Abstract

We present the first algorithm for solving the equation of radiative transfer (ERT) in the frequency domain (FD) on three-dimensional block-structured Cartesian grids (BSG). This algorithm allows for accurate modeling of light propagation in media of arbitrary shape with air-tissue refractive index mismatch at the boundary at increased speed compared to currently available structured grid algorithms. To accurately model arbitrarily shaped geometries the algorithm generates BSGs that are finely discretized only near physical boundaries and therefore less dense than fine grids. We discretize the FD-ERT using a combination of the upwind-step method and the discrete ordinates (SN) approximation. The source iteration technique is used to obtain the solution. We implement a first order interpolation scheme when traversing between coarse and fine grid regions. Effects of geometry and optical parameters on algorithm performance are evaluated using numerical phantoms (circular, cylindrical, and arbitrary shape) and varying the absorption and scattering coefficients, modulation frequency, and refractive index. The solution on a 3-level BSG is obtained up to 4.2 times faster than the solution on a single fine grid, with minimal increase in numerical error (less than 5%).

© 2010 OSA

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2010

A. D. Klose, B. J. Beattie, H. Dehghani, L. Vider, C. Le, V. Ponomarev, and R. Blasberg, “In vivo bioluminescence tomography with a blocking-off finite-difference SP3 method and MRI/CT coregistration,” Med. Phys. 37(1), 329–338 (2010).
[CrossRef] [PubMed]

2009

H. K. Kim and A. H. Hielscher, “A PDE-constrained SQP algorithm for optical tomography based on the frequency-domain equation of radiative transfer,” Inverse Probl. 25(1), 015010 (2009).
[CrossRef]

O. Gheysens and F. M. Mottaghy, “Method of bioluminescence imaging for molecular imaging of physiological and pathological processes,” Methods 48(2), 139–145 (2009).
[CrossRef] [PubMed]

2008

U. J. Netz, J. Beuthan, and A. H. Hielscher, “Multipixel system for gigahertz frequency-domain optical imaging of finger joints,” Rev. Sci. Instrum. 79(3), 034301 (2008).
[CrossRef] [PubMed]

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc. Natl. Acad. Sci. U.S.A. 105(49), 19126–19131 (2008).
[CrossRef] [PubMed]

2007

J. M. Lasker, J. M. Masciotti, M. Schoenecker, C. H. Schmitz, and A. H. Hielscher, “Digital-signal-processor-based dynamic imaging system for optical tomography,” Rev. Sci. Instrum. 78(8), 083706 (2007).
[CrossRef] [PubMed]

X. Gu, K. Ren, and A. H. Hielscher, “Frequency-domain sensitivity analysis for small imaging domains using the equation of radiative transfer,” Appl. Opt. 46(10), 1624–1632 (2007).
[CrossRef] [PubMed]

2006

J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency domain photon migration,” Med. Phys. 33(12), 4685–4700 (2006).
[CrossRef] [PubMed]

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

A. Joshi, W. Bangerth, K. Hwang, J. C. Rasmussen, and E. M. Sevick-Muraca, “Fully adaptive FEM based fluorescence optical tomography from time-dependent measurements with area illumination and detection,” Med. Phys. 33(5), 1299–1310 (2006).
[CrossRef] [PubMed]

2005

A. H. Hielscher, “Optical tomographic imaging of small animals,” Curr. Opin. Biotechnol. 16(1), 79–88 (2005).
[CrossRef] [PubMed]

A. K. Scheel, M. Backhaus, A. D. Klose, B. Moa-Anderson, U. J. Netz, K. G. Hermann, J. Beuthan, G. A. Müller, G. R. Burmester, and A. H. Hielscher, “First clinical evaluation of sagittal laser optical tomography for detection of synovitis in arthritic finger joints,” Ann. Rheum. Dis. 64(2), 239–245 (2005).
[CrossRef] [PubMed]

A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging 24(10), 1377–1386 (2005).
[CrossRef] [PubMed]

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202(1), 323–345 (2005).
[CrossRef]

M. B. Salah, F. Askri, and S. B. Nasrallah, “Unstructured control-volume finite element method for radiative heat transfer in a complex 2-D geometry,” Numer. Heat Transf. B 48(5), 477–497 (2005).
[CrossRef]

2004

2003

2002

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

1999

A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26(8), 1698–1707 (1999).
[CrossRef] [PubMed]

1998

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[CrossRef] [PubMed]

O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14(5), 1107–1130 (1998).
[CrossRef]

B. W. Pogue and G. Burke, “Fiber-optic bundle design for quantitative fluorescence measurement from tissue,” Appl. Opt. 37(31), 7429–7436 (1998).
[CrossRef] [PubMed]

