Abstract

A three-dimensional fluorescence-enhanced optical tomography scheme based upon an adaptive finite element formulation is developed and employed to reconstruct fluorescent targets in turbid media from frequency-domain measurements made in reflectance geometry using area excitation illumination. The algorithm is derived within a Lagrangian framework by treating the photon diffusion model as a constraint to the optimization problem. Adaptively refined meshes are used to separately discretize maps of the forward/adjoint variables and the unknown parameter of fluorescent yield. A truncated Gauss-Newton method with simple bounds is used as the optimization method. Fluorescence yield reconstructions from simulated measurement data with added Gaussian noise are demonstrated for one and two fluorescent targets embedded within a 512ml cubical tissue phantom. We determine the achievable resolution for the area-illumination/area-detection reflectance measurement geometry by reconstructing two 0.4cm diameter spherical targets placed at at a series of decreasing lateral spacings. The results show that adaptive techniques enable the computationally efficient and stable solution of the inverse imaging problem while providing the resolution necessary for imaging the signals from molecularly targeting agents.

© 2004 Optical Society of America

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2004 (2)

H. Quan and Z. Guo, “Fast 3-D Optical Imaging With Transient Fluorescence Signals,” Opt. Express 12(3), 449–457 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-449.
[CrossRef]

M. Huang and Q. Zhu, “Dual-mesh optical tomography reconstruction method with a depth correction that uses a priori ultrasound information,” Appl. Opt. 43(8), 1654–1662 (2004).
[CrossRef]

2003 (4)

X. Gu, Y. Xu, and H. Jiang, “Mesh-based enhancement schemes in diffuse optical tomography,” Med. Phys. 30(5), 861–869 (2003).
[CrossRef]

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901–911 (2003).
[CrossRef] [PubMed]

A. Godavarty, M. J. Eppstein, C. Zhang, S. Theru, A. B. Thompson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701–1720 (2003).
[CrossRef] [PubMed]

A. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. Boas, and R. P. Milane, “Fluorescence Optical Diffusion Tomography,” Appl. Opt. 42(16), 3061–3094 (2003).

2002 (6)

A. Godavarty, D. J. Hawrysz, R. Roy, E. M. Sevick-Muraca, and M. J. Eppstein, “Influence of the refractive index-mismatch at the boundaries measured in fluorescenceenhanced frequency-domain photon migration imaging,” Opt. Express 10(15), 650–653 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-653.

A. B. Thompson and E. M. Sevick-Muraca, “NIR fluorescence contrast enhanced imaging with ICCD homodyne detection: measurement precision and accuracy,” J. Biomed. Opt. 8, 111–120 (2002).
[CrossRef]

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three dimensional near infrared fluorescence tomography with Bayesian methodologies for image reconstruction from sparse and noisy data sets,” Proc. Nat. Acad. Sci. 99, 9619–9624 (2002).
[CrossRef] [PubMed]

M. Molinari, B. H. Blott, S. J. Cox, and G. J. Daniell, “Optimal Imaging with Adaptive Mesh Refinement in Electrical Impedence Tomography,” Physiological Measurement 23, 121–128 (2002).
[CrossRef] [PubMed]

H. Ben Ameur, G. Chavent, and J. Jaffré, “Refinement and coarsening indicators for adaptive parametrization: application to the estimation of hydraulic transmissivities,” Inverse Problems 18, 775–794 (2002).
[CrossRef]

R. Li, W. Liu, H. Ma, and T. Tang, “Adaptive finite element approximation for distributed elliptic optimal control problems,” SIAM J. Control Optim. 41, 1321–1349 (2002).
[CrossRef]

2001 (4)

M. G. Pomper, “Molecular Imaging: An Overview,” Acad. Radiol. 8, 1141–1153 (2001).
[CrossRef] [PubMed]

R. Becker and R. Rannacher, “An optimal control approach to error estimation and mesh adaptation in finite element methods,” Acta Numerica 10, 1–102 (2001).
[CrossRef]

M. Molinari, S. J. Cox, B. H. Blott, and G. J. Daniell, “Adaptive Mesh Refinement techniques for Electrical Impedence Tomography,” Physiological Measurement 22, 91–96 (2001).
[CrossRef] [PubMed]

V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,” Opt. Lett. 26(12), 893–895 (2001).
[CrossRef]

2000 (1)

R. Becker, H. Kapp, and R. Rannacher, “Adaptive Finite Element Methods for Optimal Control of Partial Differential Equations: Basic Concept,” SIAM J. Contr. Optim. 39, 113–132 (2000).
[CrossRef]

1999 (3)

R. Luce and S. Perez, “Parameter identification for an elliptic partial differential equation with distributed noisy data,” Inverse Problems 15, 291–307 (1999).
[CrossRef]

R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s Optimization Schemes for Absorption and Fluorescence Optical Tomography: part(1) theory and formulation,” Opt. Express 4(10), 353–371 (1999). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-10-353.
[CrossRef]

V. Chernomordik, D. Hattery, I. Gannot, and A. H. Gandjbakhche, “Inverse method 3-D reconstruction of localized in vivo fluorescence-application to Sjøgren syndrome,” IEEE J. Sel. Top. Quantum Electron. 54, 930–935 (1999).

1998 (4)

S. R. Arridge and M. Schweiger, “A Gradient Based Optimization Scheme for Optical Tomography,” Opt. Express 2(6), 213–225 (1998). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-2-6-213.
[CrossRef]

H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite element based algorithm and simulations,” Appl. Opt. 37(22), 5337–5343 (1998).
[CrossRef]

M. J. Eppstein, D. E. Dougherty, T. L. Troy, and E. M. Sevick-Muraca, “Biomedical optical tomography using dynamic parametrization and Bayesian conditioning on photon migration measurements,” Appl. Opt. 38(10), 2138–2150 (1998).

