Abstract

Preliminary experiments at the NIST Spectral Tri-function Automated Reference Reflectometer (STARR) facility have been conducted with the goal of providing the diffuse optical properties of a solid reference standard with optical properties similar to human skin. Here, we describe an algorithm for determining the best-fit parameters and the statistical uncertainty associated with the measurement. The objective function is determined from the profile log likelihood, including both experimental and Monte Carlo uncertainties. Initially, the log likelihood is determined over a large parameter search box using a relatively small number of Monte Carlo samples such as 2·104. The search area is iteratively reduced to include the 99.9999% confidence region, while doubling the number of samples at each iteration until the experimental uncertainty dominates over the Monte Carlo uncertainty. Typically this occurs by 1.28·106 samples. The log likelihood is then fit to determine a 95% confidence ellipse. The inverse problem requires the values of the log likelihood on many points. Our implementation uses importance sampling to calculate these points on a grid in an efficient manner. Ultimately, the time-to-solution is approximately six times the cost of a Monte Carlo simulation of the radiation transport problem for a single set of parameters with the largest number of photons required. The results are found to be 64 times faster than our implementation of Particle Swarm Optimization.

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2017 (3)

J. Hwang, H.-J. Kim, P. Lemaillet, H. Wabnitz, D. Grosenick, L. Yang, T. Gladytz, D. McClatchy, D. Allen, K. Briggman, and B. Pogue, “Polydimethylsiloxane tissue-mimicking phantoms for quantitative optical medical imaging standards,” Proc. SPIE 10056, 1005603 (2017).
[Crossref]

G. T. Kennedy, G. R. Lentsch, B. Trieu, A. Ponticorvo, R. B. Saager, and A. J. Durkin, “Solid tissue simulating phantoms having absorption at 970 nm for diffuse optics,” J. Biomed. Opt. 22, 076013 (2017).
[Crossref]

M. R. Bonyadi and Z. Michalewicz, “Particle Swarm Optimization for Single Objective Continuous Space Problems: A Review,” Evolutionary Computation 25, 1–54 (2017).
[Crossref]

2016 (2)

W. C. Vogt, C. Jia, K. A. Wear, B. S. Garra, and T. J. Pfefer, “Biologically relevant photoacoustic imaging phantoms with tunable optical and acoustic properties,” J. Biomed. Opt. 21, 101405 (2016).
[Crossref] [PubMed]

M. N. Kholodtsova, C. Dual, V. B. Loschenov, and W. C. P. M. Blondel, “Spatially and spectrally resolved Particle Swarm Optimization for precise optical property estimation using diffuse-reflectance spectroscopy,” Opt. Express 24, 12682–12700 (2016).
[Crossref] [PubMed]

2015 (3)

P. Lemaillet, J.-P. Bouchard, and D. W. Allen, “Development of traceable measurement of the diffuse optical properties of solid reference standards for biomedical optics at National Institute of Standards and Technology,” Appl. Opt. 54, 6118–6127 (2015).
[Crossref] [PubMed]

M. S. Wróbel, A. P. Popov, A. V. Bykov, M. Kinnunen, M. Jedrzejewska-Szczerska, and V. V. Tuchin, “Measurements of fundamental properties of homogeneous tissue phantoms,” J. Biomed. Opt. 20, 045004 (2015).
[Crossref] [PubMed]

P. Lemaillet, J.-P. Bouchard, J. Hwang, and D. W. Allen, “Double-integrating-sphere system at the National Institute of Standards and Technology in support of measurement standards for the determination of optical properties of tissue-mimicking phantoms,” J. Biomed. Opt. 20, 121310 (2015).
[Crossref] [PubMed]

2014 (1)

2013 (5)

S. L. Jacques, “Optical properties of biological tissues,” Phys. Med. Biol. 58, R37–R61 (2013).
[Crossref] [PubMed]

C. Zhu and Q. Liu, “Review of Monte Carlo modeling of light transport in tissues,” J. Biomed. Opt. 18, 050902 (2013).
[Crossref]

S. Tereshchenko, S. Dolgushin, and S. Titenok, “An imperfection of time-dependent diffusion models for a determination of scattering medium optical properties,” Opt. Comm. 306, 26–34 (2013).
[Crossref]

