Abstract

Imaging depth of optical microscopy has been fundamentally limited to millimeter or sub-millimeter due to strong scattering of light in a biological sample. X-ray microscopy can resolve spatial details of few microns deep inside a sample but contrast resolution is inadequate to depict heterogeneous features at cellular or sub-cellular levels. To enhance and enrich biological contrast at large imaging depth, various nanoparticles are introduced and become essential to basic research and molecular medicine. Nanoparticles can be functionalized as imaging probes, similar to fluorescent and bioluminescent proteins. LiGa5O8:Cr3+ nanoparticles were recently synthesized to facilitate luminescence energy storage with x-ray pre-excitation and subsequently stimulated luminescence emission by visible/near-infrared (NIR) light. In this paper, we propose an x-ray micro-modulated luminescence tomography (XMLT, or MLT to be more general) approach to quantify a nanophosphor distribution in a thick biological sample with high resolution. Our numerical simulation studies demonstrate the feasibility of the proposed approach.

© 2014 Optical Society of America

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2013 (6)

S. J. Shin, J. R. Beech, K. A. Kelly, “Targeted nanoparticles in imaging: paving the way for personalized medicine in the battle against cancer,” Integr Biol 5(1), 29–42 (2013).
[CrossRef] [PubMed]

10C. Ricci, L. Moroni, S. Danti, “Cancer tissue engineering: new perspectives in understanding the biology of solid tumors: A critical review,” OA Tissue Eng. 1(1), 4 (2013).

X. Cai, Y. S. Zhang, Y. Xia, L. V. Wang, “Photoacoustic Microscopy in Tissue Engineering,” Mater. Today 16(3), 67–77 (2013).
[CrossRef] [PubMed]

F. Liu, W. Yan, Y. J. Chuang, Z. Zhen, J. Xie, Z. Pan, “Photostimulated near-infrared persistent luminescence as a new optical read-out from Cr³⁺-doped LiGa₅O₈,” Sci. Rep. 3, 1554 (2013).
[CrossRef] [PubMed]

M. Chang, L. Li, Z. Q. Chen, Y. S. Xiao, L. Zhang, G. Wang, “A few-view reweighted sparsity hunting (FRESH) method for CT image reconstruction,” J. XRay Sci. Technol. 21(2), 161–176 (2013).
[PubMed]

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[CrossRef] [PubMed]

2012 (3)

J. Vila-Comamala, Y. Pan, J. J. Lombardo, W. M. Harris, W. K. Chiu, C. David, Y. Wang, “Zone-doubled Fresnel zone plates for high-resolution hard X-ray full-field transmission microscopy,” J. Synchrotron Radiat. 19(5), 705–709 (2012).
[CrossRef] [PubMed]

S. Wu, Y. Hwu, G. Margaritondo, “Hard-X-ray zone plates: recent progress,” Materials 5(12), 1752–1773 (2012).
[CrossRef]

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[CrossRef] [PubMed]

2011 (3)

C. David, S. Gorelick, S. Rutishauser, J. Krzywinski, J. Vila-Comamala, V. A. Guzenko, O. Bunk, E. Färm, M. Ritala, M. Cammarata, D. M. Fritz, R. Barrett, L. Samoylova, J. Grünert, H. Sinn, “Nanofocusing of hard X-ray free electron laser pulses using diamond based Fresnel zone plates,” Sci. Rep. 1, 57 (2011).
[CrossRef] [PubMed]

P. Timpson, E. J. McGhee, K. I. Anderson, “Imaging molecular dynamics in vivo - from cell biology to animal models,” J. Cell Sci. 124(17), 2877–2890 (2011).
[CrossRef] [PubMed]

A. Liemert, A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 83(3), 036605 (2011).
[CrossRef] [PubMed]

2010 (2)

G. Pratx, C. M. Carpenter, C. Sun, L. Xing, “X-ray luminescence computed tomography via selective excitation: a feasibility study,” IEEE Trans. Med. Imaging 29(12), 1992–1999 (2010).
[CrossRef] [PubMed]

S. D. Perrault, W. C. Chan, “In vivo assembly of nanoparticle components to improve targeted cancer imaging,” Proc. Natl. Acad. Sci. U.S.A. 107(25), 11194–11199 (2010).
[CrossRef] [PubMed]

2009 (2)

C. Weinand, J. W. Xu, G. M. Peretti, L. J. Bonassar, T. J. Gill, “Conditions affecting cell seeding onto three-dimensional scaffolds for cellular-based biodegradable implants,” J. Biomed. Mater. Res. B Appl. Biomater. 91B(1), 80–87 (2009).
[CrossRef] [PubMed]

L. V. Wang, “Multiscale photoacoustic microscopy and computed tomography,” Nat. Photonics 3(9), 503–509 (2009).
[CrossRef] [PubMed]

2008 (3)

B. Huang, W. Wang, M. Bates, X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319(5864), 810–813 (2008).
[CrossRef] [PubMed]

A. Solanki, J. D. Kim, K. B. Lee, “Nanotechnology for regenerative medicine: nanomaterials for stem cell imaging,” Nanomedicine 3(4), 567–578 (2008).
[CrossRef] [PubMed]

K. R. Chien, “Regenerative medicine and human models of human disease,” Nature 453(7193), 302–305 (2008).
[CrossRef] [PubMed]

2007 (2)

S. M. Nie, Y. Xing, G. J. Kim, J. W. Simons, “Nanotechnology applications in cancer,” Annu. Rev. Biomed. Eng. 9(1), 257–288 (2007).
[CrossRef] [PubMed]

C. Wongsrichanalai, M. J. Barcus, S. Muth, A. Sutamihardja, W. H. Wernsdorfer, “A review of malaria diagnostic tools: Microscopy and rapid diagnostic test (RDT),” Am. J. Trop. Med. Hyg. 77(6), 119–127 (2007).
[PubMed]

2006 (1)

