Abstract

The use of higher order approximations to the Radiative transport equation, through simplified spherical harmonics expansion (SPN) in optical tomography are presented. It is shown that, although the anisotropy factor can be modeled in the forward problem, its sensitivity to the measured boundary data is limited to superficial regions and more importantly, due to uniqueness of the inverse problem it cannot be determined using frequency domain data. Image reconstruction through the use of higher ordered models is presented. It is demonstrated that at higher orders (for example SP7) the image reconstruction becomes highly under-determined due to the large increase in the number of unknowns which cannot be adequately recovered. However, reconstruction of diffuse parameters, namely optical absorption and reduced scatter have shown to be more accurate where only the sensitivity matrix used in the inverse problem is based on SPN method and image reconstruction is limited to these two diffuse parameters.

© 2009 OSA

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2009 (1)

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(8), 2493–2509 (2009).
[CrossRef] [PubMed]

2008 (3)

D. R. Leff, O. J. Warren, L. C. Enfield, A. Gibson, T. Athanasiou, D. K. Patten, J. Hebden, G. Z. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: a systematic review,” Breast Cancer Res. Treat. 108(1), 9–22 (2008).
[CrossRef] [PubMed]

H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys. 35(11), 4863–4871 (2008).
[CrossRef] [PubMed]

M. E. Eames and H. Dehghani, “Wavelength dependence of sensitivity in spectral diffuse optical imaging: effect of normalization on image reconstruction,” Opt. Express 16(22), 17780–17791 (2008).
[CrossRef] [PubMed]

2007 (4)

L. C. Enfield, A. P. Gibson, N. L. Everdell, D. T. Delpy, M. Schweiger, S. R. Arridge, C. Richardson, M. Keshtgar, M. Douek, and J. C. Hebden, “Three-dimensional time-resolved optical mammography of the uncompressed breast,” Appl. Opt. 46(17), 3628–3638 (2007).
[CrossRef] [PubMed]

C. Kuo, O. Coquoz, T. L. Troy, H. Xu, and B. W. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” JBO 12, 24007 (2007).

S. Wright, M. Schweiger, and S. R. Arridge, “Reconstruction in optical tomography using the P-N approximations,” Meas. Sci. Technol. 18(1), 79–86 (2007).
[CrossRef]

B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. U.S.A. 104(29), 12169–12174 (2007).
[CrossRef] [PubMed]

2006 (3)

V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8(1), 1–33 (2006).
[CrossRef] [PubMed]

A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441–470 (2006).
[CrossRef]

G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, “In vivo mouse studies with bioluminescence tomography,” Opt. Express 14(17), 7801–7809 (2006).
[CrossRef] [PubMed]

2005 (3)

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]

R. Choe, A. Corlu, K. Lee, T. Durduran, S. D. Konecky, M. Grosicka-Koptyra, S. R. Arridge, B. J. Czerniecki, D. L. Fraker, A. DeMichele, B. Chance, M. A. Rosen, and A. G. Yodh, “Diffuse optical tomography of breast cancer during neoadjuvant chemotherapy: a case study with comparison to MRI,” Med. Phys. 32(4), 1128–1139 (2005).
[CrossRef] [PubMed]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50(4), R1–R43 (2005).
[CrossRef] [PubMed]

2004 (2)

D. A. Boas, K. Chen, D. Grebert, and M. A. Franceschini, “Improving the diffuse optical imaging spatial resolution of the cerebral hemodynamic response to brain activation in humans,” Opt. Lett. 29(13), 1506–1508 (2004).
[CrossRef] [PubMed]

J. C. Hebden, A. Gibson, T. Austin, R. M. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek, and J. S. Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49(7), 1117–1130 (2004).
[CrossRef] [PubMed]

2003 (4)

2002 (3)

E. D. Aydin, C. R. E. de Oliveira, and A. J. H. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Med. Phys. 29(9), 2013–2023 (2002).
[CrossRef] [PubMed]

A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer — Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transf. 72(5), 691–713 (2002).
[CrossRef]

C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. 4(1), 235–260 (2002).
[CrossRef] [PubMed]

2001 (3)

H. Jiang, Y. Xu, N. Iftimia, J. Eggert, K. Klove, L. Baron, and L. Fajardo, “Three-dimensional optical tomographic imaging of breast in a human subject,” IEEE Trans. Med. Img. 20(12), 1334–1340 (2001).
[CrossRef]

