Abstract

A void region exists in some biological tissues, and previous studies have shown that inaccurate images would be obtained if it were not processed. A hybrid radiosity–diffusion method (HRDM) that couples the radiosity theory and the diffusion equation has been proposed to deal with the void problem and has been well demonstrated in two-dimensional and three-dimensional (3D) simple models. However, the extent of the impact of the void region on the accuracy of modeling light propagation has not been investigated. In this paper, we first implemented and verified the HRDM in 3D models, including both the regular geometries and a digital mouse model, and then investigated the influences of the void region on modeling light propagation in a heterogeneous medium. Our investigation results show that the influence of the region can be neglected when the size of the void is less than a certain range, and other cases must be taken into account.

© 2013 Optical Society of America

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  27. J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A. 17, 1671–1681 (2000).
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    [CrossRef]
  29. J. H. Lee, S. Kim, and Y. T. Kim, “Modeling of diffuse-diffuse photon coupling via a nonscattering region: a comparative study,” Appl. Opt. 43, 3640–3655 (2004).
    [CrossRef]
  30. H. Li, J. Tian, F. P. Zhu, W. X. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Acad. Radiol. 11, 1029–1038 (2004).
    [CrossRef]
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    [CrossRef]
  32. X. Chen, J. Liang, J. Liu, H. Hu, X. Qu, F. Wang, and Y. Nie, “Multi-modality molecular imaging for gastric cancer detection,” Proc. SPIE 8311, 831115 (2011).
    [CrossRef]
  33. X. Chen, D. Yang, X. Qu, H. Hu, J. Liang, X. Gao, and J. Tian, “Comparisons of hybrid radiosity-diffusion model and diffusion equation for bioluminescence tomography in cavity cancer detection,” J. Biomed. Opt. 17, 066015 (2012).
    [CrossRef]
  34. 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]
  35. W. X. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13, 6756–6771 (2005).
    [CrossRef]
  36. X. Chen, X. Gao, X. Qu, D. Chen, X. Ma, J. Liang, and J. Tian, “Generalized free-space diffuse photon transport model based on the influence analysis of a camera lens diaphragm,” Appl. Opt. 49, 5654–5664 (2010).
    [CrossRef]
  37. B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577–587 (2007).
    [CrossRef]
  38. G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50, 4225–4241 (2005).
    [CrossRef]

2012

X. Chen, D. Yang, X. Qu, H. Hu, J. Liang, X. Gao, and J. Tian, “Comparisons of hybrid radiosity-diffusion model and diffusion equation for bioluminescence tomography in cavity cancer detection,” J. Biomed. Opt. 17, 066015 (2012).
[CrossRef]

2011

2010

2009

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation,” Phys. Med. Biol. 54, 6477–6493 (2009).
[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]

2008

J. Tian, J. Bai, X. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27, 48–57 (2008).
[CrossRef]

2007

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

W. Cong, A. Cong, H. Shen, Y. Liu, and G. Wang, “Flux vector formulation for photon propagation in the biological tissue,” Opt. Lett. 32, 2837–2839 (2007).
[CrossRef]

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577–587 (2007).
[CrossRef]

2006

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Meth. Eng. 65, 383–405 (2006).
[CrossRef]

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

2005

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

V. Ntziachristos, J. Ripoll, L. H. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23, 313–320 (2005).
[CrossRef]

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202, 323–345 (2005).
[CrossRef]

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, S. R. Arridge, and J. P. 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]

Y. Ogoshi and E. Okada, “Analysis of light propagation in a realistic head model by a hybrid method for optical brain function measurement,” Opt. Rev. 12, 264–269 (2005).
[CrossRef]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50, 4225–4241 (2005).
[CrossRef]

W. X. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13, 6756–6771 (2005).
[CrossRef]

2004

J. H. Lee, S. Kim, and Y. T. Kim, “Modeling of diffuse-diffuse photon coupling via a nonscattering region: a comparative study,” Appl. Opt. 43, 3640–3655 (2004).
[CrossRef]

H. Li, J. Tian, F. P. Zhu, W. X. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Acad. Radiol. 11, 1029–1038 (2004).
[CrossRef]

2003

2002

H. Dehghani and D. T. Delpy, “Linear single-step image reconstruction in the presence of nonscattering regions,” J. Opt. Soc. Am. A. 19, 1162–1171 (2002).
[CrossRef]

2000

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A. 17, 1659–1670 (2000).
[CrossRef]

J. Riley, H. Dehghani, M. Schweiger, S. R. Arridge, J. Ripoll, and M. Nieto-Vesperinas, “3D optical tomography in the presence of void regions,” Opt. Express 7, 462–467 (2000).
[CrossRef]

J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A. 17, 1671–1681 (2000).

