A multilevel adaptive finite element algorithm for bioluminescence tomography
Optics Express, Vol. 14, Issue 18, pp. 8211-8223 (2006)
http://dx.doi.org/10.1364/OE.14.008211
Acrobat PDF (571 KB)
Abstract
As a new mode of molecular imaging, bioluminescence tomography (BLT) has become a hot topic over the past two years. In this paper, a multilevel adaptive finite element algorithm is developed for BLT reconstruction. In this algorithm, the mesh is adaptively refined according to a posteriori error estimation, which helps not only to improve localization and quantification of sources but also to enhance the robustness and efficiency of reconstruction. In the numerical simulation, bioluminescent signals on the body surface of a heterogeneous phantom are synthesized in a molecular optical simulation environment (MOSE) that we developed to model the photon transportation via Monte Carlo simulation. The performance of the algorithm is evaluated in numerical tests involving single and multiple sources in various arrangements. The results demonstrate the merits and potential of the multilevel adaptive approach for BLT.
© 2006 Optical Society of America
1. Introduction
1 . V. Ntziachristos , J. Ripoll , L. V. Wang , and R. Weisslder , “ Looking and listening to light: the evolution of whole body photonic imaging ,” Nature Biotechnology 23 (3), 313 – 320 ( 2005 ). [CrossRef] [PubMed]
2 . C. Contag and M. H. Bachmann , “ Advances in bioluminescence imaging of gene expression ,” Annu. Rev. Bio-med. Eng. 4 , 235 – 260 ( 2002 ). [CrossRef]
3 . S. Bhaumik and S. S. Gambhir , “ Optical imaging of Renilla luciferase reporter gene expression in living mice ,” Proc. Natl. Acad. Sci. USA 99 , 377 – 382 ( 2002 ). [CrossRef]
4 . T. F. Massoud and S. S. Gambhir , “ Molecular imaging in living subjects: seeing fundamental biological processes in a new light ,” Genes Dev. 17 , 545 – 580 ( 2003 ). [CrossRef] [PubMed]
6 . G. Wang , Y. Li , and M. Jiang , “ Uniqueness theorems in bioluminescence tomography ,” Med. Phys. 31 , 2289 – 2299 ( 2004 ). [CrossRef] [PubMed]
7 . 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]
8 . Thérèse and J. W. Hastings , “ Bioluminescence ,” Annu. Rev. Cell Dev. Bi. 14 , 197 – 230 ( 1998 ). [CrossRef]
9 . W. Rice , M. D. Cable , and M. B. Nelson , “ In vivo imaging of light-emitting probes ,” J. Biomed. Opt. 6 , 432 – 440 ( 2001 ). [CrossRef] [PubMed]
6 . G. Wang , Y. Li , and M. Jiang , “ Uniqueness theorems in bioluminescence tomography ,” Med. Phys. 31 , 2289 – 2299 ( 2004 ). [CrossRef] [PubMed]
10 . M. Jiang and G. Wang , “ Image reconstruction for bioluminescence tomography ,” Proc. SPIE 5535 , 335 – 351 ( 2004 ). [CrossRef]
6 . G. Wang , Y. Li , and M. Jiang , “ Uniqueness theorems in bioluminescence tomography ,” Med. Phys. 31 , 2289 – 2299 ( 2004 ). [CrossRef] [PubMed]
11 . W. Cong , D. Kumar , Y. Liu , A. Cong , and G. Wang , “ A practical method to determine the light source distribution in bioluminescent imaging ,” Proc. SPIE 5535 , 679 – 686 ( 2004 ). [CrossRef]
12 . X. Gu , Q. Zhang , L. Larcom , and H. Jiang , “ Three-dimensional bioluminescence tomography with model-based reconstruction ,” Opt. Express 12 , 3996 – 4000 ( 2004 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-17-3996 . [CrossRef] [PubMed]
14 . A. J. Chaudhari , F. Darvas , J. R. Bading , R. A. Moats , P. S. Conti , D. J. Smith , S. R. Cherry , and R. M. Leahy , “ Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging ,” Phys. Med. Biol. 50 , 5421 – 5441 ( 2005 ). [CrossRef] [PubMed]
15 . 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] [PubMed]
15 . 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] [PubMed]
16 . W. 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 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-18-6756 . [CrossRef] [PubMed]
2. Methods
2.1. Diffusion approximation and boundary condition
16 . W. 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 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-18-6756 . [CrossRef] [PubMed]
17 . A. P. Gibson , J. C. Hebden , and S. R. Arridge , “ Recent advances in diffuse optical imaging ,” Phys. Med. Biol. 50 , R1 – R43 ( 2005 ). [CrossRef] [PubMed]
20 . S. R. Arridge , M. Schweiger , M. Hiraoka , and D. T. Delpy , “ A finite element approach for modeling photon transport in tissue ,” Med. Phys. 20 , 299 – 309 ( 1993 ). [CrossRef] [PubMed]
16 . W. 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 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-18-6756 . [CrossRef] [PubMed]
21 . 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] [PubMed]
21 . 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] [PubMed]
2.2. Reconstruction based on multilevel adaptive FEMs
23 . G. Wang , M. Jiang , J. Tian , W. Cong , Y. Li , W. Han , D. Kumar , X. Qian , H. Shen , T. Zhou , J. Cheng , Y. Lv , H. Li , and J. Luo , “ Recent development in bioluminescence tomography ,” presented in the third IEEE International Symposium on Biomedical Imaging (ISBI 2006), Virginia, USA , 6–9 Apr. 2006 .
