Determining the optical properties of a gelatin‑TiO<sub>2</sub> phantom at 780 nm

H. Günhan Akarçay; Stefan Preisser; Martin Frenz; Jaro Rička

doi:10.1364/BOE.3.000418

Determining the optical properties of a gelatin‑TiO₂ phantom at 780 nm

H. Günhan Akarçay, Stefan Preisser, Martin Frenz, and Jaro Rička

Biomedical Optics Express
Vol. 3,
Issue 3,
pp. 418-434
(2012)
•doi: 10.1364/BOE.3.000418

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H. Günhan Akarçay, Stefan Preisser, Martin Frenz, and Jaro Rička, "Determining the optical properties of a gelatin‑TiO₂ phantom at 780 nm," Biomed. Opt. Express 3, 418-434 (2012)

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Abstract

Tissue phantoms play a central role in validating biomedical imaging techniques. Here we employ a series of methods that aim to fully determine the optical properties, i.e., the refractive index n, absorption coefficient μ_a, transport mean free path $ℓ^{*}$ , and scattering coefficient μ_s of a TiO₂ in gelatin phantom intended for use in optoacoustic imaging. For the determination of the key parameters μ_a and $ℓ^{*}$ , we employ a variant of time of flight measurements, where fiber optodes are immersed into the phantom to minimize the influence of boundaries. The robustness of the method was verified with Monte Carlo simulations, where the experimentally obtained values served as input parameters for the simulations. The excellent agreement between simulations and experiments confirmed the reliability of the results. The parameters determined at 780 nm are $n = 1.359 (\pm 0.002)$ , ${μ^{'}}_{s} = 1 / ℓ^{*} = 0.22 (\pm 0.02) {mm}^{-1}$ , $μ_{a} = 0.0053(+0.0006-0.0003) {mm}^{-1}$ , and $μ_{s} = 2.86 (\pm 0.04) {mm}^{-1} .$ The asymmetry parameter g obtained from the parameters $ℓ^{*}$ and ${μ^{'}}_{s}$ is 0.93, which indicates that the scattering entities are not bare TiO₂ particles but large sparse clusters. The interaction between the scattering particles and the gelatin matrix should be taken into account when developing such phantoms.

