Abstract

We present the implementation, validation, and performance of a Neumann-series approach for simulating light propagation at optical wavelengths in uniform media using the radiative transport equation (RTE). The RTE is solved for an anisotropic-scattering medium in a spherical harmonic basis for a diffuse-optical-imaging setup. The main objectives of this paper are threefold: to present the theory behind the Neumann-series form for the RTE, to design and develop the mathematical methods and the software to implement the Neumann series for a diffuse-optical-imaging setup, and, finally, to perform an exhaustive study of the accuracy, practical limitations, and computational efficiency of the Neumann-series method. Through our results, we demonstrate that the Neumann-series approach can be used to model light propagation in uniform media with small geometries at optical wavelengths.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. A. Gibson and H. Dehghani, “Diffuse optical imaging,” Phil. Trans. R. Soc. A 367, 3055–3072 (2009).
    [CrossRef]
  4. A. A. Tanbakuchi, A. R. Rouse, and A. F. Gmitro, “Monte Carlo characterization of parallelized fluorescence confocal systems imaging in turbid media,” J. Biomed. Opt. 14, 044024 (2009).
    [CrossRef]
  5. J. Chen, V. Venugopal, and X. Ante’s, “Monte Carlo based method for fluorescence tomographic imaging with lifetime multiplexing using time gates,” Biomed. Opt. Express 2, 871–886 (2011).
    [CrossRef]
  6. M. Y. Kirillin, I. V. Meglinskii, and A. V. Priezzhev, “Effect of photons of different scattering orders on the formation of a signal in optical low-coherence tomography of highly scattering media,” Quantum Electron. 36, 247–252 (2006).
    [CrossRef]
  7. K. Takagi, H. Haneishi, N. Tsumura, and Y. Miyake, “Alternative oblique-incidence reflectometry for measuring tissue optical properties,” Opt. Rev. 7, 164–169 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. S. Srinivasan, B. W. Pogue, C. Carpenter, P. K. Yalavarthy, and K. Paulsen, “A boundary element approach for image-guided near-infrared absorption and scatter estimation,” Med. Phys. 34, 4545–4557 (2007).
    [CrossRef]
  15. A. Klose and E. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220, 441–470 (2006).
    [CrossRef]
  16. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
    [CrossRef]
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  18. E. D. Aydin, “Three-dimensional photon migration through voidlike regions and channels,” Appl. Opt. 46, 8272–8277 (2007).
    [CrossRef]
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  21. M. L. Adams and E. W. Larsen, “Fast iterative methods for discrete ordinates particle transport calculations,” Prog. Nucl. Energy 40, 3–159 (2002).
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  22. J. K. Fletcher, “A solution of the neutron transport equation using spherical harmonics,” J. Phys. A: Math. Gen. 16, 2827–2835 (1983).
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  23. K. Kobayashi, H. Oigawa, and H. Yamagata, “The spherical harmonics method for the multigroup transport equation in x-y geometry,” Ann. Nucl. Energy 13, 663–678 (1986).
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  26. S. Wright, M. Schweiger, and S. Arridge, “Reconstruction in optical tomography using the PN approximations,” Meas. Sci. Technol. 18, 79–86 (2007).
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  27. E. Aydin, C. de Oliveira, and A. Goddard, “A finite element-spherical harmonics radiation transport model for photon migration in turbid media,” J. Quant. Spectrosc. Radiat. Transfer 84, 247–260 (2004).
    [CrossRef]
  28. R. Wells, A. Celler, and R. Harrop, “Analytical calculation of photon distributions in SPECT projections,” IEEE Trans. Nucl. Sci. 45, 3202–3214 (1998).
    [CrossRef]
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  32. V. G. Peters, D. R. Wyman, M. S. Patterson, and G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
    [CrossRef]
  33. P. Gonzalez-Rodriguez and A. D. Kim, “Comparison of light scattering models for diffuse optical tomography,” Opt. Express 17, 8756–8774 (2009).
    [CrossRef]
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    [CrossRef]
  35. H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. 42, 135–146 (2003).
    [CrossRef]
  36. S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol. 13, 195–202 (2006).
  37. T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage 31, 1426–1433 (2006).
    [CrossRef]
  38. B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. USA 104, 12169–12174 (2007).
    [CrossRef]
  39. A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical tomography for imaging of rheumatoid finger joints,” Phys. Med. Biol. 49, 1147–1163 (2004).
    [CrossRef]
  40. A. H. Hielscher, “Optical tomographic imaging of small animals,” Curr. Opin. Biotechnol. 16, 79–88 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  50. J. C. Chai, H. S. Lee, and S. V. Patankar, “Finite volume method for radiation heat transfer,” J. Thermophys. Heat Transf. 8, 419–425 (1994).
  51. M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
    [CrossRef]
  52. T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Hybrid radiative-transfer-diffusion model for optical tomography,” Appl. Opt. 44, 876–886 (2005).
    [CrossRef]
  53. T. Spott and L. O. Svaasand, “Collimated light sources in the diffusion approximation,” Appl. Opt. 39, 6453–6465 (2000).
    [CrossRef]
  54. A. K. Jha, M. A. Kupinski, H. H. Barrett, E. Clarkson, and J. H. Hartman, “A three-dimensional Neumann-series approach to model light transport in non-uniform media,” J. Opt. Soc. Am. A (to be published).