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(6), 57–75 (2001).
[CrossRef]

A. Y. Bluestone, G. Abdoulaev, C. Schmitz, R. L. Barbour, and A. H. Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Express 9(6), 272–286 (2001).
[CrossRef] [PubMed]

2000 (5)

J. Riley, H. Dehghani, M. Schweiger, S. Arridge, J. Ripoll, and M. Nieto-Vesperinas, “3D optical tomography in the presence of void regions,” Opt. Express 7(13), 462–467 (2000).
[CrossRef] [PubMed]

S. Nickell, M. Hermann, M. Essenpreis, T. J. Farrell, U. Krämer, and M. S. Patterson, “Anisotropy of light propagation in human skin,” Phys. Med. Biol. 45(10), 2873–2886 (2000).
[CrossRef] [PubMed]

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(9), 1671–1681 (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(1), 252–264 (2000).
[CrossRef] [PubMed]

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. 17(9), 1659–1670 (2000).
[CrossRef]

1999 (3)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999).
[CrossRef]

A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imaging 18(3), 262–271 (1999).
[CrossRef] [PubMed]

H. B. Jiang, “Optical image reconstruction based on the third-order diffusion equations,” Opt. Express 4(8), 241–246 (1999).
[CrossRef] [PubMed]

1998 (3)

1997 (1)

E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, and C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66(1), 55–64 (1997).
[CrossRef] [PubMed]

1995 (1)

Abdoulaev, G.

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(5), 1285–1302 (1998).
[CrossRef] [PubMed]

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.

Arridge, S. R.

S. Wright, M. Schweiger, and S. R. Arridge, “Reconstruction in optical tomography using the P-N approximations,” Meas. Sci. Technol. 18(1), 79–86 (2007).
[CrossRef]

L. C. Enfield, A. P. Gibson, N. L. Everdell, D. T. Delpy, M. Schweiger, S. R. Arridge, C. Richardson, M. Keshtgar, M. Douek, and J. C. Hebden, “Three-dimensional time-resolved optical mammography of the uncompressed breast,” Appl. Opt. 46(17), 3628–3638 (2007).
[CrossRef] [PubMed]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50(4), R1–R43 (2005).
[CrossRef] [PubMed]

R. Choe, A. Corlu, K. Lee, T. Durduran, S. D. Konecky, M. Grosicka-Koptyra, S. R. Arridge, B. J. Czerniecki, D. L. Fraker, A. DeMichele, B. Chance, M. A. Rosen, and A. G. Yodh, “Diffuse optical tomography of breast cancer during neoadjuvant chemotherapy: a case study with comparison to MRI,” Med. Phys. 32(4), 1128–1139 (2005).
[CrossRef] [PubMed]

J. C. Hebden, A. Gibson, T. Austin, R. M. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek, and J. S. Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49(7), 1117–1130 (2004).
[CrossRef] [PubMed]

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. 17(9), 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(1), 252–264 (2000).
[CrossRef] [PubMed]

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(9), 1671–1681 (2000).
[CrossRef]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999).
[CrossRef]

S. R. Arridge and W. R. B. Lionheart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett. 23(11), 882–884 (1998).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, “Photon-measurement density functions. Part2: Finite-element-method calculations,” Appl. Opt. 34(34), 8026–8037 (1995).
[CrossRef] [PubMed]

Athanasiou, T.

D. R. Leff, O. J. Warren, L. C. Enfield, A. Gibson, T. Athanasiou, D. K. Patten, J. Hebden, G. Z. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: a systematic review,” Breast Cancer Res. Treat. 108(1), 9–22 (2008).
[CrossRef] [PubMed]

Austin, T.

J. C. Hebden, A. Gibson, T. Austin, R. M. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek, and J. S. Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49(7), 1117–1130 (2004).
[CrossRef] [PubMed]

Aydin, E. D.

E. D. Aydin, C. R. E. de Oliveira, and A. J. H. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Med. Phys. 29(9), 2013–2023 (2002).
[CrossRef] [PubMed]

Bachmann, M. H.

C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. 4(1), 235–260 (2002).
[CrossRef] [PubMed]

Barbour, R. L.