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B. W. Pogue and M. S. Patterson, “Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry,” J. Biomed. Opt. 11(4), 041102 (2006).
[Crossref] [PubMed]
M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77(4), 041101 (2006).
[Crossref]
K. Zell, J. I. Sperl, M. W. Vogel, R. Niessner, and C. Haisch, “Acoustical properties of selected tissue phantom materials for ultrasound imaging,” Phys. Med. Biol. 52(20), N475–N484 (2007).
[Crossref] [PubMed]
G. M. Spirou, A. A. Oraevsky, I. A. Vitkin, and W. M. Whelan, “Optical and acoustic properties at 1064 nm of polyvinyl chloride-plastisol for use as a tissue phantom in biomedical optoacoustics,” Phys. Med. Biol. 50(14), N141–N153 (2005).
[Crossref] [PubMed]
J. R. Cook, R. R. Bouchard, and S. Y. Emelianov, “Tissue-mimicking phantoms for photoacoustic and ultrasonic imaging,” Biomed. Opt. Express 2(11), 3193–3206 (2011).
[Crossref] [PubMed]
M. Jaeger, L. Siegenthaler, M. Kitz, and M. Frenz, “Reduction of background in optoacoustic image sequences obtained under tissue deformation,” J. Biomed. Opt. 14(5), 054011 (2009).
[Crossref] [PubMed]
M. Jaeger, S. Preisser, M. Kitz, D. Ferrara, S. Senegas, D. Schweizer, and M. Frenz, “Improved contrast deep optoacoustic imaging using displacement-compensated averaging: breast tumour phantom studies,” Phys. Med. Biol. 56(18), 5889–5901 (2011).
[Crossref] [PubMed]
A. Kim and B. C. Wilson, “Measurement of ex vivo and in vivo tissue optical properties: methods and theories,” in Optical-Thermal Response of Laser-Irradiated Tissue, 2nd ed. (Wiley, Springer Netherlands, 2011), Chap. 8.
T. D. Khokhlova, I. M. Pelivanov, V. V. Kozhushko, A. N. Zharinov, V. S. Solomatin, and A. A. Karabutov, “Optoacoustic imaging of absorbing objects in a turbid medium: ultimate sensitivity and application to breast cancer diagnostics,” Appl. Opt. 46(2), 262–272 (2007).
[Crossref] [PubMed]
T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol. 47(16), 2847–2861 (2002).
[Crossref] [PubMed]
N. Shah, A. Cerussi, C. Eker, J. Espinoza, J. Butler, J. Fishkin, R. Hornung, and B. Tromberg, “Noninvasive functional optical spectroscopy of human breast tissue,” Proc. Natl. Acad. Sci. U.S.A. 98(8), 4420–4425 (2001).
[Crossref] [PubMed]
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M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28(12), 2331–2336 (1989).
[Crossref] [PubMed]
S. Fantini, M. A. Franceschini, and E. Gratton, “Semi-infinite-geometry boundary problem for light migration in highly scattering media: a frequency-domain study in the diffusion approximation,” J. Opt. Soc. Am. B 11(10), 2128–2138 (1994).
[Crossref]
A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40(11), 1957–1975 (1995).
[Crossref] [PubMed]
A. Kienle and M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14(1), 246–254 (1997).
[Crossref] [PubMed]
S. L. Jacques and B. W. Pogue, “Tutorial on diffuse light transport,” J. Biomed. Opt. 13(4), 041302 (2008).
[Crossref] [PubMed]
F. Ayers, A. Grant, F. Kuo, D. J. Cuccia, and A. J. Durkin, “Fabrication and characterization of silicone-based tissue phantoms with tunable optical properties in the visible and near infrared domain,” Proc. SPIE 6870, 687007, 687007-9 (2008).
[Crossref]
T. Moffitt, Y. C. Chen, and S. A. Prahl, “Preparation and characterization of polyurethane optical phantoms,” J. Biomed. Opt. 11(4), 041103 (2006).
[Crossref] [PubMed]
B. W. Pogue, L. Lilge, M. S. Patterson, B. C. Wilson, and T. Hasan, “Absorbed photodynamic dose from pulsed versus continuous wave light examined with tissue-simulating dosimeters,” Appl. Opt. 36(28), 7257–7269 (1997).
[Crossref] [PubMed]
E. S. Thiele and R. H. French, “Light-scattering properties of representative, morphological rutile,” J. Am. Ceram. Soc. 81(3), 469–479 (1998).
[Crossref]
J. Rička and M. Frenz, “From electrodynamics to Monte Carlo simulations,” in Optical-Thermal Response of Laser-Irradiated Tissue, 2nd ed. (Wiley, Springer Netherlands, 2011), Chap. 7.
E. Moreels, W. De Ceuninck, and R. Finsy, “Measurements of the Rayleigh ratio of some pure liquids at several laser light wavelengths,” J. Chem. Phys. 86(2), 618–623 (1987).
[Crossref]
H. G. Akarçay and J. Rička, “Simulating light propagation: towards realistic tissue models,” Proc. SPIE 8088, 80880K (2011).
[Crossref]
F. Bevilacqua and C. Depeursinge, “Monte Carlo study of diffuse reflectance at source–detector separations close to one transport mean free path,” J. Opt. Soc. Am. A 16(12), 2935–2945 (1999).
[Crossref]
J. R. Mourant, J. Boyer, A. H. Hielscher, and I. J. Bigio, “Influence of the scattering phase function on light transport measurements in turbid media performed with small source-detector separations,” Opt. Lett. 21(7), 546–548 (1996).
[Crossref] [PubMed]
K. Tahir and C. Dainty, “Experimental measurements of light scattering from samples with specified optical properties,” J. Opt. A, Pure Appl. Opt. 7(5), 207–214 (2005).
[Crossref]

2011 (3)

M. Jaeger, S. Preisser, M. Kitz, D. Ferrara, S. Senegas, D. Schweizer, and M. Frenz, “Improved contrast deep optoacoustic imaging using displacement-compensated averaging: breast tumour phantom studies,” Phys. Med. Biol. 56(18), 5889–5901 (2011).
[Crossref] [PubMed]

