Abstract

The radiative transfer equation (RTE) is widely accepted to accurately describe light transport in a medium with scattering particles, and it has been successfully applied as a light-transport model, for example, in diffuse optical tomography. Due to the computationally expensive nature of the RTE, most of these applications have been in the frequency domain. In this paper, an efficient solution method for the time-domain RTE is proposed. The method is based on solving the frequency-domain RTE at multiple modulation frequencies and using the Fourier-series representation of the radiance to obtain approximation of the time-domain solution. The approach is tested with simulations. The results show that the method can be used to obtain the solution of the time-domain RTE with good accuracy and with significantly fewer computational resources than are needed in the direct time-domain solution.

© 2013 Optical Society of America

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    [CrossRef]

2012 (6)

F. Asllanaj and S. Fumeron, “Applying a new computational method for biological tissue optics based on the time-dependent two-dimensional radiative transfer equation,” J. Biomed. Opt. 17, 075007 (2012).
[CrossRef]

M. Charest, C. Groth, and Ö. Gülder, “Solution of the equation of radiative transfer using a Newton–Krylov approach and adaptive mesh refinement,” J. Comput. Phys. 231, 3023–3040 (2012).
[CrossRef]

O. Balima, Y. Favennec, J. Boulanger, and A. Charette, “Optical tomography with the discontinuous Galerkin forumulation of the radiative transfer equation in frequency domain,” J. Quant. Spectrosc. Radiat. Transfer 113, 805–814 (2012).
[CrossRef]

A. Jha, M. Kupinski, T. Masumura, E. Clarkson, A. Maslov, and H. Barrett, “Simulating photon-transport in uniform media using the radiative transport equation: a study using the Neumann-series approach,” J. Opt. Soc. Am. A 29, 1741–1756 (2012).
[CrossRef]

A. Jha, M. Kupinski, H. Barrett, E. Clarkson, and J. Hartman, “Three-dimensional Neumann-series approach to model light transport in nonuniform media,” J. Opt. Soc. Am. A 29, 1885–1898 (2012).
[CrossRef]

M. Addam, A. Bouhamidi, and K. Jbilou, “Signal reconstruction for the diffusion transport equation using tensorial spline galerking approximation,” Appl. Numer. Math. 62, 1089–1108 (2012).
[CrossRef]

2011 (4)

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230, 7364–7383 (2011).
[CrossRef]

K. Peng, X. Gao, X. Qu, N. Ren, X. Chen, X. He, X. Wang, J. Liang, and J. Tian, “Graphics processing unit parallel accelerated solution of the discrete ordinates for photon transport in biological tissues,” Appl. Opt. 50, 3808–3823 (2011).
[CrossRef]

N. Ducros, C. D’Andrea, A. Bassi, and F. Peyrin, “Fluorescence diffuse optical tomography: time-resolved versus continuous-wave in the reflectance configuration,” IRBM 32, 243–250 (2011).
[CrossRef]

M. Boffety, M. Allain, A. Sentenac, M. Massonneau, and R. Carminati, “Cramer–Rao analysis of steady-state and time-domain fluorescence diffuse optical imaging,” Biomed. Opt. Express 2, 1626–1636 (2011).
[CrossRef]

2010 (3)

M. Addam, A. Bouhamidi, and K. Jbilou, “A numerical method for one-dimensional diffusion problem using Fourier transform and the B-spline Galerkin method,” Appl. Math. Comput. 215, 4067–4079 (2010).
[CrossRef]

D. Gorpas, D. Yova, and K. Politopoulos, “A three-dimensional finite elements approach for the coupled radiative transfer equation and diffusion approximation modeling in fluorescence imaging,” J. Quant. Spectrosc. Radiat. Transfer 111, 553–568 (2010).
[CrossRef]

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. Arridge, and J. Kaipio, “Approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography,” Inverse Probl. 26, 015005 (2010).
[CrossRef]

2009 (5)

Z. Yuan, X.-H. Hu, and H. Jiang, “A higher order diffusion model for three-dimensional photon migration and image reconstruction in optical tomography,” Phys. Med. Biol. 54, 65–88(2009).
[CrossRef]

H. Gao and H. Zhao, “A fast-forward solver of radiative transfer equation,” Transp. Theory Stat. Phys. 38, 149–192 (2009).
[CrossRef]

H. Gao and H. Zhao, “A fast-forward solver of radiative transfer equation,” Transp. Theory Stat.. Phys. 38, 149–192 (2009).
[CrossRef]

S. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[CrossRef]

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

2008 (2)

Q. Zhang, H. Soon, H. Tian, S. Fernando, Y. Ha, and N. Chen, “Pseudo-random single photon counting for time-resolved optical measurement,” Opt. Express 16, 13233–13239 (2008).
[CrossRef]

T. Tarvainen, M. Vauhkonen, and S. Arridge, “Gauss–Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 109, 2767–2778 (2008).
[CrossRef]

2007 (3)

2006 (4)

K. Ren, G. Bal, and A. Hielscher, “Frequency domain optical tomography based on the equation of radiative transfer,” SIAM J. Sci. Comput. 28, 1463–1489 (2006).
[CrossRef]

I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
[CrossRef]

J. Selb, D. Joseph, and D. Boas, “Time-gated optical system for depth-resolved functional brain imaging,” J. Biomed. Opt. 11, 044008 (2006).
[CrossRef]

D. Contini, A. Torricelli, A. Pifferi, L. Spinelli, P. Taroni, V. Quaresima, M. Ferrari, and R. Cubeddu, “Multichannel time-resolved tissue oximeter for functional imaging of the brain,” IEEE Trans. Instrum. Meas. 55, 85–90 (2006).
[CrossRef]

2005 (7)

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

J. Boulanger and A. Charette, “Reconstruction optical spectroscopy using transient radiative transfer equation and pulsed laser: a numerical study,” J. Quant. Spectrosc. Radiat. Transfer 93, 325–336 (2005).
[CrossRef]

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

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, S. Arridge, and J. Kaipio, “Coupled radiative transfer equation and diffusion approximation model for photon migration in turbid medium with low-scattering and non-scattering regions,” Phys. Med. Biol. 50, 4913–4930 (2005).
[CrossRef]

E. Boman, J. Tervo, and M. Vauhkonen, “Modelling the transport of ionizing radiation using the finite element method,” Phys. Med. Biol. 50, 265–280 (2005).
[CrossRef]

S. Wright, M. Schweiger, and S. Arridge, “Solutions to the transport equation using variable order angular basis,” Proc. SPIE 5859, 585914 (2005).
[CrossRef]

J. Heiskala, I. Nissilä, T. Neuvonen, S. Järvenpää, and E. Somersalo, “Modeling anisotropic light propagation in a realistic model of the human head,” Appl. Opt. 44, 2049–2057 (2005).
[CrossRef]

2004 (2)

R. Koch and R. Becker, “Evaluation of quadrature schemes for the discrete ordinates method,” J. Quant. Spectrosc. Radiat. Transfer 84, 423–435 (2004).
[CrossRef]

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

2003 (2)

G. Abdoulaev and A. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imaging 12, 594–601 (2003).
[CrossRef]

A. Klose and A. Hielscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Probl. 19, 387–409 (2003).
[CrossRef]

2002 (3)

A. Klose, U. Netz, J. Beuthan, and A. Hielscher, “Optical tomography using the time-independent equation of radiative transfer—Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transfer 72, 691–713 (2002).
[CrossRef]

F. Gao, H. Zhao, and Y. Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt. 41, 778–791 (2002).
[CrossRef]

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

2001 (2)

A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, “Polarized pulse waves in random discrete scatterers,” Appl. Opt. 40, 5495–5502 (2001).
[CrossRef]

S. Richling, E. Meinköhn, N. Kryzhevoi, and G. Kanschat, “Radiative transfer with finite elements I. Basic method and tests,” Astron. Astrophys. 380, 776–788 (2001).
[CrossRef]

2000 (1)

F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, J. C. Hebden, and D. T. Delby, “A 32-channel time-resolved instrument for medical optical tomography,” Rev. Sci. Instrum. 71, 256–265 (2000).
[CrossRef]

1999 (5)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[CrossRef]

A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26, 1698–1707 (1999).
[CrossRef]

H. Jiang, “Optical image reconstruction based on the third-order diffusion equations,” Opt. Express 4, 241–246 (1999).
[CrossRef]

J. Tervo, P. Kolmonen, M. Vauhkonen, L. Heikkinen, and J. Kaipio, “A finite-element model of electron transport in radiation therapy and a related inverse problem,” Inverse Probl. 15, 1345–1361 (1999).
[CrossRef]

1998 (5)

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]

G. Kanschat, “A robust finite element discretization for radiative transfer problems with scattering,” East-West J. Numer. Math. 6, 265–272 (1998).

