Abstract

The bioluminescence tomography is a novel molecular imaging technology for small animal studies. Known reconstruction methods require the completely measured data on the external surface, although only partially measured data is available in practice. In this work, we formulate a mathematical model for BLT from partial data and generalize our previous results on the solution uniqueness to the partial data case. Then we extend two of our reconstruction methods for BLT to this case. The first method is a variant of the well-known EM algorithm. The second one is based on the Landweber scheme. Both methods allow the incorporation of knowledgebased constraints. Two practical constraints, the source non-negativity and support constraints, are introduced to regularize the BLT problem and produce stability. The initial choice of both methods and its influence on the regularization and stability are also discussed. The proposed algorithms are evaluated and validated with intensive numerical simulation and a physical phantom experiment. Quantitative results including the location and source power accuracy are reported. Various algorithmic issues are investigated, especially how to avoid the inverse crime in numerical simulations.

© 2007 Optical Society of America

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2007 (1)

A. D. Klose, "Transport-theory-based stochastic image reconstruction of bioluminescent sources," J. Opt. Soc. Am., A 24, 1601-1608 (2007).
[CrossRef]

2006 (1)

N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, "Iterative reconstruction method for light emitting sources based on the diffusion equation," Med. Phys. 33, 61 - 68 (2006).
[CrossRef] [PubMed]

2005 (7)

H. Dehghani, S. Davis, S. D. Jiang, B. Pogue, K. Paulsen, and M. Patterson, "Spectrally resolved bioluminescence optical tomography," Optics Letters 31, 365 - 367 (2005).
[CrossRef]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [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. Cong and G. Wang, "A finite-element-based reconstruction method for 3D fluorescence tomography," Opt. Express 13, 9847-9857 (2005).
[CrossRef] [PubMed]

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]

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

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

2004 (6)

Z. Paroo, R. A. Bollinger, D. A. Braasch, E. Richer, D. R. Corey, P. P. Antich, and R. P. Mason, "Validating bioluminescence imaging as a high-throughput, quantitative modality for assessing tumor burden," Molecular Imaging 3, 117-124 (2004).
[CrossRef] [PubMed]

G. Wang, Y. Li, and M. Jiang, "Uniqueness theorems for bioluminescent tomography," Med. Phys. 31, 2289 -2299 (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," Academic Radiology 11, 1029 - 1038 (2004).
[CrossRef] [PubMed]

X. J. Gu, Q. H. Zhang, L. Larcom, and H. B. Jiang, "Three-dimensional bioluminescence tomography with model-based reconstruction," Opt. Express 12, 3996-4000 (2004).
[CrossRef] [PubMed]

C. Q. Li and H. B. Jiang, "Imaging of particle size and concentration in heterogeneous turbid media with multispectral diffuse optical tomography," Opt. Express 12, 6313-6318 (2004).
[CrossRef] [PubMed]

R. B. Schulz, J. Ripoll, and V. Ntziachristos, "Experimental fluorescence tomography of tissues with noncontact measurements," IEEE Transactions on Medical Imaging 23, 492-500 (2004).
[CrossRef] [PubMed]

2003 (5)

M. Jiang and G. Wang, "Convergence studies on iterative algorithms for image reconstruction," IEEE Transactions on Medical Imaging 22, 569 - 579 (2003).
[CrossRef] [PubMed]

A. D. Klose and A. H. Hielscher, "Quasi-Newton methods in optical tomographic image reconstruction," Inverse Problems 19, 387-409 (2003).
[CrossRef]

A. McCaffrey, M. A. Kay, and C. H. Contag, "Advancing molecular therapies through in vivo bioluminescent imaging," Molecuar Imaging 2, 75 - 86 (2003).
[CrossRef]

A. Soling and N. G. Rainov, "Bioluminescence imaging in vivo - application to cancer research," Expert Opinion on Biological Therapy 3, 1163 - 1172 (2003).
[PubMed]

J. C. Wu, I. Y. Chen, G. Sundaresan, J. J. Min, A. De, J. H. Qiao, M. C. Fishbein, and S. S. Gambhir, "Molecular imaging of cardiac cell transplantation in living animals using optical bioluminescence and positron emission tomography," Circulation 108, 1302 - 1305 (2003).
[CrossRef] [PubMed]

2002 (4)

C. H. Contag and B. D. Ross, "It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology," J. Magn. Reson. 16, 378 - 387 (2002).
[CrossRef]

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, "Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging," Neoplasia 2, 491 - 495 (2002).
[CrossRef]

M. Jiang and G. Wang, "Development of iterative algorithms for image reconstruction," J. X-Ray Sci. Technol. 10, 77 - 86 (2002). Invited Review.