A. D. Klose, E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441–470 (2006).
[CrossRef]

2005 (3)

G. Y. Panasyuk, V. A. Markel, J. C. Schotland, “Superresolution and corrections to the diffusion approximation in optical tomography,” Appl. Phys. Lett. 87(10), 101111 (2005).
[CrossRef]

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13(18), 6756–6771 (2005).
[CrossRef] [PubMed]

W. T. Pong, C. Durkan, “A review and outlook for an anomaly of scanning tunnelling microscopy (STM): Superlattices on graphite,” J. Phys. D Appl. Phys. 38(21), R329–R355 (2005).
[CrossRef]

2004 (3)

N. Jalili, K. Laxminarayana, “A review of atomic force microscopy imaging systems: application to molecular metrology and biological sciences,” Mechatronics 14(8), 907–945 (2004).
[CrossRef]

P. Alivisatos, “The use of nanocrystals in biological detection,” Nat. Biotechnol. 22(1), 47–52 (2004).
[CrossRef] [PubMed]

X. H. Gao, Y. Y. Cui, R. M. Levenson, L. W. K. Chung, S. M. Nie, “In vivo cancer targeting and imaging with semiconductor quantum dots,” Nat. Biotechnol. 22(8), 969–976 (2004).
[CrossRef] [PubMed]

2003 (1)

2001 (1)

B. W. Rice, M. D. Cable, M. B. Nelson, “In vivo imaging of light-emitting probes,” J. Biomed. Opt. 6(4), 432–440 (2001).
[CrossRef] [PubMed]

1995 (1)

M. Schweiger, S. R. Arridge, M. Hiraoka, 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]

1994 (1)

Alivisatos, P.

P. Alivisatos, “The use of nanocrystals in biological detection,” Nat. Biotechnol. 22(1), 47–52 (2004).
[CrossRef] [PubMed]

Anderson, K. I.

P. Timpson, E. J. McGhee, K. I. Anderson, “Imaging molecular dynamics in vivo - from cell biology to animal models,” J. Cell Sci. 124(17), 2877–2890 (2011).
[CrossRef] [PubMed]

Arridge, S. R.

M. Schweiger, S. R. Arridge, M. Hiraoka, 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]

Barcus, M. J.

C. Wongsrichanalai, M. J. Barcus, S. Muth, A. Sutamihardja, W. H. Wernsdorfer, “A review of malaria diagnostic tools: Microscopy and rapid diagnostic test (RDT),” Am. J. Trop. Med. Hyg. 77(6), 119–127 (2007).
[PubMed]

Barrett, R.

C. David, S. Gorelick, S. Rutishauser, J. Krzywinski, J. Vila-Comamala, V. A. Guzenko, O. Bunk, E. Färm, M. Ritala, M. Cammarata, D. M. Fritz, R. Barrett, L. Samoylova, J. Grünert, H. Sinn, “Nanofocusing of hard X-ray free electron laser pulses using diamond based Fresnel zone plates,” Sci. Rep. 1, 57 (2011).
[CrossRef] [PubMed]

Bates, M.

B. Huang, W. Wang, M. Bates, X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319(5864), 810–813 (2008).
[CrossRef] [PubMed]

Beech, J. R.

S. J. Shin, J. R. Beech, K. A. Kelly, “Targeted nanoparticles in imaging: paving the way for personalized medicine in the battle against cancer,” Integr Biol 5(1), 29–42 (2013).
[CrossRef] [PubMed]

Bonassar, L. J.

C. Weinand, J. W. Xu, G. M. Peretti, L. J. Bonassar, T. J. Gill, “Conditions affecting cell seeding onto three-dimensional scaffolds for cellular-based biodegradable implants,” J. Biomed. Mater. Res. B Appl. Biomater. 91B(1), 80–87 (2009).
[CrossRef] [PubMed]

Bunk, O.

C. David, S. Gorelick, S. Rutishauser, J. Krzywinski, J. Vila-Comamala, V. A. Guzenko, O. Bunk, E. Färm, M. Ritala, M. Cammarata, D. M. Fritz, R. Barrett, L. Samoylova, J. Grünert, H. Sinn, “Nanofocusing of hard X-ray free electron laser pulses using diamond based Fresnel zone plates,” Sci. Rep. 1, 57 (2011).
[CrossRef] [PubMed]

Cable, M. D.

B. W. Rice, M. D. Cable, M. B. Nelson, “In vivo imaging of light-emitting probes,” J. Biomed. Opt. 6(4), 432–440 (2001).
[CrossRef] [PubMed]

Cai, X.

X. Cai, Y. S. Zhang, Y. Xia, L. V. Wang, “Photoacoustic Microscopy in Tissue Engineering,” Mater. Today 16(3), 67–77 (2013).
[CrossRef] [PubMed]

Cammarata, M.

C. David, S. Gorelick, S. Rutishauser, J. Krzywinski, J. Vila-Comamala, V. A. Guzenko, O. Bunk, E. Färm, M. Ritala, M. Cammarata, D. M. Fritz, R. Barrett, L. Samoylova, J. Grünert, H. Sinn, “Nanofocusing of hard X-ray free electron laser pulses using diamond based Fresnel zone plates,” Sci. Rep. 1, 57 (2011).
[CrossRef] [PubMed]

Carpenter, C. M.

G. Pratx, C. M. Carpenter, C. Sun, L. Xing, “X-ray luminescence computed tomography via selective excitation: a feasibility study,” IEEE Trans. Med. Imaging 29(12), 1992–1999 (2010).
[CrossRef] [PubMed]

Chan, W. C.

S. D. Perrault, W. C. Chan, “In vivo assembly of nanoparticle components to improve targeted cancer imaging,” Proc. Natl. Acad. Sci. U.S.A. 107(25), 11194–11199 (2010).
[CrossRef] [PubMed]

Chang, M.

