Abstract

A set of time-domain analytical forward solvers for Raman signals detected from homogeneous diffusive media is presented. The time-domain solvers have been developed for two geometries: the parallelepiped and the finite cylinder. The potential presence of a background fluorescence emission, contaminating the Raman signal, has also been taken into account. All the solvers have been obtained as solutions of the time dependent diffusion equation. The validation of the solvers has been performed by means of comparisons with the results of “gold standard” Monte Carlo simulations. These forward solvers provide an accurate tool to explore the information content encoded in the time-resolved Raman measurements.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2016 (1)

F. Martelli, T. Binzoni, A. Pifferi, L. Spinelli, A. Farina, and A. Torricelli, “There’s plenty of light at the bottom: statistics of photon penetration depth in random media,” Sci. Rep. 6, 27057 (2016).
[Crossref]

2015 (5)

2013 (3)

P. Matousek and N. Stone, “Recent advances in the development of Raman spectroscopy for deep non-invasive medical diagnosis,” J. Biophotonics 1(5), 7–19 (2013).
[Crossref]

S. Sundarajoo, E.L. Izake, W. Olds, B. Cletus, E. Jaatinen, and P. M. Fredericks, “Non-invasive depth profiling by space and time-resolved Raman spectroscopy, ” J. Raman Spectr. 44(7), 949–956 (2013).
[Crossref]

A. Bray, R. Chapman R, and T. Plakhotnik, “Accurate measurements of the Raman scattering coefficient and the depolarization ratio in liquid water,” Appl Opt. 52(11), 2503–2510 (2013).
[Crossref] [PubMed]

2012 (3)

2011 (2)

W. J. Olds, E. Jaatinen, P. Fredericks, B. Cletus, H. Panayiotou, and E. L. Izake, “Spatially offset Raman spectroscopy (SORS) for the analysis and detection of packaged pharmaceuticals and concealed drugs,” Forensic Sci. Int. 212(1–3), 69–77 (2011).
[Crossref] [PubMed]

D. Yang and Y. Ying, “Applications of Raman Spectroscopy in Agricultural Products and Food Analysis: A Review,” Appl. Spectrosc. Rev. 46(7), 539–560 (2011).
[Crossref]

2010 (1)

N. J. Everall, “Confocal Raman microscopy: common errors and artefacts.,” Analyst 135(10), 2512–2522 (2010).
[Crossref] [PubMed]

2009 (1)

F. Ariese, H. Meuzelaar, M. M. Kerssens, J. B. Buijs, and C. Gooijer, “Picosecond Raman spectroscopy with a fast intensified CCD camera for depth analysis of diffusely scattering media,” Analyst 134(6), 1192–1197 (2009).
[Crossref] [PubMed]

2008 (2)

M. V Schulmerich, J. H. Cole, K. A. Dooley, M. D. Morris, J. M. Kreider, S. A. Goldstein, S. Srinivasan, and B. W. Pogue, “Noninvasive Raman tomographic imaging of canine bone tissue,” J. Biomed. Opt. 13(2), 020506 (2008).
[Crossref] [PubMed]

E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration,” J. Biomed. Opt. 13(6), 060504 (2008).
[Crossref]

2007 (1)

N. Stone, R. Baker, K. Rogers, A. W. Parker, and P. Matousek, “Subsurface probing of calcifications with spatially offset Raman spectroscopy (SORS): future possibilities for the diagnosis of breast cancer,” Analyst 132(9), 899–905 (2007).
[Crossref] [PubMed]

2006 (2)

2005 (3)

2004 (1)

N. Everall, T. Hahn, P. Matousek, A. W. Parker, and M. Towrie, “Photon Migration in Raman Spectroscopy,” Appl. Spect. 58(5), 591–597 (2004).
[Crossref]

2003 (1)

2001 (2)

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, “Determining changes in NIR absorption using a layered model of the human head,” Phys. Med. Biol. 46(3), 879–896 (2001).
[Crossref] [PubMed]

N. Everall, T. Hahn, P. Matousek, A. W. Parker, and M. Towrie, “Picosecond Time-Resolved Raman Spectroscopy of Solids: Capabilities and Limitations for Fluorescence Rejection and the Influence of Diffuse Reflectance,” Appl. Spect. 55(12), 1701–1708 (2001).
[Crossref]

2000 (1)

R. A. Desiderio, “Application of the Raman scattering coefficient of water to calculations in marine optics,” Appl Opt. 39(12), 1893–1894 (2000).
[Crossref]

1999 (1)

1997 (1)

D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36(19), 4588–4599 (1997).
[Crossref]

1995 (2)

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Spect. 34(18), 3425–3430 (1995).

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Opt. 34(18), 3425–3430 (1995).
[Crossref] [PubMed]

Alerstam, E.

E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration,” J. Biomed. Opt. 13(6), 060504 (2008).
[Crossref]

Andersson-Engels, S.

E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration,” J. Biomed. Opt. 13(6), 060504 (2008).
[Crossref]

Ariese, F.

F. Ariese, H. Meuzelaar, M. M. Kerssens, J. B. Buijs, and C. Gooijer, “Picosecond Raman spectroscopy with a fast intensified CCD camera for depth analysis of diffusely scattering media,” Analyst 134(6), 1192–1197 (2009).
[Crossref] [PubMed]

Arridge, S.

Baker, R.

N. Stone, R. Baker, K. Rogers, A. W. Parker, and P. Matousek, “Subsurface probing of calcifications with spatially offset Raman spectroscopy (SORS): future possibilities for the diagnosis of breast cancer,” Analyst 132(9), 899–905 (2007).
[Crossref] [PubMed]

Binzoni, T.

F. Martelli, T. Binzoni, A. Pifferi, L. Spinelli, A. Farina, and A. Torricelli, “There’s plenty of light at the bottom: statistics of photon penetration depth in random media,” Sci. Rep. 6, 27057 (2016).
[Crossref]

Boso, G.

Bray, A.

A. Bray, R. Chapman R, and T. Plakhotnik, “Accurate measurements of the Raman scattering coefficient and the depolarization ratio in liquid water,” Appl Opt. 52(11), 2503–2510 (2013).
[Crossref] [PubMed]

Buckley, K.

K. Sowoidnich, J. H. Churchwell, K. Buckley, A. E. Goodship, A. W. Parker, and P. Matousek, “Photon migration of Raman signal in bone as measured with spatially offset Raman spectroscopy,” J. RamanSpectrosc. 47(2), 240–247 (2015).

Buijs, J. B.