S. R. Arridge and W. R. B. Lionheart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett. 23, 882–884 (1998).
[CrossRef]

A. D. Kim and A. Ishimaru, “Optical diffusion of continuos-wave, pulsed, and density waves in scattering media and comparisons with radiative transfer,” Appl. Opt. 37, 5313–5319 (1998).
[CrossRef]

O. Dorn, “A transport–backtransport method for optical tomography,” Inverse Probl. 14, 1107–1130 (1998).
[CrossRef]

1993 (1)

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]

1981 (1)

W. Martin, C. Yehnert, L. Lorence, and J. Duderstadt, “Phase-space finite element methods applied to the first order form of the transport equation,” Ann. Nucl. Energy 8, 633–646 (1981).
[CrossRef]

1941 (1)

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

Abdoulaev, G.

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

G. Abdoulaev and A. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imaging 12, 594–601 (2003).
[CrossRef]

Ackroyd, R. T.

R. T. Ackroyd, Finite Element Methods for Particle Transport: Applications to Reactor and Radiation Physics (Research Studies, 1997).

Addam, M.

M. Addam, A. Bouhamidi, and K. Jbilou, “Signal reconstruction for the diffusion transport equation using tensorial spline galerking approximation,” Appl. Numer. Math. 62, 1089–1108 (2012).
[CrossRef]

M. Addam, A. Bouhamidi, and K. Jbilou, “A numerical method for one-dimensional diffusion problem using Fourier transform and the B-spline Galerkin method,” Appl. Math. Comput. 215, 4067–4079 (2010).
[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]

Allain, M.

Arridge, S.

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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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J. Tervo, P. Kolmonen, M. Vauhkonen, L. Heikkinen, and J. Kaipio, “A finite-element model of electron transport in radiation therapy and a related inverse problem,” Inverse Probl. 15, 1345–1361 (1999).
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K. Ren, G. Abdoulaev, G. Bal, and A. Hielscher, “Algorithm for solving the equation of radiative transfer in the frequency domain,” Opt. Lett. 29, 578–580 (2004).
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A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26, 1698–1707 (1999).
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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).
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I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
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M. Addam, A. Bouhamidi, and K. Jbilou, “Signal reconstruction for the diffusion transport equation using tensorial spline galerking approximation,” Appl. Numer. Math. 62, 1089–1108 (2012).
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M. Addam, A. Bouhamidi, and K. Jbilou, “A numerical method for one-dimensional diffusion problem using Fourier transform and the B-spline Galerkin method,” Appl. Math. Comput. 215, 4067–4079 (2010).
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Kim, H. K.

H. K. Kim and A. Charette, “A sensitivity function-based conjugate gradient method for optical tomography with the frequency-domain equation of radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 104, 24–39 (2007).
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A. Klose and A. Hielscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Probl. 19, 387–409 (2003).
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A. Klose, U. Netz, J. Beuthan, and A. Hielscher, “Optical tomography using the time-independent equation of radiative transfer—Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transfer 72, 691–713 (2002).
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A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26, 1698–1707 (1999).
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R. Koch and R. Becker, “Evaluation of quadrature schemes for the discrete ordinates method,” J. Quant. Spectrosc. Radiat. Transfer 84, 423–435 (2004).
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J. Tervo, P. Kolmonen, M. Vauhkonen, L. Heikkinen, and J. Kaipio, “A finite-element model of electron transport in radiation therapy and a related inverse problem,” Inverse Probl. 15, 1345–1361 (1999).
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P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230, 7364–7383 (2011).
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Netz, U.

A. Klose, U. Netz, J. Beuthan, and A. Hielscher, “Optical tomography using the time-independent equation of radiative transfer—Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transfer 72, 691–713 (2002).
[CrossRef]

Neuvonen, T.

Nissilä, I.

I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
[CrossRef]

J. Heiskala, I. Nissilä, T. Neuvonen, S. Järvenpää, and E. Somersalo, “Modeling anisotropic light propagation in a realistic model of the human head,” Appl. Opt. 44, 2049–2057 (2005).
[CrossRef]

Noponen, T.

I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
[CrossRef]

Obrig, H.

H. Wabnitz, M. Moeller, A. Liebert, H. Obrig, J. Steinbrink, and R. Macdonald, “Time-resolved near-infrared spectroscopy and imaging of the adult human brain,” in Oxygen Transport to Tissue XXXI, E. Takahashi and D. Bruley, eds. (Springer, 2010), Vol. 662, pp. 143–148.