M. Chang, L. Li, Z. Q. Chen, Y. S. Xiao, L. Zhang, G. Wang, “A few-view reweighted sparsity hunting (FRESH) method for CT image reconstruction,” J. XRay Sci. Technol. 21(2), 161–176 (2013).
[PubMed]

Chen, Z. Q.

M. Chang, L. Li, Z. Q. Chen, Y. S. Xiao, L. Zhang, G. Wang, “A few-view reweighted sparsity hunting (FRESH) method for CT image reconstruction,” J. XRay Sci. Technol. 21(2), 161–176 (2013).
[PubMed]

Chien, K. R.

K. R. Chien, “Regenerative medicine and human models of human disease,” Nature 453(7193), 302–305 (2008).
[CrossRef] [PubMed]

Chiu, W. K.

J. Vila-Comamala, Y. Pan, J. J. Lombardo, W. M. Harris, W. K. Chiu, C. David, Y. Wang, “Zone-doubled Fresnel zone plates for high-resolution hard X-ray full-field transmission microscopy,” J. Synchrotron Radiat. 19(5), 705–709 (2012).
[CrossRef] [PubMed]

Chuang, Y. J.

F. Liu, W. Yan, Y. J. Chuang, Z. Zhen, J. Xie, Z. Pan, “Photostimulated near-infrared persistent luminescence as a new optical read-out from Cr³⁺-doped LiGa₅O₈,” Sci. Rep. 3, 1554 (2013).
[CrossRef] [PubMed]

Chung, L. W. K.

X. H. Gao, Y. Y. Cui, R. M. Levenson, L. W. K. Chung, S. M. Nie, “In vivo cancer targeting and imaging with semiconductor quantum dots,” Nat. Biotechnol. 22(8), 969–976 (2004).
[CrossRef] [PubMed]

Cong, A.

Cong, W.

Cui, Y. Y.

X. H. Gao, Y. Y. Cui, R. M. Levenson, L. W. K. Chung, S. M. Nie, “In vivo cancer targeting and imaging with semiconductor quantum dots,” Nat. Biotechnol. 22(8), 969–976 (2004).
[CrossRef] [PubMed]

Danti, S.

10C. Ricci, L. Moroni, S. Danti, “Cancer tissue engineering: new perspectives in understanding the biology of solid tumors: A critical review,” OA Tissue Eng. 1(1), 4 (2013).

David, C.

J. Vila-Comamala, Y. Pan, J. J. Lombardo, W. M. Harris, W. K. Chiu, C. David, Y. Wang, “Zone-doubled Fresnel zone plates for high-resolution hard X-ray full-field transmission microscopy,” J. Synchrotron Radiat. 19(5), 705–709 (2012).
[CrossRef] [PubMed]

C. David, S. Gorelick, S. Rutishauser, J. Krzywinski, J. Vila-Comamala, V. A. Guzenko, O. Bunk, E. Färm, M. Ritala, M. Cammarata, D. M. Fritz, R. Barrett, L. Samoylova, J. Grünert, H. Sinn, “Nanofocusing of hard X-ray free electron laser pulses using diamond based Fresnel zone plates,” Sci. Rep. 1, 57 (2011).
[CrossRef] [PubMed]

Delpy, D. T.

M. Schweiger, S. R. Arridge, M. Hiraoka, 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]

Durkan, C.

W. T. Pong, C. Durkan, “A review and outlook for an anomaly of scanning tunnelling microscopy (STM): Superlattices on graphite,” J. Phys. D Appl. Phys. 38(21), R329–R355 (2005).
[CrossRef]

Färm, E.

C. David, S. Gorelick, S. Rutishauser, J. Krzywinski, J. Vila-Comamala, V. A. Guzenko, O. Bunk, E. Färm, M. Ritala, M. Cammarata, D. M. Fritz, R. Barrett, L. Samoylova, J. Grünert, H. Sinn, “Nanofocusing of hard X-ray free electron laser pulses using diamond based Fresnel zone plates,” Sci. Rep. 1, 57 (2011).
[CrossRef] [PubMed]

Feng, T. C.

Fritz, D. M.

C. David, S. Gorelick, S. Rutishauser, J. Krzywinski, J. Vila-Comamala, V. A. Guzenko, O. Bunk, E. Färm, M. Ritala, M. Cammarata, D. M. Fritz, R. Barrett, L. Samoylova, J. Grünert, H. Sinn, “Nanofocusing of hard X-ray free electron laser pulses using diamond based Fresnel zone plates,” Sci. Rep. 1, 57 (2011).
[CrossRef] [PubMed]

Gao, X. H.

X. H. Gao, Y. Y. Cui, R. M. Levenson, L. W. K. Chung, S. M. Nie, “In vivo cancer targeting and imaging with semiconductor quantum dots,” Nat. Biotechnol. 22(8), 969–976 (2004).
[CrossRef] [PubMed]

Gill, T. J.

C. Weinand, J. W. Xu, G. M. Peretti, L. J. Bonassar, T. J. Gill, “Conditions affecting cell seeding onto three-dimensional scaffolds for cellular-based biodegradable implants,” J. Biomed. Mater. Res. B Appl. Biomater. 91B(1), 80–87 (2009).
[CrossRef] [PubMed]

Gorelick, S.

C. David, S. Gorelick, S. Rutishauser, J. Krzywinski, J. Vila-Comamala, V. A. Guzenko, O. Bunk, E. Färm, M. Ritala, M. Cammarata, D. M. Fritz, R. Barrett, L. Samoylova, J. Grünert, H. Sinn, “Nanofocusing of hard X-ray free electron laser pulses using diamond based Fresnel zone plates,” Sci. Rep. 1, 57 (2011).
[CrossRef] [PubMed]

Goy, A. S.

Grünert, J.