2011 (3)

T. Deutschmann, S. Beirle, U. Frie, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Roman, J. Pukite, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: Introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

B. F. Hutton, I. Buvat, and F. J. Beekman, “Review and current status of SPECT scatter correction,” Phys. Med. Biol. 56, R85–R112 (2011).
[CrossRef]

J. Chen, V. Venugopal, and X. Ante’s, “Monte Carlo based method for fluorescence tomographic imaging with lifetime multiplexing using time gates,” Biomed. Opt. Express 2, 871–886 (2011).
[CrossRef]

2010 (1)

2009 (5)

P. Gonzalez-Rodriguez and A. D. Kim, “Comparison of light scattering models for diffuse optical tomography,” Opt. Express 17, 8756–8774 (2009).
[CrossRef]

H. Dehghani, S. Srinivasan, B. W. Pogue, and A. Gibson, “Numerical modelling and image reconstruction in diffuse optical tomography,” Phil. Trans. R. Soc. A 367, 3073–3093(2009).
[CrossRef]

A. Gibson and H. Dehghani, “Diffuse optical imaging,” Phil. Trans. R. Soc. A 367, 3055–3072 (2009).
[CrossRef]

A. A. Tanbakuchi, A. R. Rouse, and A. F. Gmitro, “Monte Carlo characterization of parallelized fluorescence confocal systems imaging in turbid media,” J. Biomed. Opt. 14, 044024 (2009).
[CrossRef]

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol. 54, 2493–2509 (2009).
[CrossRef]

2007 (4)

S. Srinivasan, B. W. Pogue, C. Carpenter, P. K. Yalavarthy, and K. Paulsen, “A boundary element approach for image-guided near-infrared absorption and scatter estimation,” Med. Phys. 34, 4545–4557 (2007).
[CrossRef]

S. Wright, M. Schweiger, and S. Arridge, “Reconstruction in optical tomography using the PN approximations,” Meas. Sci. Technol. 18, 79–86 (2007).
[CrossRef]

B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. USA 104, 12169–12174 (2007).
[CrossRef]

E. D. Aydin, “Three-dimensional photon migration through voidlike regions and channels,” Appl. Opt. 46, 8272–8277 (2007).
[CrossRef]

2006 (5)

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol. 13, 195–202 (2006).

T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage 31, 1426–1433 (2006).
[CrossRef]

A. Klose and E. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220, 441–470 (2006).
[CrossRef]

A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrization and a boundary element method,” Inverse Probl. 22, 1509–1532 (2006).
[CrossRef]

M. Y. Kirillin, I. V. Meglinskii, and A. V. Priezzhev, “Effect of photons of different scattering orders on the formation of a signal in optical low-coherence tomography of highly scattering media,” Quantum Electron. 36, 247–252 (2006).
[CrossRef]

2005 (3)

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005).
[CrossRef]

A. H. Hielscher, “Optical tomographic imaging of small animals,” Curr. Opin. Biotechnol. 16, 79–88 (2005).
[CrossRef]

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Hybrid radiative-transfer-diffusion model for optical tomography,” Appl. Opt. 44, 876–886 (2005).
[CrossRef]

2004 (4)

K. Ren, G. S. Abdoulaev, G. Bal, and A. H. Hielscher, “Algorithm for solving the equation of radiative transfer in the frequency domain,” Opt. Lett. 29, 578–580 (2004).
[CrossRef]

M. Kim, G. Skofronick-Jackson, and J. Weinman, “Intercomparison of millimeter-wave radiative transfer models,” IEEE Trans. Geosci. Remote Sens. 42, 1882–1890 (2004).
[CrossRef]

A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical tomography for imaging of rheumatoid finger joints,” Phys. Med. Biol. 49, 1147–1163 (2004).
[CrossRef]

E. Aydin, C. de Oliveira, and A. Goddard, “A finite element-spherical harmonics radiation transport model for photon migration in turbid media,” J. Quant. Spectrosc. Radiat. Transfer 84, 247–260 (2004).
[CrossRef]

2003 (2)

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, and K. D. Paulsen, “The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,” Phys. Med. Biol. 48, 2713–2727 (2003).
[CrossRef]

H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. 42, 135–146 (2003).
[CrossRef]

2002 (2)

E. D. Aydin, C. R. de Oliveira, and A. J. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Med. Phys. 29, 2013–2023 (2002).
[CrossRef]

M. L. Adams and E. W. Larsen, “Fast iterative methods for discrete ordinates particle transport calculations,” Prog. Nucl. Energy 40, 3–159 (2002).
[CrossRef]

2001 (1)

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

2000 (2)

K. Takagi, H. Haneishi, N. Tsumura, and Y. Miyake, “Alternative oblique-incidence reflectometry for measuring tissue optical properties,” Opt. Rev. 7, 164–169 (2000).
[CrossRef]

T. Spott and L. O. Svaasand, “Collimated light sources in the diffusion approximation,” Appl. Opt. 39, 6453–6465 (2000).
[CrossRef]

1998 (3)

R. Wells, A. Celler, and R. Harrop, “Analytical calculation of photon distributions in SPECT projections,” IEEE Trans. Nucl. Sci. 45, 3202–3214 (1998).
[CrossRef]

F. Gao, H. Niu, H. Zhao, and H. Zhang, “The forward and inverse models in time-resolved optical tomography imaging and their finite-element method solutions,” Image Vis. Comput. 16, 703–712 (1998).
[CrossRef]

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef]

1997 (1)

M. Schweiger and S. R. Arridge, “The finite-element method for the propagation of light in scattering media: frequency domain case,” Med. Phys. 24, 895–902 (1997).
[CrossRef]

1995 (2)

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]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

1994 (1)

J. C. Chai, H. S. Lee, and S. V. Patankar, “Finite volume method for radiation heat transfer,” J. Thermophys. Heat Transf. 8, 419–425 (1994).