J. P. Jessee, W. A. Fiveland, L. H. Howell, P. Colella, and R. B. Pember, “An Adaptive Mesh Refinement Algorithm for the Radiative Transport Equation,” J. Comput. Phys. 139(2), 380–398 (1998).
[CrossRef]

1997

W. L. Chen, F. S. Lien, and M. A. Leschziner, “Local mesh refinement within a multi-block structured-grid scheme for genereal flows,” Comput. Methods Appl. Mech. Eng. 144(3-4), 327–369 (1997).
[CrossRef]

1995

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995).
[CrossRef] [PubMed]

1994

1993

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[CrossRef] [PubMed]

B. J. Tromberg, L. O. Svaasand, T. T. Tsay, and R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt. 32(4), 607–616 (1993).
[CrossRef] [PubMed]

1992

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37(7), 1531–1560 (1992).
[CrossRef] [PubMed]

1989

1984

M. J. Berger and J. Oliger, “Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations,” J. Comput. Phys. 53(3), 484–512 (1984).
[CrossRef]

1979

R. B. Simpson, “Automatic local refinement for irregular rectangular meshes,” Int. J. Numer. Methods Eng. 14(11), 1665–1678 (1979).
[CrossRef]

Abdoulaev, G. S.

Aikawa, E.

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc. Natl. Acad. Sci. U.S.A. 105(49), 19126–19131 (2008).
[CrossRef] [PubMed]

Alcouffe, R. E.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[CrossRef] [PubMed]

Arridge, S. R.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37(7), 1531–1560 (1992).
[CrossRef] [PubMed]

Askri, F.

M. B. Salah, F. Askri, and S. B. Nasrallah, “Unstructured control-volume finite element method for radiative heat transfer in a complex 2-D geometry,” Numer. Heat Transf. B 48(5), 477–497 (2005).
[CrossRef]

Backhaus, M.

A. K. Scheel, M. Backhaus, A. D. Klose, B. Moa-Anderson, U. J. Netz, K. G. Hermann, J. Beuthan, G. A. Müller, G. R. Burmester, and A. H. Hielscher, “First clinical evaluation of sagittal laser optical tomography for detection of synovitis in arthritic finger joints,” Ann. Rheum. Dis. 64(2), 239–245 (2005).
[CrossRef] [PubMed]

Bal, G.

Bangerth, W.

A. Joshi, W. Bangerth, K. Hwang, J. C. Rasmussen, and E. M. Sevick-Muraca, “Fully adaptive FEM based fluorescence optical tomography from time-dependent measurements with area illumination and detection,” Med. Phys. 33(5), 1299–1310 (2006).
[CrossRef] [PubMed]

Barbour, R. L.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[CrossRef] [PubMed]

Beattie, B. J.

A. D. Klose, B. J. Beattie, H. Dehghani, L. Vider, C. Le, V. Ponomarev, and R. Blasberg, “In vivo bioluminescence tomography with a blocking-off finite-difference SP3 method and MRI/CT coregistration,” Med. Phys. 37(1), 329–338 (2010).
[CrossRef] [PubMed]

Berger, M. J.

M. J. Berger and P. Colella, “Local adaptive mesh refinement for shock-hydrodynamics,” J. Comput. Phys. 82(1), 64–84 (1989).
[CrossRef]

M. J. Berger and J. Oliger, “Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations,” J. Comput. Phys. 53(3), 484–512 (1984).
[CrossRef]

Beuthan, J.

U. J. Netz, J. Beuthan, and A. H. Hielscher, “Multipixel system for gigahertz frequency-domain optical imaging of finger joints,” Rev. Sci. Instrum. 79(3), 034301 (2008).
[CrossRef] [PubMed]

A. K. Scheel, M. Backhaus, A. D. Klose, B. Moa-Anderson, U. J. Netz, K. G. Hermann, J. Beuthan, G. A. Müller, G. R. Burmester, and A. H. Hielscher, “First clinical evaluation of sagittal laser optical tomography for detection of synovitis in arthritic finger joints,” Ann. Rheum. Dis. 64(2), 239–245 (2005).
[CrossRef] [PubMed]

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

Blasberg, R.

A. D. Klose, B. J. Beattie, H. Dehghani, L. Vider, C. Le, V. Ponomarev, and R. Blasberg, “In vivo bioluminescence tomography with a blocking-off finite-difference SP3 method and MRI/CT coregistration,” Med. Phys. 37(1), 329–338 (2010).
[CrossRef] [PubMed]

Burke, G.

Burmester, G. R.