C. David, S. Gorelick, S. Rutishauser, J. Krzywinski, J. Vila-Comamala, V. A. Guzenko, O. Bunk, E. Färm, M. Ritala, M. Cammarata, D. M. Fritz, R. Barrett, L. Samoylova, J. Grünert, H. Sinn, “Nanofocusing of hard X-ray free electron laser pulses using diamond based Fresnel zone plates,” Sci. Rep. 1, 57 (2011).
[CrossRef] [PubMed]

Guzenko, V. A.

C. David, S. Gorelick, S. Rutishauser, J. Krzywinski, J. Vila-Comamala, V. A. Guzenko, O. Bunk, E. Färm, M. Ritala, M. Cammarata, D. M. Fritz, R. Barrett, L. Samoylova, J. Grünert, H. Sinn, “Nanofocusing of hard X-ray free electron laser pulses using diamond based Fresnel zone plates,” Sci. Rep. 1, 57 (2011).
[CrossRef] [PubMed]

Harris, W. M.

J. Vila-Comamala, Y. Pan, J. J. Lombardo, W. M. Harris, W. K. Chiu, C. David, Y. Wang, “Zone-doubled Fresnel zone plates for high-resolution hard X-ray full-field transmission microscopy,” J. Synchrotron Radiat. 19(5), 705–709 (2012).
[CrossRef] [PubMed]

Haskell, R. C.

Hiraoka, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, 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. A.

Huang, B.

B. Huang, W. Wang, M. Bates, X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319(5864), 810–813 (2008).
[CrossRef] [PubMed]

Hwu, Y.

S. Wu, Y. Hwu, G. Margaritondo, “Hard-X-ray zone plates: recent progress,” Materials 5(12), 1752–1773 (2012).
[CrossRef]

Jalili, N.

N. Jalili, K. Laxminarayana, “A review of atomic force microscopy imaging systems: application to molecular metrology and biological sciences,” Mechatronics 14(8), 907–945 (2004).
[CrossRef]

Jiang, M.

Keller, J. B.

Kelly, K. A.

S. J. Shin, J. R. Beech, K. A. Kelly, “Targeted nanoparticles in imaging: paving the way for personalized medicine in the battle against cancer,” Integr Biol 5(1), 29–42 (2013).
[CrossRef] [PubMed]

Kienle, A.

A. Liemert, A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 83(3), 036605 (2011).
[CrossRef] [PubMed]

Kim, A. D.

Kim, G. J.

S. M. Nie, Y. Xing, G. J. Kim, J. W. Simons, “Nanotechnology applications in cancer,” Annu. Rev. Biomed. Eng. 9(1), 257–288 (2007).
[CrossRef] [PubMed]

Kim, J. D.

A. Solanki, J. D. Kim, K. B. Lee, “Nanotechnology for regenerative medicine: nanomaterials for stem cell imaging,” Nanomedicine 3(4), 567–578 (2008).
[CrossRef] [PubMed]

Klose, A. D.

A. D. Klose, E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441–470 (2006).
[CrossRef]

Krzywinski, J.

C. David, S. Gorelick, S. Rutishauser, J. Krzywinski, J. Vila-Comamala, V. A. Guzenko, O. Bunk, E. Färm, M. Ritala, M. Cammarata, D. M. Fritz, R. Barrett, L. Samoylova, J. Grünert, H. Sinn, “Nanofocusing of hard X-ray free electron laser pulses using diamond based Fresnel zone plates,” Sci. Rep. 1, 57 (2011).
[CrossRef] [PubMed]

Kumar, D.

Larsen, E. W.

A. D. Klose, E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441–470 (2006).
[CrossRef]

Laxminarayana, K.

N. Jalili, K. Laxminarayana, “A review of atomic force microscopy imaging systems: application to molecular metrology and biological sciences,” Mechatronics 14(8), 907–945 (2004).
[CrossRef]

Lee, K. B.

A. Solanki, J. D. Kim, K. B. Lee, “Nanotechnology for regenerative medicine: nanomaterials for stem cell imaging,” Nanomedicine 3(4), 567–578 (2008).
[CrossRef] [PubMed]

Levenson, R. M.

X. H. Gao, Y. Y. Cui, R. M. Levenson, L. W. K. Chung, S. M. Nie, “In vivo cancer targeting and imaging with semiconductor quantum dots,” Nat. Biotechnol. 22(8), 969–976 (2004).
[CrossRef] [PubMed]

Li, L.

M. Chang, L. Li, Z. Q. Chen, Y. S. Xiao, L. Zhang, G. Wang, “A few-view reweighted sparsity hunting (FRESH) method for CT image reconstruction,” J. XRay Sci. Technol. 21(2), 161–176 (2013).
[PubMed]

Liemert, A.

A. Liemert, A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 83(3), 036605 (2011).
[CrossRef] [PubMed]

Liu, F.

F. Liu, W. Yan, Y. J. Chuang, Z. Zhen, J. Xie, Z. Pan, “Photostimulated near-infrared persistent luminescence as a new optical read-out from Cr³⁺-doped LiGa₅O₈,” Sci. Rep. 3, 1554 (2013).
[CrossRef] [PubMed]

Z. Pan, Y. Y. Lu, F. Liu, “Sunlight-activated long-persistent luminescence in the near-infrared from Cr3+-doped zinc gallogermanates,” Nat. Mater. 11(1), 58–63 (2012).
[CrossRef] [PubMed]

Liu, Y.

Lombardo, J. J.

J. Vila-Comamala, Y. Pan, J. J. Lombardo, W. M. Harris, W. K. Chiu, C. David, Y. Wang, “Zone-doubled Fresnel zone plates for high-resolution hard X-ray full-field transmission microscopy,” J. Synchrotron Radiat. 19(5), 705–709 (2012).
[CrossRef] [PubMed]

Lu, Y. Y.

Z. Pan, Y. Y. Lu, F. Liu, “Sunlight-activated long-persistent luminescence in the near-infrared from Cr3+-doped zinc gallogermanates,” Nat. Mater. 11(1), 58–63 (2012).
[CrossRef] [PubMed]

Margaritondo, G.

