Abstract

We present the implementation, validation, and performance of a three-dimensional (3D) Neumann-series approach to model photon propagation in nonuniform media using the radiative transport equation (RTE). The RTE is implemented for nonuniform scattering media in a spherical harmonic basis for a diffuse-optical-imaging setup. The method is parallelizable and implemented on a computing system consisting of NVIDIA Tesla C2050 graphics processing units (GPUs). The GPU implementation provides a speedup of up to two orders of magnitude over non-GPU implementation, which leads to good computational efficiency for the Neumann-series method. The results using the method are compared with the results obtained using the Monte Carlo simulations for various small-geometry phantoms, and good agreement is observed. We observe that the Neumann-series approach gives accurate results in many cases where the diffusion approximation is not accurate.

© 2012 Optical Society of America

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2012

2011

T. Deutschmann, S. Beirle, U. F. M. Grzegorski, C. Kern, L. Kritten, U. Platt, Cristina Prados-Román, Puķīte Jānis, 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. Huang, J. Mielikainen, H. Oh, and H.-L. A. Huang, “Development of a GPU-based high-performance radiative transfer model for the Infrared Atmospheric Sounding Interferometer (IASI),” J. Comput. Phys. 230, 2207–2221 (2011).
[CrossRef]

C. Gong, J. Liu, L. Chi, H. Huang, J. Fang, and Z. Gong, “GPU accelerated simulations of 3D deterministic particle transport using discrete ordinates method,” J. Comput. Phys. 230, 6010–6022 (2011).
[CrossRef]

D. S. Mishra BP, “Parallel computing environments: a review,” IETE Technical Review 28, 240–247 (2011).

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]

2010

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).
[CrossRef]

L. Szirmay-Kalos, G. Liktor, T. Umenhoffer, B. Toth, S. Kumar, and G. Lupton, “Parallel iteration to the radiative transport in inhomogeneous media with bootstrapping,” IEEE Trans. Vis. Comput. Graphics 17, 146–158 (2010).

2009

Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express 17, 20178–20190 (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]

K. Asanovic, R. Bodik, J. Demmel, T. Keaveny, K. Keutzer, J. Kubiatowicz, N. Morgan, D. Patterson, K. Sen, J. Wawrzynek, D. Wessel, and K. Yelick, “A view of the parallel computing landscape,” Commun. ACM 52, 56–67 (2009).
[CrossRef]

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

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]

2008

T. Tarvainen, M. Vauhkonen, v. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Utilizing the radiative transfer equation in optical tomography,” Piers Online 4, 655–661 (2008).

E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration,” J. Biomed. Opt. 13, 060504 (2008).
[CrossRef]

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable parallel programming with CUDA,” ACM Queue 6, 40–53(2008).

2007

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]

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]

2006

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]

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]

S. Narasimhan, M. Gupta, C. Donner, R. Ramamoorthi, S. Nayar, and H. Jensen, “Acquiring scattering properties of participating media by dilution,” ACM Trans. Graph. 25, 1003–1012 (2006).
[CrossRef]

S. Nayar, G. Krishnan, M. Grossberg, and R. Raskar, “Fast separation of direct and global components of a scene using high frequency illumination,” ACM Trans. Graph. 25, 935–944 (2006).
[CrossRef]

2005

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]

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]

2004

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]

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. Garofalakis, G. Zacharakis, G. Filippidis, E. Sanidas, D. D. Tsiftsis, V. Ntziachristos, T. G. Papazoglou, and J. Ripoll, “Characterization of the reduced scattering coefficient for optically thin samples: theory and experiments,” J. Opt. A 6, 725–735 (2004).
[CrossRef]

2003

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]

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]

A. P. Schweiger, M. Gibson, and S. R. Arridge, “Computational aspects of diffuse optical tomography,” IEEE Comput. Sci. Eng., 5, 33–41 (2003).

2002

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]

D. Boas, J. Culver, J. Stott, and A. Dunn, “Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head,” Opt. Express 10, 159–170 (2002).

2001

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

1999

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media:the ballistic to diffusive transition,” Phys. Rev. E 60, 4843–4850 (1999).
[CrossRef]

1998

G. Alexandrakis, T. J. Farrell, and M. S. Patterson, “Accuracy of the diffusion approximation in determining the optical properties of a two-layer turbid medium,” Appl. Opt. 37, 7401–7409 (1998).
[CrossRef]

F. Jacobs, E. Sundermann, B. D. Sutter, M. Christiaens, and I. Lemahieu, “A fast algorithm to calculate the exact radiological path through a pixel or voxel space,” J. Comput. Inf. Technol. 6, 89–94 (1998).