A. K. Scheel, M. Backhaus, A. D. Klose, B. Moa-Anderson, U. J. Netz, K. G. Hermann, J. Beuthan, G. A. Müller, G. R. Burmester, and A. H. Hielscher, “First clinical evaluation of sagittal laser optical tomography for detection of synovitis in arthritic finger joints,” Ann. Rheum. Dis. 64(2), 239–245 (2005).
[CrossRef] [PubMed]

Chance, B.

Chen, W. L.

W. L. Chen, F. S. Lien, and M. A. Leschziner, “Local mesh refinement within a multi-block structured-grid scheme for genereal flows,” Comput. Methods Appl. Mech. Eng. 144(3-4), 327–369 (1997).
[CrossRef]

Colella, P.

J. P. Jessee, W. A. Fiveland, L. H. Howell, P. Colella, and R. B. Pember, “An Adaptive Mesh Refinement Algorithm for the Radiative Transport Equation,” J. Comput. Phys. 139(2), 380–398 (1998).
[CrossRef]

M. J. Berger and P. Colella, “Local adaptive mesh refinement for shock-hydrodynamics,” J. Comput. Phys. 82(1), 64–84 (1989).
[CrossRef]

Cope, M.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37(7), 1531–1560 (1992).
[CrossRef] [PubMed]

de Kleine, R. H.

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc. Natl. Acad. Sci. U.S.A. 105(49), 19126–19131 (2008).
[CrossRef] [PubMed]

Dehghani, H.

A. D. Klose, B. J. Beattie, H. Dehghani, L. Vider, C. Le, V. Ponomarev, and R. Blasberg, “In vivo bioluminescence tomography with a blocking-off finite-difference SP3 method and MRI/CT coregistration,” Med. Phys. 37(1), 329–338 (2010).
[CrossRef] [PubMed]

Delpy, D. T.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37(7), 1531–1560 (1992).
[CrossRef] [PubMed]

Dorn, O.

O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14(5), 1107–1130 (1998).
[CrossRef]

Fiveland, W. A.

J. P. Jessee, W. A. Fiveland, L. H. Howell, P. Colella, and R. B. Pember, “An Adaptive Mesh Refinement Algorithm for the Radiative Transport Equation,” J. Comput. Phys. 139(2), 380–398 (1998).
[CrossRef]

Gheysens, O.

O. Gheysens and F. M. Mottaghy, “Method of bioluminescence imaging for molecular imaging of physiological and pathological processes,” Methods 48(2), 139–145 (2009).
[CrossRef] [PubMed]

Gu, X.

Haskell, R. C.

Hermann, K. G.

A. K. Scheel, M. Backhaus, A. D. Klose, B. Moa-Anderson, U. J. Netz, K. G. Hermann, J. Beuthan, G. A. Müller, G. R. Burmester, and A. H. Hielscher, “First clinical evaluation of sagittal laser optical tomography for detection of synovitis in arthritic finger joints,” Ann. Rheum. Dis. 64(2), 239–245 (2005).
[CrossRef] [PubMed]

Hielscher, A. H.

H. K. Kim and A. H. Hielscher, “A PDE-constrained SQP algorithm for optical tomography based on the frequency-domain equation of radiative transfer,” Inverse Probl. 25(1), 015010 (2009).
[CrossRef]

U. J. Netz, J. Beuthan, and A. H. Hielscher, “Multipixel system for gigahertz frequency-domain optical imaging of finger joints,” Rev. Sci. Instrum. 79(3), 034301 (2008).
[CrossRef] [PubMed]

X. Gu, K. Ren, and A. H. Hielscher, “Frequency-domain sensitivity analysis for small imaging domains using the equation of radiative transfer,” Appl. Opt. 46(10), 1624–1632 (2007).
[CrossRef] [PubMed]

J. M. Lasker, J. M. Masciotti, M. Schoenecker, C. H. Schmitz, and A. H. Hielscher, “Digital-signal-processor-based dynamic imaging system for optical tomography,” Rev. Sci. Instrum. 78(8), 083706 (2007).
[CrossRef] [PubMed]

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202(1), 323–345 (2005).
[CrossRef]

A. K. Scheel, M. Backhaus, A. D. Klose, B. Moa-Anderson, U. J. Netz, K. G. Hermann, J. Beuthan, G. A. Müller, G. R. Burmester, and A. H. Hielscher, “First clinical evaluation of sagittal laser optical tomography for detection of synovitis in arthritic finger joints,” Ann. Rheum. Dis. 64(2), 239–245 (2005).
[CrossRef] [PubMed]

A. H. Hielscher, “Optical tomographic imaging of small animals,” Curr. Opin. Biotechnol. 16(1), 79–88 (2005).
[CrossRef] [PubMed]

K. Ren, G. S. Abdoulaev, G. Bal, and A. H. Hielscher, “Algorithm for solving the equation of radiative transfer in the frequency domain,” Opt. Lett. 29(6), 578–580 (2004).
[CrossRef] [PubMed]

A. D. Klose and A. H. Hielscher, “Fluorescence tomography with simulated data based on the equation of radiative transfer,” Opt. Lett. 28(12), 1019–1021 (2003).
[CrossRef] [PubMed]

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

A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26(8), 1698–1707 (1999).
[CrossRef] [PubMed]

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[CrossRef] [PubMed]

Hiraoka, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[CrossRef] [PubMed]

Howell, L. H.