Oda, I.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[CrossRef]

Oda, M.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[CrossRef]

Oikawa, Y.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[CrossRef]

Peng, K.

Peyrin, F.

N. Ducros, C. D’Andrea, A. Bassi, and F. Peyrin, “Fluorescence diffuse optical tomography: time-resolved versus continuous-wave in the reflectance configuration,” IRBM 32, 243–250 (2011).
[CrossRef]

Pifferi, A.

D. Contini, A. Torricelli, A. Pifferi, L. Spinelli, P. Taroni, V. Quaresima, M. Ferrari, and R. Cubeddu, “Multichannel time-resolved tissue oximeter for functional imaging of the brain,” IEEE Trans. Instrum. Meas. 55, 85–90 (2006).
[CrossRef]

Politopoulos, K.

D. Gorpas, D. Yova, and K. Politopoulos, “A three-dimensional finite elements approach for the coupled radiative transfer equation and diffusion approximation modeling in fluorescence imaging,” J. Quant. Spectrosc. Radiat. Transfer 111, 553–568 (2010).
[CrossRef]

Prahl, S. A.

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, SPIE Institute Series, Vol.5, (SPIE, 1989), pp. 102–111.

Pulkkinen, A.

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230, 7364–7383 (2011).
[CrossRef]

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. Arridge, and J. Kaipio, “Approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography,” Inverse Probl. 26, 015005 (2010).
[CrossRef]

Qu, X.

Quaresima, V.

D. Contini, A. Torricelli, A. Pifferi, L. Spinelli, P. Taroni, V. Quaresima, M. Ferrari, and R. Cubeddu, “Multichannel time-resolved tissue oximeter for functional imaging of the brain,” IEEE Trans. Instrum. Meas. 55, 85–90 (2006).
[CrossRef]

Ren, K.

K. Ren, G. Bal, and A. Hielscher, “Frequency domain optical tomography based on the equation of radiative transfer,” SIAM J. Sci. Comput. 28, 1463–1489 (2006).
[CrossRef]

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

Ren, N.

Richling, S.

S. Richling, E. Meinköhn, N. Kryzhevoi, and G. Kanschat, “Radiative transfer with finite elements I. Basic method and tests,” Astron. Astrophys. 380, 776–788 (2001).
[CrossRef]

Sassaroli, A.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[CrossRef]

Schmidt, F. E. W.

F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, J. C. Hebden, and D. T. Delby, “A 32-channel time-resolved instrument for medical optical tomography,” Rev. Sci. Instrum. 71, 256–265 (2000).
[CrossRef]

Schotland, J.

S. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[CrossRef]

Schweiger, M.

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230, 7364–7383 (2011).
[CrossRef]

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. Arridge, and J. Kaipio, “Approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography,” Inverse Probl. 26, 015005 (2010).
[CrossRef]

I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
[CrossRef]

S. Wright, M. Schweiger, and S. Arridge, “Solutions to the transport equation using variable order angular basis,” Proc. SPIE 5859, 585914 (2005).
[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]

Selb, J.

Sentenac, A.

Somersalo, E.

Soon, H.

Spinelli, L.

D. Contini, A. Torricelli, A. Pifferi, L. Spinelli, P. Taroni, V. Quaresima, M. Ferrari, and R. Cubeddu, “Multichannel time-resolved tissue oximeter for functional imaging of the brain,” IEEE Trans. Instrum. Meas. 55, 85–90 (2006).
[CrossRef]

Steinbrink, J.

H. Wabnitz, M. Moeller, A. Liebert, H. Obrig, J. Steinbrink, and R. Macdonald, “Time-resolved near-infrared spectroscopy and imaging of the adult human brain,” in Oxygen Transport to Tissue XXXI, E. Takahashi and D. Bruley, eds. (Springer, 2010), Vol. 662, pp. 143–148.

Takada, M.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[CrossRef]

Tamura, M.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[CrossRef]

Taroni, P.

D. Contini, A. Torricelli, A. Pifferi, L. Spinelli, P. Taroni, V. Quaresima, M. Ferrari, and R. Cubeddu, “Multichannel time-resolved tissue oximeter for functional imaging of the brain,” IEEE Trans. Instrum. Meas. 55, 85–90 (2006).
[CrossRef]

Tarvainen, T.