A. Y. Bluestone, G. Abdoulaev, C. Schmitz, R. L. Barbour, and A. H. Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Express 9(6), 272–286 (2001).
[CrossRef] [PubMed]

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(5), 1285–1302 (1998).
[CrossRef] [PubMed]

Baron, L.

H. Jiang, Y. Xu, N. Iftimia, J. Eggert, K. Klove, L. Baron, and L. Fajardo, “Three-dimensional optical tomographic imaging of breast in a human subject,” IEEE Trans. Med. Img. 20(12), 1334–1340 (2001).
[CrossRef]

Beuthan, J.

A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer — Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transf. 72(5), 691–713 (2002).
[CrossRef]

Bluestone, A. Y.

Boas, D. A.

Brooks, D. H.

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(6), 57–75 (2001).
[CrossRef]

Brooksby, B.

Chance, B.

R. Choe, A. Corlu, K. Lee, T. Durduran, S. D. Konecky, M. Grosicka-Koptyra, S. R. Arridge, B. J. Czerniecki, D. L. Fraker, A. DeMichele, B. Chance, M. A. Rosen, and A. G. Yodh, “Diffuse optical tomography of breast cancer during neoadjuvant chemotherapy: a case study with comparison to MRI,” Med. Phys. 32(4), 1128–1139 (2005).
[CrossRef] [PubMed]

Chatziioannou, A. F.

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, K.

Choe, R.

R. Choe, A. Corlu, K. Lee, T. Durduran, S. D. Konecky, M. Grosicka-Koptyra, S. R. Arridge, B. J. Czerniecki, D. L. Fraker, A. DeMichele, B. Chance, M. A. Rosen, and A. G. Yodh, “Diffuse optical tomography of breast cancer during neoadjuvant chemotherapy: a case study with comparison to MRI,” Med. Phys. 32(4), 1128–1139 (2005).
[CrossRef] [PubMed]

Chu, M.

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(8), 2493–2509 (2009).
[CrossRef] [PubMed]

Cong, W.

Contag, C. H.

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R. Choe, A. Corlu, K. Lee, T. Durduran, S. D. Konecky, M. Grosicka-Koptyra, S. R. Arridge, B. J. Czerniecki, D. L. Fraker, A. DeMichele, B. Chance, M. A. Rosen, and A. G. Yodh, “Diffuse optical tomography of breast cancer during neoadjuvant chemotherapy: a case study with comparison to MRI,” Med. Phys. 32(4), 1128–1139 (2005).
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D. R. Leff, O. J. Warren, L. C. Enfield, A. Gibson, T. Athanasiou, D. K. Patten, J. Hebden, G. Z. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: a systematic review,” Breast Cancer Res. Treat. 108(1), 9–22 (2008).
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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(8), 2493–2509 (2009).
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H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys. 35(11), 4863–4871 (2008).
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J. Riley, H. Dehghani, M. Schweiger, S. Arridge, J. Ripoll, and M. Nieto-Vesperinas, “3D optical tomography in the presence of void regions,” Opt. Express 7(13), 462–467 (2000).
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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(9), 1671–1681 (2000).
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H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. 17(9), 1659–1670 (2000).
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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(1), 252–264 (2000).
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L. C. Enfield, A. P. Gibson, N. L. Everdell, D. T. Delpy, M. Schweiger, S. R. Arridge, C. Richardson, M. Keshtgar, M. Douek, and J. C. Hebden, “Three-dimensional time-resolved optical mammography of the uncompressed breast,” Appl. Opt. 46(17), 3628–3638 (2007).
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J. C. Hebden, A. Gibson, T. Austin, R. M. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek, and J. S. Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49(7), 1117–1130 (2004).
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H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. 17(9), 1659–1670 (2000).
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R. Choe, A. Corlu, K. Lee, T. Durduran, S. D. Konecky, M. Grosicka-Koptyra, S. R. Arridge, B. J. Czerniecki, D. L. Fraker, A. DeMichele, B. Chance, M. A. Rosen, and A. G. Yodh, “Diffuse optical tomography of breast cancer during neoadjuvant chemotherapy: a case study with comparison to MRI,” Med. Phys. 32(4), 1128–1139 (2005).
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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(6), 57–75 (2001).
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Douek, M.

Durairaj, K.

Durduran, T.

