Abstract

Diffuse reflectance spectroscopy is one of the simplest and widely used techniques for the non-invasive study of biological tissues but no exact analytical solution exists for the problem of diffuse reflectance from turbid media such as biological tissues. In this work, a general treatment of the problem of diffuse reflectance from a homogeneous semi-infinite turbid medium is presented using Monte Carlo simulations. Based on the results of the Monte Carlo method, simple semi-empirical analytical solutions are developed valid for a wide range of collection geometries corresponding to various optical detector diameters. This approach may be useful for the quick and accurate modeling of diffuse reflectance from tissues.

© 2011 OSA

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  1. L. Reynolds, C. Johnson, and A. Ishimaru, “Diffuse reflectance from a finite blood medium: applications to the modeling of fiber optic catheters,” Appl. Opt.15(9), 2059–2067 (1976).
    [CrossRef] [PubMed]
  2. R. A. J. Groenhuis, H. A. Ferwerda, and J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1: theory,” Appl. Opt.22(16), 2456–2462 (1983).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed.47(2), 131–146 (1995).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  20. V. Sharma, J. W. He, S. Narvenkar, Y. B. Peng, and H. Liu, “Quantification of light reflectance spectroscopy and its application: determination of hemodynamics on the rat spinal cord and brain induced by electrical stimulation,” Neuroimage56(3), 1316–1328 (2011).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  22. G. Zonios, A. Dimou, I. Bassukas, D. Galaris, A. Tsolakidis, and E. Kaxiras, “Melanin absorption spectroscopy: new method for noninvasive skin investigation and melanoma detection,” J. Biomed. Opt.13(1), 014017 (2008).
    [CrossRef] [PubMed]
  23. G. Zonios and A. Dimou, “Melanin optical properties provide evidence for chemical and structural disorder in vivo,” Opt. Express16(11), 8263–8268 (2008).
    [CrossRef] [PubMed]
  24. G. Zonios, A. Dimou, and D. Galaris, “Probing skin interaction with hydrogen peroxide using diffuse reflectance spectroscopy,” Phys. Med. Biol.53(1), 269–278 (2008).
    [CrossRef] [PubMed]
  25. G. Zonios and A. Dimou, “Simple two-layer reflectance model for biological tissue applications: lower absorbing layer,” Appl. Opt.49(27), 5026–5031 (2010).
    [CrossRef] [PubMed]

2011 (3)

T. Y. Tseng, C. Y. Chen, Y. S. Li, and K. B. Sung, “Quantification of the optical properties of two-layered turbid media by simultaneously analyzing the spectral and spatial information of steady-state diffuse reflectance spectroscopy,” Biomed. Opt. Express2(4), 901–914 (2011).
[CrossRef] [PubMed]

J. W. He, D. Kashyap, L. A. Trevino, H. Liu, and Y. B. Peng, “Simultaneous absolute measures of glabrous skin hemodynamic and light-scattering change in response to formalin injection in rats,” Neurosci. Lett.492(1), 59–63 (2011).
[CrossRef] [PubMed]

V. Sharma, J. W. He, S. Narvenkar, Y. B. Peng, and H. Liu, “Quantification of light reflectance spectroscopy and its application: determination of hemodynamics on the rat spinal cord and brain induced by electrical stimulation,” Neuroimage56(3), 1316–1328 (2011).
[CrossRef] [PubMed]

2010 (2)

2009 (2)

2008 (5)

G. Zonios, A. Dimou, I. Bassukas, D. Galaris, A. Tsolakidis, and E. Kaxiras, “Melanin absorption spectroscopy: new method for noninvasive skin investigation and melanoma detection,” J. Biomed. Opt.13(1), 014017 (2008).
[CrossRef] [PubMed]

G. Zonios and A. Dimou, “Melanin optical properties provide evidence for chemical and structural disorder in vivo,” Opt. Express16(11), 8263–8268 (2008).
[CrossRef] [PubMed]