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

2001 (1)

B. 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]

2000 (1)

E. A. Marengo, A. J. Devaney, and R. W. Ziolkowski, "Inverse source problem and mimnimum-energy sources," J. Opt. Soc. Am., A 17, 34 - 45 (2000).
[CrossRef]

1999 (1)

S. R. Arridge, "Optical tomography in medical imaging," Inverse Problems 15, R41 - R93 (1999).
[CrossRef]

1998 (1)

A. Sabharwal and L. C. Potter, "Convexly constrained linear inverse problems: iterative leat-squares and regularization," IEEE Transactions on Signal Processing 46, 2345 - 2352 (1998).
[CrossRef]

1997 (1)

M. Piana and M. Bertero, "Projected Landweber method and preconditioning," Inverse Problems 13, 441 - 463 (1997).
[CrossRef]

1996 (1)

R. J. Santos, "Equivalence of regularization and truncated iteration for general ill-posed problems," Linear Algebra and Its applications 236, 25-33 (1996).
[CrossRef]

1992 (2)

D. L. Snyder, T. J. Schulz, and J. A. O’Sullivan, "Deblurring subject to nonnegativity constraints," IEEE Transactions on Signal Processing 40, 1143 - 1150 (1992).
[CrossRef]

B. Eicke, "Iteration methods for convexly constrained ill-posed problems in Hilbert space," Numerical Functional Analysis and Optimization 13, 413 - 429 (1992).
[CrossRef]

1982 (1)

L. A. Shepp and Y. Vardi, "Maximum likelihood restoration for emission tomography," IEEE Transactions on Medical Imaging 1, 113 - 122 (1982).
[CrossRef] [PubMed]

1977 (1)

A. P. Dempster, N.M. Laird, and D. B. Rubin, "Maximal likelihood form incomplete data via the EM algorithm," Journal of the Royal Statistical Society. Series B. 39, 1 - 38 (1977).

Academic Radiology (1)

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," Academic Radiology 11, 1029 - 1038 (2004).
[CrossRef] [PubMed]

Annu. Rev. Biomed. Eng. (1)

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

Circulation (1)

J. C. Wu, I. Y. Chen, G. Sundaresan, J. J. Min, A. De, J. H. Qiao, M. C. Fishbein, and S. S. Gambhir, "Molecular imaging of cardiac cell transplantation in living animals using optical bioluminescence and positron emission tomography," Circulation 108, 1302 - 1305 (2003).
[CrossRef] [PubMed]

Expert Opinion on Biological Therapy (1)

A. Soling and N. G. Rainov, "Bioluminescence imaging in vivo - application to cancer research," Expert Opinion on Biological Therapy 3, 1163 - 1172 (2003).
[PubMed]

IEEE Transactions on Medical Imaging (3)

L. A. Shepp and Y. Vardi, "Maximum likelihood restoration for emission tomography," IEEE Transactions on Medical Imaging 1, 113 - 122 (1982).
[CrossRef] [PubMed]

M. Jiang and G. Wang, "Convergence studies on iterative algorithms for image reconstruction," IEEE Transactions on Medical Imaging 22, 569 - 579 (2003).
[CrossRef] [PubMed]

R. B. Schulz, J. Ripoll, and V. Ntziachristos, "Experimental fluorescence tomography of tissues with noncontact measurements," IEEE Transactions on Medical Imaging 23, 492-500 (2004).
[CrossRef] [PubMed]

IEEE Transactions on Signal Processing (2)

A. Sabharwal and L. C. Potter, "Convexly constrained linear inverse problems: iterative leat-squares and regularization," IEEE Transactions on Signal Processing 46, 2345 - 2352 (1998).
[CrossRef]

D. L. Snyder, T. J. Schulz, and J. A. O’Sullivan, "Deblurring subject to nonnegativity constraints," IEEE Transactions on Signal Processing 40, 1143 - 1150 (1992).
[CrossRef]

International Journal of Biomedical Imaging (1)

A. Cong and G. Wang, "Multispectral bioluminescence tomography: Methodology and simulation," International Journal of Biomedical Imaging 2006 (2006). Article ID 57614. doi:10.1155/IJBI/2006/57614.