R. Choe, A. Corlu, K. Lee, T. Durduran, S. D. Konecky, M. Grosicka-Koptyra, S. R. Arridge, B. J. Czerniecki, D. L. Fraker, A. DeMichele, B. Chance, M. A. Rosen, and A. G. Yodh, “Diffuse optical tomography of breast cancer during neoadjuvant chemotherapy: a case study with comparison to MRI,” Med. Phys. 32(4), 1128–1139 (2005).
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Eames, M. E.

Eggert, J.

H. Jiang, Y. Xu, N. Iftimia, J. Eggert, K. Klove, L. Baron, and L. Fajardo, “Three-dimensional optical tomographic imaging of breast in a human subject,” IEEE Trans. Med. Img. 20(12), 1334–1340 (2001).
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D. R. Leff, O. J. Warren, L. C. Enfield, A. Gibson, T. Athanasiou, D. K. Patten, J. Hebden, G. Z. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: a systematic review,” Breast Cancer Res. Treat. 108(1), 9–22 (2008).
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J. C. Hebden, A. Gibson, T. Austin, R. M. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek, and J. S. Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49(7), 1117–1130 (2004).
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Fajardo, L.

H. Jiang, Y. Xu, N. Iftimia, J. Eggert, K. Klove, L. Baron, and L. Fajardo, “Three-dimensional optical tomographic imaging of breast in a human subject,” IEEE Trans. Med. Img. 20(12), 1334–1340 (2001).
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S. Nickell, M. Hermann, M. Essenpreis, T. J. Farrell, U. Krämer, and M. S. Patterson, “Anisotropy of light propagation in human skin,” Phys. Med. Biol. 45(10), 2873–2886 (2000).
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R. Choe, A. Corlu, K. Lee, T. Durduran, S. D. Konecky, M. Grosicka-Koptyra, S. R. Arridge, B. J. Czerniecki, D. L. Fraker, A. DeMichele, B. Chance, M. A. Rosen, and A. G. Yodh, “Diffuse optical tomography of breast cancer during neoadjuvant chemotherapy: a case study with comparison to MRI,” Med. Phys. 32(4), 1128–1139 (2005).
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Gaudette, R. J.

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(6), 57–75 (2001).
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J. C. Hebden, A. Gibson, T. Austin, R. M. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek, and J. S. Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49(7), 1117–1130 (2004).
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Goddard, A. J. H.

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D. R. Leff, O. J. Warren, L. C. Enfield, A. Gibson, T. Athanasiou, D. K. Patten, J. Hebden, G. Z. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: a systematic review,” Breast Cancer Res. Treat. 108(1), 9–22 (2008).
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L. C. Enfield, A. P. Gibson, N. L. Everdell, D. T. Delpy, M. Schweiger, S. R. Arridge, C. Richardson, M. Keshtgar, M. Douek, and J. C. Hebden, “Three-dimensional time-resolved optical mammography of the uncompressed breast,” Appl. Opt. 46(17), 3628–3638 (2007).
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J. C. Hebden, A. Gibson, T. Austin, R. M. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek, and J. S. Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49(7), 1117–1130 (2004).
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Hermann, M.

S. Nickell, M. Hermann, M. Essenpreis, T. J. Farrell, U. Krämer, and M. S. Patterson, “Anisotropy of light propagation in human skin,” Phys. Med. Biol. 45(10), 2873–2886 (2000).
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A. D. Klose and A. H. Hielscher, “Fluorescence tomography with simulated data based on the equation of radiative transfer,” Opt. Lett. 28(12), 1019–1021 (2003).
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E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, and C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66(1), 55–64 (1997).
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Iftimia, N.

H. Jiang, Y. Xu, N. Iftimia, J. Eggert, K. Klove, L. Baron, and L. Fajardo, “Three-dimensional optical tomographic imaging of breast in a human subject,” IEEE Trans. Med. Img. 20(12), 1334–1340 (2001).
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Jiang, H.

H. Jiang, Y. Xu, N. Iftimia, J. Eggert, K. Klove, L. Baron, and L. Fajardo, “Three-dimensional optical tomographic imaging of breast in a human subject,” IEEE Trans. Med. Img. 20(12), 1334–1340 (2001).
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Jiang, H. B.

Keshtgar, M.

Kilmer, M.

