Abstract

As a novel modality of molecular imaging, bioluminescence tomography (BLT) is used to in vivo observe and measure the biological process at cellular and molecular level in small animals. The core issue of BLT is to determine the distribution of internal bioluminescent sources from optical measurements on external surface. In this paper, a new algorithm is presented for BLT source reconstruction based on adaptive hp-finite element method. Using adaptive mesh refinement strategy and intelligent permissible source region, we can obtain more accurate information about the location and density of sources, with the robustness, stability and efficiency improved. Numerical simulations and physical experiment were both conducted to verify the performance of the proposed algorithm, where the optical data on phantom surface were obtained via Monte Carlo simulation and CCD camera detection, respectively. The results represent the merits and potential of our algorithm for BLT source reconstruction.

© 2009 OSA

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2009

2008

J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express 16(20), 15640–15654 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-20-15640 .
[CrossRef] [PubMed]

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27(5), 48–57 (2008).
[CrossRef] [PubMed]

Y. Hou, J. Tian, Y. Wu, J. Liang, and X. He, “A new numerical method for BLT forward problem based on high-order finite elements,” Commun. Numer. Methods Eng. 6, 667–681 (2008).

2007

D. Qin, H. Zhao, Y. Tanikawa, and F. Gao, “Experimental determination of optical properties in turbid medium by TCSPC technique,” Proc. SPIE 6434, 64342E (2007).
[CrossRef]

2006

2005

2004

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. 31(8), 2289–2299 (2004).
[CrossRef] [PubMed]

M. Gurfinkel, T. S. Pan, and E. M. Sevick-Muraca, “Determination of optical properties in semi-infinite turbid media using imaging measurements of frequency-domain photon migration obtained with an intensified charge-coupled device,” J. Biomed. Opt. 9(6), 1336–1346 (2004).
[CrossRef] [PubMed]

D. Piwnica-Worms, D. P. Schuster, and J. R. Garbow, “Molecular imaging of host-pathogen interactions in intact small animals,” Cell. Microbiol. 6(4), 319–331 (2004).
[CrossRef] [PubMed]

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(9), 1029–1038 (2004).
[CrossRef] [PubMed]

2003

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).

T. F. Massoud and S. S. Gambhir, “Molecular imaging in living subjects: seeing fundamental biological processes in a new light,” Genes Dev. 17(5), 545–580 (2003).
[CrossRef] [PubMed]

2002

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]

R. Weissleder, “Scaling down imaging: molecular mapping of cancer in mice,” Nat. Rev. Cancer 2(1), 11–18 (2002).
[CrossRef] [PubMed]

1997

M. Ainsworth and B. Senior, “Aspects of an adaptive hp-finite element method: Adaptive strategy conforming approximation and efficient solvers,” Comput. Methods Appl. M 150(1-4), 65–87 (1997).
[CrossRef]

1996

M. Ainsworth, “A preconditioner based on domain decomposition for hp-finite element approximation on quasi-uniform meshes,” SIAM J. Numer. Anal. 33(4), 1358–1376 (1996).
[CrossRef]

I. Babuška and B. Guo, “Approximation properties of the hp-version of the finite element method,” Comput. Methods Appl. Mech. Eng. 133(3-4), 319–346 (1996).
[CrossRef]

1995

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995).
[CrossRef] [PubMed]

1992

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[CrossRef] [PubMed]

Ainsworth, M.

M. Ainsworth and B. Senior, “Aspects of an adaptive hp-finite element method: Adaptive strategy conforming approximation and efficient solvers,” Comput. Methods Appl. M 150(1-4), 65–87 (1997).
[CrossRef]

M. Ainsworth, “A preconditioner based on domain decomposition for hp-finite element approximation on quasi-uniform meshes,” SIAM J. Numer. Anal. 33(4), 1358–1376 (1996).
[CrossRef]

Arridge, S. R.

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(11), 1779–1792 (1995).
[CrossRef] [PubMed]

Babuška, I.

I. Babuška and B. Guo, “Approximation properties of the hp-version of the finite element method,” Comput. Methods Appl. Mech. Eng. 133(3-4), 319–346 (1996).
[CrossRef]

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]

Bai, J.

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27(5), 48–57 (2008).
[CrossRef] [PubMed]

Bao, S.

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27(5), 48–57 (2008).
[CrossRef] [PubMed]

Chan, T. F.