E. L. Hull, M. G. Nichols, and T. H. Foster, “Localization of Luminescent Inhomogeneities in Turbid Media with Spatially Resolved Measurements of CW Diffuse Luminescence Emittance,” Appl. Opt. 37, 2755–2765 (1998).
[CrossRef]

1997 (2)

J. C. Schotland, “Continuous wave diffusion imaging,” J. Opt. Soc. Am. A 14(275–279) (1997).

D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media,” Appl. Opt. 36(10), 2260–2272 (1997).
[CrossRef]

1996 (1)

1995 (1)

1994 (1)

M. A. O’Leary, D. A. Boas, B. Chance, and A. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Luminescence 60, 281–286 (1994).
[CrossRef]

1983 (1)

D. W. Kelly, J. P. d. S. R. Gago, O. C. Zienkiewicz, and I. Babuška, “A posteriori error analysis and adaptive processes in the finite element method: Part I-Error Analysis,” Int. J. Num. Meth. Engrg. 19, 1593–1619 (1983).
[CrossRef]

Adams, R. A.

R. A. Adams, Sobolev Spaces (Academic Press, 1975).

Arridge, S. R.

S. R. Arridge and M. Schweiger, “A Gradient Based Optimization Scheme for Optical Tomography,” Opt. Express 2(6), 213–225 (1998). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-2-6-213.
[CrossRef]

Babuška, I.

D. W. Kelly, J. P. d. S. R. Gago, O. C. Zienkiewicz, and I. Babuška, “A posteriori error analysis and adaptive processes in the finite element method: Part I-Error Analysis,” Int. J. Num. Meth. Engrg. 19, 1593–1619 (1983).
[CrossRef]

Bangerth, W.

A. Joshi, W. Bangerth, and E. Sevick-Muraca, “Adaptive finite element methods for fluorescence enhanced frequency domain optical tomography: Forward imaging problem,” in International Symposium on Biomedical Imaging, pp. 1103–1106 (IEEE, 2004).

W. Bangerth, “A framework for the adaptive finite element solution of large inverse problems. I. Basic techniques,” Tech. Rep. 04–39, ICES, University of Texas at Austin (2004).

W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations (Birkhäuser Verlag, 2003).

W. Bangerth, “Adaptive Finite Element Methods for the Identification of Distributed Coefficients in Partial Differential Equations,” Ph.D. thesis, University of Heidelberg (2002).

W. Bangerth, R. Hartmann, and G. Kanschat, deal.II Differential Equations Analysis Library, Technical Reference (2004). http://www.dealii.org/.

Becker, R.

R. Becker and R. Rannacher, “An optimal control approach to error estimation and mesh adaptation in finite element methods,” Acta Numerica 10, 1–102 (2001).
[CrossRef]

R. Becker, H. Kapp, and R. Rannacher, “Adaptive Finite Element Methods for Optimal Control of Partial Differential Equations: Basic Concept,” SIAM J. Contr. Optim. 39, 113–132 (2000).
[CrossRef]

Beilina, L.

L. Beilina, “Adaptive Hybrid FEM/FDM Methods for Inverse Scattering Problems,” Ph.D. thesis, Chalmers University of Technology (2002).

Ben Ameur, H.

H. Ben Ameur, G. Chavent, and J. Jaffré, “Refinement and coarsening indicators for adaptive parametrization: application to the estimation of hydraulic transmissivities,” Inverse Problems 18, 775–794 (2002).
[CrossRef]

Blott, B. H.

M. Molinari, B. H. Blott, S. J. Cox, and G. J. Daniell, “Optimal Imaging with Adaptive Mesh Refinement in Electrical Impedence Tomography,” Physiological Measurement 23, 121–128 (2002).
[CrossRef] [PubMed]

M. Molinari, S. J. Cox, B. H. Blott, and G. J. Daniell, “Adaptive Mesh Refinement techniques for Electrical Impedence Tomography,” Physiological Measurement 22, 91–96 (2001).
[CrossRef] [PubMed]

Boas, D.

A. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. Boas, and R. P. Milane, “Fluorescence Optical Diffusion Tomography,” Appl. Opt. 42(16), 3061–3094 (2003).

Boas, D. A.

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. 20, 426–428 (1996).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, and A. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Luminescence 60, 281–286 (1994).
[CrossRef]

Bouman, C. A.

A. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. Boas, and R. P. Milane, “Fluorescence Optical Diffusion Tomography,” Appl. Opt. 42(16), 3061–3094 (2003).

Brenner, S. C.

S. C. Brenner and R. L. Scott, The Mathematical Theory of Finite Elements (Springer, Berlin-Heidelberg-New York, 1994).

Chance, B.

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. 20, 426–428 (1996).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, and A. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Luminescence 60, 281–286 (1994).
[CrossRef]

Chavent, G.

H. Ben Ameur, G. Chavent, and J. Jaffré, “Refinement and coarsening indicators for adaptive parametrization: application to the estimation of hydraulic transmissivities,” Inverse Problems 18, 775–794 (2002).
[CrossRef]

Chen, A. U.

D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media,” Appl. Opt. 36(10), 2260–2272 (1997).
[CrossRef]

Chernomordik, V.

V. Chernomordik, D. Hattery, I. Gannot, and A. H. Gandjbakhche, “Inverse method 3-D reconstruction of localized in vivo fluorescence-application to Sjøgren syndrome,” IEEE J. Sel. Top. Quantum Electron. 54, 930–935 (1999).

Cox, S. J.

M. Molinari, B. H. Blott, S. J. Cox, and G. J. Daniell, “Optimal Imaging with Adaptive Mesh Refinement in Electrical Impedence Tomography,” Physiological Measurement 23, 121–128 (2002).
[CrossRef] [PubMed]

M. Molinari, S. J. Cox, B. H. Blott, and G. J. Daniell, “Adaptive Mesh Refinement techniques for Electrical Impedence Tomography,” Physiological Measurement 22, 91–96 (2001).
[CrossRef] [PubMed]

Daniell, G. J.

M. Molinari, B. H. Blott, S. J. Cox, and G. J. Daniell, “Optimal Imaging with Adaptive Mesh Refinement in Electrical Impedence Tomography,” Physiological Measurement 23, 121–128 (2002).
[CrossRef] [PubMed]

M. Molinari, S. J. Cox, B. H. Blott, and G. J. Daniell, “Adaptive Mesh Refinement techniques for Electrical Impedence Tomography,” Physiological Measurement 22, 91–96 (2001).
[CrossRef] [PubMed]

Desai, R. R.