F. Ariese, H. Meuzelaar, M. M. Kerssens, J. B. Buijs, and C. Gooijer, “Picosecond Raman spectroscopy with a fast intensified CCD camera for depth analysis of diffusely scattering media,” Analyst 134(6), 1192–1197 (2009).
[Crossref] [PubMed]

Carslaw, H. S.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Oxford University, 1959).

Chapman R, R.

A. Bray, R. Chapman R, and T. Plakhotnik, “Accurate measurements of the Raman scattering coefficient and the depolarization ratio in liquid water,” Appl Opt. 52(11), 2503–2510 (2013).
[Crossref] [PubMed]

Churchwell, J. H.

K. Sowoidnich, J. H. Churchwell, K. Buckley, A. E. Goodship, A. W. Parker, and P. Matousek, “Photon migration of Raman signal in bone as measured with spatially offset Raman spectroscopy,” J. RamanSpectrosc. 47(2), 240–247 (2015).

Clark, I. P.

Cletus, B.

S. Sundarajoo, E.L. Izake, W. Olds, B. Cletus, E. Jaatinen, and P. M. Fredericks, “Non-invasive depth profiling by space and time-resolved Raman spectroscopy, ” J. Raman Spectr. 44(7), 949–956 (2013).
[Crossref]

E. L. Izake, B. Cletus, W. Olds, S. Sundarajoo, P. M. Fredericks, and E. Jaatinen, “Deep Raman spectroscopy for the non-invasive standoff detection of concealed chemical threat agents,” Talanta 94, 342–347 (2012).
[Crossref] [PubMed]

W. J. Olds, E. Jaatinen, P. Fredericks, B. Cletus, H. Panayiotou, and E. L. Izake, “Spatially offset Raman spectroscopy (SORS) for the analysis and detection of packaged pharmaceuticals and concealed drugs,” Forensic Sci. Int. 212(1–3), 69–77 (2011).
[Crossref] [PubMed]

Cole, J. H.

M. V Schulmerich, J. H. Cole, K. A. Dooley, M. D. Morris, J. M. Kreider, S. A. Goldstein, S. Srinivasan, and B. W. Pogue, “Noninvasive Raman tomographic imaging of canine bone tissue,” J. Biomed. Opt. 13(2), 020506 (2008).
[Crossref] [PubMed]

Colombo, C.

C. Conti, C. Colombo, M. Realini, and P. Matousek, “Subsurface analysis of painted sculptures and plasters using micrometre-scale spatially offset Raman spectroscopy (micro-SORS),” J. Raman Spectrosc. 46(5), 476–482 (2015).
[Crossref]

Conti, C.

C. Conti, C. Colombo, M. Realini, and P. Matousek, “Subsurface analysis of painted sculptures and plasters using micrometre-scale spatially offset Raman spectroscopy (micro-SORS),” J. Raman Spectrosc. 46(5), 476–482 (2015).
[Crossref]

Contini, D.

Dalla Mora, A.

Dasari, R. R.

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Opt. 34(18), 3425–3430 (1995).
[Crossref] [PubMed]

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Spect. 34(18), 3425–3430 (1995).

Davis, S. C.

Del Bianco, S.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions and Software (SPIE, 2010).
[Crossref]

Demers, J.-L. H.

Desiderio, R. A.

R. A. Desiderio, “Application of the Raman scattering coefficient of water to calculations in marine optics,” Appl Opt. 39(12), 1893–1894 (2000).
[Crossref]

Dooley, K. A.

M. V Schulmerich, J. H. Cole, K. A. Dooley, M. D. Morris, J. M. Kreider, S. A. Goldstein, S. Srinivasan, and B. W. Pogue, “Noninvasive Raman tomographic imaging of canine bone tissue,” J. Biomed. Opt. 13(2), 020506 (2008).
[Crossref] [PubMed]

Draper, E. R. C.

Durduran, T.

Esmonde-White, F. W. L.

Esmonde-White, K. A.

Everall, N.

P. Matousek, I. P. Clark, E. R. C. Draper, M. D. Morris, A. E. Goodship, N. Everall, M. Towrie, W. F. Finney, and A. W. Parker, “Subsurface Probing in Diffusely Scattering Media Using Spatially Offset Raman Spectroscopy,” Appl. Spectrosc. 59(4), 393–400 (2005).
[Crossref] [PubMed]

P. Matousek, N. Everall, M. Towrie, and A. W. Parker, “Depth profiling in diffusely scattering media using Raman spectroscopy and picosecond Kerr gating,” Appl. Spectrosc. 59(2), 200–205 (2005).
[Crossref] [PubMed]

N. Everall, T. Hahn, P. Matousek, A. W. Parker, and M. Towrie, “Photon Migration in Raman Spectroscopy,” Appl. Spect. 58(5), 591–597 (2004).
[Crossref]

N. Everall, T. Hahn, P. Matousek, A. W. Parker, and M. Towrie, “Picosecond Time-Resolved Raman Spectroscopy of Solids: Capabilities and Limitations for Fluorescence Rejection and the Influence of Diffuse Reflectance,” Appl. Spect. 55(12), 1701–1708 (2001).
[Crossref]

Everall, N. J.

N. J. Everall, “Confocal Raman microscopy: common errors and artefacts.,” Analyst 135(10), 2512–2522 (2010).
[Crossref] [PubMed]

Farina, A.

Feld, M. S.

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Spect. 34(18), 3425–3430 (1995).

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Opt. 34(18), 3425–3430 (1995).
[Crossref] [PubMed]

Finney, W. F.

Folestad, S.

J. Johansson, S. Pettersson, and S. Folestad, “Characterization of different laser irradiation methods for quantitative Raman tablet assessment,” J. Pharm. Biomed. Anal. 39(3–4), 510–516 (2005).
[Crossref] [PubMed]

Fredericks, P.

W. J. Olds, E. Jaatinen, P. Fredericks, B. Cletus, H. Panayiotou, and E. L. Izake, “Spatially offset Raman spectroscopy (SORS) for the analysis and detection of packaged pharmaceuticals and concealed drugs,” Forensic Sci. Int. 212(1–3), 69–77 (2011).
[Crossref] [PubMed]

Fredericks, P. M.

S. Sundarajoo, E.L. Izake, W. Olds, B. Cletus, E. Jaatinen, and P. M. Fredericks, “Non-invasive depth profiling by space and time-resolved Raman spectroscopy, ” J. Raman Spectr. 44(7), 949–956 (2013).
[Crossref]

E. L. Izake, B. Cletus, W. Olds, S. Sundarajoo, P. M. Fredericks, and E. Jaatinen, “Deep Raman spectroscopy for the non-invasive standoff detection of concealed chemical threat agents,” Talanta 94, 342–347 (2012).
[Crossref] [PubMed]

Froud, C. A.

Goldstein, S. A.

