Quan Liu and Nirmala Ramanujam, "Scaling method for fast Monte Carlo simulation of diffuse reflectance spectra from multilayered turbid media," J. Opt. Soc. Am. A 24, 1011-1025 (2007)
A scaling Monte Carlo method has been developed to calculate diffuse reflectance from multilayered media with a wide range of optical properties in the ultraviolet–visible wavelength range. This multilayered scaling method employs the photon trajectory information generated from a single baseline Monte Carlo simulation of a homogeneous medium to scale the exit distance and exit weight of photons for a new set of optical properties in the multilayered medium. The scaling method is particularly suited to simulating diffuse reflectance spectra or creating a Monte Carlo database to extract optical properties of layered media, both of which are demonstrated in this paper. Particularly, it was found that the root-mean-square error (RMSE) between scaled diffuse reflectance, for which the anisotropy factor and refractive index in the baseline simulation were, respectively, 0.9 and 1.338, and independently simulated diffuse reflectance was less than or equal to 5% for source–detector separations from when the anisotropy factor of the top layer in a two-layered epithelial tissue model was varied from 0.8 to 0.99; in contrast, the RMSE was always less than 5% for all separations (from ) when the anisotropy factor of the bottom layer was varied from 0.7 to 0.99. When the refractive index of either layer in the two-layered tissue model was varied from 1.3 to 1.4, the RMSE was less than 10%. The scaling method can reduce computation time by more than 2 orders of magnitude compared with independent Monte Carlo simulations.
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Effect of Anisotropy Factor of Tissue Layer on RMSEa
Variable
RMSE (%) for Separation
RMSE (%) for Separation
RMSE (%) for Separation
RMSE (%) for Separation
Top anisotropy
20.0
4.2
9.5
7.4
9.5
1.7
5.1
3.8
1.7
1.7
1.0
1.2
12.3
3.4
3.7
3.6
Bottom anisotropy
0.9
4.1
2.6
2.0
0.7
1.3
1.8
1.3
1.7
1.7
1.0
1.2
3.1
3.8
0.8
1.7
RMSEs of scaled reflectance for the original two-layered tissue model relative to independently simulated reflectance for a modified two-layered epithelial tissue model where the anisotropy factor of the top or bottom layer was varied while other parameters of the modified tissue model were kept identical to those of the original tissue model. Note that the anisotropy factor was 0.9 in the original tissue model as shown in bold type.
Table 2
Effect of Refractive Index of Tissue Layer on RMSEa
Variable
RMSE (%) for Separation
RMSE (%) for Separation
RMSE (%) for Separation
RMSE (%) for Separation
Top refractive index
2.9
1.6
1.1
1.0
1.7
1.7
1.0
1.2
8.4
5.8
3.1
3.5
17.0
10.2
6.5
6.3
Bottom refractive index
0.9
1.3
3.6
4.7
1.7
1.7
1.0
1.2
2.5
5.1
8.2
8.1
1.2
6.8
20.2
19.6
RMSEs of scaled reflectance for the original two-layered tissue model relative to independently simulated reflectance for a modified two-layered epithelial tissue model where the refractive index of the top or bottom layer was varied while other parameters of the modified two-layered epithelial tissue model were kept identical to those of the original two-layered epithelial tissue model. Note that the refractive index was 1.338 in the original tissue model as shown in bold type.
Table 3
Effect of Refractive Indices of Fiber Core and Tissue Model on Specular Reflectancea
(column)\ (row)
1.3
1.4
1.5
0° incidence
1.4
0.0014
0.0012
1.5
0.0051
0.0012
0
1.6
0.011
0.0044
0.0010
Cutoff angles defined by NA 0.22
1.4
0.0014
0.0012
1.5
0.0051
0.0012
0
1.6
0.011
0.0044
0.0010
Fraction of specular reflectance for various combinations of refractive indices of the fiber core and the tissue model when (top half) the incident angle is 0° and (bottom half) the cutoff angle is defined by an NA of 0.22. represents the refractive index of the fiber core, and is the refractive index of the tissue model. The actual cutoff angles are also shown in the bottom half of the table.
RMSEs of scaled reflectance relative to independently simulated reflectance from a modified two-layered epithelial tissue model in the case that the phase function was changed from the HG function to the Mie function while other parameters of the modified two-layered epithelial tissue model were kept identical to those of the original two-layered epithelial tissue model. Note that the HG phase function was used in the baseline simulation for scaling.
Table 5
Effect of One Additional Layer and Layer Thickness on RMSEa
Variable
RMSE (%) for Separation
RMSE (%) for Separation
RMSE (%) for Separation
RMSE (%) for Separation
Top-layer thickness
1.1
0.4
0.6
1.3
1.7
1.4
0.5
2.8
2.2
1.2
0.7
1.2
RMSEs of scaled reflectance relative to corresponding independently simulated reflectance for a three-layered epithelial tissue model, whose structure and optical properties were, respectively, shown in Figs. 2b, 3. While the thickness of the top layer was varied from 50 to 250 and to , the thickness of the middle layer was changed from 450 to 250 and to accordingly to keep the total thickness of the two layers a constant . The anisotropy factor and refractive index of each layer as well as the choice of the phase function (HG) in the tissue model were identical to those in the baseline simulation for scaling.