Inverse Problems (3)

S. R. Arridge, "Optical tomography in medical imaging," Inverse Problems 15, R41 - R93 (1999).
[CrossRef]

M. Piana and M. Bertero, "Projected Landweber method and preconditioning," Inverse Problems 13, 441 - 463 (1997).
[CrossRef]

A. D. Klose and A. H. Hielscher, "Quasi-Newton methods in optical tomographic image reconstruction," Inverse Problems 19, 387-409 (2003).
[CrossRef]

J. Biomed. Opt. (1)

B. 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]

J. Magn. Reson. (1)

C. H. Contag and B. D. Ross, "It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology," J. Magn. Reson. 16, 378 - 387 (2002).
[CrossRef]

J. Opt. Soc. Am., A (2)

A. D. Klose, "Transport-theory-based stochastic image reconstruction of bioluminescent sources," J. Opt. Soc. Am., A 24, 1601-1608 (2007).
[CrossRef]

E. A. Marengo, A. J. Devaney, and R. W. Ziolkowski, "Inverse source problem and mimnimum-energy sources," J. Opt. Soc. Am., A 17, 34 - 45 (2000).
[CrossRef]

J. X-Ray Sci. Technol. (1)

M. Jiang and G. Wang, "Development of iterative algorithms for image reconstruction," J. X-Ray Sci. Technol. 10, 77 - 86 (2002). Invited Review.

Journal of the Royal Statistical Society. Series B. (1)

A. P. Dempster, N.M. Laird, and D. B. Rubin, "Maximal likelihood form incomplete data via the EM algorithm," Journal of the Royal Statistical Society. Series B. 39, 1 - 38 (1977).

Linear Algebra and Its applications (1)

R. J. Santos, "Equivalence of regularization and truncated iteration for general ill-posed problems," Linear Algebra and Its applications 236, 25-33 (1996).
[CrossRef]

Med. Phys. (2)

N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, "Iterative reconstruction method for light emitting sources based on the diffusion equation," Med. Phys. 33, 61 - 68 (2006).
[CrossRef] [PubMed]

G. Wang, Y. Li, and M. Jiang, "Uniqueness theorems for bioluminescent tomography," Med. Phys. 31, 2289 -2299 (2004).
[CrossRef] [PubMed]

Molecuar Imaging (1)

A. McCaffrey, M. A. Kay, and C. H. Contag, "Advancing molecular therapies through in vivo bioluminescent imaging," Molecuar Imaging 2, 75 - 86 (2003).
[CrossRef]

Molecular Imaging (1)

Z. Paroo, R. A. Bollinger, D. A. Braasch, E. Richer, D. R. Corey, P. P. Antich, and R. P. Mason, "Validating bioluminescence imaging as a high-throughput, quantitative modality for assessing tumor burden," Molecular Imaging 3, 117-124 (2004).
[CrossRef] [PubMed]

Nat. Biotech. (1)

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

Neoplasia (1)

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, "Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging," Neoplasia 2, 491 - 495 (2002).
[CrossRef]

Numerical Functional Analysis and Optimization (1)

B. Eicke, "Iteration methods for convexly constrained ill-posed problems in Hilbert space," Numerical Functional Analysis and Optimization 13, 413 - 429 (1992).
[CrossRef]

Opt. Express (4)

Optics Letters (1)

H. Dehghani, S. Davis, S. D. Jiang, B. Pogue, K. Paulsen, and M. Patterson, "Spectrally resolved bioluminescence optical tomography," Optics Letters 31, 365 - 367 (2005).
[CrossRef]

Phys. Med. Biol. (3)

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).
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[CrossRef]

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[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

(a) A heterogeneous mouse phantom consisting of bone (B), heart (H), lungs (L), and muscle (M). (b) A cross-section through two luminescent sources in the left lung and another source in the right lung. The four arrows show the directions of the CCD camera for data measurement.

Fig. 2.
Fig. 2.

Reconstructed results by the CL method and a cross-section at z=0cm. (a) and (b) are results from data measured at the four views. (c) and (d) are from data measured at the front view only.

Fig. 3.
Fig. 3.

(a) A cross-section through two hollow cylinders for hosting luminescent sources in the left lung. The four arrows show the direction of the CCD camera during data acquisition. (b) The measurements at the four views combined along the phantom side surface with unit µW/cm2

Fig. 4.
Fig. 4.