J. P. Jessee, W. A. Fiveland, L. H. Howell, P. Colella, and R. B. Pember, “An Adaptive Mesh Refinement Algorithm for the Radiative Transport Equation,” J. Comput. Phys. 139(2), 380–398 (1998).
[CrossRef]

Hwang, K.

A. Joshi, W. Bangerth, K. Hwang, J. C. Rasmussen, and E. M. Sevick-Muraca, “Fully adaptive FEM based fluorescence optical tomography from time-dependent measurements with area illumination and detection,” Med. Phys. 33(5), 1299–1310 (2006).
[CrossRef] [PubMed]

Jacques, S. L.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995).
[CrossRef] [PubMed]

Jessee, J. P.

J. P. Jessee, W. A. Fiveland, L. H. Howell, P. Colella, and R. B. Pember, “An Adaptive Mesh Refinement Algorithm for the Radiative Transport Equation,” J. Comput. Phys. 139(2), 380–398 (1998).
[CrossRef]

Joshi, A.

J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency domain photon migration,” Med. Phys. 33(12), 4685–4700 (2006).
[CrossRef] [PubMed]

A. Joshi, W. Bangerth, K. Hwang, J. C. Rasmussen, and E. M. Sevick-Muraca, “Fully adaptive FEM based fluorescence optical tomography from time-dependent measurements with area illumination and detection,” Med. Phys. 33(5), 1299–1310 (2006).
[CrossRef] [PubMed]

Kim, H. K.

H. K. Kim and A. H. Hielscher, “A PDE-constrained SQP algorithm for optical tomography based on the frequency-domain equation of radiative transfer,” Inverse Probl. 25(1), 015010 (2009).
[CrossRef]

Kirsch, D. G.

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc. Natl. Acad. Sci. U.S.A. 105(49), 19126–19131 (2008).
[CrossRef] [PubMed]

Klose, A. D.

A. D. Klose, B. J. Beattie, H. Dehghani, L. Vider, C. Le, V. Ponomarev, and R. Blasberg, “In vivo bioluminescence tomography with a blocking-off finite-difference SP3 method and MRI/CT coregistration,” Med. Phys. 37(1), 329–338 (2010).
[CrossRef] [PubMed]

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202(1), 323–345 (2005).
[CrossRef]

A. K. Scheel, M. Backhaus, A. D. Klose, B. Moa-Anderson, U. J. Netz, K. G. Hermann, J. Beuthan, G. A. Müller, G. R. Burmester, and A. H. Hielscher, “First clinical evaluation of sagittal laser optical tomography for detection of synovitis in arthritic finger joints,” Ann. Rheum. Dis. 64(2), 239–245 (2005).
[CrossRef] [PubMed]

A. D. Klose and A. H. Hielscher, “Fluorescence tomography with simulated data based on the equation of radiative transfer,” Opt. Lett. 28(12), 1019–1021 (2003).
[CrossRef] [PubMed]

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

A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26(8), 1698–1707 (1999).
[CrossRef] [PubMed]

Lasker, J. M.

J. M. Lasker, J. M. Masciotti, M. Schoenecker, C. H. Schmitz, and A. H. Hielscher, “Digital-signal-processor-based dynamic imaging system for optical tomography,” Rev. Sci. Instrum. 78(8), 083706 (2007).
[CrossRef] [PubMed]

Le, C.

A. D. Klose, B. J. Beattie, H. Dehghani, L. Vider, C. Le, V. Ponomarev, and R. Blasberg, “In vivo bioluminescence tomography with a blocking-off finite-difference SP3 method and MRI/CT coregistration,” Med. Phys. 37(1), 329–338 (2010).
[CrossRef] [PubMed]

Leschziner, M. A.

W. L. Chen, F. S. Lien, and M. A. Leschziner, “Local mesh refinement within a multi-block structured-grid scheme for genereal flows,” Comput. Methods Appl. Mech. Eng. 144(3-4), 327–369 (1997).
[CrossRef]

Lien, F. S.