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, “The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef]

1999

H. Dehghani, D. T. Delpy, and S. R. Arridge, “Photon migration in non-scattering tissue and the effects on image reconstruction,” Phys. Med. Biol. 44, 2897–2906 (1999).
[CrossRef]

1997

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine. 2. modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef]

1996

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783(1996).
[CrossRef]

1995

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]

Alexandrakis, G.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50, 4225–4241 (2005).
[CrossRef]

Arridge, S. R.

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

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, S. R. Arridge, and J. P. 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]

J. Riley, H. Dehghani, M. Schweiger, S. R. Arridge, J. Ripoll, and M. Nieto-Vesperinas, “3D optical tomography in the presence of void regions,” Opt. Express 7, 462–467 (2000).
[CrossRef]

J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A. 17, 1671–1681 (2000).

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, “The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef]

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A. 17, 1659–1670 (2000).
[CrossRef]

H. Dehghani, D. T. Delpy, and S. R. Arridge, “Photon migration in non-scattering tissue and the effects on image reconstruction,” Phys. Med. Biol. 44, 2897–2906 (1999).
[CrossRef]

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine. 2. modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef]

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783(1996).
[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]

Bai, J.

J. Tian, J. Bai, X. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27, 48–57 (2008).
[CrossRef]

Bao, S.

J. Tian, J. Bai, X. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27, 48–57 (2008).
[CrossRef]

Chatziioannou, A. F.

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation,” Phys. Med. Biol. 54, 6477–6493 (2009).
[CrossRef]

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577–587 (2007).
[CrossRef]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50, 4225–4241 (2005).
[CrossRef]

Chen, D.

Chen, X.

X. Chen, D. Yang, X. Qu, H. Hu, J. Liang, X. Gao, and J. Tian, “Comparisons of hybrid radiosity-diffusion model and diffusion equation for bioluminescence tomography in cavity cancer detection,” J. Biomed. Opt. 17, 066015 (2012).
[CrossRef]

X. Chen, J. Liang, J. Liu, H. Hu, X. Qu, F. Wang, and Y. Nie, “Multi-modality molecular imaging for gastric cancer detection,” Proc. SPIE 8311, 831115 (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]

X. Chen, X. Gao, X. Qu, D. Chen, X. Ma, J. Liang, and J. Tian, “Generalized free-space diffuse photon transport model based on the influence analysis of a camera lens diaphragm,” Appl. Opt. 49, 5654–5664 (2010).
[CrossRef]

Cong, A.

Cong, W.

Cong, W. X.

W. X. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13, 6756–6771 (2005).
[CrossRef]

H. Li, J. Tian, F. P. Zhu, W. X. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Acad. Radiol. 11, 1029–1038 (2004).
[CrossRef]

Dehghani, H.

H. Dehghani and D. T. Delpy, “Linear single-step image reconstruction in the presence of nonscattering regions,” J. Opt. Soc. Am. A. 19, 1162–1171 (2002).
[CrossRef]

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, “The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef]

J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A. 17, 1671–1681 (2000).

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A. 17, 1659–1670 (2000).
[CrossRef]

J. Riley, H. Dehghani, M. Schweiger, S. R. Arridge, J. Ripoll, and M. Nieto-Vesperinas, “3D optical tomography in the presence of void regions,” Opt. Express 7, 462–467 (2000).
[CrossRef]

H. Dehghani, D. T. Delpy, and S. R. Arridge, “Photon migration in non-scattering tissue and the effects on image reconstruction,” Phys. Med. Biol. 44, 2897–2906 (1999).
[CrossRef]

Delpy, D. T.