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230, 7364–7383 (2011).
[CrossRef]

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. Arridge, and J. Kaipio, “Approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography,” Inverse Probl. 26, 015005 (2010).
[CrossRef]

T. Tarvainen, M. Vauhkonen, and S. Arridge, “Gauss–Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 109, 2767–2778 (2008).
[CrossRef]

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, S. Arridge, and J. Kaipio, “Coupled radiative transfer equation and diffusion approximation model for photon migration in turbid medium with low-scattering and non-scattering regions,” Phys. Med. Biol. 50, 4913–4930 (2005).
[CrossRef]

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

Tervo, J.

E. Boman, J. Tervo, and M. Vauhkonen, “Modelling the transport of ionizing radiation using the finite element method,” Phys. Med. Biol. 50, 265–280 (2005).
[CrossRef]

J. Tervo, P. Kolmonen, M. Vauhkonen, L. Heikkinen, and J. Kaipio, “A finite-element model of electron transport in radiation therapy and a related inverse problem,” Inverse Probl. 15, 1345–1361 (1999).
[CrossRef]

Tian, H.

Tian, J.

Torricelli, A.

D. Contini, A. Torricelli, A. Pifferi, L. Spinelli, P. Taroni, V. Quaresima, M. Ferrari, and R. Cubeddu, “Multichannel time-resolved tissue oximeter for functional imaging of the brain,” IEEE Trans. Instrum. Meas. 55, 85–90 (2006).
[CrossRef]

Tsuchiya, Y.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[CrossRef]

Tsunazawa, Y.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[CrossRef]

Vauhkonen, M.

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. Arridge, and J. Kaipio, “Approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography,” Inverse Probl. 26, 015005 (2010).
[CrossRef]

T. Tarvainen, M. Vauhkonen, and S. Arridge, “Gauss–Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 109, 2767–2778 (2008).
[CrossRef]

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

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, S. Arridge, and J. Kaipio, “Coupled radiative transfer equation and diffusion approximation model for photon migration in turbid medium with low-scattering and non-scattering regions,” Phys. Med. Biol. 50, 4913–4930 (2005).
[CrossRef]

E. Boman, J. Tervo, and M. Vauhkonen, “Modelling the transport of ionizing radiation using the finite element method,” Phys. Med. Biol. 50, 265–280 (2005).
[CrossRef]

J. Tervo, P. Kolmonen, M. Vauhkonen, L. Heikkinen, and J. Kaipio, “A finite-element model of electron transport in radiation therapy and a related inverse problem,” Inverse Probl. 15, 1345–1361 (1999).
[CrossRef]

Wabnitz, H.

H. Wabnitz, M. Moeller, A. Liebert, H. Obrig, J. Steinbrink, and R. Macdonald, “Time-resolved near-infrared spectroscopy and imaging of the adult human brain,” in Oxygen Transport to Tissue XXXI, E. Takahashi and D. Bruley, eds. (Springer, 2010), Vol. 662, pp. 143–148.

Wada, Y.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[CrossRef]

Wang, X.

Welch, A. J.

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, SPIE Institute Series, Vol.5, (SPIE, 1989), pp. 102–111.

Wright, S.

S. Wright, M. Schweiger, and S. Arridge, “Solutions to the transport equation using variable order angular basis,” Proc. SPIE 5859, 585914 (2005).
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Yamada, Y.

F. Gao, H. Zhao, and Y. Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt. 41, 778–791 (2002).
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H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
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Yamashita, Y.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
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W. Martin, C. Yehnert, L. Lorence, and J. Duderstadt, “Phase-space finite element methods applied to the first order form of the transport equation,” Ann. Nucl. Energy 8, 633–646 (1981).
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Yova, D.

D. Gorpas, D. Yova, and K. Politopoulos, “A three-dimensional finite elements approach for the coupled radiative transfer equation and diffusion approximation modeling in fluorescence imaging,” J. Quant. Spectrosc. Radiat. Transfer 111, 553–568 (2010).
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Yuan, Z.

Z. Yuan, X.-H. Hu, and H. Jiang, “A higher order diffusion model for three-dimensional photon migration and image reconstruction in optical tomography,” Phys. Med. Biol. 54, 65–88(2009).
[CrossRef]

Zhang, Q.

Zhao, H.

H. Gao and H. Zhao, “A fast-forward solver of radiative transfer equation,” Transp. Theory Stat.. Phys. 38, 149–192 (2009).
[CrossRef]

H. Gao and H. Zhao, “A fast-forward solver of radiative transfer equation,” Transp. Theory Stat. Phys. 38, 149–192 (2009).
[CrossRef]

F. Gao, H. Zhao, and Y. Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt. 41, 778–791 (2002).
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Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, 1967).