Dougherty, D. E.

M. J. Eppstein, D. E. Dougherty, T. L. Troy, and E. M. Sevick-Muraca, “Biomedical optical tomography using dynamic parametrization and Bayesian conditioning on photon migration measurements,” Appl. Opt. 38(10), 2138–2150 (1998).

Eppstein, M. J.

A. Godavarty, M. J. Eppstein, C. Zhang, S. Theru, A. B. Thompson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701–1720 (2003).
[CrossRef] [PubMed]

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three dimensional near infrared fluorescence tomography with Bayesian methodologies for image reconstruction from sparse and noisy data sets,” Proc. Nat. Acad. Sci. 99, 9619–9624 (2002).
[CrossRef] [PubMed]

A. Godavarty, D. J. Hawrysz, R. Roy, E. M. Sevick-Muraca, and M. J. Eppstein, “Influence of the refractive index-mismatch at the boundaries measured in fluorescenceenhanced frequency-domain photon migration imaging,” Opt. Express 10(15), 650–653 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-653.

M. J. Eppstein, D. E. Dougherty, T. L. Troy, and E. M. Sevick-Muraca, “Biomedical optical tomography using dynamic parametrization and Bayesian conditioning on photon migration measurements,” Appl. Opt. 38(10), 2138–2150 (1998).

Feld, M. S.

Foster, T. H.

Gago, J. P. d. S. R.

D. W. Kelly, J. P. d. S. R. Gago, O. C. Zienkiewicz, and I. Babuška, “A posteriori error analysis and adaptive processes in the finite element method: Part I-Error Analysis,” Int. J. Num. Meth. Engrg. 19, 1593–1619 (1983).
[CrossRef]

Gandjbakhche, A. H.

V. Chernomordik, D. Hattery, I. Gannot, and A. H. Gandjbakhche, “Inverse method 3-D reconstruction of localized in vivo fluorescence-application to Sjøgren syndrome,” IEEE J. Sel. Top. Quantum Electron. 54, 930–935 (1999).

Gannot, I.

V. Chernomordik, D. Hattery, I. Gannot, and A. H. Gandjbakhche, “Inverse method 3-D reconstruction of localized in vivo fluorescence-application to Sjøgren syndrome,” IEEE J. Sel. Top. Quantum Electron. 54, 930–935 (1999).

Godavarty, A.

A. Godavarty, M. J. Eppstein, C. Zhang, S. Theru, A. B. Thompson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701–1720 (2003).
[CrossRef] [PubMed]

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three dimensional near infrared fluorescence tomography with Bayesian methodologies for image reconstruction from sparse and noisy data sets,” Proc. Nat. Acad. Sci. 99, 9619–9624 (2002).
[CrossRef] [PubMed]

A. Godavarty, D. J. Hawrysz, R. Roy, E. M. Sevick-Muraca, and M. J. Eppstein, “Influence of the refractive index-mismatch at the boundaries measured in fluorescenceenhanced frequency-domain photon migration imaging,” Opt. Express 10(15), 650–653 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-653.

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, Near Infrared Fluorescence Imaging and Spectroscopy in Random Media and Tissues, chap. 33, Biomedical Photonics Handbook (CRC Press, 2003).

Graves, E. E.

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901–911 (2003).
[CrossRef] [PubMed]

Grimstad, A.-A.

A.-A. Grimstad, H. Krüger, T. Mannseth, G. Nævdal, and H. Urkedal, “Adaptive selection of parameterization for reservoir history matching,” in ECMOR VIII: 8th European Conference on the Mathematics of Oil Recovery, Freiberg, Germany, pp. E–46 (European Association of Geoscientists and Engineers (EAGE), 2002).

Gu, X.

X. Gu, Y. Xu, and H. Jiang, “Mesh-based enhancement schemes in diffuse optical tomography,” Med. Phys. 30(5), 861–869 (2003).
[CrossRef]

Guo, Z.

H. Quan and Z. Guo, “Fast 3-D Optical Imaging With Transient Fluorescence Signals,” Opt. Express 12(3), 449–457 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-449.
[CrossRef]

Gurfinkel, M.

A. Godavarty, M. J. Eppstein, C. Zhang, S. Theru, A. B. Thompson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701–1720 (2003).
[CrossRef] [PubMed]

Hartmann, R.

W. Bangerth, R. Hartmann, and G. Kanschat, deal.II Differential Equations Analysis Library, Technical Reference (2004). http://www.dealii.org/.

Hattery, D.

V. Chernomordik, D. Hattery, I. Gannot, and A. H. Gandjbakhche, “Inverse method 3-D reconstruction of localized in vivo fluorescence-application to Sjøgren syndrome,” IEEE J. Sel. Top. Quantum Electron. 54, 930–935 (1999).

Hawrysz, D. J.

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three dimensional near infrared fluorescence tomography with Bayesian methodologies for image reconstruction from sparse and noisy data sets,” Proc. Nat. Acad. Sci. 99, 9619–9624 (2002).
[CrossRef] [PubMed]

A. Godavarty, D. J. Hawrysz, R. Roy, E. M. Sevick-Muraca, and M. J. Eppstein, “Influence of the refractive index-mismatch at the boundaries measured in fluorescenceenhanced frequency-domain photon migration imaging,” Opt. Express 10(15), 650–653 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-653.

Houston, J. P.

J. P. Houston, S. Ke, W. Wang, C. Li, and E. M. Sevick-Muraca, “Optical and nuclear image quality analysis with invivo NIR fluorescence and conventional gamma images acquired using a dual labeled tumor targeting probe,” J. Biomed. Opt. (submitted) (2004).

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, Near Infrared Fluorescence Imaging and Spectroscopy in Random Media and Tissues, chap. 33, Biomedical Photonics Handbook (CRC Press, 2003).

Huang, M.

