiber-based optical coherence tomography,” IEEE J. Sel. Top. Quantum Electron. 9(2), 227–233 (2003).
[Crossref]

J. Biomed. Opt. (1)

N. Bosschaart, D. J. Faber, T. G. van Leeuwen, and M. C. G. Aalders, “Measurements of wavelength dependent scattering and backscattering coefficients by low-coherence spectroscopy,” J. Biomed. Opt. 16(3), 030503 (2011).
[Crossref] [PubMed]

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

J. Phys. Chem. B (1)

P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine,” J. Phys. Chem. B 110(14), 7238–7248 (2006).
[Crossref] [PubMed]

Nano Lett. (3)

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold nanocages: bioconjugation and their potential use as optical imaging contrast agents,” Nano Lett. 5(3), 473–477 (2005).
[Crossref] [PubMed]

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Figures (4)

Fig. 1
Fig. 1 (A) Angular dependent scattering pattern of 180 nm silica nanospheres in a 5% gelatin phantom at the wavelength of 825 nm calculated by the Mie scattering theory. (B) Angular dependent scattering pattern of 75 nm gold nanocages in a 5% gelatin phantom at the wavelength of 825 nm calculated by a numerical method based on discrete dipole approximation (DDA). The scattering pattern is averaged over various nanoparticle orientations.

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Fig. 2
Fig. 2 (A) OCT images of the phantoms without nanocages (left) and with nanocages (right). (B) Intensity plots of the OCT signals on a linear scale as a function of imaging depth. (C) Ratio of the two signals in (B) on a logarithmic scale as a function of imaging depth.

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Fig. 3
Fig. 3 (A) In vivo OCT images of a mouse ear tumor (induced with A431 cell line) before (A) and after (B) intravenous injection of gold nanocages, respectively. (C) Depth-dependent OCT intensity plots from the tumor on a logarithm scale before (blue curve) and after (red curve) the administration of gold nanocages. (D) The ratio of the two curves in (C) and its moving-average results on a logarithm scale.

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Fig. 4
Fig. 4 The estimation of contrast enhancement with respect to the concentration of OCT contrast agent at different imaging depths.

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Equations (7)

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i( z )= K 1 ∫ φ=0 2π ∫ θ=0 Θ μ ′ ( θ,φ )sinθdθ dφ e − μ ext z h( z ),
i( z )= K 1 sin Θ 2 μ bs e − μ ext z h( z )=K μ bs e − μ ext z h( z ),
i ref ( z )=K μ bs ref e − μ ext ref z h ref ( z )  ( for the reference phantom with only silica nanospheres ), i test ( z )=K μ bs ref + μ bs test e −( μ ext ref + μ ext test )z h test ( z )  ( for the test phantom with both silica and test nanoparticles ).
ln i test (z)−ln i ref (z)=− μ ext test z+ 1 2 ln( μ bs test μ bs ref +1).
μ bs test =( e 2b −1 )⋅ μ bs ref .
ln i after (z)−ln i before (z)=− μ ext CA z+ 1 2 ln( μ bs CA μ bs tissue +1 ),
i after (z) i before (z) =exp[ − σ ext Au ⋅ N A ⋅ C Au ⋅z+ ln( k r (r+1) ⋅ σ ext Au ⋅ N A ⋅ C Au μ bs tissue +1 ) /2 ],

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