J. R. Cook, R. R. Bouchard, and S. Y. Emelianov, “Tissue-mimicking phantoms for photoacoustic and ultrasonic imaging,” Biomed. Opt. Express 2(11), 3193–3206 (2011).
[Crossref] [PubMed]

H. G. Akarçay and J. Rička, “Simulating light propagation: towards realistic tissue models,” Proc. SPIE 8088, 80880K (2011).
[Crossref]

2009 (1)

M. Jaeger, L. Siegenthaler, M. Kitz, and M. Frenz, “Reduction of background in optoacoustic image sequences obtained under tissue deformation,” J. Biomed. Opt. 14(5), 054011 (2009).
[Crossref] [PubMed]

2008 (2)

S. L. Jacques and B. W. Pogue, “Tutorial on diffuse light transport,” J. Biomed. Opt. 13(4), 041302 (2008).
[Crossref] [PubMed]

F. Ayers, A. Grant, F. Kuo, D. J. Cuccia, and A. J. Durkin, “Fabrication and characterization of silicone-based tissue phantoms with tunable optical properties in the visible and near infrared domain,” Proc. SPIE 6870, 687007, 687007-9 (2008).
[Crossref]

2007 (2)

T. D. Khokhlova, I. M. Pelivanov, V. V. Kozhushko, A. N. Zharinov, V. S. Solomatin, and A. A. Karabutov, “Optoacoustic imaging of absorbing objects in a turbid medium: ultimate sensitivity and application to breast cancer diagnostics,” Appl. Opt. 46(2), 262–272 (2007).
[Crossref] [PubMed]

K. Zell, J. I. Sperl, M. W. Vogel, R. Niessner, and C. Haisch, “Acoustical properties of selected tissue phantom materials for ultrasound imaging,” Phys. Med. Biol. 52(20), N475–N484 (2007).
[Crossref] [PubMed]

2006 (3)

B. W. Pogue and M. S. Patterson, “Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry,” J. Biomed. Opt. 11(4), 041102 (2006).
[Crossref] [PubMed]

M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77(4), 041101 (2006).
[Crossref]

T. Moffitt, Y. C. Chen, and S. A. Prahl, “Preparation and characterization of polyurethane optical phantoms,” J. Biomed. Opt. 11(4), 041103 (2006).
[Crossref] [PubMed]

2005 (2)

G. M. Spirou, A. A. Oraevsky, I. A. Vitkin, and W. M. Whelan, “Optical and acoustic properties at 1064 nm of polyvinyl chloride-plastisol for use as a tissue phantom in biomedical optoacoustics,” Phys. Med. Biol. 50(14), N141–N153 (2005).
[Crossref] [PubMed]

K. Tahir and C. Dainty, “Experimental measurements of light scattering from samples with specified optical properties,” J. Opt. A, Pure Appl. Opt. 7(5), 207–214 (2005).
[Crossref]

2002 (1)

T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol. 47(16), 2847–2861 (2002).
[Crossref] [PubMed]

2001 (1)

N. Shah, A. Cerussi, C. Eker, J. Espinoza, J. Butler, J. Fishkin, R. Hornung, and B. Tromberg, “Noninvasive functional optical spectroscopy of human breast tissue,” Proc. Natl. Acad. Sci. U.S.A. 98(8), 4420–4425 (2001).
[Crossref] [PubMed]

1999 (1)

F. Bevilacqua and C. Depeursinge, “Monte Carlo study of diffuse reflectance at source–detector separations close to one transport mean free path,” J. Opt. Soc. Am. A 16(12), 2935–2945 (1999).
[Crossref]

1998 (1)

E. S. Thiele and R. H. French, “Light-scattering properties of representative, morphological rutile,” J. Am. Ceram. Soc. 81(3), 469–479 (1998).
[Crossref]

1997 (2)

A. Kienle and M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14(1), 246–254 (1997).
[Crossref] [PubMed]

B. W. Pogue, L. Lilge, M. S. Patterson, B. C. Wilson, and T. Hasan, “Absorbed photodynamic dose from pulsed versus continuous wave light examined with tissue-simulating dosimeters,” Appl. Opt. 36(28), 7257–7269 (1997).
[Crossref] [PubMed]

1996 (1)