G. Zonios, A. Dimou, and D. Galaris, “Probing skin interaction with hydrogen peroxide using diffuse reflectance spectroscopy,” Phys. Med. Biol.53(1), 269–278 (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] [PubMed]

N. Rajaram, T. H. Nguyen, and J. W. Tunnell, “Lookup table-based inverse model for determining optical properties of turbid media,” J. Biomed. Opt.13(5), 050501 (2008).
[CrossRef] [PubMed]

2007 (2)

2006 (1)

2005 (1)

2004 (1)

2003 (1)

1999 (1)

1996 (1)

A. Kienle and M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol.41(10), 2221–2227 (1996).
[CrossRef] [PubMed]

1995 (1)

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed.47(2), 131–146 (1995).
[CrossRef] [PubMed]

1993 (1)

1992 (1)

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys.19(4), 879–888 (1992).
[CrossRef] [PubMed]

1983 (1)

1976 (1)

A’Amar, O.

Aarnoudse, J. G.

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] [PubMed]

Amelink, A.

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] [PubMed]

Backman, V. M.

Bard, M. P. L.

Bassukas, I.

G. Zonios, A. Dimou, I. Bassukas, D. Galaris, A. Tsolakidis, and E. Kaxiras, “Melanin absorption spectroscopy: new method for noninvasive skin investigation and melanoma detection,” J. Biomed. Opt.13(1), 014017 (2008).
[CrossRef] [PubMed]

Bigio, I. J.

Boas, D. A.

Burgers, S. A.

Chen, C. Y.

Dassel, A. C. M.

de Mul, F. F. M.

Dimou, A.

Fang, Q.

Farrell, T. J.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys.19(4), 879–888 (1992).
[CrossRef] [PubMed]

Feld, M. S.

Ferwerda, H. A.

Fitzmaurice, M.

Galaris, D.

G. Zonios, A. Dimou, and D. Galaris, “Probing skin interaction with hydrogen peroxide using diffuse reflectance spectroscopy,” Phys. Med. Biol.53(1), 269–278 (2008).
[CrossRef] [PubMed]

G. Zonios, A. Dimou, I. Bassukas, D. Galaris, A. Tsolakidis, and E. Kaxiras, “Melanin absorption spectroscopy: new method for noninvasive skin investigation and melanoma detection,” J. Biomed. Opt.13(1), 014017 (2008).
[CrossRef] [PubMed]

German, D. C.

Giller, C. A.

Graaff, R.

Groenhuis, R. A. J.

Hayashi, T.

He, J. W.

V. Sharma, J. W. He, S. Narvenkar, Y. B. Peng, and H. Liu, “Quantification of light reflectance spectroscopy and its application: determination of hemodynamics on the rat spinal cord and brain induced by electrical stimulation,” Neuroimage56(3), 1316–1328 (2011).
[CrossRef] [PubMed]

J. W. He, D. Kashyap, L. A. Trevino, H. Liu, and Y. B. Peng, “Simultaneous absolute measures of glabrous skin hemodynamic and light-scattering change in response to formalin injection in rats,” Neurosci. Lett.492(1), 59–63 (2011).
[CrossRef] [PubMed]

Ishimaru, A.

Jacques, S. L.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed.47(2), 131–146 (1995).
[CrossRef] [PubMed]

Johns, M.

Johnson, C.

Kashio, Y.

Kashyap, D.

J. W. He, D. Kashyap, L. A. Trevino, H. Liu, and Y. B. Peng, “Simultaneous absolute measures of glabrous skin hemodynamic and light-scattering change in response to formalin injection in rats,” Neurosci. Lett.492(1), 59–63 (2011).
[CrossRef] [PubMed]

Kaxiras, E.

G. Zonios, A. Dimou, I. Bassukas, D. Galaris, A. Tsolakidis, and E. Kaxiras, “Melanin absorption spectroscopy: new method for noninvasive skin investigation and melanoma detection,” J. Biomed. Opt.13(1), 014017 (2008).
[CrossRef] [PubMed]

Kienle, A.

