May  2015, 9(2): 579-589. doi: 10.3934/ipi.2015.9.579

Modulated luminescence tomography

1. 

Department of Mathematics, Purdue University, 150 N University Street, West Lafayette, IN 47907

2. 

Department of Biomedical Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, United States, United States

Received  September 2014 Revised  January 2015 Published  March 2015

We propose and analyze a mathematical model of Modulated Luminescence Tomography. We show that when single X-rays or focused X-rays are used as an excitation, the problem is similar to the inversion of weighted X-ray transforms. In particular, we give an explicit inversion in the case of Dual Cone X-ray excitation.
Citation: Plamen Stefanov, Wenxiang Cong, Ge Wang. Modulated luminescence tomography. Inverse Problems & Imaging, 2015, 9 (2) : 579-589. doi: 10.3934/ipi.2015.9.579
References:
[1]

W. Cong, Z. Pan, R. Eilkins, A. Srivastava, N. Ishaque, P. Stefanov and G. Wang, X-ray micromodulated luminescence tomography in dual-cone geometry,, J. of Biomed. Optics, 19 (2014). doi: 10.1117/1.JBO.19.7.076002. Google Scholar

[2]

W. Cong, F. Liu, C. Wang and G. Wang, X-ray micro-modulated luminescence tomography (XMLT),, Opt. Express, 22 (2014), 5572. doi: 10.1364/OE.22.005572. Google Scholar

[3]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). Google Scholar

[4]

B. Frigyik, P. Stefanov and G. Uhlmann, The X-ray transform for a generic family of curves and weights,, J. Geom. Anal., 18 (2008), 89. doi: 10.1007/s12220-007-9007-6. Google Scholar

[5]

R. A. Kruger, D. R. Reinecke and G. A. Kruger, Thermoacoustic computed tomography-technical considerations,, Med. Phys., 26 (1999), 1832. doi: 10.1118/1.598688. Google Scholar

[6]

P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography,, European J. Appl. Math., 19 (2008), 191. doi: 10.1017/S0956792508007353. Google Scholar

[7]

F. Liu, W. Yan, Y.-J. Chuang, Z. Zhen, J. Xie and Z. Pan, Photostimulated near-infrared persistent luminescence as a new optical read-out from $Cr^{3+}$-doped $LiGa_5O_8$,, Sci. Rep., 3 (2013). Google Scholar

[8]

V. Ntziachristos, Going deeper than microscopy: The optical imaging frontier in biology,, Nat. Methods, 7 (2010), 603. doi: 10.1038/nmeth.1483. Google Scholar

[9]

V. Ntziachristos, J. Ripoll, L. V. Wang and R. Weissleder, Looking and listening to light: The evolution of whole-body photonic imaging,, Nat. Biotechnol., 23 (2005), 313. doi: 10.1038/nbt1074. Google Scholar

[10]

C. C. Paige and M. A. Saunders, LSQR: An algorithm for sparse linear equations and sparse least squares,, ACM Trans. Math. Softw., 8 (1982), 43. doi: 10.1145/355984.355989. Google Scholar

[11]

G. Pratx, C. M. Carpenter, C. Sun and L. Xing, X-ray luminescence computed tomography via selective excitation: A feasibility study,, IEEE Transactions on Medical Imaging, 29 (2010), 1992. doi: 10.1109/TMI.2010.2055883. Google Scholar

[12]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. III,, Scattering Theory, (1979). Google Scholar

[13]

M. Schweiger, S. R. Arridge, M. Hiraoka and D. T. Delpy, The finite element method for the propagation of light in scattering media: Boundary and source conditions,, Med. Phys., 22 (1995), 1779. doi: 10.1118/1.597634. Google Scholar

[14]

P. Stefanov and G. Uhlmann, Linearizing non-linear inverse problems and an application to inverse backscattering,, J. Funct. Anal., 256 (2009), 2842. doi: 10.1016/j.jfa.2008.10.017. Google Scholar

[15]

________, Thermoacoustic tomography with variable sound speed,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/7/075011. Google Scholar

[16]

G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan and M. Henry, In vivo mouse studies with bioluminescence tomography,, Opt. Express, 14 (2006), 7801. doi: 10.1364/OE.14.007801. Google Scholar

[17]