S. Wu, Y. Hwu, G. Margaritondo, “Hard-X-ray zone plates: recent progress,” Materials 5(12), 1752–1773 (2012).
[CrossRef]

Markel, V. A.

G. Y. Panasyuk, V. A. Markel, J. C. Schotland, “Superresolution and corrections to the diffusion approximation in optical tomography,” Appl. Phys. Lett. 87(10), 101111 (2005).
[CrossRef]

McAdams, M. S.

McCray, P. B.

McGhee, E. J.

P. Timpson, E. J. McGhee, K. I. Anderson, “Imaging molecular dynamics in vivo - from cell biology to animal models,” J. Cell Sci. 124(17), 2877–2890 (2011).
[CrossRef] [PubMed]

McLennan, G.

Moroni, L.

10C. Ricci, L. Moroni, S. Danti, “Cancer tissue engineering: new perspectives in understanding the biology of solid tumors: A critical review,” OA Tissue Eng. 1(1), 4 (2013).

Muth, S.

C. Wongsrichanalai, M. J. Barcus, S. Muth, A. Sutamihardja, W. H. Wernsdorfer, “A review of malaria diagnostic tools: Microscopy and rapid diagnostic test (RDT),” Am. J. Trop. Med. Hyg. 77(6), 119–127 (2007).
[PubMed]

Nelson, M. B.

B. W. Rice, M. D. Cable, M. B. Nelson, “In vivo imaging of light-emitting probes,” J. Biomed. Opt. 6(4), 432–440 (2001).
[CrossRef] [PubMed]

Nie, S. M.

S. M. Nie, Y. Xing, G. J. Kim, J. W. Simons, “Nanotechnology applications in cancer,” Annu. Rev. Biomed. Eng. 9(1), 257–288 (2007).
[CrossRef] [PubMed]

X. H. Gao, Y. Y. Cui, R. M. Levenson, L. W. K. Chung, S. M. Nie, “In vivo cancer targeting and imaging with semiconductor quantum dots,” Nat. Biotechnol. 22(8), 969–976 (2004).
[CrossRef] [PubMed]

Pan, Y.

J. Vila-Comamala, Y. Pan, J. J. Lombardo, W. M. Harris, W. K. Chiu, C. David, Y. Wang, “Zone-doubled Fresnel zone plates for high-resolution hard X-ray full-field transmission microscopy,” J. Synchrotron Radiat. 19(5), 705–709 (2012).
[CrossRef] [PubMed]

Pan, Z.

F. Liu, W. Yan, Y. J. Chuang, Z. Zhen, J. Xie, Z. Pan, “Photostimulated near-infrared persistent luminescence as a new optical read-out from Cr³⁺-doped LiGa₅O₈,” Sci. Rep. 3, 1554 (2013).
[CrossRef] [PubMed]

Z. Pan, Y. Y. Lu, F. Liu, “Sunlight-activated long-persistent luminescence in the near-infrared from Cr3+-doped zinc gallogermanates,” Nat. Mater. 11(1), 58–63 (2012).
[CrossRef] [PubMed]

Panasyuk, G. Y.

G. Y. Panasyuk, V. A. Markel, J. C. Schotland, “Superresolution and corrections to the diffusion approximation in optical tomography,” Appl. Phys. Lett. 87(10), 101111 (2005).
[CrossRef]

Peretti, G. M.

C. Weinand, J. W. Xu, G. M. Peretti, L. J. Bonassar, T. J. Gill, “Conditions affecting cell seeding onto three-dimensional scaffolds for cellular-based biodegradable implants,” J. Biomed. Mater. Res. B Appl. Biomater. 91B(1), 80–87 (2009).
[CrossRef] [PubMed]

Perrault, S. D.

S. D. Perrault, W. C. Chan, “In vivo assembly of nanoparticle components to improve targeted cancer imaging,” Proc. Natl. Acad. Sci. U.S.A. 107(25), 11194–11199 (2010).
[CrossRef] [PubMed]

Pong, W. T.

W. T. Pong, C. Durkan, “A review and outlook for an anomaly of scanning tunnelling microscopy (STM): Superlattices on graphite,” J. Phys. D Appl. Phys. 38(21), R329–R355 (2005).
[CrossRef]

Pratx, G.

G. Pratx, C. M. Carpenter, C. Sun, L. Xing, “X-ray luminescence computed tomography via selective excitation: a feasibility study,” IEEE Trans. Med. Imaging 29(12), 1992–1999 (2010).
[CrossRef] [PubMed]

Psaltis, D.

Ricci, C.

10C. Ricci, L. Moroni, S. Danti, “Cancer tissue engineering: new perspectives in understanding the biology of solid tumors: A critical review,” OA Tissue Eng. 1(1), 4 (2013).

Rice, B. W.

B. W. Rice, M. D. Cable, M. B. Nelson, “In vivo imaging of light-emitting probes,” J. Biomed. Opt. 6(4), 432–440 (2001).
[CrossRef] [PubMed]

Ritala, M.

C. David, S. Gorelick, S. Rutishauser, J. Krzywinski, J. Vila-Comamala, V. A. Guzenko, O. Bunk, E. Färm, M. Ritala, M. Cammarata, D. M. Fritz, R. Barrett, L. Samoylova, J. Grünert, H. Sinn, “Nanofocusing of hard X-ray free electron laser pulses using diamond based Fresnel zone plates,” Sci. Rep. 1, 57 (2011).
[CrossRef] [PubMed]

Rutishauser, S.

C. David, S. Gorelick, S. Rutishauser, J. Krzywinski, J. Vila-Comamala, V. A. Guzenko, O. Bunk, E. Färm, M. Ritala, M. Cammarata, D. M. Fritz, R. Barrett, L. Samoylova, J. Grünert, H. Sinn, “Nanofocusing of hard X-ray free electron laser pulses using diamond based Fresnel zone plates,” Sci. Rep. 1, 57 (2011).
[CrossRef] [PubMed]

Samoylova, L.