A. Kienle and M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol.41(10), 2221–2227 (1996).
[CrossRef] [PubMed]

Koelink, M. H.

Li, J. F.

Li, Y. S.

Liang, J. M.

Liu, H.

J. W. He, D. Kashyap, L. A. Trevino, H. Liu, and Y. B. Peng, “Simultaneous absolute measures of glabrous skin hemodynamic and light-scattering change in response to formalin injection in rats,” Neurosci. Lett.492(1), 59–63 (2011).
[CrossRef] [PubMed]

V. Sharma, J. W. He, S. Narvenkar, Y. B. Peng, and H. Liu, “Quantification of light reflectance spectroscopy and its application: determination of hemodynamics on the rat spinal cord and brain induced by electrical stimulation,” Neuroimage56(3), 1316–1328 (2011).
[CrossRef] [PubMed]

Liu, H. L.

Liu, Q.

Lu, B. J.

Manoharan, R.

Mantis, G.

Narvenkar, S.

V. Sharma, J. W. He, S. Narvenkar, Y. B. Peng, and H. Liu, “Quantification of light reflectance spectroscopy and its application: determination of hemodynamics on the rat spinal cord and brain induced by electrical stimulation,” Neuroimage56(3), 1316–1328 (2011).
[CrossRef] [PubMed]

Nguyen, T. H.

N. Rajaram, T. H. Nguyen, and J. W. Tunnell, “Lookup table-based inverse model for determining optical properties of turbid media,” J. Biomed. Opt.13(5), 050501 (2008).
[CrossRef] [PubMed]

Okada, E.

Patterson, M. S.

A. Kienle and M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol.41(10), 2221–2227 (1996).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys.19(4), 879–888 (1992).
[CrossRef] [PubMed]

Peng, Y. B.

J. W. He, D. Kashyap, L. A. Trevino, H. Liu, and Y. B. Peng, “Simultaneous absolute measures of glabrous skin hemodynamic and light-scattering change in response to formalin injection in rats,” Neurosci. Lett.492(1), 59–63 (2011).
[CrossRef] [PubMed]

V. Sharma, J. W. He, S. Narvenkar, Y. B. Peng, and H. Liu, “Quantification of light reflectance spectroscopy and its application: determination of hemodynamics on the rat spinal cord and brain induced by electrical stimulation,” Neuroimage56(3), 1316–1328 (2011).
[CrossRef] [PubMed]

Perelman, L. T.

Qu, X.

Rajaram, N.

N. Rajaram, T. H. Nguyen, and J. W. Tunnell, “Lookup table-based inverse model for determining optical properties of turbid media,” J. Biomed. Opt.13(5), 050501 (2008).
[CrossRef] [PubMed]

Ramanujam, N.

Reif, R.

Ren, N. N.

Reynolds, L.

Sharma, V.

V. Sharma, J. W. He, S. Narvenkar, Y. B. Peng, and H. Liu, “Quantification of light reflectance spectroscopy and its application: determination of hemodynamics on the rat spinal cord and brain induced by electrical stimulation,” Neuroimage56(3), 1316–1328 (2011).
[CrossRef] [PubMed]

Sterenborg, H. J. C. M.

Sung, K. B.

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] [PubMed]

Ten Bosch, J. J.

Tian, J.

Trevino, L. A.

J. W. He, D. Kashyap, L. A. Trevino, H. Liu, and Y. B. Peng, “Simultaneous absolute measures of glabrous skin hemodynamic and light-scattering change in response to formalin injection in rats,” Neurosci. Lett.492(1), 59–63 (2011).
[CrossRef] [PubMed]

Tseng, T. Y.

Tsolakidis, A.