W. L. Chen, F. S. Lien, and M. A. Leschziner, “Local mesh refinement within a multi-block structured-grid scheme for genereal flows,” Comput. Methods Appl. Mech. Eng. 144(3-4), 327–369 (1997).
[CrossRef]

Masciotti, J. M.

J. M. Lasker, J. M. Masciotti, M. Schoenecker, C. H. Schmitz, and A. H. Hielscher, “Digital-signal-processor-based dynamic imaging system for optical tomography,” Rev. Sci. Instrum. 78(8), 083706 (2007).
[CrossRef] [PubMed]

McGhee, J.

J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency domain photon migration,” Med. Phys. 33(12), 4685–4700 (2006).
[CrossRef] [PubMed]

Moa-Anderson, B.

A. K. Scheel, M. Backhaus, A. D. Klose, B. Moa-Anderson, U. J. Netz, K. G. Hermann, J. Beuthan, G. A. Müller, G. R. Burmester, and A. H. Hielscher, “First clinical evaluation of sagittal laser optical tomography for detection of synovitis in arthritic finger joints,” Ann. Rheum. Dis. 64(2), 239–245 (2005).
[CrossRef] [PubMed]

Mottaghy, F. M.

O. Gheysens and F. M. Mottaghy, “Method of bioluminescence imaging for molecular imaging of physiological and pathological processes,” Methods 48(2), 139–145 (2009).
[CrossRef] [PubMed]

Müller, G. A.

A. K. Scheel, M. Backhaus, A. D. Klose, B. Moa-Anderson, U. J. Netz, K. G. Hermann, J. Beuthan, G. A. Müller, G. R. Burmester, and A. H. Hielscher, “First clinical evaluation of sagittal laser optical tomography for detection of synovitis in arthritic finger joints,” Ann. Rheum. Dis. 64(2), 239–245 (2005).
[CrossRef] [PubMed]

Nasrallah, S. B.

M. B. Salah, F. Askri, and S. B. Nasrallah, “Unstructured control-volume finite element method for radiative heat transfer in a complex 2-D geometry,” Numer. Heat Transf. B 48(5), 477–497 (2005).
[CrossRef]

Netz, U.

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

Netz, U. J.

U. J. Netz, J. Beuthan, and A. H. Hielscher, “Multipixel system for gigahertz frequency-domain optical imaging of finger joints,” Rev. Sci. Instrum. 79(3), 034301 (2008).
[CrossRef] [PubMed]

A. K. Scheel, M. Backhaus, A. D. Klose, B. Moa-Anderson, U. J. Netz, K. G. Hermann, J. Beuthan, G. A. Müller, G. R. Burmester, and A. H. Hielscher, “First clinical evaluation of sagittal laser optical tomography for detection of synovitis in arthritic finger joints,” Ann. Rheum. Dis. 64(2), 239–245 (2005).
[CrossRef] [PubMed]

Niedre, M. J.

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc. Natl. Acad. Sci. U.S.A. 105(49), 19126–19131 (2008).
[CrossRef] [PubMed]

Ntziachristos, V.

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc. Natl. Acad. Sci. U.S.A. 105(49), 19126–19131 (2008).
[CrossRef] [PubMed]

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

A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging 24(10), 1377–1386 (2005).
[CrossRef] [PubMed]

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202(1), 323–345 (2005).
[CrossRef]

Oliger, J.

M. J. Berger and J. Oliger, “Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations,” J. Comput. Phys. 53(3), 484–512 (1984).
[CrossRef]

Pan, T.

J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency domain photon migration,” Med. Phys. 33(12), 4685–4700 (2006).
[CrossRef] [PubMed]

Patterson, M. S.

Pember, R. B.

J. P. Jessee, W. A. Fiveland, L. H. Howell, P. Colella, and R. B. Pember, “An Adaptive Mesh Refinement Algorithm for the Radiative Transport Equation,” J. Comput. Phys. 139(2), 380–398 (1998).
[CrossRef]

Pogue, B. W.

Ponomarev, V.

A. D. Klose, B. J. Beattie, H. Dehghani, L. Vider, C. Le, V. Ponomarev, and R. Blasberg, “In vivo bioluminescence tomography with a blocking-off finite-difference SP3 method and MRI/CT coregistration,” Med. Phys. 37(1), 329–338 (2010).
[CrossRef] [PubMed]

Rasmussen, J. C.

J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency domain photon migration,” Med. Phys. 33(12), 4685–4700 (2006).
[CrossRef] [PubMed]

A. Joshi, W. Bangerth, K. Hwang, J. C. Rasmussen, and E. M. Sevick-Muraca, “Fully adaptive FEM based fluorescence optical tomography from time-dependent measurements with area illumination and detection,” Med. Phys. 33(5), 1299–1310 (2006).
[CrossRef] [PubMed]

Ren, K.