C. David, S. Gorelick, S. Rutishauser, J. Krzywinski, J. Vila-Comamala, V. A. Guzenko, O. Bunk, E. Färm, M. Ritala, M. Cammarata, D. M. Fritz, R. Barrett, L. Samoylova, J. Grünert, H. Sinn, “Nanofocusing of hard X-ray free electron laser pulses using diamond based Fresnel zone plates,” Sci. Rep. 1, 57 (2011).
[CrossRef] [PubMed]

Schotland, J. C.

G. Y. Panasyuk, V. A. Markel, J. C. Schotland, “Superresolution and corrections to the diffusion approximation in optical tomography,” Appl. Phys. Lett. 87(10), 101111 (2005).
[CrossRef]

Schweiger, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, 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]

Shin, S. J.

S. J. Shin, J. R. Beech, K. A. Kelly, “Targeted nanoparticles in imaging: paving the way for personalized medicine in the battle against cancer,” Integr Biol 5(1), 29–42 (2013).
[CrossRef] [PubMed]

Simons, J. W.

S. M. Nie, Y. Xing, G. J. Kim, J. W. Simons, “Nanotechnology applications in cancer,” Annu. Rev. Biomed. Eng. 9(1), 257–288 (2007).
[CrossRef] [PubMed]

Sinn, H.

C. David, S. Gorelick, S. Rutishauser, J. Krzywinski, J. Vila-Comamala, V. A. Guzenko, O. Bunk, E. Färm, M. Ritala, M. Cammarata, D. M. Fritz, R. Barrett, L. Samoylova, J. Grünert, H. Sinn, “Nanofocusing of hard X-ray free electron laser pulses using diamond based Fresnel zone plates,” Sci. Rep. 1, 57 (2011).
[CrossRef] [PubMed]

Solanki, A.

A. Solanki, J. D. Kim, K. B. Lee, “Nanotechnology for regenerative medicine: nanomaterials for stem cell imaging,” Nanomedicine 3(4), 567–578 (2008).
[CrossRef] [PubMed]

Sun, C.

G. Pratx, C. M. Carpenter, C. Sun, L. Xing, “X-ray luminescence computed tomography via selective excitation: a feasibility study,” IEEE Trans. Med. Imaging 29(12), 1992–1999 (2010).
[CrossRef] [PubMed]

Sutamihardja, A.

C. Wongsrichanalai, M. J. Barcus, S. Muth, A. Sutamihardja, W. H. Wernsdorfer, “A review of malaria diagnostic tools: Microscopy and rapid diagnostic test (RDT),” Am. J. Trop. Med. Hyg. 77(6), 119–127 (2007).
[PubMed]

Svaasand, L. O.

Timpson, P.

P. Timpson, E. J. McGhee, K. I. Anderson, “Imaging molecular dynamics in vivo - from cell biology to animal models,” J. Cell Sci. 124(17), 2877–2890 (2011).
[CrossRef] [PubMed]

Tromberg, B. J.

Tsay, T. T.

Unser, M.

Vila-Comamala, J.

J. Vila-Comamala, Y. Pan, J. J. Lombardo, W. M. Harris, W. K. Chiu, C. David, Y. Wang, “Zone-doubled Fresnel zone plates for high-resolution hard X-ray full-field transmission microscopy,” J. Synchrotron Radiat. 19(5), 705–709 (2012).
[CrossRef] [PubMed]

C. David, S. Gorelick, S. Rutishauser, J. Krzywinski, J. Vila-Comamala, V. A. Guzenko, O. Bunk, E. Färm, M. Ritala, M. Cammarata, D. M. Fritz, R. Barrett, L. Samoylova, J. Grünert, H. Sinn, “Nanofocusing of hard X-ray free electron laser pulses using diamond based Fresnel zone plates,” Sci. Rep. 1, 57 (2011).
[CrossRef] [PubMed]

Wang, C.

W. Cong, C. Wang, G. Wang, “Stored luminescence computed tomography,” Int. J. Eng. Innovative Technol. (to be published).

Wang, G.

M. Chang, L. Li, Z. Q. Chen, Y. S. Xiao, L. Zhang, G. Wang, “A few-view reweighted sparsity hunting (FRESH) method for CT image reconstruction,” J. XRay Sci. Technol. 21(2), 161–176 (2013).
[PubMed]

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13(18), 6756–6771 (2005).
[CrossRef] [PubMed]

W. Cong, C. Wang, G. Wang, “Stored luminescence computed tomography,” Int. J. Eng. Innovative Technol. (to be published).

Wang, L. V.

X. Cai, Y. S. Zhang, Y. Xia, L. V. Wang, “Photoacoustic Microscopy in Tissue Engineering,” Mater. Today 16(3), 67–77 (2013).
[CrossRef] [PubMed]

L. V. Wang, “Multiscale photoacoustic microscopy and computed tomography,” Nat. Photonics 3(9), 503–509 (2009).
[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, A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13(18), 6756–6771 (2005).
[CrossRef] [PubMed]

Wang, W.

B. Huang, W. Wang, M. Bates, X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319(5864), 810–813 (2008).
[CrossRef] [PubMed]

Wang, Y.

J. Vila-Comamala, Y. Pan, J. J. Lombardo, W. M. Harris, W. K. Chiu, C. David, Y. Wang, “Zone-doubled Fresnel zone plates for high-resolution hard X-ray full-field transmission microscopy,” J. Synchrotron Radiat. 19(5), 705–709 (2012).
[CrossRef] [PubMed]

Weinand, C.

C. Weinand, J. W. Xu, G. M. Peretti, L. J. Bonassar, T. J. Gill, “Conditions affecting cell seeding onto three-dimensional scaffolds for cellular-based biodegradable implants,” J. Biomed. Mater. Res. B Appl. Biomater. 91B(1), 80–87 (2009).
[CrossRef] [PubMed]

Wernsdorfer, W. H.