J. R. Mourant, J. Boyer, A. H. Hielscher, and I. J. Bigio, “Influence of the scattering phase function on light transport measurements in turbid media performed with small source-detector separations,” Opt. Lett. 21(7), 546–548 (1996).
[Crossref] [PubMed]

1995 (1)

A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40(11), 1957–1975 (1995).
[Crossref] [PubMed]

1994 (1)

S. Fantini, M. A. Franceschini, and E. Gratton, “Semi-infinite-geometry boundary problem for light migration in highly scattering media: a frequency-domain study in the diffusion approximation,” J. Opt. Soc. Am. B 11(10), 2128–2138 (1994).
[Crossref]

1989 (1)

M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28(12), 2331–2336 (1989).
[Crossref] [PubMed]

1987 (1)

E. Moreels, W. De Ceuninck, and R. Finsy, “Measurements of the Rayleigh ratio of some pure liquids at several laser light wavelengths,” J. Chem. Phys. 86(2), 618–623 (1987).
[Crossref]

Akarçay, H. G.

H. G. Akarçay and J. Rička, “Simulating light propagation: towards realistic tissue models,” Proc. SPIE 8088, 80880K (2011).
[Crossref]

Ayers, F.

Bevilacqua, F.

Bigio, I. J.

Bouchard, R. R.

J. R. Cook, R. R. Bouchard, and S. Y. Emelianov, “Tissue-mimicking phantoms for photoacoustic and ultrasonic imaging,” Biomed. Opt. Express 2(11), 3193–3206 (2011).
[Crossref] [PubMed]

Boyer, J.

Butler, J.

Cerussi, A.

Chance, B.

Chen, Y. C.

T. Moffitt, Y. C. Chen, and S. A. Prahl, “Preparation and characterization of polyurethane optical phantoms,” J. Biomed. Opt. 11(4), 041103 (2006).
[Crossref] [PubMed]

Choe, R.

Cook, J. R.

J. R. Cook, R. R. Bouchard, and S. Y. Emelianov, “Tissue-mimicking phantoms for photoacoustic and ultrasonic imaging,” Biomed. Opt. Express 2(11), 3193–3206 (2011).
[Crossref] [PubMed]

Cuccia, D. J.

Culver, J. P.

Dainty, C.

K. Tahir and C. Dainty, “Experimental measurements of light scattering from samples with specified optical properties,” J. Opt. A, Pure Appl. Opt. 7(5), 207–214 (2005).
[Crossref]

De Ceuninck, W.

E. Moreels, W. De Ceuninck, and R. Finsy, “Measurements of the Rayleigh ratio of some pure liquids at several laser light wavelengths,” J. Chem. Phys. 86(2), 618–623 (1987).
[Crossref]

Depeursinge, C.

Durduran, T.

Durkin, A. J.

Eker, C.

Emelianov, S. Y.

J. R. Cook, R. R. Bouchard, and S. Y. Emelianov, “Tissue-mimicking phantoms for photoacoustic and ultrasonic imaging,” Biomed. Opt. Express 2(11), 3193–3206 (2011).
[Crossref] [PubMed]

Espinoza, J.

Fantini, S.

Ferrara, D.

Finsy, R.

E. Moreels, W. De Ceuninck, and R. Finsy, “Measurements of the Rayleigh ratio of some pure liquids at several laser light wavelengths,” J. Chem. Phys. 86(2), 618–623 (1987).
[Crossref]

Fishkin, J.

Franceschini, M. A.

French, R. H.

E. S. Thiele and R. H. French, “Light-scattering properties of representative, morphological rutile,” J. Am. Ceram. Soc. 81(3), 469–479 (1998).
[Crossref]

Frenz, M.

Giammarco, J.

Grant, A.

Gratton, E.

Haisch, C.

Hasan, T.

Hielscher, A. H.

Holboke, M. J.

Hornung, R.

Jacques, S. L.

S. L. Jacques and B. W. Pogue, “Tutorial on diffuse light transport,” J. Biomed. Opt. 13(4), 041302 (2008).
[Crossref] [PubMed]

Jaeger, M.

Karabutov, A. A.

Khokhlova, T. D.

Kienle, A.

Kitz, M.

Kozhushko, V. V.

Kuo, F.