M. Huang and Q. Zhu, “Dual-mesh optical tomography reconstruction method with a depth correction that uses a priori ultrasound information,” Appl. Opt. 43(8), 1654–1662 (2004).
[CrossRef]

Hull, E. L.

Itzkan, I.

Jaffré, J.

H. Ben Ameur, G. Chavent, and J. Jaffré, “Refinement and coarsening indicators for adaptive parametrization: application to the estimation of hydraulic transmissivities,” Inverse Problems 18, 775–794 (2002).
[CrossRef]

Jiang, H.

X. Gu, Y. Xu, and H. Jiang, “Mesh-based enhancement schemes in diffuse optical tomography,” Med. Phys. 30(5), 861–869 (2003).
[CrossRef]

H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite element based algorithm and simulations,” Appl. Opt. 37(22), 5337–5343 (1998).
[CrossRef]

Joshi, A.

A. Joshi, W. Bangerth, and E. Sevick-Muraca, “Adaptive finite element methods for fluorescence enhanced frequency domain optical tomography: Forward imaging problem,” in International Symposium on Biomedical Imaging, pp. 1103–1106 (IEEE, 2004).

Kanschat, G.

W. Bangerth, R. Hartmann, and G. Kanschat, deal.II Differential Equations Analysis Library, Technical Reference (2004). http://www.dealii.org/.

Kapp, H.

R. Becker, H. Kapp, and R. Rannacher, “Adaptive Finite Element Methods for Optimal Control of Partial Differential Equations: Basic Concept,” SIAM J. Contr. Optim. 39, 113–132 (2000).
[CrossRef]

Ke, S.

J. P. Houston, S. Ke, W. Wang, C. Li, and E. M. Sevick-Muraca, “Optical and nuclear image quality analysis with invivo NIR fluorescence and conventional gamma images acquired using a dual labeled tumor targeting probe,” J. Biomed. Opt. (submitted) (2004).

Kelly, D. W.

D. W. Kelly, J. P. d. S. R. Gago, O. C. Zienkiewicz, and I. Babuška, “A posteriori error analysis and adaptive processes in the finite element method: Part I-Error Analysis,” Int. J. Num. Meth. Engrg. 19, 1593–1619 (1983).
[CrossRef]

Krüger, H.

A.-A. Grimstad, H. Krüger, T. Mannseth, G. Nævdal, and H. Urkedal, “Adaptive selection of parameterization for reservoir history matching,” in ECMOR VIII: 8th European Conference on the Mathematics of Oil Recovery, Freiberg, Germany, pp. E–46 (European Association of Geoscientists and Engineers (EAGE), 2002).

Kuwana, E.

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, Near Infrared Fluorescence Imaging and Spectroscopy in Random Media and Tissues, chap. 33, Biomedical Photonics Handbook (CRC Press, 2003).

Li, C.

J. P. Houston, S. Ke, W. Wang, C. Li, and E. M. Sevick-Muraca, “Optical and nuclear image quality analysis with invivo NIR fluorescence and conventional gamma images acquired using a dual labeled tumor targeting probe,” J. Biomed. Opt. (submitted) (2004).

Li, R.

R. Li, W. Liu, H. Ma, and T. Tang, “Adaptive finite element approximation for distributed elliptic optimal control problems,” SIAM J. Control Optim. 41, 1321–1349 (2002).
[CrossRef]

Liu, W.

R. Li, W. Liu, H. Ma, and T. Tang, “Adaptive finite element approximation for distributed elliptic optimal control problems,” SIAM J. Control Optim. 41, 1321–1349 (2002).
[CrossRef]

Luce, R.

R. Luce and S. Perez, “Parameter identification for an elliptic partial differential equation with distributed noisy data,” Inverse Problems 15, 291–307 (1999).
[CrossRef]

Luenberger, D. G.

D. G. Luenberger, Optimization by Vector Space Methods (John Wiley, 1969).

Ma, H.

R. Li, W. Liu, H. Ma, and T. Tang, “Adaptive finite element approximation for distributed elliptic optimal control problems,” SIAM J. Control Optim. 41, 1321–1349 (2002).
[CrossRef]

Mannseth, T.

A.-A. Grimstad, H. Krüger, T. Mannseth, G. Nævdal, and H. Urkedal, “Adaptive selection of parameterization for reservoir history matching,” in ECMOR VIII: 8th European Conference on the Mathematics of Oil Recovery, Freiberg, Germany, pp. E–46 (European Association of Geoscientists and Engineers (EAGE), 2002).

Milane, R. P.

A. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. Boas, and R. P. Milane, “Fluorescence Optical Diffusion Tomography,” Appl. Opt. 42(16), 3061–3094 (2003).

Milstein, A.

A. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. Boas, and R. P. Milane, “Fluorescence Optical Diffusion Tomography,” Appl. Opt. 42(16), 3061–3094 (2003).

Molinari, M.

M. Molinari, B. H. Blott, S. J. Cox, and G. J. Daniell, “Optimal Imaging with Adaptive Mesh Refinement in Electrical Impedence Tomography,” Physiological Measurement 23, 121–128 (2002).
[CrossRef] [PubMed]

M. Molinari, S. J. Cox, B. H. Blott, and G. J. Daniell, “Adaptive Mesh Refinement techniques for Electrical Impedence Tomography,” Physiological Measurement 22, 91–96 (2001).
[CrossRef] [PubMed]

Nævdal, G.

A.-A. Grimstad, H. Krüger, T. Mannseth, G. Nævdal, and H. Urkedal, “Adaptive selection of parameterization for reservoir history matching,” in ECMOR VIII: 8th European Conference on the Mathematics of Oil Recovery, Freiberg, Germany, pp. E–46 (European Association of Geoscientists and Engineers (EAGE), 2002).

Nichols, M. G.

Nocedal, J.

J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research (Springer, New York, 1999).
[CrossRef]

Ntziachristos, V.

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901–911 (2003).
[CrossRef] [PubMed]

V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,” Opt. Lett. 26(12), 893–895 (2001).
[CrossRef]

O’Leary, M. A.