C. Wongsrichanalai, M. J. Barcus, S. Muth, A. Sutamihardja, W. H. Wernsdorfer, “A review of malaria diagnostic tools: Microscopy and rapid diagnostic test (RDT),” Am. J. Trop. Med. Hyg. 77(6), 119–127 (2007).
[PubMed]

Wongsrichanalai, C.

C. Wongsrichanalai, M. J. Barcus, S. Muth, A. Sutamihardja, W. H. Wernsdorfer, “A review of malaria diagnostic tools: Microscopy and rapid diagnostic test (RDT),” Am. J. Trop. Med. Hyg. 77(6), 119–127 (2007).
[PubMed]

Wu, S.

S. Wu, Y. Hwu, G. Margaritondo, “Hard-X-ray zone plates: recent progress,” Materials 5(12), 1752–1773 (2012).
[CrossRef]

Xia, Y.

X. Cai, Y. S. Zhang, Y. Xia, L. V. Wang, “Photoacoustic Microscopy in Tissue Engineering,” Mater. Today 16(3), 67–77 (2013).
[CrossRef] [PubMed]

Xiao, Y. S.

M. Chang, L. Li, Z. Q. Chen, Y. S. Xiao, L. Zhang, G. Wang, “A few-view reweighted sparsity hunting (FRESH) method for CT image reconstruction,” J. XRay Sci. Technol. 21(2), 161–176 (2013).
[PubMed]

Xie, J.

F. Liu, W. Yan, Y. J. Chuang, Z. Zhen, J. Xie, Z. Pan, “Photostimulated near-infrared persistent luminescence as a new optical read-out from Cr³⁺-doped LiGa₅O₈,” Sci. Rep. 3, 1554 (2013).
[CrossRef] [PubMed]

Xing, L.

G. Pratx, C. M. Carpenter, C. Sun, L. Xing, “X-ray luminescence computed tomography via selective excitation: a feasibility study,” IEEE Trans. Med. Imaging 29(12), 1992–1999 (2010).
[CrossRef] [PubMed]

Xing, Y.

S. M. Nie, Y. Xing, G. J. Kim, J. W. Simons, “Nanotechnology applications in cancer,” Annu. Rev. Biomed. Eng. 9(1), 257–288 (2007).
[CrossRef] [PubMed]

Xu, J. W.

C. Weinand, J. W. Xu, G. M. Peretti, L. J. Bonassar, T. J. Gill, “Conditions affecting cell seeding onto three-dimensional scaffolds for cellular-based biodegradable implants,” J. Biomed. Mater. Res. B Appl. Biomater. 91B(1), 80–87 (2009).
[CrossRef] [PubMed]

Yan, W.

F. Liu, W. Yan, Y. J. Chuang, Z. Zhen, J. Xie, Z. Pan, “Photostimulated near-infrared persistent luminescence as a new optical read-out from Cr³⁺-doped LiGa₅O₈,” Sci. Rep. 3, 1554 (2013).
[CrossRef] [PubMed]

Zabner, J.

Zhang, L.

M. Chang, L. Li, Z. Q. Chen, Y. S. Xiao, L. Zhang, G. Wang, “A few-view reweighted sparsity hunting (FRESH) method for CT image reconstruction,” J. XRay Sci. Technol. 21(2), 161–176 (2013).
[PubMed]

Zhang, Y. S.

X. Cai, Y. S. Zhang, Y. Xia, L. V. Wang, “Photoacoustic Microscopy in Tissue Engineering,” Mater. Today 16(3), 67–77 (2013).
[CrossRef] [PubMed]

Zhen, Z.

F. Liu, W. Yan, Y. J. Chuang, Z. Zhen, J. Xie, Z. Pan, “Photostimulated near-infrared persistent luminescence as a new optical read-out from Cr³⁺-doped LiGa₅O₈,” Sci. Rep. 3, 1554 (2013).
[CrossRef] [PubMed]

Zhuang, X.

B. Huang, W. Wang, M. Bates, X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319(5864), 810–813 (2008).
[CrossRef] [PubMed]

Am. J. Trop. Med. Hyg. (1)

C. Wongsrichanalai, M. J. Barcus, S. Muth, A. Sutamihardja, W. H. Wernsdorfer, “A review of malaria diagnostic tools: Microscopy and rapid diagnostic test (RDT),” Am. J. Trop. Med. Hyg. 77(6), 119–127 (2007).
[PubMed]

Annu. Rev. Biomed. Eng. (1)

S. M. Nie, Y. Xing, G. J. Kim, J. W. Simons, “Nanotechnology applications in cancer,” Annu. Rev. Biomed. Eng. 9(1), 257–288 (2007).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

G. Y. Panasyuk, V. A. Markel, J. C. Schotland, “Superresolution and corrections to the diffusion approximation in optical tomography,” Appl. Phys. Lett. 87(10), 101111 (2005).
[CrossRef]

Biomed. Opt. Express (1)

IEEE Trans. Med. Imaging (1)

G. Pratx, C. M. Carpenter, C. Sun, L. Xing, “X-ray luminescence computed tomography via selective excitation: a feasibility study,” IEEE Trans. Med. Imaging 29(12), 1992–1999 (2010).
[CrossRef] [PubMed]

Integr Biol (1)

S. J. Shin, J. R. Beech, K. A. Kelly, “Targeted nanoparticles in imaging: paving the way for personalized medicine in the battle against cancer,” Integr Biol 5(1), 29–42 (2013).
[CrossRef] [PubMed]

J. Biomed. Mater. Res. B Appl. Biomater. (1)

C. Weinand, J. W. Xu, G. M. Peretti, L. J. Bonassar, T. J. Gill, “Conditions affecting cell seeding onto three-dimensional scaffolds for cellular-based biodegradable implants,” J. Biomed. Mater. Res. B Appl. Biomater. 91B(1), 80–87 (2009).
[CrossRef] [PubMed]