X. Chen, Y. Feng, J. Q. Lu, X. Liang, J. Ding, Y. Du, and X.-H. Hu, “Fast method for inverse determination of optical parameters from two measured signals,” Opt. Lett. 38, 2095–2097 (2013).
[Crossref] [PubMed]

B. Aernouts, E. Zamora-Rojas, R. Van Beers, R. Watté, L. Wang, M. Tsuta, J. Lammertyn, and W. Saeys, “Supercontinuum laser based optical characterization of Intralipid phantoms in the 500–2250 nm range,” Opt. Express 21, 32450–32467 (2013).
[Crossref]

2011 (3)

2010 (4)

2009 (2)

B. Cletus, R. Künnemeyer, P. Martinsen, A. McGlone, and R. Jordan, “Characterizing liquid turbid media by frequency-domain photon-migration spectroscopy,” J. Biomed. Opt. 14, 024041 (2009).
[Crossref] [PubMed]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[Crossref] [PubMed]

2008 (2)

Q. Li, B. J. Lee, Z. M. Zhang, and D. W. Allen, “Light scattering of semitransparent sintered polytetrafluoroethylene films,” J. Biomed. Opt. 13, 054064 (2008).
[Crossref] [PubMed]

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. 13, 041304 (2008).
[Crossref] [PubMed]

2007 (2)

2006 (1)

T. Moffitt, Y.-C. Chen, and S. A. Prahl, “Preparation and characterization of polyurethane optical phantoms,” J. Biomed. Opt. 14, 041103 (2006).
[Crossref]

2003 (1)

2001 (1)

2000 (2)

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, 6498–6507 (2000).
[Crossref]

M. Mascagni and A. Srinivasan, “Algorithm 806: SPRNG: A scalable library for pseudorandom number generation,” ACM Trans. Math. Software 26, 436–461 (2000).
[Crossref]

1998 (1)

1997 (1)

1996 (1)

1995 (1)

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comp. Meth. and Prog. Biomed. 47, 131–146 (1995).
[Crossref]

1994 (1)

V. V. Tuchin, S. R. Utz, and I. V. Yaroslavsky, “Tissue optics, light distribution, and spectroscopy,” Opt. Eng. 33, 3178–3188 (1994).
[Crossref]

1993 (2)

1990 (1)

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

1941 (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Aarnoudse, J.

Aernouts, B.

Alerstam, E.

Alianelli, L.

Allen, D.

J. Hwang, H.-J. Kim, P. Lemaillet, H. Wabnitz, D. Grosenick, L. Yang, T. Gladytz, D. McClatchy, D. Allen, K. Briggman, and B. Pogue, “Polydimethylsiloxane tissue-mimicking phantoms for quantitative optical medical imaging standards,” Proc. SPIE 10056, 1005603 (2017).
[Crossref]

Allen, D. W.

P. Lemaillet, J.-P. Bouchard, and D. W. Allen, “Development of traceable measurement of the diffuse optical properties of solid reference standards for biomedical optics at National Institute of Standards and Technology,” Appl. Opt. 54, 6118–6127 (2015).
[Crossref] [PubMed]

P. Lemaillet, J.-P. Bouchard, J. Hwang, and D. W. Allen, “Double-integrating-sphere system at the National Institute of Standards and Technology in support of measurement standards for the determination of optical properties of tissue-mimicking phantoms,” J. Biomed. Opt. 20, 121310 (2015).
[Crossref] [PubMed]

Q. Li, B. J. Lee, Z. M. Zhang, and D. W. Allen, “Light scattering of semitransparent sintered polytetrafluoroethylene films,” J. Biomed. Opt. 13, 054064 (2008).
[Crossref] [PubMed]

Andersson-Engels, S.

Ayers, F. R.

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[Crossref] [PubMed]

Barnes, P.

P. Barnes, E. Early, and A. Parr, “NIST Special Publication 250-48,” US Dept. of Commerce (1998).

Berger, A. J.

Bevilacqua, F.

Bigio, I. J.

Blondel, W. C. P. M.

Blumetti, C.

Bonyadi, M. R.

M. R. Bonyadi and Z. Michalewicz, “Particle Swarm Optimization for Single Objective Continuous Space Problems: A Review,” Evolutionary Computation 25, 1–54 (2017).
[Crossref]

Botwicz, M.

Bouchard, J.-P.

Boyer, J.

Briggman, K.