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(6), 57–75 (2001).
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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(8), 2493–2509 (2009).
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A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441–470 (2006).
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A. D. Klose and A. H. Hielscher, “Fluorescence tomography with simulated data based on the equation of radiative transfer,” Opt. Lett. 28(12), 1019–1021 (2003).
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A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer — Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transf. 72(5), 691–713 (2002).
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A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imaging 18(3), 262–271 (1999).
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H. Jiang, Y. Xu, N. Iftimia, J. Eggert, K. Klove, L. Baron, and L. Fajardo, “Three-dimensional optical tomographic imaging of breast in a human subject,” IEEE Trans. Med. Img. 20(12), 1334–1340 (2001).
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R. Choe, A. Corlu, K. Lee, T. Durduran, S. D. Konecky, M. Grosicka-Koptyra, S. R. Arridge, B. J. Czerniecki, D. L. Fraker, A. DeMichele, B. Chance, M. A. Rosen, and A. G. Yodh, “Diffuse optical tomography of breast cancer during neoadjuvant chemotherapy: a case study with comparison to MRI,” Med. Phys. 32(4), 1128–1139 (2005).
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S. Nickell, M. Hermann, M. Essenpreis, T. J. Farrell, U. Krämer, and M. S. Patterson, “Anisotropy of light propagation in human skin,” Phys. Med. Biol. 45(10), 2873–2886 (2000).
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C. Kuo, O. Coquoz, T. L. Troy, H. Xu, and B. W. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” JBO 12, 24007 (2007).

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(1), 441–470 (2006).
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Lee, K.

R. Choe, A. Corlu, K. Lee, T. Durduran, S. D. Konecky, M. Grosicka-Koptyra, S. R. Arridge, B. J. Czerniecki, D. L. Fraker, A. DeMichele, B. Chance, M. A. Rosen, and A. G. Yodh, “Diffuse optical tomography of breast cancer during neoadjuvant chemotherapy: a case study with comparison to MRI,” Med. Phys. 32(4), 1128–1139 (2005).
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D. R. Leff, O. J. Warren, L. C. Enfield, A. Gibson, T. Athanasiou, D. K. Patten, J. Hebden, G. Z. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: a systematic review,” Breast Cancer Res. Treat. 108(1), 9–22 (2008).
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Lin, S. P.

Lionheart, W. R. B.

Lopez, G.

E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, and C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66(1), 55–64 (1997).
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McLennan, G.

Meek, J. H.

J. C. Hebden, A. Gibson, T. Austin, R. M. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek, and J. S. Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49(7), 1117–1130 (2004).
[CrossRef] [PubMed]

Miller, E. L.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75 (2001).
[CrossRef]

Netz, U.

A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer — Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transf. 72(5), 691–713 (2002).
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Nickell, S.

S. Nickell, M. Hermann, M. Essenpreis, T. J. Farrell, U. Krämer, and M. S. Patterson, “Anisotropy of light propagation in human skin,” Phys. Med. Biol. 45(10), 2873–2886 (2000).
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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(1), 252–264 (2000).
[CrossRef] [PubMed]

Patten, D. K.

D. R. Leff, O. J. Warren, L. C. Enfield, A. Gibson, T. Athanasiou, D. K. Patten, J. Hebden, G. Z. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: a systematic review,” Breast Cancer Res. Treat. 108(1), 9–22 (2008).
[CrossRef] [PubMed]

Patterson, M. S.

S. Nickell, M. Hermann, M. Essenpreis, T. J. Farrell, U. Krämer, and M. S. Patterson, “Anisotropy of light propagation in human skin,” Phys. Med. Biol. 45(10), 2873–2886 (2000).
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Paulsen, K. D.

Pogue, B. W.

Poplack, S. P.

Qian, 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).
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E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, and C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66(1), 55–64 (1997).
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Rice, B. W.

C. Kuo, O. Coquoz, T. L. Troy, H. Xu, and B. W. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” JBO 12, 24007 (2007).

Richardson, C.

Riley, J.

Ripoll, J.

Rosen, M. A.