Ann. Nucl. Energy (1)

W. Martin, C. Yehnert, L. Lorence, and J. Duderstadt, “Phase-space finite element methods applied to the first order form of the transport equation,” Ann. Nucl. Energy 8, 633–646 (1981).
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Appl. Math. Comput. (1)

M. Addam, A. Bouhamidi, and K. Jbilou, “A numerical method for one-dimensional diffusion problem using Fourier transform and the B-spline Galerkin method,” Appl. Math. Comput. 215, 4067–4079 (2010).
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Appl. Numer. Math. (1)

M. Addam, A. Bouhamidi, and K. Jbilou, “Signal reconstruction for the diffusion transport equation using tensorial spline galerking approximation,” Appl. Numer. Math. 62, 1089–1108 (2012).
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Appl. Opt. (7)

Astron. Astrophys. (1)

S. Richling, E. Meinköhn, N. Kryzhevoi, and G. Kanschat, “Radiative transfer with finite elements I. Basic method and tests,” Astron. Astrophys. 380, 776–788 (2001).
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Astrophys. J. (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
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Biomed. Opt. Express (1)

East-West J. Numer. Math. (1)

G. Kanschat, “A robust finite element discretization for radiative transfer problems with scattering,” East-West J. Numer. Math. 6, 265–272 (1998).

IEEE Trans. Instrum. Meas. (1)

D. Contini, A. Torricelli, A. Pifferi, L. Spinelli, P. Taroni, V. Quaresima, M. Ferrari, and R. Cubeddu, “Multichannel time-resolved tissue oximeter for functional imaging of the brain,” IEEE Trans. Instrum. Meas. 55, 85–90 (2006).
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Inverse Probl. (6)

S. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
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S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
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O. Dorn, “A transport–backtransport method for optical tomography,” Inverse Probl. 14, 1107–1130 (1998).
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A. Klose and A. Hielscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Probl. 19, 387–409 (2003).
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J. Tervo, P. Kolmonen, M. Vauhkonen, L. Heikkinen, and J. Kaipio, “A finite-element model of electron transport in radiation therapy and a related inverse problem,” Inverse Probl. 15, 1345–1361 (1999).
[CrossRef]

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. Arridge, and J. Kaipio, “Approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography,” Inverse Probl. 26, 015005 (2010).
[CrossRef]

IRBM (1)

N. Ducros, C. D’Andrea, A. Bassi, and F. Peyrin, “Fluorescence diffuse optical tomography: time-resolved versus continuous-wave in the reflectance configuration,” IRBM 32, 243–250 (2011).
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J. Biomed. Opt. (3)

J. Selb, D. Joseph, and D. Boas, “Time-gated optical system for depth-resolved functional brain imaging,” J. Biomed. Opt. 11, 044008 (2006).
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I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
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F. Asllanaj and S. Fumeron, “Applying a new computational method for biological tissue optics based on the time-dependent two-dimensional radiative transfer equation,” J. Biomed. Opt. 17, 075007 (2012).
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J. Comput. Phys. (2)

M. Charest, C. Groth, and Ö. Gülder, “Solution of the equation of radiative transfer using a Newton–Krylov approach and adaptive mesh refinement,” J. Comput. Phys. 231, 3023–3040 (2012).
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P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230, 7364–7383 (2011).
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J. Electron. Imaging (1)

G. Abdoulaev and A. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imaging 12, 594–601 (2003).
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J. Opt. Soc. Am. A (2)

J. Quant. Spectrosc. Radiat. Transfer (7)

T. Tarvainen, M. Vauhkonen, and S. Arridge, “Gauss–Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 109, 2767–2778 (2008).
[CrossRef]

D. Gorpas, D. Yova, and K. Politopoulos, “A three-dimensional finite elements approach for the coupled radiative transfer equation and diffusion approximation modeling in fluorescence imaging,” J. Quant. Spectrosc. Radiat. Transfer 111, 553–568 (2010).
[CrossRef]

O. Balima, Y. Favennec, J. Boulanger, and A. Charette, “Optical tomography with the discontinuous Galerkin forumulation of the radiative transfer equation in frequency domain,” J. Quant. Spectrosc. Radiat. Transfer 113, 805–814 (2012).
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R. Koch and R. Becker, “Evaluation of quadrature schemes for the discrete ordinates method,” J. Quant. Spectrosc. Radiat. Transfer 84, 423–435 (2004).
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J. Boulanger and A. Charette, “Reconstruction optical spectroscopy using transient radiative transfer equation and pulsed laser: a numerical study,” J. Quant. Spectrosc. Radiat. Transfer 93, 325–336 (2005).
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H. K. Kim and A. Charette, “A sensitivity function-based conjugate gradient method for optical tomography with the frequency-domain equation of radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 104, 24–39 (2007).
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A. Klose, U. Netz, J. Beuthan, and A. Hielscher, “Optical tomography using the time-independent equation of radiative transfer—Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transfer 72, 691–713 (2002).
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Med. Phys. (3)