Chance, B.

M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. 50(12), 2837–2858 (2005).
[CrossRef] [PubMed]

Chatziioannou, A. F.

Cong, A.

Cong, W.

Contag, C. 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]

Delpy, D. T.

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(11), 1779–1792 (1995).
[CrossRef] [PubMed]

Douraghy, A.

Farrell, T. J.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[CrossRef] [PubMed]

Feng, J.

Gambhir, S. S.

T. F. Massoud and S. S. Gambhir, “Molecular imaging in living subjects: seeing fundamental biological processes in a new light,” Genes Dev. 17(5), 545–580 (2003).
[CrossRef] [PubMed]

Gao, F.

D. Qin, H. Zhao, Y. Tanikawa, and F. Gao, “Experimental determination of optical properties in turbid medium by TCSPC technique,” Proc. SPIE 6434, 64342E (2007).
[CrossRef]

Garbow, J. R.

D. Piwnica-Worms, D. P. Schuster, and J. R. Garbow, “Molecular imaging of host-pathogen interactions in intact small animals,” Cell. Microbiol. 6(4), 319–331 (2004).
[CrossRef] [PubMed]

Guo, B.

I. Babuška and B. Guo, “Approximation properties of the hp-version of the finite element method,” Comput. Methods Appl. Mech. Eng. 133(3-4), 319–346 (1996).
[CrossRef]

Gurfinkel, M.

M. Gurfinkel, T. S. Pan, and E. M. Sevick-Muraca, “Determination of optical properties in semi-infinite turbid media using imaging measurements of frequency-domain photon migration obtained with an intensified charge-coupled device,” J. Biomed. Opt. 9(6), 1336–1346 (2004).
[CrossRef] [PubMed]

Guven, M.

M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. 50(12), 2837–2858 (2005).
[CrossRef] [PubMed]

He, X.

Y. Hou, J. Tian, Y. Wu, J. Liang, and X. He, “A new numerical method for BLT forward problem based on high-order finite elements,” Commun. Numer. Methods Eng. 6, 667–681 (2008).

Hiraoka, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995).
[CrossRef] [PubMed]

Hoffman, E.

Hoffman, E. A.

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(9), 1029–1038 (2004).
[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).

Hou, Y.

Y. Hou, J. Tian, Y. Wu, J. Liang, and X. He, “A new numerical method for BLT forward problem based on high-order finite elements,” Commun. Numer. Methods Eng. 6, 667–681 (2008).

Intes, X.

M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. 50(12), 2837–2858 (2005).
[CrossRef] [PubMed]

Jia, K.

Jiang, M.

Kumar, D.

Li, H.

Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express 14(18), 8211–8223 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8211 .
[CrossRef] [PubMed]

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(9), 1029–1038 (2004).
[CrossRef] [PubMed]

Li, Y.

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27(5), 48–57 (2008).
[CrossRef] [PubMed]

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. 31(8), 2289–2299 (2004).
[CrossRef] [PubMed]

Liang, J.

Y. Hou, J. Tian, Y. Wu, J. Liang, and X. He, “A new numerical method for BLT forward problem based on high-order finite elements,” Commun. Numer. Methods Eng. 6, 667–681 (2008).

Liang, W.

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27(5), 48–57 (2008).
[CrossRef] [PubMed]

Liu, Y.

Lu, Y.

Luo, J.

Lv, Y.

Massoud, T. F.

T. F. Massoud and S. S. Gambhir, “Molecular imaging in living subjects: seeing fundamental biological processes in a new light,” Genes Dev. 17(5), 545–580 (2003).
[CrossRef] [PubMed]

McCray, P.

McLennan, G.

Meinel, J. F.

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).

Ntziachristos, V.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23(3), 313–320 (2005).
[CrossRef] [PubMed]

Pan, T. S.

M. Gurfinkel, T. S. Pan, and E. M. Sevick-Muraca, “Determination of optical properties in semi-infinite turbid media using imaging measurements of frequency-domain photon migration obtained with an intensified charge-coupled device,” J. Biomed. Opt. 9(6), 1336–1346 (2004).
[CrossRef] [PubMed]

Patterson, M. S.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[CrossRef] [PubMed]

Piwnica-Worms, D.