R. Choe, A. Corlu, K. Lee, T. Durduran, S. D. Konecky, M. Grosicka-Koptyra, S. R. Arridge, B. J. Czerniecki, D. L. Fraker, A. DeMichele, B. Chance, M. A. Rosen, and A. G. Yodh, “Diffuse optical tomography of breast cancer during neoadjuvant chemotherapy: a case study with comparison to MRI,” Med. Phys. 32(4), 1128–1139 (2005).
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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. U.S.A. 104(29), 12169–12174 (2007).
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Schmitz, C.

Schwartz, J. A.

Schweiger, M.

L. C. Enfield, A. P. Gibson, N. L. Everdell, D. T. Delpy, M. Schweiger, S. R. Arridge, C. Richardson, M. Keshtgar, M. Douek, and J. C. Hebden, “Three-dimensional time-resolved optical mammography of the uncompressed breast,” Appl. Opt. 46(17), 3628–3638 (2007).
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J. Riley, H. Dehghani, M. Schweiger, S. Arridge, J. Ripoll, and M. Nieto-Vesperinas, “3D optical tomography in the presence of void regions,” Opt. Express 7(13), 462–467 (2000).
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H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. 17(9), 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(1), 252–264 (2000).
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Shen, H.

Shudong, J.

Siegel, A. M.

Sinn, P.

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Thomsen, S. L.

Troy, T. L.

C. Kuo, O. Coquoz, T. L. Troy, H. Xu, and B. W. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” JBO 12, 24007 (2007).

E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, and C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66(1), 55–64 (1997).
[CrossRef] [PubMed]

Vishwanath, K.

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol. 54(8), 2493–2509 (2009).
[CrossRef] [PubMed]

Wang, G.

Wang, L. V.

Warren, O. J.

D. R. Leff, O. J. Warren, L. C. Enfield, A. Gibson, T. Athanasiou, D. K. Patten, J. Hebden, G. Z. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: a systematic review,” Breast Cancer Res. Treat. 108(1), 9–22 (2008).
[CrossRef] [PubMed]

White, B. R.

B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. U.S.A. 104(29), 12169–12174 (2007).
[CrossRef] [PubMed]

Wright, S.

S. Wright, M. Schweiger, and S. R. Arridge, “Reconstruction in optical tomography using the P-N approximations,” Meas. Sci. Technol. 18(1), 79–86 (2007).
[CrossRef]

Wyatt, J. S.

J. C. Hebden, A. Gibson, T. Austin, R. M. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek, and J. S. Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49(7), 1117–1130 (2004).
[CrossRef] [PubMed]

Xu, H.

C. Kuo, O. Coquoz, T. L. Troy, H. Xu, and B. W. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” JBO 12, 24007 (2007).

Xu, Y.

H. Jiang, Y. Xu, N. Iftimia, J. Eggert, K. Klove, L. Baron, and L. Fajardo, “Three-dimensional optical tomographic imaging of breast in a human subject,” IEEE Trans. Med. Img. 20(12), 1334–1340 (2001).
[CrossRef]

Yang, G. Z.

D. R. Leff, O. J. Warren, L. C. Enfield, A. Gibson, T. Athanasiou, D. K. Patten, J. Hebden, G. Z. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: a systematic review,” Breast Cancer Res. Treat. 108(1), 9–22 (2008).
[CrossRef] [PubMed]

Yodh, A. G.

R. Choe, A. Corlu, K. Lee, T. Durduran, S. D. Konecky, M. Grosicka-Koptyra, S. R. Arridge, B. J. Czerniecki, D. L. Fraker, A. DeMichele, B. Chance, M. A. Rosen, and A. G. Yodh, “Diffuse optical tomography of breast cancer during neoadjuvant chemotherapy: a case study with comparison to MRI,” Med. Phys. 32(4), 1128–1139 (2005).
[CrossRef] [PubMed]

Yusof, R. M.

J. C. Hebden, A. Gibson, T. Austin, R. M. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek, and J. S. Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49(7), 1117–1130 (2004).
[CrossRef] [PubMed]

Zeff, B. W.