G. Zonios, A. Dimou, I. Bassukas, D. Galaris, A. Tsolakidis, and E. Kaxiras, “Melanin absorption spectroscopy: new method for noninvasive skin investigation and melanoma detection,” J. Biomed. Opt.13(1), 014017 (2008).
[CrossRef] [PubMed]

Tunnell, J. W.

N. Rajaram, T. H. Nguyen, and J. W. Tunnell, “Lookup table-based inverse model for determining optical properties of turbid media,” J. Biomed. Opt.13(5), 050501 (2008).
[CrossRef] [PubMed]

Van Dam, J.

Wang, L. H.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed.47(2), 131–146 (1995).
[CrossRef] [PubMed]

Wilson, B.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys.19(4), 879–888 (1992).
[CrossRef] [PubMed]

Zheng, L. Q.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed.47(2), 131–146 (1995).
[CrossRef] [PubMed]

Zijistra, W. G.

Zonios, G.

Appl. Opt. (8)

L. Reynolds, C. Johnson, and A. Ishimaru, “Diffuse reflectance from a finite blood medium: applications to the modeling of fiber optic catheters,” Appl. Opt.15(9), 2059–2067 (1976).
[CrossRef] [PubMed]

R. A. J. Groenhuis, H. A. Ferwerda, and J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1: theory,” Appl. Opt.22(16), 2456–2462 (1983).
[CrossRef] [PubMed]

R. Reif, O. A’Amar, and I. J. Bigio, “Analytical model of light reflectance for extraction of the optical properties in small volumes of turbid media,” Appl. Opt.46(29), 7317–7328 (2007).
[CrossRef] [PubMed]

T. Hayashi, Y. Kashio, and E. Okada, “Hybrid Monte Carlo-diffusion method for light propagation in tissue with a low-scattering region,” Appl. Opt.42(16), 2888–2896 (2003).
[CrossRef] [PubMed]

R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijistra, A. C. M. Dassel, and J. G. Aarnoudse, “Condensed Monte Carlo simulations for the description of light transport,” Appl. Opt.32(4), 426–434 (1993).
[CrossRef] [PubMed]

G. Zonios, L. T. Perelman, V. M. Backman, R. Manoharan, M. Fitzmaurice, J. Van Dam, and M. S. Feld, “Diffuse reflectance spectroscopy of human adenomatous colon polyps in vivo,” Appl. Opt.38(31), 6628–6637 (1999).
[CrossRef] [PubMed]

G. Mantis and G. Zonios, “Simple two-layer reflectance model for biological tissue applications,” Appl. Opt.48(18), 3490–3496 (2009).
[CrossRef] [PubMed]

G. Zonios and A. Dimou, “Simple two-layer reflectance model for biological tissue applications: lower absorbing layer,” Appl. Opt.49(27), 5026–5031 (2010).
[CrossRef] [PubMed]

Biomed. Opt. Express (1)

Comput. Methods Programs Biomed. (1)

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed.47(2), 131–146 (1995).
[CrossRef] [PubMed]

J. Biomed. Opt. (3)

G. Zonios, A. Dimou, I. Bassukas, D. Galaris, A. Tsolakidis, and E. Kaxiras, “Melanin absorption spectroscopy: new method for noninvasive skin investigation and melanoma detection,” J. Biomed. Opt.13(1), 014017 (2008).
[CrossRef] [PubMed]

N. Rajaram, T. H. Nguyen, and J. W. Tunnell, “Lookup table-based inverse model for determining optical properties of turbid media,” J. Biomed. Opt.13(5), 050501 (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] [PubMed]

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

Med. Phys. (1)

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys.19(4), 879–888 (1992).
[CrossRef] [PubMed]

Neuroimage (1)

V. Sharma, J. W. He, S. Narvenkar, Y. B. Peng, and H. Liu, “Quantification of light reflectance spectroscopy and its application: determination of hemodynamics on the rat spinal cord and brain induced by electrical stimulation,” Neuroimage56(3), 1316–1328 (2011).
[CrossRef] [PubMed]