J. Hwang, H.-J. Kim, P. Lemaillet, H. Wabnitz, D. Grosenick, L. Yang, T. Gladytz, D. McClatchy, D. Allen, K. Briggman, and B. Pogue, “Polydimethylsiloxane tissue-mimicking phantoms for quantitative optical medical imaging standards,” Proc. SPIE 10056, 1005603 (2017).
[Crossref]

Bykov, A. V.

M. S. Wróbel, A. P. Popov, A. V. Bykov, M. Kinnunen, M. Jedrzejewska-Szczerska, and V. V. Tuchin, “Measurements of fundamental properties of homogeneous tissue phantoms,” J. Biomed. Opt. 20, 045004 (2015).
[Crossref] [PubMed]

Cerussi, A. E.

Chen, X.

Chen, Y.-C.

T. Moffitt, Y.-C. Chen, and S. A. Prahl, “Preparation and characterization of polyurethane optical phantoms,” J. Biomed. Opt. 14, 041103 (2006).
[Crossref]

Cheong, W.

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

Cletus, B.

B. Cletus, R. Künnemeyer, P. Martinsen, A. McGlone, and R. Jordan, “Characterizing liquid turbid media by frequency-domain photon-migration spectroscopy,” J. Biomed. Opt. 14, 024041 (2009).
[Crossref] [PubMed]

Contini, D.

A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaeli, and G. Zaccanti, “Monte Carlo procedure for investigating light propagation and imaging of highly scattering media,” Appl. Opt. 37, 7392–7400 (1998).
[Crossref]

H. Wabnitz, A. Jelzow, M. Mazurenka, O. Steinkellner, R. Macdonald, A. Pifferi, A. Torricelli, D. Contini, L. Zucchelli, L. Spinelli, R. Cubeddu, D. Milej, N. Zolek, M. Kacprzak, A. Liebert, S. Magazov, J. Hebden, F. Martelli, P. D. Ninni, and G. Zaccanti, “Performance assessment of time-domain optical brain imagers: The neuropt protocol,” in Biomedical Optics and 3-D Imaging, (Optical Society of America, 2012), p. BSu2A.4.

Cox, D. R.

D. R. Cox and E. J. Snell, Analysis of Binary Data (Chapman and Hall, 1989), p. 181.

D. R. Cox and D. V. Hinkley, Theoretical Statistics (Chapman and Hall, 1974), p. 343.

Cubeddu, R.

L. Spinelli, F. Martelli, A. Farina, A. Pifferi, A. Torricelli, R. Cubeddu, and G. Zaccanti, “Calibration of scattering and absorption properties of a liquid diffusive medium at NIR wavelengths. Time-resolved method,” Opt. Express 15, 6589–6604 (2007).
[Crossref] [PubMed]

H. Wabnitz, A. Jelzow, M. Mazurenka, O. Steinkellner, R. Macdonald, A. Pifferi, A. Torricelli, D. Contini, L. Zucchelli, L. Spinelli, R. Cubeddu, D. Milej, N. Zolek, M. Kacprzak, A. Liebert, S. Magazov, J. Hebden, F. Martelli, P. D. Ninni, and G. Zaccanti, “Performance assessment of time-domain optical brain imagers: The neuropt protocol,” in Biomedical Optics and 3-D Imaging, (Optical Society of America, 2012), p. BSu2A.4.

Cuccia, D.

Cuccia, D. J.

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[Crossref] [PubMed]

Dassel, A.

De Mul, F.

Di Ninni, P.

Ding, J.

Dolgushin, S.

S. Tereshchenko, S. Dolgushin, and S. Titenok, “An imperfection of time-dependent diffusion models for a determination of scattering medium optical properties,” Opt. Comm. 306, 26–34 (2013).
[Crossref]

Du, Y.

Dual, C.

Dunn, A. K.

Durduran, T.

Durkin, A. J.

G. T. Kennedy, G. R. Lentsch, B. Trieu, A. Ponticorvo, R. B. Saager, and A. J. Durkin, “Solid tissue simulating phantoms having absorption at 970 nm for diffuse optics,” J. Biomed. Opt. 22, 076013 (2017).
[Crossref]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[Crossref] [PubMed]

Early, E.

P. Barnes, E. Early, and A. Parr, “NIST Special Publication 250-48,” US Dept. of Commerce (1998).

Farina, A.