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).

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]

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

1997

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

A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
[CrossRef]

1993

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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).
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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).
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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).
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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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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. 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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Fang, J.

C. Gong, J. Liu, L. Chi, H. Huang, J. Fang, and Z. Gong, “GPU accelerated simulations of 3D deterministic particle transport using discrete ordinates method,” J. Comput. Phys. 230, 6010–6022 (2011).
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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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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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H. H. Barrett, B. Gallas, E. Clarkson, and A. Clough, Computational Radiology and Imaging: Therapy and Diagnostics (Springer, 1999).

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J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable parallel programming with CUDA,” ACM Queue 6, 40–53(2008).

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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. Gibson and H. Dehghani, “Diffuse optical imaging,” Phil. Trans. R. Soc. A 367, 3055–3072 (2009).
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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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A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43(2005).
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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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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).
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C. Gong, J. Liu, L. Chi, H. Huang, J. Fang, and Z. Gong, “GPU accelerated simulations of 3D deterministic particle transport using discrete ordinates method,” J. Comput. Phys. 230, 6010–6022 (2011).
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S. Narasimhan, M. Gupta, C. Donner, R. Ramamoorthi, S. Nayar, and H. Jensen, “Acquiring scattering properties of participating media by dilution,” ACM Trans. Graph. 25, 1003–1012 (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).
[CrossRef]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43(2005).
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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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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).
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M. Kim, G. Skofronick-Jackson, and J. Weinman, “Intercomparison of millimeter-wave radiative transfer models,” IEEE Trans. Geosci. Remote Sens. 42, 1882–1890 (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).

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).
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L. Szirmay-Kalos, G. Liktor, T. Umenhoffer, B. Toth, S. Kumar, and G. Lupton, “Parallel iteration to the radiative transport in inhomogeneous media with bootstrapping,” IEEE Trans. Vis. Comput. Graphics 17, 146–158 (2010).

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L. Szirmay-Kalos, G. Liktor, T. Umenhoffer, B. Toth, S. Kumar, and G. Lupton, “Parallel iteration to the radiative transport in inhomogeneous media with bootstrapping,” IEEE Trans. Vis. Comput. Graphics 17, 146–158 (2010).

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T. Tarvainen, M. Vauhkonen, v. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Utilizing the radiative transfer equation in optical tomography,” Piers Online 4, 655–661 (2008).

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Hybrid radiative-transfer-diffusion model for optical tomography,” Appl. Opt. 44, 876–886 (2005).
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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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Z. Wang, M. Yang, and G. Qin, “Neumann series solution to a neutron transport equation of slab geometry,” J. Syst. Sci. Complex. 6, 13–17 (1993).

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K. Asanovic, R. Bodik, J. Demmel, T. Keaveny, K. Keutzer, J. Kubiatowicz, N. Morgan, D. Patterson, K. Sen, J. Wawrzynek, D. Wessel, and K. Yelick, “A view of the parallel computing landscape,” Commun. ACM 52, 56–67 (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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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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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).
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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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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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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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L. Szirmay-Kalos, G. Liktor, T. Umenhoffer, B. Toth, S. Kumar, and G. Lupton, “Parallel iteration to the radiative transport in inhomogeneous media with bootstrapping,” IEEE Trans. Vis. Comput. Graphics 17, 146–158 (2010).

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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).

Inverse Probl.

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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E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration,” J. Biomed. Opt. 13, 060504 (2008).
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A. Gibson and H. Dehghani, “Diffuse optical imaging,” Phil. Trans. R. Soc. A 367, 3055–3072 (2009).
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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).
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Figures (8)

Fig. 1.
Fig. 1.

DOT setup: a 3D cuboidal phantom with a laser source along the optical axis. The figure also roughly shows the effect of different terms of the Neumann series.

Fig. 2.
Fig. 2.

Flowchart of our algorithm implementation on the GPU. Blue boxes (filled with dotted pattern) denote the code running on the CPU, orange boxes (filled with solid pattern) denote code running on the GPU, and green boxes (filled with vertical-line pattern) are the interface routines between CPU and GPU.

Fig. 3.
Fig. 3.

Summary of the parallelization scheme to implement the RTE.

Fig. 4.
Fig. 4.

Neumann-series and MC transmittance outputs in a homogeneous low-scattering medium (μs=1cm1) with (a) low-absorption coefficient (μa=0.01cm1) and (b) high-absorption coefficient (μa=1cm1).

