Abstract

Current simulation methods for light transport in biological media have limited efficiency and realism when applied to three-dimensional microscopic light transport in biological tissues with refractive heterogeneities. We describe here a technique that combines a beam propagation method valid for modeling light transport in media with weak variations in refractive index, with a fractal model of refractive index turbulence. In contrast to standard simulation methods, this fractal propagation method (FPM) is able to accurately and efficiently simulate the diffraction effects of focused beams, as well as the microscopic heterogeneities present in tissue that result in scattering, refractive beam steering, and the aberration of beam foci. We validate the technique and the relationship between the FPM model parameters and conventional optical parameters used to describe tissues, and also demonstrate the method’s flexibility and robustness by examining the steering and distortion of Gaussian and Bessel beams in tissue with comparison to experimental data. We show that the FPM has utility for the accurate investigation and optimization of optical microscopy methods such as light-sheet, confocal, and nonlinear microscopy.

© 2016 Optical Society of America

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GPU-based Monte Carlo simulation for light propagation in complex heterogeneous tissues

Nunu Ren, Jimin Liang, Xiaochao Qu, Jianfeng Li, Bingjia Lu, and Jie Tian
Opt. Express 18(7) 6811-6823 (2010)

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2016 (1)

2015 (5)

2014 (6)

A. Elmaklizi, J. Schafer, and A. Kienle, “Simulating the scanning of a focused beam through scattering media using a numerical solution of Maxwell’s equations,” J. Biomed. Opt. 19, 071404 (2014).
[Crossref]

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Llado, D. E. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an airy beam,” Nat. Methods 11, 541–544 (2014).
[Crossref]

N. Das, S. Chatterjee, S. Kumar, A. Pradhan, P. Panigrahi, I. A. Vitkin, and N. Ghosh, “Tissue multifractality and Born approximation in analysis of light scattering: a novel approach for precancers detection,” Sci. Rep. 4, 6129 (2014).
[Crossref]

J. Yi, A. J. Radosevich, Y. Stypula-Cyrus, N. N. Mutyal, S. M. Azarin, E. Horcher, M. J. Goldberg, L. K. Bianchi, S. Bajaj, H. K. Roy, and V. Backman, “Spatially resolved optical and ultrastructural properties of colorectal and pancreatic field carcinogenesis observed by inverse spectroscopic optical coherence tomography,” J. Biomed. Opt. 19, 036013 (2014).
[Crossref]

A. R. Gardner, C. K. Hayakawa, and V. Venugopalan, “Coupled forward-adjoint Monte Carlo simulation of spatial-angular light fields to determine optical sensitivity in turbid media,” J. Biomed. Opt. 19, 065003 (2014).
[Crossref]

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenek, A. C. Reymann, R. Bohme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, “Lattice light-sheet microscopy: imaging molecules to embryos at high spatiotemporal resolution,” Science 346, 1257998 (2014).
[Crossref]

2013 (7)

S. L. Jacques, “Optical properties of biological tissues: a review,” Phys. Med. Biol. 58, R37–R61 (2013).
[Crossref]

F. Devaux and E. Lantz, “3D-PSTD simulation and polarization analysis of a light pulse transmitted through a scattering medium,” Opt. Express 21, 24969–24984 (2013).
[Crossref]

A. J. Radosevich, N. N. Mutyal, J. Yi, Y. Stypula-Cyrus, J. D. Rogers, M. J. Goldberg, L. K. Bianchi, S. Bajaj, H. K. Roy, and V. Backman, “Ultrastructural alterations in field carcinogenesis measured by enhanced backscattering spectroscopy,” J. Biomed. Opt. 18, 097002 (2013).
[Crossref]

J. D. Rogers, A. J. Radosevich, J. Yi, and V. Backman, “Modeling light scattering in tissue as continuous random media using a versatile refractive index correlation function,” IEEE J. Sel. Top. Quantum Electron. 20, 7000514 (2013).