D. Piwnica-Worms, D. P. Schuster, and J. R. Garbow, “Molecular imaging of host-pathogen interactions in intact small animals,” Cell. Microbiol. 6(4), 319–331 (2004).
[CrossRef] [PubMed]

Qin, C.

Qin, D.

D. Qin, H. Zhao, Y. Tanikawa, and F. Gao, “Experimental determination of optical properties in turbid medium by TCSPC technique,” Proc. SPIE 6434, 64342E (2007).
[CrossRef]

Ripoll, J.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23(3), 313–320 (2005).
[CrossRef] [PubMed]

Schuster, D. P.

D. Piwnica-Worms, D. P. Schuster, and J. R. Garbow, “Molecular imaging of host-pathogen interactions in intact small animals,” Cell. Microbiol. 6(4), 319–331 (2004).
[CrossRef] [PubMed]

Schweiger, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995).
[CrossRef] [PubMed]

Senior, B.

M. Ainsworth and B. Senior, “Aspects of an adaptive hp-finite element method: Adaptive strategy conforming approximation and efficient solvers,” Comput. Methods Appl. M 150(1-4), 65–87 (1997).
[CrossRef]

Sevick-Muraca, E. M.

M. Gurfinkel, T. S. Pan, and E. M. Sevick-Muraca, “Determination of optical properties in semi-infinite turbid media using imaging measurements of frequency-domain photon migration obtained with an intensified charge-coupled device,” J. Biomed. Opt. 9(6), 1336–1346 (2004).
[CrossRef] [PubMed]

Stout, D.

Suter, M.

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).

Tanikawa, Y.

D. Qin, H. Zhao, Y. Tanikawa, and F. Gao, “Experimental determination of optical properties in turbid medium by TCSPC technique,” Proc. SPIE 6434, 64342E (2007).
[CrossRef]

Tian, J.

Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express 17(10), 8062–8080 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-10-8062 .
[CrossRef] [PubMed]

J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express 16(20), 15640–15654 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-20-15640 .
[CrossRef] [PubMed]

Y. Hou, J. Tian, Y. Wu, J. Liang, and X. He, “A new numerical method for BLT forward problem based on high-order finite elements,” Commun. Numer. Methods Eng. 6, 667–681 (2008).

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27(5), 48–57 (2008).
[CrossRef] [PubMed]

Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express 14(18), 8211–8223 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8211 .
[CrossRef] [PubMed]

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(9), 1029–1038 (2004).
[CrossRef] [PubMed]

Wang, G.

Wang, L.

Wang, L. V.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23(3), 313–320 (2005).
[CrossRef] [PubMed]

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(9), 1029–1038 (2004).
[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).

Weissleder, R.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23(3), 313–320 (2005).
[CrossRef] [PubMed]

R. Weissleder, “Scaling down imaging: molecular mapping of cancer in mice,” Nat. Rev. Cancer 2(1), 11–18 (2002).
[CrossRef] [PubMed]

Wilson, B.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[CrossRef] [PubMed]

Wu, Y.

Y. Hou, J. Tian, Y. Wu, J. Liang, and X. He, “A new numerical method for BLT forward problem based on high-order finite elements,” Commun. Numer. Methods Eng. 6, 667–681 (2008).

Yan, G.

Yan, X. P.

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27(5), 48–57 (2008).
[CrossRef] [PubMed]

Yang, W.

Yang, X.

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27(5), 48–57 (2008).
[CrossRef] [PubMed]

Yazici, B.

M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. 50(12), 2837–2858 (2005).
[CrossRef] [PubMed]

Zabner, J.

Zhang, X.

Zhao, H.

D. Qin, H. Zhao, Y. Tanikawa, and F. Gao, “Experimental determination of optical properties in turbid medium by TCSPC technique,” Proc. SPIE 6434, 64342E (2007).
[CrossRef]

Zhu, F.

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(9), 1029–1038 (2004).
[CrossRef] [PubMed]

Zhu, S.

Acad. Radiol.

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Figures (8)

Fig. 1
Fig. 1

p-refinement and h-refinement in three-dimensional space. (a) is the p-refinement of an element, and (b)-(d) are the h-refinement of an element.

Fig. 2
Fig. 2

The flow chat of the proposed algorithm.

Fig. 3
Fig. 3

Heterogeneous phantom. (a) A heterogeneous phantom with a single light source, composed of muscle, lungs, heart, bone, liver and source in right lung; (b) The initial mesh used in adaptive hp-FEM algorithm.