1993 (2)

Z. Wang, M. Yang, and G. Qin, “Neumann series solution to a neutron transport equation of slab geometry,” J. Syst. Sci. Complexity 6, 13–17 (1993).

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef]

1990 (2)

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

V. G. Peters, D. R. Wyman, M. S. Patterson, and G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef]

1986 (1)

K. Kobayashi, H. Oigawa, and H. Yamagata, “The spherical harmonics method for the multigroup transport equation in x-y geometry,” Ann. Nucl. Energy 13, 663–678 (1986).
[CrossRef]

1983 (1)

J. K. Fletcher, “A solution of the neutron transport equation using spherical harmonics,” J. Phys. A: Math. Gen. 16, 2827–2835 (1983).
[CrossRef]

1941 (1)

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

Abdoulaev, G. S.

Adams, M. L.

M. L. Adams and E. W. Larsen, “Fast iterative methods for discrete ordinates particle transport calculations,” Prog. Nucl. Energy 40, 3–159 (2002).
[CrossRef]

Alcouffe, R. E.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef]

A. H. Hielscher and R. E. Alcouffe, “Discrete-ordinate transport simulations of light propagation in highly forward scattering heterogenous media,” in Advances in Optical Imaging and Photon Migration, 1998 OSA Technical Digest Series (Optical Society of America, 1998), paper ATuC2.

Ante’s, X.

Arfken, G.

G. Arfken and H. Weber, Mathematical Methods for Physicists, Vol. 3 (Academic, 2005).

Arridge, S.

S. Wright, M. Schweiger, and S. Arridge, “Reconstruction in optical tomography using the PN approximations,” Meas. Sci. Technol. 18, 79–86 (2007).
[CrossRef]

Arridge, S. R.

A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrization and a boundary element method,” Inverse Probl. 22, 1509–1532 (2006).
[CrossRef]

T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage 31, 1426–1433 (2006).
[CrossRef]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005).
[CrossRef]

M. Schweiger and S. R. Arridge, “The finite-element method for the propagation of light in scattering media: frequency domain case,” Med. Phys. 24, 895–902 (1997).
[CrossRef]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef]

Austin, T.

T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage 31, 1426–1433 (2006).
[CrossRef]

Aydin, E.

E. Aydin, C. de Oliveira, and A. Goddard, “A finite element-spherical harmonics radiation transport model for photon migration in turbid media,” J. Quant. Spectrosc. Radiat. Transfer 84, 247–260 (2004).
[CrossRef]

Aydin, E. D.

E. D. Aydin, “Three-dimensional photon migration through voidlike regions and channels,” Appl. Opt. 46, 8272–8277 (2007).
[CrossRef]

E. D. Aydin, C. R. de Oliveira, and A. J. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Med. Phys. 29, 2013–2023 (2002).
[CrossRef]

Backhaus, M.

A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical tomography for imaging of rheumatoid finger joints,” Phys. Med. Biol. 49, 1147–1163 (2004).
[CrossRef]

Bal, G.

Barbour, R. L.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef]

Barrett, H. H.

B. Gallas and H. H. Barrett, “Modeling all orders of scatter in nuclear medicine,” in IEEE Nuclear Science Symposium, Vol. 3 (IEEE, 1998), pp. 1964–1968.

H. H. Barrett and K. J. Myers, Foundations of Image Science, 1st ed. (Wiley, 2004).

H. H. Barrett, B. Gallas, E. Clarkson, and A. Clough, Computational Radiology and Imaging: Therapy and Diagnostics (Springer, 1999), pp. 71–100.

A. K. Jha, M. A. Kupinski, H. H. Barrett, E. Clarkson, and J. H. Hartman, “A three-dimensional Neumann-series approach to model light transport in non-uniform media,” J. Opt. Soc. Am. A (to be published).

Beekman, F. J.

B. F. Hutton, I. Buvat, and F. J. Beekman, “Review and current status of SPECT scatter correction,” Phys. Med. Biol. 56, R85–R112 (2011).
[CrossRef]

Beirle, S.

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H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. 42, 135–146 (2003).
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D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
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H. Dehghani, S. Srinivasan, B. W. Pogue, and A. Gibson, “Numerical modelling and image reconstruction in diffuse optical tomography,” Phil. Trans. R. Soc. A 367, 3073–3093(2009).
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T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage 31, 1426–1433 (2006).
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S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol. 13, 195–202 (2006).

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T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage 31, 1426–1433 (2006).
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M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
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Jiang, S.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol. 13, 195–202 (2006).

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Kern, C.

T. Deutschmann, S. Beirle, U. Frie, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Roman, J. Pukite, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: Introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
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D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
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A. Klose and E. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220, 441–470 (2006).
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L. D. Montejo, A. D. Klose, and A. H. Hielscher, “Implementation of the equation of radiative transfer on block-structured grids for modeling light propagation in tissue,” Biomed. Opt. Express 1, 861–878 (2010).
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M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol. 54, 2493–2509 (2009).
[CrossRef]

A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical tomography for imaging of rheumatoid finger joints,” Phys. Med. Biol. 49, 1147–1163 (2004).
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S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol. 13, 195–202 (2006).

Kolehmainen, V.

A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrization and a boundary element method,” Inverse Probl. 22, 1509–1532 (2006).
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A. K. Jha, M. A. Kupinski, H. H. Barrett, E. Clarkson, and J. H. Hartman, “A three-dimensional Neumann-series approach to model light transport in non-uniform media,” J. Opt. Soc. Am. A (to be published).