Fig. 5.
Fig. 5.

Neumann-series and MC transmittance outputs in a midscattering medium (μs=4cm1) with (a) low-absorption coefficient (μa=0.01cm1) and (b) high-absorption coefficient (μa=1cm1).

Fig. 6.
Fig. 6.

Different heterogeneous tissue geometries: (a) three different tissue types scattered randomly, (b) spherical inclusion in the center of the medium, and (c) medium consisting of three layers.

Fig. 7.
Fig. 7.

Neumann-series and MC transmittance outputs in a midscattering heterogeneous medium with (a) three different tissue types scattered randomly, (b) a high-absorption spherical inclusion at the center of the medium, (c) a very-low-scattering inclusion at the center of the medium, and (d) layered medium.

Fig. 8.
Fig. 8.

The Neumann-series and MC transmittance outputs in a low-scattering heterogeneous medium with (a) three different tissue types scattered randomly, (b) a high-absorption spherical inclusion at the center of the medium, (c) a very-low-scattering inclusion at the center of the medium, and (d) layered medium.

Tables (1)

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Table 1. Speedup with GPU Implementation

Equations (25)

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[Kw](r,s^,t)=4πdΩK(s^,s^|r)w(r,s^,t),
K(s^,s^|r)=μs(r)cm4π{1g2[1+g22gcos(s^·s^)]3/2},
[Xw](r,s^,t)=1cmλ=0dλw(rs^λ,s^,t)exp[0λdλμtot(rs^λ)].
w(r,s^)=XΞ+XKXΞ+XKXKXΞ+,
w(r,s^)=l=0m=1lWlm(r)Ylm(s^),
Wlm(i,j,k)=1ΔVSijkd3rWlm(r),
w(r,s^)=i,j,k=1I,J,Kl=0Lm=llWlm(i,j,k)ψijk(r)Ylm(s^),
[DWd]lm(i,j,k)=lmWlm(i,j,k)Dlm,lm(i,j,k),
Dlm,lm(i,j,k)=cmμs(r)glδllδmm,
[AW]lm(r)=l=0m=lld3rWlm(r)Alm,lm(r,r),
Alm,lm(r,r)=1cm|rr|2Ylm*(rr|rr|)Ylm(rr|rr|)exp[λ=0|rr|dλμtot(rλrr|rr|)].
[AWd]lm(i,j,k)=i,j,k=1I,J,Kl=0Lm=llAlm,l,m(i,j,k,i,j,k)Wlm(i,j,k).
Alm,lm(i,j,k,i,j,k)=ΔVcmRvv2Ylm*(u^vv)Ylm(u^vv)exp[q=0Rvv/ΔλΔλμtot(iquxΔλΔx,jquyΔλΔy,kquzΔλΔz)].
[AW]lm(r)=1cmlm4πdΩuYlm*(u^)Ylm(u^)0dRWlm(r)exp[λ=0Rdλμtot(ru^λ)],
[ADWd]lm(i,j,k)=1cmlmWlm(i,j,k)θ=0πdθsinθϕ=02πdϕYlm*(u^)Ylm(u^){1exp[μtot(i,j,k)β(θu,ϕu)]μtot(i,j,k)},
[ADWd]lm(i,j,k)=lmWlm(i,j,k)Alm,lm(i,j,k).
Wd=Aξd+ADAξd+ADADAξd+...,
Ξ(r,s^)=αδ(s^z^)h(x,y)δ(z),
[XΞ](r,s^)=αcm0dλδ(s^z^)h(x,y)δ(zz^λ)exp[0λdλμtot(rs^λ)]=αcmδ(s^z^)h(x,y)exp[0zdλμtot(rz^λ)],
Wd=Aξd+A[n=0(DA)n](DAξd).
[KXΞ](r,s^)=αμs(r)4πh(x,y){1g2[1+g22gcos(s^·z^)]3/2}exp[0zdλμtot(rz^λ)].
[W]lm(r)=αμs(r)h(x,y)gl2l+14πexp[0zdλμtot(rz^λ)].
Wlm(i,j,k)=h(xi,yj)glΔV2l+14πk=1Kμs(i,j,k)μtot(i,j,k)exp[μtot(i,j,k)zk]{1exp[μtot(i,j,k)Δz)]},
Φij=cmAp2πdΩz^s^w(xi,yj,H,s^),
Φij,XΞ=αAph(xi,yj)exp[k=1KΔzμtot(i,j,k)].

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