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. 20, 426–428 (1996).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, and A. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Luminescence 60, 281–286 (1994).
[CrossRef]

Oh, S.

A. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. Boas, and R. P. Milane, “Fluorescence Optical Diffusion Tomography,” Appl. Opt. 42(16), 3061–3094 (2003).

Paithankar, D. Y.

D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media,” Appl. Opt. 36(10), 2260–2272 (1997).
[CrossRef]

Patterson, M. S.

D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media,” Appl. Opt. 36(10), 2260–2272 (1997).
[CrossRef]

Perez, S.

R. Luce and S. Perez, “Parameter identification for an elliptic partial differential equation with distributed noisy data,” Inverse Problems 15, 291–307 (1999).
[CrossRef]

Perleman, L.

Pogue, B. W.

D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media,” Appl. Opt. 36(10), 2260–2272 (1997).
[CrossRef]

Pomper, M. G.

M. G. Pomper, “Molecular Imaging: An Overview,” Acad. Radiol. 8, 1141–1153 (2001).
[CrossRef] [PubMed]

Quan, H.

H. Quan and Z. Guo, “Fast 3-D Optical Imaging With Transient Fluorescence Signals,” Opt. Express 12(3), 449–457 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-449.
[CrossRef]

Rannacher, R.

R. Becker and R. Rannacher, “An optimal control approach to error estimation and mesh adaptation in finite element methods,” Acta Numerica 10, 1–102 (2001).
[CrossRef]

R. Becker, H. Kapp, and R. Rannacher, “Adaptive Finite Element Methods for Optimal Control of Partial Differential Equations: Basic Concept,” SIAM J. Contr. Optim. 39, 113–132 (2000).
[CrossRef]

W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations (Birkhäuser Verlag, 2003).

Ripoll, J.

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901–911 (2003).
[CrossRef] [PubMed]

Roy, R.

A. Godavarty, D. J. Hawrysz, R. Roy, E. M. Sevick-Muraca, and M. J. Eppstein, “Influence of the refractive index-mismatch at the boundaries measured in fluorescenceenhanced frequency-domain photon migration imaging,” Opt. Express 10(15), 650–653 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-653.

R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s Optimization Schemes for Absorption and Fluorescence Optical Tomography: part(1) theory and formulation,” Opt. Express 4(10), 353–371 (1999). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-10-353.
[CrossRef]

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, Near Infrared Fluorescence Imaging and Spectroscopy in Random Media and Tissues, chap. 33, Biomedical Photonics Handbook (CRC Press, 2003).

Schotland, J. C.

J. C. Schotland, “Continuous wave diffusion imaging,” J. Opt. Soc. Am. A 14(275–279) (1997).

Schweiger, M.

S. R. Arridge and M. Schweiger, “A Gradient Based Optimization Scheme for Optical Tomography,” Opt. Express 2(6), 213–225 (1998). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-2-6-213.
[CrossRef]

Scott, R. L.

S. C. Brenner and R. L. Scott, The Mathematical Theory of Finite Elements (Springer, Berlin-Heidelberg-New York, 1994).

Sevick-Muraca, E.

A. Joshi, W. Bangerth, and E. Sevick-Muraca, “Adaptive finite element methods for fluorescence enhanced frequency domain optical tomography: Forward imaging problem,” in International Symposium on Biomedical Imaging, pp. 1103–1106 (IEEE, 2004).

Sevick-Muraca, E. M.

A. Godavarty, M. J. Eppstein, C. Zhang, S. Theru, A. B. Thompson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701–1720 (2003).
[CrossRef] [PubMed]

A. B. Thompson and E. M. Sevick-Muraca, “NIR fluorescence contrast enhanced imaging with ICCD homodyne detection: measurement precision and accuracy,” J. Biomed. Opt. 8, 111–120 (2002).
[CrossRef]

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three dimensional near infrared fluorescence tomography with Bayesian methodologies for image reconstruction from sparse and noisy data sets,” Proc. Nat. Acad. Sci. 99, 9619–9624 (2002).
[CrossRef] [PubMed]

A. Godavarty, D. J. Hawrysz, R. Roy, E. M. Sevick-Muraca, and M. J. Eppstein, “Influence of the refractive index-mismatch at the boundaries measured in fluorescenceenhanced frequency-domain photon migration imaging,” Opt. Express 10(15), 650–653 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-653.

R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s Optimization Schemes for Absorption and Fluorescence Optical Tomography: part(1) theory and formulation,” Opt. Express 4(10), 353–371 (1999). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-10-353.
[CrossRef]

M. J. Eppstein, D. E. Dougherty, T. L. Troy, and E. M. Sevick-Muraca, “Biomedical optical tomography using dynamic parametrization and Bayesian conditioning on photon migration measurements,” Appl. Opt. 38(10), 2138–2150 (1998).

D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media,” Appl. Opt. 36(10), 2260–2272 (1997).
[CrossRef]

J. P. Houston, S. Ke, W. Wang, C. Li, and E. M. Sevick-Muraca, “Optical and nuclear image quality analysis with invivo NIR fluorescence and conventional gamma images acquired using a dual labeled tumor targeting probe,” J. Biomed. Opt. (submitted) (2004).

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, Near Infrared Fluorescence Imaging and Spectroscopy in Random Media and Tissues, chap. 33, Biomedical Photonics Handbook (CRC Press, 2003).

Tang, T.

R. Li, W. Liu, H. Ma, and T. Tang, “Adaptive finite element approximation for distributed elliptic optimal control problems,” SIAM J. Control Optim. 41, 1321–1349 (2002).
[CrossRef]

Theru, S.

A. Godavarty, M. J. Eppstein, C. Zhang, S. Theru, A. B. Thompson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701–1720 (2003).
[CrossRef] [PubMed]

Thompson, A.

A. Thompson, “Development of a new optical imaging modality for detection of fluorescence enhanced disease,” Ph.D. thesis, Texas A & M University (2003).

Thompson, A. B.