M. V Schulmerich, J. H. Cole, K. A. Dooley, M. D. Morris, J. M. Kreider, S. A. Goldstein, S. Srinivasan, and B. W. Pogue, “Noninvasive Raman tomographic imaging of canine bone tissue,” J. Biomed. Opt. 13(2), 020506 (2008).
[Crossref] [PubMed]

Goodship, A. E.

Gooijer, C.

F. Ariese, H. Meuzelaar, M. M. Kerssens, J. B. Buijs, and C. Gooijer, “Picosecond Raman spectroscopy with a fast intensified CCD camera for depth analysis of diffusely scattering media,” Analyst 134(6), 1192–1197 (2009).
[Crossref] [PubMed]

Hahn, T.

N. Everall, T. Hahn, P. Matousek, A. W. Parker, and M. Towrie, “Photon Migration in Raman Spectroscopy,” Appl. Spect. 58(5), 591–597 (2004).
[Crossref]

N. Everall, T. Hahn, P. Matousek, A. W. Parker, and M. Towrie, “Picosecond Time-Resolved Raman Spectroscopy of Solids: Capabilities and Limitations for Fluorescence Rejection and the Influence of Diffuse Reflectance,” Appl. Spect. 55(12), 1701–1708 (2001).
[Crossref]

Hayward, I. P.

Ismaelli, A.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions and Software (SPIE, 2010).
[Crossref]

Itzkan, I.

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Spect. 34(18), 3425–3430 (1995).

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Opt. 34(18), 3425–3430 (1995).
[Crossref] [PubMed]

Izake, E. L.

E. L. Izake, B. Cletus, W. Olds, S. Sundarajoo, P. M. Fredericks, and E. Jaatinen, “Deep Raman spectroscopy for the non-invasive standoff detection of concealed chemical threat agents,” Talanta 94, 342–347 (2012).
[Crossref] [PubMed]

W. J. Olds, E. Jaatinen, P. Fredericks, B. Cletus, H. Panayiotou, and E. L. Izake, “Spatially offset Raman spectroscopy (SORS) for the analysis and detection of packaged pharmaceuticals and concealed drugs,” Forensic Sci. Int. 212(1–3), 69–77 (2011).
[Crossref] [PubMed]

Izake, E.L.

S. Sundarajoo, E.L. Izake, W. Olds, B. Cletus, E. Jaatinen, and P. M. Fredericks, “Non-invasive depth profiling by space and time-resolved Raman spectroscopy, ” J. Raman Spectr. 44(7), 949–956 (2013).
[Crossref]

Jaatinen, E.

S. Sundarajoo, E.L. Izake, W. Olds, B. Cletus, E. Jaatinen, and P. M. Fredericks, “Non-invasive depth profiling by space and time-resolved Raman spectroscopy, ” J. Raman Spectr. 44(7), 949–956 (2013).
[Crossref]

E. L. Izake, B. Cletus, W. Olds, S. Sundarajoo, P. M. Fredericks, and E. Jaatinen, “Deep Raman spectroscopy for the non-invasive standoff detection of concealed chemical threat agents,” Talanta 94, 342–347 (2012).
[Crossref] [PubMed]

W. J. Olds, E. Jaatinen, P. Fredericks, B. Cletus, H. Panayiotou, and E. L. Izake, “Spatially offset Raman spectroscopy (SORS) for the analysis and detection of packaged pharmaceuticals and concealed drugs,” Forensic Sci. Int. 212(1–3), 69–77 (2011).
[Crossref] [PubMed]

Jaeger, J. C.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Oxford University, 1959).

Johansson, J.

J. Johansson, S. Pettersson, and S. Folestad, “Characterization of different laser irradiation methods for quantitative Raman tablet assessment,” J. Pharm. Biomed. Anal. 39(3–4), 510–516 (2005).
[Crossref] [PubMed]

Kerssens, M. M.

F. Ariese, H. Meuzelaar, M. M. Kerssens, J. B. Buijs, and C. Gooijer, “Picosecond Raman spectroscopy with a fast intensified CCD camera for depth analysis of diffusely scattering media,” Analyst 134(6), 1192–1197 (2009).
[Crossref] [PubMed]

Kreider, J. M.

M. V Schulmerich, J. H. Cole, K. A. Dooley, M. D. Morris, J. M. Kreider, S. A. Goldstein, S. Srinivasan, and B. W. Pogue, “Noninvasive Raman tomographic imaging of canine bone tissue,” J. Biomed. Opt. 13(2), 020506 (2008).
[Crossref] [PubMed]

Laven, J.

Martelli, F.

F. Martelli, T. Binzoni, A. Pifferi, L. Spinelli, A. Farina, and A. Torricelli, “There’s plenty of light at the bottom: statistics of photon penetration depth in random media,” Sci. Rep. 6, 27057 (2016).
[Crossref]

A. Dalla Mora, D. Contini, S. Arridge, F. Martelli, A. Tosi, G. Boso, A. Farina, T. Durduran, E. Martinenghi, A. Torricelli, and A. Pifferi, “Towards next-generation time-domain diffuse optics for extreme depth penetration and sensitivity,” Biomed. Opt. Express 6(5), 1749–1760 (2015).
[Crossref]

A. Dalla Mora, E. Martinenghi, D. Contini, A. Tosi, G. Boso, T. Durduran, S. Arridge, F. Martelli, A. Farina, A. Torricelli, and A. Pifferi, “Fast silicon photomultiplier improves signal harvesting and reduces complexity in time-domain diffuse optics,” Opt. Express 23(11), 13937–13946 (2015).
[Crossref]

A. Sassaroli and F. Martelli, “Equivalence of four Monte Carlo methods for photon migration in turbid media,” J. Opt. Soc. Am. A 29(10), 2110–2117 (2012).
[Crossref]

D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36(19), 4588–4599 (1997).
[Crossref]

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions and Software (SPIE, 2010).
[Crossref]

Martinenghi, E.

Matousek, P.

C. Conti, C. Colombo, M. Realini, and P. Matousek, “Subsurface analysis of painted sculptures and plasters using micrometre-scale spatially offset Raman spectroscopy (micro-SORS),” J. Raman Spectrosc. 46(5), 476–482 (2015).
[Crossref]

K. Sowoidnich, J. H. Churchwell, K. Buckley, A. E. Goodship, A. W. Parker, and P. Matousek, “Photon migration of Raman signal in bone as measured with spatially offset Raman spectroscopy,” J. RamanSpectrosc. 47(2), 240–247 (2015).