H. Dehghani and D. T. Delpy, “Linear single-step image reconstruction in the presence of nonscattering regions,” J. Opt. Soc. Am. A. 19, 1162–1171 (2002).
[CrossRef]

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A. 17, 1659–1670 (2000).
[CrossRef]

H. Dehghani, D. T. Delpy, and S. R. Arridge, “Photon migration in non-scattering tissue and the effects on image reconstruction,” Phys. Med. Biol. 44, 2897–2906 (1999).
[CrossRef]

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783(1996).
[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]

Dogdas, B.

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577–587 (2007).
[CrossRef]

Douraghy, A.

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation,” Phys. Med. Biol. 54, 6477–6493 (2009).
[CrossRef]

Firbank, M.

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783(1996).
[CrossRef]

Gao, Q.

Gao, X.

Gibson, A. P.

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

Gorpas, 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).
[CrossRef]

Han, D.

Hayashi, T.

He, X.

Hebden, J. C.

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

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine. 2. modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef]

Herschman, H.

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation,” Phys. Med. Biol. 54, 6477–6493 (2009).
[CrossRef]

Hielscher, A. H.

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202, 323–345 (2005).
[CrossRef]

Hiraoka, M.

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]

Hoffman, E. A.

W. X. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13, 6756–6771 (2005).
[CrossRef]

H. Li, J. Tian, F. P. Zhu, W. X. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Acad. Radiol. 11, 1029–1038 (2004).
[CrossRef]

Hu, H.

X. Chen, D. Yang, X. Qu, H. Hu, J. Liang, X. Gao, and J. Tian, “Comparisons of hybrid radiosity-diffusion model and diffusion equation for bioluminescence tomography in cavity cancer detection,” J. Biomed. Opt. 17, 066015 (2012).
[CrossRef]

X. Chen, J. Liang, J. Liu, H. Hu, X. Qu, F. Wang, and Y. Nie, “Multi-modality molecular imaging for gastric cancer detection,” Proc. SPIE 8311, 831115 (2011).
[CrossRef]

Hu, X.-H.

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]

Ishimaru, A.

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

Jiang, H.

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]

Jiang, M.

Kaipio, J. P.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Meth. Eng. 65, 383–405 (2006).
[CrossRef]

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, S. R. Arridge, and J. P. 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]

Kashio, Y.

Kim, S.

Kim, Y. T.

Klose, A. D.

A. D. Klose, “The forward and inverse problem in tissue optics based on the radiative transfer equation: a brief review,” J. Quant. Spectrosc. Radiat. Transfer 111, 1852–1853 (2010).
[CrossRef]

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

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202, 323–345 (2005).
[CrossRef]

Kolehmainen, V.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Meth. Eng. 65, 383–405 (2006).
[CrossRef]

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, S. R. Arridge, and J. P. 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]

Kumar, D.

Larsen, E. W.

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

Leahy, R. M.

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577–587 (2007).
[CrossRef]

Lee, J. H.

Li, H.

H. Li, J. Tian, F. P. Zhu, W. X. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Acad. Radiol. 11, 1029–1038 (2004).
[CrossRef]

Li, J.

Li, Y.

J. Tian, J. Bai, X. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27, 48–57 (2008).
[CrossRef]

Liang, J.

Liang, W.

J. Tian, J. Bai, X. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27, 48–57 (2008).
[CrossRef]

Liu, J.

X. Chen, J. Liang, J. Liu, H. Hu, X. Qu, F. Wang, and Y. Nie, “Multi-modality molecular imaging for gastric cancer detection,” Proc. SPIE 8311, 831115 (2011).
[CrossRef]

Liu, K.

Liu, Y.

Lu, B.

Lu, Y.

K. Liu, Y. Lu, J. Tian, C. Qin, X. Yang, S. Zhu, X. Yang, Q. Gao, and D. Han, “Evaluation of the simplified spherical harmonics approximation in bioluminescence tomography through heterogeneous mouse models,” Opt. Express 18, 20988–21002 (2010).
[CrossRef]

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation,” Phys. Med. Biol. 54, 6477–6493 (2009).
[CrossRef]

Ma, X.

Machado, H. B.