Lilge, L.

Moffitt, T.

T. Moffitt, Y. C. Chen, and S. A. Prahl, “Preparation and characterization of polyurethane optical phantoms,” J. Biomed. Opt. 11(4), 041103 (2006).
[Crossref] [PubMed]

Moreels, E.

E. Moreels, W. De Ceuninck, and R. Finsy, “Measurements of the Rayleigh ratio of some pure liquids at several laser light wavelengths,” J. Chem. Phys. 86(2), 618–623 (1987).
[Crossref]

Mourant, J. R.

Niessner, R.

Oraevsky, A. A.

Patterson, M. S.

B. W. Pogue and M. S. Patterson, “Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry,” J. Biomed. Opt. 11(4), 041102 (2006).
[Crossref] [PubMed]

Pelivanov, I. M.

Pogue, B. W.

S. L. Jacques and B. W. Pogue, “Tutorial on diffuse light transport,” J. Biomed. Opt. 13(4), 041302 (2008).
[Crossref] [PubMed]

B. W. Pogue and M. S. Patterson, “Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry,” J. Biomed. Opt. 11(4), 041102 (2006).
[Crossref] [PubMed]

Prahl, S. A.

T. Moffitt, Y. C. Chen, and S. A. Prahl, “Preparation and characterization of polyurethane optical phantoms,” J. Biomed. Opt. 11(4), 041103 (2006).
[Crossref] [PubMed]

Preisser, S.

Ricka, J.

H. G. Akarçay and J. Rička, “Simulating light propagation: towards realistic tissue models,” Proc. SPIE 8088, 80880K (2011).
[Crossref]

Schweizer, D.

Senegas, S.

Shah, N.

Siegenthaler, L.

Solomatin, V. S.

Sperl, J. I.

Spirou, G. M.

Tahir, K.

K. Tahir and C. Dainty, “Experimental measurements of light scattering from samples with specified optical properties,” J. Opt. A, Pure Appl. Opt. 7(5), 207–214 (2005).
[Crossref]

Thiele, E. S.

E. S. Thiele and R. H. French, “Light-scattering properties of representative, morphological rutile,” J. Am. Ceram. Soc. 81(3), 469–479 (1998).
[Crossref]

Tittel, F. K.

Tromberg, B.

Vitkin, I. A.

Vogel, M. W.

Wang, L.

Wang, L. V.

M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77(4), 041101 (2006).
[Crossref]

Whelan, W. M.

Wilson, B. C.

Xu, M.

M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77(4), 041101 (2006).
[Crossref]

Yodh, A. G.

Zell, K.

Zharinov, A. N.

Zubkov, L.

Appl. Opt. (3)

Biomed. Opt. Express (1)

J. R. Cook, R. R. Bouchard, and S. Y. Emelianov, “Tissue-mimicking phantoms for photoacoustic and ultrasonic imaging,” Biomed. Opt. Express 2(11), 3193–3206 (2011).
[Crossref] [PubMed]

J. Am. Ceram. Soc. (1)

E. S. Thiele and R. H. French, “Light-scattering properties of representative, morphological rutile,” J. Am. Ceram. Soc. 81(3), 469–479 (1998).
[Crossref]

J. Biomed. Opt. (4)

T. Moffitt, Y. C. Chen, and S. A. Prahl, “Preparation and characterization of polyurethane optical phantoms,” J. Biomed. Opt. 11(4), 041103 (2006).
[Crossref] [PubMed]

B. W. Pogue and M. S. Patterson, “Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry,” J. Biomed. Opt. 11(4), 041102 (2006).
[Crossref] [PubMed]

S. L. Jacques and B. W. Pogue, “Tutorial on diffuse light transport,” J. Biomed. Opt. 13(4), 041302 (2008).
[Crossref] [PubMed]

J. Chem. Phys. (1)

E. Moreels, W. De Ceuninck, and R. Finsy, “Measurements of the Rayleigh ratio of some pure liquids at several laser light wavelengths,” J. Chem. Phys. 86(2), 618–623 (1987).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

K. Tahir and C. Dainty, “Experimental measurements of light scattering from samples with specified optical properties,” J. Opt. A, Pure Appl. Opt. 7(5), 207–214 (2005).
[Crossref]