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A. Klose and E. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220, 441–470 (2006).
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T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage 31, 1426–1433 (2006).
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M. Y. Kirillin, I. V. Meglinskii, and A. V. Priezzhev, “Effect of photons of different scattering orders on the formation of a signal in optical low-coherence tomography of highly scattering media,” Quantum Electron. 36, 247–252 (2006).
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Miller, E. L.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Miyake, Y.

K. Takagi, H. Haneishi, N. Tsumura, and Y. Miyake, “Alternative oblique-incidence reflectometry for measuring tissue optical properties,” Opt. Rev. 7, 164–169 (2000).
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A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical tomography for imaging of rheumatoid finger joints,” Phys. Med. Biol. 49, 1147–1163 (2004).
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Netz, U.

A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical tomography for imaging of rheumatoid finger joints,” Phys. Med. Biol. 49, 1147–1163 (2004).
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K. Kobayashi, H. Oigawa, and H. Yamagata, “The spherical harmonics method for the multigroup transport equation in x-y geometry,” Ann. Nucl. Energy 13, 663–678 (1986).
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J. C. Chai, H. S. Lee, and S. V. Patankar, “Finite volume method for radiation heat transfer,” J. Thermophys. Heat Transf. 8, 419–425 (1994).

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V. G. Peters, D. R. Wyman, M. S. Patterson, and G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
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S. Srinivasan, B. W. Pogue, C. Carpenter, P. K. Yalavarthy, and K. Paulsen, “A boundary element approach for image-guided near-infrared absorption and scatter estimation,” Med. Phys. 34, 4545–4557 (2007).
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S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol. 13, 195–202 (2006).

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, and K. D. Paulsen, “The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,” Phys. Med. Biol. 48, 2713–2727 (2003).
[CrossRef]

H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. 42, 135–146 (2003).
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V. G. Peters, D. R. Wyman, M. S. Patterson, and G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef]

Pfeilsticker, K.

T. Deutschmann, S. Beirle, U. Frie, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Roman, J. Pukite, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: Introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

Platt, U.

T. Deutschmann, S. Beirle, U. Frie, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Roman, J. Pukite, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: Introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

Pogue, B. W.

H. Dehghani, S. Srinivasan, B. W. Pogue, and A. Gibson, “Numerical modelling and image reconstruction in diffuse optical tomography,” Phil. Trans. R. Soc. A 367, 3073–3093(2009).
[CrossRef]

S. Srinivasan, B. W. Pogue, C. Carpenter, P. K. Yalavarthy, and K. Paulsen, “A boundary element approach for image-guided near-infrared absorption and scatter estimation,” Med. Phys. 34, 4545–4557 (2007).
[CrossRef]

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol. 13, 195–202 (2006).

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, and K. D. Paulsen, “The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,” Phys. Med. Biol. 48, 2713–2727 (2003).
[CrossRef]

H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. 42, 135–146 (2003).
[CrossRef]

Poplack, S. P.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol. 13, 195–202 (2006).

H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. 42, 135–146 (2003).
[CrossRef]

Prados-Roman, C.

T. Deutschmann, S. Beirle, U. Frie, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Roman, J. Pukite, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: Introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

Prahl, S. A.

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Pretorius, P. H.

M. A. King, S. J. Glick, P. H. Pretorius, R. G. Wells, H. C. Gifford, and M. V. Narayanan, Emission Tomography: The Fundamentals of PET and SPECT (Academic, 2004).

Priezzhev, A. V.

M. Y. Kirillin, I. V. Meglinskii, and A. V. Priezzhev, “Effect of photons of different scattering orders on the formation of a signal in optical low-coherence tomography of highly scattering media,” Quantum Electron. 36, 247–252 (2006).
[CrossRef]

Pukite, J.

T. Deutschmann, S. Beirle, U. Frie, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Roman, J. Pukite, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: Introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

Qin, G.

Z. Wang, M. Yang, and G. Qin, “Neumann series solution to a neutron transport equation of slab geometry,” J. Syst. Sci. Complexity 6, 13–17 (1993).

Ren, K.

Rouse, A. R.

A. A. Tanbakuchi, A. R. Rouse, and A. F. Gmitro, “Monte Carlo characterization of parallelized fluorescence confocal systems imaging in turbid media,” J. Biomed. Opt. 14, 044024 (2009).
[CrossRef]

Scheel, A. K.

A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical tomography for imaging of rheumatoid finger joints,” Phys. Med. Biol. 49, 1147–1163 (2004).
[CrossRef]

Schlaggar, B. L.

B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. USA 104, 12169–12174 (2007).
[CrossRef]

Schweiger, M.

S. Wright, M. Schweiger, and S. Arridge, “Reconstruction in optical tomography using the PN approximations,” Meas. Sci. Technol. 18, 79–86 (2007).
[CrossRef]

M. Schweiger and S. R. Arridge, “The finite-element method for the propagation of light in scattering media: frequency domain case,” Med. Phys. 24, 895–902 (1997).
[CrossRef]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef]

Sikora, J.

A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrization and a boundary element method,” Inverse Probl. 22, 1509–1532 (2006).
[CrossRef]

Skofronick-Jackson, G.

M. Kim, G. Skofronick-Jackson, and J. Weinman, “Intercomparison of millimeter-wave radiative transfer models,” IEEE Trans. Geosci. Remote Sens. 42, 1882–1890 (2004).
[CrossRef]

Soho, S.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol. 13, 195–202 (2006).

Spott, T.

Srinivasan, S.