Feng, Y.

Fortin, M.

Foschum, F.

Fuslier, T.

Gardner, A.

Garra, B. S.

W. C. Vogt, C. Jia, K. A. Wear, B. S. Garra, and T. J. Pfefer, “Biologically relevant photoacoustic imaging phantoms with tunable optical and acoustic properties,” J. Biomed. Opt. 21, 101405 (2016).
[Crossref] [PubMed]

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S. Tereshchenko, S. Dolgushin, and S. Titenok, “An imperfection of time-dependent diffusion models for a determination of scattering medium optical properties,” Opt. Comm. 306, 26–34 (2013).
[Crossref]

Torricelli, A.

L. Spinelli, F. Martelli, A. Farina, A. Pifferi, A. Torricelli, R. Cubeddu, and G. Zaccanti, “Calibration of scattering and absorption properties of a liquid diffusive medium at NIR wavelengths. Time-resolved method,” Opt. Express 15, 6589–6604 (2007).
[Crossref] [PubMed]

H. Wabnitz, A. Jelzow, M. Mazurenka, O. Steinkellner, R. Macdonald, A. Pifferi, A. Torricelli, D. Contini, L. Zucchelli, L. Spinelli, R. Cubeddu, D. Milej, N. Zolek, M. Kacprzak, A. Liebert, S. Magazov, J. Hebden, F. Martelli, P. D. Ninni, and G. Zaccanti, “Performance assessment of time-domain optical brain imagers: The neuropt protocol,” in Biomedical Optics and 3-D Imaging, (Optical Society of America, 2012), p. BSu2A.4.

Trieu, B.

G. T. Kennedy, G. R. Lentsch, B. Trieu, A. Ponticorvo, R. B. Saager, and A. J. Durkin, “Solid tissue simulating phantoms having absorption at 970 nm for diffuse optics,” J. Biomed. Opt. 22, 076013 (2017).
[Crossref]

Tromberg, B. J.

Tsuta, M.

Tuchin, V. V.

M. S. Wróbel, A. P. Popov, A. V. Bykov, M. Kinnunen, M. Jedrzejewska-Szczerska, and V. V. Tuchin, “Measurements of fundamental properties of homogeneous tissue phantoms,” J. Biomed. Opt. 20, 045004 (2015).
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[Crossref]

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V. V. Tuchin, S. R. Utz, and I. V. Yaroslavsky, “Tissue optics, light distribution, and spectroscopy,” Opt. Eng. 33, 3178–3188 (1994).
[Crossref]

Van Beers, R.

van Gemert, M. J.

Veilleux, I.

Venugopalan, V.

Vogt, W. C.

W. C. Vogt, C. Jia, K. A. Wear, B. S. Garra, and T. J. Pfefer, “Biologically relevant photoacoustic imaging phantoms with tunable optical and acoustic properties,” J. Biomed. Opt. 21, 101405 (2016).
[Crossref] [PubMed]

Wabnitz, H.

J. Hwang, H.-J. Kim, P. Lemaillet, H. Wabnitz, D. Grosenick, L. Yang, T. Gladytz, D. McClatchy, D. Allen, K. Briggman, and B. Pogue, “Polydimethylsiloxane tissue-mimicking phantoms for quantitative optical medical imaging standards,” Proc. SPIE 10056, 1005603 (2017).
[Crossref]

A. Liebert, H. Wabnitz, D. Grosenick, M. Möller, R. Macdonald, and H. Rinneberg, “Evaluation of optical properties of highly scattering media by moments of distributions of times of flight of photons,” Appl. Opt. 42, 5785–5792 (2003).
[Crossref] [PubMed]

H. Wabnitz, A. Jelzow, M. Mazurenka, O. Steinkellner, R. Macdonald, A. Pifferi, A. Torricelli, D. Contini, L. Zucchelli, L. Spinelli, R. Cubeddu, D. Milej, N. Zolek, M. Kacprzak, A. Liebert, S. Magazov, J. Hebden, F. Martelli, P. D. Ninni, and G. Zaccanti, “Performance assessment of time-domain optical brain imagers: The neuropt protocol,” in Biomedical Optics and 3-D Imaging, (Optical Society of America, 2012), p. BSu2A.4.

Wang, L.

Watté, R.

Wear, K. A.