P. Matousek and N. Stone, “Recent advances in the development of Raman spectroscopy for deep non-invasive medical diagnosis,” J. Biophotonics 1(5), 7–19 (2013).
[Crossref]

N. Stone, R. Baker, K. Rogers, A. W. Parker, and P. Matousek, “Subsurface probing of calcifications with spatially offset Raman spectroscopy (SORS): future possibilities for the diagnosis of breast cancer,” Analyst 132(9), 899–905 (2007).
[Crossref] [PubMed]

P. Matousek, E. R. C. Draper, A. E. Goodship, I. P. Clark, K. L. Ronayne, and A. W. Parker, “Noninvasive Raman Spectroscopy of Human Tissue In Vivo,” Appl. Spectrosc. 60(7), 758–763 (2006).
[Crossref] [PubMed]

P. Matousek and A. W. Parker, “Bulk Raman Analysis of Pharmaceutical Tablets,” Appl. Spectrosc. 60(12), 1353–1357 (2006).
[Crossref]

P. Matousek, I. P. Clark, E. R. C. Draper, M. D. Morris, A. E. Goodship, N. Everall, M. Towrie, W. F. Finney, and A. W. Parker, “Subsurface Probing in Diffusely Scattering Media Using Spatially Offset Raman Spectroscopy,” Appl. Spectrosc. 59(4), 393–400 (2005).
[Crossref] [PubMed]

P. Matousek, N. Everall, M. Towrie, and A. W. Parker, “Depth profiling in diffusely scattering media using Raman spectroscopy and picosecond Kerr gating,” Appl. Spectrosc. 59(2), 200–205 (2005).
[Crossref] [PubMed]

N. Everall, T. Hahn, P. Matousek, A. W. Parker, and M. Towrie, “Photon Migration in Raman Spectroscopy,” Appl. Spect. 58(5), 591–597 (2004).
[Crossref]

N. Everall, T. Hahn, P. Matousek, A. W. Parker, and M. Towrie, “Picosecond Time-Resolved Raman Spectroscopy of Solids: Capabilities and Limitations for Fluorescence Rejection and the Influence of Diffuse Reflectance,” Appl. Spect. 55(12), 1701–1708 (2001).
[Crossref]

P. Matousek, M. Towrie, A. Stanley, and A. W. Parker, “Efficient Rejection of Fluorescence from Raman Spectra Using Picosecond Kerr Gating,” Appl. Spectrosc. 53(12), 1485–1489 (1999).
[Crossref]

P. Matousek and M. D. Morris, Emerging Raman Applications and Techniques in Biomedical and Pharmaceutical Fields (Springer, 2010).
[Crossref]

Meuzelaar, H.

F. Ariese, H. Meuzelaar, M. M. Kerssens, J. B. Buijs, and C. Gooijer, “Picosecond Raman spectroscopy with a fast intensified CCD camera for depth analysis of diffusely scattering media,” Analyst 134(6), 1192–1197 (2009).
[Crossref] [PubMed]

Morris, M. D.

Obrig, H.

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, “Determining changes in NIR absorption using a layered model of the human head,” Phys. Med. Biol. 46(3), 879–896 (2001).
[Crossref] [PubMed]

Olds, W.

S. Sundarajoo, E.L. Izake, W. Olds, B. Cletus, E. Jaatinen, and P. M. Fredericks, “Non-invasive depth profiling by space and time-resolved Raman spectroscopy, ” J. Raman Spectr. 44(7), 949–956 (2013).
[Crossref]

E. L. Izake, B. Cletus, W. Olds, S. Sundarajoo, P. M. Fredericks, and E. Jaatinen, “Deep Raman spectroscopy for the non-invasive standoff detection of concealed chemical threat agents,” Talanta 94, 342–347 (2012).
[Crossref] [PubMed]

Olds, W. J.

W. J. Olds, E. Jaatinen, P. Fredericks, B. Cletus, H. Panayiotou, and E. L. Izake, “Spatially offset Raman spectroscopy (SORS) for the analysis and detection of packaged pharmaceuticals and concealed drugs,” Forensic Sci. Int. 212(1–3), 69–77 (2011).
[Crossref] [PubMed]

Panayiotou, H.

W. J. Olds, E. Jaatinen, P. Fredericks, B. Cletus, H. Panayiotou, and E. L. Izake, “Spatially offset Raman spectroscopy (SORS) for the analysis and detection of packaged pharmaceuticals and concealed drugs,” Forensic Sci. Int. 212(1–3), 69–77 (2011).
[Crossref] [PubMed]

Parker, A. W.

K. Sowoidnich, J. H. Churchwell, K. Buckley, A. E. Goodship, A. W. Parker, and P. Matousek, “Photon migration of Raman signal in bone as measured with spatially offset Raman spectroscopy,” J. RamanSpectrosc. 47(2), 240–247 (2015).

N. Stone, R. Baker, K. Rogers, A. W. Parker, and P. Matousek, “Subsurface probing of calcifications with spatially offset Raman spectroscopy (SORS): future possibilities for the diagnosis of breast cancer,” Analyst 132(9), 899–905 (2007).
[Crossref] [PubMed]

P. Matousek, E. R. C. Draper, A. E. Goodship, I. P. Clark, K. L. Ronayne, and A. W. Parker, “Noninvasive Raman Spectroscopy of Human Tissue In Vivo,” Appl. Spectrosc. 60(7), 758–763 (2006).
[Crossref] [PubMed]

P. Matousek and A. W. Parker, “Bulk Raman Analysis of Pharmaceutical Tablets,” Appl. Spectrosc. 60(12), 1353–1357 (2006).
[Crossref]

P. Matousek, I. P. Clark, E. R. C. Draper, M. D. Morris, A. E. Goodship, N. Everall, M. Towrie, W. F. Finney, and A. W. Parker, “Subsurface Probing in Diffusely Scattering Media Using Spatially Offset Raman Spectroscopy,” Appl. Spectrosc. 59(4), 393–400 (2005).
[Crossref] [PubMed]

P. Matousek, N. Everall, M. Towrie, and A. W. Parker, “Depth profiling in diffusely scattering media using Raman spectroscopy and picosecond Kerr gating,” Appl. Spectrosc. 59(2), 200–205 (2005).
[Crossref] [PubMed]

N. Everall, T. Hahn, P. Matousek, A. W. Parker, and M. Towrie, “Photon Migration in Raman Spectroscopy,” Appl. Spect. 58(5), 591–597 (2004).
[Crossref]

N. Everall, T. Hahn, P. Matousek, A. W. Parker, and M. Towrie, “Picosecond Time-Resolved Raman Spectroscopy of Solids: Capabilities and Limitations for Fluorescence Rejection and the Influence of Diffuse Reflectance,” Appl. Spect. 55(12), 1701–1708 (2001).
[Crossref]

P. Matousek, M. Towrie, A. Stanley, and A. W. Parker, “Efficient Rejection of Fluorescence from Raman Spectra Using Picosecond Kerr Gating,” Appl. Spectrosc. 53(12), 1485–1489 (1999).
[Crossref]

Perelman, L.