A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26, 1698–1707 (1999).
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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. Aydin, C. de Oliveira, and A. 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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Opt. Express (3)

Opt. Lett. (2)

Philos. Trans. R. Soc. A (1)

A. Gibson and H. Dehghani, “Diffuse optical imaging,” Philos. Trans. R. Soc. A 367, 3055–3072 (2009).
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Phys. Med. Biol. (5)

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, 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]

E. Boman, J. Tervo, and M. Vauhkonen, “Modelling the transport of ionizing radiation using the finite element method,” Phys. Med. Biol. 50, 265–280 (2005).
[CrossRef]

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, S. Arridge, and J. Kaipio, “Coupled radiative transfer equation and diffusion approximation model for photon migration in turbid medium with low-scattering and non-scattering regions,” Phys. Med. Biol. 50, 4913–4930 (2005).
[CrossRef]

Z. Yuan, X.-H. Hu, and H. Jiang, “A higher order diffusion model for three-dimensional photon migration and image reconstruction in optical tomography,” Phys. Med. Biol. 54, 65–88(2009).
[CrossRef]

Proc. SPIE (1)

S. Wright, M. Schweiger, and S. Arridge, “Solutions to the transport equation using variable order angular basis,” Proc. SPIE 5859, 585914 (2005).
[CrossRef]

Rev. Sci. Instrum. (2)

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[CrossRef]

F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, J. C. Hebden, and D. T. Delby, “A 32-channel time-resolved instrument for medical optical tomography,” Rev. Sci. Instrum. 71, 256–265 (2000).
[CrossRef]

SIAM J. Sci. Comput. (1)

K. Ren, G. Bal, and A. Hielscher, “Frequency domain optical tomography based on the equation of radiative transfer,” SIAM J. Sci. Comput. 28, 1463–1489 (2006).
[CrossRef]

Transp. Theory Stat. Phys. (1)

H. Gao and H. Zhao, “A fast-forward solver of radiative transfer equation,” Transp. Theory Stat. Phys. 38, 149–192 (2009).
[CrossRef]

Transp. Theory Stat.. Phys. (1)

H. Gao and H. Zhao, “A fast-forward solver of radiative transfer equation,” Transp. Theory Stat.. Phys. 38, 149–192 (2009).
[CrossRef]

Other (8)

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, SPIE Institute Series, Vol.5, (SPIE, 1989), pp. 102–111.

J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).

K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, 1967).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 1.

S. Chandrasekhar, Radiative Transfer (Oxford University, 1950).

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R. T. Ackroyd, Finite Element Methods for Particle Transport: Applications to Reactor and Radiation Physics (Research Studies, 1997).

H. Wabnitz, M. Moeller, A. Liebert, H. Obrig, J. Steinbrink, and R. Macdonald, “Time-resolved near-infrared spectroscopy and imaging of the adult human brain,” in Oxygen Transport to Tissue XXXI, E. Takahashi and D. Bruley, eds. (Springer, 2010), Vol. 662, pp. 143–148.

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Figures (6)

Fig. 1.
Fig. 1.

Geometry of the simulations with each material parameter domain and the light source marked. The layer defined by domain 2 is separated from the boundaries of the domain by 2 mm and has 2 mm thickness. Four circles indicate points for which temporal data will be shown.

Fig. 2.
Fig. 2.

Snapshots of fluence at t=50ps in different simulation geometries with different models. From left to right are TFSRTE, TDRTE, and TDMC. From top to bottom are homogeneous, high-scattering-layer, and low-scattering-layer geometries.

Fig. 3.
Fig. 3.

Time of flight of different simulation geometries with different models. From left to right are TFSRTE, TDRTE, and TDMC. From top to bottom are homogeneous, high-scattering-layer, and low-scattering-layer geometries.

Fig. 4.
Fig. 4.