Ripoll, J.

A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging 24(10), 1377–1386 (2005).
[CrossRef] [PubMed]

Salah, M. B.

M. B. Salah, F. Askri, and S. B. Nasrallah, “Unstructured control-volume finite element method for radiative heat transfer in a complex 2-D geometry,” Numer. Heat Transf. B 48(5), 477–497 (2005).
[CrossRef]

Scheel, A. K.

A. K. Scheel, M. Backhaus, A. D. Klose, B. Moa-Anderson, U. J. Netz, K. G. Hermann, J. Beuthan, G. A. Müller, G. R. Burmester, and A. H. Hielscher, “First clinical evaluation of sagittal laser optical tomography for detection of synovitis in arthritic finger joints,” Ann. Rheum. Dis. 64(2), 239–245 (2005).
[CrossRef] [PubMed]

Schmitz, C. H.

J. M. Lasker, J. M. Masciotti, M. Schoenecker, C. H. Schmitz, and A. H. Hielscher, “Digital-signal-processor-based dynamic imaging system for optical tomography,” Rev. Sci. Instrum. 78(8), 083706 (2007).
[CrossRef] [PubMed]

Schoenecker, M.

J. M. Lasker, J. M. Masciotti, M. Schoenecker, C. H. Schmitz, and A. H. Hielscher, “Digital-signal-processor-based dynamic imaging system for optical tomography,” Rev. Sci. Instrum. 78(8), 083706 (2007).
[CrossRef] [PubMed]

Schweiger, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[CrossRef] [PubMed]

Sevick-Muraca, E. M.

J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency domain photon migration,” Med. Phys. 33(12), 4685–4700 (2006).
[CrossRef] [PubMed]

A. Joshi, W. Bangerth, K. Hwang, J. C. Rasmussen, and E. M. Sevick-Muraca, “Fully adaptive FEM based fluorescence optical tomography from time-dependent measurements with area illumination and detection,” Med. Phys. 33(5), 1299–1310 (2006).
[CrossRef] [PubMed]

Simpson, R. B.

R. B. Simpson, “Automatic local refinement for irregular rectangular meshes,” Int. J. Numer. Methods Eng. 14(11), 1665–1678 (1979).
[CrossRef]

Soubret, A.

A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging 24(10), 1377–1386 (2005).
[CrossRef] [PubMed]

Svaasand, L. O.

Tromberg, B. J.

Tsay, T. T.

Vider, L.

A. D. Klose, B. J. Beattie, H. Dehghani, L. Vider, C. Le, V. Ponomarev, and R. Blasberg, “In vivo bioluminescence tomography with a blocking-off finite-difference SP3 method and MRI/CT coregistration,” Med. Phys. 37(1), 329–338 (2010).
[CrossRef] [PubMed]

Wang, L. H.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995).
[CrossRef] [PubMed]

Wareing, T.

J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency domain photon migration,” Med. Phys. 33(12), 4685–4700 (2006).
[CrossRef] [PubMed]

Weissleder, R.

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc. Natl. Acad. Sci. U.S.A. 105(49), 19126–19131 (2008).
[CrossRef] [PubMed]

Wilson, B. C.

Zheng, L. Q.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995).
[CrossRef] [PubMed]

Ann. Rheum. Dis.

A. K. Scheel, M. Backhaus, A. D. Klose, B. Moa-Anderson, U. J. Netz, K. G. Hermann, J. Beuthan, G. A. Müller, G. R. Burmester, and A. H. Hielscher, “First clinical evaluation of sagittal laser optical tomography for detection of synovitis in arthritic finger joints,” Ann. Rheum. Dis. 64(2), 239–245 (2005).
[CrossRef] [PubMed]

Annu. Rev. Biomed. Eng.

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

Appl. Opt.

Comput. Methods Appl. Mech. Eng.

W. L. Chen, F. S. Lien, and M. A. Leschziner, “Local mesh refinement within a multi-block structured-grid scheme for genereal flows,” Comput. Methods Appl. Mech. Eng. 144(3-4), 327–369 (1997).
[CrossRef]

Comput. Methods Programs Biomed.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995).
[CrossRef] [PubMed]

Curr. Opin. Biotechnol.

A. H. Hielscher, “Optical tomographic imaging of small animals,” Curr. Opin. Biotechnol. 16(1), 79–88 (2005).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging

A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging 24(10), 1377–1386 (2005).
[CrossRef] [PubMed]

Int. J. Numer. Methods Eng.

R. B. Simpson, “Automatic local refinement for irregular rectangular meshes,” Int. J. Numer. Methods Eng. 14(11), 1665–1678 (1979).
[CrossRef]

Inverse Probl.