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Spect. 34(18), 3425–3430 (1995).

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Opt. 34(18), 3425–3430 (1995).
[Crossref] [PubMed]

Pettersson, S.

J. Johansson, S. Pettersson, and S. Folestad, “Characterization of different laser irradiation methods for quantitative Raman tablet assessment,” J. Pharm. Biomed. Anal. 39(3–4), 510–516 (2005).
[Crossref] [PubMed]

Pifferi, A.

Plakhotnik, T.

A. Bray, R. Chapman R, and T. Plakhotnik, “Accurate measurements of the Raman scattering coefficient and the depolarization ratio in liquid water,” Appl Opt. 52(11), 2503–2510 (2013).
[Crossref] [PubMed]

Pogue, B. W.

Realini, M.

C. Conti, C. Colombo, M. Realini, and P. Matousek, “Subsurface analysis of painted sculptures and plasters using micrometre-scale spatially offset Raman spectroscopy (micro-SORS),” J. Raman Spectrosc. 46(5), 476–482 (2015).
[Crossref]

Rinneberg, H.

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, “Determining changes in NIR absorption using a layered model of the human head,” Phys. Med. Biol. 46(3), 879–896 (2001).
[Crossref] [PubMed]

Rogers, K.

N. Stone, R. Baker, K. Rogers, A. W. Parker, and P. Matousek, “Subsurface probing of calcifications with spatially offset Raman spectroscopy (SORS): future possibilities for the diagnosis of breast cancer,” Analyst 132(9), 899–905 (2007).
[Crossref] [PubMed]

Ronayne, K. L.

Sassaroli, A.

Schulmerich, M. V

M. V Schulmerich, J. H. Cole, K. A. Dooley, M. D. Morris, J. M. Kreider, S. A. Goldstein, S. Srinivasan, and B. W. Pogue, “Noninvasive Raman tomographic imaging of canine bone tissue,” J. Biomed. Opt. 13(2), 020506 (2008).
[Crossref] [PubMed]

Sowoidnich, K.

K. Sowoidnich, J. H. Churchwell, K. Buckley, A. E. Goodship, A. W. Parker, and P. Matousek, “Photon migration of Raman signal in bone as measured with spatially offset Raman spectroscopy,” J. RamanSpectrosc. 47(2), 240–247 (2015).

Spinelli, L.

F. Martelli, T. Binzoni, A. Pifferi, L. Spinelli, A. Farina, and A. Torricelli, “There’s plenty of light at the bottom: statistics of photon penetration depth in random media,” Sci. Rep. 6, 27057 (2016).
[Crossref]

Srinivasan, S.

M. V Schulmerich, J. H. Cole, K. A. Dooley, M. D. Morris, J. M. Kreider, S. A. Goldstein, S. Srinivasan, and B. W. Pogue, “Noninvasive Raman tomographic imaging of canine bone tissue,” J. Biomed. Opt. 13(2), 020506 (2008).
[Crossref] [PubMed]

Stanley, A.

Steinbrink, J.

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, “Determining changes in NIR absorption using a layered model of the human head,” Phys. Med. Biol. 46(3), 879–896 (2001).
[Crossref] [PubMed]

Stone, N.

P. Matousek and N. Stone, “Recent advances in the development of Raman spectroscopy for deep non-invasive medical diagnosis,” J. Biophotonics 1(5), 7–19 (2013).
[Crossref]

N. Stone, R. Baker, K. Rogers, A. W. Parker, and P. Matousek, “Subsurface probing of calcifications with spatially offset Raman spectroscopy (SORS): future possibilities for the diagnosis of breast cancer,” Analyst 132(9), 899–905 (2007).
[Crossref] [PubMed]

Sundarajoo, S.

S. Sundarajoo, E.L. Izake, W. Olds, B. Cletus, E. Jaatinen, and P. M. Fredericks, “Non-invasive depth profiling by space and time-resolved Raman spectroscopy, ” J. Raman Spectr. 44(7), 949–956 (2013).
[Crossref]

E. L. Izake, B. Cletus, W. Olds, S. Sundarajoo, P. M. Fredericks, and E. Jaatinen, “Deep Raman spectroscopy for the non-invasive standoff detection of concealed chemical threat agents,” Talanta 94, 342–347 (2012).
[Crossref] [PubMed]

Svensson, T.

E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration,” J. Biomed. Opt. 13(6), 060504 (2008).
[Crossref]

Torricelli, A.

Tosi, A.

Towrie, M.

Villringer, A.

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, “Determining changes in NIR absorption using a layered model of the human head,” Phys. Med. Biol. 46(3), 879–896 (2001).
[Crossref] [PubMed]

Wabnitz, H.

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, “Determining changes in NIR absorption using a layered model of the human head,” Phys. Med. Biol. 46(3), 879–896 (2001).
[Crossref] [PubMed]

Wang, Y.

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Spect. 34(18), 3425–3430 (1995).

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Opt. 34(18), 3425–3430 (1995).
[Crossref] [PubMed]

Wu, J.

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Opt. 34(18), 3425–3430 (1995).
[Crossref] [PubMed]

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Spect. 34(18), 3425–3430 (1995).

Yang, D.

D. Yang and Y. Ying, “Applications of Raman Spectroscopy in Agricultural Products and Food Analysis: A Review,” Appl. Spectrosc. Rev. 46(7), 539–560 (2011).
[Crossref]

Ying, Y.

D. Yang and Y. Ying, “Applications of Raman Spectroscopy in Agricultural Products and Food Analysis: A Review,” Appl. Spectrosc. Rev. 46(7), 539–560 (2011).
[Crossref]

Zaccanti, G.

