Abstract

We provide a methodology for accurately predicting elastic backscattering radial distributions from random media with two simple empirical models. We apply these models to predict the backscattering based on two classes of scattering phase functions: the Henyey-Greenstein phase function and a generalized two parameter phase function that is derived from the Whittle-Matérn correlation function. We demonstrate that the model has excellent agreement over all length scales and has less than 1% error for backscattering at subdiffusion length scales for tissue-relevant optical properties. The presented model is the first available approach for accurately predicting backscattering at length scales significantly smaller than the transport mean free path.

© 2010 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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2010 (1)

V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, “Characterization of Light Transport in Scattering Media at Subdiffusion Length Scales with Low-Coherence Enhanced Backscattering,” IEEE J. Sel. Top. Quantum Electron. 16(3), 619–626 (2010).
[CrossRef]

2009 (2)

2008 (2)

2007 (2)

2006 (2)

P. Guttorp and T. Gneiting, “Studies in the history of probability and statistics XLIX On the Matern correlation family,” Biometrika 93(4), 989–995 (2006).
[CrossRef]

G. M. Palmer and N. Ramanujam, “Monte Carlo-based inverse model for calculating tissue optical properties. Part I: Theory and validation on synthetic phantoms,” Appl. Opt. 45(5), 1062–1071 (2006).
[CrossRef] [PubMed]

2005 (2)

2004 (3)

2001 (1)

1999 (1)

1998 (1)

1996 (1)

1995 (1)

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml - Monte-Carlo Modeling of Light Transport in Multilayered Tissues,” Comput Meth. Prog Biol. 47(2), 131–146 (1995).
[CrossRef]

1992 (2)

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]

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]

1986 (1)

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[CrossRef] [PubMed]

A’Amar, O.

Akkermans, E.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[CrossRef] [PubMed]

Amelink, A.

Backman, V.

Bard, M. P. L.

Bevilacqua, F.

Bigio, I. J.

Burgers, S. A.

Capoglu, I. R.

Çapoglu, I. R.

Depeursinge, C.

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]

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]

Foster, T. H.

Gendron-Fitzpatrick, A.

M. C. Skala, G. M. Palmer, C. F. Zhu, Q. Liu, K. M. Vrotsos, C. L. Marshek-Stone, A. Gendron-Fitzpatrick, and N. Ramanujam, “Investigation of fiber-optic probe designs for optical spectroscopic diagnosis of epithelial pre-cancers,” Lasers Surg. Med. 34(1), 25–38 (2004).
[CrossRef] [PubMed]

Gneiting, T.

P. Guttorp and T. Gneiting, “Studies in the history of probability and statistics XLIX On the Matern correlation family,” Biometrika 93(4), 989–995 (2006).
[CrossRef]

Gomes, A. J.

Guttorp, P.

P. Guttorp and T. Gneiting, “Studies in the history of probability and statistics XLIX On the Matern correlation family,” Biometrika 93(4), 989–995 (2006).
[CrossRef]

Hayakawa, C. K.

I. Seo, C. K. Hayakawa, and V. Venugopalan, “Radiative transport in the delta-P1 approximation for semi-infinite turbid media,” Med. Phys. 35(2), 681–693 (2008).
[CrossRef] [PubMed]

Hibst, R.

Hull, E. L.

Jacques, S. L.

Jameel, M.

Kienle, A.

Kim, Y. L.

Kromine, A.

Lilge, L.

Liu, Q.

M. C. Skala, G. M. Palmer, C. F. Zhu, Q. Liu, K. M. Vrotsos, C. L. Marshek-Stone, A. Gendron-Fitzpatrick, and N. Ramanujam, “Investigation of fiber-optic probe designs for optical spectroscopic diagnosis of epithelial pre-cancers,” Lasers Surg. Med. 34(1), 25–38 (2004).
[CrossRef] [PubMed]

Liu, Y.

Marshek-Stone, C. L.

M. C. Skala, G. M. Palmer, C. F. Zhu, Q. Liu, K. M. Vrotsos, C. L. Marshek-Stone, A. Gendron-Fitzpatrick, and N. Ramanujam, “Investigation of fiber-optic probe designs for optical spectroscopic diagnosis of epithelial pre-cancers,” Lasers Surg. Med. 34(1), 25–38 (2004).
[CrossRef] [PubMed]

Maynard, R.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[CrossRef] [PubMed]

Mutyal, N. N.