J. Opt. Soc. Am. A (2)

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

Opt. Lett. (1)

Phys. Med. Biol. (5)

Proc. Natl. Acad. Sci. U.S.A. (1)

Proc. SPIE (2)

H. G. Akarçay and J. Rička, “Simulating light propagation: towards realistic tissue models,” Proc. SPIE 8088, 80880K (2011).
[Crossref]

Rev. Sci. Instrum. (1)

M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77(4), 041101 (2006).
[Crossref]

Other (4)

A. Kim and B. C. Wilson, “Measurement of ex vivo and in vivo tissue optical properties: methods and theories,” in Optical-Thermal Response of Laser-Irradiated Tissue, 2nd ed. (Wiley, Springer Netherlands, 2011), Chap. 8.

RefractiveIndex.INFO, http://refractiveindex.info/?group=CRYSTALS&material=TiO2 (retrieved Nov. 2011).

The International Association for the Properties of Water and Steam, “Release on the refractive index of ordinary water substance as a function of wavelength, temperature and pressure,” Erlangen, Germany (Sept. 1997).

J. Rička and M. Frenz, “From electrodynamics to Monte Carlo simulations,” in Optical-Thermal Response of Laser-Irradiated Tissue, 2nd ed. (Wiley, Springer Netherlands, 2011), Chap. 7.

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Fig. 1

Diagram summarizing the different stages of the work presented in this paper: the optical properties (refractive index, transport mean free path, absorption coefficient, scattering coefficient, and scattering law) of the phantom were experimentally determined by the indicated set of experiments. Monte Carlo simulations were performed ex-post to validate the experimental results.

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Fig. 2

Schematic representation of the principle of the time of flight (TOF) experiments: the light source – a picosecond laser – injects the light into the phantom at a point S; the individual photons are then subject to multiple scattering in the phantom (random photon paths are represented in red) until they exit the phantom or get absorbed in it. The photons that reach the detection point D (placed at a distance r from S) are detected and their individual TOF from S to D recorded in a histogram. This experiment is reproduced for different source-detector separations r and the obtained TOF distributions are then fitted to the diffusion model.

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Fig. 3

Detail of the setup used for the TOF measurements. (a) The gelatin phantom (5 L) is contained in a soda-lime glass aquarium. (b) and (c) In the holder-plate, the optodes can be placed at eight positions. Each pair of optode/orifice on the holder-plate corresponds to a different source-detector separation r. The tips of the needles are inside the phantom, at a depth of 40 mm beneath the phantom/air interface.

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Fig. 4

Example of the fit of the measured data (thin solid line in black) with the diffusion model (thicker line in red). The corresponding IRF is shown in green dots. For comparison, we also include the fit with the corresponding MC simulations (see Section 6) with the same parameters $ℓ^{*}$ and $ℓ_{a}$ (thicker line in blue).

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Fig. 5

The values of the transport mean free path (top) and absorption length (bottom) obtained from the TOF measurements with five different phantoms. The vertical dashed line is a “guide to the eye” indicating roughly the validity limit of the diffusion approximation.

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Fig. 6

The absorbance A of TiO₂ solutions (containing 16.7% weight percentage of gelatin) as the function of concentration c (left: sample thickness d = 0.1mm; right: d = 1mm).

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Fig. 7

Top view of the light scattering experiment. The light emitted by the laser is vertically polarized. We measured the light scattered by TiO₂ suspensions (containing 16.7% weight percentage of gelatin) of different concentrations at various angles 40°≤θ≤140°.

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Fig. 8

Estimating the scattering law of the TiO₂ scatterers immersed in gelatin: the pvgHG scattering law model is fitted to the extrapolated data (represented by “+” symbols in the main figure) corresponding to infinitesimal concentrations of TiO₂, i.e., where the single scattering regime applies. The fitting is achieved by modifying the set of parameters {g, ν}. The extrapolation procedure which gives the values corresponding to infinitesimal concentrations is indicated in the inset figure.

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Fig. 9

Screenshot from the MC simulation: the simulation tool was used to reproduce the TOF measurements presented in Section 3 (depicted in Fig. 2). The colored disc visualizes the number of photons detected by the individual detector rings.