L. V. Wang and S. Hu, Photoacoustic tomography: In vivo imaging from organelles to organs,, Science, 335 (2012), 1458. doi: 10.1126/science.1216210. Google Scholar

[18]

A. J. Welch and M. J. C. van Gemert, eds., Optical-thermal Response of Laser-Irradiated Tissue,, Springer, (2011). doi: 10.1007/978-90-481-8831-4. Google Scholar

show all references

References:
[1]

W. Cong, Z. Pan, R. Eilkins, A. Srivastava, N. Ishaque, P. Stefanov and G. Wang, X-ray micromodulated luminescence tomography in dual-cone geometry,, J. of Biomed. Optics, 19 (2014). doi: 10.1117/1.JBO.19.7.076002. Google Scholar

[2]

W. Cong, F. Liu, C. Wang and G. Wang, X-ray micro-modulated luminescence tomography (XMLT),, Opt. Express, 22 (2014), 5572. doi: 10.1364/OE.22.005572. Google Scholar

[3]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). Google Scholar

[4]

B. Frigyik, P. Stefanov and G. Uhlmann, The X-ray transform for a generic family of curves and weights,, J. Geom. Anal., 18 (2008), 89. doi: 10.1007/s12220-007-9007-6. Google Scholar

[5]

R. A. Kruger, D. R. Reinecke and G. A. Kruger, Thermoacoustic computed tomography-technical considerations,, Med. Phys., 26 (1999), 1832. doi: 10.1118/1.598688. Google Scholar

[6]

P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography,, European J. Appl. Math., 19 (2008), 191. doi: 10.1017/S0956792508007353. Google Scholar

[7]

F. Liu, W. Yan, Y.-J. Chuang, Z. Zhen, J. Xie and Z. Pan, Photostimulated near-infrared persistent luminescence as a new optical read-out from $Cr^{3+}$-doped $LiGa_5O_8$,, Sci. Rep., 3 (2013). Google Scholar

[8]

V. Ntziachristos, Going deeper than microscopy: The optical imaging frontier in biology,, Nat. Methods, 7 (2010), 603. doi: 10.1038/nmeth.1483. Google Scholar

[9]

V. Ntziachristos, J. Ripoll, L. V. Wang and R. Weissleder, Looking and listening to light: The evolution of whole-body photonic imaging,, Nat. Biotechnol., 23 (2005), 313. doi: 10.1038/nbt1074. Google Scholar

[10]

C. C. Paige and M. A. Saunders, LSQR: An algorithm for sparse linear equations and sparse least squares,, ACM Trans. Math. Softw., 8 (1982), 43. doi: 10.1145/355984.355989. Google Scholar

[11]

G. Pratx, C. M. Carpenter, C. Sun and L. Xing, X-ray luminescence computed tomography via selective excitation: A feasibility study,, IEEE Transactions on Medical Imaging, 29 (2010), 1992. doi: 10.1109/TMI.2010.2055883. Google Scholar

[12]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. III,, Scattering Theory, (1979). Google Scholar

[13]

M. Schweiger, S. R. Arridge, M. Hiraoka and D. T. Delpy, The finite element method for the propagation of light in scattering media: Boundary and source conditions,, Med. Phys., 22 (1995), 1779. doi: 10.1118/1.597634. Google Scholar

[14]

P. Stefanov and G. Uhlmann, Linearizing non-linear inverse problems and an application to inverse backscattering,, J. Funct. Anal., 256 (2009), 2842. doi: 10.1016/j.jfa.2008.10.017. Google Scholar

[15]

________, Thermoacoustic tomography with variable sound speed,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/7/075011. Google Scholar

[16]

G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan and M. Henry, In vivo mouse studies with bioluminescence tomography,, Opt. Express, 14 (2006), 7801. doi: 10.1364/OE.14.007801. Google Scholar

[17]

L. V. Wang and S. Hu, Photoacoustic tomography: In vivo imaging from organelles to organs,, Science, 335 (2012), 1458. doi: 10.1126/science.1216210. Google Scholar

[18]

A. J. Welch and M. J. C. van Gemert, eds., Optical-thermal Response of Laser-Irradiated Tissue,, Springer, (2011). doi: 10.1007/978-90-481-8831-4. Google Scholar

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