V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, “Characterization of Light Transport in Scattering Media at Subdiffusion Length Scales with Low-Coherence Enhanced Backscattering,” IEEE J. Sel. Top. Quantum Electron. 16(3), 619–626 (2010).
[CrossRef]

Palmer, G. M.

G. M. Palmer and N. Ramanujam, “Monte Carlo-based inverse model for calculating tissue optical properties. Part I: Theory and validation on synthetic phantoms,” Appl. Opt. 45(5), 1062–1071 (2006).
[CrossRef] [PubMed]

M. C. Skala, G. M. Palmer, C. F. Zhu, Q. Liu, K. M. Vrotsos, C. L. Marshek-Stone, A. Gendron-Fitzpatrick, and N. Ramanujam, “Investigation of fiber-optic probe designs for optical spectroscopic diagnosis of epithelial pre-cancers,” Lasers Surg. Med. 34(1), 25–38 (2004).
[CrossRef] [PubMed]

Patterson, M. S.

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, and B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35(13), 2304–2314 (1996).
[CrossRef]

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]

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]

Prahl, S. A.

Ramanujam, N.

G. M. Palmer and N. Ramanujam, “Monte Carlo-based inverse model for calculating tissue optical properties. Part I: Theory and validation on synthetic phantoms,” Appl. Opt. 45(5), 1062–1071 (2006).
[CrossRef] [PubMed]

M. C. Skala, G. M. Palmer, C. F. Zhu, Q. Liu, K. M. Vrotsos, C. L. Marshek-Stone, A. Gendron-Fitzpatrick, and N. Ramanujam, “Investigation of fiber-optic probe designs for optical spectroscopic diagnosis of epithelial pre-cancers,” Lasers Surg. Med. 34(1), 25–38 (2004).
[CrossRef] [PubMed]

Ramella-Roman, J. C.

Reif, R.

Rogers, J. D.

Roy, H. K.

Seo, I.

I. Seo, C. K. Hayakawa, and V. Venugopalan, “Radiative transport in the delta-P1 approximation for semi-infinite turbid media,” Med. Phys. 35(2), 681–693 (2008).
[CrossRef] [PubMed]

Sheppard, C. J. R.

Skala, M. C.

M. C. Skala, G. M. Palmer, C. F. Zhu, Q. Liu, K. M. Vrotsos, C. L. Marshek-Stone, A. Gendron-Fitzpatrick, and N. Ramanujam, “Investigation of fiber-optic probe designs for optical spectroscopic diagnosis of epithelial pre-cancers,” Lasers Surg. Med. 34(1), 25–38 (2004).
[CrossRef] [PubMed]

Steiner, R.

Sterenborg, H. J. C. M.

Taflove, A.

Turzhitsky, V.

V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, “Characterization of Light Transport in Scattering Media at Subdiffusion Length Scales with Low-Coherence Enhanced Backscattering,” IEEE J. Sel. Top. Quantum Electron. 16(3), 619–626 (2010).
[CrossRef]

Turzhitsky, V. M.

Venugopalan, V.

I. Seo, C. K. Hayakawa, and V. Venugopalan, “Radiative transport in the delta-P1 approximation for semi-infinite turbid media,” Med. Phys. 35(2), 681–693 (2008).
[CrossRef] [PubMed]

Vrotsos, K. M.

M. C. Skala, G. M. Palmer, C. F. Zhu, Q. Liu, K. M. Vrotsos, C. L. Marshek-Stone, A. Gendron-Fitzpatrick, and N. Ramanujam, “Investigation of fiber-optic probe designs for optical spectroscopic diagnosis of epithelial pre-cancers,” Lasers Surg. Med. 34(1), 25–38 (2004).
[CrossRef] [PubMed]

Wali, R. K.

Wang, L. H.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml - Monte-Carlo Modeling of Light Transport in Multilayered Tissues,” Comput Meth. Prog Biol. 47(2), 131–146 (1995).
[CrossRef]

Wang, L. V.

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]

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]

Wilson, B. C.

Wolf, P. E.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[CrossRef] [PubMed]

Zheng, L. Q.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml - Monte-Carlo Modeling of Light Transport in Multilayered Tissues,” Comput Meth. Prog Biol. 47(2), 131–146 (1995).
[CrossRef]

Zhu, C. F.