H. K. Kim and A. H. Hielscher, “A PDE-constrained SQP algorithm for optical tomography based on the frequency-domain equation of radiative transfer,” Inverse Probl. 25(1), 015010 (2009).
[CrossRef]

O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14(5), 1107–1130 (1998).
[CrossRef]

J. Comput. Phys.

M. J. Berger and J. Oliger, “Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations,” J. Comput. Phys. 53(3), 484–512 (1984).
[CrossRef]

M. J. Berger and P. Colella, “Local adaptive mesh refinement for shock-hydrodynamics,” J. Comput. Phys. 82(1), 64–84 (1989).
[CrossRef]

J. P. Jessee, W. A. Fiveland, L. H. Howell, P. Colella, and R. B. Pember, “An Adaptive Mesh Refinement Algorithm for the Radiative Transport Equation,” J. Comput. Phys. 139(2), 380–398 (1998).
[CrossRef]

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202(1), 323–345 (2005).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf.

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

Med. Phys.

A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26(8), 1698–1707 (1999).
[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(2), 299–309 (1993).
[CrossRef] [PubMed]

J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency domain photon migration,” Med. Phys. 33(12), 4685–4700 (2006).
[CrossRef] [PubMed]

A. D. Klose, B. J. Beattie, H. Dehghani, L. Vider, C. Le, V. Ponomarev, and R. Blasberg, “In vivo bioluminescence tomography with a blocking-off finite-difference SP3 method and MRI/CT coregistration,” Med. Phys. 37(1), 329–338 (2010).
[CrossRef] [PubMed]

A. Joshi, W. Bangerth, K. Hwang, J. C. Rasmussen, and E. M. Sevick-Muraca, “Fully adaptive FEM based fluorescence optical tomography from time-dependent measurements with area illumination and detection,” Med. Phys. 33(5), 1299–1310 (2006).
[CrossRef] [PubMed]

Methods

O. Gheysens and F. M. Mottaghy, “Method of bioluminescence imaging for molecular imaging of physiological and pathological processes,” Methods 48(2), 139–145 (2009).
[CrossRef] [PubMed]

Numer. Heat Transf. B

M. B. Salah, F. Askri, and S. B. Nasrallah, “Unstructured control-volume finite element method for radiative heat transfer in a complex 2-D geometry,” Numer. Heat Transf. B 48(5), 477–497 (2005).
[CrossRef]

Opt. Lett.

Phys. Med. Biol.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37(7), 1531–1560 (1992).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A.

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc. Natl. Acad. Sci. U.S.A. 105(49), 19126–19131 (2008).
[CrossRef] [PubMed]

Rev. Sci. Instrum.

J. M. Lasker, J. M. Masciotti, M. Schoenecker, C. H. Schmitz, and A. H. Hielscher, “Digital-signal-processor-based dynamic imaging system for optical tomography,” Rev. Sci. Instrum. 78(8), 083706 (2007).
[CrossRef] [PubMed]

U. J. Netz, J. Beuthan, and A. H. Hielscher, “Multipixel system for gigahertz frequency-domain optical imaging of finger joints,” Rev. Sci. Instrum. 79(3), 034301 (2008).
[CrossRef] [PubMed]

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K. E. Atkinson, An Introduction to Numerical Analysis, 2nd ed., (John Wiley & Sons, Canada, 1989).

V. D. Liseikin, Grid generation methods, Second Edition (Springer, Netherlands, 2010).

A. D. Klose, “Radiative Transfer of Luminescence in Biological Tissue”, in Light Scattering Reviews, Volume 4, A.A. Kokhanovsky (Ed.), 293–345 (Springer, Berlin, 2009).

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Figures (10)

Fig. 1
Fig. 1

General sequence in which major subroutines are executed in our algorithm.

Fig. 2
Fig. 2

Examples of the discretization of Euclidean space with structured grids. (a) The geometry the grids approximate is the cross-section of a mouse obtained from an MRI data set. (b) Discretization of mouse cross-section; “inactive” and “active” domains are represented by the light and dark gray voxels, respectively. The dark gray area represents a single structured grid. (c) A 3-level BSG fitted to the active domain. (d) A coarse grid fitted to the mouse cross-section. Decrease in grid density achieved with a BSG is evident from (b-c).

Fig. 3
Fig. 3

Visualization of algorithm for determining computational domains.

Fig. 4
Fig. 4

(a) Possible grid points on interior boundaries. The first four cases are labeled i-iv and the fifth case is specified by a black dot. (b) An interior point that is not in the active domain (black diamond).