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation,” Phys. Med. Biol. 54, 6477–6493 (2009).
[CrossRef]

McCray, P. B.

McLennan, G.

Nie, Y.

X. Chen, J. Liang, J. Liu, H. Hu, X. Qu, F. Wang, and Y. Nie, “Multi-modality molecular imaging for gastric cancer detection,” Proc. SPIE 8311, 831115 (2011).
[CrossRef]

Nieto-Vesperinas, M.

J. Riley, H. Dehghani, M. Schweiger, S. R. Arridge, J. Ripoll, and M. Nieto-Vesperinas, “3D optical tomography in the presence of void regions,” Opt. Express 7, 462–467 (2000).
[CrossRef]

J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A. 17, 1671–1681 (2000).

Ntziachristos, V.

V. Ntziachristos, J. Ripoll, L. H. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23, 313–320 (2005).
[CrossRef]

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202, 323–345 (2005).
[CrossRef]

R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nat. Med. 9, 123–128 (2003).
[CrossRef]

Ogoshi, Y.

Y. Ogoshi and E. Okada, “Analysis of light propagation in a realistic head model by a hybrid method for optical brain function measurement,” Opt. Rev. 12, 264–269 (2005).
[CrossRef]

Okada, E.

Y. Ogoshi and E. Okada, “Analysis of light propagation in a realistic head model by a hybrid method for optical brain function measurement,” Opt. Rev. 12, 264–269 (2005).
[CrossRef]

T. Hayashi, Y. Kashio, and E. Okada, “Hybrid Monte Carlo–diffusion method for light propagation in tissue with a low-scattering region,” Appl. Opt. 42, 2888–2896 (2003).
[CrossRef]

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, “The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef]

Peng, K.

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]

Qin, C.

Qu, X.

Rannou, F. R.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50, 4225–4241 (2005).
[CrossRef]

Ren, N.

Riley, J.

Ripoll, J.

V. Ntziachristos, J. Ripoll, L. H. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23, 313–320 (2005).
[CrossRef]

J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A. 17, 1671–1681 (2000).

J. Riley, H. Dehghani, M. Schweiger, S. R. Arridge, J. Ripoll, and M. Nieto-Vesperinas, “3D optical tomography in the presence of void regions,” Opt. Express 7, 462–467 (2000).
[CrossRef]

Schweiger, M.

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

J. Riley, H. Dehghani, M. Schweiger, S. R. Arridge, J. Ripoll, and M. Nieto-Vesperinas, “3D optical tomography in the presence of void regions,” Opt. Express 7, 462–467 (2000).
[CrossRef]

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A. 17, 1659–1670 (2000).
[CrossRef]

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, “The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef]

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783(1996).
[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]

Shen, H.

Stout, D.

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation,” Phys. Med. Biol. 54, 6477–6493 (2009).
[CrossRef]

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577–587 (2007).
[CrossRef]

Tarvainen, T.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Meth. Eng. 65, 383–405 (2006).
[CrossRef]

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, S. R. Arridge, and J. P. 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]

Tian, J.

X. Chen, D. Yang, X. Qu, H. Hu, J. Liang, X. Gao, and J. Tian, “Comparisons of hybrid radiosity-diffusion model and diffusion equation for bioluminescence tomography in cavity cancer detection,” J. Biomed. Opt. 17, 066015 (2012).
[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]

K. Liu, Y. Lu, J. Tian, C. Qin, X. Yang, S. Zhu, X. Yang, Q. Gao, and D. Han, “Evaluation of the simplified spherical harmonics approximation in bioluminescence tomography through heterogeneous mouse models,” Opt. Express 18, 20988–21002 (2010).
[CrossRef]

N. Ren, J. Liang, X. Qu, J. Li, B. Lu, and J. Tian, “GPU-based Monte Carlo simulation for light propagation in complex heterogeneous tissues,” Opt. Express 18, 6811–6823 (2010).
[CrossRef]

X. Chen, X. Gao, X. Qu, D. Chen, X. Ma, J. Liang, and J. Tian, “Generalized free-space diffuse photon transport model based on the influence analysis of a camera lens diaphragm,” Appl. Opt. 49, 5654–5664 (2010).
[CrossRef]

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation,” Phys. Med. Biol. 54, 6477–6493 (2009).
[CrossRef]

J. Tian, J. Bai, X. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27, 48–57 (2008).
[CrossRef]

H. Li, J. Tian, F. P. Zhu, W. X. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Acad. Radiol. 11, 1029–1038 (2004).
[CrossRef]

Vauhkonen, M.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Meth. Eng. 65, 383–405 (2006).
[CrossRef]

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, S. R. Arridge, and J. P. 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]

Wang, F.