From left to right: temporal fluence of the homogeneous, high-scattering-layer, and low-scattering-layer simulations. From top to bottom: fluence next to the light source, in the middle of the computational domain, on the opposite boundary from the source, and on top of the computational domain. The thick gray curve corresponds to TFSRTE, the dots to TDRTE, and the solid black curve to TDMC.

Fig. 5.
Fig. 5.

From left to right: relative error of TFSRTE and TDRTE with respect to MC and the relative error of TDRTE with respect to TFSRTE. From top to bottom: homogeneous, high-scattering-layer, and low-scattering-layer simulations.

Fig. 6.
Fig. 6.

Fluence as a function of time for TDRTE with various number of temporal integration steps and for TFSRTE in the homogeneous simulation case in the middle of the computational domain. TDRTE simulations are shown for 50 (dotted curve), 100 (dash–dotted curve), 200 (dashed curve), and 400 (black solid curve) temporal integration points. TFSRTE is shown with a thick gray curve.

Tables (2)

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Table 1. Absorption Coefficient μa and Scattering Coefficient μs for the Homogeneous, High-Scattering-Layer, and Low-Scattering-Layer Simulationsa

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Table 2. Computational Times and Matrix Memory Consumption for TFSRTE and TDRTEa

Equations (29)

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1cϕ(r,s^;t)t+s^·ϕ(r,s^;t)+(μs+μa)ϕ(r,s^;t)=μsSn1Θ(s^·s^)ϕ(r,s^;t)ds^,
ϕ(r,s^;t)s^·n^dSds^,
Θ(s^·s^)={12π1g2(1+g22gs^·s^),n=214π1g2(1+g22gs^·s^)3/2,n=3.
ϕ(r,s^;t)={ϕ0(r,s^;t),rϵj,s^·n^<00,rΩϵj,s^·n^<0,
f(t)=k=NωNωf(ωk)exp(iωkt),
f(ωk)=L(f(t))=1TTaTbf(t)exp(iωkt)dt,
iωkcϕ(r,s^;ωk)+s^·ϕ(r,s^;ωk)+(μs+μa)ϕ(r,s^;ωk)=μsSn1Θ(s^·s^)ϕ(r,s^;ωk)ds^
ϕ(r,s^;ωk)={ϕ0(r,s^)ϕ0(ωk),rϵj,s^·n^<00,rΩϵj,s^·n^<0.
ϕ(r,s^;t)=k=NωNωϕ(r,s^;ωk)exp(iωkt),
(iωkcA0+A1+A2+A3+A4)α(ωk)=bψ0,
A0(h,s)=Ωrteψi(r)ψj(r)drSn1ψ(s^)ψm(s^)ds^+ΩrteδSn1s^·ψj(r)ψm(s^)ψ(s^)ds^ψi(r)dr,
A1(h,s)=ΩrteSn1s^·ψj(r)ψm(s^)ψ(s^)ds^ψi(r)dr+ΩrteδSn1(s^·ψj(r)ψm(s^))(s^·ψi(r)ψ(s^))ds^dr,
A2(h,s)=Ωrteψi(r)ψj(r)dSSn1(s^·n^)+ψ(s^)ψm(s^)ds^,
A3(h,s)=Ωrte(μs+μa)ψi(r)ψj(r)drSn1ψ(s^)ψm(s^)ds^+Ωrteδ(μs+μa)Sn1s^·ψj(r)ψm(s^)ψ(s^)ds^ψi(r)dr,
A4(h,s)=Ωrteμsψi(r)ψj(r)drSn1Sn1Θ(s^·s^)ψ(s^)ds^ψm(s^)ds^ΩrteδμsSn1s^·ψj(r)ψm(s^)Sn1Θ(s^·s^)ψ(s^)ds^ds^ψi(r)dr,
b(h,s)=Ωrteψi(r)ψj(r)dSSn1(s^·n^)ψ(s^)ψm(s^)ds^,
ψ0(t)=exp(t22σ2),
Φ(r;t)=Sn1ϕ(r,s^;t)ds^.
TOF(r)=arg maxtΦ(r;t),
E(r)=Φ(r;t)ΦREF(r;t)ΦREF(r;t),
A01cα(t)t+(A1+A2+A3+A4)α(t)=bψ0(t),
α(t)t=αt+1αtΔt,
α(t)=αt+1+αt2
bψ0(t)=12b(ψt+1+ψt).
(2A0+cΔtB)αt+1=(2A0cΔtB)αt+cΔtb(ψt+10+ψt0),
Δs=lnξμs,ξUnif(]0,1[).
w=wexp(μad),
t=t+dc.
EjkEjkμaAN,

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