W. C. Vogt, C. Jia, K. A. Wear, B. S. Garra, and T. J. Pfefer, “Biologically relevant photoacoustic imaging phantoms with tunable optical and acoustic properties,” J. Biomed. Opt. 21, 101405 (2016).
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[Crossref] [PubMed]

Yang, L.

J. Hwang, H.-J. Kim, P. Lemaillet, H. Wabnitz, D. Grosenick, L. Yang, T. Gladytz, D. McClatchy, D. Allen, K. Briggman, and B. Pogue, “Polydimethylsiloxane tissue-mimicking phantoms for quantitative optical medical imaging standards,” Proc. SPIE 10056, 1005603 (2017).
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Zamora-Rojas, E.

Zhang, Z. M.

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L. Spinelli, M. Botwicz, N. Zolek, M. Kacprzak, D. Milej, P. Sawosz, A. Liebert, U. Weigel, T. Durduran, F. Foschum, and et al.., “Determination of reference values for optical properties of liquid phantoms based on Intralipid and India ink,” Biomed. Opt. Express 5, 2037–2053 (2014).
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H. Wabnitz, A. Jelzow, M. Mazurenka, O. Steinkellner, R. Macdonald, A. Pifferi, A. Torricelli, D. Contini, L. Zucchelli, L. Spinelli, R. Cubeddu, D. Milej, N. Zolek, M. Kacprzak, A. Liebert, S. Magazov, J. Hebden, F. Martelli, P. D. Ninni, and G. Zaccanti, “Performance assessment of time-domain optical brain imagers: The neuropt protocol,” in Biomedical Optics and 3-D Imaging, (Optical Society of America, 2012), p. BSu2A.4.

Zucchelli, L.

H. Wabnitz, A. Jelzow, M. Mazurenka, O. Steinkellner, R. Macdonald, A. Pifferi, A. Torricelli, D. Contini, L. Zucchelli, L. Spinelli, R. Cubeddu, D. Milej, N. Zolek, M. Kacprzak, A. Liebert, S. Magazov, J. Hebden, F. Martelli, P. D. Ninni, and G. Zaccanti, “Performance assessment of time-domain optical brain imagers: The neuropt protocol,” in Biomedical Optics and 3-D Imaging, (Optical Society of America, 2012), p. BSu2A.4.

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M. S. Wróbel, A. P. Popov, A. V. Bykov, M. Kinnunen, M. Jedrzejewska-Szczerska, and V. V. Tuchin, “Measurements of fundamental properties of homogeneous tissue phantoms,” J. Biomed. Opt. 20, 045004 (2015).
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P. Lemaillet, J.-P. Bouchard, J. Hwang, and D. W. Allen, “Double-integrating-sphere system at the National Institute of Standards and Technology in support of measurement standards for the determination of optical properties of tissue-mimicking phantoms,” J. Biomed. Opt. 20, 121310 (2015).
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D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
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G. T. Kennedy, G. R. Lentsch, B. Trieu, A. Ponticorvo, R. B. Saager, and A. J. Durkin, “Solid tissue simulating phantoms having absorption at 970 nm for diffuse optics,” J. Biomed. Opt. 22, 076013 (2017).
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W. C. Vogt, C. Jia, K. A. Wear, B. S. Garra, and T. J. Pfefer, “Biologically relevant photoacoustic imaging phantoms with tunable optical and acoustic properties,” J. Biomed. Opt. 21, 101405 (2016).
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E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. 13, 041304 (2008).
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Opt. Comm. (1)

S. Tereshchenko, S. Dolgushin, and S. Titenok, “An imperfection of time-dependent diffusion models for a determination of scattering medium optical properties,” Opt. Comm. 306, 26–34 (2013).
[Crossref]

Opt. Eng. (1)

V. V. Tuchin, S. R. Utz, and I. V. Yaroslavsky, “Tissue optics, light distribution, and spectroscopy,” Opt. Eng. 33, 3178–3188 (1994).
[Crossref]

Opt. Express (7)

F. Martelli and G. Zaccanti, “Calibration of scattering and absorption properties of a liquid diffusive medium at NIR wavelengths. CW method,” Opt. Express 15, 486–500 (2007).
[Crossref] [PubMed]