I. R. Capoglu, A. Taflove, and V. Backman, “Computation of tightly-focused laser beams in the FDTD method,” Opt. Express 21, 87–101 (2013).
[Crossref]

I. R. Capoglu, A. Taflove, and V. Backman, “Angora: a free software package for finite-difference time-domain electromagnetic simulation,” IEEE Antennas Propag. Mag. 55(4), 80–93 (2013).
[Crossref]

F. O. Fahrbach, V. Gurchenkov, K. Alessandri, P. Nassoy, and A. Rohrbach, “Self-reconstructing sectioned Bessel beams offer submicron optical sectioning for large fields of view in light-sheet microscopy,” Opt. Express 21, 11425–11440 (2013).
[Crossref]

2012 (2)

2011 (2)

J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomography-computational efficiency,” Med. Phys. 38, 5788–5798 (2011).
[Crossref]

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref]

2010 (3)

V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods 7, 603–614 (2010).
[Crossref]

H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. Biol. 55, 947–962 (2010).
[Crossref]

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[Crossref]

2009 (3)

2008 (4)

S. L. Jacques and B. W. Pogue, “Tutorial on diffuse light transport,” J. Biomed. Opt. 13, 041302 (2008).
[Crossref]

D. G. Fischer, S. A. Prahl, and D. D. Duncan, “Monte Carlo modeling of spatial coherence: free-space diffraction,” J. Opt. Soc. Am. A 25, 2571–2581 (2008).
[Crossref]

R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16, 5907–5925 (2008).
[Crossref]

M. Xu, T. T. Wu, and J. Y. Qu, “Unified Mie and fractal scattering by cells and experimental study on application in optical characterization of cellular and subcellular structures,” J. Biomed. Opt. 13, 024015 (2008).
[Crossref]

2007 (3)

2006 (2)

M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, V. Gopal, K. Badizadegan, G. D. Stoner, and M. S. Feld, “Tissue self-affinity and polarized light scattering in the born approximation: a new model for precancer detection,” Phys. Rev. Lett. 97, 138102 (2006).
[Crossref]

F. Charriere, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett. 31, 178–180 (2006).
[Crossref]

2005 (1)

2000 (2)

P. K. Milsom, “A ray-optic, Monte Carlo, description of a Gaussian beam waist-applied to reverse saturable absorption,” Appl. Phys. B 70, 593–599 (2000).
[Crossref]

J. W. Baish and R. K. Jain, “Fractals and cancer,” Cancer Res. 60, 3683–3688 (2000).

1998 (1)

1997 (2)

Q. H. Liu, “The PSTD algorithm: a time-domain method requiring only two cells per wavelength,” Microwave Opt. Technol. Lett. 15, 158–165 (1997).
[Crossref]

R. J. Crilly, W. F. Cheong, B. Wilson, and J. R. Spears, “Forward-adjoint fluorescence model: Monte Carlo integration and experimental validation,” Appl. Opt. 36, 6513–6519 (1997).
[Crossref]

1996 (3)

A. Dunn and R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Sel. Top. Quantum Electron. 2, 898–905 (1996).
[Crossref]

J. M. Schmitt and G. Kumar, “Turbulent nature of refractive-index variations in biological tissue,” Opt. Lett. 21, 1310–1312 (1996).
[Crossref]

J. Beuthan, O. Minet, J. Helfmann, M. Herrig, and G. Muller, “The spatial variation of the refractive index in biological cells,” Phys. Med. Biol. 41, 369–382 (1996).
[Crossref]

1995 (1)

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

1990 (1)

A. L. Goldberger, D. R. Rigney, and B. J. West, “Chaos and fractals in human physiology,” Sci. Am. 262(2), 42–49 (1990).
[Crossref]

1984 (1)

T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution-equations. 2. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
[Crossref]

1983 (1)

L. Thylen, “The beam propagation method-an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1983).
[Crossref]

1981 (1)

1978 (1)

1941 (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Ablowitz, M. J.

T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution-equations. 2. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
[Crossref]

Alessandri, K.

Alfano, R. R.

Al-Rubaiee, M.

Andrews, L.

L. Andrews and R. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Azarin, S. M.

J. Yi, A. J. Radosevich, Y. Stypula-Cyrus, N. N. Mutyal, S. M. Azarin, E. Horcher, M. J. Goldberg, L. K. Bianchi, S. Bajaj, H. K. Roy, and V. Backman, “Spatially resolved optical and ultrastructural properties of colorectal and pancreatic field carcinogenesis observed by inverse spectroscopic optical coherence tomography,” J. Biomed. Opt. 19, 036013 (2014).
[Crossref]

Azimipour, M.