X. Chen, J. Liang, J. Liu, H. Hu, X. Qu, F. Wang, and Y. Nie, “Multi-modality molecular imaging for gastric cancer detection,” Proc. SPIE 8311, 831115 (2011).
[CrossRef]

Wang, G.

Wang, L. H. V.

V. Ntziachristos, J. Ripoll, L. H. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23, 313–320 (2005).
[CrossRef]

Wang, L. V.

W. X. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13, 6756–6771 (2005).
[CrossRef]

H. Li, J. Tian, F. P. Zhu, W. X. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Acad. Radiol. 11, 1029–1038 (2004).
[CrossRef]

L. V. Wang and H.-I. Wu, Biomedical Optics: Principle and Imaging (Wiley, 2007).

Wang, X.

Weissleder, R.

V. Ntziachristos, J. Ripoll, L. H. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23, 313–320 (2005).
[CrossRef]

R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nat. Med. 9, 123–128 (2003).
[CrossRef]

Wright, S.

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

Wu, H.-I.

L. V. Wang and H.-I. Wu, Biomedical Optics: Principle and Imaging (Wiley, 2007).

Yan, X.

J. Tian, J. Bai, X. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27, 48–57 (2008).
[CrossRef]

Yang, D.

X. Chen, D. Yang, X. Qu, H. Hu, J. Liang, X. Gao, and J. Tian, “Comparisons of hybrid radiosity-diffusion model and diffusion equation for bioluminescence tomography in cavity cancer detection,” J. Biomed. Opt. 17, 066015 (2012).
[CrossRef]

Yang, X.

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).
[CrossRef]

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]

Zabner, J.

Zhu, F. P.

H. Li, J. Tian, F. P. Zhu, W. X. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Acad. Radiol. 11, 1029–1038 (2004).
[CrossRef]

Zhu, S.

Acad. Radiol.

H. Li, J. Tian, F. P. Zhu, W. X. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Acad. Radiol. 11, 1029–1038 (2004).
[CrossRef]

Appl. Opt.

IEEE Eng. Med. Biol. Mag.

J. Tian, J. Bai, X. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27, 48–57 (2008).
[CrossRef]

Int. J. Numer. Meth. Eng.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Meth. Eng. 65, 383–405 (2006).
[CrossRef]

J. Biomed. Opt.

X. Chen, D. Yang, X. Qu, H. Hu, J. Liang, X. Gao, and J. Tian, “Comparisons of hybrid radiosity-diffusion model and diffusion equation for bioluminescence tomography in cavity cancer detection,” J. Biomed. Opt. 17, 066015 (2012).
[CrossRef]

J. Comput. Phys.

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202, 323–345 (2005).
[CrossRef]

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

J. Opt. Soc. Am. A.

J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A. 17, 1671–1681 (2000).

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A. 17, 1659–1670 (2000).
[CrossRef]

H. Dehghani and D. T. Delpy, “Linear single-step image reconstruction in the presence of nonscattering regions,” J. Opt. Soc. Am. A. 19, 1162–1171 (2002).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

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]

A. D. Klose, “The forward and inverse problem in tissue optics based on the radiative transfer equation: a brief review,” J. Quant. Spectrosc. Radiat. Transfer 111, 1852–1853 (2010).
[CrossRef]

Meas. Sci. Technol.

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

Med. Phys.

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, “The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions,” Med. Phys. 27, 252–264 (2000).
[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]

Nat. Biotechnol.

V. Ntziachristos, J. Ripoll, L. H. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23, 313–320 (2005).
[CrossRef]

Nat. Med.

R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nat. Med. 9, 123–128 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Rev.