H. Dehghani, S. Srinivasan, B. W. Pogue, and A. Gibson, “Numerical modelling and image reconstruction in diffuse optical tomography,” Phil. Trans. R. Soc. A 367, 3073–3093(2009).
[CrossRef]

S. Srinivasan, B. W. Pogue, C. Carpenter, P. K. Yalavarthy, and K. Paulsen, “A boundary element approach for image-guided near-infrared absorption and scatter estimation,” Med. Phys. 34, 4545–4557 (2007).
[CrossRef]

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol. 13, 195–202 (2006).

Svaasand, L. O.

Takagi, K.

K. Takagi, H. Haneishi, N. Tsumura, and Y. Miyake, “Alternative oblique-incidence reflectometry for measuring tissue optical properties,” Opt. Rev. 7, 164–169 (2000).
[CrossRef]

Tanbakuchi, A. A.

A. A. Tanbakuchi, A. R. Rouse, and A. F. Gmitro, “Monte Carlo characterization of parallelized fluorescence confocal systems imaging in turbid media,” J. Biomed. Opt. 14, 044024 (2009).
[CrossRef]

Tarvainen, T.

Tosteson, T. D.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol. 13, 195–202 (2006).

Tsumura, N.

K. Takagi, H. Haneishi, N. Tsumura, and Y. Miyake, “Alternative oblique-incidence reflectometry for measuring tissue optical properties,” Opt. Rev. 7, 164–169 (2000).
[CrossRef]

Vauhkonen, M.

Venugopal, V.

Vishwanath, K.

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol. 54, 2493–2509 (2009).
[CrossRef]

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, and K. D. Paulsen, “The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,” Phys. Med. Biol. 48, 2713–2727 (2003).
[CrossRef]

Wagner, T.

T. Deutschmann, S. Beirle, U. Frie, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Roman, J. Pukite, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: Introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

Wang, L.

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]

Wang, Z.

Z. Wang, M. Yang, and G. Qin, “Neumann series solution to a neutron transport equation of slab geometry,” J. Syst. Sci. Complexity 6, 13–17 (1993).

Weber, H.

G. Arfken and H. Weber, Mathematical Methods for Physicists, Vol. 3 (Academic, 2005).

Weinman, J.

M. Kim, G. Skofronick-Jackson, and J. Weinman, “Intercomparison of millimeter-wave radiative transfer models,” IEEE Trans. Geosci. Remote Sens. 42, 1882–1890 (2004).
[CrossRef]

Welch, A. J.

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Wells, R.

R. Wells, A. Celler, and R. Harrop, “Analytical calculation of photon distributions in SPECT projections,” IEEE Trans. Nucl. Sci. 45, 3202–3214 (1998).
[CrossRef]

Wells, R. G.

M. A. King, S. J. Glick, P. H. Pretorius, R. G. Wells, H. C. Gifford, and M. V. Narayanan, Emission Tomography: The Fundamentals of PET and SPECT (Academic, 2004).

Werner, B.

T. Deutschmann, S. Beirle, U. Frie, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Roman, J. Pukite, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: Introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

White, B. R.

B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. USA 104, 12169–12174 (2007).
[CrossRef]

Wright, S.

S. Wright, M. Schweiger, and S. Arridge, “Reconstruction in optical tomography using the PN approximations,” Meas. Sci. Technol. 18, 79–86 (2007).
[CrossRef]

Wyatt, J. S.

T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage 31, 1426–1433 (2006).
[CrossRef]

Wyman, D. R.

V. G. Peters, D. R. Wyman, M. S. Patterson, and G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef]

Yalavarthy, P. K.

S. Srinivasan, B. W. Pogue, C. Carpenter, P. K. Yalavarthy, and K. Paulsen, “A boundary element approach for image-guided near-infrared absorption and scatter estimation,” Med. Phys. 34, 4545–4557 (2007).
[CrossRef]

Yamagata, H.

K. Kobayashi, H. Oigawa, and H. Yamagata, “The spherical harmonics method for the multigroup transport equation in x-y geometry,” Ann. Nucl. Energy 13, 663–678 (1986).
[CrossRef]

Yang, M.

Z. Wang, M. Yang, and G. Qin, “Neumann series solution to a neutron transport equation of slab geometry,” J. Syst. Sci. Complexity 6, 13–17 (1993).

Yusof, R. M.

T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage 31, 1426–1433 (2006).
[CrossRef]

Zacharopoulos, A. D.

A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrization and a boundary element method,” Inverse Probl. 22, 1509–1532 (2006).
[CrossRef]

Zeff, B. W.

B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. USA 104, 12169–12174 (2007).
[CrossRef]

Zhang, H.

F. Gao, H. Niu, H. Zhao, and H. Zhang, “The forward and inverse models in time-resolved optical tomography imaging and their finite-element method solutions,” Image Vis. Comput. 16, 703–712 (1998).
[CrossRef]

Zhang, Q.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Zhao, H.

F. Gao, H. Niu, H. Zhao, and H. Zhang, “The forward and inverse models in time-resolved optical tomography imaging and their finite-element method solutions,” Image Vis. Comput. 16, 703–712 (1998).
[CrossRef]

Zheng, L.

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]

Acad. Radiol. (1)

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol. 13, 195–202 (2006).