26 . M. Ainsworth and J. T. Oden , A posteriori error estimation in finite element analysis , ( Wiley , 2000 ). [CrossRef]
26 . M. Ainsworth and J. T. Oden , A posteriori error estimation in finite element analysis , ( Wiley , 2000 ). [CrossRef]
28 . Z. Wu and J. M. Sullivan Jr. , “ Multiple material marching cubes algorithm ,” Int. J. Numer. Methods Eng. 58 , 189 – 207 ( 2003 ). [CrossRef]
30 . J. Bey , “ Tetrahedral grid refinement ,” Computing 55 , 355 – 378 ( 1995 ). [CrossRef]
3. Numerical studies
15 . 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] [PubMed]
3.1. Photon transportation simulation using MOSE
31 . L. H. Wang , S. L. Jacques , and L. Q. Zheng , “ MCML-Monte Carlo modeling of photon transport in multi-layered tissues ,” Comput. Meth. Prog. Biomed. 47 , 131 – 146 ( 1995 ). [CrossRef]
32 . D. Boas , J. Culver , J. Stott , and A. Dunn , “ Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head ,” Opt. Express 10 , 159 – 169 ( 2002 ), http://www.opticsinfobase.org/abstract.cfm?URI=OPEX-10-3-159 . [PubMed]
34 . H. Li , J. Tian , F. Zhu , W. Cong , L. V. Wang , E. A. Hoffman , and G. Wang , “ A mouse optical simulation enviroment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo Method ,” Acad. Radiol. 11 , 1029 – 1038 ( 2004 ). [CrossRef] [PubMed]
3.2. Light source reconstruction
9 . W. Rice , M. D. Cable , and M. B. Nelson , “ In vivo imaging of light-emitting probes ,” J. Biomed. Opt. 6 , 432 – 440 ( 2001 ). [CrossRef] [PubMed]
9 . W. Rice , M. D. Cable , and M. B. Nelson , “ In vivo imaging of light-emitting probes ,” J. Biomed. Opt. 6 , 432 – 440 ( 2001 ). [CrossRef] [PubMed]
3.3. Spatial resolution evaluation
4. Discussions and conclusion
1 . V. Ntziachristos , J. Ripoll , L. V. Wang , and R. Weisslder , “ Looking and listening to light: the evolution of whole body photonic imaging ,” Nature Biotechnology 23 (3), 313 – 320 ( 2005 ). [CrossRef] [PubMed]
Acknowledgments
References and links
1 . | V. Ntziachristos , J. Ripoll , L. V. Wang , and R. Weisslder , “ Looking and listening to light: the evolution of whole body photonic imaging ,” Nature Biotechnology 23 (3), 313 – 320 ( 2005 ). [CrossRef] [PubMed] |
2 . | C. Contag and M. H. Bachmann , “ Advances in bioluminescence imaging of gene expression ,” Annu. Rev. Bio-med. Eng. 4 , 235 – 260 ( 2002 ). [CrossRef] |
3 . | S. Bhaumik and S. S. Gambhir , “ Optical imaging of Renilla luciferase reporter gene expression in living mice ,” Proc. Natl. Acad. Sci. USA 99 , 377 – 382 ( 2002 ). [CrossRef] |
4 . | T. F. Massoud and S. S. Gambhir , “ Molecular imaging in living subjects: seeing fundamental biological processes in a new light ,” Genes Dev. 17 , 545 – 580 ( 2003 ). [CrossRef] [PubMed] |
5 . | G. Wang , E. A. Hoffman , G. McLennan , L. V. Wang , M. Suter , and J. F. Meinel , “ Development of the first bioluminescence ct scanner ,” Radiology 229(P) , 566 ( 2003 ). |
6 . | G. Wang , Y. Li , and M. Jiang , “ Uniqueness theorems in bioluminescence tomography ,” Med. Phys. 31 , 2289 – 2299 ( 2004 ). [CrossRef] [PubMed] |
7 . | 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] |
8 . | Thérèse and J. W. Hastings , “ Bioluminescence ,” Annu. Rev. Cell Dev. Bi. 14 , 197 – 230 ( 1998 ). [CrossRef] |