D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36(19), 4588–4599 (1997).
[Crossref]

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions and Software (SPIE, 2010).
[Crossref]

Analyst (3)

N. J. Everall, “Confocal Raman microscopy: common errors and artefacts.,” Analyst 135(10), 2512–2522 (2010).
[Crossref] [PubMed]

N. Stone, R. Baker, K. Rogers, A. W. Parker, and P. Matousek, “Subsurface probing of calcifications with spatially offset Raman spectroscopy (SORS): future possibilities for the diagnosis of breast cancer,” Analyst 132(9), 899–905 (2007).
[Crossref] [PubMed]

F. Ariese, H. Meuzelaar, M. M. Kerssens, J. B. Buijs, and C. Gooijer, “Picosecond Raman spectroscopy with a fast intensified CCD camera for depth analysis of diffusely scattering media,” Analyst 134(6), 1192–1197 (2009).
[Crossref] [PubMed]

Appl Opt. (2)

R. A. Desiderio, “Application of the Raman scattering coefficient of water to calculations in marine optics,” Appl Opt. 39(12), 1893–1894 (2000).
[Crossref]

A. Bray, R. Chapman R, and T. Plakhotnik, “Accurate measurements of the Raman scattering coefficient and the depolarization ratio in liquid water,” Appl Opt. 52(11), 2503–2510 (2013).
[Crossref] [PubMed]

Appl. Opt. (2)

D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36(19), 4588–4599 (1997).
[Crossref]

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Opt. 34(18), 3425–3430 (1995).
[Crossref] [PubMed]

Appl. Spect. (3)

N. Everall, T. Hahn, P. Matousek, A. W. Parker, and M. Towrie, “Photon Migration in Raman Spectroscopy,” Appl. Spect. 58(5), 591–597 (2004).
[Crossref]

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Spect. 34(18), 3425–3430 (1995).

N. Everall, T. Hahn, P. Matousek, A. W. Parker, and M. Towrie, “Picosecond Time-Resolved Raman Spectroscopy of Solids: Capabilities and Limitations for Fluorescence Rejection and the Influence of Diffuse Reflectance,” Appl. Spect. 55(12), 1701–1708 (2001).
[Crossref]

Appl. Spectrosc. (6)

Appl. Spectrosc. Rev. (1)

D. Yang and Y. Ying, “Applications of Raman Spectroscopy in Agricultural Products and Food Analysis: A Review,” Appl. Spectrosc. Rev. 46(7), 539–560 (2011).
[Crossref]

Biomed. Opt. Express (3)

Forensic Sci. Int. (1)

W. J. Olds, E. Jaatinen, P. Fredericks, B. Cletus, H. Panayiotou, and E. L. Izake, “Spatially offset Raman spectroscopy (SORS) for the analysis and detection of packaged pharmaceuticals and concealed drugs,” Forensic Sci. Int. 212(1–3), 69–77 (2011).
[Crossref] [PubMed]

J. Biomed. Opt. (2)

M. V Schulmerich, J. H. Cole, K. A. Dooley, M. D. Morris, J. M. Kreider, S. A. Goldstein, S. Srinivasan, and B. W. Pogue, “Noninvasive Raman tomographic imaging of canine bone tissue,” J. Biomed. Opt. 13(2), 020506 (2008).
[Crossref] [PubMed]

E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration,” J. Biomed. Opt. 13(6), 060504 (2008).
[Crossref]

J. Biophotonics (1)

P. Matousek and N. Stone, “Recent advances in the development of Raman spectroscopy for deep non-invasive medical diagnosis,” J. Biophotonics 1(5), 7–19 (2013).
[Crossref]

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

J. Pharm. Biomed. Anal. (1)

J. Johansson, S. Pettersson, and S. Folestad, “Characterization of different laser irradiation methods for quantitative Raman tablet assessment,” J. Pharm. Biomed. Anal. 39(3–4), 510–516 (2005).
[Crossref] [PubMed]

J. Raman Spectr. (1)

S. Sundarajoo, E.L. Izake, W. Olds, B. Cletus, E. Jaatinen, and P. M. Fredericks, “Non-invasive depth profiling by space and time-resolved Raman spectroscopy, ” J. Raman Spectr. 44(7), 949–956 (2013).
[Crossref]

J. Raman Spectrosc. (1)

C. Conti, C. Colombo, M. Realini, and P. Matousek, “Subsurface analysis of painted sculptures and plasters using micrometre-scale spatially offset Raman spectroscopy (micro-SORS),” J. Raman Spectrosc. 46(5), 476–482 (2015).
[Crossref]

J. RamanSpectrosc. (1)

K. Sowoidnich, J. H. Churchwell, K. Buckley, A. E. Goodship, A. W. Parker, and P. Matousek, “Photon migration of Raman signal in bone as measured with spatially offset Raman spectroscopy,” J. RamanSpectrosc. 47(2), 240–247 (2015).

Opt. Express (1)

Phys. Med. Biol. (1)

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, “Determining changes in NIR absorption using a layered model of the human head,” Phys. Med. Biol. 46(3), 879–896 (2001).
[Crossref] [PubMed]

Sci. Rep. (1)

F. Martelli, T. Binzoni, A. Pifferi, L. Spinelli, A. Farina, and A. Torricelli, “There’s plenty of light at the bottom: statistics of photon penetration depth in random media,” Sci. Rep. 6, 27057 (2016).
[Crossref]

Talanta (1)

E. L. Izake, B. Cletus, W. Olds, S. Sundarajoo, P. M. Fredericks, and E. Jaatinen, “Deep Raman spectroscopy for the non-invasive standoff detection of concealed chemical threat agents,” Talanta 94, 342–347 (2012).
[Crossref] [PubMed]

Other (3)

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions and Software (SPIE, 2010).
[Crossref]

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Oxford University, 1959).

P. Matousek and M. D. Morris, Emerging Raman Applications and Techniques in Biomedical and Pharmaceutical Fields (Springer, 2010).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Diagram of a photon path trajectory in a x, z plane projection from the source (red circle) to the Raman scatter (dashed line) and from the Raman scatter to the detector (continuous line). Green circles are Tyndall interactions. The yellow circle is a Raman interaction.
Fig. 2
Fig. 2 Schematic of the parallelepiped.
Fig. 3
Fig. 3 Comparison for the TR Raman reflectance between analytical formulae and MC in absence (a, e, g) and in presence (c) of background fluorescence. For all the figure panels μ′sb = μ′sbe = 1, μabe = 7 · 10−3 mm−1, μsR = 10−3 mm−1, ni = nie = 1.4, no = noe = 1, g = 0. Specifically: (a) μab = 5 · 10−3 mm−1, μaf = 0, ρ = 20 mm; (c) μab = 5 · 10−3 mm−1, μaf = 7 · 10−3 mm−1, ηe=1 and τ = 2 ns, ρ = 20 mm; (e) μab = 5 · 10−3 mm−1, μaf = 0, ρ = 10 mm; (g) μab = 10−2 mm−1, μaf = 0, ρ = 20 mm.
Fig. 4
Fig. 4 Dependence of the temporal profile of Raman photons for different values of μabe and μ′sb. The common parameters were ρ = 10 mm, ni = nie = 1.4, no = noe = 1 and μsR = 10−6 mm−1, μab = 0.01 mm−1. Specifically: (a) μabe ∈ [0, 0.035] mm−1 and μ′sb = μ′sbe = 1mm−1 and; (b) μ′sb = μ′sbe ∈ [0.5, 2] mm−1 and μabe = 0.01 mm−1.
Fig. 5
Fig. 5 Comparison of the temporal profile of the DE Raman signal (Eq. (20)), and of the heuristic model (Eq. (12)). The common parameters were: ρ = 10 mm, ni = nie = 1.4, no = noe = 1, μsR = 10−6 mm−1, μ′sb = μ′sbe = 1mm−1, and μab = 0.01 mm−1.
Fig. 6
Fig. 6 Relative per cent error of the heuristic model (Eq. (12)) as compared to the DE solver (Eq. (20)). The common parameters were ρ = 10 mm, ni = nie = 1.4, no = noe = 1, μsR = 10−6 mm−1, μab = 0.01 mm−1 and μ′sb = 1mm−1. Specifically: (a) μabe ∈ [0, 0.02] mm−1 and μ′sbe = 1mm−1 and; (b) μ′sbe ∈ [0.5, 1] mm−1 and μabe = 0.01 mm−1.
Fig. 7
Fig. 7 Schematic of a diffusive cylinder.