Y. Ogoshi and E. Okada, “Analysis of light propagation in a realistic head model by a hybrid method for optical brain function measurement,” Opt. Rev. 12, 264–269 (2005).
[CrossRef]

Phys. Med. Biol.

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783(1996).
[CrossRef]

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, S. R. Arridge, and J. P. 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]

H. Dehghani, D. T. Delpy, and S. R. Arridge, “Photon migration in non-scattering tissue and the effects on image reconstruction,” Phys. Med. Biol. 44, 2897–2906 (1999).
[CrossRef]

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation,” Phys. Med. Biol. 54, 6477–6493 (2009).
[CrossRef]

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine. 2. modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[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]

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577–587 (2007).
[CrossRef]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50, 4225–4241 (2005).
[CrossRef]

Proc. SPIE

X. Chen, J. Liang, J. Liu, H. Hu, X. Qu, F. Wang, and Y. Nie, “Multi-modality molecular imaging for gastric cancer detection,” Proc. SPIE 8311, 831115 (2011).
[CrossRef]

Other

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

L. V. Wang and H.-I. Wu, Biomedical Optics: Principle and Imaging (Wiley, 2007).

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Figures (7)

Fig. 1.
Fig. 1.

Experimental geometries employed in the comparisons: (a) is the comparison for the classical geometry in experiment No. 1; and (b) and (c) are those for the self-designed geometries used in experiments No. 2 and No. 3, respectively.

Fig. 2.
Fig. 2.

Comparisons of results between the calculated result of the HRDM model and the simulated one of MOSE. (a)–(c) Comparisons for the three kinds of geometries.

Fig. 3.
Fig. 3.

Digital mouse used in the complicated irregular-shape-geometry-based comparison experiment.

Fig. 4.
Fig. 4.

Comparisons of the digital-mouse-based experiment between the calculated results of the HRDM model and the simulated ones of MOSE. (a)–(c) Compared curves at a height of z=16.5, 18.5, and 21.5 mm, respectively.

Fig. 5.
Fig. 5.

Compared results among the MOSE simulation, the HRDM calculated, and the DE calculated when the size of the void region was varied. (a)–(f) Comparison results of different sizes of the void region.

Fig. 6.
Fig. 6.

Variation of evaluation factor ARE versus the absolute void sizes corresponding to the different sizes of the scattering region.

Fig. 7.
Fig. 7.

Variation of evaluation factors versus the absolute size of the void region. (a) Variation of ARE versus the void sizes corresponding to the original, the first, and the second additional experiment, which have different absorption coefficients, and (b) variation of ARE versus the absolute sizes of the void regions corresponding to the original, the third, and the fourth additional experiment, which have different scattering coefficients.

Tables (4)

Tables Icon

Table 1. Geometrical Parameters and the Related Optical Properties for the Three Different Regular Shape Geometries

Tables Icon

Table 2. Error Comparisons between the Calculated Results of the HRDM Model and the Simulated Ones of MOSE

Tables Icon

Table 3. Optical Properties of the Main Organs of the Digital Mouse

Tables Icon

Table 4. Compared Results of the ARE Values of MOSE versus the DE and of MOSE versus the HRDM

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

·(D(r)Φ(r))+μa(r)Φ(r)=S(r),Φ(r)+2An(r)D(r)(υ(r)·Φ(r))=0,S0(r)=BΦ(r)2πAn(r)G(r,r)dB,G(r,r)=ξ(r,r)cosθcosθ|rr|2exp(μa|rr|),
S(r)=q(r)+S0(r),
Φ(r)=i=1Nϕiφi(r)S(r)=i=1Nsiφi(r),
MΦ=FS,
kij=ΩD(r)(φi(r))·(φj(r))dr,cij=Ωμa(r)φi(r)φj(r)dr,bij=ΩΩ12An(r)φi(r)φj(r)dr,Σij=ΩΩ12πAn(r)G(r,r)φi(r)φj(r)dSdS,fij=Ωφi(r)φj(r)dr,
ARE=i=1N|f^iMOSEf^iHRDM|N,CS=1i=1N(f^iMOSEf^iHRDM)2i=1N(f^iMOSEf^iMOSE¯)2,
ARE=i=1N|fifiMOSE|Nmax(fiMOSE),

Metrics