Table 6
Effect of Anisotropy Factor of Tissue Layer on RMSE of Estimated Optical Propertiesa
Variable
Thickness of Top Layer
RMSE (%)
Top anisotropy
10.5
45.2
Y
9.6
15.5
9.6
9.5
5.9
10.5
8.4
Bottom anisotropy
8.3
12.8
21.3
7.7
Y
9.6
9.5
5.9
12.1
6.5
RMSEs of the estimated thickness of the top layer and optical properties of the bottom layer relative to the corresponding true values for given sets of simulated diffuse reflectance spectra from the same modified two-layered epithelial tissue models as in Table 1, where the anisotropy factor of the top or bottom layer was varied while other parameters in the modified two-layered epithelial tissue models were kept identical to those in the original two-layered epithelial tissue model. The rows with bold type are the RMSEs of estimated parameters for input diffuse reflectance simulated with exactly the same anisotropy factor as in the baseline simulation for scaling. The rightmost column in the table indicates if a row contains an RMSE greater than 20% (marked by “Y”), which is a sign of inaccurate inversion.
Table 7
Effect of Refractive Index of Tissue Layer on RMSE of Estimated Optical Propertiesa
Variable
Thickness of Top Layer
RMSE (%)
Top refractive index
6.3
2.1
25.0
Y
9.6
9.5
5.9
7.8
45.4
Y
7.1
76.3
Y
Bottom refractive index
4.2
1.4
28.8
Y
9.6
9.5
5.9
6.9
2.5
6.0
14.3
RMSEs of the estimated thickness of the top layer and optical properties of the bottom layer relative to the corresponding true values for given sets of simulated diffuse reflectance spectra from the same modified two-layered epithelial tissue models as in Table 2, where the refractive index of the top or bottom layer was varied while other parameters in the modified two-layered epithelial tissue models were kept identical to those in the original two-layered epithelial tissue model. The rows with bold type are the RMSEs of estimated parameters for input diffuse reflectance simulated with exactly the same refractive index as in the baseline simulation for scaling. The rightmost column in the table indicates if a row contains an RMSE greater than 20% (marked by “Y”), which is a sign of inaccurate inversion.
Table 8
Effect of Phase Function on RMSE of Estimated Optical Propertiesa
Variable
Thickness of Top Layer
RMSE (%)
HG
9.6
9.5
5.9
Mie
31.4
5.0
Y
RMSEs of the estimated thickness of the top layer and optical properties of the bottom layer relative to the corresponding true values for given sets of simulated diffuse reflectance spectra from the same modified two-layered epithelial tissue models as in Table 4, where the phase function was changed from the HG function to the Mie function while other parameters in the modified two-layered epithelial tissue models were kept identical to those in the original two-layered epithelial tissue model. The rows with bold type are the RMSEs of estimated parameters for input diffuse reflectance simulated with exactly the same phase function as in the baseline simulation for scaling. The rightmost column in the table indicates if a row contains an RMSE greater than 20% (marked by “Y”), which is a sign of inaccurate inversion.
Tables (8)
Table 1
Effect of Anisotropy Factor of Tissue Layer on RMSEa
Variable
RMSE (%) for Separation
RMSE (%) for Separation
RMSE (%) for Separation
RMSE (%) for Separation
Top anisotropy
20.0
4.2
9.5
7.4
9.5
1.7
5.1
3.8
1.7
1.7
1.0
1.2
12.3
3.4
3.7
3.6
Bottom anisotropy
0.9
4.1
2.6
2.0
0.7
1.3
1.8
1.3
1.7
1.7
1.0
1.2
3.1
3.8
0.8
1.7
RMSEs of scaled reflectance for the original two-layered tissue model relative to independently simulated reflectance for a modified two-layered epithelial tissue model where the anisotropy factor of the top or bottom layer was varied while other parameters of the modified tissue model were kept identical to those of the original tissue model. Note that the anisotropy factor was 0.9 in the original tissue model as shown in bold type.
Table 2
Effect of Refractive Index of Tissue Layer on RMSEa
Variable
RMSE (%) for Separation
RMSE (%) for Separation
RMSE (%) for Separation
RMSE (%) for Separation
Top refractive index
2.9
1.6
1.1
1.0
1.7
1.7
1.0
1.2
8.4
5.8
3.1
3.5
17.0
10.2
6.5
6.3
Bottom refractive index
0.9
1.3
3.6
4.7
1.7
1.7
1.0
1.2
2.5
5.1
8.2
8.1
1.2
6.8
20.2
19.6
RMSEs of scaled reflectance for the original two-layered tissue model relative to independently simulated reflectance for a modified two-layered epithelial tissue model where the refractive index of the top or bottom layer was varied while other parameters of the modified two-layered epithelial tissue model were kept identical to those of the original two-layered epithelial tissue model. Note that the refractive index was 1.338 in the original tissue model as shown in bold type.