J. Biomed. Opt. (1)

B. W. Rice, M. D. Cable, M. B. Nelson, “In vivo imaging of light-emitting probes,” J. Biomed. Opt. 6(4), 432–440 (2001).
[CrossRef] [PubMed]

J. Cell Sci. (1)

P. Timpson, E. J. McGhee, K. I. Anderson, “Imaging molecular dynamics in vivo - from cell biology to animal models,” J. Cell Sci. 124(17), 2877–2890 (2011).
[CrossRef] [PubMed]

J. Comput. Phys. (1)

A. D. Klose, E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441–470 (2006).
[CrossRef]

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

J. Phys. D Appl. Phys. (1)

W. T. Pong, C. Durkan, “A review and outlook for an anomaly of scanning tunnelling microscopy (STM): Superlattices on graphite,” J. Phys. D Appl. Phys. 38(21), R329–R355 (2005).
[CrossRef]

J. Synchrotron Radiat. (1)

J. Vila-Comamala, Y. Pan, J. J. Lombardo, W. M. Harris, W. K. Chiu, C. David, Y. Wang, “Zone-doubled Fresnel zone plates for high-resolution hard X-ray full-field transmission microscopy,” J. Synchrotron Radiat. 19(5), 705–709 (2012).
[CrossRef] [PubMed]

J. XRay Sci. Technol. (1)

M. Chang, L. Li, Z. Q. Chen, Y. S. Xiao, L. Zhang, G. Wang, “A few-view reweighted sparsity hunting (FRESH) method for CT image reconstruction,” J. XRay Sci. Technol. 21(2), 161–176 (2013).
[PubMed]

Mater. Today (1)

X. Cai, Y. S. Zhang, Y. Xia, L. V. Wang, “Photoacoustic Microscopy in Tissue Engineering,” Mater. Today 16(3), 67–77 (2013).
[CrossRef] [PubMed]

Materials (1)

S. Wu, Y. Hwu, G. Margaritondo, “Hard-X-ray zone plates: recent progress,” Materials 5(12), 1752–1773 (2012).
[CrossRef]

Mechatronics (1)

N. Jalili, K. Laxminarayana, “A review of atomic force microscopy imaging systems: application to molecular metrology and biological sciences,” Mechatronics 14(8), 907–945 (2004).
[CrossRef]

Med. Phys. (1)

M. Schweiger, S. R. Arridge, M. Hiraoka, 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]

Nanomedicine (1)

A. Solanki, J. D. Kim, K. B. Lee, “Nanotechnology for regenerative medicine: nanomaterials for stem cell imaging,” Nanomedicine 3(4), 567–578 (2008).
[CrossRef] [PubMed]

Nat. Biotechnol. (2)

X. H. Gao, Y. Y. Cui, R. M. Levenson, L. W. K. Chung, S. M. Nie, “In vivo cancer targeting and imaging with semiconductor quantum dots,” Nat. Biotechnol. 22(8), 969–976 (2004).
[CrossRef] [PubMed]

P. Alivisatos, “The use of nanocrystals in biological detection,” Nat. Biotechnol. 22(1), 47–52 (2004).
[CrossRef] [PubMed]

Nat. Mater. (1)

Z. Pan, Y. Y. Lu, F. Liu, “Sunlight-activated long-persistent luminescence in the near-infrared from Cr3+-doped zinc gallogermanates,” Nat. Mater. 11(1), 58–63 (2012).
[CrossRef] [PubMed]

Nat. Photonics (1)

L. V. Wang, “Multiscale photoacoustic microscopy and computed tomography,” Nat. Photonics 3(9), 503–509 (2009).
[CrossRef] [PubMed]

Nature (1)

K. R. Chien, “Regenerative medicine and human models of human disease,” Nature 453(7193), 302–305 (2008).
[CrossRef] [PubMed]

OA Tissue Eng. (1)

10C. Ricci, L. Moroni, S. Danti, “Cancer tissue engineering: new perspectives in understanding the biology of solid tumors: A critical review,” OA Tissue Eng. 1(1), 4 (2013).

Opt. Express (1)

Phys. Rev. E Stat. Nonlinear Soft Matter Phys. (1)

A. Liemert, A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 83(3), 036605 (2011).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

S. D. Perrault, W. C. Chan, “In vivo assembly of nanoparticle components to improve targeted cancer imaging,” Proc. Natl. Acad. Sci. U.S.A. 107(25), 11194–11199 (2010).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

Science (1)

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[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

Micro-modulated luminescence tomography (MLT) system.

Fig. 2
Fig. 2

Modulated luminescence tomography simulation. (a) The true nanophosphor distribution at 2.5mm depth, (b) the reconstructed nanophosphor distribution with ~10µm resolution, and (c) the profile comparison along the vertical line at x = 160th pixel between the true and reconstructed nanophosphors concentrations. The top row images are the zoomed regions of interest in the bottom row.

Fig. 3
Fig. 3

Modulated luminescence tomography simulation. (a) The true nanophosphor distribution at 3.5mm depth, (b) the reconstructed nanophosphor distribution, and (c) the profile comparison along the horizontal centerline between the true and reconstructed nanophosphors concentrations.

Equations (7)

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μ s ( λ ) = 10 a λ b
μ a ( λ ) = S B [ x μ a H b ( λ ) + ( 1 x ) μ a H b O 2 ( λ ) ] + S W μ a W ( λ ) ,
Φ ( r ) = i = 1 N + 1 2 A i e v i r 4 π r
A i = 1 b N + 1 2 P ( λ i ) n = 1 , n i ( N + 1 ) / 2 ( λ i λ n ) .
Φ ( r ) = i = 1 N + 1 2 A i ( e v i r 1 4 π r 1 e v i r 2 4 π r 2 ) , r Ω ,
S ( r , t ) = η L ( r ) ε X ( r ) ρ ( r ) exp ( L ( r ) η t ) ,
Φ = A ρ ,

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