M. C. Skala, G. M. Palmer, C. F. Zhu, Q. Liu, K. M. Vrotsos, C. L. Marshek-Stone, A. Gendron-Fitzpatrick, and N. Ramanujam, “Investigation of fiber-optic probe designs for optical spectroscopic diagnosis of epithelial pre-cancers,” Lasers Surg. Med. 34(1), 25–38 (2004).
[CrossRef] [PubMed]

Appl. Opt. (4)

Biometrika (1)

P. Guttorp and T. Gneiting, “Studies in the history of probability and statistics XLIX On the Matern correlation family,” Biometrika 93(4), 989–995 (2006).
[CrossRef]

Comput Meth. Prog Biol. (1)

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml - Monte-Carlo Modeling of Light Transport in Multilayered Tissues,” Comput Meth. Prog Biol. 47(2), 131–146 (1995).
[CrossRef]

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

V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, “Characterization of Light Transport in Scattering Media at Subdiffusion Length Scales with Low-Coherence Enhanced Backscattering,” IEEE J. Sel. Top. Quantum Electron. 16(3), 619–626 (2010).
[CrossRef]

J. Opt. Soc. Am. A (3)

Lasers Surg. Med. (1)

M. C. Skala, G. M. Palmer, C. F. Zhu, Q. Liu, K. M. Vrotsos, C. L. Marshek-Stone, A. Gendron-Fitzpatrick, and N. Ramanujam, “Investigation of fiber-optic probe designs for optical spectroscopic diagnosis of epithelial pre-cancers,” Lasers Surg. Med. 34(1), 25–38 (2004).
[CrossRef] [PubMed]

Med. Phys. (3)

I. Seo, C. K. Hayakawa, and V. Venugopalan, “Radiative transport in the delta-P1 approximation for semi-infinite turbid media,” Med. Phys. 35(2), 681–693 (2008).
[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]

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]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. Lett. (1)

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[CrossRef] [PubMed]

Other (1)

A. Ishimaru, Wave propagation and scattering in random media (Academic Press, New York, 1978).

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Figures (8)

Fig. 1
Fig. 1

Example phase functions. (a) The Henyey-Greenstein case (m = 1.5) for various values of g. (b) The generalized Whittle-Matérn phase function for various values of m (g = 0.9).

Fig. 2
Fig. 2

Scaling relationships of P(r/ls*) from the Henyey-Greenstein phase function. (a) P(r/ls*) curves for varying values of g. The curves have variations at r/ls*<1 and converge for larger values of r/ls*. (b) Probability difference obtained by subtracting P(r/ls*) for the isotropic case (g = 0) from P(r/ls*) of a given g. (c) Probability difference curves for varying values of g with the amplitude rescaled by a coefficient that depends only on g.

Fig. 3
Fig. 3

Backscattering model for P(r/ls*) based on the Henyey-Greenstein phase function utilizing the difference method. (a) Value of amplitude coefficient as a function of g, c(g). The dots represent a least square fit to the Monte Carlo data and the solid line represents the model for c(g) from Eq. (7). The amplitude increases as a function of g follow a power law. (b) Comparison of Monte Carlo simulations (green, red, and purple curves) and model (black curves) for three values of g. g = 0.9 is not shown because the model and simulation are, by definition, identical for that case. (c) is a rescaled version of (b) shown for a smaller range of r/ls*.

Fig. 4
Fig. 4

Backscattering model for the Henyey-Greenstein case utilizing principle component analysis method. (a) Semi-log plot of Monte Carlo simulations and the PCA model for three values of g. (b) Comparison of PCA model and Monte Carlo simulation for r/ls* < 1. (c) Semi-log plot of the three principle components. Note that higher principle components have less amplitude and contribute less to P(r/ls*) prediction. (d) Plot of coefficients that multiply the principle components and their polynomial fits used to obtain the predictive model.

Fig. 5
Fig. 5

P(r/ls*) distributions for the generalized Whittle-Matérn phase function. (a) P(r/ls*) dependence on m for g = 0.9. The shape of the P(r/ls*) gradually changes with varying m. (b) Probability difference between P(r/ls*) curves and the isotropic P(r/ls*) for g = 0.9 and varying m. Unlike changes in g, alterations in m cannot be accounted for by a simple scaling of the amplitude of the probability difference curves. (d) Probability difference curves for various values of g and m = 1.8. The probability difference only changes in amplitude for a constant m and varying g. (d) The two probability difference curves used to model P(r/ls*) from Eq. (9).