Neurosci. Lett. (1)

J. W. He, D. Kashyap, L. A. Trevino, H. Liu, and Y. B. Peng, “Simultaneous absolute measures of glabrous skin hemodynamic and light-scattering change in response to formalin injection in rats,” Neurosci. Lett.492(1), 59–63 (2011).
[CrossRef] [PubMed]

Opt. Express (5)

Opt. Lett. (1)

Phys. Med. Biol. (2)

A. Kienle and M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol.41(10), 2221–2227 (1996).
[CrossRef] [PubMed]

G. Zonios, A. Dimou, and D. Galaris, “Probing skin interaction with hydrogen peroxide using diffuse reflectance spectroscopy,” Phys. Med. Biol.53(1), 269–278 (2008).
[CrossRef] [PubMed]

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Figures (9)

Fig. 1
Fig. 1

Illumination/collection geometry employed in the simulations.

Fig. 2
Fig. 2

Reflectance as a function of the reduced scattering coefficient (MC data) for various detector radii, r, and various values of the absorption coefficient. (a) μ a = 0.001 mm−1, (b) μ a = 0.01 mm−1, (c) μ a = 0.1 mm−1, and (d) μ a = 1 mm−1. Solid lines represent a fit to the model described by Eq. (1).

Fig. 3
Fig. 3

k1 and k2 parameter values obtained from the fitting results shown in Fig. 2, using Eq. (1). The solid lines represent spline interpolation between the data points.

Fig. 4
Fig. 4

Reflectance as function of the absorption coefficient for detectors with various radii. The solid lines represent fits to the MC data using Eq. (2) ( μ s = 1.8 mm−1 for the data shown here).

Fig. 5
Fig. 5

Dependence of the k1 and k2 parameters of Eq. (2) on the detector radius. The solid lines represent fits to Eqs. (3) and (4) for k1 and k2, respectively.

Fig. 6
Fig. 6

Dependence of the parameters a 1 , b 1 , a 2 and b 2 of Eqs. (3) and (4) on the reduced scattering coefficient. The solid lines represent fits to Eqs. (5)(8).

Fig. 7
Fig. 7

Diffuse reflectance scored using an optical detector with a radius of 0.5 mm showing the dependence of reflectance on (a) the absorption coefficient (reflectance shown is normalized to unity at the smallest value of the absorption coefficient), and (b) the reduced scattering coefficient. Solid lines represent fits to Eq. (9).

Fig. 8
Fig. 8

Reflectance for a detector with infinite radius and best fits using Eqs. (2) and (10). Equation (10) gives a better fit compared to Eq. (2), and this is also true for large detector diameters ( μ s = 1.8 mm−1 for the data shown here).

Fig. 9
Fig. 9

Effects of detector numerical aperture as a function of the absorption coefficient for three different values of the reduced scattering coefficient: (a) μ s = 0.6 mm−1, (b) μ s = 1.8 mm−1, and (c) μ s = 3.0 mm−1. The efficiency parameter shown in the vertical axis represents the percent ratio of reflectance scored in the 0–15 deg exit angle range over the total reflectance scored over the entire 0–90 deg exit angle range. Solid lines represent spline interpolation between data points.

Tables (2)

Tables Icon

Table 1 Optical parameter values used in the MC simulations

Tables Icon

Table 2 Model parameter values

Equations (10)

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

R= k 1 1+ k 2 μ a μ s
R= k 1 1+ k 2 μ a
k 1 = 1 a 1 + b 1 r 3/2
k 2 = 1 a 2 + b 2 r 1
a 1 = ( c 1 + d 1 / μ s ) 1
b 1 = ( e 1 + f 1 μ s 3/2 ) 1
a 2 = c 2 + d 2 μ s
b 2 = e 2 + f 2 μ s
R= k 1 μ s 1+ k 2 μ a
R= k 1 e k 2 ln(1+ μ a / μ s )

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