Tables (1)

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Table 1 Summary of the notations used for the optical properties of the medium at λ and λe in the case of a pure Raman model (column Raman) or a hybrid Raman and fluorescence model (column Raman+Fluorescence). The subscript b denotes the background, the subscript R denotes the Raman scattering and the subscript f denotes the fluorescence. In this work, we have assumed μsRe = 0 and μafe = 0. The optical parameters appearing in column Raman have to be used with Eqs. (18), (20) and (39), while optical parameters appearing in column Raman+Fluorescence have to be used with Eqs. (24), (28) and (32).

Equations (39)

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[ 1 v t + μ a D 2 ] Φ ( r , t ) = q ( r , t ) ,
[ 1 v e t + μ a e D e 2 ] Φ e ( r , t ) = q e ( r , t ) ,
q e ( r , t , λ e ) = μ s R Φ ( r s , r , μ a , μ s , n i , t , λ ) .
q e ( r , t , λ e ) = μ s R G ( r s , r , μ a , μ s , n i , t , λ ) .
[ 1 v e t + μ a e D e 2 ] Φ e ( r , t ) μ s R G ( r s , r , t ) = 0 .
Φ e ( r , t ) = 0 t V q e ( r , t ) G e ( r , r , μ a e , μ s e , n i e , t t ) d V d t ,
Φ e ( r , t ) = μ s R 0 t V G ( r s , r , μ a , μ s , n i , t ) G e ( r , r , μ a e , μ s e , n i e , t t ) d V d t .
Φ ( r , μ a = μ a b + μ s R , μ s , t , λ ) = Φ ( r , μ a = μ a b , μ s , t , λ ) exp ( μ s R v t ) .
Φ e ( r , μ a e = μ a b e , μ s e = μ s b e , t , λ e ) = Φ e ( r , μ a b , μ s b , t , λ e ) .
Φ e ( r , μ a e , μ s e , t , λ e ) = Φ ( r , μ a b , μ s b , t , λ ) Φ ( r , μ a b , μ s b , t , λ ) exp ( μ s R v t ) ,
Φ e ( r , μ a e , μ s e , t , λ e ) Φ ( r , μ a b , μ s b , t , λ ) μ s R v t .
R eHeur ( ρ , μ a e , μ s e , t , λ e ) = R ( ρ , μ a b , μ s b , t , λ ) μ s R v t ,
G ( r , r , t ) = 2 3 v L x L y L z l , m , n = 1 cos ( K l x ) cos ( K l x ) cos ( K m y ) cos ( K m y ) sin [ K n ( z + 2 A D ) ] sin [ K n ( z + 2 A D ) ] exp [ ( K l 2 + K m 2 + K n 2 ) D v ( t t ) μ a v ( t t ) ] ,
L x = L x + 4 A D , L y = L y + 4 A D , L z = L z + 4 A D ,
K l = ( 2 l 1 ) π L x , K m = ( 2 m 1 ) π L y , K n = n π L z ; l , m , n = 1 , 2 , 3 , 4 , 5 ,
Φ eRaman ( r , t ) = 2 3 μ s R v v e L x L y L z l , m , n = 1 cos ( K l x ) cos ( K m y ) sin [ K n ( z + 2 A e D e ) ] × sin [ K n ( z s + 2 A D ) ] [ ( D e v e D v ) ( K l 2 + K m 2 + K n 2 ) + ( μ a e v e μ a v ) ] 1 × { exp [ ( K l 2 + K m 2 + K n 2 ) D v t μ a v t ] exp [ ( K l 2 + K m 2 + K n 2 ) D e v e t μ a e v e t ] } .
R eRamanEPBC ( x , y , t ) = Φ eRaman ( x , y , z = 0 , t ) 2 A e .
R eRamanEBPC ( x , y , t ) = 2 2 μ s R v v e A e L x L y L z l , m , n = 1 cos ( K l x ) cos ( K m y ) sin [ K n ( 2 A e D e ) ] × sin [ K n ( z s + 2 A D ) ] [ ( D e v e D v ) ( K l 2 + K m 2 + K n 2 ) + ( μ a e v e μ a V ) ] 1 × { exp [ ( K l 2 + K m 2 + K n 2 ) D v t μ a v t ] exp [ ( K l 2 + K m 2 + K n 2 ) D e v e t μ a e v e t ] } .
R eRamanFick ( x , y , t ) = D e Φ eRaman ( x , y , z = 0 , t ) z .
R eRamanFick ( x , y , t ) = 2 3 D e μ s R v v e L x L y L z l , m , n = 1 cos ( K l x ) cos ( K m y ) K n cos [ K n ( 2 A e D e ) ] × sin [ K n ( z s + 2 A D ) ] [ ( D e v e D v ) ( K l 2 + K m 2 + K n 2 ) + ( μ a e v e μ a v ) ] 1 × { exp [ ( K l 2 + K m 2 + K n 2 ) D v t μ a v t ] exp [ ( K l 2 + K m 2 + K n 2 ) D e v e t μ a e v e t ] } .
p Fluo ( t | t ) = 1 τ exp ( t t τ ) Θ ( t t ) ,
q e ( r , t ) = 0 t p Fluo ( t | t ) η e μ a f G ( r s , r , μ a , μ s , n , t ) d t + μ s R G ( r s , r , μ a , μ s , n , t ) ,
Φ e ( r , t ) = 0 t V q e ( r , t ) G e ( r , r , μ a e , μ s e , n e , t t ) d r d t = = Φ e F l u o ( r , t ) + Φ eRaman ( r , t ) ,