Fig. 4
Fig. 4

Reconstruction results of single source simulation. (a) Result using FEM on a normal mesh; (b) Result using h-FEM; (c) Result using proposed algorithm, the red sphere denotes the actual source; (d), (e), and (f) are the cross section of (a), (b) and (c) at z = 0, respectively. The red dashed circularity denotes the actual source.

Fig. 5
Fig. 5

Dual source reconstruction results. (a) Result using FEM on a fine grid; (c) Result using the proposed algorithm; (b) and (d) are the amplified region near the actual source of (a) and (c), respectively. The red sphere is the actual source, and the green mesh denotes the tetrahedra near the actual source.

Fig. 6
Fig. 6

Physical phantom. (a) The homogeneous physical phantom; (b) The location of the single source in the phantom; (c) The middle cross section of the phantom. Four degrees show the direction of CCD camera during data acquisition.

Fig. 7
Fig. 7

The surface measured data of the homogeneous phantom. (a), (b), (c) and (d) are front view, left view, back view and right view of the cylindrical phantom on CCD camera, respectively; (e) is the photo flux density on the surface of the cylindrical phantom after mapping from 2D data.

Fig. 8
Fig. 8

Reconstruction results of phantom experiment with different permissible source region; the red cylinder is the actual source. (a), (c) Reconstruction result using FEM on a normal mesh and the proposed algorithm with a small permissible source region P 1; (e), (g) Reconstruction results using FEM on a normal mesh and the proposed algorithm with a lager permissible source region P 2; (b), (d), (f) and (h) are the amplified region near the actual source of (a), (c), (e) and(g), respectively. The green mesh denotes the tetrahedra near the actual source.

Tables (5)

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Table 1 Optical parameters of the heterogeneous phantom

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Table 2 Quantitative comparison between actual source and the reconstructed source with different methods

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Table 3 Reconstruction results with the proposed algorithm under different noise levels

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Table 4 Dual source reconstruction results with the proposed algorithm and FEM on fine grid

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Table 5 Reconstruction results in homogeneous physical phantom experiment

Equations (18)

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(D(x)Φ(x))+μa(x)Φ(x)=S(x)  (xΩ)
Φ(x)+2A(x;n,n')D(x)(v(x)Φ(x))=0  (xΩ)
Q(x)=D(x)(vΦ(x))=12An(x)Φ(x)  (xΩ)
A(x;n,n')=1+R(x)1R(x)
ΩD(x)(Φ(x))(Ψ(x))dx+Ωμa(x)Φ(x)Ψ(x)dx       +Ω12An(x)Φ(x)Ψ(x)dx=ΩS(x)Ψ(x)dx  (Ψ(x)H1(Ω))
Φ(x)Φk(x)=i=1Nϕik(x)Ψip(x)
S(x)Sk(x)=i=1Nsik(x)γip(x)
i=1N(Ω(D(x)Ψmp(x)Ψnp(x)+μaΨmp(x)Ψnp(x))dx                          +12AnΩΨmp(x)Ψnp(x)Φ(x)dx)=i=1NΩΨmp(x)γnp(x)dxS(x)
MkΦk=FkSk
[Mk11   Mk12Mk21   Mk22]{ΦkBΦkI}=[Fk11   Fk12Fk21   Fk22]{SkPerSkFor}
(Mk11Mk12(Mk22)1(Mk12)T)ΦkB=(Fk11Mk12(Mk22)1Fk21)SkPer
AkSkPer=ΦkB
minSinfSkperSsupΘ(Skper)={AkSkperΦkBΛ+λkη(SkPer)}
Θ(Skper)=AkSkperΦkBL2(Ω)+λSkPerL2(Ω)
Φ(Sper)ΦBL2(Ω)Chmin(p,t)p(t1/2)Φ(Sper)L2(Ω)C'hmin(p,t)p(t1/2)SperL2(Ω)
Θ(Sper)=ASperΦBL2(Ω)+λSper=Φ(Sper)ΦBL2(Ω)+λSperL2(Ω)           C'hmin(p,t)p(t1/2)SperL2(Ω)+λSperL2(Ω)            =C''hmin(p,t)p(t1/2)SperL2(Ω)
{Rk+1|PjRk+1 if  sPijδmax(s)}
Φkm¯=Φkm+δE

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