Ann. Nucl. Energy (1)

K. Kobayashi, H. Oigawa, and H. Yamagata, “The spherical harmonics method for the multigroup transport equation in x-y geometry,” Ann. Nucl. Energy 13, 663–678 (1986).
[CrossRef]

Appl. Opt. (4)

Astrophys. J. (1)

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

Biomed. Opt. Express (2)

Comput. Methods Programs Biomed. (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]

Curr. Opin. Biotechnol. (1)

A. H. Hielscher, “Optical tomographic imaging of small animals,” Curr. Opin. Biotechnol. 16, 79–88 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

IEEE Signal Process. Mag. (1)

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

M. Kim, G. Skofronick-Jackson, and J. Weinman, “Intercomparison of millimeter-wave radiative transfer models,” IEEE Trans. Geosci. Remote Sens. 42, 1882–1890 (2004).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

R. Wells, A. Celler, and R. Harrop, “Analytical calculation of photon distributions in SPECT projections,” IEEE Trans. Nucl. Sci. 45, 3202–3214 (1998).
[CrossRef]

Image Vis. Comput. (1)

F. Gao, H. Niu, H. Zhao, and H. Zhang, “The forward and inverse models in time-resolved optical tomography imaging and their finite-element method solutions,” Image Vis. Comput. 16, 703–712 (1998).
[CrossRef]

Inverse Probl. (1)

A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrization and a boundary element method,” Inverse Probl. 22, 1509–1532 (2006).
[CrossRef]

J. Biomed. Opt. (1)

A. A. Tanbakuchi, A. R. Rouse, and A. F. Gmitro, “Monte Carlo characterization of parallelized fluorescence confocal systems imaging in turbid media,” J. Biomed. Opt. 14, 044024 (2009).
[CrossRef]

J. Comput. Phys. (1)

A. Klose and E. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220, 441–470 (2006).
[CrossRef]

J. Phys. A: Math. Gen. (1)

J. K. Fletcher, “A solution of the neutron transport equation using spherical harmonics,” J. Phys. A: Math. Gen. 16, 2827–2835 (1983).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

T. Deutschmann, S. Beirle, U. Frie, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Roman, J. Pukite, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: Introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

E. Aydin, C. de Oliveira, and A. Goddard, “A finite element-spherical harmonics radiation transport model for photon migration in turbid media,” J. Quant. Spectrosc. Radiat. Transfer 84, 247–260 (2004).
[CrossRef]

J. Syst. Sci. Complexity (1)

Z. Wang, M. Yang, and G. Qin, “Neumann series solution to a neutron transport equation of slab geometry,” J. Syst. Sci. Complexity 6, 13–17 (1993).

J. Thermophys. Heat Transf. (1)

J. C. Chai, H. S. Lee, and S. V. Patankar, “Finite volume method for radiation heat transfer,” J. Thermophys. Heat Transf. 8, 419–425 (1994).

Meas. Sci. Technol. (1)

S. Wright, M. Schweiger, and S. Arridge, “Reconstruction in optical tomography using the PN approximations,” Meas. Sci. Technol. 18, 79–86 (2007).
[CrossRef]

Med. Phys. (5)

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

E. D. Aydin, C. R. de Oliveira, and A. J. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Med. Phys. 29, 2013–2023 (2002).
[CrossRef]

S. Srinivasan, B. W. Pogue, C. Carpenter, P. K. Yalavarthy, and K. Paulsen, “A boundary element approach for image-guided near-infrared absorption and scatter estimation,” Med. Phys. 34, 4545–4557 (2007).
[CrossRef]

M. Schweiger and S. R. Arridge, “The finite-element method for the propagation of light in scattering media: frequency domain case,” Med. Phys. 24, 895–902 (1997).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef]

Neuroimage (1)

T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage 31, 1426–1433 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Rev. (1)

K. Takagi, H. Haneishi, N. Tsumura, and Y. Miyake, “Alternative oblique-incidence reflectometry for measuring tissue optical properties,” Opt. Rev. 7, 164–169 (2000).
[CrossRef]

Phil. Trans. R. Soc. A (2)

A. Gibson and H. Dehghani, “Diffuse optical imaging,” Phil. Trans. R. Soc. A 367, 3055–3072 (2009).
[CrossRef]

H. Dehghani, S. Srinivasan, B. W. Pogue, and A. Gibson, “Numerical modelling and image reconstruction in diffuse optical tomography,” Phil. Trans. R. Soc. A 367, 3073–3093(2009).
[CrossRef]

Phys. Med. Biol. (7)

B. F. Hutton, I. Buvat, and F. J. Beekman, “Review and current status of SPECT scatter correction,” Phys. Med. Biol. 56, R85–R112 (2011).
[CrossRef]

V. G. Peters, D. R. Wyman, M. S. Patterson, and G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005).
[CrossRef]

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef]

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, and K. D. Paulsen, “The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,” Phys. Med. Biol. 48, 2713–2727 (2003).
[CrossRef]

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol. 54, 2493–2509 (2009).
[CrossRef]

A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical tomography for imaging of rheumatoid finger joints,” Phys. Med. Biol. 49, 1147–1163 (2004).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. USA 104, 12169–12174 (2007).
[CrossRef]

Prog. Nucl. Energy (1)

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Quantum Electron. (1)

M. Y. Kirillin, I. V. Meglinskii, and A. V. Priezzhev, “Effect of photons of different scattering orders on the formation of a signal in optical low-coherence tomography of highly scattering media,” Quantum Electron. 36, 247–252 (2006).
[CrossRef]

Other (8)

H. H. Barrett and K. J. Myers, Foundations of Image Science, 1st ed. (Wiley, 2004).

A. H. Hielscher and R. E. Alcouffe, “Discrete-ordinate transport simulations of light propagation in highly forward scattering heterogenous media,” in Advances in Optical Imaging and Photon Migration, 1998 OSA Technical Digest Series (Optical Society of America, 1998), paper ATuC2.

B. Gallas and H. H. Barrett, “Modeling all orders of scatter in nuclear medicine,” in IEEE Nuclear Science Symposium, Vol. 3 (IEEE, 1998), pp. 1964–1968.