Φ e F l u o ( r , t ) = 2 3 μ a f η e v v e τ L x L y L z l , m , n = 1 cos ( K l x ) cos ( K m y ) sin [ K n ( z + 2 A e D e ) ] × sin [ K n ( z s + 2 A D ) ] [ ( D e v e D v ) ( K l 2 + K m 2 + K n 2 ) + ( μ a e v e μ a v ) ] 1 × { exp [ ( K l 2 + K m 2 + K n 2 ) D v t μ a v t ] exp [ ( K l 2 + K m 2 + K n 2 ) D e v e t μ a e v e t ] } × [ ( K l 2 + K m 2 + K n 2 ) D v μ a v + 1 / τ ] 1 2 3 μ a f η e v v e τ L x L y L z l , m , n = 1 cos ( K l x ) cos ( K m y ) sin [ K n ( z + 2 A e D e ) ] × sin [ K n ( z s + 2 A D ) ] [ ( K l 2 + K m 2 + K n 2 ) D e v e + μ a e v e 1 / τ ] 1 × { exp [ t / τ ] exp [ ( K l 2 + K m 2 + K n 2 ) D e v e t μ a e v e t ] } × [ ( K l 2 + K m 2 + K n 2 ) D v μ a v + 1 / τ ] 1 .
q eFluo ( r , t ) = μ a f τ η e 0 t exp [ ( t t ) τ ] G ( r s , r , μ a , μ s , n , t ) d t = = 2 3 η e μ a f v τ L x L y L z l , m , n = 1 cos ( K l x s ) cos ( K l x ) cos ( K m y s ) cos ( K m y ) sin [ K n ( z s + 2 A D ) ] × sin [ K n ( z + 2 A D ) ] [ ( K l 2 + K m 2 + K n 2 ) D v μ a v + 1 / τ ] 1 × { exp [ ( K l 2 + K m 2 + K n 2 ) D v t μ a v t ] exp ( t / τ ) } ,
R eEBPC ( x , y , λ e , t ) = Φ eFluo ( x , y , z = 0 , λ e , t ) 2 A e + Φ eRaman ( x , y , z = 0 , λ e , t ) 2 A e ,
R eFick ( x , y , t ) = D e Φ eFluo ( x , y , z = 0 , λ e , t ) z + D e Φ eRaman ( x , y , z = 0 , λ e , t ) z = = R eFluoFick ( x , y , t ) + R eRamanFick ( x , y , t ) ,
R eFluoFick ( x , y , t ) = 2 3 D e μ a f η e v v e τ L x L y L z l , m , n = 1 cos ( K l x ) cos ( K m y ) K n cos [ K n ( 2 A e D e ) ] × sin [ K n ( z s + 2 A D ) ] [ ( D e v e D v ) ( K l 2 + K m 2 + K n 2 ) + ( μ a e v e μ a v ) ] 1 × { exp [ ( K l 2 + K m 2 + K n 2 ) D v t μ a v t ] exp [ ( K l 2 + K m 2 + K n 2 ) D e v e t μ a e v e t ] } × [ ( K l 2 + K m 2 + K n 2 ) D v μ a v + 1 / τ ] 1 2 3 D e μ a f η e v v e τ L x L y L z l , m , n = 1 cos ( K l x ) cos ( K m y ) K n cos [ K n ( 2 A e D e ) ] × sin [ K n ( z s + 2 A D ) ] [ ( K l 2 + K m 2 + K n 2 ) D e v e + μ a e v e 1 / τ ] 1 × { exp [ t / τ ] exp [ ( K l 2 + K m 2 + K n 2 ) D e v e t μ a e v e t ] } × [ ( K l 2 + K m 2 + K n 2 ) D v μ a v + 1 / τ ] 1 .
Φ e 0 Fluo ( r , t ) = μ a f η e 0 t V G ( r s , r , μ a , μ s , n , t ) G e ( r , r , μ a e , μ s e , n e , t t ) d r d t = 2 3 μ a f η e v v e L x L y L z l , m , n = 1 cos ( K l x ) cos ( K m y ) sin [ K n ( z + 2 A e D e ) ] sin [ K n ( z s + 2 A D ) ] × [ ( D e v e D v ) ( K l 2 + K m 2 + K n 2 ) + ( μ a e v e μ a v ) ] 1 × { exp [ ( K l 2 + K m 2 + K n 2 ) D v t μ a v t ] exp ( ( K l 2 + K m 2 + K n 2 ) D e v e t μ a e v e t ) } .
Φ eFluo ( r , t ) = [ Φ e 0 Fluo ( r , t ) ] * [ 1 τ exp ( t / τ ) ] .
R eFluo ( r , t ) = [ R e 0 Fluo ( r , t ) ] * [ 1 τ exp ( t / τ ) ] .
R e 0 FluoFick ( x , y , t ) = 2 3 D e μ a f η e v v e L x L y L z l , m , n = 1 cos ( K l x ) cos ( K m y ) K n cos [ K n ( 2 A e D e ) ] × sin [ K n ( z s + 2 A D ) ] [ ( D e v e D v ) ( K l 2 + K m 2 + K n 2 ) + ( μ a e v e μ a v ) ] 1 × { exp [ ( K l 2 + K m 2 + K n 2 ) D v t μ a v t ] exp [ ( K l 2 + K m 2 + K n 2 ) D e v e t μ a e v e t ] } .
μ e λ = μ a b + μ s b + μ s R + μ a f ,
w 1 = 0 μ e λ exp ( μ e λ ) d .
w 4 = 0 t p Fluo ( t | 0 ) d t .
μ e λ e = μ a b e + μ s b e .
ε ( t ) = 1 R eHeur ( ρ , t ) R eRamanFick ( ρ , t ) .
G ( r , r , t ) = 2 v π a 2 s l = 1 n = 1 J 0 ( ρ λ l ) J 0 ( ρ λ l ) J 1 2 ( a λ l ) sin [ K n ( z + 2 A D ) ] sin [ K n ( z + 2 A D ) ] exp [ λ l 2 D v ( t t ) ] exp [ K n 2 D v ( t t ) ] exp [ μ a v ( t t ) ] ,
Φ eRamanCyl ( r , t ) = 2 μ s R v v e π a 2 s l = 1 n = 1 J 0 ( ρ λ l ) J 1 2 ( a λ l ) sin [ K n ( z + 2 A e D e ) ] sin [ K n ( z s + 2 A D ) ] × [ ( D e v e D v ) ( λ l 2 + K n 2 ) + ( μ a e v e μ a v ) ] 1 × { exp [ ( λ l 2 + K n 2 ) D v t μ a v t ] exp [ ( λ l 2 + K n 2 ) D e v e t μ a e v e t ] } ,

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