L. Spinelli, F. Martelli, A. Farina, A. Pifferi, A. Torricelli, R. Cubeddu, and G. Zaccanti, “Calibration of scattering and absorption properties of a liquid diffusive medium at NIR wavelengths. Time-resolved method,” Opt. Express 15, 6589–6604 (2007).
[Crossref] [PubMed]

J.-P. Bouchard, I. Veilleux, R. Jedidi, I. Noiseux, M. Fortin, and O. Mermut, “Reference optical phantoms for diffuse optical spectroscopy. part 1–error analysis of a time resolved transmittance characterization method,” Opt. Express 18, 11495–11507 (2010).
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P. Di Ninni, F. Martelli, and G. Zaccanti, “The use of India ink in tissue-simulating phantoms,” Opt. Express 18, 26854–26865 (2010).
[Crossref]

Opt. Lett. (2)

Phys. in Med. and Biol. (1)

P. Di Ninni, F. Martelli, and G. Zaccanti, “Intralipid: towards a diffusive reference standard for optical tissue phantoms,” Phys. in Med. and Biol. 56, N21 (2010).
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Certain commercial materials and equipment are identified in order to adequately specify the experimental procedure. Such identification does not imply recommendation by the the authors’ institutions.

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Figures (13)

Fig. 1
Fig. 1

Experimental schematic for the ARS measurements. Here, R is distance between the sample first face and the detector aperture θv is the viewing angle in reflectance, θ v is the viewing angle in transmittance, and t is the thickness of the sample.

Fig. 2
Fig. 2

Search boxes for three trials with simulated data. The bounds of the search box relative to the exact answer are shown: (a) upper bound for µt, (b) upper bound for ηa, (c) lower bound for µt, and (d) lower bound for ηa. The blue, green, and red curves correspond to simulated data created with 2 · 105, 2 · 106, and 2 · 107, photons respectively, as indicated by the vertical arrows. The value Nphot corresponds to the number of photons used in each of 12 iterations, starting from 2 · 104 and doubling 11 times to 4.096 · 107. The simulation used the parameters μ s ( 0 ) = 3 mm 1 and μ a ( 0 ) = 0.01 mm 1 corresponding to μ t ( 0 ) = 3.01 mm 1 and η a ( 0 ) = 0.003322 mm 1, thickness t = 4.76 mm, index of refraction n = 1.487, and asymmetry parameter g = 0.621.

Fig. 3
Fig. 3

The quadratic approximation to the likelihood surface is obtained from a fit to the importance sampling results on a 3×3 grid including the maximum likelihood point found on the grid.

Fig. 4
Fig. 4

The log likelihood surface as calculated by the importance sampling method is shown in green tones. The quadratic approximation to the same surface is shown in peach tones. The small patch at the top of the graph is the context for the whole surface shown in Fig. 3.

Fig. 5
Fig. 5

Log likelihood surface as calculated by the explicit grid method. The region shown was calculated with 2.56 · 106 photons per grid point with the region selected on the eighth iteration building up from 2 · 104 photons per grid point over a much larger region.

Fig. 6
Fig. 6

Log likelihood surface as calculated by the Particle Swarm Optimization. Each point was calculated with 1.28 · 106 photons. All of the points evaluated at any iteration are shown. A constant number of photons is used at every iteration.

Fig. 7
Fig. 7

The convergence of the Particle Swarm Optimization algorithm is illustrated in this figure. Each region in the convex hull of the points in the swarm at a given iteration. The first iteration is purple and is placed at the bottom of the stack. Higher iterations are places upon it with the color tending toward yellow. Iterations fifteen and higher are represented in the same color yellow. The variables are plotted in the natural logarithm because the PSO operates in this space.

Fig. 8
Fig. 8

Angle-resolved scatter is given for three wavelengths, 543 nm (green), 632 nm (orange), and 805 nm (black) for the sample with a thickness t = 6.10 mm. The values of µa and µs are shown in Fig. 9. Experimental points are given as dots. For convenience, the value of 0 was assigned to the measurement at 90°. The solid line is the prediction at the best fit found by the profile log likelihood importance sampling algorithm.

Fig. 9
Fig. 9

Point estimates and 95% confidence regions for 9 measured samples. The wavelengths are indicated, and the sample thickness is 6.10 mm (red), 8.10 mm (blue), and 10.12 mm (black).