A. Godavarty, M. J. Eppstein, C. Zhang, S. Theru, A. B. Thompson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701–1720 (2003).
[CrossRef] [PubMed]

A. B. Thompson and E. M. Sevick-Muraca, “NIR fluorescence contrast enhanced imaging with ICCD homodyne detection: measurement precision and accuracy,” J. Biomed. Opt. 8, 111–120 (2002).
[CrossRef]

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, Near Infrared Fluorescence Imaging and Spectroscopy in Random Media and Tissues, chap. 33, Biomedical Photonics Handbook (CRC Press, 2003).

Troy, T. L.

M. J. Eppstein, D. E. Dougherty, T. L. Troy, and E. M. Sevick-Muraca, “Biomedical optical tomography using dynamic parametrization and Bayesian conditioning on photon migration measurements,” Appl. Opt. 38(10), 2138–2150 (1998).

Urkedal, H.

A.-A. Grimstad, H. Krüger, T. Mannseth, G. Nævdal, and H. Urkedal, “Adaptive selection of parameterization for reservoir history matching,” in ECMOR VIII: 8th European Conference on the Mathematics of Oil Recovery, Freiberg, Germany, pp. E–46 (European Association of Geoscientists and Engineers (EAGE), 2002).

Verfürth, R.

R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques (Wiley/Teubner, New York, Stuttgart, 1996).

Wang, W.

J. P. Houston, S. Ke, W. Wang, C. Li, and E. M. Sevick-Muraca, “Optical and nuclear image quality analysis with invivo NIR fluorescence and conventional gamma images acquired using a dual labeled tumor targeting probe,” J. Biomed. Opt. (submitted) (2004).

Wang, Y.

Webb, K. J.

A. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. Boas, and R. P. Milane, “Fluorescence Optical Diffusion Tomography,” Appl. Opt. 42(16), 3061–3094 (2003).

Weissleder, R.

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901–911 (2003).
[CrossRef] [PubMed]

V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,” Opt. Lett. 26(12), 893–895 (2001).
[CrossRef]

Wright, S. J.

J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research (Springer, New York, 1999).
[CrossRef]

Wu, J.

Xu, Y.

X. Gu, Y. Xu, and H. Jiang, “Mesh-based enhancement schemes in diffuse optical tomography,” Med. Phys. 30(5), 861–869 (2003).
[CrossRef]

Yodh, A.

M. A. O’Leary, D. A. Boas, B. Chance, and A. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Luminescence 60, 281–286 (1994).
[CrossRef]

Yodh, A. G.

Zhang, C.

A. Godavarty, M. J. Eppstein, C. Zhang, S. Theru, A. B. Thompson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701–1720 (2003).
[CrossRef] [PubMed]

Zhang, Q.

A. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. Boas, and R. P. Milane, “Fluorescence Optical Diffusion Tomography,” Appl. Opt. 42(16), 3061–3094 (2003).

Zhu, Q.

M. Huang and Q. Zhu, “Dual-mesh optical tomography reconstruction method with a depth correction that uses a priori ultrasound information,” Appl. Opt. 43(8), 1654–1662 (2004).
[CrossRef]

Zienkiewicz, O. C.

D. W. Kelly, J. P. d. S. R. Gago, O. C. Zienkiewicz, and I. Babuška, “A posteriori error analysis and adaptive processes in the finite element method: Part I-Error Analysis,” Int. J. Num. Meth. Engrg. 19, 1593–1619 (1983).
[CrossRef]

Acad. Radiol. (1)

M. G. Pomper, “Molecular Imaging: An Overview,” Acad. Radiol. 8, 1141–1153 (2001).
[CrossRef] [PubMed]

Acta Numerica (1)

R. Becker and R. Rannacher, “An optimal control approach to error estimation and mesh adaptation in finite element methods,” Acta Numerica 10, 1–102 (2001).
[CrossRef]

Appl. Opt. (6)

D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media,” Appl. Opt. 36(10), 2260–2272 (1997).
[CrossRef]

H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite element based algorithm and simulations,” Appl. Opt. 37(22), 5337–5343 (1998).
[CrossRef]

M. J. Eppstein, D. E. Dougherty, T. L. Troy, and E. M. Sevick-Muraca, “Biomedical optical tomography using dynamic parametrization and Bayesian conditioning on photon migration measurements,” Appl. Opt. 38(10), 2138–2150 (1998).

A. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. Boas, and R. P. Milane, “Fluorescence Optical Diffusion Tomography,” Appl. Opt. 42(16), 3061–3094 (2003).

M. Huang and Q. Zhu, “Dual-mesh optical tomography reconstruction method with a depth correction that uses a priori ultrasound information,” Appl. Opt. 43(8), 1654–1662 (2004).
[CrossRef]

E. L. Hull, M. G. Nichols, and T. H. Foster, “Localization of Luminescent Inhomogeneities in Turbid Media with Spatially Resolved Measurements of CW Diffuse Luminescence Emittance,” Appl. Opt. 37, 2755–2765 (1998).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

V. Chernomordik, D. Hattery, I. Gannot, and A. H. Gandjbakhche, “Inverse method 3-D reconstruction of localized in vivo fluorescence-application to Sjøgren syndrome,” IEEE J. Sel. Top. Quantum Electron. 54, 930–935 (1999).

Int. J. Num. Meth. Engrg. (1)

D. W. Kelly, J. P. d. S. R. Gago, O. C. Zienkiewicz, and I. Babuška, “A posteriori error analysis and adaptive processes in the finite element method: Part I-Error Analysis,” Int. J. Num. Meth. Engrg. 19, 1593–1619 (1983).
[CrossRef]

Inverse Problems (2)

H. Ben Ameur, G. Chavent, and J. Jaffré, “Refinement and coarsening indicators for adaptive parametrization: application to the estimation of hydraulic transmissivities,” Inverse Problems 18, 775–794 (2002).
[CrossRef]

R. Luce and S. Perez, “Parameter identification for an elliptic partial differential equation with distributed noisy data,” Inverse Problems 15, 291–307 (1999).
[CrossRef]

J. Biomed. Opt. (1)

A. B. Thompson and E. M. Sevick-Muraca, “NIR fluorescence contrast enhanced imaging with ICCD homodyne detection: measurement precision and accuracy,” J. Biomed. Opt. 8, 111–120 (2002).
[CrossRef]

