Analytic Model Descriptions - VirtualPhotonics/Vts.Gui.Wpf GitHub Wiki

A. R(ρ): Provides solutions for the spatially-resolved diffuse reflectance

1. Standard Diffusion (Analytic - Distributed Line Source): This option provides the R(ρ) for a line of isotropic sources that decays with depth according to the reduced total attenuation coefficient μt'. The solution is constructed from a linear superposition of the solution for a single isotropic point source developed in Haskell et al., 1994 which is arranged in a Source-Image configuration described by Spott and Svaasand, 2000.
   References: Haskell et al, ”Boundary conditions for the diffusion equation in radiative transfer”, Journal of the Optical Society of America A 11(10):2727-2741, 1994. Spott and Svaasand. “Collimated light sources in the diffusion approximation”, Applied Optics 39(34):6453-6465, 2000.

2. Standard Diffusion (Analytic - Point Source): This option provides the R(ρ) for a single isotropic point source located at a depth of the transport mean free path (l*) within the medium. The solution is constructed from a linear superposition of the infinite medium Greens function within a source-image configuration developed by Haskell et al., 1994 to satisfy a zero fluence conditions at an extrapolated boundary.
   References: Haskell et al, ”Boundary conditions for the diffusion equation in radiative transfer”, Journal of the Optical Society of America A 11(10):2727-2741, 1994.

3. Standard Diffusion (Analytic - Distributed Gaussian Source): This option provides the R(ρ) for a Gaussian beam of finite width which provides a source that decays exponentially with depth according to the reduced total atenuation coefficient μt'. The solution procedure follows that provided in Carp et al, 2004 but implements the standard diffusion rather than the delta-P1 approximation
   References: Carp, Prahl, and Venugopalan. Radiative transport in the delta-P1 approximation: accuracy of fluence rate and optical penetration depth predictions in turbid semi-infinite media”, Journal of Biomedical Optics 9(3):632-647, 2004.

B. R(ρ,t): Provides both spatially- and temporally- resolved diffuse reflectance

1. Distributed Line Source: Performs the same calculation as in case A.1. but uses the spatially- and temporally-resolved Green's function (see Kienle and Patterson, 1997)
2. Point Source: Performs the same calculation as in the case of A.2. but uses the spatially- and temporally-resolved Green's function of Kienle and Patterson, 1997) The same source image configuration used in A.2. is used to satisfy the Extrapolated Boundary Condition
3. Gaussian: Not yet implemented

References: Kienle and Patterson. ”Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium”, Journal of the Optical Society of America A 14(1):246-254, 1997.

C. R(ρ,ft): Provides spatially-resolved diffuse reflectance in the temporal frequency domain.

1. Distributed Line Source: Same as in A.1. and B.1. but uses the a frequency domain Green's function described in Kienle et al 1997, 1998
2. Point Source: Performs the same calculation as in the case of A.2. but uses the spatially- and temporally-resolved Green's function of Kienle et al, 1997, 1998) The same source image configuration used in A.2. is used to satisfy the Extrapolated Boundary Condition Kienle PMB 42 1997, Kienle AO 37(4) 1998 - Source Image solutions of infinite media Green's Function, EBC
3. Gaussian: Not yet implemented

References: Kienle and Patterson. ”Determination of the optical properties of semi-infinite turbid media from frequency-domain reflectance close to the source”, Physics in Medicine and Biology 42(9):1801-1819, 1997.
Kienle et al. “Noninvasive determination of the optical properties of two-layered turbid media”, Applied Optics 37(4):779-791, 1998.

D. R(fx): Provides results for the spatially resolved diffuse reflectance in the spatial frequency domain (i.e., the spatial Fourier transform of A.)

1. SDA: Provide the prediction developed by Cuccia et al.

References: Cuccia et al., Journal of Biomedical Optics, 14(2):024012, 2009.

E. R(fx, t): Provides results for temporally-resolved reflectance in the spatial frequency (Fourier-domain).

1. Distributed Line Source: Takes the result from B.1. for R(ρ,t) and applies a Hankel transform using a digital filter algorithm.
2. Point Source: Takes the result from B.2. for R(ρ,t) and applies a Hankel transform using a digital filter algorithm.
3. Gaussian: Not yet implemented

References: Anderson, W. L., 1979, Computer Program Numerical Integration of Related Hankel Transforms of Orders 0 and 1 by Adaptive Digital Filtering. Geophysics, 44(7):1287-1305.

F. R(fx, ft): Provides the spatially- and temporally-resolved diffuse reflectance in the spatial and temporal frequency domain

1. Modifies approach outlined in Pham et al 2000 to include spatial modulation, as described in Cuccia et al JBO March/April 2009

References: Pham et al. “Quantifying the properties of two-layer turbid media with frequency-domain diffuse reflectance”, Applied Optics 39(25):4733-4745, 2000.
Cuccia et al. “Quantitation and mapping of tissue optical properties using modulated imaging”, Journal of Biomedical Optics 14(2):024012, 2009.