Fig. 6
Fig. 6

Evaluation of P(r/ls*) predictive model accuracy. (a) PΔ model for P(r/ls*) (lines) compared to Monte Carlo data (points) for varying values of m and g = 0.9. (b) PCA model for P(r/ls*) (lines) compared to Monte Carlo data (points) for varying values of m and g = 0.9. (c) Average error of PΔ model as a function of g for r/ls*<1. (d) Average error of PCA model as a function of g for r/ls*<1.

Fig. 7
Fig. 7

P(r) for varying µa. The scattering properties of all three simulations were maintained at µs* = 100 cm−1, g = 0.9, and m = 1.6. Absorption has a minimal effect at small radial distances because of the short path lengths involved.

Fig. 8
Fig. 8

Tissue phantom for experimental measurement of P(r). (a) Comparison of the Whittle-Matérn phase function (m = 1.6 and g = 0.8) with the fit obtained from a mixture of microspheres of three sizes. (b) Resulting P(r) experimentally measured with LEBS, compared with P(r) obtained with the PCA based model and a Monte Carlo simulation. An ls* of 800μm, determined from the Mie theory calculation, was used for the model and Monte Carlo curve. The experimental LEBS measurement was scaled by the 0.75 factor to account for unpolarized light and the empirical 0.5 factor, determined from microsphere studies in prior work [20], to account for the difference observed between LEBS and Monte Carlo (see text). The average error between the experimentally measured P(r) and the model was 7.1%.

Tables (1)

Tables Icon

Table 1 Values of constants from coefficients for PΔ model [Eq. (9)]

Equations (10)

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F ( cos θ ) = 2 g ^ ( m 1 ) [ 1 2 g ^ cos θ + g ^ 2 ] m ( 1 g ^ ) 2 2 m ( 1 + g ^ ) 2 2 m g ^ = 1 1 + 4 ( k l c ) 2 1 2 ( k l c ) 2 k l c = g ^ 1 g ^ ,
F ( θ ) = 2 ( k l c ) 2 ( m 1 ) [ 1 + ( 2 k l c sin ( θ / 2 ) ) 2 ] m 1 ( 1 + ( 2 k l c ) 2 ) 1 m .
g = cos θ = { ( 1 g ^ ) 2 2 m ( 1 + g ^ 2 2 g ^ ( m 1 ) ) ( 1 + g ^ ) 2 2 m ( 1 + g ^ 2 + 2 g ^ ( m 1 ) ) 2 g ^ ( m 2 ) [ ( 1 + g ^ ) 2 2 m ( 1 g ^ ) 2 2 m ] m 2 1 + g ^ 2 2 g ^ + ln ( 1 + g ^ ) ln ( 1 g ^ ) g [ ( 1 g ^ ) 2 ( 1 + g ^ ) 2 ]                  m = 2
F ( cos θ ) = { g ^ [ 1 2 g ^ cos θ + g ^ 2 ] 1 ln ( 1 + g ^ ) ln ( 1 g ^ )    m = 1     1 / 2          g ^ = 0
g = { 1 + g ^ 2 g ^ [ ln ( 1 + g ^ ) ln ( 1 g ^ ) ]    m = 1 , g ^ 0       0          g ^ = 0 .
cos θ = 1 + g ^ 2 [ ξ ( ( 1 g ) 2 2 m ( 1 + g ) 2 2 m ) + ( 1 + g ) 2 2 m ] 1 1 m 2 g
P g = P 0 + c ( g ) [ P 0.9 P 0 ] c ( g ) = a g b ,
P g = P g = 0 + c 1 ( g ) P C 1 + c 2 ( g ) P C 2 + c 3 ( g ) P C 3 ,
P g , m = P g = 0 + c 1 ( g , m ) P 1 Δ + c 2 ( g , m ) P 2 Δ P 1 Δ = P g = 0.9 , m = 1.5 P g = 0 P 2 Δ = P g = 0.9 , m = 1.01 P g = 0
c i = a i + b i x + c i y + d i x 2 + e i y 2 + f i x 3 + g i y 3 + h i x y + i i x 2 y + j i x y 2 ,

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