B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. U.S.A. 104(29), 12169–12174 (2007).
[CrossRef] [PubMed]

Zhang, Q.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75 (2001).
[CrossRef]

Annu. Rev. Biomed. Eng. (2)

V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8(1), 1–33 (2006).
[CrossRef] [PubMed]

C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. 4(1), 235–260 (2002).
[CrossRef] [PubMed]

Appl. Opt. (5)

Breast Cancer Res. Treat. (1)

D. R. Leff, O. J. Warren, L. C. Enfield, A. Gibson, T. Athanasiou, D. K. Patten, J. Hebden, G. Z. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: a systematic review,” Breast Cancer Res. Treat. 108(1), 9–22 (2008).
[CrossRef] [PubMed]

IEEE Signal Process. Mag. (1)

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75 (2001).
[CrossRef]

IEEE Trans. Med. Imaging (1)

A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imaging 18(3), 262–271 (1999).
[CrossRef] [PubMed]

IEEE Trans. Med. Img. (1)

H. Jiang, Y. Xu, N. Iftimia, J. Eggert, K. Klove, L. Baron, and L. Fajardo, “Three-dimensional optical tomographic imaging of breast in a human subject,” IEEE Trans. Med. Img. 20(12), 1334–1340 (2001).
[CrossRef]

Inverse Probl. (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999).
[CrossRef]

J. Comput. Phys. (1)

A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441–470 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. 17(9), 1659–1670 (2000).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transf. (1)

A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer — Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transf. 72(5), 691–713 (2002).
[CrossRef]

JBO (1)

C. Kuo, O. Coquoz, T. L. Troy, H. Xu, and B. W. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” JBO 12, 24007 (2007).

Meas. Sci. Technol. (1)

S. Wright, M. Schweiger, and S. R. Arridge, “Reconstruction in optical tomography using the P-N approximations,” Meas. Sci. Technol. 18(1), 79–86 (2007).
[CrossRef]

Med. Phys. (4)

E. D. Aydin, C. R. E. de Oliveira, and A. J. H. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Med. Phys. 29(9), 2013–2023 (2002).
[CrossRef] [PubMed]

H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys. 35(11), 4863–4871 (2008).
[CrossRef] [PubMed]

R. Choe, A. Corlu, K. Lee, T. Durduran, S. D. Konecky, M. Grosicka-Koptyra, S. R. Arridge, B. J. Czerniecki, D. L. Fraker, A. DeMichele, B. Chance, M. A. Rosen, and A. G. Yodh, “Diffuse optical tomography of breast cancer during neoadjuvant chemotherapy: a case study with comparison to MRI,” Med. Phys. 32(4), 1128–1139 (2005).
[CrossRef] [PubMed]

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(1), 252–264 (2000).
[CrossRef] [PubMed]

Opt. Express (5)

Opt. Lett. (4)

Photochem. Photobiol. (1)

E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, and C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66(1), 55–64 (1997).
[CrossRef] [PubMed]

Phys. Med. Biol. (6)

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]

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(5), 1285–1302 (1998).
[CrossRef] [PubMed]

J. C. Hebden, A. Gibson, T. Austin, R. M. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek, and J. S. Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49(7), 1117–1130 (2004).
[CrossRef] [PubMed]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50(4), R1–R43 (2005).
[CrossRef] [PubMed]

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(8), 2493–2509 (2009).
[CrossRef] [PubMed]

S. Nickell, M. Hermann, M. Essenpreis, T. J. Farrell, U. Krämer, and M. S. Patterson, “Anisotropy of light propagation in human skin,” Phys. Med. Biol. 45(10), 2873–2886 (2000).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. U.S.A. 104(29), 12169–12174 (2007).
[CrossRef] [PubMed]

Other (3)

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

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near Infrared Optical Tomography using NIRFAST: Algorithms for Numerical Model and Image Reconstruction Algorithms,” Communications in Numerical Methods in Engineering (2008).

S. Cahandrasekhar, Radiative Transfer (Clarendon Press, London, 1950).

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Figures (12)

Fig. 1
Fig. 1

FEM mesh, of 43mm radius, containing 1785 nodes and 3418 linear triangular elements. Circles represent location of sources, crosses represent location of detectors.

Fig. 2
Fig. 2

Maps of sensitivity of log Amplitude data to changes in a) absorption, b) scattering and c) anisotropy

Fig. 3
Fig. 3

Maps of sensitivity of phase data to changes in a) absorption, b) scattering and c) anisotropy

Fig. 4
Fig. 4

Circular mesh with a single inclusion of 10mm radius used to test uniqueness of boundary measurements.