Backman, V.

A. J. Radosevich, A. Eshein, T. Q. Nguyen, and V. Backman, “Subdiffusion reflectance spectroscopy to measure tissue ultrastructure and microvasculature: model and inverse algorithm,” J. Biomed. Opt. 20, 097002 (2015).
[Crossref]

J. Yi, A. J. Radosevich, Y. Stypula-Cyrus, N. N. Mutyal, S. M. Azarin, E. Horcher, M. J. Goldberg, L. K. Bianchi, S. Bajaj, H. K. Roy, and V. Backman, “Spatially resolved optical and ultrastructural properties of colorectal and pancreatic field carcinogenesis observed by inverse spectroscopic optical coherence tomography,” J. Biomed. Opt. 19, 036013 (2014).
[Crossref]

A. J. Radosevich, N. N. Mutyal, J. Yi, Y. Stypula-Cyrus, J. D. Rogers, M. J. Goldberg, L. K. Bianchi, S. Bajaj, H. K. Roy, and V. Backman, “Ultrastructural alterations in field carcinogenesis measured by enhanced backscattering spectroscopy,” J. Biomed. Opt. 18, 097002 (2013).
[Crossref]

J. D. Rogers, A. J. Radosevich, J. Yi, and V. Backman, “Modeling light scattering in tissue as continuous random media using a versatile refractive index correlation function,” IEEE J. Sel. Top. Quantum Electron. 20, 7000514 (2013).

I. R. Capoglu, A. Taflove, and V. Backman, “Angora: a free software package for finite-difference time-domain electromagnetic simulation,” IEEE Antennas Propag. Mag. 55(4), 80–93 (2013).
[Crossref]

I. R. Capoglu, A. Taflove, and V. Backman, “Computation of tightly-focused laser beams in the FDTD method,” Opt. Express 21, 87–101 (2013).
[Crossref]

J. D. Rogers, I. R. Capoglu, and V. Backman, “Nonscalar elastic light scattering from continuous random media in the Born approximation,” Opt. Lett. 34, 1891–1893 (2009).
[Crossref]

I. R. Capoglu, J. D. Rogers, A. Taflove, and V. Backman, “Accuracy of the Born approximation in calculating the scattering coefficient of biological continuous random media,” Opt. Lett. 34, 2679–2681 (2009).
[Crossref]

M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, V. Gopal, K. Badizadegan, G. D. Stoner, and M. S. Feld, “Tissue self-affinity and polarized light scattering in the born approximation: a new model for precancer detection,” Phys. Rev. Lett. 97, 138102 (2006).
[Crossref]

I. R. Capoglu and V. Backman, “Validation of the born approximation in 2-D weakly-scattering biological random media using the FDTD method,” in Antennas and Propagation Society International Symposium (IEEE, 2009).

Badizadegan, K.

M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, V. Gopal, K. Badizadegan, G. D. Stoner, and M. S. Feld, “Tissue self-affinity and polarized light scattering in the born approximation: a new model for precancer detection,” Phys. Rev. Lett. 97, 138102 (2006).
[Crossref]

Baish, J. W.

J. W. Baish and R. K. Jain, “Fractals and cancer,” Cancer Res. 60, 3683–3688 (2000).

Bajaj, S.

J. Yi, A. J. Radosevich, Y. Stypula-Cyrus, N. N. Mutyal, S. M. Azarin, E. Horcher, M. J. Goldberg, L. K. Bianchi, S. Bajaj, H. K. Roy, and V. Backman, “Spatially resolved optical and ultrastructural properties of colorectal and pancreatic field carcinogenesis observed by inverse spectroscopic optical coherence tomography,” J. Biomed. Opt. 19, 036013 (2014).
[Crossref]

A. J. Radosevich, N. N. Mutyal, J. Yi, Y. Stypula-Cyrus, J. D. Rogers, M. J. Goldberg, L. K. Bianchi, S. Bajaj, H. K. Roy, and V. Backman, “Ultrastructural alterations in field carcinogenesis measured by enhanced backscattering spectroscopy,” J. Biomed. Opt. 18, 097002 (2013).
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Supplementary Material (3)

NameDescription
» Supplement 1: PDF (991 KB)      Supplemental document.
» Visualization 1: AVI (2108 KB)      Propagation of Gaussian beam through fractal tissue model.
» Visualization 2: AVI (3047 KB)      Propagation of Bessel beam through fractal tissue model.