9 . | W. Rice , M. D. Cable , and M. B. Nelson , “ In vivo imaging of light-emitting probes ,” J. Biomed. Opt. 6 , 432 – 440 ( 2001 ). [CrossRef] [PubMed] |
10 . | M. Jiang and G. Wang , “ Image reconstruction for bioluminescence tomography ,” Proc. SPIE 5535 , 335 – 351 ( 2004 ). [CrossRef] |
11 . | W. Cong , D. Kumar , Y. Liu , A. Cong , and G. Wang , “ A practical method to determine the light source distribution in bioluminescent imaging ,” Proc. SPIE 5535 , 679 – 686 ( 2004 ). [CrossRef] |
12 . | X. Gu , Q. Zhang , L. Larcom , and H. Jiang , “ Three-dimensional bioluminescence tomography with model-based reconstruction ,” Opt. Express 12 , 3996 – 4000 ( 2004 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-17-3996 . [CrossRef] [PubMed] |
13 . | C. Kuo , O. Coquoz , D. G. Stearns , and B. W. Rice , “ Diffuse luminescence tomography of in vivo bioluminescence markers using multi-spetral data ,” in Proceedings of the 3rd International Meeting of the Society ( MIT Press , 2004 ), p. 227 . |
14 . | A. J. Chaudhari , F. Darvas , J. R. Bading , R. A. Moats , P. S. Conti , D. J. Smith , S. R. Cherry , and R. M. Leahy , “ Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging ,” Phys. Med. Biol. 50 , 5421 – 5441 ( 2005 ). [CrossRef] [PubMed] |
15 . | 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] [PubMed] |
16 . | W. 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 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-18-6756 . [CrossRef] [PubMed] |
17 . | A. P. Gibson , J. C. Hebden , and S. R. Arridge , “ Recent advances in diffuse optical imaging ,” Phys. Med. Biol. 50 , R1 – R43 ( 2005 ). [CrossRef] [PubMed] |
18 . | S. Holder , Electrical Impedance Tomography , ( Institute of Physics Publishing, Bristol and Philadelphia , 2005 ). |
19 . | A. Joshi , W. Bangerth , and E. M. Sevick-Muraca , “ Adaptive finite element based tomography for fluorescence optical imaging in tissue ,” Opt. Express 12 , 5402 – 5417 ( 2004 ), www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5402 . [CrossRef] [PubMed] |
20 . | S. R. Arridge , M. Schweiger , M. Hiraoka , and D. T. Delpy , “ A finite element approach for modeling photon transport in tissue ,” Med. Phys. 20 , 299 – 309 ( 1993 ). [CrossRef] [PubMed] |
21 . | 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] [PubMed] |
22 . | S. S. Rao , The finite element method in engineering , ( Butterworth-Heinemann, Boston , 1999 ). |
23 . | G. Wang , M. Jiang , J. Tian , W. Cong , Y. Li , W. Han , D. Kumar , X. Qian , H. Shen , T. Zhou , J. Cheng , Y. Lv , H. Li , and J. Luo , “ Recent development in bioluminescence tomography ,” presented in the third IEEE International Symposium on Biomedical Imaging (ISBI 2006), Virginia, USA , 6–9 Apr. 2006 . |
24 . | W. Bangerth and R. Rannacher , Adaptive finite element methods for differential equations , ( Birkhäuser Verlag , 2003 ). |
25 . | P. E. Gill , W. Murray , and M. Wright , Practical optimization , ( Academic Press, New York , 1981 ). |
26 . | M. Ainsworth and J. T. Oden , A posteriori error estimation in finite element analysis , ( Wiley , 2000 ). [CrossRef] |
27 . | W. E. Lorensen and H. E. Cline , “ Marching cubes: a high resolution 3D surface construction algorithm ,” in Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques ( ACM Press , 1987 ), pp. 163 – 169 . |