H. H. Barrett, B. Gallas, E. Clarkson, and A. Clough, Computational Radiology and Imaging: Therapy and Diagnostics (Springer, 1999), pp. 71–100.

E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Chelsea, 1955).

G. Arfken and H. Weber, Mathematical Methods for Physicists, Vol. 3 (Academic, 2005).

M. A. King, S. J. Glick, P. H. Pretorius, R. G. Wells, H. C. Gifford, and M. V. Narayanan, Emission Tomography: The Fundamentals of PET and SPECT (Academic, 2004).

A. K. Jha, M. A. Kupinski, H. H. Barrett, E. Clarkson, and J. H. Hartman, “A three-dimensional Neumann-series approach to model light transport in non-uniform media,” J. Opt. Soc. Am. A (to be published).

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Figures (13)

Fig. 1.
Fig. 1.

A schematic illustrating the effect of the various terms of the Neumann series as light propagation occurs through the biological tissue.

Fig. 2.
Fig. 2.

Setup for diffuse optical imaging. The defined coordinate system is shown in the top-right corner.

Fig. 3.
Fig. 3.

(a) The function γ p decreases as Neumann-series iteration advances; (b) comparison between the actual convergence curve β p and the predicted convergence curve λ p evaluated by the distribution function at the transmitted plane.

Fig. 4.
Fig. 4.

Flowchart of our implementation for the Neumann-series form of the RTE.

Fig. 5.
Fig. 5.

Illustration of the setup for Experiment 1.

Fig. 6.
Fig. 6.

Comparison results between the RTE solver and MC simulation with different spatial grid size Δ x : (a)  Δ x = 0.04 mfp ; (b)  Δ x = 0.05 mfp ; (c)  Δ x = 0.1 mfp ; (d)  Δ x = 0.2 mfp .

Fig. 7.
Fig. 7.

Average percent error of the RTE solver with respect to MC as a function of grid size Δ x .

Fig. 8.
Fig. 8.

Comparison between the results from the RTE, MC, and DE in (a) transmitted flux and (b) reflected flux. The optical parameters are μ a = 0.1 cm 1 , μ s H = 5.0 , g = 0 . The voxel size Δ x is 0.1 mfp. The fluxes are plotted in arbitrary units (a.u.).

Fig. 9.
Fig. 9.

Computed (a) transmitted flux and (b) reflected flux shown as a function of the number of Neumann-series terms. We observe that these flux values converge as the number of Neumann-series terms increases.

Fig. 10.
Fig. 10.

Convergence of the Neumann-series RTE with spherical harmonics: (a) transmitted flux with g = 0 and (b) transmitted flux with g = 0.8 . In each figure, the MC result is also plotted.

Fig. 11.
Fig. 11.

Comparison of the Neumann-series RTE ( L = 3 ) and MC results for the transmitted and reflected fluxes, respectively, with different values of μ s H . The different cases are (a), (b)  μ s H = 5 ; (c), (d)  μ s H = 6 ; (e), (f)  μ s H = 8 ; (g), (h)  μ s H = 9 ; (i), (j)  μ s H = 10 . The value of Δ x is 0.2 mfp for these experiments, and the optical parameters of the medium are μ a = 0.1 cm 1 and g = 0 .

Fig. 12.
Fig. 12.

Error between the RTE and MC results as a function of the optical length of the medium. The L = 3 approximation and Δ x = 0.2 mfp are used in the simulation. The dotted lines are superimposed on the errors to demonstrate the observation that the error varies approximately linearly with the optical length.

Fig. 13.
Fig. 13.

(a) Plot of computation time versus number of voxels based on Fig. 6 simulations. (b) Plot of computation time versus number of spherical harmonics based on Fig. 10 simulations.

Tables (1)

Tables Icon

Table 1. Computation Time Required by the RTE Implementation for the Various Simulations Performed in this Papera

Equations (53)