J. Luminescence (1)

M. A. O’Leary, D. A. Boas, B. Chance, and A. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Luminescence 60, 281–286 (1994).
[CrossRef]

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

J. C. Schotland, “Continuous wave diffusion imaging,” J. Opt. Soc. Am. A 14(275–279) (1997).

Med. Phys. (2)

X. Gu, Y. Xu, and H. Jiang, “Mesh-based enhancement schemes in diffuse optical tomography,” Med. Phys. 30(5), 861–869 (2003).
[CrossRef]

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901–911 (2003).
[CrossRef] [PubMed]

Opt. Express (4)

H. Quan and Z. Guo, “Fast 3-D Optical Imaging With Transient Fluorescence Signals,” Opt. Express 12(3), 449–457 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-449.
[CrossRef]

R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s Optimization Schemes for Absorption and Fluorescence Optical Tomography: part(1) theory and formulation,” Opt. Express 4(10), 353–371 (1999). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-10-353.
[CrossRef]

A. Godavarty, D. J. Hawrysz, R. Roy, E. M. Sevick-Muraca, and M. J. Eppstein, “Influence of the refractive index-mismatch at the boundaries measured in fluorescenceenhanced frequency-domain photon migration imaging,” Opt. Express 10(15), 650–653 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-653.

S. R. Arridge and M. Schweiger, “A Gradient Based Optimization Scheme for Optical Tomography,” Opt. Express 2(6), 213–225 (1998). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-2-6-213.
[CrossRef]

Opt. Lett. (3)

Phys. Med. Biol. (1)

A. Godavarty, M. J. Eppstein, C. Zhang, S. Theru, A. B. Thompson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701–1720 (2003).
[CrossRef] [PubMed]

Physiological Measurement (2)

M. Molinari, S. J. Cox, B. H. Blott, and G. J. Daniell, “Adaptive Mesh Refinement techniques for Electrical Impedence Tomography,” Physiological Measurement 22, 91–96 (2001).
[CrossRef] [PubMed]

M. Molinari, B. H. Blott, S. J. Cox, and G. J. Daniell, “Optimal Imaging with Adaptive Mesh Refinement in Electrical Impedence Tomography,” Physiological Measurement 23, 121–128 (2002).
[CrossRef] [PubMed]

Proc. Nat. Acad. Sci. (1)

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A. Thompson, “Development of a new optical imaging modality for detection of fluorescence enhanced disease,” Ph.D. thesis, Texas A & M University (2003).

A.-A. Grimstad, H. Krüger, T. Mannseth, G. Nævdal, and H. Urkedal, “Adaptive selection of parameterization for reservoir history matching,” in ECMOR VIII: 8th European Conference on the Mathematics of Oil Recovery, Freiberg, Germany, pp. E–46 (European Association of Geoscientists and Engineers (EAGE), 2002).

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A. Joshi, W. Bangerth, and E. Sevick-Muraca, “Adaptive finite element methods for fluorescence enhanced frequency domain optical tomography: Forward imaging problem,” in International Symposium on Biomedical Imaging, pp. 1103–1106 (IEEE, 2004).

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J. P. Houston, S. Ke, W. Wang, C. Li, and E. M. Sevick-Muraca, “Optical and nuclear image quality analysis with invivo NIR fluorescence and conventional gamma images acquired using a dual labeled tumor targeting probe,” J. Biomed. Opt. (submitted) (2004).

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E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, Near Infrared Fluorescence Imaging and Spectroscopy in Random Media and Tissues, chap. 33, Biomedical Photonics Handbook (CRC Press, 2003).

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Figures (6)

Fig. 1.
Fig. 1.

Adaptive tomography algorithm. GN stands for Gauss-Newton; see Section 2.4 for a description of symbols.

Fig. 2.
Fig. 2.

Area illumination and area detection geometry employed by Thompson et al. [23]

Fig. 3.
Fig. 3.

Single target reconstruction: A black wire-frame depicts the actual target and colored blocks represent the reconstruction. Top 10% of the contour levels of µaxf are shown.

Fig. 4.
Fig. 4.

Adaptive mesh evolution for state/adjoint (left) and parameter discretization (right). Meshes are shown at 1st, 11th and 22nd Gauss-Newton iterations.

Fig. 5.
Fig. 5.

Dual target reconstructions: A black wire-frame depicts the actual targets and colored blocks represent the reconstruction. Top 10% of the contour levels of µaxf are shown. Edge to edge spacing: (a) 1.0142cm, (b) 0.6607cm, (c) 0.3071cm, and (d) 0.1657cm.

Fig. 6.
Fig. 6.

Dual target reconstruction for 0.1cm target separation.

Tables (1)

Tables Icon

Table 1. Summary of results for dual fluorescent target reconstructions. d is the edge to edge target separation in cm; Iter. is the Gauss-Newton iteration for which the other results are reported; ‖q-q true‖2 is the error in reconstructed parameter; 1 2 v z Σ 2 is the meausurement error; Nq is the number of elements (unknowns) in the parameter mesh.

Equations (34)