28 . | Z. Wu and J. M. Sullivan Jr. , “ Multiple material marching cubes algorithm ,” Int. J. Numer. Methods Eng. 58 , 189 – 207 ( 2003 ). [CrossRef] |
29 . | M. Garland and P. S. Heckbert , “ Surface simplification using quadric error metrics ,” in Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques ( ACM Press , 1997 ), pp. 209 – 216 . |
30 . | J. Bey , “ Tetrahedral grid refinement ,” Computing 55 , 355 – 378 ( 1995 ). [CrossRef] |
31 . | L. H. Wang , S. L. Jacques , and L. Q. Zheng , “ MCML-Monte Carlo modeling of photon transport in multi-layered tissues ,” Comput. Meth. Prog. Biomed. 47 , 131 – 146 ( 1995 ). [CrossRef] |
32 . | D. Boas , J. Culver , J. Stott , and A. Dunn , “ Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head ,” Opt. Express 10 , 159 – 169 ( 2002 ), http://www.opticsinfobase.org/abstract.cfm?URI=OPEX-10-3-159 . [PubMed] |
33 . | User’s Manual (release 3.0) of TracePro (Software for Opto-Mechanical Modeling), Lambda Research Corporation, Littleton, MA. |
34 . | H. Li , J. Tian , F. Zhu , W. Cong , L. V. Wang , E. A. Hoffman , and G. Wang , “ A mouse optical simulation enviroment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo Method ,” Acad. Radiol. 11 , 1029 – 1038 ( 2004 ). [CrossRef] [PubMed] |
35 . | J. Huang , X. Huang , D. Metaxes , and D. Banerjee , “ 3D tumor shape reconstruction from 2D bioluminescence images ,” presented in the third IEEE International Symposium on Biomedical Imaging (ISBI 2006), Virginia, USA , 6–9 Apr. 2006 . |
OCIS Codes
(110.6960) Imaging systems : Tomography
(170.3010) Medical optics and biotechnology : Image reconstruction techniques
(170.6280) Medical optics and biotechnology : Spectroscopy, fluorescence and luminescence
ToC Category:
Medical Optics and Biotechnology
History
Original Manuscript: March 16, 2006
Revised Manuscript: August 17, 2006
Manuscript Accepted: August 18, 2006
Published: September 1, 2006
Virtual Issues
Vol. 1, Iss. 10 Virtual Journal for Biomedical Optics
Citation
Yujie Lv, Jie Tian, Wenxiang Cong, Ge Wang, Jie Luo, Wei Yang, and Hui Li, "A multilevel adaptive finite element algorithm for bioluminescence tomography," Opt. Express 14, 8211-8223 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-18-8211
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References
- V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weisslder, "Looking and listening to light: the evolution of whole body photonic imaging," Nature Biotechnology 23(3),313-320 (2005). [CrossRef] [PubMed]
- C. Contag and M. H. Bachmann, "Advances in bioluminescence imaging of gene expression," Annu. Rev. Biomed. Eng. 4,235-260 (2002). [CrossRef]
- S. Bhaumik and S. S. Gambhir, "Optical imaging of Renilla luciferase reporter gene expression in living mice," Proc. Natl. Acad. Sci. USA 99,377-382 (2002). [CrossRef]
- T. F. Massoud and S. S. Gambhir, "Molecular imaging in living subjects: seeing fundamental biological processes in a new light," Genes Dev. 17,545-580 (2003). [CrossRef] [PubMed]
- G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. F. Meinel, "Development of the first bioluminescence ct scanner," Radiology 229(P),566 (2003).