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Figures (7)

Fig. 1.
Fig. 1. (a) Simulation geometry for the BPM. A lens focuses the light beam at a half-angle, θ, into a tissue volume of length Lz at a focal depth zf. The step size along the optical axis is given by Δz. (b) Lateral cross section of the tissue volume with the beam focus at the center. The lateral grid resolutions are given by Δx and Δy.
Fig. 2.
Fig. 2. Representative cross sections for two fractal media are shown: (a) medium 1, with Df=3.25, σn2=0.25×104 and lc=1.00  μm; and (b) medium 2, with Df=3.67, σn2=0.50×104, and lc=2.00  μm. Corresponding histograms of all (c) refractive index values and (d) the PSDs for medium 1 and medium 2. The histograms contain labels to indicate how the fractal parameters (Df, and lc, n0, and σn2) influence the shape of the plotted distributions. In addition, labels are provided to illustrate how cellular components typically contribute to the refractive index and spatial-frequency profiles.
Fig. 3.
Fig. 3. (a)–(d) xz cross sections of the beam intensity are shown as a function of focal depth, zf, for a focused Gaussian beam propagating through in silico fractal medium 2. For each panel, the result for a single simulation is displayed on top, with the corresponding averaged result over N=100 randomly generated fractal media displayed on the bottom. For visualization, all images are self-normalized to a maximum value of 1.
Fig. 4.
Fig. 4. (a) Exponential decay of the on-axis peak intensity is plotted for both medium 1 and medium 2, where the slope of the line is used to calculate the scattering coefficient, μs. (b) μs plotted as a function of σn2 for lc=0.50, 1.00, or 2.00 μm and Df=3.25. The shaded regions indicate the simulation results (95% confidence intervals) and the solid lines indicate the theoretical values (based on analytical equations) for scattering in a fractal medium. (c) Corresponding results for Df=3.67. Medium 1 and medium 2 are highlighted in (b) and (c).
Fig. 5.
Fig. 5. (a) Configuration for assessing the steering and distortion of a Gaussian beam (FWHM=1.2  μm), for both experiments and simulations. (b) Corresponding setup for the Bessel beam (FWHM=1.0  μm). For both setups, Lz=zf=75  μm and λ=550  nm. The unperturbed and perturbed wavefronts are shown by the solid lines in both (a) and (b).
Fig. 6.
Fig. 6. (a) FPM simulations of the intensity of a Gaussian beam (FWHM=1.2  μm) propagating through a realistic tissue model, shown in three dimensions. A zoom-in of the beam focus at the back end of the tissue (75 μm deep), is shown for (b) one simulation in comparison to (c) one experimental image. (d)–(f) Corresponding results for the Bessel beam (FWHM=1.0  μm). The main lobe of the Bessel beam is less sensitive to beam steering and aberrations compared with the focus of the Gaussian beam (see text for details). See Visualization 1 and Visualization 2.
Fig. 7.
Fig. 7. (a), (b) Beam centroid coordinates (calculated from the images shown in Fig. 6 for N=100 simulations) plotted in comparison to the experimental results. (c) Beam-distortion metric plotted for the simulated Gaussian and Bessel beams in comparison to experiments.

Tables (1)

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Table 1. Summary of Experimentally Measured Fractal Tissue Parameters

Equations (7)

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E(kx,ky,z+Δz/2)=F[E(x,y,z)]eikx2+ky2k¯Δz2,
E(x,y,z+Δz/2)=F1[E(kx,ky,z+Δz/2)]eik0ΔnΔz,
E(x,y,z+Δz)=F1[F[E(x,y,z+Δz/2)]eikx2+ky2k¯Δz2,
Bn(x,y,z)=An(x2+y2+z2lc)(Df3)/2×K(Df3)/2(x2+y2+z2lc),
An=σn22(Df5)/2|(Df3)/2|.
Φ(kx,ky,kz)=Anlc3Γ(Df/2)π322(5Df)/2[1+(kx2+ky2+kz2)lc2](Df/2),
Δn(x,y,z)=Re[F1(RΦ(kx,ky,kz)8π3NxNyNz)]+Im[F1(RΦ(kx,ky,kz)8π3NxNyNz)],

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