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Fig. 10

Examples of the MC simulations where we varied the scattering law and its parameters. The curves in the inset of the bottom figure are polarization averaged. The data were generated in an overnight simulation including 60 million photons for each scattering law (total number of scattering events: 47 × 10⁹). The simulated curves show the raw, non-normalized and unsmoothed data with a time-resolution of 2 ps. At small source-detector separations (top), the limitation of the diffusion model (red line) is clearly visible but the differences between the scattering laws are hardly discernable in the semi-log plot. These differences are significant at short times, as shown in the linear plot in the inset of the top figure. (The small bump at 25 ps is probably a commensurability artifact due to the finite resolution of the ring detector.) At large source-detector separations (bottom), the applicability of the diffusion model becomes apparent. At both distances, the diffusion model is shifted by τ_d ≈50ps.

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Tables (2)

Table 1 The optical properties (at λ = 780 nm) assigned to the three different materials used in the experiment: the quantities listed here correspond to the experimental results and were provided as input parameters to the MC simulation program

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Table 2 Parameters obtained by fitting the diffusion model to the simulated data with $ℓ_{i n p u t}^{*} = 4.5 m m, ℓ_{a_i n p u t} = 190 m m$

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Equations (9)

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\frac{\partial P (r, t)}{\partial t} - D_{p} \nabla^{2} P (r, t) = - α_{p} P (r, t) + S (r, t)

D_{p} = \frac{1}{3} c_{l} ℓ^{*} and α_{p} = \frac{c_{l}}{ℓ_{a}} .

P (r, t) = \frac{1}{{(4 π D_{p} t)}^{3 / 2}} \cdot \exp (\frac{- r^{2}}{4 D_{p} t}) \cdot \exp (- α_{p} t)

μ_{s} (c) = c σ_{s_T i O 2}

ℓ^{*} = \frac{1}{{μ^{'}}_{s}} = \frac{1}{(1 - g) μ_{s}}

I_{n} (θ) = \frac{I_{m e a s} (θ) - I_{b l a n k} (θ)}{I_{t o l} (θ)}

I_{n} (θ) = \frac{μ_{s}}{μ_{s_t o l}} \frac{p (θ)}{p_{t o l} (θ)}

p (θ | c) = \frac{3}{8 π} \frac{μ_{s_t o l}}{μ_{s} (c)} I_{n} (θ)

F (q | L, ν) \propto \frac{1}{{[1 + {(L | q |)}^{2}]}^{ν + 3 / 2}} where ν > - 1

Parameters	Phantom	Glass Aquarium	Air
Refractive index	1.359	1.52	1.0
Transport mean free path $ℓ_{i n p u t}^{*}$	4.5 mm	∞	∞
Absorption length $ℓ_{a_i n p u t}$	190 mm	∞ (no absorption)	∞ (no absorption)
Scattering coefficient	2.86 mm⁻¹	∞ (no scattering)	∞ (no scattering)

Optode distance  r [mm]	Shift $τ_{d}$ [ps]	Transport mean  path $ℓ^{*}$ [mm]	Absorption length   $ℓ_{a}$ [mm]
24	50 ± 5	5.5 ± 0.5	214 ± 16
36	50 ± 6	4.8 ± 0.3	186 ± 8
48	50 ± 14	4.6 ± 0.2	179 ± 5
55	50 ± 20	4.6 ± 0.2	178 ± 3

Parameters	Phantom	Glass Aquarium	Air
Refractive index	1.359	1.52	1.0
Transport mean free path $ℓ_{i n p u t}^{*}$	4.5 mm	∞	∞
Absorption length $ℓ_{a_i n p u t}$	190 mm	∞ (no absorption)	∞ (no absorption)
Scattering coefficient	2.86 mm⁻¹	∞ (no scattering)	∞ (no scattering)

Optode distance  r [mm]	Shift $τ_{d}$ [ps]	Transport mean  path $ℓ^{*}$ [mm]	Absorption length   $ℓ_{a}$ [mm]
24	50 ± 5	5.5 ± 0.5	214 ± 16
36	50 ± 6	4.8 ± 0.3	186 ± 8
48	50 ± 14	4.6 ± 0.2	179 ± 5
55	50 ± 20	4.6 ± 0.2	178 ± 3