Table 3
Effect of Refractive Indices of Fiber Core and Tissue Model on Specular Reflectancea
(column)\ (row)
1.3
1.4
1.5
0° incidence
1.4
0.0014
0.0012
1.5
0.0051
0.0012
0
1.6
0.011
0.0044
0.0010
Cutoff angles defined by NA 0.22
1.4
0.0014
0.0012
1.5
0.0051
0.0012
0
1.6
0.011
0.0044
0.0010
Fraction of specular reflectance for various combinations of refractive indices of the fiber core and the tissue model when (top half) the incident angle is 0° and (bottom half) the cutoff angle is defined by an NA of 0.22. represents the refractive index of the fiber core, and is the refractive index of the tissue model. The actual cutoff angles are also shown in the bottom half of the table.
RMSEs of scaled reflectance relative to independently simulated reflectance from a modified two-layered epithelial tissue model in the case that the phase function was changed from the HG function to the Mie function while other parameters of the modified two-layered epithelial tissue model were kept identical to those of the original two-layered epithelial tissue model. Note that the HG phase function was used in the baseline simulation for scaling.
Table 5
Effect of One Additional Layer and Layer Thickness on RMSEa
Variable
RMSE (%) for Separation
RMSE (%) for Separation
RMSE (%) for Separation
RMSE (%) for Separation
Top-layer thickness
1.1
0.4
0.6
1.3
1.7
1.4
0.5
2.8
2.2
1.2
0.7
1.2
RMSEs of scaled reflectance relative to corresponding independently simulated reflectance for a three-layered epithelial tissue model, whose structure and optical properties were, respectively, shown in Figs. 2b, 3. While the thickness of the top layer was varied from 50 to 250 and to , the thickness of the middle layer was changed from 450 to 250 and to accordingly to keep the total thickness of the two layers a constant . The anisotropy factor and refractive index of each layer as well as the choice of the phase function (HG) in the tissue model were identical to those in the baseline simulation for scaling.
Table 6
Effect of Anisotropy Factor of Tissue Layer on RMSE of Estimated Optical Propertiesa
Variable
Thickness of Top Layer
RMSE (%)
Top anisotropy
10.5
45.2
Y
9.6
15.5
9.6
9.5
5.9
10.5
8.4
Bottom anisotropy
8.3
12.8
21.3
7.7
Y
9.6
9.5
5.9
12.1
6.5
RMSEs of the estimated thickness of the top layer and optical properties of the bottom layer relative to the corresponding true values for given sets of simulated diffuse reflectance spectra from the same modified two-layered epithelial tissue models as in Table 1, where the anisotropy factor of the top or bottom layer was varied while other parameters in the modified two-layered epithelial tissue models were kept identical to those in the original two-layered epithelial tissue model. The rows with bold type are the RMSEs of estimated parameters for input diffuse reflectance simulated with exactly the same anisotropy factor as in the baseline simulation for scaling. The rightmost column in the table indicates if a row contains an RMSE greater than 20% (marked by “Y”), which is a sign of inaccurate inversion.
Table 7
Effect of Refractive Index of Tissue Layer on RMSE of Estimated Optical Propertiesa
Variable
Thickness of Top Layer
RMSE (%)
Top refractive index
6.3
2.1
25.0
Y
9.6
9.5
5.9
7.8
45.4
Y
7.1
76.3
Y
Bottom refractive index
4.2
1.4
28.8
Y
9.6
9.5
5.9
6.9
2.5
6.0
14.3
RMSEs of the estimated thickness of the top layer and optical properties of the bottom layer relative to the corresponding true values for given sets of simulated diffuse reflectance spectra from the same modified two-layered epithelial tissue models as in Table 2, where the refractive index of the top or bottom layer was varied while other parameters in the modified two-layered epithelial tissue models were kept identical to those in the original two-layered epithelial tissue model. The rows with bold type are the RMSEs of estimated parameters for input diffuse reflectance simulated with exactly the same refractive index as in the baseline simulation for scaling. The rightmost column in the table indicates if a row contains an RMSE greater than 20% (marked by “Y”), which is a sign of inaccurate inversion.
Table 8
Effect of Phase Function on RMSE of Estimated Optical Propertiesa
Variable
Thickness of Top Layer
RMSE (%)
HG
9.6
9.5
5.9
Mie
31.4
5.0
Y
RMSEs of the estimated thickness of the top layer and optical properties of the bottom layer relative to the corresponding true values for given sets of simulated diffuse reflectance spectra from the same modified two-layered epithelial tissue models as in Table 4, where the phase function was changed from the HG function to the Mie function while other parameters in the modified two-layered epithelial tissue models were kept identical to those in the original two-layered epithelial tissue model. The rows with bold type are the RMSEs of estimated parameters for input diffuse reflectance simulated with exactly the same phase function as in the baseline simulation for scaling. The rightmost column in the table indicates if a row contains an RMSE greater than 20% (marked by “Y”), which is a sign of inaccurate inversion.