Fig. 5
Fig. 5

Error maps of SP7 data with varying optical properties with arbitrary units of error. (a) map of log(Amplitude) with varying µa (y axis) and µs (x axis), (b) same as (a) but for phase and (c) error map of combined log(Amplitude) and Phase data.

Fig. 6
Fig. 6

Same as Fig. 5, but for (a) map of log(Amplitude) data with varying g and µa, (b) same as (a) but for phase, (c) sum of (a) and (b)

Fig. 7
Fig. 7

Same as Fig. 6, but for (a) map of log(Amplitude) data with varying g and µs, (b) same as (a) but for phase, (c) sum of (a) and (b)

Fig. 8
Fig. 8

Maps of residuals between (a) μa1, (b) μa2, (c) μa3 and (d) μa4 for a range of scattering coefficients and anisotropy factors as compared to a set of reference values. The range of anisotropy factors is listed on the x-axes whilst the y-axes represent the range of scattering coefficients.

Fig. 9
Fig. 9

FEM model with one highly absorbing target and one highly scattering target.

Fig. 10
Fig. 10

Reconstructed optical maps using a) SP1, b) SP3, c) SP5, d) SP7 based reconstruction algorithms

Fig. 11
Fig. 11

Recovered optical map generated using SP5 reconstruction with prior information.

Fig. 12
Fig. 12

Diffusion based images from SP5 data using (a) SP5 based Jacobian and (b) SP1 based Jacobian.

Tables (1)

Tables Icon

Table 1 Reconstructed values of SPN image reconstruction algorithms where the diffusion terms κn = 1/(Aμan) where A is a constant.

Equations (14)

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

. 1 3 μ a 1 ϕ 1 + μ a ϕ 1 = Q + 2 3 μ a ϕ 2 8 15 μ a ϕ 3 + 16 35 μ a ϕ 4
. 1 7 μ a 3 ϕ 2 + ( 4 9 μ a + 5 9 μ a 2 ) ϕ 2 = 2 3 Q + 2 3 μ a ϕ 1 + ( 16 45 μ a + 4 9 μ a 2 ) ϕ 3 ( 32 105 μ a + 8 21 μ a 2 ) ϕ 4
. 1 11 μ a 5 ϕ 3 + ( 64 225 μ a + 16 45 μ a 2 + 9 25 μ a 4 ) ϕ 3 = 8 15 Q 8 15 μ a ϕ 1 + ( 16 45 μ a + 4 9 μ a 2 ) ϕ 2 + ( 128 525 μ a + 32 105 μ a 2 + 54 175 μ a 4 ) ϕ 4
. 1 15 μ a 7 ϕ 4 + ( 256 1225 μ a + 64 245 μ a 2 + 324 1225 μ a 4 + 13 49 μ a 6 ) ϕ 4 = 16 35 Q + 16 35 μ a ϕ 1 ( 32 105 μ a + 8 21 μ a 2 ) ϕ 2 + ( 128 525 μ a + 32 105 μ a 2 + 54 175 μ a 4 ) ϕ 3
μ a n ( x ) = μ t ( x ) μ s ( x ) g n
φ = ϕ 1 2 3 ϕ 2 + 8 15 ϕ 3 16 35 ϕ 4
χ 2 = { i = 1 N M ( Φ M Φ C ) 2 + λ j = 1 N N ( μ j μ 0 ) 2 } μ min
δ μ = ( J T J + λ ¯ I ) 1 J T δ Φ
J = [ δ ln I 1 δ μ 1 δ ln I 1 δ μ 2 δ ln I 1 δ μ N N δ θ 1 δ μ 1 δ θ 1 δ μ 2 δ θ 1 δ μ N N δ ln I 2 δ μ 1 δ ln I 2 δ μ 2 δ ln I 2 δ μ N N δ θ 2 δ μ 1 δ θ 2 δ μ 2 δ θ 2 δ μ N N δ ln I N M δ μ 1 δ ln I N M δ μ 2 δ ln I N M δ μ N N δ θ N M δ μ 1 δ θ N M δ μ 2 δ θ N M δ μ N N ]
J = [ J μ a ; J μ s ; J g ]
J μ s = [ δ log I δ μ s ; δ θ δ μ s ]
J μ a = [ δ log I δ μ a ; δ θ δ μ a ]
J g = [ δ log I δ g ; δ θ δ g ]
δ = | ( φ r e f φ μ ) |

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