- G. Wang, Y. Li, and M. Jiang, "Uniqueness theorems in bioluminescence tomography," Med. Phys. 31,2289- 2299 (2004). [CrossRef] [PubMed]
- 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]
- Thérése and J. W.Hastings, "Bioluminescence," Annu. Rev. Cell Dev. Bi. 14,197-230 (1998). [CrossRef]
- W. Rice, M. D. Cable, and M. B. Nelson, "In vivo imaging of light-emitting probes," J. Biomed. Opt. 6,432-440 (2001). [CrossRef] [PubMed]
- M. Jiang and G. Wang, "Image reconstruction for bioluminescence tomography," Proc. SPIE 5535,335-351 (2004). [CrossRef]
- W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, "A practical method to determine the light source distribution in bioluminescent imaging," Proc. SPIE 5535,679-686 (2004). [CrossRef]
- X. Gu, Q. Zhang, L. Larcom, and H. Jiang, "Three-dimensional bioluminescence tomography with model-based reconstruction," Opt. Express 12,3996-4000 (2004). [CrossRef] [PubMed]
- C. Kuo, O. Coquoz, D. G. Stearns, and B.W. Rice, "Diffuse luminescence tomography of in vivo bioluminescence markers using multi-spetral data," in Proceedings of the 3rd International Meeting of the Society (MIT Press, 2004), p. 227.
- A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, "Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging," Phys. Med. Biol. 50,5421-5441 (2005). [CrossRef] [PubMed]
- 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] [PubMed]
- W. 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] [PubMed]
- A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50,R1-R43 (2005). [CrossRef] [PubMed]
- S. Holder, Electrical Impedance Tomography, (Institute of Physics Publishing, Bristol and Philadelphia, 2005).
- A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, "Adaptive finite element based tomography for fluorescence optical imaging in tissue," Opt. Express 12,5402-5417 (2004). [CrossRef] [PubMed]
- S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, "A finite element approach for modeling photon transport in tissue," Med. Phys. 20,299-309 (1993). [CrossRef] [PubMed]
- 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] [PubMed]
- S. S. Rao, The finite element method in engineering, (Butterworth-Heinemann, Boston, 1999).
- G. Wang, M. Jiang, J. Tian, W. Cong, Y. Li, W. Han, D. Kumar, X. Qian, H. Shen, T. Zhou, J. Cheng, Y. Lv, H. Li, and J. Luo, "Recent development in bioluminescence tomography," presented in the third IEEE International Symposium on Biomedical Imaging (ISBI 2006), Virginia, USA, 6-9 Apr. 2006.
- W. Bangerth and R. Rannacher, Adaptive finite element methods for differential equations, (Birkh¨auser Verlag, 2003).
- P. E. Gill, W. Murray, and M. Wright, Practical optimization, (Academic Press, New York, 1981).
- M. Ainsworth and J. T. Oden, A posteriori error estimation in finite element analysis, (Wiley, 2000). [CrossRef]
- W. E. Lorensen and H. E. Cline, "Marching cubes: a high resolution 3D surface construction algorithm," in Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques (ACM Press, 1987), pp. 163-169.
- Z. Wu and J. M. Sullivan, Jr, "Multiple material marching cubes algorithm," Int. J. Numer. Methods Eng. 58,189-207 (2003). [CrossRef]
- M. Garland and P. S. Heckbert, "Surface simplification using quadric error metrics," in Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques (ACM Press, 1997), pp. 209-216.
- J. Bey, "Tetrahedral grid refinement," Computing 55,355-378 (1995). [CrossRef]
- L. H. Wang, S. L. Jacques, and L. Q. Zheng, "MCML-Monte Carlo modeling of photon transport in multi-layered tissues," Comput. Meth. Prog. Biomed. 47,131-146 (1995). [CrossRef]
- D. Boas, J. Culver, J. Stott, and A. Dunn, "Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head," Opt. Express 10,159-169 (2002). [PubMed]
- User’s Manual (release 3.0) of TracePro (Software for Opto-Mechanical Modeling), Lambda Research Corporation, Littleton, MA.
- H. Li, J. Tian, F. Zhu, W. 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] [PubMed]
- J. Huang, X. Huang, D. Metaxes, and D. Banerjee, "3D tumor shape reconstruction from 2D bioluminescence images," presented in the third IEEE International Symposium on Biomedical Imaging (ISBI 2006), Virginia, USA, 6-9 Apr. 2006.
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