Fig. 5
Fig. 5

(a) Depiction of grid points on the fine/coarse grid boundary. The first four cases are labeled i-iv while the fifth case is specified by the black dot. (b) The point denoted by the black triangle is a point on a fine/coarse boundary; the fine and coarse grids are to its right and left, respectively. The black dot is the virtual point that must be created when the differencing scheme in the x-direction requires a backward Euler-step. Solutions at the four white dots surrounding the black dot are averaged to create the virtual point.

Fig. 6
Fig. 6

Numerical phantoms with homogeneous backgrounds and embedded fluorophores: (a) disk, (b) two-dimensional mouse cross section, (c) cylinder, and (d) three-dimensional mouse model. The positions of boundary sources are shown for (a-b) as black dots, while detectors are assigned to every boundary point. The position of the boundary source and detectors are shown for the cylinder (c) as an arrow and black dots, respectively. (d) Sources for the mouse are located within the interval defined by x = [16,18], y = [43,46], and z = 0. Similarly, detectors are defined within x = [16,18], y = [43,46], and z = 15.

Fig. 7
Fig. 7

(a-c) Examples of single structured grids fitted to the disk, cylinder, and mouse phantom, respectively. (d-f) 2-level BSGs fitted to same three phantoms. Three-dimensional BSGs and their interior structure are visualized by showing only a section of the full three-dimensional shape (e,f).

Fig. 8
Fig. 8

(a,b) Fluorescence excitation and emission from the two-dimensional mouse phantom on a single grid. (c,d) Fluorescence excitation and emission computed on a 2-level BSG fitted to the same phantom.

Fig. 9
Fig. 9

Partial current measurements at the boundary of the disk phantom during fluorescence excitation (a-b) and emission (c-d). (a,c) Partial current on single grids with increasing Δx. The benchmark solution is computed with Δx = 2/256 cm. (b,d) Partial current with 1-, 2-, and 3-level BSGs with Δx = 2/256 cm.

Fig. 10
Fig. 10

Three-dimensional representations of excitation (a) and emission (b) fluence on the cylindrical phantom. (d) Sample excitation on the three-dimension mouse phantom.

Tables (7)

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Table 1 List of related variables [4]

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Table 2 Properties of embedded fluorescent probes

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Table 3 Summary of parameter values used for simulations

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Table 4 Computation time, MPE, and RSU on a disk

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Table 5 Computation time, MPE, and RSU on a mouse cross-section

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Table 6 Computation time, MPE, and RSU on a cylindrical phantom

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Table 7 Computation time, MPE, and RSU on a three-dimensional mouse phantom

Equations (15)

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( Ω + μ t ( r ) + i ω m v ) ψ ^ ( r , Ω , ω m ) = μ s ( r ) 4 π p ( Ω Ω ' ) ψ ^ ( r , Ω ' , ω m ) d Ω ' + Q ( r , ω m ) 4 π ,
ψ ^ ( r , Ω , ω m ) = S ^ ( r , Ω , ω m ) + R ( Ω ' n ) ψ ^ ( r , Ω ' , ω m ) , Ω n < 0.
ϕ ^ ( r , ω m ) = 4 π ψ ^ ( r , Ω , ω m ) d Ω ,
J + ( r , ω m ) = Ω n > 0 [ 1 R ( Ω n ) ] ( Ω n ) ψ ( r , Ω , ω m ) d Ω .
S ^ ( r , Ω , ω m ) = S 0 ( r , Ω ) e i ω m t ,
μ t = μ a e x + μ s e x + μ a e x e m .
Q ( r , ω m ) = η μ a e x e m ( r ) ϕ ^ e x ( r , ω m ) 1 + i ω m τ .
μ t = μ a e m + μ s e m .
4 π ψ ( r , Ω ) d Ω = k = 1 K ω k ψ k ( r ) ,
Ω x [ ψ ] i j l k [ ψ ] ( i 1 ) j l k Δ x + Ω y [ ψ ] i j l k [ ψ ] i ( j 1 ) l k Δ y + Ω z [ ψ ] i j l k [ ψ ] i j ( l 1 ) k Δ z + [ μ t ] i j l [ ψ ] i j l k + ω m i ν [ ψ ] i j l k = [ μ s ] i j l k = 1 K ω k p k k [ ψ ] i j l k + Q i j l 4 π ,
[ ψ ] i j l k = S k + R ( Ω k n i j l ) [ ψ ] i j l k , Ω k n i j l < 0.
p k k = 1 g 2 4 π ( 1 + g 2 2 g Ω k Ω k ) 3 / 2 ,
ϕ i j l = k = 1 K ω k ψ i j l k ,
RSU = T F T B S G T B S G
MPE = ( 1 n i = 1 n | f i a f i b | f i b ) × 100 ,

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