Representative results reconstructed by the EM algorithm. (a) and (c) are the sources reconstructed by the EMalgorithm from the data measured in the four views and in the front view only, respectively. (b) and (d) are cross-sections at z=0 cm of the sources in (a) and (c), respectively.

Tables (8)

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Table 1. Optical parameters for the phantom

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Table 2. Finite element information for the simulation

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Table 3. Quantitative results for the reconstructed locations of the three sources at S1=(-0.90, 0.25, 0), S2=(-0.90, -0.25, 0) and S3=(0.90, 0.25, 0), respectively. The unit is cm.

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Table 4. Quantitative results for the reconstructed source integrals of the sources. The sources are listed in the order as in Table 3. Their true values are 105.1, 97.4 and 105.1, respectively. The unit is nW.

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Table 5. Quantitative results for the reconstructed source moments of the sources. The sources are listed in the order as in Table 3. Their true values are 125.5, 116.5 and 125.5, respectively. The unit is nW.

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Table 6. Quantitative results for the reconstructed locations of the two sources at S1=(-0.90,0.15,0) and S2=(-0.90,-0.15,0), respectively. The unit is cm.

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Table 7. Quantitative results for the reconstructed source integrals of the sources. The sources are listed in the order as in Table 6. Their true values are 105.1 and 97.4, respectively. The unit is nW.

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Table 8. Finite element information for the physical phantom experiment

Equations (71)