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d w d t = [ w t ] abs + [ w t ] em + [ w t ] prop + [ w t ] sc .
[ w t ] abs = c m μ a ( r ) w ( r , s ^ , t ) ,
[ w t ] em = Ξ ( r , s ^ , t ) .
[ w t ] prop = c m s ^ · w ( r , s ^ , t ) .
[ w t ] sc = c m μ s ( r ) w ( r , s ^ , t ) + 4 π d Ω K ( s ^ , s ^ | r ) w ( r , s ^ , t ) ,
K ( s ^ , s ^ | r ) = c m μ s ( r ) 4 π { 1 g 2 [ 1 + g 2 2 g ( s ^ · s ^ ) ] 3 / 2 } .
[ K w ] ( r , s ^ , t ) = 4 π d Ω K ( s ^ , s ^ | r ) w ( r , s ^ , t ) .
d w d t = c m μ tot ( r ) w ( r , s ^ , t ) + Ξ ( r , s ^ , t ) c m s ^ · w ( r , s ^ , t ) + K w ( r , s ^ , t ) ,
s ^ · w ( r , s ^ , ν ) + [ μ tot + i ν c m ] w ( r , s ^ , ν ) = 1 c m [ Ξ ( r , s ^ , ν ) + K w ( r , s ^ , ν ) ] .
s ^ · w ( r , s ^ ) + μ tot w ( r , s ^ ) = 1 c m [ Ξ ( r , s ^ ) + K w ( r , s ^ ) ] .
w ( r , s ^ ) = 1 c m 0 d Λ Ξ ( r s ^ Λ , s ^ ) exp [ 0 Λ d Λ μ tot ( r s ^ Λ ) ] + 1 c m 0 d Λ [ K w ] ( r s ^ Λ , s ^ ) exp [ 0 Λ d Λ μ tot ( r s ^ Λ ) ] .
w = X Ξ + X K w ,
[ X w ] ( r , s ^ ) = 1 c m 0 d Λ w ( r s ^ Λ , s ^ ) exp [ 0 Λ d Λ μ tot ( r s ^ Λ ) ] .
[ I X K ] w = X Ξ .
w = X Ξ + X K X Ξ + X K X K X Ξ +
w ( r , s ^ ) = l = 0 m = l l W l m ( r ) Y l m ( s ^ ) ,
W l m ( r ) = 4 π d Ω Y l m * ( s ^ ) w ( r , s ^ ) .
[ D W ] l m ( r ) = l , m D l m , l m ( r ) W l m ( r ) ,
D l m , l m ( r ) = 4 π 2 l + 1 k l ( r ) δ l l δ m m ,
k l ( r ) = c m μ s ( r ) 4 π ( 2 l + 1 ) g l .
[ A W ] l m ( r ) = l m d 3 r A l m , l m ( r , r ) W l m ( r ) ,
A l m , l m ( r , r ) = 1 c m 1 | r r | 2 Y l m * ( r r | r r | ) Y l m ( r r | r r | ) exp [ 0 | r r | d t μ tot ( r t r r | r r | ) ] .
W = A ξ + A D A ξ + A D A D A ξ + ,
W l m ( r ) = i W l m i ϕ i ( r ) .
W l m i = 1 Δ V S i d 3 r W l m ( r ) ,
D l m , l m = c m μ s g l δ l l δ m m .
v l m i = 1 Δ V l m i u l m i S i d 3 r S i d 3 r A l m , l m ( r , r ) .
A l m i , l m i = 1 Δ V S i d 3 r S i d 3 r A l m , l m ( r , r ) .
W d = A ξ d + A D A ξ d + A D A D A ξ d +
Ξ ( r , s ^ ) = α δ ( z ) h ( x , y ) δ ( s ^ z ^ ) .
[ X Ξ ] ( r , s ^ ) = α c m h ( x , y ) δ ( s ^ z ^ ) exp ( μ tot z ) .
W d = A ξ d + A [ n = 0 ( D A ) n ] ( D A ξ d ) .
[ K X Ξ ] ( r , s ^ ) = K ( s ^ , z ^ ) α c m h ( x , y ) exp ( μ tot z ) .
[ K X Ξ ] ( r , s ^ ) = μ s α 4 π { 1 g 2 [ 1 + g 2 2 g ( s ^ · z ^ ) ] 3 / 2 } h ( x , y ) exp ( μ tot z ) .
[ D A ξ ] l m ( r ) = μ s α 4 π h ( x , y ) exp ( μ tot z ) 4 π d Ω 1 g 2 [ 1 + g 2 2 g cos θ ] 3 / 2 Y l m * ( s ^ ) .
1 g 2 [ 1 + g 2 2 g cos θ ] 3 / 2 = l = 0 ( 2 l + 1 ) g l P l ( cos θ ) .
Y l m ( s ^ ) = ( 1 ) m 2 l + 1 4 π ( l m ) ! ( l + m ) ! P l m ( cos θ ) exp ( i m ϕ ) .
[ D A ξ ] l m ( r ) = μ s α 4 π h ( x , y ) exp ( μ tot z ) ( 1 ) m 2 l + 1 4 π ( l m ) ! ( l + m ) ! × l = 0 ( 2 l + 1 ) g l θ = 0 π d θ P l ( cos θ ) P l m ( cos θ ) sin θ ϕ = 0 2 π exp ( i m ϕ ) d ϕ .
ϕ = 0 2 π exp ( i m ϕ ) d ϕ = { 2 π , if m = 0 0 , if m 0 .
[ D A ξ ] l m ( r ) = μ s α h ( x , y ) exp ( μ tot z ) 2 l + 1 4 π l = 0 g l δ l l = μ s α h ( x , y ) exp ( μ tot z ) 2 l + 1 4 π g l .
[ W ] l m i = g l 2 l + 1 4 π α μ s Δ V μ tot h ( x i , y i ) { exp [ μ tot ( z i + Δ z ) ] exp [ μ tot z i ] } ,
η p = k , l , m [ ( D A ) p 1 ξ d ] l m k ,
β p = j = 1 p η j j = 1 η j .
γ p = η p j = 1 p η j .
λ p = 1 γ p 1 r .
Q i = P d 2 r 2 π d Ω 0 τ d t d i ( r , s ^ , t ) w ( r , s ^ , t ) ( n ^ · s ^ ) ,
Φ i = c m Δ x Δ y 2 π d Ω ( z ^ · s ^ ) w ( x i , y i , H , s ^ ) ,
Φ i = c m Δ x Δ y 0 π / 2 d θ sin θ cos θ 0 2 π d ϕ l m W l m i Y l m ( θ , ϕ ) ,
Φ i = 2 π c m Δ x Δ y l W l 0 i 2 l + 1 4 π 0 π / 2 d θ sin θ cos θ P l ( cos θ ) ,
Φ i = 2 π c m Δ x Δ y l W l 0 i 2 l + 1 4 π 0 1 d u P l ( u ) u .
A l m , l m ( r , r ) = 1 c m 1 | r r | 2 Y l m * ( r r | r r | ) Y l m ( r r | r r | ) exp ( μ tot | r r | ) .
q i = Φ i α Δ x Δ y ,
ϵ RTE , MC = i | Φ i , RTE Φ i , MC | Φ i , MC × 100 ,

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