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. [ D x ( r ) u ( r , ω ) ] + k x u ( r , ω ) = 0 ,
[ D m ( r ) v ( r , ω ) ] + k m v ( r , ω ) = β xm u ( r , ω ) ,
2 D x u n + γ u + S ( r ) = 0 , 2 D m v n + γ v = 0 ,
A ( q ; [ u , v ] ) ( [ ζ , ξ ] ) = 0 ζ , ξ H 1 ,
A ( q ; [ u , v ] ) ( [ ζ , ξ ] ) = ( D x u , ζ ) Ω + ( k x u , ζ ) Ω + γ 2 ( u , ζ ) Ω + 1 2 ( S , ζ ) Ω
+ ( D m v , ξ ) Ω + ( k m v , ξ ) Ω + γ 2 ( v , ξ ) Ω ( β xm u , ξ ) Ω .
J ( q , v ) = 1 2 v z Σ 2 + β r ( q ) ,
min q , u , v J ( q , v ) subject to A ( q ; [ u , v ] ) ( [ ζ , ξ ] ) = 0 .
L ( [ u , v ] , [ λ ex , λ em ] , q ) = J ( q , v ) + A ( q ; [ u , v ] ) ( [ λ ex , λ em ] ) .
L x ( x ) ( y ) = 0 y = { φ ex , φ em , ψ ex , ψ em , χ } ,
L u ( x ) ( φ ex ) = A u ( q ; [ u , v ] ) ( φ ex ) ( [ λ ex , λ em ] ) = 0 ,
L v ( x ) ( φ em ) = J v ( q , v ) ( φ em ) + A v ( q ; [ u , v ] ) ( φ em ) ( [ λ ex , λ em ] ) = 0 ,
L λ ex ( x ) ( ψ ex ) = A ( q ; [ u , v ] ) ( [ ψ ex , 0 ] ) = 0 ,
L λ em ( x ) ( ψ em ) = A ( q ; [ u , v ] ) ( [ 0 , ψ em ] ) = 0 ,
L q ( x ) ( χ ) = J q ( q , v ) ( χ ) + A q ( q ; [ u , v ] ) ( χ ) ( [ λ ex , λ em ] ) = 0 .
L xx ( x k ) ( δ x k , y ) = L x ( x k ) ( y ) y ,
A u ( q k ; [ u k , v k ] ) ( ϕ e x ) ( [ δ λ k e x , 0 ] ) + A u ( q k ; [ u k , v k ] ) ( ϕ e x ) ( [ 0 , δ λ k e m ] ) + A u q ( q k ; [ u k , v k ] ) ( ϕ e x , δ q k ) ( [ λ e x , λ e m ] ) = L u ( x k ) ( ϕ e x ) ,
J v v ( q k , v k ) ( δ v k , ϕ e m ) + A v ( q k ; [ u k , v k ] ) ( ϕ e m ) ( [ δ λ k e x , 0 ] ) + A v ( q k ; [ u k , v k ] ) ( ϕ e m ) ( [ 0 , δ λ k e m ] ) + J v q ( q k , v k ) ( δ q k , ϕ e m ) + A v q ( q k ; [ u k , v k ] ) ( ϕ e m , δ q k ) ( [ λ e x , λ e m ] ) = L v ( x k ) ( ϕ e m ) ,
A u ( q k ; [ u k , v k ] ) ( δ u k ) ( [ ψ e x , 0 ] ) + A v ( q k ; [ u k , v k ] ) ( δ v k ) ( [ ψ e x , 0 ] ) + A q ( q k ; [ u k , v k ] ) ( δ q k ) ( [ ψ e x , 0 ] ) = L λ e x ( x k ) ( ψ e x ) ,
A u ( q k ; [ u k , v k ] ) ( δ u k ) ( [ 0 , ψ e m ] ) + A v ( q k ; [ u k , v k ] ) ( δ v k ) ( [ 0 , ψ e m ] ) + A q ( q k ; [ u k , v k ] ) ( δ q k ) ( [ 0 , ψ e x ] ) = L λ e m ( x k ) ( ψ e m ) ,
A q u ( q k ; [ u k , v k ] ) ( δ u k , χ ) ( [ λ e x , λ e m ] ) + A q v ( q k ; [ u k , v k ] ) ( δ v k , χ ) ( [ λ e x , λ e m ] ) + J q v ( q k , v k ) ( δ v k , χ ) + J q q ( q k , v k ) ( δ q k , χ ) + A q ( q k ; [ u k , v k ] ) ( χ ) ( [ δ λ e x , 0 ] ) + A q ( q k ; [ u k , v k ] ) ( χ ) ( [ 0 , δ λ e m ] ) = L q ( x k ) ( χ ) .
x k + 1 = x k + α k δ x k .
[ M 0 P T 0 R C T P C 0 ] [ δ p k δ q k δ d k ] = [ F 1 F 2 F 3 ] ,
M = [ 0 0 0 ( φ i , φ j ) Σ ] i j , R = [ β r " ( q k , χ i , χ j ) ] ij , P T = [ A global ex B global ex em 0 A global em ] .
C 1 = ( D x ( q k ) q u k ψ i , χ j ) ij + ( k x ( q k ) q u k ψ i , χ j ) ij ,
C 2 = ( D m ( q k ) q v k ψ i , χ j ) ij + ( k m ( q k ) q v k ψ i , χ j ) ij ( β xm ( q k ) q u k ψ i , χ j ) ij .
F 1 = [ ( D x ( q k ) λ k ex , φ i ) ( k x ( q k ) λ k ex , φ i ) γ 2 ( λ k ex , φ i ) Ω + ( β xm ( q k ) λ k em , φ i ) ( v k z , φ i ) Σ ( D m ( q k ) λ k em , φ i ) ( k m ( q k ) λ k em , φ i ) γ 2 ( λ k em , φ i ) Ω ] i ,
F 2 = [ β r ( q k , χ i ) ( D x ( q k ) q u k λ k ex , χ i ) ( k x ( q k ) q u k λ k em , χ i ) ( D m ( q k ) q v k . λ k em , χ i ) ( k m ( q k ) q v k λ k em , χ i ) + ( β xm ( q k ) q u k λ k em , χ i ) ] i ,
F 3 = [ D x ( q k ) ψ i , u k ) ( k x ( q k ) ψ i , u k ) 1 2 ( S x , ψ i ) Ω γ 2 ( ψ i , u k ) Ω D m ( q k ) ψ i , v k ) ( k m ( q k ) ψ i , v k ) γ 2 ( ψ i , v k ) Ω + ( β xm ( q k ) ψ i , u k ) ] i .
{ R + C T P T M P 1 C } δ q k = F 2 C T P T F 1 + C T P T M P 1 F 3 ,
P δ p k = F 3 C δ q k ,
P T δ d k = F 1 M δ p k .
u u h C ( u ) h 2 ,
η K u = h 24 n u h K 2 , η K v = h 24 n v h K 2 , η K = α η K u + ( 1 α ) η K v ,

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