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1 c u t ( x , θ , t ) + θ · x u ( x , θ , t ) + μ ( x ) u ( x , θ , t ) = μ s ( x ) S 2 η ( θ · θ ) u ( x , θ , t ) d θ + q ( x , θ , t )
u ( x , θ , 0 ) = 0 , x Ω , θ S 2 ,
u ( x , θ , t ) = g ( x , θ , t ) , x Γ , θ S 2 , ν ( x ) · θ < 0 , t > 0 ,
g ( x , t ) = S 2 ν ( x ) · θ u ( x , θ , t ) d θ , x = Γ , t 0 .
u 0 ( x , t ) = 1 4 π S 2 u ( x , θ , t ) d θ .
μ s = ( 1 η ¯ ) μ s ,
D ( x ) = 1 3 ( μ a ( x ) + μ s ( x ) ) ,
q 0 ( x , t ) = 1 4 π S 2 q ( x , θ , t ) d θ .
1 c u 0 ( x , t ) t · ( D ( x ) u 0 ( x , t ) ) + μ a ( x ) u 0 ( x , t ) = q 0 ( x , t ) , x Ω , t > 0
u 0 ( x , 0 ) = 0 , x Ω ,
u 0 ( x , t ) + 2 D ( x ) u 0 ν ( x , t ) = g ( x , t ) , x Γ , t > 0 .
g ( x , t ) = D ( x ) u 0 ν ( x , t ) .
· ( D ( x ) u 0 ( x ) ) + μ a ( x ) u 0 ( x ) = q 0 ( x ) , x Ω ,
u 0 ( x ) + 2 D ( x ) u 0 ν ( x ) = g ( x ) , x Γ .
BLT ( P ) { · ( D u 0 ) + μ a u 0 = q 0 , in Ω , u 0 + 2 D u 0 ν = g , on Γ , D u 0 ν = g , on Γ P .
γ 0 [ u ] = u Γ , and γ 1 [ u ] = D u ν Γ
L [ u ] = · ( D u ) + μ a u .
L [ w 1 ] = 0 , in Ω ,
γ 0 [ w 1 ] = f , on Γ P
γ 0 [ w 1 ] + 2 γ 1 [ w 1 ] = g , on Γ Γ P .
N Γ P [ f ] = γ 1 [ w 1 ] Γ P .
L [ w 2 ] = q 0 , in Ω ,
γ 0 [ w 2 ] = 0 , on Γ P ,
γ 0 [ w 2 ] + 2 γ 1 [ w 2 ] = 0 , on Γ Γ P .
Λ Γ P [ q 0 ] = γ 1 [ w 2 ] Γ P .
L [ u ] = q 0 , in Ω ,
γ 0 [ u ] = g + 2 g , on Γ P ,
γ 0 [ u ] + 2 γ 1 [ u ] = g , on Γ Γ P .
L [ v ] = q 0 , in Ω .
γ 0 [ v ] = g + 2 g , on Γ P ,
γ 0 [ v ] + 2 γ 1 [ v ] = g , on Γ Γ P .
g = γ 1 [ u ] = γ 1 [ w 1 ] + γ 1 [ w 2 ] = N Γ P [ g + 2 g ] Λ Γ P [ q 0 ] , on Γ P .
Λ Γ P [ q 0 ] = N Γ P [ g + 2 g ] + g , on Γ P .
Ω [ v · L [ w ] w · L [ v ] ] dx = Γ [ v γ 1 [ w ] w γ 1 [ v ] ] d Γ .
ϕ = T Γ P [ ψ ] ,
L [ ϕ ] = 0 , in Ω ,
γ 0 [ ϕ ] = ψ , on Γ P ,
γ 0 [ ϕ ] + 2 γ 1 [ ϕ ] = 0 , on Γ Γ P .
q 0 , T Γ P [ ψ ] L 2 ( Ω ) = Λ Γ P [ q 0 ] , ψ L 2 ( Γ P ) ,
Λ Γ P * = T Γ P .
𝒩 [ Λ Γ P ] = 𝓡 [ Λ Γ P * ] = 𝓡 [ T Γ P ] .
H 0 , Γ P 2 ( Ω ) = { p H 2 ( Ω ) : γ 0 [ p ] Γ P = 0 , γ 1 [ p ] Γ P = 0 , and γ 0 [ p ] + 2 γ 1 [ p ] Γ Γ P = 0 } .
𝓡 [ T Γ P ] = L [ H 0 , Γ P 2 ( Ω ) ] .
q , v L 2 ( Ω ) = Ω v L [ p ] d x = Γ [ v γ 1 [ p ] p γ 1 [ v ] ] d Γ + Ω L [ v ] p d x = 0 ,
L [ w 2 ] = q , in Ω γ 0 [ w 2 ] = 0 , on Γ P γ 0 [ w 2 ] + 2 γ 1 [ w 2 ] = 0 , on Γ \ Γ P , γ 1 [ w 2 ] = 0 , on Γ P .
N Γ P [ g + 2 g ] + g H 1 2 ( Γ P ) ,
q 0 ( y ) = s = 1 S a s δ ( y y s ) .
q 0 ( y ) = s = 1 S g s ( y x s ) χ B r 0 s , r 1 s ( x s )
φ ( r ) = 1 ,
φ ( r ) = sinh ( μ a D r ) μ a D r .
r 0 s r 1 s r N 1 φ C ( s ) ( r ) g s ( r ) dr = R 0 τ ( s ) R 1 τ ( s ) r N 1 φ C ( s ) ( r ) G τ ( s ) ( r ) dr , for s = 1 , , S ,
Λ Γ P [ q 0 ] = b .
F [ q 0 ] = Γ P { b log Λ Γ P [ q 0 ] Λ Γ P [ q 0 ] } d Γ ,
arg max F q 0 0 [ q 0 ] .
f ( t ) = F [ q 0 + tv ] , for t around 0 ,
F [ q 0 ] = Λ Γ P * [ b Λ Γ P [ q 0 ] 1 ] L 2 ( Ω ) .
q 0 · Λ Γ P * [ b Λ Γ P [ q 0 ] 1 ] = 0 .
L [ ϕ 1 ] = 0 , in Ω ,
γ 0 [ ϕ 1 ] = 1 , on Γ P ,
γ 0 [ ϕ 1 ] + 2 γ 1 [ ϕ 1 ] = 0 , on Γ Γ P .
q 0 = 1 ϕ 1 q 0 · T Γ P [ b Λ Γ P [ q 0 ] ] .
q 0 ( n + 1 ) = 1 ϕ 1 q 0 ( n ) · T Γ P [ b Λ Γ P [ q 0 ( n ) ] ] .
𝓒 = { q 0 : q 0 satisfies some convex constraints . } ,
q 0 ( n + 1 ) = P 𝓒 { q 0 ( n ) + λ n Λ Γ P * [ b Λ Γ P [ q 0 ( n ) ] ] } ,
arg min q 0 𝓒 1 2 b Λ Γ P [ q 0 ] L 2 ( Γ ) 2 .
Ω [ μ a u q 0 ] dx = Γ g d Γ .
Ω q 0 ( 0 ) dx = Γ g d Γ + Ω μ a u dx .
Ω q 0 ( 0 ) dx Γ g d Γ + Ω μ a w 1 dx .
q 0 ( 0 ) = Q 0 χ Ω 0
Q 0 Ω 0 Γ P g d Γ + Ω μ a w 1 dx ,
q i ( x ) = A i χ Ω i ( x ) , Ω = { x x x 0 < r } ,

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