A. Scaled Monte Carlo - Basic (g=0.8, n=1.4): We provide Monte Carlo results by running a single Monte Carlo simulation with μa = 0, μs' = 1/mm, and g = 0.8 for a refractive-index mismatch of 1.4 and use the technique described by Kienle and Patterson, 1996 to rescale the results for the time and/or locations of interest. The reference Monte Carlo database results from the simulation of 100 million photons and spans ρ in [0,40]mm and t in [0,4]ns. The simulation is binned uniformly in space and time at 0.2mm and 5ps intervals, respectively. Linear interpolation is used to obtain the model results at desired values of the independent variable(s).

1. Spatially-resolved Reflectance R(ρ): Provides solutions for the spatially-resolved diffuse reflectance by summing the R(ρ,t) result over all t.
2. Spatially- and Temporally-resolved Reflectance R(ρ,t): Provided by applying the procedure described by Kienle and Patterson, 1996.
3. Spatially-resolved, temporal frequency domain Reflectance R(ρ,ft): Takes the Fourier transform of the result provided by A.2.
4. Spatial frequency domain Reflectance R(fx): Provides the diffuse reflectance in the spatial frequency domain by summing the R(fx,t) result provided by A.5. below over all t.
5. Temporally-resolved spatial frequency domain Reflectance R(fx,t): Provides the time-resolved diffuse reflectance in the spatial frequency domain by applying a Hankel transform to the R(ρ,t) provided by A.2.
6. Spatial and Temporal frequency domain Reflectance R(fx,ft): Provides the spatially- and temporally-resolved diffuse reflectance in the spatial and temporal frequency domain by taking the Fourier transform of the R(fx,t) result provided in A.5.

References: Kienle and Patterson. ”Determination of the optical properties of turbid media from a single Monte Carlo simulation”, Physics in Medicine and Biology 41(10):2221-2227, 1996.

Partial Derivatives: partial R/partial μa, partial R/partial μs', partial R/partial g, partial R/partial n: Provides estimates for the derivative of reflectance with respect to the optical properties μa, μ's, g, refractive index (n), or the independent variable (the variable on the x-axis of the plot) .... by applying a first-order finite difference approximation.

B. Scaled Monte Carlo - NURBS (g=0.8, n=1.4): This implements a variation on the Monte Carlo - Basic algorithm described above where the Monte Carlo reference simulation is the same but adaptive binning is used to ensure the same number of detected photons reside in each bin. This serves to ensure that the relative variance is uniform across all bins. Additional constraints were added to ensure that the change in reflectance between adjacent bins would not exceed 10% and that the mean distance/time between bins did not exceed 0.2mm and 20ps, respectively. This resulted in a database with 436 bins in ρ and 826 bins in time. The Monte Carlo database used spans ρ in [0,100]mm and t in [0,20]ns. Moreover, interpolation using NURBS (non-uniform rational b-splines) is used to obtain the model results at desired values of the independent variable(s). Both the adaptive binning and NURBS interpolation serve to provide more accurate results. This algorithm is described in Martinelli et al., 2011.

1. Spatially-resolved Reflectance R(ρ): Provides solutions for the spatially-resolved diffuse reflectance by analytical integration of the NURBS curves along the time dimension.
2. Spatially- and Temporally-resolved Reflectance R(ρ,t): Provided by applying the procedure described by Kienle and Patterson, 1996, on the adaptively binned database using NURBS interpolation.
3. Spatially-resolved, temporal frequency domain Reflectance R(ρ,ft)}: Takes the Fourier transform of the result provided by B.2.
4. Spatial frequency domain Reflectance R(fx): Provides the diffuse reflectance in the spatial frequency domain by summing the R(fx,t) result provided by B.5. below over all t.
5. Temporally-resolved spatial frequency domain Reflectance R(fx,t): Provides the time-resolved diffuse reflectance in the spatial frequency domain by applying the procedure described by Kienle and Patterson on the adaptively binned database obtained from a photon by photon Hankel transform of the photons biographies generated with the reference MC simulation. The reference reflectance "R(fx,t)" covers the range [0-10] /mm in spatial frequency and the range [0-20] ns in time.
6. Spatial and Temporal frequency domain Reflectance R(fx,ft): Provides the spatially- and temporally-resolved diffuse reflectance in the spatial and temporal frequency domain by taking the Fourier transform of the R(fx,t) result provided in B.5.

References: M. Martinell, A. Gardner, D. Cuccia, C. Hayakawa, J. Spanier and V. Venugopalan. ”Analysis of single Monte Carlo methods for prediction of reflectance from turbid media”, Optics Express 19(20):19627-42, 2011.