Fig. 10
Fig. 10

Effect of the choice of the random seed is shown. The point estimates and 95% confidence regions shown with solid red circles and solid red lines are replotted from Fig. 9 for the sample thickness 6.10 mm. The hollow red circles and black lines represent other solutions which differ only in the seed of the random number generator.

Fig. 11
Fig. 11

The effect of the finite detector radius is shown. Pairs of results are generated with the same seed for for λ = 543 nm, 632 nm, and 805 nm with the detector at a finite radius (solid circles and 95% confidence contours) and at an effectively infinite radius (hollow circles and dashed contours). The results are given for samples of two thicknesses, 6.10 mm and 10.12 mm as marked in the figure.

Fig. 12
Fig. 12

The effect of changing the thickness and the index of refraction is given for a wavelength of 805 nm. The red and black symbols refer to the sample with a thicknesses of 6.10 mm and 10.12 mm, respectively. The solid dots and ellipses are carried over from Fig. 11. The solid and dashed arrows describe calculations with the the index of refraction and the sample thickness respectively changed at the k = 2 level as quoted in the text. The tails of the arrows represent lowering the central value at the k = 2 level and the heads of the arrows represent raising the central value at the k = 2 level. The shaft of the arrow is a guide to the eye.

Fig. 13
Fig. 13

Comparison of maximum likelihood estimates (large green circles) with ground truth (small red circles) for several values of µa and µs with fixed parameters g = 0.621, t = 6.10 mm, and n = 1.5. The ground truth was created with the forward program using 212 · 104 = 40 960 000 photons, vs. 27 · 104 = 1 280 000 photons in the final round for the inverse problem.

Tables (3)

Tables Icon

Table 1 Importance sampling weighting factors for various processes, based on the probability density function (PDF) or its integral, the cumulative density function (CDF). The superscript (0) refers to a reference value which is sampled with Monte Carlo. Note that all weights are 1 for the reference value. Here, is a distance a particle propagates in the medium whether it stops in the interior or at the boundary.

Tables Icon

Table 2 High-level flowcharts for importance-sampling-based inverse Monte Carlo algorithm for the determination of optical parameters. [49] The variables are defined in the text. Without the vector of weights, the forward model is very close to that of MCML. [40]

Tables Icon

Table 3 The run time is given for Nphot = 1.28 · 106 Monte Carlo trials, which is our principal operating point in the paper. for the sample with t=6.10 mm and λ = 805 nm. The number of processors used is Nproc. The notation “Fwd” means that only the forward problem is solved (i.e., the ARS is found for one set of parameters) “Inv” refers to the inverse problem (i.e., solving for a set of parameters). The number of iterations is quoted as the number necessary to reach convergence. The number of times the 1.28 106 Monte Carlo trials were performed in the final iteration is reported as Neval. In the·case of PSO, this number of evaluations is performed at every iteration. In the present methods, the number of Monte Carlo trials increases from 2 · 104 to the final value by doubling at each iteration. The times refer to runs on a Dell Optiplex 9010 AIO computer with an Intel i5-3570S quad-core processor running at 3.10 GHz. The present explicit grid method is estimated because it was implemented in a scripting language on a different computer.

Equations (12)

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

A R S = P s Ω P i
p H G ( θ ; g ) = 1 g 2 2 ( 1 + g 2 2 g cos θ ) 3 / 2 ,
r = α v ^ + r 0
α = v ^ 0 r 0 + [ ( v ^ 0 r 0 ) 2 + R 2 r 0 2 ] 1 / 2
y i = α i ( Θ ) + ϵ i α ^ i ( Θ ) = α i ( Θ ) + δ i
y i α ^ i ( Θ ) ~ Normal ( 0 , σ ϵ 2 + σ δ 2 )
( Θ , μ s , V ) = N 2 log ( 2 π ) N 2 log ( V ) 1 2 V i = 1 N [ y i α ^ i ( Θ ) ] 2
V = 1 N i = 1 N [ y i α ^ i ( Θ ) ] 2 ,
P ( μ a , μ s ) = N 2 { log [ 1 N i = 1 N [ y i α ^ i ( Θ ) ] 2 ] + log ( 2 π ) + 1 } .
P ( x m l e ) P ( x ) 1 2 ( χ v 2 ) 1 ( p )
V i + i = w i V i + c 1 1 ( X cl ( opt ) X i ) + c 2 2 ( X cl ( opt ) X i )
X i + i = X i + V i + 1

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